Question 1:

Let the line L intersect the lines:

x − 2 = −y = z − 1, 2(x + 1) = 2(y − 1) = z + 1, and be parallel to the line: (x − 2)/3 = (y − 1)/1 = (z − 2)/2. Then which of the following points lies on L?

1. (1/3, -1, 1)
2. (-1/3, 1, -1)
3. (-1/3, -1, -1)
4. (-1/3, -1, 1)

Question 2:

The parabola y² = 4x divides the area of the circle x² + y² = 5 in two parts. The area of the smaller part is:

1. 2/3 + 5 sin⁻¹(2√5)
2. 1/3 + 5 sin⁻¹(2√5)
3. 1/3 + √5 sin⁻¹(2√5)
4. 2/3 + √5 sin⁻¹(2√5)

Question 3:

The solution curve of the differential equation 2y (dy/dx) + 3 = 5 (dy/dx), passing through the point (0, 1), is a conic whose vertex lies on the line:

1. 2x + 3y = 9
2. 2x + 3y = −9
3. 2x + 3y = −6
4. 2x + 3y = 6

Question 4:

A ray of light coming from P(1, 2) reflects from Q on the x-axis and passes through R(4, 3). If S(h, k) makes PQRS a parallelogram, then hk² equals:

1. 80
2. 90
3. 60
4. 70

Question 5:

For λ, μ ∈ R, if the system of equations 3x + 5y + λz = 3, 7x + 11y − 9z = 2, 97x + 155y − 189z = μ has infinitely many solutions, then μ + 2λ equals:

1. 25
2. 24
3. 27
4. 22

Question 6:

The coefficient of x⁷⁰ in x²(1 + x)⁹⁸ + x³(1 + x)⁹⁷ + ... + x⁵⁴(1 + x)⁴⁶ is (99Cp) − (46Cq). A possible value of p + q is:

1. 55
2. 61
3. 68
4. 83

Question 7:

If ∫ (2 − tanx)/(3 + tanx) dx = (1/2)(αx + loge|β sinx + γ cosx|) + C, then α + γ/β equals:

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

Question 8:

A variable line passes through (3, 5) and intersects the positive axes at A and B. The minimum area of triangle OAB is:

1. 30
2. 25
3. 40
4. 35

Question 9:

If |cosθ cos(60° − θ) cos(60° + θ)| ≤ 1/8, θ ∈ [0, 2π], then the sum of all θ where cos3θ attains its maximum value is:

1. 9π
2. 18π
3. 6π
4. 15π

Question 10:

Let OA = 2a, OB = 6a + 5b, and OC = 3b. If the area of the parallelogram with sides OA and OC is 15 sq. units, the area of quadrilateral OABC is:

1. 38
2. 40
3. 32
4. 35

Question 11:

If the domain of the function f(x) = sin⁻¹((x−1)/(2x+3)) is R−(α, β), then 12αβ is equal to:

1. 36
2. 24
3. 40
4. 32

Question 12:

If the sum of the series 1/(1 · (1 + d)) + 1/((1 + d)(1 + 2d)) + ... + 1/((1 + 9d)(1 + 10d)) is equal to 5, then 50d is equal to:

1. 20
2. 5
3. 15
4. 10

Question 13:

Let f(x) = ax³ + bx² + cx + 41 be such that f(1) = 40, f'(1) = 2, and f''(1) = 4. Then a² + b² + c² is equal to:

1. 62
2. 73
3. 54
4. 51

Question 14:

Let a circle passing through (2, 0) have its center at the point (h, k). Let (xc, yc) be the point of intersection of the lines 3x + 5y = 1 and (2 + c)x + 5c²y = 1. If h = lim c→1 xc and k = lim c→1 yc, then the equation of the circle is:

1. 25x² + 25y² − 20x + 2y − 60 = 0
2. 5x² + 5y² − 4x − 2y − 12 = 0
3. 25x² + 25y² − 2x + 2y − 60 = 0
4. 5x² + 5y² − 4x + 2y − 12 = 0

Question 15:

The shortest distance between the lines (x−3)/4 = (y+7)/−11 = (z−1)/5 and (x−5)/3 = (y−9)/−6 = (z+2)/1 is:

1. 187√563
2. 178√563
3. 185√563
4. 179√563

Question 16:

The frequency distribution of the age of students in a class of 40 students is given below:

If the mean deviation about the median is 1.25, then 4x + 5y is equal to:

1. 43
2. 44
3. 47
4. 46

Question 17:

The solution of the differential equation (x² + y²)dx − 5xy dy = 0, y(1) = 0, is:

1. |x² − 4y²|⁵ = x²
2. |x² − 2y²|⁶ = x
3. |x² − 4y²|⁶ = x
4. |x² − 2y²|⁵ = x²

Question 18:

Let three vectors a = αî + 4ĵ + 2k̂, b = 5î + 3ĵ + 4k̂, c = xî + yĵ + zk̂, form a triangle such that c = a − b and the area of the triangle is 5√6. If α is a positive real number, then |c|² is:

1. 16
2. 14
3. 12
4. 10

Question 19:

Let α, β be the roots of the equation x² + 2√2x − 1 = 0. The quadratic equation whose roots are α⁴ + β⁴ and 1/10(α⁶ + β⁶) is:

1. x² − 190x + 9466 = 0
2. x² − 195x + 9466 = 0
3. x² − 195x + 9506 = 0
4. x² − 180x + 9506 = 0

Question 20:

Let f(x) = x² + 9, g(x) = x/(x − 9), and a = f(g(10)), b = g(f(3)). If e and ℓ denote the eccentricity and the length of the latus rectum of the ellipse x²/a² + y²/b² = 1, then 8e² + ℓ² is equal to:

1. 16
2. 8
3. 6
4. 12

Question 21:

Let a, b, c denote the outcomes of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that ax² + bx + c = 0 has all real roots is m/n, where gcd(m, n) = 1, then m + n is equal to:

1. 19

Question 22:

The sum of the square of the modulus of the elements in the set: {z = a + ib : a, b ∈ Z, z ∈ C, |z − 1| ≤ 1, |z − 5| ≤ |z − 5i|} is:

1. 9

Question 23:

Let the set of all positive values of λ, for which the point of local minimum of the function:
f(x) = (1 + x(λ² − x²)) satisfies x² + x + 2 / x² + 5x + 6 < 0, be (α, β). Then α² + β² is equal to:

1. 39

Question 24:

Let the following limit be equal to π/k, where k is an integer. Then k² is equal to:
Given:
lim n→∞ [n√(n⁴ + 1) − 2n(n² + 1)√(n⁴ + 1) + n√(n⁴ + 16) + ...]

1. 32

Question 25:

The remainder when 428²⁰²⁴ is divided by 21 is:

1. 1

Question 26:

Let f : (0, π) → R be a function given by:
f(x) =
8/7 * tan(8x) / tan(7x), 0 < x < π/2 a − 8, x = π/2 (1 + |cot(x)|) * (b/a) * tan(|x|), π/2 < x < π
where a, b ∈ Z. If f is continuous at x = π/2, find a² + b².

1. 81

Question 27:

Let A be a non-singular matrix of order 3. If:
det(3adj(2adj((detA)A))) = 3⁻¹³ · 2⁻¹⁰, and:
det(3adj(2A)) = 2^m · 3^n, then |3m + 2n| is equal to:

1. 14

Question 28:

Let the center of a circle, passing through the points (0, 0), (1, 0), and touching the circle x² + y² = 9, be (h, k). Then for all possible values of the coordinates of the center (h, k), 4(h² + k²) is equal to:

1. 9

Question 29:

If a function f satisfies f(m + n) = f(m) + f(n) for all m, n ∈ N, and f(1) = 1, then the largest natural number λ such that:
Σ_k=1²⁰²² f(λ + k) ≤ (2022)², is equal to:

1. 1010

Question 30:

Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on A × B by:
(a₁, b₁) R (a₂, b₂) ⇔ a₁ + a₂ = b₁ + b₂. Then the number of elements in R is:

1. 25