13-07-2025

1103CJA101021250028 JA

PHYSICS

SECTION-I

1) The current through an inductor of '1H' is given by i = 3t sin t. The magnitude of the voltage
across the inductor of 1H is :-

(A) 3 sin t + 3 cos t

(B) 3 cos t + t sin t
(C) 3 sin t + 3t cos t
(D) None of these

2) A solid metal cube of edge length 2 cm is moving in a positive y direction at a constant speed of 6
m/s. There is a uniform magnetic field of 0.1 T in the positive z-direction. The potential difference
between the two faces of the cube perpendicular to the x-axis, is :

(A) 6 mV
(B) 1 mV
(C) 12 mV
(D) 2 mV

3) The figure shows a rod of length with points A and B on it. The rod is moved in a uniform
magnetic field (B0) in different ways as shown. In which case potential difference (VA – VB) between A
& B is minimum?

(A)

(B)

(C)
(D)

4) A conducting wire frame is placed in a magnetic field which is directed into the paper. The
magnetic field is increasing at a constant rate. The directions of induced currents in wires AB and

CD are

(A) B to A and D to C
(B) A to B and D to C
(C) A to B and C to D
(D) B to A and C to D

5) A circular coil of 500 turns encloses an area of 0.04 m2. A uniform magnetic field of induction 0.25
Wb/m2 is applied perpendicular to the plane of the coil. The coil is rotated by 90° in 0.1 second at a
constant angular velocity about one of its diameters. A galvanometer of resistance 25Ω was
connected in series with the the coil. The total charge that will pass through the galvanometer is -

(A) 0.4 C
(B) 1 C
(C) 0.2 C
(D) Zero

6) Radius of a circular ring is changing with time and the coil is placed in uniform magnetic field
perpendicular to its plane. The variation of ' r ' with time ' t ' is shown in the figure. Then induced

e.m.f. e with time will be best represented by :

(A)
(B)

(C)

(D)

7) A rod PQ is connected to the capacitor plates. The rod is placed in a uniform magnetic field (B)
directed into the plane of the paper. If the rod is pulled out of magnetic field with velocity as
shown in figure,

(A) Plate M will be positively charged

(B) Plate N will be positively charged
(C) Both plates will be similarly charged
(D) No charge will be collected on plates.

8) A 0.1 m long conductor carrying a current of 50 A is perpendicular to a magnetic field of 1.25 mT.
The mechanical power to move the conductor with a speed of 1 ms-1, in a direction perpendicular to
length and field is

(A) 0.25 mW
(B) 6.25 mW
(C) 0.625 W
(D) 1 W

9) A wire of length 50 cm moves with a velocity of 300 m/min perpendicular to a magnetic field. If
the emf included in the wire is 2V, the magnitude of field in tesla is:

(A) 2
(B) 5
(C) 0.8
(D) 2.5

10) A conducting loop moves in xy-plane with velocity as shown. A uniform magnetic field exists
in whole region and a uniform magnetic field exists in addition for all x > 0. (At shown instant)

List–I List–II

(P) , (1) potential

(Q) , (2) potential

(R) , (3) no induced current

(S) , (4) induced current flows from Q to P

(5) induced current flows from P to Q

(A) P → 1,3 ; Q → 2,3 ; R → 1,2 : S → 1,4
(B) P → 2,3 ; Q → 1,3 ; R → 2,5 : S → 1,4
(C) P → 2,3 ; Q → 2,5 ; R → 1,4 : S → 1,2
(D) P → 1,2 ; Q → 4,5 ; R → 2,3 : S → 5

11) A metal rod moves at a constant velocity in a direction perpendicular to its length. A constant
uniform magnetic field exists in space in a direction perpendicular to the rod as well as its velocity.
Select the correct statement from the following:

(A) The entire rod is at the same electric potential

(B) There is an electric field in the rod
(C) The electric potential is highest at the centre of the rod and decrease towards its ends
(D) The electric potential is lowest at the centre of the rod and increases towards its ends.

12) A magnetic flux through stationary loop with resistance R varies during the time interval t = 0 to
t = ν as ϕ = at (ν – t). The amount of heat generated in the loop during this time is :-

(A)
(B)

(C)

(D)

13) The mutual inductance between two coils is 1.25 henry. If the current in the primary changes at
the rate of 80 ampere/second, then the induced e.m.f. in the secondary is

(A) 12.5 V
(B) 64.0 V
(C) 0.016 V
(D) 100.0 V

14) A coil of N = 100 turns carries a current I = 5 A and creates a magnetic flux ϕ = 10–5 Tm2 per
turn. The value of its inductance L will be

(A) 0.05 mH
(B) 0.10 mH
(C) 0.15 mH
(D) 0.20 mH

15)

In gravity free region a body of mass 1 kg moving on the x-axis, suddenly explodes into fragments of
mass 1/8 kg and 7/8 kg. An instant later, the smaller fragment is 0.14 m above the x-axis. The
position of the heavier fragment is :-

(A) 0.02 m above x-axis.

(B) 0.02 m below x-axis
(C) 0.14 m below x-axis
(D) 0.14 m above x-axis

16) Centre of mass of 3 particles of 10 kg, 20 kg and 30kg is at (0, 0, 0). Where should a particle of
40 kg be placed so that the combined centre of mass will be at (3, 3, 3) ?

(A) (0, 0, 0)
(B) (7.5, 7.5, 7.5)
(C) (1, 2, 3)
(D) (4, 4, 4)

17) Block of mass "2kg" is moving with 10 m/s. Later on the "2kg" block climbs on the wedge having
mass of 3 kg placed on smooth surface as shown in figure. Find work done by the normal force
acting by the block on the wedge till 2kg block comes to rest with respect to wedge.:
(A) 0 joule
(B) 12 joule
(C) –8 joule
(D) 24 joule

18) Mass m1 strikes m2 which is at rest. The ratio of masses m1/m2 for which they will collide again.
(Collisions between ball and wall are elastic. Coefficient of restitution between m1 and m2 is e and all

the surfaces are smooth)

(A)

(B)

(C)

(D) 1

19) The figure shows a square loop L of side 5 cm which is connected to a network of resistances.
The whole setup is moving towards right with a constant speed of 1 cms–1. At some instant, a part of
L is in a uniform magnetic field of 1T, perpendicular to the plane of the loop. If the resistance of L is
1.7 Ω, the current in the loop at that instant will be close to :

(A) 115 μA
(B) 170 μA
(C) 60 μA
(D) 150 μA

20) A right angled triangular loop travelling with constant velocity (as shown in the figure) enters in
uniform magnetic field (at right angle to the boundary of the field) directed into the paper. Draw the
graph between induced emf e and the distance along the perpendicular to the boundary of the field
(say X) along which loop moves

(A)

(B)

(C)
(D)

SECTION-II

1) The friction coefficient between the horizontal surface and each of the block shown in the figure is
0.2. The collision between the blocks is perfectly elastic. Find the separation (in cm) between them

when they come to rest. Take g = 10 m/s2.

2) A uniform chain of length L and mass M is lying on a smooth table and of its length is hanging
down over the edge of the table. If g is acceleration due to gravity, then the work done to pull the

hanging part on the table is then find the value of N.

3)

The square loop has sides of length 2cm. A magnetic field points out of the page and its magnitude is
given by B = (4t2y) T. Emf induced in loop is given by at t = 2.5 s. Find the value of x.

4) Two identical superconducting circular coils A and B carry identical currents i as shown. Initially
their planes are mutually perpendicular with line joining centres perpendicular to plane of B. Now
the loops are oriented parallel to each other, very close together as shown. What is the modulus of
the work done (in J) in the process if their self inductance is L each ? (L = 2H, i = 2A)
5) A wire PQ of mass 10 g is at rest on two parallel metal rails. The separation between the rails is
4.9 cm. A magnetic field of 0.80 tesla is applied perpendicular to the plane of the rails, directed
inwards. The resistance of the circuit is slowly decreased. When the resistance decreases to below
20 ohm, the wire PQ begins to slide on the rails. The coefficient of friction between the wire and the

rails is found to be , where α and β are single digit integers. Then find the value of α + β.

CHEMISTRY

SECTION-I

1) NaOH(aq.), HCl(aq.) and NaCl(aq.) have concentration of 10–3 M each. Their pH will be
respectively

(A) 11, 7, 3
(B) 11, 3, 7
(C) 3, 11, 7
(D) 11, 3, 3

2)

Which of the following solutions has the maximum pH value ?

(A) 0.2 M HNO3

(B) 0.2 M HCl
(C) 0.2 M CH3COOH
(D) 0.2 M CH3COONa
3)

Spinal is an important class of oxide having two types of metal ions. A type of spinal have ccp
arrangement to O2– ions in which Fe2+ cation occupy 1/8th of tetrahedral voids and Fe3+ cation
occupies half of the octahedral voids. If 'a' be the edge of unit cell then. What is formula of spinal -

(A) Fe2O3
(B) Fe3O4
(C) FeO
(D) Fe4O5

4) Crystal AB shows ZnS (zinc blende) type structure if the radius of B– is 50 pm then the radius of
A+ will be, if edge length of unit cell is 160 pm.

(A) 11.25 pm
(B) 27.9 pm
(C) 15.25 pm
(D) 19.2 pm

5) 8 : 8 co-ordination of CsCl is found to change into 6 : 6 co-ordination on:

(A) applying normal pressure

(B) increasing temperature
(C) both (a) and (b)
(D) none of these

6) The appearance of colour in solid alkali metal halide is generally due to:

(A) Frenkel defect

(B) interstitial position
(C) F-centres
(D) Schottky defect

7) A fcc element (atomic mass = 60) has a cell edge of 400 pm. Its density is:

(A) 6.23 gcm-3

(B) 6.43 gcm-3
(C) 6.53 gcm-3
(D) 6.63 gcm-3

8) A solid XY has NaCl structure. If radius of X + is 100 pm. What is the radius of Y- ion?

(A) 120 pm
(B) 136.6 to 241.6 pm
(C) 136.6 pm
(D) 241.6 pm
9) Which of the following statement is not true about the hexagonal close packing?

(A) The coordination number is 12

(B) It has 74% packing efficiency
(C) Tetrahedral voids of the second laycr are covered by the spheres of the third layer
(D) In this arrangement spheres of the fourth layer are exactly aligned with those of the first layer

10) Suitable reagent for above reaction.

(A) Zn-Hg/HCl
–
(B) N2H4/OH / Δ
(C) RedP/HI
(D) All of these

11) Match the list

Column-I (Reaction / conversion) Column-II (Reagent used)

(A) (P) NaBH4 + SOCl2

(B) (Q) DlBAL-H(–78ºC)

NBS/Peroxide + KOH(alc)
(C) (R)
+ OsO4 + NaIO4

(D) (S) LiAlH4/MeOH

(A) A → P;B → Q;C → R;D → S

(B) A → S;B → P;C → Q;D → R
(C) A → S;B → P;C → R;D → Q
(D) A → Q;B → S;C → P;D → R
12) The water gas on reacting with cobalt as a catalyst forms

(A) Ethanol
(B) Methanoic acid
(C) Methanal
(D) Methanol

13)

(A)

(B)

(C)

(D) No reaction

14)
A and B are respectively :

(A)

(B)

(C)
in both cases

(D)
in both cases
15)
Number of functional group reduced by NaBH4 and LiAlH4 respectively are :-

(A) 2, 2
(B) 2, 3
(C) 3, 3
(D) 1, 3

16) For weak monobasic acid 'HA' the plot of α versus is given below. α = degree of dissociation
of acid (assume 'α' to be small as compared to unity), C = concentration of acid. What is the pH of a

0.01M solution of this acid 'HA'.

(A) 4
(B) 3.7
(C) 4.7
(D) 4.3

17) At 25°C, the solubility product of Mg(OH)2 is 1.0 × 10–11. At which pH, will Mg2+ ions start
precipitating in the form of Mg(OH)2 from a solution of 0.001 M Mg2+ ions ?

(A) 8
(B) 9
(C) 10
(D) 11

18) The major product ‘P’ formed in the given reaction is

(A)

(B)

(C)

(D)

19) Hex-4-ene-2-ol on treatment with PCC gives 'A'. 'A' on reaction with sodium hypoiodite gives 'B',
which on further heating with soda lime gives 'C'. The compound 'C' is

(A) 2- pentene
(B) proponaldehyde
(C) 2-butene
(D) 4-methylpent-2-ene

20) In the following sequence of reactions, the final product D is :

(A)

(B) CH3 – CH = CH – CH2 – CH2 – CH2 – COOH

H3C – CH = CH – CH(OH) – CH2-CH2-CH3
(C)
H3C – CH = CH – CH(OH) – CH2 – CH2 –CH3
(D)

SECTION-II

1)

BA crystallizes in CsCl type structure and KCl in NaCl type structure. Edge-length of unit cell of KCl
is twice that of BA and molecular weight of BA is twice that of KCl. Calculate ratio of density of BA to
that of KCl.

2) For ABC ABC ABC .... packing distance between two succesive tetrahedral void is X and distance

between two successive octahedral void is y in an unit cell, then is .

3)

An element 'M' crystallizes in ABAB....type packing if adjacent layer A & B are pm apart, then
calculate radius of largest sphere which can be fitted in the void. (in pm) without disturbing the
lattice arrangement (Given : )
Fill your answer as sum of digits (excluding decimal places) till you get the single digit
answer.

4) How many maximum moles of HIO4 will be consumed on oxidation of one mole of D-Glucose.

5)
Degree of unsaturation in (B)

MATHEMATICS

SECTION-I

1) For the pair of straight lines x2 – 4αxy + y2 = 0, if sum of slopes is four times product of slopes,
then α is

(A) –2
(B) 2
(C) 1
(D) –1
2) A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of

this line on the coordinate axes is . Three stones A, B and C are placed at the points (1,1), (2, 2) and
(4, 4) respectively. Then which of these stones is / are on the path of the man?

(A) A only
(B) C only
(C) All the three
(D) B only

3) The point P (a,b) undergoes the following three transformations successively :

(a) reflection about the line y = x.
(b) translation through 2 units along the positive direction of x-axis.

(c) rotation through angle about the origin in the anti-clockwise direction.

If the co-ordinates of the final position of the point P are then the value of 2a + b is
equal to :

(A) 13
(B) 9
(C) 5
(D) 7

4) If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is (α, β), then the
quadratic equation whose roots are α + 4β and 4α + β, is

(A) x2 – 19x + 90 = 0
(B) x2 – 18x + 80 = 0
(C) x2 – 22x + 120 = 0
(D) x2 – 20x + 99 = 0

5) The equation y = sinx sin(x + 2) – sin2(x+1) represents a straight line lying in :

(A) second and third quadrants only

(B) third and fourth quadrants only
(C) first, third and fourth quadrants
(D) first, second and fourth quadrants

6) If the line ℓx + my + n = 0 is tangent to the circle x2 + y2 – 2ax = 0, then

(A) n2 + 2aℓn – a2m2 = 0

(B) n2 + 2aℓn + a2m2 = 0
(C) n2 – 2aℓn + a2m2 = 0
(D) a2m2 + 2aℓn – n2 = 0

7) Circles C1 : x2 + y2 = 9 and C2 : x2 + y2 – 4x = 0. Common chord PQ of C1 & C2 intersects the direct

common tangent at R. If RP + RQ = , where a & b are relatively prime. Then (a + b) equals-

(A) 65
(B) 63
(C) 71
(D) 79

8)

The locus of the centers of circles which cut the circles x2 + y2 + 4x – 6y + 9 = 0 and
x2 + y2 – 5x + 4y – 2 = 0 orthogonally is

(A) 9x + 10y – 7 = 0
(B) x – y + 2 = 0
(C) 9x – 10y + 11 = 0
(D) 9x + 10y + 7 = 0

9)

A circle with centre at (15, –3) is tangent to at a point in the first quadrant. The radius of the
circle is equal to

(A)
(B)
(C)
(D)

10) A circle is drawn circumscribing the square with vertices A, B, C, D (in order) represented by |x
+ y| + |x – y| = 2. If tangents at A and B to this circle meets at P, then PA2 + PC2 is equal to

(A) 4
(B) 8
(C) 12
(D) 20

11) Value of is
(where [.] is greatest integer function)

(A)

(B)

(C)
(D)

12)

is equal to

(A)

(B)

(C)

(D)

13) If the value of the integral is , then k is equal to :

(A)
(B)
(C)
(D)

14) is equal to :

(A) π2
(B) 2π2
(C)

(D)

15) Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that
and P(x) leaves remainder 5 when it is divided by (x – 2). Then the value of 9(b + c) is equal to:

(A) 9
(B) 15
(C) 7
(D) 11
16) If , then :

(A)

(B)

(C)

(D)

17) Let f and g be continuous functions on [0, a] such that f(x) = f(a–x) and g(x)+g(a–x)=4, then

is equal to :-

(A)

(B)

(C)

(D)

18) If , then f′(1/2) is :

(A)

(B)

(C)

(D)

19) Let ƒ : R → [4, ∞) be an onto quadratic function whose leading coefficient is 1, such that ƒ'(x) +

ƒ'(2 – x) = 0, then the value of is equal to

(A)

(B)

(C)

(D)

20) Let f(x) be a differentiable function satisfying Then the

value of f(2) is

(A) e2

(B)

(C) e + e2
(D) 2e2

SECTION-II

1) Let f(x) = max{|x + 1|, |x + 2|, ..., |x + 5|}. Then is equal to _____________.

2) The value of b > 3 for which

, is equal to

3) How many tangents to the circle are there which are normal to the ellipse

4) The circle x2 + y2 = 4 cuts the line joining the points A(1, 0) and B(3, 4) in two points P and Q. Let

and , then |αβ| is equal to

5) The value of λ, if the lines 3x – 4y – 13 = 0, 8x – 11y – 33 = 0 and 2x – 3y – λ = 0 are concurrent is

:
 ANSWER KEYS

PHYSICS

SECTION-I

Q. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A. C C C A C B A B C B B B D D B B D C B C

SECTION-II

Q. 21 22 23 24 25
A. 5.00 72.00 8.00 4.00 3.00

CHEMISTRY

SECTION-I

Q. 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
A. B D B D B C A B D D D D A B B C C D C D

SECTION-II

Q. 46 47 48 49 50
A. 4.00 2.00 2.00 5.00 3.00

MATHEMATICS

SECTION-I

Q. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
A. C D B D B A D C D C D A A A C A B B D B

SECTION-II

Q. 71 72 73 74 75
A. 21.00 6.00 0.00 7.00 7.00
 SOLUTIONS

PHYSICS

1)

v=–L

v=–1H
v = 3t cos t + 3 sin t

2) Potential difference between two faces perpendicular to x-axis will be

3) For (A):

For (B):

For (C): VA – VB = 0

For (D):

4)

∴ e2 > e1 therefore direction of induced current will be as shown figure.

5) Induced current I = ⇒ =

⇒q=
Total charge

q= = 0.2 C
OR
Induced emf,

e= = = = 50 volt

q = it = Δt

∴ q= × 0.1 = 0.2 C

6)

e= = =

7) Consider the force on an electron in PQ. This electron experiences a force towards Q. Free
electrons in PQ tend to move towards N. So M will be positively charged.

8)

9) When a wire of length l moves with velocity v, perpendicular to a magnetic field B. The
induced emf is produced. The magnitude of incluced emf is given by :
|E| = B/v

E = 2V, B = = 0.8 tesla

10) Induced current is only due to component of fields along z-axis.

11)
emf induced

12)
13)
= (1.25)(80) = 100 V

14) L =

15) Question Explanation :

In the absence of external froce, COM at doesnot mmove in the direction where there is no
force present.

Concept :
This question is based on Motion of COM

Solution :
m1d1 + m2d2 = 0

⇒ (0.14) + d2 = 0
⇒ d2 = –0.02 m

Hence,
Option (2) is correct : 0.02 m below x-axis

16)

In the first case,

or 10x1 + 20x2 + 30x3 = 0

In the second case,

or ⇒ x4 = 7.5 units
Similarly y4 = z4 = 7.5 units

17) 2 × 10 = 5 × v
v = 4 m/s

ωN = × 3 × 16 = 24 J
18)
Just before 1st collision
using momentum conservation
m1 u = – m1v1 + m2v2
⇒ m1u = m2v2 – m1v1

v2 + v1 = eu → (2)

v1 = eu – v2 = eu –

After collision with wall only dirn of velocity changes.

For again collision with balls.

⇒ e m2 – m1 > m1 + e m1

19) Since it is a balanced wheatstone bridge, its equivalent resistance =

ε = Bℓν = 5 × 10–4 V
So total resistance

R= + 1.7 3Ω

170μA

20)
21)

By momentum conservation.

Seperation from

Total seperation =

22)

23) dϕ = 4t2yℓdy

ϕ=

e=

24)

Uf =
 25) Wire PQ begins to slide when magnetic force is just equal to the force of friction i.e., μ mg
= iℓ B sinθ (θ = 90°)

Here

so α + β = 3

CHEMISTRY

26)

11, 3, 7

27)

Explanation
"Which of the given electrolytes has the highest pH"?
Given data:
Given solutions with concentration are-
0.2M HNO3, 0.2 M HCl, 0.2 M CH3COOH, 0.2 M CH3COONa
HNO3 and HCl : These are strong and will have a pH less than 7.
• CH3COOH: This is a weak acid and will have a pH less than 7 , but higher than the strong
acids.
• CH3COONa: This is the salt of a weak acid (CH3COOH) and a strong base (NaOH). It will
undergo hydrolysis to form a basic solution with a pH greater than 7.
Therefore the solution with the highest pH will be the basic one which is 0.2 M CH3COONa.

Final Answer. 0.2 M CH3COONa

correct Ans. 4

28)

Fe3+ = 4 ×

Fe2+ = 8 × = 1
O2– = 4
Formula = Fe3O4

29)
19.2 pm

30)

increasing temperature

31)

F-centres

32) Density =
Given, at.mass = 60
= 4 × 102 pm = 4 × 102 ×10-12 m
= 4 × 10-10 ×102 cm = 4 × 10-8 cm
( )

Density = = 6.23g cm3

33) The for NaCl = 0.414 to 0.732 (due to fcc structure)

r- = 241.54 to 136.6 pm

34)

In this arrangement spheres of the fourth layer are exactly aligned with those of the first layer.

35)

36) A → Q;B → S;C → P;D → R

37)

38)

39)

H2/Ni reduce aldehyde as well as give addition reaction. on alcohol.

NaBH4 only reduce aldehyde group.
(B) is correct.

40) NaBH4 can not reduced ester.

41)

The pH of a 0.01M solution of this acid 'HA' is 4.7.

42)

10

43) KMnO4 oxidises benzylic carbon containing atleast one a-hydrogen atom to –COOH.

44)
 45)

46)

d=

47)

48)

Distance between two layers =

⇒ R = 5 pm

for octahedral void

r = 0.414 × 5 = 2.07 pm

49)

The maximum moles of HIO4 will be consumed on oxidation of one mole of D-Glucose is 5.00

50)
D.U = no of rings + no of bond
D.U = 1 + 2 = 3

MATHEMATICS
51) m1 + m2 = 4α & m1m2 = 1
m1 + m2 = 4m2m2
4α = 4
α=1

52) Let the line be y = mx + c

x-intercept :
y-intercept : c
A.M of reciprocals of the intercepts :

line : y = mx + 2(1 – m) = c
⇒ (y – 2) – m(x – 2) = 0
⇒ line always passes through (2, 2)
Ans. 4

53) Image of A(a,b) along y = x is B(b,a). Translating it 2 units it becomes C(b + 2, a).
Now, applying rotation theorem

⇒b–a+2=–1 ....(i)
and b + 2 + a = 7 ....(ii)
⇒ a = 4; b = 1
⇒ 2a + b = 9

54)
Here mBH × mAC = –1

β– 3 = 2α – 4
β = 2α – 1
mAH × mBC = –1

2β – 4 = α – 1
⇒ 2(2α – 1) = α + 3 ⇒
3α = 5

⇒
x2 – 20x + 99 = 0

55) 2y = 2sinx sin(x+2) – 2sin2(x+1)

2y = cos2 – cos(2x+2) – (1–cos(2x+2))
= cos2 – 1

2y =

y=

56) A line will be tangent to a circle, if length of perpendicular drawn from centre (a, 0) upon
the line ℓx + my + n = 90 is equal to radius = a

57)

(RT)2 = RP × RQ

Let RP = x ⇒

RP + RQ =
58)

9x – 10y + 11 = 0

59)

t = 3 satisfying
P (3, 3) C (15, –3)

60)

61)

62)
63)

Let x–2 – 1 = t2 ⇒ x–3 dx = – tdt

64)

65)

⇒
...(1)
P(2) = 5
4 + 2b + c = 5
2b + c = 1 ...(2)
From (1) & (2)
b= &c=
9(b + c) = 7

66)

∴ ƒ(x) is decreasing in (1,2)

67)

68) =?
Differentiate w.r.t. 'x'
f(x) = 2x + 0 – x2 f(x)

f(x) = ⇒ f'(x) =

f'(x) =

69)
⇒ ƒ(x) – ƒ(2 – x) = λ put x = 1 we get λ = 0
∴ ƒ(x) = ƒ(2 – x) ⇒ ƒ(x) is symmetrical about line x = 1
∴ ƒ(1) = 4
y = ƒ(x) = (x – 1)2 + 4
70)

...(1)

...(1)

in (1) put x = 0 f(0) = –1

71) f(x) = max{|x+1|, |x+2|, |x + 3|, |x + 4|, |x + 5|}

72)

b=6

73) Equation of normal at is

Now it is tangent to circle so

But
∴ No such θ exist.

74) Equation of line joining A(1, 0) and B(3, 4) is y = 2x – 2 this cuts the circle x2 + y2 = 4 at

Q(0, –2) and .

, , ,

75)

The given lines are concurrent if

⇒ 3(11λ – 99) + 4(–8λ + 66) – 13(–24 + 22) = 0

⇒λ–7=0
⇒λ=7