JEE MAIN 2025 (Online) 5th April Morning Shift

Question 1:

Let a, b ∈ C. Let α, β be the roots of the equation x² + ax + b = 0. If β – α = √11 and β² – α² = 3i√11, then (β³ – α³)² is equal to:

Question 2:

Let the sum of the first n terms of an A.P. be 3n² + 5n. Then the sum of squares of the first 10 terms of the A.P. is:

Question 3:

Let A be a 3 × 3 matrix such that A[0; 0; 1] = [1; 3; 1]. If det(A) = 1, then det(adj(A² + A)) is equal to:

Question 4:

Consider the system of linear equations in x, y, z:
x + 2y + tz = 0,
6x + y + 5tz = 0,
3x + t²y + f(t)z = 0,
where f : R → R is a differentiable function. If this system has infinitely many solutions for all t ∈ R then f:

Question 5:

∑ₙ₌₁¹⁰ (528/(n(n+1)(n+2))) is equal to:

Question 6:

Let tan A, tan B, where A, B ∈ (-π/2, π/2), be the roots of the quadratic equation x² – 2x – 5 = 0. Then 20 sin²((A + B)/2) is equal to:

Question 7:

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:

Question 8:

The mean deviation about the mean for the given data (xᵢ : 5, 7, 9, 10, 12, 15 and fᵢ : 6, 8, 2, 2, 2, 6) is equal to:

Question 9:

Let a focus of the ellipse E : x²/a² + y²/b² = 1 be S(4, 0) and its eccentricity be 4/5. If the point P(3, a) lies on E and O is the origin, then the area of triangle POS is equal to:

Question 10:

Let P be a moving point on the circle x² + y² – 6x – 8y + 21 = 0. Then, the maximum distance of P from the vertex of the parabola x² + 6x + y + 13 = 0 is equal to:

Question 11:

In an equilateral triangle PQR, let the vertex P be at (3, 5) and the side QR be along the line x + y = 4. If the orthocentre of the triangle PQR is (α, β), then 9(α + β) is equal to:

Question 12:

The sum of all the integral values of p such that the equation 3sin 2x + 12cos x – 3 = p, x ∈ R has at least one solution, is:

Question 13:

The square of the distance of the point P(5, 6, 7) from the line (x-2)/2 = (y-5)/3 = (z-2)/4 is equal to:

Question 14:

Let a⃗ = √7î + ĵ – k̂ and b⃗ = ĵ + 2k̂. If r⃗ is a vector such that r⃗ × a⃗ + a⃗ × b⃗ = 0⃗ and r⃗ · a⃗ = 0, then |3r⃗|² is equal to:

Question 15:

The square of the distance of the point of intersection of the lines r⃗ = (î + ĵ – k̂) + λ(aî – ĵ), a ≠ 0 and r⃗ = (4î – k̂) + μ(2î + ak̂) from the origin is:

Question 16:

The area of the region R = {(x, y) : xy ≤ 27, 1 ≤ y ≤ x²} is equal to:

Question 17:

The product of all possible values of a, for which lim(x→0) [(1 – cos(ax)cos((a+1)x)cos((a+2)x))/(sin²((a+1)x))] = 2, is:

Question 18:

The value of the integral ∫₀^∞ (logₑx)/(x² + 4) dx is:

Question 19:

Let f : R → R be a differentiable function such that f((x + y)/3) = (f(x) + f(y))/3 for all x, y ∈ R and f′(0) = 3. Then the minimum value of the function g(x) = 3 + eˣf(x) is:

Question 20:

The value of the integral ∫_{π/6}^{π/3} (4 – csc²x)/(cos⁴x) dx is: ________

Question 21:

Let A = {1, 2, 3, 4, 5, 6}. The number of one-one functions f : A → A such that f(1) ≥ 3, f(3) ≤ 4 and f(2) + f(3) = 5, is ________

Question 22:

Two players A and B play a series of games of badminton. The player, who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player A wins the series is ________

Question 23:

If the sum of the coefficients of x⁷ and x¹⁴ in the expansion of (1/x³ – x⁴)ⁿ, x ≠ 0, is zero, then the value of n is ________

Question 24:

If π/4 + ∑ₚ₌₁¹¹ tan⁻¹(2^(p-1)/(1 + 2^(2p-1))) = α then tan α is equal to ________

Question 25:

Let y = y(x) be the solution of the differential equation x sin y dy = (y sin x – x) dx, y(1) = π/2 and let a = cos(y(e¹²)/e¹²). Then the number of integral values of p, for which the equation x² + y² – 2px + 2py + a + 2 = 0 represents a circle of radius r ≤ 6, is ________