JEE MAIN 2025 (Online) 8th April Evening Shift

Question 1:

Consider the relation R on the set {-2, -1, 0, 1, 2} defined by (a, b) ∈ R if and only if 1 + ab > 0. Then, among the statements:
I. The number of elements in R is 17
II. R is an equivalence relation

Question 2:

The number of values of z ∈ ℂ, satisfying the equations |z – (4 + 8i)| = √10 and |z – (3 + 5i)| + |z – (5 + 11i)| = 4√5, is:

Question 3:

If the system of linear equations: x + y + z = 6, x + 2y + 5z = 10, 2x + 3y + λz = μ has infinitely many solutions, then the value of λ + μ equals:

Question 4:

Let A = [α 1 2; 2 3 0; 0 4 5] and B = [1 0 0; 0 -5α 0; 0 4α -2α] + adj(A). If det(B) = 66, then det(adj(A)) equals:

Question 5:

Let α = 3 + 4 + 8 + 9 + 13 + 14 + … upto 40 terms. If (tan β)^(α/1020) is a root of the equation x² + x – 2 = 0, β ∈ (0, π/2), then sin²β + 3cos²β is equal to:

Question 6:

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car are 2/5, 1/5, 2/5 respectively. The probabilities that the candidate reaches late at the examination centre are 1/3, 1/4, 1/5 if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

Question 7:

A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to:

Question 8:

If 26(2³/3(¹²C₂) + 2⁵/5(¹²C₄) + 2⁷/7(¹²C₆) + … + 2¹³/13(¹²C₁₂)) = 3¹³ – α, then α is equal to:

Question 9:

A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is:

Question 10:

If a straight line drawn through the point of intersection of the lines 4x + 3y – 1 = 0 and 3x + 4y – 1 = 0, meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is:

Question 11:

Let O be the vertex of the parabola y² = 4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is:

Question 12:

Let α = 3sin⁻¹(6/11) and β = 3cos⁻¹(4/9), where inverse trigonometric functions take only the principal values. Given below are two statements:
Statement I: cos(α + β) > 0
Statement II: cos(α) < 0
In the light of the above statements, choose the correct answer from the options given below:

Question 13:

For the function f(x) = e^(sin|x|) – |x|, x ∈ R, consider the following statements:
Statement I: f is differentiable for all x ∈ R
Statement II: f is increasing in (-π, -π/2)
In the light of the above statements, choose the correct answer from the options given below:

Question 14:

Let a⃗ = 4î – ĵ + 3k̂, b⃗ = 10î + 2ĵ – k̂ and a vector c⃗ such that 2(a⃗ × b⃗) + 3(b⃗ × c⃗) + 3(c⃗ × a⃗) = 0⃗. If a⃗ · c⃗ = 15, then c⃗ · (î + ĵ – 3k̂) is equal to:

Question 15:

Let the foot of perpendicular from the point (x, 2, 3) on the line (x-4)/1 = (y-9)/2 = (z-5)/1 be the point (1, μ, 2). Then the distance between the lines (x-1)/2 = (y-2)/3 = (z+4)/6 and (x-λ)/2 = (y-μ)/3 = (z+5)/6 is equal to:

Question 16:

The value of the integral ∫₀² [√(x(x² + x + 1)) / (√(x+1) · √(x⁴ + x² + 1))] dx is equal to:

Question 17:

Let y = y(x) be the solution of the differential equation x√(1-x²)dy + (y√(1-x²) – x·cos⁻¹x)dx = 0, x ∈ (0, 1), lim(x→1⁻) y(x) = 1. Then y(1/2) equals:

Question 18:

Let f : (1, ∞) → R be a function defined as f(x) = (x-1)/(x+1). Let f^(i+1)(x) = f(f^i(x)), i = 1, 2, …, 25, where f¹(x) = f(x). If g(x) + f²⁶(x) = 0, x ∈ (1, ∞), then the area of the region bounded by the curves y = g(x), 2y = 2x – 3, y = 0 and x = 4 is:

Question 19:

Let f(x) = {1/3, x ≤ π/2; b(1 – sin x)/(π – 2x)², x > π/2}. If f is continuous at x = π/2, then the value of ∫₀^(3b-6) |x² + 2x – 3| dx is:

Question 20:

Let x²/f(a² + 7a + 3) + y²/f(3a + 15) = 1 represent an ellipse with major axis along y-axis, where f is a strictly decreasing positive function on R. If the set of all possible values of a is R – [α, β], then α² + β² is equal to:

Question 21:

The sum of squares of all the real solutions of the equation log₍ₓ₊₁₎(2x² + 5x + 3) = 4 – log₍₂ₓ₊₃₎(x² + 2x + 1) is equal to

Question 22:

If ∫₍₋π/₄₎^(π/4) (cot(x – π/3)·cot(x + π/3) + 1) dx = α·logₑ(√3 – 1), then 9α² is equal to

Question 23:

Let a line L₁ pass through the origin and be perpendicular to the lines L₂ : r⃗ = (3+t)î + (2t-1)ĵ + (2t+4)k̂ and L₃ : r⃗ = (3+2s)î + (3+2s)ĵ + (2+s)k̂, t, s ∈ R. If (a, b, c), a ∈ Z, is the point on L₃ at a distance of √17 from the point of intersection of L₁ and L₂, then (a + b + c)² is equal to

Question 24:

Consider the circle C : x² + y² – 6x – 8y – 11 = 0. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle x² + y² – αx – βy – γ = 0, then α + β + 2γ is equal to

Question 25:

Let f be a polynomial function such that f(x) + f(1/x) = f(x)·f(1/x) for all x > 0 and f(6) = 37. Then ∑ₙ₌₁¹⁰ f(n) is equal to