Question 1:

If the function f(x) = { (72x − 9x − 8x + 1) / √(2 − √(1 + cosx)), x ≠ 0; a ln e² ln e³, x = 0 } is continuous at x = 0, then the value of a² is equal to:

1. 968
2. 1152
3. 746
4. 1250

Question 2:

If λ > 0, and θ is the angle between vectors a = î + λĵ − 3k̂ and b = 3î − ĵ + 2k̂, such that a + b and a − b are mutually perpendicular, then the value of (14 cos θ)² is equal to:

1. 25
2. 20
3. 50
4. 40

Question 3:

Let C be a circle of radius √10 units centered at the origin. The line x + y = 2 intersects the circle at points P and Q. If MN is a chord of length 2 units and slope −1, then the distance between the chords PQ and MN is:

1. 2 − √3
2. 3 − √2
3. √2 − 1
4. √2 + 1

Question 4:

Let a relation R on N × N be defined as: (x₁, y₁)R(x₂, y₂) if and only if x₁ ≤ x₂ or y₁ ≤ y₂. Consider the two statements:

(I) R is reflexive but not symmetric.
(II) R is transitive.
Which one of the following is true?

1. Only (II) is correct.
2. Only (I) is correct.
3. Both (I) and (II) are correct.
4. Neither (I) nor (II) is correct.

Question 5:

Let three real numbers a, b, c be in arithmetic progression, and a + 1, b, c + 3 in geometric progression. If a > 10 and their arithmetic mean is 8, the cube of their geometric mean is:

1. 120
2. 312
3. 316
4. 128

Question 6:

Let A = [[1, 2], [0, 1]] and B = I + adj(A) + (adj(A))² + ... + (adj(A))¹⁰. The sum of all elements in B is:

1. -110
2. 22
3. -88
4. -124

Question 7:

Evaluate 1×2² + 2×3² + ... + 100×(101)² / (12×22 + 22×32 + ... + 100²×101):

1. 306
2. 305
3. 32
4. 31

Question 8:

Let f(x) = ∫₀ˣ (t + sin(1 − eᵗ)) dt, x ∈ R. Find lim(x→0) f(x) / x³:

1. 1/6
2. -1/6
3. -2/3
4. 2/3

Question 9:

The area of the region {(x, y): y² ≤ 2x, y ≥ 4x − 1} is:

1. 11/32
2. 8/9
3. 11/12
4. 9/32

Question 10:

The area of the region S = {z ∈ C : |z − 1|² ≤ 4, z + z̅ ≥ 2, Im(z) ≥ 0} is:

1. π/3
2. 3π/2
3. 17π/8
4. 7π/4

Question 11:

If the value of the integral ∫_-1¹ (cos(αx) / (1 + 3x)) dx = (2 / π), then a value of α is:

1. π / 6
2. π / 2
3. π / 3
4. π / 4

Question 12:

Let f(x) = ∛(x − 2) + √(4 − x). If α and β are the minimum and maximum values of f respectively, then α² + 2β² is equal to:

1. 44
2. 42
3. 24
4. 38

Question 13:

If the coefficients of x⁴, x⁵, and x⁶ in the expansion of (1 + x)ⁿ are in arithmetic progression, then the maximum value of n is:

1. 14
2. 21
3. 28
4. 7

Question 14:

Consider a hyperbola H centered at the origin with foci on the x-axis. Let C₁ be a circle touching the hyperbola and centered at the origin, and C₂ touching the hyperbola at its vertex and centered at one focus. If the areas of C₁ and C₂ are 36π and 4π, respectively, the length of the latus rectum of H is:

1. 28/3
2. 14/3
3. 10/3
4. 11/3

Question 15:

If the mean of the following probability distribution is 46/9, the variance is:

1. 581/81
2. 566/81
3. 173/27
4. 151/27

Question 16:

Let PQ be a chord of the parabola y² = 12x, with the midpoint of PQ at (4, 1). Which of the following points lies on the line passing through P and Q?

1. (3, -3)
2. (3/2, -16)
3. (2, -9)
4. (1/2, -20)

Question 17:

Given cos⁻¹x − sin⁻¹y = α, −π/2 ≤ α ≤ π. The minimum value of x² + y² + 2xy sinα is:

1. -1
2. -1/2
3. 0
4. 1/2

Question 18:

Let y = y(x) be the solution of the differential equation (x² + 4)² dy + (2x³y + 8xy − 2) dx = 0, with y(0) = 0. Then y(2) is equal to:

1. π/8
2. π/16
3. 2π
4. π/32

Question 19:

Let a = î + ĵ + k̂, b = 2î + 4ĵ − 5k̂, and c = xî + 2ĵ + 3k̂. If d is a unit vector in the direction of b + c such that a·d = 1, then (a × b) · c is equal to:

1. 9
2. 6
3. 3
4. 11

Question 20:

Let P be the intersection of the lines x−2 = (y−4)/5 = z−2 and x−3 = (y−2)/3 = (z−3)/2. The shortest distance of P from the line 4x = 2y = z is:

1. 5√14/7
2. √14/7
3. 3√14/7
4. 6√14/7

Question 21:

Let S = {sin²(2θ) : (sin⁴θ + cos⁴θ)x² + (sin²θ)x + (sin⁶θ + cos⁶θ) = 0 has real roots}. If α and β are the smallest and largest elements of the set S respectively, then 3((α−2)² + (β−1)²) equals:

1. 2
2. 3
3. 4
4. 5

Question 22:

If ∫csc⁵x dx = α cot(x)csc(x)(csc²x + 3/2) + β log|tan(x/2)| + C, where α, β ∈ R and C is a constant of integration, then the value of 8(α + β) equals:

1. 1
2. 2
3. 3
4. 4

Question 23:

Let f: R → R be a thrice differentiable function such that f(0) = 0, f(1) = 1, f(2) = −1, f(3) = 2, and f(4) = −2. The minimum number of zeros of (3f′f′′ + ff′′′)(x) is:

1. 3
2. 4
3. 5
4. 6

Question 24:

Consider the function f(x) = (2x/√(1 + 9x²)). If the 10th composition of f, f(f(f(...f(x)))), is written as 2¹⁰x/√(1 + 9αx²), the value of √(3α + 1) is:

1. 256
2. 512
3. 1024
4. 2048

Question 25:

Let A be a 2×2 symmetric matrix such that A[1 1] = [3 7], and the determinant of A is 1. If A⁻¹ = αA + βI, where I is the identity matrix, then α + β equals:

1. 3
2. 4
3. 5
4. 6

Question 26:

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is:

1. 3600
2. 4525
3. 5626
4. 6400

Question 27:

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as 1/3 and 2/3, respectively. Let x be the number of matches that the team wins, and y be the number of matches that the team loses. If the probability P(|x − y| ≤ 2) is p, then 39p equals:

1. 72
2. 84
3. 96
4. 108

Question 28:

Consider a triangle ABC having the vertices A(1, 2), B(α, β), and C(γ, δ) and angles ∠ABC = π/6 and ∠BAC = 2π/3. If the points B and C lie on the line y = x + 4, then α² + γ² is equal to:

1. 15
2. 20
3. 25
4. 30

Question 29:

Consider a line L passing through the points P(1, 2, 1) and Q(2, 1, −1). If the mirror image of the point A(2, 2, 2) in the line L is (α, β, γ), then α + β + 6γ is equal to:

1. 9
2. 12
3. 15
4. 18

Question 30:

Let y = y(x) be the solution of the differential equation (x + y + 2)² dx = dy, y(0) = −2. Let the maximum and minimum values of the function y = y(x) in (0, π/3) be α and β, respectively. If (3α + π)² + β² = γ + δ√3, where γ, δ ∈ Z, then γ + δ equals:

1. 7
2. 8
3. 9
4. 10

Question 31:

The translational degrees of freedom (f_t) and rotational degrees of freedom (f_r) of CH₄ molecule are:

1. f_t = 2 and f_r = 2
2. f_t = 3 and f_r = 3
3. f_t = 3 and f_r = 2
4. f_t = 2 and f_r = 3

Question 32:

A cyclist starts from the point P of a circular ground of radius 2 km and travels along its circumference to the point S. The displacement of the cyclist is:

1. 6 km
2. √8 km
3. 4 km
4. 8 km

Question 33:

The magnetic moment of a bar magnet is 0.5 Am². It is suspended in a uniform magnetic field of 8 × 10^-2 T. The work done in rotating it from its most stable to most unstable position is:

1. 16 × 10^-2 J
2. 8 × 10^-2 J
3. 4 × 10^-2 J
4. Zero

Question 34:

Which of the diode circuits shows correct biasing used for the measurement of dynamic resistance of a p-n junction diode:

1. Option 1
2. Option 2
3. Option 3
4. Option 4

Question 35:

Arrange the following in ascending order of wavelength:
(A) Gamma rays (λ₁)
(B) X-rays (λ₂)
(C) Infrared waves (λ₃)
(D) Microwaves (λ₄)

1. λ₄ < λ₃ < λ₁ < λ₂
2. λ₄ < λ₃ < λ₂ < λ₁
3. λ₁ < λ₂ < λ₃ < λ₄
4. λ₂ < λ₁ < λ₄ < λ₃

Question 36:

Identify the logic gate given in the circuit:

1. NAND gate
2. OR gate
3. AND gate
4. NOR gate

Question 37:

The width of one of the two slits in a Young’s double-slit experiment is 4 times that of the other slit. The ratio of the maximum to the minimum intensity in the interference pattern is:

1. 9:1
2. 16:1
3. 1:1
4. 4:1

Question 38:

Correct formula for the height of a satellite from Earth’s surface is:

1. (T²R²g/4π²)^1/2 − R
2. (T²R²g/4π²)^1/3 − R
3. (T²R²/4π²g)^1/3 − R
4. (T²R²/4π²)^1/3 + R

Question 39:

Match List I with List II:

List I:
A. Purely capacitive circuit
B. Purely inductive circuit
C. LCR series circuit at resonance
D. General LCR series circuit

List II:
I. Voltage lags current by 90°
II. Voltage leads current by 90°
III. Voltage and current are in phase
IV. Voltage and current are out of phase

1. A-I, B-IV, C-III, D-II
2. A-IV, B-I, C-III, D-II
3. A-IV, B-I, C-II, D-III
4. A-I, B-II, C-III, D-IV

Question 40:

Given below are two statements:

Statement I: The contact angle between a solid and a liquid depends on the material properties of the solid and liquid.
Statement II: The rise of a liquid in a capillary tube does not depend on the radius of the tube.

Choose the correct answer:

1. Both Statement I and Statement II are false
2. Statement I is false but Statement II is true
3. Statement I is true but Statement II is false
4. Both Statement I and Statement II are true

Question 41:

A body of mass m kg slides from rest along the curve of a vertical circle from point A to B in a frictionless path. The velocity of the body at B is:
Given: R = 14 m, g = 10 m/s², √2 = 1.4

1. 19.8 m/s
2. 21.9 m/s
3. 16.7 m/s
4. 10.6 m/s

Question 42:

An electric bulb rated 50 W – 200 V is connected across a 100 V supply. The power dissipation of the bulb is:

1. 12.5 W
2. 25 W
3. 50 W
4. 100 W

Question 43:

A 2 kg brick begins to slide over a surface inclined at an angle of 45° with respect to the horizontal. The coefficient of static friction between their surfaces is:

1. 1
2. 1/√3
3. 0.5
4. 1.7

Question 44:

In simple harmonic motion, the total mechanical energy of a system is E. If the mass of the oscillating particle is doubled, the new energy of the system for the same amplitude is:

1. E/√2
2. E
3. E√2
4. 2E

Question 45:

Assertion (A): Number of photons increases with increase in frequency of light.
Reason (R): Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation.
Choose the correct answer:

1. Both A and R are correct, and R is NOT the correct explanation of A.
2. A is correct but R is not correct.
3. Both A and R are correct, and R is the correct explanation of A.
4. A is not correct but R is correct.

Question 46:

According to Bohr’s theory, the moment of momentum of an electron revolving in the 4th orbit of a hydrogen atom is:

1. 8h/π
2. h/π
3. 2h/π
4. h/2π

Question 47:

A sample of gas at temperature T is adiabatically expanded to double its volume. The adiabatic constant for the gas is γ = 3/2. The work done by the gas in the process is:

1. RT[√2 − 2]
2. RT[1 − 2√2]
3. RT[2√2 − 1]
4. RT[2 − √2]

Question 48:

A charge q is placed at the center of one surface of a cube. The flux linked with the cube is:

1. q/4ε₀
2. q/2ε₀
3. q/8ε₀
4. Zero

Question 49:

Applying the principle of homogeneity of dimensions, determine which one is correct:

1. T² = 4π²r/GM²
2. T² = 4π²r²r³
3. T² = 4π²r³/GM
4. T² = 4π²r²/GM

Question 50:

A 90 kg body placed at 2R distance from the surface of the Earth experiences a gravitational pull of:
Given: R = Radius of Earth, g = 10 m/s²

1. 300 N
2. 225 N
3. 120 N
4. 100 N

Question 51:

The displacement of a particle executing SHM is given by x = 10 sin(ωt + π/3) m. The time period of motion is 3.14 s. The velocity of the particle at t = 0 is:

1. 5 m/s
2. 10 m/s
3. 15 m/s
4. 20 m/s

Question 52:

A bus moving along a straight highway with a speed of 72 km/h is brought to a halt within 4 seconds after applying the brakes. The distance travelled by the bus during this time is:

1. 20 m
2. 40 m
3. 60 m
4. 80 m

Question 53:

A parallel plate capacitor of capacitance 12.5 pF is charged by a 12.0 V battery. After disconnecting the battery, a dielectric slab (ε_r = 6) is inserted between the plates. The change in potential energy after inserting the dielectric slab is:

1. 500 × 10⁻¹² J
2. 600 × 10⁻¹² J
3. 750 × 10⁻¹² J
4. 800 × 10⁻¹² J

Question 54:

In a system of two particles, m₁ = 3 kg and m₂ = 2 kg are placed at a certain distance. m₁ is moved 2 cm towards the center of mass. To keep the center of mass at the original position, m₂ should move by:

1. 2 cm
2. 3 cm
3. 4 cm
4. 5 cm

Question 55:

The disintegration energy Q for the nuclear fission ²³⁵U → ¹⁴⁰Ce + ⁹⁴Zr + n is (in MeV):
Given: Atomic masses - ²³⁵U = 235.0439 u, ¹⁴⁰Ce = 139.9054 u, ⁹⁴Zr = 93.9063 u, n = 1.0086 u, c² = 931 MeV/u

1. 200 MeV
2. 205 MeV
3. 208 MeV
4. 212 MeV

Question 56:

A light ray is incident on a glass slab of thickness 4√3 cm and refractive index √2. The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of the ray after passing through the glass slab is:

1. 3 cm
2. 4 cm
3. 2√3 cm
4. 2 cm

Question 57:

A rod of length 60 cm rotates with a uniform angular velocity 20 rad/s about its perpendicular bisector, in a uniform magnetic field 0.5 T. The direction of the magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is:

1. 0.24 V
2. 0.36 V
3. 0.18 V
4. 0.12 V

Question 58:

Two wires A and B are made up of the same material and have the same mass. Wire A has a radius of 2.0 mm and wire B has a radius of 4.0 mm. The resistance of wire B is 2 Ω. The resistance of wire A is:

1. 16 Ω
2. 24 Ω
3. 32 Ω
4. 40 Ω

Question 59:

Two parallel long current-carrying wires separated by a distance 2r are shown in the figure. The ratio of magnetic field at A to the magnetic field produced at C is x/7. The value of x is:

1. 3
2. 4
3. 5
4. 6

Question 60:

Mercury is filled in a tube of radius 2 cm up to a height of 30 cm. The force exerted by mercury on the bottom of the tube is:
Given: Atmospheric pressure = 10⁵ N/m², density of mercury = 1.36 × 10⁴ kg/m³, g = 10 m/s², π = 22/7

1. 1530 N
2. 1630 N
3. 1830 N
4. 1930 N

Question 61:

The equilibrium constant for the reaction:
SO₃(g) ⇌ SO₂(g) + (1/2)O₂(g)
is K_c = 4.9 × 10⁻². The value of K_c for the reaction given below is:
2SO₂(g) + O₂(g) ⇌ 2SO₃(g)

1. 4.9
2. 41.6
3. 49
4. 416

Question 62:

Find out the major product formed from the following reaction:

1. Product A
2. Product B
3. Product C
4. Product D

Question 63:

When MnO₂ and H₂SO₄ are added to a salt (A), the greenish-yellow gas liberated indicates that salt (A) is:

1. NaBr
2. CaI₂
3. KNO₃
4. NH₄Cl

Question 64:

The correct statements about hydrogen bonding are:

1. A only
2. A, D, E only
3. A, B, D only
4. A, B, C only

Question 65:

In the above chemical reaction sequence, "A" and "B" respectively are:

1. O₃, Zn/H₂O and NaOH (alc)/I₂
2. H₂O, H⁺ and NaOH (alc)/I₂
3. H₂O, H⁺ and KMnO₄
4. O₃, Zn/H₂O and KMnO₄

Question 66:

Common name of Benzene-1,2-diol is:

1. Quinol
2. Resorcinol
3. Catechol
4. o-Cresol

Question 67:

Consider the above reactions. Identify product B and product C:

1. B = C = 2-Propanol
2. B = 2-Propanol, C = 1-Propanol
3. B = 1-Propanol, C = 2-Propanol
4. B = C = 1-Propanol

Question 68:

The adsorbent used in adsorption chromatography is/are:

1. Silica gel
2. Alumina
3. Quick lime
4. Magnesia

Question 69:

The reaction shown below leads to the formation of a major product "P":

1. Product A
2. Product B
3. Product C
4. Product D

Question 70:

Correct order of stability of carbanions is shown below:

1. c > b > d > a
2. a > b > c > d
3. d > a > c > b
4. d > c > b > a

Question 71:

The correct order of the first ionization enthalpy is:

1. Al > Ga > Tl
2. Ga > Al > B
3. B > Al > Ga
4. Tl > Ga > Al

Question 72:

If an iron (III) complex with the formula [Fe(NH₃)_x(CN)_y]³⁺ has no electron in its e_g orbital, then the value of x + y is:

1. 5
2. 6
3. 3
4. 4

Question 73:

A fuel cell, using hydrogen and oxygen as fuels:

1. A, B, C only
2. A, B, D only
3. A, B, D, E only
4. A, D, E only

Question 74:

Choose the incorrect statement about Dalton's Atomic Theory:

1. Compounds are formed when atoms of different elements combine in any ratio.
2. All the atoms of a given element have identical properties including identical mass.
3. Matter consists of indivisible atoms.
4. Chemical reactions involve reorganization of atoms.

Question 75:

Match List I with List II:

List I:

• A. α-Glucose and α-Galactose
• B. α-Glucose and β-Glucose
• C. α-Glucose and α-Fructose
• D. α-Glucose and α-Ribose

List II:

• I. Functional isomers
• II. Homologous
• III. Anomers
• IV. Epimers

1. A-III, B-IV, C-II, D-I
2. A-III, B-IV, C-I, D-II
3. A-IV, B-III, C-I, D-II
4. A-IV, B-III, C-II, D-I

Question 76:

Given below are two statements:

Statement I: The correct order of first ionization enthalpy values of Li, Na, F, and Cl is Na < Li < Cl < F.
Statement II: The correct order of negative electron gain enthalpy values of Li, Na, F, and Cl is Na < Li < F < Cl.

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is false but Statement II is true.
4. Statement I is true but Statement II is false.

Question 77:

For a strong electrolyte, a plot of molar conductivity against √(concentration) is a straight line with a negative slope. The correct unit for the slope is:

1. S cm² mol^-3/2 L^1/2
2. S cm² mol^-1 L^1/2
3. S cm² mol^-3/2 L
4. S cm² mol^-3/2 L^-1/2

Question 78:

A first-row transition metal in its +2 oxidation state has a spin-only magnetic moment value of 3.86 BM. The atomic number of the metal is:

1. 25
2. 26
3. 22
4. 23

Question 79:

The number of unpaired d-electrons in [Co(H₂O)₆]³⁺ is:

1. 4
2. 2
3. 0
4. 1

Question 80:

The number of species from the following that have pyramidal geometry around the central atom is:

• S₂O₃²⁻
• SO₄²⁻
• SO₃²⁻
• S₂O₇²⁻

1. 4
2. 3
3. 1
4. 2

Question 81:

The maximum number of orbitals which can be identified with n = 4 and m_l = 0 is:

1. 4
2. 3
3. 5
4. 2

Question 82:

The number of compounds/species from the following with non-zero dipole moment is:

• BeCl₂
• BCl₃
• NF₃
• XeF₄
• CCl₄
• H₂O
• H₂S
• HBr
• CO₂
• H₂
• HCl

1. 3
2. 5
3. 7
4. 4

Question 83:

Three moles of an ideal gas are compressed isothermally from 60 L to 20 L using a constant pressure of 5 atm. Heat exchange Q for the compression is:

1. 200 Lit. atm
2. 150 Lit. atm
3. 100 Lit. atm
4. 50 Lit. atm

Question 84:

From 6.55 g of aniline, the maximum amount of acetanilide that can be prepared will be ___ × 10^-1 g.

1. 85
2. 95
3. 75
4. 65

Question 85:

A first-order reaction has a rate constant (k) of 4.6 × 10^-2 s^-1. The time taken for the concentration of reactant to drop from 1 M to 0.1 M is:

1. 50 s
2. 60 s
3. 45 s
4. 55 s

Question 86:

Phthalimide is made to undergo the following sequence of reactions:

(a) KOH

(b) Benzyl chloride

Total number of π-bonds present in product ‘P’ is:

1. 4
2. 5
3. 6
4. 7

Question 87:

The total number of σ and π bonds in 2-oxohex-4-ynoic acid is:

1. 18 σ, 2 π
2. 16 σ, 4 π
3. 20 σ, 3 π
4. 15 σ, 5 π

Question 88:

A first-row transition metal with the highest enthalpy of atomization forms Cr₂O₃ with a spin-only magnetic moment of:

1. 5.92 BM
2. 4.9 BM
3. 3.87 BM
4. 6.1 BM

Question 89:

2.7 kg each of water and acetic acid are mixed. The freezing point of the solution will be -x °C.

1. 8.37 °C
2. 7.28 °C
3. 9.14 °C
4. 6.85 °C

Question 90:

The vanillin compound obtained from vanilla beans has a total sum of oxygen atoms and π electrons of:

1. 16
2. 14
3. 12
4. 18