27-04-2025

1001CJA101021240059 JA

PART-1 : PHYSICS

SECTION-I

1) The vessel volume is V, the initial pressure is P0. The process is assumed to be isothermal, and the
evacuation rate is constant & equal to C and independent of pressure. [The evacuation rate is the
gas volume being evacuated per unit time with that volume being measured under the gas pressure
attained by that moment]. Choose the CORRECT option(s).

(A) Pressure P of gas at time ‘t’ will be

(B)
Pressure P of gas at time ‘t’ will be
(C) No. of moles evacuated in time ‘t’ will be linearly proportional to time t.
(D) No. of moles evacuated in time ‘t’ will be exponentially dependent on time t.

2) A glass partially filled with water is fastened to a wedge that slides, without friction, down a large
plane inclined at an angle α (as shown in the figure). The mass of the inclined plane is M, the
combined mass of the wedge, the glass and the water is m. If there were no motion the water
surface would be horizontal. Choose the CORRECT option(s).

If the inclined plane is fixed, then the angle free surface of water will make with the inclined
(A)
plane in steady state will be zero.
If the inclined plane is fixed, then the angle surface of water will make with the inclined plane
(B)
in steady state will depend on m.
If the inclined plane can move freely in the horizontal direction, then the angle surface of water
(C)
will make with the inclined plane in steady state will be zero.
If the inclined plane can move freely in the horizontal direction, then the angle surface of water
(D)
will make with the inclined plane in steady state will depend on m & M.

3) In an old record found in a laboratory, a process 1→2→3 was shown. Over the time, ink faded and
it became impossible to see the pressure and volume axes. However, descriptions given there reveal
that the states 1 and 3 lie on an isochore corresponding to a volume V, amount of heat supplied
during the whole process 1→2→3 is zero and the gas involved is one mole of helium, then choose the
CORRECT option(s).

(A) Volume occupied by the gas in the state 2 will depend on pressure at 2.
(B) Volume occupied by the gas in the state 2 is independent of pressure at 2.
(C) Volume occupied by the gas in the state 2 is 4V.
(D) Heat supplied will be positive if we complete cycle 1→2→3→1.

4) A point mass of 1 kg moving at constant speed of 5 m/s on an elliptical path experiences

a centripetal force of 20N when at either end point of the major axis and a similar force of 2.5 N at
each end of minor axis. Then :-

(A) length of semi major axis is 5 m

(B) length of semi major axis is 10 m
(C) length of semi minor axis is 2.5 m
(D) length of semi minor axis is 5 m

SECTION-II (i)

Common Content for Question No. 1 to 2

There are three sound sources producing sound with slight different frequency. The equation of
sound waves when received by detector is given by

Where ΔP is change in pressure and v is speed of sound in medium. Let the detector is placed at x =
0 and intensity due to each source is I0 at the place of detector.

1) Number of maxima local as well as global of loudness in every one second will be :

2) If maximum intensity of resulting sound is 9I0 and detector cannot record an intensity lower than
4I0, If the duration of time for which detector remains idle during first one second is τ(sec), find 3τ.

Common Content for Question No. 3 to 4

A flat conducting sheet A is suspended by an insulating thread at distance ‘s’ from both the surfaces
formed by the bend conducting sheet B as shown in the figure. The entire space between bent sheet
B is filled with medium of dielectric constant K. The sheets A and B are oppositely charged, the
difference in potential is constant and equal to ΔV. This causes a force F, pulling A downward.
(Neglect gravity)

3) If the modulus of work done by external agent needed to slowly increase the inserted distance by

Δy is . Find the value of P.

4) An expression for the difference in potential ΔV in terms of F and relevant dimensions shown in

the figure is . Find the value of P.

Common Content for Question No. 5 to 6

Hydrogen atoms in ground state are excited by means of a monochromatic radiation of wavelength
969·69 Å. After absorbing the energy of radiation, hydrogen atoms go to the excited state. After 10–8
s, the hydrogen atom will come to the ground state by emitting the absorbed energy. Different
wavelengths are possible in the spectrum. (Take : R = 1.1 × 107 / m)

5) The electron of excited hydrogen atom is in which of the energy state n?

6) Number of different wavelengths present in the spectrum is :-

SECTION-II (ii)

1) A very small circular object is kept infront of an optical device as shown in figure. The plane of
object is parallel to the optical device. One quadrant of circle is hatched as shown. Match the images
as seen by the observer (ignoring magnification).

List-I (Device) List-II (Probable Image)

(P) Plane mirror (The observer is at A) (1)

Concave mirror (The observer is at A) object between F

(Q) (2)
and 2F

(R) Convex mirror (The observer is at A) (3)

Convex lens (The observer is at B) object between F and

(S) (4)
optical centre

2) Match statement of List-I with List-II and select the correct answer using the code given below
the list.

List-I List-II
(P) (1) 0

Graph-1 and Graph-2 represents the displacement time graph of two identical particles, performing SHM with same total energy. The value of
a1/a2 is ____ .

(Q) (2) 1

A circuit contains two capacitors as shown in the figure. Initially switch S1 is closed and S2 is open. At t = 0, S2 is closed keeping S1 closed, and a
charge Q1 is supplied by the battery to the circuit. After long time, S1 is opened keeping S2 closed. Now battery supplied charge Q2. The value of
Q2/Q1 is

(R) (3) 2

A massive rope of uniform mass density and length ℓ is suspended from points A and B in vertical plane as shown in the figure. Point C is the
lowest point of the chord. Given length of rope from A to C = ℓ1 and length of rope from B to C = ℓ2. The value of ℓ2/ℓ1 is

(S) A particle performs SHM along a straight line between the points M and N about O(mean position) as shown in figure. t and t are the time (4) 3
1 2
taken by the particle from M to P and P to N respectively. Find t1/t2 (given OP = PN)

SECTION-III

1) Parallel conducting rails are located horizontally at a distance L from each other and placed in a
uniform magnetic field whose induction vector B is directed at an angle of 37° to their plane as
shown. The rails are connected by a resistor of high resistance R. Far from the resistor on the rails
lies massive conducting rod of mass m and length L. It is perpendicular to the rails. What force (in
N) do you need to exert on the rod in the horizontal direction so that it slides along the rails with
constant speed v? The surface is rough with coefficient of friction µ = 0.5 and B = 0.5 T, L = 30 cm,

m = 1 kg, R = 27 mΩ, v = 10 m/s, g = 10 m/s2.

2) An ac voltmeter with large impedance is connected one by one across the inductor, the capacitor,
and the resistor in a series circuit having an alternating emf of 125 V (rms); the meter gives the
same reading in volts in each case. The angular frequency of the AC source is rad/s. Now the
circuit is disconnected from the AC source, the capacitor is separately charged and reconnected to
the inductor and the resistor, all three elements in series. The circuit is found to undergo
underdamped oscillations. What is the angular frequency of oscillation in rad/s ? Round off to
nearest integer if necessary.

3) 4He with energy E0 bombard stationary 7Li nuclei. They become composite nuclei 11B after
completely inelastic collision. It further splits into 10B and neutron 1n. The reaction equation is 4He +
7
Li → (11B) → 10B + 1n. Assume that the energy E0 is minimum. What is the corresponding neutron
kinetic energy En (in MeV)? Mass of helium = 4.0026 u, Mass of Lithium = 7.0160 u, Mass of 11B =
11.0093 u, mass of neutron = 1.0087 u, Mass of 10B = 10.0129 u, 1 amu = 930 MeV. Find 50 En.
Round off to nearest integer if required.

4) Intensity observed in an interference pattern is I = I0 sin2 θ. At θ = 30°, intensity I = 5 ± 0.0020

W/m2. If I0 = 20 W/m2 and percentage error in angle θ is , then the value of n is:

5) A simple pendulum of inextensible string of length ℓ and mass m is hinged at point A in a vertical
plane as shown in the figure. The ball is released from the same horizontal level as that of A and at a

distance from point ‘A’. At the lowest point bob hits a block of identical mass m elastically. Find

the maximum compression in the spring ( and ground is smooth). Height of A

from ground is equal to ℓ.

6) In the given circuit, the value of R (in Ω) so that thermal power generated in R will be maximum.

PART-2 : CHEMISTRY

SECTION-I
1)
Select correct statements for all above alkenes.

(A) Compound ‘e’ shows the highest heat of combustion.

(B) Compound ‘d’ is the thermodynamically least reactive towards catalytic hydrogenation.
(C) On ozonolysis using O3/Me2S, only 2 alkenes give smallest non–cyclic ketone.
Only 2 alkenes among these give chiral alcohol product(s) on reactive with B2H6/THF followed
(D)
by H2O2/OH–.

2) How many of these solutions (of equal volume) on mixing create a Buffer solution?

(A) 0.1 M NaOH, 0.3 M HCN

– +
(B) 0.3 M H2SO4, 0.8 M CH3COO Na
(C) 0.2 M Ba(OH)2, 0.4 M HCN
(D) 0.3 M H2CO3, 0.1 M NaOH

3) Among the following solutions, which two solutions will show ‘highest’ and ‘lowest’ increase in
temperature on mixing?
Assume that initially all solutions at 25°C and density of solutions are identical.

(A) 0.4 M, 50 mL NaOH with 0.5 M, 100 mL HNO3

(B) 0.2 M, 75 mL H2SO4 with 0.6 M, 25 mL NaOH
(C) 0.3 M, 20 mL HBr with 0.3 M, 80 mL Ba(OH)2
(D) 0.9 M, 40 mL Ba(OH)2 with 0.7 M, 60 mL HI

4) Select correct for metallurgical process which is/are involve in extraction metal from galena.

(A) Reduction of roasted ore with fresh galena

(B) Copper impurities can be remove by liquation
(C) Alkali metal cynide selectively prevent galena coming to the froth
(D) Oxidation of silver impurity in molten metal by O2

SECTION-II (i)
Common Content for Question No. 1 to 2
The following amino acids are used to form dipeptides :

1) A dipeptide ‘X’ is formed which has maximum possible degree of unsaturation. The number of
π–bonds in that dipeptide is 2n. Find the value of n.

2) Another dipeptide ‘Y’ with only one Sulphur atom and two nitrogen atoms is formed. Let ‘P’ is the
number of oxygen atoms in ‘Y’ and ‘Q’ is the number of chiral centres in ‘Y’. Then report the value of
(P + Q)

Common Content for Question No. 3 to 4

Consider the following graph of ‘Bond order’ vs ‘total number of protons’ in diatomic species ‘a’ to ‘f’
According to molecular orbital theory, if bond order is zero at ‘a’, ‘c’ and 'g' (protons in ‘g' = 20’)
then species is highly unstable & most likely non–existent.

3) Among the following species, how many will either relate to ‘d’ or ‘e’?

4) If X2 diatomic species relates to ‘b’ and Y2 relates to ‘f’, then total number of electrons in σ*2s of
both X2 & Y2 species for ground state only.

Common Content for Question No. 5 to 6

16 g of Sulphur is reacted with ‘X’ g of O2 to produce both SO2 & SO3 gases. The actual proportion of
SO2 & SO3 produced depends on value of ‘X’ (relative to mass of Sulphur taken). It is best to
associate combustion of Sulphur to SO2 & SO3 using two separate chemical equations. (S = 32, O =
16)

5) Find mass % of SO2 in product mixture if X = 25 (refer initial data in comprehension) and no
sulphur is left behind.

6) If value of X = 20, then the ratio of moles of SO2 & SO3 formed as products, if none of the reactant
is left behind is a : 1. The value of 'a' is.

SECTION-II (ii)

1) Correct matching from the two lists.

List-I List-II

(P) (1) Positional isomers

(Q) CH3CH2CH2OH & CH3OCH2CH3 (2) Metamers

(R) (3) Chain isomers

(S) (4) Functional isomers

2) Correct matching from the two lists.

List-I List-II (Type of π–bonds)

(P) NO3– (1) Only pπ–dπ bonding

(Q) (CH3)2O (2) No pπ–dπ and pπ–pπ bonding

(R) SO3 (3) Only pπ–pπ bonding

(S) SO42– (4) Both pπ–pπ and pπ–dπ bonding

SECTION-III

1) How many of these do not give self–aldol reaction?

(i) CH3CHO (ii) CH3COCH2CH3 (iii) CD3COCD3 (iv)

(v) Ph–CHO (vi) HCHO (vii) (viii)

(ix)

2) How many of these can exhibit geometrical isomerism?

(i) (ii) (iii)

(iv) (v) (vi) (CH3)2C=CHMe

(vii) (viii) (ix)

3) A weak acid HA aqueous solution is found to have conductivity value of 4.5 × 10–4 ohm–1cm–1. Its
concentration was found to be 0.01 molar and conductivity of water is found to be 3 × 10–4 Scm–1. If
Ka of this acid HA is y × 10–6, report value of ‘y’? (Answer to the nearest integer)

Given: [Limiting molar conductance]

4) A sample containing fixed amount of ideal gas mass, is subjected to 50% increase in T & 25%
decrease in pressure. If the ratio of initial density over final density of gas sample is x : 1, the value
of ‘x’ is?

5) If reaction has K = 6.93 × 10–3 min–1 at 20°C, then half-life of ‘A’ at 70°C
is Find value of ‘x’, if temperature coefficient of reaction is assumed to have a constant value of 2 ?
6) SO3 decomposes into SO2 & O2 according to this equation in a rigid closed vessel of volume 1 litre.

At time = 20 min, rate of appearance of O2 was 20 g L–1 per minute and at that instant is

reported as Then value of ‘x’ is? [Atomic mass of S = 32]

PART-3 : MATHEMATICS

SECTION-I

1) Which of the following is/are correct

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

2) Let –x + y + 2 = 0 and 2x – y + 6 = 0 be tangent of parabola having focus at (1, 3) then :

(A) Equation of directrix of parabola is 3x + 4y = 11

(B)
Length of latus rectum of parabola is
(C) Equation of tangent at vertex is 3x + 4y – 13 = 0
(D) Equation of axis of parabola is 4x – 3y + 5 = 0

3) Which of the following is (are) correct (where [.] represent GIF and {.} represent fractional part
function)

(A)
Let then number of points where f(x) is discontinuous is 120
Let then number of points where f(x) is discontinuous is
(B)
18

(C)

(D)
Let then number of real roots of equation f(x) = 0 is 3

4) If Ai is the area bounded by where and , a1

= 0, b1= 32 then

(A) A3 = 128
(B) A3 = 256

(C)

(D)

SECTION-II (i)

Common Content for Question No. 1 to 2

Let is a factor of then answer the following questions :

1) The value of ‘2a+b’ is :

2) Number of real roots of equation is:

Common Content for Question No. 3 to 4

Let ‘z1’ and ‘z2’ be any two distinct complex number which satisfy z120 = 1.

3) The probability that is (where p and q are co-prime) then the value of q
– 2p is :

4) The number of possible ordered pair (z1, z2) such that

Common Content for Question No. 5 to 6

Consider all the permutation of the word ‘MAHAKUMBH’

5) Number of permutation of the word in which vowels are in alphabetical order is P then the value

of is :

6) Number of permutation of the word if no two alike letters are together is N then total number of
divisor of N is :

SECTION-II (ii)

1) Let A be any 3 × 3 matrix such that |A| = 2, then match the column

List-I List-II

(P) (1)

(Q) (2)
 (R) (3)

(S) (4)

2) Match the following

List-I List-II

If , angle between each pair of vectors is and

(P) (1) 3
,then is equal to

If is perpendicular to is perpendicular to is

(Q) perpendicular and then (2) 2

is equal to

and
(R) (3) 4
, then is equal to

If and then
(S) (4) 5
is equal to

SECTION-III

1) If , then the value of [8L] ( where [ . ] is represent G.I.F )

2) Let then number of solutions of the equation is :

3) Number of solution of the equation is 3 , then sum of all possible integers value
of ‘k’ is :

4) The value of integral dx is :

5) Let and then total number of even divisor of N is

(where [.] represents greatest integer function).
6) The value of is equal to :
 ANSWER KEYS

PART-1 : PHYSICS

SECTION-I

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

SECTION-II (i)

Q. 5 6 7 8 9 10
A. 2 2 1 1 4 6

SECTION-II (ii)

Q. 11 12
A. 1314 3143

SECTION-III

Q. 13 14 15 16 17 18
A. 6 6 7 4 7 3

PART-2 : CHEMISTRY

SECTION-I

Q. 19 20 21 22
A. A,B,C A,B,D C,D A,B

SECTION-II (i)

Q. 23 24 25 26 27 28
A. 5 5 2 2 0 1

SECTION-II (ii)

Q. 29 30
A. 3442 3241

SECTION-III

Q. 31 32 33 34 35 36
A. 5 5 9 2 2 5

PART-3 : MATHEMATICS
 SECTION-I

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

SECTION-II (i)

Q. 41 42 43 44 45 46
A. 8 3 3 2400 504 48

SECTION-II (ii)

Q. 47 48
A. 1342 2413

SECTION-III

Q. 49 50 51 52 53 54
A. 6 3 3 2 8 6
 SOLUTIONS

PART-1 : PHYSICS

2)

The forces acting on the wedge are its weight mg and a force K, perpendicular to the inclined
plane; the magnitude of the latter may change with time. As a result of these two forces, the
only component of the wedge’s acceleration a parallel to the inclined plane is g sin α (as
measured in an inertial frame). Newton’s equations of motion remain valid in an accelerating
frame of reference fixed to the wedge only if an ‘inertial force’ –m’a is added to the forces
actually causing the motion of a body. Here m’ is the mass of the body under examination (e.g.
that of a small volume of water).

The resultant of the gravitational and inertial forces acting on the mass m’ must be
perpendicular to the inclined plane as the components parallel to it cancel each other. The
bodies on the wedge (the glass and the water in it) ‘feel’ as if they were in a gravitational field
perpendicular to the inclined plane, with the consequence that the surface of the water lies
parallel to the plane.
This statement does not depend on the motion of the plane; it can be fixed or move freely or
even–as the result of a small force – be shaken to and fro. As long as the friction between the
inclined plane and the wedge is negligible and the wedge does not rise off the plane, the shape
of the water surface cannot be other than a plane parallel to the inclined surface.

3)
V2 – V = 3V
V2 = 4V

4)

Let ‘2a’ & ‘2b’ be the major axis & minor axis length.

....(i)

....(ii)
We get a = 5, b = 2.5
5)
As observed from the graph we get 2 maxima in every one second.

6)

we get,

7)

W.D.

We get work done =

8)

We get, F =

10)

12) For (P) :

For (Q) :
Initial condition Q1 = 3C

Q2 = 0
For (R) :
T1 cos θ1 = T2 cos θ2 = T ….(1)
T1 sin θ2 = m1g ….(2)
T2 sin θ2 = m2g ….(3)

⇒
For (S) :

13)

N + 45° sin 53° = mg = 40

N=6
F = 5 cos 53° + µN

14)
VL = VC = resonance
V = iR ⇒

V = iωL ⇒

15)

4u = 11v

16)

17)
 ⇒ ℓ = 7m

18)

PART-2 : CHEMISTRY

19)

(A) Compound ‘e’ has maximum carbon content, so maximum heat of combustion.
(B) Compound ‘d’ is most stable alkene. (Maximum hyperconjugable H)

(C)
 (D)

20)

(A)

(B)
(C) CN– only → imme. Conju. not formedNo buffer formed.

(D)

21) Heat of neutralization Q ∝ nacid/base (depends on LR)

VSolution n/V

(A) n1 = 50 m mol n2 = 20 m mol 150 mL 20/150

(B) n1 = 30 m mol n2 = 15 m mol 100 mL 15/100

(C) n1 = 6 m mol n2 = 48 m mol 100 mL 6/100 (lowest)

(D) n1 = 42 m mol n2 = 72 m mol 100 mL 42/100 (highest)

TFinal of 4 solutions: D > B > A > C

23) (Trp–Trp) is dipeptide with maximum DOU and total ‘10’ π–bonds.

24) Met–Pro or Pro–Met is the dipeptide ‘Y’.

P=3
Q=2
25)

26) X2 = Li2 ⇒
Y2 = F2 ⇒ [KK]

27) Assume ‘x’ grams of ‘S’ converting to SO2 gas, then

30)

31)

(i), (ii), (iii), (iv) will give self–aldol reaction.

(v), (vi), (vii), (viii) and (ix) will not give self–aldol reaction.

32) (i), (iii), (iv), (vii) and (ix) exhibit geometrical isomerism.

33) KSolution – KWater = KSolute or Electrolyte)

KHA = KSolution – ⇒ (4.5 × 10–4 – 3 × 10–4) = 1.5 × 10–4 Scm–1.
 (3% dissociation)

34)

Initial Final
P1 = P P2 = (3/4)P = 0.75P
T1 = T T2 = (3/2)T = 1.5T
d1 = d d2 = ?

35) At 20°C

Hence = 2.

36)

PART-3 : MATHEMATICS

37)

Compare coefficient of x60 and x30

Both side in expansion of to get A and B for (C) compare coefficient of

x30 in

38)
Reflection of (1, 3) in the tangent lines lie on directrix

Directrix passes through (5, –1) and (–3, 5)

So equation of directrix is

3x + 4y = 11

39) (A) Number of points of discontinuity = 100 + 60 – 40 = 120

Number of points of discontinuity is 19

40)

a1 = 0, b1 = 32,

Area of ith loop (square) =

41)

α + β = 1, αβ = –1

….(1)

….(2)

Hence a= 1 × 3 × 7 = 21
Also

= = –34
have three real and distinct roots.

42)

α + β = 1, αβ = –1
….(1)
….(2)

⇒
 Hence a= 1 × 3 × 7 = 21

Also

= = –34
have three real and distinct roots.

43)

⇒
⇒

No of favourable cases
h(s) = 120C2

P(E) =

44)
= 2 sin 15°
⇒ θ ≤ 30°

45)
= 126 × 120

46)

P : A are together
Q : M are together

R : H are together

47)
(P)

(Q)

(R)

(S)
=
= 312 . 28

48)

(P)
⇒
∴
(Q) is perpendicular to
⇒ ....(i)
is perpendicular to
⇒ ....(ii)
is perpendicular to
⇒ ...(iii)
From (i), (ii) and (iii), we get
49)

8L = 2π

50) 7 – 3x = y2

Now

⇒
⇒
⇒
⇒
⇒
⇒

⇒ , (Rejected)

51)
, k < 4 and
⇒ k=3

52)

Let , 3x2 dx = dt
53)

Also

⇒ P < 2505

54) Let
Defined as

Defined as

= (4 – 0) + 2 = 6