08-01-24_JEE-MAIN_Q’P

PHYSICS MAX.MARKS: 100

SECTION – I
(SINGLE CORRECT ANSWER TYPE)
This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its answer,
out of which ONLY ONE option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -1 if not correct.
1. A wire frame in the shape of an equilateral triangle is hinged at one vertex so that it can
swing freely in a vertical plane, with the plane of the triangle always remaining vertical.
1
The side of the frame is m . The time period in seconds of small oscillations of the
3

frame will be

  
A) B)  2 C) D)
2 6 5

2. The dimensions of angular momentum are

A) M 0 L1T 0 B) M 0 L2T 0 C) M 1LT 2 D) M 1L2T 1

3. There are two identical small holes on the opposite sides of a tank containing liquid. The
tank is open at the top. The difference in height between the two holes is h. As the liquid
comes out of the two holes, the tank will experience a net horizontal force proportional
to.

3
A) h B) h C) h 2
D) h 2

4. One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to
a mass less spring of force constant K. A mass m hangs freely from the free end of the
spring. The area of cross-section and Young’s modulus of the wire are A and Y
respectively. If the mass is slightly pulled down and released, it will oscillate with a time
period T equal to

m YA  KL  mYA mL
A) 2 m / K B) 2 C) 2 D) 2
YAK KL YA
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5. Internal energy of the gas as it expands according to the graph AB which is a
recatangular hyperbola

A) Increasing continuously B) decreasing continuously

C) Always constant D) Initially Increasing then decreasing.

6. There are four concentric shells A,B,C and D of radii a,2a,3a and 4a respectively. Shells B
& D are given charges +q & -q respectively. Shell C is now earthed. The potential
1
difference VA  VC is___(take K)
4 0

Kq Kq Kq Kq
A) B) C) D)
6a 2a 3a 4a

7. Two Identical discs initially at rest are in contact on a table. A third disc of same mass
but of double radius strikes them symmetrically and itself comes to rest after impact.
The co-efficient of restitution is:

9 3 1 1
A) B) C) D)
16 4 2 16

8. Find out the value of current through 3 resistance for the given circuit

A) 10 amp B) 6 amp C) 4 amp D) zero

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9. One plate of a capacitor is connected to a spring as shown in figure. Area of both the
plates is A. In steady state separation between the plates is 0.8d (spring was unstretched
and the distance between the plates was d when the capacitor was uncharged). The force
constant of the spring is approximately

4 0 AE 2 2 0 AE 6 0 E  0 AE 3
A) B) C) D)
d3 d2 Ad 3 2d 3
10. The current through 3 mH inductor in steady state after closing switch S is

1 2 3
A) ampere B) ampere C) 1 ampere D) ampere
3 3 2
11. STATEMENT-1 : Time constants of the circuits shown in the figure are same.

AND
STATEMENT-2 : Instantaneous current through the capacitor branch is same at any
instant for both the circuits, if batteries are inserted in the circuits at t=0.
A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for
Statement-1
B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation
for Statement-1
C) Statement-1 is True, Statement-2 is False
D) Statement-1 is False, Statement-2 is True
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12. A small sphere A of mass m and radius r rolls without slipping inside a large fixed
hemispherical bowl of radius R(>> r) as shown in figure. If the sphere starts from rest at
the top point of the hemisphere. Find the normal force exerted by the small sphere on the
hemisphere when it is at the bottom B of the hemisphere.

10 17 5 7
A) mg B) mg C) mg D) mg
7 7 7 5

7
13. Refractive index of a prism is and the angle of prism is 600 . The minimum angle of
3
incidence of a ray that will be transmitted through the prism is ______
A) 300 B) 400 C) 600 D) 900
14. The radius of the conducting loop shown in figure is R. Magnetic field is decreasing at a
constant rate  . Resistance per unit length of the loop is  . Then current in wire AB is (
AB is one of the diameters)

R R 2R
A) from Ato B B) from B to A C) from Ato B D) zero
2 2 

15. In YDSE, coherent monochromatic light having wavelength 600 nm has fallen on slits.
First order bright fringe is at 4.84 mm from central maxima. Determine the wavelength
for which the first order dark fringe will be observed at same location on screen ? Take
D  3m

A) 600 nm B) 1200 nm C) 300 nm D) 900 nm

16. Two capacitors having capacitance C1 & C2 are connected in series and a potential
difference V is applied across them. Then:
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V1 , V2 & U1 , U 2 be the potentials drop & energy store in C1 & C2 respectively.

Match the entries in Column-I with Column-II :

Column-I Column-II
a) V1  V2 p) C1  C2

b) U1  U 2 q) C1  C2

c) V1 r) C1V /  C1  C2 

d) V2 s) C2V /  C1  C2 

A)  a  q  ;  b  q  ;  c  s  ;  d  r  B)  a  r  ;  b  q  ;  c  p  ;  d  s 

C)  a  q  ;  b  p  ;  c  r  ;  d  s  D)  a  s  ;  b  r  ;  c  p  ;  d  q 
17. A changing electric field produces magnetic field. The direction of this magnetic field is
A) In the direction of electric field
B) In the direction opposite to the electric field
C) Perpendicular to the direction of electric field
D) Independent of the direction of electric field
18. A photon collides with a stationary hydrogen atom in ground state ineleastically. Energy
of the colliding photon is 10.2 eV. After a time interval of the order of microsecond
another photon collides with same hydrogen atom inelastically with an energy of 15 eV.
What will be observed by the detector?
A) 2 photons of energy 10.2 eV
B) 2 photons of energy 1.4 eV
C) one photon of energy 0.2 eV and an electron of energy 1.4 eV
D) one photon of energy 10.2 eV and an electron of energy 1.4 eV
19. Two short bar magnets of magnetic moment M each are placed at a distance 2 d apart.
The magnetic field. Midway between them at P is

 0 3M 0 M 5 0 2M 0 M
A) B) C) D)
4 d 3 4 d 3 4 d 3 4 d 3
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20. Graph shows a hypothetical speed distribution for a ample of N gas particles (for
dN
V  V0 ,  0 ). If Vave and Vrms are the average speed and rms speed of the gas
dV
molecules, then

V0 V0
A) Vave  B) Vrms  C) Vave : Vrms  3: 2 D) Vave : Vrms  3 : 2
3 2

SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the Answer in the
decimals , Mark nearest Integer only. Have to Answer any 5 only out of 10 questions and question will be
evaluated according to the following marking scheme:
Marking scheme: +4 for correct answer, -1 in all other cases.
21. A half section of thin uniform pipe of mass m and radius r is released from rest. Pipe
nmgR
rolls without slipping. The change in PE of pipe when it has rolled through 900 is .

Then the value of n is_

22. The system is pushed by a force F as shown in figure. All surfaces are smooth except
between B and C friction coefficient between B and C is  .Minimum value of F to
 n 
prevent block B from downward slipping is   mg . Then the value of n is _____.
 2 
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23. A uniform stick of length l and mass m lies on a smooth table. It rotates with angular
velocity  about an axis perpendicular to the table and though one end of the stick. The
ml 2
angular momentum of the stick about the end is .Find the value of n_______.
n

24. A uniform ring of mass m is lying at a distance 3a from centre of a sphere of mass M
just over the sphere where a is the radius of ring as well as that of sphere. Then,
3GMm
gravitational force exerted is then the value of n is ____
na 2

25. Two tuning forks A & B when sounded together produces 4 beats/s. If B is loaded with
wax then also beat frequency remains same. Frequency of A is 242 Hz, find frequency
of B?
26. The relative density of ice is 0.9 and that of sea water is 1.125. The fraction of the whole
1
volume of an iceberg appears above the surface of the sea is then the value of x is ___
x
27. An electric current is flowing through a circular coil of radius R. The ratio of the
magnetic field at the centre of the coil and that at a distance 2 2 R from the centre of the
coil and on its axis is

28. When a metallic surface is illuminated by a light of frequency 8 1014 Hz , photoelectron of

maximum energy 0.5 eV is emitted. When the same surface is illuminated by light of
frequency12 1014 Hz , photoelectron of maximum energy 2 eV is emitted. The work
10
function is  ev  .Then the value of n _____
n
29. An unstable heavy nucleus at rest breaks in to two nuclei which moves away with
velocities in the ratio of 8:27. The ratio of the radii of the nuclei (assume to be spherical)
is x. Then y = 2x is______
30. In a meter bridge experiment null point is obtained at 20 cm from one end of the wire
when resistance ‘x’ is balanced against another resistance y. If x  y , then where will be
the new position of the null point from the same end, if one decides to balance a
resistance of 4 x against ' y ' in ___cm.
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CHEMISTRY MAX.MARKS: 100
SECTION – I
(SINGLE CORRECT ANSWER TYPE)
This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its answer,
out of which ONLY ONE option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -1 if not correct.
31. Which of the following is not the name of an element with atomic number 104?

A) Nobelium B) Rutherfordium C) Kurchatovium D) Unnilqauadium

32. During halogen test, sodium fusion is boiled with con.HNO3 to

A) remove unreacted sodium

B) decompose cyanide or sulphide of sodium

C) extract halogen from organic compound

D) Maintain the p H of extract

33. In the ionization of N 2 to N 2 , the electron is lost from a

* *
A)   orbital B)   orbial C)   orbial D)   orbial

34. A certain substance ‘A’ tetramerises in water to the extent of 80%. A solution of 2.5g of
‘A’ in 100g of water lowers the freezing point by 0.30 C . The m.wt of ‘A’ is
K f  1.86 k.kg.mol 1 
A) 122 B) 31 C) 344 D) 62
35. Which technique among the following is most appropriate in separation of a mixture of
100 mg of
p-nitrophenol and picric acid?
A) Steam distillation B) Distillation under reduced pressure.
C) Sublimation D) Thin layer chromatography.
36. The equilibrium constants K p and K p for the reactions X  2Y and Z  P  Q
1 2

respectively are in the ratio of 1: 9 . If the degree of dissociation of X and Z be equal then
the ratio of total pressure at these equilibria is
A) 1: 36 B) 1:1 C) 1: 3 D) 1: 9
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37. Transition metal complex with the highest value of crystal field splitting energy   0 
will be
3 3 3 3
A) Cr  H 2O6   B)  Mo  H 2O6  C)  Fe  H 2O6   D) Os  H 2O6 

38. The major product of the reaction is

I I

Br SPh PhS F

Cl Cl

A) B) NO2 C) D) NO2

39. When an aldehyde is heated with Fehilings solution, a reddish brown precipitate is
formed which is

A) CuO B) Cu C) Cu2O D) Cu  C  C  Cu

40. Which of the following cannot be prepared by using willamson’s synthesis?

A) Methoxy benzene B) Benzyl p-nitro phenyl

ether

C) t-butyl methyl ether D) Ditertiarybutyl ether

41. Among the following the metal with the highest melting point will be

A) Hg B) Ag C) Ga D) Cs

42. The element of group 15 which can form a strong bond with hydrogen is

A) Nitrogen B) Phosphorous C) Arsenic D) Antimony

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43. Which of the following is not a mineral of fluorine?

A) Fluorspar B) Cryolite C) Fluoroapetite D) Carnallite

44. Consider the following statements:

According to Werner’s theory

i) Secondary valencies are non-ionisable and directional

ii) Secondary valencies are satisfied by neutral molecules or negative ions .

iii) Ligands form coordinate bonds with metal ions or atoms.

Iv) The charge on the complex ion is always equal to the oxidation state of the metal
atom.

A) all are correct B) (i), (ii) and (iii) are correct

C) (i), (ii) and (iv) are correct D) (iii) and (iv) are correct

45. Zinc and mercury do not show variable valency like other d-block elements because

A) They are soft

B) Their (n-1) d-shells are completely filled

C) They have only two electrons in the outermost shell

D) Their d-shells are incompletely filled.

46. Assertion (A): The purple colour of KMnO 4 is due to charge transfer transition

Reason (R): The intense colour, in most of the transition metal complexes is due to d – d
transition.

A) Both A and R are true and R is the correct explanation of A

B) Both A and R are true but R is not the correct explanation of A

C) A is true but R is false

D) A is false but R is true

47. Which of the following doesn’t undergo Friedal-craft’s reaction?

A) Xylene B) Nitrobenzene C) Cumene D) Toluene

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48. Match the transition element ions given in Column I with the characteristic (s) of
products given in Column II.
Column –I Column – II
a) Cu 2 P) Form amphoteric oxide
b) Zn 2 Q) Diamagnetic and colourless compounds
c) Cr 3 R) Coloured hydrated transition metal ion
d) Ni 2 S) Paramagnetic
A) a – RS, b – PQ, c – PRS, d – QRS
B) a – PQ, b – RS, c – PRS, d – QRS
C) a – PRS, b – RS, c – PQ, d – QRS
D) a – RS, b – PQ, c – QRS, d – PRS
49. Oxidation state of iodine in H 4 IO6 is
A) +7 B) +5 C) +1 D) -1
50. Which of the following does not give a white precipitate when dilute hydrochloric acid
is added?
A) Ag  B) Ba 2 C) Pb 2 D) Hg 22

SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the Answer in the
decimals , Mark nearest Integer only. Have to Answer any 5 only out of 10 questions and question will be
evaluated according to the following marking scheme:
Marking scheme: +4 for correct answer, -1 in all other cases.
51. The wave length of an electron and a neutron will become equal when the velocity of
electron is x times the velocity of neutron. The value of x is (nearest integer)

 Mass of electron  9.110 31

kg  ,  Mass of neutron  1.6  1027 kg 

52. The major product in the following reaction is

The degree of unsaturation of product P is _______

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53. The volume of 0.15M H 2 SO4 solution required to neutralise 15 ml of 0.5M KOH in
presence of phenolphthalein indicator is ___ ml.
54. An known amine A with benzene sulphonyl chloride yields a derivative which dissolves
in aq.KOH solution. Compound A on reaction with nitrous acid gives an alcohol B
which responds to iodoform test. If hydrochloride of A contains 37.2% of chlorine by
weight, the total number of carbon atoms present in compounds A and B is ___.

55.

If the molecular weight of tripeptide formed by X is 189, the value of n is ________

56. Amongst the following, the number of species with only two lone pairs of electrons on
central atom is ______.
SF6 , XeF4 , CF4 and H 2O

57. 50 g of ethylene glycol is dissolved in 170.3 g of water and is cooled to 9.30 C . The
amount of water in g, separated as ice  K f  1.86 K / molal 

58. How many of the following can exhibit geometrical isomerism?

 3  
Co  en 2 Cl2  CrCl2  C2O4 2 
,   2  4  2  ,   3 2   4  ,
 Fe H O OH   Fe NH CN 
,
I II III IV
2 2
Co  en 2  NH 3  Cl  
,   3  4  2   
Co NH H O Cl 
V VI

59. 10 ml of a gaseous organic compound ( vapour density = 23 ) containing C,H and O only
was mixed with 100 ml of oxygen and exploded under conditions which allowed the
water formed to condense. The volume of gas after explosion was 90 ml. On treatment
with potash solution, a further contraction of 20 ml in volume was observed. The number
of moles of CO2 formed by 1 mole of organic compound is ____
60. How many moles of  CH3CO 2 O are required to react with 1 mole of sucrose?
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MATHEMATICS MAX.MARKS: 100

SECTION – I
(SINGLE CORRECT ANSWER TYPE)
This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its answer,
out of which ONLY ONE option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -1 if not correct.
61. If a denotes the number of permutation of n different things taken all at a time, b the
number of permutation of n-2 different things taken 10 at a time and c, the number of
permutations of n-12 different things taken all at a time such that a  182bc , then value of
n is:

A) 10 B) 12 C) 14 D) 18

62. STATEMENT – 1: f  x   x  x  is discontinuous at all Integers , where [.] denotes G.I.F

STATEMENT – 2: If a function is non-differentiable at a point then it may be

continuous at that point

A) Statement 1 is true, statement 2 is true

B) Statement 1 is false, statement 2 is false

C) Statement 1 is true, statement 2 is false

D) Statement 1 is false, statement 2 is true

x  a2 ab ac
63. If a,b,c are real, then f  x   ab xb 2
bc is decreasing in:
ac bc x  c2

  a 2  b2  c2  
A)    a 2  b 2  c 2  , 0  B)  0,  a 2  b 2  c 2   C)  0,
2 2
 D) Never decreases
 3   3   3 
 

64. If pth , q th , r th , terms of a G.P are the positive numbers a,b,c respectively then angle

between the vectors log a3 i  log b3 j  log c3 k and  q  r  i   r  p  j   p  q  k is:

   1 
A) B) C) 0 D) sin 1  
2 3  p 2  q 2  r 2 

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65. Consider the data on X taking the values 0, 2, 4, 8, …. , 2n with frequencies
728
n
C0 ,n C1 ,n C2 ,..., n Cn respectively. If the mean of this data is , then n is equal to ___.
2n

A) 15 B) 8 C) 4 D) 6

66. Complete set of real values of ‘a’ for which the equation x 4  2ax 2  x  a 2  a  0 has all its
roots real

A)  ,  
3
B) 1,   C)  2,   D)  0,  
4 

20
67. The value of  sin x  cos x  dx is: (where . denotes greatest integer function)
20

A) 10 B) 20 C) 20 D) 10

68. Statement-1: The number of non-negative integral solutions of 2x + y + z = 21 is 132

Statement-2: For n  N ,  n 2 ! is divisible by (n!)n .

A) Statement 1 is true, statement 2 is true

B) Statement 1 is false, statement 2 is false

C) Statement 1 is true, statement 2 is false

D) Statement 1 is false, statement 2 is true

69. Of all the functions that can be defined from the set A : 1, 2,3, 4  B : 5, 6, 7,8, 9 a
mapping is randomly selected. The chance that the selected mapping is strictly
monotonic is

2 4 5
A) 1105 B) C) D)
125 4096 2048

70. Three distinct numbers a1 , a2 , a3 are in increasing G.P a12  a22  a32  364 and a1  a2  a3  26 ,
then the value of a10 if an is the nth term of the given G.P is:

A) 2.39 B) 39 C) 2.310 D) 312

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Let x   5 2  7  , then x  x  x denotes the fractional part of x  is equal to:

19
71.

A) 219 B) 319 C) 0 D) 1
 /2
dx
72. If I a  
0
2 cos x  sin x  a
, then the value of I a for

Column – I Column – II

3
A) a = 1 P) ln  
2

1
B) a = 3 Q) log 3
2

2  1 3 1 
C) a = 2 R)  tan  tan 1 
11  11 11 

S) tan 1  
1
D) a = 4
3

A) A – Q ; B – P ; C – S ; D – R B) A – Q ; B – S ; C – R ; D – P

C) A – Q ; B – S ; C – P ; D – R D) A – Q ; B – P ; C – Q ; D – S

73. A curve passing through  2,3 and satisfying the differential equation
x

 ty  t  dt  x y  x  ,  x  0  is :
2

9 x2 y 2
A) x 2  y 2  13 B) y 2  x C)  1 D) xy  6
2 8 18

     
tan     tan     tan    
4  4  4  . Then 12 sin 2     15sin 2     7 sin 2   
74. Let      
5 3 2

is equal to

1 1
A)  B) C) 1 D) 0
2 2
75. A 13 ft. ladder is leaning against a wall when its base starts to slide away. At the instant
when the base is 12 ft. away from the wall, the base is moving away from the wall at the
rate of 5 ft/sec. The rate at which the angle  between the ladder and the ground is
changing is

12 13 10
A)  rad / sec B) 1 rad / sec C)  rad / sec D)  rad / sec
13 12 13

76. Two aero planes I and II bomb a target in succession. The probabilities of I and II
scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if
the first misses the target. The probability that the target is hit by the second plane is

A) 0.06 B) 0.14 C) 0.312 D) 0.70

16  x 2
77. The area of the region bounded by the curve y  and y  sec 1   sin 2 x 
4

( where . denotes greatest integer function is:)

1 3 3 8 3 8 1
A)  4    2
B) 8  4    2
C)  4    2
D)  4    2
3 3 3

e   3
x x
78. If f  x   , then: (where . represents greatest integer function)
 x  x  1

A) lim f  x   2 B) lim f  x   0 C) lim f  x   2 D) lim f  x  exist

x0 x0 x0 x0

1 1
79. Let g  x   f  2 x 2  1  f 1  x 2  x  R , where f ''  x   0x  R, g  x  is necessarily
4 2
increasing in the interval

 2 2  2   2 
A)   ,  B)   ,0  ,
 3 3   3   3 

C)  1,1 D) None of these

The largest integral value of x satisfying the inequality  tan 1  x    4  tan 1  x    3  0 is:
2
80.

A) 0 B) 1 C) 2 D) 3
 08-01-24_JEE-MAIN_Q’P
SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the Answer in the
decimals , Mark nearest Integer only. Have to Answer any 5 only out of 10 questions and question will be
evaluated according to the following marking scheme:
Marking scheme: +4 for correct answer, -1 in all other cases.
   
81. Let O be an interior point of ABC such that OA  2OB  3OC  0 then ratio of the area of
ABC to the area of AOC is _______.

82. Find number of integral values of k for which the line 3x  4 y  k  0 , lies between the
circles x 2  y 2  2 x  2 y  1  0 and x 2  y 2  18 x  12 y  113  0, without cutting a chord on
either of circle.
  
83. Let a  3i  2 j  4k , b  2  i  k  and c  4i  2 j  3k . If the equation
  
 
xa  yb  zc   xi  y j  zk has a non-trivial solution, then find the sum of all distinct

possible values of  .
84. For each positive integer n, consider the point P with abscissa n on the curve y 2  x 2  1 . If
d n represents the shortest distance from the point P to the line y  x then lt  n.d n  has
n 

1
value , then K is
K K
x
85. Let f  x    3t  3t  4   x  t  dt  x  0  . If x  a is the point where f  x  attains it’s local
0

minimum value then value of 3a is ;

y 3 y 3
86. The distance between x    0 & 2 x  y  4  0 is  , and distance between x    0 &
2 2 2 2
3 y 15
3x    0 is  , then find  2   2
2 2

87. The number of points of intersection of Z   4  3i   2 and Z  Z  4  6, Z  C is

88. Defined a “good word” as a sequence of letters that consists only of the letters A, B and
C and in which A never immediately followed by B, B is never immediately followed by
C, and C is never immediately followed by A. If the number of n-letter “good words” is
384, find n.
ab
89. If 2ab 2  4ab  6b 2 , a  0, b  0 then find the minimum value of .
6
x2  4 1  x  4 
2
1
90. If  x 4  16 a b
dx  tan 
 ax b 
  c then a  b is __.
Sec: MODEL-A,B&C Date: 10-01-24
Time: 3 HRS JEE-MAIN Max. Marks: 300

KEY SHEET

PHYSICS
1 D 2 D 3 B 4 B 5 C
6 A 7 A 8 D 9 A 10 B
11 C 12 B 13 A 14 D 15 B
16 A 17 C 18 D 19 B 20 D
21 2 22 5 23 3 24 8 25 246
26 5 27 27 28 4 29 3 30 50

CHEMISTRY
31 A 32 B 33 A 34 D 35 D
36 A 37 D 38 A 39 C 40 D
41 B 42 A 43 D 44 B 45 B
46 B 47 B 48 A 49 A 50 B
51 1758 52 3 53 25 54 6 55 1
56 2 57 9 58 6 59 2 60 8

MATHEMATICS
61 C 62 D 63 A 64 A 65 D
66 A 67 B 68 A 69 B 70 A
71 D 72 C 73 D 74 D 75 B
76 C 77 C 78 A 79 B 80 B
81 3 82 30 83 7 84 2 85 7
86 1 87 2 88 8 89 4 90 4
 08-01-24_JEE-MAIN_KEY&SOL
SOLUTIONS
PHYSICS
1
1. T  2
mgd
2. Angular momentum is defined by the equation L=MVR

3. Let height of liquid above second hole (2) be ‘H’

 v1  2 g  H  h 
Thus the force experienced by the tank at point
d
‘A’ is F1   AV12
dt
Similarly F2  AV22
V2 is the velocity of liquid coming out at second hole (2).
At point B V2  2 gH
  
 The net force on tank is, F  F2  F1
 A V22  V12   A 2 gh  A  2 g  h
Fnet h
4.

YA
K'
L
KYA
KK ' L  KYL
 K eq  
K  K ' K  YA KL  YA
L
m
 T  2
K eq

m  KL  YA 
 2
KYA
5. The given graph represent isothermal process and for isothermal process internal energy is
constant.

6.
 08-01-24_JEE-MAIN_KEY&SOL

+q -q
A
B
C
D
Kq KQ Kq
  0
3a 3a 4 a
Q  q
4
q
Kq K 4 Kq Kq
VA    
2a 3a 4a 6a
Kq
VA  VC 
6a
1 8
7. sin   , cos  
3 3
According to conservation of momentum mu  2mv cos 
8
u  2v
3
v 9
e 
u cos  16
8. As the branch of the circuit containing 3 resistor is open so no current flows through it.
9. In equilibrium electrostatic attraction between the plates = spring force
q2
  kx
2 0 A
 CE 
2

  k  d  0.8d 
2 0 A
2
 0 A  2
  E
  0.8d   0.2dk
2 0 A
 0 AE 2 4 0 AE 2
 k 
0.256d 3 d3
10. As magnetic flux is same through inductors, L1i1  L2i2

 i1  2i2 …..(i)
 08-01-24_JEE-MAIN_KEY&SOL
 10V
Also, i1  i2    1A …..(ii)
R 10Ω
(i) and (ii)
2
 i1  A
3
R1R2
11. Time constants of both the circuits   C but that does not ensure same charge at any time on
R1  R2
capacitors or same current in branches of the capacitor.
mv 2
12. At the bottom, N  mg 
r
13. r2  c ; A  r1   c

r1  A  c
Sin r1  sin  A   c 
sin i
 sin  A   c 

sin i    sin A cos c  cos A sin  c 
7 3 3 3 1 1 1
 1    1  
3 2 7 7 2 2 2
1
sin i  or i  300
2
14. According to Lenz’s law e.m.fs of the same magnitude in the clockwise direction are induced in the
two loops into which the figure is divided. So, current is induced in the clockwise direction in the
outer boundary but no current in wire AB.
1D
15.  4.84mm
d
D
Let required wavelength is 2 the according to given information 2  4.84mm
2d
1
  1  2  1200mm
2 / 2
16.
 08-01-24_JEE-MAIN_KEY&SOL

17. The field due to current (either conventional or displacement) is normal to the direction of current.
18. The first photon will excite the hydrogen atom ( in ground state) in first excited state
 E2  E1  10.2 eV  . Hence, during de-excitation a photon of 10.2eV will be released. The second
photon of energy 15eV can ionize the atom. Hence the balance energy 1.4eV is retained by the
electron.
 2M
19. The magnetic fields at P due to horizontal and vertical magnets are respectively 0 3 towards
4 d
0 M
right and upwards.
4 d 3
 M  M 5
Their resultant is 0 3 2 2  12  0
4 d 4 d 3
20.

21.

 2R  
PEmax  mgR   mg  R 
   

 mgR  mgR  mg  
2R
  
 PEmax  mg  
2R
  
 08-01-24_JEE-MAIN_KEY&SOL
F F
22. Horizontal acceleration of the system is a   .
2 m  m  2 m 5m
Let N be the normal reaction between B and C.
Free body diagram of C gives

Now B will not slide downwards if  N  mB g

    F   mg
2
or
5 
5
or F mg
2
5
So, minimum value of F is mg
2
ml 2
23. Angular momentum L  I   .
3
24. For all point out the sphere we can treat it as point mass at its centre so, effectively it will be the
force between ring and point mass.
Gravitation field at x  3a on axis of ring is

E
Gmx

Gm  3a 
R  x2  a  3a 2 
3 3
2 2 2 2

3Gm

8a 2
3GMm
F  ME 
8a 2

26. Let V is total volume of iceberg & n is the fraction of the volume of iceberg that appears above the
surface of sea water. According to principle of floatation
V  0.9  103  g  1  n  V  1.125  103 g
 n  0.2
i 0iR 2
27. Bcentre  0 & Bdis tan ce  28. 8  1014 h  0  0.5
 
3
2R

2 2
2 R2  2 2R

12 1014 h  0  2
12 0  2
Dividing, we get 
8 0  0.5
3 0  2

2 0  0.5
30  1.5  20  4 or 0  2.5eV
29. m1v1  m 2 v2
 08-01-24_JEE-MAIN_KEY&SOL
m1 v 2

m 2 v1
m1 27

m2 8
4
p  r13
3 27

4
p  r13 8
3
r1 3

r2 2
3
x
2
y3

R1 l1 l1
30.  
R2 l2 100  l1
x 20 1 4x l
   1 and    2
y 80 4 y 100  l
From (1) and (2) we get l  50 cm

CHEMISTRY
31. CONCEPTUAL
32. CONCEPTUAL
33. M.O configuration of N 2 is

 
* *
1s 2 1s 2 2 s 2 2 s 2  22 px   22p y 2p 2
z
1 i
34.  and  f  ik f m
1 1
n
35. TLC is a technique used to isolate non-volatile mixtures.
X  2Y Z  PQ
36. 1 0 1 0 0
1   2 1  
4 2 P1  2 P2
KP  KP 
1 1  1 2
2
2
K P1 1 4 P1 P 1
   1 
K P2 9 P2 P2 36

37. 5d series member has more  0 than 3d and 4d series.

38. -I is a good leaving group and also bonded to benzylic carbon.
39. Cu 2 is reduced to Cu 
CH3 CH3 CH3
| | |
CH 3  C  Cl  CH 3  C  O  Na   CH 3  C  CH 2
40. | |
CH 3 CH 3

 major 
 08-01-24_JEE-MAIN_KEY&SOL

41. CONCEPTUAL
42. CONCEPTUAL
43. CONCEPTUAL
44. The charge on the complex ion is not always equal to the oxidation state of the metal atom. It
actually
depends on the nature of ligands and oxidation state of the metal atom.
45. CONCEPTUAL
46. The purple colour of KMnO 4 is due to charge transfer transition
The intense colour, in most of the transition metal complexes is due to d – d transition
47. CONCEPTUAL
48. CONCEPTUAL
49. CONCEPTUAL
50. CONCEPTUAL
h h
51. e  and n 
me ve mn vn
52.

H 2 SO4 KOH
53.
N1V1  N 2V2
0.3  V1  0.5 15
0.5 15
V1   25ml
0.3

CH 3  CH 3 
CH 3 
CH  NH 2 , CH 3 
CH  NH 3 Cl 
54. , wt % of Cl  37.2%
A
55. Tripeptide  2 H 2O  3 moles of Amino Acid
 NH| 2 
 
  CH 2   COOH  3  189  36
 
 n
 n 1
NH 2
|
X is CH 2  COOH

56. H 2O and XeF4

57. Tf  K f  m
58. ALL
59.
 08-01-24_JEE-MAIN_KEY&SOL


60. CONCEPTUAL

MATHS
61. a  n!

62.
is continuous at

1 0 0 x  a2 ab ac x  a2 ab ac
63. f '  x   ab xb 2
bc  0 1 0  ab xb 2
bc
ac bc x  c2 ac bc x  c2 0 0 1

64. a  A  R  p 1log a  log A   p  1 log R

65.
 08-01-24_JEE-MAIN_KEY&SOL

xi  observation  0 2 22 2n
f i  frequency  n
C0 n
C1 n
C2 n
Cn

x
fxi i

f i

0  C0  2 n C1  22 n C2 ......2n n Cn 3n  1 728
n
 n  n
n
C0  n C1  n C1.......n Cn 2 2
 3n  36
n6
66. a 2  a  2 x 2  1  x 4  x  0
2 x 2  1   2 x  1
a
2
ax x
2
a  x2  x  1
a1
4
a3
4  x  R 
 
 1 0 x
2

 0  3
x
 2 4
 3
 1  x 
     4
67. sin x  cos x    2 sin  x     
  4   3
2  x
 2
 3 7
 1 x
 2 2
 7
0  x  2
 4

68. Statement I : Req. number of solutions

10
  21 x  2 1 C21  132.
x0
Statement II : Is true by the definition of n!.

69. n  s   54
n  A   2.5 C4  10
10 2
P  A  
625 125
a
70.  a  ar  26 .....  i 
r
 08-01-24_JEE-MAIN_KEY&SOL

f  5 2  7 
19
71. Let
x  f  an integer   x    x  f  an integer
  x  f  an integer,but -1<   x  f  1  f
So, x  x  x. f  119  1
x 2dt
72. If t  tan , then dx 
2 1 t2
1
dt
A) a  1, I1  
0 2 1  t   2t  1  t 2 
2

1 1
dt dt 1
I1  2   2  log 3
0 4   t  1
0
t  2t  3
2 2
2
Similarly, for others

73. xy  x   x 2 y '  x   2 xy  x  , xy  x   x 2 y '  x   0

dy
x  y  0 , xy  c
dx
 
tan    
74. Taking first two 4 5
  3
tan    
 4 
 08-01-24_JEE-MAIN_KEY&SOL

75.
y

x
db
b  5 ft / sec , when b  12 then l  5
dt
h2  l 2  b2
b  h cos 
db d
  h sin 
dt dt
d
5  l
dt
d
 1 rad / sec
dt

76. Required probability = P  I ' P  II   P  I ' P  II '  P  I ' P  II   .......

 0.7  0.2  0.7  0.8  0.2  ......
0.14
  0.312
1  0.56
77.
 08-01-24_JEE-MAIN_KEY&SOL
y

Q
P

y  sec 1   sin 2 x   
a
 16  x 2  1
A  a  4
    dx where a  2  4    2

e0h  3
78. lim f  x   lim  2
x 0 x 0 0  x  1

e 1 h  3 e1 h  3
lim f  x   lim  lim  
x 0 h  0 1  h  1 h 0 h
79. f ''  x   0
 f ' is inc.fn
To find : where g is nec.Inc
G is inc  g '  0
1 1
 . f '  2 x 2  1  4 x   f ' 1  x 2   2 x   0
4 2
 
 x f '  2 x  1  f ' 1  x 2   0
2

  
Case I: x  0  1 f ' 2 x 2  1  f ' 1  x 2 
 2 x2  1  1  x2
 2  2 
 x   ,   ,     2
 3   3 
 2 
1   2   x   ,   .........  3
 3 
Case II: x  0   3 f '  2 x 2  1  f ' 1  x 2 
 2x2  1  1  x2
 2 2
 x    ,    4 
 3 3
 2 
 3   4     , 0    6 
 3 
 g is inc in x   5    6 
 08-01-24_JEE-MAIN_KEY&SOL
 2   2 
 x    , 0    ,  
 3   3 
80. tan  x   t
1

t 2  4t  3  0
t   ,1   3,  

tan 1  x     ,1
 2 
x   , tan1
Largest integral x 1
81.

82.

83. Equating the components

3 x  2 y  3 z   x , 2 x  2 z   y and 4 x  2 y  3 z   z
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Hence,
3    x  2 y  4z  0
2x   y  2z  0
4x  2 y  3    z  0
3  2 4
For non-trivial solution 2  2 0
4 2 3 
Hence   1 or 8  sum  7
84. Curve is rectangular hyperbola

 
P n, n 2  1  d n 
n  n2  1
2
1
 lt n.d n  K 2
n 2 2
85. f ' x  0
x
f '  x    3t  3t  4  dt  x.3x  3x  4   3x.x  3x  4 
0

1
f ' x 
2 ln 3
 32 x  8.3x  7 
1
f ' x 
2ln 3
 3x  1 3x  7 

x  log3 7 is the point of minima

 3a  3log3 7  7
86. 4x  2 y  6  0
4x  2 y  8  0
86 2 1
  
16  4 20 5
6x  3 y  9  0
6 x  3 y  15  0
15  9 6 6 2
   
36  9 45 3 5 5
1 4
 2   2    1
5 5
88. There are three choices for the first letter and two choices for each subsequent letters. Hence using
fundamental principal.
No. of good words  3.2 n 1  384  2n 1  128  n  8
89. 2ab 2  4ab  6b 2  ab  2a  3b   a  3 b  2a
2a 2a 2
b  ab  K
a 3 a 3
dk 2  a  6a 
2

  0 a  0, 6
 a  3
2
da
dk 2
At a  6,  Ve
da 2
When a  6 & b  4, ab is min imum
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6 4

ab
6
| min 
6
4

1 4 2 4  4
90.   4 x2 dx ; x  x  t  1  x2  dx  dt
 x   8
 x
1  t  1  x  4 
2
dt 1 1
2  tan    c  tan  c
 
2
t  2 2 2 2  2 2  2 2  2 x 2 
a  2, b  2  a  b  4