Question 1:

The domain of
f(x) = log_x+1(x − 2) / e^{2 log_xx − (2x + 3)}, x ∈ R is:

(1) R − {−1, 3}

(2) (2, ∞) − {3}

(3) (−1, ∞) − {3}

(4) R − {3}

Question 2:

Let f : R → R be a function such that
f(x) = (x² + 2x + 1) / (x + 1).

(1) f(x) is many-one in (−∞, −1)

(2) f(x) is many-one in (1, ∞)

(3) f(x) is one-one in [1, ∞) but not in (−∞, ∞)

(4) f(x) is one-one in (−∞, ∞)

Question 3:

For two non-zero complex numbers z₁ and z₂, if
Re(z₁z₂) = 0 and Re(z₁ + z₂) = 0,
then which of the following are possible?

1. (A) Im(z₁) > 0 and Im(z₂) > 0
2. (B) Im(z₁) < 0 and Im(z₂) > 0
3. (C) Im(z₁) > 0 and Im(z₂) < 0
4. (D) Im(z₁) < 0 and Im(z₂) < 0

Choose the correct answer from the options given below:

(1) B and D

(2) B and C

(3) A and B

(4) A and C

Answer: (2) B and C

Question 4:

Let λ ≠ 0 be a real number. Let α, β be the roots of the equation
14x² − 3λx + 3λ = 0,
and α, γ be the roots of the equation
35x² − 53x + 4λ = 0.
Then 3α/β and 4α/γ are the roots of the equation:

(1) 7x² + 245x − 250 = 0

(2) 7x² − 245x + 250 = 0

(3) 49x² − 245x + 250 = 0

(4) 49x² + 245x + 250 = 0

Question 5:

Consider the following system of equations:

• αx + 2y + z = 1
• 2αx + 3y + z = 1
• 3x + αy + 2z = β

For some α, β ∈ R. Then which of the following is NOT correct:

(1) It has no solution if α = −1 and β ≠ 2.

(2) It has no solution for α = −1 and for all β ∈ R.

(3) It has no solution for α = 3 and for all β ≠ 2.

(4) It has a solution for all α ≠ −1 and β = 2.

Answer: (2)

Question 6:

Let α and β be real numbers. Consider a 3 × 3 matrix A such that:
A² = 3A + αI,
A⁴ = 21A + βI.
Then:

(1) α = 1

(2) α = 4

(3) β = 8

(4) β = −8

Answer: (4) β = −8

Question 7:

Let x = 2 be a root of the equation x² + px + q = 0 and
f(x) = {
  1 − cos(x² − 4px + q − 8q² + 16) / (x − 2p)², x ≠ 2p,
  0, x = 2p.
}
Then
lim_x→2p [f(x)],
where [·] denotes the greatest integer function, is:

(1) 2

(2) 1

(3) 0

(4) −1

Answer: (3) 0

Question 8:

Let
f(x) = x + a / (π / 2 − 4 sin x) + b / (π / 2 − 4 cos x), x ∈ R
be a function which satisfies
f(x) = x + ∫₀^π/2 sin(x + y)f(y) dy.
Then (a + b) is equal to:

(1) −π(π + 2)

(2) −2π(π + 2)

(3) −2π(π − 2)

(4) −π(π − 2)

Answer: (2) −2π(π + 2)

Question 9:

Let
A = {(x, y) ∈ R² : y ≥ 0, 2x ≤ y ≤ π/4 − (x − 1)²}
B = {(x, y) ∈ R² : 0 ≤ y ≤ min{2x, π/4 − (x − 1)²}}.
Then the ratio of the area of A to the area of B is:

(1) (π−1)/(π+1)

(2) π/(π−1)

(3) π/(π+1)

(4) (π+1)/(π−1)

Answer: (1) (π−1)/(π+1)

Question 10:

Let Δ be the area of the region
{(x, y) ∈ R² : x² + y² ≤ 21, y² ≤ 4x, x ≥ 1}.
Then
(1/2)Δ − 21 sin⁻¹(2/√7) is equal to:

(1) 2√3 − 1/3

(2) √3 − 2/3

(3) 2√3 − 2/3

(4) √3 − 4/3

Answer: (4) √3 − 4/3

Question 11:

A light ray emits from the origin making an angle of 30 degrees with the positive x-axis. After getting reflected by the line x + y = 1, if this ray intersects the x-axis at Q, then the abscissa of Q is:

(1) (√2)/3 − 1

(2) 2/3 + √3

(3) 2/3 − √3

(4) √3/2 (√3 + 1)

Answer: (2) 2/3 + √3

Question 12:

Let B and C be the two points on the line y + x = 0 such that B and C are symmetric with respect to the origin. Suppose A is a point on y − 2x = 2 such that triangle ABC is an equilateral triangle. Then, the area of triangle ABC is:

(1) √3

(2) 2√3

(3) √8/3

(4) √10/3

Answer: (3) √8/3

Question 13:

Let the tangents at the points A(4, −11) and B(8, −5) on the circle x² + y² − 3x + 10y − 15 = 0 intersect at the point C. Then the radius of the circle, whose center is C and the line joining A and B is its tangent, is equal to:

(1) 3√3 / 4

(2) 2√13

(3) 13

(4) 2√13 / 3

Answer: (4) 2√13 / 3

Question 14:

Let [x] denote the greatest integer. Consider the function f(x) = max{x², 1 + [x]}, where [x] denotes the greatest integer ≤ x. Then the value of the integral ∫₂⁰ f(x) dx is:

(1) 5 + 4√2/3

(2) 8 + 4√2/3

(3) 1 + 5√2/3

(4) 4 + 5√2/3

Answer: (1) 5 + 4√2/3

Question 15:

If the vectors a = λi + μj + 4k, b = −2i + 4j − 2k, and c = 2i + 3j + k are coplanar, and the projection of a on vector b is √54 units, then the sum of all possible values of λ + μ is equal to:

(1) 0

(2) 6

(3) 24

(4) 18

Answer: (3) 24

Question 16:

Fifteen football players of a club are given 15 T-shirts with their names written on the back. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is:

(1) 5/24

(2) 2/15

(3) 1/6

(4) 5/36

Answer: (1) 5/24

Question 17:

Let f(θ) = 3 sin^4(3π/2 − θ) + sin^4(3π + θ) − 2(1 − sin^2(2θ)), and S = {θ ∈ [0, π] : f′(θ) = −√3/2}. If 4β = Σθ∈S θ, then f(β) is equal to:

(1) 11/8

(2) 5/4

(3) 9/8

(4) 3/2

Answer: (2) 5/4

Question 18:

If p, q, and r are three propositions, then which of the following combinations of truth values of p, q, and r makes the logical expression {(p ∨ q) ∧ ((¬p) ∨ r)} → ((¬q) ∨ r) false?

(1) p = T, q = F, r = T

(2) p = T, q = T, r = F

(3) p = F, q = T, r = F

(4) p = T, q = F, r = F

Answer: (3) p = F, q = T, r = F

Question 19:

Three rotten apples are accidentally mixed with seven good apples, and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If μ and σ² represent the mean and variance of X, respectively, then 10(μ² + σ²) is equal to:

(1) 20

(2) 250

(3) 25

(4) 30

Answer: (1) 20

Question 20:

Let y = f(x) be the solution of the differential equation y(x + 1) dx − x² dy = 0, y(1) = e. Then limₓ→0⁺ f(x) is equal to:

(1) 0

(2) 1/e

(3) e²

(4) 2e

Answer: (1) 0

Question 21:

Let the coordinates of one vertex of triangle ABC be A(0, 2, α) and the other two vertices lie on the line x + α/5 = (y - 1)/2 = (z + 4)/3. For α ∈ Z, if the area of triangle ABC is 21 square units and the line segment BC has length 2√21 units, then α² is equal to:

(1) 9

(2) 16

(3) 25

(4) 36

Answer: (3) 25

Question 22:

Let the equation of the plane P containing the line x + 10 = (8 - y)/2 = z be ax + by + 3z = 2(a + b), and the distance of the plane P from the point (1, 27, 7) be c. Then a² + b² + c² is equal to:

(1) 355

(2) 200

(3) 150

(4) 400

Answer: (1) 355

Question 23:

Suppose f is a function satisfying f(x + y) = f(x) + f(y) for all x, y ∈ N and f(1) = 1/5. If the sum from n = 1 to m of [f(n) / (n(n + 1)(n + 2))] equals 1/12, then m is equal to:

(1) 5

(2) 10

(3) 15

(4) 20

Answer: (2) 10

Question 24:

Let a1, a2, a3, ... be a geometric progression (GP) of increasing positive numbers. If the product of the fourth and sixth terms is 9 and the sum of the fifth and seventh terms is 24, then a1a9 + a2a4a9 + a5 + a7 is equal to:

Answer: 60

Question 25:

Let a, b, and c be three non-zero, non-coplanar vectors. Let the position vectors of four points A, B, C, and D be a - b + c, λa - 3b + 4c, -a + 2b - 3c, and 2a + 4b + 6c respectively. If vectors AB, AC, and AD are coplanar, then λ is:

Answer: 2

Question 26:

If all the six-digit numbers x1x2x3x4x5x6 with 0 < x1 < x2 < x3 < x4 < x5 < x6 are arranged in increasing order, then the sum of the digits in the 72nd number is:

Answer: 32

Question 27:

Let f : R → R be a differentiable function that satisfies the relation f(x + y) = f(x) + f(y) − 1 for all x, y ∈ R. If f′(0) = 2, then |f(−2)| is equal to:

(1) 1

(2) 2

(3) 3

(4) 4

Answer: (3) 3

Question 28:

If the coefficient of x⁹ in (a x³ + 1/(β x¹¹)) and the coefficient of x⁻⁹ in (a x − 1/(β x³))¹¹ are equal, then (αβ)² is equal to:

(1) 1

(2) 4

(3) 9

(4) 16

Answer: (1) 1

Question 29:

Suppose the coefficients of three consecutive terms in the binomial expansion of (1 + 2x)ⁿ are in the ratio 2 : 5 : 8. Then the coefficient of the term which is in the middle of these three terms is:

(1) 560

(2) 1120

(3) 1680

(4) 2240

Answer: (2) 1120

Question 30:

Five-digit numbers are formed using the digits {1, 2, 3, 5, 7} with repetitions allowed, and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is:

(1) 1430

(2) 1436

(3) 1500

(4) 1600

Answer: (2) 1436

Question 31:

Match List-I with List-II:

Choose the correct answer from the options given below:

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

Question 53:

A body cools from 60°C to 40°C in 6 minutes. If the temperature of the surroundings is 10°C, then after the next 6 minutes, its temperature will be —— °C.

Question 54:

A solid sphere of mass 2 kg is making pure rolling on a horizontal surface with kinetic energy 2240 J. The velocity of the center of mass of the sphere will be —– ms⁻¹.

Question 55:

A 0.4 kg mass takes 8 seconds to reach the ground when dropped from a certain height P above the surface of Earth. The loss of potential energy in the last second of fall is —— J. (Take g = 10 m/s²)

Question 56:

A tennis ball is dropped onto the floor from a height of 9.8 m. It rebounds to a height of 5.0 m. The ball comes in contact with the floor for 0.2 s. The average acceleration during contact is —— m/s². (Given g = 10 m/s²)

Question 57:

A point charge q₁ = 4q₀ is placed at the origin. Another point charge q₂ = −q₀ is placed at x = 12 cm. The proton is placed on the x-axis so that the electrostatic force on the proton is zero. In this situation, the position of the proton from the origin is — cm.

Question 58:

In a metre bridge experiment, the balance point is obtained if the gaps are closed by 2 Ω and 3 Ω. A shunt of X Ω is added to the 3 Ω resistor to shift the balancing point by 22.5 cm. The value of X is — Ω.

Question 59:

A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field B = 0.8 T. When released, the radius of the loop starts shrinking at a constant rate of 2 cm/s. The induced emf in the loop at an instant when the radius of the loop is 10 cm will be — mV.

Question 60:

Three identical polaroids P₁, P₂, and P₃ are placed one after another. The pass axis of P₂ and P₃ are inclined at angles of 60° and 90° with respect to the axis of P₁. The source S has an intensity of 256 W/m². The intensity of light at point O is —— W/m².

Question 61:

For 1 mol of gas, the plot of pV vs p is shown below. p is the pressure and V is the volume of the gas. What is the value of compressibility factor at point A?

(1) 1 − ^a/_RT V

(2) 1 + ^b/_V

(3) 1 − ^b/_V

(4) 1 + ^a/_RT V

Question 62:

The shortest wavelength of hydrogen atom in the Lyman series is λ. The longest wavelength in the Balmer series of He⁺ is:

(1) ⁵/₉ λ

(2) ⁹/₅ λ

(3) ³⁶/₅ λ

(4) ⁵/₉ λ

Question 63:

Which of the following salt solutions would coagulate the colloid solution formed when FeCl₃ is added to NaOH solution, at the fastest rate?

(1) 10 mL of 0.2 mol dm⁻³ AlCl₃

(2) 10 mL of 0.1 mol dm⁻³ Na₂SO₄

(3) 10 mL of 0.1 mol dm⁻³ Ca₃(PO₄)₂

(4) 10 mL of 0.15 mol dm⁻³ CaCl₂

Question 64:

The bond dissociation energy is highest for:

(1) Cl₂

(2) I₂

(3) Br₂

(4) F₂

Question 65:

The reaction representing the Mond process for metal refining is:

(1) Ni + 4CO Δ → Ni(CO)₄

(2) 2K[Au(CN)₂] + Zn Δ → K₂[Zn(CN)₄] + 2Au

(3) Zr + 2I₂ Δ → ZrI₄

(4) ZnO + C Δ → Zn + CO

Question 66:

Which of the given compounds can enhance the efficiency of a hydrogen storage tank?

(1) Li/P₄

(2) SiH₄

(3) NaNi₅

(4) Di-isobutylaluminium hydride

Question 67:

The correct order of hydration enthalpies is:

(A) K+
(B) Rb+
(C) Mg2+
(D) Cs+
(E) Ca2+

Choose the correct answer from the options given below:

1. (1) C > A > E > B > D
2. (2) E > C > A > B > D
3. (3) C > E > A > D > B
4. (4) C > E > A > B > D

Question 68:

The magnetic behavior of Li2O, Na2O2, and KO2, respectively, is:

(1) diamagnetic, paramagnetic, and diamagnetic

(2) paramagnetic, paramagnetic, and diamagnetic

(3) paramagnetic, diamagnetic, and paramagnetic

(4) diamagnetic, diamagnetic, and paramagnetic

Question 69:

"A" obtained by Ostwald's method involving air oxidation of NH3, upon further air oxidation produces "B". "B" on hydration forms an oxoacid of nitrogen along with evolution of "A". The oxoacid also produces "A" and gives a positive brown ring test.

(1) NO2, N2O5

(2) NO2, N2O4

(3) NO, NO2

(4) N2O3, NO2

Question 70:

The standard electrode potential (M3+/M2+) for V, Cr, Mn, and Co are -0.26 V, -0.41 V, +1.57 V, and +1.97 V, respectively. The metal ions which can liberate H2 from a dilute acid are:

(1) V2+ and Mn2+

(2) Cr2+ and Co2+

(3) V2+ and Cr2+

(4) Mn2+ and Co2+

Question 71:

Correct statement about smog is:

(1) NO₂ is present in classical smog

(2) Both NO₂ and SO₂ are present in classical smog

(3) Photochemical smog has a high concentration of oxidizing agents

(4) Classical smog also has a high concentration of oxidizing agents

Question 72:

Chiral complex from the following is:

(1) cis–[PtCl₂(en)₂]²⁺

(2) trans–[PtCl₂(en)₂]²⁺

(3) cis–[PtCl₂(NH₃)₂]

(4) trans–[Co(NH₃)₄Cl₂]⁺

Question 73:

Identify the correct order for the given property for the following compounds:

(The exact compounds and property details are not fully shown in the question excerpt, but the final conclusion is provided.)

Choose the correct answer from the options given below:

(1) (B), (C), and (D) only

(2) (A), (C), and (E) only

(3) (A), (C), and (D) only

(4) (A), (B), and (E) only

Question 74:

The increasing order of pKa for the following phenols is:

(1) 2,4-Dinitrophenol

(2) 4-Nitrophenol

(3) 2,4,5-Trimethylphenol

(4) Phenol

(5) 3-Chlorophenol

Answer: (2) 2,4-Dinitrophenol < 4-Nitrophenol < 3-Chlorophenol < Phenol < 2,4,5-Trimethylphenol

Question 75:

Match the reactions in List-I with the reagents in List-II:

(1) (A) – III, (B) – IV, (C) – II, (D) – I

(2) (A) – II, (B) – IV, (C) – I, (D) – III

(3) (A) – III, (B) – IV, (C) – I, (D) – II

(4) (A) – II, (B) – I, (C) – III, (D) – IV

Question 76:

The major product 'P' for the following sequence of reactions is:

(1) Some other derivative

(2) Yet another derivative

(3) Ph-CH2-CH2-CH2-CH2-NH2

(4) A substituted ring compound

Question 77:

During the borax bead test with CuSO4, a blue-green colour of the bead was observed in the oxidizing flame due to the formation of:

1. (1) Cu3B2
2. (2) Cu
3. (3) Cu(BO2)2
4. (4) CuO

Question 78:

Match List I with List II:

1. (1) (A) – III, (B) – I, (C) – II, (D) – IV
2. (2) (A) – I, (B) – II, (C) – IV, (D) – III
3. (3) (A) – II, (B) – I, (C) – IV, (D) – III
4. (4) (A) – III, (B) – I, (C) – IV, (D) – II

Answer: (1) (A) – III, (B) – I, (C) – II, (D) – IV

Question 79:

Number of cyclic tripeptides formed with 2 amino acids A and B is:

1. (1) 2
2. (2) 3
3. (3) 5
4. (4) 4

Question 80:

Compound that will give positive Lassaigne's test for both nitrogen and halogen is:

1. (1) N2H4·HCl
2. (2) CH3NH2·HCl
3. (3) NH4Cl
4. (4) NH2OH·HCl

Question 81:

Millimoles of calcium hydroxide required to produce 100 mL of the aqueous solution of pH 12 is x × 10⁻¹. The value of x is —— (Nearest integer). Assume complete dissociation.

Question 82:

The number of molecules or ions from the following, which do not have an odd number of electrons, are ——:

• (A) NO₂
• (B) ICl₄⁻
• (C) BrF₃
• (D) ClO₂
• (E) NO₂⁺
• (F) NO

Question 83:

Consider the following reaction approaching equilibrium at 27°C and 1 atm pressure:

A + B ⇌ C + D

K_f = 10³, K_r = 10²

The standard Gibbs energy change (ΔrG°) at 27°C is (–) —— kJ mol⁻¹ (Nearest integer).

(Given: R = 8.3 J K⁻¹ mol⁻¹ and ln(10) = 2.3)

Answer: 6

Question 84:

Solid lead nitrate is dissolved in 1 litre of water. The solution was found to boil at 100.15°C. When 0.2 mol of NaCl is added to the resulting solution, it was observed that the solution froze at –0.8°C. The solubility product of PbCl₂ formed is — × 10⁻⁶ at 298 K (Nearest integer).

Given: K_b = 0.5 K kg mol⁻¹, K_f = 1.8 kg mol⁻¹

Answer: 13

Question 85:

Water decomposes at 2300 K:

2 H2O(g) ⇌ 2 H2(g) + O2(g)

The percent of water decomposing at 2300 K and 1 bar is —— (Nearest integer). Equilibrium constant for the reaction is 2 × 10^-3 at 2300 K.

Answer: 2

Question 86:

The following figure shows the dependence of molar conductance of two electrolytes on concentration. Lambda°m is the limiting molar conductivity. The number of incorrect statements from the following is ——.

(A) Lambda°m for electrolyte A is obtained by extrapolation.
(B) For electrolyte B, sqrt(c) vs. Lambda(m) graph is a straight line with intercept equal to Lambda°m.
(C) At infinite dilution, the value of degree of dissociation approaches zero for electrolyte B.
(D) Lambda°m for any electrolyte A or B can be calculated using lambda° for individual ions.

Answer: 2

Question 87:

For a certain chemical reaction X → Y, the rate of formation of the product is plotted against time as shown in the figure. The number of correct statements from the following is ——.

(A) Overall order of this reaction is one.
(B) Order of this reaction cannot be determined.
(C) In region-I and III, the reaction is of first and zero order respectively.
(D) In region-II, the reaction is of first order.
(E) In region-II, the order of the reaction is in the range of 0.1 to 0.9.

Answer: 2

Question 88:

The sum of bridging carbonyls in W(CO)6 and Mn2(CO)10 is ——.

Answer: 0

Question 89:

Following chromatogram was developed by adsorption of compound A on a 6 cm TLC glass plate. The retardation factor of the compound A is —— × 10⁻¹.

Answer: 6

Question 90:

17 mg of a hydrocarbon (Molecular Formula: C10H16) takes up 8.40 mL of H2 gas measured at 0°C and 760 mmHg. Ozonolysis of the hydrocarbon yields certain products. The number of double bonds present in the hydrocarbon is ——.

Answer: 3