PRACTICE MODULE - 1

FOR

JEE (MAIN & ADVANCED)

CLASS - XII

 PHYSICS

 CHEMISTRY

 MATHEMATICS
 Contents – Physics

SR. NO. Electrostatics PAGE NO.

1. EXERCISE – 1 : Concept building questions P-1 to P-5
2. EXERCISE – 2 : Single choice correct with multiple options P-6 to P-9
3. EXERCISE – 3 : Multiple choice correct with multiple options P-10 to P-13
4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
P-13 to P-17
• Section – 2 : Match the Column
• Section – 3 : Comprehension
5. EXERCISE – 5 : Revision exercise (Moderate to Tough) P-17 to P-18
6. EXERCISE – 6
• Section – I : JEE (Advanced) questions Previous Years P-19 to P-30
• Section – II : JEE (Main) questions Previous Years
7. ANSWER KEY P-31 to P-32

SR. NO. Current Electricity PAGE NO.

1. EXERCISE – 1 : Concept building questions P-33 to P-38
2. EXERCISE – 2 : Single choice correct with multiple options P-39 to P-47
3. EXERCISE – 3 : Multiple choice correct with multiple options P-47 to P-52
4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
P-52 to P-55
• Section – 2 : Match the Column
• Section – 3 : Comprehension
5. EXERCISE – 5 : Revision exercise (Moderate to Tough) P-55 to P-57
6. EXERCISE – 6
• Section – I : JEE (Advanced) questions Previous Years P-57 to P-69
• Section – II : JEE (Main) questions Previous Years
7. ANSWER KEY P-70 to P-73

SR. NO. Magnetic Effects of Electric Current and Magnetism PAGE NO.
1. EXERCISE – 1 : Concept building questions P-74 to P-80
2. EXERCISE – 2 : Single choice correct with multiple options P-80 to P-89
3. EXERCISE – 3 : Multiple choice correct with multiple options P-90 to P-93
4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
P-93 to P-99
• Section – 2 : Match the Column
• Section – 3 : Comprehension
5. EXERCISE – 5 :
P-99 to P-104
Revision exercise – Subjective questions (Moderate to Tough)
6. EXERCISE – 6
• Section – I : JEE (Advanced) questions Previous Years P-104 to P-119
• Section – II : JEE (Main) questions Previous Years
7. ANSWER KEY P-120 to P-121
 Contents – Chemistry
SR. NO. Solutions & Colligative Properties PAGE NO.
1. EXERCISE – 1 : Concept building questions C-1 to C-3
2. EXERCISE – 2 : Single choice correct with multiple options C-4 to C-7
3. EXERCISE – 3 : Multiple choice correct with multiple options C-7 to C-9
4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
C-9 to C-12
• Section – 2 : Match the Column
• Section – 3 : Comprehension
5. EXERCISE – 5 : Revision exercise (Moderate to Tough) C-13 to C-15
6. EXERCISE – 6
• Section – I : JEE (Advanced) questions Previous Years C-15 to C-23
• Section – II : JEE (Main) questions Previous Years
7. ANSWER KEY C-24 to C-25

SR. NO. Electrochemistry PAGE NO.

1. EXERCISE – 1 : Concept building questions C-26 to C-30
2. EXERCISE – 2 : Single choice correct with multiple options C-30 to C-34
3. EXERCISE – 3 : Multiple choice correct with multiple options C-34 to C-36
4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
C-36 to C-39
• Section – 2 : Match the Column
• Section – 3 : Comprehension
5. EXERCISE – 5 : Revision exercise (Moderate to Tough) C-39 to C-41
6. EXERCISE – 6
• Section – I : JEE (Advanced) questions Previous Years C-42 to C-51
• Section – II : JEE (Main) questions Previous Years
7. ANSWER KEY C-52 to C-53

SR. NO. Optical Isomerism PAGE NO.

1. EXERCISE – 1 : Concept building questions C-54 to C-57
2. EXERCISE – 2 : Single choice correct with multiple options C-58 to C-65
3. EXERCISE – 3 : Multiple choice correct with multiple options C-65 to C-69
4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
C-69 to C-73
• Section – 2 : Match the Column
• Section – 3 : Comprehension
5. EXERCISE – 5 : Revision exercise (Moderate to Tough) C-74 to C-77
6. EXERCISE – 6
• Section – I : JEE (Advanced) questions Previous Years C-77 to C-82
• Section – II : JEE (Main) questions Previous Years
7. ANSWER KEY C-83

SR. NO. Haloalkanes and Haloarenes PAGE NO.

1. EXERCISE – 1 : Concept building questions C-84 to C-89
2. EXERCISE – 2 : Single choice correct with multiple options C-89 to C-97
3. EXERCISE – 3 : Multiple choice correct with multiple options C-98 to C-102
4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
C-102 to C-108
• Section – 2 : Match the Column
• Section – 3 : Comprehension
5. EXERCISE – 5 : Revision exercise (Moderate to Tough) C-108 to C-110
6. EXERCISE – 6
• Section – I : JEE (Advanced) questions Previous Years C-111 – 117
• Section – II : JEE (Main) questions Previous Years
7. ANSWER KEY C-118
 Contents – Mathematics
SR. NO. Matrices PAGE NO.

1. EXERCISE – 1 : Concept building questions M-1 to M-5

2. EXERCISE – 2 : Single choice correct with multiple options M-5 to M-11

3. EXERCISE – 3 : Multiple choice correct with multiple options M-11 to M-12

4. EXERCISE – 4

• Section – 1 : Numerical Value/Subjective Type Questions

M-12 to M-16
• Section – 2 : Match the Column

• Section – 3 : Comprehension

5. EXERCISE – 5 : Revision exercise (Moderate to Tough) M-16 to M-17

6. EXERCISE – 6

• Section – I : JEE (Advanced) questions Previous Years M-17 to M-31

• Section – II : JEE (Main) questions Previous Years

7. ANSWER KEY M-32 to M-34

SR. NO. Determinants PAGE NO.

1. EXERCISE – 1 : Concept building questions M-35 to M-38

2. EXERCISE – 2 : Single choice correct with multiple options M-38 to M-44

3. EXERCISE – 3 : Multiple choice correct with multiple options M-44 to M-46

4. EXERCISE – 4

• Section – 1 : Numerical Value/Subjective Type Questions

M-46 to M-52
• Section – 2 : Match the Column

• Section – 3 : Comprehension

5. EXERCISE – 5 : Revision exercise (Moderate to Tough) M-53 to M-57

6. EXERCISE – 6

• Section – I : JEE (Advanced) questions Previous Years M-57 to M-62

• Section – II : JEE (Main) questions Previous Years

7. ANSWER KEY M-63

SR. NO. Functions & Relations PAGE NO.

1. EXERCISE – 1 : Concept building questions M-64 to M-67

2. EXERCISE – 2 : Single choice correct with multiple options M-67 to M-70

3. EXERCISE – 3 : Multiple choice correct with multiple options M-70 to M-72

4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
M-72 to M-74
• Section – 2 : Match the Column
• Section – 3 : Comprehension

5. EXERCISE – 5 : Revision exercise (Moderate to Tough) M-74 to M-77

6. EXERCISE – 6

• Section – I : JEE (Advanced) questions Previous Years M-77 to M-82

• Section – II : JEE (Main) questions Previous Years

Relations

7. • Section – I : Concept building questions M-83 to M-87

• Section – II : Single choice correct with multiple options

8. ANSWER KEY M-88 to M-91

SR. NO. Inverse Trigonometry Functions PAGE NO.

1. EXERCISE – 1 : Concept building questions M-92 to M-94

2. EXERCISE – 2 : Single choice correct with multiple options M-94 to M-96

3. EXERCISE – 3 : Multiple choice correct with multiple options M-97 to M-99

4. EXERCISE – 4
• Section – 1 : Numerical Value/Subjective Type Questions
M-99 to M-102
• Section – 2 : Match the Column
• Section – 3 : Comprehension

5. EXERCISE – 5 : Revision exercise (Moderate to Tough) M-102 to M-103

6. EXERCISE – 6

• Section – I : JEE (Advanced) questions Previous Years M-103 to M-108

• Section – II : JEE (Main) questions Previous Years

7. ANSWER KEY M-109 to M-110

 PHYSICS

ELECTROSTATICS
 P-1

Exercise – 1
Concept Building Questions

Properties of charge & Coulomb’s law and Continuous Charged Distribution

1. A substance of mass 1 g consists of 5 1021 molecules. If from 0.01 % molecules of the substance 1
electron is removed, then total electric charge on the substance is
(a) 0.08 C (b) 0.8 C (c) +0.008 C (d) +0.16 C
2. If two unlike charge of same magnitude are placed at certain distance the force between them is F. If 25%
electric charge is moved from one charge to another charge then the force between them is
4 15F 9
(a) F (b) F (c) (d) F
5 16 16
3. Two particles of mass 5 g and charge 10 7 C are placed on horizontal table at distance 10 cm. When both
the particle are in limiting equilibrium position, the co-efficient of static friction s =
(a) 0.15 (b) 0.19 (c) 0.18 (d) 0.2
4. The linear charge density on the circumference of a circle of radius a
varies as    0 cos 2  . The total charge on it is

a  0
(a) (b) Zero
2
(c) 2a 0 (d) a 0
5. The charge density on a sphere of radius R is given by equation   r    r 2 , then the total charge on the
sphere is
2R3 4R3 4R5 2 R 5
(a) (b) (c) (d)
3 3 5 5
6. A non-conducting square surface having side length a has electric charge distribution of surface charge
density   0 xy, then total electric charge on the square with respect to the Cartesian Co-ordinate
system placed at the centre of the square is
 a4
(a) 0 (b) 4 0 a 2 (c) 20 a 2 (d) Zero
4
Electric field and its application
7. A coin of mass 1.6 g is placed in an electric field of intensity 109 NC–1 in vertical upward direction. For
equilibrium of coin, the number of electrons to be removed from the coin is
(a) 9.8  107 (b) 6.25  109 (c) 1.6 1019 (d) 4.25 1010
8. Mass of a bob of simple pendulum is 80 mg and the charge on it is 20 nC. It is suspended by a string in a
horizontal electric field of intensity 2 104 NC–1. In equilibrium, the angle made with vertical direction
and tension force arising in string is
(a) 60º, 2.4 × 10–2 N (b) 45º, 1.57 × 10–3 N (c) 27º, 8.9 × 10–4 N (d) 37º, 4.5 × 10–4 N
9. If a particle of mass 1 g and charge 5C is moved with velocity 20 ms–1 in direction opposite of electric
field of intensity 2 × 105 NC–1 then how much distance is travelled by the particle before coming to rest?
(a) 1 m (b) 0.4 m (c) 10 cm (d) 0.2 m
10. 108 C and 108 C charges are placed on any two vertices of equilateral triangle, then the intensity of
electric field at third vertex is ____________ NC–1. (length of sides of equilateral triangle is 0.1 m).
(a) 3.6  104 (b) 7.2  104 (c) 9 103 (d) 3  109
11. 64 small drops of radius 0.02 m and charge 5 C are combined to form one big drop. If the charge is not
leak then the ratio of initial and final charge density on the drop is
(a) 2 : 4 (b) 1 : 2 (c) 1 : 4 (d) 1 : 1

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12. A charged wire is bent in half circle arc of radius a. If the linear charge density is  , then the electric
field at the center of arc is
   
(a) (b) (c) (d)
2 0 a 2  0 a 2 2
4  0 a 0 a
Electric flux & Gauss theorem
13. An uncharged sphere of metal is placed in between two charged plates as shown in figure. The lines of
force look like.

(a) (b) (c) (d)

(a) a (b) b (c) c (d) d
14. Three positive charges of equal value q are placed at the vertices of an equilateral triangle. Which of the
following will be resultant electric field lines?

O O

(a) (b) (c) (d)

O

O
O

15. Let there be a spherically symmetric charge distribution with charge density varying as
5 r 
  r   0    up to r = R and   r   0 For r > R, where r is the distance from the origin. The
4 R
electric field at a distance r  r  R  from the origin is given by
4 0 r  5 r  0 r 5 r  4  0 r 5 r  0 r 5 r 
(a)  (b) 4  R  (c) 3  R  (d) 3  R
3 0  4 R  3 0   3 0   4 0  

16. A hollow cylinder of radius 1 cm is placed in a uniform electric field of magnitude E  2  104 NC1 in
such a way that its axis is parallel to electric field, then flux linked with cylinder is
(a) 2  104  Vm (b) 2  102  Vm (c) 0.02  103 NC–1 (d) zero
17. The electric field in a region is given by the following equation:
  3 4 
E   i  j   2  103 NC–1
5 5 
The flux passing through a rectangular of 0.2 m2 area placed in yz plane inside the electric field is
(a) 240 Nm2C–1 (b) 120 Nm2C–1 (c) 1.4 × 102 Nm2C–1 (d) 3 × 103 Nm2C–1
18. A copper wire having linear charge density  is passed through a cube of length a, then the maximum
flux linked with the cube is
a 2 a 6 a 2 3a
(a) (b) (c) (d)
0 0 0 0
19. The inward and outward electric flux for a closed surface are respectively 5 105 and 4 105 MKS unit.
The charge inside the closed surface is

(a) 8.85  107 C (b) 8.85  107 C (c) 8.85 107 C (d) 6.85 107 C

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 P-3

20. Electric charge is uniformly distributed on a long straight wire of radius 1 mm. The charge per 1 cm
length of wire is Q C. Another cylindrical surface of radius 50 cm and length I m symmetrically encloses
the wire. The total electric flux passing through the cylindrical surface is
Q 100Q 10Q 100Q
(a) (b) (c) (d)
0 0  0 0
Electric potential & potential energy
21. Two parallel conducting plates lie at a distance 20 mm. Potential of upper plate is +24000 V wrt lower
plate. If an electron is released at lower plate then how much time does it take to reach upper plate?
e
 1.8  1011 Ckg–1
m
(a) 2s (b) 1.4ns (c) 1.7 ms (d) 2.7 s
22. In Millikan experiment, a drop of charge Q remains stationary between the two plates having 2400 V
potential difference. If the radius of drop becomes half and potential difference 600 V then for
equilibrium, the charge on drop is
Q Q 3Q
(a) (b) (c) Q (d)
4 2 2

23. Electric field in a region is E  30 x 2 i, then potential difference VA – V0 equals to where
V0 = Potential at origin point, VA = Potential at point A located at x = 2m
(a) 80 V (b) –80 V (c) 120 V (d) –120V
24. The work done to move a charge 5 C from point A to point B against electric field is 20 J. If the electric
potential at point A is 10 V, then what is the potential at point B ?
(a) Zero (b) 6 V (c) 14 V (d) 2.5 V
25. The work done to move charge q from point x to point y as shown in figure.
+Q x r y –Q

a a
2kqQr kQq Qq 2kQqa
(a) (b) (c) (d)
a a  r  r  2a r r r  a 
26. The potential on “n” point drops having equal magnitude is V volt. They join together and form a bigger
drop, then the potential on it, is
1 2
V
(a) (b) Vn (c) Vn 2 (d) Vn 3
n
27. As shown in figure, a square having side a has charges +q, +q, –q A D
+q –q
and –q on vertices of square ABCD. It E is the midpoint of side
BC, then the amount of work to be done to move a charge e from
centre of square O to point E o
qe a
(a) (b) Zero
4 0

(c)
qe
4 0

4 2 1  (d)
qe  1

 0 a  5

 1

+q
B E C
–q

28. Q charge is uniformly distributed on a thin ring of radius R. If, initially, electron is at rest at point A
which is quite far from centre and axis of ring, then the velocity when electron passes through the centre
of ring is
2kQe kQe kme kQe
(a) (b) (c) (d)
mR m QR mR

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29. The charge per unit length of an arc ring forming an angle of  in radian at center having radius R is .
The electric potential at its centre is
k k
(a) (b) (c) 2k (d) k
4 2
30. The total charge on an insulator ring of radius 0.5 m is 1.11 × 10–10 C, which is distributed unequally on
 0  
circumference. The value of   E.d (l = 0 is the centre of ring) is
 
(a) –1 V (b) +2 V (c) –2 V (d) Zero
31. The radius of a conducting hollow sphere is a. If the potential difference between two points, one at
distance a from centre and other at distance 3 a from centre is V, then electric field at distance 3 a from
centre is
V V V V
(a) (b) (c) (d)
6a 3a 4a 2a
32. The distance between two points A and B is 2 L, +q and –q charges are placed at points A and B
respectively. The midpoint of distance AB is C. The work done for +Q charge to move from C to infinity
is
qQ  qQ qQ
(a) 0 (b) (c) (d)
6  0 L 6  0 L 4 0 L
33. The distance between two thin rings of radius R is d. The charge on these rings are +Q and –Q
respectively. The potential difference between centre of these two rings is
Q 1 1 
(a) Zero (b)   
40  R R 2  d 2 
QR 1 1 Q
(c)   (d) 
2  0 d 2  R 20
R 2  d 2 
34. Point charge ‘q’ moves from point P to point S along the path Y 
P E
PQRS in a uniform electric field E pointing co-parallel to the
positive direction of the x-axis. The co-ordinates of the points P,
S
Q, R and S are (a, b, 0), (2a, 0, 0), (a, –b, 0) and (0, 0, 0) X
Q
respectively. The work done by the field in the above process is
(a) qEa (b) –qEa
R
(c) qEa 2 (d) qE  2a 2  b2
 A
35. An electric field is represented by E  3 i in a region. Then magnitude of electric potential in this region
x
is given by (assume the electric potential zero at infinite distance)
2A A A
(a) 2 (b) 2
(c) Zero (d) 3
x 2x x

36.  
What is the electric potential of electric field E  yi  xi ?
(a) V = – xy + C (b) V = – (x + y) + C

(c) V = – x 2  y 2 + C  (d) V = C
37. The electric potential at point  x, y, z  is V =  x 2 y  xz3 + 4. Electric field intensity at this point, is given
by
 
  
(a) E  2 xyi  x 2  y 2 j  3xy  y 2 k  (b) E  z 3 i  xyz j  z 2 k
 
 
(c) E  2 xy  z 3 i  xy 2 j  3z 2 xk  
(d) E  2 xy  z 3 i  x 2 j  3xz 2 k

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38. On the vertices of an isosceles right angled triangle Q, +q and +q Q

charges are placed as in figure. If the total electrostatic energy of

the whole system is zero, then Q equals to
q 2q
(a) (b)
1 2 2 2
+q +q
(c) 2q (d) q a

39 On the vertices of a cube of sides b, –q charges are placed. The electric potential energy of +q charge
placed at the centre of cube is
4 2 q 2 8 2q 2 8 2 q 2 4 q 2
(a) (b) (c) (d)
 0 b 4 0 b 0 b 3 0 b
40. Two equal point charges placed on x-axis at distance x = – a and x = + a. A point charge Q is at origin.
When the charge Q travel a very small distance x on x-axis the electric potential energy difference is
proportional to
1
(a) x3 (b) x2 (c) (d) x
x
41. An insulated solid sphere of radius R have positive charge density  . The potential energy difference to
bring a charge q from centre to the surface of sphere is
R 2 q R 2 q R 2 q R 2 q
(a) (b) (c) (d)
6 0 0 4 0 3 0
Electric dipole
42. An electric dipole placed in a uniform electric field of intensity 4 × 105 NC–1 at angle 60º with the electric
field, experiences torque 8 3 Nm. If length of dipole of 4 cm then magnitude of charge will be
(a) 3C (b) 1 mC (c) 2 C (d) 2 mC
43. Dipole moment of an electric dipole is 2 108 Cm. The electric field intensity at a point of distance 1 m
and make an angle 60º with the centre of dipole is
(a) 300 N/C (b) 238.1 N/C (c) 4295 N/C (d) 255.2 N/C
44. An electric dipole consists of two opposite charge 1C, each separated by a distance 2 cm is placed in an
electric field of 105 Vm–1. The work done for rotation of this dipole from equilibrium to 180º is
(a) 4 103 J (b) 2 103 J (c) 10 3 J (d) 5  103 J
45. Three point charges +q, –2q and +q are situated at points (0, a, 0), (0, 0, 0), (a, 0, 0) respectively. The
magnitude and direction of the dipole moment consisting this charges is
(a) 2 qa, in + y direction
(b) 2 qa, In direction of line joining points (0, 0, 0) and (a, a, 0)
(c) qa, In the direction of the line joining points (0, 0, 0) and (a, 0, a)
(d) 2 qa, in + x direction
46. A quadruple is shown in figure. The potential at a point which is at distance r from the axis of quadruple
–2q
+q a a +q p

r
2 2
2qa 2qa 2qa 2 qa 2
(a) (b) (c) (d)

40 r r 2  a 2  
40 r r 2  a 2  4 0 r 3 4 0 r 3

47. Two dipoles of dipole moment 5  1012 Cm are placed in such a way that their axis are parallel to co-
ordinate axis and intersect at the origin. Then potential at point 20 cm away and making an angle 30º with
axis is
(a) 1.536 V (b) 1.12 V (c) 1.25 V (d) 2.12 V

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 P-6

Exercise – 2
Single choice correct with multiple option

1. Three point charges are placed at the corners of an equilateral triangle. Assume that only electrostatic
forces are acting. Then
(a) the system will be in equilibrium if the charges have the same magnitude but not all have the same
sign
(b) the system will be in equilibrium if the charges have different magnitudes and not all have the same
sign
(c) the system will be in equilibrium if the charges rotate about the centre of the triangle
(d) the system can never be in equilibrium
2. A charge +q is placed at each of the points x = x0, x = 3x0, x = 5x0, …… ad infinitum on the x-axis, and a
charge – q is placed at each of the points x = 2x0, x = 4x0, x = 6x0, …. ad infinitum. Here, x0 is a positive
constant. Then, the potential at the origin due to the above system of charges is
q q ln 2
(a) 0 (b) (c)  (d)
80 x 0 ln 2 4 0 x 0
3. A solid sphere of radius R is charged uniformly. The electrostatic potential V is plotted as a function of
distance r from the centre of the sphere. Which of the following best represents the resulting curve?

(a) (b) (c) (d)

4. A large solid sphere with uniformly distributed positive charge has a smooth narrow tunnel through its
centre. A small particle with negative charge, initially at rest far from the sphere, approaches it along the
line of the tunnel, reaches its surface with a speed v, and passes through the tunnel. Its speed at the centre
of the sphere will be
(a) 0 (b) v (c) 2v (d) 1.5v
5. A large flat metal surface has a uniform charge   . An electron of mass m and charge e leaves the
surface at point A with speed u, and returns to it at point B. Disregard gravity. The maximum value of AB
is
u 2 m 0 u 2 m 0 2u 2 m0 2u 2 m0
(a) (b) (c) (d)
e 2 e e e
6. An electric dipole is placed at the origin and is directed along the x-axis. At a point P, for away from the
dipole, the electric field is parallel to the y-axis. OP makes an angle  with the x-axis. Then
1
(a) tan   3 (b) tan   2 (c)   45o (d) tan  
2
7. A spherical charged conductor has surface charge density  . The electric field on its surface is E and
electric potential of conductor is V. Now the radius of the sphere is halved keeping the charge to be
constant. The new values of electric field and potential would be
(a) 2E, 2V (b) 4E, 2V (c) 4E, 4V (d) 2E, 4V
–5
8. An insulated ring of radius 4m having a charge q1 = 2 × 10 C is uniformly distributed over it. A
negatively charged particle having charge q2 = 4 × 10–4C is released from rest along its axis at distance
x = 3m from its centre. Mass of both ring and the particle is 1 kg each. Neglect gravitational effects. If the
ring is free to move then maximum speed of particle will be
(a) 4.4 m/s (b) 3.1 m/s (c) 5.2 m/s (d) 6.1 m/s
9. The magnitude of dipole moment of a system of charge +q distributed uniformly on an arc of radius R
subtending an angle  / 2 at its centre where another charge –q is placed is
2 2qR 2qR qR 2qR
(a) (b) (c) (d)
   

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10. A charged particle of charge Q is held fixed and another charged particle of mass m and charged (of the
same sign) is released from a distance r. The impulse of the force exerted by the external agent on the
fixed charge by the time when distance between Q and q becomes 2r is
Qq Qqm Qqm Qqm
(a) (b) (c) (d)
40 mr 40 r 0 r 20 r
11. A radioactive source, in the form of a metal sphere of radius 10–2m, emits  -particles at rate of 5 × 1010
particles per second. The source is electrically insulated. Assume that 40% of the emitted  -particles
escape the source and it is found that in time ‘t’ the potential of the sphere will become 2V. The value of t
is
(a) 0.252 ms (b) 0.694 ms (c) 2.52 ms (d) 6.94 ms
12. A uniform non-conducting rod of mass m and length l, with the charge density  as shown in the figure,
is hinged at mid point at origin O so that it can rotate in a horizontal plane without any friction. A uniform
electric field E exists in right direction in the entire region.

The period of small oscillations of the rod is given by

m m m 3m
(a) 2 (b)  (c) 2 (d) 2
E E 3E E
21. A nonconducting ring of mass m and radius R, with charge per unit
length  , is shown in figure. It is then placed on a rough non-
condcuting horizontal plane. At time t = 0, a uniform electric field
E  E i is switched on and the ring starts rolling without sliding.


The magnitude of friction force acting on the ring is

RE o
(a) REo (b)
2
3RE o 2REo
(c) (d)
2 3
14. Two isolated metallic solid spheres of radii R and 2R are charged such that both of these have same
charge density  . The spheres are located for away from each other and connected by a thin conducting
wire. The new charge density on the bigger sphere
 3 2 5
(a) (b) (c) (d)
6 5 5 6
15. A positive charge +Q is fixed at a point A. Another positively
charged particle of mass m and charge +q is projected from a point
B with velocity u as shown in figure. Point B is at a large distance
from A and at distance d from the line AC. The initial velocity is
parallel to the line AC. The point C is at a very large distance from
A. Take Qq = 40 mu 2 d and d  ( 2  1)m . The minimum
distance (in meter) of +q from +Q during the motion is
(a) d( 2  1) (b) d( 3  1)
(c) d( 3  1) (d) d( 2  1)
16. A charged dust particle of radius 5 × 10–17 m is located in a horizontal electric field of magnitude
6.28 × 105 N/C. The surrounding medium is air with coefficient of viscosity 1.6 × 10–5 kgm–1s–1. Neglect
gravity. If this particle moves with a uniform speed of 0.02 m/s then the number of electrons on it is
(a) 10 (b) 20 (c) 30 (d) 40

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17. A ball of mass 10–2 kg and charge 3 × 10–6C is tied at one end of a 1 m long thread. The other end of the
thread is fixed and a charge of –3 × 10–6C is placed at this end. The ball can move in the circular orbit of
radius 1 m in the vertical plane. Initially the ball is at the bottom. Take g = 10 m/s2, the minimum value of
horizontal speed provided to the ball at the bottom so that it will be able to complete the full circle in
vertical plane is
(a) 4.6 m/s (b) 7.6 m/s (c) 8.6 m/s (d) 10.6 m/s
18. A copper sphere has mass 2.1 g and if the charge on sphere is 2C then fraction of electrons removed
(a) 4.32 × 10–10 (b) 4.32 × 10–11 (c) 2.16 × 10–10 (d) 2.16  1011
19. Two point charges placed at distance of 20 cm in air attracts each other with certain force. When a
dielectric slab of thickness 8 cm and dielectric constant K is introduced between these two charges force
of interaction becomes half of its previous value. Then the magnitude of K is approx.
(a) 1 (b) 4 (c) 2 (d) 2
20. Two point charges q and 2q are placed in air at distance d. If third electric charge Q is kept on the line
joining two charges such that the resultant force on q and 2q becomes zero, then the distance of charge Q
from charge q is

(a)
d
2 1

(b) 2  1 d  (c)
d
3 1

(d) 3  1 d 
21. As shown in figure the component of electric field Y
produced due to a charge inside a cube is 0.1 m 0.1 m
1
E x  600 x 2 , Ey  0 and Ez  0 then the charge
inside the cube is, approx. O
(a) 600 C (b) 60 C X
(c) 7 C (d) 6 C
Z
a
22. A disk of radius having a uniformly distributed charge of 6C is placed in the x-y plane with its centre at
4
  a 
 ,0,0  . A rod of length a carrying a uniformly distributed charge 8 C is placed on the x-axis from
 2 
a 5a  a a   3a 3a 
x to x  x. Two point charges –7C and 3C are placed at  , ,0  and  , ,0  ,
4 4 4 4   4 4 
a a a
respectively. Consider a cubical surface formed by six surfaces x   , y   , z   . The electric
2 2 2
flux through this cubical surface is
2C 2C 10C 12C
(a) (b) (c) (d)
0 0 0 0
23. 8q charge is placed on any one vertex of a cube. The flux linked with this cube is
q q q q
(a) (b) (c) (d)
8 0 4 0 6 0 0
24. An infinitely long wire of linear charge distribution  is passing through any side of cube of length “a”,
then the total flux passing through cube is
a a a a
(a) (b) (c) (d)
0 2 0 4 0 6 0
25. The electric charge density on two parallel very long straight wire is 2 104 Cm–1 respectively. If the
distance between these two wire is 0.2 m, then due to charge of first wire the force on unit length of
second wire is
(a) 72  102 N (b) 8.4 × 102 N (c) 9 × 102 N (d) 36 × 102 N

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26. The linear charge density on infinitely long straight wire is  Cm–1. If an electron moving round in
perpendicular plane to the wire and its centre is on the wire then the kinetic energy of electron is
e e e e
(a) (b) (c) (d)
4  0 2 0 0 8 0

27. Two wires of linear charge density  passing through a sphere of radius R and a cube of sides R so that
the flux linked with them is maximum. Then the ratio of flux of sphere to the cube is
1 2 3
(a) 2 (b) (c) (d)
2 3 2
28. An electric dipole is prepared by taking two electric charges of 5nC separated by distance 2 mm. The

dipole is kept near a line charge distribution having density 4.5  104 Cm–1 in such a way that the negative
electric charge of the dipole is at a distance 2.5 cm perpendicular to the wire. The force acting on the
dipole
(A) 0.12 N (b) 0.5 N (c) 1.5 N (d) 0.25 N
29. A solid sphere of radius R is uniformly charged. The distance from the surface at which the electric
potential is half of the potential at centre of sphere
R R 4R
(a) (b) R (c) (d)
2 3 3
30. Potential at a point on the bisector of a thin rod (charge per unit length of the rod is  ) of length 2 l and
at a distance “a” from the centre of thin rod is
 l 2  a2  l 2  a2  1
(a) ln 2 (b) ln
 0 l  a2 4 0 l 2  a2  1
 l 2  a2  l 2  a2  1
(c) ln (d)
ln
 0 l 2  a2 2  0 l 2  a2  1
31. The electric potential at centre for half sphere having radius R and surface charge density  is
R R 6R R
(a) (b) (c) (d)
4 0 0 30 2 0
32. As shown in figure A, B and C are concentric shells of radius a, b and c C
B
respectively. (a < b < c). Their surface charge densities are ,  and 
A
respectively. The electric potential on the surface of shell A is a
  b
(a) a  b  c (b) a  b  c 
0 0 – c
  
(c)  b  a  c (d)  a  b  c
0 0

33. The radius of two concentric metal shells are R1 and R2 respectively and electric charge on them are Q1
and Q 2 . The surface charge density  of both the shell is equal. The electric potential at the centre, is
  R1      R2 
(a)   (b)  R1  R 2  (c)  R1  R 2  (d)  
0  R 2  0 0 0  R1 

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Exercise – 3
Multiple choice correct with multiple options

1. Two plane parallel conducting plates 1.5 cm apart are held horizontally one above the other in air. The
upper plate is maintained at a positive potential of 1.5 kV while the other plate is earthed. It is observed
that a small oil drop of mass 4.9 × 10–15 kg is at rest between the plates. Now the potential of the upper
plate is suddenly changed to –1.5 kV. Neglect the density of air.
Take g = 9.8 m/s2, Radius of the drop = 5.0 × 10–6m, Coefficient of viscosity of air = 1.8 × 10–5 kg m–1s–1.
On the basis of above information select the correct option:
(a) The number of electrons on the drop in 3
(b) Initial upward acceleration of the drop immediately after the upper plate is negatively charged is
19.6 m/s2
(c) Initial upward acceleration of the drop immediately after the upper plate is negatively charged is 9.8
m/s2
(d) The terminal speed of the drop is 5.7 × 10–5 m/s.
A
2. An annular spherical region a < r < b carries a charge per unit volume of   where A is constant. At
r
the center of the enclosed cavity, a point charge q is placed. For certain value of A, it is found that the
electric field ‘E’ inside the region a < r < b, is independent of ‘r’. Then
q q q q
(a) A  (b) A  (c) E  (d) E 
2a 2 2b2 40 a 2
40 b2
3. Three charged particles are in equilibrium under their electrostatic forces only, if
(a) the particles must be collinear
(b) all the charges must have the same magnitude
(c) all the charges cannot have the same sign
(d) the equilibrium is unstable
4. Four charges, all of the same magnitude, are placed at the four corners of a square. At the centre of the
square, the potential is V and the field is E. By suitable choices of the signs of the four charges, which of
the following can be obtained?
(a) V = 0, E = 0 (b) V = 0, E  0 (c) V  0, E  0 (d) V  0, E  0
5. Two identical charges +Q are kept fixed some distance apart. A small particle P with magnitude of charge
q is placed midway between them. If P is given a small displacement x, it will not perform simple
harmonic motion if
(a) q is positive and x is along the line joining the charges
(b) q is positive and x is perpendicular to the line joining the charges
(c) q is negative and x is perpendicular to the line joining the charges
(d) q is positive and x is along the line joining the charges
6. A positively charges thin metal ring of radius R is fixed in the xy plane, with its centre at the origin O. A
charged particle P is released from rest at the point (0, 0, z0), where z0 > 0. Then the motion of P is
(a) periodic, for all value of z0 satisfying 0  z 0   if P is negative
(b) away from ‘O’, for all values of z0 satisfying 0  z 0  R if P is positive
(c) approximately simple harmonic, provided z 0  R if P is negative
(d) such that P crosses O and continues to move along the negative z-axis towards z   and then it
returns back if P is positive
7. Charges Q1 and Q2 lie inside and outside respectively of a closed surface S. Let E be the field at any point
on S and  be the flux of E over S. Then
(a) if Q1 changes, both E and  will change
(b) if Q2 changes, E will change but  will not change
(c) if Q1  0 and Q 2  0 and E  0 but   0
(d) if Q1  0 and Q2 = 0 then E  0 and   0

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8. Charges Q1 and Q2 are placed inside and outside respectively of an uncharged conducting shell at the
distance r. Then
QQ
(a) the force on Q1 is zero (b) the force on Q1 is k 1 2 2
r
Q1Q 2
(c) the force on Q2 is k 2 (d) the force on Q2 is zero
r
9. A, B and C are three concentric metallic shells. Shell A is the innermost and shell C is the outermost. A is
given some charge. Then
(a) the inner surfaces of B and C will have the same charge
(b) the inner surfaces of B and C will have the same charge density
(c) the outer surfaces of A, B and C will have the same charge
(d) the outer surfaces of A, B and C will have the same charge density
10. S1 and S2 are two equipotential surfaces on which the potentials are not equal. Then
(a) S1 and S2 cannot intersect
(b) S1 and S2 can be plane surfaces
(c) In the region between S1 and S2, the field is maximum where they are closest to each other
(d) A line of force from S1 and S2 must be perpendicular to both
11. A simple pendulum of length l has a bob of mass m, with a charge q on it. A vertical sheet of charge, with
charge  per unit area, passes through the point of suspension of the pendulum. At equilibrium, the string
makes an angle  with the vertical. Its time period of oscillation is T in this case for small displacement.
Then
q q
(a) tan   (b) tan   (c) T  2  / g (d) T  2  / g
2 0 mg  0 mg
12. Choose the correct statements from the following
(a) If the electric field is zero at a point, the electric potential must also be zero at that point
(b) If electric potential is constant in a given region of space, the electric field must be zero in that
region
(c) Two different equipotential surfaces can never intersect
(d) Electrons move from a region of lower potential to a region of higher potential

13. A conducting sphere A of radius a, with charge Q, is placed concentrically

inside a conducting shell B of radius b. B is earthed. C is the common
centre of A and B.
Q
(a) The field at a distance r from C, where a  r  b, is k 2
r
Q
(b) The potential at a distance r from C, where a  r  b , is k
r
 1 1 
(c) The potential difference between A and B is kq   
 a b 
1 1 
(d) The potential at a distance r from C, where a  r  b , is kQ    .
r b
14. Two concentric shells have radii R and 2R charges qA and qB and potentials

2V and (3/2)V respectively. Now shell B is earthed and let charges on them

become qA and qB . Then

(a) qA/qB = 1/2 (b) qA / qB  1

(c) potential of A and B both will change after earthing
(d) potential difference between A and B remains same after earthing

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15. A particle of charge q and mass m moves rectilinearly under the action of an electric field E    x .
Here,  and  are positive constants and x is the distance from the point where the particle was initially
at rest. Then
(a) the motion of the particle is oscillatory but not SHM

(b) the amplitude of the particle is


(c) the maximum speed of the particle is at x 

q
(d) the maximum acceleration of the particle is
m
16. A particle of mass m and charge q is fastened to one end of a string of
length l fixed at point O. The whole system lies on a frictionless horizontal
plane. Initially, the mass is at rest at A. A uniform electric field in the
direction shown in then switched on. The magnitude of field is E and
directed towards right. Then
2qE
(a) the speed of the particle when it reaches B is
m
qE
(b) the speed of the particle when it reaches B is
m
(c) the tension in the string when particle reaches at B is 2qE
(d) the tension in the string when the particle reaches at B is qE

17. A block of mass m is attached to a spring of force constant k.

Charge on the block is q. A horizontal electric field E is acting
in the direction as shown. Block is released with the spring in
unstretched position. Then
(a) block will execute SHM
(b) time period of oscillation depends on the charge ‘q’ and
electric field E
qE
(c) amplitude of oscillation is
k
(d) block will oscillate but not simple harmonically

18. A uniform electric field E0 exists in a region at angle 45o with x-axis.
There are two point A(a, 0) and B(0, b) having potential VA and VB,
respectively, then
(a) VA > VB if a > b (b) VA = VB if a = b
(c) VA > VB if a < b (d) VA < VB if a > b
19. A non-conducting solid sphere of radius R is uniformly charged. The magnitude of the electric field due
to the sphere at a distance r form its centre
(a) increases as r increases for r < R (b) decreases as r increases 0  r  
(c) decreases as r cross O increase for R  r   (d) is discontinuous at r = R
  
20. A point charge q is placed at origin. Let E A , E B and E C be the electric field at three points A(1, 2, 3),
B(1, 1, –1), and C(2, 2, 2) due to charge q. Then
       
(a) EA  EB (b) E A || E B (c) | EB |  4 | EC | (d) | E B |  16 | E C |

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21. An ellipsoidal cavity is carved within a perfect conductor. A

positive charge q is placed at the centre of the cavity. The point A
and B are on the cavity surface as shown in the fig. Then
(a) electric field near A in the cavity  electric field near B in the
cavity
(b) charge density at A  charge density at B
(c) potential at A = potential at B
(d) total electric field flux through the surface of the cavity is q / 0

Exercise – 4
Section – I : Numerical Value / Subjective Type Questions

1. Two small equally charged spheres, each of mass m, are suspended from the same point by silk thread of
length l. The distance between the spheres x << l. The rate dq/dt with which the charge leaks off each
dq
sphere if their approach velocity varies as v  a / x , where a is a constant, and the rate with which
dt
  0 mg
the charge leaks off each sphere is given by a then the value of      is
 
2. A thin wire ring of radius r has an electric charge q. The increment of the force stretching the wire if a
q q0
point charge q0 is placed at the ring’s center is then find the value of [K], where [ ] represents
K 0 r 2
maximum integer.
3. A point charge q is located at the centre of a thin ring of radius R with uniformly distributed charge –q.
The magnitude of the electric field strength vector at the point lying on the axis of the ring at a distance x
from its centre, if x << R, is inversely proportional to xn then the value of n is
4. A system consists of a thin charged wire ring of radius R and a very long uniformly charged thread
oriented along the axis of the ring, with one of its ends coinciding with the centre of the ring. The total
charge of the ring is equal to q. The charge of the thread (per unit length) is equal to . . The interaction
q
force between the ring and the thread, is given by then the value of  is
0 R

5. A thread carrying a uniform charge  per unit length has the

configurations shown in figure a and b. Assuming a
curvature radius R to be considerably less than the length of
the thread, let the electric field at point O in case a is E1 and O O
E R
in case b is E2 then 2 is
E1 (a) (b)
6. Suppose the surface charge density over a sphere of radius R depends on a polar angle  as   0 cos ,
0
where 0 is a positive constant. The magnitude of the electric field inside the sphere is given by
 0
then the value of  is, is
7. A very long uniformly charged thread oriented along the axis of a circle of radius R rests on its centre
with one of the ends. The charge of the thread per unit length is equal to  . The flux of the vector E
R
across the circle area, is given by then the value of k is
k 0

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8. Two point charges q and –q are separated by the distance 2l (figure as

shown). The flux of the electric field strength vector across a circle of radius R

q O +q –q
R= 3 l, is given by then the value of k is l l
k 0

9. 
The electric field strength depends only on the x and y coordinates to the law E =  a  xi  yj  / x 2  y 2 , 
where a is a constant, i and j are the unit vectors of the x and y axes. The flux of the vector E through a
sphere of radius R with its centre at the origin of coordinates, is given by Ra then the value of [] is
2
10. The field potential inside a charged ball depends only on the distance from its centre as   ar  b,
where a and b are constants. Find the space charge distribution  r  inside the ball, is given by  0 a
then the value of  is
11. Figure shows three concentric thin spherical conducting shells A, B and C of
radii a, 2a and 3a respectively. The shells A and C are given charges q and –q
respectively and the shell B is earthed. Let the surface charge density of inner
surface of B is B and the surface charge density of outer surface of shell C is
 
C then B  , the value of    is
C 
12. Three identical metal plates with large surface areas are kept parallel to
each other as shown in figure. The left most plate is given a charge Q, the
rightmost a charge –2Q and the middle is neutral. If the charge appearing
Q
on the outer surface of the rightmost plate is  then the value of K is
K
13. Certain amount of charge is uniformly distributed over the volume of a ball. If the energy stored in the
V
ball is V1 and the energy stored in the surrounding space is V2 then 2 
V1
14. Two point charges +q1 and –q2 are placed at A and B, respectively.
A line of force emanates from q1 at an angle  with the line AB and
  q1 
terminates at  at –q2. If   x sin 1 sin  then the value of
 y q 2 

(x + y) is

15. Four concentric hollow spherical conductors of ratio of radii 1 : 2 :

3 : 4 are given the charges as shown in figure
Innermost conductor and third conductor are connected by
connecting wire. Also second conductor and outermost conductor
are connected at the same time by connecting wire. Let the
potential of conductor 1 is V1 and the potential of conductor 2 is V2
V1 
after connection then is found to be then the value of
V2 

   is

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Section – II : Column Match Type

1. Four large parallel identical conducing plates of area A are

arranged as shown. The charges on each plate are given in
diagram and separation between plates is d. Surfaces of
plates are numbered (1), (2), up to (8).
Column – I Column – II
(A) The charge on surface (4) is (p) – 3Q
(B) The charge on surface (6) is (q) –2Q
(C) The charge on surface (2) is (r) 7Q
(D) The charge on surface (8) is (s) Zero

2. Column I gives a situation in which two dipoles of dipole moment pi

and 3p j are placed at origin. A circle of radius R with centre at origin

is drawn as shown in figure. Column II gives coordinates of certain

positions on the circle. Match the statements in Column I with the
statements in Column II.
Column – I Column – II
 R 3R 
(A) The coordinate(s) of point on circle where potential is maximum (p)  , 
2 2 
 R 3R 
(B) The coordinate(s) of point on circle where potential is zero (q)   ,  
 2 2 
 3R R
(C) The coordinate(s) of point on circle where magnitude of electric field (r)   , 
 2 2 
1 4p
Intensity is
4 0 R 3
 3R R
(D) The coordinate(s) of point on circle where magnitude of electric field (s)  ,  
 2 2
1 2p
intensity is
4 0 R 3

Section – III : Comprehension

Comprehension # (1-2)
A thin nonconducting ring of radius R has a linear charge density    0 cos , where  0 is a constant, 
is the azimuthal angle.
1. Let the electric potential on the axis of the ring at distance ‘x’ measured from center is V then which of
the following graph will represent the variation of V with ‘x’
V
V

(a) (b)

x x

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V V

(c) (d)

x x
2. The electric field on the axis of the ring is given by for x >> R
0R 2 0R 2 2 0 R 2 4 0 R 2
(a) (b) (c) (d)
4 0 x 3 2 0 x 3 0 x 3 0 x 3
Comprehension # (3-4)
A ball of radius R carries a positive charge whose volume density depends only on a separation r from the
ball’s centre as   0 1  r / R , where 0 is a constant. Assuming the permittivities of the ball and the
environment to the equal to unity.
3. The electric field as a function of the distance r is
(a) continues decreases
(b) first increases and then decreases
(c) inside constant and outside the sphere decreases
(d) inside increases and outside decreases
4. The maximum possible value of electric field is
 R  R  R 0 R
(a) 0 (b) 0 (c) 0 (d)
3 0 6 0 9 0 12 0

Comprehension # (5-6)
A rigid insulated wire frame in the form of a right angled triangle
 
ABC  A   is set in vertical plane as shown in figure.
 2
Two point charges P & Q of equal mass but different charges are
connected by an insulated cord, are at rest. Neglect friction.
5. The angle made by the cord with AB is given by
(a) 30o (b) 60o (c) 37o (d) 45o
N1
6. Let normal reaction on the particle P is N1 and on the particle Q is N2 then 
N2
1 2 3
(a) (b) (c) 3 (d)
3 3 2
Comprehension # (7-8)
A non-conducting disc of radius a and uniform positive surface charge density  is placed on the ground,
with its axis vertical. A particle of mass m and positive charge q is dropped, along the axis of the disc,
q 40 g
from a height H with zero initial velocity. The particle has  .
m 
7. The value of H if the particle just reaches the disc.
4a 5a 2a
(a) (b) (c) (d) 2a
3 3 3
8. The potential energy of the particle as a function of its height
(a) decreases (b) first decreases and then increases
(c) first increases and then decreases (d) increases

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Comprehension # (9-10)
27 3 3
Four point charges  8C,  1C,  1C and 8 C, are fixed at the points,  m,  m,  m and
2 2 2
27
 m respectively on the y-axis. A particle of mass 6 × 10–4 kg and of charge 0.1C moves along
2
the x-direction. Its speed at x   is v0. Assume that space is gravity free. Given :
1/(4  0 )  9  10 9 Nm 2 / C 2
9. The least value of v0 for which the particle will cross the origin
(a) 2 m/s (b) 2.5 m/s (c) 3 m/s (d) 3.5 m/s
10. The kinetic energy of the particle at the origin.
(a) 0 (b) 125 J (c) 200 J (d) 250 J

Exercise - 5
Revision Exercise (Moderate to tough)

1. Figure shows a rod of length L which is uniformly charged with linear

charge density  kept on a smooth horizontal insulated surface. Right
end of rod is in contact with a vertical fixed insulated wall. A block of
mass m and charge q is projected with a velocity v from a point very
far from rod in the line of rod. Find the distance of closest approach
20 mv 2
between the block and the left end A of the rod, if  n2 .
q
2. A conducting sphere S1 of radius r is attached to an insulating handle. Another conducting sphere S2 of
radius R is mounted on an insulating stand. S2 is initially uncharged. S1 is given a charge Q, brought into
contact with S2 and removed. S1 is then recharged such that the charge on it is again Q and it is again
brought into contact with S2 and removed. This procedure is repeated n times.
(a) Find the electrostatic energy of S2 after n such contacts with S1.
(b) What is the limiting value of this energy as n   ?
3. A triangle is made from thin insulating rods of different lengths, and the linear charge density on each rod
is uniform and the same for all three rods. Find a particular point in the plane of the triangle at which the
electric field strength is zero.
4. A square of side d, made from a thin insulating plate, is
uniformly charged and carries a total charge of Q. A point
charge q is placed on the symmetrical normal axis of the
square at a distance d/2 from the plate. How large is the force
acting on the point charge?

5. A very thin disc is uniformly charged with surface charge density   0. Find the electric field intensity E
on the axis of this disc at the point from which the disc is seen at an angle  .
6. Find the potential at the edge of a thin disc of radius R with a charge uniformly distributed over one of its
sides with the surface density  .
7. An uncharged metallic sphere of radius R is placed into an external uniform field, as a result of which an
induced charge appears on the sphere with surface density   0 cos , where 0 is a positive constant
and  is a polar angle. Find the magnitude of the resultant electric force acting on like charges.
8. A point charge q is at a distance l from an infinite conducting plane. Find the density of surface charges
induced on the plane as a function of the distance r from the base of the perpendicular dropped from the
charge q onto the plane.

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9. A small charged ball is in state of equilibrium at a

height h above a large horizontal uniformly
charged dielectric plate having surface charge
density of  C / m 2 .
(a) Find the acceleration of the ball if a disc of
radius r(<<h) is removed from the plate
directly underneath the ball.
(b) Find the terminal speed (V0) acquired by the
falling ball. Assume that mass of the ball is
m, its radius is x and coefficient of viscosity
of air is  . Neglect buoyancy and assume
that the ball acquires terminal speed within a
short distance of its fall.
10. A soap bubble of radius R = 1 cm is charged with the maximum charge for which breakdown of air on its
surface does not occur. Calculate the electrostatic pressure on the surface of the bubble. It is know that
dielectric breakdown of air takes place when electric field becomes larger then E0 = 3 × 106 V/m.
11. On a horizontal table, there is a smooth circular groove
of mean radius R. The walls of the groove are non
conducting. Two metal balls (each having mass m and
radius r) are placed inside the groove with their centers
R apart. The balls just fit inside the groove. The two
balls are given charge +3q and –q and released from
state of rest. Ignore the non-uniformity in charge
distribution as the balls come close together and
collide. The collision is elastic. Find maximum speed
acquired by each ball after they collide for the first
time.
12. A thin insulating rod of mass m and length L is hinged at its upper end (O)
so that it can freely rotate in vertical plane. The linear charge density on the
rod varies with distance (y) measured from upper end as
L
  y 2 ; 0  y 
2
n L
 y ;  y  L
2
where  and  both are positive constants. When a horizontal electric
field E is switched on the rod is found to remain stationary.
(a) Find the value of constant  in terms of  . Also find n.
(b) Find the force applied by the hinge on the rod, if EaL3 = 45 mg.

13. A and B are two large identical thin metal plates placed
parallel to each other at a small separation. Plate A is
given a charge Q.
(i) Find the amount of charge on each of the two faces of
A and B.
(ii) Another identical plate C having charge 3Q is
inserted between plate A and B such that distance of C
from B is twice its distance from A. Plate A and B is
shorted using a conducting wire. Find charge on all six
faces of plates A, B and C.
(iii) In the situation described in (ii) the plate A is
grounded. Now write the charge on all six faces.

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Exercise – 6
Section – I : JEE (Advanced) Questions Previous Years

1.

2.

3.

4.

5.

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6.

7.

8.

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9.

10.

11.

12.

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13.

14.

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15.

16.

17.

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18.

(A) P  5; Q  3,4; R  1; S  2 (B) P  5; Q  3; R  1,4; S  2

(C) P  5; Q  3; R  1,2; S  4 (D) P  4; Q  2,3; R  1; S  5

Question Stem for Question Nos. 19 and 20 [JEE (Adv.) 2021, P-1]
Two point charges –Q and +Q/ 3 are placed in the xy-plane at
the origin (0, 0) and a point (2, 0), respectively, as shown in the
figure. This results in an equipotential circle of radius R and
potential V = 0 in the xy-plane with its center at (b, 0). All lengths
are measured in meters.
19. The value of R is ______ meter.
20. The value of b is ______ meter.
21. In the following circuit C1  12 F, C 2  C3  4 F and

C4  C5  2 F. The charge stored in C3 is _____ C.

[JEE (Adv.) 2022, P-1]

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22. A medium having dielectric constant K > 1 fills the space between the
plates of a parallel plate capacitor. The plates have large area, and the
distance between them is d. The capacitor is connected to a battery of
voltage V, as shown in Figure (a). Now, both the plates are moved by a
d
distance of from their original positions, as shown in Figure (b).
2
In the process of going from the configuration depicted in Figure (a) to
that in Figure (b), which of the following statement(s) is(are) correct?
(A) The electric field inside the dielectric material is reduced by a
factor of 2𝐾.
1
(B) The capacitance is decreased by a factor of .
k 1
(C) The voltage between the capacitor plates is increased by a factor
of (𝐾 + 1).
(D) The work done in the process DOES NOT depend on the
presence of the dielectric material.

23. Six charges are placed around a regular hexagon of side length
𝑎 as shown in the figure. Five of them have charge 𝑞, and the
remaining one has charge 𝑥. The perpendicular from each
charge to the nearest hexagon side passes through the center O
of the hexagon and is bisected by the side.
[JEE (Adv.) 2022, P-1]
Which of the following statement(s) is(are) correct in SI units?
(A) When 𝑥 = 𝑞, the magnitude of the electric field at O is zero.
(B) When 𝑥 = −𝑞, the magnitude of the electric field at O is
q
.
6 0 a 2
7q
(C) When 𝑥 = 2𝑞, the potential at O is .
4 3 0 a
3q
(D) When 𝑥 = −3𝑞, the potential at O is  .
4 3 0 a

24. A charge 𝑞 is surrounded by a closed surface consisting of an inverted

cone of height ℎ and base radius 𝑅, and a hemisphere of radius 𝑅 as

shown in the figure. The electric flux through the conical surface is

nq
(in SI units). The value of 𝑛 is _______. [JEE (Adv.) 2022, P-2]
60

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25. In the figure, the inner (shaded) region A represents a sphere of radius
rA = 1, within which the electrostatic charge density varies with the radical
distance r from the center as A  kr, where k is positive. In the spherical
shell B of outer radius rB, the electrostatic charge density varies as
2k
B  . Assume that dimensions are taken care of. All physical
r
quantities are in their SI units. [JEE (Adv.) 2022, P-2]
Which of the following statement(s) is(are) correct?
3
(A) If rB = , then the electric field is zero everywhere outside B.
2
3 k
(B) If rB = , then the electric potential just outside B is .
2 0
(C) If rB = 2 then the total charge of the configuration is 15k.
5
(D) If rB = , then the magnitude of the electric field just outside B is
2
13k
.
0
26. A disk of radius R with uniform positive charge density 𝜎 is placed on the xy plane with its center at the
origin. The Coulomb potential along the z-axis is [JEE (Adv.) 2022, P-2]

V(z) 

2 0
 
R 2  z2  z .

A particle of positive charge 𝑞 is placed initially at rest at a point on the z axis with 𝑧 = 𝑧0 and 𝑧0 > 0. In

addition to the Coulomb force, the particle experiences a vertical force F  ckˆ with 𝑐 > 0. Let 𝛽 =
2c 0
. Which of the following statement(s) is(are) correct?
q
1 25
(A) For 𝛽 = and 𝑧0 = 𝑅, the particle reaches the origin.
4 7
1 3
(B) For 𝛽 = and 𝑧0 = 𝑅, the particle reaches the origin.
4 7
1 R
(C) For 𝛽 = and 𝑧0 = , the particle returns back to 𝑧 = 𝑧0.
4 3
(D) For 𝛽 > 1 and 𝑧0 > 0, the particle always reaches the origin.
27. An electric dipole is formed by two charges +q and
–q located in xy-plane at (0, 2) mm and (0, –2) mm,
respectively, as shown in the figure. The electric
potential at point P (100, 100) mm due to the dipole
is V0. The charges +q and –q are then moved to the
points (–1, 2) mm and (1, –2) mm, respectively.
What is the value of electric potential at P due to the
new dipole? [JEE (Adv.) 2023, P-2]
(A) V0/4 (B) V0/2
(C) V0/ 2 (D) 3V0/4

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Section – II : JEE (Main) Questions Previous Years

11.

2.

3.

4. A cube of side ‘a’ has point chares +Q located at each of is vertices

except at the origin where the charge is – Q. The electric field at
the centre of cube is: [JEE Main 2021]

(1)
2Q
3 3 0 a 2
 x  y  z  (2)
Q
3 3 0 a 2
 x  y  z 
(3)
2Q
3 3 0 a 2
 x  y  z  (4)
Q
3 3 0 a 2
 x  y  z 
5. Two electrons each are fixed at a distance ‘2d’. A third charge proton placed at the midpoint is displaced
slightly by a distance x(x << d) perpendicular to the line joining the two fixed charges. Proton will
execute simple harmonic motion having angular frequency: (m = mass of charged particle)
[JEE Main 2021]
1 1 1 1
 q2 2  0 md3  2  20 md3  2  2q 2  2
(1)  3
(2)  2  (3)  2  (4)  3 
 20 md   2q   q   0 md 

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6. A vertical electric field of magnetic 4.9 × 105 N/C just prevents a water droplet of a mass 0.1 g from
falling. The value of charge on the droplet will be : (Given = 9.8 m/s2)
[JEE Main, 24 June 2022 - Shift 1]
(1) 1.6 × 10–9 C (2) 2.0 × 10–9 C (3) 3.2 × 10–9 C (4) 0.5 × 10–9 C
7. In the figure, a very large plane sheet of positive charge is shown.
P1 and P2 are two points at distance l and 2l from the charge
distribution. If  is the surface charge density, then the magnitude
of electric fields E1 and E2 at P1 and P2 respectively are :
[JEE Main, 25 June 2022 - Shift 1]
(1) E1   / 0 , E 2   / 20 (2) E1  2 /  0 , E 2   / 0
(3) E1  E 2   / 20 (4) E1  E 2   / 0
8. Sixty four conducting drops each of radius 0.02 m and each carrying a charge of 5 C are combined to
form a bigger drop. The ratio of surface density of bigger drop to the smaller drop will be:
[JEE Main, 26 June 2022 - Shift 2]
(1) 1:4 (2) 4:1 (3) 1:8 (4) 8:1
9. If a charge q is placed at the centre of a closed hemispherical non-
conducting surface, the total flux passing through the flat surface
would be : [JEE Main, 27 June 2022 - Shift 2]
q
q q
(1) (2)
0 2 0
q q
(3) (4)
4 0 20
10. Two point charges A and B of magnitude +8 × 10–6 C and –8 × 10–6 C respectively are placed at a
distance d apart. The electric field at the middle point O between the charges is 6.4 × 104 NC–1. The
distance ‘d’ between the point charges A and B is : [JEE Main, 28 June 2022 - Shift 2]
(1) 2.0 m (2) 3.0 m (3) 1.0 m (4) 4.0 m
11. A positive charge particle of 100 mg is thrown in opposite direction to a uniform electric field of strength
1 × 105 NC–1. If the charge on the particle is 40 C and the initial velocity is 200 ms–1, how much
distance it will travel before coming to the rest momentarily: [JEE Main, 29 June 2022 - Shift 1]
(1) 1m (2) 5m (3) 10 m (4) 0.5 m
12. The volume charge density of a sphere of radius 6 m is 2 C cm–3. The number of lines of force per unit
surface area coming out from the surface of the sphere is _____ × 1010 NC–1. (Given : Permittivity of
vacuum 0  8.85  1012 C2 N 1  m2 ) [JEE Main, 25 July 2022 - Shift 1]
13. Three point charges of magnitude 5C,0.16C and 0.3C are located at the vertices A, B, C of a right
angled triangle whose sides are AB = 3 cm, BC = 3 2 cm and CA = 3 cm and point A is the right corner.
Charge at point A experiences _______ N of electrostatic force due to the other two charges.
[JEE Main, 26 July 2022 - Shift 2]
14. A charge of 4 C is to be divided into two. The distance between the two divided charges is constant. The
magnitude of the divided charges so that the force between them is maximum, will be:
[JEE Main, 27 July 2022 - Shift 2]
(1) 1C and 3 C (2) 2 C and 2 C (3) 0 and 4 C (4) 1.5 C and 2.5 C

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15. Two identical metallic spheres A and B when placed at certain distance in air repel each other with a
force of F. Another identical uncharged sphere C is first placed in contact with A and then in contact with
B and family placed at midpoint between spheres A and B. The force experienced by sphere C will be :
[JEE Main, 29 July 2022 - Shift 2]
(1) 3 F/2 (2) 3 F/4 (3) F (4) 2F
16. If two charges q1 and q2 are separated with distance ‘d’ and placed in a medium of dielectric constant K.
What will be the equivalent distance between charges in air for the same electrostatic force?
[JEE Main, 24 Jan. 2023 - Shift 1]
(1) d k (2) k d (3) 1.5d k (4) 2d k

17. A uniform electric field of 10 N/C is created between two

parallel charged plates (as shown in figure). An electron
enters the field symmetrically between the plates with a
kinetic energy 0.5 eV. The length of each plate is 10 cm. The
angle () of deviation of the path of electron as it comes out
of the field is (in degree).
[JEE Main, 25 Jan. 2023 - Shift 1]
18. In a cuboid of dimension 2L × 2L × L, a charge q is placed at the centre of the surface ‘S’ having area of
4 L2. The flux through the opposite surface to ‘S’ is given by [JEE Main, 29 Jan. 2023 - Shift 1]
q q q q
(1) (2) (3) (4)
120 30 20 60

19. A point charge 2 × 10–2 C is moved from P to S in a uniform electric field of 30 NC–1 directed along
positive x-axis. If coordinates of P and S are (1, 2, 0) m and (0, 0, 0) m respectively, the work done by
electric field will be [JEE Main, 29 Jan. 2023 - Shift 2]
(1) 1200 mJ (2) 600 mJ (3) –600 mJ (4) –1200 mJ
20. As shown in the figure, a point charge Q is placed at the centre of
conducting spherical shell of inner radius a and outer radius b. The
electric field due to charge Q in three different regions I, II and III
is given by : (I : r < a, II : a < r < b, III : r > b)
[JEE Main, 30 Jan. 2023 - Shift 2]
(1) EI = 0, EII = 0, EIII ≠ 0 (2) EI ≠ 0, EII = 0, EIII ≠ 0
(3) EI ≠ 0, EII = 0, EIII = 0 (4) EI = 0, EII = 0, EIII = 0
 V
21. Expression for an electric field is given by E  4000x 2 ˆi . The electric flux through the cube of side 20
m
cm when placed in electric field (as shown in figure) is ____ V cm. [JEE Main, 31 Jan. 2023 - Shift 1]

22. A cubical volume is bounded by the surfaces x = 0, x = a, y = 0, y = a, z =


ˆ Where E0 =
0, z = a. The electric field in the region is given by E  E 0 xi.
4 × 104 NC–1 m–1. If a = 2 cm, the charge contained in the cubical volume
is Q × 10–14C. The value of Q is _____. (Take 0 = 9 × 10–12 C2/Nm2)
[JEE Main, 01 Feb. 2023 - Shift 2]

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23. A dipole comprises of two charged particles of identical magnitude and opposite in nature. The mass m of
the positive charged particle is half of the mass of the negative charged particle. The two charges are

separated by a distance l. If the dipole is placed in a uniform electric field E ; in such a way that dipole

axis makes a very small angle with the electric field, E . The angular frequency of the oscillations of the
dipole when released is given by: [JEE Main, 06 April 2023 - Shift 2]
3qE 8qE 4qE 8qE
(1) (2) (3) (4)
2ml ml ml 3ml
24. An electric dipole of dipole moment is 6.0 × 10–6 C m placed in a uniform electric field of 1.5 × 103 N C–1
in such a way that dipole moment is along electric field. The work done in rotating dipole by 180º in this
field will be_______ mJ. [JEE Main, 08 April 2023 - Shift 1]
25. Three concentric spherical metallic shells X, Y and Z of radius a, b and c respectively [a < b < c] have
surface charge densities 1, –  and , respectively. The shells X and Z are at same potential. If the radii
of X & Y are 2 cm and 3 cm, respectively. The radius of shell Z is _____ cm.
[JEE Main, 10 April 2023 - Shift 1]
26. As shown in the figure, a configuration of two equal point charges
(q0 + 2µC) is placed on an inclined plane. Mass of each point
charge is 20 g. Assume that there is no friction between charge and
plane. For the system of two point charges to be in equilibrium (at
rest) the height h = x × 10–3 m. The value of x is (Take
1
 9  109 N m 2 C –2 ,g  10m s –2 )
40
[JEE Main, 11 April 2023 - Shift 1]
27. Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : If an electric dipole of dipole moment 30 × 10–5 C mis enclosed by a closed surface, the net
flux coming out of the surface will be zero.
Reason R : Electric dipole consists of two equal and opposite charges. In the light of above, statements,
choose the correct answer from the options given below. [JEE Main, 12 April 2023 - Shift 1]
(1) Both A and R are true and R is the correct explanation of A
(2) A is false but R is true
(3) A is true but R is false
(4) Both A and R are true and R is NOT the correct explanation of A

28. Three point charges q, –2q and 2q are placed on axis at a

3
distance x = 0, x = R and x = R respectively from
4
origin as shown. If q = 2 × 10–6 C and R = 2 cm, the
magnitude of net force experienced by the charge –2q is
N. [JEE Main, 13 April 2023 - Shift 2]

29. The electric field due to a short electric dipole at a large distance (r) from center of dipole on the
equatorial plane varies with distance as : [JEE Main, 15 April 2023 - Shift 1]
1 1 1
(1) r (2) 2
(3) 3
(4)
r r r

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 P-31

ANSWER KEY
EXERCISE – 1
1. (a) 2. (d) 3. (c) 4. (d) 5. (c)
6. (d) 7. (a) 8. (c) 9. (d) 10. (c)
11. (c) 12. (a) 13. (c) 14. (c) 15. (d)
16. (d) 17. (a) 18. (d) 19. (a) 20. (b)
21. (b) 22. (b) 23. (b) 24. (c) 25. (a)
26. (d) 27. (b) 28. (a) 29. (d) 30. (b)
31. (a) 32. (a) 33. (d) 34. (b) 35. (b)
36. (a) 37. (d) 38. (b) 39 (d) 40. (b)
41. (a) 42. (b) 43. (b) 44. (a) 45. (b)
46. (b) 47. (a)
EXERCISE – 2

1. (d) 2. (d) 3. (c) 4. (d) 5. (a)

6. (b) 7. (b) 8. (b) 9. (a) 10. (b)
11. (b) 12. (c) 13. (a) 14. (d) 15. (d)
16. (c) 17. (b) 18. (d) 19. (b) 20. (b)
21. (c) 22. (a) 23. (d) 24. (c) 25. (d)
26. (a) 27. (c) 28. (a) 29. (c) 30. (b)
31. (d) 32. (d) 33. (c)

EXERICSE - 3

1. (a, b, d) 2. (a, c) 3. (a, b, d) 4. (a, b, c, d) 5. (b, d)

6. (a, b, c) 7. (a, b, c, d) 8. (a, c) 9. (a, c) 10. (a, b, c, d)
11. (a, c) 12. (b, c, d) 13. (a, c, d) 14. (a, c, d) 15. (b, c, d)
16. (b, c) 17. (a, c) 18. (b, c, d) 19. (a, c) 20. (a, c)
21. (a, b, c, d)

EXERCISE – 4
Section - I

1. 7 2. 78 3. 4 4. 16 5. 0
6. 3 7. 2 8. 2 9. 12 10. 6
11. 31 12. 2 13. 5 14. 4 15. 8

Section - II
1. (A)  (q); (B)  (s); (C)  (p); (D)  (r)
2. (A)  (p); (B)  (r,s); (C)  (p,q); (D)  (r,s)

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 P-32

Section – III

1. (b) 2. (a) 3. (b) 4. (c) 5. (b)

6. (c) 7. (a) 8. (b) 9. (c) 10. (d)

EXERCISE – 5
1. L
2
Q2    R Q2 R
2. (a)   (1   n 2
) . Here   (b)
8 0 R  1    Rr 80 r 2

Qq  R
3. Incentre of the triangle. 4. 5. E 6.
6 0 d 2 4 0  0

20 R 2 q gr 2 mgr 2
7. 8.  9. (a) (b)
40 2  (  r 2 ) 3 / 2
2
2h 2 12 xh 2

q 2 (4R  7r) 
10. 39.6 N/m2 11. 12. (a)   , n  2 (b) 2mg
80 mrR 15

13. (i) (ii)

(iii)

EXERCISE – 6
Section - I
1. (c) 2. (a,b,c,d) 3. (a) 4. (c) 5. (d)
6. (a,b,c) 7. (b,d) 8. (c) 9. (c) 10. (c)
11. (a) 12. 6 13. (c) 14. (d) 15. (c,d)
16. (a,b) 17. 2.00 18. (b)
19. R = 01.73 20. b = 03.00 21. Range (7.9 – 8.1) 22. (B) 23. (A,B,C)
24. 3 25. (B) 26. (A,C,D) 27. (B)
Section - II
1. (3) 2. (4) 3. (3) 4. (3) 5. (1) 6. (2) 7. (3)
8. (2) 9. (2) 10. (2) 11. (4) 12. (45) 13. (17) 14. (2)
15. (2) 16. 1 17. 45 18. 4 19. 3 20. 2 21. 640
22. 288 23. (1) 24. 18 25. 5 26. 300 27. (1) 28. 5440
29. (3)

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 CURRENT
ELECTRICITY
 P-33

Exercise – 1
Concept Building Questions

1. If a wire is stretched to double its length, find the new resistance if the original resistance of the wire was R.
2. The wire is stretched to increase the length by 1%. Find the percentage change in the resistance.
3. The voltage-current graphs for two resistors of the same
material and the same radius with lengths, L1 and L2 are
shown in figure. If L1 > L2, state with reason which of
these graphs represents voltage-current change for L1.

4. A hollow cylinder of length l, has inner and outer radius

a and b respectively. If the resistivity of material of the
cylinder is  . Then find the resistance of the cylinder.
(i) across its ends (across M and N)
(ii) across its inner and outer surface (across P and Q)
5. In a hydrogen discharge tube, the number of protons drifting across a cross section per second is 1.0 × 1018,
while the number of electrons drifting in the opposite direction across the same cross section is 2.7 × 1018 per
second. Find the current flowing in the tube.
6. You need to produce a set of cylindrical copper wires 2.5 m long that will have a resistance of 0.125  each.
What will be the mass of each of these wires? (Density of copper is 8.9 × 103 kgm–3 and resistivity of copper is
1.72 × 10–8  m) .
7. Consider a wire of length l, area of cross section A, and resistivity  with resistance 10 . Its length is increased
by applying a force, and it becomes four times of its original value. Find the changed resistance of the wire.
8. A wire of mass m, length l, density d, and area of cross section A is stretched in such a way that its length
increases by 10% of its original value. Express the changed resistance in percentage.
9. A uniform copper wire of mass 2.23 × 10–3 kg carries a current of 1 A when 1.7 V is applied across it. Calculate
the length and the area of cross section. If the wire is uniformly stretched to double its length, calculate the new
resistance. Density of copper is 8.92 × 103 kgm–3 and resistivity is 1.7 × 10–8  m .
10. A cylindrical wire has length l and area of cross section A. The
conductivity of wire material changes with distance from left end as

  0 , where 0 is a constant. Calculate electric field inside the
x
conductor as a function of x, when a battery of emf V is connected across
its ends.
11. Draw a colour code for 42k  10% carbon resistance.
12. What is resistance of following resistor.

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 P-34
13. A voltage of 30 V is applied across a colour coded carbon resistor with first, second and third rings of blue,
black and yellow colours. What is the current flowing through the resistor?
14. A copper wire has a square cross section of 6 mm on a side. The wire is 10 m long and carries a current of 3.6
A. The density of free electrons is 8.5 × 1028/m3. Find the magnitude of (a) the current density in the wire; (b)
the electric field in the wire. (c) How much time is required for an electron to travel the length of the wire?
(  , electrical resistivity, is 1.72 × 10–8  m)
15. Consider a wire of length 0.1 m with an area of cross section 1 mm2 connected to 5 V. Find the current flowing
through the metallic wire where   5 106 m2 V1s1 , e  1.6 1019 C, and n = 8 × 1028m–3.
16. Find the approximate total distance traveled by an electron in the time interval in which its displacement is one
meter along the wire. Given that the value of drift velocity is 1 mm s–1.
17. How much time will be taken by an electron to move a distance l = 1 km in a copper wire of cross section
A= 1 mm2 it carries a current I = 4.5 A?
18. Current is flowing from a conductor of nonuniform cross-sectional area.
If A1 > A2, then find relation between
(a) i1 and i2
(b) j1 and j2
(c) (vd)1 and (vd)2 (drift velocity) where i is current, j is current density,
and v is drift velocity.
19. Consider a conductor of length 40 cm where a potential difference of 10 V is maintained between the ends of
the conductor. Find the mobility of the electrons provided the drift velocity of the electrons is 5 × 10–6 ms–1.
20. A copper coil has a resistance of 20.0  at 0oC and a resistance of 26.4 at 80oC. Find the temperature
coefficient of resistance of copper.
21. A metallic wire has a resistance of 120  at 20oC. Find the temperature at which the resistance of same metallic

wire rises to 240 where the temperature coefficient of the wire is 2  104 C 1 .
22. The I-V characteristics of a resistor is observed to
deviate from a straight line for higher value of
current as shown in figure. Why?

23. In the circuit given, find the potential difference

across the resistances.

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 P-35
24. In the circuit given, find the currents I, I1 and I2 in
the circuit.

25. The V-I graphs for two resistors and their series
combination are shown in figure. Which one of
these graphs represents the series combination of
the two resistors? Give reason for your answer.

26. Two students perform experiments on series and

parallel combinations of two given resistors R1 and
R2 and plot the following V-I graphs. Which of the
graphs is/are correctly labeled in terms of the words
‘series’ and ‘parallel’? Justify your answer.
27. Four identical resistances each having value R are
arranged as shown in figure. Find the equivalent
resistance between A and B.

28. In the given circuits, if all the resistance has value equal to R. Find the equivalent resistance across A and B.

29. In the network in figure find the equivalent

resistance across the points M and M  .

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 P-36
30. In figure, the resistances are connected as

shown. Given R 1  10  and R 2  20  .

Determine the equivalent resistance

between points A and D.

31. In the given circuits (fig. (a) and (b),

calculate the resistance between points A

and B.

32. Figure shows a network of currents. The

magnitude of the current is shown here. Find
the current i.

33. Figure shows three resistances are

connected with switch S initially the switch
is open. When the switch S is closed find the
current passed through it.

34. Find the current through 12 resistor in

figure.

35. The figure given below is a part of a circuit.

Calculate the current through 4 resistance
and also find the potential difference VA –
VD?
36. In the circuit shown in figure, find the current
through the branch BD.

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 P-37
37. In the circuit shown in figure, determine the

voltage drop between A and D.

38. In the circuit shown in figure, E, F, G, and H

are cells of emf 2, 1, 3, and 1 V, respectively.
The resistance 2 , 1, 3  and 1 are their
respective internal resistances.
(i) Find the potential difference between B
and D.
(ii) Calculate the potential differences across
the terminals of each of the cells G and H.
39. Twelve identical resistances are arranged on
all edges of a cube. The resistors are all the
same. Then find the equivalent resistance
between the edges A and B as shown in
figure.
40. Resistances R1, R2¸ R3, R4, R5, and R6 are connected with a 6 V battery with internal resistance 1 as shown in
figure. Find (a) equivalent resistance and (b) current in each resistance.
R 1  10 
R2  6
R3  5
R 4  3
R 5  20 
R 6  16 

41. In a mixed grouping of identical cells, five rows are connected in parallel and each now contains 10 cell. This
combination sends a current i through an external resistance of 20  . If the emf and internal resistance of each
cell is 1.5 V and 1 , respectively, then find the value of i.
42. n identical cells, each of emf E and internal resistance r, are joined in series to form a closed circuit. Find the
potential difference across any one cell.
43. Three cells of emf 3V, 4 V, and 6 V are connected in parallel. If their
internal resistances are 1, 2  and 1 , find the  eff , reff , and the current
in the external load R  1.6 

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 P-38
44. 100 cells each of emf 5 V and internal resistance 1 are to be arranged to produce maximum current in a 25
resistance. Each row contains equal number of cells. Find the number of rows.
45. In figure, given below find the current in the wire BD.

(a) (b) (c)

46. Find out the potential difference between the points x and y in figure.

47. In the diagram shown in figure, find the potential difference between
the points A and B and between the points B and C in the steady state.

48. In the given circuit, the switch S is closed at time t = 0. The charge Q
on the capacitor at any instant t is given by Q(t)  Q0 (1  et ). Find the
value of Q0 and  in terms of given parameters as shown in the circuit
in figure.

49. A part of a circuit is in steady state along with the current flowing in

the branches. Value of each resistance is shown in figure. Calculate the

energy stored in the capacitor C(4 F) .

50. If a battery of emf 8 V and negligible internal resistance is connected

between terminal P and Q of the circuit shown in figure, calculate the
current through 2.5 resistance and hence calculate the equivalent
resistance of the circuit.

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 P-39

Exercise – 2
Single choice correct with multiple option

1. I-V characteristic of a copper wire of length L and area of cross-section

A is shown in the figure. The slope of the curve becomes
(a) More if the experiment is performed at higher temperature
(b) More if a wire of steel of the same dimension is used
(c) More if the length of the wire is increased
(d) Less if the length of the wire is increased
2. The resistance R1 of a conductor varies with temperature t as shown in
the figure. If the variation is represented by R t  R 0 [1  t  t 2 ], then
(a)  and  are both negative
(b)  and  are both positive
(c)  is positive and  is negative
(d)  is negative and  is positive
3. A cylindrical conductor has uniform cross-section. Resistivity of its material increased linearly from the left end
to the right end. If a constant current is flowing through it and at a section distance x from left end, magnitude
of electric field intensity is E, which of the following graphs is correct

(a) (b) (c) (d)

4. A steady current flow in a metallic conductor of non-uniform cross-section. The quantity/quantities constant
along the length of the conductor is/are
(a) current, electric field and drift speed (b) drift speed only
(c) current and drift speed (d) current only
5. Two wires of resistance R1 and R2 have temperature coefficient of resistance 1 and  2 , respectively. These
are joined in series. The effective temperature coefficient of resistance is
1   2 1R 1   2 R 2 R 1R 2 1 2
(a) (b) 1 2 (c) (d)
2 R1  R 2 R 12  R 22
6. Length of a hollow tube is 5 m, it’s outer diameter is 10 cm and thickness of it’s wall is 5 mm. If resistivity of
the material of the tube is 1.7 × 108  m then resistance of tube will be

(a) 5.6  10 5  (b) 2  105  (c) 4  105  (d) None of these

7. Two wires each of radius of cross section r but of different materials are connected together end to end (i.e. in
series). If the densities of charge carriers in the two wires are in the ratio 1 : 4, the drift velocity of electrons in
the two wires will be in the ratio
(a) 1 : 2 (b) 2 : 1 (c) 4 : 1 (d) 1 : 4
8. A wire has a non-uniform cross-section as shown in figure.
A steady current flow through it. The drift speed of electrons
at point P and Q is vP and vQ
(a) vP - vQ (b) vP < vQ

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 P-40
(c) vP > vQ (d) data insufficient
9. A current I flow through a cylindrical rod of uniform cross-
section area A and resistivity  . The electric flux through the
shaded cross-section of rod as shown in the figure is
I
(a) (b) I

I A
(c) (d)
A I
10. If each resistance in the figure is of 9 , then reading of
ammeter is
(a) 5 A (b) 8 A
(c) 2 A (d) 9 A

11. The potential difference between points A and B of

adjoining figure is
2 8
(a) V (b) V
3 9
4
(c) V (d) 2 V
3
12. Thirteen resistances each of resistance R are connected in
the circuit as shown in the following figure. The effective
resistance between A and B is
4R
(a) 2R  (b) 
3
2R
(c)  (d) R
3
13. A wire of resistance 10 is bent to form a circle. P and Q
are points on the circumference of the circle dividing it into
a quadrant and are connected to a battery of 3 V and internal
resistance 1 as shown in the figure. The currents in the
two parts of the circle are
6 18 5 15
(a) A and A (b) A and A
23 23 26 26
4 12 3 9
(c) and A (d) A and A
25 25 25 25
14. The resistance between the terminal points A and B of the
given infinitely long circuit will be
(a) ( 3  1) (b) (1  3)

(c) (1  3) (d) (2  3)

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 P-41
15. Find equivalent resistance between A and B
(a) R
3R
(b)
4
R
(c)
2
(d) 2R
16. A wire has resistance of 24  is bent in the following shape.
The effective resistance between A and B is
(a) 24  (b) 10

16
(c)  (d) None of these
3
17. Find the equivalent resistance of the circuit between points
A and B shown in figure is (each branch is of resistance =
1)
22 12
(a)  (b) 
25 25
22 12
(c)  (d) 
35 35
18. The figure shows a network of resistor each heaving value
12 . Find the equivalent resistance between point A and B.

12
(a) 9 (b) 
5
11
(c) 8 (d) 
3
19. Five 1 resistance are connected as shown in the figure. The resistance in the connecting wires is negligible.
The equivalent resistance between A and B is

(a) 5 (b) 2.5 (c) 1 (d) 0.5

20. In the diagram resistance between any two junctions is R.
Equivalent resistance across terminals A and B is
11R 18R
(a) (b)
7 11
7R 11R
(c) (d)
11 18

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 P-42
21. A finite square grid, each link having resistance r is fitted in a
resistance less conducting circular wire. Determine equivalent
80
resistance between A and B. If r   , find the resistance
7
between A and B.
11
(a) 6 (b) 
7
15
(c)  (d) none of these
7
22. The given infinite grid consists of hexagonal cell of six resistors
each of resistance R. Then R12 is equal to
R 2R
(a) (b)
3 3
4R 3R
(c) (d)
3 4

23. The resistance of all the wires between any two adjacent dots is r.
The equivalent resistance between A and B as shown in figure is
7 7
(a) R (b) R
3 6
14
(c) R (d) None of these
8
24. 12 cells each having same emf are connected in series with some cells wrongly connected. The arrangement is
connected in series with an ammeter and two cells which are in series. Current is 3A when cells and battery aid
each other and is 2A when cells and battery oppose each other. The number of cells wrongly connected is
(a) 4 (b) 1 (c) 3 (d) 2
25. A battery of 24 cells each of emf 1.5 V and internal resistance 2 is to be connected in order to send the
maximum current through a 12 resistor. The correct arrangement of cells will be
(a) 2 rows of 12 cells connected in parallel (b) 3 rows of 8 cells connected in parallel
(c) 4 rows of 6 cells connected in parallel (d) all of these
26. Consider the circuits shown in the figure. Both the circuits
are taking the same current from battery but current through
1
R in the second circuit is th of current through R in the
10
first circuit. If R is 11 , the value of R1
(a) 9.9 (b) 11
(c) 8.8 (d) 7.7
27. In the circuit of adjoining figure the current through 12
resister will be
1
(a) 1 A (b) A
5
2
(c) A (d) 0 A
5

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 P-43
28. Eight cells marked 1 to 8, each of emf 5 V and internal
resistance 0.2 are connected as shown. What is the reading
of ideal voltmeter?
(a) 40 V (b) 20 V
(c) 5 V (d) zero
29. n identical cells are joined in series with its two cells A and B in the loop with reversed polarities. EMF of each
shell is E and internal resistance r. Potential difference across cell A or B is (here n > 4)
2E  1 4E  2
(a) (b) 2E 1   (c) (d) 2E 1  
n  n n  n
30. For what value of R in circuit, current through 4 resistance
is zero?
(a) 4 (b) 1
(c) 3 (d) 5
31. In the circuit shown, calculate the current through 6 V
battery?
(a) (1/4) A
(b) (1/8) A
(c) (1/2) A
(d) none of these
32. In the circuit shown in figure potential difference between
point A and B is 16 V. Find the current passing through 2
resistance.
(a) 1.5 A (b) 3.5 A
(c) 1.0 A (d) 2.5 A
33. In the circuit shown the currents in 4 and 8 resistances
are
(a) 2.5 a, 0 A (b) 5 A, 0 A
40 80
(c) 2.5A, A (d) 5A, A
3 3

34. The reading of the ideal voltmeter in the adjoining diagram

will be
(a) 4V (b) 8 V
(c) 12 V (d) 14 V

35. Voltmeter shown reads 18 V. The resistance of voltmeter is

(a) 75 (b) 120 

800 900
(c)  (d) 
7 7

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 P-44
36. A potentiometer is an ideal device of measuring potential difference because
(a) it uses a sensitive galvanometer
(b) it does not disturb the potential difference it measures
(c) it is an elaborate arrangement
(d) it has a long wire hence heat developed is quickly radiated
37. A resistance of 4 and a wire of length 5 metres and resistance
5 are joined in series and connected to a cell of e.m.f. 10 V
and internal resistance 1 . A parallel combination of two
identical cells is balanced across 300 cm of the wire. The e.m.f.
E of each cell is
(a) 1.5 V (b) 3.0 V
(c) 0.67 V (d) 1.33 V
38. In the Wheatstone bridge (Shown in the figure) X = Y and A >
B. The direction of the current between ab will be
(a) From a to b
(b) From b to a
(c) From b to a through c
(d) From a to b through c

39. In the circuit shown, a meter bridge is in its balanced state. The
meter bridge wire has a resistance 0.1 ohm/cm. The value of
unknown resistance X and the current drawn from the battery
of negligible resistance is
(a) 6 , 5A (b) 10 , 0.1A
(c) 4 , 1.0 A (d) 12 , 0.5A
40. In the shown arrangement of the experiment of the meter bridge
if AC corresponding to null deflection of galvanometer is x,
what would be its value if the radius of the wire AB is doubled
(a) x (b) x/4
(c) 4x (d) 2x
41. In the following circuit, 5 resistor develops 45 J/s due to
current flowing through it. The power developed per second
across 12 resistor is
(a) 16 W (b) 192 W
(c) 36 W (d) 64 W

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42. In the circuit diagram, all the bulbs are identical. Which bulb
will be the brightest?
(a) A (b) B
(c) C (d) D
43. Three 60 W, 120 V light bulbs are connected across a 120 V
power source. If resistance of each bulb does not change with
current then find out total power delivered to the three bulbs

(a) 180 W (b) 20 W (c) 40 W (d) 60 W

44. For what value of R, current in the galvanometer is zero
(a) 1 (b) 2
(c) 5 (d) 7

45. An ideal ammeter (of zero resistance) is connected as shown. The

reading of the ammeter is
E
(a) 0 (b)
2R
E E
(c) (d)
5R 7R

46. A battery of internal resistance 4 is connected to the

network of resistance as shown. In order that the maximum
power can be delivered to the network, the value of R in 
should be
4 8
(a) (b) 2 (c) (d) 18
9 3

47. For ensuring dissipation of same energy in all three resistors

(R1, R2 and R3) connected as shown in figure, their values must
be related as
(a) R1 = R2 = R3 (b) R2 = R3 and R1 = 4R2
(c) R2 = R3 and R2 = 4R1 (d) R1 = R2 + R3
48. Four infinite ladder network containing identical
resistances of R  each, are combined as shown in
figure. The equivalent resistance between A and B is
RAB and between A and C is RAC. Then the value of
R AB
is
R AC
3 4
(a) (b)
4 3
1
(c) 2 (d)
2

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49. Equivalent resistance between A and B in figure is

(a) 8r/7 (b) 7r/8

(c) 3r/4 (d) r

50. Each of the resistors shown in figure has resistance R. Find

the equivalent resistance between A and B.
7R 5R
(a) (b)
4 4

9R 11R
(c) (d)
4 4
51. The circuit shown has resistors of equal resistance R. Find
the equivalent resistance between A and B, when the key
is closed.
11R 13R
(a) (b)
12 12
R 15R
(c) (d)
5 12
52. The equivalent resistance of the combination across AB (figure) is

 3  17  3  17
(a)   (b) 3  17 (c) (d) 2(3  17)
 4  2
53. In the given circuit (figure), the potential difference across the
capacitor is 12 V. Each resistance is of 3 . The cell is ideal.
The emf of the cell is
(a) 15 V (b) 9 V
(c) 12 V (d) 24 V
54. The equivalent resistance between A and B in the arrangement of
resistances as shown is
(a) 4r (b) 3r
(c) 2.5 r (d) r

55. In the circuit as shown, VA – VB = V. The resistance of each wire

AB, AC, etc. are shown in the figure. Find the current in AC.

(a) 2V/3R (b) 4V/11R

(c) V/6R (d) none of these

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56. In the given circuit, find the value of R:

(a) 20  (b) 25

(c) 15 (d) 10

57. The potential of the point O is

(a) 3.5 V (b) 6.5 V

(c) 3V (d) 6V

Exercise - 3
Multiple choice correct with multiple options

1. When some potential difference is maintained between A and

B, current I enters the network at A and leaves at B.

(a) The equivalent resistance between A and B is 8 .

(b) C and D are at the same potential
(c) No current flows between C and D
(d) Current 3I/5 flows from D to C.
2. In the circuit shown in figure, some potential difference
is applied between A and B. the equivalent resistance
between A and B is R.

(a) No current flows through the 5 resistor (b) R  15 

18
(c) R  12.5  (d) R 
5
3. A battery of emf E and internal resistance r is connected across a resistance R. Resistance R can be adjusted to
any value greater than or equal to zero. A graph is plotted between the current passing through the resistance (I)
and potential difference (V) across it. Select the correct alternatives.

(a) Internal resistance of the battery is 5

(b) Emf of the battery is 10 V
(c) Maximum current that can be taken from the battery is 2 A
(d) V-I graph can never be a straight lines as shown in figure

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4. In the given circuit (figure),
(a) the current through the battery is 5 A
(b) P and Q are at the same potential
(c) P is 2 V higher than Q
(d) Q is 2 V higher than P

5. For the batteries shown in figure, R1, R2 and R3 are the internal
resistances of E1, E2 and R3 respectively. Then, which of the
following is/are correct?
(a) The equivalent internal resistance R of the system is given
by (R1R2R3)/(R1R2 + R2R3 + R3R1)
(b) If R3 = (E1R2 + E2R1) / (R1 + R2), the equivalent emf of the
batteries will be equal to E3.
(c) The equivalent emf of the battery is equal to E = (E1 + E2 +
E3)/3
(d) The equivalent emf of the battery not only depends upon
values of E1, E2 and E3 but also depends upon values of R1,
R2 and R3.
6. A single battery is connected to three resistances as shown in
figure.
(a) The current through 7  resistance is 4 A
(b) The current through 3 resistance is 4 A
(c) The current through 6 resistance is 2 A
(d) The current through 7  resistance is 0

7. The charge flowing in a conductor varies with time as Q = at – bt2. Then, the current

(a) decreases linearly with time (b) reaches a maximum and then decreases

(c) falls to zero at time t = a/2b (d) changes at a rate – 2b

8. The potential difference between points A and B in the circuit

shown in figure is 16 V. Then

(a) the current through the 2 resistance is 3.5 A

(b) the current through the 4 resistance is 2.5 A
(c) the current through the 3 resistance is 1.5 A
(d) the potential difference between the terminals of the 9 V battery is 7 V
9. In the circuit shown in figure, mark the correct options.

(a) Potential drop across R1 is 3.2 V

(b) Potential drop across R2 is 5.4 V

(c) Potential drop across R1 is 7.2 V

(d) Potential drop across R2 is 4.8 V

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10. In the given circuit (as shown in figure),

(a) the equivalent resistance between C and G is 3k

(b) the current provided by the source is 4 mA
(c) the current provided by the source is 8 mA
(d) voltage across points G and E is 4 V

11. Study the following circuit diagram in figure and mark the correct
options
(a) The potential of point a with respect to point b in the figure
when switch S is open is –6V
(b) The points a and b are at the same potential, when S is opened
(c) The charge flowing through switch S when it is closed is 54 C
(d) The final potential of b with respect to ground when switch S
is closed is 8 V

12. Consider a conductor of variable cross section in which current is

flowing from cross section 1 to 2. Then
(a) current passing through both the cross sections is the same
(b) current through 1 is less than that through 2
(c) drift velocity of electrons at 1 is less than that at 2
(d) drift velocity is same at both the cross sections
13. For the circuit shown in figure, select the correct statements from the
following options
(a) x and y are equipotential points
(b) Effective resistance between A and B is 2
(c) Effective resistance between A and B is 1
(d) None of the above
14. In the given circuit (figure),
(a) the current through the battery is 5.0 A
(b) P and Q are at the same potential
(c) P is 2.5 V higher than Q
(d) Q is 2.5 V higher than P
15. In the circuit shown in figure, E1 and E2 are two ideal sources of
unknown emfs. Some currents are shown. Potential difference
appearing across 6 resistance is VA  VB  10 V .
(a) The current in the 4.00  resistor is 5 A
(b) The unknown emf E1 is 36 V
(c) The unknown emf E2 is 54 V
(d) The resistance R is equal to 9
16. Current through the battery in the circuit shown in the figure
(a) immediately after the switch S is closed is  / R
(b) immediately after the switch S is closed is  / 2R
(c) after long time is  / 2R
(d) after long time is 3 / 4R

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17. In the circuit shown in figure,
5
(a) i  A when S1 is closed and S2 is open
3
(b) i = 5 A when S1 is open and S2 is closed
(c) i = 2A when S1 and S2 both are open
(d) i = 10 A when both S1 and S2 are closed
18. In the given circuit, mark the correct statement/statements.

(a) The current through 10 V battery is 35/4 A

(b) The current through 5 V battery is 5/4 A

(c) The current through upper conducting wire is 15/8 A

(d) The current through lower conducting wire is 25/8 A

19. A voltmeter reads the potential difference across the terminals of an

old battery as 1.40 V, while a potentiometer reads its voltage to be
1.55 V. The voltmeter resistance is 280
(a) The emf of the battery is 1.4 V
(b) The emf of the battery is 1.55 V
(c) The internal resistance r of the battery is 30 
(d) The internal resistance r of the battery is 5
20. In the circuit shown in figure, the cell is ideal with emf 9 V. If the
resistance of the coil of galvanometer is 1 , then
(a) no current flows in the galvanometer
(b) charge flowing through 8 F is 40 C
(c) potential difference across 10 F is 5 V
(d) potential difference across 10 F is 4 V
21. Figure shows a balanced Wheatstone bridge
(a) If P is slightly increased, the current in the galvanometer flows
from C to A
(b) If P is slightly increased, the current in the galvanometer flows
from A to C
(c) If Q is slightly increased, the current in the galvanometer flows
from C to A
(d) If Q is slightly increased, the current in the galvanometer flows
from A to C

22. A battery of emf 0  5V and internal resistance 5 is connected

across a long uniform wire AB of length 1 m and resistance per unit
length 5 m1 . Two cells of 1  1V and  2  2 V are connected as
shown in the figure
(a) The null point is at A
(b) If the jockey is touched to point B, the current in the
galvanometer will be going towards B
(c) When jockey is connected to point A, no current flows through
1 V battery
(d) The null point is at distance of 8/15 m from A

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23. All the resistance in the given Wheatstone bridge have different
values and the current through the galvanometer is zero. The current
through the galvanometer will still be zero if
(a) the emf of the battery is doubled
(b) all resistance are doubled
(c) resistance R1 and R2 are interchanged
(d) the battery and the galvanometer are interchanged
24. A beam of electrons emitted from the electron gun G is accelerated by an electric field E. the area of cross-
section of the beam remains constant. As the beam moves away from G
(a) the speed of the electrons increases
(b) the current constituted by beam increases
(c) the number of electrons per unit volume in the beam decreases
(d) the number of electrons per unit volume in the beam increases
25. When some potential difference is maintained between A and
B, current I enters the network at A and leaves at B
(a) C and D are at the same potential
(b) the equivalent resistance between A and B is 8
(c) no current flows between C and D
(d) current 3 I/5 flows from D to C
26 In the given circuit
(a) the points A and B are at same potential
(b) the current through the battery is 5.0 A
(c) A is 2.5 V higher than B
(d) B is 2.5 V higher than A

27. Two cells of unequal emfs E1 and E2 and internal resistance r1

and r2 are joined as shown in fig. VP and VQ are the potential at P
and Q respectively
(a) The potential difference across both the cells will be equal
(b) One cell will continuous supply energy to the other
(c) The potential difference across one cell will be greater than its
emf
E1r2  E 2 r1
(d) VP  VQ 
r1  r2
28. Three voltmeters, all having different resistances are joined as
shown in fig. When some potential difference is applied across
P and Q, their readings are V1, V2 and V3 respectively. Then
(a) V1 = V2 (b) V1  V2
(c) V1 + V2 = V3 (d) V1 + V2 > V3
29. In the potentiometer arrangement shown in fig., the driving cell
B2 has emf E and internal resistance r. The cell B1, whose emf is
to be measured, has emf E/2 and internal resistance 2r. The
potentiometer wire is 100 cm long. If balance is obtained, the
length AJ = l. Then
(a) l = 50 cm
(b) l > 50 cm
(c) balance will be obtained only if resistance of wire AB is
greater than r
(d) balance cannot be obtained

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30. In the circuit shown in the fig. the current
(a) through the 3 resistance is 1 A
(b) through the 3 resistance is 0.5 A
(c) through the 4 resistance is 0.5 A
(d) through the 4 resistance is 0.25 A

Exercise – 4
Section – I : Numerical Value / Subjective Type Questions

1. The figure shows a network of resistor each heaving value 12 .

Find the equivalent resistance between points A and B (in  ).

2. Find the current (in A) in 4 resistance in circuit shown in

figure.

3. In the given network in the figure, find the charge on the capacitor
(in C )

4. Find circuit in the branch CD (in A) of the circuit shown in figure.

5. Nine wires each of resistance r  5  are connected to make a

prim as shown in figure. Find the equivalent resistance of the

arrangement across AB in  ).

6. Find the potential difference (in V) between points A and B

shown in the circuit.

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7. In the circuit shown in figure, find the steady state charge in 6 F

capacitor (in C) .

8. In the circuit shown in figure, the internal resistance of the cell is

negligible. For the value of R  40 / x  , no current flows

through the galvanometer. What is x?

9. A 5 m potentiometer wire having 3 resistance per meter is connected to a storage cell of steady emf 2V and
internal resistance 1 . A primary cell is balanced against 3.5 m of it. When a resistance of 32/n  is put in
series with the storage cell, the null point shifts to the center of the last wire, i.e. 4.5 m. What is ‘n’?
10. In the circuit shown, R  10  and V = 100 volt. With switch S open
the reading of ammeter is one third its reading when ‘S’ is closed.
Calculate the resistance of the ammeter (in ) .

11. A heating coil is rated 100 W, 220 V. The coil is cut in half and two pieces are joined in parallel to the same
source. Now what is the energy (in × 102J) liberated per second?
12. Three identical resistors are connected in series. When a certain potential difference is applied across the
combination, the total power dissipated is 27 W. How many times the power would be dissipated if the three
resistors were connected in parallel across the same potential difference?
13. Two circular rings of identical radii and resistance of 36  each are placed
in such a way that they cross each other’s centre C1 and C2 as shown in figure.
Conducting joints are made at intersection points A and B of the rings. An
ideal cell of emf 20 V is connected across A and B. Find the power delivered
by the cell (in 102 W).

14. An electric current of 2.0 A passes through a wire of resistance 25 . How much heat will be developed in 1
min? (in × 103 J)
15. A 500 W heater is designed to operate at 200 V potential difference. If it is connected across 160 V line, find
the heat (in kJ) it will produce in 20 minutes.

Section – II : Column Match Type

1. Column I Column II
(A) The series combination of cells is for (P) more current
(B) The parallel combination of cells is for (Q) more voltage
(C) In series combination of n cells, each of emf  , (R) 
the effective voltage is
(D) In parallel combination of n cells, each of emf  , (S) n 
the effective voltage is

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2. Column I Column II
(A) The unit of electrical resistivity is (P) m 2s 1V 1
(B) The unit of current density is (Q)  1m 1
(C) The unit of electrical conductivity is (R) Am 2
(D) The unit of electric mobility is (S)  m
3. A skeleton cube is made of 12 identical metal wires each of resistance r, then
Column I Column II
7
(A) resistance across the longest diagonal is (p) less than r
5
7
(B) resistance across the face diagonal (q) greater than r
12
7
(C) resistance across the side is (r) r
12
7
(D) resistance across the side of the cube which is (s) r
5
In broken state is
4. Column I Column II
(A) Rheostat (p) constantan
(B) Standard resistance coil (q) Manganin
(C) Bulb filament (r) Nichrome
(D) Heater wire (s) tungsten

Section – III : Comprehension

Comprehension # (1-3)
Consider the circuit shown in figure. The circuit is in steady state.
1. The value of i1 is
(a) 7/9 A (b) 14/13 A
(c) 14/3 A (d) 17/23 A
2. The potential of point B is
(a) 27/34 V (b) 46/13 V
(c) 1/2 V (d) 61/49 V
3. The charge in capacitor is
(a) 2 C (b) 4 C
(c) 6 C (d) 8 C

Comprehension # (5-6)
The circuit shown is in a steady state.
4. The charge in capacitor C1 is
(a) 20 C (b) 30 C
(c) 40 C (d) 10 C
5. The charge in capacitor C2 is
(a) 30 C (b) 10 C
(c) 20 C (d) 40 C
6. The charge in capacitor C3 is
(a) 10 C (b) 30 C
(c) 20 C (d) 40 C

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Comprehension # (7-8)
In the circuit shown in the figure
7. The value of current I1 is
(a) 2A (b) zero
(c) 2.5 A (d) none of these
8. The value of current I4 is
(a) 2A (b) zero
(c) 2.5 A (d) none of these

Exercise - 5
Revision Exercise (Moderate to tough)

1. At the temperature 0oC the electric resistance of conductor 2 is  times that of conductor 1. Their temperature
coefficients of resistance are equal to  2 and 1 respectively. Find the temperature coefficient of resistance of
a circuit segment consisting of these two conductors when they are connected
(a) in series (b) in parallel
2. Find the resistance of a wire frame shaped as a cube, when measured
between points
(a) 1–7
(b) 1–2
(c) 1–3
The resistance of each edge of the frame is R
3. At what value of the resistance Rx in the circuit shown in Fig. will
the total resistance between points A and B be independent of the
number of cells?
4. There is an infinite wire grid with square cells. The resistance of
each wire between neighbouring joint connections is equal to R0.
Find the resistance R of the whole grid between points A and B.

5. A metal ball of radius a is surrounded by a thin concentric metal shell of radius b. The space between these
electrodes is filled up with a poorly conducting homogeneous medium of resistivity  . Find the resistance of
the interelectrode gap. Analyse the obtained solution at b   .
6. Find a potential difference 1   2 between points 1 and 2 of the circuit shown
in Fig. If R1  10 , R 2  20  , E1 = 5.0 V, and E2 = 2.0 V. The internal
resistances of the current sources are negligible

7. N sources of current with different emf’s are connected as shown in fig. The
emf’s of the sources are proportional to their internal resistances, i.e. E =  R ,
where  is an assigned constant. The lead wire resistance is negligible. Find
(a) the current in the circuit
(b) the potential difference between points A and B dividing the circuit in n
and N – n links.

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8. In the circuit shown in fig. the sources have emf’s E1 = 1.0 V and E2 = 2.5 V
and the resistances have the values R 1  10  and R 2  20  . The internal

resistances of the sources are negligible. Find a potential difference A  B

between the plates A and B of the capacitor C.

9. In the circuit shown in figure, the emf of the source is equal to E = 5.0 V and
the resistances are equal to R 1  4.0  and R 2  6.0  . The internal
resistance of the source equals R R  0.10  . Find the currents flowing
through the resistances R1 and R2.
10. In figure show, illustrates a potentiometric circuit by means of which we can
vary a voltage V applied to a certain device possessing a resistance R. The
potentiometer has a length l and a resistance R0 and voltage V0 is applied to
its terminals. Find the voltage V fed to the device as a function of distance x.
Analyse separately the case R >> R0.
11. Find the magnitude and direction of the current flowing through the
resistance R in the circuit shown in Fig. if the emf’s of the sources are equal
to E1= 1.5 V and E2 = 3.7 V and the resistances are equal to
R 1  10 , R 2  20 , R  5.0  . The internal resistances of the sources are
negligible.
12. In the circuit shown in fig. the sources have emf’s E1= 1.5 V, E2= 2.0 V,
e3 = 2.5 V, and the resistances are equal to R 1 10 , R 2  20 , R 3  30  .
The internal resistances of the sources are negligible. Find :
(a) the current flowing through the resistance R1;

(b) a potential difference A  B between the points A and B.

13. Find a potential difference A  B between the plates of a capacitor C in the circuit shown in fig. If the sources
have emf’s E1 = 4.0 V and E2 = 1.0 V and the resistances are equal to R 1  10 , R 2  20  and R 3  30  . The
internal resistances of the sources are negligible.

14. Find the current flowing through the resistance R1 of the circuit shown in
fig. If the resistances are equal to R 1  10 , R 2  20  and R 3  30 
and the potentials of points 1, 2 and 3 are equal to 1  10 V, 2  6V and

3  5 V .
15. Find the resistance between points A and B of the circuit shown in fig.

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16. In a circuit shown in fig. resistances R1 and R2 are known, as well as emf’s E1 and E2. The internal resistances
of the sources are negligible. At what value of the resistance R will the thermal power generated in it be the
highest? What is it equal to?
17. Find the charge on each of the capacitors 0.20 ms after
the switch S is closed in figure.

18. The switch S shown in figure is kept closed for a long

time and is then opened at t = 0. Find the current in the
middle 10 resistor at t = 1.0 m.s

19. Consider the situation shown in figure. The switch is closed at t = 0

when the capacitors are uncharged. Find the charge on the capacitor
C1 as a function of time t.

20. A capacitor of capacitance C is given a charge Q. At t = 0, it is connected to an uncharged capacitor of equal

capacitance through a resistance R. Find the charge on the second capacitor as a function of time.

Exercise – 6
Section – I : JEE (Advanced) Questions Previous Years

1.

2.

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3.

4.

5.

6.

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7.

8.

9.

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Paragraph for Question Nos. 10 to 11

10.

11.

12.

13.

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Question Stem for Question Nos. 14 and 15 [JEE (Advanced) 2021, P-1]
Question Stem
In the circuit shown below, the switch S is connected to
position P for a long time so that the charge on the
capacitor becomes 𝑞1 μC. Then S is switched to position
Q. After a long time, the charge on the capacitor is
𝑞2 μC.
14. The magnitude of 𝑞1 is ______ .
15. The magnitude of 𝑞2 is ______ .
16. In order to measure the internal resistance 𝑟1 of a cell of
emf 𝐸, a meter bridge of wire resistance 𝑅0 = 50 Ω, a
resistance 𝑅0/2, another cell of emf 𝐸/2 (internal resistance
𝑟) and a galvanometer G are used in a circuit, as shown in
the figure. If the null point is found at 𝑙 = 72 cm, then the
value of 𝑟1 = ___ Ω.
[JEE (Advanced) 2021, P-2]
17. The figure shows a circuit having eight resistances of 1 Ω
each, labelled R1 to R2, and two ideal batteries with
voltages ℰ1 = 12 𝑉 and ℰ2 = 6 𝑉.
[JEE (Advanced) 2022, P-1]
Which of the following statement(s) is(are) correct?
(A) The magnitude of current flowing through 𝑅 is 7.2 A.
(B) The magnitude of current flowing through 𝑅 is 1.2 A.
(C) The magnitude of current flowing through 𝑅 is 4.8 A.
(D) The magnitude of current flowing through 𝑅 is 2.4 A.

18. Two resistances R1 = X  and R2 = 1  are

connected to a wire AB of uniform resistivity, as
shown in the figure. The radius of the wire varies
linearly along its axis from 0.2 mm at A to 1 mm at
B. A galvanometer (G) connected to the center of the
wire, 50 cm from each end along its axis, shows zero
deflection when A and B are connected to a battery.
The value of X is _______.
[JEE (Advanced) 2022, P-2]
19. In Circuit-1 and Circuit-2 shown in the figures,
𝑅1 = 1 Ω, 𝑅2 = 2 Ω and 𝑅3 = 3 Ω. 𝑃1 and 𝑃2 are the
power dissipations in Circuit-1 and Circuit-2 when
the switches S1 and S2 are in open conditions,
respectively. 𝑄1 and 𝑄2 are the power dissipations in
Circuit-1 and Circuit-2 when the switches S1 and S2
are in closed conditions, respectively.
[JEE (Advanced) 2022, P-2]

Which of the following statement(s) is(are) correct?

(A) When a voltage source of 6 𝑉 is connected across A and B in both circuits, 𝑃1 < 𝑃2.
(B) When a constant current source of 2 𝐴𝑚𝑝 is connected across A and B in both circuits, 𝑃1 > 𝑃2.
(C) When a voltage source of 6 𝑉 is connected across A and B in Circuit-1, 𝑄1 > 𝑃1.
(D) When a constant current source of 2 𝐴𝑚𝑝 is connected across A and B in both circuits, 𝑄2 < 𝑄1.

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 P-62
20. In a circuit shown in the figure, the capacitor C is initially uncharged
and the key K is open. In this condition, a current of 1 A flows through
the 1 Ω resistor. The key is closed at time t = t0. Which of the following
statement(s) is(are) correct? [Given: e–1 = 0.36]
(A) The value of the resistance of R is 3 Ω
(B) For t < t0, the value of current I1 is 2A
(C) At t = t0 + 7.2 s, the current in the capacitor is 0.6 A.
(D) For t , the charge on the capacitor is 12 C.
[JEE (Advanced) 2023, P-1]

Section – II : JEE (Main) Questions Previous Years

1.

2.

3.

4.

5.

6.

7.

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 P-63
8.

9.

10.

11.

12.

13.

14.

15.

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 P-64
16.

17.

18.

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 P-65
19.

20.

21.

25. A cell E1 of emf 6 V and internal resistance 2 is connected with

another cell E2 of emf 4 V and internal resistance 8 (as shown in
the figure). [JEE (Main) 2021, 24 February, Shift-1]
(1) 3.6 V (2) 10.0 V
(3) 5.6 V (4) 2.0 V

23. A current through a wire depends on time as i = α0t + βt2 where α0 = 20 A/s and β = 8As−2. Find the charge
crossed through a section of the wire in 15 s. [JEE (Main) 2021, 24 February, Shift-1]
(1) 2100 C (2) 260 C (3) 2250 C (4) 11250 C
7
24. A cylindrical wire of radius 0.5 mm and conductivity 5 × 10 S/m is subjected to an electric field of 10mV/m.
The expected value of current in the wire will be x3πmA. The value of x is
[JEE (Main) 2021, 24 February, Shift-2]
25. Five equal resistances are connected in a network as shown in figure.
The net resistance between the points A and B is :
[JEE (Main) 2021, 26 February, Shift-1]
(1) 3R/2 (2) R/2
(3) R (4) 2R

26. A conducting wire of length 'l, area of cross section A and electric resistivity  is connected between the
terminals of a battery. A potential difference V is developed between its ends, causing an electric current. If the
length of the wire of the same material is doubled and the area of cross-section is halved, the resultant current
would be : [JEE (Main) 2021, 16 March, Shift-1]
1 VA 3 VA 1 l VA
(1) (2) (3) (4) 4
4 l 4 l 4 VA l
27. A current of 10 A exists in a wire of cross-sectional area of 5 mm2 with a drift velocity of 2 × 10–3 ms–1. The
number of free electrons in each cubic meter of the wire is [JEE (Main) 2021, 17 March, Shift-1]
6 25 25
(1) 2 × 10 (2) 625 × 10 (3) 2 × 10 (4) 1 × 1023
28. In the experiment of Ohm's law, a potential difference of 5.0 V is applied across the end of a conductor of length
10.0 cm and diameter of 5.00 mm. The measured current in the conductor is 2.00 A. The maximum permissible
percentage error in the resistivity of the conductor is :- [JEE (Main) 2021, 18 March, Shift-1]
(1) 3.9 (2) 8.4 (3) 7.5 (4) 3

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 P-66
29. Two wires of same length and thickness having specific resistances 6  cm and 3  cm respectively are
connected in parallel. The effective resistivity is   cm . The value of  to the nearest integer, is
[JEE (Main) 2021, 19 March, Shift-1]
30. In a potentiometer arrangement, a cell gives a balancing point at 75 cm length of wire. This cell is now replaced
by another cell of unknown emf. If the ratio of the emf’s of two cells respectively is 3 : 2, the difference in the
balancing length of the potentiometer wire in above two cases will be _____ cm.
[JEE (Main) 2022, 24 June, Shift-1]
31. The total current supplied to the circuit as shown in figure by the 5V battery is ________ A

[JEE (Main) 2022, 25 June, Shift-1]

32. An aluminium wire is stretched to make its length, 04% larger. Then percentage increase in the resistance is:
[JEE (Main) 2022, 26 June, Shift-1]
(1) 0.4% (2) 0.2% (3) 0.8% (4) 0.6%
33. A cell, shunted by a 8 resistance, is balanced across a potentiometer wire of length 3m. The balancing length is
2 m when the cell is shunted by 4 resistance. The value of internal resistance of the cell will be _____  .
[JEE (Main) 2022, 27 June, Shift-1]
34. Current measured by the ammeter  in the reported circuit when no

current flows through 10 resistance will be __________ A.

[JEE (Main) 2022, 28 June, Shift-1]

35. Which of the following physical quantities have the same dimensions? [JEE (Main) 2022, 25 July, Shift-1]

(1) Electric displacement (D) and surface charge density
(2) Displacement current and electric field
(3) Current density and surface charge density
(4) Electric potential and energy
36. The current I in the given circuit will be :
[JEE (Main) 2022, 26 July, Shift-1]
(1) 10 A (2) 20 A
(3) 4 A (4) 40 A

37. A direct current of 4 A and an alternating current of peak value 4 A flow through resistance of 3  and 2 
respectively. The ratio of heat produced in the two resistances in same interval of time will be:
[JEE (Main) 2022, 27 July, Shift-1]
(1) 3 : 2 (2) 3 : 1 (3) 3 : 4 (4) 4 : 3

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 P-67
38. As shown in the figure, a potentiometer wire of resistance 20  and length
300 cm is connected with resistance box (R.B.) and a standard cell of emf
4 V. For a resistance ‘R’ of resistance box introduced into the circuit, the
null point for a cell of 20 mV is found to be 60 cm. The value of ‘R’
is _______  . [JEE (Main) 2022, 28 July, Shift-1]
39. The current I flowing through the given circuit will be ________ A.

[JEE (Main) 2022, 29 July, Shift-1]

40. A hollow cylindrical conductor has length of 3.14 m, while its inner and outer diameters are 4 mm and 8 mm
respectively. The resistance of the conductor n × 10–3 Ω. If the resistivity of the material is 2.4 × 10–8 Ωm. The
value of n is [JEE (Main) 2023, 24 Jan. Shift-1]
41. If a copper wire is stretched to increase its length by 20%. The percentage increase in resistance of the wire
is____________ %. [JEE (Main) 2023, 24 Jan. Shift-2]
42. A uniform metallic wire carries a current 2 A. when 3.4 V battery is connected across it. The mass of uniform
metallic wire is 8.92 × 10–3 kg. density is 8.92 × 103 kg/m3 and resistivity is 1.7 × 10–8 Ω –m. The length of wire
is : [JEE (Main) 2023, 25 Jan. Shift-1]
(1) l = 6.8 m (2) l = 10 m (3) l = 5 m (4) l = 100 m
43. With the help of potentiometer, we can determine the value of a given cell. The sensitivity of the potentiometer
is [JEE (Main) 2023, 29 Jan. Shift-2]
(A) directly proportional to the length of the potentiometer wire
(B) directly proportional to the potential gradient of the wire
(C) inversely proportional to the potential gradient of the wire
(D) inversely proportional to the length of the potentiometer wire
Choose the correct option for the above statements:
(1) B and D only (2) A and C only (3) A only (4) C only
44. In the following circuit, the magnitude of current I1, is ______ A.

[JEE (Main) 2023, 30 Jan. Shift-1]

45. Two identical cells, when connected either in parallel or in series gives same current in an external resistance
5Ω. The internal resistance of each cell will be _________Ω. [JEE (Main) 2023, 31 Jan. Shift-1]

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 P-68
46. For the given circuit, in the steady state,

|VB – VD| = ________ V.

[JEE (Main) 2023, 31 Jan. Shift-2]

47. Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : For measuring the potential difference across a resistance of 600 Ω, the voltmeter with resistance
1000 Ω will be preferred over voltmeter with resistance 4000 Ω.
Reason R : Voltmeter with higher resistance will draw smaller current than voltmeter with lower resistance.
In the light of the above statements, choose the most appropriate answer from the options given below.
(1) A is not correct but R is correct [JEE (Main) 2023, 1 Feb. Shift-2]
(2) Both A and R are correct and R is the correct explanation of A
(3) Both A and R are correct and R is not the correct explanation of A
(4) A is correct but R is not correct
I1  I3
48. In the given circuit the value of is:
I2
[JEE (Main) 2023, 1 Feb. Shift-2]

49. A student is provided with a variable voltage source V, a test resistor RT = 10Ω, two identical galvanometers G1

and G2 two additional resistors, R1 = 10MΩ and R2 = 0.001Ω. For conducting an experiment to verify ohm's

law, the most suitable circuit is: [JEE (Main) 2023, 06 April, Shift-2]

(1) (2)

(3) (4)

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 P-69
50. As shown in the figure the voltmeter reads 2V across 5Ω resistor. The
resistance of the voltmeter is _____ Ω.
[JEE (Main) 2023, 06 April, Shift-2]

51. The equivalent resistance between A and B as shown in figure is: [JEE (Main) 2023, 8 April, Shift-2]

(1) 10 kΩ (2) 5 kΩ (3) 20 kΩ (4) 30 kΩ

52. 10 resistors each of resistance 10 Ω can be connected in such as to get maximum and minimum equivalent
resistance. The ratio of maximum and minimum equivalent resistance will be________.
[JEE (Main) 2023, 10 April, Shift-1]
53. Two identical heater filaments are connected first in parallel and then in series. At the same applied voltage, the
ratio of heat produced in same time for parallel to series will be: [JEE (Main) 2023, 11 April, Shift-1]
(1) 1 : 4 (2) 4:1 (3) 2 : 1 (4) 1:2
54. A wire of resistance 160 Ω is melted and drawn in a wire of one-fourth of its length. The new resistance of the
wire will be [JEE (Main) 2023, 12 April, Shift-1]
(1) 16 Ω (2) 10 Ω (3) 640 Ω (4) 40 Ω
55. When a resistance of 5 Ω is shunted with a moving coil galvanometer, it shows a full-scale deflection for a

current of 250 mA, however when 1050 Ω resistance is connected with it in series, it gives full scale deflection

for 25 volt. The resistance of galvanometer is _______ Ω. [JEE (Main) 2023, 13 April, Shift-1]

56. A network of four resistances is connected 9 V to battery, as shown in

figure. The magnitude of voltage difference between the points A and B

is __________ V.

[JEE (Main) 2023, 15 April, Shift-1]

___________
________
____

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 P-70

ANSWER KEY
EXERCISE - 1

1. 4R 2. 2%
3. Graph B represents voltage-current change for L1.
  b
4. (i) R MN  (ii) R PQ  ln 5. 0.529A
(b  a )
2 2
2 a
6. 7.654 g 7. 160  8. 21%
2V
9. 6.8 10. x
2
11. Yellow, Red, Orange and Silver 12. 470  5%  13. 0.5 × 10–4A
14. (a) 105Am–2 (b) 1.72 × 10–3Vm–1 (c) 1.36 × 106s 15. 3.2A
3 9 6
16. 10 s, Distance 10 m 17. 3 × 10 s
18. (a) i1 = i2 (b) j1 < j2 (c) (vd)1 < (vd)2 19. 2  107 m 2 V 1s 1 20. 4 × 10–3 oC–1
21. 5020oC
22. For higher value of current, the resistor gets heated and consequently its resistance increases. The resistor becomes
non-ohmic due to which the I-V characteristic deviates from straight line thereby showing lesser current for the
same voltage.
23. V1 = 4V, V2 = 6V 24. I = 6A, I1 = 4A and I2 = 2A
25. Graph 1 represents the series combination of the two resistors.
26. Both the graphs are property labeled.
27. 2R 28. (i) R/2 (ii) R/4 (iii) R/2 (iv) 6R/5
6R 2R 4R
29. 30. 10 31. (a) (b)
7 3 5
157
32. 23 A 33. 4.5 A 34. A
180
35. Current in 4 should be 3A towards B, VA  VD  12V
36. 5A 37. VA – VD = 0V
10 27 17
38. (i) V (ii) Potential differences across the cell G is V across H is V
13 13 13
5
39. R
6
40. (a) 4 (b) Current in R1 and R2 will be 3A. The current in R3 and R4 = 6A, Current in R6 = 6A

2
41. 0.68 42. 0 43. 2.8 V,  , 1.4 A
5
5
44. n = 50, m = 2 45. (a) 5A (b) 4 A (c) A current will flow from B to D.
2
CVR 2 R  R2
46. Vx  Vy  7 V 47. 25 V, 75 V 48. Q0  ,  1
R1  R 2 CR1R 2
49. 8 × 10–4 J 50. Current in 2.5 resistance = 0, RPQ = 4

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 P-71

EXERCISE – 2

1. (d) 2. (b) 3. (b) 4. (d) 5. (c)

6. (a) 7. (c) 8. (c) 9. (b) 10. (a)
11. (c) 12. (c) 13. (a) 14. (c) 15. (c)
16. (b) 17. (c) 18. (a) 19. (d) 20. (d)
21. (a) 22. (b) 23. (b) 24. (b) 25. (a)
26. (a) 27. (d) 28. (d) 29. (d) 30. (b)
31. (c) 32. (b) 33. (a) 34. (b) 35. (d)
36. (b) 37. (b) 38. (b) 39. (c) 40. (a)
41. (b) 42. (c) 43. (c) 44. (a) 45. (d)
46. (b) 47. (c) 48. (a) 49. (b) 50. (d)
51. (c) 52. (a) 53. (a) 54. (d) 55. (c)
56. (b) 57. (d)

EXERCISE - 3

1. (a, b, d) 2. (a, d) 3. (a,b,c) 4. (a,d) 5. (a,b,d)

6. (b,c,d) 7. (a,c,d) 8. (a,c,d) 9. (c,d) 10. (a,b,d)
11. (a,c) 12. (a,c) 13. (a,c) 14. (a,d) 15. (a,b,c,d)
16. (a,c) 17. (b,c,d) 18. (b,c) 19. (b,c) 20. (a,b,d)
21. (b,c) 22. (a,b) 23. (a,b,d) 24. (a,c) 25. (a,b,d)
26. (b,d) 27. (a,b,c,d) 28. (b,c) 29. (b,c) 30. (a,b,d)

EXERCISE – 4
Section – I

1. 9 2. 3 3. 60 4. 15 5. 3
6. 6 7. 4 8. 9 9. 7 10. 10
11. 4 12. 9 13. 1 14. 6 15. 384
Section – II
1. A-Q, B-P, C-S, D-R 2. A-R, B-Q, C-P, D-S
3. A-p,q; B-p,q; C-r;, D-s 4. A-p,q; B-p,q; C-s; D-r

Section – III
1. (b) 2. (b) 3. (d) 4. (d) 5. (c)
6. (b) 7. (b) 8. (a)

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 P-72
EXERCISE – 5

(1   2 ) (  1 ) 5 7 3
1. (a)   (b)   2 2. (a) R (b) R (c) R
(1  ) (1  ) 6 12 4

R0
3. R x  R( 3  1) 4.
2

(b  a)  (E1  E 2 )R1

5. R in the case of b   . R  6. 1  2   E1  4V
4ab 4a (R1  R 2 )

(E1  E 2 )R1
7. (a) I   (b)  A  B  0 8. A   B   0.5V
(R1  R 2 )

ER 2 IR
9. I1  = 1.2A, I 2  1 1  0.8A
(RR1  R1R 2  R 2 R) R2

V0 Rx V0 x
10. V for R >> R0 V 
x
[R  R 0 (  x) ]



(R1E 2  R 2 E1 )
11. I  0.02A
(RR1  R1R 2  R 2 R)

[R 3 (E1  E 2 )  R 2 (E1  E 2 )]
12. (a) I1   0.06A (b) A   B  E1  I1R 1  0.9 V
(R1R 2  R 2 R 3  R 3 R1 )

[E 2 R 3 (R1  R 2 )  E1R1 (R 2  R 3 )]
13. A   B   0.1V
R 1R 2  R 2 R 3  R 3 R 1

[R 3 (1  2 )  R 2 (1  2 )] r(r  3R)

14. I1   0.2A 15. R AB 
R 1 R 2  R 2 R 3  R 3 R1 R  3r)

R 1R 2
16. R 17. 9.2 C
(R1  R 2 )

C1C2
18. 11 Ma 19. q  C(1  e1/ rC ), where C 
C1  C2

Q
20. (1  e 2 t / RC )
2

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 P-73
EXERCISE – 6

Section - I

1. 5 2. (a,b,c,d) 3. (b,d) 4. (a,b,d) 5. (c)

6. (b) 7. (b) 8. 1 9. (b,c) 10. (b)

11. (a) 12. 5.56 or 5.55 13. (2, 4) 14. 01.33 15. 00.67

16. 3 17. (A,B,C,D) 18. 5 19. (A,B,C)

Section - II
1. (2) 2. (3) 3. (3) 4. (4) 5. (4)

6. (3) 7. (4) 8. (4) 9. (1) 10. (3)

11. (2) 12. (4) 13. (4) 14. (1) 15. (4)

16. (2) 17. (3) 18. (4) 19. (1) 20. (3)

21. (3) 22. (3) 23. (4) 24. (5) 25. (3)

26. (1) 27. (2) 28. (1) 29. (4) 30. (25)

31. (2) 32. (3) 33. (8) 34. (10) 35. (1)

36. (1) 37. (2) 38. (780) 39. (2) 40. (2)

41. (44) 42. (2) 43. (2) 44. (1) 45. (5)

46. (1) 47. (1) 48. (2) 49. (1) 50. (20)

51. (2) 52. (100) 53. (2) 54. (2) 55. (50)

56. (3)

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MAGNETIC EFFECT
OF CURRENT
AND MAGNETISM
 P-74

EXERCISE - 1
Concept Building Questions (MCQ With Single Correct Answer)

1. If a particle of charge 10–12C moving along the x-direction with a velocity of 105 m/s experiences a force
of 10–10 N in y-direction due to magnetic field, then the minimum value of magnetic field is
(a) 6.25 × 103 tesla in Z-direction (b) 10–15 tesla in z-direction
(c) 6.25 × 10–3 tesla in z-direction (d) 10–3 tesla in Z-direction
2. A wire placed along north-south direction carries a current of 5A from south to north. What is the
magnetic field due to a 1 cm piece of wire at a point 200 cm north east from the piece?
(a) 4.4 × 10–10 T vertically upwards (b) 4.4 × 10–10 T vertically downwards
(c) 8.8 × 10–10T vertically upwards (d) 8.8 × 10–10 T vertically downwards
3. If B1 is the magnetic field induction at a point on the axis of a circular coil of radius R situated at a

distance R 3 and B2 is the magnetic field at the centre of the coil, then ratio of B1/B2 is equal to
(a) 1/3 (b) 1/8 (c) 1/4 (d) ½
4. A coil carrying a heavy current and having large number of turns is mounted in a N-S vertical plane. A
current flows in the clockwise direction. A small magnetic needle at its centre will have its north pole in
(a) east-north direction (b) west-north direction (c) east-south direction (d) west-south direction
5. Magnetic field at the centre of a circular loop of area A is B. The magnetic moment of the loop will be
BA2 BA3 / 2 BA3 / 2 2BA 3 / 2
(a) (b) (c) (d)
0  0  0 1/ 2  0 1/ 2

a a
6. A straight section PQ of a circuit lies along the X-axis from x  to x  and carries a steady current
2 2
I. The magnetic field due to the section PQ at a distance x = +a will be
(a) proportional to a (b) proportional to 1/a (c) proportional to a2 (d) zero
7. A horizontal long straight wire placed east-west carries a current 3.6A. What is the distance of neutral
point from the wire if the horizontal component of the earth’s magnetic field from south to north is :
BH  3.6105 T
(a) 1 × 10–2 m (b) 2 × 10–2 m (c) 1.5 × 10–2 m (d) 3 × 10–2 m
8. The wire loop PQRS formed by jointing two semicircular wires of radii R1
and R2 carries a current I as shown. The magnitude of magnetic induction at
the centre O is
(a) ( 0 / 4)I(R 1 / R 2  1/ R 1 )

(b) ( 0 / 4)I(1/ R 1  1/ R 2 )

(c)  0 I(1/ R 2  1/ R 1 )

(d)  0 I(1/ R 1 )

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 P-75
9. Two straight long conductors AOB and COD are perpendicular to each other and carry currents I1 and I2.
The magnitude of the magnetic induction at a point P at a distance a from the point O in a direction
perpendicular to the plane ABCD is
0 0  0 2 2 1/ 2 0 I1I 2
(a) (I1  I2 ) (b) (I1  I 2 ) (c) (I1  I2 ) (d)
2a 2a 2a 2a (I1  I 2 )

10. The magnetic field induction at the centre O, in the arrangement

shown in fig. is

0 I 0I
(a) (4  ) (b) (3  )
4r 4r
0 I 0 I
(c) (2  ) (d) (1  )
4r 4 r
11. At the mid point along the length of a long solenoid, the magnetic field is equal to B. If the length of
solenoid is doubled and the current is reduced to half, the magnetic field at the new mid point will nearest
to
(a) 2 B (b) B (c) B/4 (d) B/2
12. The magnetic field normal to the plane of a wire of n turns and radius r which carries a current I is
measured on the axis of the coil at a small distance h from the centre of the coil. This is smaller than the
magnetic field at the centre by the fraction
(a) (2/3)r2/h2 (b) (3/2)r2/h2 (c) (2/3)h2/r2 (d) (3/2)h2/r2
13. A pair of stationary and infinite long bent wires are placed in the x-
y plane. The wires carrying currents of 10 A each as shown in
figure. The segments L and M are parallel to x-axis. The segments
P and Q are parallel to y-axis, such that OS = OR = = 0.02 m. The
magnetic field induction at the origin O is
(a) 10–3 T (b) 4 × 10–3T
(c) 2 × 10–4T (d) 10–4T
14. A proton, a deuteron and an  -particle with the same K.E. enter a region of uniform magnetic field,
moving at right angle to B. What is the ratio of the radius of their circular paths?
(a) 1: 2 :1 (b) 1: 2 : 2 (c) 2 :1:1 (d) 2 : 2 :1
15. A particle of charge q and mass m starts moving from the origin under the action of an electric field
5
E  E 0 i and B  B0 i with a velocity v  v 0 j . The speed of the particle will become
  
v 0 after a time
2
mv 0 mv 0 3mv0 5mv0
(a) (b) (c) (d)
qE 2qE 2qE 2qE

16. An electron and a proton enter a magnetic field perpendicularly. Both have same kinetic energy. Which of
the following is true?
(a) Trajectory of electron is less curved (b) Trajectory of proton is less curved
(c) Both trajectories are equally curved (d) Both move on straight line path

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 P-76
17. Two particles X and Y having equal charges, after being accelerated through the same potential
difference, enter a region of uniform magnetic field and describes circular path of radius R1 and R2
respectively. The ratio of mass of X to that of Y is
(a) (R1 / R 2 )1/ 2 (b) R 2 / R1 (c) (R 1 / R 2 ) 2 (d) R1 / R 2

18. A conducting rod of length L and mass is moving down a smooth

inclined plane of inclination  with constant speed v. A current I is
flowing in the conductor perpendicular to the paper inwards. A

vertically upward magnetic field B exists in space there. The magnitude

of magnetic field B is
mg mg mg mg
(a) sin  (b) cos  (c) tan  (d)
IL IL IL ILsin 
19. A current I1 carrying wire AB is placed near an another long wire CD carrying
current I2. If free to move, wire AB will have
(a) rotational motion only
(b) translational motion only
(c) rotational as well as translation motion
(d) neither rotational nor translational motion

20. The unit vector i, j and k are as shown in figure. What is the magnetic field induction at the point O due
to the current I through the conductors?

0 I 0 I 0I 0 I
(a)  (   2)k (b)  (  2)k (c)  (   2)j (d)  (   2)j
4r 4r 4r 4 r
21. An infinitely long wire carrying current I is along Y-axis such that its one end is at point (0, b) while the
wire extends upto  . The magnitude of magnetic field strength at point P(a, 0) is

0 I  b  0 I  b 
(a) 1  2  (b) 1  2 
4a  a b 
2 4a  a b 
2

0 I  a  0 I  b 
(c) 1  2  (d)  2 
4 a  a  b2  4a  a  b 2 

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 P-77
22. There is an infinite long V shaped wire carrying current I in x-y plane as shown in figure. The magnitude
of magnetic field induction at P is

0 I  0 I  0 I  0 I 
(a) sin (b) cos (c) tan (d) cot
2 d 2 2 d 2 2 d 2 4 d 2
23. There is an infinite long wire carrying current I in x-y plane as shown in figure. The magnetic field
induction at P is

0 I I  0 I 
(a) tan along + z direction (b) tan along – z direction
4 d 2 4 d 2
0 I  0 I 
(c) cot along + z direction (d) cot along – z direction
4 d 2 4 d 2
24. In a square loop PQRS made with a wire of cross-section as shown in figure, current I enters from point P
and leaves from point S. The magnitude of magnetic field induction at the centre O of the square is

 0 2 2I  0 4 2I  0 4 2I
(a) (b) (c) (d) Zero
4 a 4 a 4 a
25. A wire along x-axis carries a current 3.5 A. Find the force in newton on a 1 cm section of the wire exerted

by a magnetic field B  (0.74j  0.36k)

 tesla.


(a) (1.26 k  2.59j)102 N (b) (1.26 k  2.59j)  102 N

(c) (2.59k  1.26j)

 102 N (d) (2.59k  1.26j) 102 N
26. A thin disc having radius r and charge q distributed uniformly over the disc is rotated n rotations per
second about its axis. The magnetic field at the centre of the disc is
 0 qn  0 qn  0 qn 3 0 qn
(a) (b) (c) (d)
2r r 4r 4r

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 P-78
27. An electron is revolving around a proton in a circular path of diameter 0.1 nm. It produces a magnetic
field 14 tesla at a proton. Then the angular speed of the electron is
(a) 8.8 × 1016 rad s–1 (b) 4.4 × 1016 rad s–1 (c) 2.2 × 1016 rad s–1 (d) 1.1 × 1016 rad s–1
28. An electron having kinetic energy T is moving in a circular orbit of radius R perpendicular to a uniform

magnetic field induction B . If kinetic energy is doubled and magnetic field induction is tripped, the
radius will become
(a) R 9/4 (b) R 3/ 2 (c) R 2/9 (d) R 4/3
29. A square frame side 1 m carries a current I, produces a magnetic field B at its centre. The same current is
passed through a circular coil having the same perimeter as the square. The magnetic field at the centre of
the circular coil is B . The ratio B/ B is
8 8 2 16 16
(a) (b) (c) (d)
2 2 2 2 2

30. The magnetic field at the centre of a loop of a circular wire of radius r carrying current I may be taken as
B0. If a particle of charge q moving with speed v passes the centre of a semicircular wire, as shown in
figure, along the axis of the wire, the force on it due to the current is

1 1
(a) zero (b) qB 0 v (c) qB 0 v (d) qB0v
4 2
31. A current I flows through a closed loop shown in figure. The magnetic field induction at the centre O is

0 I 0 I 0 I 0 I
(a)  (b) (  sin ) (c) (     sin ) (d) (     tan )
4R 4R 4R 2R
32. An equilateral triangle of side length l is formed from a piece of wire of uniform resistance. The current I
is fed as shown in the figure. The magnitude of the magnetic field at its centre O is

3 0 I 3 3 0 I 0 I
(a) (b) (c) (d) zero
2 2 2

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 P-79
33. What is the magnetic field induction at the centre O of the arc in the figure

0i  0 2I    0 2I 0 I  
(a) [ 2  ] (b) 2  (c)  2   (d) 2 
4r 4 r   4 4 r   
4 r  4
34. Two parallel wires carrying equal currents in opposite directions are placed at x   r parallel to y-axis
with z = 0. Magnetic field at origin O is B1 and at P(3, r, 0, 0) is B2. Then the ratio B1/B2 is
1 1
(a) – 8 (b)  (c)  (d) 0
4 2
35. A conductor AB of length L carrying a current I2 is placed perpendicular to a long straight conductor XY
carrying current I1, as shown in figure. The force on AB in magnitude is

3 0 I1I 2  0 I1I 2  0 I1I 2 2 0 I1I 2

(a) (b) log e 2 (c) log e 3 (d)
2 2 2 3
36. A square loop ABCD, carrying a current I2, is placed near and coplanar with a long straight conductor XY
carrying a current I1, as shown in the figure. The net force on the loop will be

 0 I1I 2  0 I1I2 L 2 0 I1 I 2 L 2 0 I1I 2

(a) (b) (c) (d)
2 2 3 3
37. A particle with specific charge s is fired with a speed v towards a wall at a distance d, perpendicular to the
wall. What minimum magnetic field must exist in this region for the particle not to hit the wall?
v v v 2v
(a) (b) (c) (d)
4s d 2s d sd sd
38. An equilateral triangular loop is of each side l an is carrying a current I. It is

placed in a uniform magnetic field B with is acting perpendicular to the plane of
loop. The net magnetic force acting on the loop is

3
(a) 3ILB (b) IB
2
(c) ILB (d) zero

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 P-80
39. A coil having N turns is wound tightly in the form of a spiral with inner and outer radii a and b
respectively. When a current I passes through the coil, the magnetic field at the centre is
 0 NI 2 0 NI  0 NI b  0 IN b
(a) (b) (c) ln (d) ln
b a 2(b  a) a 2(b  a) a
40. The length of a magnetized steel wire is l and its magnetic moment is M. It is bent
into the shape of L with two sides equal. The magnetic moment now will be
M
(a) (b) 2 M
2
M
(c) 2M (d)
2
41. The magnetic potential due to a magnetic dipole at a point on its axis distant 40 cm from its centre is
found to be 2.4 × 10–5 J/A-m. The magnetic moment of the dipole will be
(a) 28.6 A-m2 (b) 32.2 A-m2 (c) 38.4 A-m2 (d) none of these
42. A magnetized wire of magnetic moment M is bent into an arc of a circle that subtends an angle of 60o at
the centre. The equivalent magnetic moment is
M 3M 2M 4M
(a) (b) (c) (d)
   
2
43. Two small magnets, each of magnetic moment 10 A m are placed in end on position 0.1 m apart from
their centres. The force acting between them is
(a) 0.6 N (b) 0.6 × 107 N (c) 0.6 × 10–7 N (d) none of these
44. The angle of dip at the poles and at the equator respectively are
(a) 30o, 60o (b) 90o, 0o (c) 30o, 90o (d) 0o, 0o
45. If a magnet is suspended at an angle of 30o to the magnetic meridian, the dip needle makes an angle of 45o
with the horizontal. The real dip is

(a) tan 1 ( 3/ 2) (b) tan 1 ( 3) (c) tan 1 ( 3 / 2) (d) tan 1 (2/ 3)

EXERCISE – 2
Single Choice Correct With Multiple Options

1. A proton is moving along Z-axis in a magnetic field. The magnetic field is along X-axis. The proton will
experience a force along
(a) X-axis (b) Y-axis (c) Z-axis (d) Negative Z-axis
2. A particle of charge q and mass m moving with a velocity v along the x-axis enters the region x > 0 with

uniform magnetic field B along the k direction. The particle will penetrate in this region in the x-
direction up to a distance d equal to
mv 2mv
(a) Zero (b) (c) (d) Infinity
qB qB

3. An electron, a proton, a deuteron and an alpha particle, each having the same speed, are in a region of
constant magnetic field perpendicular to the direction of the velocities of the particles. The radius of the
circular orbits of these particles are respectively Re, Rp, Rd and Ra. If follows that
(a) Re = Rp (b) Rp = Rd (c) Rd = R  (d) Rp  R

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 P-81
4. A uniform magnetic field B and a uniform electric field E act in a common region. An electron is entering
this region of space. The correct arrangement for it to escape undeviated is

(a) (b)

(c) (d)

5. In the given figure, an electron of mass m, charge –e, and low (negligible) speed enters the region
between two plates of potential difference V and plate separation d, initially headed directly toward the
top plate. A uniform magnetic field of magnitude B is normal to the plane of the figure. Find the
minimum value of B such that the electron will not strike the top plate.

2mV mV mV mV
(a) 2
(b) 2
(c) 2
(d) 2
ed 2ed ed ed 2
6. Three identical charge particles A, B and C are projected perpendicular to the uniform magnetic field with
velocities v1, v2 and v3(v1 < v2 < v3), respectively such that T1, T2 and T3 are their respective time period of
revolution and r1, r2 and r3 are respective radii of circular path described. Then :
r1 r2 r r r r
(a)   3 (b) T1 < T2 < T3 (c) 1  2  3 (d) r1 = r2 = r3
T1 T2 T3 T1 T2 T3
7. A charged particle enters into a uniform magnetic field with velocity v0 perpendicular to it. The length of
magnetic field is x  3R / 2 , where R is the radius of the circular path of the particle in the field. The
magnitude of change in velocity of the particle when it comes out of the field is

3v 0
(a) 2v0 (b) v0/2 (c) (d) v0
2

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 P-82
8. A charged particle of mass m and charge q enters along AB at point A in a uniform magnetic field
existing in the rectangular region of size a × b. The particle leaves the region exactly at corner point c.
What is the speed v of the particle?

qB(a 2  b 2 ) qB(a 2  b 2 ) qB(a 2  b 2 ) qB(a 2  b 2 )

(a) (b) (c) (d)
2mb 2ma 2ma 2mb
9. A particle having mass m, charge q enters a cylindrical region having
uniform magnetic field B in the inward direction as shown. If the
particle is deviated by 60o as it emerges out of the field, then what is
the time spent by it in the field?

2 m 2 m
(a) (b)
qB 3qB

m
(c) (d) It depends on the speed of particle
3qB

10. A particle of mass m having negative charge –q is projected

an angle  with x-axis with velocity v. There exists uniform
electric field E and a uniform magnetic field B along x-axis.
The particle will return to its initial point once if (n is some
integer)
Bv cos  2Bv cos 
(a) n  (b) n 
E E
Bv sin  2Bvsin 
(c) n (d) n 
E E
11. An arbitrary shaped closed coil is made of a wire of length L and a current I ampere is flowing in it. If the

plane of the coil is perpendicular to magnetic field B , the force on the coil is
1
(a) Zero (b) IBL (c) 2IBL (d) IBL
2
12. An infinitely long, straight conductor AB is fixed and a current is
passed through it. Another movable straight wire CD of finite length
and carrying current is held perpendicular to it and released. Neglect
weight of the wire
(a) The rod CD will move upwards parallel to itself
(b) The rod CD will move downward parallel to itself
(c) The rod CD will move upward and turn clockwise at the same time
(d) The rod CD will move upward and turn anti-clockwise at the same time.

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 P-83
13. A wire carrying current i is placed in a uniform magnetic field in the form of the curve
 x 
y  a sin   0  x  2L . The force acting on the wire is
 L 

iBL
(a) (b) iBL (c) 2iBL (d) Zero

14. A square loop of side a hangs from an insulating hanger of spring

balance. The magnetic field of strength B occurs only at the lower

edge. It carries a current I. Find the change in the reading of the

spring balance if the direction of current is reversed.

IaB 3
(a) IaB (b) 2IaB (c) (d) IaB
2 2
15. Same current i = 2A is flowing in a wire frame as shown in the figure. The frame is a combination of two
equilateral triangles ACD and CDE of side 1 m. It is placed in uniform magnetic field B = 4T acting
perpendicular to the plane of frame. The magnitude of magnetic force acting on the frame is

(a) 24 N (b) Zero (c) 16 N (d) 8 N

16. A horizontal metallic rod of mass ‘m’ and length ‘l’ is supported by two
vertical identical springs of spring constant ‘K’ each and natural length l0.
A current ‘i’ is flowing in the rod in the direction shown. If the rod is in
equilibrium, then the length of each spring in this state is

iB  mg iB  mg
(a) 0  (b)  0 
K 2K
mg  iB mg  iB
(c) 0  (d)  0 
2K K

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 P-84
17. A conducting wire bent in the form of a parabola y2 = 2x carries a current i = 2A as shown in figure. This
wire is placed in a uniform magnetic field B  4k Tesla. The magnetic force on the wire is (in newton)


(a) 16i (b) 32i (c) 32i (d) 16i

18. A cylindrical rod of mass m, radius r and length L rests on two
rails. A uniform magnetic field B acts vertically downwards. The
rods carries a current i and can roll along the rails without slipping.
If the rod starts from rest, calculate its linear speed as a function of
its linear displacement x.
1/ 2 1/ 2 1/ 2 1/ 2
 4BiLx   4BiLx   BiLx   BiLx 
(a) v  (b) v    (c) v  (d) v 
 3m   m   3m   m 
19. A long, horizontal wire AB rests on the surface of a table and
carries a current I. A horizontal wire CD is vertically above wire
AB and is free to slide up and down on two vertical metal guides C
and D (as shown in figure). Wire CD is connected through the
sliding contacts to another wire that also carries a current I,
opposite in direction to the current in wire AB. The mass per unit
length of the wire CD is  . To what equilibrium height h will wire
CD rise, assuming that magnetic force on it is due wholly to the current in wire AB?
30 I2 0 I2 5 0 I 0 I2
(a) (b) (c) (d)
2g 3g 2g 2g
20. A wire PQRST carrying current I = 5A is placed in uniform
magnetic field B = 2T as shown in figure. In the length of part
QR = 4 cm and SR = 6 cm, then the magnetic force on SR edge
of the wire is

(a) 0.4 N (b) 0.6 N

(c) zero (d) 6 N
21. A long straight wire carrying current I1 is placed in the plane of a ribbon
carrying current I2 parallel to the wire. The width of ribbon is b. The straight
conductor is placed at a distance ‘a’ from the nearer edge of ribbon. Find the
force of attraction per unit length between the two.

 0 I1I2 ab  0 I1I2 a

(a) log (b) log
2a b 2b ab
0 I1I2 a  0 I1I 2 (a  b)
(c) log (d) log
2b b 2b a

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 P-85
22. A wire of length L metre carrying a current of I ampere is bent in the form of a circle. Its magnitude of
magnetic moment will be
IL IL2 I 2 L2 I2 L
(a) (b) (c) (d)
4 4 4 4
23. A toroid of mean radius a, cross-section radius r and total number of turns N carries current i . The torque
experienced by the toroid if a uniform magnetic field of strength B is applied
(a) is zero (b) is BiNr 2
(c) is BiNa 2 (d) depends on the direction of magnetic field
24. A uniform constant magnetic field is directed at an angle of 45o to the x-axis in the xy-plane, PQRS is a
rigid square wire frame carrying a steady current I0, with its centre at the origin O. At time t = 0, the frame
is at rest in the position shown in figure, with its sides parallel to the x and y-axes. Each side of the frame
is of mass M and length L.


The torque  about O acting on the frame due to the magnetic field will be
BI L2 BI L2
(a)   0 ( i  j) (b)   0 (i  j)
 

2 2
BI0 L2   BI L2
(d)   0 ( i  j)
 
(c)  (i  j)
2 2
25. A spherical shell of radius R and uniformly charged with charge Q is rotating about its axis with
frequency  . Find the magnetic moment of the sphere.
QR 2  2QR 2  QR 2  QR 2 
(a) (b) (c) (d)
2 3 4 3
26. Figure shows a square current-carrying loop ABCD of side 10 cm and current i = 10 A. The magnetic

moment M of the loop is

(a)   m2
(0.05)(i  3k)A (b)   m2
(0.05)(i  k)A
(c) (0.05)( 3i  k)A
  m2 (d) (i  k)A
  m2
27. A current i is flowing in a straight conductor of length L. The magnetic induction at a point distance L/4
from its centre will be
4 0i 0i 0i
(a) (b) (c) (d) Zero
5L 2L 2L

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 P-86
28. AB and CD are long straight conductor, distance d apart, carrying a current I. The
magnetic field at the midpoint of BC is
 0 I   0 I 
(a) k (b) k
2 d d
 0 I   0 I 
(c) k (d) k
4 d 8 d
29. In the following figure a wire bent in the form of a regular polygon of n sides is
inscribed in a circle of radius a. Net magnetic field at centre will be
  / n
 0i   0 ni 
(a) tan (b) tan
2a n 2a n
2 ni  ni 
(c)  0 tan (d)  0 tan
 a n 2a n
30. The field normal to the plane of a wire of n turns and radius r which carries a current i is measured on the
axis of the coil at a small distance h from the centre of the coil. This is smaller than the field at the centre
by the fraction
3 h2 2 h2 3 r2 2 r2
(a) 2
(b) 2
(c) 2
(d)
2r 3r 2h 3 h2
31. In the given figure, the net magnetic field at O will be

2 0 i 0i
(a) 4  2 (b) 4  2
3a 3a
2 0 i 2 0i
(c) 4  2 (d) (4  2 )
3a 2 3a
32. In the figure shown ABCDEFA is a square loop of side l, but is folded in
two equal parts so that half of it lies in xz plane and the other half lies in the
yz plane. The origin O is centre of the frame also. The loop carries current
i. The magnetic field at the centre is
 0 i    0 i  
(a) (i  j) (b) ( i  j)
2 2 4 
2 0 i   0 i  
(c) (i  j) (d) (i  j)
 2
33. A square frame of side l carries a current i and produces a magnetic field B at its centre. If the same
current is passed through a circular coil having the same perimeter as the square frame, it produces a field
B at its centre. The B/ B will be
8 8 2 16 16
(a) (b) (c) (d)
2 2 2 22
34. All straight wires are very long. Both AB and CD are arcs of the same circle,
both subtending right angles at the centre O. Then the magnetic field at O is
 0 i  0 i
(a) (b) 2
4 R 4 R

 0i  0 i
(c) (d) (   1)
2 R 2 R

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35. A long thin hollow metallic cylinder of radius ‘R’ has a current I ampere. The magnetic induction ‘B’
away from the axis at a distance r from the axis varies as shown in figure.

(a) (b)

(c) (d)

36. From a cylinder of radius R, a cylinder of radius R/2 is removed, as shown. Current flowing
in the remaining cylinder is I. Magnetic field strength is
(a) zero at point A
(b) zero at point B
0 I
(c) at point A
3R
0 I
(d) at point B
3R
37. A cylindrical conductor of radius R carries a current ‘i’. The value of magnetic field at a point which is
R/4 distance inside from the surface is 10 T. Find the value of magnetic field at point which is 4R distance
outside from the surface.
4 8 40 80
(a) T (b) T (c) T (d) T
3 3 3 3
38. Two particles of charge +Q and –Q are projected from the same point with a velocity v in a region of
uniform magnetic field B such that the velocity vector makes an angle  with the magnetic field. Their
masses are M and 2M, respectively. Then, they will meet again for the first time at a point whose distance
from the point of projection is
2Mv cos  8Mv cos  Mv cos  4Mv cos 
(a) (b) (c) (d)
QB QB QB QB
39. A thin beam of charged particles is incident normally on the
boundary of a region containing a uniform magnetic field as
shown. The beam comprises of monoenergetic  and  particles,
each possessing a kinetic energy T. If masses of  particle and 
particles are me and m  respectively, Neglecting interaction
between particles of the beam, find the separation between the 
and  particles when they emerge out of the magnetic field.
2T  m   2 2T  m 
(a)   me  (b)   me 
eB  2  eB  2 

(c)
2 2T
eB
 m  me  (d)
2 T
eB
 m   me 

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40. A block of mass m and charge q is released on a long smooth inclined plane.
Magnetic field B is constant, uniform, horizontal and parallel to surface as shown.
Find the time from start when block loses contact with the surface.
m cos  m cosec 
(a) (b)
qB qB
m cot 
(c) (d) None of these
qB

41. Two large conducting planes carrying current perpendicular to x-axis are
placed at (d, 0) and (2d, 0) as shown in figure. Current per unit width in
both the planes is same and current is flowing in the outward direction.
The variation of magnetic induction (taken as positive if it is in positive y-
direction) as function of x(0  x  3d) is best represented by

(a) (b)

(c) (d)

42. A long horizontal wire carries a current of 10A. A charged particle of mass 1 mg moves parallel to the
wire with a constant velocity of magnitude 10 m/s. The distance of the charge from the wire is 1 cm. The
magnitude of the charge is
(a) 5 mC (b) 5 C (c) 500 C (d) 5 C
43. B1 is the magnetic field due to bigger coil, B2 is the magnetic field due to
smaller coil and Bnet is the net magnetic field at the centre of two
concentric coils. If Bnet < B1 and B2 < B1, then decide the direction of
current I1 and I2 in the two coils:

(a) both clockwise (b) both anticlockwise

(c) both opposite to each other (d) it can’t be predicted
44. In a bent wire shown in figure, a current I is passed. Find the value of B at the common centre.

 0 (b  a)(2  )  0 a(2  )

(a) (b)
4ab 4b
 0 b(2  )  0 ab
(c) (d)
4a 4

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45. Figure below shows the cross-sectional view of the hollow cylindrical conductor with inner radius ‘R’ and
outer radius ‘2R’, cylinder carrying uniformly distributed current along its axis. The magnetic induction at
point ‘P’ at a distance 3R/2 from the axis of the cylinder will be

5 0 i 7 0 i 5 0 i
(a) Zero (b) (c) (d)
72R 18R 36R
46. Two wires are wrapped over a wooden cylinder to form two co-axial
loops carrying currents i1 and i2. If i2 = 8i1, the value of x for B = 0 at the
origin O is

(a) ( 7  1)R (b) 5R

(c) 3R (d) 7R

47. Consider a set of six infinite long straight parallel wires

arranged perpendicular to the plane of paper in a hexagon as
shown. The length of the each side of the hexagon is 3 cm.
What is the magnitude and direction of the magnetic field at
point P?
(a) 20 T (b) 40 T
(c) 60 T (d) 80 T
48. Two infinitely long parallel wires are a distance d apart and carry equal parallel currents I in the same
direction as shown in the figure. If the wires are located on y-axis (normal to x-y plane) at y = d/2 and
y = –d/2, then the magnitude of x-coordinate of the point on x-axis where the magnitude of magnetic field
is maximum is (consider points on x-axis only)

(a) 2d (b) d / 2 (c) 2d (d) d/2

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EXERCISE - 3
Multiple Choice Correct With Multiple Options
1. A charged particle goes undeflected in a region containing electric and magnetic fields. It is possible that
     
(a) E || B, v || E (b) E is not parallel to B
       
(c) v || B but E is not parallel to B (d) E || B but v is not parallel to E
2. If a charged particle goes unaccelerated in a region containing electric and magnetic fields, then
   
(a) E must be perpendicular to B (b) v must be perpendicular to E
 
(c) v must be perpendicular to B (d) E must be equal to vB
  
3. The force F experienced by a particle of charge q moving with a velocity v in a magnetic field B is
  
given by F  q(v B). Which pairs of vectors are at right angles to each other?
        
(a) F and v (b) F and B (c) B and v (d) F and (v B)
4. Which of the following statements is correct?
(a) If a moving charged particle enters into a region of magnetic field from outside, it does not complete
a circular path.
(b) If a moving charged particle traces a helical path in a uniform magnetic field, the axis of the helix is
parallel to the magnetic field
(c) The power associated with the force exerted by a magnetic field on a moving charged particle is
always equal to zero
(d) If in a region a uniform magnetic field and a uniform electric field both exist, a charged particle
moving in this region cannot trace a circular path.
5. An electron is moving along the positive x-axis. You want to apply a magnetic field for a short time so
that the electron may reverse its direction and move parallel to the negative x-axis. This can be done by
applying the magnetic field along
(a) y-axis (b) z-axis (c) y-axis only (d) z-axis only
6. In previous problem, if the pitch of the helical path is equal to the maximum distance of the particle from
the x-axis, then which of the following are not correct?
1 1 1
(a) cos   (b) sin   (c) tan   (d) tan   
  

A proton is fired from origin with velocity v  v 0 j  v 0 k in a uniform magnetic field B  B0 j . In the
 
7.
subsequent motion of the proton
(a) its z-coordinate can never be negative
(b) its x-coordinate can never be positive
(c) its x and z-coordinates cannot be zero at the same time
(d) its y-coordinate will be proportional to its time of flight
8. Velocity and acceleration vector of a charged particle moving in a magnetic field at some instant are

v  3i  4j and a  2i  x j . Select the correct options:

 

(a) x = –1.5 (b) x = 3

(c) Magnetic field is along z-direction (d) Kinetic energy of the particle is constant

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 
9. Let E and B denote the electric and magnetic fields in a certain region of space. A proton moving with a
velocity along a straight line enters the region and is found to pass through it undeflected. Indicate which
of the following statements are consistent with the observations:
 
(a) E  0 and B  0
 
(b) E  0 and B  0
    
(c) E  0 and B  0 and both E and B are parallel to v
   
(d) E is parallel to v but B is perpendicular to v
10. When a current carrying coil is placed in a uniform magnetic field with its magnetic moment anti-parallel
to the field.
(a) torque on it is maximum (b) torque on it is zero
(c) potential energy is maximum (d) dipole is in unstable equilibrium
11. A particle having a mass of 0.5 g carries a charge of 2.5 × 10–8C. The particle is given an initial horizontal
velocity of 6 × 104 ms–1. To keep the particle moving in a horizontal direction
(a) the magnetic field may be perpendicular to the direction of the velocity
(b) the magnetic field should be along the direction of the velocity
(c) magnetic field should have a minimum value of 3.27 T
(d) no magnetic field is required
12. A conductor ABCDE, shaped as shown, carries current I. It is
placed in the x-y plane with the ends A and E on the x-axis. A
uniform magnetic field of magnitude B exists in the region. The
force acting on it will be
(a) zero, if B is in the x-direction
(b) BI in the z-direction, if B is in the y-direction
(c) BI in the negative y-direction, if B is I the z-direction
(d)  aBI, if B is in the x-direction
13. In previous problem, if the current is I and the magnetic field at D has magnitude B, then
I I
(a) B  0 (b) B  0
2 2 2 3
(c) B is parallel to the z-axis (d) B makes an angle of 45o with the x-y plane
14. A particle of charge q and mass m enters normally (at point P) in a
region of magnetic field with speed v. It comes out normally from Q
after time T as shown in figure. The magnetic field B is present only in
the region of radius R and is uniform. Initial and final velocities are
along radial direction and they are perpendicular to each other. For this
to happen, which of the following expression(s) is/are correct?

mv R m
(a) B (b) T  (c) T (d) None of these
qR 2v 2qB

A proton is fired from origin with velocity v  v 0 j  v 0 k in a uniform magnetic field B  B0 j . In the
 
15.
subsequent motion of the proton
(a) its z co-ordinate can never be negative
(b) its x co-ordinate can never be positive
(c) its x and z co-ordinates cannot be zero at the same time
(d) its y co-ordinate will be proportional to its time to flight

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16. A steady electric current is flowing through a cylindrical conductor. Then,
(a) the electric field at the axis of he conductor is zero
(b) the magnetic field at the axis of the conductor is zero
(c) the electric field in the vicinity of the conductor is zero
(d) the magnetic field in the vicinity of the conductor is zero
17. Two thin long wires carry currents I1 and I2 along x and y-axes, respectively, as
shown in figure. Consider the points only in x-y plane.

(a) Magnetic field is zero at least at one point in each quadrant

(b) Magnetic field can be zero somewhere in the first quadrant
(c) Magnetic field can be zero somewhere in the second quadrant
(d) Magnetic field is non-zero in second quadrant
18. In the loops shown in figure, all curved sections are either semicircles or quarter circles. All the loops
carry same current. The magnetic fields at the centers have magnitudes B1, B2, B3, and B4. Then,

(a) B4 is maximum (b) B3 is minimum

(c) B4 > B1 > B2 > B3 (d) B1 > B4 > B3 > B2
19. Two circular coils of radii 5 cm and 10 cm carry equal currents of 2A. The coils have 50 and 100 turns,
respectively, and are placed in such a way that their planes as well as their centers coincide. Magnitude of
magnetic field at the common center of coils is
(a) 8  104 T if current in the coils are in same sense
(b) 4  104 T if current in the coils are in opposite sense
(c) zero if current in the coils are in opposite sense
(d) 8  104 T if current in the coils are in opposite sense
20. A long straight wire carries the current along +ve x-direction. Consider four points in space A(0, 1, 0),
B(0, 1, 1), C(1, 0, 1), and D(1, 1, 1). Which of the pairs will have the same magnitude of magnetic field?
(a) A and B (b) A and C (c) B and C (d) B and (D)
21. Figure shows cross section of two large parallel metal sheets carrying electric currents along their
surfaces. The current in each sheet is 10 A/m along the width. Consider two points A and B, as shown
in the figure with their positions.

(a) Magnetic field at A is 4 T along x-direction

(b) Magnetic field at a is 4 T along negative x-direction
(c) Magnetic field at B is zero
(d) Magnetic field at B is 2 T along x-direction

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22. Two long conducting wires carrying equal currents are placed parallel to
the z-axis. A charged particle is made to move in a circular path of radius
a with centre of the path at point (0, 2a) in clockwise direction. During
the course of motion, it passes through four points P, Q, R and S having
coordinates (0, a), (–a, 2a), (0, 3a) and (a, 2a), respectively. The
magnitude of force exerted on the particle by the magnetic field created
by the wires is maximum when it passes through point.
(a) P (b) Q
(c) R (d) S
23. A straight wire carrying current is parallel to the y-axis as shown in figure.
The
(a) magnetic field at the point P is parallel to the x-axis
(b) magnetic field at P is along z-axis
(c) magnetic field are concentric circle with the wire passing through their
common centre
(d) magnetic fields to the left and right of the wire are oppositely directed
24. Two circular coils A and B with their centres lying on the same axis have same number of turns and carry
equal currents in the same sense. They are separated by a distance, have different diameters but subtend
same angle at a point P lying on their common axis. The coil B lies exactly midway between coil A and
the point P. The magnetic field at point P due to coils A and B is B1 and B2 respectively
B1 B1 1
(a) B1 > B2 (b) B1 < B2 (c) 2 (d) 
B2 B2 2

Exercise – 4
Section – I : Numerical Value / Subjective Type Questions

1. Two circular coils made of similar wires but of radii 20 cm and 40 cm are connected in parallel. Find the
ratio of the magnetic fields at their centers.
2. A wire carrying current i has the configuration shown in figure. Two semi-
infinite straight sections, each tangent to the same circle, are connected by
a circular arc, of angle  , along the circumference of the circle, with all
sections lying in the same plane. What must  (in rad) be in order for B to
be zero at the center of circle?

3. Two circular coils X and Y, having equal number of

turns and carrying currents in the same sense, subtend
same solid angle at point O. If the smaller coil X is
midway between O and Y and if we represent the
magnetic induction due to bigger coil Y at O as By and that due to smaller coil X at O as Bx, then fid the
ratio Bx/By.

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4. A square loop of side a = 6 cm carries a current I = 1A. Calculate magnetic induction B (in T) at point

P, lying on the axis of loop and at a distance x  7 cm from the center of loop.
5. An elevator carrying a charge of 0.5 C is moving down with a velocity of 5 × 103 ms–1. The elevator is 4
m from the bottom and 3 m horizontally from P as shown in figure. What magnetic field (in T) does it
produce at point P?

6. A system consists of two parallel planes carrying currents producing a uniform magnetic field of
induction B between the planes. Outside this space there is no magnetic field. The magnetic force acting
2
per unit area of each plane is found to be B / N0 . Find N.
7. Two parallel, long wires carry current i1 and i2 with i1 > i2. When the currents are in the same direction,
the magnetic field at a point midway between the wire is 10 mT. If the direction of i2 is reversed, the field
becomes 30 mT. Find the ratio i1/i2.
8. An infinitely long conductor PQR is bent to form a right angle as shown. A current I flows through PQR.
The magnetic field due to this current at the point M is B1. Now, another infinitely long straight conductor
QS is connected at Q so that the current is I/2 in QR as well as in QS, the current in PQ remaining
unchanged. The magnetic field at M is now B2 = 4 mT. Find the value of B1 (in mT).

9. Consider three long straight parallel wires as shown in figure. The force experienced by a 25 cm length of

wire C is N × 10–4N. Find the value of N?

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 P-95
Section – II : Column Match Type

1. Three wires are carrying same constant current i in different directions. Four loops enclosing the wires in

different manners are shown in figure. The direction of d is shown in the figure.

Column I Column II
i. Along closed loop 1 a.  

 Bd  0i
ii. Along closed loop 2 b.  

 B.d  0i
iii. Along closed loop 3 c. 

 B.d  0
iv. Along closed loop 4 d. Network done by the magnetic force to
move a unit charge along the loop is zero
2. A circular current carrying loop having 100 turns and radius 10 cm is placed in

 tesla is
x-y plane as shown in figure. A uniform magnetic field B  ( i  k)


present in the region. If current in the loop is 5 A, then match the following:

Column I Column II
i. Magnitude and direction of moment (in A-m2) a. Zero
of the loop are
ii. Magnitude and direction of torque (in N-m) on b. 5
the loop are
iii. Magnitude and direction of net force (in N) on c. Along positive z-axis
the current loop are
iv. Direction of magnetic field of loop at the d. Along negative y-axis
center is
3. Column II gives four situations in which three or four semi-infinite current carrying wires are placed in x-
y plane. The magnitude of the direction of current is shown in each figure. Column I gives statements
regarding the x and components of magnetic field at point P whose coordinates are (0, 0, d). Match the
statements in Column I with the corresponding figures in Column II.

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Column I Column II
i. The x-component of magnetic field at point P a.
is zero in

ii. The z-component of magnetic field at point P b.

is zero in

iii. The magnitude of magnetic field at point P is c.

0i
in
4d

iv. The magnitude of magnetic field at point P is d.

i
less than 0 in
2d

Section – III : Comprehension

Comprehension # (1-3)
A particle of mass m and charge q is accelerated by a potential difference V volt and made to enter a magnetic
field region at an angle  with the field. At the same moment, another particle of same and charge is projected in
the direction of the field from the same point. Magnetic field of induction is B.
1. What would be the speed of second particle so that both particles meet again and again after a regular
interval of time, which should be minimum?
qV 2qV qV qV
(a) cos  (b) cos  (c) sin  (d) cos 
m m m 2m

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2. Find the time interval after which they meet.
2m m m 3m
(a) (b) (c) (d)
qB 2qB qB 2qB

3. Find the distance travelled by the second particle during that interval mentioned in the above problem.
Vm 2 2Vm 2  2Vm 2  2 Vm 
(a) cos  (b) cos  (c) cos  (d) cos 
q B 3q B q B 3 q m

Comprehension # (4-5)
A charged particle carrying charge q = 10 C moves with velocity v1= 106 ms–1 at angle 45o with x-axis in the xy

plane and experiences a force F1  5 2 mN along the negative z-axis. When the same particle moves with

velocity v2  10 ms along the z-axis, it experiences a force F2 in y direction.

6 1


4. Find the magnetic field B .

(a) (103 T)(i  j) (b) (2  103 T)i (c) (103 T)i (d) (2  103 T)(i  j)
5. Find the magnitude of the force F2.
(a) 10–2 N (b) 10–3 N (c) 10–4N (d) 10–5 N
Comprehension # (6-7)
A conducting bar with mass m and length L slides over horizontal rails that
are connected to a voltage source. The voltage source maintains a constant
current I in the rails and bar, and a constant, uniform, vertical magnetic

field B fills the region between the rails (as shown in figure).

6. Find the magnitude and direction of the net force on the conducting bar. Ignore friction, air resistance and
electrical resistance.
(a) ILB, to the right (b) ILB, to the left (c) 2ILB, to the right (d) 2ILB, to the left
7. If the bar has mass m, find the distance d that the bar must move along the rails from rest to attain speed v.
3v 2 m 5v 2 m v2 m v2 m
(a) (b) (c) (d)
2ILB 2ILB ILB 2ILB
Comprehension # (8-10)
3
 q  10
A charged particle with charge to mass ratio    C kg 1 enters a uniform magnetic field
m
  19

B  20i  30j  50k T at time t = 0 with velocity V  (20i  50j  30k)

 m/s. Assume that magnetic field exists in
 

large space.
8. During the further motion of the particle in the magnetic field, the angle between the magnetic field and
velocity of the particle
(a) remains constant (b) increases
(c) decreases (d) may increase or decrease

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9. The frequency (in Hz) of the revolution of the particle in cycles per second will be
103 10 4 104 104
(a) (b) (c) (d)
 38  38  19 2 19
10. The pitch of the helical path of the motion of the particle will be
(a)  /100 m (b)  /125m (c)  / 215m (d)  / 250 m
Comprehension # (11-12)
According to Biot-Savart’s law, magnetic field due to a straight current carrying wire at a poit
0 I
at a distance r from it is given by B  (sin 1  sin 2 ) . The direction of magnetic field
4r
being perpendicular to the plane containing the wire and that point.

11. Figure shows a closed loop AOCBA in which current I is flowing as shown. Given OA = OB = OC = a.
Find the magnetic field at point B due to this loop.

0 I 0 I 0 I  
(a)  (i  k)
 (b)  (j  k)
 (c) ( j  k) (d) None of these
4 2a 4 2a 2 2a
12. Find magnetic field at point O in figure.
0 I   0 I    0 I   0 I  
(a)  (i  k) (b)  ( j  k) (c) ( j  k) (d) ( j  k)
4a 4a 2a 2 2a
Comprehension # (13-15)
An infinite cylindrical wire of radius R and having current density varying with its radius r as, J = J0[1 – (r/R)].
Then answer the following questions.

13. Position where magnetic field strength is maximum

(a) R (b) 3R/4 (c) R/2 (d) R/4
14. Magnitude of maximum magnetic field is
J0 R 3 5 5
(a) 0 (b) J 0 R 0 (c) J 0 R 0 (d) J 0 R 0
6 16 16 6

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15. Graph between the magnetic field and radius is

(a) (b)

(c) (d)

Comprehension # (16-18)
A current I amperes flows through a loop abcdefgha along the edge of a cube of
width l metres as shown in figure. One corner a of the loop lies at origin.

16. This current path (abcdefgha) can be treated as a superposition of three

square loops carrying current I. Choose the correct option.
(a) fghaf, fabef, ebcde (b) fghaf, fabef, fgdef
(c) fghaf, abcha, ebcde (d) fgdef, fabef, ebcde
17. The unit vector in the direction of magnetic field at the centre of cube abcdefgh of width  is given by

2i  j
(a) i (b) j (c) (d) k
5

Now if a uniform external magnetic field is B  B0 j is switched on, then the unit vector in the direction of

18.

torque due to external magnetic field (B) acting on the current carrying loop (abcdefgha) is

2i  j
(a) k (b) i (c) (d) None of these
5

EXERCISE - 5
Revision Exercise Subjective Questions (Moderate To Tough)
1. A particle having mass m and charge q is released from the origin in a region in which electric field and

magnetic field are given by B   B0 j and E  E 0 k . Find the speed of the particle as a function of its z-
  

coordinate.

A charge q = – 4 C has an instantaneous velocity v  (2i  3j  k)

  10 6 ms 1 in a uniform magnetic field

2.

B  (2i  5j  3k)

  10 2 T . What is the force on the charge?


3. A charged particle is having charge q  50 C is projected vertically upward with a speed of 5.0 × 104
km/s in a region where a magnetic field of magnitude 2.0 T exists in the direction south to north. Find the
magnetic force that acts on the  -particle.

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4. An electron is moving northwards with a velocity 107 ms–1 in a magnetic field of 3 T directed eastwards.
Calculate the instantaneous force on the electron. Given that charge on electron = 1.6 × 10–19C.

5. 
A magnetic field of (5.0  103 k)T exerts a force of (2.0i  1.5j)  1010 N on a particle having a charge of

1.0  10 9 C and going in the x-y plane. Find the velocity of the particle.

6. A charge particle is projected in a magnetic field of (5.0i  2.0j)  103 T. If the acceleration of the particle

is found to be (xi  5.0j)  106 m / s 2 . What should be the value of x?

7. A charged sphere of mass m and charge q starts sliding from rest on a vertical
fixed circular track of radius R from the position as shown in figure. There
exists a uniform and constant horizontal magnetic field of induction B. Find
the maximum force exerted by the track on the sphere.

8. Doubly-ionized helium ions are projected with a speed of 10 km s–1 in a direction perpendicular to a
uniform magnetic field of magnitude 1.0 T. Find (a) the force acting on an ion, (b) the radius of the circle
in which it circulates and (c) the time taken by an ion to complete the circle.
9. A stream of protons and deuterons in a vacuum chamber enters a uniform magnetic field. Both protons
and deuterons have been subjected to same accelerating potential, hence the kinetic energies of the
particles are the same. If the ion-stream is perpendicular to the magnetic field and the protons move in a
circular path of radius 15 cm, find the radius of the path traversed by the deuterons. Given that mass of
deuteron is twice that of a proton.
10. An   particle is accelerated by a potential difference of 104V. Find the change in its direction of motion,
if it enters normally in a region of thickness 0.1 m having transverse magnetic induction of 0.1 tesla.
(Given : mass of   particle 6.4 × 10–27 kg).
11. A charge particle of mass m and charge q is accelerated through a potential difference of V volts. It enters
a region of uniform magnetic field which is directed perpendicular to the direction of motion of the
particle. Find the radius of circular path moved by the particle in magnetic field.
12. A proton (charge 1.6 × 10–19C, mass = 1.60 × 10–27 kg) is shot with a speed 5 × 106 ms–1 at an angle of 37o
with the X-axis. A uniform magnetic field B = 0.30 T exists along the X-axis. Show that path of the
proton is a helix. Find the radius and pitch of the helix.
13. A non-relativistic charge q of mass m originates at a point A

lying on x-axis and moves with velocity v at an angle  to the

x-axis. A screen is located at a distance l from A. Find the

distance r from the x-axis to the point on the screen into which

the charge strikes.

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14. A loop of wire has the shape of a right triangle and carries a current of I =
5.0 A. A uniform magnetic field is directed parallel to side AB and has a
magnitude of 2.0 T.
(a) Find the magnitude and direction of the magnetic force exerted on each
side of the triangle.
(b) Determine the magnitude of the net force exerted on the triangle.
15. Two conducting rails in the drawing are tilted upward so they each make an angle of 37o with respect to
the ground. The vertical magnetic field has a magnitude of 0.050 T. The 0.20-kg aluminium rod
(length = 2.0 m) slides without friction down the rails at a constant velocity. How much current flows
through the rod.

16. A metal wire PQ of mass 10 g lies at rest on two horizontal metal rails separated by 5 cm. A vertically
downward magnetic field of magnitude 0.800 T exists of the circuit is slowly decreased and it is found
that when the resistance goes below 20 , the wire PQ starts sliding on the rails. Find the coefficient of
friction.

17. Figure (a) shows a rod PQ of length 20 cm and mass 200 g suspended through a fixed point O by two
threads of lengths 20 cm each. A magnetic field of strength 0.500 T exists in the vicinity of the wire PQ as
shown in the figure.

The wire connecting PQ with the battery are loose and exert no force on PQ.
(a) Find the tension in the threads when the switch S is open.
(b) A current of 2 A is established when the switch S is closed. Find the tension in the threads now.

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18. A square loop of edge 1 and carrying a current i, is placed with its edges

parallel to the XY axes. There is present a non-uniform magnetic field in the


 x 
region B  B0 1   (  k). Find the magnitude of the net magnetic force
 

experienced by the loop.

19. In figure, a semicircular wire loop is placed in uniform magnetic field B = 1.0 T. The plane of the loop is
perpendicular to the magnetic field. Current i = 2A A flows in the loop in the direction shown. Find the
magnitude of the magnetic force in both the cases (a) and (b). The radius of the loop is 1 m.

20. A conductor (rod) of mass m, length l carrying a current i is subjected to a magnetic field of induction B.
If the coefficients of friction between the conducting rod and rail is  , find the value of i if the rod start
sliding.

21. A wire of 60 cm length and mass 16 gm is suspended by a

pair of flexible leads in a magnetic field of induction 0.40 T.

What are the magnitude and direction of the current required

to remove the tension in the supporting leads?

22. A conductor carries a constant current I along the closed path abcdefgha involving 8 of the 12 edges of
length l. Find the magnetic dipole moment of the closed path.
23. A circular coil of wire 8 cm in diameter has 12 turns and carries a current of 5A. the coil is in a field
where the magnetic induction is 0.6 T.
(a) What is the maximum torque on the coil?
(b) In what position would the torque be half as great as in (i)?

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24. A particle of mass m and charge +q enters a region of magnetic field with a velocity v, as shown in figure.

(a) Find the angle subtended by the circular arc described by it in the magnetic field.
(b) How long does the particle stay inside the magnetic field?
25. Find the magnitude and direction of magnetic field at point P due to the current carrying wire as shown in
figure.

26. A square loop of wire, edge length a, carries a current i,. Compute the magnitude of the magnetic field
produced at a point on the axis of the loop at a distance x from the center.
27. A current i flows in a long straight wire with cross-section having the form of a thin semicircular-ring of
radius R as shown in figure. Find the magnetic field at the point O, situated on the axis of the wire.

28. A straight conductor of length r0 carrying a current i is placed perpendicular to a long straight conductor
which carries a current i0 as shown in the figure. Find the force of interaction between them.

29. A non-conducting thin disc of radius R charged uniformly over one side with surface density s rotates
about its axis with an angular velocity  . Find
(a) the magnetic induction at the center of the disc;
(b) the magnetic moment of the disc.

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30. A square frame carrying a current I = 0.90 A is located in the same plane
as a long straight wire carrying a current I0 = 5A. The frame side has a
length a = 9 cm. The axis of the frame passing through the mid-points of
the opposite sides is parallel to the wire and is separated from it by a
distance h = 15 times greater then the side of the frame. Find :

(a) ampere force acting on the frame.

(b) the mechanical work to be performed in order to turn the frame through 180o about its axis, with the
current maintained constant.
31. A circular loop of radius R is bent along the diameter and given a shape as shown in figure. One of the
semicircles (KNM) lies in the x-z plane and the other one (KLM) in the y-z plane with their centers at the
origin. Current I is flowing through each of the semicircles as shown in the figure.

(a) A particle of charge q is released at the origin with a velocity v   v 0 i . Find the instantaneous force

F on the particle. Assume that space is gravity free.
(b) If an external uniform magnetic field Bj is applied determined the force F1 and F 2 on the
 


semicircles KLM and KNM due to this field and the net force F on the loop.

Exercise – 6
Section – I : JEE (Advanced) Questions Previous Years

1. Consider the motion of a positive point charge in a region where there are simultaneous uniform electric
  
and magnetic fields E  E 0 ˆj and B  B0 ˆj . At time t = 0, this charge has velocity v in the x-y plane,

making an angle  with x-axis. Which of the following option(s) is(are) correct for time t > 0 ?
(A) If  = 0°, the charge moves in a circular path in the x-z plane. [IIT-JEE-2012, Paper-1; 4/70]
(B) If  = 0°, the charge undergoes helical motion with constant pitch along the y-axis.
(C) If  = 10°, the charge undergoes helical motion with its pitch increasing with time, along the y-axis.
(D) If  = 90°, the charge undergoes linear but accelerated motion along the y-axis.
2. A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a
shown in the figure. Both the cylinder and the cavity are infinitely long. A
uniform current density J flows along the length. If the magnitude of the
N
magnetic field at the point P is given by 0aJ, then the value of N is :
12

[IIT-JEE-2012, Paper-1; 4/70]

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3. A loop carrying current  lies in the x-y plane as shown in the figure.
The unit vector k̂ is coming out of the plane of the paper. The
magnetic moment of the current loop is :
[IIT-JEE-2012, Paper-2; 3/66, –1]

   
(A) a 2 kˆ (B)   1 a 2 kˆ (C)    1 a 2 kˆ (D) (2  1)a 2 kˆ
2  2 
4. An infinitely long hollow conducting cylinder with inner radius R/2 and outer radius R carries a uniform

current density along is length. The magnitude of the magnetic field, B as a function of the radial
distance r from the axis is best represented by : [IIT-JEE-2012, Paper-2; 3/66, –1]

(A) (B)

(C) (D)

 ˆ 1 , enters a region
5. A particle of mass M and positive charge Q, moving with a constant velocity u1  4ims
of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from
x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side

 
after 10 milliseconds with a velocity u  2 3iˆ  ˆj ms–1. The correct statement(s) is (are) :
2

[JEE (Advanced)-2013, 4/60, –1]

(A) The direction of the magnetic field is –z direction.
(B) The direction of the magnetic field is +z direction
50M
(C) The magnitude of the magnetic field units.
3Q
100M
(D) The magnitude of the magnetic field is units.
3Q
6. A steady current  flows along an infinitely long hollow cylindrical conductor of radius R. This cylinder is
placed coaxially inside an infinite solenoid of radius 2R. The solenoid has n turns per unit length and
carries a steady current . Consider a point P at a distance r from the common axis. The correct
statement(s) is (are) : [JEE (Advanced) 2013, 3/60, –1]
(A) In the region 0 < r < R, the magnetic field is non–zero.
(B) In the region R < r < 2R, the magnetic field is along the common axis.
(C) In the region R < r < 2R, the magnetic field is tangential to the circle of radius r, centered on the axis.
(D) In the region r > 2R, the magnetic field is non–zero.

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7. Two parallel wires in the plane of the paper are distance X0 apart. A point charge is moving with speed u
between the wires in the same plane at a distance X1 from one of the wires. When the wires carry current
of magnitude  in the same direction, the radius of curvature of the path of the point charge is R1. In
contrast, if the currents  in the two wires have direction opposite to each other, the radius of curvature of
x0 R
the path is R2. If = 3, the value of 1 is. [JEE (Advanced) 2014, P-1, 3/60]
x1 R2

Paragraph for Questions 8 to 9

The figure shows a circular loop of radius a with two long parallel wires (numbered 1 and 2) all in the
plane of the paper. The distance of each wire from the centre of the loop is d. The loop and the wires are
carrying the same current . The current in the loop is in the counterclockwise direction if seen from
above. [JEE (Advanced) 2014, P-2, 3/60, –1]

8. When d  a but wires are not touching the loop, it is found that the net magnetic filed on the axis of the
loop is zero at a height h above the loop. In that case [JEE (Advanced) 2014, 3/60, –1]
(A) current in wire 1 and wire 2 is the direction PQ and RS, respectively and h  a
(B) current in wire 1 and wire 2 is the direction PQ and SR, respectively and h  a
(C) current in wire 1 and wire 2 is the direction PQ and SR, respectively and h  1.2 a
(D) current in wire 1 and wire 2 is the direction PQ and RS, respectively and h  1.2 a
9. Consider d >> a, and the loop is rotated about its diameter parallel to the wires by 30º from the position
shown in the figure. If the currents in the wires are in the opposite directions, the torque on the loop at its
new position will be (assume that the net field due to the wires is constant over the loop)
[JEE (Advanced) 2014, 3/60, –1]
02a 2  2a 2 3 0  2 a 2 3 0  2 a 2
(A) (B) 0 (C) (D)
d 2d d 2d
10. A conductor (shown in the figure) carrying contant current  is kept in the x-y plane in a uniform magnetic

field B . If F is the magnitude of the total magnetic force acting on the conductor, then the correct
statement(s) is (are) : [JEE (Advanced) 2015 ; 4/88, –2]

 
(A) If B is along ẑ , F (L + R) (B) If B is along x̂ , F = 0
 
(C) If B is along ŷ , F (L + R) (D) If B is along ẑ , F = 0

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Paragraph for Questions 11 to 12

In a thin rectangular metallic strip a constant current  flows along the positive x-direction, as shown in
the figure. The length, width and thickness of the strip are l, w and d, respectively.

A uniform magnetic field B is applied on the strip along the positive y-direction. Due to this, the charge
carriers experience a net deflection along the z-direction. This results in accumulation of charge carriers
on the surface PQRS and is appearance of equal and opposite charges on the face opposite to PQRS. A
potential difference along the z-direction is thus developed. Charge accumulation continues until the
magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the
cross section of the strip and carried by electrons. [JEE (Advanced) 2015 ; P-2, 4/88, –2]

11. Consider two different metallic strips (1 and 2) fo the same material. Their lengths are the same, width are
w1 and w2 and thicknesses are d1 and d2, respectively. Two points K and M are symmetrically located on
the opposite faces parallel to the x-y plane (see figure). V1 and V2 are the potential differences between K
and M in strips 1 and 2, respectively. Then, for a given current  flowing through them in a given
magnetic field strength B, the correct statement(s) is (are)
(A) If w1 = w2 and d1 = 2d2, then V2 = 2V1 (B) If w1 = w2 and d1 = 2d2, then V2 = V1
(C) If w1 = 2w2 and d1 = d2, then V2 = 2V1 (D) If w1 = 2w2 and d1 = d2, then V2 = V1
12. Consider two different metallic strips (1 and 2) of same dimensions (length l, width w and thickness d)
with carrier densities n1 and n2, respectively. Strip 1 is placed in magnetic field B1 and strip 2 is placed in
magnetic field B2, both along positive y-directions. Then V1 and V2 are the potential differences
developed between K and M in strips 1 and 2, respectively. Assuming that the current  is the same for
both the strips, the correct option(s) is (are)
(A) If B1 = B2 and n1 = 2n2, then V2 = 2V1 (B) If B1 = B2 and n1 = 2n2, then V2 = V1
(C) If B1 = 2B2 and n1 = n2, then V2 = 0.5V1 (D) If B1 = 2B2 and n1 = n2, then V2 = V1
Answer Q.13. Q.14 and Q.15 by appropriately matching the information given in the three columns
of the following table.
A charged particle (electron or proton) is introduced at the origin (x = 0, y = 0, z = 0) with a given initial
   
velocity v . A uniform electric field E and a uniform magnetic field B exist everywhere. The velocity v ,
 
electric field E and magnetic field B are given in column 1, 2 and 3, respectively. The quantities E0, B0
are positive in magnitude.
Column-1 Column-2 Column-3
 E  
(I) Electron with v  2 0 xˆ (i) E  E 0 zˆ (P) B  –B0 xˆ
B0
 E  
(II) Electron with v  0 yˆ (ii) E  –E 0 yˆ (Q) B  B0 xˆ
B0
  
(III) Electron with v  0 (iii) E  –E 0 xˆ (R) B  B0 yˆ
 E  
(IV) Electron with v  2 0 xˆ (iv) E  E0 xˆ (S) B  B0 zˆ
B0
13. In which case will the particle move in a straight line with constant velocity?
[JEE (Advanced) 2017 ; P-1, 3/61, –1]
(A) (IV) (i) (S) (B) (III) (ii) (R) (C) (II) (iii) (S) (D) (III) (iii) (P)

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14. In which case will the particle describe a helical path with axis along the positive z direction ?
[JEE (Advanced) 2017 ; P-1, 3/61, –1]
(A) (IV) (i) (S) (B) (II) (ii) (R) (C) (III) (iii) (P) (D) (IV) (ii) (R)
15. In which case would the particle move in a straight line along the negative direction of y-axis (i.e, move
along – ŷ ) ? [JEE (Advanced) 2017 ; P-1, 3/61, –1]
(A) (III) (ii) (R) (B) (IV) (ii) (S) (C) (III) (ii) (P) (D) (II) (iii) (Q)
16. A symmetric star shaped conducting wire loop is carrying a steady state current  as shown in the figure.
The distance between the diametrically opposite vertices of the star is 4a. The magnitude of the magnetic
field at the center of the loop is : [JEE (Advanced) 2017 ; P-2, 3/61, –1]



4a

0 0 0 0 

(A) 6  3  1 (B) 6  3  1 (C) 3  2  3  (D)3  3  1
4a  4a  4a  4a 
3R
17. A uniform magnetic field B exists in the region between x = 0 and x = (region 2 in the figure)
2
pointing normally into the plane of the paper. A particle with charge +Q and momentum p directed along
x-axis enters region 2 from region 1 at point P1 (y = –R). Which of the following option(s) is/are correct ?
[JEE (Advanced) 2017 ; P-2, 4/61, –2]
y
Region 1 Region 2 Region 3
× × ×
B
× × ×
× × ×
× × ×

O × × × x
× P2
× ×
+Q P1
× × ×
(y=–R) ×
× ×
× × ×
3R/2

(A) When the particle re-enters region 1 through the longest possible path in region 2, the magnitude of
the change in its linear momentum between P1 and the farthest point from y-axis is p / 2 .
(B) For a fixed B, particles of same charge Q and same velocity v, the distance between the point P1 and
the point of re-entry into region 1 is inversely proportional to the mass of the particle.
8 p
(C) For B  , the particle will enter region 3 through the point P2 on x-axis
13 QR
2 p
(D) For B  , the particle will re-enter region 1.
3 QR

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18. Two infinitely long straight wires lie in the xy-plane along the lines x = ±R. The wire located at x = +R
carries a constant current 1 and the wire located at x = – R carries a constant current 2. A circular loop of

radius R is suspended with its centre at (0, 0, 3R ) and in a plane parallel to the xy-plane. This loop
carries a constant current  in the clockwise direction as seen from above the loop. The current in the wire

is taken to be positive if it is in the + ĵ direction. Which of the following statements regarding the

magnetic field B is (are) true? [JEE (Advanced) 2018 ; P-1, 4/60, –2]

(A) If 1 =  2, then B cannot be equal to zero at the origin (0, 0, 0)

(B) If 1 > 0 and 2 < 0, then B can be equal to zero at the origin (0, 0, 0)

(C) If 1 < 0 and 2 > 0, then B can be equal to zero at the origin (0, 0, 0)
  
(D) If 1 = 2, then the z-component of the magnetic field at the centre of the loop is   0 
 2R 

19. In the xy-plane, the region y > 0 has a uniform magnetic field B1 k̂ and the
region y < 0 has another uniform magnetic field B2 k̂ . A positively
charged particle is projected from the origin along the positive y-axis with
speed v0 = ms−1 at t = 0, as shown in the figure. Neglect gravity in this
problem. Let t = T be the time when the particle crosses the x-axis from
below for the first time. If B2 = 4B1, the average speed of the particle, in
ms−1, along the x-axis in the time interval T is __________.
[JEE (Advanced) 2018 ; P-1, 3/60]

20. A long straight wire carries a current, l = 2 ampere. A semi-circular

conducting rod is placed beside it on two conducting parallel rails of
negligible resistance. Both the rails are parallel to the wire. The wire, the
rod and the rails lie in the same horizontal plane, as shown in the figure.
Two ends of the semi-circular rod are at distances 1 cm and 4 cm from the
wire. At time t = 0, the rod starts moving on the rails with a speed v = 3.0
m/s (see the figure). [JEE (Advanced) 2021 ; P-1]
A resistor R = 1.4  and a capacitor C0 = 5.0 F are connected in series
between the rails. At time t = 0, C0 is uncharged. Which of the following
statement(s) is(are) correct? [0 = 4 × 10–7 SI units. Take ln 2 = 0.7]
(A) Maximum current through R is 1.2 × 10–6 ampere
(B) Maximum current through R is 3.8 × 10–6 ampere
(C) Maximum charge on capacitor C0 is 8.4 × 10–12 coulomb
(D) Maximum charge on capacitor C0 is 2.4 × 10–12 coulomb

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22. An α-particle (mass 4 amu) and a singly charged sulfur ion (mass 32 amu) are initially at rest. They are
accelerated through a potential V and then allowed to pass into a region of uniform magnetic field which is
normal to the velocities of the particles. Within this region, the α-particle and the sulfur ion move in circular orbits
r 
of radii rα and rs, respectively. The ratio  s  is _____. [JEE (Advanced) 2021 ; P-2]
 r 
21. Two concentric circular loops, one of radius R and the other of radius 2R, lie in the
xy-plane with the origin as their common center, as shown in the figure. The
smaller loop carries current I1 in the anti-clockwise direction and the larger loop

carries current I2 in the clockwise direction, with I2 > 2I1. B (x, y) denotes the
magnetic field at a point (x, y) in the xy-plane. Which of the following statement(s)
is(are) correct?

(A) B(x, y) is perpendicular to the xy-plane at any point in the plane [JEE (Advanced) 2021 ; P-2]

(B) B(x, y) depends on x and y only through the radial distance r  x 2  y2

(C) B(x, y) is non-zero at all points for r < R

(D) B(x, y) points normally outward from the xy-plane for all the points between the two loops
Paragraph
A special metal S conducts electricity without any resistance. A closed wire loop, made of S, does not
allow any change in flux through itself by inducing a suitable current to generate a compensating flux.
The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a
magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of
radius a, with its center at the origin. A magnetic dipole of moment m is brought along the axis of this
loop from infinity to a point at distance r (>> a) from the center of the loop with its north pole always
facing the loop, as shown in the figure below. [JEE (Advanced) 2021 ; P-2]
 m
The magnitude of magnetic field of a dipole m, at a point on its axis at distance r, is 0 3 , where μ0 is the
2r
permeability of free space. The magnitude of the force between two magnetic dipoles with moments, m1
km m
and m2, separated by a distance r on the common axis, with their north poles facing each other, is 14 2 ,
r
where k is a constant of appropriate dimensions. The direction of this force is along the line joining the
two dipoles.

22. When the dipole m is placed at a distance r from the center of the loop (as shown in the figure), the current
induced in the loop will be proportional to
(A) m/r3 (B) m2/r2 (C) m/r2 (D) m2/r
23. The work done in bringing the dipole from infinity to a distance r from the center of the loop by the given process
is proportional to
(A) m/r5 (B) m2/r5 (C) m2/r6 (D) m2/r7

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24. Consider an LC circuit, with inductance 𝐿 = 0.1 𝐻 and capacitance 𝐶 = 10–3 𝐹, kept on a plane. The area of the
circuit is 1 𝑚2. It is placed in a constant magnetic field of strength 𝐵0 which is perpendicular to the plane of the
circuit. At time 𝑡 = 0, the magnetic field strength starts increasing linearly as 𝐵 = 𝐵0 + 𝛽𝑡 with 𝛽 = 0.04 𝑇𝑠–1. The
maximum magnitude of the current in the circuit is _____ mA. [JEE (Advanced) 2022 ; P-2]

Section – II : JEE (Main) Questions Previous Years

1. A current  flows in an infinitely long wire with cross-section in the form of a semicircular ring of radius
R. The magnitude of the magnetic induction on its axis is : [AIEEE - 2011, 1-May, 4/120, –1]
0 0 0 0
(1) (2) (3) (4)
2 R 2 2 R 2 R 4 R

2. An electric charge +q moves with velocity v  3iˆ  4ˆj  kˆ , in an electromagnetic field given by :
 
E  3iˆ  ˆj  2kˆ and B  ˆi  ˆj  3kˆ . The y-component of the force experienced by + q is :
[AIEEE 2011, 11 May; 4/120, –1]
(1) 7 q (2) 5 q (3) 3 q (4) 2 q
3. A thin circular disk of radius R is uniformly charged with density  > 0 per unit area. The disk rotates
about its axis with a uniform angular speed . The magnetic moment of the disk is :
[AIEEE 2011, 11 May; 4/120, –1]
R 4 R 4
(1) R4  (2)  (3)  (4) 2R4 
2 4
4. A charge Q is uniformly distributed over the surface of non-conducting disc of radius R. The disc rotates
about an axis perpendicular to its plane and passing through its centre with an angular velocity . As a
result of this rotation a magnetic field of induction B is obtained at the centre of the disc. if we keep both
the amount of charge placed on the disc and its angular velocity to be constant and vary the radius of the
disc then the variation of the magnetic induction at the centre of the disc will be represented by the figure :
[AIEEE 2012 ; 4/120, –1]

(1) (2) (3) (4)

5. Two short bar magnets of length 1 cm each have magnetic moments 1.20 Am2 and 1.00 Am2 respectively.
They are placed on a horizontal table parallel to each other with their N poles pointing towards the South.
They have a common magnetic equator and are separated by a distance of 20.0 cm. The value of the
resultant horizontal magnetic induction at the mid - point O of the line joining their centres is close to
(Horizontal component of earth’s magnetic induction is 3.6× 10–5 Wb/m2)
[JEE (Main) 2013 ; 4/120, –1]
(1) 3.6 × 10–5 Wb/m2 (2) 2.56 × 10–4 Wb/m2 (3) 3.50 × 10–4 Wb/m2 (4) 5.80 × 10–4 Wb/m2

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6. A conductor lies along the z-axis at –1.5  z < 1.5 m and carries a fixed current of 10.0 A

in – â z direction (see figure). For a field B = 3.0 × 10–4e–0.2 x â y T, find the power required to move the

conductor at constant speed to x = 2.0 m, y = 0 m in 5 × 10–3s. Assume parallel motion along the x-axis
[JEE (Main) 2014, 4/120, –1]

(1) 1.57 W (2) 2.97 W (3) 14.85 W (4) 29.7 W


7. Two coaxial solenoids of different radii carry current I in the same direction. Let F1 be the magnetic force

on the inner solenoid due to the outer one and F2 be the magnetic force on the outer solenoid due to the
inner one. Then : [JEE (Main) 2015; 4/120, –1]
   
(1) F1 = F2 = 0 (2) F1 is radially inwards and F2 is radially outwards
   
(3) F1 is radially inwards and F2 = 0 (4) F1 is radially outwards and F2 = 0
8. Two long current carrying thin wires, both with current , are held by insulating threads of length L and are
in equilibrium as shown in the figure, with threads making an angle '' with the vertical. If wires have mass 
per unit length then the value of  is (g = gravitational acceleration) [JEE (Main) 2015; 4/120, –1]

gL gL gL gL

(1) sin  (2) 2sin  (3) 2 tan  (4) tan 
 0 cosθ  0 cosθ 0 0

9. A rectangular loop of sides 10 cm and 5 cm carrying a current I of 12 A is placed in different orientations

as shown in the figures below : [JEE (Main) 2015 ; 4/120, –1]

(a) (b) (c) (d)

If there is a uniform magnetic field of 0.3 T in the positive z direction, in which orientations the loop
would be in (i) stable equilibrium and (ii) unstable equilibrium ?
(1) (a) and (b), respectively (2) (a) and (c), respectively
(3) (b) and (d), respectively (4) (b) and (c), respectively

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10. Two identical wires A and B, each of length l, carry the same current l. Wire A is bent into a circle of
radius R and wire B is bent to form a square of side 'a'. If BA and BB are the values of magnetic field at the
BA
centres of the circle and square respectively, then the ratio is : [JEE (Main) 2016 ; 4/120, –1]
BB

2 2 2 2
(1) (2) (3) (4)
16 2 16 8 2 8
11. Hysteresis loops for two magnetic materials A and B are given below : [JEE (Main) 2016 ; 4/120, –1]
B B

H H

(A) (B)

These materials are used to make magnets for electric generators, transformer core and electromagnet
core. Then it is proper to use :
(1) A for electromagnets and B for electric generators.
(2) A for transformers and B for electric generators.
(3) B for electromagnets and transformers.
(4) A for electric generators and transformers.
12. A magnetic needle of magnetic moment 6.7 × 10–2 Am2 and moment of inertia 7.5 × 10–6 kg m2 is
performing simple harmonic oscillations in a magnetic field of 0.01 T. Time taken for 10 complete
oscillations is : [JEE (Main) 2017 ; 4/120, –1]
(1) 8.76 s (2) 6.65 s (3) 8.89 s (4) 6.98 s
13. An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of
radii re, rp, r respectively in uniform magnetic field B. The relation between re, rp, r is :
[JEE (Main) 2018; 4/120, –1]
(1) re < rp< r (2) re < r< rp (3) re > rp = r (4) re < rp = r
14. The dipole moment of a circular loop carrying a current , is m and the magnetic field at the centre of the
loop is B1. When the dipole moment is doubled by keeping the current constant, the magnetic field at the
B
centre of loop is B2. The ratio 1 is : [JEE (Main) 2018; 4/120, –1]
B2
1
(1) 2 (2) (3) 2 (4) 3
2
15. A current loop, having two circular arcs joined by two radical lines is shown in the figure. It carries a
current of 10 A. The magnetic field at point O will be close to: [JEE (Main) 2019, 9 January, Shift-1]

(1) 1.0 × 10–7 T (2) 1.5 × 10–7 T (3) 1.5 × 10–5 T (4) 1.0 × 10–5 T

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 P-114
16. A bar magnet is demagnetized by inserting it inside a solenoid of length 0.2 m, 100 turns, and carrying a
current of 5.2 A. The coercivity of the bas magnet is: [JEE (Main) 2019, 9 January, Shift-1]
(1) 285 A/m (2) 2600 A/m (3) 520 A/m (4) 1200 A/m
17. In an experiment, electrons are accelerated, from rest, by applying a voltage of 500 V. Calculate the radius
of the path if a magnetic field 100 mT is then applied. [Charge of the electron = 1.6 × 10–19 C Mass of the
electron = 9.1 × 10–31 kg] [JEE (Main) 2019, 11 January, Shift-1]
(1) 7.5 × 10–3 m (2) 7.5 × 10–2 m (3) 7.5 m (4) 7.5 × 10–4 m
18. A paramagnetic substance in the form of a cube with sides 1 cm has a magnetic dipole moment of
20 × 10–6 J/T when a magnetic intensity of 60 × 103 A/m is applied. Its magnetic susceptibility is :
[JEE (Main) 2019, 11 January, Shift-2]
(1) 3.3 × 10–2 (2) 4.3 × 10–2 (3) 2.3 × 10–2 (4) 3.3 × 10–4
19. A proton and an α-particle (with their masses in the ratio of 1 : 4 and charges in the ratio 1 : 2) are
accelerated from rest through a potential difference V. If a uniform magnetic field (B) is up perpendicular
to their velocities, the ratio of the radii rp : rα of the circular paths described by them will be:
[JEE (Main) 2019, 12 January, Shift-1]
(1) 1: 2 (2) 1:2 (3) 1:3 (4) 1: 3
20. A circular coil having N turns and radius r carries a current I. It is held in the XZ plane in a magnetic field
ˆ The torque on the coil due to the magnetic field is :
Bi. [JEE (Main) 2019, 8 April, Shift-1]

Br 2 I Br 2 I
(1) (2) Br 2 I N (3) (4) Zero
N N
21. A moving coil galvanometer has a coil with 175 turns and area 1 cm2. It uses a torsion band of torsion
constant 10–6 N-m/rad. The coil is placed in a magnetic field B parallel to its plane. The coils deflects by
1º for a current of 1mA. The value of B (in Tesla) is approximately:
[JEE (Main) 2019, 9 April, Shift-2]
(1) 10–4 (2) 10–2 (3) 10–1 (4) 10–3
22. The magnitude of the magnetic field at the center of an equilateral triangular loop of side 1 m which is
carrying a current of 10 A is : [Take 0  4  107 NA 2 ] [JEE (Main) 2019, 10 April, Shift-2]
(1) 18 T (2) 9 T (3) 3 T (4) 1 T
23. A thin ring of 10 cm radius carries a uniformly distributed charge. The ring rotates at a constant angular
speed of 40  rad s–1 about its axis, perpendicular to its plane. If the magnetic field at its centre is
3.8 × 10–9 T, then the charge carried by the ring is close to (0  4  107 N / A 2 ).
[JEE (Main) 2019, 12 April, Shift-1]
(1) 2 × 10–6 C (2) 3 × 10–5 C (3) 4 × 10–5 C (4) 7 × 10–6 C
24. Consider a circular coil of wire carrying constant current I, forming a magnetic dipole. The magnetic flux
through an infinite plane that contains the circular coil and excluding the circular coil area is given by i .
The magnetic flux through the area of the circular coil area is given by 0 . Which of the following option
is correct? [JEE (Main) 2020, 7 January, Shift-1]
(1) i   0 (2) i   0 (3) i  0 (4) i   0

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25. The figure gives experimentally measured B vs. H variation in a ferromagnetic material. The retentivity,
co-ercivity and saturation, respectively, of the material are: [JEE (Main) 2020, 7 January, Shift-2]

(1) 50 A/m, 1T, 1.5 T (2) 1.5 T, 50 A/m, 1T

(3) 1T, 50 A/m, 1.5 T (4) 50 A/m, 1.5 T, 1T
26. A long, straight wire of radius a carries a current distributed uniformly over its cross-section. The ratio of
a
the magnetic fields due to the wire at distance and 2a, respectively from the axis of the wire is :
3
[JEE (Main) 2020, 9 January, Shift-1]
3 2 1 4
(1) (2) (3) (4)
5 3 2 3
27. A beam of protons with speed 4 × 105 ms–1 enters a uniform magnetic field of 0.3 T at an angle of 60º to
the magnetic field. The pitch of the resulting helical path of protons is close to (Mass of the
proton = 1.67 × 10–27 kg, charge of the proton = 1.69 × 10–19 C) [JEE (Main) 2020, 2 Sept., Shift-1]
(1) 2 cm (2) 12 cm (3) 5 cm (4) 4 cm
6 –1
28. An electron is moving along + × direction with a velocity 6 × 10 ms . It enters a region of uniform
electric field of 300 V/cm pointing along +y direction. The magnitude and direction of the magnetic field
set up in this region such that the electron keeps moving along the x direction will be
[JEE (Main) 2020, 6 Sept., Shift-1]
–3 –3
(1) 5 × 10 T, along +z direction (2) 5 × 10 T, along –z direction
–4
(3) 3 × 10 T, along +z direction (4) 3 × 10–4T, along –z direction
29. A soft ferromagnetic material is placed in an external magnetic field. The magnetic domains :
(1) decrease in size and changes orientation. [JEE (Main) 2021, 24 Feb, Shift-2]
(2) may increase or decrease in size and change its orientation.
(3) increase in size but no change in orientation.
(4) have no relation with external magnetic field.
30. Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : When a rod lying freely is heated, no thermal stress is developed in it. Reason R : On
heating, the length of the rod increases. In the light of the above statements, choose the correct answer
from the options given below : [JEE (Main) 2021, 25 Feb, Shift-1]
(1) is true but is false
(2) Both A and R are true and R is the correct explanation of R
(3) Both A and R are true but R is NOT the correct explanation of R
(4) A is false but R is true

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31. Magnetic fields at two points on the axis of a circular coil at a distance of 0.05 m and 02 m from the
centre are in the ratio 8: 1. The radius of coil is [JEE (Main) 2021, 25 Feb, Shift-1]
(1) 0.15 m (2) 0.2 m (3) 0.1 m (4) 1.0 m
32. In a ferromagnetic material, below the curie temperature, a domain is defined as:
[JEE (Main) 2021, 25 Feb, Shift-2]
(1) a macroscopic region with consecutive magnetic diploes oriented in opposite direction.
(2) a macroscopic region with zero magnetization.
(3) a macroscopic region with saturation magnetization.
(4) a macroscopic region with randomly oriented magnetic dipoles.
  
33. A charge Q is moving dI distance in the magnetic field B . Find the value of work done by B .
[JEE (Main) 2021, 16 March, Shift-2]
(1) 1 (2) Infinite (3) Zero (4) –1
34. A hairpin like shape as shown in figure is made by bending a long current carrying wire. What is the
magnitude of a magnetic field at point which lies on the centre of the semicircle?
[JEE (Main) 2021, 17 March, Shift-2]

0 I 0 I 0 I 0 I
(1)  2   (2)  2   (3)  2   (4)  2  
4r 4r 2r 2r
35. A loop of flexible wire of irregular shape carrying current is placed in an external magnetic field. Identify
the effect of the field on the wire. [JEE (Main) 2021, 18 March, Shift-1]
(1) Loop assumes circular shape with its plane normal to the field.
(2) Loop assumes circular shape with its plane parallel to the field.
(3) Wire gets stretched to become straight.
(4) Shape of the loop remains unchanged.
36. The magnetic field at the centre of a circular coil of radius r, due to current I flowing through it, is B. The
r
magnetic filed at a point along the axis at a distance from the centre is:
2
[JEE (Main) 2022, 24 June, Shift-1]
3 3
 2   2 
(1) B/2 (2) 2B (3)   B (4)   B
 5  3
37. A long solenoid carrying a current produces a magnetic field B along its axis. If the current is doubled and
the number of turns per cm is halved, the new value of magnetic field will be equal
[JEE (Main) 2022, 25 June, Shift-2]
B
(1) B (2) 2B (3) 4B (4)
2

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38. A metal surface is illuminated by a radiation of wavelength 4500 Å. The ejected photo-electron enters a
constant magnetic field of 2 mT making an angle of 90º with the magnetic field. If it starts revolving in a
circular path of radius 2 mm, the work function of the metal is approximately:
[JEE (Main) 2022, 26 June, Shift-2]
(1) 1.36 eV (2) 1.69 eV (3) 2.78 eV (4) 2.23 eV
39. A charged particle moves along circular path in a uniform magnetic field in a cyclotron. The kinetic
energy of the charged particle increases to 4 times its initial value. What will be the ratio of new radius to
the original radius of circular path of the charged particle : [JEE (Main) 2022, 29 June, Shift-1]
(1) 1:1 (2) 1:2 (3) 2:1 (4) 1:4
40. Two charged particles, having same kinetic energy, are allowed to pass through a uniform magnetic field
perpendicular to the direction of motion. If the ratio of radii of their circular paths is 6 : 5 and their
respective masses ratio is 9 : 4. Then, the ratio of their charges will be :
[JEE (Main) 2022, 25 July, Shift-1]
(1) 8:5 (2) 5:4 (3) 5:3 (4) 8:7
41. BX and BY are the magnetic field at the centre of two coils of two coils X and Y respectively, each
carrying equal current. If coil X has 200 turns and 20 cm radius and coil Y has 400 turns and 20 cm
radius, the ratio of BX and BY is [JEE (Main) 2022, 26 July, Shift-1]
(1) 1:1 (2) 1:2 (3) 2:1 (4) 4:1
42. A triangle shaped wire carrying 10A current is placed in a uniform magnetic field of 0.5 T, as shown in
figure. The magnetic force on segment CD is (Given BC = CD = BD = 5 cm).
[JEE (Main) 2022, 28 July, Shift-2]

(1) 0.126 N (2) 0.312 N (3) 0.216 N (4) 0.245 N

43. A closely wounded circular coil of radius 5 cm produces a magnetic field of 37.88 × 10–4 T at its center.
The current through the coil is _____________ A. [Given, number of turns in the coil is 100 and  = 3.14]
[JEE (Main) 2022, 29 July, Shift-1]
44. Two long straight wires P and Q carrying equal current 10A each were kept parallel to each other at 5 cm
distance. Magnitude of magnetic force experienced by 10 cm length of wire P is F1. If distance between
wires is halved and currents on them are doubled, force F2 on 10 cm length of wire P will be:
[JEE (Main) 2023, 24 Jan, Shift-1]
(1) 8 F1 (2) 10F1 (3) F1/8 (4) F1/10
–1
45. A long solenoid is formed by winding 70 turns cm . If 2.0 A current flows, then the magnetic field
produced inside the solenoid is ________(0 = 4 × 10–7 TmA–1) [JEE (Main) 2023, 24 Jan, Shift-2]
(1) 1232 × 10–4 T (2) 176 × 10–4 T (3) 352 × 10–4 T (4) 88 × 10–4 T
46. A circular loop of radius r is carrying current I A. The ratio of magnetic field at the centre of circular loop
and at a distance r from the center of the loop on its axis is: [JEE (Main) 2023, 24 Jan, Shift-1]
(1) 1: 3 2 (2) 3 2:2 (3) 2 2:1 (4) 1: 2

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47. A single turn current loop in the shape of a right angle triangle with sides 5 cm, 12 cm, 13 cm is carrying a

current of 2A. The loop is in a uniform magnetic field of magnitude 0.75 T whose direction is parallel to

the current in the 13 cm side of the loop. The magnitude of the magnetic force on the 5 cm side will be

x
N. The value of x is [JEE (Main) 2023, 24 Jan, Shift-2]
130

48. For a moving coil galvanometer, the deflection in the coil is 0.05 rad when a current of 10 mA is passed

through it. If the torisional constant of suspension wire is 4.0 × 10–5 Nm rad–1, the magnetic field is 0.01 T

and the number of turns in the coil is 200, the area of each turn (in cm2) is:

[JEE (Main) 2023, 25 Jan, Shift-2]

(1) 2.0 (2) 1.0 (3) 1.5 (4) 0.5

49. As shown in the figure, a long straight conductor with semicircular are of radius m is carrying current
10
1 = 3A. The magnitude of the magnetic field. at the center O of the arc is:
(The permeability of the vacuum = 4 × 10–7 NA–2) [JEE (Main) 2023, 01 Feb, Shift-2]

(1) 6T (2) 1T (3) 4T (4) 3T

50. A long straight wire of circular cross-section (radius a) is carrying steady current I. The current I is
uniformly distributed across this cross-section. The magnetic field is [JEE (Main) 2023, 06 Apr, Shift-1]
(1) inversely proportional to r in the region r < a and uniform throughout in the region r > a
(2) directly proportional to r in the region r < a and inversely proportional to r in the region r > a
(3) Zero in the region r < a and inversely proportional t r in the region r > a
(4) uniform in the region r < a and inversely proportional to distance r from the axis, in the region r > a
51. Two identical circular wires of radius 20cm and carrying current 2A are placed in perpendicular planes
as shown in figure. The net magnetic field at the centre of the circular wires is _______ × 10–8T.

[JEE (Main) 2023, 06 Apr, Shift-1]

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 P-119
52. A proton with a kinetic energy of 2.0 eV moves into a region of uniform magnetic field of magnitude

 10 –3 T. The angle between the direction of magnetic field and velocity of proton is 60º. The pitch of
2
the helical path taken by the proton is _____ cm. (Take, mass of proton = 1.6×10–27 kg and charge on
proton = 1.6×10–19 C). [JEE (Main) 2023, 06 Apr, Shift-2]
53. A charge particle moving in magnetic field B, has the components of velocity along B as well as
perpendicular to B. The path of the charge particle will be [JEE (Main) 2023, 08 Apr, Shift-1]
(1) helical path with the axis perpendicular to the direction of magnetic field B
(2) helical path with the axis along magnetic field B
(3) circular path
(4) straight along the direction of magnetic field B
54. The magnetic intensity at the centre of a long current carrying solenoid is found to be 1.6 × 103 A m–1. If
the number of turns is 8 per cm, then the current flowing through the solenoid is …. A.
[JEE (Main) 2023, 08 Apr, Shift-1]
55. An electron is allowed to move with constant velocity along the axis of current carrying straight solenoid.
[JEE (Main) 2023, 11 Apr, Shift-2]
(A) The electron will experience magnetic force along the axis of the solenoid.
(B) The electron will not experience magnetic force.
(C) The electron will continue to move along the axis of the solenoid.
(D) The electron will follow parabolic path-inside the solenoid.
Choose the correct answer from the option given below:
(1) B, C and D only (2) A and D only (3) B and C only (4) B and E only

___________
________
_____

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ANSWER KEY
EXERCISE - 1
1. (d) 2. (d) 3. (b) 4. (d) 5. (d)
6. (d) 7. (b) 8. (b) 9. (c) 10. (d)
11. (d) 12. (d) 13. (d) 14. (a) 15. (b)
16. (b) 17. (c) 18. (c) 19. (c) 20. (b)
21. (b) 22. (c) 23. (d) 24. (d) 25. (b)
26. (b) 27. (b) 28. (c) 29. (d) 30. (c)
31. (d) 32. (d) 33. (b) 34. (a) 35. (c)
36. (d) 37. (c) 38. (d) 39. (c) 40. (d)
41. (c) 42. (b) 43. (a) 44. (b) 45. (c)
EXERCISE – 2
1. (b) 2. (b) 3. (c) 4. (c) 5. (b)
6. (d) 7. (d) 8. (a) 9. (c) 10. (a)
11. (a) 12. (c) 13. (b) 14. (b) 15. (b)
16. (b) 17. (b) 18. (a) 19. (d) 20. (a)
21. (a) 22. (b 23. (a) 24. (a) 25. (d)
26. (a) 27. (a) 28. (b) 29. (b) 30. (a)
31. (b) 32. (c) 33. (b) 34. (c) 35. (a)
36. (d) 37. (b) 38. (d) 39. (b) 40. (c)
41. (d) 42. (a) 43. (c) 44. (a) 45. (d)
46. (d) 47. (b) 48. (d)
EXERCISE - 3
1. (a, b) 2. (a, b) 3. (a, b) 4. (a,b,c,d) 5. (a,b)
6. (a,b,c) 7. (b, d) 8. (a,b,c) 9. (a,b,c) 10. (b,c,d)
11. (a,c) 12. (a,b,c) 13. (a,d) 14. (a,b,c) 15. (b,d)
16. (b, c) 17. (b, d) 18. (a, b, c) 19. (a, c) 20. (b, d)
21. (a, c) 22. (b, d) 23. (b,c,d) 24. (b, d)
EXERCISE – 4
Section – 1
Numerical Value Type
1. 4 2. 2 3. 1 4. 9 5. 6
6. 2 7. 2 8. 3 9. 3

Section – 2
Column Match Type
1. i.  b,d; ii.  a,d; iii  c,d; iv.  c,d
2. i.  b,c; ii.  b,d; iii  a; iv.  c
3. i.  a,b,c; ii.  a,b,c,d; iii  c; iv.  a,b,c,d
Section – 3
Paragraph Type
1. (b) 2. (a) 3. (c) 4. (c) 5. (a)
6. (a) 7. (d) 8. (a) 9. (b) 10. (d)
11. (b) 12. (c) 13. (b) 14. (b) 15. (d)
16. (a) 17. (b) 18. (d)

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 P-121
EXERCISE – 5
  102 N
16(i  2j  4k)
2qEz
1. v 2. 3. 5 kN towards west direction.
m
 v  ( 40i  30j) m / s

4. 4.8 × 10–12 (k)N 5. 6. x=2
mv 2
7. N  mg sin   qvB 8. (a) 3.2 × 10–15 N (b) 2.1 × 10–4 m (c) 1.32 × 107 s
R
2mV 8
9. 0.212 m 10.   30o 11. R 12. 0.10 m, m
qB2 30
2mvsin   qB  
13. r sin   14. (a) 40.0 N (b) 0 N 15. 14A
qB  2m vcos  
2.2
16. 0.12 17. (a) 1.13 N (b) N 18. iB0  19. (a) 0 (b) 8 m
3
mg
20. i 21. 0.41 A left to right 22.  2 Ij
B(cos    sin )
23. (a) 0.3 Am2 (b) The normal to the coil is at 30o is the field.
T 0 I 4 0 ia 2
24. (180o - 2  ) (b) t  (  2) 25. ( 3  1) 26.
2 8R (4x 2  a 2 ) 4x 2  2a 2
0i  0 ii 0 0 1
27. 28. ln 2 29. (a) r (b) R 4
2 R 2 r0 2 4
q 0 Iv0 
30. (a) 40 nN (b) 9.6 nJ 31. (a)  k (b) 4BIRi
4R
EXERCISE – 6
Section – I : JEE (Advanced) Questions Previous Years
1. (C,D) 2. 5 3. (B) 4. (D) 5. (A,C)
6. (A,D) 7. 3 8. (C) 9. (B) 10. (A,B,C)
11. (A,D) 12. (A,C) 13. (C) 14. (A) 15. (A)
16. (A) 17. (C,D) 18. (A,B,D) 19. 2.00 20. (A,C)
22. (4) 21. (A,B) 22. (A) 23. (C) 24. (3.98 – 4.02)

Section – II : JEE (Main) Questions Previous Years

1. (1) 2. (1) 3. (3) 4. (1) 5. (2) 6. (2) 7. (4)
8. (2) 9. (3) 10. (3) 11. (3) 12. (2) 13. (4) 14. (1)
15. (4) 16. (2) 17. (4) 18. (4) 19. (1) 20. (2) 21. (4)
22. (1) 23. (2) 24. (3) 25. (3) 26. (2) 27. (4) 28. (1)
29. (2) 30. (3) 31. (3) 32. (4) 33. (3) 34. (2) 35. (1)
36. (3) 37. (1) 38. (1) 39. (3) 40. (2) 41. (2) 42. (3)
43. (3) 44. (1) 45. (2) 46. (3) 47. 9 48. (2) 49. (4)
50. (2) 51. 628 52. 40 53. (2) 54. 2 55. (3)

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 CHEMISTRY

SOLUTIONS
AND COLLIGATIVE
PROPERTIES
 C-1
Exercise - 1
Concept Building Questions

Raoult’s law

1. At 250C, the vapour pressure of methyl alcohol is 96.0 torr. What is the mole fraction of CH3OH in a
solution in which the (partial) vapour pressure of CH3OH is 23.0 torr at 250C?

2. The vapour pressure of pure liquid solvent A is 0.80 atm. When a nonvolatile substance B is added to the
solvent its vapour pressure drops to 0.60 atm. What is the mole fraction of component B in the solution?

3. The vapour pressure of pure water at 260C is 25.21 torr. What is the vapour pressure of a solution which
contains 20.0g glucose, C6H12O6, In 70g water?

4. The vapour pressure of pure water at 250C is 23.76 torr. The vapour pressure of a solution containing
5.40g of a nonvolatile substance in 90.0 g water is 23.32 torr. Compute the molecular weight of the solute.

Raoult’s law in combination with Dalton’s law of P.P. and V.P. lowering

5. The vapour pressure of ethanol and methanol are 44.5 mm and 88.7 mm Hg respectively. An ideal
solution is prepared at the same temperature by mixing 60g of ethanol with 40g of methanol. Calculate
total vapour pressure of the solution.

6. Calculate the mole fraction of toluene in the vapour phase which is in equilibrium with a solution of
benzene and toluene having a mole fraction of toluene 0.50. The vapour pressure of pure benzene is 119
torr, that of toluene is 37 torr at the same temperature.

7. At 900C, the vapour pressure of toluene is 400 torr and that of o-xylene is 150 torr. What is the
composition of the liquid mixture that boils at 900C, when the pressure is 0.50 atm? What is the
composition of vapour produced?

8. Two liquid A and B form an ideal solution at temperature T. When the total vapour pressure above the
solution is 400 torr, the mole fraction of A in the vapour phase is 0.40 and in the liquid phase 0.75. What
are the pressure of pure A and pure B at temperature T?

9. Calculate the relative lowering in vapour pressure if 100g of a nonvolatile solute (mol.wt. 100) are
dissolved in 432 g water.

10. What weight of the non-volatile solute, urea needs to be dissolved in 100g of water, in order to decrease
the vapour pressure of water by 25%? What will be the molality of the solution?

11. The vapour pressure of pure benzene at 250C is 639. 7 mm of Hg and the vapour pressure of a solution of
a solute in C6H6 at the same temperature is 631.7 mm of Hg. Calculate molality of solution.

12. The vapour pressure of pure benzene at a certain temperature is 640 mm of Hg. A nonvolatile
nonelectrolyte solid weighing 2.175g is added to 39.0 of benzene. The vapour pressure of the solution is
600 mm of Hg. What is molecular weight of solid substance?

13. Benzene and toluene form two ideal solution A and B at 313K, Solution A (total pressure PA) contains
equal mole of toluene and benzene. Solution B contains equal masses of both (total pressure PB) . The
vapour pressure of benzene and toluene are 160 and 60mm Hg resepectively at 313K. Calculate the value
of PA/PB.

Chemistry – Solutions & Colligative Properties Toll Free Number : 1800 103 9888
 C-2
Boiling point elevation and freezing point depression

14. When 10.6g of a nonvolatile substance is dissolved in 740g of ether, its boiling point is raised by 0.2840C.
What is the molecular weight of the substance? Molal boiling point elevation constant for ether is
2.110C.kg/mol.

15. A solution containing 3.24g of a nonvolatile nonelectrolyte and 200 g of water boils at 100.130ºC at
1 atm. What is the molecular weight of the solute? (Kb for water 0.5130C/m)

16. The molecular weight of an organic compound is 58.0 g/mol. Compute the boiling point of a solution
containing 24.0 g of the solute and 600 g of water, when the barometric pressure is such that pure water
boils at 99.725ºC.

17. An aqueous solution of a nonvolatile solute boils at 100.17ºC. At what temperature will this solution
freeze? [Kf for water 1.86º C/m].

18. Pure benzene freeze at 5.45ºC. A solution containing 7.24 g of C2H2Cl4 in 115.3 g of benzene was
observed to freeze at 3.55ºC. What is the molal freezing point depression constant of benzene?

19. The freezing point of a solution containing 2.40 g of a compound in 60.0g of benzene is 0.10ºC lower than
that of pure benzene. What is the molecular weight of the compound? (Kf is 5.12ºC/m for benzene)

20. Calculate the molal elevation constant, Kb for water and the boiling point of 0.1 molal urea solution.
Latent heat of vaporistion of water is 9.72 kcal mol–1 at 373.15K.

21. Calculate the amount of ice that will separate out on cooling a solution containing 50g of ethylene glycol
in 200 g water to –9.3ºC. (Kf for water = 1.86 K mol–1 kg).

22. A solution of 0.643 g of an organic compound in 50ml of benzene (density; 0.879 g/ml) lowers its
freezing point from 5.51ºC to 5.03ºC. If Kf for benzene is 5.12 K/m calculate the molecular weight of the
compound.

Osmotic pressure

23. Find the freezing point of glucose solution whose osmotic pressure at 25ºC is found to be 30 atm.
Kf(water) =1.86 kg. mol–1.K.

24. At 300 K, two solutions of glucose in water of concentration 0.01 M and 0.001 M are separated by
semipermeable membrane. Pressure needs to be applied on which solution, to prevent osmosis? Calculate
the magnitude of this applied pressure.

25. At 10ºC, the osmotic pressure of urea solution is 500 mm. The solution is diluted and the temperature is
raised to 25ºC, when the osmotic pressure is found to be 105.3 mm. Determine extent of dilution.

26. The osmotic pressure of blood is 7.65 atm at 37º C. How much glucose should be used per L for an
intravenous injection that to have the same osmotic pressure as blood?

27. A 250 mL water solution containing 48.0 g of sucrose, C12H22O11, at 300 K is separated from pure water
by means of a semipermeable membrane. What pressure must be applied above the solution in order to
just prevent osmosis?

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 C-3
28. A 5% solution (w/v) of cane-sugar (Mol. Weight = 342) is isotonic with 0.877% (w/v) of urea solution.
Find molecular weight of urea.

29. 10gm of solute A and 20gm of solute B are both dissolved in 500 ml water. The solution has the same
osmotic pressure as 6.67 gm of A and 30 gm of B dissolved in the same amount of water at the same
temperature. What is the ratio of molar masses of A and B?

Van’t Hoff factor & colligative properties

30. A storage battery contains a solution of H2SO4 38% by weight. What will be the Van’t Hoff factor if the 
Tf(experiment) in 29.08 K [Given Kf = 1.86 K mol–1 Kg]

31. A certain mass of a substance, when dissolved in 100 g C6H6, lowers the freezing point by 1.280C. The
same mass of solute dissolved in 100g water lowers the freezing point by 1.400C. If the substance has
normal molecular weight in benzene and is completely ionized in water, into how many ions does it
dissociate in water? Kf for H2O and C6H6 are 1.86 and 5.12K kg mol–1.

32. 2.0g of benzoic acid dissolved in 25.0g of benzene shows a depression in freezing point equal to 1.62K.
Molal depression constant (Kf) of benezene is 4.9 K. kg.mol–1. What is the percentage association of the
acid?

33. A decimolar solution of potassium ferrocyanide is 50% dissociated at 300K. Calculate the osmotic
pressure of the solution. (R = 8.314 JK–1 mol–1)

34. The freezing point of a solution containing 0.2 g of acetic acid in 20.0g of benzene is lowered by 0.45ºC.
Calculate the degree of association of acetic acid in benzene. (Kf for benzene = 5.12 K mol–1 kg)

35. 0.85% aqueous solution of NaNO3 is apparently 90% dissociated at 27º C. Calculate its osmotic pressure.
(R = 0.082 ltr atm K–1 mol–1)

36. A 1.2% solution (w/v) of NaCl is isotonic with 7.2% solution (w/v) of glucose. Calculate degree of
ionization and Van’t Hoff factor of NaCl.

37. If relative decrease in V.P. is 0.4 for solution containing 1 mol NaCl in 3 mol of H2O. Calculate %
ionization of NaCl.

38. 2.56 gm of sulphur in 100 gm of CS2 has depression in F.P. of 0.010ºC, Kf =0.1ºC m–1. Calculate
atomicity of sulphur in CS2.

Henry’s Law:

39. 1 kg of water under a nitrogen pressure of 1 atmosphere dissolves 0.02 gm of nitrogen at 293K. Calculate
Henry’s law constant.

40. Calculate the amount of oxygen at 0.20 atm dissolved in 1 kg of water at 293 K. The Henry’s law constant
for oxygen is 4.58 x 104 atmosphere at 293K.

Chemistry – Solutions & Colligative Properties Toll Free Number : 1800 103 9888
 C-4
Exercise - 2
Single choice correct with multiple options

1. For an ideal binary liquid solution with PAo  PBo , which relation between XA (mole fraction of A in liquid
phase) and YA (mole fraction of A in vapour phase) is correct?
YA X A YA X A
(A) YA  YB (B) X A  X B (C)  (D) 
YB X B YB X B

2. Mole fraction of A vapours above the solution in mixture of A and B (XA =0.4) will be
[Given : PAo  100mm Hg and PBo  200 mm Hg ]
(A) 0.4 (B) 0.8 (C) 0.25 (D) none of these

3. The exact mathematical expression of Raoult’s law is

P 0  Ps n P 0  Ps N P 0  Ps n P 0  Ps
(A) 0
 (B)  (C)  (D) nxN
P N P0 n Ps N P0

4. A mixture contains 1 mole of volatile liquid A (PAo  100 mm Hg) and 3 moles of volatile liquid B
(PAo  80 mm Hg) . If solution behaves ideally, the total vapour pressure of the distillate is
(A) 85 mm Hg (B) 85.88 mm Hg (C) 90 mm Hg. (D) 92 mm Hg

5. The vapour pressure of a solvent decreased by 10 mm of Hg when a non-volatile solute was added to the
solvent. The mole fraction of solute in solution is 0.2, what would be mole fraction of the solvent if
decrease in vapour pressure is 20 mm of Hg.
(A) 0.2 (B) 0.4 (C) 0.6 (D) 0.8

6. Two liquids A & B form an ideal solution. What is the vapour pressure of solution containing 2 moles of
A and 3 moles of B at 300 K? [Given: At 300 K, Vapour pr. of pure liquid A ( PAo ) =100 torr, Vapour
pressure of pure liquid B ( PBo ) = 300 torr]
(A) 200 torr (B) 140 torr (C) 180 torr (D) None of these

7. The solubility of common salt is 36.0 gm in 100 gm of water at 20º C. If systems I, II and III contain 20.0,
18.0 and 15.0 g of the salt added to 50.0 gm of water in each case, the vapour pressure would be in the
order.
(A) I< II< III (B) I > II > III (C) I = II > III (D) I = II< III

8. 18 g of glucose (C6H12O6) is added to 178.2 g of water. The vapour pressure of water for this aqueous
solution at 100ºC is
(A) 7.60 torr (B) 76.00 torr (C) 752.40 torr (D) 759.00 torr

9. Benzene and toluene form nearly ideal solutions, At 20º C, the vapour pressure of benzene is 75 torr and
that of toluence is 22 torr. The partial vapour pressure of benzene at 20º C for a solution containing 78g of
benzene and 46 g of toluene in torr is:
(A) 50 (B) 25 (C) 37.5 (D) 53.5

10. At 300 K, the vapour pressure of an ideal solution containing 3 mole of A and 2 mole of B is 600 torr. At
the same temperature, if 1.5 mole of A & 0.5 mole of C (non-volatile) are added to this solution the
vapour pressure of solution increases by 30 torr. What is the value of PBo ?
(A) 940 (B) 405 (C) 90 (D) None of these

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 C-5
11. Which of the following plots represents an ideal binary mixture?
(A) Plot of Ptotal v/s 1/XB is linear (XB = mole fraction of ‘B’ in liquid phase).
(B) Plot of Ptotal v/s YA is linear (YB = mole fraction of ‘A’ in vapour phase)
1
(C) Plot of v/s YA is linear
Ptotal
1
(D) Plot of v/s YB is non linear
Ptotal

12. The vapour pressure of a solution of a non-volatile electrolyte B in a solvent A is 95% of the vapour
pressure of the solvent at the same temperature. If the molecular weight of the solvent is 0.3 times the
molecular weight of solute, the weight ratio of the solvent and solute are
(A) 0.15 (B) 5.7 (C) 0.2 (D) 4.0

13. At a given temperature, total vapour pressure in Torr of a mixture of volatile components A and B is given
by P Total= 120 – 75 XB hence, vapour pressure of pure A and B respectively (In Torr) are
(A) 120, 75 (B) 120, 195 (C) 120, 45 (D) 75, 45

14. Which of the following aqueous solution will show maximum vapour pressure at 300 K?
(A) 1 M NaCl (B) 1 M CaCl2 (C) 1 M AlCl3 (D) 1 MC12 H22O11

15. The Van’t Hoff factor for a dilute aqueous solution of glucose is
(A) zero (B) 1.0 (C) 1.5 (D) 2.0

16. The correct relationship between the boiling points of very dilute solution oif AlCl3 (T1K) and CaCl2
(T2K) having the same molar concentration is
(A) T1 = T2 (B) T1>T2 (C) T2>T1 (D) T2 < T1

17. A 0.001 molal solution of a complex [MA8] in water has the freezing point of – 0.0054ºC. Assuming
100% ionization of the complex salt and Kf for H2O = 1.86 km–1, write the correct representation for the
complex.
(A) [MA8] (B) [MA7]A (C) [MA6]A2 (D) [MA5]A3

18. Assuming each salt to be 90% dissociated, which of the following will have highest boiling point?
(A) Decimolar Al2 (SO4)3
(B) Decimolar BaCl2
(C) Decimolar Na2SO4
(D) A solution obtained by mixing equal volumes of (B) and (C)

19. Elevation of boiling point of 1 molar aqueous glucose solution (density = 1.2 g/ml) is
(A) Kb (B) 1.20 Kb (C) 1.02 Kb (D) 0.98Kb

20. What will be the molecular weight of CaCl2 determined in its aq. Solution experimentally from depression
of freezing point?
(A) 111 (B) < 111 (C) > 111 (D) data insufficient

21. 1.0 molal aqueous solution of an electrolyte A2 B3 is 60% ionized. The boiling point of the solution at
1 atm is (K b(H2O) = 0.52 K kg mol–1)
(A) 274.76K (B) 377K (C) 376.4K (D) 374.76K

22. The Freezing point depression of a 0.1 M aq. Solution of weak acid (HX) is – 0.200C. What is the value
of equilibrium constant for the reaction?
 H  (aq)  X   aq 
HX(aq) 
[Given: Kf for water = 1.8 kg mol–1 K & Molality = Molarity]
(A) 1.46 x 10–4 (B) 1.35 x 10–3 (C) 1.21 x 10 –2 (D) 1.35 x 10–4

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 C-6
23. The van’t Hoff factor for 0.1 M Ba (NO3)2 solution is 2.74. The degree of dissociation is
(A) 91.3% (B) 87% (C) 100% (D) 74%
24. The vapour pressure of an aqueous solution is found to be 750 torr at certain temperature ‘T’ is the
temperature at which pure water boils under atmospheric pressure and same solution show elevation in
boiling point  Tb = 1.04K, find the atmosphemic pressure (Kb = 0.52 K kg mol–1 )
(A) 777 (B) 779 (C) 782 (D) 746
25. A 0.2 molal aqueous solution of a weak acid (HX) is 20 percent ionized. The freezing point of this
solution is (Given Kf = 1.860C kg mol–1 for water):
(A) –0.440C (B) –0.900C (C) –0.310C (D) –0.530C

26. In a 0.2 m aqueous solution of a weak acid HX, the degree of ionization is 0.3 Taking Kf for water as
1.85, The freezing point of the solution will be nearest to:
(A) –0.4800C (B) –0.3600C (C) –0.2600C (D) +0.4800C

27. Freezing point of an aqueous solution is –0.186ºC. Elevation of boiling point of the same solution is
Kb = 0.5120C m–1, Kf = 1.860C m–1. The increase In boiling point is:
(A) 0.1860C (B) 0.05120C (C) 0.00920C (D) 0.23720C

28. Which one of following aqueous solutions will exhibit highest boiling point?
(A) 0.01 M Na2SO4 (B) 0.01 M KNO3 (C) 0.015 M urea (D) 0.015 M glucose

29. If  is the degree of dissociation of Na2SO4, the vant Hoffs factor (i) used for calculating the molecular
mass is
(A) 1 +  (B) 1–  (C) 1+2  (D) 1–2 

30. Which has the highest boiling point?

(A) 0.1 M Na2SO4 (B) 0.1 M C6H12O6 (glucose)
(C) 0.1 M MgCl2 (D) 0.1 M Al (NO3)3
31. Which combination of (I) vapour pressure, (II) intermolecular forces and (III) Hvap (latent heat of
vaporisation) is matched correctly
I II III
[A] low strong small
[B] high strong large
[C] low weak large
[D] high weak small

32. The molecular weight of benzoic acid in benzene as determined by depression in freezing point of the
solution is due to:
(A) ionization of benzoic acid (B) dimerization of benzoic acid
(C) trimerization of benzoic acid (D) solvation of benzoic acid

33. If liquids A and B form an ideal solution, the

(A) enthalpy of mixing is zero
(B) entropy of mixing is zero
(C) free energy of mixing is zero
(D) free energy as well as the entropy of mixing are each zero

34. Equimolar solutions in the same solvent have.

(A) same boiling point but different freezing point (B) same freezing point but different boiling point
(C) same boiling and same freezing points (D) different boiling and freezing points.

35. 1 mol each of following solutes are taken in 9 mol water,

(i) NaCl (ii) K2SO4 (iii) Na3PO4 (iv) glucose
Relative decrease in vapour pressure will be in order:
(A) i < ii < iii < iv (B) iv < iii < ii > i (C) iv < i < ii < iii (D) equal

Chemistry – Solutions & Colligative Properties Toll Free Number : 1800 103 9888
 C-7
36. In the depression of freezing point experiment, it is found that
(i) The vapour pressure of the solution is less than that of pure solvent.
(ii) The vapour pressure of the solution is more than that of pure solvent.
(iii) Only solute molecules solidify at the freezing point.
(iv) Only solvent molecules solidify at the freezing point.
(A) I, II (B) II, III (C) I, IV (D) I, II, III

37. There are some of the characteristics of the supersaturated solution

(i) Equilibrium exists between solutions and solid solute
(ii) If a crystal of solute is added to supersaturated solution, crystallization occurs
(iii) Supersaturated solutions contain more solute than they should at a particular temperature correct
characteristics of supersaturated solutions are;
(A) I, II, III (B) II, III (C) I, III (D) I, II

38. Select correct statements:

(A) The fundamental cause of colligative properties is the higher entropy of the solution relative to that
of the pure solvent
(B) The freezing point of hydrofluoride solution is larger than that of equimolal hydrogen chloride
solution
(C) 1 M glucose solution and 0.5 M NaCl solution are Isotonic at a given temperature
(D) All are correct statements

39. In cold countries, ethylene glycol is added to water in the radiators of cars during winters. It results in
(A) lowering of boiling point (B) reduced viscosity
(C) reduced specific heat (D) lowering of freezing point

40. A liquid mixture having composition corresponding to point z in the figure shown is subjected to
distillation at constant pressure. Which of the following statement is correct about the process

(A) The composition of distillate differs from the mixture

(B) The boiling point goes on changing
(C) The mixture has highest vapour pressure than for any other composition.
(D) Composition of an azeotrope alters on changing the external pressure.

Exercise - 3
Multiple choice correct with multiple options

1. In a mixture A and B components show negative deviation as:

(A)  Vmix > 0
(B)  Vmix < 0
(C) A – B interaction is stronger than A–A and B–B interaction
(D) None of the above reasons in correct.

2. Acetone and carbon disulphide form binary liquid solution showing positive deviation from Raoult law.
The normal boiling point (Tb) for pure acetone is less than that of pure CS2. Pick out the incorrect
statements among the following
(A) Boiling temperature of mixture is always less than boiling temperature of acetone
(B) Boiling temperature of Azeotorpic mixture is always less than boiling temperature of pure CS2
(C) When a small amount of CS2 (Less volatile component) is added to excess of acetone boiling point of
resulting mixture increases
(D) A mixture of CS2 and CH3COCH3 can be completely separated by simple fractional distillation.

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3. Which one of the following statements are true?
(A) Raoult’s law states that the vapour pressure of a component over a solution is proportional to its mole
fraction.
(B) The osmotic pressure (  ) of a solution is given by the equation  = MRT, where M is the molarity
of the solution.
(C) The correct order of osmotic pressure for 0.01 M aqueous solution of each compound is
BaCl2 > KCl > CH3COOH > sucrose.
(D) Two sucrose solution of same molality prepared in different solvents will have the same freezing
point

4. The vapour pressure of a dilute solution of a solute is influenced by:

(A) Temperature of solution (B) Mole fraction of solute
(C) Mole fraction of solvent (D) Degree of dissociation of solute

5. According to Henry’s law, the partial pressure of gas (Pg) is directly proportional to mole fraction of gas
in dissolved state, i.e., Pgas = KH Xgas where KH is Henry’s constant. Which are correct?
(A) KH is characteristic constant for a given gas solvent system
(B) Higher is the value of KH lower is solubility of gas for a given partial pressure of gas
(C) KH has temperature dependence
(D) KH increase with temperature

6. 1 kg of aqueous solution has 0.40 kg NaOH. Thus, this solution is

(A) 66.67% by weight of water (B) 40.00% by weight of aqueous solution
(C) 16.67 molal (D) 16.67 molar

7. Mole fraction of ethanol in ethanol-water solution is 0.25. Thus, this solution is

(A) 46% by mass of solution (B) 85.2% by mass of water solvent
(C) 54% by mass of solution (D) 46% by mass of solvent

8. Select the correct statement(s).

(A) Gases which form ions in the solvent are soluble in that solvent
(B) Gases which are liquefied easily are more soluble in water than the other gases which are not
liquefied easily
(C) Higher the value of KH at a given pressure, lower is the solubility of the gas in the liquid
(D) Higher the value of KH at a given pressure, larger the solubility of the gas in the liquid

9. Select the correct statement(s).

(A) KH (Henry’s law constant) is a function of nature of gases
(B) Higher the value of KH at a given pressure, the lower is the solubility of the gas in the liquid
(C) KH increases with increases in temperature
(D) KH increases with increases in partial pressure

10. Total vapour pressure of mixture of 1 mole of a volatile component A ( P oA = 100 mm Hg) and 3 moles of
volatile component B( P oB = 60 mm Hg) is 75 mm Hg. For such case,
(A) there is positive devlation from Raoult’s law
(B) there is negative deviation from Raoult’s law
(C) boiling point is lowered
(D) force of attraction between A and B is smaller than that between A and A, and B and B

11. In the depression of freezing point experiment, it is found that the

(A) vapour pressure of the solution is less than that of pure solvent
(B) vapour pressure of the solution is more than that of pure solvent
(C) only solute molecules solidify at the freezing point
(D) only solvent molecules solidify at the freezing point

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12. A solution containing 3.5 g of solute X is 50.0g of water has a volume of 52.5 mL and a freezing point of
–0.86ºC. Thus, (Kf of H2O = 1.86º mol–1 kg)
(A) molality of the solution is 0.46 mol kg–1 (B) molarity of the solution is 0.44 mol L–1
(C) molar mass of solute X is 152 g mol–1 (D) mole fraction of solute X is 0.0082

13. Out of the following, colligative properties are

(A) osmotic pressure (B) boiling point
(C) depression in freezing point (D) relative lowering of vapour pressure

14. 10% aqueous solution of a non-electrolyte solute of molar mass (m) has elevation in boiling point Xº, then

B 
(A) graphically log(x)

A log (Kb/m)
(B) AB = 2 and tan  = 1
(C) x is dependent on (Kb/m)
K 
(D) log  b   0, when boiling point of the solution is twice the boiling point of water
 m 

15. In which case, van’t Hoff factor (i) remains unchanged?

(A) PtCl4 reacts with KCl
(B) aq. ZnCl2 reacts with aq. NH3 (excess)
(C) aq. FeCl3 reacts with aq. K4[Fe(CN)6].
(D) HgCl2 reacts with excess of KI

Exercise – 4
Section - I : Numerical Value/Subjective Type Questions

1. Assuming 50% dissociation, Van’t Hoffs factor for Na2SO4 is _?

2. A 0.2 percent aqueous solution of a non- volatile solute exerts a vapour pressure of 1.004 bar at 1000C.
What is the molar mass of the solute? (Give your answer in nearest interger)
(Give:Vapour pressure of pure water at 1000 C is 1.013 bar and molar mass of water is 18g mol–1)

3. Calculate the mass of ascorbic acid (C6H8O6) to be dissolved in 75g of acetic acid to lower its melting
point by 1.50C. [for acetic acid Kf = 3.9 K kg mol–1] (Give your answer in nearest integer)

4. The molal freezing point depression constant of benzene (C6H6) is 4.90 K kg mol–1. Selenium exists as a
polymer of the type Sex. When 3.26 g of selenium is dissolved in 226 g of benzene, the observed freezing
point is 0.1120C lower than for pure benzene. Deduce the molecular formula of selenium. (At. Mass of
Se = 78.8 g mol–1)

5. Moles of K2SO4 to be dissolved in the 12 mol water to lower its vapour pressure by 10 mm Hg at a
temperature at which vapour pressure of pure water is 50 mm Hg is:

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Section - II : Match the Column
6. Match the following:
Column – I Column – II
(A) 10 vol H2O2 1. Perhydrol
(B) 20 vol H2O2 2. 5.358 N
(C) 30 vol H2O2 3. 1.785 N
(D) 100 vol H2O2 4. 3. 03%
The correct match is :
(A) A – 4, B – 3, C – 2, D – 1 (B) A – 1, B – 2, C – 3, D – 4
(C) A – 1, B – 3, C – 2, D – 4 (D) A – 4, B – 2, C – 3, D - 1

7. Match the solutions in Column-1 with the van’t Hoff factors in Column-II:
Column –I Column – II
(a) K4[Fe(CN)6] (p) 1 + 2
(b) Al2 (SO4)3 (q) Greater than 1
O
(c)  (r) (1+ 4  )
NH 2  C  NH 2
(d) CaCl2 (s) 1
 = Degree of ionization.

8. Match the solutions in Column – 1 with their nature in Column – II

Column – I Column – II
(a) Benzene + toluene (p) Non-ideal solution
(b) Ethanol + water (q) Ideal solution
(c) Benzene + chloroform (r) H mix  0
(d) Carbon tetrachloride + chloroform (s) H mixing  0

9. Match the solution in Column- I with their nature in Column – II:

Column – I Column – II
(a) n-hexane + n heptane (p) Can be perfectly separated by distillation
(b) Acetone + chloroform (q) Maximum boiling azeotrope
(c) Acetone + aniline (r) Cannot be perfectly separated by distillation
(d) Ethanol + water (s) Nearly ideal

10. Match the solutions in Column – I with their osmotic properties in column – II:
Column – I Column – II
(a) S1: 0.1 M glucose, S2: 0.1 M urea (p) S1 and S2 are isotonic
(b) S1: 0.1 M NaCl, S2: 0.1 M Na2SO4 (q) No migration of solvent across the membrane
(c) S1: 0.1 M NaCl, S2: 0.1 M KCI (r) S1 is hypertonic to S2
(d) S1: 0.1 M CuSO4, S2: 0.1 M sucrose (s) S1 is hypotonic to S2
[Note: Assume that the electrolytes are completely ionized.]

11. Match the solutions in Column-I with their colligative properties in Column –II.
Column – I Column – II
(a) 0.1 M Ca3 (PO4)2 (p) Solution with highest boiling point
(b) 0.1 M NaCl (q) Solution with van’t Hoff factor greater than
(c) 0.1 M glucose (r) Solution with lowest osmotic pressure
(d) 0.1 M CaCl2 (s) Solution with lowest freezing point
[Note: Assume that the electrolytes are completely ionized.]

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12. Match the concentration terms of List –I with their informations in List-II:
List – I List – II
(a) Molarity (p) Number of gram formula mass of solute dissolved
per litre of solution
(b) Molality (q) Number of moles of solute dissolved per kg of solvent
(c) Formality (r) Depends on temperature
(d) Strength of solution (s) Number of moles of solute dissolved per litre of solution.

Section - III : Comprehension

Comprehension#1 (13 to 18)

A 1.24M aqueous solution of KI has density of 1.15 g/cm3

Answer the following questions about this solutions:

13. Percentage composition of solute in the solution:

(A) 17.89 (B) 27.89 (C) 37.89 (D) 47.89

14. Molality of this solution will be:

(A) 2.61 (B) 1.31 (C) 4.12 (D) 3.12

15. What is the freezing point of the solution if the KI is completely dissociated in the solution?
(A) –4.870C (B) –3.220C (C) –1.220C (D) None of these

16. Experimental freezing point of the solution is – 4.460C. What % of KI is dissociated?

(A) 82% (B) 90% (C) 83% (D) None

17. Normality of this solution is:

(A) 0.62 (B) 1.24 (C) 2.48 (D) 3.72

Comprehension#2 (18 to 23)

The colligative properties of electrolytes require a slightly different approach than the one used for the
colligative properties of is the number of solute particles that determines the colligative properties of a
solution. The electrolyte solutions, therefore, show abnormal colligative properties. To account for this
effect we define a quantity called the van’t Hoff factor, given by:
Actual number of Particles in solution after dissociation
i. =
Number of formula units initally dissolved in solution
i. =1(for non – electrolytes);
i. > 1( for electrolytes, undergoing dissociation)
I < 1 (for solutes, undergoing association).

Answer the following questions:

18. Benzoic acid undergoes dimerisation in benzene solution. The van’t Hoff factor ‘i’ is related to the degree
of association ‘  ’ of the acid as:
 
(A) i = 1 –  (B) i = 1 +  (C) i  1  (D) i  1 
2 2

19. A substance trimerises when dissolved in a solvent A. The van’t Hoff factor ‘i’ for the solution is:
(A) 1 (B) 1/3 (C) 3 (D) unpredictable

20. For a solution of a non-electrolyte in water, the van’t Hoff factor is:
(A) always equal to 0 (B) < 1 (C) always equal to 2 (D) > 1 but < 2

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21. 0.1 M K4 [Fe(CN)6] is 60% ionized. What will be its van’t Hoff factor ?
(A) 1.4 (B) 2.4 (C) 3.4 (D) 4.4
i 1 il
[Hint:   , n = 5 since, K4 [Fe (CN)6] gives 5 ions in the solution 0.6  i = 3.4]
n 1 5l

22. A solution of benzoic acid dissolved in benzene such that it undergoes molecular association and its molar
mass approaches 244. Benzoic acid molecules will exist as:
(A) dimer (B) monomer (C) tetramer (D) trimer

23. The molar mass of the solute sodium hydroxide obtained from the measurement of the osmotic pressure of
its aqueous solution at 270C is 25 g mol–1. Therefore, its dissociation percentage in this solution is:
(A) 75 (b) 60 (C) 80 (D) 70
Normal molar mass
[Hint: i=
Abnormal molar mass
il

nl
i= 1+  for binary electrolyte
40
1+  = ,  = 0.6
25
% ionization = 60]

Comprehension#3 (24 to 26)

Many chemical and biological process depend on osmosis which is, the selective passage of solvent
molecules through a porous membrane from a dilute solution to a more concentrated one. The osmotic
pressure  depends on molar concentration of the solution (  = CRT). If two solutions are of equal solute
concentration and, hence, have the same osmotic pressure, they are said to be isotonic. If two solutions are
of unequal osmotic pressures, the more concentrated solutions is said to be hypertonic and the more
diluted solution is described as hypotonic.
Osmosis is the major mechanism for transporting water upward in the plants. Transpiration in the leaves
supports the transport mechanism of water. The osmotic pressure of seawater is about 30 atm; this is the
pressure that must be applied to the seawater (separated from pure water using a semi permeable
membrane) to get drinking water.

Answer the following questions:

24. A plant cell shrinks when it is kept in:

(A) hypotonic solution (B) hypertonic solution (C) isotonic solution (D) pure water

25. 4.5% solution of glucose would be isotonic with respect to …………….. solution of urea.
(A) 4.5% (B) 13.5% (C) 1.5% (D) 9%

26. Glucose solutions to be injected into the bloodstream must have same ………. As that of the bloodstream.
(A) molarity (B) vapour pressure (C) osmotic pressure (D) viscosity

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Exercise – 5
Revision exercise (Moderate to Tough)

1. An aqueous solution containing 288 gm of a non-volatile compound having the stoichiometric

composition CXH2XOX in 90 gm water boils at 101.240C at 1.00 atmospheric pressure. What is the
molecular formula?
Kb(H2O) = 0.512 K mol–1 kg; Tb (H2O) = 1000C
2. The degree of dissociation of Ca (NO3)2 in a dilute aqueous solution containing 7 gm of the salt per 100
gm of water at 1000 C is 70%. If the vapour pressure of water at 100ºC is 760 mm of Hg. Calculate the
vapour pressure of the solution.
3. The addition of 3 gm of substance to 100 gm CCl4 (M= 154 gm mol–1) raises the boiling point of CCl4 by
0.60ºC. Kb of (CCl4) is 5.03 kg mole–1 K. Calculate.
(i) the freezing point depression (ii) the relative lowering of vapour pressure
(iii) the osmotic pressure at 298K (iv) the molar mass of the substance
4. A 10% solution of cane sugar has undergone partial inversion according to the reaction:
Sucrose + water  Glucose + Fructose. If the boiling point of solution is 100.270C.
(i) What is the average mass of the dissolved materials?
(ii) What fraction of the sugar has inverted? Kb(H2O) = 0.512 K mol–1 kg
5. 1.5 g of a monobasic acid when dissolved in 150g of water lowers the freezing point by 0.1650C. 0.5 g of
the same acid when titrated, after dissolution in water, requires 37.5 ml of N/10 alkali. Calculate the
degree of dissociation of the acid (Kf for water = 1.860C mol–1).
6. Sea water is found to contain 5.85% NaCl and 9.50% MgCl2 by weight of solution. Calculate its normal
boiling point assuming 80% ionization for NaCl and 50% ionization of MgCl2
[Kb (H2O) = 0.51 kgmol–1K].
7. The vapour pressure of an aqueous solution of glucose is 750 mm Hg at 373 K. Calculate molality and
mole fraction of solute.
8. A complex is represented as CoCl3 . × NH3. It’s 0.1 molal solution in aq. Solution shows  Tf = 0.5580C.
Kf for H2O is 1.86 K mol–1 kg. Assuming 100% ionization of complex and coordination no. of Co is six,
calculate formula of complex.
9. The molar volume of liquid benzene (density = 0.877 g ml–1) increases by a factor of 2750 as it vaporizes
at 200C and that of liquid toluene (density=0.867 gml–1) increases by a factor of 7720 at 200C. Solution of
benzene & toluene has a vapour pressure of 46.0 torr. Find the mole fraction of benzene in the vapour
above the solution.
10. At 1000C, benezene & toluene have vapour pressure of 1375 & 558 Torr respectively. Assuming these
two form an ideal binary solution, calculate the composition of the solution that boils at 1 atm & 1000C.
What is the composition of vapour issuing at these conditions?
11. Calculate the boiling point of a solution containing 0.61 g of benzoic acid in 50g of carbon disulphide
assuming 84% dimerization of the acid. The boiling point and Kb of CS2 are 46.20C and 2.3 kg mol–1,
respectively.

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12. The cryoscopic constant for acetic is 3.6 K kg/mol. A solution of 1 g of a hydrocarbon in 100 g of acetic
acid freezes at 16.140C instead of the usual 16.600C. The hydrocarbon contains 92.3% carbon. What is the
molecular formula?
13. Phenol associate in benzene to a certain extent to form a dimer. A solution containing 20 x 10–3 kg phenol
in 1 kg of benzene has its freezing point depressed by 0.69 K. Calculate the fraction of phenol that has
dimerised. Kf for benzene = 5.12 kg mol–1K.
14. 30 ml of CH3OH (d=0.7980 gm cm–3) and 70 ml of H2O (d=0.9984 gm cm–3) are mixed at 250C to form a
solution of density 0.9575 gm cm–3. Calculate the freezing point of the solution. Kf(H2O) is 1.86 kg mol–1
K. Also calculate its molarity.
15. Dry air was drawn through bulbs containing a solution of 40 grams of urea in 300 grams of water, then
through bulbs containing pure water at the same temperature and finally through a tube in which pumice
moistened with strong H2SO4 was kept. The water bulbs lost 0.0870 grams and the sulphuric acid tube
gained 2.036 grams. Calculate the molecular weight of urea.
16. Vapour pressure of C6H6 and C7H8 mixture at 500C is given by P (mm Hg) = 180 XB + 90, where XB is
the mole fraction of C6H6. A solution is prepared by mixing 936g benzene and 736g toluene and if the
vapours over this solution are removed and condensed into liquid and again brought to the temperature of
500C, what would be mole fraction of C6H6 in the vapour state?
17. When the mixture of two immicible liquids (water and nitrobenzene) boils at 372K and the vapour
pressure at this temperature are 97.7 k Pa (H2O) and 3.6 kPa (C6H5NO2). Calculate the weight % of
nitrobenzene in the vapour.
18. The osmotic pressure of a solution of a synthetic polyisobutylene in benzene was determined at 250C. A
sample containing 0.20 g of solute/100 cm3 of solution developed a rise of 2.4 mm at osmotic equilibrium.
The density of the solution was 0.88 g/cm3. What is the molecular weight of the polyisobutylene?
19. The vapour pressure of a saturated solution of sparingly soluble salt (XCl3) was 17.20 mm Hg at 270C. if
the vapour pressure of pure H2O is 17.25 mm Hg at 300 K. Calculate the solubility of sparingly soluble
salt XCl3.
20. If the apparent degree of ionization of KCl (KCl=74.5 gm mol–1) in water at 290 K is 0.86. Calculate the
mass of KCl which must be made up to 1 dm3 of aqueous solution to the same osmotic pressure as the
4.0% solution of glucose at that temperature.
21. An ideal solution was prepared by dissolving some amount of cane sugar (non-volatile) in 0.9 moles of
water. The solution was then cooled just below its freezing temperature (271 K), where some ice get
separated out. The remaining aqueous solution registered a vapour pressure of 700 torr at 373K. Calculate
the mass of ice separated out, if the molar heat of fusion of water is 6 kJ.
22. The specific conductivity of a 0.5 M aq. Solution of monobasic acid HA at 270C is 0.006 Scm–1. It’s molar
conductivity at infinite dilution is 200 S cm2 mol–1. Calculate osmotic pressure (in atm) of 0.5 M HA (aq)
atmL
solution at 270C. Given R = 0.08 .
molK
23. What would be the osmotic pressure at 170C of an aqueous solution containing 1.75 g of sucrose
(C12H22O11) per 150 cm3 of solution?
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24. The vapour pressure of two pure liquids, A and B that form an ideal solution are 300 and 800 torr
respectively, at temperature T. A mixture of the vapour of A and B for which the mole fraction of A is
0.25 is slowly compressed at temperature T. Calculate
(i) the composition of the first drop of the condensate,
(ii) the total pressure when this drop is formed,
(iii) the composition of the solution whose normal boiling point is T,
(iv) the pressure when only the last bubble of vapour remains
(v) the composition of the last bubble.
25. Tritium, T (an isotope of H) combines with fluorine to form weak acid TF, which ionizes to give T.
Tritium is radioactive and is a  –emitter. A freshly prepared aqueous solution of TF has pT (equivalent of
pH) of 1.5 and freezes at –0.3720C. If 600 ml of freshly prepared solution were allowed to stand for 24.8
years. Calculate
(i) ionization constant of TF.
(ii) Number of  –particles emitted. (Given Kf for water = 1.86 kg mol K–1, t1/2 for tritium =12.4 years)

Exercise –6
Section – I : JEE (Advanced) Questions Previous Years

1. During depression of freezing point in a solution, the following are in equilibrium: [JEE 2003]
(A) liquid solvent-solid solvent (B) liquid solvent-solid solute
(C) liquid solute-solid solute (D) liquid solute-solid solvent
2. A 0.004 M solution of Na2SO4 is isotonic with a 0.010 M solution of glucose at same temperature. The
apparent degree of dissociation of Na2SO4 is [JEE 2004]
(A) 25% (B) 50% (C) 75% (D) 85%
3. The elevation in boiling point, when 13.44 g of freshly prepared CuCl2 are added to one kilogram of
water, is [Some useful data, Kb (H2O) = 0.52 kg K mol–1, mol. Wt. of CuCl2  134.4 gm ] [JEE 2005]
(A) 0.05 (B) 0.1 (C) 0.16 (D) 0.21
4. When 20g of naphtholic acid (C11H8O2) is dissolved in 50 g of benzene (Kf= 1.72 K kg mol–1), a freezing
point depression of 2 K is observed. The van’t Hoff factor (i) is [JEE 2007]
(A) 0.5 (B) 1 (C) 2 (D) 3
5. The Henry’s law constant for the solubility of N2 gas in water at 298 K is 1.0 x 105 atm. The mole
fraction of N2 in air is 0.8. The number of moles of N2 from air dissolved in 10 moles of water at 298 K
and 5 atm pressure is [JEE 2009]
(A) 4.0 x 10–4 (B) 4.0 x 10–5 (C) 5.0 x 10–4 (D) 4.0 x 10–5

Paragraph for Question No. 6 to 8

Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute
molecules are added to get homogeneous solution. These are called colligative properties. Applications of
colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol
and water mixture as anti-freezing liquid in the radiator of automobiles.
A solution M is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixture is 0.9.

water
Given: Freezing point depression constant of water. K f 
= 1.86 K kg mol–1


ethanol
Freezing point depression constant of ethanol K f 
= 2.0 K kg mol–1

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 water
Boiling point elevation constant of water K b  = 0.52K kg mol –1

Boiling point elevation constant of ethanol  K  = 1.2 K kg mol

ethanol –1
b

Standard freezing point of water = 273K

Standard freezing point of ethanol = 155.7K
Standard boiling point of water =373K
Standard boiling point of ethanol =351.5K
Vapour pressure of pure water = 32.8 mm Hg
Vapour pressure of pure ethanol = 40 mm Hg
Molecular weight of water = 18g mol–1
Molecular weight of ethanol = 46g mol–1
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be
non-volatile and non- dissociative.
6. The freezing point of the solution M is [JEE 2008]
(A) 268.7 K (B) 268.5 K (C) 234. 2 K (D) 150. 9 K
7. The vapour pressure of the solution M is [JEE 2008]
(A) 39.3 mm Hg (B) 36.0 mm Hg (C) 29.5 mm Hg (D) 28.8 mm Hg
8. Water is added to the solution M such that the mole fraction of water in the solution becomes 0.9. The
boiling point of this solutions is [JEE 2008]
(A) 380.4 K (B) 376.2 K (C) 375.5K (D) 354.7 K
9. The Freezing point (in 0C) of a solution containing 0.1 g of K3 [Fe (CN)6] (Mol. Wt. 329) in 100 g of
water (Kf = 1.86 K kg mol–1) is [JEE 2011]
(A) –23 × 10–2 (B) –5.7 × 10–2 (C) –5.7 × 10–3 (D) –1.2 × 10–2
10. For a dilute solution containing 2.5 g of a non-volatile non – electrolyte solute in 100g of water, the
elevation in boiling point at 1 atm pressure is 2ºC. Assuming concentration of solute is much lower than
the concentration of solvent, the vapour pressure (mm of Hg) of the solution is (take Kb = 0.76 K kg
mol–1) [JEE 2012]
(A) 724 (B) 740 (C) 736 (D) 718
11. Benzene and naphthalene form an ideal solution at room temperature. For this process, the true statement
(s) is (are) [JEE Advanced - 2013]
(A)  G is positive (B)  Ssystem is positive
(C)  S surroundings = 0 (D) H=0
12. The vapour pressure of two miscible liquids (A) and (B) are 300 and 500 mm of Hg respectively. In a
flask 10 mole of (A) is mixed with 12 mole of (B). However, as soon as (B) is added, (A) starts
polymerising into a completely insoluble solid. The polymerization follows first-order kinetics. After 100
minute, 0.525 mole of a solute is dissolved which arrests the polymerization completely.
The final vapour pressure of the solution is 400 mm of Hg. Estimate the rate constant of the
polymerization reaction. Assume negligible volume change on mixing and polymerization and ideal
behaviour for the final solution. [JEE 2001]

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13. Match the boiling point with Kb for x, y and z, if molecular weight for x, y and z are same.
b.pt. Kb [JEE 2003]
x 100 0.68
y 27 0.53
z 253 0.98
14. 1.22 g of benzoic acid is dissolved in (i) 100 g acetone (Kb for acetone = 1.7) and (ii) 100g benzene (Kb
for benzene = 2.6). the elevation in boiling points Tb is 0.170C and 0.130C respectively. [JEE 2004]
(i) What are the molecular weights of benzoic acid in both the solutions?
(ii) What do you deduce out of it in terms of structure of benzoic acid?
15. 72.5g of phenol is dissolved in 1 kg of a solvent (Kf =14) which leads to dimerization of phenol and
freezing point is lowered by 7 kelvin. What is the percent of dimerization of phenol. [JEE 2006]
16. MX2 dissociates into M2+ and X– ions in an aqueous solution, with a degree of dissociation (  ) of 0.5. The
ratio of the observed depression of freezing point of the aqueous solution to the value of the depression of
freezing point in the absence of ionic dissociation is [JEE Advanced- 2014]
17. If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride ammonia complex (which
behaves as a strong electrolyte) is – 0.05580C, the number of chloride (s) in the coordination sphere of the
complex is [Kf of water = 1.86 K kg mol–1] [JEE Advanced- 2015]
0
18. Mixture (s) showing positive deviation from Raoult’s law at 35 C is (are): [JEE-Advanced-2016]
(A) Carbon tetrachloride + methanol (B) carbon disulphide + acetone
(C) benzene + toluene (D) phenol + anlline
19. Liquids A and B form ideal solution for all compositions of A and B at 25℃. Two such solutions with
0.25 and 0.50 mole fractions of A have the total vapor pressures of 0.3 and 0.4 bar, respectively. What is
the vapor pressure of pure liquid B in bar? [JEE Advanced, 2020]
Question stem for Question Nos. 20 and 21
Question Stem
The boiling point of water in a 0.1 molal silver intrate solution (soltuion A) is xºC. To this solution A, an
equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution B. The
difference in the boiling points of water in the two solutions A and B is y × 10–2 ºC.
(Assume: Densities of the solutions A and B are the same as that of water and the soluble salts dissociate
completely.
Use: Molal elevation constant (Ebullioscopic constant), Kb = 0.5 K kg mol–1; Boiling point of pure water
as 100ºC.) [JEE Advanced, 2021]
20. The value of x is ______.
21. The value of |y| is _____.
22. An aqueous solution is prepared by dissolving 0.1 mol of an ionic salt in 1.8 kg of water at 35 ºC. The salt
remains 90% dissociated in the solution. The vapour pressure of the solution is
59.724 mm of Hg. Vapor pressure of water at 35 ºC is 60.000 mm of Hg. The number of ions present per
formula unit of the ionic salt is _______. [JEE Advanced, 2022]

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 C-18
23. 50 mL of 0.2 molal urea solution (density = 1.012 g mL–1 at 300 K) is mixed with 250 mL of a solution
containing 0.06 g of urea. Both the solutions were prepared in the same solvent. The osmotic pressure (in
Torr) of the resulting solution at 300 K is _____. [JEE-Advanced-2023, P-2]
[Use: Molar mass of urea = 60 g mol–1, gas constant, R = 62 L Torr K–1 mol–1;

Assume,  mix H  0,  mix V  0 ]

Section – II : JEE (Main) Questions Previous Years

Chemistry – Solutions & Colligative Properties Toll Free Number : 1800 103 9888
 C-19

Chemistry – Solutions & Colligative Properties Toll Free Number : 1800 103 9888
 C-20

Chemistry – Solutions & Colligative Properties Toll Free Number : 1800 103 9888
 C-21
21. There are two beakers (I) having pure volatile solvent and (Il) having volatile solvent and non-volatile
solute. If both beakers are placed together in a closed container then: [JEE Main, 7 Jan 2020 - Shift 2]
(1) Volume of solvent beaker will decrease and solution beaker will increase
(2) Volume of solvent beaker will increase and solution beaker will also increase
(3) Volume of solvent beaker will decrease and solution beaker will also decrease
(4) Volume of solvent beaker will increase and solution beaker will decrease
22. A flask contains a mixture of isohexane and 3-methylpentane. One of the liquids boils at 63°C while the
other boils at 60°C . What is the best way to seperate the two liquids and which one will be distilled out
first ? [JEE Main, 8 Jan 2020 - Shift 1]
(1) Simple distillation and isohexane. (2) Fractional distillation and isohexane.
(3) Simple distillation and 3-Methylpantane. (4) Fractional distillation and 3-Methylpantane.
23. How much amount of NaCl should be added to 600 g for water ( ρ = 1.00 g/mL) to decrease the freezing
point of water to – 0.2°C’?...... [JEE Main, 9 Jan 2020 - Shift 1]
(The freezing point depression constant of water = 2 K kg mol–1,
24. An open beaker of water in equilibrium with water vapour is in a sealed container. When a few grams of
glucose are added to the beaker of water, the rate at which water molecules
[JEE Main, 2 Sep 2020 - Shift 1]
(1) leaves the solution increases (2) leaves the vapour increases
(3) leaves the vapour decreases (4) leaves the solution decreases
25. If 250 cm3 of an aqueous solution containing 0.73g of a protein A is isotonic with one litre of another
aqueous solution containing 1.65g of a protein B, at 298K, the ratio of the molecular masses of A and B
is ___ × 10–2 (to the nearest integer). [JEE Main, 3 Sep 2020 - Shift 2]
26. At 300K, the vapour pressure of a solution containing 1 mole of n-hexane and 3 moles of n-heptane is 550
mm of Hg. At the same temperature, if one more mole of n-heptane is added to this solution, the vapour
pressure of the solution increases by 10mm of Hg. What is the vapour pressure in mm Hg of n-heptane in
its pure state [JEE Main, 4 Sep 2020 - Shift 1]
27. A set of solutions is prepared using 180g of water as a solvent and 10g of different nonvolatile solutes A,
B and C. The relative lowering of vapour pressure in the presence of these solutes are in the order [Given,
molar mass of A = 100gmol–1; B = 200gmol–1 C = 10, 000gmol–1] [JEE Main, 6 Sep 2020 - Shift 2]
(1) A>C>B (2) C>B>A (3) A>B>C (4) B>C>A
28. When 9.45 g of ClCH2COOH is added to 500 mL of water, its freezing point drops by 0.5°C. The
dissociation constant of ClCH2COOH is x × 10–3. The value ofx is __ (Rounded off to the nearest integer)
 K f (H O)  1.86K kg mol1  [JEE Main, 24 Feb 2021 - Shift 1]
 2 
29. 1 molal aqueous solution of an electrolyte A2 B3 is 60% ionised. The boiling point of the solution at 1 atm
is _____ K. (Rounded-off to the nearest integer) [Given Kb for (H2O) = 0.52 K kg mol−1]
[JEE Main, 25 Feb 2021 - Shift 1]
30. When 12.2 g of benzoic acid is dissolved in of 100 g water, the freezing point of solution was found to be
–93ºC (Kf H2O) = 1.86 K kg mol−1). The number (n) of benzoic acid molecules associated (assuming
100% association) is [JEE Main, 24 Feb 2021 - Shift 1]

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 C-22
31. At 363 K, the vapour pressure of A is 21kPa and that of B is 18kPa. One mole of A and 2 moles of B are
mixed. Assuming that this solution is ideal, the vapour pressure of the mixtureis___ kPa. (Round of to the
Nearest Integer). [JEE Main, 16 March 2021 - Shift 2]
32. A 1 molal K4Fe (CN)6 solution has a degree of dissociation of 0.4. Its boiling point 1s equal to that of
another solution which contains 18.1 weight percent of a non electrolytic solute A. The molar mass of
Ais___u. (Round off to the Nearest Integer). [Density of water = 1.0 g cm–3]
[JEE Main, 17 March 2021 - Shift 2]
33. A solute a dimerizes in water. The boiling point of a 2 molar solution of A is 100.52°C. The percentage
association of A is ________ (Round off to the Nearest integer) [JEE Main, 18 March 2021 - Shift 2]
[Use : Kb, for water = 0.52 K kg mol–1 Boiling point of water = 100°C]
34. The osmotic pressure of blood is 7.47 bar at 300 K. To inject glucose to a patient intravenously, it has to
be isotonic with blood. The concentration of glucose solution in gL–1 is (Molar mass of glucose = 180 g
mol–1 R = 0.083 L bar K–1 mol–1) (Nearest integer) [JEE Main, 24 June 2022 - Shift 1]
35. 1 L aqueous solution of H2SO4 contains 0.02 m mol H,SO,. 50% of this solution is diluted with deionized
water to give 1 L solution (A). In solution (A), 0.01 m mol of H2SO4 are added. Total m mols of H2SO4 in
the final solution is _____ × 10° m mols. [JEE Main, 25 June 2022 - Shift 1]
36. The osmotic pressure exerted by a solution prepared by dissolving 2.0 g of protein of molar mass 60 kg
mol–1 in 200 mL of water at 27°C is Pa. [integer value]
(use R = 0.083 L bar mol–1 K–1) [JEE Main, 26 June 2022 - Shift 1]
–3 –3
37. A solution containing 2.5 × 10 kg of a solute dissolved in 75 × 10 kg of water boils at 373.535 K. The
molar mass of the soluteis _____ g mol–1. [nearest integer] (Given: Kb (H2O) = 0.52 K Kg mol–1, boiling
point of water = 373.15K) [JEE Main, 27 June 2022 - Shift 2]
38. The vapour pressures of two volatile liquids A and B at 25°C are 50 Torr and 100 Torr, respectively. If the
liquid mixture contains 0.3 mole fraction of A, then the mole fraction of liquid B in the vapour phase is
x
. The value of x is _________ [JEE Main, 28 June 2022 - Shift 1]
17
39. Elevation in boiling point for 1.5 molal solution of glucose in water is 4K. The depression in freezing
point for 4.5 molal solution of glucose in water is 4K. The ratio of molal elevation constant to molal
depression constant (Kb/Kf) is. [JEE Main, 29 June 2022 - Shift 2]
40. Two solutions A and B are prepared by dissolving 1 g of non-volatile solutes X and Y respectively in 1 kg
of water. The ratio of depression in freezing points for A and B is found to be 1 : 4. The ratio of molar
masses of X and Y is: [JEE Main, 25 July 2022 - Shift 2]
(1) 1:4 (2) 1 : 0.25 (3) 1 : 0.20 (4) 1:5
41. The elevation in boiling point for 1 molal solution of non-volatile solute A is 3K. The depression in
freezing point for 2 molal solution of A in the same solvent is 6 K. The ratio of Kb and Kg i.e., Kb/Kf is
1 : X. The value of X is [nearest integer] [JEE Main, 26 July 2022 - Shift 2]
42. Boiling point of a 2% aqueous solution of a non-volatile solute A is equal to the boiling point of 8%
aqueous solution of a non-volatile solute B. The relation between molecular weights of A and B is.
[JEE Main, 27 July 2022 - Shift 1]
(1) MA = 4MB (2) MB = 4MA (3) MA = 8MB (4) MB = 8 MA
43. If O2 gas is bubbled through water at 303 K, the number of millimoles of O, gas that dissolve in 1 litre of
water is ________. (Nearest Integer)
(Given : Henry's Law constant for O2, at 303 K is 46.82 k bar and partial pressure of O2 = 0.920 bar)
(Assume solubility of O2 in water is too small, nearly negligible) [JEE Main, 29 July 2022 - Shift 1]

44. In the depression of freezing point experiment [JEE Main, 24 Jan 2023 - Shift 1]
A. Vapour pressure of the solution is less than that of pure solvent
B. Vapour pressure of the solution is more than that of pure of solvent
C. Only solute molecules solidify at the freezing point

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 C-23
D. Only solvent molecules solidify at the freezing point
(1) A and D only (2) B and C only (3) A and C only (4) A only
45. The Total pressure observed by mixing two liquid A and B is 350 mm Hg when their mole fractions are
0.7 and 0.3 respectively.
The Total pressure becomes 410 mm Hg if the mole fractions are changed to 0.2 and 0.8 respectively for
A and B. The vapour pressure of pure A is _______ mm Hg. (Nearest integer) Consider the liquids and
solutions behave ideally. [JEE Main 24 Jan 2023 - Shift-2]
46. A solution containing 2 g of a non-volatile solute in 20 g of water boils at 373.52 K. The molecular mass
of the solute is _______ gmol–l. (Nearest integer) [JEE Main 30 Jan 2023 - Shift -1]
–1
Given, water boils at 373 K, Kb for water = 0.52 K kg mol
47. Lead storage battery contains 38% by weight solution of H2SO4. The van’t Hoff factor is 2.67 at this
concentration. The temperature in Kelvin at which the solution in the battery will freeze is ______
[JEE Main 30 Jan-2023 - Shift -2]
(Nearest integer).
Given Kf = 1.8 kg mol–1
48. At 27ºC, a solution containing 2.5 g of solute in 250.0mL of solution exerts an osmotic pressure of 400
Pa. The molar mass of the solute is ____ g mol–1 (Nearest integer) [JEE Main 24 Jan– 2023 -Shift -1]
–1 –1
(Given: R = 0.083 L bar K mol )
49. Mass of Urea (NH2 CONH2) required to be dissolved in 100 g of water in order to reduce the vapour
pressure of water by 25% is ______g. (Nearest integer) [JEE Main 06 Apr-2023- Shift -1]
Given: Molar mass of N, C, O and H are 14, 12, 16 and 1g mol–1 respectively.
50. Consider the following pairs of solution which will be isotonic at the same temperature. The number of
pairs of solution is/are ______ [JEE Main 06 Apr-2023- Shift -2]
(1) 1 M aq. NaCl and 2 M aq. urea (2) 1 M aq. CaCl2 and 1.5 M aq. KCl
(3) 1.5 M aq. AlCl3 and 2 M aq. Na2SO4 (4) 2.5 M aq. KCl and 1 M aq. Al2(SO4)3
51. If the boiling points of two solvents X and Y (having same molecular weights) are in the ratio 2:1 and
their enthalpy of vaporizations are in the ratio 1 : 2, then the boiling point elevation constant of X is m
times the boiling point elevation constant of Y. The value of m is (nearest integer).
[JEE Main 08 Apr-2023 - Shift -2]
52. What weight of glucose must be dissolved in 100 g of water to lower the vapour pressure by 0.20 mm
Hg? (Assume dilute solution is being formed)
Given: Vapour pressure of pure water is 54.2 mm Hg at room temperature.
Molar mass of glucose is 180g mol–1 [JEE Main 11 Apr-2023- Shift -2]
(1) 3.59 g (2) 3.69 g (3) 4.69 g (4) 2.59 g
53. The vapour pressure of 30% (w/v), aqueous solution of glucose is ______ mm Hg at 25ºC.
[Given: The density of 30% (w/v), aqueous solution of glucose is 1.2 g cm–3 and vapour pressure of pure
water is 24 mm Hg.] (Molar mass of glucose is 180g mol–1) [JEE Main 15 Apr-2023- Shift -1]

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 C-24

ANSWER KEY
Exercise-1
1. 0.24 2. 0.25 3. 24.5 torr 4. 57.24 g/mol
5. 66.13 mm Hg 6. 0.237 7. 92 mol% toluene; 96.8 mol% toluene
8. PAo  213.33torr, PBo  960.0 torr 9. 0.04 10. 111. 1g, 18.52 molal
11. 0.162m 12. 65.25 13. 0.964 14. 106g/mol
0 0
15. 64.0 g/mol 16. 100.079 C 17. –0.62 C. 18. 5.080C/m
19. 2050 g/mol 20. Kb=0.512 kg mol K–1, Tb = 373.20 K 21. 38.71g
22. 156.06 23. Tf =–2.280C 24. P= 0.2217 atm should be applied
25. (Vfinal = 5.Voriginal) 26. 54.2g 27. 13.8 atm 28. 59.99
29. MA/MB = 0.33 30. i=2.5 31. 3 ions 32. a = 99.2%
33. 7.482 x 105 Nm–2 34. 94.5% 35. 4.64 atm 36. 0.95; 1.95
37. 100 38. 8 39. 7.7 x 104 atm 40. 7.77 x 10–6 kg

Exercise-2
1. (C) 2. (C) 3. (C) 4. (B) 5. (C) 6. (D) 7. (D)
8. (C) 9. (A) 10. (C) 11. (C) 12. (B) 13. (C) 14. (D)
15. (B) 16. (B) 17. (C) 18. (A) 19. (D) 20. (B) 21. (D)
22. (B) 23. (B) 24. (A) 25. (A) 26. (A) 27. (B) 28. (A)
29. (C) 30. (D) 31. (D) 32. (B) 33. (A) 34. (C) 35. (C)
36. (C) 37. (B) 38. (D) 39. (D) 40. (D)

Exercise-3
1. (BC) 2. (ACD) 3. (ABC) 4. (ABCD) 5. (ABCD) 6. (ABC) 7. (AB)
8. (ABC) 9. (ABC) 10. (ACD) 11. (AD) 12. (ABCD) 13. (ACD) 14. (ABCD)
15. (B)

Exercise-4
Section-I
1. 2 2. 4 3. 5 4. 8 5. 1
Section-II
6. A – 4, B – 3, C – 2, D – 1 7. (a – q, r); (b – q, r); (c – s ); (d – p, q )
8. (a – q, s); (b – p, r); (c – p); (d – s ) 9. (a – p, s); (b – q, r); (c – q, r); (d – r )
10. (a – p, q); (b – s ); (c – p, q); (d – r) 11. (a – p, q, s); (b – q); (c – r); (d – q)
12. (a – r, s); (b – q); (c – p, r); (d – r)
Section-III
13. (A) 14. (B) 15. (A) 16. (C) 17. (B) 18. (C) 19. (B)
20. (B) 21. (C) 22. (A) 23. (B) 24. (B) 25. (C) 26. (C)

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Exercise-5
1. C44 H88O44 2. 746.24 mm/Hg 3. (i) 3.79ºC, (ii) 0.018, (iii) 4.65 atm, (iv) 251.5

4. (i) 210.65, (ii) 62.35% 5. 18.34% 6. Tb = 102.3º 7. 0.741m, 0.013

8. [Co(NH3)5Cl]Cl2 9. 0.73 10. Xb = 0.2472, Yb =0.4473
11. 46.33ºC12. C6H6 13. a = 0.7333 14. –19.91ºC, 7.63M 15. M = 53.8
16. 0.93 17. 20.11% 18. 2.4 × 105 g/mol 19. 4.04 × 10–2 20. 8.9 gm
21. 12.54 gm 22. 12.72 23. 0.81 atm

24. (i) 0.47, (ii) 565 torr, (iii) XA=0.08, XB = 0.92, (iv) 675 torr, (v) X'A  0.11,X'B  0.89
25. (i) Ka = 7.3 × 10–3 (ii) 4.55 × 1022

Exercise – 6
Section – I
1. (A) 2. (C) 3. (C) 4. (A) 5. (A) 6. (D) 7. (B)
8. (B) 9. (A) 10. (A) 11. (BCD)
12. 1.0 × 10–4 13. Kb(X) = 0.68, Kb (y) = 0.53, Kb (Z) = 0.98
14. (i) 122, (ii) it means that benzoic acid remains as it is in acetone while it dimerises in benzene as

15. 70% 16. 2 17. 1 18. (AB) 19. 0.2 20. (100.1)

21. (2.5) 22. (5) 23. 682

Section - II

1. (1) 2. (3) 3. (2) 4. (3) 5. (4) 6. (2) 7. (2)

8. (1) 9. (3) 10. (1) 11. (4) 12. (1) 13. (4) 14. (2)

15. (4) 16. (4) 17. (3) 18. (3) 19. (3) 20. (3) 21. (1)

22. (2) 23. (1.76) 24. (4) 25. (177) 26. (600) 27. (3) 28. (35)

29. (375) 30. (2) 31. (19) 32. (85) 33. (100) 34. (54) 35. (15)

36. (415) 37. (45) 38. (14) 39. (3) 40. (2) 41. (1) 42. (2)

43. (1) 44. (1) 45. 314 46. 100g 47. 243 48. 62250 49. 1111

50. (4) 51. 8 52. (2) 53. 23

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ELECTROCHEMISTRY
 C-26
Exercise - 1
Concept Building Questions

GALVANIC CELL:
Representation of Cell diagrams, complete and half cell reactions:

1. Make complete cell diagrams of the following cell reactions:

(A) C d 2  ( a q )  Z n (s )  Z n 2  ( a q ) C d (s )
(B) 2 A g  ( a q )  H 2 ( g )  2 H  ( a q )  2 A g (s )
(C) H g 2 C  2 (s )  C u (s )  C u 2  ( a q )  2 C   ( a q )  2 H g (  )
(D) C r2 O 72  ( a q )  1 4 H  ( a q )  6 F e 2  ( a q )  6 F e 3  ( a q )  7 H 2 O ( l)

2. Write cell reaction of the following cells:

 2 2 3
(A) Ag Ag (aq) Cu (aq) cu (B) Pt Fe ,Fe MnO4 ,Mn2,H Pt
   2
(C) Pt,CI2 CI (aq) Ag (aq) Ag (D) Pt,H2 H (aq) Cd (aq) Cd

3. Write half cells of each cell with following cell reactions:

(A) Z n (s )  2 H  ( a q )  Z n 2  ( a q )  H 2 ( g )
(B) 2 F e 3  ( a q )  S n 2  ( a q )  2 F e 2  (a q )  S n 4  ( a q )
(C) M n O 4 ( a q )  8 H  ( a q )  5 F e 2  ( a q )  5 F e 3  (a q )  M n 2  (a q )  4 H 2 O ( l)
(D) P b ( s )  B r2 ( l)  P b 2  ( a q )  2 B r  ( a q )

Electrode potential and standard electrode potential:

4. For the cell reaction 2 C e 4   C o  2 C e 3  C o 2 

E oc ell is 1.89 V. If E o is –0.28 V, what is the value of E o ?

Co 2  lCo C e 4  lC e 3 

5. Determine the standard reduction potential for the half reaction:

Cl2+2e–  2Cl –
Given Pt2+ +2Cl–  Pt + Cl2, E oc ell = – 0.15V.
Pt + 2e  Pt
2+ –
Eo = 1.20 V

6. What is E oCell if
2 C r  3H 2 O  3O C l –  2C r 3  3C l  6 O H 
C r 3  3e   C r, E o   0 .7 4 V
O C I  H 2O  2e  C I  2O H  , E o  0 .9 4 V

G o ,EoCell and K eq :

7. Is 1.0 M H+ solution under (H2SO4 at 1.0 atm capable of oxidizing silver metal in the presence
of 1.0 M Ag+ ion?
Eo  0.80 V , E o  0.0 V
Ag  l Ag H  lH 2 ( Pt )

Cu 2   e   Cu  E o  0.15V
8. If for the half cell reactions
Cu 2   2e   Cu E o  0.34V
Calculate Eo of the half cell reaction
Cu   e   C u
+
also predict whether Cu undergoes disproportionation or not.

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 C-27
9. If E o  0.44V, E o  0.77V. Calculate E o
Fe 2  lFe Fe3 lFe 2  Fe3 lFe
10. Calculate the equilibrium constant for the reaction
o
given: ECe4 /Ce3 1.44V;Eo
F e 2   C e 4   F e 3  C e 3 ,
Fe3 /Fe2 0.68V]
11. Calculate the EMF of a Daniel cell when the concentration of ZnSO4 and CuSO4 are 0.001 M and 0.1 M
respectively. The standard potential of the cell is 1.1V
12. Calculate the equilibrium constant for the reaction Fe + CuSO4  FeSO4 + Cu at 25ºC.
Give = E0 (Fe/Fe2+) = 0.44V, E0 (Cu/Cu2+) = – 0.337V.
2 
13. For a cell Mg(s) Mg (aq) Ag (aq) Ag, calculate the equilibrium at 25ºC. Also find the maximum
work that can be obtained by operating the cell.
E o ( M g 2  / M g )   2 .3 7 V , E o ( A g  / A g )  0 .8 V .
14. The standard reduction potential at 25ºC for the reduction of water
2 H 2 O  2 e   H 2  2 O H  is – 0.8277 volt. Calculate the equilibrium constant for the reaction
2 H 2O  H 3O   O H  at 25º C.
15. At 25ºC the value of K for the equilibrium Fe3+ + Ag  Fe2+ + Ag + is 0.531mol/litre. The standard
electrode potential for Ag+ + e– Ag is 0.799V. What is the standard potential for Fe3+ + e– Fe2+ ?
16. The EMF of the cell M|Mn+(0.02M)||H+ (1M) | H2 (g) (1 atm), Pt at 25ºC is 0.81V. Calculate the valency
of the metal if the standard oxidation potential of the metal is 0.76V.
17. Equinormal Solutions of two weak acids, HA (pKa = 3) and HB (pKb =5) are each placed in contact with
equal pressure of hydrogen electrode at 25ºC. When a cell is constructed by interconnecting them through
a salt bridge, find the emf of the cell.
18. In two vessels each containing 500 ml water, 0.5mmol of aniline (Kb= 10–9) and 25 mmol of HCl are
added separately. Two hydrogen electrodes are constructed using these solutions. Calculate the emf of cell
made by connecting them appropriately.
19. Calculate E o and E for the cell Sn | Sn2+(1M) || Pb2+ (10–3M) | Pb, E0(Sn2+| Sn) = –0.14V, E0 (Pb2+| Pb) = –
0.13V. Is cell representation is correct?
20. At what concentration of Cu2+ in a solution of CuSO4 will the electrode potential be zero at 25ºC?
Give:Eo (Cu|Cu2+) = 0.34V.
21. A zinc electrode is placed in a 0.1 M solution at 25ºC. Assuming that the salt is 20% dissociated at this
dilutions calculate the electrode potential. E o ( Z n 2  Z n )   0 .7 6 V .
22. From the standard potentials shown in the following diagram, calculate the potentials E 1o a n d E o2 .

E1o

1
BrO 3 0.54 V BrO 0.45V Br 2 1.07 V .
Br 
2

E o2

23. For the reaction, 4Al(S) + 3O2(g) + 6H2O () + 4OH– (aq)  4 [ A l( O H ) 4  ( a q )] ; E co e ll  2 .7 3 V . If
 G of ( O H  )   1 5 7 k J m o l  1 a n d  G of ( H 2 O )   2 3 7 .2 k j m o l 1
, d e t e r m in e  G of [ A l( O H ) 4  ] .
24. Calculate the EMF of the following cell Zn|Zn2+(0.01M)||Zn2+(0.1M)| Zn at 298K.
25. Calculate the EMF of the cell, Zn – Hg (c1 M) Zn–Hg(c1M)|Zn2+ (aq)| Hg –Zn (c2M) at 25ºC, if the
concentration of the zinc amalgam are: c1 = 10g per 100g of mercury and c2 = 1g per 100 g of mercury.

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26. Calcualte pH using the following cell:
Pt (H2) |H+(x M) || H+ (1M) | Pt (H2) if Ecell = 0.2364 V.
1 atm 1atm

27. Calculate the EMF of following cell at 25ºC.

P t , H 2 ( 2 atm ) H C l H 2 (1 0 atm ), P t.

28. EMF of the cell Z n Zn S O 4 (a 1  0 .2) Z n S O 4 (a 2 ) zn is – 0.0088V at 25ºC. Calculate the value of a2

ELECTROLYTIC CELL:

29. Calculate the no. of electrons lost or gained during electrolysis of

(A) 3.55 gm of Cl– ions (B) 1 gm Cu2+ ions (C) 2.7 gm of Al3+ ions

30. How many faradays of electricity are involved in each of the case.
(A) 0.25 mole Al3+ is converted to Al.
(B) 27.6 gm of SO3 is converted to S O 32 
(C) The Cu2+ in 1100 ml of 0.5 M Cu2+ is converted to Cu.

31. 0.5 Mole of electron is passed through two electrolytic cells in series. One contains silver ions, and the
other zinc ions. Assume that only cathode reaction in each cell is the reduction of the ion to the metal.
How many gm of each metals will be deposited.

32. The electrosynthesis of MnO2 is carried out from a solution of MnSO4 in H2SO4 (aq). If a current of 25.5
ampere is used with a current efficiency of 85%, how long would it take to produce 1 kg of MnO2?

33. A constant current of 30A is passed through an aqueous solution of NaCl for a time of 1.0 hr. How many
grams of NaOH are produced? What is volume of Cl2 gas at S.T.P. produced?

34. If 0.224 litre of H2 gas is formed at the cathode, how much O2 gas is formed at the anode under identical
conditions?

35. If 0.224 litre of H2 gas is formed at the cathode of one cell at S.T.P., how much of Mg is formed at the
cathode of the other electrolytic cell.

36. Assume 96500 C as one unit of electricity. If cost of electricity of producing x gm A is Rs x, what is the
cost of electricity of producing x gm Mg?

37. Chromium netal can be plated out from an acidic solution containing CrO3 according to following
equation: CrO3 (aq) + 6H+(aq) + 6 e–Cr (s) + 3H2O
Calculate:
(i) How many grams of chromium will be plated out by 24000 coulombs and
(ii) How long will it take to plate out 1.5 gm of chromium by using 12.5 ampere current

38. Calculate the quantity of electricity that would be required to reduce 12.3 g of nitrobenzene to aniline, if
the current efficiency for the process is 50 percent. If the potential drop across the cell is 3.0 volts, ‘how
much energy will be consumed?

39. How long a current of 2 A has to be passed through a solution of AgNO3 to coat a metal surface of 80 cm2
with 5  m thick layer? Denisty of silver = 10.8 g/cm3.

40. 3A current was passed through an aqueous solution of a unknown salt of Pdn+for 1 Hr. 2.977 g of Pd was
deposited at cathode. Find n.

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41. 50 mL of 0.1 M CuSO4 solution is electrolyzed with current of 0.965 A for a period of 200 sec. The
reactions at electrodes are: Cathode: C u 2   2 e   C u (s ) Anode: 2H2O  O2 + 4H+ + 4e.
Assuming no change in volume during electrolysis, calculate the molar concentration of Cu2+, H+ and
SO42– at the end of electrolysis.

42. A metal is known to form fluoride MF2. When 10A of electricity is passed through a molten salt for 330
sec, 1.95g of metal is deposited. Find the atomic weight of M. What will be the quality of electricity
required to deposit the same mass of Cu from CuSO4?

43. 10g fairly concentrated solution of CuSO4 is electrolyzed using 0.01F of electricity. Calculate:
(a) The weight of resulting solution (b) Equivalents of acid or alkali in the solution.

44. An electric current is passed through electrolysis cells in series one containing Ag (NO3) (aq) and other
H2SO4 (aq). What volume of O2 measured at 25ºC and 750 mm Hg pressure would be liberated from
H2SO4 if (a) one mole of Ag+ is deposited from AgNO3 solution
(b) 8 × 1022 ions of Ag+ are deposited from AgNO3 solution.

45. Cadmium amalgam is prepared by electrolysis of a solution of CdC2 using a mercury cathode. How long
should be a current of 5 A be passed in order to prepare 12% Cd – Hg amalgam on a cathode of 2gm Hg
(Cd = 112.4)

46. After electrolysis of NaCl solution with inert electrodes for a certain period of time. 600 mL of the
solution was left. Which was found to be 1N in NaOH. During the same time, 31.75g of Cu was deposited
in the copper voltameter in series with the electrolytic cell. Calculate the percentage yield of NaOH
obtained.

47. Three electrolytic cells A, B, C containing solution of ZnSO4, AgNO3 and CuSO4, respectively are
connected in series. A steady current of 2 ampere was passed through them until 1.08 g of silver deposited
at the cathode of cell B. How long did the current flow? What mass of copper and of zinc were deposited?

48. Copper sulphate solution (250 mL) was electrolysed using a platinum anode and a copper cathode. A
constant current of 2 mA was passed for 16 minutes. It was found that after electrolysis the concentration
of the solution was reduced to 50% of its original value. Calculate the concentration of copper sulphate in
the original solution.

49. A solution of Ni (NO3)2 is electrolysed between platinum electrodes using a current of 5 ampere for 20
minutes. What mass of Ni is deposited at the cathode?
50. A current of 3.7A is passed for 6 hrs. between Ni electrodes in 0.5L of 2 M solution of Ni (NO3)2. What
will be the molarity of solution at the end of electrolysis?

CONDUCTANCE:
Conductivities and cell constant:

51. The resistance of a conductivity cell filled with 0.01 N solution of NaCl is 210 ohm at 18ºC. Calculate the
equivalent conductivity of the solution. The cell constant of the conductivity cell is 0.88 cm–1.

52. The molar conductivity of 0.1 M CH3COOH solution is 4.6 S cm2 mole–1. What is the specific
conductivity and resistivity of the solution?

53. The conductivity of pure water in a conductivity cell with electrodes of cross sectional area 4 cm2 and 2
cm apart is 8 × 10–7 S cm–1
(i) What is resistance of conductivity cell?
(ii) What current would flow through the cell under an applied potential difference of 1 Volt?

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54. Resistivity of 0.1 M KCI solution is 213 ohm cm in a conductivity cell. Calculate the cell constant
resistance is 330 ohm.
55. Resistance of a 0. 1M KCl solution in a conductance cell is 300 ohm and specific conductance of 0. KCl is
1.29 × 10 –2 ohm –1 cm–1. The resistance of 0.1M NaCl solution in the same cell is 380 of Calculate the
equivalent conductance of the 0.1M NaCl solution.
56. For 0.01N KCl, the resistivity 709.22 ohm cm. Calculate the conductivity and equivalent conductance.
57. A solution containing 2.08 g of anhydrous barium chloride is 400 cc of water has a specific conduction
0.0058 ohm–1cm–1. What are molar and equivalent conductivities of this solution.

Application of Kohlrausch’s law:

58. Equivalent conductance of 0.01 N Na2SO4 solution is 112.4 ohm–1 cm2 eq–1. The equivalent conductance
at infinite dilution is 129.9 ohm–1 cm2 eq–1. What is the degree of dissociation in 0.01 N Na2SO4

59. Specific conductance of a saturated solution of AgBr is 8.486 × 10–7 ohm–1 cm–1 at 25ºC. Specific
conductance of pure water at 25ºC is 0.75 x 10–6 ohm–1 cm–2. Am for KBr, agNO3 and KNO3 are 137,
133, 131 (S cm2 mol–1) respectively. Calculate the solubility of AgBr in gm/litre.

60. Saturated solution of AgCl at 25ºC has specific conductance of 1.12 × 10–6 ohm–1 cm–1The
–1 2
  A g  a n d   C l  are 54.3 and 65.5 ohm cm / equi respectively. Calculate the solubility product of
AgCl at 25ºC.

61. Hydrofluoric acid is weak acid. At 25ºC, the molar conductivity of 0.002M HF is 176.2 ohm–1 cm2 moles.
If its Am = 405.2 ohm–1 cm2 mole–1, calculate its degree of dissociation and equilibrium constants the
given concentration.

62. The value of m for HCl. NaCl and CH3CO2 Na are 426.1, 126.5 and 91 S cm2 mol–1 respectively
calculate the value of m for acetic acid. If the equivalent conductivity of the given acetic acid is 48.15 at
25ºC calculate its degree of dissociation.

63. Calculate the specific conductance of a 0.1 M aqueous solution of NaCl at room temperature, given that
the mobilities of Na+ and Cl– ions at this temperature are 4.26 × 10–8 and 6.80 × 10–8 m2 V–1s–1
respectively.

64. For the strong electrolytes NaOH, NaCl and BaC2 the molar ionic conductivities at infinite dilution are
248.1 × 10–4, 126.5 × 10–4 amd 280.0 × 10–4 mho cm2 mol–1 respectively. Calculate the molar conductivity
of Ba (OH)2 at infinite dilution.

65. At 25ºC,   ( H  )  3 .4 9 8 2  1 0  2 S m 2 m o l  1 and   ( O H  )  1 .9 8  1 0  2 S m 2 m o l1 Given: Sp.

Conductance = 5.7 × 10–6 S m –1 for H2O, determine, pH and Kw.

Exercise - 2
Single choice correct with multiple options

1. A standard hydrogen electron has zero electrode potential because

(A) hydrogen is easier to oxidise
(B) this electrode potential is assumed to be zero
(C) hydrogen atom has only one electron
(D) hydrogen is the lighest element.

2. If the pressure of H2 gas is increased from 1 atom to 100 atm keeping H+ concentration constant at 1 M.
the change in reduction potential of hydrogen half cell at 25ºC will be
(A) 0.059V (B) 0.59V (C) 0.0295V (D) 0.118V
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3. The equilibrium constant for the reaction Sr (s) + Mg+2 (aq)  Sr+2 (aq) + Mg (s) is 2.69 × 1012 at 25ºC
The Eo for a cell made up of the Sr/Sr+2 and Mg+2/Mg half cells
(A) 0.3667V (B) 0.7346V (C) 0.1836V (D) 5V

4. When ∆ G = –100 kJ mol–1 and n = 1, the potential of the cell is

(A) 2 V (B) 1 V (C) 10 V (D) 5 V

5. For a cell reaction involving a two electron change, the standard emf of the cell is found to be 0.295V at
25ºC. The equilibrium constant of the reaction at 25ºC will be
(A) 1 × 10–10 (B) 29.5 × 10–2 (C) 10 (D) 1 × 1010

6. Standard electode potential of three metals A, B and C are + 0.5V, –3.0V and –1.2V respectively. The
reducting power of these metals are
(A) B > C > A (B) A > B > C (C) C > B > A (D) A > C > B

7. Consider the following Eo values:

E0   0.77 V ; E 0   0.14V
Fe 3  |Fe 2  Sn 2  |Sn
Under standard conditions the potential for the reaction is
Sn(s) + 2Fe3+ (aq)  2Fe2+ (aq) + Sn2+ (aq)
(A) 1.68 V (B) 1.40 V (C) 0.91V (D) 0.63V

8. The standard e.m.f of a cell, involving one electron change is found to be 0.591 V at 25ºC. The
equilibrium constant of the reaction is (F=96, 500 C mol–1, R = 8.314 JK–1 mol–1)
(A) 1.0 × 101 (B) 1.0 × 105 (C) 1.0 × 1010 (D) 1.0 × 1030

9. In a cell that utilizes the reaction Zn (s) + 2H+ (aq)  Zn2+ (aq) + H2 (g) addition of H2SO4 to cathode
compartment, will
(A) lower the E and shift equilibrium to the left
(B) lower the E and shift equilibrium to the right
(C) increase the E and shift the equilibrium to the right
(D) increase the E and shift the equilibrium to the left

10. The E o values for Cr, Mn Fe and Co are-0.41, + 1.57, 0.77 and + 1.97V respectively. For which
M 3  |M 2 
one of these metals the change in oxidation state from +2 to +3 is easiest?
(A) Cr (B) Mn (C) Fe (D) Co

11. Reduction potential of silver wire dipped in 0.1 M HC solution saturated with AgCl is 0.25 V. If
E oA g |A g   –0.799 V, the Ksp of AgCl in pure water will be

(A) 2.95 × 10–11 (B) 5.1 × 10–11 (C) 3.95 × 10–11 (D) 1.95 × 10–11

12. Consider the reaction of extraction of gold from its ore

1 1
Au(s)+2CN–(aq.) + O 2 (g)  H 2 O (l)  Au (CN) 2 ( a q )  O H  ( a q )
4 2
Use the following data to calculate ∆Gº for reaction Kf (Au (CN) 2 = X
O2 + 2H2O + 4e–  4OH– ; Eo = + 0.41 volt
Au3+ + 3e–  Au ; Eo = + 1.5 volt
Au3+ + 2e–  Au+ ; Eo = + 1.4 volt
(A) –RTλnX + 1.29F (B) –RTλnX – 2.11F
1
(C) –RTλn + 2.11 F (D) –RTλnX – 1.29F
X

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13. Consider the following Galvanic cell.

By what value the cell voltage change when concentration of ions in anodic and cathodic compartments
both increased by factor of 10 at 298K.
(A) +0.0591 (B) –0.0591 (C) –0.041 (D) none

14. For the cell

The measured potential at 25ºC is
P t | H 2 ( 0 .4 a t m ) | H  ( p H  1) || H  ( p H  2 ) | H 2 ( 0 .1 a tm ) | P t
(A) –0.1V (B) –0.5 (C) –0.041 (D) none

15. For the fuel cell reaction 2H2(g)  2H2O(l) ; ∆ f H o2 9 8 ( H 2 O , l ) = – 285.5 kJ/mol What is  S o2 9 8 for the
given fuel cell reaction?
Given : O2 (g) + 4H+ (aq) + 4e–  2H2O (l) Eo = 1.23 V
(A) –0.322 J/K (B) –0.635 kJ/K (C) 3.51 kJ/K (D) –0.322 kJ/K

16. The standard reduction potentials of Cu2+/Cu and Cu2+/ Cu+ are 0.337 and 0.153V respectively. The
standard electrode potential of Cu+/Cu half cell is
(A) 0.184V (B) 0.827V (C) 0.521V (D) 0.490V

17. A hydrogen electrode X was placed in a buffer solution of sodium acetate and acetic acid in the ratio a : b
and another hydrogen electrode Y was placed in a buffer solution of sodium acetate and acetic acid in the
ratio b : a. If reduction potential values for two cells are found to be E1 and E2 respectively w.r.t. standard
hydrogen electrode, the pKa value of the acid can be given as
E  E2 E  E1  (E1  E 2 ) E  E2
(A) 1 (B) 2 (C) (D) 1
0.118 0.118 0.118 0.118

18. Given the data at 25ºC,

A g  l   A g l  e  ; E o  0 .1 5 2 V
  RT 
Ag  Ag  e  ; E o  – 0.800 V  2.309  0.059V 
 F 
What is the value of log Ksp for Agl ?
(A) –8.12 (B) + 8.613 (C) –37.83 (D) –16.13

19. For the electrochemical cell, M|M+||X–|X, Eo (M+/M)= 0.44V. and Eo (X/X–) = 0.33V. From this data can
deduce that
(A) M + X  M+ + X– is the spontaneous reaction
(B) M+ + X–  M + X is the spontaneous reaction
(C) Ecell = 0.77 V
(D) Ecell = – 0.77 V

20. How many faradays are required to reduce 1 mol of Br O 3 to Br– ?

(A) 3 (B) 5 (C) 6 (D) 4

21. One gm metal M+2 was discharged by the passage of 1.81 × 1022 electrons. What is the atomic weight of
metals?
(A) 33.35 (B) 133.4 (C) 66.7 (D) 55

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22. Aluminum oxide may be electrolysed at 1000ºC to furnish aluminium metal.
The cathode reaction is Al3+ + 3e–  Al. To prepare 5.12 kg of aluminium metal by this method would
require.
(A) 5.49 × 107 C of electricity (B) 1.83 × 107 C of electricity
4
(C) 5.49 × 10 C of electricity (D) 5.49 × 1010 C of electricity

23. One mole of electron passes through each of the solution of AgNO3, CuSO4 and AlCl3 when Ag, Cu and
Al are deposited at cathode. The molar ratio of Ag, Cu and Al deposited are
(A) 1 : 1 : 1 (B) 6 : 3 : 2 (C) 6 : 3 : 1 (D) 1 : 3 : 6

24 Salts of A (atomic weight =7), B (atomic weight = 27) and C (atomic weight = 48) were electrolysed
under identical conditions using the same quantity of electricity. It was found that when 2.1 g of A was
depositied, the weight of B and C deposited were 2.7 and 7.2g. The valencies of A, B and C respectively
are
(A) 3,1 and 2 (B) 1, 3 and 2 (C) 3, 1 and 3 (D) 2, 3 and 2

25. The density of Cu is 8.94 g cm–3. The quantity of electricity needed to plate an area 10 cm × 10 cm to a
thickness of 10–2 cm using CuSO4 solution would be
(A) 13586C (B) 27172 C (C) 40758C (D) 20348C

26. During electrolysis of an aqueous solution of sodium sulphate, 2.4 L of oxygen at STP was liberated at
anode. The volume of hydrogen at STP, liberated at cathode would be
(A) 1.2L (B) 2.4L (C) 2.6L (D) 4.8L

27. During electrolysis of an aqueous solution of CuSO4 using copper electrodes, if 2.5g of Cu is deposited at
cathode, then at anode
(A) 890 ml of Cl2 at STP is liberated
(B) 445 ml of O2 at STP is liberated
(C) 2.5 g of copper is deposited
(D) a decrease of 2.5g of mass takes place

28. An aqueous solution containing one mole per litre each of Cu(NO3)2, AgNO3, Hg2 (NO3)2, Mg(NO3)2 is
being electrolysed by using inert electodes. The value of standard potentials are
Eo  0.80V,Eo  0.79V,Eo  0.34VandEo  2.3V
Ag |Ag Hg22 |Hg cu2 |cu Mg2 |Mg
With increasing voltage, the sequence of deposition of metals on the cathode will be
(A) Ag, Hg, Cu, Mg (B) Mg, Cu, Hg, Ag
(C) Ag, Hg, Cu (D) Cu, Hg, Ag

29. The charge required for the oxidation of one mole Mn3O4 into MnO 24  in presence of alkaline medium is
(A) 5 × 96500 C (B) 96500 C
(C) 10 × 96500 C (D) 2 × 96500 C

30. The cost at 5 paise /KWH of operating an electric motor for 8 hours which takes 15 amp at 110 V is
(A) Rs. 66 (B) 66 paise (C) 37 paise (D) Rs. 6.60

31. A solution of sodium sulphate in water is electrolysed using inert electrodes. The products at the cathode
and anode are respectively.
(A) H2,O2 (B) O2, H2 (C) O2, Na (D) none

32. When an aqueous solution of lithium chloride is electrolysed using graphite electrodes
(A) Cl2 is liberated at the anode.
(B) Li is deposited at the cathode.
(C) as the current flows, pH of the solution around the cathode remains constant
(D) as the current flows, pH of the solution around the cathode decreases.

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33. The highest electrical conductivity of the following aqueous solutions is of
(A) 0.1 M acetic acid (B) 0.1 M chloroacetic acid
(C) 0.1 M fluoroacetic acid (D) 0.1 M difluoroacetic acid

34. The value of molar conductance of HCl is greater than that of NaCl at a given temperature and
concentration because
(A) ionic mobility of HCI is greater than that of NaCl
(B) the dipole moment of NaCl is greater than that of HCl
(C) NaCl is more ionic than HCl
(D) HCl is Bronsted acid and NaCl is a salt of a strong acid and strong base.

35. Resistance of a conductivity cell filled with a solution of an electrolyte of concentration 0.1 M is 100 .
The conductivity of this solution is 1.29 Sm–1. Resistance of the same cell when filled with 0.2 M of the
same solution is 520 . The molar conductivity of 0.2 M solution of the electrolyte will be
(A) 124 × 10–1 Sm2 mol–1 (B) 1240 × 10–4 Sm2 mol–1
(C) 1.24 S m2 mol–1 (D) 12.4 × 10–4 S m2 mol–1
o
36. The limiting molar conductivities mfor NaCl, KBr and KCl are 126, 152 and 150 S cm2 mol–1
o
respectively. The  mfor NaBr is
(A) 128 S cm2 mol–1 (B) 176 S cm2 mol–1 (C) 278 S cm2 mol–1 (D) 302 S cm2 mol–1

37. The resistance of 0.5 M solution of an electrolyte in a cell was found to be 50. If the electrodes in the
cell are 2.2 cm apart and have an area of 4.4 cm2 then the molar conductivity (in Sm2 mol–1) of the
solutions is
(A) 0.2 (B) 0.02 (C) 0.002 (D) None of these

38 Equivalent conductance of 0.1 M HA (weak acid) solution is 10 Scm2 equivalent–1 and that at infinite
dilution is 200 Scm2 equivalent–1 Hence pH of HA solution is
(A) 1.3 (B) 1.7 (C) 2.3 (D) 3.7

39. If X is specific resistance of the electrolyte solution and y is the molarity of the solution, then m is given
by :
y xy
(A) 1 0 0 0 x (B) 1000 (C) 1000 (D)
y x xy 1000

Exercise - 3
Multiple choice correct with multiple options

1. During discharging of lead storage battery, which of the following is/are true?
(A) H2SO4 is produced (B) H2O is consumed
(C) PbSO4 is formed at both electrodes (D) Density of electrolytic solution decreases

2. Which of the following arrangement will produce oxygen at anode during electrolysis?
(A) Dilute H2SO4 solution with Cu electrons
(B) Dilute H2SO4 solution with inert electrodes.
(C) Fused NaOH with inert electrodes.
(D) Dilute NaCl solution with inert electrodes.

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3. If 270.0 g of water is electrolysed during an experiment performed by Mr. Tom with 75% current
efficiency then
(A) 168L of O2 (g) will be evolved at anode at 1 atm & 273K
(B) Total 504 L gasses will be producted at 1 atm & 273 K.
(C) 336 L of H2 (g) will be evolved at anode at 1 atm & 273K
(D) 45 F electricity will be consumed

4. In an electrochemical process, a salt bridge is used

(A) to maintain electroneutrality in each solution
(B) to complete the circuit so that current can flow
(C) as an oxidizing agent
(D) as a colour indicator

5. When galvanic cell started, with passage of time

(A) spontaneity of the cell reaction decreases; Ecell decreases
(B) reaction quotient Q decreases; Ecell increases
(C) reaction quotient Q increases; Ecell decreases
(D) at equilibrium Q = Kc; Ecell = 0

6. Which one is/are correct among the followings?

Given, the half cell emf’s E 0  2  0.337, E 0  0.521
Cu |C u C u  |C u

(A) Cu+ disproportionates

(B) Cu and Cu2+ comproportionates (reverse of disproportionation into Cu+)
(C) E 0 2  E
0

is positive
C u |C u C u |C u

(D) All of these

7. Pick out the correct statements among the following from inspection of standard reduction potentials
(Assume standards state conditions).
Cl2 (aq.)  2e 
 2Cl (aq.) Eo  1.36volt
Cl2 /Cr 

Br2 (aq.)  2e 

 2Br  (aq.) Eo  1.09volt
Br2 /Br 

l2 (s)  2e 
 2l (aq.) Eo   0.54volt
l2 /l

S2O82 (aq.)  2e 

 2SO24 aq.) Eo  2.00volt
S2O82 /SO24

(A) Cl2 can oxidize SO 2

4
from solution
(B) Cl2 can oxidize B r  an d l from aqueous solution
(C) S 2 O 82  c a n o x id is e C l  , B r  a n d l  from aqueous solution
(D) S2O 8 2
i s a d d e d s lo w ly , B r  can be reduce in presence of Cl

8. The EMF of the following cell is 0.22 volt.

A g (s ) | A g C l | ( s ) | K C l(1M ) | H  (1M ) | H 2 ( g ) (1 a tm ) | p t (s ).
Which of the following will decrease the EMF of cell.
(A) increasing pressure of H2 (g) from 1 atm to 2 atm.
(B) increasing Cl– concentration in Anodic compartment.
(C) increasing H+ concentration in cathodic compartment
(D) decreasing KCl concentration in Anodic compartment.

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9. Which of the following units is correctly matched?
(A) Sl units of conductivity (k) is S m–1
(B) Sl units molar conductivity is S m2 mol–1
(C) Sl unit of ionic mobility is m V–1s–1
(D) All of these

10. The dissociation constant of n-butyric acid is 1.6 × 10–5 and the molar conductivity at infinite dilution is
380 × 10 –4 Sm2 mol–1 The specific conductance of the 0.01 M acid solution is
(A) 1.52 × 10–5 Sm–1 (B) 1.52 × 10–2 Sm –1
–4 –1
(C) 1.52 × 10 Scm (D) None

11. The molar conductivities oNaOAc and oHCl at infinite dilution in water at 25ºC are 91.0 and 426.2 S
cm2/mol respectively. To calculate oHOAc the additional value required is/are
(A)  o H 2 O (B)  o KCl (C)  o N aO H (D)  o N aC l

Exercise –4
Section - I : Numerical Value/Subjective Type Questions

1. All the energy from the reaction X  Y, ∆r Go = – 193 kJ mol–1 is used for oxidizing
M+ as M+  M3+ +2e– Eo = – 0.25 V. Under standard conditions, the number of moles of M+ oxidized
when one mole of X is converted to Y is [F = 96500 C mol–1]

2. The molar conductivity of a solution of weak acid HX (0.01 M) is 10 times smaller than the molar
conductivity of a solution of a weak acid HY (0.10 M). If  ox   oy , the difference in their
pKa values, pKa (HX) – pKa (HY) is (Consider degree of ionization of both acids to be <<1)

3. For the following electrochemical cell at 298 K,

Pt(s) H2 (g, 1bar) H (aq,1M) M4 (aq),M2 (aq) Pt(s)
[M2 (aq)]
Ecell  0.092V when  10.
[M4 (aq)]
RT
Given : E o  0.151V, 2.303  0.059V
M 4  /M 2  F
The value of x is:

4. The equivalent conductance of 0.10 N solution of MgCl2 is 97.1 mho cm2 equi–1 at 25ºC. a cell with
electrode that are 1.5 cm2 in surface area and 0.5 cm apart is filled with 0.1 N MgCl2 solution. How much
current will flow when potential difference between the electrodes is 5 volt, write the answer by dividing
0.01456.

5. A dilute aqueous solution of KCI was placed between two electrodes 10 cm apart, across which a
potential of 6 volt was applied. How far would the K+ ion move in 2 hours at 25ºC? Ionic conductance of
K+ ion at infinite dilution at 25ºC is 73.52 ohm–1 cm2 mole–1, write your answer by dividing 3.29.

6. When a solution of specific conductance 1.342 ohm–1 metre–1 was placed in a conductivity cell with
parallel electrodes, the resistance was found to be 170.5 ohm. Area of electrodes is 1.86 × 10–4 m2
Calculate separation of electrodes. If answer is X .25 ×10–2 metre then find X.

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7. The specific conductance at 25ºC of a saturated solution of SrSO4 is 1.482 × 10–4 ohm–1 cm–1 while that
of water used is 1.5 × 10–6 mho cm–1. Determine at 25ºC the solubility in gm per litre of SrSO4 in water,
Molar ionic conductance of Sr2+ and SO42– ions at infinite diluti8on are 59.46 and 79.8 ohm–1 cm2 mole–1
respectively. [Sr = 87.6, S = 32, O = 16], write the answer by dividing 0.01934.

8. Calculate the solubility and solubility product of Co2 [Fe(CN)6] in water at 25ºC from the following data:
conductivity of a saturated solution of Co2 [Fe(CN)6] is 2.06 × 10–6 –1 cm–1and that of water used
4.1×10–7–1 cm–1. The ionic molar conductivities of Co2+ and Fe (CN)64– are 86.0 –1 cm2 mol–1 and
444.0–1 cm2 mol–1, if answer is X .682 × 10–17, Find X.

9. We have taken a saturated solution of AgBr.KSP of Ag Br is 12 × 10–14. If 10–7 mole of AgNO3 are added 1
litre of this solution find conductivity (specific conductance) of this solution in terms of 10–7 Sm–1 mol–1
Given:  0( A g  )  6  1 0  3 S m 2 m o l  1 ;  0( B r  )  8  1 0  3 S m 2 m o l  1 ;  0   7  10  3 S m 2 m ol  1
( NO3 )

Give your answer by dividing 11.

Section - II : Match the Column

10. Column I Column II

(Electrolysis product using inert electrode)
(A) Dilute solution of HCl (P) O2 evolved at anode
(B) Dilute solution of NaCl (Q) H2 evolved cathode
(C) Concentrated solution of NaCl (R) Cl2 evolved at anode
(D) AgNO3 solution (S) Ag deposition at cathode

11. Column I Column II

(Electrolysis product using inert electrode)
(A) Z n | Z n  2 ( c 1 ) || M g 2  ( c 1 ) | M g (P) E 0c e l l  0
(B) Z n | Z n  2 ( c 1 ) || A g  ( c 1 ) | A g (Q) E 0c e ll  0
(C) A g | A g  || A g  | A g (R) E 0c e ll   v e
(D) F e | F e 2  || A g  | A g (S) E 0c e ll   v e

12. Column I Column II

(Factors on which dependency exist)
(A) Molar conductance (P) Temperature
(B) Emf of a concentration cell (Q) Concentration of species involved
(C) Electrode potential (R) Nature of substance involved
(D) Standard reduction potential (S) Pressure of gas

Section - III : Comprehension

Comprehension#1 (13 to 15)

A sample of water from a large swimming pool has a resistance of 10000  at 25ºC. when placed in a
certain conductance cell. When filled with 0.02 M KCl solution, the cell has a resistance of 100 at 25ºC.
585gm of NaCl were dissolved in the pool, which was thoroughly stireed. A sample of this solution gave a
resistance of 8000 .

13. Cell constant (in cm–1) conductance cell is

(A) 4 (B) 0.4 (C) 4 × 10–5 (D) None of these

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14. Conductivity (Scm–1) of H2O is
(A) 4 × 10–2 (B) 4 × 10–3 (C) 4 × 10–5 (D) None of these

15. Volume (in Litres) of water in the pool is

(A) 1.25 × 105 (B) 1250 (C) 12500 (D) None of these

Comprehension#2 (16 to 18)

Tollen’s reagent is used for the detection of aldehyde when a solution of AgNO3 is added to glucose with
NH4OH then gluconic is formed
Ag  e  Ag ; E 0re d  0 .8 V
C 6 H12 O 6  H 2 O  C 6 H12 O 7 (Gluconic acid) + 2H+ + 2e– ; E 0red   0 .0 5 V

Ag  NH3 2  e   Ag (s)  2NH 3 ; Eo = – 0.337 V
RT F
[Use 2.303 ×  0.0592 and  38.92 at 298K]
F RT

16. Ag+ + C6H12O6 + H2O  2 Ag (s) + C6H12O7 + 2H+

Find in K of this reaction
(A) 66.13 (B) 58.38 (C) 28.30 (D) 46.29

17. When ammonia is added to the solution, pH is raised to 11. Which half-cell reaction is affected by pH and
by how much?
(A) Eoxd will increase by a factor of 0.65 from E oo x d
(B) Eoxd will decrease by a factor of 0.65 from E o
oxd

(C) Ered will increase by a factor of 0.65 from E ore d

(D) Ered will decrease by a factor of 0.65 from E ore d

18. Ammonia is always is added in this reaction. Which of the following must be incorrect?
(A) NH3 combines with Ag+ to form a complex.
(B) Ag(NH3) 2 is a weaker oxidizing reagent than Ag+
(C) In absence of NH3 silver salt of gluconic acid is formed.
(D) NH3 has a affected the standard reduction potential of glucose/gluconic acid electrode.

Comprehension#3 (19 to 21)

Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules
(approximately 6.023 × 1023) are present in a few grams of any chemical compound varying with their
atomic/ molecular masses. To handle such large number convenlently, the mole concept was introduced.
This concept has implications in diverse areas such as analytical chemistry, biochemistry,
electrochemistry and radiochemistry. The following example illustrates a typical case, involving
chemical/electrochemical reaction, which requires a clear understanding of the mole concept.
A 4.0 molar aqueous solution of NaCl is prepared and 500 mL of this solution is electrolysed. This leads
to the evolution of chlorine gas at one of the electrodes (atomic mass : Na = 23, Hg = 200; 1
Faraday=96500 coulombs)

19. The total number of moles of chlorine gas evolved is

(A) 0.5 (B) 1.0 (C) 2.0 (D) 3.0

20. If the cathode is a Hg electrode, the maximum weight (g) of amalgam formed from this solution is
(A) 200 (B) 225 (C) 400 (D) 446

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21. The total charge (coulombs) required for complete electrolysis is
(A) 24125 (B) 48250 (C) 96500 (D) 193000

Comprehension#4 (22 to 23)

Redox reactions play a pivoted role in chemistry and biology. The values of standard redox potential (Eo)
of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a
Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell
reactions (acidic medium) along with their Eo (V with respect to normal hydrogen electrode) values. Using
this data obtain the correct explanations to Questions 14-16.
l2 + 2e–  2l– Eo = 0.54
Cl2 + 2e– 2Cl– Eo = 1.36
Mn + e  Mn
3+ – 2+
Eo = 1.50
Fe3+ + e–  Fe2+ Eo = 0.77
O2 + 4H + 4e  2H2O
+ –
Eo = 1.23

22. Among the following, identify the correct statement.

(A) Chloride ion is oxidised by O2 (B) Fe2+ is oxidised by iodine
(C) Iodine ion is oxidised by chlorine (D) Mn2+ is oxidised by chlorine

23. While Fe3+ is stable, Mn3+ is not stable in acid solution because
(A) O2 Oxidises Mn2+ to Mn3+ (B) O2 oxidises both Mn2+ to Mn3+ and Fe2+ to Fe3+
(C) Fe3+ oxidises H2O to O2 (D) Mn3+ oxidises H2O to O2

Exercise –5
Revision exercise (Moderate to Tough)

1. The emf of the cells obtained by combining Zn and Cu electrode of a Daniel cell with N calomel electrode
in two different arrangements are 1.083V and 0.018V respectively at 25ºC. If the standard reduction
potential of N calomel electrode is 0.28V and that of Zn is –0.76V. Find the emf of Daniel cell.

2. Given the standard reduction potential TI+ + e–  Tl, Eo = – 0.34 V and TI3+ + 2e–  TI+, Eo = 1.25V.
Examine the spontaneity of the reaction, 3Tl+  2TI + TI3+. Also find Eo for this disproportination

3. The emf of the cell A g A g l K l(0 .0 5 M ) A g N O 3 (0 .0 5 M ) A g is 0.788V. Calculate the solubility product of
Agl.

4. The cell Pt, H2 (1atm) |H+ (pH = ×) || Normal calomel Electrode has an EMF of 0.67V at 25ºC. Calculate
the pH of the solution. The oxidation potential of the calomel electrode on hydrogen scale is –0.28V.

5. Estimate the cell potential of a Daniel cell having 1 M Zn++ & originally having 1 M Cu ++ after sufficient
NH3 has been added to the cathode compartment to make NH3 concentration 2 M. K f f o r[ C u ( N H 3 ) 4 ] 2 
= 1× 1012, Eo for the reaction, Z n  C u 2   Z n 2   C u is 1.I V.

6. Consider the cell Ag|AgBr(s) Br–(aq)||Cl–(aq)|AgCl(s)|AgCl(s)Ag at 25ºC. The solubility prouct constants
of AgBr & AgCl are respectively 5 × 10–13 & 1 × 10–10. For what ratio of the concentrations of Br– & Cl–
ions would the emf of the cell be zero?

7. The pKsp of Agl is 16.07. If the Eo value for Ag+Ag is 0.7991 V. Find the Eo for the half cell – reaction
Agl (s) + e–  Ag + I–.

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8. Calculate the potential of an indicator electrode versus the standard hydrogen electrode, which originally
contained 0.1 M MnO4– and 0.8M H+ and which was treated with 90% of the Fe2+ necessary to reduce all
the MnO4- to Mn+2.
MnO4– + 8H+ + 5e  Mn2+ + 4H2O, Eo = 1.51V

9. Kd for complete dissociation of [Ag(NH3)2]+ into Ag+ and 2NH3 is 6 × 10–8. Calculate Eo for the following
half reaction; Ag (NH3)2+ + e–  Ag + 2NH3 Ag+ + e–  Ag, Eo = 0.799 V

10. For the galvanic cell: A g A g C l(s ) K C I(0 .2 M ) K B r (0 .0 0 1M ) A g B r (s ) A g , Calculate the EMF generated
and assign correct polarity to each electrode for a spontaneous process after taking into account the cell
reaction at 25ºC. [Ksp (AgCλ) = 2.8 × 10–10; Ksp(AgBr)=3.3 × 10–13]

11. Determine the degree of hydrolysis and hydrolysis constant of aniline hydrochloride in M/32 solution of
salt at 298 K from the following cell data at 298 K.

Pt H 2 (1 a t m ) H (1 M ) M / 3 2 C 6 H 5 N H 3 C l H 2 (1 a t m ) P t ; E c e ll   0 .1 8 8 V .

12. o
Given, E = –0.268V for the Cl C l  P b C l2 Pb couple and –0.126 v for the Pb2+Pb couple, determine Ksp
for PbCl2 at 25ºC?

13. Calculate the equilibrium constant for the reaction:

3S n (s )  2 C r2 O 7 2 –  2 8 H   3S n 4   4 C r 3   1 4 H 2 O
2
E o
fo r S n / S n  0 .1 3 6 V E o fo r S n 2  / S n 4    0 .1 5 4 V

Eo for Cr 2O72 / Cr3(aq)  1.33V

14. Calculate the voltage of the cell at 25

Mn(s)|Mn(OH)2 (s) | Mn2+ (× M). OH– (1.00 × 10–4M)|| Cu2+ (0.675M) | Cu (s)
given that Ksp = 1.9 × 10–13 for Mn (OH)2 (s) Eo (Mn2+/Mn) = – 1.18V

15. Calculate the voltage, E, of the cell

Ag(s) AglO3 (s) Ag (M),HIO3(0.300M) Zn2 (0.175M) Zn(s) if Ksp = 3.02 × 10–8 for AglO3(s) and
Ka = 0.162 for HIO3.

16. The voltage of the cell

Pb(s) PbSO 4 (s) NaHSO 4 (0.600M) Pb 2  (2.50  10 5 M) Pb(s) , is E  0.061 V. Calculate K 2

 [H  ][SO 4 2  ] / .[HSO 4  ],
the dissociation constant for HSO4, Given: Pb (s)+ SO42– (aq)  PbSO4 (s) + 2e– (Eo = 0.356), Eo
(Pb2+/Pb) = –0.126V.

2  3 2 4
17. The voltage of the cell Zn(s) Zn(CN)4 (0.450M),CN (2.65 10 M) Zn (3.84 10 M) Zn(s) is
E = + 0.099 V. Calculate the constant Kf for Zn2+ + 4CN– Zn (CN)42– , the only Zn2+ +| CN–
complexation reaction of importance.

18. An external current source giving a current of 5.0 A was joined with Daniel cell and removed after 10 hrs.
Before passing the current the LHE and RHE contained 1L each of 1M Zn2+ and Cu2+ respectively. Find
the EMF supplied by the Daniel cell after removel of the external current source. Eo of Zn2+/Zn and Cu2+/
Cu at 25ºC is –0.76 and +0.34V respectively.

19. Same quantity of electricity is being used to liberate iodine (at anode) and a metal X (at cathode). The
mass of X deposited is 0.617g and the iodine is completely reduced by 46.3 cc of 0.124M sodium
thiosulphate. Find the equivalent mass of x.

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20. The standard reduction potential values, Eo(Bi3+|Bi) and Eo(Cu2+|Cu) are 0.226V and 0.344V respectively
A mixture of salts of bismuth and copper at unit concentration each is electrolysed at 25ºC. to what value
can [Cu2+] be brought down before bismuth starts to deposit, in electrolysis.

21. In a fuel cell, H2 & O2 react to produce electricity. In the process, H2 gas is oxidized at the anode & O2 at
the cathode. If 67.2 litre of H2 at STP react in 15 minutes, what is the average current produced? If the
entire current is used for electrode depostition of Cu form Cu (II) solution, how many grams of Cu will be
deposited? Anode H2 + 2OH- 2H2O + 2e– Cathode : O2 + 2H2O + 4e –  4 OH–

22. One of the methods of preparation of per disulphuric acid, H2S2O8, involve electrolytic oxidation of
H2SO4 at anode (2H2SO4  H2S2O8 + 2H+ +2e–) with oxygen and hydrogen as by-products. In such an
electrolysis 9.722 L of H2 and 2.35 L of O2 were generated at STP. What is the weight of H2S2O8 formed?

23. During the discharge of a lead storage battery the density of sulphuric acid fell from 1.294 to 1.139 g. ml–
1
. H2SO4 of density 1.294 g mL–1 is 39% and that of density 1.139 g mL–1 is 20% by weight. The battery
holds 3.5L of acid and the volume practically remains constant during the discharge. Calculate the number
of ampere hours for which the battery must have been used. The discharging reactions are:
Pb + SO42–  PbSO4 + 2e– (anode)
Pb O2 + 4H + SO4 + 2e  PbSO4 + 2H2O
+ 2– –
(cathode)

24. A lead storage cell is discharged which causes the H2SO4 electrolyte to change from a concentration of
34.6% by weight (density 1.261g ml–1 at 25ºC) to 27% by weight. The original volume of electrolyte is
one litre. Calculate the total charge released at anode of the battery. Note that the water is produced by the
cell reaction as H2SO4 is used up. Over all reaction is Pb (s)+PbO2(s)+2H2SO4(l)2PbSO4(s) +2H2O(l)

25. A current of 3 amp was passed for 2 hour through a solution of CuSO4, 3g of Cu2+ ions were deposited as
Cu at cathode. Calculate percentage current efficiency of the process.

26. An acidic solution of Cu2+ salt containing 0.4 g of Cu2+ is electrolyzed until all the copper is deposited.
The electrolysis is continued for seven more minutes with the volume of solution kept at 100 ml and the
current at 1.2 amp. Calculate the volume of gases evolved at STP during the entire electrolysis.

27. In the refining of silver by electrolytic method what will be the weight of 100 gm Ag anode if 5 ampere
current is passed for 2 hours? Purity of silver is 95% by weight.

28. Dal lake has water 8.2 × 1012 litre approximately. A power reactor produces electricity at the rate of
1.5 × 106 coulomb per second at an appropriate voltage. How many years would it take to electrolyse the
lake?

29. Assume that impure copper contains only iron, silver and a gold as impurities. After passage of 140A, for
482.5s of the mass of the anode decreased by 22.260g and the cathode increased in mass by 22.011 g.
Estimate the % iron and % copper originally present. (Some amount of hydrogen evolved at cathode)

30. 100 ml CuSO4 (aq) was electrolyzed using inert electrodes by passing 0.965 A till the pH of the resulting
solutions was 1. The solution after electrolysis was neutralized, treated with excess KI and titrated with
0.04M Na2S2O3. Volume of Na2S2O3 required was 35 ml. Assuming no volume change during
electrolysis, calculate: (a) duration of electrolysis if current efficiency is 80% (b) initial concentration (M)
of CuSO4

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Exercise –6
Section – I : JEE (Advanced) Questions Previous Years
1. Standard electrode data are useful for understanding the suitability of an oxidant in a redox titration some
half cell reactions and their standard potential are given below:
M n O 4 ( a q )  8 H  ( a q )  5 e   M n 2  ( a q )  4 H 2 O () ; E = 1.51V
o

C r2 O 72  ( a q )  1 4 H  ( a q )  6 e   2 C r 3  (a q )  7 H 2 O () ; E = 1.38V
o

F e 3  ( a q )  e   F e 2  (a q ); E o  0 .7 7 V

C l 2 ( g )  2 e   2 C l  (a q ); E o  1 .4 0 V [JEE 2002]
Identify the only incorrect statement regarding quantitative estimation of aqueous Fe(NO3)2
(A) Mn O 4 can be used in aqueous HCl (B) C r2 O 72  can be used in aqueous HCl
(C) Mn O 4 can be used in aqueous H2SO4 (D) C r2 O 72  can be used in aqueous H2SO4
2. In the electrolytic cell, flow of electrons is from: [JEE 2003]
(A) Cathode to anode in solution (B) Cathode to anode through external supply
(C) Cathode to anode through internal supply (D) Anode to cathode through internal supply.

3. Z n | Z n 2  ( a  0 .1M ) || F e 2  ( a  0 .0 1M ) | F e. The emf of the above cell is 0.2905V. Equilibrium constant

for the cell reaction is [JEE 2004]
(A) 1 0 0 .3 2 / 0 .0 59 1 (B) 1 0 0.32 / 0.029 5
(C) 1 0 0.26 / 0.029 5
(D) e 0.32/ 0.295

4. The half cell reactions for rusting of iron are: [JEE 2005]
1
2H   O 2  2e   H 2 O;E 0  1.23V, Fe2   2e   Fe; E0 = – 0.44 V ∆ G0 (in kJ) for the reaction is:
2
(A) –76 (B) –322 (C) –122 (D) –176

5. Electrolysis of dilute aqueous NaCl solution was carried out by passing 10 milli ampere current. The time
required to liberate 0.01 mol of H2 gas at the cathode is (1 Faraday = 96500 C mol–1) [JEE 2008]
4 4 4 4
(A) 9.65 × 10 sec (B) 19.3 × 10 sec (C) 28.95 × 10 sec (D) 38.6 × 10 sec

6. For the reaction of N O 3 ion in an aqueous solution, Eo is + 0.96 V. Values of Eo for some metal ions are
given below
V 2 (aq )  2e   V E o   1.19 .V

F e 3  ( a q )  3e   F e E o   0.04V
A u 3  ( a q )  3e   A u E o   1 .4 0 V
2 
Hg (a q )  2 e  Hg E o   0 .0 8 6 V
The pairs (s) of metal that is (are) oxidized by N O 3 in aqueous solution is (are) [JEE 2009]
(A) V and Hg (B) Hg and Fe (C) Fe and Au (D) Fe and V

7. The concentration of potassium ions inside a biological cell is at least twenty times higher than the
outside. The resulting potential difference across the cell is important in several process such as
transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration
cell involving a metal M is:
M (s) M (s ) | M  ( a q ; 0 .0 5 m o la r ) || M  ( a q ; | m o la r ) | M (s )
For the above electrolytic cell the magnitude of the cell potential |Ecell| = 70mV. [JEE 2010]
For the above cell:-
(A) Ecell < 0; ∆ G > 0 (B) Ecell > 0; ∆ G < 0 (C) Ecell <0 ; ∆Gº > 0 (D) Ecell > 0 ; ∆Gº < 0

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8. If the 0.05 molar solution of M+ is replaced by a 0.0025 molar M+ solution, then the magnitude of the cell
potential would be:-
(A) 35 mV (B) 70 mV (C) 140 mV (D) 700 mV

9. Consider the following cell reaction

2 F e (s )  O 2 (g )  4H 
(aq )  2 F e 2  (aq )  2 H 2 O (l) E o  1 .67 V
At [Fe2+] = 10–3 M, P(O2) = 0.1 atm and pH= 3, the cell potential at 25ºC is
(A) 1.47V (B) 1.77 V (C) 1.87 V (D) 1.57 V

10. AgNO3 (aq.) was added to an aqueous KCl solution gradually and the conductivity of the solution was
measured. The plot of conductance (A) versus the volume of AgNO3 is:

(A) P (B) Q (C) R (D) S

Paragraph for Questions Nos. 11 to 12

The electrochemical cell shown below is a concentration cell.

MlM2+ (saturated solution of a sparingly soluble salt, MX2) || M 2  ( 0 .0 0 1m o ld m  3 ) || M The emf of the cell
depends on the difference in concentration of M2+ ions at the two electrodes. The emf of the cell at 298 is
0.059V [JEE 2012]

11. The solubility product (Ksp ; mol3 dm–9) of MX2 at 298 based on the information available the given
concentration cell is (take 2.303 × R × 298/F = 0.059 V)
(A) 1 × 10–15 (B) 4 × 10–15 (C) 1 × 10–12 (D) 4 × 10–12

12. The value of ∆G(kJ mol–1) for the given cell is (take 1F=96500 C mol–1)
(A) –5.7 (B) 5.7 (C) 11.4 (D) –11.4

13. An aqueous solution of X is added slowly to an aqueous solution of Y as shown in List I. the variation in
conductivity of these reactions is given in List. II. Match List I with List II and select the correct answer
using the code given below the lists: [JEE (Adv.) 2013]
List – I List – II
(P) (C2H5 )3 N  CH3 COOH (1) Conductivity decreases and then increases
x Y

(Q) Kl(0.1M)  Ag N O 3 (0.01M) (2) Conductivity decreases and then does not change much
X Y
(R) CH 3 C OOH  K O H (3) Conductivity increase and
X Y
then does not change much
(S) Na O H  H l (4) Conductivity does not change much and then increases
X Y
Codes:
P Q R S
(A) 3 4 2 1
(B) 4 3 2 1
(C) 2 3 4 1
(D) 1 4 3 2

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14. The standard reduction potential data at 25ºC is given below. [JEE (Adv.) 2013]
E o ( F e 3  , F e 2  )   0 .7 7 V ;
E o ( F e 2  , F e )   0 .4 4 V
E o ( C u 2  , C u )   0 .3 4 V ;
E o ( C u  , C u )   0 .5 2 V ;
E o [ O 2 ( g )  4 H   4 e   2 H 2 O ]   1 .2 3 V ;
E o [ O 2 (g )  2 H 2 O  4 e   4 O H  ]   0 .4 0 V
E o ( C r 3  , C r )   0 .7 4 V ;
E o ( C r 2  , C r )   0 .9 1V
Match Eo of the redox pair in List 1 with the values given in List II and select the correct answer using the
code given below the lists:
List-I List –II
(P) E o ( F e 3  , F e ) 1. – 0.18V
 4H   4OH  )
(Q) E o (4H 2 O  2. – 0.4 V
(R) E o ( C u 2   C u  2C u  ) 3. – 0.04 V
(S) E o ( C r 3  , C r 2  ) 4. – 0.83 V

Codes:
P Q R S
(A) 4 1 2 1
(B) 4 3 4 1
(C) 1 2 3 4
(D) 3 4 1 2

15. In a galvanic cell, the salt bridge [JEE (Adv.) 2014]

(A) does not participate chemically in the cell reaction.
(B) Stops the diffusion of ions from one electrode to another.
(C) is necessary for the occurrence of the cell reaction.
(D ensures mixing of the two electrolytic solutions.

16. Copper is purified by electrolytic refining of blister copper. The correct statements (s) about this process is
(are) [JEE (Adv.) 2015]
(A) Impure Cu strip is used as cathode
(B) Acidified aqueous CuSO4 is used as electrolyte
(C) Pure Cu deposits at cathode
(D) Impurities settle as anode-mud

17. Some standard electrode potentials at 298 K are given below:

Pb2+/Pb –0.13 V
2+
Ni /Ni –0.24 V
2+
Cd /Cd –0.40 V
Fe2+/Fe –0.44 V
To a solution containing 0.001 M of X2+ and 0.1 M of Y2+, the metal rods X and Y are inserted (at 298K)
and connected by a conductng wire. This resulted in dissolution of X. The correct combination (s) of X
and Y, respectively, is(are) [JEE (Adv.) 2021, P-2]
Given: Gas constant, R = 8.314 JK mol , Faraday constant, F = 96500 C mol–1)
–1 –1

(A) Cd and Ni (B) Cd and Fe (C) Ni and Pb (D) Ni and Fe

At 298 K, the limiting molar conductivity of a weak monobasic acid is 4 × 102 S cm2 mol–1. At 298 K, for
an aqueous solution of the acid the degree of dissociation is  and the molar conductivity is y × 102 S cm2
mol–1

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18. The value of  is ___________. [JEE (Adv.) 2021, P-2]
19.  
The reduction potential E 0 , in V of MnO 4 (aq) / Mn (s) is_____________. [JEE (Adv.) 2022, P-1]

[Given : E 0  1.68V; E 0  1.21V; E 0   1.03V ]

( M nO 
4 (aq ) / M nO 2 (s )) ( M nO 2 (s ) / M n 2  (aq )) ( M n 2  ( aq ) / M n (s ))

20. Consider the strong electrolytes ZmXn, UmYp and VmXn. Limiting molar conductivity (0) of UmYp and
VmXn are 250 and 440 S cm2 mol–1, respectively. The value of (m + n + p) is _________.
Given:

Ion Zn+ Up+ Vp+ Xm– Ym–

 (S cm mol )
0 2 –1 50.0 25.0 100.0 80.0 100.0
 is the limiting molar conductivity of ions
0

The plot of molar conductivity () of ZmXn vs C1/2 is given below. [JEE (Adv.) 2022, P-2]

21. The correct option(s) about entropy (S) is (are) [JEE (Adv.) 2014]
[R = gas constant, F = Faraday constant, T = Temperature]
dE cell R
(A) For the reaction, M(s) + 2H+ (aq)  H2(g) + M2+(aq), if  , then the entropy change of the
dT F
reaction is R (assume that entropy and internal energy changes are temperature independent).
(B) The cell reaction, Pt (s) | H2(g, 1 bar)|H+ (aq, 0.01M) || H+ (aq, 0.1M) | H2(g, 1 bar) | pt(s), is an
entropy driven process.
(C) For racemization of an optically active compound,  S > 0.
(D) S > 0, for [Ni(H2O)6]2+ + 3 en  [Ni(en)3]2+ + 6H2O (where en = ethylenediamine).

22. Plotting 1/m against cm for aqueous solutions of a monobasic weak acid (HX) resulted in a straight line
with y-axis intercept of P and slope of S. The ratio P/S is [JEE (Adv.) 2023, P-1]
[m = molar conductivity]
 0m = limiting molar conductivity
c = molar concentration
Ka = dissociation constant of HX]
2.303RT 0
(A) Ka  0.059V (B) Kam / 2
F
(C) 2Kam
0

(D) 1/ K a  0m 
Section – II : JEE (Main) 2019, 2020 Questions Previous Years
1. The anodic half-cell of lead-acid battery is reacharged using electricity of 0.05 Faraday. The amount of
PbSO4 electrolyzed in g during the process is:
(Molar mass of PbSO4 = 303 g mol–1)
(1) 22.8 (2) 15.2 (3) 7.6 (4) 11.4

2. If the standard electrode potential for a cell is 2 V at 300 K, the equilibrium constant (K) for the reaction
 Zn2+ (aq) + Cu(s) at 300 K is approximately (R=8 JK–1 mol–1; F = 96000 C mol–1)
Zn(s)+ Cu2+ (aq) 
(1) e–80 (2) e–160 (3) e320 (4) e160

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3. In the cell Pt (s)| H2(g) (1 bar)/ HCl (aq)||AgCl(s)/Ag(s)|Pt(s), the cell potential is 0.92 V when a 10–6
molal HCl solution is used. The standard electrode potential of (AgCl/Ag, Cl–) electrode is:
 2.303RT 
Given :  0.06V at 298K
 F 
(1) 0.94 V (2) 0.76 V (3) 0.40 V (4) 0.20 V

4. For the cell Zn(s) | Zn2+ (aq) || Mx+ (aq) | M (s), different half cells and their standard electrode potentials
are given below:

Mx+ (aq)/ M(s) Au3+(aq)/ Au(s) Ag+(aq)/ Ag(s) Fe3+(aq)/ Fe2+ (aq) Fe2+(aq)/Fe(s)
EoMX /M /(V) 1.40 0.80 0.77 –0.44

o
If E zn2 /Zn = – 0.76V, which cathode will give a maximum value of Eocell per electron transferred?
(1) Ag+/Ag (2) Fe3+/Fe2+ (3) Au3+/Au (4) Fe2+/Fe

5. Given the equilibrium constant:

KC of the reaction:
Cu(s) + 2 Ag+ (aq)  Cu2+ (aq) + 2Ag(s) is
 RT 
10 × 1015 calculate the E0 cell of this reaction at 298K 2.303 at 298 K  0.059 V
 F 
(1) 0.04736 mV (2) 0.4736 mV (3) 0.4736 V ((4) 0.04736 V

6. Electrochemistry
 dE 0 
The standard electrode potential E0 and its temperature coefficient   for a cell are 2 V and –5×10
–4

 dT 
VK–1 at 300 K respectively. The cell reaction is:
Zn(s) + Cu2+ (aq) Zn2+ (aq) + Cu
The standard reaction enthalpy (rH0) at 300 K in kJ mol–1 is,
[Use R = 8 JK–1 mol–1 and F = 96,000 C mol–1]
(1) –412.8 (2) –384.0 (3)192.0 (4) 0.75
2 –1
7.  0m for NaCl, HCl and NaA are 126.4, 425.9 and 100.5 S cm mol , respectively. If the conductivity of
0.001 M HA is 5 × 10–5 S cm–1, degree of dissociation of HA is:
(1) 0.50 (2) 0.25 (3) 0.125 (4) 0.75

8. Given that E
o
O 2 /H2O  1.23V;
E So2 O 2 /SO 2  2.05V EoBr2 / Br  1.09V EoAu3 /Au  1.4V
8 4

The strongest oxidizing agent is:

(1) Au3+ (2) O2 (3) S2O 82  (4) Br2

9. Calculate the standard cell potential (in V) of the cell in which following reaction takes place:
Fe2+(aq)+ Ag+ (aq)  Fe3+ (aq) + Ag(s)
Give that
E 0 Ag /Ag  xV
E0Fe2 /Fe  yV
E0Fe3 /Fe  zV
(1) x – z (2) x – y (3) x + 2y – 3z (4) x + y – z

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10. The standard Gibbs energy for the given cell reaction in KJ mol–1 at 298 K is:
Zn(s) + Cu2+ (aq)  Zn2+ (aq) + Cu(s),
Eo = 2 V at 298 K
(Faraday’s constant, F = 96000 C mol–1)
(1) –384 (2) 384 (3) 192 ((4) –192

11. A solution of Ni (NO3)2 is electrolysed between platinum electrodes using 0.1 Faraday electricity. How
many mode of Ni will be deposited at the cathode?
(1) 0.05 (2) 0.20 (3) 0.15 (4) 0.10

12. Consider the statements S1 and S2:

S1: Conductivity always increases with decrease in the concentration of electrolyte.
S2: Molar conductivity always increases with decrease in the concentration of electrolyte.
The correct option among the following is:
(1) Both S1 and S2 are wrong (2) S1 is wrong and S2 is correct
(3) Both S1 and S2 are correct (4) S1 is correct and S2 is wrong

13. Which one of the following graphs between molar conductivity (Am) versus C is correct?

(1) (2) (3) (4)

14. Given :
Co3++ e–  Co2+; Eo = + 1.81 V
Pb4+ + 2e–Pb2+ ; Eo = + 1.67 V
Ce4+ + e–Ce3+; Eo = + 1.67 V
Bi3+ 3e– Bi; Eo = + 0.20 V
oxidizing power of the species will increase in the order:
(1) Ce4+ < Pb4+ < Bi3+ < Co3+ (2) Bi3+ < Ce4+ < Pb4+ < Co3+
3+ 4+ 3+ 4+
(3) Ce < Ce < Bi < Pb (4) Co3+ < Pb4+ < Ce4+ < Bi3+

15. The Gibbs energy change (in J) for the given reaction at  Cu 2    Sn 2   = 1M and 298K is:

Cu(s) + Sn2+ (aq.)  Cu2+ (aq.) + Sn(s)

E 0
S n 2
  0 .1 6 V , E C0 u 2  |C u  0 .3 4 V T ake F  9 6 5 0 0 C m o l – 1 
16. 
For the disproportionation reaction 2Cu+ (aq)   Cu(s) + Cu2+ (aq) at 298K. In K
(Where K is the equilibrium constant) is × 10–1 Given:
 0 RT 
 ECu2 /Cu  0.16VECu/Cu  0.52V  0.025 
0

 F 
17. Let C N a C l and C BaSO be the conductances (in S) measured for saturated aqeous
4

Solutions of NaCl and BaSO4, respectively, at a temperature T. Which of the following is False?
(1) CBaSO4 (T2) > C BaSO (T1) for T2 > T1
4

(2) C N a C l (T2) > CNaCl (T1) for T2 > T1

(3) C N a C l >> C BaSO at a given T
4

(4) Ionic mobilities of ions from both salts increase with T.

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18 The photoelectric current from Na (work function, WO = 2.3e V) is stopped by the output voltage of the
cell
Pt (s) | H2(g, 1bar)| HCl (aq, pH = 1)| AgCl(s)| Ag(s)
The pH of aq. HCl required to stop the photoelectric current from K (wo = 2.25eV), all other conditions
remaining the same, is × 10–2 (to the nearest integer).
RT
Given, 2.303  0.06V; E 0AgClAglCC –  0.22V
F
19. An acidic solution of dichromate is electrolyzed for 8 minutes using 2A current. As per the following
equation
Cr2O 72  + 14H+ + 6e–  2Cr3+ + 7H2O
The amount of Cr3+ obtained was 0.104g. The efficiency of the process (in\%) is (Take: F = 96000C, At.
Mass of chromium = 52)

20.

E 0Cu 2 |Cu   0.34V

E 0Zn 2 |Zn   0.76V
Identify the incorrect statement from the option below for the above cell:
(1) If Eext = < 1.1V, Zn dissolves at anode and Cu deposits at cathode
(2) If Eext = 1.1V, no flow of e– or current Occurs
(3) If Eext > 1.1V, e– flows from Cu to Zn
(4) If Eext > 1.1V, Zn dissolves at Zn electrode and Cu deposits at Cu electrode

21. 250 mL of waste solution obtained from the workshop of a goldsmith contains 0.1M AgNO3 and 0.1 M
AuCl. The solution was electrolyzed at 2V by passing a current of 1A for 15 minutes. The metal/metals
electrodeposited will be
E 0   0.80 V , E 0A u  / Au  1.69 V 
 Ag / Ag 
(1) Silver and gold in equal mass proportion (2) Silver and gold in proportion to their atomic weights
(3) Only gold (4) Only silver

22. An oxidation reduction reaction in which 3 electrons are transferred has a

Gº of 17.37kJ mol–1 at 25ºC. The value of E 0ce ll (in V) is × 10–2 (1F = 96,500 Cmol–1)

23 The variation of molar conductivity with concentration of an electrolyte (X) in aqueous solution is shown
in the given figure.

The electrolyte X is
(1) NaCl (2) HCl (3) CH3COOH (4) KNO3

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24. For the given cell;
Cu(s)|Cu2+ (C1M)|| Cu2+ (C2M) | Cu(S) change in Gibbs energy (G) is negative, if
C1
(1) C2 = 2C1 (2) C2 = (3) C1 = 2C2 (4) C1 = C2
2
25. Given that the standard potentials (E0) of Cu2+/Cu and Cu+/Cu are 0.34 V and 0.522V respectively, the
(E0) of Cu2+/ Cu+ is :
(1) 0.158 V (2) –0.158 V (3) 0.182V (4) –0.182V
26. Which of the following is incorrect?
(1)  0m N a C l   0m N a B r   0m K C l   0m K B r
(2)  0m H 2 O   0m H C l   m0 N a O H   0m N a C l
(3) \Lambda _{m} {\circ} NaBr -  0m N a l =  0m K B r –\Lambda_{m} {\circ} NaBr
(4)  0m N a C l   0m K C l   0m N a B r   0m K B r
27. What would be the electrode potential for the given half cell reaction at pH = 5?
Given : 2H2O  O2 + 4H+ + 4e– E 0re d = – 1.23 V (R = 8.314 J mol –1 K –1; Temp = 298 K; oxygen under
std. pressure of 1 bar) Multiply your answer with - 100
Sn2 
2+ 2+
28. For an electrochemical cell Sn (s) | Sn (aq, 1 M) ||Pb (aq, 1M) | Pb (s) the ratio when this cell
Pb2 
attains equilibrium is …………
 0 2.303RT 
 ESn2 /Sn  0.14V, Epb2 /pb  0.13V,  0.06 
0

 F 
29. The electrode potential of M2+/M of 3 d – series elements shows positive value for:
[JEE (Main) 24 Feb 2021 Shift- 1]
(1) Zn (2) Co (3) Fe (4) Cu
 2 4
30. The magnitude of the change in oxidizing power of the MnO4 / Mn coupleis x  10 V, if the H+

concentration is decreased from 1 M to 10–4 M at 25ºC. (Assume concentration of MnO4 and Mn 2 to be
same on change in H+ concentration). The value of x is ______ (Rounded off to the nearest integer)
2303RT
[Given:  0.059 ] [JEE (Main) 24 Feb 2021 Shift-2]
F
31. Consider the following reaction [JEE (Main) 26 Feb 2021 Shift- 1]
MnO4  8H  5e–  Mn2  4H2O,E0  1.51V

The quantity of electricity required in Faraday to reduce five moles of MnO4 is
32. A 5. 0 m moldm –3 aqueous solution of KCl has a conductance of 0.55 mS when measured in a cell
constant: 1.3 cm–1. The molar conductivity of this solutions is ______ mSm2 mol–1
(Round off to the Nearest Integer) [JEE (Main) 16 Mar 2021 Shift- 2]
33. A KC solution of conductivity 0.14 S m shows a resistance of 4.19  in a conductivity cell. If the same
–1

cell is filled with an HCl solution, the resistance drops to 1.03. The conductivity of the HC1 solution is
___________ × 10–2 S m–1. (Round to the Nearest Integer). [JEE (Main) 17 March 2021 Shift- 2]
34. The molar conductivities at infinite dilution of barium chloride, sulphuric arid and hydrochloric acid are
280, 860 and 426Scm2 mol–1 respectively. The molar conductivity at infinite dilution of barium sulphate
is ________.
Scm2 mol–l (Round off to the Nearest Integer). [JEE (Main) 18 Mar 2021 Shift- 2]

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35. The cell potential for the following cell Pt|H2 (g) | H+ (aq) || Cu2+ (0.01M) | Cu(s) is 0.576V at 298K. The
pH of solution is _____. [JEE (Main) 24 Jun 2022 Shift- 1]
36. The resistance of conductivity cell contining 0.01 M KCl solution at 298 K is 1750 . If the conductively
of 0.01 M KCl solution at 298 K is 0.152 × 10–3 S cm–1, then the cell constant of the conductivity cells is
____________ × 10–3 cm–1. [JEE (Main) 24 Jun 2022 Shift- 2]
2
37. The quantity of electricity in Faraday needed to reduce 1 mol of Cr2O7 is ______.
[JEE (Main) 28 Jun 2022 Shift- 1]
2+ x+
38. The cell potential for Zn|Zn (aq)||Sn |Sn is 0.801 V at 298 K. The reaction quotient for the above
reaction is 10–2. The number of electrons involved in the given electrochemical cell reaction is………
2.303RT
(Given E 0 2    0.763V, E 0 x    0.008V and 0.06V)
Zn |Zn Sn |Sn F
[JEE (Main) 25 Jul 2022 Shift- 1]
39. The amount of charge in F (Faraday) required to obtain one mole of iron from Fe3O4 is ______.
[JEE (Main) 26 Jul 2022 Shift- 1]
40. For a cell, Cu(s) | Cu2+ (0.001M | | Ag+(0.01M) | Ag(s) the cell potential is found to be 0.43 V at 298 K.
The magnitude of standard electrode potential for Cu2+/Cu is _______ × 10–2 V.
  2.303RT  [JEE (Main) 29 Jul 2022 Shift- 2]
 G ive : E Ag  / Ag  0.80V and F
 0.06V 
 
41. At 298 K, a 1 litre solution contining 10 mmol of Cr2O72– and 100 mmol of Cr3+ shows a Ph OF 3.0.
2 0 2.303RT
Given: CrO7 Cr3;E 1.330V and  0.059V
F
The potential for the half cell reaction is X × 10–3 V. The value of x is [JEE (Main) 24 Jan 2023 Shift- 1]
42. Choose the correct representation of conductometric titration of bezoic acid vs sodium hydroxide.
[JEE (Main) 24 Jan 2023 Shift- 2]

(1) (2)

(3) (4)

43. Pt (s) H2 (g) (1 bar) | H+ (aq) (1M)|M3+ (aq), M+(aq)|Pt(s) The Ecell for the given cell is 0.1115 V at 298 K
 M   aq  
When    10a The value of a is _________ [JEE (Main) 25 Jan 2023 Shift- 2]
 M 3 (aq) 
 
2.303RT
Given : E  M 3 / M   0.2V  0.059V
F
44. The standard electrode potential (M3+/M2) for V, Cr, Mn & Co are -0.26V, –0.41 V, +1.57V and +1.94V,
respectively. The metal ions which can liberate H2 from a dilute acid are
[JEE (Main) 29 Jan 2023 Shift- 1]
(1) V2+ and M2+ (2) Cr2+ and CO2+ (3) V2+ and Cr2+ (4) Mn2+ and CO2+

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45. Following figure shows dependence of molar conductance of two electrolytes on
0
concentration.  m is the limiting molar conductivity.
[JEE (Main) 29 Jan 2023 Shift- 1]
The number of Incorrect Statement (s) from the following is _______.
0
(1)  m for electrolyte A is obtained by extrapolation
0
(2) For electrolyte B, vx  m vs graph is a straight line with intercept equal to  m
(3) At infinite dilution, the value of degree of dissociation approach zero for electrolyte B.
0
(4)  m for any electrolyte A or B can be calculated using º for individual ions.
46. The standard electrode potential of M+/M in aqueous solution does not depend on
[JEE (Main) 06 Apr 2023 Shift-1]
(1) Hydration of a gasesou metal ion (2) Sublimation of a solid metal
(3) Ionisation of a solid metal atom (4) Ionisation of a gaseous metal atom
47. The product, which is not obtained during the electrolysis of brine solution is
[JEE (Main) 06 Apr 2023 Shift-2]
(1) H2 (2) HCl (3) NaOH (4) Cl2
48. The specific conductance of 0.0025M acetic acid is 5×10 S cm–1 at a certain temperature. The
–5

dissocation constant of acetic acid is _______ × 10–7. [JEE (Main) 10 Apr 2023 Shift-2]
(Nearest integer) Consider limiting molar conductivity of CH3COOH as 400 S cm2 mol–1
In an electrochemical reaction of lead, at standard temperature, if Eº  mVolt andEº
49.
 Pb2 /Pb  Pb4 /Pb = n
Volt, then the value of Eº(Pb2+/Pb4+) is given by m – xn. The value of x is ____. (Nearest integer)
[JEE (Main) 11 Apr 2023 Shift-1]

50. The number of correct statement from the following is…….. [JEE (Main) 11 Apr 2023 Shift-2]
(1) Ecell is an intensive parameter
(2) A negative Eº means that the redox couple is a stronger reducing agent than the H+/H2 couple.
(3) The amount of electricity required for oxidation or reduction depends on the stoichiometry of the
electrode reaction.
(4) The amount of chemical reaction which occurs at any electrode during electrolysis by a current is
proportional to the quantity of electricity passed through the electrolyte.

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ANSWER KEY
Exercise – 1
2 2  
1. (a) Zn(s) Zn (aq) Cd (aq) Cd(s) (b) Pt H2(g) H (aq) Ag (aq) Ag(s)
2  2 3 2 3
(c) Cu(s) Cu (aq) Cl (aq) Hg2Cl2(s) Hg() (d) Pt Fe (aq) ,Fe (aq) Cr2O7 (aq) ,Cr (aq) Pt
2. (a) 2 A g  C u 2   2 A g   C u , (b) M n O 4  5 F e 2   8 H   M n 2   5 F e3  4 H 2 O
(c) 2 C l   2 A g   2 A g  C l 2 , (d) H 2  C d 2   C d  2 H 
3. Anode Cathode
2
(a) Zn Zn H 
H 2 Pt

2 4
(b) Pt Sn ,Sn F e3 , F e 2 P t

2 3
(c) Pt Fe ,Fe M n O 4 , M n 2  P t

2
(d) Pb Pb B r 2, B r  P t

4. 1.61V 5. 1.35V 6. 1.68V

7. –0.80V, NO 8. 0.53 V, disproportionation 9. –0.0367V
10. Kc = 7.6 × 1012 11. E = 1.159V 12. Kc =1.96 × 1026
107
13. Kc = 1.864 × 10 , ∆Gº = – 611.8 kJ
14. Kw≈ 10–14 15. Eo = 0.7826 V 16. n = 2
17. E = 0.059 18. E = 0.395 V
2 3 2
19. E oc e ll   0 .0 1V , E c e ll   0 .0 7 8 5 V , correct representation is Pb Pb (10 M) Sn (1M) Sn
20. [ C u 2  ]  2 .9 7  1 0 – 1 2 M fo r E  0 21. E = –0.81V
22. 0.52 V, 0.61 V 23. –1.30 × 103 kJ mol–1
24. 0.0295V 25. 0.0295V 26. pH = 4
27. E = – 0.0206 V 28. a2 = 0.1006 M
29. (A ) 6.02 × 1022 electrons lost, (b) 1.89 × 1022 electrons gained, (c) (b) 1.80 × 1023 electrons gained
30. (A) 0.75F, (b) 0.69F, (c) 1.1 F 31. (i) 54 gm, (ii) 16.35 gm
5
32. (a) 1.023 × 10 sec 33. 1.12 mol, 12.535 litre
34. 0.112 litre 35. 0.24 gms
36. Rs. 0.75 × 37. (i) 2.1554 gm; (ii) 1336. 15 sec
38. 115800C, 347.4 kJ 39. t = 193 sec 40. n = 4
41. Cu2+ = 0.08M, H+ = 0.04M, S O 24  = 0.1M
42. 114, Q = 5926.8C 43. Final weight = 9.6g, 0.01 Eq of acid
44. (a) V(O2) = 6.2L, (b) V (O2) = 0.824L 45. t = 93.65 sec. 46. 60%
47. (i) 482.5 sec (ii) 0.3175 gm (iii) 0.327 gm
48. 7.958 × 10–5M 49. 1.825 g 50. 2M
51. 419 S cm2 equivalent–1 52. 0.00046 S cm–1; 2174 ohm cm
53. (i) 6.25 × 105 ohm, (ii) 1.6 × 10–6 amp 54. 1.549 cm–1
–1 2
55. 101.8 ohm cm / gm equivalent 56. (i) 0.0141 mho g equiv–1 m2, (ii) 0.141 mho m–1
57. (i) 232 Mho cm mol , (ii) 116 Mho cm equivalent–1
2 –1 2

58. 0.865 59. 1.33 × 10–4 gm/litre

60. 8.74 × 10–11 mole2/litre2 61.   0 .4 3 5 , k  6 .7  1 0  4
2 –1
62. (i) 390.6 S cm mol (ii) 12.32% 63. 1.067 S m–1
64. 523.2 × 10–4 mho cm2 mol–1 65. (i) 6.98 (ii) 1.08 × 10–14

Exercise –2
1. (B) 2. (A) 3. (A) 4. (B) 5. (D) 6. (A) 7. (C)
8. (C) 9. (C) 10. (A) 11. (B) 12. (A) 13. (C) 14. (C)
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15. (D) 16. (C) 17. (C) 18. (D) 19. (B) 20. (C) 21. (C)
22. (A) 23. (B) 24. (B) 25. (B) 26. (D) 27. (D) 28. (C)
29. (C) 30. (B) 31. (A) 32. (A) 33. (D) 34. (A) 35. (D)
36. (A) 37. (C) 38. (C) 39. (C)

Exercise –3
1. (CD) 2. (BCD) 3. (AB) 4. (AB) 5. (ACD) 6. (AC) 7. (BC)
8. (AD) 9. (AB) 10. (BC) 11. (D)

Exercise –4
Section - I
1. 4 2. 3 3. 2 4. 10 5. 1 6. 4 7. 10
8. 7 9. 5

Section - II
10. A P,Q; B  P,Q; C  Q,R; D  P, S 11. A Q, S; B  Q, R; C  P; D  Q, R
12. A P,Q,R ; B  P,Q, S; C  P,Q,R,S; D  P, R

Section - III
13. (B) 14. (C) 15. (A) 16. (A) 17. (A) 18. (D)
19. (B) 20. (D) 21. (D) 22. (C) 23. (D)

Exercise – 5
1. E = 1.1 V 2. Eo = – 1.59V, non-spontaneous
3. Ksp = 1.1 × 10–16 4. pH = 6.61
5. Eo = 0.71V 6. [Br–] : [Cl–] = 1 : 2007. Eo = –0.1511V
8. 1.39V 9. 0.373V 10. – 0.037V
–5
11. h = 2.12 × 10–2, Kh= 1.43 × 10 M 12. 1.536 × 10–5 M3
268
13. K = 10 14. 1.66V 15. –1.188V 16. 10–2
16
17. 5.24 × 10 18. 1.143V 19. Eq. wt. = 107.3 20. [Cu2+] = 10–4M
21. 643.33amp, 190,5g 22. 43.456g 23. 265 Amp. hr.
5
24. 1.21 × 10 coulomb 25. 42.2%
26. V(O2) = 99.68 mL, V (H2) = 58.46 mL, Total vol. = 158.1 mL
27. 57.5894gm 28. 1.9 million year
29. Cu = 98.88%, Fe = 0.85% 30. 1250s, 0.064 M

Exercise – 6
Section - I
1. (A) 2. (C) 3. (B) 4. (B) 5. (B) 6. (ABD) 7. (B)
8. (C) 9. (D) 10. (D) 11. (B) 12. (D) 13. (A) 14. (D)
15. (A) 16. (BCD) 17. (ABC) 18. 0.215 19. –1.03V 20. 7 21. (BCD)
22. (A)
Section - II
1. (3) 2. (4) 3. (4) 4. (1) 5. (3) 6. (1) 7. (3)
8. (3) 9. (3) 10. (1) 11. (1) 11. (2) 12. (1) 13. (2)
14. 96500 15. 144 16. Bonus 17 142 18. 60 19. (4) 20. (3)
21. –6 22. (3) 23. (1) 24. (1) 25. (3) 26. 9 27. 215
28. 5.67 29. (4) 30. 3776 31. 25 32. 14.3 33. 57. 34. 288
35. 5 36. 266 37. 6 38. 4 39. 3 40. 34 41. 9
42. (2) 43. 3 44. (3) 45. (2) 46. (3) 47. (2) 48. 66
49. 2 50. (4)

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 OPTICAL
ISOMERISM
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Exercise - 1
Concept Building Questions

1. Which of the following compounds is optically active?

(A) 1 - Bromobutane (B) 2 - Bromobutane
(C) 1- Bromo-2-methylpropane (D) 2-Bromo-2-methylpropane

2. The process of converting d form to l or vice-versa is known as

(A)Walden inversion (B) Resolution (C) Racemisation (D) None of these

3. Which is incorrect about enantiomorphs

(A) They have different biological properties (B) They have identical physical properties
(C) They can rotate the plane polarised light (D) They have same shape of crystal

4. Diastereomers can be separated by

(A) Simple distillation (B) Fractional Distillation
(C) Electrophoresis (D) None of these

5. The least number of carbon atoms for an alkane to show stereoisomerism is

(A) 2 (B) 3 (C) 4 (D) 5

6. Which of the following is threo isomer

     
     

(A) Only I (B) Only II (C) Only III (D) All of the above
CH3
7. Which of the following operations on the Fischer formula H OH does not change its absolute
C2H5
configuration?
(A) Exchanging groups across the horizontal bond
(B) Exchanging groups across the vertical bond
(C) Exchanging groups across the horizontal bond and also across the vertical bond
(D) Exchanging a vertical and horizontal group

8. Which of the following compound shows optical isomerism?

(A) CH3CH2CH3 (B) CH2OHCHOHCH2OH
(C) CH3CHOHC2H5 (D) CCI2F2

9. Which one of the following is chiral?

(A) 1, 1- Dibromo-1-chloropropane (B) 1, 3- Dibromo-1-chloropropane
(C) 1, 1-Dibromo-3-chloropropane (D) 1, 3-Dibromo-2-chloropropane

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10. Identify number of chiral carbons in the following compounds.
H H

(I) H3C CH CH CH2 (II) H3C H

OH OH OH CI H
(III) H3C CH CH CH2 CH3

CH3 CH3
(A) 1, 2, 1 (B) 1, 1, 2 (C) 2, 0, 1 (D) 2, 1, 1

11. Which of the following statements is true?

(A) A mixture of enantiomers can be separated on the basis of difference in their boiling points (by a
method called fractional distillation).
(B) A mixture of enantiomers can be separated on the basis of difference in their solubility in any solvent.
(C) A mixture of enantionmers can be separated by converting them into diastereomers by reacting them
with an optically active reagent
(D) A mixture of enantiomers can be separated by passing plane polarized through their solution

12. Which of the following compounds may not exist as enantiomers?

(A) CH3CH(OH)CO2H (B) CH3CH2CH(CH3)CH2OH
(C) C6H5CH2CH3 (D) C6H5CH(Cl)CH2CH3

13. Which of the following have the same value of optical rotation?
Me C C C

(I) H C (II) Et H (III) Me H (IV) H H

Et Me Et CH2CH2CH3

(A) I, IV (B) I, II (C) III, IV (D) I, III

14. The compound which has maximum number of chiral centres is:
C OH OH
(A) (B)

HO
C OH
(C) (D)
C

15. Which of the following compounds can exhibit enantiomerism?

(A) 3-hydroxy propanoic acid (B) 3-hydroxy butanoic acid
(C) 4-hydroxy butanoic acid (D) none of these

16. Which of the following is a chiral molecule?

(A) 1-chloro propane (B) 2-chloro propane (C) 1-chloro butane (D) 2-chloro butane

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17. A molecule is said to be chiral, only if:
(A) it is superimposable on its mirror image
(B) it is non superimposable on its mirror image
(C) it possesses stereogenic centers
(D) it can have different configuration

18. The number of chiral centres present in 3, 4-dibromo-2-pentanol is:

(A) 1 (B) 2 (C) 3 (D) 4

(+)-2-butanol has     13.90. A sample of 2-butanol containing both the enantiomers was found to
25
19.
have a specific rotation value of –3.5º under similar condition. The percentage of the (+) and (–)
enantiomer present in the sample are, respectively:
(A) 37.4% and 62.6% (B) 62.6% and 37.4%
(C) 42.2% and 57.8% (D) 35.5% and 64.5%

20. How many different stereoisomers are possible for the following compound?
H
ClHC = HC – C – CH = CHCl
Cl
(A) 1 (B) 2 (C) 3 (D) 4

21. The following compounds are best described as:

(R) – PhCH(OH)CH3 and (S) – PhCH(OH)CH3
(A) enantiomers
(B) diastereomers
(C) not stereoisomers
(D) conformational isomers (differing by single bond rotation)

22. Rank the following substituent groups in order of decreasing priority according to the
Cahn – Ingold – Prelog system:
–CH(CH3)2 –CH2Br –CH2CH2Br
1 2 3
(A) 2 > 3 > 1 (B) 1 > 3 > 2 (C) 3 > 1 > 2 (D) 2 > 1 > 3

23. The structure of (S) –2– fluorobutane is best represented by:

(A) (B) (C) (D)

24. Which one of the following is chiral?

(A) 1, 1–Dibromo–1–chloropropane (B) 1, 3–Dibromo–1–chloropropane
(C) 1, 1–Dibromo–3–chloropropane (D) 1, 3–Dibromo–2–chloropropane

25. The separation of a racemic mixture into pure enantiomers is termed as:
(A) Racemization (B) Isomerization (C) Resolution (D) Equilibration

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26. Stereoisomers have different
(A) Molecular formula (B) Structural formula (C) Configuration (D) Molecular Mass

27. Chiral molecules are

(A) Superimposable on their mirror image (B) Not Superimposable on their mirror image
(C) Unstable molecules (D) Capable of showing geometrical isomerism

28. Stereoisomers which are not mirror image of each other are called
(A) Enantiomers (B) Tautomers (C) Meso (D) Diastereomers

29. The instrument which can be used to measure optical activity i.e. specific rotation
(A) Refractometer (B) Photometer (C) Voltmeter (D) Polarimeter

30. Racemic mixture is formed by mixing two

(A) Isomeric compounds (B) Chiral compounds
(C) Meso compounds (D) Optical isomers

31. Enantiomers have

(A) Similar physical properties
(B) Similar chemical properties with optical active compounds
(C) Same absolute value of specific rotation
(D) Different configuration.

32. Which of the following compounds is not chiral ?

(A) 1-Chloropentane (B) 2-Choloropentane
(C) 1-Chloro-2-methylpentane (D) 3-Chloro-2-methylpentane

33. Number of chiral carbon atoms in the compound x, y and z respectively would be :

(A) 0, 2,1 (B) 1,0,1 (C) 1,2 ,1 (D) 1,2 0

34. Which of the following pair of isomers can’t be separated by fractional distillation
(A) Maleic acid and Fumaric acid (B) (+)-Tartaric acid and meso-tartaric acid
(C) (+)- Lactic acid and (-)-Lactic acid (D) CH3CH(NH2)COOH& NH2CH2CH2COOH

35. What is the number of isomers (including stereoisomers) possibly formed on free radical
monochlorination of 2-methylbutane?
(A) 4 (B) 5 (C) 6 (D) 7

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Exercise - 2
Single choice correct with multiple options

1. An enantiomerically pure acid is treated with racemic mixture of an alcohol having one chiral carbon. The
ester formed will be
(A) Optically active mixture (B) Pure enantiomer
(C) Meso compound (D) Racemic mixture

2. How are the following compounds related?

(A) Diastereomers (B) Enantiomers (C) Meso compounds (D) Identical

3. Rank of the following groups in order of R, S precedence (IV is highest):

(1) –CH(CH3)2 (2) –CH2 CH2 Br (3) –CH2Br (4) –C(CH3)3
I II III IV I II III IV
(A) 3 2 4 1 (B) 1 4 2 3
(C) 3 4 1 2 (D) 3 4 2 1

4. Which of the following compound is non – resovable compounds?

(A) (B) (C) (D) All

5. Which of the following compound are meso forms?

CH3 CH3 CH3
H OH H C
H OH C H

3 CH3
CH2CH3 CH3
1 2
(A) 1 only (B) 3 only (C) 1 and 2 (D) 2 and 3

6. Hydrogenation of the adjoining compound in the presence of poisoned palladium catalyst gives
Me
Me H

Me H
H
(A) An optically active compound (B) An optically inactive compound
(C) A racemic mixture (D) A diastereomeric mixture

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7. How many stereoisomers of the following molecule are possible?
HOOC.CH=C=CH.COOH
(A) two optical isomers (B) two geometrical isomers
(C) two optical and two geometrical isomers (D) None

8. Out of the following which are chiral?

H 3C H 3C C H3C C

(I) (II) (III) (IV)

H3C CH3
CH3 CH3 H 3C CH3

(A) I, II, III (B) I, III, IV (C) II, III (D) II, III, IV

CH3

H
9. HO H

C2 H 5
The compound with the above configuration is called
(A) (2S, 3S)-2-chloro-3-hydroxypentane
(B) (2S, 3R)-2-chloro-3-hydroxypentane
(C) (2R, 3R)-2-chloro-3-hydroxypentane
(D) (2R, 3S)-2-chloro-3-hydroxypentane

10. Which of the following statements regarding the projections shown below is true?
CH3
H 3C C C
H

H CH3
.
H C C CH3
H
(A) (B)
(A) ‘A’ and ‘B’ both represent the same configuration
(B) Both ‘A’ and ‘B’ are optically active
(C) ‘B’ alone is optically active
(D) ‘A’ alone is optically active

11. Among the following, the optically inactive compound is:

N P
CH3CH2 H 3C
(A) H3 C (B)

Ph C
(C) (D) C C C
H H 3C Br
COOH
HO

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12. A natural occurring substance has the constitution shown below. How many isomers may have this
constitution?
O
HO CH2CI

HO CH = CHCH = CHCH2CH2CH3
(A) 2 (B) 8 (C) 16 (D) 64

13. Which of the following will not show optical isomerism as well as geometrical isomerism?

H3C H3 C
CH 3 CH 3
(A) (B)
H CH 3

H3C H 3C
CH 3 CH3
(C) (D)
H CH3
H3C H 3C

14. Identify number of chiral carbons in the following compounds.

(I) (II) (III) (IV)

(A) 0, 2, 2, 4 (B) 2, 2, 0, 4 (C) 1, 2, 2, 4 (D) 2, 2, 2, 4

15. Which of the following will not show optical isomerism?

(A) C ─ CH ═ C ═ CH ─ C (B) Br ─ CH ═ C ═ CH ─ Br
H
COOH C H

OH
(C) H (D)
H OH
Me C
Me
COOH

16. Which type of symmetry is present in the following molecule?

H

C
C

H
(A) Plane of symmetry (B) Centre of symmetry
(C) Both of these (D) None

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C2H5

Ha (X)
Hb
17.
Ha Hb (Y )

C 2H5
Relation between (X) and (Y) is:
(A) Enantiomers (B) Identical (C) E and Z isomer (D) Constitutional isomer

18. Among the following, the Newmann projections of meso-2, 3-butanediol are:

Me Me Me Me
OH OH HO Me Me H
H H
(P) (Q) (R) (S)

HO H H OH HO H H OH
Me Me H OH
(A) P,Q (B) P, R (C) R, S (D) Q, S.

19. The number of meso form of the given compound is:

CH3 CH CH CH CH3

OH OH OH
(A) 2 (B) 3 (C) 4 (D) 8

20. Which one of the following statements regarding the projection shown below is correct?
C
H Ph C C

Ph H Ph H H Ph
C

I II
(A) Both the projections represent the same configuration
(B) Both (I) and (II) are optically active
(C) Only (I) is optically active
(D) Only (II) is optically active

21. Which of the following is erythro form and optically inactive?

COOH CH3 CH3 CH3

HO H H OH H Br HO H
(A) (B) (C) (D)
HO H Br H H Br H Br

CH3 CH3 CH3 CH3

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22. The structures A and B represent:
COOH COOH

H OH CH3 OH
HO CH3 HO H

COOH COOH
(A) enantiomers (B) diastereomers (C) homomers (D) racemic mixture

23. The compound has:

Br F

H CH3
H 3C H

F Br
(A) plane of symmetry (B) axis of symmetry
(C) center of symmetry (D) no symmetry

24. Which of the following compounds is chiral?

H C H C H Br C Br
(A) (B) (C) (D)

O O
25. Which of the following dienes is chiral?
(A) CH3 ─ CH ═ C ═ CH2 (B) CH3 ─ CH ═ CH ─ CH ═ CH2
(C) CH3 ─ CH ═ C ═ CH ─ CH3 (D) CH2 ═ CH ─ CH2 ─ CH ═ CH2
26. The following structures represent a pair of:
CH3 C

H C Br CH3

Br H
(A) enantiomers (B) diastereomers (C) meso compound (D) homomers

27. Which of the following compounds possesses chiral carbon?

(A)

(B)

(C)

(D)

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28. Among the following compounds, the one which can exhibit chirality is

(A) (B)

(C) (D)

29. C8H16 that can form cis-trans isomerism and also chiral center is:

(A) (B)

(C) both of these (D) none of these

30. Which of the following compounds are meso forms?

CH3 CH 3
CH3
H OH H C
H OH C H

C 2 H5 CH3 CH3
1 2
3
(A) 1 only (B) 3 only (C) 1 and 2 (D) 2 and 3

31. A naturally occurring substance has the constitution shown. How many stereoisomers may have this
constitution?
O
HO CH2OH

HO CH CH CH CH CH2CH2CH3
(A) 2 (B) 8 (C) 16 (D) 64

32. Identify relation between these two compounds:

.
Ph H

H Ph
(A) homomers (B) enantiomers (C) diastereomers (D) different compounds

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33. The correct statement about compounds (P), (Q) and (R):
COOCH3 COOH COOH

H OH H OH H OH
H OH H OH HO H

COOH COOCH3 COOCH 3

(P) (Q) (R)
(A) P and Q are identical (B) P and Q are diastereomers
(C) P and R are enantiomers (D) P and Q are enantiomers

34. Which of the following molecules is chiral?

(A) CH3CH2CH ═ C ═ CH2 (B) CH3 ─ CH ═ CH ─ CH ═ CH2
(C) CH3CH ═ C ═ CHCH3 (D) CH2 ═ CH ─ CH2 ─ CH ═ CH2

35. Total number of stereoisomers of the compound

CH3 CH CH CH CH CH CH3 is

OH
(A) 2 (B) 3 (C) 4 (D) 8

36. Consider the following pairs of compounds:

CHO CHO

H OH HO H
HO H H OH
.
C6 H 5 C6H5
(I) (II)
Which among the following statements is correct?
(1) Both are enantiomers (2) Both are in threo form
(3) Bothe are diastereomers (4) Both are in erythro form
(A) 1 and 2 (B) 1, 2 and 3 (C) 2 and 3 (D) 3 and 4
37. Compound X can exist in how many orientations?

CH3 CH2CH3
X=H C CH = C
OH CH3
(A) 1 (B) 2 (C) 3 (D) 4

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38. How many chiral centers are present in tetracycline?
CH3
H OH H N
CH3
OH
C NH2
OH O
O
(A) 6 (B) 4 (C) 8 (D) 5

39. How many chiral centers are present in cholesterol?

H3C

H3C H

H H
HO H H
(A) 7 (B) 8 (C) 9 (D) 5

40. Find out no, of sterogenic center in following compound:

H
NH2
S
N
N
OO
HO COOH
(A) 4 (B) 5 (C) 3 (D) 6

Exercise - 3
Multiple choice correct with multiple options

1. Which of the following represents a pair of enantiomers?

(A) (B) H2C ═ C ═ CH2

(C) (D)

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2. Which of the following represents a pair of enantiomers?

(A) (B)

(C) (D)

3. Which of the following are optically active?

C C2 H 5 H3 C H
(A) C C C C C (B)
H H H CH3

H 3C C H3
(C) CH3 CH COOH (D) C C C C
H H
D

CH3 CH3 CH3 CH3

H OH HO H
4. H OH HO H
H OH HO H HO H H OH
C2 H 5 C2 H 5 C2H5 C2 H 5
(I) (II) (III) (IV)
Which of the following statements are true about these isomers?
(A) I and II are a pair of enantiomers
(B) III and IV are a pair of enantiomers
(C) II is the diastereomer of III and IV
(D) I and III are homomers

5.

Which of the following statements are correct about these molecules?

(A) I is a meso compound (B) I and II are identical
(C) II and IV are a pair of enantiomers (D) II and III are diastereomers

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6. Which of the following compounds are optically inactive but exhibit geometrical isomerism?

CH–Me

(A) (B) H3C CH3

CH–Me
CI
H 3C CH3 CH3
(C) C C C C (D) C C C
H H CH3
H

7. Enantiomers have:
(A) all physical properties same except their action on plane polarized light which is equal in magnitude
but opposite in direction
(B) all chemical properties same except when reagent is chiral in that case, reactivity of enantiomer will
be different.
(C) opposite configuration of all chiral centers according to CIP rule.
(D) superimposable image of each other.

H H H
O
8. O
O Ph Ph Ph
I II III
Which of the following statements are true about these molecules?
(A) I and II are a pair of enantiomers
(B) III is metamer of I and II
(C) III is diastereomer of I and II
(D) III is not stereoisomer of I and II

9. Which of the following will show optical isomerism?

Br C
(A) (B) H3C CH 3
H CH3

H3 C C
(C) (D) C C C C C
H OH

10. The correct statements about the compound given below.

(A) Compound is optically active (B) Compound possess center of symmetry

(C) Compound possess plane of symmetry (D) Compound possess axis of symmetry

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11. Find the correct statements regarding following molecules.

(A) I, II, III are meso (B) II and III are enantiomers
(C) I and IV are diasteromers (D) IV are diasteromers

12. Which of the following statements is/are correct?

(A) A meso compound has chiral centres but exhibits no optical activity
(B) A meso compound has no chiral centres and thus are optically inactive.
(C) A meso compound has molecules which are superimposable on their mirror images even though they
contain chiral centres
(D) A meso compound is optically inactive because the rotation caused by any molecule is cancelled by
an equal and opposite rotation caused by another molecule that is the mirror image of the first.

13. Which of the following statements is/are not correct for D-(+) glyceraldehyde?
(A) The symbol D indicates the dextrorotatory nature of the compound
(B) The sign (+) indicates the dextrorotatory nature of the compound
(C) The symbol D indicates that hydrogen atom lies left to the chiral centre in the Fischer projection
diagram
(D) The symbol D indicates that hydrogen atom lies right to the chiral centre in the Fischer projection
diagram

14. Which of the following operations on chiral carbonin the Fischer formulachange its absolute
configuration?
(A) Exchanging groups across the horizontal bond
(B) Exchanging groups across the vertical bond
(C) Exchanging groups across the horizontal bond and also across the vertical bond
(D) Exchanging a vertical and horizontal group

15. Which of the following statements for a meso compound is/are correct?
(A) The meso compound has either a plane or a centre of symmetry
(B) The meso compound has at least one pair of similar stereocenters
(C) The meso compound is achiral
(D) The meso compound is formed when equal amounts of two enantiomers are mixed

16. Which of the following states are correct:

(A) Any chiral compound with a single asymmetric carbon must have a positive optical rotation if the
compound has the R configuration
(B) If a structure has no plane of symmetry it is chiral
(C) All asymmetric carbons are stereocentres.
(D) Alcohol and ether are functional isomers

17. The correct statement(s) about the compound H3C(HO) HC – CH = CH – CH (OH) CH3 (X) is (are)
(A) The total number of stereoisomers possible for X is 6
(B) The total number of diastereomers possible for X is 3
(C) If the stereochemistry about the double bond in X is trans, the number of enantiomers possible for
X is 4
(D) If the stereochemistry about the double bond in X is cis, the number of enantiomers possible for X is 2

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18. Which of the following compounds are optically active?
(A) CH3CHOH CH2 CH3 (B) H2C ═ CH. CH2 CH ═ CH2
COOH
O2 N
H C
(C) C C C (D)
C H
NO2 COOH

19. Which of the following statements is/are correct?

(A) A meso compound has chiral centres but exhibits no optical activity.
(B) A meso compound has no chiral centres and thus are optically inactive
(C) A meso compound has molecules which are superimposable on their mirror images even though they
contain centres.
(D) A meso compound is optically inactive because the rotation caused by any molecule that is the mirror
image of the first.

20. Choose correct statements regarding following compounds:

and
H3 C CH3 H3 C CH 3

(A) The boiling point of both compounds are same

(B) Both are optically active
(C) Equal mixture of both compounds are optically inactive
(D) Both are diastereomers

Exercise –4
Section - I : Numerical Value/Subjective Type Questions

1. (i) 1, 2–dichlorocyclopropane = w (ii) 1, 3–dimethylcyclobutane = x

(iii) 2–bromo–3–chlorobutane = y (iv) 1, 3–dimethyl cyclohexane = z
Calculate total number of stereoisomer of the above compounds. Sum of w + x + y + z =…….….….….

2. The compound, whose stereo-chemical formula is written below, exhibits x geometrical isomers and y
optical isomers
CH3 H OH
C=C
H CH2 CH2 C CH 3
H
The sum of x and y _____

3. Calculate the number of chiral center in the molecule Ethyl 2,2-dibromo-4-ethyl-6-methoxy cyclohexane
carboxylate.

4. How many monochlorinated products of methyl cyclohexane are optically active.

5. How many enantiomers are possible on monochlorination of isopentane.

6. The number of optically active compounds in the isomers of C4H9Br is

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7. A pure sample of 2-chlorobutane shows rotation of PPL by 30° in standard conditions. When above
sample is made impure by mixing its opposite form, so that the composition of the mixture becomes
87.5% d-form and 12.5% l-form, then what will be the observed rotation for the mixture.

8. When an optically active compound is placed in a 10 dm tube is present 20 gm in a 200 ml solution rotates
the PPL by 30°. Calculate the angle of rotation & specific angle of rotation if above solution is diluted to 1
Litre.

9. How many optically active stereoisomers are possible for butane-2, 3-diol?

10. On monochlorination of 2-methylbutane, the total number of chiral compounds formed is

11. Total number of stereoisomers possible for the following compound is.

Section - II : Match the Column

12. Match the Column (I) and (II). (MATRIX)

Column (I) - Molecule Column (II) -Property

(A)
(P) Chiral centers containing compound

(B)
(Q) Presence of stereocenter

(C)

(R) Optically active compound

(D) Compound containing plane of

(S)
symmetry

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13. Column (I) Column (II)
(A) HO Br P. Plane of symmetry
C C C
CH3 C
CH3 Br

(B) Q. Meso

CH3 OH
H 3C CH3
(C) C C C C R. Optically active
C C
CH3

H
(D) H 3C H S. Geometrical isomerism

14. Column (I) Column (II)

C H
(A) C C C C P. Polar molecule
H C
H H
(B) C C C Q. Optically active
H C
(C) C C R. Optically inactive

H
H F
(D) S. Symmetry element
H H
H

Section - III : Comprehension

Comprehension#1 (15 to 17)

A line which bisects a compound in two equal parts and both parts appear to be the mirror image of each
other, such kind of symmetry is known as plane of symmetry.
Any molecule that has a internal mirror plane of symmetry cannot be chiral, even though it may contain

asymmetric carbon atoms.

C C

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15. Which of the following compounds does not contain plane of symmetry?
CH3
CH3
H OH
(A) (B) (C) (D)
H OH

CH3

16. Which of the following compounds is optically active?

CH3
CH3
H Br Br
(A) H3C OH (B) (C) (D)

H CH3
CH3

17. Which of the following compounds is not chiral?

Br
CH3
(A) Br OH (B)

H
CH3
C CH3
CH3
(C) Br C (D)
C C
HO OH

Comprehension#2 (18 to 20)

R, S-configuration is a useful tool for determination of enantiomers, diastereomers and homomers. If

configuration of all chiral centers are opposite then structures are enantiomers, if all chiral centers have
same configuration then they are homomers and if some have same configuration and some have opposite
configuration then they are diastereomers.

CH3 CH3
CH3 CH3
H H
H CH3
H NH2 H3C NH2 H3C NH2 H3C H
CH3 H
H NH2
I II III IV
18. Among above structures find out enantiomeric structures:
(A) II and III (B) I and II, II and IV (C) I and IV (D) III and IV

19. Find out homomers:

(A) I and II (B) II and IV (C) I and IV (D) III and IV

20. Which of the following is not diastereomer?

(A) I and III (B) II and III (C) III and IV (D) II and IV

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Comprehension#3 (21 to 23)

Isomers which are non superimposable mirror images of each other are called enantiomers. All optically
active compounds exhibit enantiomerism. The stereoisomers which are not mirror images of each other
are called diastereomers. Enantiomers are always chiral molecules whereas diastereomers may or may not
be chiral, configuration of the compound having no element of symmetry is always chiral. Chiral
molecule may or may not contain chiral carbon.

21. Which of the following compound are chiral?

CHO H3C C 2H5
(A) H OH (B) C C C
CH2OH H H

Br
CH3
(C) (D) All of these
H
Cl
22. Which of the following pairs are diastereomers?
COOH COOH
CHO CHO
H Br Br H
(A) H OH and HO H (B) and
Br H H Br
CH3 CH3
C 6H5 C 6H5
H3C CH3 H3C H
(C) and (D) All of these
H H H CH3

23. Which one of the following pairs is correctly matched?

enantiomers Diastereomers:
CHO CHO CHO CHO
(A) H OH HO H (B) H Br Br H
and and
H OH H OH H Br Br H
CH3 CH3 C 6 H5 C 6 H5

meso form Homomer

COOH COOH COOH COOH
(C) H OH and HO H (D) H OH H OH
and
H OH HO H HO H HO H
COOH COOH COOH COOH

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Exercise –5
Revision exercise (Moderate to Tough)
COOH
HOOC 2
2 3 H HO H
C—C 3
1. H OH H OH
HO COOH
COOH
I II
Which of the following statement(s) is/are correct for the above compounds ?
(A) the compound I posses an axis of symmetry.
(B) the compound II posses a centre of symmetry.
(C) compound I and II both are optically active and enantiomers of each other.
(D) Diastereomers of both compounds are identical and meso.

2. Which of the following statement are incorrect ?

(A) Every chiral compound has a diastereomer.
(B) Diastereomers are always optically active.
(C) If a compound has an enantiomer, it must be chiral.
(D) Every molecule containing one or more asymmetric carbons is chiral

3. Which of the following has two-fold axis of symmetry ?

(A) Ethylene (B) Cyclopropane (C) Cyclobutane (D) Benzene

4. Which of the following pair of compounds can be separated by fractional distillation ?

D H Me Me
O O
|| ||
(A) H O–C– Me and D O–C– OH

Me OH H H
Cl
O Cl O
(B) and
Cl
Cl
COOH COOH

(C) H OH and HO H
HO H HO H

COOH COOH
Me Me
Me
(D) and

Me

CH3
5. COOH + (S) 2–Butanol
Fractional
Y
H X distillation
(no. of product) (no. of fraction)
C2H5

(Racemic mixture)
Report your answer as XY :

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6. Which one of the following is optically inactive
(A) Cis-1,2-Dibromocyclobutane (B) Trans-1,2-Dibromocyclobutane
(C) Trans-1,3-Dibromo-1-chlorocyclobutane (D) Trans-1,3-Dibromocyclobutane

7. Identify which of the the given substance is/are chiral :

H H
H HH C=C H DH C=C
C=C D C=C D
(A) H C D (B) D C H
H C=C D C=C
C=C C=C
D D
D HD H HH
H H
D HD C=C H HD C=C
C=C D C=C D
(C) H C D (D) H C D
H C=C D C=C
C=C C=C
D D
H DH H DD

8. Which of the following is/are correct statement :

(A) N - Methylethanamine exists as nonisolable racemic mixture.
(B) N - Methylethanamine react with excess of DCl and give configurational enantiomer.
(C) Two conformational isomers of butane have equal energy (degenerate) and they are chiral.
(D) Penta-2,3-diene exist in optically active isomeric forms

9. Observe the compound 'M'

Cl

M=
HO OH
If in this compound
X = Total number of asymmetric C atoms
Y = Number of similar asymmetric C atoms
Z = Number of optically active stereoisomers
W = Number of optically inactive isomers
R = Number of geometrical orientations in space
Report your answer as : X + Y + Z + W + R

10. Which of the following compounds will show optical activity?

(A) (B) (C) (D)

11. The following molecules are:

(A) Enantiomers (B) Diastereomers (C) Identical (D) Conformers

12. Which o the following is/are chiral?

(A) (B)

(C) (D)

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13. The given compound (X) has:

(A) chirality
(B) superimposability on its mirror image isomer
(C) plane of symmetry
(D) C2 axis of symmetry

14. The major product (ester) of the following reaction is

(A) A single stereoisomer (optically active)

(B) A mixture of diastereomers (both optically active)
(C) A racemic mixture (optically inactive)
(D) A mixture of four stereoisomers (two racemic mixtures)

15. Which statement9s) is/are correct for the given reaction and compounds

(A) Two esters are formed (B) All the esters are chiral
(C) Both esters are diastereomers (D) Racemic mixture is formed as a product

16. Which of the following compounds can show optical isomerism as well as geometrical isomerism?

(A) (B)

(C) (D)

17. Which of the following compounds can show geometrical isomerism

(A) (B) (C) (D)

18. How many stereoisomers of a drug for healing the wounds are possible & how many of them are optically
active?

(A) 4, 2 (B) 4, 4 (C) 8, 4 (D) 16, 4

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19. Which of the following compounds is capable of showing geometrical, optical and conformational
isomerism

(A) (B) H 2C = CH—CH —CH 3

|
OH
(C) CH3—CH = CH—CH2—CH2—OH (D) H2C = CH—CH2—OH

Exercise –6
Section – I : JEE (Advanced) Questions Previous Years

1. The maximum number of isomers (including stereoisomers) that are possible on mono-chlorination of the
following compound, is [JEE (Advanced) 2011 P-2]

2. The number of optically active products obtained from the complete ozonolysis of the given compound is:
[JEE (Advanced) 2012 P-1]

(A) 0 (B) 1 (C) 2 (D) 4

3. Which of the given statement(s) about N, O, P and Q with respect to M is (are) correct?

(A) M and N are non-mirror image stereoisomers [JEE (Advanced) 2012 P-2]
(B) M and O are identical
(C) M and P are enantiomers
(D) M and Q are identical

4. The total number of stereoisomers that can exist for M is [JEE (Advanced) 2015 P-1]

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5. Compound(s) that on hydrogenation produce(s) optically inactive compound(s) is(are)
[JEE (Advanced) 2015 P-1]

(A) (B)

H Br
H2C
(C) CH3 (D)
CH3

6. In the following monobromination reaction, the number of possible chiral products is

[JEE (Advanced) 2016 P-1]

7. For the given compound X, the total number of optically active stereoisomer is _______.
[JEE (Advanced) 2018 P-2]
HO
HO

HO
HO
X
This type of bond indicates that the configuration at the specific carbon and the geometry of the
double bond is fixed
This type of bond indicates that the configuration at the specific carbon and the geometry of the
double bond is NOT fixed

8. Total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the
molecular formula C4H8O is __________. [JEE (Advanced) 2019 P-2]

9. The Fischer projection of D-erythrose is shown below [JEE (Advanced) 2020 P-1]

D-Erythrose and its isomers are listed as P, Q, R and S in Column-I. Choose the correct relationship of P,
Q, R and S with D-erythrose from Column II

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Column-I Column II

P. 1. Diastereomer

Q. 2. Identical

R. 3. Enantiomer

S.

(A) P → 2, Q → 3, R → 2, S → 2 (B) P → 3, Q → 1, R → 1, S → 2
(C) P → 2, Q → 1, R → 1, S → 3 (D) P → 2, Q → 3, R → 3, S → 1
10. An organic compound (C8H10O2) rotates plane-polarized light. It produces pink color with neutral FeCl3
solution. What is the total number of all the possible isomers for this compound?
[JEE (Advanced) 2020 P-2]
11. Among the following, the conformation that corresponds to the most stable conformation of meso-butane-
2,3-diol is [JEE Advanced, 2021, P-1]

(A) (B) (C) (D)

12. The total number of chiral molecules formed from one molecule of P on complete ozonolysis
(O3, Zn/H2O) is ________. [JEE Advanced, 2022, P-2]

Section – II : JEE (Main) Questions Previous Years

13. 3-Methyl-pent-2-ene on reaction with HBr in presence of peroxide forms an addition product. The

number of possible stereoisomers for the product is: [JEE (Main) 2017]

(1) Two (2) Four (3) Six (4) Zero

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14. The total number of optically active compounds formed in the following reaction is: [JEE (Main) 2018]

O HBr

(1) Two (2) Four (3) Six (4) Zero

15. Which of the following compounds is not aromatic? [JEE (Main) 9 Jan, 2019, E]

(1) (2) (3) (4)

16. Which compound (s) out of the following is/are not aromatic? [JEE (Main) 11 Jan, 2019, M]

(1) (B), (C) and (D) (2) (C) and (D)

(3) (B) (4) (A) and (C)
17. In the following compound, the favourable site/s for protonation is/are :
[JEE (Main) 11 Jan, 2019, E]

(1) (a) and (e) (2) (b), (c) and (d) (3) (a) and (d) (4) (a)
18. The increasing order of reactivity of the following compound towards reaction with alkyl halides
directly is: [JEE (Main) 12 Jan, 2019, M]

(1) (B) < (A) < (C) < (D) (2) (A) < (B) < (C) < (D)
(3) (B) < (A) < (D) < (C) (4) (A) < (C) < (D) < (B)
19. Which of the following compounds will show the maximum ‘enol’ content?
[JEE (Main) 8 April, 2019, E]
(1) CH3COCH2COOC2H5 (2) CH3COCH2COCH3
(3) CH3COCH3 (4) CH3COCH2CONH2

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21. The decreasing order of electrical conductivity of the following aqueous solutions is :
0.1 M Formic acid (A), [JEE (Main) 12 April, 2019, E]
0.1 M Acetic acid (B),
0.1 M Benzoic acid (C).
(1) A>C>B (2) C>B>A (3) A>B>C (4) C>A>B
22. Number of chiral centers in chloramphenicol is : [JEE (Main) 7 Jan, 2020, M]
23. Number of chiral centres in Pencillin is [JEE (Main) 8 Jan, 2020, M]
24. The number of chiral carbons present in the molecule given below is
[JEE (Main) 2 Sep, 2020, M]

25. The number of chiral carbon(s) present in peptide, 1 e-Arg-Pro, is [JEE (Main) 5 Sep, 2020, M]
26. Which of the following compounds shows geometrical isomerism?
[JEE (Main) 6 Sep, 2020, M]
(1) 2 -methylpent-1-ene (2) 4-methylpent-1-ene
(3) 2 -methylpent-2-ene (4) 4-methylpent-2-ene
27. Compound with molecular formula C3H6O can show: [JEE (Main) 18 March, 2021, Shift-1]
(1) Positional isomerism
(2) Both positional isomerism and metamerism
(3) Metamerism
(4) Functional group isomerism
28. Number of electrophilic centre in the given compound is [JEE (Main) 24 June, 2022, Shift-1]

29. Phenol on reaction with dilute nitric acid, gives two products. Which method will be most
effective for large scale separation? [JEE (Main) 25 June, 2022, Shift-1]
(1) Chromatographic separation (2) Fractional Crystallisation
(3) Steam distillation (4) Sublimation

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30. Compound 'P' on nitration with dil. HNO3 yields two isomers (A) and (B). These isomers can be
separated by steam distillation. Isomers (A) and (B) show the intramolecular and intermolecular
hydrogen bonding respectively. Compound (P) on reaction with conc. HNO3 yields a yellow
compound 'C', a strong acid. The number of oxygen atoms is present in compound 'C' ________.
[JEE (Main) 26 June, 2022, Shift-1]
31. Optical activity of an enantiomeric mixture is +12.6º and the specific rotation of (+) isomer is
+30º. The optical purity is ______% [JEE (Main) 27 July, 2022, Shift-1]
32. Given below are two statements. [JEE (Main) 27 July, 2022, Shift-1]

Statement I : The compound is optically active.

Statement II : is mirror image of above compound A.

In the light of the above statement, choose the most appropriate answer from the options given
below.
(1) Both Statement I and Statement II correct
(2) Both Statement I and Statement II are incorrect.
(3) Statement I is correct but Statement II is incorrect.
(4) Statement I is incorrect but Statement II is correct.
33. Compound ‘A’ undergoes following sequence of reactions to give compound ‘B’. The correct structure
and chirality of compound ‘B’ is: [where Et is– C2H5] [JEE (Main) 29 July, 2022, E]

(1) (2)

(3) (4)

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ANSWER KEY
Exercise-1
1. (B) 2. (A) 3. (D) 4. (B) 5. (A) 6. (B) 7. (C)
8. (C) 9. (B) 10. (C) 11. (C) 12. (C) 13. (D) 14. (C)
15. (B) 16. (D) 17. (B) 18. (C) 19. (A) 20. (D) 21. (A)
22. (D) 23. (C) 24. (B) 25. (C) 26. (C) 27. (B) 28. (D)
29. (D) 30. (D) 31. (ACD) 32. (A) 33. (A) 34. (C) 35. (B)

Exercise-2
1. (A) 2. (A) 3. (C) 4. (D) 5. (B) 6. (B) 7. (A)
8. (C) 9. (A) 10. (C) 11. (A) 12. (D) 13. (D) 14. (A)
15. (C) 16. (B) 17. (B) 18. (B) 19. (A) 20. (D) 21. (C)
22. (C) 23. (C) 24 (B) 25. (C) 26. (D) 27. (D) 28. (D)
29. (A) 30. (B) 31. (D) 32. (B) 33. (D) 34. (C) 35. (C)
36. (A) 37. (C) 38. (D) 39. (B) 40. (C)

Exercise- 3
1. (AC) 2. (AB) 3. (ABC) 4. (ABC) 5. (ABCD) 6. (ABC) 7. (ABC)
8. (ABD) 9. (ACD) 10. (AD) 11. (AC) 12. (AC) 13. (AD) 14. (ABD)
15. (ABC) 16. (CD) 17. (AD) 18. (ACD) 19. (AC) 20. (ABC)

Exercise-4
Section-I
1. 12 2. 4 3. 3 4. 8 5. 4 6. 2 7. 22.5
8. 36º 9. 2 10. 4 11. 8
Section-II
12. A – Q, R B – Q, S C – P, Q, R D – Q, S
13. A–R B – P, Q C – P, Q, S D – P, Q
14. A – R, S B – P, R C – R, S D – P, R, S
Section-III
15. (D) 16. (C) 17. (B) 18. (B) 19. (C) 20. (D) 21. (D)
22. (C) 23. (C)

Exercise-5
1. (ACD) 2. (ABD) 3. (ABCD) 4. (BCD) 5. 22 6. (ACD) 7. (AC)
8. (ABCD) 9. 12 10. (CD) 11. (A) 12. (CD) 13. (A) 14. (A)
15. (A) 16. (B) 17. (D) 18. (B) 19. (A)

Exercise – 6
Section – I
1. 8 2. (A) 3. (ABC) 4. 2 5. (BD) 6. 5 7. 7
8. 10 9. (C) 10. 6 11. (B) 12. 2
Section – II
13. (2) 14. (2) 15. (1) 16. (1) 17. (2) 18. (1) 19. (2)
20. (4) 21. (1) 22. (02.00) 23. (03.00) 24. (5) 25. (4) 26. (4)
27. (4) 28. (3) 29. (3) 30. (7) 31. (42) 32. (3) 33. (3)

Chemistry – Optical Isomerism Toll Free Number : 1800 103 9888

HALOALKANES
AND
HALOARENES
 C-84
Exercise - 1
Concept Building Questions

1. Which of the following statements are correct for nucleophile?

(A) All negatively charged species are nucleophiles
(B) Nucleophiles arc Lewis bases
(C) Alkenes, alkynes, benzene and pyrrole are nucleophiles
(D) All are correct

2. Arrange the following nucleophiles in the order of their nucleophilic strength:

       
(A) OH  CH3COO  OCH3  C6 H5O (B) CH3COO  C6 H5O  OCH3  OH
       
(C) C6 H5O  CH3COO  CH3O  OH (D) CH3COO  C6 H5O  OH  CH3O

3. Correct order of leaving group tendency is:

(A) I   Br   Cl   F  (B) F   Cl   Br   I 
(C) Cl   F   Br   I  (D) I   Cl   Br   F 

ZnX
4. In reaction C2 H 5OH  HX 
2  C H X  H O the order of reactivity of HX is:
2 5 2
(A) HBr > HI > HCl (B) HI > HCl > HBr (C) HCl > HBr > HI (D) HI > HBr > HCl

5. Which of the following leads to the formation of an alkyl halide?

Re d P  Br SOCl
(A) C2 H 5OH 
2 (B) C2 H 5OH 
2

KBr Conc. H SO
(C) C2 H5OH 
2 4 (D) All of these

6. The S N 2 reactivity order for alkyl halides is:

(A) R — F > R — Cl > R— Br > R — I (B) R — I> R — Br> R — Cl> R — F
(C) R — Br> R — I> R — Cl> R — F (D) R — Cl> R — Br> R — F> R — I

7. In S N 1 reaction, the first step involves the formation of :

(A) free radical (B) carbanion (C) carbocation (D) final product

8. The rate law for the reaction, RCl  NaOH  aq.  ROH  NaCl is given by, rate  k1  RCl . The rate of
the reaction will be:
(A) doubled on doubling the concentration of sodium hydroxide
(B) halved on reducing the concentration of alkyl halide to half
(C) decreased on increasing the temperature of the reaction
(D) unaffected by increasing the temperature of the reaction

9. Vinylic halides are unreactive towards nucleophilic substitution because of the following except:
(A) C-halogen bond is strong
(B) The halogen in bonded to sp2 carbon
(C) A double bond character is developed in the carbon-halogen bond by resonance
(D) Repulsion of Nucleophile from π e– cloud

10. Which of the following will not undergo nucleophilic aromatic substitution?
D Cl Cl
Cl Cl D CH3 H 3C CH3
(I) (II) (III) (IV)
D D
CH3 CH3
(A) I, II and III (B) II and IV (C) III and IV (D) Only IV

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-85
11. Arrange the following alkyl chlorides in order of decreasing reactivity in an SN1 reaction:
(I) isopropyl bromide (II) propyl bromide
(III) tert-butyl bromide (IV) methyl bromide
(A) (III) > (I) > (II) > (IV) (B) (I) > (III) > (IV) > (II)
(C) (IV) > (III) > (II) > (I) (D) (I) > (II) > (III) > (IV)

12. Which one of the following is most reactive towards nucleophilic substitution reaction?
(A) CH2 CH — Cl (B) C6H5Cl
(C) CH3CH CH — Cl (D) ClCH2 — CH CH2

13. Alkyl halides can be converted to ethers through:

(A) Rankland reaction (B) Williamson synthesis
(C) Fittig reaction (D) Grignard reaction

14. Which of the following reactions will go faster if the concentration of the nucleophile is increased?
CH3 Br H OCH3

(A) + CH3O–

Br + CH3S– SCH3
(B)
Br O–CO–CH3

(C) +CH3COO–

(D) No comparison between these reactions

15. The order of reactivity of alkyl halide in the reaction R  X  Mg   RMgX is:
(A) RI  RBr  RCl (B) RCl  RBr  RI (C) RBr  RCl  RI (D) RBr  RI  RCl

16. Which alkyl halide will react fastest with aqueous methanol?
(A) Me3 C — Br (B) Me3C — Cl (C) Me2 CH — Br (D) CH 3 — CH 2 — CH 2 — Br

17. When the concentration of alkyl halide is triple and concentration of OH is reduced to half, the rate of
SN2 reaction increased by:
(A) 3 times (B) 1.5 times (C) 2 times (D) 6 times

18. The compound which undergoes fastest reaction with aq. NaOH solution is:
(A) C6H5 — CH — OCH3 (B) C6H 5 — CH — CH 3
Cl Cl
(C) C6H5 — CH2 — CH2 — Cl (D) C6H 5 — CH — CH2CH3
Cl
19. The rate of SN1 reaction is fastest with:

(A) CH (B) CH
Br Br

(C) CH NO2 (D) CH2 Br

Br

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-86
20. In the following reaction the most probable product will be:
Br
H CH3 OH
H 3C H S N2

C2H 5
OH CH3 C2H5 CH3
H CH3 H OH HO H
(A) (B) (C) H CH3 (D)
H 3C H H 3C H H 3C H H 3C H
C2H5 C2 H 5 OH C2 H 5
21. In the following reaction find the correct product:
OH
H C2H5 NaI/Acetone
H CH3
ClCl
OH I OH I
(A) H C2 H 5 (B) H CH3 (C) H C2H5 (D) H C2 H 5
H CH3 H C2 H 5 H 3C H H CH3
Cl OH I OH

22. The correct decreasing order of relative reactivity of the following chlorides toward aqueous KOH
solution:
Cl Cl Cl
Cl HO
(P) (Q) (R) (S)
(A) P  Q  R  S (B) R  P  Q  S (C) S  R  Q  P (D) R  S  Q  P

23. The relative reactivity of following halides toward SN2 reaction follows the order:
O
O Cl CH2 Cl
Cl Cl
(P) (Q) (R) (S)
(A) Q > S > R > P (B) P > S > R > Q (C) S > R > Q > P (D) P > R > S > Q

24. Rate of SN1 reaction is:

Br CH2 Br

(P) (Q)
CH3
CH2 CH2 Br CH
Br
(R) (S)
(A) S > Q > R > P (B) S > R > P > Q (C) P > Q > R > S (D) S > R > Q > P
Arrange the following in decreasing order of SN2 reaction (from question no. 25-33)
Cl
25. CH3Cl CH3CH2 Cl CH3CH2CH2Cl CH3 CH CH3
(P) (Q) (R) (S)

(A) P > Q >S>R (B) P > Q > R > S (C) S> R >Q>P (D) S> Q >R>P

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-87

Cl
26. Cl Cl Cl

(P) (Q) (R) (S)

(A) P > Q > R > S (B) R > P > Q > S (C) Q > R > P > S (D) R > Q > P > S

Cl
27. Cl Cl
Cl
(P) (Q) (R) (S)

(A) S > R > P > Q (B) S > R > Q > P (C) R > S > Q > P (D) S > P > R > Q

Cl Cl Cl Cl

CH3 H 3C CH3
28.

(P) (Q) (R) (S)

(A) P > R > S > Q (B) P > Q > R > S (C) P > R > Q > S (D) R > Q > S > P

Cl
N C CH2 Cl
29. Cl Cl
Cl
(P) (Q) (R) (S)
(A) P > Q > R > S (B) Q > P > S > R (C) Q > P > R > S (D) R > S > Q > P
Br H3C Br H2C CH Br
30. Br
(P) (Q) (R) (S)
(A) S > P > Q > R (B) Q > S > R > P (C) Q > R > P > S (D) R > Q > P > S

31. H 3C O CH2 Cl H3C C CH2 Cl H 3C CH2 Cl H 3C Cl

O
(P) (Q) (R) (S)
(A) Q > P > S > R (B) P > Q > R > S (C) S > R > P > Q (D) S > R > Q > P

O
32. CH2 Cl H3C CH2 Cl H 3C CH 2 CH 2 Cl CH3 C CH2 Cl

(P) (Q) (R) (S)

(A) P > R > Q > S (B) S > P > Q > R (C) Q > R > P > S (D) Q > S > P > R
Br Br Br Br

33.

OCH3 CH3 NO2

(P) (Q) (R) (S)
(A) Q > R > P > S (B) R > Q > S > P (C) P > Q > R > S (D) S > P > R > Q

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-88
Arrange the following in decreasing order of reactivity toward SN1 reaction. (From question no. 34-39)
Cl
Cl
34. CH3 Cl
Cl
(P ) (Q ) (R ) (S )
(A) S > R > Q > P (B) S > Q > R > P (C) R > S > Q > P (D) P > Q > R > S

Cl
Cl Cl
35. Cl
(P) (Q ) ( R) (S)
(A) P > Q > R > S (B) S > R > Q > P (C) S > Q > R > P (D) R > Q > S > P

36. Cl Cl Cl Cl

(P) (Q ) (R ) (S)
(A) S > Q > P > R (B) S > P > Q > R (C) Q > P > S > R (D) S > R > Q > P
Br
Br Br
37.
Br
(P ) ( Q) (R) ( S)
(A) P > Q > R > S (B) Q > R > S > P (C) Q > P > R > S (D) Q > R > P > S

Br CH3 O

C Br H 3C C CH Br Br
38.
CH3 CH3
(P) (Q) (R ) (S )
(A) P > Q > S > R (B) Q > P > S > R (C) Q > P > R > S (D) R > S > P > Q

Br Br Br
Br

39.

(P) (Q) (R) (S)

(A) P > Q > R > S (B) S > P > Q > S (C) R > Q > P > S (D) R > P > Q > S

40. Cl NaBr / DMSO

 
Cl

(A) Cl (B) Br (C) Br (D)

Br Br Cl Cl

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-89
41. The reaction
H OH Cl H
Pyridine
+ SOCl2 + SO2 + HCl
proceed by the mechanism:
(A) SN1 (B) SN2 (C) SE2 (D) SNi

42. The order of decreasing nucleophilicites of the following species is:

(A) CH3S > CH3O > CH3COO > CH3OH (B) CH3COO > CH3S > CH3O > CH3OH
(C) CH3OH > CH3S > CH3COO > CH3O (D) CH3O > CH3OH > CH3COO > CH3S

43. Which of the following cannot undergo E2 reaction?

Br
(A) (B) (C) (D) None of these
Br
Br
44. In which of the following reactions, regioselectivity can be observed?
Cl Cl
Cl Cl
(A) (B) (C) (D)
alc. KOH alc. KOH alc. KOH alc. KOH

45. Which of the following will undergo fastest elimination with alcoholic KOH?
Br Br
Br H 3C Br
(A) (B) (C) (D)

CH3 CH3 H 3C

Exercise - 2
Single choice correct with multiple options

1. Which of the following reaction is possible?

(A) C6 H5OH  HBr 
 C6 H 5 Br  H 2O
(B)  CH 3 3 CCl  NaOCH 3 
  CH 3 3 COCH 3  NaCl
Cl OMe

(C) + CH3ONa  CH OH
 3 
Cl Cl
H3O
(D) + C6H5MgBr   C6 H 5CH 2 C  CH 3 2
|
O OH

CH3
Hoffman HBr
2. H3 C — CH — CH — CH3 (B) (C)
Elimination
Cl
HBr/hv
A
D
Correct order of rate of S N 2 for A, C and D will be:
(A) A  C  D (B) C  D  A (C) A  D  C (D) C  A  D

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-90
3. The following reaction is described as:

NaOH 
Br
O
OH CH3
(A) SN1 reaction with racemisation
(B) Intramolecular SN2 reaction with walden inversion
(C) Intramolecular SN2 reaction with retention of configuration
(D) Intramolecular SN2 reaction with racemisation

Cl Cl
Cl
Cl
4. Cl KOH (aq.)

Cl
Cl
Cl
HO OH HO OH
OH OH
OH OH
(A) HO (B) Cl

OH OH
OH OH
OH Cl
O OH
HO
Cl Cl
OH OH
(C) Cl (D) Cl

Cl Cl
OH OH
Cl Cl

5. In the following compound, arrange the reactivity of different bromine atoms toward NaSH in
decreasing order:
Br (R)
Br (S) Br (Q)
Br (P)
O
O
(A) P > Q > R > S (B) S > Q > P > R (C) Q > S > P > R (D) P > S > Q >R

6. Rate of reaction with aqueous ethanol follows the order:

Br Br
Br CH3

O CH2 Br

(P) (Q) (R) (S)

(A) P > Q > S > R (B) Q > P > R > S (C) P > R > Q > S (D) R > P > S > Q

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-91
Br Br

CH2 Br Br
7.

(P) (Q) (R) (S)

(A) P > Q > R > S (B) S > R > P > Q (C) Q > S > P > R (D) Q > S > R > P

Cl Cl Cl Cl

8.

CH3 CH2CH3 CH CH3 H3C C CH3

CH3 CH3
(P) (Q) (R) (S)
(A) S > R > Q > P (B) R > S > Q > P (C) Q > R > S > P (D) P > Q > R > S

Cl Cl Cl Cl

9.

CH3 OH NO2
(P) (Q) (R) (S)
(A) S > P > Q > R (B) R > Q > P > S (C) Q > R > P > S (D) S > Q > P > R

OH

TsCl LiAlH 4
10.   X   Y; identify major product ‘Y’ :

OTs

(A) (B) (C) (D)

H F Acetone
11. + NaI (1 mole)   Major product :
Cl H

H F I F
(A) (B)
I H H H

H I H H
(C) (D)
Cl H Cl I

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-92
12. Find the major product of the following reaction:
CH2—Br
KSH
 
Br
CH2—Br
(A) (B) S
SH
CH2—SH CH2—SH
(C) (D)
SH Br

OTs
NaCN
13.   Product:

CN
CN
(A) (B) * (C) Both (A) and (B) (D) None of these

CH 3

NBS KCN
14. 
(2 Mole )
 A 
(1 Mole )
 B; Product B is:

CH 2CH 3
Br CN Br Br

(A) (B) (C) (D)

Br Br CN

15. Consider the following reaction

Et20
+ HC CNa
Br H
Which of the following products is not expected to form?
(A) (B) (C) (D)
H C CH HC C H

16. The product formed in the reaction:

Br Br Na
 
Br Br ether , 
Br Br Br
(A) (B)
Br Br Br Br
Br
(C) (D)
Br

17. The order of decreasing nucleophilicity of the following is:

(A) H2O > OH > CH3COO > CH3O (B) CH3O > OH > CH3COO > H2O
(C) CH3COO > CH3O > OH > H2O (D) OH > CH3O > CH3COO > H2O

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-93
18. Consider the following nucleophiles
O O O

CF3 S O, Ph S O, Ph O, CH3 C O

O O
I II III IV
when attached to sp3-hybridized carbon, their leaving group ability in nucleophilic substitution
reactions decreases in the order:
(A) I > II > III > IV (B) I > II > IV > III (C) IV > I > II > III (D) IV > III > II > I

19. The most probable product in the following reaction is:

Br
CH3CH 2OK

Br

(A) (B) (C) (D)

Br

20. Major product of the following reaction is:

Br
CH3NH 2


NH—CH3 NH2

(A) (B) (C) (D)

Cl
Cl Cl
alc. KOH
21.

Cl Cl
Cl
Cl Cl

(A) (B) (C) (D) No reaction

Cl Cl Cl Cl
Cl Cl

22. Arrange the following in decreasing order of E2 reaction:

Cl Cl Cl
Cl Cl Cl Cl Cl Cl

Cl Cl Cl Cl Cl Cl
Cl Cl Cl
(X) (Y) (Z)
(A) X > Y > Z (B) X > Z > Y (C) Z > X > Y (D) Y > X > Z

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-94
23. Which of the following reactions will undergo an elimination reaction and an alkene will be
formed in the product?
Br Br
Br
Zn OH, 
(A)  (B)
OH Me
N
Conc. H2SO4 OH, 
(C)  (D)

E2
24. elimination
+
Saytzeff Hofmann
X product product
In the above reaction, maximum Saytzeff product will be obtained where X is:
(A) —I (B) —Cl (C) —Br (D) —F

25. In the above reaction (Q. No. 24) Hofmann product is major when base is:
CH3
H 3C
(A) CH 3O (B) CH3CH2O (C) H3C C O (D) CH O
H 3C
CH3
Me
Me
N
O 
26. Major product:
CH3

(A) (B) (C) (D) None of these

Me
Me
N
27. O 

CH3

(A) (B) (C) (D) None of these

OH
28. NMe3 

(A) (B) (C) (D)

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-95
CH3
Ph H alc. KOH
29. Major product:
Ph Br
CH3
Ph Ph Ph CH3 Ph CH3 Ph CH3
(A) (B) (C) (D)
H 3C CH3 H 3C Ph Ph CH3 Ph CH 2

CH3
D H NaNH2
30.
Br D
CH3
H 3C CH3 H 3C D D CH3 H 3C D
(A) (B) (C) (D)
D D D CH3 D CH3 H CH3

31.

Identify the major product Y:

(A) (B) (C) (D)

N
NMe3
Me Me

H Br
H3C D
32. alc. KOH Major product:
H H
OH
H 3C D D D

(A) H (B) H (C) (D)

33. CH3I (excess) Ag2O/H2O, 

X Y
Identify ‘X’ :

(A) (B) (C) (D)

N N N N
H H H H

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-96
Cl
Br

34. Zn,  Major product:

Br
Cl Cl
Br

(A) (B) (C) (D) None of these

Br

CH3
Ph Cl Zn, 
35.
Ph Cl
CH3
Ph Me Ph Ph Ph Me
(A) C C (B) C C (C) C C
Me Ph Me Me Ph Me
(D) No reaction

Ph
36. H3 C Cl NaI/acetone
Cl CH 3 
Ph
Ph Ph Ph CH3
(A) C C (B) C C
H 3C CH3 H 3C Ph
Ph CH3
(C) C C (D) No reaction
Ph CH3

N
Ag2O/H2O
37.  Major product:
H
D

(A) (B) (C) (D) H2C CH 2

D
HD
38. Arrange the following in decreasing order of E2 reaction:
CH 3 CH 3 CH3
Br Br Br

CH 3 CH 3 CH3
A (X) (Y) (Z)

(A) X > Y > Z (B) X > Z > Y (C) Y > Z > X (D) Z > Y > X
Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-97
Br

39. Major product:
N

CH3

(A) Br (B)
N N Br

CH3 CH3
Br

(C) (D) No reaction

N

CH3

40. Find the nature of following reaction:

H H

NaCN
Br
EtOH

(A) SN2 (B) SN1 (C) E2 (D) E1cb

41. Identify the major product of following reaction:

H 2O
Br

(A) (B) (C) OH (D) OH

OH
OH

42. Find the major product of the following reaction:

C H OH

2 
5 


Br

(A) (B) (C) (D)

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-98

Exercise - 3
Multiple choice correct with multiple options
1. Which of the following compounds will not give SN2 reaction?
Cl

(A) (B) H 2C CH Cl (C) Br (D) Br

2. Which of the following compounds will give SN1 reaction?

(A) H 3C—CH 2—Br (B) Br

Ph
(C) H3C—O—CH2—Cl (D) H3C I
CH2CH3

3. Which of the following are correct for the given reaction?

Ag2O, H2O

N

(A) Major product of reaction is

N

(B) Major product of reaction is

N
(C) Hofmann’s alkene is major product of reaction
(D) Reaction is unimolecular

4. Which of the following halides undergo E2 elimination?

Br

CMe 3
(A) (B)
CMe 3
Cl

(C) CMe3 (D) Br

Br
CMe3

1. Li Y
5. X 2, 7-dimethyl octane, then
Br 2. CuI
(A) X is (B) X is
CuLi 2
CuLi
Br
(C) Y is (D) Y is
Br

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-99
6. Choose the correct among the following statements:
(A) I will react more readily than I forSN2 reaction
(B) Cl will react more readily than Br for SN2 reaction

(C) Br will react more readily than Br for SN2 reaction

(D) CH2Br will react more readily than Br for SN2 reaction

7. Consider the following statements and pick up the correct statements:

(A) MeO CH2—Br will react more readily than

O 2N CH2Br for SN1 reaction

Br Br

(B) will react more readily than for SN1 reaction

Cl
Cl

(C) will react more readily than for SN1 reaction

N
N
H
(D) SN1 reaction occurs in polar protic solvent

8. H 2C CH Cl can undergo:
(A) addition reaction (B) elimination reaction
(C) substitution reaction (D) electrophilic substitution reaction

9. Which of the following reactions take place by SN2 reaction?

O NaI OTs NaI
(A) Br acetone (B) acetone
O
O CH3
NaI
(C) acetone (D) + H3C—C—Br
O O CH2S
CH3
OTs

10. Which of the following compounds will give racemic mixture by SN1 reaction?
CH3 CH3
(A) Et Br (B) H2C CH CH
Br
H

(C) CH2—Br (D) H3C CH CH3

CH2 Br

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11. Pick the correct orders of reactivity:
Cl
(A) SN1  > > Cl >
Ph Cl Cl
O
Cl Cl
(B) SN2  > H 3 CO Cl > Ph Cl >
Ph
(C) E2  Cl > Cl > Cl
O
Cl
(D) SN1  N Cl > > >
Cl Cl

12. Which of the following reactions are not feasible?

Cl Br

KNH2 NaNH2
(A) liq. NH3 (B) liq. NH3

Br Cl
O
NaNH2
NaNH2
(C) liq. NH3 (D) H3C
liq. NH3

13. Which of the following are SN2 reactions?

Cl
(A) + I (B) Br + CN

(C) Br + OH (aq.) (D) Br + OH (alc.)

14. Identify the compounds which may give NGP reaction:

CH3
S Cl
O
(A) (B) H3C Br

CH3
(C) (D) H 3C—CH—CH2—Cl

CH2—CH2—Cl
15. Which of the following reactions involve benzyne intermediate?
Cl Br
KNH2 Mg, THF
(A) NH3(liquid) (B)

Cl
NH2
CH3Br
NaNO2+HCl
(C) (D) AlCl3
O 
C
O

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16. Among the following pair of reactions in which pair second reaction is more reactive than first
for SN1 reaction?
(A) Me3C — Cl  H 2 O and Me3C — Br  H 2O
(B) Me3C — Cl  CH 3OH and Me3C — Cl  H 2 O
(C) Me3C — Cl  H 2 O and Me3C — Cl  H 2 O
(1M) (2 M)
(D) Me3C — Cl  CH 3 SH and Me3 C — Cl  CH 3OH

17. For the reaction

NMe3
OH, 
CH3—CH3—CH—CH3 +
Major Minor
Choose the correct statements:
(A) The reaction is E1 elimination
(B) The reaction is E2 elimination
(C) Transition state has carbanion like character
(D) Transition state has carbocation like character

18. Which of the following are correct order of nucleophilicity in CH3OH?

(A) NH3 < NH2 — NH2
O
(B) CH3CH2O > OH > CH3—C—O
(C) F > Cl > Br > I
O
(D) H3CO O > H3C—C O

19. Which of the following can give E1cb reaction in basic medium?
OH H

(A) CH3 CH C N (B) Cl C CF 3

Cl
O OH

(C) (D) HCF2 CCl3

20. Choose the correct comparison of reactivity toward E2 reaction:

Br

(A) < (B) Br >

Br
Br

(C) > (D) >

Br Cl I
Br

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21. Which of the following compounds cannot give E2 reaction with strong base?
OTs F

(A) (B)

Br
(C) (D) CH
Br

22. Which of the following are better leaving group than

O
O S OCH3?
O
O O
(A) O S CF3 (B) O S Br

O O
(C) N N (D) F

Exercise –4
Section - I : Numerical Value/Subjective Type Questions

O
(1) 'X' CH3MgBr
1. Cl C OC2H5 3º Alcohol. Find out value of ‘X’.
NH4Cl

SH SH
OC2 H5 (1) 'X' CH3 MgBr
2. HO (2) H /H2 O HO
O HO

Find out value of ‘X’.

O
HO
Cl (1) 'X' PhMgBr
3. 3º Alcohol
(2) H /H 2O
HO

4. How many set of carbonyl compound and RMgX can produce 3º alcohol.
OH
C Ph
CH 3

5. Find out numbers of possible E1 products from following reaction.

CH3OH

Br

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6. Identify number of substrate those can give SN1 and SN2 reaction both.
CH3 CH 3 H3 C Br
CH2 Br , H3 C Cl , ,
H3 C Br Ph
CH 3
Br OTf

Cl , , , CH3 CH2 Cl
O
H3 C O
Cl Br

, ,
Cl

7. Examine the ten structures shown below and select those that satisfy each of the following condition.
Br
CH3
Br
(A) (B) H 3C Cl (C) (D) CH3 I
CH3
Cl
Br Cl
(e) (f) (g) Cl (h)

Cl

(i) Br (j)

(i) How many compounds give SN2 reaction on treatment with NaSH?
(ii) How many compounds give E2 reaction on treatment with alcoholic KOH?
(iii) How many compounds do not each react under either of the previous reaction conditions?

8. Examine the ten structures shown below and select those that satisfy each of the following condition.
Cl
H 3C Cl
(a) (b) CH2 Br (c) CH3 I (d) C C
H 3C H
H Br
Cl Br

(e) (f) (g) Cl (h) H

CH3
H
CH2 —Br Br

(i) (j)
H CH3
H H
CH3 CH3
(i) How many compounds give substitution reaction with CH3SNa?
(ii) How many compounds give elimination reaction with Na CN?
(iii) How many compounds do not react with NaOH?

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9. How many substrates will show rearrangement during SN1 reaction?
Cl
Cl
, , , ,
Br Cl I
Br
Br Cl
, Cl ,
Cl , ,

10. Find out number of reactions those proceed with retention of configuration.
CH3 CH3
HOH OH/H2O
(a) H (b) H
Ph Ph
Br Br
CH3
HOH Ph
H
(c) Ph (d) D Cl KCN
OH
H

aq. KOH aq. KOH

(e) OCH3 H (f) OCH3 Br

H Br H H
CH 3 Ph
NaI KSH
SOCl2
(g) H Cl acetone (h) D OH Pyridine
Ph CH 3

Section - II : Match the Column

11. Column (I) Column (II)

CH3
Ph CH3
H ,
(A) Ph C CH CH3 C C (P) Elimination
H 3C CH3
CH3 OH
CH3
H 3C H
H ,
(B) H 3C C CH 2 CH 3 C C (Q) Rearrangement
H 3C CH3
OH
alc. KOH
(C) (R) Unimolecular
Br
O O

(D) O H2C CH2 + CH3 C OH (S) Bimolecular

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12. Column (I) Column (II)
H KNH2
(A) (P) -elimination
Ph Cl
CH3
SOCl2
(B) H OH (Q) SN2
Ph
CH3
(C) Ph N CH3 NaOH (R) -elimination
CH3
O
CH2CH2CH 3
(D) H 3C N CH 2CH 3 (S) SNi
CH2 CH2 Ph

13. Column (I) Column (II)

AlCl3
(A) + Ph CH2 Cl (P) Nucleophilic substitution

AlCl3
(B) + Ph Cl (Q) Electrophilic substitution

Br
KOH (aq.)
(C) (R) Cation intermediate
CH3
Br2,h
(D) H 3C C CH 3 (S) Free radical substitution

CH3

14. Column (I) Column (II)

Ph
(A) H Br Zn, CH3OH (P) Anti elimination
H Br
Ph
CH3
H Br NaI, acetone
(B) 
(Q) Rearrangement
Br H
CH3
OH
Conc. H2SO4
(C)  (R) Carbocation
CH3
Ph
Br

H EtO Na
(D) CH3  (S) Transition state

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Section - III : Comprehension

Comprehension#1 (15 to 17)

Williamson synthesis is an important method for the preparation of symmetrical and unsymmetrical ether.
In this method halide is allowed to react with alcohol in presence of Na or K metal.
Na or K
R OH + R' X R O R'
Mechanism:
Na or 1
R O H R ONa/K+ H2
K 2

R O + R' X R O R'

3º halides and aromatic halides do not give this reaction.

15. Find out the product of the following reaction:

NaH or
OH + CH3 —CH 2—Br
KH

(A) ONa (B) O CH 2CH3

(C) (D) H 2C CH2

Na
16. A+B O—CH3

Find out A and B :

(A) Br and CH 3OH (B) OH and CH3—F

(C) OH and CH3—Br (D) + CH 3OH

CH3
K
17. CH3CH2OH + CH3 C Cl Major product:
CH3
CH3 CH3
(A) H 2C C (B) H 3C C O CH 2CH 3
CH3
CH3
(C) CH3CH2OK (D) None of these

Comprehension#2 (18 to 20)

Aliphatic nucleophilic substitution is mainly of two type SN1 and SN2. SN2 reaction proceed with strong
nucleophile in polar aprotic solvent. 3º halides do not give SN2 reaction. Inverted products are obtained in
this reaction and mechanism of reaction occurs through the formation of transition state.
SN1 reaction proceed through the formation of carbocation in polar aprotic solvent. Solvent itself acts as
nucleophile in this reaction. Racemization takes place in SN1 reaction.

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18. Which of the following compounds will give SN1 reaction?
I
CH3
(A) Ph C CH 2CH 3 (B) H3C Cl (C) H 3C CH 2 Br (D)
Cl
OCH3

19. Which one of the following will give racemised product in C2H5OH?
H 3C Br
CH3 CH3
(A) H3C CH Br (B) Ph C Cl (C) Ph C Cl (D)
CH3 CH3 H

20. Which one of the following will give SN2 reaction?

CH3
(A) H 3C C Cl (B) CH2 Cl
CH3
CH3
(C) Br (D) H 2C CH Cl

Comprehension#1 (21 to 23)

Type of elimination reaction in which least substituted alkene is major product known as Hofmann’s
elimination. Such reaction occur in following conditions:
(X) when base is bulky
(Y) when leaving group is very poor such as fluoride, ammonium group ( NR3) etc.
(Z) when alkyl halide contain one or more double bonds.

21. What is the major product of the following reaction?

CH3 CH3
Me3CO K
H 3C CH CH C CH 3
Br CH3
CH3 CH3 CH3 CH3
(A) H2C CH CH C CH3 (B) H3C CH C C CH3
CH3 CH3
CH2 CH3
(C) H3C H2C C C CH3 (D) None of these
CH3

22. Which of the following will not produce Hofmann’s alkene as major product on reaction with strong
base?
F Cl
(A) (B) H3 C N CH3 (C) S (D)
Ph H3C CH3

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CH3
C2H5O
23. CH2 CH CH CH3 
Br

CH3 CH3
(A) CH CH CH (B) CH2 CH CH
CH3 CH3

CH3 CH 3
(C) CH2 CH CH (D) CH 2 CH2 C CH3
CH3
OC2H 5 OC2 H5

Exercise –5
Revision exercise (Moderate to Tough)

1. The reactions ‘x’ and ‘y’ give the same products which is given below. If they differ by solvents, the
correct reaction coordinate diagram is

OH
(x) CH3 —CH2 —Br 
C2H5OH
 CH3 —CH2 —OH

OH
(y) CH3 —CH 2 —Br 
H 2O
 CH3 —CH2 —OH

y x
G x y
(A) (B) G

Reaction coordinates
Reaction coordinates

x
(C) G y (D) G x
y

Reaction coordinates Reaction coordinates

2. The more possible product is

(A) (B)

(C) (D)

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3. Which of the following compounds will you expect to be more reactive in an SN2 reaction?

(A) y is more reactive than x (B) x is more reactive than y

(C) both are equally reactive (D) both will not undergo substitution

4.

The major product of the above reaction is:

(A) (B)

(C) (D) Both B and C are equally possible

5.

The major product of the above reaction is

(A) (B)

(C) (D)

6. The order of nucleophilicity of the following pair is

(A) (p) x (q) x (r) y (B) (p) x (q) x (r) x (C) (p) y (q) x (r) y (D) (p) y (q) y (r) x

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
 C-110

7.

The product of the above reaction is

(A) (B)

OH

(C) (D) CH3O OH

NO2

+
8.

From the above reaction products, the correct statements are:

(A) attack of nucleophile from the back side to the leaving group is fast in contact ion pair
(B) 83% of S-enantiomer + 17% R-enantiomer is the product
(C) 17% R-enantiomer + 83% S-enantiomer is the product
(D) 66% S-Enantiomer + 34% racemic mixture

9.
The possible products of the above reaction are

(A) (B)

(C) (D)

10.

The major product of the above reaction is

(A) (B) (C) (D)

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Exercise –6
Section – I : JEE (Advanced) Questions Previous Years

1. KI in acetone, undergoes SN2 reaction with each P, Q, R and S. The rates of the reaction vary as
[JEE (Advanced) 2013 P-1]
O
Cl
H3C—Cl Cl Cl
P Q R S
(A) P>Q>R>S (B) S>P>R>Q (C) P>R>Q>S (D) R>P>S>Q

2. For the following compounds, the correct statement(s) with respect to nucleophilic substitution reactions
is(are) [JEE (Advanced) 2017, P-2]
CH3 CH3
Br Br H3C C Br Br

CH3
I II III IV
(A) I and III follow SN1 mechanism
(B) I and II follow SN2 mechanism
(C) Compound IV undergoes inversion of configuration
(D) The order of reactivity for I, III and IV is: IV > I > III

3.

(A) P  1; Q  2; R  5; S  3 (B) P  2; Q  1; R  3; S  5
(C) P  1; Q  2; R  5; S  4 (D) P  2; Q  4; R  3; S  5
[JEE (Advanced) 2023, P-1]

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4. In the following reactions, P,Q, R, and S are the major products.

The correct statement about P, Q, R, and S is

(A) P is a primary alcohol with four carbons.
(B) Q undergoes Kolbe’s electrolysis to give an eight-carbon product
(C) R has six carbons and it undergoes Cannizzaro reaction.
(D) S is a primary amine with six carbons. [JEE (Advanced) 2023, P-2]

Section – II : JEE (Main) Questions Previous Years

1. The increasing order of the reactivity of the following halides for the SN1 reaction is: [JEE (Main) 2017]

CH 3CHCH 2CH 3 CH3CH2CH2Cl p−H3CO−C6H4−CH2Cl

|
Cl

(I) (II) (III)

(1) (I) < (III) < (II) (2) (II) < (III) < (I) (3) (III) < (II) < (I) (4) (II) < (I) < (III)

2. The major product of the following reaction is: [JEE (Main) 2017]

KOH, CH OH
CH3CHCH 2CHCH 2CH3 
heat
3

| |
Br Br

(1) CH2= CHCH2CH = CHCH3 (2) CH2= CHCH = CHCH2CH3

(3) CH3CH = C = CHCH2CH3 (4) CH3CH = CH−CH = CHCH3

3. The major product of the following reaction is: [JEE (Main) 2017]
CH 3
|
C 2H5ONa
C6 H 5CH 2 —C—CH 2 —CH 3 
C 2 H5OH
|
Br
CH3
|
(1) C6 H5CH 2 —C—CH 2 CH3 (2) C6 H5CH = C—CH 2 CH 3
| |
OC 2 H 5 CH3
(3) C6 H5CH 2 —C = CHCH3 (4) C6 H5CH 2 —C = CH 2
| |
CH3 CH 2CH3

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4. The major product of the following reaction is: [JEE (Main) 2018]

(1) (2) (3) (4)

5. The major product formed in the following reaction is: [JEE (Main) 2018]

(1) (2) (3) (4)

6. Which of the following will most readily give the dehydrohalogenation product? [JEE (Main) 2018]

(1) (2) (3) (4)

7. The major product of the following reaction is: [JEE (Main) 2018]

(1) (2) (3) (4)

8. Increasing order of reactivity of the following compounds for SN1 substitution is: [JEE (Main) 2019]

(1) B<C<D<A (2) A<B<D<C (3) B<A<D<C (4) B<C<A<D

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
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9. Which of the following potential energy (PE) diagrams represents the SN1 reaction? [JEE (Main) 2019]

(1) (2) PE
PE

Progress of reaction
Progress of reaction

(3) PE (4) PE

Progress of reaction Progress of reaction

10. The major product of the following reaction is: [JEE (Main) 2019]

KOH alc. (excess)


Δ


(1) (2) (3) (4)

11. An ‘Assertion’ and a 'Reason' are given below. Choose the correct answer from the following options.
Assertion (A): Vinyl halides do not undergo nucleophilic substitution easily.
Reason (R): Even though the intermediate carbocation is stabilized by loosely held p-electrons, the
cleavage is difficult because of strong bonding.
(1) Both (A) and (R) are correct statements but (R) is not the correct explanation of (A)
(2) Both (A) and (R) are wrong statements
(3) Both (A) and (R) are correct statements and (R) is the correct explanation of (A)
(4) (A) is a correct statement but (R) is a wrong statement.
12. The major product of the following reaction is: [JEE (Main) 2019]

CO 2CH 2CH3
|
(1) CH3CH 2C = CH 2 (2) CH 3C = CHCH 3
|
CO 2CH 2CH 3

(3) (4)

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13. The major product of the following reaction is: [JEE (Main) 2019]
(i) KOH alc.
CH3CH2CH—CH2
(ii) NaNH 2
in liq NH3
Br Br
(1) CH3CH = C = CH2 (2) CH3CH2CH——CH2
NH2 NH2
(3) CH3CH = CHCH2NH2 (4) CH3CH2C ≡ CH
14. The product formed in the first step of the reaction of

with excess Mg/Et2O (Et = C2H5) is: [JEE (Main) 24 Feb, 2021, M]

(1) (2)

(3) (4)

15. Ammonolysis of Alkyl halides followed by the treatment with NaOH solution can be used to prepare
primary, secondary and tertiary amines. The purpose of NaOH in the reaction is:
[JEE (Main) 16 March, 2021, E]
(1) to remove basic impurities (2) to activate NH3 used in the reaction
(3) to remove acidic impurities (4) to increase the reactivity of alkyl halide

16. [JEE (Main) 17 March, 2021, M]

The above reaction requires which of the following reaction condition?

(1) 573K, Cu, 300 atm (2) 623 K, Cu, 300 atm
(3) 573 K, 300 atm (4) 623 K, 300 atm
17. The major product in the reaction

is [JEE (Main) 25 June, 2022, M]

(1) t-Butyl ethyl ether (2) 2,2-Dimethyl butane

(3) 2- Methyl pent-1-ene (4) 2-Methyl prop-1-ene

18.

Considering the above reactions, the compound ‘A’ and compound ‘B’ respectively are:
[JEE (Main) 29 July, 2022, M]

(1) (2)

(3) (4)

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19. Assertion A: Hydrolysis of an alkyl chloride is a slow reaction but in the presence of Nal, the rate of the
hydrolysis increases. [JEE (Main) 24 Jan, 2023, M]
Reason R: I is a good nucleophile as well as a good leaving group.
In the light of the above statements, choose the
Correct answer from the options given below.
(1) A is false but R is true (2) A is true but R is false
(3) Both A and R are true and R is the correct
(4) Both A and R are true but R is NOT the correct Explanation of A
20. Number of moles of AgCl formed in the following reaction is _____ [JEE (Main) 24 Jan, 2023, M]

21. Maximum number of isomeric monochloro derivatives which can be obtained from 2, 2, 5, 5-
tetramethylhexane by chlorination is ________ [JEE (Main) 24 Jan, 2023, E]
22. Decreasing order twards SN1 reaction for the following compounds is: [JEE (Main) 30 Jan, 2023, E]

(1) a > c > d > b (2) a>b>c>d (3) b>d>c>a (4) d>b>c>a
23. For the reaction
Acetone
RCH2Br + I–   RCH 2 I  Br –
major

The correct statement is [JEE (Main) 06 Apr, 2023, M]

(1) Br– can act as competing nucleophile. (2) The reaction can occur in acetic acid also.
(3) The transition state formed in the above reaction is less polar than the localized anion.
(4) The solvent used in the reaction solvates the ions formed in rate determining step.
24. Choose the halogen which is most reactive towards SN1 reaction in the given compounds (A, B, C & D)
[JEE (Main) 08 Apr, 2023, M]

(A) (B)

(C) (D)

(1) A – Br(b); B – I(a); C – Br(a); D – Br(a) (2) A – Br(b); B – I(b); C – Br(b); D – Br(b)
(3) A – Br(a); B – I(a); C – Br(b); D – Br(a) (4) A – Br(a); B – I(a); C – Br(a); D – Br(a)

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25. The correct order of reactivity of following haloarenes towards nucleophilic substitution with aqueous
NaOH is: [JEE (Main) 08 Apr, 2023, E]

Choose the correct answer from the option given below:

(1) A>B>D>C (2) C>A>D>B (3) D > C > B > A (4) D > B > A > C
26. The major product formed in the Friedel-Craft acylation of chlorobenzene is
[JEE (Main) 15 Apr, 2023, M]

(1) (2)

(3) (4)

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ANSWER KEY
Exercise-1
1. (A) 2. (B) 3. (A) 4. (D) 5. (D) 6. (B) 7. (C)
8. (B) 9. (C) 10. (D) 11. (A) 12. (D) 13. (B) 14. (A)
15. (A) 16. (A) 17. (B) 18. (A) 19. (A) 20. (B) 21. (C)
22. (B) 23. (D) 24. (A) 25. (B) 26. (D) 27. (A) 28. (C)
29. (B) 30. (C) 31. (A) 32. (B) 33. (D) 34. (A) 35. (C)
36. (B) 37. (D) 38. (B) 39. (C) 40. (A) 41. (D) 42. (A)
43. (C) 44. (A) 45. (A)

Exercise-2
1. (D) 2. (D) 3. (B) 4. (C) 5. (C) 6. (A) 7. (C)
8. (D) 9. (B) 10. (B) 11. (B) 12. (D) 13. (C) 14. (B)
15. (D) 16. (C) 17. (B) 18. (B) 19. (A) 20. (B) 21. (C)
22. (B) 23. (D) 24. (A) 25. (C) 26. (C) 27. (C) 28. (A)
29. (B) 30. (A) 31. (B) 32. (A) 33. (C) 34. (C) 35. (A)
36. (A) 37. (D) 38. (D) 39. (C) 40. (A) 41. (B) 42. (D)

Exercise-3
1. (ABCD) 2. (CD) 3. (BC) 4. (AB) 5. (BC) 6. (ACD)
7. (ACD) 8. (AB) 9. (ABC) 10. (AB) 11. (AB) 12. (BCD)
13. (AC) 14. (ABC) 15. (ABC) 16. (ABC) 17. (BC) 18. (ABD)
19. (ABC) 20. (ABD) 21. (BCD) 22. (ABC)

Exercise-4
Section-I
1. 3 2. 4 3. 4 4. 3 5. 4 6. 6
7. (i) 5 C D E F H (ii) 3 B C F (iii) 4 A G I J
8. (i) 5 A B C H I (ii) 2 G J (iii) 3 D E F 9. 6 10. 4ACEG

Section-II
11. (A) – P Q R ; (B) – P R ; (C) – P S ; (D) – P R
12. (A) – R ; (B) – S ; (C) – Q ; (D) – P
13. (A) – Q R ; (B) – Q R ; (C) – P ; (D) – S
14. (A) – P S ; (B) – P S ; (C) – Q R ; (D) – P S

Section-III
15. (B) 16. (C) 17. (A) 18. (A) 19. (C) 20. (B) 21. (A)
22. (D) 23. (A)

Exercise-5
1. (D) 2. (A) 3. (B) 4. (B) 5. (A) 6. (A) 7. (C)
8. (ABD) 9. (ABCD) 10. (B)

Exercise-6
Section-I
1. (B) 2. (ABCD) 3. (B) 4. (B)
Section-II
1. (4) 2. (2) 3. (2) 4. (2) 5. (3) 6. (4) 7. (4)
8. (3) 9. (4) 10. (2) 11. (4) 12. (2) 13. (4) 14. (3)
15. (3) 16. (4) 17. (4) 18. (3) 19. (3) 20. 2 21. 3
22. (3) 23. (3) 24. (3) 25. (4) 26. (2)

Chemistry – Haloalkanes and Haloarenes Toll Free Number : 1800 103 9888
MATHEMATICS

MATRICES
 M-1
Exercise - 1
Concept Building Questions

Formation of matrices:
 i  j 2
1. For 2 × 3matrix A  aij  whose elements are given by a ij  , then A is equal to
2

x  2 5   4 y  3
2. If   3 
, then find x, y, z
 2 x  1  z

Sum and difference of the matrices and scalar multiple of a matrix

 1 2   3 3 
3. If A–2B =   and 2A–3B =   , then B =
3 0   1 1

1 0  0 1  cos  sin  
4. If I =  , B =   and C    , then prove that C = B sin θ + I cos θ
0 1  1 0    sin  cos  

1 2  4 3
5. If A    ,B    and 2A + C = B, then C =
2 5 0 2

0 2   0 3a 
6. If A =   , kA =   , then find value of k, a, b
 3 4   2b 24 

Multiplication of matrices
7. If (1 2 3) B = (3 4) then order of the matrix B is

8. If a matrix has 13 elements, then the possible dimensions (orders) of the matrix are

 2 3
 1 2 3  
9. If A =   and B =  4 5 , then find AB, BA
 4 2 5   2 1

m 
10. If A   m n      25 and m < n, m,n  N then (m, n) =
n 

 ab b2 
11. If A   n
 and A = 0 then the minimum value of ‘n’ is
2
 a ab 

a h  x
12. If A   x, y , B    ,C    , then ABC =
h b   y

13. If A = diagonal (3,3,3), then A4 =

 1 2   3 6
14. If A    andf(t) = t2 – 3t + 7, thenf(A) +  
4 5   12 9 

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-2
0 1
15. If A   , then A5 =
1 0 
1 2 2
16. If A   2 1 2  , then A2 – 4A is equal to
 2 2 1 

 1  tan   1 tan    a  b 
17. If     , find a and b.
 tan  1   tan  1   b a 

a 2 ab ac 
 
18. If A   ab b 2 bc  and a2 + b2 + c2 = 1, then A2 =
 2
 ac bc c 

19. A fruit shop has 5 dozen oranges, 3 dozen mangoes, 6 dozen bananas their selling prices are Rs 60, Rs 40,
Rs. 30 each respectively. Using matrix algebra the value of the fruits in the shop is

 1 3 2  1 
20. Find the value of x such that [1 x 1]  2 5 1   2  = 0.
15 3 2   x 
 0  tan   / 2  
21. Let A =   and I be the identity matrix of order 2.
 tan   / 2  0 
cos   sin  
Show that I + A = (I – A)  .
 sin  cos  

Problems based on induction

x x
22. If A    , then A = …….., n  N
n

 x x 

 i 0 0
23.  0 i 0  , then A4n 1  ....,n  N
 
 0 0 i 

1 0  1 0
24. If A    and I    , Prove by the principle of mathematical induction that
1 1  0 1 
A n  nA   n  1 I

3n 1 3n 1 3n 1 
1 1 1  
25. If A = 1 1 1 , prove that An = 3n 1 3n 1 3n 1  for all positive integers n.
 n 1 n 1 n 1 
1 1 1 3 3 3 

Trace of matrix

26. If Tr(A) = 6 Tr(4A) =

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-3

27. If Tr (A) = 3, Tr(B) = 5, then Tr(AB) =

1 2 3  1 0 0 
28. If A   4 5 6  , B   0 3 0  , Tr (BA) = ….
 
7 1 0   0 4 5 

Transpose and properties of transpose of matrix

29. If A is a 3 × 4 matrix and B is matrix such that ATB and BAT are both define, then order of B is

T
r  4 6   5 r  2
30.  3   , then r =
 r  3  r  5 4 

 1 18 
 7 10 17  T  
 and2B–3A =  4 6  , then B =
T
31. If 3A + 4B = 
 0 6 31   5 7 
 

 3 4  T T
32. If 5A =   and AA = A A = I, then x =
 4 x 

2 x  3 x  2
33. If A =  3 2 1  is a symmetric matrix, then x =

 4 1 5 

 2 3 5
 
34.
 4 1 2  = P + Q, where P is a symmetric and Q is a skew-symmetric, then Q =
1 2 1
 

 2 2 4 
35. If A =  1 3 4  , then A is

 1 2 3
(A) An idempotent matrix (B) Nilpotent matrix
(C) Involutary (D) Orthogonal matrix

36. If A and B are skew-symmetric matrices of the same order, prove that
(i) AB + BA is a symmetric matrix
(ii) AB – BA is a skew-symmetric matrix

 3 2 3
37.
 
Express the matrix A = 4 5 3 as the sum of a symmetric and a skew-symmetric matrix.
 
 2 4 5

Adjoint and Inverse of a matrix

12 22 32 
 2 
38. If A =  2 32 42  , then Adj A =
 2 
 3 42 52 

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-4
 4 0 0
39.
 
If A is a square matrix such that A  (Adj A) = 0 4 0 , then det (Adj A) =
 
 0 0 4
 
40. If A is a 3x3 matrix and |adj A| = 16, then |A| =

41. If det (A3×3) = 6, then det (adj 2A)=

1 1 x 
42. If  1 x 1  has no inverse, then the real value of x is

 x 1 1 

1 1
 1  tan    1  tan   cos   sin  
43. If    1 
   , then  
 tan  1   tan   sin  cos  

 1 a b  1  a  b 
The inverse of  0  is   , then x =
 x 0  0 1 0 
44.
 0 0 1  0 0 1 

45. If A is non zero square matrix of order n with det (I + A)  0 and A3 = 0, where I, O are unit and null matrices of
1
order n x n respectively, then  I  A  

46. If A  0 and (A – 2I) (A – 3I) = 0, then A–1 =

47. Find the inverse of the following matrices, if they exist, by using elementary operations:
1 3 3   1 3 2
 2 3    
(i)   (ii) 1 4 3 (iii) 3 0 5
   
 1 2  1 3 4   2 5 0 

48. Prove that

(i)   
If A be any square matrix, then A  adj  A   A  I n  adj  A   A
n 1
(ii) If A be any square matrix of order n, then adj  A   A
(iii) If |A| = 0, then |adj (A) | = 0
i.e. if A is singular, then adj (A) is also singular
(iv) adj (AT) = (adj (A) T
(v) adj (AB) = (adj B) (adj A)
(vi) If A is a non-singular matrix, then adj (adj (A) ) = |A|n–2. A

(vii) |adj (adj (A) )| = A

 n 12

(viii) adj (KA) = kn–1 (adj A)

1 1 1
 
49. If A =  2 1 0 , show that A–1 = A2.
1 0 0
 2 1   3 2   1 0 
50. Find the matrix A satisfying the matrix equation  A  .
 3 2   5 3 0 1 

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-5

 2 1 1 
 
51. If A =  1 2 1 . Verify that A3 – 6A2 + 9A – 4I = O and hence find A–1.
 1 1 2 
Solution of simultaneous equation non homogeneous linear equations
52. The solution of 2x + y + z = 1, x – 2y – 3z = 1, 3x + 2y + 4z = 5 is

53. Solve
2x + 6y + 11 = 0, 6y – 18z + 1 = 0, 6x + 20y – 6z + 3 = 0

54. The value of ‘a’ for which the equations 3x – y + az = 1, 2x + y + z = 2, x + 2y – az = –1 fail to have unique solution
is

55. The system of equations 3x + 2y + z = 6, 3x + 4y + 3z = 14 and 6x + 10y + 8z = a, has infinite number of solutions,
if a is equal to

56. The number of solutions of the equation 3x + 3y – z = 5, x + y + z = 3, 2x + 2y – z = 3 is ……

 3 4 2 
57. (i)
 
A = 2 3 5 , find A–1 and hence solve the following system of equation:
 
 1 0 1 
3x – 4y + 2z = 1, 2x + 3y + 5z = 7, x + z = 2

1 2 0  7 2 6 
(ii)
 
A = 2 1 3 and B =
 2 1 3 , find AB. Hence, solve the system of equations :
   
0 2 1   4 2 5 
x – 2y = 10, 2x + y + 3z = 8 and – 2y + z = 7

Homogeneous linear equations

58. The number of non-trivial solutions of the system x – y + z = 0, x + 2y – z = 0, 2x + y + 3z = 0 is

59. If x, y, z not all zeros and the equations x = cy + bz, y = az + cx, z = bx + ay are consistent then a relation among a,
b, c is

60. If x, y, z not all zeros and the equations x + y + z = 0, (1 + a) x + (2 + a) y – 8z = 0, x – (1 + a) y + (2 + a) z = 0 have

non-trivial solution then a =

61. If a  b  c  1 and the system ax + y + z = 0, x + by + z = 0, x + y + cz = 0 have non trivial solutions then a

+ b + c – abc = ……….

Exercise - 2
Single choice correct with multiple options

1 2   1 4   0 1
1. If A    ,B    and C =   , then 5A – 3B + 2C is equal to:
3 0   2 3  1 0 
 8 7 8 20  8 20  8 20 
(A)   (B)   (C)   (D)  
 20 9  7 9   7 9  7 9 

 1 0
2. If A2 = 8A + kI, where A=   , then k is:
 1 7 
(A) 7 (B) –7 (C) 1 (D) –1

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-6
 2 3
1 1 1  
3. If A   ,B   1 5 , then AB is equal to:
 3 3 3   4 1 
 3 1 3 1 3 1  3 1
(A)   (B) 9 3 (C)   (D)  
 9 3    9 3 9 3

 0 1 0 
If A   and B    such that A = B, then  is:
2
4. 
 1 1   5 1 
(A) 1 (B) –1 (C) 4 (D) none of these

 1 2  1 0 
5. If A    ,B    then which of the following is correct?
 3 0   2 3
(A) A2 = A (B) B2 = B (C) AB = BA (D) None of these

 0 4 1 
6. If A   2  3 , then A–1 exists (i.e., A is invertible), if:
1 2 1
(A)   4 (B)   8 (C)   4 (D) none of these

 1 2 3
7. If A   2 1 2  , then A is:
 3 2 1 
(A) a symmetric matrix (B) a skew symmetric matrix
(C) a singular matrix (D) non-singular matrix

0 1 
8. If A   2
 and I is the unit matrix of order 2, then (aI + bA) , where a and b are given constants, is
 0 0 
equal to:
(A) a 2 I  b 2 A (B) a 2 I  2abA (C) a 2 I  abA (D) a 2  b 2 A

 cos x  sin x 0 
9. If f(x) =  sin x cos x 0  , then f (x + y) is equal to:
 0 0 1 
(A) f(x) + f(y) (B) f (x) – f (y) (C) f (x) . f (y) (D) none of these

 x 0   2 1   2 5  3 4 
10. If      , then ordered pair (x, y) is:
 1 y   3 4   4 3  0 1 
(A) (1, –2) (B) (–1, 2) (C) (1, 2) (D) (–1, –2)

 cos  sin   1 0 
11. If A    and A (adj (A) ) =    , then  is equal to:
  sin  cos   0 1
(A) 1 (B) 2 (C) 3 (D) 1/2

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-7
 1  2   2 1 1
 2
    
12. If A     1  , B = 2 1  and C     , where  is the complex cube root of 1, then
 2    2 
  1     2 1   
(A + B) C is equal to:
0 1 0 0 1 1

(A) 0  (B)  0 1 0  (C) 0 (D) 1
0  0 0 1  1 1

 1 2   x   5 x 2  y2
13. If  
    , then the value of is equal to:
 2 1   y   4 xy
5 3 13 13
(A) (B) – (C) (D) 
2 2 6 6

 2 1
14. If A   2
 and A – 4A – nI = 0, where I is the unit matrix of order 2, then:
 1 2 
1 1
(A) n   (B) n  (C) n = 3 (D) n = –3
3 3

1 2   6 0 
15. If A be a matrix such that A ×    , then A is:
1 4   0 6 
2 4  1 1  4 2
(A)   (B)   (C)   (D) none of these
1 1  4 2 1 1

 ab b2  n 0 0 
16. If A    and A    , then minimum value of ‘n’ is equal to:
 a
2
ab  0 0 
(A) 2 (B) 4 (C) 6 (D) 3

17. If A and B are square matrices such that AB = B and BA = A, then A2 + B2 is always
(A) 2AB (B) 2BA (C) A + B (D) AB

 3 4 
If A    , then A is equal to  n  N  :
n
18.
 1  1 
3n 4n   3n  4 n  n  2 5  n 
(A)   (B)  (C)  (D) none of these
 n n  1  1 
n
 n  n 


1 0   1 1 
19. If A + B =   and A – 2B =   , then A is equal to:
1 1   0 1
 1 1  2 / 3 1/ 3  1/ 3 1/ 3  1 1/ 3
(A)   (B)   (C)   (D) 
 2 1  1/ 3 2 / 3  2 / 3 1/ 3 2 / 3 1 

20. If ‘A’ is a symmetric matrix of order n, then maximum number of non-zero elements in ‘A’ is:
n  n  1 n  n  1
(A) n2 (B) n(n – 1) (C) (D)
2 2

Mathematics –Matrices Toll Free Number : 1800 103 9888

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21. The minimum number of zero elements in an upper triangular matrix of order n × n is:

(A)
n  n  1
(B)
n  n  1
(C)
n 2

1
(D)
n 2

1
2 2 2 2

1 2 1 4 4 8
22. If A   ,B=   and A . B . C =   , then C is equal to:
3 0   2 3 3 7
 9 /11 5 / 3  9 /11 5/ 3  9 /11 5 / 3 
(A)   (B)   (C)   (D) none of these
1/ 22 1/ 6   1/ 22 1/ 6   1/ 22 1/ 6 

1 2 3
23. If A = 0 1 2  , then A–1 is equal to:
0 0 1 
1 2 1  1 0 0 1 2 1 
(A)  0 1 
2  (B)  2 1 0  (C) 0 1 2  (D) none of these
 0 0 1   1 2 1 0 0 1 

24. If for the matrices A and B, AB = A and BA = B, then A2 is equal to:

(A) I (B) A (C) B (D) none of these

1 a 2
25. The matrix 1 2 5  will be non-singular, if:
 2 1 1 
(A) a = 1 (B) a  1 (C) a = 9 (D) a  9

 1 0 1  x  1 
26. If  1 1 0   y    1  , then ordered triplet (x, y, z) is:
 0 1 1   z   2 
(A) (0, –1, 2) (B) (–1, 0, 2) (C) (–1, 2, 0) (D) (0, 2, –1)

 1 2  4 1 
27. If A     , then A is equal to:
 3 1  7 7 
 1 1 1 1 1 1 1 1
(A)   (B)   (C)   (D)  
2 3  2 3  2 3 2 3 
 x  1 
   
28. The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system A  y   0 has
 z  0
exactly two solutions, is
(A) 0 (B) 29 – 1 (C) 168 (D) 12
 cos  sin   T
29. If A  
 sin  cos   , then A  A is equal to:
 
 0 0 1 1 1 0  0 1
(A)   (B)   (C)   (D)  
 0 0 1 1 0 1 1 0 

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-9
1 1
30. If A    , then A , n  N , is equal to:
n

 0 1
 n n  1 n   n 1 1 1 
(A)   (B)   (C)   (D)  
0 n 0 1   0 1 0 n 
31. If ‘A’ is a matrix of order m × n and Im, In represent the unit matrices of order ‘m’ and ‘n’ respectively,
then:
(A) I m A  A  AIn (B) AI m  A  I n A (C) AI m  A  AI n (D) I m A  A  In A

32.  
If A is non-singular matrix, then det A 1 is equal to :
 1  1 1 1
(A) det  2 (B) (C) det   (D)
A   
det A 2
A det  A

33. If A = [aij]n×nand aij = i2 –j2, then ‘A’ is necessarily:

(A) a unit matrix (B) a zero matrix
(C) a symmetric matrix (D) a skew symmetric matrix

 cos x1  sin x1 0   cos x 2 0 sin x 2 

34. If A  x1    sin x1 cos x1 0  and A (x2) =  0
 1 0  , then(A(x1) . A(x2))–1 is equal to:
 0 0 1    sin x 2 0 cos x 2 
(A) A  x1   A  x 2  (B) A  x1   A  x2  (C) A  x 2   A  x1  (D) A  x 2   A  x1 

 x  x
 1  tan   1 tan 
2 2
35. If A   ,B    , then A  B1 is equal to:
 tan x 1    tan x 1 
 2 
 
 2 
cos x sin x    cos x sin x   cos x  sin x   cos x sin x 
(A)   (B)   (C)   (D)  
 sin x cos x   sin x cos x   sin x cos x    sin x cos x 

i  j, i  j
36. If A   a ij , where aij =  , then A–1 is equal to:
2 2
 i  j, i  j
 0 1/ 3   2 / 9 1/ 3  0 1/ 3 
(A)   (B)   (C)   (D) none of these
1/ 3 0   1/ 3 0  1/ 3 2 / 9 

 1 2 1   x   0 
37. If  0 1 0   y    2  , then ordered triplet (x, y, z) is:
 3 1 1   z   1 
 9 7 7 9 9 7 7 9
(A)   , 2,  (B)  ,2,   (C)  , 2,  (D)  , 2, 
 4 4 4 4 4 4 4 4

38. Every square matrix can be expressed as the sum of:

(A) two symmetric matrices
(B) two skew symmetric matrices
(C) a symmetric matrix and a skew symmetric matrix
(D) a singular matrix and a non-singular matrix

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-10
39. If A is symmetric matrix and B is a skew symmetric matrix, then for any n  N , which of the following is
not correct?
(A) An is symmetric (B) An is skew symmetric, if n is odd
n
(C) B is skew symmetric, if n is odd (D) Bn is symmetric, if n is even

 a  ib c  id 
40. If a2 + b2 + c2 + d2 = 1 and A =  –1
 , then A is equal to:
  c  id a  ib 
a  ib c  id  a  ib c  id   a  ib c  id   a  ib  c  id 
(A)   (B)   (C)   (D)  
 c  id a  ib   c  id a  ib   c  id a  ib   c  id a  ib 

41. Which of the following statements is correct?

(A) A skew symmetric matrix of odd order is always invertible.
(B) A skew symmetric matrix of odd order is always non-invertible.
(C) A skew symmetric matrix of even order is always non-invertible.
(D) None of the above

 1 2   3 2
42. If A – B =   and A + B =  2 0  , then AB is equal to:
2 4   
2 4  2 4  2 4  2 4
(A)   (B)   (C)   (D)  
 4 4 4 4  4 4  4 4 

1 2 2 
1
43. If A   2 1 2  is an orthogonal matrix, then the value of x + y is equal to:
3
 x 2 y 
(A) 3 (B) –3 (C) 0 (D) 1

 3 1 
  1 1
2 2 
44. If P   ,A   T T 2005
 and Q = PAP , then P (Q ) P is equal to:
 1 3 0 1
 
 2 2 
 3   1 2005  3 
1 2005  2005 1
(A)   (B) 2 (C)  3  (D) 2 

0 1     1  
 1 0   2  0 2005

45. If A5 = 0 such that A n  I for l  n  4, then (I – A)–1 is equal to:

(A) A4 (B) A3 (C) I + A (D) none of these

 1  i 3 1  i 3 
 
2i 2i 
46. If A =  ,i  1 and f(x) = x2 + 2,then f (A) is equal to:
 1 i 3 1 i 3 
 
 2i 2i 
1 0   3  i 3  1 0 
(A)   (B)    
0 1   2  0 1 
 5  i 3  1 0  1 0 
(C)     
(D) 2  i 3   
 2  0 1  0 1

Mathematics –Matrices Toll Free Number : 1800 103 9888

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47. The characteristic of an orthogonal matrix A is :
(A) A1  A  I (B) A  A1  I (C) A T A T  A 1  I (D) A  AT  I
48. If A is an orthogonal matrix, then A–1 is equal to :
(A) A (B) AT (C) A2 (D) none of these
2 3 4 
49. If A =  5 3 8  , then trace of A is:
 
 9 2 16 
(A) 17 (B) 25 (C) 8 (D) 15
50. If A is a square matrix of order n × n, then adj (adj A) is equal to :
n n 1 n 2 n 3
(A) A A (B) A A (C) A A (D) A A

51. If A is square matrix, then adj AT – (adj A)T is equal to :

(A) 2 A (B) 2 A I (C) null matrix (D) unit matrix

52. If A is an orthogonal matrix, then :

(A) A  0 (B) A  1 (C) A  2 (D) none of these
1 2 3
53. The matrix 1 2 3  is :

 
 1 2 3
(A) idempotent (B) nilpotent (C) involutory (D) orthogonal

Exercise - 3
Multiple choice correct with multiple options

1 1 1
1. If A = 1 1 1 then
 
1 1 1
(A) A3 = 9A (B) A3 = 27A (C) A + A = A2 (D) A–1 does not exist
2. A square matrix A with elements from the set of real numbers is said to be orthogonal if AT = A–1. If A is
an orthogonal matrix, then
(A) AT is orthogonal (B) A–1is orthogonal (C) adj A = AT (D) |A–1| = 1
1 2 2
3. Let A   2 1 2  , then
 
 2 2 1 
1
(A) A2 – 4A – 5I3 = 0 (B) A–1= (A – 4I3) (C) A3 is not invertible (D) A2 is invertible
5
1 0 
4. Let A =   . Then which of following is true?
1 1 
1  0 0 1  0 0
(A) lim 2 A  n    (B) lim A  n   
n  n  1 0  n  n  1 0 
 1 0
(C) A–n =   (D) none of these
n 1

 3 3 4 
5. If A   2 3 4  , then
 
 0 1 1 
(A) adj (adj A) = A (B) |adj (adj (A) | = 1 (C) |adj (A) | = 1 (D) None of these

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-12
6. If B is an idempotent matrix and A = I – B, then
(A) A2 = A (B) A2 = I (C) AB = O (D) BA = O
7. If A is non-singular matrix, then
(A) A–1is symmetric if A is symmetric (B) A–1 isskew symmetric if A is symmetric
1 1 1
(C) A A (D) A  A
8. Let A and B are two matrices such that AB = BA, then for every n  N
(A) An B  BAn
(B)  AB  A n Bn
n

(C)  A  B  n C0 An  n C1A n 1B  ...  n Cn Bn

n

(D) A
2n
 
 B2n  An  Bn An  Bn 
9. If A and B are 3 × 3 matrices and |A|  0, which of the following are true?
(A) AB  0  B  0 (B) AB  0  B  0
1 1
(C) A  A (D) A  A  2 A
2
10. If A is a matrix of order m × m such that A + A + 2I = O, then

(A) A is non-singular (B) A is symmetric (C) A  0 (D) A 1  (A  I)
2
11. If A2 – 3A + 2I = 0, then A is equal to
 3 2   3 1
(A) I (B) 2I (C)   (D)  
1 0   2 0 
12. If A and B are two matrices such that their product AB a null matrix, then
(A) det A  0 B must be a null matrix
(B) det B  0 A must be a null matrix
(C) atleast one of the two matrices must be singular
(D) if neither det A nor det B is zero, then the given statement is not possible
13. If D1 and D2 are two 3 × 3 diagonal matrices where none of the diagonal elements is zero, then
(A) D1D2 is a diagonal matrix (B) D1D2 = D2D1
2 2
(C) D1  D2 is a diagonal matrix (D) None of the above
C2 0 k 0
14. Let, Ck = nCk for 0  k  n and Ak =  k 1  for k  1 and A1  A 2  A 3  ...  A n   1 , then
 0 C 2k  0 k 2 

(A) k1  k2 (B) k1  k2  2 (C) k1  2n Cn  1 (D) k 2  2n Cn 1

Exercise –4
Section - I : Numerical Value/Subjective Type Questions

2
1. If A  3A  2I and A8  pA  qI, then p + q is ……….

2 0 7 x 14x 7x 
1
2. The value of for which the matrix product  0 1 0   0 1 0  equals an identity matrix is
x 
 1 2 1   x 4x 2x 
___.

1 2  a b 
3. Let A    and B    be two matrices such that they are commutative and c  3b, then the value
3 4  c d 
a d
of is ____ .
3b  c

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 1 1 a 1 
4. If A    ,B   2 2 2
 and (A + B) = A + B , then the value of |a + b| is _____ .
 2  1  b  1

5. If A, B and C are n × n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value of [det(A2BC–
1
)] (where [.] represents greatest integer function) is ______ .

a b c 
6. If matrix A   b c a  , where a, b, c are real positive numbers, abc = 1 and AT A = I, then the greatest
 c a b 
value of a + b3 + c3 is _____ .
3

1 2 
7. If M    and M2  M  I2  0, then 6 must be ……..
 2 3

a b c 
8. Given a matrix A   b c a  where a, b, c are real positive numbers, abc = 1 and ATA = I, then find the
 c a b 
3 3 3
value of a + b + c .

a 0 7
a 0 
9. Let ar = r (7Cr), br = (7 – r) (7Cr) and Ar =  r  . If A =  A r    , then the value of a + b must be
0 br  r 0 0 b 

10 0 
10. If any 2 × 2 matrix A, if A (adj . A) =  3
 , then the value of |A| must be ……………. .
 0 10 

11. Let S denote the sum of all the value of  for which the system of equations
1    x1  x2  x3  1
x1  1    x 2  x3  
x 1  x 2  1    x 3   2 is inconsistent. Then, the value of |S| is ______ .

12. Suppose A is a 4 × 4 matrix such that |A| = 4, then the value of |adjA| is ………… .

a b 
13. Let A   3 2
 be such that A = 0. Then sum of all the elements of A , is …………. .
 c d 

14. Let A be 3 × 3 matrix such that A 'A  I and |A| = I, find the value of |A – I|.

1 2 1 
15. Let A   0 1 1 . Then, the sum of all the values of  for which there exists a column vector X  0
 
 3 1 1 
such that AX =  X, is……. .

16. Let S denote the set of all values of  for which the system of equations
 x1  x 2  x 3  1
x1   x 2  x 3  1
x1  x 2   x 3  1
Is inconsistent, then the value of |  |, is …….. .
S

Mathematics –Matrices Toll Free Number : 1800 103 9888

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17. Let A be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries
n
are 1 and four of them are 0. If n is the number of such matrices, then , is ………… .
2

1 1 1  4 2 2
18. Let A   2 1  3  and (10) B   5 0   . If B is the inverse of A, then the value of , is.... .
 

 1 1 1   1 2 3 

 2 3
19. Let A   –1
 . If A = xA + yI, then the value of 2 (x + y), is …….. .
 1 5 

20. The number of values of , for which the homogeneous system of equations
 a    x  by  cz  0
bx   c    y  az  0
cx  ay   b    z  0
has a non-trivial solution, are …………. .

Section - II : Match the Column

21. Match the columns :

Column-I Column-II
(A, B, C are matrices)
a. If |A| = 2, then |2A–1| = (where A is of order 3) p. 1
1
b. If |A| = , then |adj . (adj .(2A))| = (where A is of order 3) q. 4
8
c. If (A + B)2 = A2 + B2, and |A| = 2, then |B| = r. 24
(where A and B are of odd order)
d. |A2×2| = 2, |B3×3| = 3 and |C4×4| = 4, then |ABC| is equal to : s. 0

22. Match the columns:

List-I List-II
1 1 1
a. If a, b and c are all different from zero such that    0, then p. symmetric
a b c
1  a 1 1 

the matrix A   1 1  b 1  is :
 1 1 1  c 
b. If , ,  are three real numbers, then the matrix q. singular
 1 cos      cos      
 
A  cos      1 cos      
 cos      cos      1 
c. If A, B and C are the angles of a triangle, then the matrix r. non-singular
 sin 2A sin C sin B 
A   sin C sin 2B sin A  is:
 sin B sin A sin 2C 
s. invertible
t. non-invertible

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23. Match the following
Column-I Column–II
a. If A and B are orthogonal, then AB is p. Zero matrix

b. If A and B are nilpotent matrices of order r and s and A and B q. Nilpotent matrix
commute, then(AB)r is

c. If A is a hermitian matrix such that A2 = 0, then A is r. Unitary matrix

d. If A and B are unitary matrices, then AB is s. Orthogonal

24. Match the statements/expressions in Column-I with the Statements/Expressions in Column-II and indicate
your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
Column-I Column -II
x 2  2x  4
a. The minimum value of is p. 0
x2
b. Let A and B be 3 × 3 matrices of real number, where A is symmetric, q. 1
B is skew-symmetric, and (A + B) (A – B) = (A – B) (A + B).
If (AB)t = (–1)k AB, where (AB)t is the transpose of the matrix AB,
then the possible values of k are

c.
 k3a 
Let a = log3 log32. An integer k satisfying 1  2  2, must r. 2
be less than
1 
d. If sin   cos , then the possible values of       are s. 3
 2

Section - III : Comprehension

Comprehension#1 (25 to 27)

Let p be an odd prime number and Tp be the following set of 2 × 2 matrices
 a b  
Tp  A    : a, b,c 0,1,2,.....p  1
 c a  
25. The number of A in Tp such that A is either symmetric or skew symmetric or both and let det (A)
divisible by p is
(A) (p – 1)2 (B) 2 (p – 1)
(C) (p – 1)2 + 1 (D) 2p – 1
26. The number of A is Tp such that the trace of A is not divisible by p put det (A) is divisible by p is
(A) (p – 1) (p2 – p + 1) (B) p3 – (p – 1)2
2
(C) (p – 1) (D) (p – 1) (p2 – 2)
27. The number of A is Tp such that det (A) is not divisible by p is
(A) 2p2 (B) p3 – 5p (C) p3 – 3p (D) p3 – p2
Comprehension#2 (28 to 30)
Let A be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries
are 1 and four of them are 0.
28. The number of matrices in A is
(A) 12 (B) 6 (C) 9 (D) 3
 x  1 
29. The number of matrices A in A for which the system linear equations A  y    0  has a unique solution,
   
 z  0 
is
(A) less than 4 (B) at least 4 but less than 7
(C) at least 7 but less than 10 (D) at least 10

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 x  1 
30. The number of matrices A in A for which the system of linear equations A  y    0  is inconsistent, is
   
 z  0 
(A) 0 (B) more than 2 (C) 2 (D) 1

Exercise –5
Revision exercise (Moderate to Tough)

1. If A and B are two square matrices of the order 3, then the value of 998 tr(I) – 999 tr(AB) + 999 tr(BA) is
……...
 12 1 2 1 3 
  2 2 2
2. The matrix A    2 1  22  2  3  is idempotent if 1  2  3  k where 1, 2 , 3 are non-zero real
 
  31  3 2  32 

numbers. Then the value of (10 + k)2 is …..

3. Find all solutions of the matrix equationX2 = I, where I is the 2-rowed unit matrix, and X is a real matrix
i.e. a matrix all of whose elements are real.
 1 1 
 2 2 
4. The matrix A   is
 1 1 
 2 2 

(1) Unitary (2) Orthogonal (3) Nilpotent (4) Involutary

5. Let A be an idempotent matrix and (I + A)100 = I + (220k – 1)A, then k =

 
6. Let A   3 2
 such that A = 0, then sum of all the elements of A is …….
   

1 2 5
7. Consider the two matrices A and B where A    : B    . Let n(A) denotes the number of elements
4 3  3 
in A. When the two matrices X and Y are not conformable for multiplication then n(XY) = 0. If

C   AB B'A ;D  (B'A)(AB) then, find the value of 

  
 n  C D 2  n  D  

.
 n  A   n  B 
 
a b 
8. If A   2
 then prove that value of f and g satisfying the matrix equationA + fA + gI = 0 are equal
c d 

1 0  0 0 
to –tr (A) and determinant of A respectively. Given a, b, c, d are non zero reals an I    ;O   .
0 1  0 0 

9. A3×3 is a matrix such that |A| = a, B = (adj A) such that |B| = b. Find the value of (ab2 + a2b + 1)S where
1 a a 2 a3
S   3  5  ....... up to , and a = 3.
2 b b b

10. Find the number of 2 × 2 matrix satisfying :

2 2 2 2
(A) aij is 1 or –1 ; (B) a11  a12  a 21  a 22 2; (C) a11a21 + a12a22 = 0

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11. If A is a skew symmetric matrix and 1 + A is non singular, then prove that the matrix B = (I – A) (I + A)–1
 0 5
is an orthogonal matrix. Use this to find a matrix B given A   .
 5 0
 2 1  9 3
12. Given A    ;B    . I is a unit matrix of order 2. Find all possible matrix X in the following
 2 1  3 1
cases.
(A) AX = A (B) XA = I (C) XB = O but BX  O
 3 2 1   x   b 
13. Determine the values of a and b for which the system  5 8 9   y    3 
 2 1 a   z   1
(A) has a unique solution (B) has no solution and (C) has infinitely many solutions
14. If A is an orthogonal matrix and B = AP where P is a non singular matrix then show that the matrix PB–1
is also orthogonal.
n
1 2 a  1 18 2007 
15. If 0 1 4  0 1
  36  then find a + n.
0 0 1  0 0 1 

a b   p  0 
16. Let A    and P       . Such that AP = P and a + d = 5050. Find the value of (ad – bc).
c d  q  0 

cos   sin  0   cos  0 sin  

If F      sin  cos  0 and G      0
  0  , then  F    G     is equal to
1
17. 1
 0 0 1    sin  0 cos 

(A) F    G      
(B) F  1 G 1 (C) G    F       
(D) G 1 F  1

 1 2 5 
18. The rank of the matrix  2  4 a  4  , is

 2  2 a  1 

(A) 2, if a = – 6 (B) 2, if a = 1 (C) 1, if a = 2 (D) 1, if a = – 6

Exercise –6
Section – I : JEE (Advanced) Questions Previous Years

1. Let M be a 3 × 3 matrix satisfying [IIT-JEE – (P-2) 2011]

0   1 1 1 1  0 
M 1    2  , M  1   1  , and M 1   0  .
       
0   3   0   1 1 12 

Then the sum of the diagonal entries of M is

2. Let M and N be two 3 × 3 non-singular skew-symmetric matrices such that MN = NM. If PT denotes the
transpose of P, then M2N2 (MT N) –1 (MN–1)T is equal to [IIT-JEE – (P-1) 2011]
(A) M2 (B) –N2 (C) –M2 (D) MN

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Paragraph Type questions (3-5)
Let a, b and c be three real numbers satisfying [IIT-JEE – (P-1) 2011]
1 9 7 
a b c 8 2 7   0 0 0 …….. (E)
7 3 7 

3. If the point P(a, b, c) with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is
(A) 0 (B) 12 (C) 7 (D) 6

4. Let  be a solution x3 – 1 = 0 with Im   > 0. If a = 2 with b and c satisfying (E), then the value of
3 1 3
a
 b
 is equal to
  c
(A) –2 (B) 2 (C) 3 (D) –3
5. Let b = 6, with a and c satisfying (E). If  and  are the roots of the quadratic equation ax2 + bx + c = 0,
 n
 1 1
then     is
n 0    
6
(A) 5 (B) 7 (C) (D) 
7

6. If P is 3 × 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 × 3 identity matrix,
 x  0
then there exists a column matrix X =  y   0  such that [IIT-JEE – (P-2) 2012]
 z  0 
0 
(A) PX  0  (B) PX = X (C) PX = 2X (D) PX = –X
0 
1 4 4 
7. If the adjoint of a 3 × 3 matrix P is  2 1 7  , then the possible value(s) of the determinant of P is (are)
1 1 3
[IIT-JEE – (P-2) 2012]
(A) –2 (B) –1 (C) 1 (D) 2

8. Let P = [aij] be a 3 × 3 matrix and let Q = [bij], where bij =2i+jaijfor 1  i, j  3. If the determinant of P is 2,
then the determinant of the matrix Q is [IIT-JEE – (P-1) 2012]
(A) 210 (B) 211 (C) 212 (D) 213

i j
9. Let  be a complex cube root of unity with   1 and P = [pij] be a n × n matrix with pij =  . Then
P2  0, when n = [IIT-JEE – (P-2) 2013]
(A) 57 (B) 55 (C) 58 (D) 56

10. For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct?
[IIT-JEE – (P-1) 2013]
(A) NT M is symmetric or skew symmetric, according as M is symmetric or skew symmetric
(B) M N – N M is skew symmetric for all symmetric matrices M and N
(C) M N is symmetric for all symmetric matrices M and N
(D) (adj M) (adj N) = adj (M N) for all invertible matrices M and N

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11. Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible if
(A) the first column of M is the transpose of the second row of M [IIT-JEE – (P-1) 2014]
(B) the second row of M is the transpose of the first column of M
(C) M is a diagonal matrix with nonzero entries in the main diagonal
(D) the product of entries in the main diagonal of M is not the square of an integer

2
12. Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M  N andM2 = N4, then
(A) determinant of (M2 + MN2) is 0 [IIT-JEE – (P-1) 2014]
(B) there is a 3 × 3 non-zero matrix U such that (M2 + MN2)U is the zero matrix
(C) determinant of (M2 + MN2)  1
(D) for a 3 × 3 matrix U, if (M2 + MN2) U equals the zero matrix then U is the zero matrix

13. Let X and Y be two arbitrary, 3 × 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 × 3, non-
zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
[IIT-JEE – (P-1) 2015]
(A) Y3Z4 – Z4Y3 (B) X44+ Y44 (C) X4Z3 – Z3X4 (D) X23 + Y23
 3 1 2 
14. Let P   2 0   , where   R . Suppose Q = [qij] is a matrix such that PQ = kl, where k  R , k  0
 3 5 0 
k k2
and l is the identity matrix of order 3. If q 23   and det (Q) = , then
8 2
[IIT-JEE – (P-1) 2016]
(A)   0, k  8 (B) 4   k  8  0
(C) det  P adj Q    29 (D) det  Q adj  P    213

1  3i   z  rz 2s 
15. Let z  , where i = 1 , and r, s  (1,2,3). Let P =   and I be the identity matrix of
2  z 2s r 
z 
2
order 2. Then the total number of ordered pairs (r, s) for which p = –1 is[IIT-JEE – (P-1) 2016]

 1 0 0
16. Let P =  4 1 0  and I be the identity matrix of order 3. If Q = [qij] is a matrix such that P50 – Q = 1,
16 4 1 
q q
then 31 32 equals [IIT-JEE – (P-2) 2016]
q 21
(A) 52 (B) 103 (C) 201 (D) 205
17. Let a, ,  R. Consider the system of linear equations [IIT-JEE – (P-2) 2016]
ax + 2y = 
3x – 2y = 
Which of the following statement(s) is(are) correct
(A) If a = –3, then the system has infinitely many solutions for all values of  and 
(B) If a  3, then the system has a unique solution for all values of  and 
(C) If     0, then the system has infinitely many solutions for a = –3
(D) If     0, then the system has no solution for a = – 3

18. How many 3 × 3 matrices M with entries from [0, 1, 2] are there, for which the sum of the diagonal entries
of MT M is 5? [IIT-JEE – (P-2) 2017]
(A) 126 (B) 198 (C) 162 (D) 135

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19. Which of the following is(are) NOT the square of a 3 × 3 matrix with real entries?
[IIT-JEE – (P-1) 2017]
1 0 0   1 0 0  1 0 0  1 0 0 

(A) 0 1 0   
(B)  0 1 0   
(C) 0 1 0  (D) 0 1 0 
0 0 1  0 0 1 0 0 1  0 0 1

1  2  x   1 
 
20. For a real number , if the system   1    y    1 of linear equations, has infinitely many
 2  1   z   1 

2
solutions, then 1     [IIT-JEE – (P-1) 2017]

 b1 
21. Let S be the set of all column matrices  b 2  such that b1, b2, b3  R and the system of equations (in real
 b3 
variables) [JEE (Advanced) 2018, P-2]
 x  2y  5z  b1
2x  4y  3z  b 2
x  2y  2z  b3
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one
 b1 
solution for each  b 2   S?
 b3 
(A) x + 2y + 3z = b1, 4y + 5z = b2 and x + 2y + 6z = b3
(B) x + y + 3z = b1, 5x + 2y + 6z = b2 and –2x – y – 3z = b3
(C) –x + 2y – 5z = b1, 2x – 4y + 10z = b2 and x – 2y + 5z = b3
(D) x + 2y + 5z = b1, 2x + 3z = b2 and x + 4y – 5z = b3

22. Let P be a matrix of order 3 × 3 such that all the entries is P are from the set (–1, 0, 1). Then, the
maximum possible value of the determinant of P is _____. [JEE (Advanced) 2018, P-2]
 1 1 1   2 x x 
23. Let x  R and let P   0 2 2  , Q   0 4 0  and R = PQP–1. Then which of the following options
   
 0 0 3   x x 6 
is/are correct? [JEE (Advanced) 2019, P-2]
1  1 
(A) For x = 0, if R a  6  a  , then a + b = 5
 
   
 b   b 
  0
(B) For x = 1, there exists a unit vector i  j   k for which R     0
   
   0
2 x x
(C) det R = det  0 4 0  + 8, for all x  R
 
 x x 5 
(D) There exists a real number x such that PQ = QP

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1 0 0  1 0 0  0 1 0 0 1 0 0 0 1  0 0 1 
24. Let P1  1  0 1 0 , P2  0 0 1 P3  1 0 0 P4  0 0 1 , P5  1 0 0 , P6   0 1 0 
         
           
 0 0 1   0 1 0   0 0 1   1 0 0   0 1 0   1 0 0 

6
 2 1 3
and X   Pk  1 0 2  PkT where PkT denotes the transpose of the matrix Pk. Then which of the following
 
k 1
 3 2 1 
options is/are correct? [JEE (Advanced) 2019, P-2]
(A) X – 30I is an invertible matrix (B) X is a symmetric matrix
1 1
(C) The sum of diagonal entries of X is 18 (D) If X 1   1 , then  = 30
 
 
1 1
 sin 4  1  sin 2  
   l   M , where   () and   () are real numbers, and I is the 2
1
25. Let M  
2
1  cos  cos 4  

 
× 2 identity matrix. If * is the minimum of the set   :  0,2 and * is the minimum of the set

  : 0,2 , then the value of *  * is [JEE (Advanced) 2019, P-1]

37 31 17 29
(A)  (B)  (C)  (D) 
16 16 16 16
 0 1 a   1 1  1
26. Let M  1 2 3 and adj M   8 6 2  where a and b area real numbers. Which of the following
 3 b 1   5 3 1
options is/are correct? [JEE (Advanced) 2019, P-1]

    1
(A) (adj M) –1 + adj M–1 = –M (B) If M      2 , then       3
    3
(C) det(adj M2) = 81 (D) a + b = 3
27. The trace of a square matrix is defined to be the sum of its diagonal entries. If A is 2 × 2 matrix such that
the trace of A is 3 and the trace of A3 is –18, then the value of the determinant of A is _____.
[JEE (Advanced) 2020, P-2]
28. Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If
M–1 = adj(adj M), then which of the following statements is/are ALWAYS TRUE?
[JEE (Advanced) 2020, P-1]
(A) M = I (B) det M = 1 (C) M2 = I (D) (ad M)2 = I
Question Stem for Question Nos. 29 and 30
Question Stem
Let ,  and  be real numbers such that the system of linear equations [JEE (Advanced) 2021, P-1]
x + 2y + 3z = 
4x + 5y + 6z = 
7x + 8y + 9z =   1
is consistent. Let |M| represent the determinant of the matrix
  2 
M    1 0 
 1 0 1 
Let P be the plane containing all those (, ,  ) for which the above system of linear equations is
consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.
29. The value of |M| is _________.
30. The value of D is _____.

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31. For any 3 × 3 matrix M, let |M| denote the determinant of M. Let [JEE (Advanced) 2021, P-1]
1 2 3  1 0 0  1 3 2 
E   2 3 4  , P   0 0 1  and F  8 18 13
   
8 13 18 0 1 0  2 4 3 
If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is(are) TRUE?
1 0 0 
(A) F = PEP and P = 0 1 0
2

0 0 1 
(B) EQ  PFQ 1  EQ  PFQ1
2
(C) (EF)3  EF
(D) Sum of the diagonal entries of P–1EP + F is equal to the sum of diagonal entries of E + P–1FP
32. For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identify matrix. Let E and
F be two 3 × 3 matrices such that (I – EF) is invertible. If G = (I – EF)–1, then which of the following
statements is (are) TRUE? [JEE (Advanced) 2021, P-1]
(A) |FE| = |I – FE| |FGE| (B) (I – FE) (I + FGE) = I
(C) EFG = GEF (D) (I – FE) (I – FGE) = I
33. Let 𝛽 be a real number. Consider the matrix [JEE (Advanced) 2022, P-2]
 0 1 
 
A   2 1 2  .
 3 1 2 
 
If 𝐴7 − (𝛽 − 1)𝐴6 − 𝛽𝐴5 is a singular matrix, then the value of 9𝛽 is ___________.
 5 3 
 2 2 
34. If M    , then which of the following matrices is equal to 𝑀2022 ? [JEE (Advanced) 2022, P-2]
 3  1 
 
 2 2
 3034 3033   3034 3033   3033 3032   3032 3031 
(A)   (B)   (C)   (D)  
 3033 3032   3033 3032   3032 3031   3031 3030 
35. Let α, β and γ be real numbers. Consider the following system of linear equations
x + 2y + z = 7 [JEE (Advanced) 2023, P-1]
x + αz = 11
2x – 3y + βz = γ
Match each entry in List-I to the correct entries in List-II.
Column-I Column–II
1
a. If   (7  3) and γ = 28, then the system has (1) A unique solution
2
1
b. If   (7  3) and γ ≠ 28, then the system has (2) No solution
2
1
c. If   (7  3) where α = 1 and γ ≠ 28, then the (3) Infinitely many solution
2
system has
1
d. If   (7  3) where α = 1 and γ = 28, then the (4) x = 11, y = –2 and z = 0 as a solution
2
system has (5) x = –15, y = 4 and z = 0 as a solution
The correct option is:
(A) (P) → (3) (Q) → (2) (R) → (1) (S) → (4)
(B) (P) → (3) (Q) → (2) (R) → (5) (S) → (4)
(C) (P) → (2) (Q) → (1) (R) → (4) (S) → (5)
(D) (P) → (2) (Q) → (1) (R) → (1) (S) → (3)

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36. Let M = (aij), i, j {1, 2, 3}, be the 3 × 3 matrix such that aij = 1 if j + 1 is divisible by i, otherwise aij = 0.
Then which of the following statements is(are) true? [JEE (Advanced) 2023, P-2]
(A) M is invertible
 a1   a1   a1 
     
(B) There exists a nonzero column matrix  a 2  and such that M  a2  =  a 2 
a  a   a 
 3  3  3

0
 
(C) The set {X   : MX = 0} ≠ {0}, where 0 =  0 
3

0
 

(D) The matrix (M – 2I) is invertible, where I is the 3 × 3 identity matrix

Section – II : JEE (Main) Questions Previous Years

1 1 1
1  
1. Let  be a root of equation x2 + x + 1 = 0 and the matrix A = 1   2  , then the matrix A31 is
3
1 
2
 4 
equal to: (JEE Main 2020-7 Jan, Morning)
2 3
(1) A (2) A (3) A (4) A4
2. If the system of linear equations (JEE Main 2020-7 Jan, Morning)
2x + 2ay + az = 0
2x + 3by + bz = 0
2x + 4cy + cz = 0,
Where a, b, c  R are non-zero and distinct ; has a non-zero solution, then
1 1 1
(1) a + b + c = 0 (2) a, b, c are in A.P. (3) , , are in A.P. (4) a, b, c in G.P.
a b c
3. Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j–2)aji, where i, j = 1, 2, 3. If the
determinant of B is 81, then the determinant of A is: (JEE Main 2020-7 Jan, Evening)
1 1 1
(1) (2) 3 (3) (4)
9 81 27
4. If the system of linear equations, (JEE Main 2020-7 Jan, Evening)
x+y+z=6
x + 2y + 3z = 10
3x + 2y + z  
has more than two solutions, then   2 is equal to ………….
5. For which of the following ordered pairs  ,  , the system of linear equations

x + 2y + 3z = 1
x + 4y + 5z = 
4x + 4y + 4z = 
is inconsistent? (JEE Main 2020-8 Jan, Morning)
(1) (4, 6) (2) (3, 4) (3) (4, 3) (4) (1, 0)
6. The number of all 3 × 3 matrices A, with entries from the set {–1, 0, 1} such that the sum of the diagonal
elements of AAT is 3, is ……………….. (JEE Main 2020-8 Jan, Morning)

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 M-24
2 2 1 0 –1
7. If A =   and I =   , then 10A is equal to: (JEE Main 2020-8 Jan, Evening)
 9 4   0 1 
(1) 4I – A (2) 6I – A (3) A – 4I (4) A – 6I
8. The system of linear equations
 x  2y  2z  5
2 x  3y  5z  8 (JEE Main 2020-8 Jan, Evening)
4x  y  6z  10 has
(1) Infinite solutions when   8 (2) Infinite solutions when   2
(3) no solutions when   8 (4) no solutions when   2
1 1 2 
adj B
9. If A = 1 3 4  , B = adj (A) and C = 3A then ? (JEE Main 2020-9 Jan, Morning)
C
1 1 3 
(1) 8 (2) 4 (3) 2 (4) 16
10. Let a – 2b + c = 1,
 x  a x  2 x  1
If f  x    x  b x  3 x  2  , then: (JEE Main 2020-9 Jan, Evening)
 x  c x  4 x  3 
(1) f (–50) = –1 (2) f (50) = 1 (3) f (50) = –501 (4) f (–50) = 501
11. The following system of linear equations
7x + 6y – 2z = 0
3x + 4y + 2z = 0
x – 2y – 6z = 0 has (JEE Main 2020-9 Jan, Evening)
(1) infinitely many solution (x, y, z) satisfying y = 2 z
(2) infinitely many solution, (x, y, z) satisfying x = 2 z
(3) Only the trivial solution,
(4) no solution
12. Let S be the set of all   R for which the system of linear equations
2x – y + 2z = 2
x – 2y +  z = – 4
x + y + z = 4 has no solution. Then the set S (JEE Main 2020-2 Sep, Morning)
(1) is a singleton (2) contains more than two elements
(3) is an empty set (4) contains exactly two elements
13. Let A be a 2 × 2 real matrix with entries from {0,1} and |A|  0. Consider the following two statements :
(P) If A  I2, then |A| = –1 (JEE Main 2020-2 Sep, Morning)
(Q) If |A| = 1, then tr(1) = 2,
where I2 denotes 2 × 2 identify matrix and tr(1) denotes the sum of the diagonal entries of A. Then:
(1) (P) is true and (Q) is false (2) Both (P) and (Q) are false
(3) Both (P) and (Q) are true (4) (P) is false and (Q) is true
a b c
 
14. Let a, b, c  R be all non-zero and satisfy a + b + c = 2. If the matrix A =  b c a  satisfies ATA = 1,
3 3 3

 c a b
 
then a value of abc can be: (JEE Main 2020-2 Sep, Evening)
1 1 2
(1) 3 (2) (3)  (4)
3 3 3

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 M-25
1 2 1
15. Let A = {X = (x, y, z) : PX = 0 and x + y + z = 1), where P =  2 3 4  , then the set A :
T 2 2 2

 1 9 1
(JEE Main 2020-2 Sep, Evening)
(1) is an empty set (2) contains more than two elements
(3) contains exactly two elements (4) is a singleton
x 1
 , x  R and A = [aij]. If a11 = 109, then a22 is equal to (JEE Main 2020-3 Sep, Morning)
4
16. Let A = 
 1 0 
 2 1 1 
 
 
T
17. Let A be a 3 × 3 matrix such that adj A =  1 0 2  and B = adj(adj A). If |A| =  and B1  
 1 2 1
then the ordered pair,   ,   is equal to (JEE Main 2020-3 Sep, Evening)
 1  1  1
(1) (3, 81) (2)  9,  (3)  3,  (4)  9, 
 9  81   81 
18. Let S be the set of all integer solutions, (x, y, z), of the system of equations
x – 2y + 5z = 0
– 2x + 4y + z = 0 (JEE Main 2020-3 Sep, Evening)
– 7x + 14y + 9z = 0
such that 15  x 2  y2  z2  150. Then, the number of elements in the set S is equal to
 cos  isin      5 a b 
19. If A    ,     and A    where i  1, then which one of the following is not
i sin  cos    24  c d 
true? (JEE Main 2020-4 Sep, Morning)
1
(1) a 2  b 2  (2) a 2  c 2  1 (3) a 2  d 2  0 (4) 0  a 2  b 2  1
2
20. If the system of equations (JEE Main 2020-4 Sep, Morning)
x – 2y + 3z = 9
2x + y + z = b
x – 7y + az = 24, has infinitely many solutions, then a – b is equal to
21. If the system of equations
x+y+z=2
2x + 4y – z = 6
3x + 2y + z  
has infinitely many solutions, then: (JEE Main 2020-4 Sep, Morning)
(1) 2   5 (2)   2  5 (3)   2  14 (4) 2    14
22. Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the
vector b on the right side is equal to b1, b2 and b3 respectively. (JEE Main 2020-4 Sep, Evening)
1  
0  
0  
1  
0  0
           
If x1  1 , x 2   2  , x 3  0  , b1  0 , b 2   2  and b3   0  , then the determinant of A is equal to:
1 1  1  0 0   2 
1 3
(1) 4 (2) (3) 2 (4)
2 2
23. Let   R . The system of linear equations
2x1  4x 2  x 3  1
x1  6x 2  x 3  2
x1  10x 2  4x 3  3
is inconsistent for : (JEE Main 2020-5 Sep, Morning)
(1) exactly two values of  (2) exactly one positive value of 
(3) every value of  (4) exactly one negative value of 

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24. If the system of linear equations (JEE Main 2020-5 Sep, Evening)
x + y + 3z = 0
x + 3y + k2z = 0
3x + y + 3z = 0
y
has a non-zero solution (x, y, z) for some k  R then x +   is equal to
z
(1) 9 (2) 3 (3) – 9 (4) –3
25. The value of  and  for which the system of linear equations
x+y+z=2
x + 2y + 3z = 5
x + 3y + z  
has infinitely many solutions are, respectively (JEE Main 2020-6 Sep, Morning)
(1) 5 and 7 (2) 6 and 8 (3) 4 and 9 (4) 5 and 8
26. The sum of distinct values of  for which the system of equations (JEE Main 2020-3 Sep, Evening)
   1 x   3  1 y  2z  0
   1 x   4  2  y     3 z  0
2x   3  1 y  3    1 z  0
has non-zero solutions, is
27. The system of linear equations
x+y+z=2
2x + 3y = 2z = 5
2x + 3y + (a2 – 1) z = a + 1 (JEE Main 2019-9 Jan, Morning)
(1) is inconsistent for a = 4 (2) has a unique solution for a  3
(3) has infinitely many solutions for a = 4 (4) is inconsistent when a  3
 cos   sin   
28. If A    , then the matrix A–50 when   , is equal to:
 sin  cos   12
(JEE Main 2019-9 Jan, Morning)
 1 3  3 1  3 1   1 3
         
(1)  2 2  (2)  2 2 (3)  2 2  (4)  2 2 
 3 1   1 3  1 3  3 1 
       
 2 2   2 2   2 2   2 2 
29. If the system of linear equations
x – 4y + 7z = g
3y – 5z = h
–2x + 5y – 9z = k is consistent, then : (JEE Main 2019-9 Jan, Evening)
(1) g + 2h + k = 0 (2) g + h + 2k = 0 (3) 2g + h + k = 0 (4) g + h + k = 0
e t  t  t 
e cos t e sin t
 
30. If e t e t cos t  e  t sin t e t sin t  e  t cos t  then A is: (JEE Main 2019-9 Jan, Evening)
 t 
e 2e t sin t 2e t cos t 
(1) invertible for all t  R (2) invertible only if t  

(3) not invertible for any t  R (4) invertible only if t 
2
31. If the system of equations (JEE Main 2019-10 Jan, Morning)
x+y+z=5
x + 2y + 3z = 9
x + 3y + z  
has infinitely many solutions, then    equals:
(1) 21 (2) 8 (3) 18 (4) 5

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32. The number of values of   0,   for which the system of linear equations
x + 3y + 7z = 0
–x + 4y + 7z = 0
(sin 3)x + (cos 2)y + 2z = 0
has a non-trivial solution, is: (JEE Main 2019-10 Jan, Evening)
(1) three (2) two (3) four (4) one

33. If the system of linear equations (JEE Main 2019-11 Jan, Morning)
2x + 2y + 3z = a
3x – y + 5z = b
x – 3y + 2z = c
where, a, b, c are non-zero real numbers, has more than one solution, then :
(1) b – c + a = 0 (2) b – c – a = 0 (3) a + b + c = 0 (4) b + c – a = 0
 0 2q r 
 
34. Let A   p q r  . If AAT = 13, then |p| is : (JEE Main 2019-11 Jan, Morning)
 p q r 
 
1 1 1 1
(1) (2) (3) (4)
5 3 2 6
35. An ordered pair  , for which the system of linear equations
1    x  y  z  2
x  1    y  z  3
x  y  2z  2
has a unique solution, is : (JEE Main 2019-12 Jan, Morning)
(1) (2, 4) (2) (–3, 1) (3) (–4, 2) (4) (1, –3)
1 0 0 
  q  q31
36. Let P = 3 1 0 and Q = [qij] be two 3 × 3 matrices such that Q – P5 = I3. Then 21 is equal to:
q32
9 3 1
(JEE Main 2019-12 Jan, Morning)
(1) 10 (2) 135 (3) 15 (4) 9
 1 sin  1 
  3 5 
37. If A    sin  1 sin  ; then for all   ,  , det (A) lies in the interval :
 4 4 
 1  sin  1 
(JEE Main 2019-12 Jan, Evening)
  5  5   3  3 
(1) 1,  (2)  , 4  (3)  0,  (4)  ,3
 2 2   2 2 
38. The set of all values of  for which the system of linear equations
x – 2y – 2z =  x
x + 2y + z = y
– x – y = z
has a non-trivial solution : (JEE Main 2019-12 Jan, Evening)
(1) is a singleton (2) contains exactly two elements
(3) is an empty set (4) contains more than two elements
39. The greatest value of c  R for which the system of linear equations
x – cy – cz = 0
cx – y + cz = 0
cx + cy – z = 0
has a non-trivial solution, is : (JEE Main 2019-8 April, Morning)
1
(1) –1 (2) (3) 2 (4) 0
2

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-28
 cos   sin    0 1
40. Let A    ,    R  such that A 32    . Then a value of is :
 sin  cos   1 0 
(JEE Main 2019-8 April, Morning)
  
(1) (2) 0 (3) (4)
32 64 16
1 1 1
 
41. Let the numbers 2, b, c be in an A.P. and A   2 b c  . If det(A)  [2, 16] then c lies in the interval:
4 b2 c 2 

(JEE Main 2019-8 April, Evening)
(1) [2, 3) (2) (2 + 23/4, 4) (3) [4, 6] (4) [3, 2 + 23/4]
42. If the system of linear equations (JEE Main 2019-8 April, Evening)
x – 2y + kz = 1
2x + y + z = 2
3x – y – kz = 3
has a solution (x, y, z), z  0, then (x, y) lies on the straight line whose equation is :
(1) 3x – 4y – 1 = 0 (2) 4x – 3y – 4 = 0 (3) 4x – 3y – 1 = 0 (4) 3x – 4y – 4 = 0
1 1  1 2   1 3  1 n  1  1 78  1 n 
43. If   .  .  ............     , then the inverse of   is:
 0 1  0 1   0 1  0 1  0 1  0 1 
(JEE Main 2019-9 April, Morning)
 1 0  1 13  1 12   1 0
(1)   (2)   (3)  (4) 
12 1  0 1  0 1  
13 1 
 0 2y 1 
 
44. The total number of matrices A   2x y 1 ,  x, y  R, x  y  for which ATA = 3I3 is :
 2x  y 1 
 
(JEE Main 2019-9 April, Evening)
(1) 2 (2) 3 (3) 6 (4) 4
45. If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trivial solution
x y z
(x, y, z), then    k is equal to : (JEE Main 2019-9 April, Evening)
y z x
3 1 1
(1) (2) (3)  (4) – 4
4 2 4
46. If the system of linear equations (JEE Main 2019-10 April, Morning)
x+y+z=5
x + 2y + 2z = 6
x + 3y + z  ,  ,  R  has infinitely many solutions, then the value of    is :
(1) 12 (2) 9 (3) 7 (4) 10
47. Let  be a real number for which the system of linear equations: (JEE Main 2019-10 April, Evening)
x+y+z=6
4x + y  z    2
3x + 2y – 4z = –5
has infinitely many solutions. Then  is a root of the quadratic equation:
(1)  2  3  4  0 (2)  2  3  4  0 (3)  2    6  0 (4)  2    6  0
2 3 
48. If A is a symmetric matrix and B is a skew-symmetrix matrix such that A – B =   , then AB is
 5 1
equal to: (JEE Main 2019-12 April, Morning)
 4 1  4 2  4 2   4 2 
(1)   (2)   (3)   (4)  
 1 4   1 4  1 4   1 4
Mathematics –Matrices Toll Free Number : 1800 103 9888
 M-29
49. If the system of linear equations (JEE Main - 2018)
x + ky + 3z = 0
3x + ky – 2z = 0
2x + 4y – 3z = 0
xz
has a non-zero solution (x, y, z), then is equal to:
y2
(1) 30 (2) –10 (3) 10 (4) –30
1 2  2
50. Let A be a matrix such that A   is a scalar matrix and |3A| = 108. Then A equals :
 0 3 
(JEE Main - 2018)
36 32   4 0  4 32  36 0 
(1)   (2)   (3)   (4)  
0 4   32 36  0 36   32 4 
51. If the system of linear equations : (JEE Main - 2018)
x + ay + z = 3
x + 2y + 2z = 6
x + 5y + 3z = b
has no solution, then :-
(1) a = –1, b = 9 (2) a   1, b  9 (3) a  1, b  9 (4) a   1, b  9
52. The number of values of k for which the system of linear equations. (JEE Main - 2018)
(k + 2)x + 10y = k
kx + (k + 3) y = k – 1
has no solution is :
(1) infinitely many (2) 1 (3) 2 (4) 3
1 0 0
 
53. Let A = 1 1 0 and B = A20. Then the sum of the elements of the first column of B is :
1 1 1
(JEE Main - 2018)
(1) 211 (2) 251 (3) 231 (4) 210
 3 1 2 
 
54. Let P =  2 0   , where   R. Suppose Q = [qij] is a matrix satisfying PQ = kI3 for some non-zero
 3 5 0 
k k2
k  R. If q23 = – and |Q| = , then  2  k 2 is equal to (JEE Main – 2021, 24 Feb, Shift-1)
8 2
55. Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for
which the sum of diagonal elements of MT M is seven, is (JEE Main – 2021, 24 Feb, Shift-1)
56. Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then
the system of linear equations (A2 B2 –B2 A2) X = O, where X is a 3 × 1 column matrix of unknown
variables and O is a 3 × 1null matrix, has: (JEE Main – 2021, 24 Feb, Shift-2)
(1) a unique solution (2) exactly two solutions
(3) infinitely many solutions (4) no solution
   
 0  tan   
 2  a b
57. If A =  and (I2 + A) (I2 – A) –1 =  2 2
 , then 13 (a + b ) is equal to
    b a 
 tan   0
  2  
(JEE Main – 2021, 25 Feb, Shift-1)
 i  i    8
x
58. Let A =   ,i  –1. Then, the system of linear equations A8      has:
 i i   y   64 
(JEE Main – 2021, 16 Mar, Shift-1)
(1) A unique solution (2) Infinitely many solutions
(3) No solution (4) Exactly two solutions

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-30
59. The total number of 3 × 3 matrices A having entries from the set (0, 1, 2, 3) such that the sum of all the
diagonal entries of AAT is 9, is equal to______ (JEE Main – 2021, 16 Mar, Shift-1)
60. If A = 
2 3 
 0  1

 , then the value of det  A   det A   Adj(2A) 
4 10
 10
________

(JEE Main – 2021, 17 Mar, Shift-1)

61. The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to
(JEE Main – 2021, 17 Mar, Shift-1)
(1) 0 (2) 1 (3) –1 (4) –2
62. If x, y, z are in arithmetic progression with common difference d, x  3d, and the determinant of the
3 4 2 x 
 
matrix  4 5 2 y  is zero, then the value of k2 is (JEE Main – 2021, 17 Mar, Shift-2)
5 k 
z
 
(1) 72 (2) 12 (3) 36 (4) 6
63. Let , ,  be the real roots of the equation, x + ax + bx + c = 0, (a, b, c  R and a, b  0) If the system
3 2

of equation (in, u, v, w) given by u + v + w = 0, u + v + w = 0 u + v + w = 0 has non-trivial

a2
solution, then the value of is (JEE Main – 2021, 18 Mar, Shift-1)
b
(1) 5 (2) 3 (3) 1 (4) 0
64. Let A be a 3 × 3 matrix having entries from. The set {–1, 0, 1}. The number of all such matrices A having
sum of all the entries equal to 5, is ___________. (JEE Main – 2022, 25 June, Shift-1)
 2  2    1 2 
65. Let A =   and B =   . Then the number of elements in the set
 1  1    1 2 
{(n, m)} : n, m  {1, 2, ….., 10} and nAn + mBm = 1} is _____ (JEE Main – 2022, 25 June, Shift-2)
66. Let A be a 3 × 3 invertible matrix. If |adj (24A)| = |adj (3adj (2A))|, then | A |2 is equal to:
(JEE Main – 2022, 26 June, Shift-1)
(1) 66 (2) 212 (3) 26 (4) 1
 2 1 1
  3 i 1
67. Let A = 1 0 1 and B = A – I. If  = , then the number of elements in the set {n  {1, 2,
2
1 1 0 

……,100} : An + (B)n = A + B} is equal to _____. (JEE Main – 2022, 25 July, Shift-1)

68. Let A be a 2 × 2 matrix with det (A) = – 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the
diagonal elements of A can be: (JEE Main, 2022, 26 July, Shift-1)
(1) –1 (2) 2 (3) 1 (4) – 2
a b 
 , where a, b, c, d  {–1, 0, 1, 2, 3, ………, 10}, such that A = A ,
–1
69. The number of matrices A = 
 c d 
is ______. (JEE Main, 2022, 26 July, Shift-2)
70. If A and B are two non-zero n × n matrices such that A2 + B = A2 B, then
(JEE Main, 2023, 24 Jan., Shift-1)
2 2
(1) AB = I (2) A B = I (3) A = I or B = I (4) A2B = BA2
 1 3 
 10 10   1 i 
71. Let A   and B    , where i  1. If M = A BA, then the inverse of the matrix
T
 3 1   0 1 
 10 
 10 
2023 T
AM A is (JEE Main, 2023, 25 Jan., Shift-2)
1 2023i   1 0  1 0 1 2023i 
(1)   (2)   (3)   (4) 
0 1   2023i 1   2023i 1  0 1 

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-31
72. Let α and β be real numbers. Consider a 3 × 3 matrix A such that A2 = 3A + αI. If A4 = 21A + βI, then
(JEE Main, 2023, 29 Jan., Shift-1)
(1)   1 (2)   4 (3)   8 (4)   8
T
73. If P is a 3 × 3 real matrix such that P = aP + (a – 1) I, where a > 1, then
(JEE Main, 2023, 30 Jan., Shift-2)
(1) P is a singular matrix (2) Adj P  1

1
(3) Adj P  (4) Adj P  1
2
74. Let A = [aij], aij  Z  [0, 4],1  i, j  2. The number of matrices A such that the sum of all entries is a

prime number p  (2,13) is __________. (JEE Main, 2023, 31 Jan., Shift-2)

1 1 3
75. If A    , then : (JEE Main, 2023, 01 Feb., Shift-2)
2   3 1 

(1) A30 – A25 = 2I (2) A30 + A25 + A = I (3) A30 + A25 – A = I (4) A30 = A25
76. Let A = a ij  where a ij  0 for all i, j and A2 = I, Let a be the sum of all diagonal elements of A and
22 '

b = |A| Then 3a2 + 4b2 is equal to (JEE Main 2023, 06 Apr. Shift-1)
(1) 4 (2) 14 (3) 7 (4) 3
 3 1 
  1 1 a b 
2 2 
77. Let P =  ,A= T T 2007
 0 1 and Q = PAP . If P Q P=  c d  then 2a + b – 3c – 4d is equal to
 1 3    
 
 2 2 
(JEE Main 2023, 08 Apr. Shift-1)
(1) 2004 (2) 2005 (3) 2007 (4) 2006
5! 6! 7!
1 
78. If A = 6! 7! 8! , then |adj(adj(2A))| is equal to (JEE Main 2023, 06 Apr. Shift-1)
5!6!7! 
7! 8! 9!

(1) 220 (2) 28 (3) 212 (4) 216

79. The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
is (JEE Main 2023, 13 Apr. Shift-1)
(1) 610 (2) 106 (3) 910 (4) 109
80. Let the determinant of a square matrix A of order m be m – n, where m and n satisfy 4m + n = 22 and
17m + 4n = 93. If det (n adj (adj(m4))) = 3a5b6c, then a + b + c is equal to
(JEE Main 2023, 15 Apr. Shift-1)
(1) 84 (2) 96 (3) 101 (4) 109

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-32

ANSWER KEY
Exercise – 1
 9 
2 2
8
 5 7 
1.   2. x = 2, z = –2, y = 2 3.   4. I cos   B sin 
9 8
25   5  1 
 2 2 
 2 1
5.  4  8  6. –6, –4, –9 7. 3×2 8. 1 × 13 or 13 × 1
 

 10 2 21
 0 4   16 2 37 
9. AB =   , BA =   10. (3, 4) 11. 2
10 3   2 2 11 

0 0
12. (ax2 + 2 hxy + by2) 13. 27 A 14. 0 0
 
15. A 16. 5I3 17. a = sec2 θ, b = 0 18. A

 2n 1 x n  i 0 0
2n 1 x n  0 i 0 
19. 7200 20. x = –2 or –14 22.  n 1 n  23.
 
 2 x 2n 1 x n   0 0 i 

26. 24 27. Cannot say 28. 40 29. 3×4 30. 1

 1 
 0 2
2
 1 3  
   1 0  35.
 1 0 
31. 32. 3 33. 6 34. 0 (A)
 2 
 2 4  
 
 2 0 0
 
 

 3 3 5/ 2 
 3 5 7 / 2   P
37. 
5/ 2 7 / 2 5 

 0 1 1/ 2 
 1 0 1/ 2  Q

 1/ 2 1/ 2 0 

Thus, P is symmetric and Q is skew-symmetric

38. 64 39. 16 40. ±4 41. 32 × 28 42. 1 43. 0 44. 1

5I  A
45. I – A + A2 46.
6

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-33
 2 3
 1  5  5
 7 3 3   
2 3 2 4 11 
47. (i)   (ii)  1 1 0  
(iii) 
1 2   5 25 25 
 1 0 1   
 3 1 9 
 5 25 25 

 2 1  3 2   6  5 4  3  1 1 
50. A = B–1 C–1 =       .
 3 2   5 3   9  10 6  6  1 0 

 3 1 1
1
51.  1 3 1 
4
 1 1 3 

52. 1,–3,2 53. in inconsistent 54. –7/2 55. 36 56. infinite

57. (i) x = 3, y = 2, z = –1, (ii) x = 4, y = –3, z = 1
58. 0 59. a2 + b2 + c2 + 2abc = 1 60. 5  2 2 61. 2

Exercise – 2
1. (2) 2. (2) 3. (2) 4. (4) 5. (4) 6. (2) 7. (4)
8. (2) 9. (3) 10. (A) 11. (A) 12. (A) 13. (A) 14. (4)
15. (3) 16. (A) 17. (3) 18. (4) 19. (3) 20. (A) 21. (A)
22. (2) 23. (3) 24. (2) 25. (2) 26. (2) 27. (4) 28. (A)
29. (3) 30. (2) 31. (A) 32. (4) 33. (D) 34. (4) 35. (3)
36. (A) 37. (2) 38. (3) 39. (2) 40. (3) 41. (2) 42. (A)
43. (2) 44. (A) 45. (4) 46. (4) 47. (4) 48. (2) 49. (4)
50. (3) 51. (3) 52. (2) 53. (2)

Exercise – 3
1. (AD) 2. (ABD) 3. (ABD) 4. (BC) 5. (ABC) 6. (ACD) 7. (AD)
8. (ABCD) 9. (AC) 10. (ACD) 11. (ABCD) 12. (CD) 13. (ABC) 14. (AC)

Exercise – 4
Section –I
1. 1 2. 5 3. 1 4. 5 5. 2 6. 4
7. 1296 8. 4 9. 896 10. 1000 11. 3 12. 64 13. 0
14. 0 15. 3 16. 3 17. 4.5 18. 1 19. 12/13 20. 3

Section – II
21. a  q; b  p; c  s; d  r 22. a  p, r, s; b  p, q, t; c  p, q, t
23. a  s; b  p, q; c  p, q; d  r 24. a  r; b  q, s; c  r, s; d  p, r

Mathematics –Matrices Toll Free Number : 1800 103 9888

 M-34

Section – III
25. (4) 26. (3) 27. (4) 28. (A) 29. (2) 30. (2)

Exercise – 5
   2
1. 2994 2. 121 3. x = ± I or X  
   , where     1 4. 3
 

1  12 5 
5. 5 6. 0 7. 650 9. 225 10. 8 11.
13  5 12 

 a b 
12. (1) X    for a, b  R
 2  2a 1  ab 
(2) X does not exist
a  3a 
(3) X   a, c  R and 3a + c  0
c  3c 

13. (A) a   3; b  R ; (2) a   3, b  1 / 3 (3) a = –3, b = 1/3

15. 200 16. 5049 17. (3) 18. (BD)

Exercise – 6
Section – I
1. 9 2. (ABCD) 3. (4) 4. (A) 5. (2) 6. (4)
7. (AD) 8. (4) 9. (BCD) 10. (CD) 11. (CD) 12. (AB)
13. (CD) 14. (BC) 15. 1 16. (2) 17. (BCD) 18. (2)
19. (AB) 20. 1 21. (AD) 22. 4 23. (AC) 24. (BCD)
25. (D) 26. (ABD) 27. 5 28. (BCD) 29. (1) 30. (1.50)
31. (A,B,D) 32. (A,B,C) 33. (3) 34. (A) 35. (A) 26. (B,C)
Section – II
1. (3) 2. (3) 3. (1) 4. 13 5. (3) 6. 672 7. (4)
8. (4) 9. (1) 10. (2) 11. (2) 12. (4) 13. (4) 14. (2)
15. (3) 16. 10 17. (3) 18. 8 19. (1) 20. 5 21. (4)
22. (3) 23. (4) 24. (4) 25. (4) 26. 3 27. (4) 28. (3)
29. (3) 30. (1) 31. (2) 32. (2) 33. (2) 34. (3) 35. (1)
36. (1) 37. (4) 38. (1) 39. (2) 40. (3) 41. (3) 42. (2)
43. (2) 44. (4) 45. (2) 46. (4) 47. (4) 48. (2) 49. (3)
50. (1) 51. (4) 52. (2) 53. (3) 54. 17 55. 540 56. (3)
57. 13 58. (3) 59. 766 60. 16 61. (4) 62. (1) 63. (2)
64. 414 65. 1 66. (3) 67. 17 68. (2) 69. 50 70. (4)
71. (4) 72. (4) 73. (4) 74. (196) 75. (3) 76. (1) 77. (2)
78. (4) 79. (2) 80. (2)

Mathematics –Matrices Toll Free Number : 1800 103 9888

DETERMINANTS
 M-35
Exercise - 1
Concept Building Questions

Minor, Cofactors, Adjoint

1 i i
1. Find the minors of the diagonal elements of the determinant i 1 i .
1 i i

5 3 8
2. Using cofactors of the elements of second row, evaluate   2 0 1 .
1 2 3

1 x yz
3. Using cofactors of elements of third row, evaluate   1 y z  x .
1 z xy

Evaluate the value of determinant &prove the identities

4. Evaluate the following determinant:
1 3 9 27
102 18 36
3 9 27 1
(vii) (viii) 1 3 4
9 27 1 3
17 3 6
27 1 3 9

5. Without expanding, show that the value of each of the following determinants in zero:
1/ a a2 bc a  b 2a  b 3a  b
2
(iv) 1/ b b ac (v) 2a  b 3a  b 4a  b
1/ c c2 ab 4a  b 5a  b 6a  b

12 22 32 42
1 a a 2  bc
22 32 42 52
(vi) 1 b b2  ac (x)
32 42 52 62
1 c c 2  ab
42 52 62 72

 2 x  2 x   2 x  2 x 
2 2
1
sin  cos  cos     
3x  3 x  3x  3 x 
2 2
(xii) 1 (xiii) sin  cos  cos     
sin  cos  cos     
 4 x  4 x   4 x  4 x 
2 2
1

cos  x  y   sin  x  y  cos 2y

(xv) sin x cos x sin y
 cos x sin x  cos y

sin 2 A cot A 1
(xvii) sin 2 B cot B 1 , where A, B, C are angles of ABC.
sin 2 C cot C 1

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-36

1 x x2 1 1 1
2
6. If   1 y y , 1  yz zx xy , then prove that   1  0.
1 z z2 x y z

Prove the following identities:

a b c
7. a  b b  c c  a = a3 + b3 + c3 – 3abc
bc ca ab

ab bc ca a b c

8. bc ca ab 2 b c a
ca a b bc c a b

a  b  2c a b
3
9. c b  c  2a b  2  a  b  c
c a c  a  2b

a bc 2a 2a
10. 2b bca 2b = (a + b + c)3
2c 2c cab

2
1 a bc 1 a a
11. 1 b ca  1 b b2
1 c ab 1 c c2

z x y x y z x2 y2 z2
12. z2 x2 y2  x 2 y2 z2  x 4 y4 z 4 = xyz (x – y) (y – z) (z – x) (x + y + z).
z4 x4 y4 x4 y4 z4 x y z

a2 bc ac  c 2
13. a 2  ab b2 ac = 4a2b2c2
ab b2  bc c2

1 1 p 1 p  q
14. 2 3  2p 4  3p  2q  1
3 6  3p 10  6p  3q

a bc cb
15. a c b c  a = (a + b – c) (b + c – a) (c + a – b)
a b ba c

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-37

a2 2ab b2
 
2
16. b2 a2 2ab  a 3  b3
2ab b2 a2

a2 1 ab ac
2
17. ab b 1 bc = 1 + a2 + b2 + c2
ca cb c2  1

1 a a2
18. a2 1 a = (a3 – 1)2
a a2 1

a bc c b
19. c abc a = 2 (a + b) (b + c) (c + a)
b a abc

b 2  c2 ab ac
20. ba c2  a 2 bc = 4a2 b2 c2
ca cb a 2  b2

a 2  b2
c c
c
b2  c2
21. a a = 4 abc
a
c2  a 2
b b
b

 bc b 2  bc c2  bc
22. a 2  ac ac c 2  ac = (ab + bc + ca)3
a 2  ab b2  ab ab

x  2x 2x
23. 2x x   2x  (5x  )(  x)2
2x 2x x  


a b 2  c 2  a 2  2b3 2c3

24. 2a 3 
b c2  a 2  b2  2c3 = abc (a2 + b2 + c2)3

2a 3 2b3 
c a 2  b 2  c 2 

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-38
a b c x y z y b q
25. Without expanding, prove that x y z  p q r  x a p .
p q r a b c z c r
p b c
p q r
26. If a q c  0, find the value of   , p  a,q  b, r  c.
pa qb rc
a b r

1 a 1 1
1 1 1
27. If a, b and c are all non-zero and 1 1 b 1  0, then prove that    1  0.
a b c
1 1 1 c
a by cz
a b c
28. If a  x b c  z  0, then using properties of determinants, find the value of   , where
x y z
ax by c
x, y, z  0.

29. If x, y, z are not all zero such that

ax + y + z = 0
x + by + z = 0
x + y + cz = 0,
1 1 1
then prove that    1.
1 a 1 b 1 c

30. Solve each of the following system of homogeneous linear equations:

1. x + y – 2z = 0 2. 2x + 3y + 4z = 0 3. 3x + y + z = 0
2x + y – 3z = 0 x+y+z=0 x – 4y + 3z = 0
5x + 4y – 9z = 0 2x + 5y – 2z = 0 2x + 5y – 2z = 0

31. If a, b, c are non-zero real numbers and if the system of equations

(a – 1) x = y + z
(b – 1) y = z + x
(c – 1) z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc.

Exercise - 2
Single choice correct with multiple options

Determinant of matrix, Properties of determinant, Triangle properties, Factor property, Applications of

determinant, Minor and cofactor, Differentiation

1 a a 2  bc
1. 1 b b 2  ac 
1 c c2  ab
(A) 0 (B) a3 + b3 + c3 – 3abc (C) 3abc (D) (a + b + c)3

1 4 20
2. The roots of the equation 1 2 5 = 0 are
1 2x 5x 2
(A) –1, –2 (B) –1, 2 (C) 1, –2 (D) 1, 2

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-39

0 x a xb
3. If a ≠ b ≠ c, the value of x which satisfies the equation x  a 0 x  c = 0, is
xb xc 0
(A) x = 0 (B) x = a (C) x = b (D) x = c

ab a  2b a  3b
4. a  2b a  3b a  4b 
a  4b a  5b a  6b
(A) a2 + b2 + c2 – 3abc (B) 3ab (C) 3a + 5b (D)0

bc a a
5. b ca b 
c c ab
(A) abc (B) 2abc (C) 3abc (D) 4abc

b 2  c2 a2 a2
6. b2 c2  a 2 b2 
c2 c2 a 2  b2
(A) abc (B) 4abc (C) 4a2b2c2 (D) a2b2c2

x 3 7
7. If –9 is a root of the equation 2 x 2 = 0, then the other two roots are
7 6 x
(A) 2, 7 (B) –2, 7 (C) 2, –7 (D) –2, –7

8. If a, b, c are unequal what is the condition that the value of the following determinant is zero
a a2 a3  1
  b b2 b3  1
c c2 c3  1
(A) 1 + abc = 0 (B) a + b + c + 1 = 0
(C) (a – b) (b – c) (c – a) = 0 (D) None of these

 2  3  1   3
4 3 2
9. If p  q  r  s  t    1 2     4 , the value of t is
 3   4 3
(A) 16 (B) 18 (C) 17 (D) 19

x2  x x 1 x2
10. If 2x 2  3x  1 3x 3x  3 , = Ax – 12, then the value of A is
x 2  2x  3 2x  1 2x  1
(A) 12 (B) 24 (C) –12 (D) –24

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-40
6i 3i 1
11. Let 4 3i 1 = x + iy, then
20 3 i
(A) x = 3, y = 1 (B) x = 0, y = 0 (C) x = 0, y = 3 (D) x = 1, y = 3

a1 b1 c1 a1  pb1 b1  qc1 c1  ra1

12. Suppose D = a 2 b2 c2 and D′ = a 2  pb 2 b2  qc2 c2  ra 2 , then
a3 b3 c3 a 3  pb3 b3  qc3 c3  ra 3
(A) D′ = D (B) D′ = D(1 – pqr)
(C) D′ = D(1 + p + q + r) (D) D′ = D(1 + pqr)

3 x 6 3
13. A root of the equation 6 3 x 3 = 0 is
3 3 6  x
(A) 6 (B) 3 (C) 0 (D) None of these

bc bc' b'c b'c'

14. ca ca ' c'a c'a ' is equal to
ab ab' a 'b a 'b'
(A)  ab  a 'b' bc  b'c' ca  c'a ' (B)  ab  a 'b ' bc  b 'c' ca  c'a '
(C)  ab' a 'b  bc' b'c  ca ' c'a  (D)  ab' a 'b  bc' b'c  ca ' c'a 

xnsin x cos x
n n dn
15. If  (x)  n! sin cos , then the value of  (x) at x = 0 is
2 2 dx n
a a2 a3
(A) –1 (B) 0 (C) 1 (D) Dependent of a

Determinants and its properties

2i 2i
16. The value of
1 i 1 i
is  i 2  1  
(A) A complex quantity (B) real quantity
(C) 0 (D) cannot be determined

18 40 89 
17. Det  40 89 198  = ……
 
89 198 440 
(A) –8 (B) –6 (C)–1 (D) 0

b2  c2 a2 a2
18. b2 c2  a 2 b2 
c2 c2 a 2  b2
(A) a2b2c2 (B) 4abc (C) 4a2b2c2 (D) 2a2b2c2

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-41
abc 2a 2a
19. 2b bca 2b 
2c 2c ca b
(A) 2 (a+b+c)3 (B) (a–b–c)3 (C) 2(a–b–c)3 (D) (a+b+c)3

2  3   1   3
20. If  1 2     4  p 4  q3  r 2  s  t then t =
3   4 3
(A) 16 (B) 17 (C) 18 (D) 19

Adjoint and inverse of matrix, singular and non-singular matrix and invertible matrix

a c b
21. If A   b a c  then the cofactor of a32 in A + AT is
 c b a 


(A)  2a  b  c    b  c 
2
 (B) ac – b2

(D) 2a  a  c    a  c 
2
(C) a2 – bc

 1 0 2   5 a 2
22. Adj  1 1 2    1 1 0   a b  
 0 2 1   2 2 b 
(A) [–4 1] (B) [–4 –1] (C) [4 1] (D) [4 –1]

23. If k is a scalar and I is a unit matrix of order 3, then adj(kI)

(A) k3I (B) k2I (C) –k3I (D) –k2I

24. What of the following is/are true?

(i) Adjoint of a symmetric matrix is symmetric
(ii) Adjoint of a unit matrix is a unit matrix
(iii) A(adj A) = (adj A) A = |A|I and
(iv) Adjoint of a diagonal matrix is a diagonal matrix, is/are incorrect
(A) (i) (B) (ii) (C) (iii) & (iv) (D) All of these

1 1 1 
 
25. The adjoint 1 2 3 is
 2 1 3 
 3 9 5  3 4 5  3 4 5 
 4 1 3     
(A)   (B)  9 1 4  (C)  9 1 4  (D) None of these
 5 4 1   5 3 1   5 3 1

1 2 3 
 
26. The inverse of  0 1 2  is
 0 0 1 
1 2 1  1 2 1  1 2 1 
0 1 2     
(A)   (B) 0 1 2  (C)  0 1 2  (D) None of these
0 0 0  0 0 1   0 0 1 

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-42
27. If A and B are non-singular square matrices of same order, then adj(AB) is equal to
(A) (adj A) (adj B) (B) (adj B) (adj A) (C) (adj B–1) (adj A–1) (D) (adj A–1) (adj B–1)

28. If I3 is the identity matrix of order 3, then I31 is

(A) 0 (B) 3I3 (C) I3 (D) Does not exist

29. For two invertible A and B of suitable orders, the value of (AB)–1 is
(A) (AB)–1 (B) B–1A–1 (C)A–1B–1 (D) (AB′)–1
cos 2  sin 2
30. Inverse of the matrix   is
 sin 2 cos 2 
cos 2  sin 2 cos 2 sin 2  cos 2 sin 2   cos 2 sin 2 
(A)   (B)   (C)   (D)  
 sin 2 cos 2   sin 2  cos 2  sin 2 cos 2   sin 2 cos 2

1 0 0  1 0 0
1
31. A =  0 1 1  ; I   0
  1 0  , A 1   A 2  cA  dI  , where c, d  R, then pair of values (c, d)
6 
 0 2 4   0 0 1 
(A) (6, 11) (B) (6, –11) (C) (–6, 11) (D) (–6, –11)

Applications of determinants and matrices, Cramer’s rule and consistent system and inconsistent system

32. x + ky – z = 0, 3x – ky – z = 0 and x – 3y + z = 0 has non-zero solution for k =

(A) –1 (B) 0 (C) 1 (D) 2

a1 b1 c1
33. If a1x + b1y + c1z = 0, a2x + b2y + c2z = 0, a3x + b3y + c3z = 0 and a 2 b2 c 2 = 0, then the given system
a3 b3 c3
has
(A) One trivial and one non-trivial solution (B) No solution
(C) One solution (D) Infinite solution

34. If the system of equations, x + 2y – 3z = 1, (k + 3) z = 3, (2k + 1) x + z = 0 is inconsistent, then the value

of k is
(A) –3 (B) 1/2 (C) 0 (D) 2

35. The number of values of k for which the system of equations (k + 1) x + By = 4k, kx + (k + 3) y = 3k – 1
has infinitely many solutions, is
(A) 0 (B) 1 (C) 2 (D) infinite

36. The existence of the unique solution of the system x + y + z = λ, 5x – y + μz = 10, 2x + 3y – z = 6 depends
on
(A) μ only (B) λ only (C) λ and μ both (D) Neither λ nor μ

37. The value of λ for which the system of equations 2x – y – z = 12, x – 2y + z = –4, x + y + λz = 4 has no
solution is
(A) 3 (B) –3 (C) 2 (D) –2

38. The equation x + 2y + 3z = 1, 2x + y + 3z = 2, 5x + 5y + 9z = 4 have

(A) Unique solution (B) Infinitely many solutions
(C) Inconsistent (D) Finitely many solutions

39. Equations x + y = 2, 2x + 2y = 3 will have

(A) Only one solution (B) Many finite solutions
(C) No solution (D) Finitely many solutions

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-43

40. Let a, b, c be positive real numbers. The following system of equation in x, y and z
x 2 y2 z 2 x 2 y2 z2 x 2 y2 z2
   1,    1,     1 has
a 2 b2 c2 a 2 b2 c2 a 2 b 2 c2
(A) No solution (B) Unique solution
(C) Infinitely many solutions (D) Finitely many solutions

41. The number of solution of the following equations x2 – x3 = 1, –x1 + 2x3 = –2, x1 – 2x2 = 3 is
(A) Zero (B) One (C) Two (D) Infinite

42. The number of solutions of the system of equations 2x + y – z = 7, x – 3y + 2z = 1, x + 4y – 3z = 5 is

(A) 3 (B) 2 (C) 1 (D) 0

 1 0 1   x  1 
43. The solution of the equation  1 1 0   y   1  is (x, y, z) =
 0 1 1   z   2 
(A) (1, 1, 1) (B) (0, –1, 2) (C) (–1, 2, 3) (D) (–1, 0, 2)

 1 2 3   x   4 2 
2
44. If  3 1 2   y    0 6    , then (x, y, z) =
1
 2 3 1   z   1 2   
(A) (–4, 2, 2) (B) (4, –2, –2) (C) (4, 2, 2) (D) (–4, –2, –2)

1 1 1   x  0  x 
45. If 1 2 2   y    3  , then  y  is equal to
     
1 3 1   z   4   z 
1 1 1 1

(A) 1  
(B)  2  (C)  2  (D)  2 
1  3   1   3

46. x + ay + a2z = 0
x + by + b2z = 0
x + cy + c2z = 0
then given system of equations
(A) Has no solution
(B) Has infinite solution
(C) Has unique solution
(D) Has unique or infinite solution depending on values of a, b, c

47. Let x + 2y + z = 0
3x – y + z = 0
x + 4y – z = 0
If x = 1 is its one of solution then is equal to
5 1 5
(A) 0 (B) (C) (D)
13 13 13

48. Of A is a square matrix of order 3, then the true statement is (where I is unit matrix)
(A) det (–A) = –det A (B) det A = 0
(C) det (A +I) = 1 + det A (D) det 2A = 2 det A

49. Let the determinant of a 3 × 3 matrix A be 2, then B is a matrix defined by B = 5A2. Then det of B is
(A) 20 (B) 10 (C) 40 (D) 500

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-44

ai bi  
50. If Ai  
bi
 and a  1, b  1, then  det  Ai  is equal to
ai  i 1

a2 b2 a 2  b2
(A)  (B)
1  a 2 1  b 2 1  a 2 1  b2 
a2 b2 a2 b2
(C)  (D) 
1  a 2 1  b2  1  a 2 1  b2 
1 0  1 0 
51. If A =   and I    , then which one of the following holds for all n  1, (by the principle of
1 1  0 1 
mathematical induction)
(A)An = nA + (n – 1)I (B)An = 2n–1 A + (n – 1) I
n
(C)A = nA – (n – 1) I (D) An = 2n–1 A – (n – 1) I

 2 
52. If A    and A  125, then  =
3
 2  
(A)± 3 (B) ± 2 (C)± 5 (D) 0

Exercise - 3
Multiple choice correct with multiple options

a2  x2 ab ac
1. The determinant   ab b2  x 2 bc is divisible by
ac bc c2  x 2
(A) x (B) x2 (C) x3 (D) x4

6 2i 3 6
2. The value of the determinant 12 3  8i 3 2  6i , is (where i = 1 )
18 2  12i 27  2i

(A) complex (B) real (C) irrational (D) rational

1
2k 1 sin k
k  k  1 n
3. If D k  x y z , then  Dk is equal to
k1
 n 1 n
n
sin    sin 
2n  1  2  2
n 1 
sin
2
(A) 0 (B) independent of n
(C) independent of θ (D) independent of x, y and z

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-45
a b a  b
4. The determinant b c b  c is equal to zero, if
a  b b   c 0
(A) a, b and c are in AP (B) a, b, c, are in GP
(C) a, b and c are in HP (D) (x – α) is a factor of ax2 + 2bx + c

2cos x 1 0
5. Let f(x) = 1 2 cos x 1 , then
0 1 2cos x
   
(A) f     1 (B) f '    3 (C) 0 f (x)dx  0 (D)  f (x)dx  0
3 3 

x 2  5x  3 2x  5 3
6. If ∆(x) = 3x 2  x  4 6x  1 9 = ax3 + bx2 + cx + d, then
7x 2  6x  9 14x  6 21
(A) a = 0 (B) b = 0 (C) c = 0 (D) d = 47

7. If a, b and c are the sides of a triangle and A, B and C are the angles opposite to a, b and c respectively,
a2 bsin A csin A
then is   bsin A 1 cos A independent of
csin A cos A 1
(A) a (B) b (C) c (D) A, B, C

a a2 0
8. Let f(a, b) = 1 (2a  b) (a  b)2 , then
0 1 (2a  3b)

(A) (a + b) is a factor of f(a, b) (B) (a + 2b) is a factor of f(a,b)

(C) (2a + b) is factor of f(a,b) (D) a is a factor of f(a, b)

sec2 x 1 1
9. If f(x) = cos2 x cos 2 x cosec 2 x , then
1 cos2 x cot 2 x
/4 1 
(A)  / 4 f ( x)dx  16  3  8 (B) f '    0
2
(C) maximum value of f(x) is 1 (D) minimum value of f(x) is 0

a a  x2 a  x 2  x4
10. If 2a 3a  2 x 2 4a  3x 2  2 x4 =a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7 and
3a 6a  3 x 2 10a  6 x 2  3 x4
f(x) = a0 x2 + a3 x + a6, then
(A) f(x)≥ 0, x  R if a > 0 (B) f(x) = 0, only if a = 0
(C) f(x) = 0, has two equal roots (D)f(x) = 0, has more than two root if a = 0

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-46

4x  4 (x  2)2 x3
11. If (x)  8x  4 2 (x  2 2)2 (x  1)3 , then
12x  4 3 (x  2 3)2 (x  1)3

(A) term independent of x in ∆(x) = 16(5 – 2 – 3)

(B) coefficient of x in ∆(x) = 48 (1 + 2 – 3)
(C) coefficient of x in ∆(x) = 16 (5 + 2 – 3 )
(D) coefficient of x in ∆(x) is divisible by 16

3 3x 3 x 2  2a 2
12. If f ( x)  3x 3 x 2  2a 2 3 x3  6 a 2 x , then
3x 2  2a3 3x3  6a 2 x 3x 4  12a 2 x 2  2a 4
(A) f '( x )  0 (B) y = f(x) is a straight line parallel to X-axis
2
(C) 0 f ( x) dx  32a 4 (D) None of the above

13. If a > b > c and the system of equations ax + by + cz = 0, bx + cy + ax = 0, cx + ay + bz = 0 has a non-

trivial solution, then both the roots of the quadratic equation at2 + bt + c, are
(A) real (B) of opposite sign (C) positive (D) complex

14. The values of λ and b for which the equations x + y + z = 3, x + 3y + 2z = 6 and x + λy + 3z = b have
(A) a unique solution, if λ ≠ 5, b R (B) no solution if λ ≠ 5, b = 9
(C) infinite many solution λ = 5, b = 9 (D) None of the above

15. Let λ and α be real. Let S denote the set of all values of λ for which the system of linear equations
λx + (sin α) y + (cos α) z = 0
x + (cos α) y + (sin α) z = 0
– x + (sin α) y – (cos α) z = 0
has a non-trivial solution, then S contains
(A) (–1, 1) (B) [– 2 , – 1] (C) [1, 2] (D) (–2, 2)

Exercise –4
Section - I : Numerical Value/Subjective Type Questions

(b  c)2 a2 a2
10
1. If b2 (c  a)2 b2  abc  a  b  c  , then the value of
3
 r must be………….
r 1
c2 c2 (a  b)2

x2  x x 1 x2
2. If 2x 2  3x  1 3x 3x  3 = Ax – 12, then the value of A2 must be ………. .
x 2  2x  3 2x  1 2x  1

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-47

1  b2  x 1  c2  x
1  a2 x

3. If a2 + b + c = –2 andf(x) = 1  a 2  x 1  b 2 x 1  c 2  x , then f(x) is polynomial of degree …...

2 2

1  a2  x 1  b2  x 1  c 2 x

4. If the system of equations 3x – 2y + z = 0, λx – 14y + 15z = 0, x + 2y + 3z = 0 has non-trivial solution,

then the value of λ2 must be ……. .

sin 2 A cot A 1
5. If A + B + C = π and   sin 2 B cot B 1 , then the value of ∆ + 5, is …….. .
2
sin C cot C 1

e 2iA eiC eiB

6. If A, B and C are angles of a triangle, and   eiC e 2iB eiA . Then the value of  is ……….
eiB eiA e 2iC

1 3cos x 1
7. If f(x) =   sin x 1 3cos x , the maximum value of f(x) is ……… .
1 sin x 1

cos x x 1
f '( x )
8. Let f(x) = 2sin x  x 2 2 x . The value of lim is
x 0 x
tan x  x 1

2cos 2 x sin (2 x)  sin x

1 /2
9. Let f(x) = sin (2 x) 2sin 2 x cos x . The value of  [ f ( x)  f '( x)] dx is …………
 0
sin x  cos x 0

 
cos x  x 2  
sin x  x 2 
 cos x  x 2 
10. If f(x) = sin  x  x 2  cos  x  x 2  
sin x  x 2  . The value of f′(0) is …….. .

sin  2 x  0  
sin 2 x 2

tan A 1 1
11. If A, B and C are angles of non-right angled triangle ABC, then the value of 1 tan B 1 is …. .
1 1 tan C

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-48

1 e i / 3 e i / 4
12. If eiθ = cos θ + isin θ, and D = e i / 3 1 ei2  / 3 , then the value of |[D]| (where [.] represents
e  i / 4 e i2 / 3 1
greatest integer function) is …… .

1 1 1 1
1 2 3 4
13. The value of the determinant is ……. .
1 3 6 10
1 4 10 20

3a a  b a  c
14. If  b  a 3b  b  c = k (a + b + c) (ab + bc + ca) , then the value of k is ……. .
c  a c  b 3c

15. The value of [k/2] (where [.] represents the greatest integer function) for which the system of equations
x + ky + 3z = 0; 3x + ky – 2z = 0 ; 2x + 3y – 4z = 0 possesses a non-trivial solution is ………. .

16. Let  (x) = x + a1, 2 (x) = x2 + b1x + b2, x1 = 2, x2 = 3 and x3 = 5 and

1 1 1
∆ = 1  x1  1  x 2  1  x 3  . The value of ∆ is ……….. .
2  x1  2  x 2  2  x 3 

2r  1 6n  n  2  f n
17. Let f(n), g(n) and h(n) be polynomials in n and let  r  2r 1 3.2n 1  6 g  n  . The value of
r  n  r  1 n  n  1 n  2 h  n 
n
 r is ……… .
r 1

18. If the system of equations

ax + hy + g = 0 …(i)
hx + by + f = 0 …(ii)
and ax2 + 2hxy + by2 + 2gx + 2fy + c + t = 0 …(iii)
abc  2 fgh  af 2  bg 2  ch 2
has a unique solution and  8, the value of ‘t’ is ………. .
h 2  ab

1 x x2 x3  1 0 x  x4
19. If x x2 1  3, then the value of 0 x  x4 x 3  1 is …….. .
x2 1 x x  x4 x3  1 0

p b c
p q r
20. If a ≠ p, b ≠ q, c ≠ r and a q c  0, then the value of   is ……….. .
pa qb rc
a b r

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-49
Section - II : Match the Column

12 22 32
21. Let , ,  be the roots ofx3–3x2 + 3x – 1 = 22 32 42 .
32 42 52

List-I List–II
(p) The value of  is (q) 0
 1  1  1
(q) The value of   is (2) 32
 1  1  1
(r) The value of (α – 1) (β– 1) (γ – 1) is (3) –8
 1  1  1
(s) The value of   1   1   1 is (4) –7
 1  1  1

Match the List-I and List-II and choose the correct option from given codes.
Codes :
(p) (q) (r) (s)
(A) (2) (1) (4) (3)
(B) (4) (1) (2) (3)
(C) (4) (3) (2) (1)
(D) (4) (2) (3) (1)

22. Consider the system of equations

λx + y + z = 1;
x + λy + z = λ;
and x + y + λz = λ2
List-I List–II
(p) λ=1 (1) Unique solution
(q) λ≠1 (2) Infinite solution
(r) λ ≠ 1, λ ≠ –2 (3) No solution
(s) λ = –2
Match the List-I and List-II and choose the correct option from given codes.
Codes :
(p) (q) (r) (s)
(A) (2) (1) (4) (3)
(B) (2) (1) (2) (3)
(C) (2) (3) (2) (1)
(D) (4)(1), (3) (1) (3)

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-50
23. Match the following lists:
List-I List–II
x 2 x 3 x5
(p) The value of the x  4 x  6 x  9 is (1) 1
x  8 x  11 x  15

7 6 x 2  13
If one of the roots of the equation 2 x 2  13 2 =0
(q) (2) –6
x 2  13 3 7

is x = 2, then sum of all other five roots is

6 2i 3 6
(r) The value of 12 3  8i 3 2  6i is (3) 2
18 2  12i 27  2i

cos2  cos  sin   sin 

(s) If f(θ) = cos  sin  sin 2  cos  then f(π/3) = (4) –2
sin   cos  0

Match the List-I and List-II and choose the correct option from given codes.
Codes :
(p) (q) (r) (s)
(A) (2) (1) (4) (3)
(B) (3) (4) (2) (1)
(C) (4) (3) (2) (1)
(D) (4) (2) (3) (1)

Section - III : Comprehension

Paragraph # 1 (24 to 26)

A 3 × 3 determinant has entries either 1 or –1. Let S3 be set of all determinants, which contain product of
elements of any row or any column as –1.
1 1 1
For example, 1 1 1 is an element of the set S3.
1 1 1

24. Number of elements of the set S3 is

(A) 10 (B) 16 (C) 12 (D) 18

25. Number of elements of the set Sn is

2
(A) 2n (B) 2n–1 (C) 22n (D) 2(n 1)

26. The minimum value out of all elements of the set S3 is

(A) –4 (B) –6 (C) 4 (D) 6

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-51
Paragraph # 2 (27 to 29)
x  c1 x  a xa
If f(x) = x  b x  c2 x  a , and g(x) = (c1 – x) (c2 – x) (c3 – x), then
xb xb x  c3

27. Coefficient of x in f(x) is

g (a)  f (b) g (a)  g (b) g (a)  g (b)
(A) (B) (C) (D) None of these
ba ba ba

28. Which of the following is not a constant term is f(x) ?

bg (a )  ag (b) bg ( a )  af (b) bf (  a )  ag (b)
(A) (B) (C) (D)none of these
b  a  b  a  b  a 

29. Which of the following is not true?

(A) f(–a) = g(A) (B) f(–a) = g(–a) (C) f(A) = g(–a) (D) none of these

Paragraph # 3 (30 to 32)

a2  x ab ac
2
Consider the function f(x) = ab b x bc . Then
2
ac bc c x

30. Which of the following is true?

(A) f(x) = 0 and f′(x) = 0 have one positive common root.
(B) f(x) = 0 and f′(x) = 0 have one negative common root.
(C) f(x) = 0 and f′(x) = 0 have no common root.
(D) None of these

31. Which of the following is true?

(A) f(x) has one +ve point of maxima. (B) f(x) has one –ve point of minima
(C) f(x) = 0 has three distinct roots (D) local minimum value of f(x) is zero

32. In which of the following interval f(x) is strictly increasing?

(A) (–∞, ∞) (B) (–∞, 0) (C) (0, ∞) (D) none of these

Paragraph # 4 (33 to 35)

Consider the system of equations

x+y+z=6;
x + 2y + 3z = 10;
and x + 2y + z = 

33. The system has unique solution if

(A)  ≠ 3 (B)  = 3,  = 10 (C)  = 3,  ≠ 10 (D) none of these

34. The system has infinite solutions if

(A)  ≠ 3 (B)  = 3,  = 10 (C)  = 3,  ≠ 10 (D) none of these

35. The system has no solution if

(A)  ≠ 3 (B)  = 3,  = 10 (C)  = 3,  ≠ 10 (D) none of these

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-52
Paragraph # 5 (36 to 38)

Let α,β be the roots of the equation ax2 + bx + c = 0.

3 1  S1 1  S2
n n
Also, let Sn =    for n ≥ 1 and ∆ = 1  S1 1  S2 1  S3 .
1  S2 1  S3 1  S4
36. If ∆< 0 then the equation ax2 + bx + c = 0 has
(A) positive real roots (B) negative real roots
(C) equal roots (D) imaginary roots

37. If a, b, c are rational and one of the roots of the equation is 1 + 2 , then the value of ∆ is
(A) 8 (B) 12 (C) 30 (D) 32

38. If b2 – 4ac > 0, then

(A) f(1) > 0 (B) f(1) < 0
(C) f(1) = 0 (D) cannot say anything about f(1)

Paragraph # 6 (39 to 41)

 bc b 2  bc c 2  bc
Let ∆ = a 2  ac ac c2  ac and root of the equation px3 + qx2 + rx + s = 0 be a, b, c,where
a 2  ab b2  ab ab
a, b, c R+.

39. The value of ∆ is

(A) r2/p2 (B) r3/p3 (C) –s/p (D) none of these

40. The value of ∆

(A)  9r2/p2 (B) ≥ 27s2/p2 (C)  27s3/p3 (D) none of these

41. If ∆ = 27 and a2 + b2 + c2 = 3, then

(A) 3p + 2q = 0 (B) 4p + 3z = 0 (C) 3p + q = 0 (D) none of these

Paragraph # 7 (42 to 44)

Consider the following polynomial function

1  x a 1  2 x b 1
f(x) = 1 1  x a 1  2 x b a, b being positive integers, then
1  2 x b 1 1  x a

42. The constant term in f(x) is

(A) 2 (B) 1 (C) –1 (D) 0

43. The coefficient of x in f(x) is

(A) 2a (B) 2a – 3.2b + 1 (C) 0 (D) none of these

44. Which of the following is true?

(A) All the roots of the equation f(x) = 0 are positive.
(B) All the roots of the equation f(x) = 0 are negative.
(C) At least one root of the equation f(x) = 0 is repeating.
(D) None of these

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-53
Exercise –5
Revision exercise (Moderate to Tough)

 a2  ab  ab
1. Let  ab  a 2  ab . Prove that ∆ is non-negative and establish the relation between a, b and c if ∆ =
 ab  ab  a 2
0.

0 a2 b2 c2 0 a b c
2 2 2
a 0   a 0 c b
2. Prove that 
b 2
 2
0  2 b c 0 a
2 2 c b a 0
c  2 0

pa qb rc a b c
3. If p + q + r = 0, prove that qc ra pb  pqr c a b
rb pc qa b c a

a2  s  a 2  s  a 2
If 2s = a + b + c, prove that  s  b 
2
4. b2  s  b 2 = 2s3 (s – a) (s – b) (s – c).
 s  c 2  s  c 2 c2

5. (A) Without expanding as far as possible, evaluate

sin 2 A sin Acos A cos2 A
sin 2 B sin Bcos B cos 2 B
sin 2 C sin CcosC cos 2 C
cos  A  B  cos  B  C  cos  C  A 
(B) If A, B, C be the angles of a triangle and cos  A  B  cos  B  C  cos  C  A  = 0,
sin  A  B  sin  B  C  sin  C  A 
then prove that the triangle is an isosceles triangle.

1 1 1
6. (A) If A, B, C be the angles of a triangle and 1  sin A 1  sin B 1  sin C =0
2 2 2
sin A  sin A sin B  sin B sin C  sin C
then prove that ∆must be isosceles.
1 cos  cos 2 
   
(B) Prove that 1 cos  cos 2  = 2 sin sin sin [sin ( – ) + sin ( – ) + sin
2 2 2
1 cos  cos 2 
( – )]

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-54
x x x x x 1 x 2
Cr Cr 1 Cr  2 Cr Cr 1 Cr  2
y y y y y1 y 2
7. (A) Show that Cr Cr 1 Cr  2  Cr Cr 1 Cr  2
z z z z z 1 z2
Cr Cr 1 Cr  2 Cr Cr 1 Cr  2
x n 1 n 1
Cr Cr Cr 1
x 1 n n
(B) If both n and r be greater than 1, find the value of x if   Cr Cr Cr 1  0
x 2 n 1 n 1
Cr Cr Cr 1
n n n
Cp Cp1 Cp  2
n 1 n 1 n 1
(C) If n and p be two +ive integers such that n ≥ p + 2 and D(n,p) Cp Cp1 Cp 2 then
n2 n 1 n2
Cp Cp1 Cp  2
n 2
C3
prove that D (n, p) = p2
D (n – 1, p – 1)
C3

n 2 n 2 n 2
n
Cr 2 Cr 1 Cr
If n > 2 then sum the series   2 
r
8. (A) 3 1 1
r 2 2 1 0

1 1 1
n n 3 n 6
(B) If 3n is a factor of the determinant C1 C1 C2 , then the maximum value of n is 3.
n n 3 n 6
C2 C2 C2

9. Suppose three digit numbers A28, 3B9 and 62C where A, B and C are integers between 0 and 9, are
A 3 6
divisible by a fixed integer k. Prove that determinant 8 9 C is also divisible by k.
2 B 2
n!  n  1!  n  2 !  D 
10. (A) For a fixed positive integer n, if D =  n  1!  n  2 !  n  3! then show that  3  4  is
  n! 
 n  2 !  n  3 !  n  4 !
divisible by n.
 n  1!  n  2 !  n  3!/ n  n  1
(B) For a fixed +ive integer n, let D =  n  1!  n  3!  n  5 !/  n  2  n  3
 n  3!  n  5 !  n  7 !/  n  4  n  5
D
then is equal to
 n  1! n  1! n  3 !
(A) – 8 (B) –16 (C) –32 (D) –64

a x x x
x b x x
11. If f(x) = (x – a) (x – b) (x – c) (x – d) then prove that   = f(x) – xf′(x).
x x c x
x x x d

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 M-55
12. If f(x) is a polynomial of degree < 3, prove that
2
1 a f a /  x  a 1 a a
f  x
1 b f  b  /  x  b   1 b b2 
1 c f  c  /  x  c  1 c c2
 x  a  x  b  x  c 

xk x k 2 x k 3
1 1 1
13. If y k yk 2 y k 3 = (x – y) (y – z) (z – x)     , then k =
x y z
zk zk 2 z k 3
(A) – 3 (B) 3 (C) – 1 (D)1

 r3 r 2  3 
If a, b, c are all different and the points 
 r  1 r  1 
14. , where r = a, b, c are collinear, then prove that
 
3 (a + b + c) = ab + bc + ca – abc

 a  x 2  a  y 2  a  z 2 1  ax 2 1  ay 2 1  az 2
   b  x   b  y  b  z or 1  bx  1  by  1  bz 
2 2 2 2 2 2
15. Express as the product of two
 c  x  2  c  y 2  c  z  2 1  cx 2 1  cy 2 1  cz 2
determinants and evaluate it.

16. If (ar, br), r = 1, 2, 3 be the vertices of a triangle, then prove that

a 2  a3 b 2  b3 a1  a 2  a 3   b1  b 2  b3 
a 3  a1 b3  b1 a 2  a 3  a1   b 2  b3  b1   0
a1  a 2 b1  b 2 a1  a1  a 2   b3  b1  b 2 
and hence prove that the altitudes of a triangle are concurrent.

1 cos      cos 
17. If α and β are the roots of the equation x – px + q = 0, then evaluate cos     
2
1 cos 
cos  cos  1

1 a a2
18. The parameter, on which the value of the determinant cos  p  d  x cos px cos  p  d  x does not
sin  p  d  x sin px sin  p  d  x

depend upon is
(A) a (B) p (C) d (D) x

cos      cos      cos     

19. Show that the value of the determinant sin      sin      sin      is independent of θ.
sin      sin      sin     

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-56
cos       sin      cos 2
20. Show that the determinant sin  cos  sin  is independent of θ or φ or both.
 cos  sin  cos 

21. If fr(x), gr(x), hr(x), r = 1,2,3 are polynomials in x such that fr(A) = gr (A) = hr (A) , r = 1, 2, 3 and
f1  x  f 2  x  f3  x 
F(x) = g1  x  g 2  x  g3  x  , then F′ at x = ais …….
h1  x  h2  x  h3  x 

22. If u, v, w be functions of x and if suffixes denote differentiation w.r.t. x, prove that

u1 v1 w1 u1 v1 w1
d
u 2 v2 w 2  u 2 v2 w 2
dx
u 3 v3 w 3 u 4 v 4 w 4

f g h
23. If f, g and h are differentiable functions of x and    xf  '  xg  '  xh  '
 x2 f  ''  x2 g  ''  x2h  ''
f g h
prove that  '  f' g' h'

 x3 f '' '  x3 g '' '  x3h '' '

1  x a1b1 1  x a1b2 1  x a1b3
24. If f(x) = 1  x a2b1 1  x a2b2 1  x a2b3 then prove that coefficient of x in f(x) = 0.
1  x a3b1 1  x a3b2 1  x a3b3

25. Let α be a repeated root of quadratic equation f(x) = 0 and A (x), B (x), C (x) be polynomials of degree 3, 4
A x  B  x C  x 
and 5 respectively, then show that A    B    C    is divisible by f(x), where dash denotes the
A '    B '() C '   
derivative.

a11  b11 a1 2  b12 a13  b13

26. Prove that a 21  b 21 a 2 2  b 22 a 2 3  b23 = 0.
a 31  b31 a 3 2  b32 a 33  b33

cos  A  P  cos  A  Q  cos  A  R 

27. (A) For all values of A, B, C and P, Q, R show that cos  B  P  cos  B  Q  cos  B  R  = 0
cos  C  P  cos  C  Q  cos  C  R 
(B) If A, B, C are real numbers, then prove without expansion that  = 0 where
1 cos  B  A  cos  C  A 
  cos  A  B  1 cos  C  B 
cos  A  C  cos  B  C  1

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-57
2 a bcd ab  cd
28. Evaluate a  b  c  d 2  a  b  c  d  ab  c  d   cd  a  b 
ab  cd ab  c  d   cd  a  b  2abcd

1 1 1 1 s0 s1 s2 s3
    s s2 s3 s4
29. By squaring the determinant show that 1
2 2 2 2 s2 s3 s4 s5
3 3 3 3 s3 s4 s5 s6
2
                           where s r   r  r   r  r .

yz  x 2 zx  y 2 xy  z 2 r2 u2 u2
30. (A) Prove zx  y 2 xy  z 2 yz  x 2 = u 2 r2 u 2 where r2 = x2 + y2 + z2and
xy  z 2 yz  x 2 zx  y 2 u2 u2 r2
u2 = yz + zx + xy.
a 2  2 ab  c ca  b  c b
(B) Prove that ab  c b2   2 bc  a c  a = γ3 (γ2 + a2 + b2 + c2)3.
ca  b bc  a c2   2 b a 

sec x cos x sec 2 x  cot x.cos ecx

/2  8 
31. (A) Let f(x) = cos 2 x cos2 x cosec 2 x then 0 f ( x)dx     
 4 15 
1 cos 2 x cos 2 x

4sin 2  1 1
Evaluate    sin   1  sin   2 2  sin   12
2
(B)
 sin   12  sin   12 sin 2 

Hence show that  / 2/ 2  d  4

 / 2 A2 A3
 / 2 1  cos 2nx
32. If An = 0 1  cos 2x
dx, then evaluate   A4 A5 A6 .
A7 A8 A9

Exercise –6
Section – I : JEE (Advanced) QuestionsPrevious Years

1   2 1  2 2 1  3 2
1. Which of the following values of  satisfy the equation  2   2  2  2 2  2  3 2 = –648 ?
 3   2  3  2 2  3  3 2
[IIT-JEE – (P-1) 2015]
(A) –4 (B) 9 (C) –9 (D) 4

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-58

x x2 1  x3
2. The total number of distinct x R for which 2x 4x 2 1  8x3 = 10 is [IIT-JEE – (P-1) 2016]
3x 9x 2 1  27x 3

 n n 
 k  n
Ck k 2 
n n
 k 0  Ck
 = 0 holds for some positive integer n. Then  k  1 equals
k 0
3. Suppose det 
n n
 Ck 3k  k 0
 
n n
Ck k

 k 0 k 0 
______. [IIT-JEE – (P-2) 2019]

Section – II : JEE (Main) Questions Previous Years

4. The number of values of  (0, ) for which the system of linear equations
x + 3y + 7z = 0
– x + 4y + 7z = 0
(sin 3) x + (cos 2) y + 2z = 0 has a non-trivial solution, is: (JEE Main – 2019, 10 Jan, Shift-1)
(1) three (2) two (3) four (4) one
2 b 1
 2  det  A 
5. Let A   b b  1 b where b > 0. Then the minimum value of is:
1  b
 b 2
(JEE Main 2019-10 Jan, Evening)
(1) 2 3 (2) 2 3 (3)  3 (4) 3
a bc 2a 2a
6. If 2b bca 2b = (a + b + c) (x + a + b + c)2, x  0 and a + b + c  0, then x is equal to:
2c 2c ca b
(JEE Main 2019-11 Jan, Evening)
(1) abc (2) – (a + b + c) (3) 2 (a + b + c) (4) – 2 (a + b + c)
7. Let A and B be two invertible matrices of order 3 × 3. If det (ABAT) = 8 and det (AB–1) = 8, then det
(BA–1 BT) is equal to: (JEE Main 2019-11 Jan, Evening)
1 1
(1) (2) 1 (3) (4) 16
4 16
 1  
8. Let  and  be the roots of the equation x2 + x + 1 = 0. Then for   0 in R,    1 is equal
 1 
to (JEE Main 2019-09April, Morning)

(1)   2  1  
(2)    3 2
 (3)  3
(4) 3  1
x sin  cos  x sin 2 cos 2

9. If 1   sin  x 1 and  2   sin 2 x 1 , x  0; then for all    0,  :
 2
cos  1 x cos 2 1 x
(JEE Main 2019-09April, Evening)
(1) 1   2  2x 3
(2) 1   2  x  cos 2  cos 4


(3) 1   2  2 x 3  x  1  (4) 1   2  2x 3

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-59
 5 2 1 
 
10. If B   0 2 1  is the inverse of a 3 × 3 matrix A, then the sum of all values of α for which
 3 1
det (A) + 1 = 0, is (JEE Main 2019-10April, Morning)
(1) 0 (2) –1 (3) 1 (4) 2
x 6 1
11. The sum of the real roots of the equation 2 3x x  3  0, is equal to :
3 2x x  2
(JEE Main 2019-10April, Evening)
(1) 6 (2) 0 (3) 1 (4) –4
x2 2x  3 3x  4
12. If   2x  3 3x  4 4x  5 = Ax3 + Bx2 + Cx + D, then B + C is equal to:
3x  5 5x  8 10x  17
(JEE Main 2020-3Sep, Morning)
(1) 9 (2) –1 (3) 1 (4) –3
 
13. If the minimum and the maximum values of the function f :  ,   R, defined by
4 2
 sin 2  1  sin 2  1
f     cos  1  cos  1 are m and M respectively, then the ordered pair (m, M) is equal to
2 2

12 10 2
(JEE Main 2020-5Sep, Morning)

(1) 0, 2 2  (2) (0, 4) (3) (–4, 4) (4) (–4, 0)
x a  y xa
14. If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then y b  y y  b
z c y zc
is equal to : (JEE Main 2020-5Sep, Evening)
(1) y(b – a) (2) y(a – b) (3) y(a – c) (4) 0
15. Let m and M be respectively the minimum and maximum values of (JEE Main 2020-6Sep, Morning)
cos2 x 1  sin 2 x sin 2x
1  cos2 x sin 2 x sin 2x Then the ordered pair (m, M) is equal to :
cos2 x sin 2 x 1  sin 2x
(1) (1, 3) (2) (–3, –1) (3) (–4, –1) (4) (–3, 3)
  cos  sin  
16. Let   and A    . If B = A + A4 then det (B) : (JEE Main 2020-6Sep, Evening)
5   sin  cos  
(1) lies in (2, 3) (2) is zero (3) is one (4) lies in (1, 2)
17. The system of linear equations
3x–2y –kz = 10
2x – 4y – 2z = 6
x + 2y – z = 5m is inconsistent if: (JEE Main 2021-24 Feb, Shift-1)
4 4 4
(1) k = 3, m = (2) k  3, m  R (3) k  3, m  (4) k  3, m 
5 5 5
18. For the system of liner equations: (JEE Main 2021-24 Feb, Shift-2)
x – 2y = 1, x – y + kz = – 2, ky + 4z = 6, k  R consider the following statements:
(A) The system has unique solution if k  2, k  – 2.
(B) The system has unique solution if k = – 2
(C) The system has unique solution if k = 2.

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-60
(D) The system has no-solution if k = 2.
(E) The system has infinite number of solutions if k  –2.
Which of the following statements are correct?
(1) (B) and (E) only (2) (C) and (D) only (3) (A) and (D) only (4) (A) and (E) only
19. If the system of equations (JEE Main 2021-25 Feb, Shift-1)
kx + y + 2z = 1
3x – y – 2z = 2
–2x – 2y – 4z = 3
has infinitely many solutions, then k is equal to
20. Let A be a 3 × 3 matrix with det (A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by
performing the operation R2  2R2 + 5R3 on 2 A, then det (B) is equal to:
(JEE Main 2021-25 Feb, Shift-2)
(1) 64 (2) 19 (3) 80 (4) 128
21. The maximum value of (JEE Main 2021-16 March, Evening)
sin 2 x 1  cos 2 x cos 2x
f(x) = 1  sin 2 x cos 2 x cos 2x , x  R is :
sin 2 x cos 2 x sin 2x
3
(1) 7 (2) (3) 5 (4) 5
4
22. The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to:
(JEE Main 2021-17 March, Shift-1)
(1) 0 (2) 1 (3) –1 (4) –2
23. If x, y, z are in arithmetic progression with common difference d, x  3d, and the determinant of the
3 4 2 x 
 
matrix  4 5 2 y  is zero, then the value of k2 is (JEE Main 2021-17March, Shift-2)
5 k z
 
(1) 72 (2) 12 (3) 36 (4) 6
24. The number of values of  for which the system of equations:
x+y+z=
x + 2y + 3z = – 1
x + 3y + 5z = 4
is inconsistent, is (JEE Main 2022-24 June, Shift-1)
(1) 0 (2) 1 (3) 2 (4) 3
25. The system of equations
– kx + 3y – 14z = 25
– 15x + 4y – kz = 3
–4x + y + 3z = 4 is consistent for all k in the set (JEE Main – 2022, 25 June, Shift-2)
(1) R (2) R – {– 11, 13} (3) R – {13} (4) R – {–11, 11}
26. If the system of equations x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = , has infinitely many solutions,
then the ordered pair (,) is equal to: (JEE Main 2022-26 June, Shift-2)
(1) (1, –3) (2) (–1, 3) (3) (1, 3) (4) (–1, – 3)
27. Let the system of linear equations x + 2y + z = 2, x + 3y – z = , – x + y + 2z = – be inconsistent.
Then  is equal to: (JEE Main 2022-27 June, Shift-1)
5 5 7 7
(1) (2)  (3) (4) 
2 2 2 2
28. The number of   (0, 4) for which the system of linear equations
3 (sin 3) x – y + z = 2
3 (cos 2) x + 4y + 3z = 3
6x + 7y + 7z = 9 has no solution is: (JEE Main – 2022, 25 July, Shift-1)
(1) 6 (2) 7 (3) 8 (4) 9

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 M-61
29. The number of real values , such that the system of linear equations
2x – 3y + 5z = 9
x + 3y – z = – 18
3x – y + (2 – | |) z = 16 has no solution, is:- (JEE Main – 2022, 25 July, Shift-2)
(1) 0 (2) 1 (3) 2 (4) 4
30. If the system of equations (JEE Main – 2022, 25 July, Shift-2)
x+y+z=6
2x + 5y + z = 
x + 2y + 3z = 14 has infinitely many solutions, then  +  equal to:
(1) 8 (2) 36 (3) 44 (4) 48
31. If the system of equations (JEE Main 2023, 24 Jan., Shift-2)
x + 2y + 3z = 3
4x + 3y – 4z = 4
8x + 4y – λz = 9 + µ
has infinitely many solutions, then the ordered pair (λ, µ) is equal to
 72 21   72 21   72 21   72 21 
(1)  ,  (2)  ,  (3)  ,  (4)  , 
 5 5  5 5   5 5   5 5
32. Let the system of linear equations (JEE Main 2023, 30 Jan., Shift-1)
x + y + kz = 2
2x + 3y – z = 1
3x + 4y + 2z = k
have infinitely many solutions. Then the system
(k + 1) x + (2k – 1) y = 7
(2k + 1) x + (k + 5) y = 10 has :
(1) infinitely many solutions (2) unique solution satisfying x – y = 1
(3) no solution (4) unique solution satisfying x + y = 1
33. For the system of linear equations (JEE Main 2023, 31 Jan., Shift-1)
x+y+z=6
αx + βy + 7z = 3
x + 2y + 3z = 14,
which of the following is NOT true?
(1) If α = β = 7, then the system has no solution
(2) If α = β and α ≠ 7 then the system has no unique solution.
(3) There is a unique point (α, β) on the line x + 2y + 18 = 0 for which the system has infinitely many
solutions
(4) For every point (α, β) ≠ (7, 7) on the line x – 2y + 7 = 0, the system has infinitely may solutions.

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-62
34. For the system of linear equations ax + y + z = 1, x + ay + z = 1, x + y + az = β, which one of the
following statements is NOT correct? (JEE Main 2023, 01 Feb., Shift-2)
(1) It has infinitely many solutions if α = 2 and β = –1
(2) It has no solution if α = –2 and β = 1
3
(3) x + y + z = if α = 2 and β = 1
4
(4) It has infinitely many solutions if α = 1 and β = 1
35. If the system of equations
x + y + az = b
2x + 5y + 2z = 6
x + 2y + 3z = 3
has infinitely many solutions, then 2a + 3b is equal to (JEE Main 2023, 06 Apr., Shift-1)
(1) 25 (2) 20 (3) 23 (4) 28
36. Let S be the set of values of λ, for which the system of equations (JEE Main 2023, 10 Apr., Shift-2)
6λx − 3y + 3z = 4λ2, 2x + 6λy + 4z = 1 and 3x + 2y + 3λz = λ has no solution. Then, 12  S |  | is equal
to __________.
1 2k 2k  1
37. Let D k  n n 2  n  2 n2 . If  nk 1 D k  96, then n is equal to ________.
n n2  n n2  n  2

(JEE Main 2023, 12 Apr., Shift-1)

38. Let the system of linear equations (JEE Main 2023, 15 Apr., Shift-1)
–x + 2y − 9z = 7
−x + 3y + 7z = 9
−2x + y + 5z = 8
−3x + y + 13z = λ
has a unique solution x = α, y = β, z = γ. Then the distance of the point (α, β, γ) from the plane
2x − 2y + z = λ is
(1) 11 (2) 7 (3) 9 (4) 13

Mathematics –Determinants Toll Free Number : 1800 103 9888

 M-63

ANSWER KEY
Exercise – 1
1 i 1 i
1. M 22   i  i  2i and M33   1  i2  1  1  0
1 i i 1
2. 7 3. 0 4. (i) 512000 (ii) 0 26. 2 28. 2
30. 1. x = k, y = k, z = k, where kR
2. x = 0, y = 0, z = 0
3. x = –7k, y = 8k, z = 13k, where kR

Exercise – 2
1. (A) 2. (B) 3. (A) 4. (D) 5. (D) 6. (C) 7. (A)
8. (A) 9. (B) 10. (B) 11. (B) 12. (D) 13. (C) 14. (C)
15. (B) 16. (A) 17. (C) 18. (C) 19. (D) 20. (C) 21. (A)
22. (BC) 23. (B) 24. (D) 25. (B) 26. (B) 27. (B) 28. (C)
29. (B) 30. (D) 31. (C) 32. (C) 33. (D) 34. (A) 35. (B)
36. (A) 37. (D) 38. (A) 39. (C) 40. (D) 41. (A) 42. (D)
43. (D) 44. (A) 45. (D) 46. (C) 47. (B) 48. (A) 49. (D)
50. (B) 51. (C) 52. (A)

Exercise – 3
1. (ABCD) 2. (BD) 3. (ABCD) 4. (BD) 5. (ACD)
6. (ABC) 7.(ABCD) 8. (ABD) 9. (ABCD) 10. (ACD)
11. (ABD) 12. (AB) 13. (AB) 14. (AC) 15. (ABC)

Exercise – 4
Section - I
1. 2046 2. 576 3. 2 4. 841 5. 5 6. 4 7. 10
8. 2 9. 1 10. 2 11. 2 12. 3 13. 1 14. 3
15. 8 16. 6 17. 0 18. 8 19. 9 20. 2
Section - II
21. (C) 22. (D) 23. (C)
Section - III
24. (B) 25. (D) 26. (A) 27. (C) 28. (D) 29. (B) 30. (D)
31. (D) 32. (C) 33. (A) 34. (B) 35. (C) 36. (D) 37. (D)
38. (D) 39. (B) 40. (B) 41. (C) 42. (D) 43. (C) 44. (C)

Exercise – 5
1. a  0 or a 2  ab  0 7. (B) n, n – 1 10. (D) 13. (C) 18. (B)
15. 2 (x – y) (y – z) (z – x) (z – x) (a – b) (b – c) (c – a) 17. 0 28. 0 32. 0

Exercise – 6
Section - I
1. (BC) 2. 2 3. 6.20
Section – II
4. (2) 5. (1) 6. (4) 7. (3) 8. (3) 9. (4) 10. (3)
11. (2) 12. (4) 13. (4) 14. (2) 15. (2) 16. (4) 17. (4)
18. (3) 19. 21 20. (1) 21. (3) 22. (4) 23. (1) 24. (2)
25. (4) 26. (3) 27. (4) 28. (2) 29. (3) 30. (3) 31. (3)
32. (4) 33. (4) 34. (1) 35. (3) 36. (24) 37. (6) 38. (2)

Mathematics –Determinants Toll Free Number : 1800 103 9888

FUNCTIONS
AND
RELATIONS
 M-64
Exercise - 1
Concept Building Questions

Definition of function, Domain and Range :

1. Let g(x) be a function defined on [–1, 1]. If the area of the equilateral triangle with two of its vertices at
3
(0, 0) and (x, g(x)) is sq. units, then the function g(x) may be
4

(A) g(x) =  1  x 2 (B) g(x) = 1  x 2 (C) g(x) = – 1  x 2 (D) g(x) = 1  x 2

2. Represent all possible functions defined {, } to {1, 2}
3. Find the domain of each of the following functions :

x3  5x  3 sin 1 x 1
(i) f(x) = 2
(ii) f(x) = (iii) f(x) = (iv) f(x) = ex+sinx
x 1 x x x
4. Find the domain of each of the following functions :
1  3x  1 
(i) f(x) =  x2 (ii) f(x) = 1  2x  3sin 1  
log10 1  x   2 
1 x1  1 
(iii) f(x) = 2sin  (iv) f(x) = log x log 2  
x2  x  1/ 2 
5. Find the domain of definitions of the following functions :

(i) f(x) = 3  2 x  21 x (ii) f(x) = 1  1  x 2

x2 1 x
(iii) f(x) = (x2 + x + 1)–3/2 (iv) f(x) = +
x2 1 x
6. Find the domain of definitions of the following functions :
1
(i) f(x) = tan x  tan 2 x (ii) f(x) =
1  cos x
 5x  x 2 
(iii) f(x) =og1/ 4   (iv) f(x) = log10 (1 – log10(x2 – 5x + 16))
 4
 
7. Find the domain and the range of each of the following functions :
1 x2  9
(i) f(x) = (ii) f(x) = x ! (iii) f(x) = (iv) f(x) = sin2(x3) + cos2(x3)
4  3sin x x 3
8. Find the range of each of the following functions :
x x4
(i) f(x) = x  3 (ii) f(x) = (iii) f(x) = 16  x 2 (iv) f(x) =
1  x2 x4
9. Find the range of each of the following functions :
1
(i) f(x) = 5 + 3 sin x + 4 cos x (ii) f(x) = (iii) f(x) = n (sin–1x)
1 x
10. Find the range of each of the following functions :
(i) f(x) = 2 – 3x – 5x2 (ii) f(x) = 3 |sin x| – 4 |cos x|
sin x cos x
(iii) f(x) = 
1  tan 2 x 1  cot 2 x

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-65
Algebraic operations on functions, Equal and Identical functions, composite functions :

11. If f(x) = x and g(x) = sin x ; find f(x) + g(x) and f(x)/g(x).
x  1 x  0  x 2  1 x  1
12. If f(x) =  ; g(x) = 
 2x x  0  x  1 x  1
find f(x) – g(x).
13. Check whether following pairs of functions are identical or not?

 x
2
(i) f (x)  x 2 and g(x)  (ii) f(x) = sec (sec–1 x) and g(x) = cosec (cosec–1 x)

1  cos2x
(iii) f(x) = and g(x) = cos x (iv) f(x) = x and g(x) = enx
2
14. Find fog and gof, if
(i) f(x) = ex ; g(x) = n x (ii) f(x) = |x| ; g(x) = sin x
1
(iii) f(x) = sin–1 x ; g(x) = x2 (iv) f(x) = x2 + 2 ; g(x) = 1 – ,x 1
1 x
15. Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog  gof
16. If f(x) = n(x2 – x + 2) ; R+ R and
g(x) = {x} + 1 ; [1, 2]  [1, 2], where {x} denotes fractional part of x.
Find the domain and range of f(g(x)) when defined.
1  x 2 ; x 1
17. If f(x) =  and g(x) = 1 – x ; – 2 ≤ x ≤ 1, then define the function fog(x).
 x  1; 1  x  2

Injective, Surjective and Bijective functions :

18. Let f : D → R, where D is its domain. Find whether the following functions are into/onto.
1  x6 1
(i) f(x) = 3
(ii) f(x) = x cos x (iii) f(x) = (iv) tan (2 sin x)
x sin | x |
19. Find whether the following functions are one-one or many-one.
(i) f(x) = |x2 + 5x + 6| (ii) f(x) = |log x|
   1
(iii) f(x) = sin 4x, x    ,  (iv) f(x) = x + , x   0,  
 8 8  x
20. Find whether the following functions are one-one or many-one.
1 
 1 3x 2
(i) f(x) = 1  e x  (ii) f(x) =  cos x (iii) f(x) = sin–1 x – cos–1 x
4
21. Let f : A A where A = {x : –1 ≤ x ≤ 1}. Find whether the following functions are bijective.
x
(i) x – sin x (ii) x |x| (iii) tan (iv) x4
4
22. Classify the following functions f(x) defined in R R as injective, surjective, both or none.
x2
(i) f(x) = x |x| (ii) f(x) = x2 (iii) f(x) = 2
(iv) f(x) = x3 – 6x2 + 11x – 6
1 x

Even and odd functions :

23. Determine whether the following functions are even or odd or neither even nor odd:
(i) tan x (ii) cos x (iii) sin (x2 + 1) (iv) x + x2

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-66
24. Determine whether the following functions are even or odd or neither even nor odd :

(i) x – x3
 ax 1 
(ii) f(x) = x  x
 a  1 
 

(iii) f(x) = log x  x 2  1 
(iv) f(x) = sin x + cosx (v) f(x) = (x2 – 1) |x|
 x 2  sin x , 0  x  1
25. If f(x) = 
 x  e x , x 1
then extend the definition of f(x) for x   ,0  such that f(x) becomes
(i) an even function (ii) an odd function

Periodic functions :
26. Find the fundamental period of the following functions: (where [.] denotes G.I.F.)
(i) f(x) = 2 + 3 cos (x – 2) (ii) f(x) = sin 3x + cos2 x + |tan x|
x x 3 2
(iii) f(x) = sin + sin (iv) f(x) = cos x – sin x
4 3 5 7
27. Prove that the following functions are not periodic :
(i) f(x) = sin x (ii) f(x) = x + sin x
28. Find the fundamental period of the following functions: (where [.] denotes G.I.F.)
1
(i) f(x) = [sin 3x] + |cos 6x| (ii) f(x) =
1  cos x
sin12x
(iii) f(x) = 2
(iv) f(x) = sec3x + cosec3x
1  cos 6x

Inverse of a functions :
29. Let f : D R, where D is the domain of f. Find the inverse of f, if it exists, where

(i) f(x) = 1 – 2–x 

(ii) f(x) = 4   x  7  
3 1/ 5

(iii) f(x) = n x  1  x 2 
30. Let f : R R be defined by
e 2x  e 2x
f(x) = . Is f(x) invertible ? If yes, then find its inverse.
2
  
31. Let f :   ,  B defined by f(x) = 2 cos2x + 3 sin2x + 1. Find B such that f –1
exists. Also find
 3 6
f –1 (x).

Miscellaneous :
1 1
32. Let f(x) be a polynomial function satisfying the relation f(x). f   = f(x) + f   x R – {0} and
x
  x
f(3) = –26. Determine f ′(1).
10
33. If f(x + y) = f(x) . f(y), x & y N and f(1) = 2, then find  f (n).
n 1

4x
34. If f(x) = , then show that f(x) + f(1 – x) = 1.
4x  2


 2 tan x   cos 2x  1 sec x  2 tan x
2

35. If f   , then find f(x).
 1  tan 2 x  2

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-67
36. The graph of the function y = f(x) is symmetrical about the line x = 2, then :
(A) f(x + 2) = f(x – 2) (B) f(2 + x) = f(2 – x)
(C) f(x) = f(–x) (D) f(x) = –f(–x)
37. The greatest value of the function f(x) = (sin–1 x)3 + (cos–1 x)3 is:

3 3 33 73
(A) (B) (C) (D)
32 8 8 8
38. If x and y satisfy the equation y = 2 [x] + 3 and y = 3 [x – 2] simultaneously, where [.] denotes the greatest
integer function, then [x + y] is equal to
(A) 21 (B) 9 (C) 30 (D) 12
1  x 
39. If f(x) = cos (nx), then f(x) f(y) –  f    f  xy   has the value
2  y 
1
(A) –1 (B) (C) –2 (D) None of these
2
40. Let f(x) = |x – 1|, then
(A) f(x2) = (f(x))2 (B) f(x + y) = f(x) + f(y)
(C) f(|x|) = |f(x)| (D) none of these

Exercise - 2
Single choice correct with multiple options

1. The domain of definition of f(x) = sin–1 (|x – 1| – 2) is:

(A) [–2, 0]  [2, 4] (B) (–2, 0)  (2, 4)
(C) [–2, 0]  [1, 3] (D) (–2, 0)  (1, 3)
2. The function f(x) = cot–1  x  3 x + cos–1 x 2  3x  1 is defined on the set S, where S is equal to:
(A) {0, 3} (B) (0, 3) (C) {0, - 3} (D) [- 3, 0]
3. Range of f(x) = 4x + 2x + 1 is
(A) (0,) (B) (1, ) (C) (2, ) (D) (3, )
4. Range of f(x) = log 5  
2  sin x  cos x   3 is

 3
(A) [0, 1] (B) [0, 2] (C)  0,  (D) none of these
 2
5. Let f(x) be a function whose domain is [– 5, 7]. Let g(x) = |2x + 5|. Then domain of (fog) (x) is
(A) [– 4, 1] (B) [– 5, 1] (C) [– 6, 1] (D) none of these
ax  b
6. If f(x) = , then (fof) (x) = x, provided that
cx  d
(A) d + a = 0 (B) d – a = 0 (C) a = b = c = d = 1 (D) a = b = 1
7. If ‘f’ and ‘g’ are bijective functions and gof is defined, then gof must be:
(A) injective (B) surjective (C) bijective (D) into only

 1  sin x 
8. The function f(x) = log   is
 1  sin x 
(A) even (B) odd
(C) neither even nor odd (D) both even and odd

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-68
1
9. The function f(x) = [x] + , x I is a/an (where [ . ] denotes greatest integer function)
2
(A) Even (B) odd
(C) neither even nor odd (D) none of these
2x 2  x  5
10. Let f : R R be a function defined by f(x) = , then f is:
7x 2  2x  10
(A) one-one but not onto (B) onto but not one-one
(C) onto as well as one-one (D) neither onto nor one-one
11. Let f : R R be a function defined by f(x) = x3 + x2 + 3x + sin x. Then f is:
(A) one-one and onto (B) one-one and into
(C) many one and onto (D) may one and into
12. If f(x) = sin  a x  (where [ . ] denotes the greatest integer function) has as its fundamental period,
then -
(A) a = 1 (B) a = 9 (C) a[1, 2) (D) a[4, 5)
e x  e x
13. The inverse of the function f(x) = is
e x  e x
1 1 x 1 2 x 1 1 x
(A) n (B) n (C) n (D) 2 n(1  x)
2 1 x 2 2x 2 1 x
14. If f(1) = 1 and f(n + 1) = 2 f(n) + 1 if n  1, then f(n) is equal to
(A) 2n + 1 (B) 2n (C) 2n – 1 (D) 2n–1 – 1
15. A function f : R R satisfies the condition x2 f(x) + f(1 – x) = 2x – x4. The f(x) is:
(A) – x2 – 1 (B) – x2 + 1 (C) x2 – 1 (D) – x4 + 1
1
16. The domain of the function f(x) = is:
| x | 1 cos  2x  1  tan 3x
1

 
(A) (–1, 0) (B) (–1, 0) –   
 6
     
(C) (–1, 0] –  ,   (D)   ,0 
 6 2  6 
17. If q – 4 p r = 0, p > 0, then the domain of the function f(x) = log (p x3 + (p + q) x2 + (q + r) x + r) is:
2

 q    q 
(A) R –   (B) R – (, 1]   
 2p    2p 
  q 
(C) R – (, 1]    (D) none of these
  2p 
18. The image of the interval R under the mapping f: R R given by f(x) = cot–1 (x2 – 4x + 3) is
  3     3 
(A)  ,  (B)  ,   (C) (0, ) (D)  0, 
4 4  4   4
x   x
19. Let f(x) = ,x  R. Then range of f(x), where [.] denotes greatest integer function, is:
1  x   x
 1  1  1  1
(A)  0,  (B)  0,  (C)  0,  (D)  0, 
 2  2  2  2
20.  
The range of the function f(x) = log 2 2  log 2 16sin 2 x  1 is 
(A) (  ,1) (B) (  , 2) (C) (  ,1] (D) (  , 2]

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
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21. If [2 cos x] + [sin x] = 3, then the range of the function, f(x) = sin x + 3 cos x in [0, 2] is: (where [ ]
denotes greatest integer function)
(A) [–2, –1) (B) (–2, –1] (C) (–2, –1) (D) [2,  3)
22. If domain of f(x) is (–  , 0], then domain of f(6{x} – 5 {x} + 1) is (where {  } represents fractional part
2

function).
 1 1  1 
(A)   n  , n   (B) (  , 0) (C)   n  ,n  1 (D) none of these
nI 
3 2 nI 
6 
4a  7 3
23. If f(x) = x + (a – 3) x2 + x + 5 is one-one function, then
3
(A) 2  a  8 (B) 1  a  2 (C) 0  a  1 (D) none of these
24. If f(x) = 2 [x] + cos x, then f: R R is: (where [ ] denotes greatest integer function)
(A) one-one and onto (B) one-one and into
(C) many-one and into (D) many-one and onto
x x
25. The function f(x) = x   1 is
e 1 2
(A) an odd function (B) an even function
(C) neither an odd nor an even function (D) a periodic function
a x 1
26. If the graph of the function f(x) = is symmetric about y-axis, then n is equal to:

xn a x  1 
(A) 2 (B) 2 / 3 (C) 1 / 4 (D) –1 / 3
 x 
27. The period of sin [x] + cos + cos [x], where [ . ] denotes the integral part of x, is
4 2 3
(A) 8 (B) 12 (C) 24 (D) Non-periodic
 1  2
28. The fundamental period of function f(x) = [x] + x   + x   – 3x + 15, where [ . ] denotes greatest
3  3   
integer function, is:
1 2
(A) (B) (C) 1 (D) non-periodic
3 3
x
29. Let f : (2, 4) (1, 3) be a function defined by f(x) = x –   (where [ . ] denotes the greatest integer
2
function), then f–1 (x) is equal to :
x
(A) 2x (B) x +   (C) x + 1 (D) x – 1
2 
30. The number of solution(s) of the equation [x] + 2{–x} = 3x, is/are (where [ ] represents the greatest
integer function and {x} denotes the fractional part of x) :
(A) 1 (B) 2 (C) 3 (D) 0
31. The domain of definition of the function, y(x) given by the equation, 2x + 2y = 2 is:
(A) 0 < x ≤ 1 (B) 0 ≤ x ≤ 1 (C) –∞ < x ≤ 0 (D) –∞ < x < 1
1
32. If f: [1 ∞]  [2 ∞] is given by f(x) = x + , then f–1 (x) equals:
x
x  x2  4 x x  x2  4
(A) (B) 2
(C) (D) 1  x 2  4
2 1 x 2
33. Let E = {1, 2, 3, 4} and F = {1, 2}. Then the number of onto functions from E to F is
(A) 14 (B) 16 (C) 12 (D) 8
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x
34. Let f(x) = , x ≠ – 1.Then for what value of  is f(f(x)) = x?
x 1
(A) 2 (B)  2 (C) 1 (D) – 1
35. Let function f : R  R be defined by f(x) = 2x + sin x for xR. Then f is
(A) one to one and onto (B) one to one but not onto
(C) onto but not one to one (D) neither one to one nor onto
36. If the functions f(x) and g(x) are defined on R  R such that
 0 , x  rational  0 , x  irrational
f(x) =  , g(x) =  , then (f – g) (x) is
x , x  irrational x , x  rational
(A) one-one and onto (B) neither one-one nor onto
(C) one-one but not onto (D) onto but not one-one
 n 1
 2 , where n is odd
37. A function f from the set of natural numbers to integers defined by f(n) =  is :
  n , when n is even
 2
(A) One-one but not onto (B) onto but not one-one
(C) One-one and onto both (D) Neither one-one nor onto
38. If f : R  S, defined by f(x) = sin x – 3 cos x + 1, is onto, then the interval of S is:
(A) [0, 3] (B) [–1, 1] (C) [0, 1] (D) [–1, 3]
7–x
39. The range of the function f(x) = Px–3 is :
(A) {1,2,3} (B) {1,2,3,4,5,6} (C) {1,2,3,4} (D) {1,2,3,4,5}
40. A real valued function f(x) satisfies the functional equation f(x – y) = f(x) f(y) – f(a – x) f(a + y), where a
is a given constant and f(0) = 1, then f(2a – x) is equal to :
(A) f(–x) (B) f(A) + f(a – x) (C) f(x) (D) –f(x)

Exercise - 3
Multiple choice correct with multiple options

1. For the function f(x) = n (sin–1og2 x),

1   
(A) Domain is  ,2 (B) Range is  , n 
2   2
(C) Domain is (1, 2] (D) Range is R
2. The mapping f : R R given by f (x) = x3 + ax2 + bx + c is a bijection if
(A) b2  3a (B) a 2  3b (C) a 2  3b (D) b2  3a
sin [x]
3. If F (x) = , then F (x) is: (where { . } denotes fractional part function and [ ] denotes greatest
{x}
integer function and sgn (x) is a signum function.
(A) periodic with fundamental period 1 (B) even
 {x} 
(C) range is singleton (D) identical to sgn  sgn  –1
 {x} 

4. Let f : [–1, 1]  [0, 2] be a linear function which is onto, then f(x) is/are
(A) 1 – x (B) 1 + x (C) x – 1 (D) x + 2
5. In the following functions defined from [–1, 1] to [–1, 1], then functions which are not bijective are
2
(A) sin (sin–1 x) (B) sin–1 (sin x) (C) (sgn x) n ex (D) x3 sgn x


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6. Function f(x) = sin x + tan x + sgn (x2 – 6x + 10) is
(A) periodic with period 2 (B) periodic with period 
(C) Non-periodic (D) periodic with period 4
7. A function ' f ' from the set of natural numbers to integers defined by
 n 1
 2 , when n is odd
f (n) =  , then f is:
 n , when n is even
 2
(A) one-one (B) many-one (C) onto (D) into
8. Which of the following functions are not periodic (where [.] denotes greatest integer function)
(A) y = [x + 1] (B) y = sin x2 (C) y = sin2 x (D) y = sin–1 x
9. Let D  [–1, 1] is the domain of the following functions, state which of them are injective.
(A) f(x) = x2 (B) g(x) = x3 (C) h(x) = sin 2x (D) k(x) = sin (x/2)
4 4
10. The period of the function f(x) = sin 3x + cos 3x is:
(A)  / 6 (B)  / 3 (C)  / 2 (D)  /12
 
11. If f: R [–1, 1], where f(x) = sin  x , (where [.] denotes greatest integer function), then
2 
(A) f(x) is onto (B) f(x) is into (C) f(x) is periodic (D) f(x) is many one
12. Identify the statement(s) which is/are incorrect?
(A) the function f(x) = cos (cos–1 x) is neither odd nor even
(B) the fundamental period of f(x) = cos (sin x) + cos (cos x) is 
(C) the range of the function f(x) = cos (3 sin x) is [–1, 1]
(D) none of these
13. Which of the statements is/are true
(A) Every real valued function can be expressed as the sum of an even function and an odd function.
(B) There is no function which is even as well as odd
(C) Constant functions are not periodic functions
(D) All invertible functions are one-one and onto
14. If f(x) = cos [π2] x + cos [–π2] x, where [x] is the greatest integer function, then

(A) f    1 (B) f(x) is periodic with period 2π
2

(C) f(–π) = 0 (D) f   = 2
4
15. Let D ≡ [–1, 1] is the domain of the following functions, state which of them are injective.
 1 1
tan x0
(A) f(x) =  x (B) g(x) = x3
 1 x 0
x 
(C) h(x) = sin 2x (D) k(x) = sin  
 2 
16. Let f(x) = x135 + x125 – x115 + x5 + 1. If f(x) is divided by x3– x, then the remainder is some function of x
say g(x). Then g(x) is an :
(A) one-one function (B) many one function
(C) into function (D) onto function
17. f : N  N where f(x) = x – (–1)x then f is :
(A) one-one (B) many-one (C) onto (D) into

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18. Which one of the following pair of functions are NOT identical?
(A) e(nx)/2 and x
(B) tan (tanx) and cot (cotx)
(C) cos2x + sin4x and sin2x + cos4x
x
(D) and sgn (x), where sgn(x) stands for signum function.
x
 x 2  1
19. If f : [–2, 2]  R where f(x) = x3 + tan x +   is an odd function, then the value of parametric P,
 P 
where [.] denotes the greatest integer function, can be
(A) 5 < P < 10 (B) P < 5 (C) P > 5 (D) P = 15
20. Let f : RR and g :RR be two one-one and onto functions such that they are mirror images of each
other about the line y = a. If h(x) = f(x) + g(x), then h(x) is
(A) one-one (B) into (C) onto (D) many-one

Exercise –4
Section - I : Numerical Value/Subjective Type Questions

4
1. The fundamental period of ecos x x [x]cos x is _____. (where [ ] denotes the greatest integer function)
2. The number of mappings from the set {1, 2, 3} into the set {1, 2, 3, 4, 5} such that f(i) ≤ f(j) whenever
i < j is _______.
3. Let A be a set of n distinct elements. Then the total number of distinct functions from A to A is aa and out
of these b! are onto functions. If a + b = λn, then n is equal to
4. The maximum value of the function f(x) = 2x3 – 15x2 + 36x – 48 on the set A = {x |x2 + 20 ≤ 9x} is
[IIT-JEE 2009, P-2, (4, –1), 80]
x
5. If the function f(x) = x +3
e2 and g(x) = f–1 (x), then the value of g′ (1) is
[IIT-JEE 2009, P-2, (4, –1), 80]
x 3  2x 2  3x  2
6. Number of integers in the range of the function f(x) ; x  R  {0} is :
x 3  2x 2  2x  1
7. Range of the function f(x) = |sin x |cos x| + cos x|sinx|| is [a, b] then (a + b) is equal to
4a  7 3
8. If f(x) = x + (a – 3) x2 + x + 5 is a one-one function, then number of possible integral values of a is
3
2
9. Number of solutions of the equation e sin x = tan2x in [0, 10π] is
10. Let f be a one-one function with domain {21, 22, 23} and range {x,y,z}. It is given that exactly one of the
following statements is true and the remaining two are false. f(21) = x; f(22) ≠ x ; f(23) ≠ y. Then f–1(x) is:
11. The number of real solutions of the equation x3 + 1 = 2 3 2x  1 , is :
12. If f(x) = ax7 + bx3 + cx – 5 ; a, b, c are real constants and f(–7) = 7 then maximum value of
|f(7) + 17 cos x| is
1
13. If f(x) = , g(x) = f (f(x)), h(x) = f(f(f(x))), then the absolute value of f(x) . g(x) . h(x), where x ≠ 0, 1,
1 x
is
14. If f and g are two distinct linear functions defined on R such that they map [-1,1] onto [0, 2] and
f (x)
h : R – {–1, 0, 1}→ R defined by h(x) = , then |h(h(x)) + h(h(1/x))| > n. Then maximum integral
g(x)
value of n is:

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15. Let f(x) = ([a]2 – 5[a] +4)x3 – (6{a}2 – 5 {a} + 1) x – (tan x) sgn (x) be an even function x R, then the
sum of all possible values of ′3a′ is
[where [.] denotes G.I.F and {.} fractional part function]

Section - II : Match the Column

x 2  6x  5
16. Let f(x) = [IIT-JEE 2007, P-2, (6, 0), 81]
x 2  5x  6
Match of expression/statements in Column-1 with expression/statements in Column-2
Column-I Column–II
(A) If – 1 , x < 1, then f(x) satisfies (p) 0 < f(x) < 1
(B) If 1 < x < 2, then f(x) satisfies (q) f(x) < 0
(C) If 3 < x < 5, then f(x) satisfies (r) f(x) > 0
(D) If x > 5, then f(x) satisfies (s) f(x) < 1

17.
Column-I Column–II
The period of the function
(A) (p) ½
y = sin (2πt + π/3) + 2 sin (3πt + π/4) + 3 sin 5πt
y = {sin (πx)} is a many one function for
(B) (q) 8
x(0, a) where {x} denotes fractional part of x, then a may be
The fundamental period of the function
(C) 1  sin(  / 4)x sin(  / 4)x  (r) 2
y=   
2  cos( / 4)x cos( / 4)x 
If f: [0, 2] → [0, 2] is bijective function defined by
(D) f(x) = ax2 + bx + c, where a, b, c are non-zero real (s) 0
numbers, then f(2) is equal to

18. Let f(x) = sin–1 x, g(x) = cos–1 x and h(x) and tan–1 x. For what interval of variation of x the following are
true.
Column-I Column–II

(A) f  x   g x   / 2 (p) [0, ∞)

(B) f(x) + g  1 x   0
2
(q) [0, 1]

 1  x2 
(C) g
 1  x2   2h  x  (r) (–∞, 1)
 

1 x 
(D) h(x) + h(1) = h   (s) [–1, 0]
1 x 

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Section - III : Comprehension

Comprehension#1 (19 to 21)

x3 x 2
Let f(x) =   ax  b x  R
3 2
19. Least value of 'a' for which f(x) is injective function, is
1 1 1
(A) (B) 1 (C) (D)
4 2 8
20. If a = –1, then f(x) is
(A) bijective (B) many-one and onto
(C) one-one and into (D) many-one and into
21. f(x) is invertible iff

(A) a   ,   , b  R (B) a   ,   , b  R (C) a   ,  , b  R (D) a   ,  , b  R

1 1 1 1
4  8   4  4

Comprehension#2 (22 to 24)

x 
tan    1  2[x]   [x] 
Given a function f1(x)= e 4
 cos     sin   whose fundamental period is p, (where
 2   2 
{.} and [.] represent fractional part and greatest integral part functions respectively) and y =
p
2p  [x]  [x]2 the domain of y is [q, r),
2
2  x, x  0
one another function f2(x) =  ,
2  x, x  0
then on the basis of above information answer the following :
22. The period p of f1(x) is :
(A) An irrational number (B) Prime number
(C) A composite number (D) Neither prime nor a composite number
23. Value of r – q – 1 is equal to
(A) 6 (B) 7 (C) 8 (D) 9
24. Range of f2 (f2 (x)) in terms of p, q, r is :
(A) [p, ∞) (B) [q, ∞) (C) [r, ∞) (D) (– ∞, –r)  (p, ∞)

Exercise –5
Revision exercise (Moderate to Tough)

 2x  1 
1. The domain of the function f(x) =  log x  4  log 2  is
 3 x 
5

(A) (–4, –3)  (4, ∞) (B) (–∞, –3)  (4,∞)

(C) (–∞, –4)  (3, ∞) (D) None
2. Let f(x) = (x12 – x9 + x4 – x + 1)–1/2. The domain of the function is
(A) (1, ∞) (B) (–∞, –1) (C) (–1, 1) (D) (–∞, ∞)

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3. Find domain of the following functions

(i) f(x) =  
log1/ 3 log4 [x]2  5 , where [.] denotes greatest integer function.

1
(ii) f(x) = , where [x] denotes the greatest integer not greater than x.
[| x  1|]  |12  x |  11

x 2 2x 3
log 0.5+x 
(iii) f(x) = (x + 0.5) 4x 2 4x 3

1
cos x 
(iv) f(x) = 2
6  35x  6x 2
(v) f(x) =
5 1 2
 3sin x 
 7x  1! , where [.] denotes greatest integer function.
 x  1 x 1
 2 
4. Find the domain and range of the following functions.
(i) f(x) = x 1  2 3  x
|x|
(ii) f(x) = cos 1 log x  , where [.] denotes the greatest integer function.
x

(iii) f(x) = log1/ 2 log2  x 2  4x  5 , where [.] denotes the greatest integer function.
 
  x 2 
(iv) f(x) = sin 1 log 2    , where [.] denotes the greatest integer function.
  2 
 
(v) f(x) = log[x 1] sin x, where [.] denotes the greatest integer function.
1
(vi) f(x) = tan–1 [x]  [  x]  2  | x |  , where [.] denotes the greatest integer function.
x2
5. It is given that f(x) is a function defined on N, satisfying f(1) = 1 and for any x N ; f(x + 5)  f(x) + 5
and f(x + 1) ≤ f(x) + 1. If g(x) = f(x) + 1 – x, then g(2016) equals.
 x, y  R, y ≠ 0, then prove that f(x) . f   = 1
f (x) 1
6. If f(x ×f(y)) =
y x
7. If f : R  R is an odd function such that
(ii) x2f   = f(x), x ≠ 0
1
(i) f(1 + x) = 1 + f(x)
x
Prove that f(x) = x  x  R
8. Solve for x, where [.] represents greatest integer function and {.} represent fractional part function.
(i) |[x] – 2x| = 4 (ii) [x – 1] + [1 – x] + x – {x} > 0
9. If f(x) + f(y) + f(xy) = 2 + f(x) . f(y), for all real values of x and y and f(x) is a polynomial function with
f(4) = 17 and f(1) ≠ 1, then find the value of f(5).
10. Find fundamental period of following functions :
1  sin x 1  secx   
(i) f(x) = (ii) f(x) = sin  2 x   + 2 sin  3x   + 3 sin (5 x)
1  cos x 1  cosecx   3  4
x x
(iii) f(x) = sin 3  sin 5
2 5
11. Draw rough sketch of the following functions
(i) y = cos (sin x) (ii) y = n x 2  x
(iii) y = min (x – [x], – x – [–x]), where [.] represents greatest integer function

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(iv) y = (sin 2x) 1  tan 2 x (v) y = sin x + |sin x|

2
(ln x)
(vi) y = ln x –
| ln x |
 1 
12. If f : R+ R+ a function such that f(x) f(y) = f(xy) + 2  x   1 , then find f(x).
 y 
 x , x 1
 2
13. If f(x) =  x , 1  x  4, then find f–1 (x).

8 x , x4
 0 for x  0

 
14. Let f(x) =  x 2 sin   for  1  x  1(x  0), then:
 x
 x | x | for x  1 or x  1
(A) f(x) is an odd function (B) f(x) is an even function
(C) f(x) is neither odd nor even (D) f ′(x) is an even function
15. Let f(x) = x (2 – x), 0 ≤ x ≤ 2. If the definition of ‘f’ is extended over the set,
R – [0, 2] by f(x + 2) = f(x), then ‘f’ is a :
(A) periodic function of period 1 (B) non-periodic function
(C) periodic function of period 2 (D) periodic function of period ½
16. Find the value of ‘a’ in the domain of the definition of the function, f(A) = 2a 2  a for which the roots
of the equation, x2 + (a + 1) x + (a – 1) = 0 lie between – 2 & 1.
(A) (–1/2, 0]  [1/2, 1) (B) (–2/3, 0] [1/3, 1)
(C) (–1/2, –1/4]  [0, 1) (D) none of these
5  9  8x 5
17. Solve 2x2 – 5x + 2 = , where x <
4 4
18. Let f(x, y) be a periodic function satisfying f(x, y) = f(2x + 2y, 2y – 2x) for all x, y; Define g(x) = f(2x, 0).
Show that g(x) is a periodic function with period 12.
19. The fundamental period of the function,
f(x) = x + a – [x + b] + sin x + cos 2x + sin 3x + cos 4x +……+ sin (2n – 1) x + cos 2 nx for every
a, b  R is: (where [ ] denotes the greatest integer function)
(A) 2 (B) 4 (C) 1 (D) 0
2x  sin x  tan x 
20. Function f(x) = is even, odd on neither
 x  2 
2 3
  
21. Let ‘f’ be a real valued function defined for all real numbers x such that for some positive constant ‘a’ the
1
+ f  x    f  x   holds for all x. Prove that the function f is periodic
2
equation f(x + a) =
2
22. Find the period of f(x) satisfying the condition :
(i) f(x + p) = 1 + {1 – 3 f(x) + 3 f2(x) – f3 (x)}1/3, p > 0
(ii) f(x – 1) + f(x + 3) = f(x + 1) + f(x + 5)
 1  2 1  2x 
23. Determine a function f satisfying the functional relation f(x) + f   .
 1  x  x 1  x 
24. Let f(x) = Ax2 + Bx + C, where A, B, C are real numbers. Prove that if f(x) is an integer whenever x is
integer, then the numbers 2A, A + B and C are all integers. Conversely, prove that if the numbers 2A, A +
B and C are all integer then f(x) is an integer whenever x is an integer.
25. Find the integral solutions to the equation [x] [y] = x + y. Show that all the non-integral solutions lie on
exactly two lines. Determine these lines. Here [.] denotes G.I.F.

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
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Consider the function g(x) defined as g(x) = (x + 1) (x2 + 1) (x4 + 1) ……  x 2 

2010
26.  1  – 1, (|x| ≠ 1).
 
Then the value of g(2) is equal to
27. Let f(x) = log2 log3 log4 log5 (sin x + a2). Find the set of values of a for which domain of f(x) is R.
 x 
28. The fundamental period of sin [x] + cos + cos [x], where [.] denotes integral part of x, is
4 2 3
29. Find range of the function f(x) = log2 [3x – [x + [x + [x]]]. (where [.] denotes G.I.F.
30. Find the number of functions f : A  B, where n(A) = m, n (B) = t, which are non-decreasing.

Exercise –6
Section – I : JEE (Advanced) Questions Previous Years

1.

2.

3.

4.

5.

6.

7.

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-78
8. Let the function f : [0, 1]   be defined by [JEE (Advanced) 2020; P-2]

4x
f (x)  .
4x  2

 1   2   3   39   1 
Then the value of f    f    f    ...  f    f   is ________
 40   40   40   40   2 

Section – II : JEE (Main) Questions Previous Years

1 1
1. For x  R – {0, 1}, let f1 (x) = , f2 (x) = 1 – x and f3 (x) = be there given functions. If a function, J
x 1 x
(x) satisfies (f2oJof1) (x) = f3(x) then J(x) is equal to: [JEE (Main) 9 Jan 2019; Shift-M]
1
(1) f3 ( x) (2) f3 ( x) (3) f 2 ( x) (4) f1 ( x )
x
2 403 k
2. If the fractional part of the number is , then k is equal to: [JEE (Main) 9 Jan 2019; Shift-M]
15 15
(1) 6 (2) 8 (3) 4 (4) 14
x
3. Let f : R  R be defined by f ( x)  , x  R. Then the range of f is :
1  x2
[JEE (Main) 11 Jan 2019; Shift-M]
 1 1  1 1
(1)  2 , 2  (2) R   1,1 (3) R   ,  (4) ( 1,1)  {0}
   2 2

 1 x   2x 
4. If f ( x)  log e   , x  1, then f   is equal to: [JEE (Main) 8 April 2019; Shift-M]
1 x   1  x2 

(A) 2 f ( x) (B) 2 f ( x2 ) (C) ( f ( x))2 (D) 2 f ( x)

x2
5. If the function f : R – {1, –1}  A defined by f ( x)  , is surjective, then A is equal to:
1  x2
[JEE (Main) 9 April 2019; Shift-M]
(1) R – {–1} (2) [0, ) (3) R – [–1, 0) (4) R – (–1, 0)

6. Let f ( x )  log e (sin x ), (0  x   ) and g ( x)  sin 1 (e x ),( x  0). If α is a positive real number such that
a  ( fog )'() and b  ( fog )(), then: [JEE (Main) 10 April 2019; Shift-E]

(1) a2  b  a  0 (2) a2  b  a  1

(3) a2  b  a  0 (4) a2  b  a  2a2

 3 1  x2 
7. For x   0,  , let f ( x)  x , g( x)  tan x and h( x)  2
. If ( x )  (( hof )og )( x), then    is equal
 2 1 x 3
to : [JEE (Main) 12 April 2019; Shift-M]
 11 7 5
(1) tan (2) tan (3) tan (4) tan
12 12 12 12
Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-79
5
8. If g ( x)  x 2  x  1 and ( gof )( x)  4 x2  10 x  5, then f   is equal to.
4
[JEE (Main) 7 Jan 2020; Shift-M]
1 1 1 1
(1) (2)  (3)  (4)
2 2 3 3
x[ x]
9. Let f : (1,3)  R be a function defined by f ( x)  , where [x] denotes the greatest integer  x.
1  x2
Then the range of f is : [JEE (Main) 8 Jan 2020; Shift-E]
 1   3 7  2 1   3 4  2   4  1   2 4
(1)  0,    ,  (2)  ,  ,  (3)  ,1    1,  (4)  0,    , 
 2   5 5  5 2   5 5  5   5  3  5 5

10. The number of distinct solutions of the equation, log1/2 sin x  2  log1/2 cos x in the interval [0, 2 ], is

[JEE (Main) 9 Jan 2020; Shift-M]

1  | x | 5 
11. The domain of the function f ( x)  sin  2  is ( , a ]  [a,  ). Then a is equal to :
 x 1 
[JEE (Main) 2 Sep 2020; Shift-M]
1  17 17 17  1 17
(1) (2) 1 (3) (4)
2 2 2 2

12. Let [t] denote the greatest integer  t. Then the equation in x,[ x]2  2[ x  2]  7  0 has:
[JEE (Main) 4 Sep 2020; Shift-M]
(1) exactly two solutions (2) infinitely many solutions
(3) exactly four integral solutions (4) no integral solution

13. If f ( x  y )  f ( x ) f ( y ) and  x1 f ( x)  2, x, y  N where N is the set of all natural numbers, then the

f (4)
value of is: [JEE (Main) 6 Sep 2020; Shift-M]
f (2)

1 4 1 2
(1) (2) (3) (4)
9 9 3 3
14. Suppose that a function f : R  R satisfies f ( x  y )  f ( x ) f ( y ) for all x, y  R and f (1)  3. If

i1 f (i)  363, the n is equal to

n
[JEE (Main) 6 Sep 2020; Shift-E]

1
f ( x)  f  
1   x
15. If a    1, b    2 and af ( x )  f    bx  , x  0, then the value of the expression
 x x 1
x
x
is [JEE (Main) 24 Feb 2021; Shift-2]
2 2
16. The number of roots of the equation, (81)sin x  (81)cos x  30 in the interval [0, ] is equal to:
[JEE (Main) 16 March 2021; Shift-1]
(1) 3 (2) 4 (3) 8 (4) 2

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-80
cos ec 1 x
17. The real valued function f ( x)  , where [x] denotes the greatest integer less than or equal to x,
x  [ x]
is defined for all x belonging to: [JEE (Main) 18 March 2021; Shift-1]
(1) all real except integers (2) all non-integers except the interval [–1,1]
(3) all integers except 0, –1, 1 (4) all reals except the Interval [–1,1]
18. If the functions are defined as f ( x)  x and g ( x)  1  x , then what is the common domain of the
f ( x)
following functions: f  g , f  g , f / g , g / f where  f  g  ( x)  f ( x)  g ( x),( f / g )( x ) 
g ( x)
[JEE (Main) 18 March 2021; Shift-1]
(1) 0  x  1 (2) 0  x  1 (3) 0  x  1 (4) None of these
x2
19. Let f : R  {3}  R  {1} be defined by f ( x)  . Let g : R  R be given as g ( x)  2 x  3. Then, the
x 3
1 1 13
sum of all the values of x for which f ( x)  g ( x)  is equal to
2
[JEE (Main) 18 March 2021; Shift-2]
(1) 7 (2) 2 (3) 5 (4) 3
20.  
The number of solutions of the equation log x 1 2 x 2  7 x  5  log(2 x 5)  x  1  4  0 x > 0, is
2

[JEE (Main) 20 July 2021; Shift-2]

21. Let [x] denote the greatest integer less than or equal to x. Then, the values of x  R satisfying the
2
equation e x   e x  1  3  0 lie in the interval: [JEE (Main) 22 July 2021; Shift-1]
   
 1
(1)  0,  (2)  loge 2,loge 3 (3) [1, e) (4) [0, loge 2)
 e
22. The number of real solutions of the equation, x2 – |x| – 12 = 0 is : [JEE (Main) 25 July 2021; Shift-2]
(1) 2 (2) 3 (3) 1 (4) 4
23. Let f : N  N be a function such that f(m + n) = f(m) + f(n) for every m, n  N. If f(6) = 18, then
f (2)  f (3) is equal to : [JEE (Main) 31 Aug 2021; Shift-2]
(1) 6 (2) 54 (3) 18 (4) 36
2
24. Let f (x) be a polynomial of degree 3 such that f (k )   for k = 2, 3, 4, 5. Then value of 52 – 10f(10) is
k
equal to : [JEE (Main) 01 Sep 2021; Shift-2]
 x  5x  6 
2
cos 1 
 x 2  9 
25. The domain of the function f ( x)    is [JEE (Main) 24 June 2022; Shift-1]
2
log e ( x  3 x  2)
(1) ( ,1)  (2, ) (2) (2,  )
 1   1   3  5 3  5 
(3)   2 ,1   (2,  ) (4)  ,1
 2    (2,  )   , 
   2 2 
1
  x25   50
26. Let f : R  R be a function defined by f (x) =  2 1 
  
2 
 
2  x 25  . If the function g (x) = f(f(f(x)))

  
+ f(f(x)), the greatest integer less than or equal to g (1) is _______. [JEE (Main) 25 June 2022; Shift-1]
x 1
27. Let f (x)  , x  R  {0, 1,1). If f n 1 (x)  f (f n (x)) for all n  N, then f6(6) + f7 (7) is equal to
x 1
[JEE (Main) 26 June 2022; Shift-1]
7 3 7 11
(1) (2)  (3) (4) 
6 2 12 12

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-81

2e2 x
. Then f 
1   2   3   99 
28. Let f : R  R be a function defined f ( x)  2x  f   f    .....  f  
e e  100   100   100   100 
is equal to __________. [JEE (Main) 27 June 2022; Shift-1]
29. Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x2 – 2x, and g(f(x))
= 4x2 + 6x + 1, then the value of f(2) + g(2) is _________. [JEE (Main) 29 June 2022; Shift-2]
30. The total number of functions, f : {1, 2, 3, 4}  {1, 2, 3, 4, 5, 6} such that f(1) + f(2) = f(3), is equal to :
[JEE (Main) 25 July 2022; Shift-1]
(1) 60 (2) 90 (3) 108 (4) 126
1
31. Let f(x) be a quadratic polynomial with leading coefficient 1 such that f(0) = p, p  0 and f(1) = . If the
3
equation f(x) = 0 and fofofof(x) = 0 have a common real root, then f(–3) is equal to ……………..
[JEE (Main) 28 July 2022; Shift-1]
32. Let ,  and  be three positive real numbers. Let f (x)  x  x 3  x, x  R and g : R  R be such
5

that g(f(x)) = x for all x  R. If a1, a2, a3,….., an be in arithmetic progression with mean zero, then the
 1 n 
value of f  g   f  a i    is equal to : [JEE (Main) 28 July 2022; Shift-1]
 n 
  i 1 
(1) 0 (2) 3 (3) 9 (4) 27
  x2  x  
33. The number of elements in the set S   x   : 2cos    4x  4 x  is:
 6 
   
[JEE (Main) 29 July 2022; Shift-2]
(1) 1 (2) 3 (3) 0 (4) infinite
1  x  3x  2 
2
The domain of the function f (x)  sin  2
 x  2x  7 
34. is : [JEE (Main) 29 July 2022; Shift-2]
 
(1) [1, ) (2) (–1, 2] (3) [–1, ) (4) (–, 2]
22x  1   2   2022 
35. If f(x) = , x  R, f f   ......  f   then is equal to
22x  2  2023   2023   2023 
[JEE (Main) 24 Jan. 2023; Shift-2]
(1) 2011 (2) 1010 (3) 2010 (4) 1011
1/3
 x 7
36. For some a, b, c, let f(x) = ax – 3 and g(x) = xb + c, x. If (fog)–1 (x) =   then (fog)
 2 
(ac) + (gof) (b) is equal to _______. [JEE (Main) 25 Jan. 2023; Shift-1]
 
37. Let a, b and c be three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and
              
D be a  b  c, a  3b  4c,  a  2b  3c and 2a  4b  6c respectively. If AB, AC and AD are
coplanar, then  is : [JEE (Main) 29 Jan. 2023; Shift-1]
38. The range of the function f(x) = 3  x  2  x is [JEE (Main) 30 Jan. 2023; Shift-2]
(1)  5, 10  (2)  2 2, 11  (3)  5, 13  (4)  2, 7 
       
[x]
39. If the domain of the function f(x) = , where [x] is greatest integer  x, is (2, 6), then its range is
1  x2
[JEE (Main) 31 Jan. 2023; Shift-1]
 5 2   9 27 18 9   5 2
(1)  ,  , , ,  (2)  , 
 26 5   29 109 89 53   26 5 
 5 2   9 27 18 9   5 2
(3)  ,  , , ,  (4)  , 
 37 5   29 109 89 53   37 5 

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-82
 1 
40. Let f : R – {0, 1} → R be a function such that f(x) + f   = 1 + x. Then f(2) is equal to :
1 x 
[JEE (Main) 01 Feb. 2023; Shift-2]
9 9 7 7
(1) (2) (3) (4)
2 4 4 3
1 1 2
41. Let 5 f(x) + 4f   = + 3, x > 0. Then 18  f (x) dx is equal to [JEE (Main) 06 Apr. 2023; Shift-1]
x x 1

(1) 5 loge 2 + 3 (2) 10 loge 2 + 3 (3) 10 loge 2 – 6 (4) 5 loge 2 – 3

1
42. Let the sets A and B denote the domain and range respectively of the function f (x)  , where [x]
(x)  x
denotes the smallest integer greater than or equal to x. Then among the statements
(S1) : A  B  (1, )   and [JEE (Main) 06 Apr. 2023; Shift-2]
(S2) : A  B  (1,  )
(1) Only (S2) is true (2) Only (S1) is true
(3) Neither (S1) nor (S2) is true (4) Both (S1) nor (S2) is true
 2x 
43. If the domain of the function f(x)= sec–1   is [, )  (  , ], then 3  10(   )  21 is equal
 5x  3 
to________. [JEE (Main) 10 Apr. 2023; Shift-2]
  6  2log3 x  
44. Let D be the domain of the function f(x) = sin–1  log3x    . If the range of the function
  5x 
5
g : D →  defined by g(x) = x – [x], [x] is the greatest integer function) is (, ), then  2  is equal to

[JEE (Main) 12 Apr. 2023; Shift-1]
(1) 135 (2) 45 (3) 46 (4) 136
 x 2
45. The range of f(x) = 4 sin–1  2  is [JEE (Main) 13 Apr. 2023; Shift-2]
 x 1
(1) [0, 2] (2) [0, ] (3) [0, 2) (4) [0, )

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 M-83
RELATIONS
Section – I : Concept Building Questions

1. Let A be the set of all human beings in a town at a particular time. Determine whether eachof the
following relations are reflexive, symmetric and transitive:
(i) R = {(x, y):x and y work at the same place}
(ii) R = {(x, y):x and y live in the same locality}
(iii) R = {(x, y): xis wife of y}
(iv) R = {(x, y): xis father of y}

2. Test whether the following relations R1, R2, and R3 are (i) reflexive (ii) symmetric and (iii) transitive:
(i) R1 on Q0 defined by (a,b)  R1 a=1/b
(ii) R2 on Z defined by (a, b)  R2 |a – b| ≤5
(iii) R3 on R defined by (a,b)  R3 a2 — 4 ab + 3b2 =0.

3. The following relations are defined on the set of real numbers:

(i) aRb ifa – b>0
(ii) aRbiff 1+ab>0
(iii) aRb if|a| ≤ b
Find whether these relations are reflexive, symmetric or transitive.

4. Check whether the relation R defined on the set A = {1,2, 3, 4,5, 6} as R = {(a, b) :b =a +1} is reflexive,
symmetric or transitive.

5. (i) Check whether the relation R on R defined by R = {(a, b):a ≤ b3}is reflexive, symmetricor
transitive.
(ii) Check if the relation R in the set of all real numbers defined as R = {(a, b) : a <b} is
(i) symmetric (ii) transitive.

6. If A={1, 2, 3, 4}, define relations on A which have properties of being

(i) reflexive, transitive but not symmetric.
(ii) symmetric but neither reflexive nor transitive.
(iii) reflexive, symmetric and transitive.

7. Let R be a relation defined on the set of natural numbers N as

R={(x,y):x,yN,2x+y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric, (iii) transitive.

8. Let W denote the set of words in the English dictionary. Define the relation R byR ={(x, y) W × W such
that x and y have at least one letter in common}. Show that thisrelation R is reflexive and symmetric but
not transitive.

9. Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number ofordered pairs so
that the enlarged relation is symmetric, transitive and reflexive.

10. Let A ={a,b, c} and the relation R be defined on A as follows: R ={(a, a), (B, c), (a, b)}. Then,write
minimum number of ordered pairs to be added in R to make it reflexive andtransitive.

11. Show that the relation R defined by R ={(a,b):a – b is divisible by 3; a,b Z} is anequivalence relation.

12. Prove that the relation R on Z defined by

(a,b) R a – bis divisible by 5is anequivalence relation on Z.

13. Show that the relation R on the set A = {x  Z ; 0 ≤x ≤12}, given by R = {(a,b): a = b}, is anequivalence
relation. Find the set of all elements related to 1.

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 M-84
14. Show that the relation R, defined on the set A of all polygons as
R = {(P1, P2): P1 and P2 have same number of sides},
is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with
sides 3, 4 and 5?

15. Let R be the relation defined on the set A = {1, 2, 3, 4,5, 6, 7}by R = {(a, b): both a and b are either odd or
even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3,5,
7}are related to each other and all the elements of the subset{2, 4, 6} are related to each other, but no
element of the subset {1, 3,5,7}is related to an element of the subset {2, 4, 6}.

16. Check whether the relation R on the set N of natural numbers given by R ={(a,b):a is divisor of b} is
reflexive, symmetric or transitive. Also, determine whether R is an equivalence relation.

17. Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0 be
defined as follows:
(a,b) R (c,d)  ad = bc for all (a,b), (c, d) Z × Z0
Prove that R is an equivalence relation on Z × Z0.

18. If R and Sare relations on a set A, then prove the following:

(i) R and Sare symmetric R Sand R  Sare symmetric
(ii) R is reflexive and Sis any relation R  Sis reflexive.

19. If R and Sare transitive relations on a set A, then prove that R  S may not be a transitive relation on A.

20. Let C be the set of all complex numbers and C0 be the set of all non-zero complex numbers. Let a relation
R on C0 be defined as
z z
z1 R z2 1 2 is real for all z1, z2 C0.
z1  z 2
Show that R is an equivalence relation.

21. Show that the relation R in the set A= {1,2, 3,4, 5} given by R = {(a, b): |a – b| is even} is an equivalence
relation. Also, show that all elements of {1, 3, 5} are related to each other and all the elements of {2, 4}
are related to each other, but no element of{1, 3, 5} is related to any element of (2, 4}.

22. Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P
from the origin is same as the distance of the point Q from the origin} is an equivalence relation. Further,
show that the set of all points related to a point P ≠ (0,0) is the circle passing through P with origin as
centre.
23. If R1 and R2 are equivalence relation in a set A, then show that R1 R2 is also an equivalence relation.
Also, give an example to show that the union of two equivalence relations on a set A need not be an
equivalence relation on a set A.

24. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R, be a relation on X given by R1 = {(x y): x – y is divisible by 3}
and R2 be another relation on X given by R2 = {((x y): {x, y}(1, 4, 7} or {x, y}  {2, 5, 8} or
{x y}  {3, 6, 9}}. Show thatR1 = R2.

25. If N denotes the set of all natural numbers and R is the relation on N × N defined by (a, b) R (c, d), if
ad(b + c) = bc(a + d). Show that Ris an equivalence relation.

26. In the set of natural numbers N, define a relation R as follows n, m N, n Rm, if on division by 5 each
of the integers n and m leaves the remainder less than 5, i.e. one of numbers 0, 1, 2, 3 and 4. Show that R
is an equivalence relation. Also, obtain the pair wise disjoint subset determined by R.

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Section – II : Single choice correct with multiple options

1. Consider the following two binary relations on the set

A = {a, b, c}:
R1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and
R2 = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}. Then
(A) R1 is not symmetric but it is transitive (B) both R1 and R2 are transitive
(C) R2 is symmetric but it is not transitive (D) both R1 and R2 are not symmetric

2. Let N denote the set of all natural numbers. Define two binary relations on N as
R1 = {(x, y)  N × N : 2x + y = 10}
And R2 = {(x, y)  N × N : 2x + 2y = 10}. Then :
(A) Both R1 and R2 are symmetric relations (B) Range of R1 is {2, 4, 8}
(C) Both R1 and R2 are transitive relations (D) Range of R2 is {1, 2, 3, 4}

3. Let R be the set of real numbers.

Statement – 1 : A = {(x, y)R × R : y – x is an integer} is an equivalence relation on R.
Statement – 2 :B = {(x, y)  R × R : x=αy for some rational number α} is an equivalence relation on R.
(A) Statement – 1 is true, Statement – 2 is false
(B) Statement – 1 is false, Statement – 2 is true
(C) Statement – 1 is true, Statement – 2 is true; Statement – 2 is a correct explanation for Statement – 1.
(D) Statement – 1 is true, Statement – 2 is true; Statement–2 is not a correct explanation for Statement–1.

4. Consider the following relations:

R = {(x, y) |x, y are real numbers and x = wy for some rational number w};
 m p 
S =  ,  m, n, p and q are integers such that n, q ≠ 0 and qm = pn}. Then
 n q 
(A) R is an equivalence relation but S is not an equivalence relation
(B) neither R nor S is an equivalence relation
(C) S is an equivalence relation but R is not an equivalence relation
(D) R and S both are equivalence relations

5. Let R be the real line. Consider the following subsets of the plane R × R :
S = {(x, y) : y = x + 1 and 0 < x < 2}
T = {(x, y) : x – y is an integer}.
Which one of the following is true?
(A) T is an equivalence relation on R but S is not
(B) Neither S nor T is an equivalence relation on R
(C) Both S and T are equivalence relations on R
(D) S is an equivalence relation on R but T is not

6. Let W de note the words in the English dictionary. Define the relation R by :
R = {(x, y)  W × W | the words x and y have at least one letter in common}.
Then R is
(A) not reflexive, symmetric and transitive (B) reflexive, symmetric and not transitive
(C) reflexive, symmetric and transitive (D) reflexive, not symmetric and transitive

7. Let R = {(3, 3) (6, 6) (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9,
12}. The relation is
(A) reflexive and symmetric only (B) an equivalence relation
(C) reflexive only (D) reflexive and transitive only

8. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is
(A) not symmetric (B) transitive (C) a function (D) reflexive

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9. Let R be a relation on the set N given by R = {(a, b) : a = b – 2, b > 6}. Then,
(A) (2, 4)  R (B) (3, 8)  R (C) (6, 8)  R (D) (8, 7)  R

10. Which of the following is not an equivalence relation on Z?

(A) a R b a + b is an even integer (B) a R b a – b I an even integer
(C) a R b  a < b (D) a R b  a = b

11. R is relation on the set Z of integers and it is given by (x, y)  R |x – y| ≤ 1. Then, R is
(A) reflexive and transitive (B) reflexive and symmetric
(C) symmetric and transitive (D) and equivalence relation

12. Let R be the relation over the set of all straight lines in a plane such that l1 R l2l1l2.Then, R is
(A) symmetric (B) reflexive (C) transitive (D) an equivalence relation

13. If A = {a, b, c}, then the relation R = {(b, c)} on A is

(A) reflexive only (B) symmetric only
(C) transitive only (D) reflexive and transitive only

14. Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3) which are reflexive and
symmetric but not transitive is
(A) 1 (B) 2 (C) 3 (D) 4

15. The relation ‘R’ in N × N such that (a, b) R (c, d)  a + d = b + c is

(A) reflexive but not symmetric (B) reflexive and transitive but not symmetric
(C) an equivalence relation (D) none of these

16. A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y  x is relatively prime to y. Then,
domain of R is
(A) {2, 3, 5} (B) {3, 5} (C) {2, 3, 4} (D) {2, 3, 4, 5}

17. A relation  from C to R is defined by x  y  |x| = y. Which one is correct?

(A) (2 + 3 i) 13 (B) 3  (– 3) (C) (1 + i)  2 (D) i 1

18. R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x – 3. Then, R–1 is
(A) {(8, 11), (10, 13)} (B) {(11, 8), (13, 10)}
(C) {(10, 13), (8, 11), (8, 10)} (D) none of these

19. Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = {a, b, c}. Then, R is
(A) identity relation (B) reflexive (C) symmetric (D) equivalence

20. If R is the largest equivalence relation on set A and S is any relation on A, then
(A) R  S (B) S  R (C) R = S (D) none of these

21. If R is a relation on the set A = {1, 2, 3} given by R = (1, 1), (2, 2), (3, 3), then R is
(A) reflexive (B) symmetric (C) transitive (D) all the three options

22. If A = {a, b, c, d}, then a relation R = {(a, b), (b, a), (a, a)} on A is
(A) symmetric and transitive only (B) reflexive and transitive only
(C) symmetric only (D) transitive only

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23. Let R be the relation on the set A = {1, 2, 3, 4} given by
R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then,
(A) R is reflexive and symmetric but not transitive
(B) R is reflexive and transitive but not symmetric
(C) R is symmetric and transitive but not reflexive
(D) R is an equivalence relation

24. Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is
(A) 1 (B) 2 (C) 3 (D) 4

25. S is a relation over the set R of all real numbers and it is given by (a, b) S ab 0. Then, S is
(A) symmetric and transitive only (B) reflexive and symmetric only
(C) antisymmetric relation (D) an equivalence relation

26. Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then, R is
(A) reflexive but not symmetric (B) reflexive but not transitive
(C) symmetric and transitive (D) neither symmetric nor transitive

27. The relation S defined on these set R of al real number by the rule a Sb iff a  b is
(A) an equivalence relation (B) reflexive, transitive but not symmetric
(C) symmetric, transitive but not reflexive (D) neither transitive nor reflexive but symmetric

28. Let R be relation on the set N of natural numbers defined by n R m iff n divides m. Then, R is
(A) Reflexive and symmetric (B) Transitive and symmetric
(C) Equivalence (D) Reflexive, transitive but not symmetric

29. Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as a Rb if a is
congruent to be for all a, b  T. Then, R is
(A) reflexive but not transitive (B) transitive but not symmetric
(C) equivalence (D) none of these

30. If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is

(A) reflexive (B) transitive (C) symmetric (D) none of these

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ANSWER KEY
Exercise – 1
1. (BC)
 1  1  1  1
2. (i) (ii) (iii) (iv)
 2  2  2  2
3. (i) R – {–1, 1} (ii) [–1, 1] – {0} (iii) (0, ∞) (iv) R
(ii)   ,  (iv)  ,1    1, 
1 1 1 3
4. (i) [–2, 0)  (0, 1) (iii) 
 3 2 2   2
5. (i) [0, 1] (ii) [–1, 1] (iii) R (iv) 
 
6. (i)   n,n   (ii) R – {2n}, nI (iii) (0, 1]  [4, 5) (iv) (2, 3)
nI 
4
1
7. (i) Domain : R, Range :  y 1 (ii) Domain : N  {0}, Range : (n! : n = 0, 1, 2,…}
7
(iii) Domain R – {3}, Range : R – {6} (iv) Domain : R, Range : {1}
 1 1
8. (i)  0,  (ii)  ,  (iii) [0, 4] (iv) {–1, 1}
2 2
 
9. (i) [0, 10] (ii) (0, 1] (iii) (  , n  / 2]
 49 
10. (i)  , (ii) [–4, 3] (iii) [–1, 1]
 20 
11. f(x) + g(x) = x + sin x for all real x
x
f(x)/g(x) = for all real x except x = nπ; ninteger
sin x
 3x  1 x0

12. f(x) – g(x) = 2  x  1 0  x  1
 2
 x x x 1
13. (i) No (ii) Yes (iii) No (iv) No
14. (i) fog = x, x > 0 ; gof = x, x  R (ii) |sin x|, sin |x|
3x 2  4x  2 x 2  2
(iii) sin–1 (x2), (sin–1x)2 (iv) ,
 x  12 x 2  1
16. Domain : [1, 2] : Range : [n2, n4)
 2  2x  x 2 , 0  x  1
17. f(g(x)) = 
 2  x, 1  x  0
18. (i) into (ii) onto (iii) into (iv) onto
19. (i) many-one (ii) many-one (iii) one-one (iv) many-one
20. (i) one-one (ii) many-one (iii) one-one
21. (i) No (ii) Yes (iii) Yes (iv) No
22. (i) bijective (injective as well as surjective) (ii) neither injective nor surjective
(iii) neither surjective nor injective (iv) surjective but not injective
23. (i) odd (ii) even (iii) even (iv) neither even nor odd
24. (i) odd (ii) even (iii) odd (iv) neither even nor odd
(v) even
 x 2  sin x, 1  x  0   x 2  sin x, 1  x  0
25. (i) f(x) =  (ii) f(x) = 
  x  ex , x  1  x  e x , x  1

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26. (i) 2 (ii) 2 (iii) 24 (iv) 70
2
28. (i) (ii) 2 (iii)  / 6 (iv) 2
3

29. (i) f –1 Does not exists (ii) f –1 : R R ; f –1 = 7 + (4 – x5)1/3

ex  e x
(iii) f –1 : R R ; f –1 =
2
30.
1

f –1 : R → R, f –1(x) = n x  x 2  1
2 
1  x 2 
31. B = [0, 4] ; f –1 (x) =  sin 1   
2  2  6
32. –3 33. 2046 35. f(x) = 1 + x 36. (B)
37. (D) 38. (C) 39. (D) 40. (D)

Exercise – 2
1. (A) 2. (C) 3. (B) 4. (B) 5. (C) 6. (A) 7. (A)
8. (B) 9. (B) 10. (D) 11. (A) 12. (D) 13. (A) 14. (C)
15. (B) 16. (D) 17. (B) 18. (D) 19. (C) 20. (D) 21. (D)
22. (A) 23. (A) 24. (C) 25. (B) 26. (D) 27. (C) 28. (A)
29. (C) 30. (C) 31. (D) 32. (A) 33. (A) 34. (D) 35. (A)
36. (A) 37. (C) 38. (D) 39. (A) 40. (D)

Exercise – 3
1. (BC) 2. (B) 3. (ABCD) 4. (AB) 5. (BCD) 6. (AD) 7. (AC)
8. (ABD) 9. (BD) 10. (ABC) 11. (BCD) 12. (ABC) 13. (AD) 14. (ABC)
15. (BC) 16. (AD) 17. (AC) 18. (ABD) 19. (ACD) 20. (BD)

Exercise – 4
Section - I
1. 2 2. 35 3. 2 4. 7 5. 2 6. 0 7. 1
8. 7 9. 20 10. 22 11. 3 12. 34 13. 1 14. 2
15. 35
Section - II
16. (A)  (p, r, s); (B)  (q, s); (C)  (q, s); (D)  (p, r, s)
17. (A)  (q, r); (B)  (q, r); (C)  (q); (D)  (s)
18. (A)  (q); (B)  (s); (C)  (p); (D)  (r)
Section - III
19. (A) 20. (B) 21. (A) 22. (C) 23. (A) 24. (A)

Exercise – 5
1. (A)
2. (D)
3. (i) [–3, –2]  [3, 4) (ii) R – {(0, 1)  {1,2,….., 12}  (12, 13)}
 1 1 1  3   1    5 
(iii)   ,    ,1    ,   (iv)   ,    ,6 
 2 2 2  2   6 3  3 
n 
(v)  , n  I, 1  n  6 
7 
4. (i) D : [1, 3] ; R :  2, 10 
(ii) D : [2, ∞) ; R : {π/2}
(iii)   
D :  2  2 ,  3 ,   1, 2  2 ; R{0}]

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 M-90
 
(iv)   2

D :  8, 1  1, 8 ; R :   , 0, ]
2

(v) D : [3, )    2n, 2n    ; R : (–∞, 0]}
n 1

D : {–2, –1, 1, 2} ; R :  , 2 
1
(vi)
4 
5. 1
 9 7
x 1  [2, )
8. (i)  4,  , 4,  (ii)
 2 2
9. 26
10. (i) π (ii) 2 (iii) 10π

11. (i)

(ii)

½
(iii)
–1 0 1 2 3

(iv) –/2 /2 3/2 5/2

–

–2

(v)
–2 – 0  2 3 4 5

(vi)
0 e–1/2
–1 1
–1


 x , x 1

12. f(x) = x + 2 13. f 1 (x)   x , 1  x  16 14. (AD)
 2
x , x  16
 64

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 M-91
3 5
15. (C) 16. (A) 17. x 19. (A) 20. Odd
2
x 1
21. (i) 2p (ii) 8 23.
x 1
25. Integral solutions (0, 0) ; (0, 2). x + y = 0, x + y = 6 26. 2
27.   
a ,  626  626,   28. 24 29. {0, 1} 30. m + t –1
Cm

Exercise – 6
Section – I
1. 7 2. 2 3. (A) 4. (B) 5. (AB) 6. (ABC) 7. 119
8. 19.00
Section – II
1. (1) 2. (2) 3. (1) 4. (1) 5. (3) 6. (2) 7. (2)
8. (2) 9. (2) 10. (8) 11. (1) 12. (2) 13. (2) 14. (5)
15. (2) 16. (2) 17. (2) 18. (3) 19. (3) 20. (1) 21. (4)
22. (1) 23. (2) 24. (26) 25. (4) 26. (2) 27. (2) 28. (99)
29. (18) 30. (2) 31. 25 32. (1) 33. (1) 34. (3) 35. (4)
36. (2039) 37. (2) 38. (1) 39. (4) 40. (2) 41. (3) 42. (3)
43. (24) 44. (1) 45. (3)

Relations
Section-I
1. (i) Reflexive, symmetric and transitive
(ii) Reflexive, symmetric and transitive
(iii) Neither reflexive, nor symmetric but transitive
(iv) neither reflexive nor symmetric nor transitive
2. (i) R1is symmetric but it is neither reflexive nor transitive
(ii) R2 is reflexive and symmetric but it is not transitive
(iii) R3 is reflexive but it is neither symmetric nor transitive.
3. (i) Transitive
(ii) Reflexive and symmetric but not transitive
(iii) Transitive neither reflexive nor symmetric.
4. Neither reflexive nor symmetric nor transitive
5. (i) Neither reflexive nor symmetric nor transitive.
(ii) Transitive but not symmetric.
6. (i) R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2)}
(ii) R = {(1, 2), (2, 1)}
(iii) R = { (1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}
7. Domain R = {1, 2, 3, ..., 19, 20}, Range R = {39, 37, 35, ..., 7, 5, 3, 1}.
R is neither reflexive nor symmetric and is not transitive.
9. (1, 1), (2, 2), (3, 3), (1, 3), (2, 1), (3, 2), (3, 1)
10. (b,b), (c, c), (a, c) 13. {1} 14. Set of all triangles 16. Equivalence
Section-II
1. (C) 2. (D) 3. (A) 4. (C) 5. (A) 6. (B) 7. (D)
8. (A) 9. (C) 10. (C) 11. (B) 12. (A) 13. (C) 14. (A)
15. (C) 16. (D) 17. (D) 18. (A) 19. (B) 20. (B) 21. (D)
22. (C) 23. (B) 24. (B) 25. (B) 26. (A) 27. (B) 28. (D)
29. (C) 30. (B)

Mathematics –Functions & Relations Toll Free Number : 1800 103 9888
 INVERSE
TRIGONOMETRIC
FUNCTIONS
 M-92
Exercise - 1
Concept Building Questions

1. Evaluate each of the following function:

 3  2   3 1 1 1
(i) sin 1    (ii) sin 1  cos  (iii) sin 1   (iv) sin 1  2sin 1
 2   3   2 2  2 2
  3   1  4   3 
(v) sin 1 cos  sin 1   (vi) cos  sin  (vii) cos 1  tan 
  2    3   4 

 1  1  5    13  
(viii) tan 1 1  cos 1     sin 1    (ix) tan 1  tan   cos 1 cos  
 2  2  6    6 
1  2 
(x) cot 1  cosec 1  2   sec 1  
3  3
2. Find the domain of the following functions:
(i) f  x   sin 1  2x  3 (ii) f  x   sin 1 x  sin 1 2x
(iii) f  x   2cos 1 2x  sin 1 x (iv) 
f  x   cos 1 x 2  4 
(v) f  x   sec x  tan x
1 1
(vi) f  x   cot x  cot x 1

1 1 1 1
3. If sin x  sin y  sin z  sin t  2 then find the value of x2 + y2 + z2 + t2

     
3
2 2 2
4. If sin 1 x  sin 1 y  sin 1 z  2 , find the value of x2 + y2 + z2
4
n 
5. Find the minimum value of n for which tan 1  , n  N
 4
3 x y z
6. If cosec 1 x  cosec 1 y  cosec 1 z   , find the value of  
2 y z x
7. Express in the simplest form
 cos x   
(i) tan 1  ,   x 
 1  sin x  2 2
 cos x  sin x   
(ii) tan 1  ,  x 
 cos x  sin x  4 4
 1  cos x  1  cos x   x 
8. Prove that: tan 1     , If 0 < x <
 1  cos x  1  cos x  4 2 2

 1  sin x  1  sin x   x 
9. Prove that: cot 1     , If  x  
 1  sin x  1  sin x  2 2 2
 1  x 2  1  x 2   1
10. Prove that: tan 1  1 2
   cos x , 1  x  1
2 2 4 2
 1  x  1  x 
3 4  3 
11. Simplify: cos 1  cos x  sin x  , where  x
5 5  4 4
 sin x  cos x   5
12. Simplify: cos 1   , where 4  x  4
 2 
13. Evaluate the following:
(i) sin 1  sin 5  (ii) sin 1  sin10  (iii) cos 1  cos10  (iv) tan 1  tan  6  
14. Write in simplest form:
 x  1  x 2  1 1
sin 1  ,   x 
 2  2 2

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-93
15. Write in simplest form:
 1  x  1  x 
sin 1   ,0  x  1
 2 
16. Write in simplest form:
 1  x 
sin  2 tan 1 
 1  x 
17. Prove that
(i)   
sec 2 tan 1 2  cosec2 cot 1 3  15  (ii)   
tan 2 sec1 2  cot2 cosec 1 3  11
18. Prove that

  
x2 1
cos  tan 1 sin cot 1 x   2
  x 2
19.    
If sin cot 1  x  1  cos tan 1 x , then find x.
20. Solve the following equation for x:

  

3
(i) cos tan 1 x  sin  cot 1 
4

 
(ii) tan cos 1 x  sin  cot 1 

1
2
 1  x 2  1  x 2  2
21. If tan 1    , then prove that x  sin 2
2 2
 1  x  1  x 
4 1 1
22. If tan 1 x  tan 1 y  , find cot x  cot y
5
 1 1 
23. If sin  sin  cos 1 x   1, then find the value of x
 5 
24. Prove that:
1 1 1 1 
(i) tan 1  tan 1  tan 1  tan 1  (ii) cot 1 7  cot 1 8  cot 1 18  cot 1 3
5 7 3 8 4
25. Simplify the following:
 a cos x  b sin x    a
tan 1   ,   x  , tan x  1
 b cos x  a sin x  2 2 b
1 1  x 1 y yx
26. Prove that: tan  tan 1  sin 1
1 x 1 y 1  x 2 1  y2
 ab  1  1  bc  1  1  ca  1 
27. If a > b > c > 0 prove that cot 1    cot    cot  
 a  b   b  c   ca 
x y x 2 2xy y2
28. If cos 1  cos 1  , prove that 2  cos   2  sin 2 
a b a ab b
1 1 1 2 2 2
29. If cos x  cos y  cos z  , prove that x  y  z  2xyz  1
1  1 2x 1 1  y 
2
 xy
30. Prove that: tan sin  cos 2
 , if |x| < 1, y > 0 &xy< 1
2  1 x 2
1  y  1  xy
31. Solve for x: 2 tan 1  cos x   tan 1  2cosecx 

32. Solve for x: sin 1 1  x   2sin 1 x 
2
  1 a   1 a  2b
33. Prove that: tan   cos 1   tan   cos 1  
4 2 b 4 2 b a

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-94
52
   
2 2
34. If tan 1 x  cot 1 x  , then find x.
8
    cos x 
2 2
35. Find the greatest and least values of sin 1 x 1

36. Solve the following:


(i) sin 1 x   cos 1 x (ii) 4sin 1 x    cos1 x
6
(iii) 5 tan–1 x + 3 cot–1 x = 2 
37. Sum the following series to infinity:
1 1 1
tan–1 2
+ tan–1 2
+ tan–1 + ……..
11 1 1 2  2 1  3  32
38. If ci> 0for i  1, 2,...., n, prove that
c x  y   c  c   cc3  cc2  1 x
tan11 1
tan tan11 2 1 tan
tan tan11 3 2 ..... tan11 tan
.....tan tan11
cc11yy xx 11 cc22cc11 111 cc232cc22  ccnn yy
 cos   cos   1   
39. Prove that: cos 1    2 tan  tan tan 
 1  cos  cos    2 2
      sin  cos  
40. Show that: 2 tan 1  tan tan      tan 1  
 2  4 2   cos   sin  
3 1   3 
41. Prove that:
2 2  2
1
cosec 2  tan 1   sec 2  tan 1         2  2
2 
 
 cos 2 sec 2  cos 2 sec 2 
42. Prove that: tan 1 
 2 
1 2 2

  tan tan      tan       tan 1
1

 ab   a cos   b 
43. Prove that: 2 tan 1  tan   cos1  
 ab 2  a  b cos  

Exercise - 2
Single choice correct with multiple options

  2 3 12 
1. The value of sin–1  cot  sin 1  cos 1  sec1 2   is
  4 4 
  
 
(A) 0 (B) (C) (D) None of these
2 3
2.  
If cosec 1  cosecx  & cosec cosec 1x are equal functions then the maximum range of value of x is

         
(A)   2 , 1  1, 2  (B)   2 , 0    0, 2 
       
(C)  , 1  1,   (D)  1,0   0,1
3. For the equation cos–1 x + cos–1 2x +  = 0, the number of real solution is
(A) 1 (B) 2 (C) 0 (D) 
1 2 1 2
4. The number of real solutions of the equation tan x  3x  2  cos 4x  x  3   is
(A) 1 (B) 2 (C) 0 (D) infinite
1 x 1 x
5. If 22  /sin  2  a  2  2 /sin  8a  0 for at least on real x, then
1  1
(A) a2 (B) a<2 (C) a  R  2 (D) a   0,    2,  
8  8

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-95

6.
1 1 1

If sin x  sin y  sin z  n, then x 4  y 4  z 4  4x 2 y 2 z 2  k x 2 y 2  y 2 z 2  z 2 x 2 , where k is equal to 
(A) 1 (B) 2 (C) 3 (D) none of these
7. If the equation x  bx  cx  1  0,  b  c  has only one real root , then the value of 2 tan 1  cosec  
3 2


 tan 1 2sin  sec 2  is 
 
(A)  (B)  (C) (D) 
2 2

8. The sum of the series sec 1

2  sec 1 10
 sec 1
50
 ....  sec 1
n 2

 1 n 2  2n  2  is
n 
2
3 7 2
 n 1

(A) tan 1 1 (B) tan1 n (C) tan 1  n  1 (D) tan 1  n  1

n
 2m 
9.  tan 1  m 4  m 2  2  is equal to
m 1

 n2  n   n2  n 
(A) tan 1  2  (B) tan 1  2 
 n n 2  n n 2
 n2  n  2 
(C) tan 1  2  (D) none of these
 n n 
10. If cot 1  
cos   tan 1  
cos   x, then sin x is
  
(A) tan 2 (B) cot 2 (C) tan  (D) cot
2 2 2

11. If cot–1 x + cot–1 y + cot–1 z = , x,y,z > 0 &xy< 1 then x + y + z is also equal to
2
1 1 1
(A)   (B) xyz (C) xy + yz + zx (D) none of these
x y z

12. The number of solutions of the equation tan 1 1  x   tan 1 1  x   is
2
(A) 2 (B) 3 (C) 1 (D) 0
13.  1 
  tan 1 2 tan 2   tan 1  tan   then tan  
 3 
(A) –2 (B) –1 (C) 2/3 (D) 2
2
1 x
14. The solution set of the equation sin 1 1  x 2  cos 1 x  cot 1  sin 1 x is
x
(A) [–1, 1] – {0} (0, 1]  {–1}
(B) (C) [–1,0)  {1} (D) [–1, 1]
4 4
 1 x  y
15. If sin 1 x  sin 1 y  , then 2 is equal to
2 x  x 2 y2  y2
1
(A) 1 (B) 2 (C) (D) none of these
2
16. If a sin 1 x  b cos1 x  c, then asin 1 x  bcos1 x is
ab  c  b  a   ab  c  a  b 
(A) 0 (B) (C) (D)
ab 2 ab
17. For 0 << 2, sin–1 (sin ) > cos–1 (sin ) is true when
   3    3   3 
(A)  ,   (B)  ,  (C)  ,  (D)  , 2 
4   2  4 4   4 

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-96
m
18. The number of solution of equation sin 1 x  n sin 1 1  x   , where n > 0, m  0, is
2
(A) 3 (B) 1 (C) 2 (D) none of these
19. If f  x   x  x  x  x  1 &f(sin (sin 8)) = α, where  is a constant, then f(tan–1 (tan 8)) =
11 9 7 3 –1

(A)  (B)  – 2 (C)   2 (D) 2  

    tan x  1  3sin 2x 
20. If x    ,  , then the value of tan 1    tan   is
 2 2   4   5  3cos 2x 
(A) x/2 (B) 2x (C) 3x (D) x
n  2 r 1 
21.  tan 1  1  22r 1  
r 1  
 
(A) tan 1 2n   (B) tan 1 2n 
4
 
(C) tan 1 2n 1  
(D) tan 1 2n 1 4

22. If sin 1 x    cos 1 x   0, where ‘x’ is a non-negative real number and [.] denotes the greatest integer
function, then complete set of values of x is :
(A)  cos1,1 (B)  1,cos1 (C)  sin1,1 (D)  cos1,sin1
23. The value of sin 1  sin12   cos1  cos12  is equal to:
(A) zero (B) 24  2  (C) 4   24 (D) none of these
24. 2
Complete solution set of tan sin x  1 is:  1

 1   1   1 1 
(A)  1,   ,1 (B)  ,   0
 2  2   2 2
(C) (–1,1)  0 (D) none of these
n2
 
2
25. Total number of positive integral value of ‘n’ such that the equations cos 1 x  sin 1 y  and
4
2
sin y 
1 2
 cos 1 x  are consistent, is equal to:
16
(A) 1 (B) 4 (C) 3 (D) 2
x 
26. If tan 1  , x  N , then the maximum value of x is:
 3
(A) 2 (B) 5 (C) 7 (D) none of these
 n  10n  21.6  
2
27. If cot 1    ,n  N, then the minimum value of n is
   6
(A) 1 (B) 2 (C) 3 (D) none of these
1 2x 2x
28. If sin 2
 tan 1 , then the value of x is
1 x 1 x2
 1 1 
(A)  1,1 (B)  1,1 (C)   ,  (D) none of these
 2 2
 4x  1   x 
29. If sin 1  2   2 tan   is independent of x, then value of x is
x 4  2 
(A) (–2, 2) (B)  2,2 (C)  1,1 (D)  1,1
1 6x 
30. If cos 2
   2 tan 1 3x, then find the value of x
1  9x 2
1   1 1   1
(A) 3 ,  (B)  , 
3
(C)  ,  (D)  , 
   3   3

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-97
Exercise - 3
Multiple choice correct with multiple options

1 x  1 1
1. If the equation 22  / cos   a   2 / cos x  a 2  0 has only one real root, then
 2
(A) 1 a  3 (B) a 1 (C) a  3 (D) a 3
2. Indicate the relation which can hold in their respective domain for infinite values of x.
(A) tan |tan–1 x| = |x| (B) cot |cot–1 x| = |x| (C) tan–1 |tan x| = |x| (D) sin |sin–1 x| = |x|
3. Equation 1 + x2 + 2x sin (cos–1 y) = 0 is satisfied by
(A) exactly one value of x (B) exactly two values of x
(C) exactly one value of y (D) exactly two values of y
 n 2  10n  21.6  
4. If cot 1    ,n  N, then n can be
 
  6
(A) 3 (B) 2 (C) 4 (D) 8

5. If Sn  cot 1  3  cot 1  7   cot 1 13  cot 1  21  ...... n terms, then

5  4
(A) S10  tan 1 (B) S  (C) S6  sin 1 (D) S20  cot 1 1.1
6 4 5

 1  1
6. If z  sec 1  x    sec 1  y   , where xy< 0, then the possible values of z is (are)
 x  y

8 7 9 21
(A) (B) (C) (D)
10 10 10 20
 36  4 8
7. Let   sin 1   ,   cos 1   and   tan 1   . Then
 85  5  15 
(A) cot   cot   cot   cot  cot  cot  (B) tan  tan   tan  tan   tan  tan   1
(C) tan   tan   tan   tan  tan  tan  (D) cot  cot   cot  cot   cot  cot   1
8. The value of k (k > 0) such that the length of the longest interval in which the function
f(x) = sin–1 |sin kx| + cos–1 (cos kx) is constant is  / 4 is/are
(A) 8 (B) 4 (C) 12 (D) 16

9.   
2 tan tan 1  x   tan 1 x 3 , where x  R  1,1 , is equal to

2x
(A) (B) tan(2 tan–1 x)
1 x2
(C) tan (cot–1 (–x) – cot–1 (x)) (D) 
tan 2cot 1 x 
10. Which of the following pair of function/functions has same graph?
1  x2
(A)  
y  tan cos 1 x ; y 
x
(B)   1x
y  tan cot 1 x ; y 

y  sin  tan 1 x  ; y  y  cos  tan 1 x  ; y  sin  cot 1 x 

x
(C) (D)
1  x2

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-98
 4x  4x 3   2x  
If   tan 1  ,   2sin 1  and tan  k, then
 1  6x 2  x 4 
11. 2
  1 x  8

 1
(A)      for x  1,  (B)    for x   k, k 
 k
 1
(C)      for x  1,  (D)     0 for x   k, k 
 k

12. If sin 1 x  sin 1 y  and sin 2x  cos 2y, then
2
 1 2 1 2 
(A) x   (B) y  
8 2 64 2 64 12
 1 2 1 2 
(C) x   (D) y  
12 2 64 2 64 8
13.   
If tan 1 x 2  3 x  4  cot 1 4  sin 1 sin14   
2
, then the value of sin–1 sin 2x is

(A) 6  2  (B) 2   6 (C)   3 (D) 3  

14. If cos x + cos y + cos–1 z = , then
–1 –1

(A) x 2  y 2  z 2  2xyz  1

(B)  
2 sin 1 x  sin 1 y  sin 1 z  cos1 x  cos1 y  cos1 z
(C) xy + yz + zx = x + y + z – 1
 1  1  1
(D) x   y   z    6
 x  y  z
2x
15. If 2 tan 1 x  sin 1 is independent of x, then
1  x2
(A) x > 1 (B) x < –1 (C) 0<x<1 (D) –1 < x < 0
 a 2 a3 
16. If sin 1  a 
 3

9   
 ...   cos 1 1  b  b 2  ...  , then
2
 
2a  3 3a  2 3 2
(A) b  (B) b  (C) a  (D) a
3a 2a 2  3b 3  2b
x N1 y N2
17.   
If sin 1 x  sin 1 w sin 1 y  sin 1 z  2 , then D =
z N3 w N4
(N1, N2, N3, N4 N)

(A) has a maximum value of 2 (B) has a minimum value of 0

(C) 16 different D are possible (D) has a minimum value of –2
18. If x < 0, then tan–1 x is equal to
1 x 1 1  x2
(A)   cot 1 (B) sin 1 (C)  cos 1 (D)  cosec1
x 1  x2 1  x2 x
19. If – 1 < x < 0, then cos–1 x is equal to
1
(A) sec 1 (B)   sin 1 1  x 2
x
1  x2 x
(C)   tan 1 (D) cot 1
x 1  x2

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
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1
1   14  
20. Which of the following is/are the value of cos  cos  cos     ?
2   5  
 7    2   3 
(A) cos    (B) sin   (C) cos   (D)  cos  
 5   10   5   5 
21. Which of the following quantities is/are positive?
(A) cos(tan–1 (tan 4)) (B) sin(cot–1 (cot 4))
–1
(C) tan(cos (cos 5)) (D) cot(sin–1 (sin 4))
–1
22. 2 tan (–2) is equal to
 3  3   3  3
(A)  cos 1   (B)   cos 1 (C)   tan 1    (D)   cot 1   
 5  5 2  4  4
23. Which of the following is/are a rational number ?
 1  3
(A) sin  tan 1 3  tan 1  (B) cos   sin 1 
 3 2 4
 1 63   1 5
(C) log 2  sin  sin 1  (D) tan  cos 1 

 4 8   2 3 

24. If ,       are the roots of the equation 6x 2  11x  3  0, then which of the following are real?

(A) cos 1  (B) sin 1 

(C) cosec1  (D) Both cot 1  and cot 1 
 3x 2  1   1  3x 2 
Value of cot 1    cot 1   tan 1 6x is
 x 
25.
 x 
   
1 1 1 1
(A) (B) – (C) (D) –
2 2 3 3

Exercise –4
Section - I : Numerical Value/Subjective Type Questions

1. If the domain of the function f  x   3cos 1  4x    is [a, b], then the value of (4a + 64b) is _____

2. The number of integral value of x satisfying the equation tan–1 (3x) + tan–1 (5x) = tan–1 (7x) + tan–1 (2x)
is___.
3. If range of the functions f  x   sin1 x  2 tan 1 x  x 2  4x  1 is [p, q], then the value of (p + q) is _____.

4.    
The number of solutions of cos  2sin 1 cot tan 1 sec 6cosec 1x

     1  0 where x > 0 is _____.
5. Sum of all integers in the domain of f  x   cot 1  x  3 x  cos1 x 2  3x  1 is _____.

  5 5   k 5
6. sin 2  sin 1  cos 1   is equal to then k = ________.
  3 3   81

7. If the area enclosed by the curves f  x   cos 1  cos x  and g  x   sin 1  cos x  in x 9 / 4,15 / 4 is

92 / b (where a and b are coprime), then the value of b is _______.

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-100

8.  
If the equation sin 1 x 2  x  1  cos 1  x  1 
2
has exactly two solutions for  a,b , then the value

of a + b is _____.
9. If cos–1 (x) + cos–1 (y) + cos–1 (z) = π(sec2 (u) + sec4 (v) + sec6 (w)) where u, v, w are least non-negative
36
angles such that u < v < w, then the value of x2000 + y2002 + z2004 + is _____.
uvw
10. The least value of (1 + sec–1 x) (1 + cos–1 x) is _____.
 3  3 6
11. If tan 1  x    tan 1  x    tan 1 , then the value of x 4 is ____.
 x  x x
12. If 0 < cos–1 x < 1 and 1 + sin(cos–1 x) + sin2(cos–1 x) + sin3 (cos–1 x) + ….   2, then the value of 12x2 is
_____.

13.   
Number of solutions of equation sin cos 1 tan sec1 x   1  x is/are _____.

14. Let cos–1(x) + cos–1(2x) + cos–1 (3x) be . If x satisfies the equation ax3 + bx2 + c = 0, then the value of
(b + a + c) is ______ .
15. If the roots of the equation x3 – 10x + 11 = 0 are u, v & w, then the value of
3 cosec2 (tan–1 u + tan–1 v + tan–1w) is _______.

Section - II : Match the Column

16. Column-I Column -II

4 1
a. sin 1  2 tan 1  p. /6
5 3
12 4 63
b. sin 1  cos 1  tan 1  q. /2
13 5 16
x 3  2x   
c. If A  tan 1 , B  tan 1   then the value of A – B is r. /4
2  x   3 
1 1
d. tan 1  2 tan 1  s. 
7 3

17. Match the following lists:

List-I List-II
x
a. Number of roots of the equation sin 1  sin x   is p. 2
4
x
b. Number of roots of the equation cos 1  cos x   is q. 3
4
x
c. Number of roots of the equation tan  tan x   is
1
r. 4
4
x
d. Number of roots of the equation cot 1  cot x   is s. 5
4
Codes
a b c d
(A) s r q p
(B) q s r p
(C) s r p q
(D) q r q p

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-101
18. List-I List-II
2
    sin y 
2 2 3 3
a. If sin 1 x 1
 , then x  y can be p. 0
2
 cos x    cos y   2 , then x  y can be
1 2 1 2 5 5 2
b. q. –2
4

 sin x   cos y  
1 2 1 2
c. then x – y can be r. 2
4
d. sin 1 x  sin 1 y   then x – y can be s. –1
Codes
a b c d
(A) r q p s
(B) s r q p
(C) q s p r
(D) s r q p

19. Column-I Column-II

   
a. Range of f  x   sin 1 x  cos 1 x  cot 1 x is p.  0, 2    2 ,  
   
  3 
b. Range of f  x   cot 1 x  tan 1 x  cosec 1 x is q. 2, 2 
 
c. Range of f  x   cot 1 x  tan 1 x  cos 1 x is r. 0,
 3 5 
d. Range of f  x   sec 1 x  cosec 1 x  sin 1 x is s. 4 , 4
 

Section - III : Comprehension

Comprehension # 1 (20-22)
For x, y, z, t  R, sin–1 x + cos–1 y + sec–1 z  t2 – 2 t  3
20. The value of x + y + z is equal to
(A) 1 (B) 0 (C) 2 (D) –1
21. The principal value of cos–1 (cos 5t2) is
3   2
(A) (B) (C) (D)
2 2 3 3
22. The value of cos–1(min {x, y, z}) is
 
(A) 0 (B) (C)  (D)
2 3
Comprehension # 2 (23-25)
 
ax  b sec tan 1 x    c and ay  b  sec  tan y    c
1

23. The value of xy is

2ab c2  b2 c2  b2
(A) 2 (B) (C) (D) none of these
a  b2 a 2  b2 a 2  b2
24. The value of x + y is
2ac c2  b2 c2  b2
(A) 2 (B) (C) (D) none of these
a  b2 a 2  b2 a 2  b2
xy
25. The value of is
1  xy
2ab 2ac c2  b2
(A) (B) (C) (D) none of these
a  c2
2
a  c2
2
a 2  b2

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-102
Comprehension # 3 (26-28)
 
Let cos 1 4x 3  3x  a  b cos1 x
 1
26. If x   1,   , then the value of a  b  is
 2
(A) 2  (B) 3 (C)  (D) –2
 1 1  a
27. If x    ,  , then the principal value of sin 1  sin  is
 2 2  b
   
(A)  (B) (C)  (D)
3 3 6 6
1 
28. If x   ,1 , then the value of lim b cos  y  is
2  y a

1
(A) –1/3 (B) –3 (C) (D) 3
3

Exercise –5
Revision exercise (Moderate to Tough)

1. Let f  x   sin x  cos x  tan x  sin 1 x  cos 1 x  tan 1 x. Then find the maximum & minimum value of
f(x)
2. Find the sum cosec1 10  cosec1 50  cosec1 170  ... n terms
 
3. If tan–1 y = 4 tan–1 x  x  tan  , find y as an algebraic function of x, and hence prove that tan  / 8 is a
 8
root of the equation x 4  6x 2  1  0
1 1 y 3
4. Find the number of positive integral solutions of the equation tan x  cos  sin 1
1 y 2 10
 
5. Solve the equation sin 1 cos x  cos 1 sin x  sin 1 cos x  cos 1 sin x ,  x
2 2

    cos x 
3 3
6. Find ‘a’ so that the equation sin 1 x 1
 a3 has a solution

7. Find the range of f  x   cot 1  2x  x  2

 1  x2 
8. Find the domain of f  x   sin 1  
 2x 
9. 
Solve 2cos 1 x  sin 1 2x 1  x 2 
3 16 1 7
10. Find the value of 2cos 1  cos 1  cos 1
3 63 2 25
x 1  
11. Find the set of value of x for which the equation cos 1 x  cos 1   3  3x 2   holds good
2 2  3
x x
12. Solve sec 1  sec 1  sec1 b  sec 1 a, a  1, b  1, a  b
a b
ax bx
13. If a 2  b2  c2 ,c  0, then find the non-zero solution of the equation sin 1  sin 1  sin 1 x
c c
1 1 4x
14. Find x for 2 tan 2x  sin
1  4x 2

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-103
 4x  1  x 
15. If sin 1  2   2 tan    is independent of x, find the value of x.
x 4  2

16.   1 3
If A  2 tan 1 2 2  1 & B  3sin 1  sin 1 , then which is greater?
3 5
17. Find the value
n 1  k  1 k  k  1 k  2 
lim
n 
 cos 1 
 k  k  1 
k 2  
 
18.
 2 
7
If x   0,  , then show that cos 1  1  cos 2x  
2
sin 2
 
x  48cos 2 x  sin x   x  cos 1  7 cos x 

 x   1  3
19. Find the number of solutions of the equation tan 1  2 
 tan 1  3  
1 x  x  4
  2  2r  1 
20. Find the sum  tan 1  
r 1 
 
4  r 2 r 2  2r  1  


21.  
If equation sin 1 4sin 2   sin   cos 1  6sin   1 
2
has 10 solutions for  0,n , then find the

minimum value of n.

Exercise –6
Section – I : JEE (Advanced)Previous Years

1
1. Find the value of cos (2 cos 1 x  sin 1 x) at x  , where 0  cos 1 x   and   / 2  sin 1 x   / 2.
5
[IIT-JEE 1981, 2M]
 2 
2. The principal value of sin 1  sin  is [IIT-JEE 1986, 2M]
 3 
2 2  5
(A)  (B) (C) (D)
3 3 3 3

3. The number of real solutions of tan 1 x(x  1)  sin 1 x 2  x  1  is [JEE (Advanced) 1999, 2M]
2
(A) zero (B) one (C) two (D) infinite

 x2 x3   x4 x6  
4. If sin 1  x    .....   cos 1  x 2    ....  = , for 0  x  2, then x equals
 2 4   2 4  2
[IIT-JEE 2001, 1M]
(A) 1/2 (B) 1 (C) –1/2 (D) –1

1 1 x2 1
5. Prove that cos tan [sin(sin cot x)]  [IIT-JEE 2002, 5M]
x2  2
6. The value of x for which sin [cot 1 (1  x)]  cos(tan 1 x) is [IIT-JEE 2004, 1M]

1 1
(A) (B) 1 (C) 0 (D) 
2 2

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-104

7. Let (x, y) be such that sin 1 (ax)  cos 1 (y)  cos 1 (bxy)  [IIT-JEE 2007]
2
Column I Column II
A If a = 1 and b = 0, then (x, y ) p. lies on the circle x 2  y 2  1

B. If a = 1 and b = 1, then (x, y) q. lies on (x 2  1)(y 2  1)  0

C. If a = 1 and b =2, then (x, y) r. lies on y = x

D. If a = 2 and b = 2, then (x, y) s. lies on (4x 2  1)(y 2  1)  0
1/ 2

 
1  x 2  x cos  cot 1 x   sin  cot 1 x   1
2
8. If 0  x  1, then is equal to [IIT-JEE 2008, 3M]
 
x
(A) (B) x (C) x 1 x2 (D) 1 x2
1 x 2

9. f : [0, 4  ]  [0,  ] be defined by f (x)  cos 1 (cos x). Then, the number of points x  [0, 4  ] satisfying
10  x
the equation f (x)  , is [JEE (Advanced) 2014]
10
6 1  4 
10. If   3sin 1   and   3cos   , where the inverse trigonometric functions take only the principal
 11  9
values, then the correct option (s) is/are [JEE (Advanced) 2015]
(A) cos   0 (B) sin   0 (C) cos(  )  0 (D) cos   0
x    x  
11. Let E1  {x  R : x  1and  0} and E 2   x  E1 : sin 1  log e    is a real number  (Here,
x 1    x 1   
  
the inverse trigonometric function sin 1 x assumes values in   ,  . Let f : E1  R be the function
 2 2
 x  1   x 
defined by (x) = loge   and g : E 2  R be the function defined by g (x) = sin  log e  x  1  
 x 1   
[JEE (Advanced) 2018]
List I List II
P The range of f is 1.  1   e 
 ,  ,
 1 e   e 1 
Q. The range of g contains 2. (0, 1)
R. The domain of f contains 3.  1 1
 , 
 2 2
S. The domain of g is 4. (  , 0)  (0,  )
5.  e 
 , 
 e 1 
6. 1 e 
(, 0)   , 
 2 e 1 

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-105
  i1 
x 
i

The number of real solutions of the equation sin   x  x    

1
12.
 i 1 i 1  2  


    x i    1 1
=  cos        (x)  lying in the interval   ,  is ……………….
1 i

2  i 1  2  i 1   2 2
 
  
(Here, the inverse trigonometric functions sin 1 x and cos 1 x assume values in   ,  and [0,  ],
 2 2
respectively.) [JEE (Advanced) 2018]
13. Match List I with List II and select the correct answer using the code given below the lists.
List-I List-II
1/2
 1  cos(tan 1 y)  y sin(tan 1 y) 2  1 5
P  2   y 4
 takes value 1.
 y  cot(sin 1 y)  tan(sin 1 y)   2 3
 
If cos x  cos y  cos z  0  sin x  sin y  sin z, then 2 possible
Q. xy 2 2
value of code is
2
 
If cos   x  cos 2x + sin x sin 2x sec x = cos x sin 2x sec x + cos
4  1
R. 3.
  2
  x  cos 2x, then possible value of sec x is
4 

S.

If cot sin
1
  
1  x 2  sin  tan 1 x 6  . X = 0. Then possible

value of x is
Codes
P Q R S
(A) 4 3 1 2
(B) 4 3 2 1
(C) 3 4 2 1
(D) 3 4 1 2
14. Considering only the principal values of the inverse trigonometric functions, the value of
3 2 1 2 2 2
cos 1  sin 1  tan 1 is __________ . [JEE (Advanced) 2022; P-1]
2 2 2
4 2 2


Section – II : JEE (Main)Previous Years

 1 4 1  2 
1. The value of tan cos    tan    is [AIEEE 1983]
 5  3 
6 17 16
(1) (2) (3) (4) none of these
17 6 7
2. If x, y and z are in AP and tan 1 x, tan 1 y and tan 1 z are also in AP, then [AIEEE 2013]
(1) xyz (2) 2x  3y  6z (3) 6x  3y  2z (4) 6x  4 y  3z

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-106
 19
1 
n 
3. The value of cot 
   cot 1  2p   is [JEE (Main) 10 Jan 2019]
 n 1  p1  
23 21 19 22
(1) (2) (3) (4)
22 19 21 23
4. Considering only the principal values of inverse functions, the set
 
A=  x  0 : tan 1 2x  tan 1 (3x)   [JEE (Main) 2019, 12 Jan I]
 4
(1) is an empty set (2) is a singleton
(3) contains more than two elements (4) contains two elements
3 1 
5. If   cos 1   ,   tan 1   , where 0  ,   , then    is equal to
5  3 2
[JEE (Main) 2019, 8 April II]
 9   9  1  9  1  9 
(1) tan 1   cos1 
(2)  (3) tan  14  (4) sin  
 5 10   5 10     5 10 
6. The Let f (x) = loge (sin x), (0 < x <  ) and g (x) = sin –1 (e –x), (x  0 ) if  is a positive real number
such that a = (fog)′ (  ) and b = (fog) (  ), then [JEE (Main) 10 April 2019]
(1) a 2  b  a  0 (2) a 2  b   a  1
(3) a 2  b  a   2 2 (4) a 2  b   a  0
y y
7. If cos 1 x  cos 1  , where 1  x  1,  2  y  2, x  , then for all x, y, 4x 2  4xy cos   y 2
2 2
is equal to [JEE (Main) 2019, 10 April II]
(1) 2 sin 2  (2) 4 cos   2x y 2
2 2

(3) 4 sin 2  (4) 4 sin 2   2x 2 y 2

8. If x  sin 1 (sin10) and y  cos 1 (cos10), then y  x is equal to [JEE (Main) 2019, 10 April II]
(1) 0 (2) 10 (3) 7π (4) π
 2x  1
9. If tan 1 y  tan 1 x  tan 1  2 
, where x  Then, the value of y is
 1 x  3
[JEE (Main) 2019, 10 April II]
3x  x 3
3x  x 2
3x  x 3
3x  x 3
(1) (2) (3) (4)
1  3x 2 1  3x 2 1  3x 2 1  3x 2
 23  n

10. The value of cot  cot 1  1   2k   is [JEE (Main) 2019, 10 April II]
 n 1  r 1  
23 25 23 24
(1) (2) (3) (4)
25 23 24 23
 12  1  3 
11. The value of sin 1    sin   is equal to [JEE (Main) 2019, 12 April I]
 13  5
 63    56    9   33 
(1)   sin 1   (2)  sin 1   (3)  cos 1   (4)   cos 1  
 65  2  65  2  65   65 
12. Let f (x) = (sin (tan–1x) + sin (cot–1 x))2 –1, | x | > 1. If
dy 1 d

dx 2 dx

sin –1  f  x   and y   3   6 , then y   3  is equal to: [JEE (Main) 2020, 8 Jan, M]

5   2
(1) (2) (3) (4)
6 6 3 3

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-107
3 4  dy

6
13. If y = k cos –1  cos kx  sin kx  , then
k 1
at x  0 is [JEE (Main) 2020, 2 Sep, E]
5 5  dx
 4 5 16 
14. 2 –  sin –1  sin –1  sin –1  is equal to: [JEE (Main) 2020, 3 Sep, M]
 5 13 65 
 7 3 5
(1) (2) (3) (4)
2 4 2 4
15. If S is the sum of the first 10 terms of the series [JEE (Main) 2020, 5 Sep, M]
1 1 1  1 
tan 1    tan 1    tan 1    tan –1    ……… then tan (S) is equal to:
 3 7  13   21 
6 5 10 5
(1)  (2) (3) (4)
5 11 11 6
 n  1 
16. lim x  tan   r 1 tan 1  2 
is equal to [JEE (Main) 2021, 24 Feb, Shift-1]
  1  r  r 
1 63 
17. A possible value of tan  sin –1  is: [JEE (Main) 2021, 24 Feb, Shift-2]
4 8 
1 1
(1) (2) (3) 7 1 (4) 2 2  1
2 2 7
  4 
18. cos ec  2 cot 1 (5)  cos –1    is equal to: [JEE (Main) 2021, 25 Feb, Shift-2]
  5 
75 65 56 65
(1) (2) (3) (4)
56 56 33 33
2
x x2
19. Let f ( x )  sin  1 x and g ( x )  2 . If g (2)  lim x2 g ( x), then the domain of the function fog
2x  x  6
is : [JEE (Main) 26 Feb 2021; Shift-2]
 4 
(1) (, 2]    ,   (2) (  , 1]   2,  
 3 
 3 
(3) (  , 2]   1,   (4) (, 2]    ,  
 2 
 6x 
Let Sk   r 1 tan 1  2x 1 2x 1  . Then limk Sk equal to:
k
20. [JEE (Main) 2021, 16 March, Shift-1]
2 3 
 3   3
(1) tan–1   (2) (3) cot–1   (4) tan–1(3)
 2 2  2
21. Given that the inverse trigonometric functions take principal values only. Then, the number of real values
 3x   4x 
of x which satisfy sin–1    sin 1    sin 1 x is equal to: [JEE (Main) 2021, 16 March, Shift-2]
 5   5 
(1) 2 (2) 1 (3) 3 (4) 0
22. If cot () = cot 2 + cot 8 + cot 18 + cot 32 + …….. upto 100 terms, then  is :
–1 –1 –1 –1 –1

[JEE (Main) 2021, 17 March, Shift-1]

(1) 1.01 (2) 1 (3) 1.02 (4) 1.03
  15  
 cos  4   1 
23. The value of tan–1     is equal to [JEE (Main) 2022, 25 June, Shift-2]
   
 sin   
 4 
  5 4
(1)  (2)  (3)  (4) 
4 8 12 9

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-108
 2   7   3 
24. sin 1  sin   cos –1  cos   tan –1  tan  is equal to: [JEE (Main) 2022, 27 June, Shift-1]
 3   6   4 
1 17 31 3
(1) (2) (3) (4) 
12 12 12 4
 50  1  
25. The value of cot   tan 1  2 
is [JEE (Main) 2022, 27 June, Shift-2]
 n 1  1  n  n 
26 25 50 52
(1) (2) (3) (4)
25 26 51 51
  1 
26.

1
50 tan  3tan 1    2cos 1 
2  5 

1

1 

  4 2 tan  tan 2 2  is [JEE (Main) 2022, 29 June, Shift-1]
2 
27. S = {  R : y2 = 1 – x}, then 16
S
3
is equal to ……… [JEE (Main) 2022, 25 July, Shift-2]

 1 5 1
28. tan  2 tan 1  sec 1  tan 1  is equal to: [JEE (Main) 2022, 26 July, Shift-1]
 5 2 8
1 5
(1) 1 (2) 2 (3) (4)
4 4
29. Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the
equation cos 1 (x)  2sin 1 (x)  cos 1 (2x) is equal to: [JEE (Main) 2022, 28 July, Shift-1]
 
(1) 0 (2) 1 (3) (4) 
2 2
 1 3   84 3 
30. tan 1  1
  sec   is equal to [JEE (Main) 2023, 24 Jan., Shift-1]
3 3   63 3 
 
   
(1) (2) (3) (4)
4 2 3 6
31. Let a1 = 1, a2, a3, a4, ………. be consecutive natural numbers. [JEE (Main) 2023, 30 Jan., Shift-2]
 1   1   1 
Then tan–1  –1
 + tan 
–1
 +……+ tan   is equal to
 1  a1a 2   1  a1a 2   1  a 2021a 2022 
   
(1)  cot 1 (2022) (2) cot 1 (2022)  (3) tan 1 (2022)  (4)  tan 1 (2022)
4 4 4 4
32. If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then
the sum of common ratios of all such GPs is [JEE (Main) 2023, 31 Jan., Shift-1]
9
(1) 7 (2) (3) 3 (4) 14
2
 1 1
33. Let S be the set of all solutions of the equation cos 1 (2x)  2cos 1 ( 1  x 2 )  , x    ,  . Then
 2 2
 2sin (x  1) is equal to
1 2
[JEE (Main) 2023, 01 Feb., Shift-1]
xS

2  3  3
(1) 0 (2) (3)   sin 1   (4)   2sin 1  
3  4   4 
  x 1   x   
34. If S =  x   : sin 1  1
  sin     then
 2 2 4 
  x  2x  2   x 1 
 
 
 
  

 xS  sin  x 2  x  5   cos x 2  x  5   is equal to _____. [JEE (Main) 2023, 13 Apr, Shift-1]
2 
35. For x  ( 1,1], the number of solutions of the equation sin–1 x = 2 tan–1 x is equal to
[JEE (Main) 2023, 13 Apr, Shift-2]

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-109

ANSWER KEY
Exercise – 1
  5   5
1. (i)  (ii)  (iii) (iv)  (v) (vi) (vii) 
3 6 12 3 6 6
3 2
(viii) (ix) 0 (x)
4 3
 1 1  1 1
2. (i) x 1,2 (ii) x    ,  (iii) x    , 
 2 2  2 2

(iv) x    5,  3    3, 5  (v) x   , 1  1, 

(vi) x  R  n : n  z

3. 4 4. 3 5. n=4 6. 3
 x  4 
7. (i)  (ii) x 11. x  tan 1 12. x
4 2 4 3 4
13. (i) 5 – 2 (ii) 3 – 10 (iii) 4 – 10 (iv) 2 – 6
  1 1
14.  sin 1 x 15.  cos1 x 16. 1  x2 19. x=–
4 4 2 2

3 5
20. (i) x=  (ii) x= 
4 3
1 a 
22. /5 23. x 25. tan 1 x 31. x  n  , n  z
5 b 4
1
32. x 34. x = –1
2
5 2
35. Greatest value =
4

2
Least value =
8


36. (i) x  3 / 2 (ii) x  1 / 2 (iii) x = 1 37.
4

Exercise – 2

1. (A) 2. (A) 3. (C) 4. (C) 5. (D) 6. (B) 7. (A)

8. (B) 9. (A) 10. (A) 11. (B) 12. (C) 13. (A) 14. (C)

15. (B) 16. (D) 17. (C) 18. (D) 19. (D) 20. (D) 21. (B)

22. (D) 23. (A) 24. (A) 25. (A) 26. (B) 27. (C) 28. (A)

29. (B) 30. (C)

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888
 M-110
Exercise – 3
1. (BC) 2. (ABCD) 3. (AC) 4. (AC) 5. (ABD) 6. (CD) 7. (AB)
8. (B) 9. (ABC) 10. (ABCD) 11. (AB) 12. (AD) 13. (AB) 14. (AB)
15. (AB) 16. (AC) 17. (ACD) 18. (ABC) 19. (ABCD) 20. (BCD) 21. (ABC)
22. (ABC) 23. (ABC) 24. (BCD) 25. (CD)

Exercise – 4
Section-I
1. 7 2. 1 3. 4 4. 3 5. –3 6. 8 7. 8
8. 1 9. 9 10. 25 11. 9 12. 9 13. 0 14. 3
15. 6
Section-II
16. a  q; b  s; c  p; d  r 17. (D) 18. (C)
19. a  s; b  p; c  q; d  r
Section-III
20. (D) 21. (B) 22. (C) 23. (B) 24. (A) 25. (B) 26. (C)
27. (A) 28. (D)

Exercise – 5

1. Minimum value:  cos1  sin1  tan1
4
3
Maximum value:  cos1  sin1  tan1
4

4x 1  x 2  
2. tan 1
 n  1   / 4 3. y= 4 2
4. 2 5. 
x  6x  1 2
 1 7    1 
6. a ,  7.  4 ,   8. 1,1 9. x ,1 10. 
 32 8     2 
1  1 1
11. x   ,1 12. x = ab 13. x  1 14.  x 15. x  2,2
2  2 2
 
16. A>B 17. 19. No solution 20. 21. 9
6 2

Exercise – 6
Section-I
2 6
1. 2. (C) 3. (C) 4. (B) 6. (D)
5
7. (A) → (p); (B) → (q); (C) → (p); (D) → (s) 8. (C) 9. (3) 10. (BCD)
11. (P) → (4); (Q) → (2); (R) → (1); (S) → (1) 12. 2 13. (B) 14. (2.36)
Section-II
1. (2) 2. (1) 3. (2) 4. (2) 5. (4) 6. (2) 7. (3)
8. (4) 9. (1) 10. (2) 11. (2) 12. (1) 13. 91 14. (3)
15. (4) 16. (1) 17. (2) 18. (2) 19. (1) 20. (3) 21. (3)
22. (1) 23. (2) 24. (1) 25. (1) 26. 29 27. 130 28. (2)
29. (1) 30. (3) 31. (3) 32. (1) 33. (2) 34. (4) 35. (2)

Mathematics – Inverse Trigonometry Functions Toll Free Number : 1800 103 9888