25-05-2025

1001CJA101021250010 JA

PART-1 : PHYSICS

SECTION-I (i)

1) Consider the shown uniform solid insulating sphere with a short and light electric dipole of dipole
moment (embedded at its centre) placed at rest on a horizontal surface. An electric field is
suddenly switched on in the region such that the sphere instantly begins rolling without slipping.
Speed of the sphere when the dipole moment becomes horizontal for the first time is (m = Mass of

the sphere)

(A)

(B)

(C)

(D) zero

2) A light ray incident along vector strikes on the x-z plane from medium of refractive
index and enters into medium II of refractive index is μ2. The value of μ2 for which the value of
0
angle of refraction becomes 90 , is

(A)

(B)

(C)

(D)

3) A rod of mass M & length ℓ is hinged at one of the ends. At another end, disk of mass M and
radius R is attached and it can freely rotate about its centre (that is point of attachment with rod).
For small oscillation about hinge what is time period of the system :-

(A)

(B)

(C)

(D) None of the above

4)

If i = 0.25 amp. in figure then value of R is:

(A) 48Ω
(B) 12Ω
(C) 120Ω
(D) 42Ω

SECTION-I (ii)
1) In the circuit shown in figure, calculate the following

(A) Potential difference between points a and b i.e. Va – Vb , when switch S is open is 12 Volt
(B) current through the switch S when it is closed is 3A from a to b.
(C) current through the switch S when it is closed is 3A from b to a.
(D) potential of point ‘a’ after closing the switch is 18 Volt.

2) The minimum and maximum distances of satellite from the center of the earth are 2R and 4R
respectively, where R is the radius of earth and M is the mass of the earth.

(A) Radius of curvature of the path of the particle at apogee is 8R/3

(B) Radius of curvature of the path of the particle at apogee is 2R/3
(C) Radius of curvature of the path of the particle at perigee is 4R/3
(D) Radius of curvature of the path of the particle at perigee is 8R/3

3) A prism of R.I. and apex angle A is (in air) shown. Light is incident from PQ side at angle of

incidence i. (0 < i < 90°). Choose the correct options.

(A) If A = 40° then light incident at all angles will be refracted from surface PR.
If A = 80° then light incident at some angles will be refracted and at some other angles light
(B)
will be reflected.
(C) If A = 92° then light incident at all angles will be reflected back.
(D) Whatever is the value of A, light definitely emerge from the surface PR.

4) A block of mass m is placed on another block of mass 2m as shown. A horizontal force F acts on
the block m at height '2b' without any relative motion between blocks. If Fmin is minimum force for
topple the block, a is acceleration of 2m and f is friction between blocks then
(A) Fmin = 3mg
(B) Fmin = mg/3

(C)
a=

(D)

5) A conducting wire has a non uniform cross section as shown in figure. A steady current flows

through it. Which of the following is/are true :

(A) Drift speed of electrons at A is less than at B.

(B) Drift speed of electron at A is more than at B.
(C) Magnitude of electric field at A is less than magnitude of electric field at B.
(D) Electric field is zero at every point, inside of wire.

6) In the network shown points, A, B and C are at potential of 70 V, zero and 10 V respectively.

(A) Point D is at a potential of 40 V

(B) The currents in the sections AD, DB, DC are in the ratio 3 : 2 : 1
(C) The currents in the sections AD, DB, DC are in the ratio 1 : 2 : 3
(D) The network draws a total power of 200W

SECTION-III
1)

The current density at a point is . Find the rate of charge flow (in A) through a

cross sectional area

2) A particle is projected from point A, that is at a distance 4R from the centre of the Earth, with
speed v1 in a direction making 30° with the line joining the centre of the Earth and point A, as
shown. Find the speed v1 if particle passes grazing the surface of the earth. Consider gravitational

interaction only between these two. (use = 6.4 × 107 m2/s2) Express your answer in the form

m/s and write the value of X.

3) Four concentric conducting shells A, B, C, D are of radii a, 2a, 3a, 4a respectively. B and D carry
charges +Q and –Q whereas A & C are neutral. Now the shell C is earthed. Potential of shell A after

C is earthed is , find x

4) A charge 2 × 10–3 C and mass 1 gm is projected from level ground with a velocity of 10 m/s at an
angle of 37° with the horizontal (x direction). The electric potential in space is given by V = 3x + 4y.
What is the speed of the charge (in m/s) when its y coordinate is maximum (g = 10m/s in vertical

downward direction)?

5) A ray of light enters a spherical water droplet, and after three internal reflections it travels into
its original direction as shown in the figure. The angle of incidence of the ray when it entered into

the droplet is α and corresponding angle of refraction is β. Find value of (The refractive
index of water is n=4/3).

6) Four uniform rods of equal length ℓ and mass m are rigidly joined together at their ends to form a
square framework ABCD. The framework is free to rotate in a vertical plane about a fixed smooth
horizontal axis passing through A. The framework is slightly displaced from rest from its position of
unstable equilibrium. If the maximum angular velocity reached in subsequent motion of its centre

moves in its own plane is given by . Fill the value of x.

7) Half ring of mass m and radius R is released from the position shown in diagram. A small point
mass of same mass is also fixed at the end as shown in figure. If the initial acceleration of point mass

m is then find the value of 2β.

8) For what value of R in circuit, current through 4Ω resistance is zero.

PART-2 : CHEMISTRY
 SECTION-I (i)

1)
Major product (A) of above reaction :

(A)

(B)

(C)

(D)

2) In which of the following process oxidation state of underlined atom changes :

(A) N2O4

(B) BCl3

(C) P4O10
(D) None of these

3) A solution of x moles of sucrose in 100 grams of water freezes at –0.2ºC. The amount of ice that
will separate out when it is cooled upto –0.25°C is :-

(A) 18 grams
(B) 20 grams
(C) 25 grams
(D) 23 grams

4) A certain mass of an ideal gas absorbs 80 kJ heat and simultaneously the gas is expanded from 2L
to 10L against a constant external pressure of 25 bar. What is ΔU for gas in the process. (1 bar - L =
100J) ?

(A) 280 kJ
(B) –120 kJ
(C) 60 kJ
(D) 100 kJ

SECTION-I (ii)

1) In which product formation takes place according to Hoffmann's rule

(A)
(B)

(C)

(D)

2) Correct statement about monochlorinated products of above reaction :

(A) Total 10 monochlorinated products are formed

(B) 6 resolvable products are formed
(C) 4 enantiomeric pair are obtained
(D) 7 fractions are obtained after fractional distillation of product mixture

3)

The cyanide process of gold extraction involves leaching out gold from its ore with CN⊝ in the
presence of Q in water to form R. Subsequently, R is treated with T to obtain Au and Z. Choose the
correct option(s).

(A) T is Zn
⊝
(B) R is [Au(CN)4]
2⊝
(C) Z is [Zn(CN)4]
(D) Q is O2

4) π bonding is not involved in

(A) Ferrocene
(B) Dibenzene chromium
(C) Zeise’s salt
(D) Grignard reagent

5) Select the correct statement(s):

The molecularity of an elementary reaction indicates the number of reactant molecules or ions
(A)
that collide with each other.
 The rate law of an elementary reaction can be predicted simply from the stoichiometry of the
(B)
reaction.
The slowest elementary step in sequence of the reactions determines the overall rate of
(C)
formation of product in a complex reaction.
(D) No intermediate species are involved in an elementary reaction.

6) Which of the following statements is/are wrong about a reaction at equilibrium ?

(A) A catalyst increases equilibrium concentration of products if used with a promoter.

(B) Addition of catalyst speeds up the forward reaction more than the backward reaction.
(C) Equilibrium constant of an exothermic reaction decreases with increase of temperature.
(D) Kp is always greater than Kc for a gaseous reaction.

SECTION-III

1) Number of correct statements: (1)

(2)

(3)

(4)

(5)

(6)

2) Find total number of compounds which can produced by Wurtz reaction in good yeild

(i) (ii) CH3 – CH3 (iii) (iv)

(v) (vi) (vii)

3) How many reaction have CORRECT major product?

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

(5)

4) How many ore are reduced by self reduction method ?

Cinnabar, Chalcocite, Pyrolusite, Calamine, Epsom salt, Sphalerite

5)

Count the no. of ions which can form both low spin & high spin complexes when co-ordination no. 6
Co+3, Ni+2 , Cr+3 , Fe+2, Fe+3, Cu+2, Ti+3, Co+2

6) How many of the following metallurgies given below involve leaching ?

Al2O3 —→ Al, ;
Ag2S —→ Ag ;
Au(ore) —→ Au ;
PbS —→ Pb
MgCl2 —→ Mg ;
FeCO3 —→ Fe ;
HgS —→ Hg

7) 0.05 moles of N2 gas dissolves completely in 900 gm water at 27°C, as shown in figure.
The available space between piston and liquid solution is 16.42 L. If number of moles of N2 in

gaseous phase, above solution is 'N' then find nearest integer value of ? (KH for N2 gas = 7.2 ×

104 atm.) (R = 0.0821 atm-L/mole-K)

8) For the first order reaction : A → 2B, following plot is observed experimentally. Calculate time
required for 75% completion of reaction in hours : [Given : log 2 = 0.3, log 3 = 0.48]
 PART-3 : MATHEMATICS

SECTION-I (i)

1) where C is a constant, then at x = 1 is equal to

(A)

(B)

(C)

(D)

2) If , then family of lines always passes through a fixed

point whose coordinates are (a > 0, b > 0, c > 0)

(A) (–1,2)

(B)

(C)

(D) (1,–2)

3) The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = {x ∈ R : x2 + 30
≤ 11x} is :

(A) 122
(B) –222
(C) –122
(D) 222
4) In a problem of differentiation of , one student writes the derivative of as and he
finds the correct result. If g(x) = x2 and , then function f(x) is

(A)

(B)

(C)

(D)

SECTION-I (ii)

1) Consider f(x) = (x – 1) tan–1x – . Identify which of the following statement(s)

is(are) correct ?

(A)

(B) f'(x) is increasing in [1,∞)

(C) f(x + 4) – f(x) > 2π ∀ x ∈ [1,∞)
(D) f(x + 4) – f(x) < 2π ∀ x ∈ [1,∞)

2) If
Where p ∈ N, q ∈ I and C is the constant of integration then

(A) p = 2
(B) p = 3
(C) q = –2
(D) q = 2

3) Which of the following option(s) is/are correct ?

(A)

(B)

(C)
(D)

4) If family of straight lines ax + by + c = 0 always passes through a fixed point , then

equation 36ax2 + 8bx + 2c = 0 has

(A) at least one root in [0,1]

(B)
at least one root in
(C) at least one root in [–1,2]

(D)
at least one root in

5)

Let , then find the value of is

greater than

(A) 0
(B) 2
(C) 4
(D) 6

6) let P(x) be polynomial of degree 4 and it vanishes at x = 0. Given P(–1) = 55 and P(x) has relative
maximum and relative minimum at x= 1, 2, 3. Which of the following is/are true ?

(A) P(1) = –9
(B) Area of the triangle formed by the points of extremum of P(x) is 1.
(C) Number of negative integers in the range of y = P(x) is 9.
(D) Area of the triangle formed by the points of extremum of P(x) is 2.

SECTION-III

1) Let α, β ∈ be such that . Then 2α + 6β equals

2) If
(where C is constant of integration and α ∈ N), then value of α is
3) If the minimum area of the triangle formed by the line with co-ordinate
axes is 'a' sq. units, then the value of a is {Where b ∈ [0,8]}

4) The fuel charges for running a train are proportional to the square of the speed generated in
km/hr and the costs Rs. 48/- per hour at 16km/hr. If the most economical speed when the fixed

charges are amount to Rs 300/- per hour is 'v' km/hr then value of is

5) If the tangent to the curve 4x3 = 27y2 at the point (3,2) meet the curve again at the point (a,b)
then (|a| + |b|) equals

6) If and f(1) = 0, then the value of |3f(e2)| is equal to

7) f : R → (0,∞), f(x) is twice differentiable function such that f '(x) = xf(x) for x = 1,2,3,4, then find
minimum number of roots of the equation f(x).f "(x) = (f '(x))2 + (f(x))2

8) If the angle between the pair of straight lines formed by joining the points of intersection of x2 +

y2 = 4 and y = 3x + c to the origin is a right angle, then is

 ANSWER KEYS

PART-1 : PHYSICS

SECTION-I (i)

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

SECTION-I (ii)

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

SECTION-III

Q. 11 12 13 14 15 16 17 18
A. 6 8 4 6 4 5 3 1

PART-2 : CHEMISTRY

SECTION-I (i)

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

SECTION-I (ii)

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

SECTION-III

Q. 29 30 31 32 33 34 35 36
A. 6 3 5 2 4 3 8 4

PART-3 : MATHEMATICS

SECTION-I (i)

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

SECTION-I (ii)

Q. 41 42 43 44 45 46
A. A,B,D A,D A,D A,B,C,D A,B,C,D A,B,C
 SECTION-III

Q. 47 48 49 50 51 52 53 54
A. 3 7 2 4 1 8 3 4
 SOLUTIONS

PART-1 : PHYSICS

1)

or

or

5) Potential difference between points a and b i.e. Va – Vb , when switch S is open is –12 Volt.
current through the switch S when it is closed is 3A from b to a.

6) (a) Applying conservation of angular momentum

Mv1(2R) = mv2 (4R)

V1 = 2v2 ……….(1)
From C.O.E

……………..(2)
Solving Eqs. (1) and (2)

V2 = , V1 =
(b) if r is the radius of curvature at point B

V2 = (putting value of v2)

8) For no relative motion

For topping about P w.r.t. 2m
= 2m × g/9

9) by i = nAeVd
⇒ AB > AA ⇒ Vd|B < Vd|A

11)

The rate of flow of charge = current = I =

⇒I= = (2 × 104)

12)

Conserving angular momentum :

m(v1 cos60°)4R = mv2R ⇒ = 2.

Conserving energy of the system :

⇒ Þ

14)
15)
If incidence angel is a and b is angle of refraction the according to law.
(α – β) + 3(180º – 2β) + (α – β) = 360º,
That is

So,

16)

17)
τ = I0α

18) Ans. 1 Ω

0 – 2i – Ri – 4 = 0
 10 – 2i = 6 i=2

PART-2 : CHEMISTRY

19)

20)

(A)

(B)

(C)

21)

Ans : → Given : → moles of sucrose = x moles

mass of water (solvent) = 100 g

malality = = 10 x mol/kg
Freezing point = –0.2°C
Depression in freezing point. = ΔTf = 0.2°C
ΔTf = Kf m

Kf = = =
⇒ New temperature = –0.25°C
= 0.25°C

KF = , mass of solvent left = Wg

moles of sucrose = x moles
= 0.25 = Kf × m

0.25 =

W= = 80 g
Mass of ice separated = initial mass of solvent – mass of solvent left
= 100 – 80 = 20g – .

22) W = – PΔV = –25 × 8

= – 200 bar- L
= – 20 kJ
ΔU = Q + W
= 80 – 20 = 60 kJ

23) (A) Major product is CH2 = CH – CH2 – CH3 E2 with characters of E1CB. (B) Major
product is CH3– CH = CH – CH3 E2
(C) Major products is CH2 = CH – CH2 – CH3 E2
(D) Major products is CH2 = CH – CH2 – CH3 E2

24)

25)

4Au(s) + 8CN⊝(aq) + 2H2O(aq) + O2(g) → 4[Au(CN)2]⊝ (aq) + 4OH⊝(aq)

(Q)
2[Au(CN)2] (aq) + Zn(s) → [Zn (CN)4]2⊝(aq) + 2Au(s)
⊝

(R) (T) (Z)

26)

27)

(A) The molecularity of an elementary reaction indicates how many reactant species take part
in the step. Thus in unimolecular reaction, only one molecule participates in the step and in
bimolecular reaction, two molecules participate in the step.
(B) The rate law of an elementary relation can be predicted by simply seeing the stoichiometry
of reaction. Thus, for the reaction 2A + B → C, the rate law cannot be written as rate =
k[A]2[B]
(C) The slowest elementary step in sequence of the reactions governs the overall rate of
formation of product. This step is called rate determining step.

28)

Equilibrium constant of an exothermic reaction decreases with increase of temperature.

 30) i, ii, v

34) Al2O3 + 2NaOH + 2H2O 2NaAlO2 + 3H2O —→ leaching

⊝ ⊝ 2⊝
Ag2S + 2CN —→ [Ag(CN)2] + S —→ leaching
Au + 2CN + 2H2O + O2 —→ [Au(CN)2]⊝ + 4OH⊝ —→ leaching
⊝

PbS—→ No leaching ;
MgCl2 —→ No leaching —→ No leaching
FeCO3 —→ No leaching
HgS —→ No leaching

35) = 7.2 × 104 ×

× 16.42 = n × 0.0821 × 300
moles = 48

36) A —→ 2B
t=0 C 0
t = 72 C – x 2x
At t = 20 min
C – x = 2x
∴ x = C/3

∴ [A]t at t = 72 minutes =

For 75% completion, [A]t =

PART-3 : MATHEMATICS

37)
= f(x) ex + C

Where

38)

or

39)

S = {x ∈ R : x2 + 30 – 11x ≤ 0}
= {x ∈ R, 5 ≤ x ≤ 6}
Now f(x) = 3x3 – 18x2 + 27x – 40
⇒ f'(x) = 9(x – 1)(x – 3)
which is positive in [5, 6]
⇒ f(x) increasing in [5, 6]
Hence maximum value = f(6) = 122

40)

by wrong calculation

⇒ x2 f '(x) = 2x f'(x) – 4 f(x)

On integrating
∴

41)

f'(x) > 0 ∀ x > 1

⇒ f(x) is increasing function
f "(x) > 0
⇒ f(x) is increasing function ∀ x > 1

Using LMVT in [x, x + 4], x > 1

for c ∈ (x,x + 4)
∴ f(x + 4) – f(x) < 2π ∀ x ∈ [1,∞)

42)

x3 – 1 = t2
3x2dx=2tdt

43)
4 – sin2x = t
–2 cos2x dx = dt

Now
2(cos2 – cos3)

44)

ax + by + c = 0 passes through
⇒ 3a + 2b + 2c = 0
Let f '(x) = 36ax2 + 8bx +2c
f(x) = 12ax3 + 4bx2 + 2cx + d
f(x) is continuous & differentiable

f(0) = d, f a+b+c+d=d

f(0) = f ⇒ at least one root in

45)

cos–1sin(cos–1x) + sin–1(cos(sin–1x))

⇒ f(x) = 1

46)

P'(x) = λ(x – 1)(x – 2)(x – 3)

P'(x) = λ(x3 – 6x2 + 11x – 6)

P(0) = 0 ⇒ t = 0

P(–1) = 55 ⇒
λ=4
P(x) = x4 – 8x3 + 22x2 – 24x
P(1) = –9
A(1, –9), B(2, –8), C ≡ (3, –9)
area (ΔABC)

Range : [–9, ∞)
47)

If α ≠ 1, then
∴α=1

⇒
⇒ 6β = 1

48)
Put cosecx + cot x = z

cosecx – cot x =

49)

y = mx + 4 (m > 0)

Δ is minimum if m is maximum
∵

⇒b=4
so, at b = 4 value of

so, Area

50)

Fuel charges per hour = kv2 ⇒ 48 = k. 162

⇒ Fuel charges per hour =

Charges per hour =

Expenses of journey
where v = speed s = distance

Maximum occures when

(∵ ax + , a,b, > 0, x > 0, has minimum when ax = )

v2 = 16.100
v = 40

51)

52)

Put ℓnx = t ⇒ dx = etdt

∴ f(1) = 0 + C
∴

53)

Consider
H(1), H(2), H(3), H(4) = 0 by Rolle's Theorem

has atleast 3 roots in (1,4)

54)

x2 + y2 – 4(1)2 = 0

(use principle of homogenisation)

(coefficient of x2 + coeff. of y2 = 0)