Question 1:

Considering only the principal values of inverse trigonometric functions, the number of positive real values of x satisfying arctan(x) + arctan(2x) = π/4 is:

1. More than 2
2. 1
3. 2
4. 0

Question 2:

Consider the function f: (0,2) → ℝ defined by f(x) = x² + 2/x, and the function g(x) by min{f(t)} for 0 < t ≤ x, 0 < x ≤ 1 and g(x) = (3/2 + x) for 1 < x < 2. Which statement about g is correct?

1. g is continuous but not differentiable at x = 1
2. g is not continuous for all x in (0,2)
3. g is neither continuous nor differentiable at x = 1
4. g is continuous and differentiable for all x in (0,2)

Question 3:

Let R be the region between lines 3x − y + 1 = 0 and x + 2y − 5 = 0 containing the origin. Find values of a for which (a², a + 1) lie in R:

1. (−3, −1) ∪ (−1/3, 1)
2. (−3, 0) ∪ (−1/3, 1)
3. (−3, 0) ∪ (2/3, 1)
4. (−3, −1) ∪ (−1/3, 1)

Question 4:

The 20th term from the end of the progression 20, 19.25, 18.5, 17.75, ..., −129.25 is:

1. −118
2. −110
3. −115
4. −100

Question 5:

Let f: ℝ − {−1/2} → ℝ and g: ℝ − {−5/2} → ℝ be defined as f(x) = (2x + 3)/(2x + 1) and g(x) = |x| + 1/(2x + 5). What is the domain of f ∘ g?

1. ℝ − {−5/2}
2. ℝ
3. ℝ − {−7/4}
4. ℝ − {−5/2, −7/4}

Question 6:

For 0 < a < 1, find the value of the integral ∫₀^π dx / (1 − 2a cos(x) + a²):

1. π / (1 − a²)
2. π / (1 + a²)
3. π² / (π + a²)
4. π² / (π − a²)

Question 7:

Let g(x) = 3f(x/3) + f(3 − x) where fʺ(x) > 0 for all x in (0,3). If g is decreasing in (0, α) and increasing in (α,3), what is 8α?

1. 24
2. 0
3. 18
4. 20

Question 8:

If lim(x→0) [(3 + α sin(x) + β cos(x) + log(1 − x)) / (3 tan²(x))] = 1/3, then 2α − β is equal to:

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

Question 9:

If α and β are the roots of x² − x − 1 = 0, and Sₙ = 2023αⁿ + 2024βⁿ, then:

1. 2S₁₂ = S₁₁ + S₁₀
2. S₁₂ = S₁₁ + S₁₀
3. 2S₁₁ = S₁₂ + S₁₀
4. S₁₁ = S₁₀ + S₁₂

Question 10:

Let A and B be two finite sets with m and n elements, respectively. If the total subsets of set A are 56 more than B’s subsets, then the distance of the point P(m, n) from Q(−2, −3) is:

1. 10
2. 6
3. 4
4. 8

Question 11:

The values of α for which the determinant |1 3 2; α+3 2 1; 1 1 3| = 0 lie in the interval:

1. (−2, 1)
2. (−3, 0)
3. (−3/2, 3/2)
4. (0, 3)

Question 12:

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and the second draw gives all black balls is:

1. 5/256
2. 5/715
3. 3/715
4. 3/256

Question 13:

The integral ∫(x⁸ − x²) dx / ((x¹² + 3x⁶ + 1) arctan(x³ + 1/x³)) is equal to:

1. log(arctan(x³ + 1/x³)^(1/3) + C)
2. log(arctan(x³ + 1/x³)^(1/2) + C)
3. log(arctan(x³ + 1/x³) + C)
4. log(arctan(x³ + 1/x³)³ + C)

Question 14:

If 2tan²(θ) − 5sec(θ) = 1 has exactly 7 solutions in the interval [0, nπ/2] for the least value of n ∈ ℕ, then Σ(k=1 to n) [k / 2k] is equal to:

1. 1/215(214 − 14)
2. 1/214(215 − 15)
3. 1 − 15/213
4. 1/213(214 − 15)

Question 15:

The position vectors of the vertices A, B, and C of a triangle are A = 2i − 3j + 3k, B = 2i + 2j + 3k, and C = −i + j + 3k respectively. If ℓ is the length of the angle bisector AD of ∠BAC, then 2ℓ² equals:

1. 49
2. 42
3. 50
4. 45

Question 16:

If y = y(x) is the solution curve of the differential equation (x² − 4) dy − (y² − 3y) dx = 0, x > 2, y(4) = 3/2, and the slope of the curve is never zero, then y(10) equals:

1. 3/(1 + 8^(1/4))
2. 3/(1 + 2√2)
3. 3/(1 − 2√2)
4. 3/(1 − 8^(1/4))

Question 17:

If e₁ is the eccentricity of the hyperbola x²/16 − y²/9 = 1 and e₂ is the eccentricity of the ellipse x²/a² + y²/b² = 1, which passes through the foci of the hyperbola and satisfies e₁e₂ = 1, then the length of the chord of the ellipse parallel to the x-axis and passing through (0,2) is:

1. 4√5
2. 8√5/3
3. 10√5/3
4. 3√5

Question 18:

Let α = (4!)! / (4!)³! and β = (5!)! / (5!)⁴!. Then:

1. α ∈ ℕ and β ∉ ℕ
2. α ∉ ℕ and β ∈ ℕ
3. α ∈ ℕ and β ∈ ℕ
4. α ∉ ℕ and β ∉ ℕ

Question 19:

Let the position vectors of vertices A, B, and C of a triangle be 2i + 2j + k, i + 2j + 2k, and 2i + j + 2k respectively. Let ℓ₁, ℓ₂, and ℓ₃ be lengths of perpendiculars from the orthocenter to sides AB, BC, and CA. Then ℓ₁² + ℓ₂² + ℓ₃² equals:

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

Question 20:

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking, it was found that an observation was read as 10 instead of 12. If μ and σ² denote the mean and variance of the correct observations, then 15(μ + μ² + σ²) is equal to:

Question 21:

The values of α for which lines 2x − y + 3 = 0, 6x + 3y + 1 = 0, and ax + 2y − 2 = 0 do not form a triangle are:

1. (−2, 1)
2. (−3, 0)
3. (−3/2, 3/2)
4. (0, 3)

Question 22:

If the area of the region {(x, y) : 0 ≤ y ≤ min(2x, 6x − x²)} is A, then 12A is equal to:

1. 304
2. 594
3. 128
4. 360

Question 23:

Let Λ be a 2×2 real matrix and I be the identity matrix of order 2. If the roots of the equation |Λ − xI| = 0 are −1 and 3, then the sum of the diagonal elements of the matrix Λ² is:

1. 10
2. 6
3. 14
4. 7

Question 24:

If the sum of squares of all real values of α, for which the lines 2x − y + 3 = 0, 6x + 3y + 1 = 0, and ax + 2y − 2 = 0 do not form a triangle is p, then the greatest integer less than or equal to p is:

1. 32
2. 64
3. 29
4. 41

Question 25:

The coefficient of x^2012 in the expansion of (1 − x)^2008(1 + x + x²)^2007 is:

1. 0
2. 1
3. −1
4. 2

Question 26:

If the solution curve of the differential equation dy/dx = (x + y − 2)/(x − y) passes through the point (2, 1), the value of y(10) equals:

1. 3/(1 + (8)^(1/4))
2. 3/(1 + (8)^(1/3))
3. 1/(3 + (8)^(1/4))
4. 2/(3 + (8)^(1/4))

Question 27:

Let f(x) = ∫₀ˣ g(t) log((1 − t)/(1 + t)) dt, where g is a continuous odd function. If ∫ from −π/2 to π/2 of [f(x) x² cos(x)/(1 + e^x)] dx = π/2 − α, then α is equal to:

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

Question 28:

Consider a circle (x − α)² + (y − β)² = 50, where α, β > 0. If the circle touches the line y + x = 0 at point P, whose distance from the origin is 4√2, then (α + β)² is equal to:

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

Question 29:

The lines (x − 2)/1 = (y − 1)/(−1) = (z − 7)/8 and (x + 3)/4 = (y + 2)/3 = (z + 2)/1 intersect at the point P. If the distance of P from the line (x + 1)/2 = (y − 1)/3 = (z − 1)/1 is ℓ, then 14ℓ² is equal to:

1. 108
2. 120
3. 100
4. 150

Question 30:

Let the complex numbers α and 1/α lie on the circles |z − z₀| = 2 and |z − z₀| = 4 respectively, where z₀ = 1 + i. Then, the value of 100|α|² is:

1. 20
2. 10
3. 30
4. 15

Question 31:

The equation of state of a real gas is given by P + (a/V²)(V − b) = RT, where P, V, and T are pressure, volume, and temperature. The dimensions of a/b² are similar to:

1. P
2. PV
3. RT
4. R

Question 32:

The total kinetic energy of 1 mole of oxygen at 27°C is:

1. 6845.5 J
2. 5942.0 J
3. 6232.5 J
4. 5670.5 J

Question 33:

The primary side of a transformer is connected to a 230 V, 50 Hz supply. Turns ratio of primary to secondary winding is 10:1. Load resistance on the secondary side is 46 Ω. The power consumed in it is:

1. 12.5 W
2. 10.0 W
3. 11.5 W
4. 12.0 W

Question 34:

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of Cp/Cv for the gas is:

1. 5/3
2. 3/2
3. 7/5
4. 9/7

Question 35:

The threshold frequency of a metal with work function 6.63 eV is:

1. 16 × 10¹⁵ Hz
2. 16 × 10¹² Hz
3. 1.6 × 10¹² Hz
4. 1.6 × 10¹⁵ Hz

Question 36:

A current of 200 µA deflects the coil of a moving coil galvanometer through 60°. The current to cause deflection through π/10 radians is:

1. 30 µA
2. 120 µA
3. 60 µA
4. 180 µA

Question 37:

The atomic mass of 6C¹² is 12.000000 u and that of 6C¹³ is 13.003354 u. The required energy to remove a neutron from 6C¹³, if the mass of the neutron is 1.008665 u, will be:

1. 62.5 MeV
2. 6.25 MeV
3. 4.95 MeV
4. 49.5 MeV

Question 38:

A ball suspended by a thread swings in a vertical plane so that its acceleration in the extreme and lowest position are equal. The angle (θ) of deflection in the extreme position is:

1. tan⁻¹(√2)
2. 2tan⁻¹(1/2)
3. tan⁻¹(1/2)
4. 2tan⁻¹(1/√5)

Question 39:

Three voltmeters are joined as shown. When a potential difference is applied across A and B, their readings are V₁, V₂, and V₃. Choose the correct option:

1. V₁ = V₂
2. V₁ = V₃ − V₂
3. V₁ + V₂ > V₃
4. V₁ + V₂ = V₃

Question 40:

The total kinetic energy of 1 mole of oxygen at 27°C is:

1. 6845.5 J
2. 5942.0 J
3. 6232.5 J
4. 5670.5 J

Question 41:

Given that the angular speed of the moon in its orbit about the earth is greater than that of the earth around the sun, identify the reason.

1. Shorter period of orbit
2. Larger radius
3. Higher mass of the moon
4. Higher gravitational pull

Question 42:

The primary side of a transformer is connected to 230 V, 50 Hz supply with a turns ratio of 10:1. Load resistance is 46 Ω. The power consumed is:

1. 12.5 W
2. 10 W
3. 11.5 W
4. 12 W

Question 43:

During an adiabatic process, the pressure of a gas is proportional to the cube of its absolute temperature. The ratio of Cp/Cv is:

1. 5/3
2. 3/2
3. 7/5
4. 9/7

Question 44:

The threshold frequency of a metal with work function 6.63 eV is:

1. 16 × 10¹⁵ Hz
2. 16 × 10¹² Hz
3. 1.6 × 10¹² Hz
4. 1.6 × 10¹⁵ Hz

Question 45:

A current of 200 µA deflects the coil of a moving coil galvanometer through 60°. The current to cause deflection through π/10 radians is:

1. 30 µA
2. 120 µA
3. 60 µA
4. 180 µA

Question 46:

The atomic mass of 6C¹² is 12.000000 u and that of 6C¹³ is 13.003354 u. The required energy to remove a neutron from 6C¹³, if the mass of the neutron is 1.008665 u, will be:

1. 62.5 MeV
2. 6.25 MeV
3. 4.95 MeV
4. 49.5 MeV

Question 47:

A ball suspended by a thread swings in a vertical plane so that its acceleration in the extreme and lowest position are equal. The angle (θ) of deflection in the extreme position is:

1. tan⁻¹(√2)
2. 2tan⁻¹(1/2)
3. tan⁻¹(1/2)
4. 2tan⁻¹(1/√5)

Question 48:

Three voltmeters are joined as shown. When a potential difference is applied across A and B, their readings are V₁, V₂, and V₃. Choose the correct option.

1. V₁ = V₂
2. V₁ = V₃ − V₂
3. V₁ + V₂ > V₃
4. V₁ + V₂ = V₃

Question 49:

The total kinetic energy of 1 mole of oxygen at 27°C is:

1. 6845.5 J
2. 5942.0 J
3. 6232.5 J
4. 5670.5 J

Question 50:

The primary side of a transformer is connected to 230 V, 50 Hz supply with a turns ratio of 10:1. Load resistance is 46 Ω. The power consumed is:

1. 12.5 W
2. 10 W
3. 11.5 W
4. 12 W

Question 51:

The magnetic field at the centre of a wire loop formed by two semicircular wires of radii R1 = 2 m and R2 = 4 m carrying current I = 4 A as per figure given below is α×10⁻⁷ T. The value of α is:

Question 52:

Two charges of −4µC and +4µC are placed at the points A(1, 0, 4)m and B(2, −1, 5)m in an electric field E = 0.20 i V/cm. The magnitude of the torque acting on the dipole is 8√α×10⁻⁵ Nm, where α = :

Question 53:

A closed organ pipe 150 cm long gives 7 beats per second with an open organ pipe of length 350 cm, both vibrating in fundamental mode. The velocity of sound is:

Question 54:

A body falling under gravity covers two points A and B separated by 80 m in 2 s. The distance of upper point A from the starting point is:

Question 55:

The reading of a pressure meter attached with a closed pipe is 4.5×10⁴ N/m². On opening the valve, water starts flowing and the reading of pressure meter falls to 2.0×10⁴ N/m². The velocity of water is found to be √V m/s. The value of V is:

Question 56:

A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is:

Question 57:

A parallel beam of monochromatic light of wavelength 5000 Å is incident normally on a single narrow slit of width 0.001 mm. The light is focused by a convex lens on a screen placed on its focal plane. The first minima will be formed for the angle of diffraction of:

Question 58:

The electric potential at the surface of an atomic nucleus (Z = 50) of radius 9×10⁻¹³ cm is ×10⁶ V:

Question 59:

If Rydberg’s constant is R, the longest wavelength of radiation in Paschen series will be α×7R, where α = :

Question 60:

A series LCR circuit with L = 100π mH, C = 10⁻³ F, and R = 10 Ω is connected across an ac source of 220 V, 50 Hz supply. The power factor of the circuit would be:

Question 61:

The order of relative stability of the contributing structures is:

1. I > II > III
2. II > I > III
3. I = II = III
4. III > II > I

Question 62:

Which among the following halide(s) will not show SN1 reaction:
(A) H₂C=CH−CH₂Cl
(B) CH₃−CH=CH−Cl
Choose the most appropriate answer from the options given below:

1. (A), (B), and (D) only
2. (A) and (B) only
3. (B) and (C) only
4. (B) only

Question 63:

Which of the following statements is not correct about rusting of iron?

1. Coating of iron surface by tin prevents rusting, even if the tin coating is peeled off.
2. When pH lies above 9 or 10, rusting of iron does not take place.
3. Dissolved acidic oxides SO₂, NO₂ in water act as catalyst in the process of rusting.
4. Rusting of iron is envisaged as setting up of electrochemical cell on the surface of iron object.

Question 64:

Given below are two statements: Statement (I): In the Lanthanides, the formation of Ce⁴⁺ is favored by its noble gas configuration. Statement (II): Ce⁴⁺ is a strong oxidant reverting to the common +3 state. Choose the correct option:

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

Question 65:

Choose the correct option having all the elements with d¹⁰ electronic configuration from the following:

1. Zn, Co²⁺, Ni, Fe²⁺, Cr
2. Cu, Zn, Ag, Cd
3. Pd, Ni, Fe²⁺, Cr
4. Ni, Zn, Fe²⁺, Cu

Question 66:

Phenolic group can be identified by a positive:

1. Phthalein dye test
2. Lucas test
3. Tollen’s test
4. Carbylamine test

Question 67:

The molecular formula of second homologue in the homologous series of mono carboxylic acids is:

1. C₃H₆O₂
2. C₂H₄O₂
3. CH₂O
4. C₂H₂O₂

Question 68:

The technique used for purification of steam volatile water immiscible substance is:

1. Fractional distillation
2. Fractional distillation under reduced pressure
3. Distillation
4. Steam distillation

Question 69:

The final product A, formed in the following reaction sequence is a mixture of:
Correct Answer: (4)

Question 70:

Match List-I with List-II:
Correct Matching: (1) (A)-(IV), (B)-(I), (C)-(III), (D)-(II)

Question 71:

Major product formed in the following reaction is a mixture of:
Correct Answer: (4)

Question 72:

The bond line formula of HOCH(CN)₂ is:
Correct Answer: (4)

Question 73:

Given below are two statements:
(I) Oxygen, being the first member of group 16, exhibits only −2 oxidation state.
(II) Down group 16, stability of +4 oxidation state decreases and +6 oxidation state increases.
Choose the most appropriate answer:

1. Statement I is correct but Statement II is incorrect
2. Both Statement I and II are correct
3. Both Statement I and II are incorrect
4. Statement I is incorrect but Statement II is correct

Question 74:

Identify from the following species in which d²sp³ hybridization is shown by the central atom:

1. [Co(NH₃)₆]³⁺
2. BrF₅
3. [PtCl₄]²⁻
4. SF₆

Question 75:

Identify the product formed in the following reaction: Cl-(CH₂)₄-Cl + excess NH₃ → A, then NaOH → B + H₂O + NaCl
Correct Answer: (2)

Question 76:

The quantity which changes with temperature is:

1. Molarity
2. Mass percentage
3. Molality
4. Mole fraction

Question 77:

Which structure of protein remains intact after coagulation of egg white on boiling?

1. Primary
2. Tertiary
3. Secondary
4. Quaternary

Question 78:

Which of the following cannot function as an oxidising agent?

1. N³⁻
2. SO₄²⁻
3. BrO₃⁻
4. MnO₄⁻

Question 79:

The incorrect statement regarding conformations of ethane is:

1. Ethane has an infinite number of conformations
2. The dihedral angle in staggered conformation is 60°
3. Eclipsed conformation is the most stable conformation
4. The conformations of ethane are inter-convertible to one another

Question 80:

Identify the incorrect pair from the following:

1. Photography - AgBr
2. Polythene preparation - TiCl₄, Al(CH₃)₃
3. Haber process - Iron
4. Wacker process - PtCl₂

Question 81:

Total number of ions from the following with noble gas configuration is:

Question 82:

The number of non-polar molecules from the following is:

Question 83:

Time required for completion of 99.9% of a first-order reaction is times of half life (t₁/₂) of the reaction:

Question 84:

The spin-only magnetic moment value of square planar complex [Pt(NH₃)₂Cl(NH₂CH₃)]Cl is B.M. (Nearest integer):

Question 85:

For a certain thermochemical reaction M → N at T = 400 K, ΔH° = 77.2 kJ/mol, ΔS° = 122 J/K, log equilibrium constant (log K) is x×10⁻¹.

Question 86:

Volume of 3 M NaOH (formula weight 40 g/mol) which can be prepared from 84 g of NaOH is x×10⁻¹ dm³.

Question 87:

1 mole of PbS is oxidized by “X” moles of O₃ to get “Y” moles of O₂. X + Y = ?

Question 88:

The hydrogen electrode is dipped in a solution of pH = 3 at 25°C. The potential of the electrode will be − x×10⁻² V.

Question 89:

9.3 g of aniline is reacted with excess acetic anhydride to prepare acetanilide. The mass of acetanilide produced, if the reaction is 100% complete, is x×10⁻¹ g.

Question 90:

Total number of compounds with chiral carbon atoms from the following is: