09-02-2025

7501CJA101001240079 JA

PHYSICS

SECTION-I

1) If a body is moving with constant speed, then its acceleration :-

(A) Must be zero

(B) May be variable
(C) May be uniform
(D) Both (2) & (3)

2) A shell is fired vertically upwards with a velocity v1 from the deck of a ship travelling at a speed of
v2. A person on the shore observes the motion of the shell as parabola whose horizontal range is
given by

(A)

(B)

(C)

(D)

3) A student is cleaning a blackboard by moving a light duster up and down on it. The coefficient of
static friction between the duster and the board is µs. The duster will not move, no matter how large

the force is if (ignore gravity) :-

(A) tan β > µs

(B) tan β < µs
tan β >
(C)

tan β <
(D)
4) In the arrangement of rigid links of equal length , they can freely rotate about the joined ends as
shown in the figure. If the end U is pulled horizontally with constant speed 20 m/s, find the approx.

speed of end P when the angle SUT is 90°.

(A) 5 m/s
(B) 10 m/s
(C) 7.1 m/s
(D) 14.12 m/s

5) A mass m starting from A reaches B of a frictionless track. On reaching B, it pushes the track with

a force equal to x times its weight, then the the applicable relation is

(A)

(B)

(C) h = r

(D)

6) A particle experiences a variable force in a horizontal x-y plane. Assume distance

in meters and force is newton. If the particle moves from point (1, 2) to point (2, 3) in the x-y plane,
the Kinetic Energy changes by

(A) 50.0 J
(B) 12.5 J
(C) 25.0 J
(D) 0 J

7) A bomb of mass m is projected from the ground with speed v at angle θ with the horizontal. At
the maximum height from the ground it explodes into two fragments of equal mass. If one fragment
comes to rest immediately after explosion, then the horizontal range of centre of mass is :-

(A)
(B)

(C)

(D)

8) A bullet of mass m moving with velocity v strikes a suspended wooden block of mass M. If the
block rises to height h, then the initial velocity v of the bullet must have been :-

(A)

(B)

(C)

(D)

9) A small object of uniform density rolls up a rough curved surface with an initial velocity v while

doing pure rolling. It reaches upto a maximum height of w.r.t. the initial position. The object is

(A) Ring
(B) Solid sphere
(C) Hollow sphere
(D) Disc

10) A slender uniform rod of mass M and length ℓ is pivoted at one end so that it can rotate in a
vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically
above the pivot and then released. The angular acceleration of the rod when it makes an angle θ

with the vertical is :

(A)

(B)

(C)
(D)

11) Two spheres each of mass M and radius R/2 are connected with a massless rod as shown in the
figure. What will be the moment of inertia of the system about an axis passing through the centre of

one of the spheres and perpendicular to the rod.

(A)

(B)

(C)

(D)

12) The work done in increasing the size of a rectangular soap film with dimensions 8 cm × 3.75 cm
to 10 cm × 6 cm is 2 × 10–4 J. The surface tension of the film in N/m is:

(A) 1.65 × 10–2

(B) 3.3 × 10–2
(C) 6.6 × 10–2
(D) 8.25 × 10–2

13) A vertical U-tube of uniform inner cross section contains mercury in both sides of its arms. A
glycerin (density = 1.3 g/cm3) column of length 10 cm is introduced into one of its arms. Oil of
density 0.8 gm/cm3 is poured into the other arm until the upper surfaces of the oil and glycerin are
in the same horizontal level. Find the length of the oil column, (Density of mercury = 13.6 g/cm3) :-

(A) 10.4 cm
(B) 8.2 cm
(C) 7.2 cm
(D) 9.6 cm

14) A tank is filled upto a height h with a liquid and is placed on a platform of height h from the
ground. To get maximum range xm a small hole is punched at a distance of y from the free surface of
the liquid. Then :-

(A) xm = 3h
(B) xm = 1.5 h
(C) y = h
(D) y = 0.75 h

15) A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing
a certain mass of gas. The cylinder is kept with mass of gas. The cylinder is kept with its axis
horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically.
The period of oscillation will be :-

(h = initial length of cylinder)

(A)

(B)

(C)

(D)

16) Vertical displacement of a plank with a body of mass 'm' on it is varying according to law y = sin
ωt + cos ωt. The minimum value of w for which the mass just breaks off the plank and the
moment it occurs first after t = 0 are given by :( y is positive vertically upwards)

(A)

(B)

(C)
(D)

17) A tuning fork of frequency 240 Hz vibrates at one end of the tube shown in the figure. If sound is

not heard at other end of the tube, the velocity of the waves will be-

(A) 274 m/s

(B) 672 m/s
(C) 168 m/s
(D) 240 m/s

18) Here given figure (i) shows snap shot at t = T/4 and figure (ii) shows motion of

particle Then the possible equations of the wave will be .

(A)

(B)

(C)
Both

(D)
Both

19) A pendulum clock loses 12s a day if the temperature is 40°C and gains 4s a day if the
temperature is 20°C. The temperature at which the clock will show correct time, and the co-effecient
of linear expansion (α) of the metal of the pendulum shaft are respectively :-

(A) 55°C ; α = 1.85 × 10–2 / °C

(B) 25°C ; α = 1.85 × 10–5 / °C
(C) 60°C ; α = 1.85 × 10–4 / °C
(D) 30°C ; α = 1.85 × 10–3 / °C

20) A cyclic process ABCA is shown in PT diagram. When presented on PV, it would :-
(A)

(B)

(C)

(D)

SECTION-II

1) An ideal gas heat engine operates in a cornot cycle between 227°C and 127°C. It absorbs 6 kcal at
the higher temperature. The amount of heat (in kcal) converted into work is equal to :–

2) Ice starts forming in a lake with water at 0°C when the atmospheric temperature is –10°C. If the
time taken for the first 1 cm of ice to be formed is 7hour, then the time taken for the thickness of ice
to change from 1 cm to 2 cm is [in hours]

3) Two spherical vessel of equal volume, are connected by an arrow tube. The apparatus contains an
ideal gas at one atmosphere and 300K. Now if one vessel is immersed in a bath of constant
temperature 600K and the other in a bath of constant temperature 300K. Then the common pressure

(in atm) will be :

4) A body takes 4 min. to cool from 61oC to 59oC. If the temperature of surrounding is 30oC, the time
taken by the body to cool from 51oC to 49oC is T min. The value of T is ________.

5) For a certain organ pipe three successive resonance frequencies are observed at 425 Hz, 595 Hz
and 765 Hz respectively. If the speed of sound in air is 340 m/s, then the length (in metre) of the
pipe is
 CHEMISTRY

SECTION-I

1) The correct order of second ionisation potential of C, N, O and F is:

(A) C > N > O > F

(B) O > N > F >C
(C) O > F > N > C
(D) F > O > N > C

2) The electronic configuration of an element is 1s2 2s2 2p6 3s2 3p4. The atomic number and the group
number of the element ‘X’ which is just below the above element in the periodic table are
respectively.

(A) 24 & 6
(B) 24 & 15
(C) 34 & 16
(D) 34 & 8

3) Which of the following is not correctly matched

(A) [Xe] 4f14 5d10 6s2 → Transition element

(B) [Rn] 5f14 6d1 7s2 → Inner transition element
(C) [Xe] 4f14 5d10 6s2 6p6 7s2 → Representative element
(D) [Xe] 4f14 5d2 6s2 → d-block element

4) Identify the pair in which the geometry of the species is T-shape and square pyramidal,
respectively

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

5) Which of the following is having non-zero dipole moment?

(A) BF3
(B) CH4
–
(C) I3
(D) NF3

6) The correct order of bond dissociation energy among is shown in which of the
following arrangements?
(A)
(B)
(C)
(D)

7) Order of Covalent bond:

A. KF > KI; LiF > KF
B. KF < KI; LiF > KF
C. SnCl4 > SnCl2; CuCl > NaCl
D. LiF > KF; CuCl < NaCl
E. KF < KI; CuCl > NaCl

(A) C, E only
(B) B, C only
(C) B, C, E only
(D) A, B only

8)

S1 : Potential energy of the two opposite charge system increases with the decrease in distance.
S2 : When an electron make transition from higher orbit to lower orbit it's kinetic energy increases.
S3 : When an electron make transition from lower energy to higher energy state its potential energy
increases.
S4 : 11 eV photon can free an electron from the 1st excited state of He+ -ion.

(A) T T T T
(B) F T T F
(C) T F F T
(D) F F F F

9) 1 mol of nitrogen is mixed with 3 moles of hydrogen in a litre container where 66.67 % of
nitrogen is converted in to ammonia by the following reaction :

N2(g) + 3H2(g) → 2NH3(g) , then the value of KC for the reaction will be -

(A) 2 M
(B) 1/2 M
(C) 4 M
(D) 1/4 M

10) What is the mass ratio of ethylene glycol (C2H6O2, molar mass = 62 g/mol) required for making
0.25 molal aqueous solution (containing 500 gm water) and 250 mL of 0.25 molar another aqueous
solution ?

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

11) A weak acid HX (Ka = 10–5) forms a salt NaX when reacted with NaOH. What is % hydrolysis of
0.1 M NaX aqueous solution :

(A) 0.0001 %
(B) 0.001 %
(C) 0.01 %
(D) 0.1 %

12) According to Charle’s law which is incorrect

(A)

(B)

(C)

(D)

13) 0.789 g of crystalline barium hydroxide is dissolved in water. For the neutralization of this

solution. 20 mL of HNO3 is required. How many g mole of water are present in one g mole of this
base?
(Ba = 137.4, O = 16, N = 14, H = 1)

(A) 12
(B) 8
(C) 23
(D) 4

14)

Give the correct acidic order of hydrogen for following rounded.

(A) b > c > d > a

(B) c > b > a > d
(C) b > a > c > d
(D) c > b > d > a
15) Correct order of stability:

(A)

(B)

(C)

(D)

16) Arrange in the order of basic character in aqueous solution.

(A) a > b > c > d

(B) b > c > d > a
(C) b > a > c > d
(D) d > c > b > a

17)

Which of the following are correct :-

(A)
(position isomer)

(B)

(functional group)
(C)

(position isomer)
(D) All are incorrect.

18) Which is most stable conformer ?

(A)

(B)

(C)

(D)

19) Which one of the following is Z isomer :

(A)

(B)

(C)
(D)

20) Which is correct statement for inductive effect?

(A) It involves delocalisation of π-bond

(B) It is temporary effect
(C) It involves delocalisation of σ-bond
(D) It is permanent effect

SECTION-II

1) The maximum number of C-atom, which lie in same plane in C2(CN)4 molecule is _________.

2) For the standardization of Ba(OH)2 solution, 0.204 g of potassium acid phthalate was weighed
which was then titrated with Ba(OH)2 solution. The titration indicated equivalence at 25.0 ml of
Ba(OH)2 solution. The reaction involved is
KHC8H4O4 + Ba(OH)2 H2O + K+ + Ba2+ + C8H4O42–
The molarity of the base solution is (K = 39)

3) What is the density (in gm/litre) of moist air with 76% relative humidity under the condition of 1
atm pressure and 27°C. The vapour pressure of water at 27°C is 30 torr and dry air has 76% N2 and
24% O2.

4) Optically active among the following :

5)
How many geometrical isomers are possible for the above compound ?

MATHEMATICS

SECTION-I

1) Find the complete solution set of the inequality

(A)

(B)

(C)

(D)

2) The number of ordered pairs (m,n), m,n ∈ {1, 2....100} such that 7m + 7n is divided by 5 is

(A) 1250
(B) 2000
(C) 2500
(D) 5000

3) In a class of 140 students numbered 1 to 140, all even numbered students opted mathematics
course, those whose number is divisible by 3 opted Physics course and those whose number is
divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the
three courses is :

(A) 102
(B) 42
(C) 1
(D) 38

4) If α,β,γ are roots of equation 9x3 – 7x + 6 = 0, then the equation whose roots are 3α + 2, 3β + 2,
3γ + 2 is-

(A) x3 – 6x2 – 5x – 24 = 0
(B) x3 + 6x2 – 5x + 24 = 0
(C) x3 – 6x2 + 5x + 24 = 0
(D) x3 – 6x2 + 5x – 24 = 0

5) Let α and β be the roots of x2 – 6x – 2 = 0. If an = αn – βn for n ≥ 1, then the value of is

___.

(A) 1
(B) 2
(C) 3
(D) 4

6) If (1 + x)n =

(A)

(B)

(C)

(D)

7) The greatest value of the term independent of x in the expansion of ; where (α

∈ R) is-

(A) 25

(B)

(C)

(D) 1

8) If z1 = 1 + 2i and z2 = 3 + 5i, then Re =

(A)

(B)

(C)

(D)
9) The roots of the cubic equation (z + ab)3 = a3, such that a ≠ 0, represent the vertices of a triangle
of sides of length

(A)

(B)
(C)
(D) |a|

10) C1 is a circle of radius 1 touching x-axis and the y-axis. C2 is another circle of radius > 1 and
touching the axis as well as the circle C1 then radius of C2 is :–

(A) 3 –
3+
(B)

3+
(C)

(D) None of these

11) Let x, y, z are integers such that x ≥ 0, y ≥ 2, z ≥ 3 and x + y + z = 20. If k is the number of
values of ordered triplex (x,y,z), then k is divisible by

(A) 3
(B) 6
(C) 7
(D) 8

12) The slope of the line touching both the parabola y2 = 8x and x2 = –64y is:

(A)

(B)

(C)

(D)

13) If the foci of the ellipse coincide with the foci of the hyperbola , then
b2 is equal to :

(A) 9
(B) 8
(C) 10
(D) 7
14) The locus of the point of intersection of the straight lines, tx – 2y – 3t = 0, x – 2ty + 3 = 0 (t ∈R),
is :

(A) an ellipse with the length of major axis 6

(B) a hyperbola with the length of conjugate axis 3
(C) a hyperbola with eccentricity

(D)
an ellipse with eccentricity

15) The area of the triangle formed by the lines y = ax, x + y – a = 0 and the y-axis is equal to

(A)

(B)

(C)

(D)

16) The natural numbers are grouped as follows {1}, {2, 3, 4}, {5, 6, 7, 8, 9}, .... , then the first
element of the nth group is-

(A) n2 – 1
(B) n2 + 1
(C) (n – 1)2 – 1
(D) (n – 1)2 + 1

17) The sum of the infinite series

is equal to :-

(A)

(B)

(C)

(D)

18) The solution of equation is

(A) x = 0

(B)

(C) no real solution

(D) none of the above

19) If Find θ.

(A) 2
(B) 1
(C) 3
(D) 4

20) A vertical lamp-post of height 9 metres stands at the corner of a rectangular field. The angle of
elevation of its top from the farthest corner is 30°, while from another corner it is 45°.The area of
the field is :-

(A)
(B) m2
(C) m2
(D) m2

SECTION-II

1) The number of integral values of k, for which one root of the equation 2x2 – 8x + k = 0 lies in the
interval (1, 2) and its other root lies in the interval (2, 3) is ___.

2)

If , , and , then value of is

equal to:

3) Circles C1 and C2 are externally tangent and they are both internally tangent to the circle C3. The
radii of C1 and C2 are 4 and 10, respectively and the centres of the three circles are collinear. A
chord of C3 is also a common internal tangent of C1 and C2. Given that the length of the chord is

where m, n and p are positive integers, m and p are relatively prime and n is not divisible by
the square of any prime, find the value of (m + n + p).

4) then the real value of x is :

5) The value of a for which system of equations and has a

solution 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. D C C C A C D B D C A B D C A A A D B C

SECTION-II

Q. 21 22 23 24 25
A. 1.2 21.00 1.34 6.00 1.00

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. C C A A D C C B D C C A B A D C C B D D

SECTION-II

Q. 46 47 48 49 50
A. 6.00 0.02 1.15 to 1.17 600.00 16.00

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. D C D C B D C D B B D D D B D D A C B A

SECTION-II

Q. 71 72 73 74 75
A. 1.00 216.00 19.00 4.00 1.00
 SOLUTIONS

PHYSICS

3)
Duster will not move
If fL ≥ F cos β
⇒ µsN = µsF sin β ≥ F cos β

tan β ≥

7) C.O.M. follow the same parabolic path.

8) mV = (M + m)V'

⇒ V' = ,h=

h= ⇒V=

9) and ω = v/R (rolling)

⇒ ⇒ disc

10)
Taking torque about pivot τ = Ια

mgsinθ =α
⇒α= sinθ

11)

12) Change in surface energy = 2 × 10–4 J

ΔA = 10 × 6 – 8 × 3.75 = 30 cm2
= 30 × 10–4 m2
Work done W = T × 2 × (Change in area)
Now, Change in surface energy = Work done
2 × 10–4 = T × 2 × 30 × 10–4
∴ T = 3.3 × 10–2 N/m

13) P0 + ρglyg(10) = ρ0 + ρoilgh + ρHgg(10–h)

(1.3) (10) = (0.8) (h) + (13.6) (10–h)
h 9.6 cm

14)

Velocity of liquid through orifice, n = and time taken by liquid to reach the ground

t= =
∴ Horizontal distance covered by liquid

x = v.t. = =
​⇒ x2 = 4y(2h – y)

for x to be maximum, =0
∴ 8h – 8y = 0 or h = y
so xm = = 2h

15) at equillibrium position

pA – patm A = 0
when disturbed by x
pAh = p' A(h – x)

p' =
net force

(Fnet) = (p' – patm) A = A

 = ≈ x<<h k=

=ω=

T = 2π

16) y = 1 sin ωt + cosωt

y = 2sin (ωt + )
For breaking off plank

ωmin =

At t = 0, ϕ0 = , It leaves when ϕ =

Δϕ =

25) 425 : 595 : 765 = 5 : 7 : 9

hence it is a closed organ pipe

CHEMISTRY

34)
1 3

35) Assume : Mass of solvent Mass of solution

Case I :-

Case II :-
36)

=
= 10–2 = 0.01 %

40)

41) In gaseous state, basicity proportional to number of EDG.

lone pair will be delocalized.

46)
All C-atoms are present in same plane.

47)

neq KHC8H4O4 = neqBa(OH)

or

48) The average molecular weight is

Mavg = 0.76 × 28 + 0.24 × 32 = 21.28 + 7.68 = 28.96 g mol–1
The relative humidity is

Relative humidity =

where density is

and = = 0.03 atm.

Now, Mmoist air = (28.96 × 0.97) + (18 × 0.03) = 28.63 g mol–1.
Substituting these values, we get the density as

= 1.163 g L–1

49) (i), (ii), (iii), (iv), (v), (vii) are optically active.
MATHEMATICS

51) For domain

Case - I:

Case - II:

52)

53)

We have numbers 1,2,.....140.

Even numbers are 2,4,6,....140, i.e., 70
Numbers divisible by 3 are 3,6,9,...,138, i.e., 46
Number divisible by 5 are 5,10,....,140, i.e., 28
Even numbers divisible by 3 are 6,12,..., 138, i.e., 23
Even numbers divisible by 5 are 10,20,..., 140, i.e., 14.
Numbers divisible by 3 and 5 are 15,30,..., 135, i.e., 9
Even numbers divisible by 3 and 5 are 30,60,90,120, i.e., 4.
Let n(M) = Number of students opted Mathematics = 70,
n(P) = Number of students opted Physics = 46,
n(C) = Number of students opted Chemistry = 28
∴ n(M ∩ P) = 23, n(P ∩ C) = 9, n(M ∩ C) = 14, n(M ∩ P ∩ C) = 4,
Now, n(M ∪ P ∪ C) = n(M) + n(P) + n(C) – n(M ∩ P)
– n(P ∩ C) – n(M ∩ C) + n(M ∩ P ∩ C)
= 70 + 46 + 28 – 23 – 9 – 14 + 4 = 120
So, number of students who did not opt for any course.
= Total number of students –n(M ∪ P ∪ C)
= 140 – 102 = 38.

54) Put y = 3x + 2

⇒ y3 – 6y2 + 5y + 24 = 0

55) (Reference to Ex.4b, Q.31)

(α2 – 6α – 2 = 0) × α8
(β2 – 6β – 2 = 0) ×β8
(since, a10 = α10 – β10, a8 = α8 – β8, a9 = α9 – β9)
α10 – 6α9 – 2α8 = 0 .... (i)
β10 – 6β9 – 2β8 = 0 .... (ii)
From (i) + (ii)

56)

57) r = = 5 ⇒ T6 = 10C5 (x sin α)5

⇒ T6 = 10C5 (sin α cos α)5 = 10C5

T6 is maximum when sin 2α = 1 so maximum value is

58) Given, z1 = 1 + 2i, z2 = 3 + 5i and = 3 – 5i

Then Re

59)

Taking cube roots of both sides, we get

z + ab = a(1)1/3 = a, aω, aω2
where

ω=– +i , ω2 = – – i
∴ z1 = a – ab, z2 = aω – ab, z3 = aω2 – ab
|z1 – z2| = |a(1 – ω)|

= |a|

= |a|

= |a|
Similarly,
|z2 – z3| = |z3 – z1| =

60) From fig.

61) x + y + z = 20, x ≥ 0, y ≥ 2, z ≥ 3
⇒ if x ≥ 0, y ≥ 0, z ≥ 0
x + y + z = 15
⇒ non negative integral solutions 15+2C2 = 136

62) The tangent of slope m of parabola y2 = 8x is

y = mx +
if this is also a tangent to x2 = –64y then
 = 16m2 ⇒ m3 = ⇒m=

63)

foci ±ae =

Eq. of hyperbola =

foci ±ae = ±3
foci of ellipse & hyperbola coincide

∴ = ±3
b2 = 7

64) tx – 2y – 3t = 0 ... (1)

x – 2ty + 3 = 0 ... (2)

... (3)
Substiting t = tanθ,
we get the cocos of the POI of the two given st. lines as
2y = 3 (–tan2θ) ... (4)

x = –3sec2θ ... (5)

2θ 2θ
using sec2 – tan2 = 1

a = 3, b = hyperbola

e=

65) Intersection point of lines

y = ax & x + y – a = 0 is
Area of ΔBOC is

66) Let Ist element of nth group is = Tn

1, 2, 5, 10, ....., Tn
= (02 + 1), (12 + 1), (22 + 1), ....., [(n – 1)2 + 1]
∴ Tn = (n – 12) + 1

67)

Tn =

Tn =

68)

Put
Given equation becomes t2 - 4t - 1 = 0

Either

or

as e or as is (-ve) and log is not defined for (-ve) values.

sinx > 1 no solution.

69)
Multiplying and dividing with sin1°
= cot1° cosec1°

70) Clearly, one side = 9 cot 45°

=9m

& diagonal = 9 cot 30°

= m

∴ Other side =
∴ Area =
= 81 m2

71) (Reference to Jee-Main 2023)

α + β = 4, αβ = k/2
f(1) f(2) < 0
(k – 6) (k – 8) < 0
f(2) f(3) < 0
(k – 8) (k – 6) < 0

k ϵ (6, 8)
k = 7 is only one solution

72)

2 × 3 × 4 × 9 = 216
73)

Let the radius of bigger circle is R.

Then distance c2c3 = R – 10 ...(1)
distance c1c3 = R – 4 ...(2)
now (1) + (2) ⇒ c1c3 + c2c3 = 2R – 14
or c1c2 = 2R – 14
or (4 + 10) = 2R – 14
since R = 14 or
therefore, TC3 = 14
or TC1 + C1S + SC3 = 14

or 4 + 4 + SC3 = 14 ⇒

from Δ QSC3, (SQ)2 = (QC3)2 – (SC3)2

= (14)2 – (6)2 = 160

or
Now length of chord PQ = 2SQ
=
or PQ =

or
here, m = 8, n = 10, p = 1
then m + n + p = 8 + 10 + 1 = 19 Ans.

74)
75)

Adding given equations, we get

⇒ a2 + 3a – 4 = 0
⇒ (a + 4)(a – 1) = 0
⇒ a = 1 (as a = –4 is rejected)