JEE Main 2025 · 4 Apr 2025 Shift 2

Let the domains of the functions f(x) = log₄ log₃ log₇(8 − log₂(x² + 4x + 5)) and g(x) = sin⁻¹((7x + 10)/(x − 2)) be (α, β) and [γ, δ], respectively. Then α² + β² + γ² + δ² is equal to :

1. A13
2. B14
3. C15
4. D16

Let A = {−3, −2, −1, 0, 1, 2, 3} and R be a relation on A defined by xRy if and only if 2x − y ∈ {0, 1}. Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then l + m + n is equal to :

1. A15
2. B16
3. C17
4. D18

Let the product of ω₁ = (8 + i)sinθ + (7 + 4i) cosθ and ω₂ = (1 + 8i)sinθ + (4 + 7i) cosθ be α + iβ, i = √−1. Let p and q be the maximum and the minimum values of α + β respectively. Then p + q is equal to :

1. A130
2. B140
3. C150
4. D160

Let the matrix A = [[1, 0, 0], [1, 0, 1], [0, 1, 0]] satisfy Aⁿ = Aⁿ⁻² + A² − I for n ≥ 3. Then the sum of all the elements of A⁵⁰ is :

1. A52
2. B53
3. C44
4. D39

If the sum of the first 20 terms of the series (4·1)/(4 + 3·1² + 1⁴) + (4·2)/(4 + 3·2² + 2⁴) + (4·3)/(4 + 3·3² + 3⁴) + (4·4)/(4 + 3·4² + 4⁴) + .... is m/n, where m and n are coprime, then m + n is equal to :

1. A420
2. B421
3. C422
4. D423

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of A.P’s in A and B respectively such that D = d + 3, d > 0. If (p + q)/(p − q) = 19/5, then p − q is equal to

1. A540
2. B600
3. C630
4. D450

If 1²(¹⁵C₁) + 2²(¹⁵C₂) + 3²(¹⁵C₃) + ... + 15²(¹⁵C₁₅) = 2ᵐ · 3ⁿ · 5ᵏ, where m, n, k ∈ N, then m + n + k is equal to :

1. A18
2. B19
3. C20
4. D21

Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, a, b be 4 and √2 respectively. Then the mean deviation about the mode of these observations is :

1. A2
2. B1/2
3. C3/4
4. D1