JEE-Main 2025 - Mathematics

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

• 6
• 7
• 5
• 4

Q2: Let ƒ : R→R be a twice differentiable function such that ƒ(x + y) = ƒ(x) ƒ(y) for all x, y  R. If ƒ'(0) = 4a and ƒ stais fies ƒ''(x) – 3a ƒ'(x) – ƒ(x) = 0, a > 0, then the area of the region R = {(x,y) | 0 ≤ y ≤ ƒ(ax), 0 ≤ x ≤ 2} is :

• e² – 1
• e⁴ + 1
• e⁴ – 1
• e² + 1

Q3: If the centroid of \u0394PQR is the point ( \u03b1, \u03b2), then 15( \u03b1 – \u03b2) is equal to :

• 24
• 19
• 21
• 22

Q4: Let z₁, z₂ and z₃ be three complex numbers on the circle |z| = 1 with \u03c0=1 arg(z )4 , arg(z₂) = 0 and \u03c0=3 arg(z )4 . If z₁z₂ + z₂z₃ + z₃z₁ =\u03b1+\u03b22 , \u03b1, \u03b2 \u2208 Z, then the value of \u03b1² + \u03b2² is :

• 24
• 41
• 31
• 29

Q5: Using the principal values of the inverse trigonometric functions the sum of the maximum and the minimum values of 16((sec–1x)² + (cosec–1x)²) is :

• 24 \u03c0²
• 18\u03c0²
• 31 \u03c0²
• 22 \u03c0²

Q6: A coin is tossed three times. Let X denote the number of times a tail follows a head. If \u03bc and \u03c3² denote the mean and variance of X, then the value of 64(\u03bc + \u03c3²) is :

• 51
• 48
• 32
• 64

Q7: Let a₁, a₂, a₃..... be a G.P. of increasing positive terms. If a₁a₅ = 28 and a₂ + a₄ = 29, the a₆ is equal to

• 628
• 526
• 784
• 812

Q8: Let L: (x-1)/2 = (y-2)/3 = (z-3)/4 and L: (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 L₁ and L₂ ?

• (-5, 7,13)
• (12,3,3)
• (-81, 1,33)
• (-14, 22, 3,33)

Q9: The product of all solutions of the equation e^(5 log x) + 3 = 8e^x , x > 0, is :

• e^(8/5)
• e^(6/5)
• e²
• e

Q10: If T\u2099 =[(2n-1)(2n+1)(2n+3)(2n+5)] / 64 , then lim (n->\u221e) \u03a3 (r=1 to n) 1/Tᵣ is equal to :

• 1
• 0
• 2/3
• 1/3

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 :

• 14950
• 6084
• 4356
• 5148

Q12: Let x = x(y) be the solution of the differential equation (y²+1)dx - xdy = 0. If x(1)=1, then x(e) is :

• e/2
• (e+1)/2
• (e+3)/2
• 3 – e

Q13: Let the parabola y = x² + 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 \u0394PQR is :

• 4
• 6
• 7
• 5

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 ( \u03b1, \u03b2), then 3\u03b2 – 2\u03b1 is equal to :

• 15
• 14
• 12
• 10

Q15: Let for ƒ(x) = 7tan⁸x + 7tan⁶x – 3tan⁴x – 3tan²x, I₁ = \u222b (from 0 to \u03c0/4) ƒ(x)dx and I₂ = \u222b (from 0 to \u03c0/4) xƒ(x)dx. Then 7I₁ + 12I₂ is equal to :

• 2\u03c0
• \u03c0
• 1
• 2