Question 1:

All values of a for which the inequality:

(1/√a) ∫₁ᵃ [ (3/2)√x + 1 - 1/√x ] dx < 4

is satisfied, lie in the interval:

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

Question 2:

For any integer n:

∫₀ᵖⁱ e^(cos²x) · cos³((2n + 1)x) dx has the value.

1. π
2. 1
3. 0
4. 3π/2

Question 3:

Let f be a differential function with:

lim_x→∞ f(x) = 0. If y' + y f'(x) - f(x) f'(x) = 0, lim_x→∞ y(x) = 0, then:

1. y + 1 = e^-f(x) + f(x)
2. y + 1 = e^-f(x) + f(x)
3. y + 2 = e^-f(x) + f(x)
4. y - 1 = e^-f(x) + f(x)

Question 4:

If xy' + y - e^x = 0, y(a) = b, then:

1. e + 2ab - e^a
2. e² + ab - e^-a
3. e - ab + e^a
4. e + ab - e^a

Question 5:

The area bounded by the curves x = 4 - y² and the Y-axis is:

1. 16 square units
2. 32/3 square units
3. 16/3 square units
4. 32 square units

Question 6:

If f(x) = cos(x) - 1 + x²/2, x ∈ ℝ, then f(x) is:

1. Decreasing function
2. Increasing function
3. Neither increasing nor decreasing
4. Constant for x > 0

Question 7:

Let y = f(x) be any curve on the X-Y plane and P be a point on the curve. Let C be a fixed point not on the curve. The length PC is either a maximum or a minimum. Then:

1. PC is perpendicular to the tangent at P.
2. PC is parallel to the tangent at P.
3. PC meets the tangent at an angle of 45°.
4. PC meets the tangent at an angle of 60°.

Question 8:

If a particle moves in a straight line according to the law x = a sin(√t + b), then the particle will come to rest at two points whose distance is:

1. a
2. a/2
3. 2a
4. 4a

Question 9:

A unit vector in the XY-plane making an angle of 45° with (i + j) and an angle of 60° with (3i - 4j) is:

1. 13/14 i + 1/14 j
2. 1/14 i + 13/14 j
3. 13/14 i - 1/14 j
4. 1/14 i - 13/14 j

Question 10:

Let f: R → R be given by f(x) = |x² - 1|. Then:

1. f has a local minimum at x = 1 but no local maximum.
2. f has a local maximum at x = 0, but no local minimum.
3. f has a local minimum at x = ±1 and a local maximum at x = 0.
4. f has neither any local maximum nor any local minimum.

Question 11:

Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If a_n and b_n are the n-th terms of A.P. and G.P. respectively, then:

1. a_n > b_n for all n > 2
2. a_n < b_n for all n > 2
3. a_n = b_n for some n > 2
4. a_n = b_n for some odd n

Question 12:

If for the series a_1, a_2, a_3, ..., the difference a_(n+1) - a_n bears a constant ratio with a_n + a_(n+1), then the series a_1, a_2, a_3, ... is:

1. A.P.
2. G.P.
3. H.P.
4. Any other series

Question 13:

If z_1 and z_2 are roots of the equation z^2 + az + b = 0, a^2 < 4b, then the origin, z_1, and z_2 form an equilateral triangle if:

1. a^2 = 3b^2
2. a^2 = 3b
3. b^2 = 3a
4. a^2 = b^2

Question 14:

If cos(θ) + i sin(θ) (θ ∈ R) is a root of the equation:

a_0x^n + a_1x^(n-1) + ... + a_n = 0,

then the value of a_1 sin(θ) + a_2 sin(2θ) + ... + a_n sin(nθ) is:

1. 2n
2. n
3. 0
4. n + 1

Question 15:

If (x^2 log(x)) log_9(x) = x + 4, then the value of x is:

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

Question 16:

If P(x) = ax^2 + bx + c and Q(x) = -ax^2 + dx + c, where ac ≠ 0, then P(x) · Q(x) = 0 has:

1. 2 real roots
2. At least two real roots
3. 4 real roots
4. No real roots

Question 17:

Let N be the number of quadratic equations with coefficients from {0, 1, 2, ..., 9} such that 0 is a solution of each equation. Then the value of N is:

1. 29
2. 39
3. 90
4. 81

Question 18:

If a, b, c are distinct odd natural numbers, then the number of rational roots of ax^2 + bx + c = 0 is:

1. Must be 0
2. Must be 1
3. Must be 2
4. Cannot be determined from the given data

Question 19:

The numbers 1, 2, ..., m are arranged in random order. The number of ways this can be done, so that 1, 2, ..., r (r < m) appear as neighbors is:

1. (m - r)!
2. (m - r + 1)!
3. (m - r)! r!
4. (m - r + 1)! r!

Question 20:

If A = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]] and θ = 2π/7, then A^100 is:

1. [[cos(2θ), -sin(2θ)], [sin(2θ), cos(2θ)]]
2. [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]]
3. [[1, 0], [0, 1]]
4. [[0, -1], [1, 0]]

Question 21:

If (1 + x + x² + x³)⁵ = Σₖ₌₀¹⁵ aₖ xᵏ, then Σₖ₌₀⁷ (-1)ᵏ · a₂ₖ is equal to:

Options:

1. 2⁵
2. 4⁵
3. 0
4. 4⁴

Question 22:

The coefficient of a¹⁰ b⁷ c³ in the expansion of (bc + ca + ab)¹⁰ is:

Options:

1. 140
2. 150
3. 120
4. 160

Question 23:

Given the determinant, determine the value of k:

Options:

1. k = -3
2. k = 3
3. k = 1
4. k = -1

Question 24:

If

Then A is:

• (A) [ 1 & 1 ] [ 1 & 0 ]
• (B) [ 1 & 1 ] [ 0 & 1 ]
• (C) [ 1 & 0 ] [ 1 & 1 ]
• (D) [ 0 & 1 ] [ 1 & 1 ]

Question 26:

In ℝ, a relation p is defined as follows: For a, b ∈ ℝ, a p b holds if a² - 4ab + 3b² = 0. Then:

Options:

1. p is an equivalence relation
2. p is only symmetric
3. p is only reflexive
4. p is only transitive

Question 27:

Let f: ℝ → ℝ be a function defined by f(x) = (e^|x| - e^(-x)) / (e^x + e^(-x)), then:

Options:

1. f is both one-to-one and onto
2. f is one-to-one but not onto
3. f is onto but not one-to-one
4. f is neither one-to-one nor onto

Question 28:

Let A be the set of even natural numbers that are < 8 and B be the set of prime integers that are < 7. The number of relations from A to B is:

Options:

1. 3²
2. 2⁹ - 1
3. 9²
4. 2⁹

Question 29:

Two smallest squares are chosen one by one on a chessboard. The probability that they have a side in common is:

Options:

1. 1/9
2. 2/7
3. 1/18
4. 5/18

Question 30:

Two integers r and s are drawn one at a time without replacement from the set {1, 2, ..., n}. Then P(r ≤ k / s ≤ k) is:

Options:

1. k/n
2. k/(n-1)
3. (k-1)/n
4. (k-1)/(n-1)

Question 31:

A biased coin with probability p (where 0 < p < 1) of getting head is tossed until a head appears for the first time. If the probability that the number of tosses required is even is 2/5, then p =:

Options:

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

Question 32:

The expression cos²θ + cos²(θ + φ) - 2cosθ cos(θ + φ) is:

Options:

1. independent of θ
2. independent of φ
3. independent of θ and φ
4. dependent on θ and φ

Question 33:

If 0 < θ < π/2 and tan 30° ≠ 0, then tan θ + tan 2θ + tan 3θ = 0 if tan θ * tan 2θ = k, where k =:

Options:

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

Question 34:

The equation r cos θ = 2a sin² θ represents the curve:

Options:

1. x³ = y²(2a + x)
2. x² = y²(2a + x)
3. x³ = y²(2a - x)
4. x³ = y²(2a + x)

Question 35:

If (1, 5) is the midpoint of the segment of a line between the lines 5x - y - 4 = 0 and 3x + 4y - 4 = 0, then the equation of the line will be:

Options:

1. 83x + 35y - 92 = 0
2. 83x - 35y + 92 = 0
3. 83x - 35y - 92 = 0
4. 83x + 35y + 92 = 0

Question 36:

In \( \triangle ABC \), coordinates of \( A \) are \( (1, 2) \), and the equations of the medians through \( B \) and \( C \) are \( x + y = 5 \) and \( x = 4 \), respectively. Then the midpoint of \( BC \) is:

Options:

1. \( \left( 5, \frac{1}{2} \right) \)
2. \( \left( \frac{11}{2}, 1 \right) \)
3. \( \left( 11, \frac{1}{2} \right) \)
4. \( \left( \frac{11}{2}, \frac{1}{2} \right) \)

Question 37:

A line of fixed length \( a + b \), moves so that its ends are always on two fixed perpendicular straight lines. The locus of a point which divides the line into two parts of length \( a \) and \( b \) is:

Options:

1. A parabola
2. A circle
3. An ellipse
4. A hyperbola

Question 38:

With origin as a focus and \( x = 4 \) as the corresponding directrix, a family of ellipses are drawn. Then the locus of an end of the minor axis is:

Options:

1. A circle
2. A parabola
3. A straight line
4. A hyperbola

Question 39:

Chords AB & CD of a circle intersect at right angle at the point P. If the lengths of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is:

Options:

1. 4 units
2. \( \frac{\sqrt{65}}{2} \) units
3. \( \frac{\sqrt{67}}{2} \) units
4. \( \frac{\sqrt{66}}{2} \) units

Question 40:

The plane \( 2x - y + 3z + 5 = 0 \) is rotated through 90° about its line of intersection with the plane \( x + y + z = 1 \). The equation of the plane in the new position is:

Options:

1. 3x + 9y + z + 17 = 0
2. 3x + 9y + z = 17
3. 3x - 9y - z = 17
4. 3x + 9y - z = 17

Question 41:

If the relation between the direction ratios of two lines in R^3 are given by l + m + n = 0, 2lm + 2mn - ln = 0, then the angle between the lines is:

Options:

1. π/6
2. 2π/3
3. π/2
4. π/4

Question 42:

△ OAB is an equilateral triangle inscribed in the parabola y² = 4ax, a > 0 with O as the vertex. Then the length of the side of △ OAB is:

Options:

1. 8a√3 units
2. 8a units
3. 4a√3 units
4. 4a units

Question 43:

For every real number x ≠ -1, let f(x) = x / (x + 1). Write f₁(x) = f(x) and for n ≥ 2, fₙ(x) = f(fₙ₋₁(x)). Then f₁(-2), f₂(-2), ... , fₙ(-2) must be:

Options:

1. 2n / (3 * 1 * 5 ... (2n - 1))
2. 1
3. (1 / 2) * (2n / n)
4. 2n / n

Question 44:

If U_n (n = 1, 2) denotes the n-th derivative (n = 1, 2) of U(x) = (Lx + M) / (x² - 2Bx + C) (L, M, B, C are constants), then P U₂ + Q U₁ + R U = 0 holds for:

Options:

1. P = x² - 2B, Q = 2x, R = 3x
2. P = x² - 2Bx + C, Q = 4(x - B), R = 2
3. P = 2x, Q = 2B, R = 2
4. P = x, Q = x, R = 3

Question 45:

The equation 2x⁵ + 5x = 3x³ + 4x⁴ has:

Options:

1. No real solution
2. Only one non-zero real solution
3. Infinitely many solutions
4. Only three non-negative real solutions

Question 46:

Consider the function f(x) = (x - 2) log x. Then the equation x log x = 2 - x has:

Options:

1. At least one root in (1, 2)
2. Has no root in (1, 2)
3. Is not solvable
4. Has infinitely many roots in (-2, 1)

Question 47:

If α, β are the roots of the equation ax² + bx + c = 0, then:

Options:

1. (α - β)²
2. (1/2)(α - β)²
3. (a² / 4)(α - β)²
4. (a² / 2)(α - β)²

Question 48:

If f(x) = e^x / (1 + e^x), I₁ = ∫ from -a to a x g(x(1 - x)) dx and I₂ = ∫ from -a to a g(x(1 - x)) dx, then the value of I₂/I₁ is:

Options:

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

Question 49:

Let f: R → R be a differentiable function and f(1) = 4. Then the value of lim (x → 1) ∫ from 4 to f(x) (2t / (x - 1)) dt is:

Options:

1. 16
2. 8
3. 4
4. 2

Question 50:

If ∫ (log(x + √(1 + x²))) / (1 + x²) dx = f(g(x)) + c, then:

Options:

1. f(x) = x² / 2, g(x) = log(x + √(1 + x²))
2. f(x) = log(x + √(1 + x²)), g(x) = x² / 2
3. f(x) = x², g(x) = log(x + √(1 + x²))
4. f(x) = log(x - √(1 + x²)), g(x) = x²

Question 51:

Let

I(R) = ∫₀ᴿ e^(-R sin x) dx, R > 0.

Which of the following is correct?

• (A) I(R) > (π / 2R) (1 - e^(-R))
• (B) I(R) < (π / 2R) (1 - e^(-R))
• (C) I(R) = (π / 2R) (1 - e^(-R))
• (D) I(R) and (π / 2R) (1 - e^(-R)) are not comparable

Question 52:

Consider the function

f(x) = x(x - 1)(x - 2)...(x - 100)

Which one of the following is correct?

• (A) This function has 100 local maxima.
• (B) This function has 50 local maxima.
• (C) This function has 51 local maxima.
• (D) Local minima do not exist for this function.

Question 53:

In a plane, ???? and b are the position vectors of two points A and B respectively. A point P with position vector r moves on that plane in such a way that

|r - a| - |r - b| = c
(real constant). The locus of P is a conic section whose eccentricity is:

• (A) |a - b| / c
• (B) |a + b| / c
• (C) |a - b| / 2c
• (D) |a + b| / 2c

Question 54:

Five balls of different colors are to be placed in three boxes of different sizes. The number of ways in which we can place the balls in the boxes so that no box remains empty is:

• (A) 160
• (B) 140
• (C) 180
• (D) 150

Question 55:

Let

A = [1, -1, 0], [0, 1, -1], [1, 1, 1], B = [2, 1, 7] For the validity of the result AX = B, X is:

• (A) [ -1, 1, 7 ]
• (B) [ 1, 2, 4 ]
• (C) [ 3, -1, -1 ]
• (D) [ 4, 2, 1 ]

Question 56:

If a₁, a₂, ... , aₙ are in A.P. with common difference θ, then the sum of the series:

sec a₁ sec a₂ + sec a₂ sec a₃ + ... + sec aₙ₋₁ sec aₙ = k(tan aₙ - tan a₁), where k = ?

• (A) sin θ
• (B) cos θ
• (C) sec θ
• (D) csc θ

Question 57:

For the real numbers x and y, we write x P y iff x - y + √2 is an irrational number. Then the relation P is:

• (A) Reflexive
• (B) Symmetric
• (C) Transitive
• (D) Equivalence relation

Question 58:

Let

A = [0, 0, -1], [0, -1, 0], [-1, 0, 0]

Which of the following is true?

• (A) A is a null matrix
• (B) A is skew-symmetric
• (C) A^-1 does not exist
• (D) A² = I

Question 59:

If 1000! = 3ⁿ × m, where m is an integer not divisible by 3, then n = ?

• (A) 498
• (B) 298
• (C) 398
• (D) 98

Question 60:

If A and B are acute angles such that sin A = sin² B and 2 cos² A = 3 cos² B, then (A, B) is:

• (A) (π / 6, π / 4)
• (B) (π / 6, π / 6)
• (C) (π / 4, π / 6)
• (D) (π / 4, π / 4)

Question 61:

If two circles which pass through the points (0, a) and (0, -a) and touch the line y = mx + c cut orthogonally, then:

• (A) c² = a²(1 + m²)
• (B) c² = a²(2 + m²)
• (C) c² = 2a²(1 + 2m²)
• (D) 2c² = a²(1 + m²)

Question 62:

The locus of the midpoint of the system of parallel chords parallel to the line y = 2x to the hyperbola 9x² - 4y² = 36 is:

• (A) 8x - 9y = 0
• (B) 9x - 8y = 0
• (C) 8x + 9y = 0
• (D) 9x - 4y = 0

Question 63:

The angle between two diagonals of a cube will be:

• (A) cos^-1(1/3)
• (B) sin^-1(1/3)
• (C) (π/2) - cos^-1(1/3)
• (D) (π/2) - sin^-1(1/3)

Question 64:

If y = tan^-1 [log_e (e/x²)] / log_e (e x²) + tan^-1 [(3 + 2 log_e x)/(1 - 6 log_e x)], then d²y/dx² = ?

• (A) 2
• (B) 1
• (C) 0
• (D) -1

Question 65:

Evaluate:

lim_{n → ∞} [1/n^k+1] (2^k + 4^k + 6^k + ... + (2n)^k)

• (A) (n + 1) 2ⁿ
• (B) 3ⁿ
• (C) (n + 1) 2ⁿ⁺¹
• (D) 2^k / (k+1)

Question 66:

The acceleration f (in ft/sec²) of a particle after a time t seconds starting from rest is given by:

f = 6 - √(1.2t)

• (A) T = 20 sec
• (B) v = 60 ft/sec
• (C) T = 30 sec
• (D) v = 40 ft/sec

Question 67:

Let Γ be the curve y = be^-x/a and L be the straight line: x/a + y/b = 1, where a, b ∈ ℝ. Then:

• (A) L touches the curve Γ at the point where the curve crosses the axis of y.
• (B) L does not touch the curve at the point where the curve crosses the axis of y.
• (C) Γ touches the axis of x at a point.
• (D) Γ never touches the axis of x.

Question 68:

If n is a positive integer, the value of:

(2n + 1) binom{n}{0} + (2n - 1) binom{n}{1} + (2n - 3) binom{n}{2} + ... + 1 · binom{n}{n}

• (A) (n + 1) · 2ⁿ
• (B) 3ⁿ
• (C) f'(2) where f(x) = xⁿ⁺¹
• (D) (n + 1) · 2ⁿ⁺¹

Question 69:

If the quadratic equation ax² + bx + c = 0 (a > 0) has two roots α and β such that α < -2 and β > 2, then:

• (A) c < 0
• (B) a + b + c > 0
• (C) a - b + c < 0
• (D) a - b + c > 0

Question 70:

If a_i, b_i, c_i ∈ ℝ (i = 1, 2, 3) and x ∈ ℝ, and:

det{a₁ + b₁ x, a₁ x + b₁, c₁; a₂ + b₂ x, a₂ x + b₂, c₂; a₃ + b₃ x, a₃ x + b₃, c₃ } = 0, then:

• (A) x = 1
• (B) x = -1
• (C) det{a₁, b₁, c₁; a₂, b₂, c₂; a₃, b₃, c₃} = 0
• (D) x = 2

Question 71:

The function f: ℝ → ℝ defined by f(x) = e^x + e^-x is:

• (A) One-one
• (B) Onto
• (C) Bijective
• (D) Not bijective

Question 72:

A square with each side equal to a lies above the x-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle α (0 < α < π/4) with the positive direction of the x-axis. The equation of the diagonals of the square is:

• (A) y (cos α - sin α) = x (sin α + cos α)
• (B) y (cos α + sin α) = x (cos α - sin α)
• (C) y (sin α + cos α) + x (cos α - sin α) = a
• (D) y (cos α - sin α) + x (cos α + sin α) = a

Question 73:

If △ABC is an isosceles triangle and the coordinates of the base points are B(1, 3) and C(-2, 7), the coordinates of A can be:

• (A) (1, 6)
• (B) (-1/8, 5)
• (C) (5/6, 6)
• (D) (-7, -1/8)

Question 74:

The points of extremum of

∫₀^x² (t² - 5t + 4) / (2 + e^t) dt are:

• (A) ±1
• (B) ±2
• (C) ±3
• (D) ±√2

Question 75:

Choose the correct statement:

• (A) x + sin 2x is a periodic function
• (B) x + sin 2x is not a periodic function
• (C) cos(√x + 1) is a periodic function
• (D) cos(√x + 1) is not a periodic function