JEE MAIN 2025 (Online) 5th April Evening Shift

Question 1:

Let α, β be the roots of the equation x² – x + p = 0 and γ, δ be the roots of the equation x² – 4x + q = 0; p, q ∈ Z. If α, β, γ, δ are in G.P., then |p + q| equals:

Question 2:

Let z₁, z₂ ∈ C be the distinct solutions of the equation z² + 4z – (1 + 12i) = 0. Then |z₁|² + |z₂|² is equal to:

Question 3:

If f : N → Z is defined by
f(n) = |n -1 -5; -2n² 3(2k+1) 2k+1; -3n³ 3k(2k+1) 3k(k+2)+1|, k ∈ N,
and ∑ₙ₌₁ᵏ f(n) = 98, then k is equal to:

Question 4:

Let M be a 3×3 matrix such that M[1; 1; 0] = [2; 3; 2], M[0; 1; 1] = [1; 2; 3], M[0; 0; 1] = [-1; 1; 1]. If M[x; y; z] = [1; 7; 11], then x + y + z equals:

Question 5:

If the sum of the first 10 terms of the series
1/(1 + 1⁴ × 4) + 2/(1 + 2⁴ × 4) + 3/(1 + 3⁴ × 4) + 4/(1 + 4⁴ × 4) + …
is m/n, gcd(m, n) = 1, then m + n is equal to:

Question 6:

Let A₁, A₂, A₃, …, A₃₉ be 39 arithmetic means between the numbers 59 and 159. Then the mean of A₂₅, A₂₈, A₃₁ and A₃₆ is equal to:

Question 7:

The coefficient of x² in the expansion of (2x² + 1/x)¹⁰, x ≠ 0, is:

Question 8:

The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is:

Question 9:

A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is:

Question 10:

A variable X takes values 0, 0, 2, 6, 12, 20, …, n(n-1) with frequencies ⁿC₀, ⁿC₁, ⁿC₂, ⁿC₃, ⁿC₄, ⁿC₅, …, ⁿCₙ respectively. If the mean of this data is 60, then its median is:

Question 11:

Let the point P be the vertex of the parabola y = x² – 6x + 12. If a line passing through the point P intersects the circle x² + y² – 2x – 4y + 3 = 0 at the points R and S, then the maximum value of (PR + PS)² is:

Question 12:

Let the directrix of the parabola P : y² = 8x cut the x-axis at the point A. Let B(α, β), α > 1, be a point on P such that the slope of AB is 3/5. If BC is a focal chord of P, then six times the area of triangle ABC is:

Question 13:

Let the eccentricity e of a hyperbola satisfy the equation 6e² – 11e + 3 = 0. If the foci of the hyperbola are (3, 5) and (3, 4), then the length of its latus rectum is:

Question 14:

Let a triangle PQR be such that P and Q lie on the line (x+3)/8 = (y-4)/2 = (z+1)/2 and are at a distance of 6 units from R(1, 2, 3). If (α, β, γ) is the centroid of triangle PQR, then α + β + γ is equal to:

Question 15:

If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line x/1 = (y-1)/1 = (z-2)/2 is 4, then the sum of all possible values of a is equal to:

Question 16:

Let O be the origin, OP⃗ = a⃗ and OQ⃗ = b⃗. If R is the point on OP such that OP⃗ = 5OR⃗, and M is the point such that OQ⃗ = 5RM⃗, then PM⃗ is equal to:

Question 17:

Let f(x) = lim(y→0) [(1 – cos(xy))tan(xy)]/y³. Then the number of solutions of the equation f(x) = sin x, x ∈ R:

Question 18:

Let (2^(1-a) + 2^(1+a)), f(a), (3^a + 3^(-a)) be in A.P. and α be the minimum value of f(a). Then the value of the integral ∫_{logₑ(α-1)}^{logₑ(α)} dx/(e^(2x) – e^(-2x)) is:

Question 19:

Let f : [1, ∞) → R be a differentiable function defined as f(x) = ∫₁ˣ f(t)dt + (1 – x)(logₑx – 1) + e. Then the value of f(f(1)) is:

Question 20:

Let f(x) and g(x) be twice differentiable functions satisfying f″(x) = g″(x) for all x ∈ R, f′(1) = 2g′(1) = 4 and g(2) = 3f(2) = 9. Then f(25) – g(25) is equal to:

Question 21:

Let A = {1, 4, 7} and B = {2, 3, 8}. Then the number of elements, in the relation R = {((a₁, b₁), (a₂, b₂)) ∈ ((A × B) × (A × B)) : a₁ + b₂ divides a₂ + b₁} is

Question 22:

From the point (-1, -1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 9 and cx + dy = 7, a, b, c, d ∈ Z, then the value of ad + bc is

Question 23:

If S = {θ ∈ [-π, π] : cos θ cos(5θ/2) = cos 7θ cos(7θ/2)} then n(S) is equal to

Question 24:

Let f : R → R be a function such that f(x) + 3f(π/2 – x) = sin x, x ∈ R. Let the maximum value of f on R be α. If the area of the region bounded by the curves g(x) = x² and h(x) = βx³, β > 0, is α², then 30β³ is equal to

Question 25:

Let y = y(x) be the solution of the differential equation (tan x)(dy/dx) = (sec³x – (tan x)²y), 0 < x < π/2, y(π/4) = 6√2/5. If y(π/3) = (4/5)α, then α⁴ equals