JEE Main 2025 · 8 Apr 2025 Shift 2

Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements (S₁): The number of elements in R is 18, and (S₂): The relation R is symmetric but neither reflexive nor transitive

1. Aboth are true
2. Bboth are false
3. Conly (S₁) is true
4. Donly (S₂) is true

The sum of the squares of the roots of |x − 2|² + |x − 2| − 2 = 0 and the squares of the roots of x² − 2|x − 3| − 5 = 0, is

1. A24
2. B26
3. C30
4. D36

Let A = {θ ∈ [0, 2π] : 1 + 10 Re((2 cos θ + i sin θ)/(cos θ − 3i sin θ)) = 0}. Then ∑₍θ∈A₎ θ² is equal to

1. A8π²
2. B(27/4)π²
3. C6π²
4. D(21/4)π²

Let A = [[2, 2 + p, 2 + p + q], [4, 6 + 2p, 8 + 3p + 2q], [6, 12 + 3p, 20 + 6p + 3q]]. If det(adj(adj(3A))) = 2ᵐ · 3ⁿ, m, n ∈ N, then m + n is equal to

1. A20
2. B22
3. C24
4. D26

Let α be a solution of x² + x + 1 = 0, and for some a and b in R, [4 a b] [[1, 16, 13], [−1, −1, 2], [−2, −14, −8]] = [0 0 0]. If 4/α⁴ + m/αᵃ + n/αᵇ = 3, then m + n is equal to ______

1. A8
2. B11
3. C7
4. D3

If 1/1⁴ + 1/2⁴ + 1/3⁴ + ...∞ = π⁴/90, 1/1⁴ + 1/3⁴ + 1/5⁴ + ...∞ = α, 1/2⁴ + 1/4⁴ + 1/6⁴ + ...∞ = β, then α/β is equal to

1. A14
2. B15
3. C18
4. D23

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

1. A210
2. B220
3. C200
4. D230

The number of integral terms in the expansion of (5¹⁄² + 7¹⁄⁸)¹⁰¹⁶ is

1. A129
2. B127
3. C128
4. D130