CUSAT CAT 2022 Question Paper PCM Shift 3

101 Test in Physics Chemistry and Mathematics (Shift 3)

1. When there is an electric current through a conducting wire along its length, then an
electric field must exist

(A) outside the wire but normal to it
(B) outside the wire but parallel to it
(C) inside the wire but normal to it
(D) inside the wire but parallel to it

2. A moving charge produces electric field along x-direction and magnetic field along
y-direction. Then what is the direction of its velocity?

(A) x direction
(B) y direction
(C) z direction
(D) Can be in any direction

3. Phase difference between voltage and current of a purely inductive circuit is

π
(A)
4
π
(B)
12
π
(C)
8
π
(D)
2

4. When two coherent sources of same intensity interfere, the resultant intensity will be

(A) 4 times of the initial intensity
(B) 8 times of the initial intensity
(C) 2 times of the initial intensity
(D) Equal to initial intensity

5. Wavelength of source is 6000 Å and the diameter of object is 100 inch, so the limit of
resolution is

(A) 2.9 × 10–7 radians
(B) 2.8 × 10–7 radians
–7
(C) 2.5 × 10 radians
(D) 2.8 × 10–5 radians

6. The SI units of radioactivity is

(A) Curie
(B) Fermi
(C) Becquerel
(D) Joule

7. The mean life for particle decay is

(A) 1.145 times greater than half life
(B) 1.445 times greater than half life
(C) 1.465 times greater than half life
(D) 1.345 times greater than half life

8. Which of the following electromagnetic waves have lowest wavelength?

(A) Green light
(B) X-rays
(C) Gamma rays
(D) Ultraviolet rays

9. Fermi energy level of the intrinsic semiconductor is located

(A) just below the valence band
(B) just above the conduction band
(C) either below or above the conduction band
(D) half way between the valence and conduction band

10. Zener diode is always used in the

(A) Forward bias condition
(B) Reverse bias condition
(C) Zero bias condition
(D) All the above

11. At room temperature, the n-type semiconductor will have

(A) more of electrons
(B) more of ions
(C) more of holes
(D) equal number of electrons and holes

12. The SI base unit for forces is

(A) mkgs–2
(B) N
2
(C) mkgs
2 –2
(D) m kgs

13. When light rays enter into a medium having different optical density,
there will be change in

(A) its speed and frequency
(B) its speed and wavelength
(C) its frequency and wavelength
(D) its speed, frequency and wavelength

14. An electric dipole placed in a non uniform electric field experiences

(A) no force and no torque
(B) a force and a torque
(C) no force but a torque
(D) a force but no torque

15. Which of this experiment proves that particle has wave nature?

(A) Davisson-Germer experiment
(B) Millikan experiment
(C) Faraday's experiment
(D) Newton rings experiment

16. A proton and an alpha particle are accelerated by a constant electric field. Their
acceleration will be in the ratio of

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

17. Superconductor exhibits

(A) paramagnetism
(B) ferromagnetism
(C) diamagnetism
(D) ferrimagnetism

18. At what distance from the point of equilibrium, the kinetic energy equals the potential
energy for a simple harmonic oscillator of amplitude A?

A
(A)
4
A2
(B)
4
A
(C)
2
A
(D)
2

19. For a given material, the Young's modulus is 6 times its rigidity modulus.
Its Poisson's ratio is

(A) 0.2
(B) 2
(C) 4
(D) 0.4

20. The gravitational potential of a solid sphere is minimum

(A) at the surface of the sphere
(B) at a point outside the sphere
(C) at midpoint between the centre and surface of the sphere
(D) at the centre of the sphere

21. Adding detergent to the water, increases its

(A) surface tension
(B) viscosity
(C) wetting action
(D) angle of contact

22. A 2 kg box sits on a 3 kg box which sits on a 5 kg box. The 5 kg box rests on a table
top. What is the normal force exerted on the 5 kg box by the table top?

(A) 29.4 N
(B) 49 N
(C) 98 N
(D) 19.6 N

23. A Decoration of mass M is suspended by a string from the ceiling inside an elevator.
The elevator is travelling upward with a constant speed. The tension in the string is

(A) equal to Mg
(B) less than Mg
(C) greater than Mg
(D) impossible to tell without knowing the speed

24. If the pressure of an ideal gas in a closed chamber is doubled, then the volume of the gas

(A) become two times
(B) becomes half
(C) remain constant
(D) become four times

25. The work done by pseudo forces is

(A) Positive
(B) Negative
(C) Zero
(D) Infinite

26. In an isothermal process, the specific heat of the gas is

(A) finite
(B) one
(C) zero
(D) infinite

27. Which theory explains that every point on a wavefront may be considered as a source
of secondary spherical wavelets?

(A) Huygen's wave theory
(B) Corpuscular theory
(C) Electromagnetic theory
(D) Quantum theory

28. Which light source is used in long distance optical fiber communication?

(A) Metal Halide light
(B) Incandescent light
(C) LED source
(D) Laser

29. The NAND gate output will be low if the two inputs are

(A) 01
(B) 00
(C) 10
(D) 11

30. Cp and Cv are specific heats at constant pressure and constant volume respectively.
It is observed that Cp – Cv = a for oxygen gas and Cp – Cv = b for nitrogen gas.
The correct relation between a and b is

(A) 8a = 7b
(B) 7a = 8b
(C) a=b
(D) a = −b

31. The temperature of a body falls from 40°C to 36°C in 5 minutes when placed in a
surrounding of constant temperature 16°C. The time taken by the body temperature to
fall from 36°C to 32°C is

(A) 8 min
(B) 6.1 min
(C) 4.2 min
(D) 5 min

32. 2 kg ice at –20°C is mixed with 5 kg water at 20°C, then final amount of water in
the mixture will be (specific heat of ice is = 0.5 cal/gm°C; specific heat of water
is = 1 cal/gm°C; latent heat of fusion is 80 cal/gm)

(A) 6 kg
(B) 7 kg
(C) 3.5 kg
(D) 5 kg

33. The minimum orbital angular momentum of an electron in hydrogen atom is

(A) h
h
(B)
2
h
(C)
2π
h
(D)
λ

34. Cooking gas containers are kept in a lorry moving with uniform speed.
The temperature of the gas molecules inside will be

(A) increase
(B) decrease
(C) remain same
(D) increases for some while decreases for others

35. An electric dipole is placed at an angle of 30° in a non-uniform electric field. The
dipole will experience

(A) a translational force only in the direction of the field
(B) a translational force only in a direction normal to the direction of the field
(C) a torque as well as a translational force
(D) a torque only

36. Which of the following phenomena is NOT common to sound and light waves?

(A) Interference
(B) Diffraction
(C) Coherence
(D) Polarization

37. If the binding energy of the electron in a hydrogen atom is 13.6 eV, the energy
2+
required to remove the electron from the first excited state of Li is

(A) 30.6 eV
(B) 13.6 eV
(C) 3.4 eV
(D) 122.4 eV

38. Heat transfer in air occurs mainly due to

(A) Conduction
(B) Convection
(C) Radiation
(D) Radiation and Conduction

39. A body weighs 40 g in air. If its volume is 10 cc, then in water it will weigh

(A) 30 g
(B) 40 g
(C) 50 g
(D) 33 g

40. The strength of the magnetic field around a straight wire is

(A) Same everywhere around the wire
(B) Obeys inverse square law
(C) Directly proportional to square of the distance from the wire
(D) Directly proportional to the distance from the wire

41. The electrostatic pressure on a charged surface having a surface charge density σ is

(A) σ/2ε0
2
(B) σ /2ε0

(C) σ/ε0
2
(D) σ /ε0

42. According to Joule’s law, if the potential difference across a conductor having a
material of specific resistance ‘ρ’ remains constant, then heat produced in the
conductor is directly proportional to

(A) ρ
(B) ρ2
(C) 1 / ρ
(D) 1 / σ

43. A circular coil of radius R carries a current I, for which the magnetic field at its centre
is B. At what distance x from the centre on the axis of the coil, the magnetic field will
be B / 8

(A) 2R
(B) 3R
(C) 2R
(D) 3R

44. If a magnet is enclosed in a box made up of iron, then the magnetic field outside the
box will be

(A) very high but finite
(B) infinity
(C) low value but finite
(D) zero

45. A bar magnet is dropped vertically down through a wire loop held horizontally.
The magnet will fall

(A) with acceleration 'g'
(B) with acceleration greater than 'g'
(C) with uniform acceleration less than 'g'
(D) with non-uniform acceleration less than 'g'

46. The mutual inductance between a pair of coils does not depend on

(A) number of turns in the coil
(B) separation between the coils
(C) relative orientation of the coil
(D) rate of change of current with coils

47. Which of the following statement is NOT correct?

(A) The magnification produced by a convex mirror is always less than one
(B) A virtual, erect, same-sized image can be obtained using a plane mirror
(C) A virtual, erect, magnified image can be formed using a concave mirror
(D) A real, inverted, same-sized image can be formed using a convex mirror

48. A beam of light consisting of red, green and blue colours is incident on AB of a right
angled prism. The refractive index of the material of the prism for red, green and blue
are 1.39, 1.44 and 1.47 respectively. The prism will

(A) separate red colour from the green and blue colour
(B) separate blue colour from the red and green colour
(C) separate all the colours from one another
(D) all colours propagate along same path

49. Light from the constellation Virgo is observed to increase in wavelength by 0.4%.
With respect to the earth the constellation is

(A) moving away with velocity 1.2 × 106 m/s

(B) coming close with velocity 1.2 × 106 m/s

(C) moving away with velocity 4 × 106 m/s

(D) coming close with velocity 4 × 106 m/s

50. When light passes from one medium into another medium, which of the physical
property does not change?

(A) Velocity
(B) Wavelength
(C) Frequency
(D) Refractive index

51. In hydrogen atom, if the difference in the energy of the electron in n = 2 and n = 3
orbits is E, the ionization energy of hydrogen atom is

(A) 13.2 E
(B) 7.2 E
(C) 5.6 E
(D) 3.2 E

52. Nucleus with same neutron number but different atomic number is called as

(A) isobars
(B) isotones
(C) isotopes
(D) isotherm

234
53. The mass of one Curie of U is

(A) 3.7 × 1010 gm

(B) 2.348 × 10 23 gm

(C) 1.48 × 10 −11 gm

(D) 6.25 × 10−24 gm

54. A voltmeter with resistance 150 Ω is connected across a 150 V source having an
internal resistance of 0.8 Ω, then the voltmeter will read

(A) 100.4
(B) 149.2
(C) 120
(D) 178.6

55. Electrical conductivity of a semiconductor

(A) decreases with the rise in its temperature
(B) increases with the rise in its temperature
(C) does not change with the rise in its temperature
(D) first increases and then decreases with the rise in its temperature

56. In an insulator, the energy gap between the valance band and conduction band is of
the order of

(A) > 5 eV
(B) 2 eV
(C) 3.6 eV
(D) 4.1 eV

57. Three small spheres, each carrying a positive charge Q, are placed on the
circumference of a circle of radius ‘r’ to form an equilateral triangle. The electric field
intensity at the center of the circle will be

(A) 3Q/r
(B) 3Q/r2

(C) Q/2r2
(D) zero

58. In an LCR series a.c circuit, the voltage across each of the components, L, C, and R is
50 V. The voltage across the LC combination will be

(A) 100 V
(B) 50√2 V
(C) 50 V
(D) 0 V (zero)

59. A telescope has a magnifying power of 10. If one looks at a tree of height 15 meters
through the telescope, then the tree appears

(A) 10 times taller
(B) 10 times farther
(C) 10 times nearer
(D) 15 times nearer

60. If the radius of the Earth’s orbit is made one-fourth, the duration of one year will become

(A) 8 times
(B) 4 times
1
(C) times
4
1
(D) times
8

61. In which of the following thermodynamic processes, there is no flow of heat between
the system and the surroundings?

(A) Isobaric
(B) Isochoric
(C) Adiabatic
(D) Isothermal

62. Entropy remains constant in

(A) isothermal process
(B) adiabatic process
(C) cyclic process
(D) isobaric process

63. While charging the lead storage battery

(A) PbSO4 at cathode is reduced to Pb
(B) PbSO4 at anode is reduced to Pb
(C) PbSO4 at cathode is oxidised to Pb
(D) PbSO4 at anode is oxidised to Pb

64. Axial ratios of hexagonal will be

(A) a=b=c
(B) a=b≠c
(C) a≠b≠c
(D) a = 2b = 3c

65. Structure of cesium chloride crystal is

(A) face centred cubic
(B) body centred cubic
(C) simple cubic
(D) hexagonal close packing

66. What type of stoichiometric defect is shown by ZnS?

(A) Schottky defect
(B) Frenkel defect
(C) Both Frenkel and Schottky defects
(D) Plane defect

67. Which one of the following is NOT applicable to catalytic action?

(A) Catalyst reduces energy of activation
(B) Catalyst will be most effective in the finely divided state
(C) Catalyst can alter the position of equilibrium of reversible reactions
(D) Catalyst cannot initiate a reaction

68. If dispersed phase and dispersion medium are gas and liquid, respectively,
then the name of the colloidal system is

(A) aerosol
(B) solid foam
(C) foam
(D) emulsion

69. The electric charge on a colloidal particle is observed by

(A) Brownian movement
(B) Electrolysis
(C) Electrodialysis
(D) Electrophoresis

70. ‘All four quantum numbers cannot be the same for any two electrons in an atom’.
This principle is known as

(A) Aufbau principle
(B) Hund’s rule
(C) Pauli’s Exclusion principle
(D) Plank’s rule

71. 50 ml of 0.1 M acetic acid is mixed with 50 ml 0.1 M sodium acetate. The pH of the
solution is (pKa of acetic acid is 4.26)

(A) 13.0
(B) 7.0
(C) 4.26
(D) 1.0

72. One mole of an ideal gas undergoes expansion from 24.6 l to 246.0 l against a
constant pressure of 1 atmosphere at 300 K. The work done is

(A) 221.4 l atm
(B) 24.6 l atm
(C) 9.0 l atm
(D) 0.082 l atm

73. Aqueous solution of CuSO4 is electrolyzed between the Pt electrodes. At anode
2+
(A) Cu is oxidized to Cu
2+
(B) Cu is reduced to Cu
(C) H2 gas is evolved
(D) O2 gas is evolved

74. IUPAC name of the following compound is

(A) 4-formyloctan-2-one
(B) 2-(2-oxopropyl)hexanal
(C) 2-butyl-4-oxopentanal
(D) 2-butyl-4-ketopentanal

75. Carbon monoxide (CO) acts as a

(A) strong π-donor and weak π-acceptor
(B) weak π-donor and strong π-acceptor
(C) weak σ-donor and strong π-acceptor
(D) strong σ-donor and good π-acceptor

76. Methoxypropane and Ethoxyethane constitute a pair of

(A) functional group isomers
(B) metamers
(C) position isomers
(D) regioisomers

77. The stability order of methyl, ethyl, isopropyl and tert-butyl carbocations is

(A) methyl > ethyl > isopropyl > tert-butyl
(B) methyl < ethyl < isopropyl < tert-butyl
(C) methyl ≈ ethyl > isopropyl > tert-butyl
(D) methyl < ethyl ≈ isopropyl < tert-butyl

78. Compare the steam volatility of 2-hydroxybenzaldehyde with that of
4-hydroxybenzaldehyde.

(A) both are equally steam volatile
(B) both are not steam volatile
(C) 2-hydroxybenzaldehyde is much more steam volatile than 4-hydroxybenzaldehyde
(D) 4-hydroxybenzaldehyde is much more steam volatile than 2-hydroxybenzaldehyde

79. The active stationary phase in Partition chromatography over chromatographic paper is

(A) cellulose
(B) starch
(C) hemicellulose
(D) water trapped in chromatographic paper

80. Kolbe’s electrolytic method is suitable for the generation of which among the
following gases in the pure form?

(A) Methane
(B) Ethane
(C) Propane
(D) Isobutane (2-methylpropane)

81. But-2-yne is converted to (E)-but-2-ene by

(A) catalytic hydrogenation using Lindlar’s catalyst
(B) using Raney nickel
(C) copper catalyzed partition reaction with n-butane
(D) reduction using controlled amount of sodium in liquid ammonia

82. Cyclic compounds with alternating single and double bonds can be

(A) aromatic or antiaromatic
(B) aromatic or nonaromatic
(C) antiaromatic or nonaromatic
(D) aromatic, antiaromatic or nonaromatic

83. Which among the following methods is NOT a suitable one for the preparation of
benzene?

(A) Passing ethyne under pressure through an iron tube heated up to 873 K
(B) Soda lime distillation of benzoic acid
(C) Treatment of chlorobenzene with sodium metal in dry ether
(D) Passing phenol in the gaseous state over a heated bed of zinc dust

84. Which among the following halides will undergo fastest SN1 solvolysis in water?

(A) 2-fluoro-2-methylpropane
(B) 2-chloro-2-methylpropane
(C) 2-bromo-2-methylpropane
(D) 2-iodo-2-methylpropane

85. Which among the following methods is NOT suitable for selective generation of
butan-1-ol?

(A) Reaction of butylmagnesium bromide with water
(B) Reaction of propylmagnesium bromide with formaldehyde
(C) Reaction of ethylmagesium bromide with ethylene oxide
(D) Hydroboration of but-1-ene followed by oxidation with hydrogen peroxide in
the presence of aqueous sodium hydroxide

86. Predict the major products formed in the acidic hydrolysis of cumene hydroperoxide.

(A) Phenol and acetone
(B) Benzoic acid and methanol
(C) Acetophenone and ethanol
(D) Phenol and propan-2-ol

87. Which of the following nuclides is most radioactive?
108
(A) 47 Ag
66
(B) 30 Zn
37
(C) 17 Cl
31
(D) 15 P

88. Which radical can be tested with Nessler’s reagent?

(A) K
(B) NH4+
+
(C) Na
(D) Fe3+

56
89. The observed mass of 26Fe is 55.9375 amu. Using the mass of proton and
neutron = 1.00732 amu and 1.00866 amu respectively, calculate the mass defect?

(A) 0.6234
(B) 0.6753
(C) 0.5678
(D) 0.5126

90. The thermal stability of hydrides is in the order of

(A) HF > HCl > HBr > HI
(B) HF > HI > HCl > HBr
(C) HF > HBr > HI > HCl
(D) HF < HI < HBr > HCl

91. Which complex is called as outer orbital complex?

(A) [Fe(CO)5]
2+
(B) [Fe(H2O)6]
(C) [Ni(CO)4]
(D) [Fe(CN)6]2−

92. How many atoms are present in a FCC unit cell?

(A) 3
(B) 4
(C) 2
(D) 6

93. How many elements are present in the sixth period of the modern periodic table?

(A) 18
(B) 22
(C) 36
(D) 32

94. In which of the following pairs, Dipole-induced dipole interaction is present?

(A) HCl and He atoms
(B) Cl2 and CCl4
(C) SiF4 and He atoms
(D) H2O and alcohol

95. In which of the following compounds, the maximum covalent character is shown?

(A) MgCl2
(B) FeCl2
(C) SnCl2
(D) AlCl3

96. Which of the following is Paramagnetic?

(A) CO
(B) CN−
(C) NO+
(D) O2

97. Choose the correct formula for borax.

(A) Na2B4O7.5H2O
(B) Na2B4O7.10H2O
(C) Na2B4O7.3H2O
(D) Na2B4O7.H2O

98. Diagonal relationship between beryllium and aluminium is due to the fact that
2+
(A) the ionic radius and charge/radius ratio of Be is nearly the same as that of
3+
Al ion.
(B) like aluminium, beryllium is readily attacked by acids
(C) the chlorides of beryllium and aluminium are not soluble in organic solvents
and are strong Lewis bases
(D) beryllium and aluminium ions have no tendency to form complexes

99. Identify the CORRECT statement

(A) Sodium carbonate is also called as baking soda
(B) Sodium carbonate is a white crystalline solid which exists as a decahydrate,
Na2CO3.10H2O
(C) Sodium carbonate is generally prepared by Castner-Kellner process
(D) Anhydrous sodium carbonate is called as soda lime

100. Considering the elements F, Cl, O and N, the correct order of their chemical reactivity
in terms of oxidizing property is

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

(1 − x)n − 1
101. If the value of lim is 100, then n is equal to
x→0 x

(A) 100
(B) −100
(C) 99
(D) −99

102. Let f ( x) be a polynomial of degree 2 satisfying f (0) = 1, f '(0) = −2 and f ''(0) = 6.
2
Then ∫ f ( x) dx is equal to
−1

(A) 6
(B) 0
(C) 9
(D) 3

103. The domain of the derivative of the function f ( x) = tan −1 x, if x ≤ 1 and
1
f ( x) =
2
( x − 1) , if x > 1, is
(A) − {0}
(B) − {1}
(C) − {−1}
(D) − {−1,1}

1 2
104. The function f : − {0} → given by f ( x) = − 2x can be made continuous at
x e −1
x = 0 by defining f (0) as

(A) 0
(B) 1
(C) 2
(D) −1

105. Rolle’s theorem is applicable to the function

(A) f ( x ) = x , −1 ≤ x ≤ 1
(B) f ( x) = tan x, 0 ≤ x ≤ π
(C) f ( x) = sin 2 x, 0 ≤ x ≤ π
(D) f ( x) = x3 − 3x + 3, 0 ≤ x ≤ 1

π
106. If sin −1 x + sin −1 y = , then the value of cos −1 x + cos −1 y is
2

π
(A)
3
π
(B)
2
(C) π
2π
(D)
3

107. The number of different salads that can be made from cucumber, tomatoes, onions,
beetroot and carrots is

(A) 16
(B) 28
(C) 31
(D) 32

108. The total number of subsets of a finite set A has 56 more elements than the total
number of subsets of another finite set B. The number of elements in the set A is
(A) 5
(B) 6
(C) 7
(D) 8

109. If 5 and − 5 are two roots of the polynomial x3 + 3x 2 − 5 x − 15, then its third root is

(A) 3
(B) −3
(C) 5
(D) −5

110. The value(s) of p such that the lines, represented by the pair of linear equations
3x − y − 5 = 0 and 6x − 2y − p = 0, be parallel is

(A) all real values except 10
(B) 10
5
(C)
2
1
(D)
2

111. If A = [aij ] is a symmetric matrix of order 2×2 such that A = −15 and cij represents
the cofactor of aij then the value of a21c12 + a22c22 is

(A) 17
(B) 18
(C) 19
(D) −15

112. One ticket is selected at random from 50 tickets numbered 00, 01, 02, . . . 49. Then
the probability that the sum of the digits on the selected ticket is 8, given that the
product of these digits is zero, equals

5
(A)
14
1
(B)
50
1
(C)
7
1
(D)
14

113. Let x + 8 y − 22 = 0, 5 x + 2 y − 34 = 0, 2 x − 3 y + 13 = 0 be the three sides of a triangle.
Then the area of the triangle is

(A) 19 square unit
(B) 36 square unit
(C) 42 square unit
(D) 72 square unit

114. For any two real numbers θ and φ , we define θ R φ if and only if sec2 θ − tan 2 θ = 1.
The relation R is

(A) reflexive but not transitive
(B) symmetric but not reflexive
(C) both reflexive and symmetric but not transitive
(D) an equivalence relation

f (3 x)
115. Let f : R → R be a positive increasing function with lim x→∞ = 1. Then
f ( x)
f (2 x)
lim x→∞ =
f ( x)

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

116. From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to
be selected and arranged in a row on a shelf so that the dictionary is always in the
middle. Then the number of such arrangement is

(A) at least 750 but less than 1000
(B) at least 1000
(C) at least 500 but less than 750
(D) less than 500

2z +1
117. If the imaginary part of is −1, then the locus of the point representing z in the
iz + 1
complex plane is

(A) a straight line
(B) a parabola
(C) a circle
(D) an ellipse

118. The minimum value of 27 tan 2 θ + 3cot 2 θ

(A) lies between 1 and 17
(B) greater than or equal to 18
(C) less than 18
(D) lies between 2 and 12

3π
119. If sin −1 x + sin −1 y + sin −1 z = , then the value of
2
9
x100 + y100 + z100 − 101 101 101 is
x +y +z

(A) −1
(B) 0
(C) 1
(D) 3

2
120. Let x be an integer such that x −3x < 4. Then the number of possible values of x is

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

121. The product of all solutions of the equation ( x − 2)2 − 3 x − 2 + 2 = 0 is

(A) 0
(B) −2
(C) 2
(D) −4

122. If log 0.5 sin x = 1 − log 0.5 cot x, then the number of solutions of x ∈ [−2π , 2π ] is

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

123. The value of 3. cosec 20° − sec 20° is equal to

(A) 1
(B) 3sin 20°
(C) 4
(D) 4 cos 40°

α + β 
tan  
124. If 5sin α = 7 sin β , then  2  is equal to
α − β 
tan  
 2 

(A) 6
(B) 8
(C) 10
(D) 12

125. The minimum value of P = bcx + cay + abz, when xyz = abc is

(A) abc
(B) 2abc
(C) 3abc
(D) 6abc

b d
126. If 3 + i = (a + ib)(c + id), then tan −1 + tan −1 is equal to
a c

π
(A) + 2nπ for some integer n
2
π
(B) − + nπ for some integer n
3
π
(C) − + 2nπ for some integer n
6
π
(D) + nπ for some integer n
6

127. If three positive unequal numbers x, y, z are in Harmonic Progression, then

(A) x2 + y2 > z2
(B) x − y > z
2 2 2
(C) x + z > 2y
(D) x2 + y2 + z2 > 1

128. Let a be positive and let M and N be the arithmetic and geometric means of the roots
2 2
of x − 2ax + a respectively. Then

(A) M = 2N
(B) M = −N
(C) M=N
(D) M = −2N

dy
129. Let y = f x3 , z = g x5 , f '( x) = tan x and g '( x) = sec x. Then
( ) ( ) is equal to
dz

3 tan x3
(A) .
5x 2
sec x5
5 x 2 sec x5
(B) .
3 tan x3
3x 2 tan x3
(C) .
5 sec x5
5 sec x5
(D) 2
.
3x tan x3

130. Let y = sin −1 ( x − ax − a − ax ) . Then dy
dx
is equal to

1
(A)
sin a − ax
(B) sin x .sin a
1
(C)
2 x 1− x
(D) 0

dy
131. If 3x + 3 y = 3x + y , then equals
dx

3x + 3 y
(A)
3x − 3 y
3x + 3 y
(B)
1 + 3x+ y
3x + y − 3 y
(C)
3y
 3y −1 
(D) 3x − y  x
 1− 3 

132. If P(1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then

(A) a = 2, b = 4
(B) a = 3, b = 4
(C) a = 2, b = 3
(D) a = 3, b = 5

133. The area of the triangle formed by joining the origin to the points of intersection of
the line 5 x + 2 y = 3 5 and the circle x 2 + y 2 = 10, is

(A) 6 sq units
(B) 5 sq units
(C) 4 sq units
(D) 3 sq units

134. The lines x − y − 2 = 0, x + y − 4 = 0 and x + 3y = 6 meet at the common point

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

2 2
135. The equation of the chord of the circle x + y − 4x = 0, whose mid point is (1, 0), is

(A) y=2
(B) y=1
(C) x=2
(D) x=1

2 2
136. One of the diametrical chord of the circle x + y − 12x + 4y + 6 = 0 is

(A) x+y=0
(B) x + 3y = 0
(C) x=y
(D) 3x + 2y = 0

2 2
137. The equation of the line which is tangent to both the circle x + y = 5 and the
2
parabola y = 40x is

(A) 2x − y ± 5 = 0
(B) 2x ± y + 5 = 0
(C) 2x ± y − 5 = 0
(D) 2x + y ± 5 = 0

138. Let the points (1, 2) and (k, −1) be conjugate points with respect to the ellipse
2 2
2x + 3y = 6. Then the value of k is

(A) 2
(B) 4
(C) 6
(D) 8

139. If f ( x + y, x − y) = xy, then the arithmetic mean of f ( x, y) and f ( y, x) is

(A) y
(B) x
(C) 0
(D) xy

140. If f ( x) is an odd periodic function with period 2, then f ( 4) equals

(A) −4
(B) 4
(C) 2
(D) 0

2 x − x2
141. lim x is equal to
x →2 x − 2 2

log 2 − 1
(A)
log 2 + 1
log 2 + 1
(B)
log 2 − 1
(C) 1
(D) −1

∞ xn
142. If f ( x) = ∑ (log a ) n , then at x = 0, f ( x)
n =0 n !

(A) has no limit
(B) is discontinuous
(C) is continuous but not differentiable
(D) is differentiable

143. Let f ( x + y) = f ( x) f ( y) and f ( x) = 1 + sin 2x.g ( x), where g ( x) is continuous.
Then f ′(x) equals

(A) f ( x) g (0)
(B) 2f ( x) g (0)
(C) 2g(0)
(D) g(0)

2
144. The area of the triangle formed by a tangent to the curve 2xy = a
and the coordinate axes is

(A) 2a 2
(B) a2
(C) 3a 2
(D) 4a 2

3 2
145. ∫ sin x ⋅ cos x dx is equal to
sin 5 x sin 3 x
(A) − +c
5 3
sin 5 x sin 3 x
(B) + +c
5 3
cos5 x cos3 x
(C) − +c
5 3
cos5 x cos3 x
(D) + +c
5 3

sin(2 x)
146. ∫ 1 + cos2 x dx is equal to
1
(A) − log 1 + cos 2 x + c
( )
2

(B) 2 log 1 + cos 2 x + c
( )
1
(C) log (1 + cos 2 x ) + c
2

(D) c − log 1 + cos 2 x
( )

x3 + 3 x 2 + 3x + 1
147. ∫ ( x + 1)5
dx is equal to

1
(A) − +c
x +1
1
(B) log( x + 1) + c
5
(C) log( x + 1) + c

(D) tan −1 x + c

dy x(2 log x + 1)
148. The solution of the differential equation = is
dx sin y + y cos y

x2
(A) y sin y = x 2 log x + +c
2
(B) y cos y = x 2 (log x + 1) + c
x2
(C) y cos y = x 2 log x + +c
2
(D) y sin y = x 2 log x + c

149. The solution of the differential equation ydx + x − y 3 dy = 0 is
( )
1 3
(A) xy = y +c
3
(B) xy = y 4 + c
(C) y 4 = 4 xy + c
(D) 4 y = y3 + c

150. The ratio in which iˆ + 2 ˆj + 3kˆ divides the join of −2iˆ + 3 ˆj + 5kˆ and 7iˆ − kˆ is

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

151. A vector perpendicular to the plane containing the points
A(1, −1, 2), B(2, 0, −1) and C(0, 2, 1) is

(A) 4iˆ + 8 ˆj − 4kˆ
(B) 8iˆ + 4 ˆj + 4kˆ
(C) 3iˆ + ˆj + 2kˆ
(D) iˆ + ˆj − kˆ

x + 1 y −1 z − 2
152. Let the angle θ between the line = = and the plane
1 2 2
1
2 x − y + λ z + 4 = 0 be such that sin θ = . Then the value of λ is
3

4
(A) −
3
3
(B)
4
3
(C) −
5
5
(D)
3

153. The order of the element 4 in the group ( *11, 11 ) is
(A) 1
(B) 3
(C) 10
(D) 11

th th th
154. Let a, b, c be the l m and n powers of a GP and all are positive.
log a l 1
Then log b m 1 equals
log c n 1

(A) 0
(B) 1
(C) 2
(D) 3

r r r r r r
155. The torque about the point 3i − j + 3k of a force 4i + 2 j + k through the point
r r r
5i + 2 j + 4k , is
r r r
(A) i + 2 j − 8k
r r r
(B) i + 2 j + 8k
r r r
(C) i − 2 j − 8k
r r r
i +2j −k
(D)
3

156. If i = −1, then the value of i + i 22 + i 23 + i 24 + i 25 is

(A) i
(B) −1
(C) 1
(D) −i

157. If f '( x) = x and f (1) = 2, then f ( x) is

2
(A) −
3
(
x x +2 )
2
(B)
3
(
x x +2 )
1 2
(C) −
2
(
x +3 )
1 2
(D)
2
(
x +3 )

158. If f ( x) = x 2 and g ( x) = x , then

(A) ( g o f )( −2) = 2
(B) ( f o g )( 2) = 4
(C) ( g o f )( 2) = 4
(D) ( f o g )(3) = 6

 kx 2 , x ≤ 2
159. The value of k, such that the function f ( x) =  is continuous at x = 2, is
3, x > 2

(A) 2
(B) 1.75
(C) 0.75
(D) 2.75

160. The missing vertex of a triangle whose other two vertices
 1 
(−3, −9), (−1, 6) and centroid  − , 2  is
 3 

(A) (3, 9)
(B) (2, 3)
(C) (3, 4)
(D) (4, 3)

161. The function f ( x) = x − 1 is not differentiable at

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

 π
162. If ω (≠ 1) be a cube root of unity, then the value of tan  ω10 + ω 20 π +  is
( )
 4

(A) 1
(B) −1
1
(C)
3
1
(D) −
3

163. The two circles z − 1 − i = 2 and z = 2 intersect at

(A) no point
(B) one point
(C) two points
(D) four points

th th
164. In the binomial expansion of (a − b)n , n ≥ 5, the sum of the 5 and 6 terms is zero.
a
Then equals
b

n−5
(A)
6
n−4
(B)
5
5
(C)
n−4
6
(D)
n−5

x2 y 2
165. The radius of the circle passing through the foci of the ellipse + = 1, and
16 9
having its centre at (0, 3) is

(A) 4
(B) 3
(C) 12
7
(D)
2

166. The harmonic mean of the roots of the equation ( 5 + 2 ) x2 − ( 4 + 5 ) x + ( 8 + 2 5 ) = 0 is

(A) 2
(B) 4
(C) 6
(D) 8

167. The point at which the tangent to the curve y = 4 x − 3 −1 has its slope
2 is
3
(A) (1, 2)
(B) (3, 2)
(C) (2, 3)
(D) (2, 1)

168. The solution set of the inequality
x + 3 ≥ 0 is
x−4

(A) (−∞, −3] ∪ (4, ∞)
(B) (−∞, −3] ∪ [4, ∞)
(C) (−∞, −3) ∪ (1, ∞)
(D) 3 and −4

3
169. If f ( x) = A ( 2 x ) + B, where f '(1) = 2 and ∫ f ( x)dx = 7, then which of the following
0
statements are not correct?

1
(i) A=
log 2
7  2 
(ii) B=
2 (log 2) − 1
3(log 2)
7
(iii) B=
log 2
7  2 
(iv) A=
2 (log 2) − 1
3(log 2)

(A) (i) only
(B) (i) and (iii)
(C) All are correct
(D) (i) and (ii)

170. The area of the portion of the circle x2 + y 2 = 64 which is exterior to the parabola
y 2 = 12 x is equal to

16 (8π + 3)
(A)
3
8 (8π + 3)
(B)
3
16 (8π − 3)
(C)
3
(D) (8π + 3)

2 5 −3
171. If ∫ f ( x)dx = 2 and ∫ [ 5 + f ( x)] dx = 9, then the value of the integral ∫ f ( x)dx is
−3 2 5

(A) 3
(B) 2
(C) 5
(D) 4

172. The relation R in the set A = {1, 2, 3} is given by
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is

(A) symmetric
(B) asymmetric
(C) neither symmetric nor transitive
(D) either symmetric or reflexive

173. The locus of the poles of the focal chords of a parabola is the

(A) axis
(B) directrix
(C) tangent at the vertex
(D) circle

1999 2
174. The eccentricity of the hyperbola
3 ( )
x − y 2 = 1 is

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

175. The sum of distances of any point on the ellipse 3x 2 + 4 y 2 = 24 from its foci is

(A) 8 2
(B) 4 2
(C) 24 2
(D) 16 2

176. Two dice are thrown. The probability that the numbers appeared have a sum 8 if it is
known that the second dice always exhibits 4, is

5
(A)
6
6
(B)
5
1
(C)
6
2
(D)
3

177. Which of the following is not a group?

(A) (Zn , +n)
(B) (Z, +)
(C) (Z, .)
(D) (R, +)

178. If the value of mode and mean is 60 and 66 respectively, then the value of median is

(A) 60
(B) 64
(C) 68
(D) 63

log e (π + x)
179. The function f ( x) = is
log e (e + x)

(A) increasing on (0, ∞)
 π π 
(B) decreasing on  0,  , increasing on  , ∞ 
 e e 
(C) decreasing on (0, ∞)
 π π 
(D) increasing on  0,  , decreasing on  , ∞ 
 e e 

dy
180. The integrating factor of the differential equation cos x + y sin x = 1 is
dx

(A) sin x
(B) cos x
(C) tan x
(D) sec x

181. The circles x 2 + y 2 − 10 x + 16 = 0 and x 2 − y 2 = r 2 intersect each other in two distinct
points if

(A) r < 2
(B) r > 8
(C) 2 < r < 8
(D) 2 ≤ r ≤ 8

182. Suppose A1, A2 ,..., A30 are thirty sets, each with five elements and B1, B2 ,..., B30 are n
30 n
sets each with three elements. Let U i =1 Ai =U j =1 B j = S . If each element of S

belongs to exactly ten of the Ai 's and exactly nine of the B j 's then n =

(A) 45
(B) 35
(C) 40
(D) 25

183.
 ( )
The function f ( x) = sec log x + 1 + x 2  is

(A) even
(B) odd
(C) constant
(D) neither even nor odd

184. A solution of the equation | z | − z = 1 + 2i is

3
(A) − 2i
2
3
(B) + 2i
2
3
(C) 2− i
2
3
(D) 2+ i
2

2 2
185. The equation z − 1 + z + 1 = 4 represents on the Argand plane

(A) a straight line
(B) an ellipse
(C) a circle with centre origin and radius 2
(D) a circle with centre origin and radius unity

186. If A is a singular matrix, then adj A is

(A) non-singular
(B) singular
(C) symmetric
(D) antisymmetric

xp + y x y
187. The determinant yp + z y z = 0 if
0 xp + y yp + z

(A) x, y, z are in A.P
(B) x, y, z are in G.P
(C) x, y, z are in H.P
(D) xy, yz , zx are in A.P

188. A telegraph has 5 arms and each arm is capable of 4 distinct positions, including the
position of rest. The total number of signals that can be made is

(A) 473
(B) 1023
(C) 1173
(D) 423

189. A club consists of members whose ages are in A.P, the common difference being
3 months. If the youngest member of the club is just 7 years old and the sum of the
ages of all the members is 250 years, then the number of members in the club is

(A) 15
(B) 25
(C) 20
(D) 30

x 2 − 9 x + 20
190. lim
x→5 x − [ x]

(A) is 1
(B) is 0
(C) does not exist
(D) cannot be determined

3− f ( x)
191. If f (9) = 0 and f '(9) = 1, then lim is equal to
x→9 3− x

(A) 0
(B) 1
(C) −1
(D) 3

192. Let f ( x + y ) = f ( x) ⋅ f ( y ) for all x, y where f (0) ≠ 0. If f (5) = 2 and f '(0) = 3,
then f '(5) is equal to

(A) 6
(B) 0
(C) 1
(D) −1

1 x −x
193. The shortest distance of the point (0, 0) from the curve y =
2
( e + e ) is
(A) 2
(B) 1
(C) 3
(D) 0

194. A determinant is chosen at random from the set of all determinants of order 2 with
elements 0 or 1 only. The probability that the value of the determinant chosen is
positive is

16
(A)
81
7
(B)
16
3
(C)
16
16
(D)
3

r r r r r r
195. If a is a unit vector and ( x + a ).( x − a ) = 24, the x must be

(A) 3
(B) 4
(C) 5
(D) 6

r r
196. The angle between two vectors a and b with magnitudes 5 and 4 respectively
rr
and a.b = 2 5 is

π
(A)
3
π
(B)
6
π
(C)
2
5π
(D)
12

 3 
197. The length of perpendicular from the point 1, , 2  to the plane 2 x − 2 y + 4 z + 5 = 0 is
 2 

(A) 3
(B) 6
(C) 5
(D) 7

198. P is a point on the line segment joining the points A(3, 2, 1) and B (6, 2, − 2) . If the
x − coordinate of P is 5, then the z − coordinate of P is

(A) −2
(B) 1
(C) −1
(D) 2

1 1
199. The probability of A and B solving a problem correctly is and respectively. If
3 4
the probability of their making a common error is 1/20 and they obtain the same
answer, then the probability of their answer to be correct is

1
(A)
12
1
(B)
40
13
(C)
120
10
(D)
13

200. A pair of dice is thrown 4 times. If getting a doublet is considered to be a success, the
probability of 2 successes is

25
(A)
216
29
(B)
216
31
(C)
216
33
(D)
216

KEY
SI Ke SI Ke SI Ke SI Ke SI Ke SI Ke SI Ke
No. y No. y No. y No. y No. y No. y No. y
1 D 31 B 61 C 91 B 121 A 151 B 181 C
2 C 32 A 62 B 92 B 122 B 152 D 182 A
3 D 33 C 63 A 93 D 123 C 153 C 183 A
4 C 34 C 64 B 94 A 124 A 154 A 184 A
5 A 35 C 65 B 95 D 125 C 155 A 185 D
6 C 36 D 66 B 96 D 126 D 156 A 186 B
7 B 37 A 67 C 97 B 127 C 157 D 187 B
8 C 38 B 68 C 98 A 128 C 158 A 188 B
9 D 39 A 69 D 99 B 129 A 159 C 189 B
10 B 40 A 70 C 100 B 130 C 160 A 190 C
11 A 41 B 71 C 101 B 131 D 161 A 191 B
12 A 42 C 72 A 102 C 132 C 162 A 192 A
13 B 43 B 73 D 103 D 133 B 163 C 193 B
14 B 44 D 74 B 104 B 134 C 164 B 194 C
15 A 45 D 75 D 105 C 135 D 165 A 195 C
16 A 46 D 76 B 106 B 136 B 166 B 196 B
17 C 47 D 77 B 107 C 137 B 167 B 197 C
18 C 48 A 78 C 108 B 138 C 168 A 198 A
19 B 49 A 79 D 109 B 139 C 169 D 199 D
20 D 50 C 80 B 110 A 140 D 170 C 200 A
21 C 51 B 81 D 111 D 141 A 171 D
22 C 52 B 82 D 112 D 142 D 172 C
23 A 53 C 83 C 113 A 143 B 173 B
24 B 54 B 84 D 114 D 144 B 174 A
25 C 55 B 85 A 115 C 145 C 175 B
26 D 56 A 86 A 116 B 146 D 176 C
27 A 57 D 87 A 117 C 147 A 177 C
28 D 58 D 88 B 118 B 148 D 178 B
29 D 59 C 89 D 119 B 149 C 179 C
30 A 60 D 90 A 120 D 150 A 180 D