Question 1:

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

1. (1) R − {−1, 3}
2. (2) (2, ∞) − {3}
3. (3) (−1, ∞) − {3}
4. (4) R − {3}

Question 2:

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

1. (1) f(x) is many-one in (−∞, −1)
2. (2) f(x) is many-one in (1, ∞)
3. (3) f(x) is one-one in [1, ∞) but not in (−∞, ∞)
4. (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. (1) B and D
2. (2) B and C
3. (3) A and B
4. (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. (1) 7x² + 245x − 250 = 0
2. (2) 7x² − 245x + 250 = 0
3. (3) 49x² − 245x + 250 = 0
4. (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. (1) It has no solution if α = −1 and β ≠ 2.
2. (2) It has no solution for α = −1 and for all β ∈ R.
3. (3) It has no solution for α = 3 and for all β ≠ 2.
4. (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. (1) α = 1
2. (2) α = 4
3. (3) β = 8
4. (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. (1) 2
2. (2) 1
3. (3) 0
4. (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. (1) −π(π + 2)
2. (2) −2π(π + 2)
3. (3) −2π(π − 2)
4. (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. (1) (π−1)/(π+1)
2. (2) π/(π−1)
3. (3) π/(π+1)
4. (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. (1) 2√3 − 1/3
2. (2) √3 − 2/3
3. (3) 2√3 − 2/3
4. (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. (1) (√2)/3 − 1
2. (2) 2/3 + √3
3. (3) 2/3 − √3
4. (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. (1) √3
2. (2) 2√3
3. (3) √8/3
4. (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. (1) 3√3 / 4
2. (2) 2√13
3. (3) 13
4. (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. (1) 5 + 4√2/3
2. (2) 8 + 4√2/3
3. (3) 1 + 5√2/3
4. (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. (1) 0
2. (2) 6
3. (3) 24
4. (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. (1) 5/24
2. (2) 2/15
3. (3) 1/6
4. (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. (1) 11/8
2. (2) 5/4
3. (3) 9/8
4. (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. (1) p = T, q = F, r = T
2. (2) p = T, q = T, r = F
3. (3) p = F, q = T, r = F
4. (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. (1) 20
2. (2) 250
3. (3) 25
4. (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. (1) 0
2. (2) 1/e
3. (3) e²
4. (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. (1) 9
2. (2) 16
3. (3) 25
4. (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. (1) 355
2. (2) 200
3. (3) 150
4. (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. (1) 5
2. (2) 10
3. (3) 15
4. (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) 1
2. (2) 2
3. (3) 3
4. (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) 1
2. (2) 4
3. (3) 9
4. (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. (1) 560
2. (2) 1120
3. (3) 1680
4. (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. (1) 1430
2. (2) 1436
3. (3) 1500
4. (4) 1600

Answer: (2) 1436