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