JEE MAINS 2025

Q1. The number of non-empty equivalence relations on the set ${1,2,3}$ is :

Q2. Let $f : R → R$ be a twice differentiable function such that $f(x + y) = f(x) f(y)$ for all $x, y ∈ R$. If $f'(0) = 4a$ and $f$ satisfies $f''(x) - 3af'(x) - f(x) = 0$, $a > 0$, then the area of the region $R = {(x,y) | 0 ≤ y ≤ f(ax), 0 ≤ x ≤ 2}$ is :

Q3. Let the triangle PQR be the image of the triangle with vertices $(1,3)$, $(3,1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $ΔPQR$ is the point $(α, β)$, then $15(α - β)$ is equal to :

Q4. Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $arg(z_1) = -π/4$, $arg(z_2) = 0$ and $arg(z_3) = π/4$. If $|z_1overline{z}_2 + z_2overline{z}_3 + z_3overline{z}_1|^2 = α + β√2$, $α, β ∈ Z$, then the value of $α^2 + β^2$ is :

Q5. Using the principal values of the inverse trigonometric functions the sum of the maximum and the minimum values of $16(sec^{-1}x)^2 + (cosec^{-1}x)^3$ is:

Q6. A coin is tossed three times. Let X denote the number of times a tail follows a head. If $μ$ and $σ^2$ denote the mean and variance of X, then the value of $64(μ + σ^2)$ is:

Q7. Let $a_1, a_2, a_3, ..., a_n$ be a G.P. of increasing positive terms. If $a_1 a_3 = 28$ and $a_2 + a_4 = 29$, then $a_5$ is equal to

Q8. Let $L_1: (x-1)/2 = (y-2)/3 = (z-3)/4$ and $L_2: (x-2)/3 = (y-4)/4 = (z-5)/5$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$?

Q9. The product of all solutions of the equation $e^{5(ln x)^2 + 3} = x^8$, $x > 0$, is:

Q10. If $Σ_{i=1}^{n} T_i = [(2n-1)(2n+1)(2n+3)(2n+5)]/64$, then $lim_{n→∞} Σ_{i=1}^{n} (1/T_i)$ is equal to:

Q11. From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:

Q12. Let $x = x(y)$ be the solution of the differential equation $y^2 dx + (x - 1/y) dy = 0$. If $x(1) = 1$, then $x(1/2)$ is:

Q13. Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(-1, -1)$ passes through the points P, Q and R, then the area of $ΔPQR$ is:

Q14. A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2, 5)$ and intersects the circle C at exactly two points. If the set of all possible values of r is the interval $(α, β)$, then $3β - 2α$ is equal to:

Q15. Let for $f(x) = 7tan^4x + 7tan^6x - 3tan^4x - 3tan^2x$, $I_1 = ∫_0^{π/4} f(x) dx$ and $I_2 = ∫_0^{π/4} x f(x) dx$. Then $7I_1 + 12I_2$ is equal to: