27-04-2025

1402CJA101021250020 JA

PART-1 : PHYSICS

SECTION-I (i)

1) A satellite is moving with a constant speed 'V' in a circular orbit about the earth. An object of
mass 'm' is ejected from the satellite such that it just escapes from the gravitational pull of the earth.
At the time of its ejection, the kinetic energy of the object is

(A)

(B)

(C)

(D)

2) A small charged ball 'q' of mass 'm' is suspended on an insulating string of length ℓ. Another
identical charged ball is slowly moved by a student towards the first one from a large distance.
Eventually, the second ball is placed at the original location of the first one as shown in figure. At
that moment, the first ball is elevated a small distance h above its original position and remains at

rest. Then

cosα =
(A)

(B)
cosα =

(C)
cosα =
cosα =
(D)

3) Two closed vessels of equal volume contain air at 105 kPa, 300 K and are connected through a
narrow tube. If one of the vessels is now maintained at 300 K and the other at 400 K, what will be
the pressure in the vessels ?
(A) 100 kPa
(B) 120 kPa
(C) 150 kPa
(D) 160 kPa

4) A cylinder is filled with a liquid of refractive index m. The radius of the cylinder is decreasing at a
constant rate K. The volume of the liquid inside the container remains constant at V. The observer
and the object O are in a state of rest and at a distance L from each other. The apparent velocity of

the object as seen by the observer, (when radius of cylinder is r)

(A)

(B)

(C)

(D)

SECTION-I (ii)

1) Consider a uniform spherical planet of mass M and radius R. Two parallel tunnels are dug at

perpendicular distance symmetrically from centre. A particle is to be thrown from the starting of
one tunnel such that is enters in another tunnel without making collision with tunnel wall and
continues its motion as shown in figure. Tunnels are friction less particle just fits inside the tunnel.
Consider the planet to be at rest. Choose the correct options.

(A)
Speed with which the particle is to be thrown at starting of one tunnel is

(B)
Minimum speed of particle during entire motion is

(C)
Maximum height achieved by the particle during subsequent motion is

(D)
Speed with which the particle is to be thrown at starting of one tunnel is

2) A positive point charge q is located inside a neutral hollow spherical conducting shell. The shell
has inner radius a and outer radius b ; b – a is not negligible. The shell is centred on the origin.
Which of the following is correct graph of electric field vs radial distance x or electric potential vs
radial distance x. The point charge can be located anywhere inside shell on x-axis.

(A)
(B)

(C)

(D)

3)

Figure shows a battery, a conductor and a switch. After the switch is closed

(A) The electric field is set up inside the conductor.

(B) The time taken by the current to travel from A to B is

[vd is drift speed of electrons]
(C) The random motion of the electron stops and they start drifting to the left.
(D) The random motion of the electron continues along with the drifting.

4) A charged particle q is placed at a distance d from the centre of conducting sphere of radius R
(<d), then in static condition at the centre of sphere :

(A)
magnitude of electric field due to induced charge is

(B)
magnitude of electric field due to induced charge is
(C) magnitude of electric field due to induced charge is zero.

(D)
magnitude of electric field due to charge q is (where )
5) A charged ball of charge +q1 is revolving around another charge +q as shown in a conical

pendulum. The motion is in a horizontal plane :-

(A) Tension in the string is greater than the weight of the ball.
(B) The tension in the string is greater than the electrostatic repulsive force.
(C) If the charge is removed, the speed of ball has to be increased to maintain the angle.
(D) If the charge is removed, the speed of ball has to be decreased to maintain the angle.

6) An observer is standing in front of a plane mirror at a distance ‘d’. An object is at a distance 2d

from plane mirror and it is given a speed v0 in upward direction under uniform gravity as
shown. Choose correct statement(s):- (Neglect height of observer) consider only one vertical motion

in upward and downward direction.

(A)
The time interval for which the observer can see the image of object in plane mirror is .

(B)
The time interval for which the observer can see the image of object in plane mirror is .
The time interval for which the observer can not see the image of object in plane mirror is
(C)
.
The time interval for which the observer can not see the image of object in plane mirror
(D)
is .

7) A cyclic process ABCD is shown in the P–V diagram. (BC and DA are isothermal)

Which of the following curves represents the same process?

(A)

(B)

(C)

(D)

8) Two point charges +Q, –2Q are placed at point O and O' respectively. Consider a sphere of radius

R and centre C as shown in figure, CO = CO' = R/2.

Flux through the surface of upper hemisphere (dotted line) is

(A)

(B)
Flux through the surface of lower hemisphere (solid thick line) is
Flux through the circular flat surface (solid thin line) is
(C)

(D)
Flux through the surface of lower hemisphere (solid thick line) is

SECTION-II
1) In a region, the potential is represented by V (x, y, z) = 6x – 8xy – 8y + 6yz, where V is in volts and
x, y, z are in meters. The electric force experienced by a charge of 2 coulomb situated at point (1, 1,
1) is N , what is N?

2) The height above the surface of earth (radius Re) at which the gravitational field intensity is
reduced to 1% of its value on the surface of earth is nRe. Find n.

0
3) The density inside a solid sphere of radius R = 5 m is given by where ρ is the density at
the surface and r is the distance from the centre. The gravitational field due to this sphere at
distance 2R from its centre is . Find K.

4) A plane mirror is placed 25 cm away from a concave spherical mirror perpendicularly to the
principal axis of the concave mirror. What should be the distance (in cm) in front of concave mirror,
where we place a candle if its images formed by the two mirror independently are at the same
distances from the candle? The radius of the concave mirror is 40 cm. (Consider images formed by
single reflection only.)

5) A rod of mass m (uniformly distributed) and length ℓ is hinged about point O in a uniform vertical
electric field as shown. Linear charge density of rod varies with x as λ = λ0x. Electric field for which

rod remain horizontal is . Find value of n.

6) A diatomic ideal gas undergoes a thermodynamic change according to the P-V diagram shown in
Figure. The total heat given to the gas is xP0V0 (ln 2 = 0.7), find 'x'

PART-2 : CHEMISTRY

SECTION-I (i)

1) Which of the following represent pair of enantiomers

(A)

(B)

(C)

(D)

2) The two compounds given below are :

(A) Enantiomers
(B) Diastereomers
(C) Optically inactive
(D) Identical

3) Which of the following have maximum dipole moment.

(A)

(B)
(C)

(D)

4) Consider the following reaction :

The values of x, y and z in the reaction are respectively :-

(A) 5 , 2 and 16
(B) 2 , 5 and 8
(C) 2 , 5 and 16
(D) 5 , 2 and 8

SECTION-I (ii)

1) The correct statement(s) about the compound given below is (are) :

(A) The compound is optically active

(B) The compound possesses centre of symmetry
(C) The compound possesses plane of symmetry
(D) The compound possesses axis of symmetry

2) Which will show geometrical isomerism :-

(A) C6H5–CH=NOH

(B)

(C) C6H5 –N=N–C6H5

(D)

3)

Identify correct statements.

(A) II and IV are identical and optically active

(B) I and III both are optically inactive & diastereomers
(C) I and II are diastereomer & both are optically active
(D) II & III are diastereomers & one of them is optically active.

4) Which of the following compound is/are meso?

(A)

(B)

(C)

(D)
5) Which of the following statements is/are correct?

(A)

(B)

(C) (singlet) > (triplet) (stability order)

(D) (Basic strength in polar solvent)

6) C—C and C=C bond lengths are unequal in :

(A) Benzene
(B) 1,3-buta-di-ene
(C) 1,3-cyclohexa-di-ene

(D)

7) Cycloalkanes are not planar except cyclopropane. They exist in various arrangement by ring
flipping & these arrangements are known as conformation. Cyclohexane has various conformations

known as chair, Boat, Twist boat, Half chair etc.

Conformation corresponding to 'Q' can be :

(A)

(Twist boat form)

(B)

(Chair form)

(C)

(Boat form)

(D)

(Half chair form)

8) Which of the following is correct order of EA.

(A) N < C < O < F

(B) F > Cl > Br > I
(C) Cl > F > Br > I
(D) C < N < O < F

SECTION-II

1) Number of compound can show G.I. and O.I. both?

2)

Find the number of compound which can show geometrical isomerism

3) A sample of hard water contains 324 ppm of Ca(HCO3)2. How many gm of CaO is needed to
remove all bicarbonates from 100 L hard water ?

4) Number of chiral centres in [X] & [Y] is a & b respectively. The value of (a–b) is :

5) Given for a first ordered gaseous reaction :

X(g) —→ 3Y(g)
Total pressure after time 70 min : Pt = 400 mmHg
Total pressure after a very long time : P∞ = 600 mmHg
Calculate rate constant for appearance of Y in minute–1 [Given ln2 = 0.7]

6) A complex of potassium, iron and cyanide ions Kx [Fe(CN)y ] is 100% ionized at 1 molal. If its
elevation in boiling points is 2.08°C, find out the value of 'x' in the complex: (Given Kb = 0.52°C
mol-1)

PART-3 : MATHEMATICS

SECTION-I (i)

1) For every even continuous function ƒ(x), equation ƒ(x) = ƒ(x + 2) has

(A) atleast one root in (0,2)

(B) atleast one root in (–1,0)
(C) atleast one root in (0,1)
(D) atleast one root in (–2,0)

2) If (where a,b,c ∈ ) is continuous for all x ∈ R, then the value of

2(a – 2b) + ec is

(A) 5
(B) 6
(C) 7
(D) 8

3) Let S be the set of all real numbers. Then, the relation R = {(a, b) : 1 + ab > 0} on S is

(A) Reflexive and symmetric but not transitive

(B) Reflexive and transitive but not symmetric
(C) Symmetric and transitive but not reflexive
(D) Reflexive, symmetric and transitive

4) 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

SECTION-I (ii)

1) Given then

(A) f is continuous but not derivable at x = 0

(B) f is differentiable at x = 0
(C) f¢(x) is continuous for every
(D) f¢(x) is not differentiable at x = 0

2)

The function
(A) continuous at x = –1
(B) neither continuous nor derivable at x = –1
(C) derivable at x = –1
(D) Not derivable at x = –1

3) If y = , then is equal to :

(A)

(B)

(C)

(D)

4) If x + |y| = 2y, then y as a function of x is

(A) defined for all real x

(B) continuous at x = 0
(C) Differentiable for all x

(D)
Such that for x < 0

5) Let for x ≠ 1. Then

(A) limx→1 f(x) = 0

+

(B) limx→1 f(x) = 0

–

(C) limx→1 f(x) does not exist

+

(D) limx→1 f(x) does not exist

–

6) 2 tan(tan–1(x) + tan–1(x3)), where x ∈ R –{–1, 1}, is equal to

(A)

(B) tan(2 tan–1x)

(C) tan(cot–1(–x) – cot–1 (x))
(D) tan(2 cot–1x)

7) If z1 = 5 + 12i and |z2| = 4, then

(A) maximum (|z1 + iz2|) = 17

(B) minimum (|z1 + (1 + i)z2|) = 13 –
minimum
(C)

maximum
(D)

8) Consider three distinct lines x + λy + 6 = 0, 2x + y – 3 = 0 & λx + 2y + 5 = 0 let m denotes

number of possible real values of λ for which given lines are concurrent and n denotes number of
possible real values of λ for which given lines do not form a triangle, then

(A) m = 2
(B) m = 3
(C) n = 6
(D) n = 7

SECTION-II

1) If the function f(x) defined as f(x) = is continuous but not derivable at x = 0

then sum of all non-negative integral value(s) of 'n' is

2) If f(x) = x + ; then value of f(2022) . (2022) is :-

3) The derivative of

at x = is

4) If the equation sin–1(x2 + x + 1) + cos–1(λx + 1) = has exactly two solutions of λ∈ [a, b), then the
value of a + b is

5) Let n ∈ N.
Find the value of

6) The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y =
9 and y = mx + 1 is also an integer, is :
 ANSWER KEYS

PART-1 : PHYSICS

SECTION-I (i)

Q. 1 2 3 4
A. B C B A

SECTION-I (ii)

Q. 5 6 7 8 9 10 11 12
A. A,B,C A,B,D A,D A,D A,B,C B,D A,B A,C,D

SECTION-II

Q. 13 14 15 16 17 18
A. 4.00 9.00 2.50 10 3.00 3.9

PART-2 : CHEMISTRY

SECTION-I (i)

Q. 19 20 21 22
A. C A A C

SECTION-I (ii)

Q. 23 24 25 26 27 28 29 30
A. A,D A,B,C,D A,B,D A,B,C,D A,B,D B,C,D A A,C

SECTION-II

Q. 31 32 33 34 35 36
A. 7.00 5.00 11.20 2.00 0.03 3.00

PART-3 : MATHEMATICS

SECTION-I (i)

Q. 37 38 39 40
A. D C A C

SECTION-I (ii)

Q. 41 42 43 44 45 46 47 48
A. B,C,D B,D B,D A,B,D B,C A,B,C A,B,D A,C
 SECTION-II

Q. 49 50 51 52 53 54
A. 1 2022.00 0.50 1 770 2.00
 SOLUTIONS

PART-1 : PHYSICS

1)

2)

3) Let the initial pressure, volume and temperature in each vessel be p0(= 105 kPa), V0 and
T0(= 300 K). Let the number of moles in each vessel be n. When the first vessel is maintained
at temperature T0 and the other is maintained at T′ = 400 K, the pressures change. Let the
common pressure become p' and the number of moles in the two vessels become n1 and n2. We
have

p0V0 = nRT0 ........(i)

p′V0 = n1RT0 ........(ii)
 p′V0 = n2RT′ .......(iii)
and n1 + n2 = 2n. .......(iv)
Putting n, n1 and n2 from (i), (ii) and (iii) in (iv),

or,

or, = .

5) R = a

6) (A) Option is electric field v//s x graph and

(B) option for potential when point charge is kept at centre.
(D) Option is for potential v/s x when point charge is not at centre.
(C) Option is incorrect become magnitude of electric field at x = + b & x = –b must be same.

7) Although not steady but current is setup just after closing the switch. More over electron
does not forget its random motion.
8)

⇒ Esp =

9)

T cos α = Mg

T sin α – Fe =

10)
From similar triangle

Total time of flight

Time for which object will not be seen

Time for which object will be seen

11)

A → B constant pressure B→C T = constant

C → D constant Volume D→A T = constant
∴ clearly, option A and B are constant

12) Flux through the flat face

Flux through the upper hemisphere

Flux through the lower hemisphere

13)

So

14)

15) New Ans. 0.5 by Anshul Sir

16) Distance of image of candle in plane mirror from candle = 50 – 2x.

For image in concave mirror, .

Distance of image from the candle =

As per question . Solving for x; x = 10

17) For rotational equilibrium torque of electric field

should be equal and opposite to torque of weight
Torque on small element dq.
dτ = Edq x
 τ=

τ=

& Put τ =

PART-2 : CHEMISTRY

19)

20) and
Enantiomers (I & II are mirror image of each other)

21)
23)

24) All show geometrical isomerism

26) meso compounds are achiral despite the presence of chirality centres due to presence of
either plane of symmetry or centre of symmetry.

27) More resonating structure of conjugate base, more stronger is the acid.
Bridge head carbocations are least stable.
Singlet carbene is vacent orbital, so it is less stable

29) A is conformation corresponding to Q

30)

EA ⇒ Cl > F,
N<C

31)

ii, iii, iv, v, vi, vii, ix

33) Ca(HCO3)2 = 324 ppm

106g hard water contains 324g Ca(HCO3)2
1g hard water contains Ca(HCO3)2
105 g or 100 by or 1002 H2O contains = 32.4 g Ca(HCO3)2

moles of Ca(HCO3)2 = = 0.2 moles

Ca(HCO3)2 + 2CaO → Ca(OH)2 + 2CaCO3↓
0.2 mol
* moles of CaO required 0.2 mole or 11.2 g

35) Reaction belongs to first order kinitcs

X(g) —→ 3Y(g)
0
P 0
0
P –P 3P Pt
0 0
3P = 600 ⇒ P = 200
0
P – P + 3P = Pt = 100
 Kx =
KY = 3 × Kx = 3 × 0.01 = 0.03 min–1

36)

i=4

PART-3 : MATHEMATICS

37) Let g(x) = ƒ(x) – ƒ(x + 2)

g(0) = ƒ(0) – ƒ(2)
g(–2) = ƒ(–2) – ƒ(0) = ƒ(2) – ƒ(0)
(∵ ƒ(–2) = ƒ(2))
∵ g(0) and g(–2) are opposite in sign
∴ atleast one root of g(x) = 0 lies in (–2,0)

38) f(x) is continuous at x = 0

⇒ = b ⇒ 2a – 3b = 0 .... (1)
f(x) is continuous at x = 1
⇒ 7 + b = ec ⇒ ec – b = 7 .... (2)
Adding (1) & (2)
2a – 4b + ec = 7
⇒ 2(a – 2b) + ec = 7

39) R is reflexive as 1 + a × a = 1 + a2 > 0

R is symmetric because, if 1 + ab > 0 ⇒ 1 + ba > 0
But R is not transitive in all case
For example

as

and R (–4) is not possible as = –1 < 0

Hence, R is not transitive.

40) m1 + m2 = 4α & m1m2 = 1

m1 + m2 = 4m2m2
4α = 4
α=1

42)

For x = − 1, we have
f(x) = 1 − 2x + 3x2 − 4x3 + .... = (1 + x)−2

Now,
f(x) is not continous at x = −1
So, f(x) is not derivable at x = −1
Hence f(x) is neither continuous nor derivable at x = −1

44) x + |y| = 2y
3y = x if y < 0
y = x is y ≥ 0
y = 1/3x
y=x y>0
(A) Domain and range of function is set of real numbers so (A) is true
(B) f(0)= L.H.L. = R.H.L.
So (B) is true (D) L.H.D. = 1/3 and R.H.D = 1 so (D) is true

45)

number lying between –2 and 2

Hence, limit does not exist.

46) Let tan–1x = α and tan–1x3 = β

tan α = x and tan β = x3

∴ 2 tan (α + β) =

Also = tan2α = tan(2 tan–1x)

=
= tan(π – cot–1 x cot–1x)
= tan(cot–1(–x) – cot–1(x))

47) z1 = 5 + 12i, |z2| = 4

|z1 + iz2| ≤ |z1| + |z2| = 13 + 4 = 17
∴ |z1 + (1 + i)z2| ≥ ||z1| – |1 + i| |z2|| = 13 –
∴ min (|z1 + (1 + i)z2|) = 13 –

∴ and

48)

For concurrency
⇒ 3λ2 + 16λ – 35 = 0 ⇒ (λ + 7) (3l – 5) = 0

⇒
for not forming triangles either lines are parallel or concurrant

49)

f(0–) = f(0) = 0, f(0+) =

f(x) is continuous at x = 0 → f(0+) = 0

If n = 0 → f(0+) does not exit

If n < 0 → f(0+) does not exist
+
If n > 0 → f(0 ) is equal to 0
∴ for f(x) to be continuous at x = 0 n needs to be positive. (n > 0)

Lf'(0) = 0, Rf'(0) =
Rf'(0) = =
∴ f(x) is not differentiable at x = 0 → Lf'(0) Rf'(0)

for Rf'(0),
if n > 1 → Rf'(0) is equal to 0
if n = 1 → Rf'(0) does not exist
if n < 1 → Rf'(0) does not exist
∴ for f(x) to be not differentiable. at x = 0

→ only integer value of n = 1

52) sin–1 (x2 + x + 1) + cos–1(λx + 1) =

⇒ x2 + x + 1 = λx + 1
⇒ x2 + (1 – λ)x = 0
⇒ x = 0 to x = λ – 1
Also, sin–1 (x2 + x + 1) with exist if –1 ≤ x2 + x + 1 ≤ 1.
⇒ x2 + x ≤ 0 ⇒ – 1 ≤ x < 0
For exactly two solution, we have
–1 ≤ λ – 1 < 0
⇒0≤λ<1
∴ a = 0 and b = 1

53) Put

So,

= =

Now, = 770

54)

Given equation of the lines are

3x + 4y = 9 and y = mx + 1
3x + 4y = 9
3x + 4(mx + 1) = 9
x(3 + 4m) = 9 - 4

Clearly x will attain integer values when

3 + 4m = 5, -5, 1, -1
4m = 5 - 3, 4m = -5 -3, 4m = 1 - 3

4m = -1 -3
m = -1

So number of integer values of m is 2