BITSAT 2024 Question Paper

BITSAT 2024 Question Paper

Time Allowed :3 Hour Maximum Marks :390 Total Questions :130

General Instructions
Read the following instructions very carefully and strictly follow them:
1. Mode: Computer-based online test
2. Duration: 3 hours (180 minutes)
3. Sections: The exam consists of four parts:
(a) Part I: Physics (30 questions)
(b) Part II: Chemistry (30 questions)
(c) Part III: English Proficiency (10 questions) and Logical Reasoning (20
questions)
(d) Part IV: Mathematics (40 questions) or Biology (for B.Pharm candidates)
4. Total Marks: 390
5. Marking Scheme: Each correct answer awards 3 marks, and 1 mark is deducted
for each incorrect answer
6. Subjects:
(a) Physics: Mechanics, Electromagnetism, Thermodynamics, Modern Physics
(b) Chemistry: Organic, Inorganic, and Physical Chemistry
(c) Mathematics: Calculus, Algebra, Geometry (or Biology for B.Pharm
candidates)
(d) English Proficiency: Reading Comprehension, Vocabulary
(e) Logical Reasoning: Analytical and Problem-solving skills

1

Physics
√ as A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2 m. We should
1. You measure two quantities
report the correct value for AB as:
(A) 1.4 m ± 0.4 m
(B) 1.41 m ± 0.15 m
(C) 1.4 m ± 0.3 m
(D) 1.4 m ± 0.2 m

2. The dimensional formula of latent heat is:
(A) [M 0 LT −2 ]
(B) [M LT −2 ]
(C) [M 0 L2 T −2 ]
(D) [M L2 T −2 ]

3. The dimensions of the coefficient of self-inductance are:
(A) [M L2 T −2 A−2 ]
(B) [M L2 T −2 A−1 ]
(C) [M LT −2 A−2 ]
(D) [M LT −2 A−1 ]

4. A particle is moving in a straight line. The variation of position x as a function
of time t is given as:
x = t3 − 6t2 + 20t + 15
The velocity of the body when its acceleration becomes zero is:
(A) 6 m/s
(B) 10 m/s
(C) 8 m/s
(D) 4 m/s

5. The distance travelled by a particle starting from rest and moving with an
acceleration 43 ms−2 , in the third second is:
(A) 6 m
(B) 4 m
(C) 10
3 m
19
(D) 3 m

6. A projectile is projected with velocity of 40 m/s at an angle θ with the
horizontal. If R is the horizontal range covered by the projectile and after t
seconds its inclination with horizontal becomes zero, then the value of cot θ is:
[Take, g = 10 m/s2 ]
R
(A) 20t 2
R
(B) 10t 2

(C) 5R
t2

2

(D) tR2

7. A rigid body rotates about a fixed axis with variable angular velocity
ω = α − βt at time t, where α, β are constants. The angle through which it rotates
before it stops is:
α2
(A) 2β
2 2
−β
(B) α 2α
2 2
−β
(C) α 2β
(D) (α−β)α
2

8. The range of a projectile projected at an angle of 15◦ with the horizontal is 50
m. If the projectile is projected with the same velocity at an angle of 45◦ , then its
range will be:
(A) 50√m
(B) 50 2 m
(C) 100 √
m
(D) 100 2 m

9. A particle of mass m is projected with a velocity u making an angle of 30◦ with
the horizontal. The magnitude of angular momentum of the projectile about the
point
√ of projection when the particle is at its maximum height h is:
3 mu3
(A) 16 g
√
3 mu2
(B) 2 g
3
(C) mu
√
2g
(D) zero

10. A body is thrown with a velocity of 9.8 m/s making an angle of 30◦ with the
horizontal. It will hit the ground after a time:
(A) 3.0 s
(B) 2.0 s
(C) 1.5 s
(D) 1.0 s

11. A light string passing over a smooth light pulley connects two blocks of
masses m1 and m2 (where m2 > m1 ). If the acceleration of the system is √g2 , then
the ratio of the masses mm2 is:
1
√
(A) √2−1
2+1 √
(B) 1+
√ 5
5−1 √
1+ 5
(C) √ 2−1

3

√
(D) √3+1
2−1

12. A block of mass 1 kg is pushed up a surface inclined to horizontal at an angle
of 60◦ by a force of 10 N parallel to the inclined surface. When the block is pushed
up by 10 m along the inclined surface, the work done against frictional force is:

[Given: 2
√ g = 10 m/s , µs = 0.1]
(A) 5 3 J
(B) 5 J
(C) 5 × 103 J
(D) 10 J

13. A person of mass 60 kg is inside a lift of mass 940 kg. The lift starts moving
upwards with an acceleration of 1.0 m/s2 . If g = 10 m/s2 , the tension in the
supporting cable is:
(A) 8600 N
(B) 9680 N
(C) 11000 N
(D) 1200 N

14. A force of F = 0.5 N is applied on the lower block as shown in the figure. The
work done by the lower block on the upper block for a displacement of 3 m of the
upper block with respect to the ground is (Take, g = 10 m/s2 ):

(A) −0.5 J
(B) 0.5 J
(C) 2 J

4

(D) −2 J

15. A pendulum of mass 1 kg and length l = 1 m is released from rest at an angle
θ = 60◦ . The power delivered by all the forces acting on the bob at angle θ = 30◦
will be (Take, g = 10 m/s2 ):
(A) 13.4 W
(B) 20.4 W
(C) 24.6 W
(D) zero

16. An ideal massless spring S can be compressed 1 m by a force of 100 N in
equilibrium. The same spring is placed at the bottom of a frictionless inclined
plane inclined at 30◦ to the horizontal. A 10 kg block M is released from rest at
the top of the incline and is brought to rest momentarily after compressing the
spring by 2 m. If g = 10 m/s2 , what is the speed of the mass just before it touches
the spring?

√
(A) √ 20 m/s
(B) √30 m/s
(C) √10 m/s
(D) 40 m/s

17. A spherical ball of mass 20 kg is stationary at the top of a hill of height 100 m.
It rolls down a smooth surface to the ground, then climbs up another hill of
height 30 m and finally rolls down to a horizontal base at a height of 20 m above
the ground. The velocity attained by the ball is:
(A) 20 m/s
(B) 40√m/s
(C) 10 30 m/s
(D) 10 m/s

18. A particle of mass 2 kg is on a smooth horizontal table and moves in a
circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If
the angular speed of the particle is 12 rad/s, the magnitude of its angular
momentum about a point on the ground right under the center of the circle is:
(A) 14.4 kg m2 s−1
(B) 8.64 kg m2 s−1
(C) 20.16 kg m2 s−1

5

(D) 11.52 kg m2 s−1

19. A ball falling freely from a height of 4.9 m/s hits a horizontal surface. If e = 34 ,
then the ball will hit the surface the second time after:
(A) 1.0 s
(B) 1.5 s
(C) 2.0 s
(D) 3.0 s

20. Two bodies of mass 1 kg and 3 kg have position vectors î + 2ĵ + k̂ and
−3î − 2ĵ + k̂ respectively. The magnitude of the position vector of the center of
mass of this system will be similar to the magnitude of which vector?
(A) î − 2ĵ + k̂
(B) −3î − 2ĵ + k̂
(C) −2î + 2k̂
(D) −2î − ĵ + 2k̂

21. The moment of inertia of a cube of mass m and side a about one of its edges
is equal to:
(A) 32 ma2
(B) 43 ma2
(C) 3ma2
(D) 83 ma2

22. A body which is initially at rest at a height R above the surface of the Earth
of radius R, falls freely towards the Earth. The velocity on reaching the surface of
the Earth is:
√
(A) 2gR
√
(B) qgR
3
(C) 2 gR
√
(D) 4gR

23. The distance between the Sun and Earth is R. The duration of a year if the
distance
√ between the Sun and Earth becomes 3R will be:
(A) 3 years
(B) 3 years
(C) 9 √
years
(D) 3 3 years

24. For a particle inside a uniform spherical shell, the gravitational force on the
particle is:
(A) Infinite

6

(B) Zero
(C) −Gmr2
1 m2

(D) Gmr12m2

25. The kinetic energy of a satellite in its orbit around Earth is E. What should
be the kinetic energy of the satellite to escape Earth’s gravity?
(A) 4E
(B) 2E
√
(C) 2E
(D) E

26. Two wires of the same material (Young’s modulus Y ) and same length L but
radii R and 2R respectively, are joined end to end and a weight W is suspended
from the combination. The elastic potential energy in the system is:

2
3W L
(A) 4πR 2Y
3W 2 L
(B) 8πR2 Y
5W 2 L
(C) 8πR 2Y
W 2L
(D) πR2 Y

27. With rise in temperature, the Young’s modulus of elasticity:
(A) Changes erratically
(B) Decreases

7

(C) Increases
(D) Remains unchanged

28. Young’s modules of materials of a wire of Length ’ L ’ and cross-sectional area
A is Y. If the length of the wire is doubled and cross-sectional area is halved then
Young’s modules will be:
(A) Y /4
(B) 4Y
(C) Y
(D) 2Y

29. Pressure inside two soap bubbles are 1.01 and 1.02 atmosphere, respectively.
The ratio of their volumes is:
(A) 4:1
(B) 0.8:1
(C) 8:1
(D) 2:1

30. A cube of ice floats partly in water and partly in kerosene oil. The radio of
volume of ice immersed in water to that in kerosene oil (specific gravity of
Kerosene oil = 0.8, specific gravity of ice = 0.9)

(A) 8:9
(B) 5:4
(C) 9:10
(D) 1:1

31. A solid metallic cube having total surface area 24 m2 is uniformly heated. If
its temperature is increased by 10◦ C, calculate the increase in volume of the cube.
Given: α = 5.0 × 10−4 C −1
(A) 2.4 × 106 cm3
(B) 1.2 × 105 cm3
(C) 6.0 × 104 cm3
(D) 4.8 × 105 cm3

32. In the given cycle ABCDA, the heat required for an ideal monoatomic gas

8

will be:

(A) p0 V0
(B) 13
2 p0 V0
11
(C) 2 p0 V0
(D) 4p0 V0

33. A gas can be taken from A to B via two different processes ACB and ADB.
When path ACB is used, 60J of heat flows into the system and 30J of work is
done by the system. If path ADB is used, the work done by the system is 10J.
The heat flow into the system in path ADB is:

(A) 40J
(B) 80J
(C) 100J
(D) 20J

34. A source supplies heat to a system at the rate of 1000W . If the system
performs work at the rate of 200W , the rate at which internal energy of the
system increases is:
(A) 1200W
(B) 600W

9

(C) 500W
(D) 800W

35. On Celsius scale, the temperature of a body increases by 40◦ C. The increase
in temperature on Fahrenheit scale is:
(A) 70◦ F
(B) 68◦ F
(C) 72◦ F
(D) 75◦ F

36. In a mixture of gases, the average number of degrees of freedom per molecule
is 6. The RMS speed of the molecule of the gas is c. Then the velocity of sound
in the gas is:
(A) √c3
(B) √c2
(C) 2c
3
(D) 3c
3

37. The temperature of an ideal gas is increased from 200 K to 800 K. If the
RMS speed of gas at 200 K is v0 , then the RMS speed of the gas at 800 K will be:
(A) v0
(B) 4v0
(C) v40
(D) 2v0

38. Two vessels A and B are of the same size and are at the same temperature. A
contains 1 g of hydrogen and B contains 1 g of oxygen. PA and PB are the
pressures of the gases in A and B respectively, then PPBA is:
(A) 8
(B) 16
(C) 32
(D) 4

39. Five identical springs are used in the three configurations as shown in figure.
The time periods of vertical oscillations in configurations (a), (b) and (c) are in
the ratio:

10

√
(A) 1 : 2 : √12
√
(B) 2 : 2 : √12
(C) √12 : 2 : 1
(D) 2 : √12 : 1

40. A particle executes simple harmonic motion between x = −A and x = +A. If
the time taken by the particle to go from x = 0 to A2 is 2 s, then the time taken by
the particle in going from x = A2 to A is:
(A) 3 s
(B) 2 s
(C) 1.5 s
(D) 4 s

41. A simple pendulum doing small oscillations at a place R height above the
Earth’s surface has a time period of T1 = 4 s. T2 would be its time period if it is
brought to a point which is at a height 2R from the Earth’s surface. Choose the
correct relation [R = radius of Earth]:
(A) T1 = T2
(B) 2T1 = 3T2
(C) 3T1 = 2T2
(D) 2T1 = T2

42. The speed of sound in oxygen at STP will be approximately:(Given, R =
8.3J(K)1 , = 1.4)
(A) 315 m/s
(B) 333 m/s
(C) 341 m/s
(D) 325 m/s

43. A plane progressive wave is given by y = 2 cos 2π(330t − x) m. The frequency of

11

the wave is:
(A) 165 Hz
(B) 330 Hz
(C) 660 Hz
(D) 340 Hz

44. An oil drop of radius 1 µm is held stationary under a constant electric field of
3.65 × 104 N/C due to some excess electrons present on it. If the density of the oil
drop is 1.26 g/cm3 , then the number of excess electrons on the oil drop
approximately is:
Take, g = 10 m/s2
(A) 7
(B) 12
(C) 9
(D) 8

45. The potential of a large liquid drop when eight liquid drops are combined is
20 V. Then, the potential of each single drop was:
(A) 10 V
(B) 7.5 V
(C) 5 V
(D) 2.5 V

46. A dust particle of mass 4 × 10−12 mg is suspended in air under the influence of
an electric field of 50 N/C directed vertically upwards. How many electrons were
removed from the neutral dust particle? [Take, g = 10 m/s2 ]
(A) 15
(B) 8
(C) 5
(D) 4

47. The electric field at point (30, 30, 0) due to a charge of 0.008 µC placed at the
origin will be: (coordinates are in cm)
(A) 8000 N/C î + 8000 N/C ĵ
(B) 4000(
√î + ĵ) N/C
(C) 200 √2(î + ĵ) N/C
(D) 400 2(î + ĵ) N/C

48. If two charges q1 and q2 are separated with distance ’d’ and placed in a
medium of dielectric constant K. What will be the equivalent distance between
charges
√ in air for the same electrostatic force?
(A) d √K
(B) K d

12

√
(C) 1.5d
√ K
(D) 2d K

49. Electric potential at a point ’P’ due to a point charge of 5 × 10−9 C is 50 V.
The distance of ’P’ from the point charge is:
1
(Assume, 4πϵ 0
= 9 × 109 Nm2 C−2 )
(A) 3 cm
(B) 9 cm
(C) 90 cm
(D) 0.9 cm

50. Five charges +q, +5q, -2q, +3q and -4q are situated as shown in the figure.
The electric flux due to this configuration through the surface S is:

(A) 5q
ϵ0
(B) 4q
ϵ0
3q
(C) ϵ0
(D) ϵq0

51. A parallel plate capacitor with plate area A and plate separation d = 2 m has
a capacitance of 4µF . The new capacitance of the system if half of the space
between them is filled with a dielectric material of dielectric constant K = 3 (as
shown in the figure) will be:

(A) 2µF
(B) 32µF
(C) 6µF

13

(D) 8µF

52. In the given circuit, E1 = E2 = E3 = 2V and R1 = R2 = 4Ω, then the current
flowing through the branch AB is:

(A) 0
(B) 2A from A to B
(C) 2A from B to A
(D) 5A from B to B

53. In the following circuit diagram, when the 3Ω resistor is removed, the
equivalent resistance of the network:

(A) Increases
(B) Decreases
(C) Remains the same
(D) None of these

54. A conducting wire is stretched by applying a deforming force, so that its
diameter decreases to 40% of the original value. The percentage change in its
resistance will be:
(A) 0.9%
(B) 0.12%
(C) 1.6%
(D) 0.5%

14

55. A wire of resistance 160Ω is melted and drawn into a wire of one-fourth of its
length. The new resistance of the wire will be:
(A) 10Ω
(B) 640Ω
(C) 40Ω
(D) 16Ω

56. Five cells each of emf E and internal resistance r send the same amount of
current through an external resistance R whether the cells are connected in
parallel or in series. Then the ratio Rr is:
(A) 2
(B) 12
(C) 15
(D) 1

57. The straight wire AB carries a current I. The ends of the wire subtend angles
θ1 and θ2 at the point P as shown in the figure. The magnetic field at the point P
is:

µ0 I
(A) 4πd (sin θ1 − sin θ2 )
µ0 I
(B) 4πd (sin θ1 + sin θ2 )
µ0 I
(C) 4πd (cos θ1 − cos θ2 )
µ0 I
(D) 4πd (cos θ1 + cos θ2 )

15

58. A long straight wire of radius a carries a steady current I. The current is
uniformly distributed across its cross-section. The ratio of the magnetic field at
a/2 and 2a from the axis of the wire is:
(A) 1 : 4
(B) 4 : 1
(C) 1 : 1
(D) 3 : 4

59. The electrostatic force F1 and magnetic force F2 acting on a charge q moving
with velocity v can be written as:
(A) F⃗1 = q⃗v · E,
⃗ F⃗2 = q(B
⃗ · ⃗v )
(B) F⃗1 = q B,
⃗ F⃗2 = q(B⃗ × ⃗v )
(C) F⃗1 = q E,
⃗ F⃗2 = q(⃗v × B)
⃗
(D) F⃗1 = q E,
⃗ F⃗2 = q(B⃗ × ⃗v )

60. Inside a solenoid of radius 0.5 m, the magnetic field is changing at a rate of
50 × 10−6 T/s. The acceleration of an electron placed at a distance of 0.3 m from
the axis of the solenoid will be:
(A) 23 × 106 m/s2
(B) 26 × 106 m/s2
(C) 1.3 × 109 m/s2
(D) 26 × 109 m/s2

61. There are two long co-axial solenoids of the same length l. The inner and
outer coils have radii r1 and r2 and the number of turns per unit length n1 and n2 ,
respectively. The ratio of mutual inductance to the self-inductance of the inner
coil is:
(A) nn21
(B) nn21 · rr12
2
(C) nn21 · rr22
1
(D) nn21

62. A rectangular loop of length 2.5 m and width 2 m is placed at 60◦ to a
magnetic field of 4 T. The loop is removed from the field in 10 sec. The average
emf induced in the loop during this time is:
(A) −2 V
(B) +2 V
(C) +1 V
(D) −1 V

63. Find the average value of the current shown graphically from t = 0 to t = 2 s.

16

(A) 3 A
(B) 5 A
(C) 10 A
(D) 4 A

64. In an AC circuit, an inductor, a capacitor, and a resistor are connected in
series with XL = R = XC . The impedance of this circuit is:
(A) 2R2
(B) Zero
(C) R √
(D) R 2

65. An alternating voltage V (t) = 220 sin 100πt volt is applied to a purely resistive
load of 50Ω. The time taken for the current to rise from half of the peak value to
the peak value is:
(A) 5 ms
(B) 3.3 ms
(C) 7.2 ms
(D) 2.2 ms

66. A parallel plate capacitor consists of two circular plates of radius R = 0.1 m.
They are separated by a short distance. If the electric field between the capacitor
plates changes as:

dE V
= 6 × 1013
dt m·s
then the value of the displacement current is:
(A) 15.25 A
(B) 6.25 A
(C) 16.67 A
(D) 4.69 A

67. Electromagnetic waves travel in a medium with speed 1.5 × 108 m/s. The

17

relative permeability of the medium is 2.0. The relative permittivity will be:
(A) 5
(B) 1
(C) 4
(D) 2

68. Power of a biconvex lens is P diopter. When it is cut into two symmetrical
halves by a plane containing the principal axis, the ratio of the power of two
halves is:
(A) 1:2
(B) 2:1
(C) 1:4
(D) 1:1

69. The magnifying power of a telescope is 9. When adjusted for parallel rays,
the distance between the objective and eyepiece is 20 cm. The ratio of the focal
length of the objective lens to the focal length of the eyepiece is:
(A) 8
(B) 7
(C) 9
(D) 12

70. In normal adjustment, for a refracting telescope, the distance between the
objective and eyepiece is 30 cm. The focal length of the objective, when the
angular magnification of the telescope is 2, will be:
(A) 20 cm
(B) 30 cm
(C) 10 cm
(D) 15 cm

71. If the distance between an object and its two times magnified virtual image
produced by a curved mirror is 15 cm, the focal length of the mirror must be:
(A) 10
3 cm
(B) −12 cm
(C) −10 cm
(D) 15 cm

72. Young’s double slit experiment is performed in a medium of refractive index
1.33. The maximum intensity is I0 . The intensity at a point on the screen where
the path difference between the light coming out from slits is λ/4, is:
(A) 0
(B) I20
(C) 3I80

18

(D) 2I30

73. In YDSE, monochromatic light falls on a screen 1.80 m from two slits
separated by 2.08 mm. The first and second order bright fringes are separated by
0.553 mm. The wavelength of light used is:
(A) 520 nm
(B) 639 nm
(C) 715 nm
(D) None of these

74. A microwave of wavelength 2.0 cm falls normally on a slit of width 4.0 cm.
The angular spread of the central maxima of the diffraction pattern obtained on a
screen 1.5 m away from the slit will be:
(A) 60°
(B) 45°
(C) 15°
(D) 30°

75. The property of light which cannot be explained by Huygen’s construction of
a wavefront is:
(A) Refraction
(B) Reflection
(C) Diffraction
(D) Origin of spectra

76. When a light ray incidents on the surface of a medium, the reflected ray is
completely polarized. Then the angle between reflected and refracted rays is:
(A) 45°
(B) 90°
(C) 120°
(D) 180°

77. Which figure shows the correct variation of applied potential difference (V )
with photoelectric current (I) at two different intensities of light (I1 < I2 ) of same
wavelengths:

19

78. The acceptor level of a p-type semiconductor is 6 eV. The maximum
wavelength of light which can create a hole would be: Given hc = 1242 eV nm.

20

(A) 407 nm
(B) 414 nm
(C) 207 nm
(D) 103.5 nm

79. When light is incident on a metal surface, the maximum kinetic energy of
emitted electrons:
(A) Varies with intensity of light
(B) Varies with frequency of light
(C) Varies with speed of light
(D) Varies irregularly

80. If the kinetic energy of a free electron doubles, its de-Broglie wavelength
changes by the factor:
(A) 2
(B) 21
√
(C) 2
(D) √12

81. Which of the following transitions of He+ ion will give rise to a spectral line
that has the same wavelength as the spectral line in a hydrogen atom?
(A) n = 4 → n = 2
(B) n = 6 → n = 5
(C) n = 6 → n = 3
(D) None of these

82. The ratio of the shortest wavelength of the Balmer series to the shortest
wavelength of the Lyman series for the hydrogen atom is:
(A) 4:1
(B) 1:2
(C) 1:4
(D) 2:1

83. The minimum excitation energy of an electron revolving in the first orbit of
hydrogen is:
(A) 3.4 eV
(B) 8.5 eV
(C) 10.2 eV
(D) 13.6 eV

84. The atomic mass of 6 C 12 is 12.000000 u and that of 6 C 13 is 13.003354 u. The
required energy to remove a neutron from 6 C 13 , if the mass of the neutron is

21

1.008665 u, will be:
(A) 62.5 MeV
(B) 6.25 MeV
(C) 4.95 MeV
(D) 49.5 MeV

85. The nucleus having highest binding energy per nucleon is:
(A) 16
8 O
56
(B) 26 F e
(C) 208
84 P b
(D) 42 He

86. Identify the correct output signal Y in the given combination of gates for the
given inputs A and B shown in the figure.

22

87. Identify the logic gate given in the circuit:

(A) NAND gate
(B) OR gate
(C) AND gate
(D) NOR gate

88. A reverse biased zener diode when operated in the breakdown region works
as:
(A) an amplifier
(B) an oscillator
(C) a voltage regulator
(D) a rectifier

89. Identify the logic operation performed by the following circuit.

(A) OR
(B) AND
(C) NOT
(D) NAND

23

90. One main scale division of a vernier caliper is equal to m units. If the mth
division of main scale coincides with the (n + 1)th division of vernier scale, the
least count of the vernier caliper is:
n
(A) (n+1)
m
(B) (n+1)
1
(C) (n+1)
m
(D) n(n+1)

Chemistry

1. A 1 L closed flask contains a mixture of 4 g of methane and 4.4 g of carbon
dioxide. The pressure inside the flask at 27C is (Assume ideal behaviour of gases):
(A) 8.6 atm
(B) 2.2 atm
(C) 4.2 atm
(D) 6.1 atm

2. In which mode of expression, the concentration of a solution remains
independent of temperature?
(A) Molarity
(B) Normality
(C) Formality
(D) Molality

3. The degeneracy of hydrogen atom that has energy equal to − R9H is (where RH
= Rydberg constant)
(A) 6
(B) 8
(C) 5
(D) 9

4. If the de-Broglie wavelength of a particle of mass (m) is 100 times its velocity,
then its
p mvalue in terms of its mass (m) and Planck constant (h) is:
1
(A) 10
qh
h
(B) 10 m
q
1 h
(C) 10
pm
(D) 10 m h

5. The energy of the second orbit of a hydrogen atom is −5.45 × 10−19 J. What is
the energy of the first orbit of Li2+ ion (in J)?
(A) −1.962 × 10−18

24

(B) −1.962 × 10−17
(C) −3.924 × 10−17
(D) −3.924 × 10−18

6. A photon of wavelength 3000 Å strikes a metal surface. The work function of
the metal is 2.13 eV. What is the kinetic energy of the emitted photoelectron?
(h = 6.626 × 10−34 Js)
(A) 4.0 eV
(B) 3.0 eV
(C) 2.0 eV
(D) 1.0 eV

7. A stream of electrons from a heated filament was passed between two charged
plates at a potential difference V volt. If e and m are the charge and mass of an
electron, then the value of λh is:
√
(A) √ meV
(B) 2meV
(C) meV
(D) 2meV

8. Electron affinity is positive when:
(A) O changes into O−
(B) O− changes to O2−
(C) O changes into O+
(D) O changes to O2+

9. The ionic radii in (Å) of N3− , O2− and F− are respectively.
(A) 1.71, 1.40 and 1.36
(B) 1.71, 1.36 and 1.40
(C) 1.36, 1.40 and 1.71
(D) 1.36, 1.71 and 1.40

10. Intramolecular hydrogen bonding is found in
(A) o-nitrophenol
(B) m-nitrophenol
(C) p-nitrophenol
(D) phenol

11. The hybridisation scheme for the central atom includes a d-orbital
contribution in
(A) I−
3
(B) PCl3

25

(C) NO− 3
(D) H2 Se

12. In the following species, how many species have the same magnetic moment?
(i) Cr2+ (ii) Mn3+ (iii) Ni2+ (iv) Sc2+ (v) Zn2+ (vi) V3+ (vii) Ti4+
(A) 1
(B) 3
(C) 2
(D) 4

13. The spin only magnetic moment of Fe3+ ion (in BM) is approximately.
(A) 4
(B) 5
(C) 6
(D) 7

14. Which one of the following compounds is having maximum ’lone pair-lone
pair’ electron repulsions?
(A) ClF3
(B) IF5
(C) SF4
(D) XeF2

15. Identify the species having one π-bond and maximum number of canonical
forms from the following:
(A) SO3
(B) O2
(C) SO2
(D) CO2−
3

16. sp3 d2 hybridisation is not displayed by:
(A) BrF5
(B) SF6
(C) [CrF6 ]3−
(D) PF5

17. What would be the amount of heat absorbed in the cyclic process shown
below?

26

(A) 5π J
(B) 15π J
(C) 25π J
(D) 100π J

18. The bond dissociation energy of X2 , Y2 and XY are in the ratio of 1 : 0.5 : 1.
∆H for the formation of XY is -200 kJ/mol. The bond dissociation energy of X2
will be
(A) 200 kJ/mol
(B) 100 kJ/mol
(C) 400 kJ/mol
(D) 800 kJ/mol

19. Which of the following relation is not correct?
(A) ∆H = ∆U − P ∆V
(B) ∆U = q + W
(C) ∆Ssys + ∆Ssurr ≥ 0
(D) ∆G = ∆H − T ∆S

20. The standard Gibbs energy (∆G◦ ) for the following reaction is

A(s) + B 2+ (aq) ⇀
↽ A2+ (aq) + B(s), Kc = 1012 at

(Kc = equilibrium constant)
(A) -150 kJ/mol
(B) -96.80 kJ/mol
(C) -68.47 kJ/mol
(D) -100 kJ/mol

21. The combustion of benzene (L) gives CO2 (g) and H2 O (L). Given that heat
of combustion of benzene at constant volume is -3263.9 kJ/mol at 25°C, heat of
combustion (in kJ/mol) of benzene at constant pressure will be: (R = 8.314JK −1
mol−1 )
(A) 4152.6
(B) 452.46

27

(C) 3260
(D) -3267.6

22. Choose the correct option for free expansion of an ideal gas under adiabatic
condition from the following:
(A) q = 0, ∆T ̸= 0, w = 0
(B) q = 0, ∆T < 0, w ̸= 0
(C) q ̸= 0, ∆T = 0, w = 0
(D) q = 0, ∆T = 0, w = 0

23. Le-Chatelier’s principle is not applicable to
(A) H2 (g) + I2 (g) ⇀
↽ 2HI(g)
(B) Fe(s) + S(s) ⇀↽ FeS(s)
(C) N2 (g) + 3H2 (g) ⇀
↽ 2NH3 (g)
⇀
(D) N2 (g) + O2 (g) ↽ 2NO(g)

24. The ratio Kp
Kc for the reaction
1
CO(g) + O2 (g) ⇀
↽ CO2 (g)
2
is:
(A) (RT )1/2
(B) RT
(C) 1
(D) √ 1
RT

25. The pH of 1 N aqueous solutions of HCl, CH3 COOH and HCOOH follows the
order:
(A) HCl > HCOOH > CH3 COOH
(B) HCl = HCOOH > CH3 COOH
(C) CH3 COOH > HCOOH > HCl
(D) CH3 COOH = HCOOH > HCl

26. 20 mL of 0.1 M acetic acid is mixed with 50 mL of potassium acetate. Ka of
acetic acid = 1.8 × 10−5 at 27°C. Calculate the concentration of potassium acetate
if the pH of the mixture is 4.8.
(A) 0.1 M
(B) 0.04 M
(C) 0.03 M
(D) 0.02 M

27. What is the stoichiometric coefficient of SO2 in the following balanced
reaction?

28

MnO− 2+ −
4 (aq) + SO2 (g) → Mn (aq) + HSO4 (aq)
(in acidic solution)
(A) 5
(B) 4
(C) 3
(D) 2

28. Volume of M/8 KMnO4 solution required to react completely with 25.0 cm3
of M/4 FeSO4 in acidic medium is:
(A) 8.0 mL
(B) 5.0 mL
(C) 15.0 mL
(D) 10.0 mL

29. Which of the following is only a redox reaction but not a disproportionation
reaction?
(A) 4H3 P O3 → 3H3 P O4 + P H3
(B) 2H2 O2 → 2H2 O + O2
(C) P4 + 3N aOH + 3H2 O → 3N aH2 P O2 + P H3
(D) P4 + 8SOCl2 → 4P Cl3 + 2S2 Cl2 + 4SO2

30. Among the following, the correct statements are:
I. LiH, BeH2 and MgH2 are saline hydrides with significant covalent character
II. Saline hydrides are volatile
III. Electron - precise hydrides are Lewis bases
IV. The formula for chromium hydride is CrH
(A) I, III only
(B) II, IV only
(C) I, IV only
(D) III, IV only

31. In which of the following reactions of H2 O2 acts as an oxidizing agent (either
in acidic, alkaline, or neutral medium)?
(i) 2F e2+ + H2 O2 →
(ii) 2M nO4− + 6H + + 5H2 O2 →
(iii) I2 + H2 O2 + 2OH − →
(iv) M n2+ + H2 O2 →
(A) (ii), (iii)
(B) (i), (iv)
(C) (i), (iii)
(D) (ii), (iv)

29

32. The strongest reducing agent among the following is:
(A) SbH3
(B) NH3
(C) BiH3
(D) PH3

33. The correct order of melting points of the following salts is:
LiCl (I)
LiF (II)
LiBr (III)
(A) I > II > III
(B) II > I > III
(C) III > II > I
(D) II > III > I

34. Which among the following is used in detergent?
(A) Sodium acetate
(B) Sodium stearate
(C) Calcium stearate
(D) Sodium lauryl sulphate

35. Thermal decomposition of lithium nitrate gives:
(A) LiO2 , O2 , NO2
(B) Li2 O, O2 , N2 O
(C) Li2 O, O2 , N2
(D) Li2 O, O2 , NO2

36. The number of geometrical isomers possible for the compound, CH3 CH = CH
- CH = CH2 is:
(A) 2
(B) 3
(C) 4
(D) 6

37. Correct order of stability of carbanion is:

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

30

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

38. Which of the following is not correct about Grignard reagent?
(A) It is a nucleophile
(B) Forms new carbon-carbon bond
(C) Reacts with carbonyl compounds
(D) It is an organomanganese compound

39. The IUPAC name of the following molecule is:

(A) 2-Methyl-5-nitro-1-chlorobenzene
(B) 3-Chloro-4-methyl-1-nitrobenzene
(C) 2-Chloro-1-methyl-4-nitrobenzene
(D) 2-Chloro-4-nitro-1-methylbenzene

40. Choose the correct stability order of the given free radicals.

(A) I > II > III > IV
(B) II > I > IV > III

31

(C) II = I > IV > III
(D) III > IV > II > I

41. Which of the following is the strongest Bronsted base?

42. The major product X in the following given reaction is:

32

43. The major product of the reaction between CH3 CH2 ONa and (CH3 )3 CCl in
ethanol is:
(A) CH3 CH2 OC(CH3 )3
(B) CH2 = C(CH3 )2
(C) CH3 CH2 C(CH3 )3
(D) CH3 CH = CHCH3

44. Dinitrogen is a robust compound, but reacts at high altitude to form oxides.
The oxide of nitrogen that can damage plant leaves and retard photosynthesis is:
(A) NO
(B) NO−
3
(C) NO2
(D) NO−2

45. A decimolar solution potassium ferrocyanide is 50% dissociated at 300 K. The

33

osmotic pressure of solution is (R = 8.314 J K−1 mol−1 ):
(A) 7.48 atm
(B) 4.99 atm
(C) 3.74 atm
(D) 6.23 atm

46. 58.5 g of NaCl and 180 g of glucose were separately dissolved in 1000 mL of
water. Identify the correct statement regarding the elevation of boiling point of
the resulting solution.
(A) NaCl solution will show higher elevation of boiling point.
(B) Glucose solution will show higher elevation of boiling point.
(C) Both solutions will show equal elevation of boiling point.
(D) None will show boiling point elevation.

47. One molar concentration of a solution represents:
(A) 1 mole of solute in 1 kg of solution.
(B) 1 mole of solute in 1 L of solution.
(C) 1 mole of solvent in 1 kg of solution.
(D) 1 mole of solvent in 1 L of solution.

48. Which of the following substances show the highest colligative properties?
(A) 0.1M BaCl2
(B) 0.1M AgNO3
(C) 0.1M urea
(D) 0.1M (NH4 )3 PO4

49. The pH of 0.5 L of 1.0 M NaCl solution after electrolysis for 965 s using 5.0 A
current is:
(A) 1.0
(B) 12.7
(C) 1.30
(D) 13.0

50. Calculate the molarity of a solution containing 5 g of NaOH dissolved in the
product of H2 – O2 fuel cell operated at 1 A current for 595.1 hours.(Assume F =
96500C/mol of electron and molecular weight of NaOH as 40 g/mol).
(A) 0.625 M
(B) 0.05 M
(C) 0.1 M
(D) 6.25 M

51. When the same quantity of electricity is passed through the aqueous solutions
of the given electrolytes for the same amount of time, which metal will be

34

deposited in maximum amount on the cathode?
(A) ZnSO4
(B) FeCl3
(C) AgNO3
(D) NiCl2

⇀ 2SO3 , the rate of disappearance of O2 is
52. For the reaction 2SO2 + O2 ↽
−4 −1 −1
2 × 10 mol L s . The rate of appearance of SO3 is:
(A) 2 × 10−4 mol L−1 s−1
(B) 4 × 10−4 mol L−1 s−1
(C) 1 × 10−1 mol L−1 s−1
(D) 6 × 10−4 mol L−1 s−1

53. If for a first-order reaction, the value of A and Ea are 4 × 1013 s−1 and 98.6 kJ
mol−1 respectively, then at what temperature will its half-life be 10 minutes?
(A) 330 K
(B) 300 K
(C) 330.95 K
(D) 311.15 K

54. In the chemical reaction A → B, what is the order of the reaction? Given
that, the rate of reaction doubles if the concentration of A is increased four times.
(A) 2
(B) 1.5
(C) 0.5
(D) 1

55. Calculate the activation energy of a reaction, whose rate constant doubles on
raising the temperature from 300 K to 600 K.
(A) 3.45 kJ/mol
(B) 6.90 kJ/mol
(C) 9.68 kJ/mol
(D) 19.6 kJ/mol

56. In the reaction, A → products, if the concentration of the reactant is doubled
but the rate of reaction remains unchanged, what is the order of the reaction
with respect to A?
(A) 1
(B) 2
(C) 0.5
(D) 0

35

57. In a first-order reaction, the concentration of the reactant decreases from 0.8
M to 0.4 M in 15 minutes. The time taken for the concentration to change from
0.1 M to 0.025 M is:
(A) 7.5 minutes
(B) 15 minutes
(C) 30 minutes
(D) 60 minutes

58. The charge on colloidal particles is due to:
(A) Presence of electrolyte
(B) Very small size of particles
(C) Adsorption of ions from the solution
(D) Can’t be determined

59. The chemical composition of ’slag’ formed during the smelting process in the
extraction of copper is:
(A) Cu2 O + FeS
(B) FeSiO3
(C) CuFeS2
(D) Cu2 S + FeO

60. Calamine, malachite, magnetite, and cryolite, respectively, are:
(A) ZnCO3 , CuCO3 ·Cu(OH)2 , Fe3 O4 , Na3 AlF6
(B) ZnSO4 , Cu(OH)2 , Fe3 O4 , Na3 AlF6
(C) ZnSO4 , CuCO3 , Fe2 O3 , AlF3
(D) ZnCO3 , CuCO3 , Fe2 O3 , Na3 AlF6

61. In which of the following molecules, all bond lengths are not equal?
(A) SF6
(B) PCl5
(C) BCl3
(D) CCl4

62. The sol formed in the following unbalanced equation is:

As2 O3 + H2 S → ?
(A) As2 S2
(B) As2 S3
(C) As
(D) S

63. Which of the following has least tendency to liberate H2 from mineral acids?

36

(A) Cu
(B) Mn
(C) Ni
(D) Zn

64. The metal that shows highest and maximum number of oxidation states is:
(A) Fe
(B) Mn
(C) Ti
(D) Co

65. Hybridisation and geometry of [Ni(CN)4 ]2− are:
(A) sp3 and tetrahedral
(B) sp3 and square planar
(C) sp3 and tetrahedral
(D) dsp2 and square planar

66. Match List I with List II.
List I (Complex) List II (Oxidation Number of Metal)
A. Ni(CO)4 I. +1
B. [Fe(H2 O)5 NO]2+ II. Zero
C. [Co(CO)5 ]2− III. -1
D. [Cr2 (CO)10 ]2− IV. -2
(A) A-II, B-I, C-IV, D-III
(B) A-II, B-IV, C-I, D-III
(C) A-II, B-III, C-I, D-IV
(D) A-I, B-II, C-IV, D-III

67. Which of the following is the correct order of ligand field strength?
(A) CO < N H3 < en < C2 O42− < S 2−
(B) S 2− < C2 O42− < N H3 < en < CO
(C) N H3 < en < CO < S 2− < C2 O42−
(D) S 2− < N H3 < en < CO < C2 O42−

68. The correct statement among the following is:
(A) Ferrocene has two cyclohexadiene rings coordinated to iron atom.
(B) Ferrocene has two cyclopentadienyl anion rings bonded to iron (II) ion.
(C) Perxenate ion is [XeO2 F2 ]2− .
(D) Perxenate ion is tetrahedral in shape.

69. The type of isomerism present in nitropentammine chromium (III) chloride is:
(A) Optical

37

(B) Linkage
(C) Ionization
(D) Polymerization

70. Identify, from the following, the diamagnetic, tetrahedral complex:
(A) [Ni(Cl)4 ]2−
(B) [Co(C2 O4 )3 ]3−
(C) [Ni(CN)4 ]2−
(D) [Ni(CO)4 ]

71. Ferrocene is:
(A) F e(η 5 − C5 H5 )2
(B) F e(η 2 − C5 H5 )2
(C) Cr(η 5 − C5 H5 )5
(D) Os(η 5 − C5 H5 )2

72. The chemical name of calgon is:
(A) Sodium hexametaphosphate
(B) Potassium hexametaphosphate
(C) Calcium hexametaphosphate
(D) Sodium hexametaphosphate

73. The complex with the highest magnitude of crystal field splitting energy (∆0 )
is:
(A) [Cr(OH2 )6 ]3+
(B) [T i(OH2 )6 ]3+
(C) [F e(OH2 )6 ]3+
(D) [M n(OH2 )6 ]3+

74. IUPAC name of [P t(N H3 )2 Cl(N H2 CH3 )]Cl is:
(A) (Amino methane) chloro (diammine) platinum (II) chloride.
(B) Chlorodiammine (methanamine) platinum (II) chloride.
(C) Diamminechloro (methanamine) platinum (II) chloride.
(D) Diamminechloro (methylamine) platinum (IV) chloride.

75. Which of the following complexes will exhibit maximum attraction to an
applied magnetic field?
(A) [Zn(H2 O)6 ]2+
(B) [Co(H2 O)6 ]2+
(C) [Co(en)3 ]3+
(D) [N i(H2 O)6 ]2+

38

76. In an SN 2 substitution reaction of the type:
R − Br + Cl− −→ R − Cl + Br−
Which one of the following has the highest relative rate?
(A) CH3 − CH − CH2 Br (with a CH3 group attached to the second carbon)
(B) CH3 − CH(CBr) − CH3 (with two CH3 groups attached to the second carbon)
(C) CH3 CH2 Br
(D) CH3 CH2 CH2 Br

77. The final product in the following reaction Y is:

78. In the Victor-Meyer test, the color given by 1°, 2°, and 3° alcohols are
respectively:
(A) Red, colorless, blue
(B) Red, blue, colorless
(C) Colorless, red, blue

39

(D) Red, blue, violet

79. What is X in the following reaction?
X
CO + 2H2 −→ CH3 OH
(A) 623 K / 300 atm
(B) KMnO4 /H
(C) Zn /
(D) ZnO - Cr2 O3 , 200 – 300 atm, 573 – 673 K

80. An unknown alcohol is treated with ”Lucas reagent” to determine whether
the alcohol is primary, secondary, or tertiary. Which alcohol reacts fastest and by
what mechanism?
(A) Secondary alcohol by SN 1
(B) Tertiary alcohol by SN 1
(C) Secondary alcohol by SN 2
(D) Tertiary alcohol by SN 2

81. Which of the following compounds will undergo self aldol condensation in the
presence of cold dilute alkali?
(A) CH2 = CH − CHO
(B) CH ≡ C − CHO
(C) C6 H5 CHO
(D) CH3 CH2 CHO

82. An alkene X on ozonolysis gives a mixture of Propan-2-one and methanal.
What is X?
(A) Propene
(B) 2-Methylpropene
(C) 2-Methylbut-1-ene
(D) 2-Methylbut-2-ene

83. Cheilosis and digestive disorders are due to deficiency of:
(A) Vitamin A
(B) Thiamine
(C) Riboflavin
(D) Ascorbic acid

84. A tetrapeptide is made of naturally occurring alanine, serine, glycine, and
valine. If the C-terminal amino acid is alanine and the N-terminal amino acid is
chiral, the number of possible sequences of the tetrapeptide is:
(A) 4
(B) 8

40

(C) 6
(D) 12

85. Which one of the following is a water-soluble vitamin that is not excreted
easily?
(A) Vitamin B2
(B) Vitamin B1
(C) Vitamin B6
(D) Vitamin B12

86. Glycosidic linkage between C1 of α-glucose and C2 of β-fructose is found in:
(A) maltose
(B) sucrose
(C) lactose
(D) amylose

87. The naturally occurring amino acid that contains only one basic functional
group in its chemical structure is:
(A) arginine
(B) lysine
(C) asparagine
(D) histidine

88. Which of the following is not a semi-synthetic polymer?
(A) Cis-polyisoprene
(B) Cellulose nitrate
(C) Cellulose acetate
(D) Vulcanised rubber

89. Zinc acetate - antimony trioxide catalyst is used in the preparation of which
polymer?
(A) High-density polyethylene
(B) Teflon
(C) Terylene
(D) PVC

90. ........ is a potent vasodilator.
(A) Histamine
(B) Serotonin
(C) Codeine
(D) Cimetidine

41

Mathematics
1. Roots of the equation x2 + bx − c = 0 (b, c > 0) are:
(A) Both positive
(B) Both negative
(C) Of opposite sign
(D) None of the above

2. Rational roots of the equation 2x4 + x3 − 11x2 + x + 2 = 0 are:
(A) 21 , 2
(B) 13 , 2, −2
(C) 12 , 2, 3, 4
(D) 12 , 2, 3, −2

tan 15◦ and tan 30◦ are the roots of the equation x2 + px + q = 0, then pq =:
3. If √
(A) 6 √3+10
3√
(B) 10−6
3√
3

(C) 10+6
3√
3

(D) 10−6
√ 3
3

4. The points represented by the complex numbers 1 + i, −2 + 3i, 35 i on the Argand
plane are:
(A) Vertices of an equilateral triangle
(B) Vertices of an isosceles triangle
(C) Collinear
(D) None of the above

5. The modulus of the complex number z such that |z + 3 − i| = 1 and arg(z) = π is
equal to:
(A) 3
(B) 2
(C) 9
(D) 4

√
6. If z, z̄,
√ −z, −z̄ forms a rectangle of area 2 3 square units, then one such z is:
1
(A) √2 + √ 3i
5+ 3i
(B) 4√
(C) 2√+ √23i
3

(D) 3+2 11i

7. If z1 , z2 , . . . , zn are complex numbers such that |z1 | = |z2 | = · · · = |zn | = 1, then

42

|z1 + z2 + · · · + zn | is equal to:
(A) |z1 ||z2 | . . . |zn |
(B) |z1 | + |z2 | + · · · + |zn |
(C) |z11 | + |z12 | + · · · + |z1n |
(D) n

8. If |z1 | = 2, |z2 | = 3, |z3 | = 4 and |2z1 + 3z2 + 4z3 | = 4, then the absolute value of
8z2 z3 + 27z3 z1 + 64z1 z2 equals:
(A) 24
(B) 48
(C) 72
(D) 96

9. A person invites a party of 10 friends at dinner and places so that 4 are on one
round table and 6 on the other round table. Total number of ways in which he
can arrange the guests is:
(A) 10!
6!
(B) 10!
24
9!
(C) 24
(D) None of these

10. How many different nine-digit numbers can be formed from the number
223355888 by rearranging its digits so that the odd digits occupy even positions?
(A) 16
(B) 36
(C) 60
(D) 100

11. If 22Pr+1 : 20Pr+2 = 11 : 52, then r is equal to:
(A) 3
(B) 5
(C) 7
(D) 9

12. At an election, a voter may vote for any number of candidates not exceeding
the number to be elected. If 4 candidates are to be elected out of the 12
contested in the election and voter votes for at least one candidate, then the
number of ways of selections is:
(A) 793
(B) 298
(C) 781
(D) 1585

43

13. The number of arrangements of all digits of 12345 such that at least 3 digits
will not come in its position is:
(A) 89
(B) 109
(C) 78
(D) 57

14. If a > 0, b > 0, c > 0 and a, b, c are distinct, then (a + b)(b + c)(c + a) is greater than:
(A) 2(a + b + c)
(B) 3(a + b + c)
(C) 6abc
(D) 8abc

15. If nk=1 k(k + 1)(k − 1) = pn4 + qn3 + tn2 + sn, where p, q, t, s are constants, then
P
the value of s is equal to:
(A) −1/4
(B) −1/2
(C) 1/2
(D) 1/4

16. There are four numbers of which the first three are in GP and the last three
are in AP, whose common difference is 6. If the first and the last numbers are
equal, then the two other numbers are:
(A) -2, 4
(B) 4, 2
(C) 2, 6
(D) None of the above

17. If A = 1 + ra + r2a + r3a + . . . ∞ and B = 1 + rb + r2b + r3b + . . . ∞, then ab is equal.
(A) logb (A)
(B) log1−b (1 − A)
(C) log b−1 A−1


b A
(D) None of these

18. The sum of the infinite series 1 + 65 + 12 22 35
62 + 63 + 64 + . . . is equal to:
425
(A) 216
429
(B) 216
(C) 288
125
(D) 280
125

 
19. If tan−1 1
+ tan−1 1
+ . . . + tan−1 1
= tan−1 (x), then x is equal



1+1·2 1+2·3 1+n(n+1)

44

to:
1
(A) n+1
n
(B) n+1
1
(C) n+2
n
(D) n+2

20. If the arithmetic mean of two distinct positive real numbers a and b (where
a > b) is√twice their
√ geometric mean, then a : b is:
(A) 2 + √ 3 : 2 − √ 3
(B) 2 + √5 : 2 − √5
(C) 2 + 2 : 2 − 2
(D) None of these

21. If
     
−1 1 −1 1 −1 1
y = tan + tan + tan + · · · (to n terms)
x2 + x + 1 x2 + 3x + 3 x2 + 5x + 7
dy
, then dx is:
1
(A) x2 +n2 − x21+1
1 1
(B) (x+n) 2 +1 − x2 +1
1 1
(C) x2 +(n+1)2 − x2 +1

(D) None of these

 10
1 12 − 14
22. The coefficient of x2 term in the binomial expansion of 3x + x is:
70
(A) 243
60
(B) 423
(C) 50
13
(D) None of these

23. The coefficient of xn in the expansion of
e7x + ex
e3x
is: n−1
·(−2)n
(A) 4 n!
n n
(B) 4 −1·(2)
n! n
n
(C) 4 +(−2)
n!
n n−1
(D) 4 −1·(−2)
n!

24. The
√ coefficient of
√ the highest power of x in the expansion of
2 8 2 8
(x + x − 1) + (x − x − 1) is:
(A) 64

45

(B) 128
(C) 256
(D) 512

25. If the 17th and the 18th terms in the expansion of (2 + a)50 are equal, then the
coefficient of x35 in the expansion of (a + x)−2 is:
(A) −35
(B) 3
(C) 36
(D) −36

26. Let A, B and C are the angles of a triangle and tan A2 = 1/3, tan B2 = 23 . Then,
tan C2 is equal to:
(A) 97
(B) 29
(C) 13
(D) 23

27. The sum of all values of x in [0, 2π], for which
sin(x) + sin(2x) + sin(3x) + sin(4x) = 0 is equal to:
(A) 8π
(B) 11π
(C) 12π
(D) 9π

28. Number of solutions of equations sin(9θ) = sin(θ) in the interval [0, 2π] is:
(A) 16
(B) 17
(C) 18
(D) 15

29. The range of (8 sin(θ) + 6 cos(θ))2 + 2 is:
(A) (0,2)
(B) [2,102]
(C) (−∞, ∞)
(D) (2,1)

30. The locus of the point of intersection of the lines x = a(1 − t2 )/(1 + t2 ) and
y = 2at/(1 + t2 ) (t being a parameter) represents:
(A) Circle
(B) Parabola
(C) Ellipse

46

(D) Hyperbola

31. If the straight line 2x + 3y − 1 = 0, x + 2y − 1 = 0 and ax + by − 1 = 0 form a
triangle with origin as orthocentre, then (a, b) is equal to:
(A) (6,4)
(B) (-3,3)
(C) (-8,8)
(D) (0,7)

33. The distance from the origin to the image of (1, 1) with respect to the line
x + y√+ 5 = 0 is:
(A) 7√ 2
(B) 3√2
(C) 6 √2
(D) 4 2

34. A(3,2,0), B(5,3,2), C(-9,6,-3) are three points forming a triangle. AD, the
bisector of angle BAC meets BC in D. Find the coordinates of D:
(A) 19 57 57


, ,
8 15 15

(B) 19 57 17
8 , 16 , 16
(C) (2,3,0)
(D) (4,5,6)

35. The locus of the mid-point of a chord of the circle x2 + y 2 = 4 which subtends
a right angle at the origin is:
(A) x + y = 2
(B) x2 + y 2 = 1
(C) x2 + y 2 = 2
(D) x + y = 1

36. If p and q be the longest and the shortest distance respectively of the point
(-7,2) from any point (α, β) on the curve whose equation is

x2 + y 2 − 10x − 14y − 51 = 0

then √the geometric mean (G.M.) of p is:
(A) 2√ 11
(B) 5 5
(C) 13
(D) 11

37. From a point A(0,3) on the circle

(x + 2)2 + (y − 3)2 = 4

47

a chord AB is drawn and extended to a point Q such that AQ = 2AB. Then the
locus of Q is:
(A) (x + 4)2 + (y − 3)2 = 16
(B) (x + 1)2 + (y − 3)2 = 32
(C) (x + 1)2 + (y − 3)2 = 4
(D) (x + 1)2 + (y − 3)2 = 1

38. If the focus of the parabola
(y − k)2 = 4(x − h)
always lies between the lines x + y = 1 and x + y = 3 then:
(A) 0 < h + k < 2
(B) 0 < h + k < 1
(C) 1 < h + k < 2
(D) 1 < h + k < 3

39. Let L1 be the length of the common chord of the curves
x2 + y 2 = 9 and y 2 = 8x
and let L2 be the length of the latus rectum of y 2 = 8x. Then:
(A) L1 > L2
(B) L1 = L2
(C) L1 < L
√2
L1
(D) L2 = 2

40. The foci of the hyperbola
4x2 − 9y 2 − 1 = 0
are: √
(A) 
(± √13, 0)
(B) ± 613 , 0
 √ 
(C) 0, ± 63
(D) None of these

41. Given a real-valued function f such that:
 2
tan {x}
 x2 −⌊x⌋2 , for x > 0


f (x) = 1, for x = 0

p{x} cot{x}, for x < 0


Then:
(A) LHL = 1√
(B) RHL = cot 1

48

(C) lim f (x) exists
x→0
(D) lim f (x) does not exist
x→0

42. Let f (x) = sin x, g(x) = cos x, and h(x) = x2 . Then, evaluate:

f (g(h(x))) − f (g(h(1)))
lim
x→1 x−1
(A) 0
(B) −2 sin 1 cos(cos 1)
(C) ∞
(D) −2 sin 1 cos 1

43. The Boolean expression:

∼ (p ∨ q) ∨ (∼ p ∧ q)
is equivalent to:
(A) p
(B) q
(C) ∼ q
(D) ∼ p

44. If p: 2 is an even number, q: 2 is a prime number, and r: 2 + 2 = 22 , then the
symbolic statement p → (q ∨ r) means:
(A) 2 is an even number and 2 is a prime number or 2 + 2 = 22
(B) 2 is an even number then 2 is a prime number or 2 + 2 = 22
(C) 2 is an even number or 2 is a prime number then 2 + 2 = 22
(D) If 2 is not an even number then 2 is a prime number α = 2 + 2 = 22

45. Consider the following statements:
A: Rishi is a judge.
B: Rishi is honest.
C: Rishi is not arrogant.
The negation of the statement ”If Rishi is a judge and he is not arrogant, then he
is honest” is:
(A) B → (A ∨ C)
(B) (∼ B) ∧ (A ∧ C)
(C) B → ((∼ A) ∨ (∼ C))
(D) B → (A ∧ C)

46. If p: It is raining today, q: I go to school, r: I shall meet my friends, and s: I
shall go for a movie, then which of the following represents:

49

”If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie?”

(A) ∼ (p ∧ q) ⇒ (r ∧ s)
(B) ∼ (p∧ ∼ q) ⇒ (r ∧ s)
(C) ∼ (p ∧ q) ⇒ (r ∨ s)
(D) None of these

47. Let p, q, r be three logical statements. Consider the compound statements:

S1 : (∼ p ∨ q) ∨ (∼ p ∨ r)

S2 : p → (q ∨ r)
Which of the following is NOT true?
(A) If S2 is true, then S1 is true
(B) If S2 is false, then S1 is false
(C) If S2 is false, then S1 is true
(D) If S1 is false, then S2 is false

48. Consider the following two propositions:

P1 :∼ (p →∼ q)

P2 : (p∧ ∼ q) ∧ ((∼ p) ∨ q)
If the proposition p → ((∼ p) ∨ q) is evaluated as FALSE, then:
(A) P1 is TRUE and P2 is FALSE
(B) P1 is FALSE and P2 is TRUE
(C) Both P1 and P2 are FALSE
(D) Both P1 and P2 are TRUE

49. If the variance of the data 2, 3, 5, 8, 12 is σ 2 and the mean deviation from the
median for this data is M , then σ 2 − M is:
(A) 10.2
(B) 5.8
(C) 10.6
(D) 8.2

50. The mean of n items is X. If the first item is increased by 1, second by 2, and
so on, the new mean is:
(A) X̄ + x2
(B) X̄ + x
(C) X̄ + n+12

50

(D) None of these

51. The variance of 20 observations is 5. If each observation is multiplied by 2,
then the new variance of the resulting observation is:
(A) 23 × 5
(B) 22 × 5
(C) 2 × 5
(D) 24 × 5

52. If the function f (x), defined below, is continuous on the interval [0, 8], then:

2
x + ax + b, 0 ≤ x < 2

f (x) = 3x + 2, 2≤x≤4

2ax + 5b, 4<x≤8


(A) a = 3, b = −2
(B) a = −3, b = 2
(C) a = −3, b = −2
(D) a = 3, b = 2

53. From the top of a cliff 50 m high, the angles of depression of the top and
bottom of a tower are observed to be 30◦ and 45◦ . The height of the tower is:
(A) 50√m
(B) 50 √3 m
(C) 50( 3 −√1)m
(D) 50 1 − 33 m

54. ABC is a triangular park with AB = AC = 100 m. A TV tower stands at the
midpoint of BC. The angles of elevation of the top of the tower at A, B, C are
45◦ , 60◦ , 60◦ respectively. The height of the tower is:
(A) 50√m
(B) 50√3 m
(C) 50 2 m√
(D) 50(3 − 3) m

55. In a statistical investigation of 1003 families of Calcutta, it was found that 63
families have neither a radio nor a TV, 794 families have a radio, and 187 have a
TV. The number of families having both a radio and a TV is:
(A) 36
(B) 41
(C) 32
(D) None of these

51

56. Let R be the relation ”is congruent to” on the set of all triangles in a plane.
Is R:
(A) Reflexive only
(B) Symmetric only
(C) Symmetric and reflexive only
(D) Equivalence relation

57. Number of subsets of set of letters of word ’MONOTONE’ is:
(A) 8
(B) 256
(C) 64
(D) 32

58. In an examination, 62% of the candidates failed in English, 42% in
Mathematics and 20% in both. The number of those who passed in both the
subjects is:
(A) 11
(B) 16
(C) 18
(D) None of these

 
1 2 2
59. If A = 13 2 1 −2 is an orthogonal matrix, then
a 2 b
(A) a = −2, b = −1
(B) a = 2, b = 1
(C) a = 2, b = −1
(D) a = −2, b = 1

 
3 −2 4
60. If matrix A = 1 2 −1 and A−1 = k1 adj(A), then k is
0 1 1
(A) 7
(B) -7
(C) 15
(D) -11

61. If A and B are symmetric matrices of the same order such that AB + BA = X
and AB − BA = Y , then (XY )T =
(A) XY
(B) X T Y T
(C) −Y X
(D) −Y T X T

52

   
1 0 1 1
62. If A = ,P = and X = AP AT , then AT X 50 A is:
0 −1 0 1
 
0 1
(A)
1 0
 
2 1
(B)
0 −1
 
25 1
(C)
1 −25
 
1 50
(D)
0 1

63. If A is a square matrix of order 3, then |Adj(Adj A2 )| is:
(A) |A|2
(B) |A|4
(C) |A|8
(D) |A|16

64. Suppose p, q, r ̸= 0 and the system of equations:

(p + a)x + by + cz = 0
ax + (q + b)y + cz = 0
ax + by + (r + c)z = 0
has a non-trivial solution, then the value of
a b c
+ +
p q r
is:
(A) −1
(B) 0
(C) 1
(D) 2

65. If x is a complex root of the equation

1 x x 1−x 1 1
x 1 x + 1 1−x 1 = 0,
x x 1 1 1 1−x
then x2007 + x−2007 is:
(A) 1
(B) −1
(C) −2
(D) 2

53

66. The system of equations:

x − y + 2z = 4
3x + y + 4z = 6
x+y+z =1
has:
(A) unique solution
(B) infinitely many solutions
(C) no solution
(D) two solutions

67. If the system of linear equations:

2x + y − z = 7
x − 3y + 2z = 1
x + 4y + δz = k
has infinitely many solutions, then δ + k is:
(A) −3
(B) 3
(C) 6
(D) 9

  
68. If cot(cos−1 x) = sec tan−1 √b2a−a2 , then:
(A) √2b2b−a2
√
(B) bab−a
2 2

(C) √2ba2 −a2
√
b2 −a2
(D) a

69. If cos cot−1 1
= cot(cos−1 x), then the value of x is:


2
(A) √16
(B) √−1
12
(C) √26
−2
(D) √ 6

70. Let [x] denote the greatest integer ≤ x. If f (x) = [x] and g(x) = |x|, then the
value of:
     
8 −8
f g −g f
5 5

54

is:
(A) 2
(B) −2
(C) 1
(D) −1

71. The number of real solutions of
p
5 − log2 |x| = 3 − log2 |x|
is:
(A) 1
(B) 2
(C) 3
(D) 4

72. The function
cos x
f (x) =  2x  1 ,
π + 2
where x is not an integral multiple of π and ⌊·⌋ denotes the greatest integer
function, is:
(A) an odd function
(B) an even function
(C) neither odd nor even
(D) None of these

73. The function f: R→ R is defined by
x
f (x) = √
1 + x2
is:
(A) surjective but not injective
(B) bijective
(C) injective but not surjective
(D) neither injective nor surjective

74. If f : R → R, g : R → R are defined by f (x) = 5x − 3, g(x) = x2 + 3, then g ◦ f −1 (3)
is equal to
(A) 25
3
(B) 111
25
9
(C) 25
25
(D) 111

75. The domain of the real-valued function

55

s
2x2 − 7x + 5
f (x) =
3x2 − 5x − 2
is:
(A) (−∞, − 13 ) ∪ [1, 2) ∪ [ 25 , ∞)
(B) (−∞, 1) ∪ (2, ∞)
(C) (− 13 , 52 )
(D) (−∞, − 31 ] ∪ [ 52 , ∞)

76. If a function f : R \ {1} → R \ {m} defined by f (x) = x+3
x−2 is a bijection, then
3/l + 2m =
(A) 10
(B) 12
(C) 8
(D) 14

77. Given that f (x) = sin x + cos x and g(x) = x2 − 1, find the conditions under
which g(f (x)) is invertible.
(A) − π4 ≤ x ≤ π4
(B) 0 ≤ x ≤ π
(C) − π4 ≤ x ≤ π
(D) 0 ≤ x ≤ π2

78. Let the function g : (−∞, −0) → (− π2 , π2 ) be given by g(u) = 2 tan−1 (eu ) − π2 .
Determine the properties of g.
(A) Even and is strictly increasing in (0, ∞)
(B) Odd and is strictly decreasing in (−∞, 0)
(C) Odd and is strictly increasing in (−∞, ∞)
(D) Neither even nor odd, but is strictly increasing in (−∞, ∞)

79. Let f be the function defined by:
x2 −1
(
x2 −2|x−1|−1 , if x ̸= 1,
f (x) = 1
2, if x = 1.
The function is continuous at:
(A) The function is continuous for all values of x
(B) The function is continuous only for x > 1
(C) The function is continuous at x = 1
(D) The function is not continuous at x = 1

56

80. If ( 2
x log(cos x)
log(1+x) , x ̸= 0
f (x) =
0, x=0
then at x = 0, f (x) is .
(A) not continuous
(B) continuous but not differentiable
(C) differentiable
(D) not continuous, but differentiable

81. If f (x) is defined as follows:
 √
4, if − ∞ < x < − 5,
√ √

f (x) = x2 − 1, if − 5 ≤ x ≤ 5,
 √
4, if 5 ≤ x < ∞.


If k is the number of points where f (x) is not differentiable, then k − 2 =
(A) 2
(B) 1
(C) 0
(D) 3

√ √ dy
82. If x 1 + y + y 1 + x = 0, then find dx .
1
(A) x + x
1
(B) 1+x
1
(C) − (1+x) 2
x
(D) 1+x

√ 
x−x
83. If y = tan−1 1+x3/2
, then y ′ (1) is equal to:
(A) 0
(B) 12
(C) -1
(D) − 14

2 √
84. At x = π4 , dx
d
tan−1 (cos x) + sec−1 (ex ) =



(A) q π12 − π1
e 2 −1
(B) 4 + q 1 π2
π
eπ2 +e 2
1 2 π


(C) q π 2
+ π cot 2
eπ2 +e 2
(D) √1eπ + π1

57

85. The maximum area of a rectangle inscribed in a circle of diameter R is:
(A) R2
2
(B) R2
2
(C) R4
2
(D) R8

86. Consider the function f (x) = |x−1|
x2 . Then f (x) is:
(A) Increasing in (0, 1) ∪ (2, ∞)
(B) Increasing in (−∞, 0) ∪ (0, 1)
(C) Decreasing in (−∞, 0) ∪ (2, ∞)
(D) Decreasing in (0, 1) ∪ (2, ∞)

87. The maximum volume (in cu. units) of the cylinder which can be inscribed in
a sphere
√ of radius 12 units is:
(A) 384√ 3π
(B) 768 3π
√
(C) 768π/ √ 3
(D) 1152π/ 3

88. If the angle made by the tangent at the point (x0 , y0 ) on the curve
x = 12(t + sin t cos t), y = 12(1 + sin t)2 , with 0 < t < π2 , with the positive x-axis is π3 ,
then y0 is √equal to:
(A) 6(3 + 2√ 2)
(B) 3(7 + 4 3)
(C) 27
(D) 48

89. The altitude of a cone is 20 cm and its semi-vertical angle is 30◦ . If the
semi-vertical angle is increasing at the rate of 2◦ per second, then the radius of
the base is increasing at the rate of:
(A) 30 cm/sec
(B) 160
3 cm/sec
(C) 10 cm/sec
(D) 160 cm/sec

90. The point of inflexion for the curve y = (x − a)n , where n is odd integer and
n ≥ 3, is:
(A) (a, 0)
(B) (0, a)
(C) (0, 0)
(D) None of these

58

91. The population p(t) at time t of a certain mouse species satisfies the
differential equation:

dp(t)
= 0.5p(t) − 450.
dt
If p(0) = 850, then the time at which the population becomes zero is:
(A) 2 ln 18
(B) ln 9
(C) 12 ln 18
(D) ln 18

92. Evaluate the integral:
x3 − 1
Z
dx
x3 + x
(A) x + log |x| + 12 log(x2 + 1) + sin−1 (x) + c
(B) x − log |x| + 12 log(x2 + 1) − sin−1 (x) + c
(C) x + log |x| − 12 log(x2 + 1) + tan−1 (x) + c
(D) x − log |x| + 12 log(x2 + 1) − tan−1 (x) + c

93. Evaluate the integral: Z q p
x + x2 + 2 dx.
√ √
(A) 23 (x + x2 + 2)3/2 − 2(x + x2 + 2)1/2 + C
√ √
(B) 31 (x + x2 + 2)3/2 − 2(x + x2 + 2)1/2 + C
√ √
(C) (x +√ x2 + 2)−3/2 − 2(x + x2 + 2)1/2 + C
2
√ x √+2) −6 + C
2
(D) (x+
3 x+ x2 +2

94. The value of etan θ (sec θ − sin θ)dθ is:
R

(A) etan θ sec θ + c
(B) etan θ sin θ + c
(C) etan θ (tan θ + sin θ) + c
(D) etan θ cos θ + c

R∞ dx
95. The value of 0 (x2 +a2 )(x2 +b2 ) is:
πab
(A) a+b
ab
(B) 2(a+b)
π
(C) 2ab(a+b)
(D) π(a+b)
2ab

R π/2
96. The value of definite integral 0 log(tan x)dx is:
(A) 0

59

(B) π4
(C) π2
(D) π

97. Evaluate the integral:
Z 9
log 3x2
dx
5 log 3x2 + log(588 − 84x + 3x2 )
(A) 2
(B) 1
(C) 12
(D) 4

98. Evaluate the integral:
x2 (x sec2 x + tan x)
Z
dx
(x tan x + 1)2
2
(A) − x tanx x+1
(B) 2 loge |x sin x + cos x| + C
2
(C) − x tanx x+1 + 2 loge |x sin x + cos x| + C
2
(D) − x tanx x+1 − 2 loge |x sin x + cos x| + C

99. Evaluate the following limit:
n
Y r3 − 8
lim .
n→∞ r3 + 8
r=3

(A) 27
(B) 37
(C) 47
(D) 67

R π2 sin( π4 +x)+sin( 3π4 +x)
100. The value of 0 cos x+sin x dx is:
(A) √π2
π
(B) 2√ 2
π
(C) 3√ 2
π
(D) 4√ 2

101. The line y = mx bisects the area enclosed by lines x = 0, y = 0, and x = 23 and
the curve y = 1 + 4x − x2 . Then, the value of m is:
(A) 13
6
(B) 13
2
(C) 13
5

60