JEE MAIN 2025 (Online) 6th April Evening Shift

Question 1:

Let f : R → R be defined as f(x) = (2x² – 3x + 2)/(3x² + x + 3). Then f is:

Question 2:

Consider the quadratic equation (n² – 2n + 2)x² – 3x + (n² – 2n + 2)² = 0, n ∈ R. Let α be the minimum value of the product of its roots and β be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is 1 and the common ratio is α/β is:

Question 3:

Let S = {z ∈ C : z² + √6·iz – 3 = 0}. Then ∑_{z∈S} z⁸ is equal to:

Question 4:

The sum of all possible values of θ ∈ [0, 2π], for which the system of equations: x·cos 3θ – 8y – 12z = 0, x·cos 2θ + 3y + 3z = 0, x + y + 3z = 0 has a non-trivial solution, is equal to:

Question 5:

Let A = [1 0 0; 3 1 0; 9 3 1] and B = [bᵢⱼ], 1 ≤ i, j ≤ 3. If B = A⁹⁹ – I, then the value of (b₃₁ – b₂₁)/b₃₂ is:

Question 6:

The sum 1 + (1/2)(1² + 2²) + (1/3)(1² + 2² + 3²) + … upto 10 terms is equal to:

Question 7:

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to:

Question 8:

Let the mean and the variance of seven observations 2, 4, α, 8, β, 12, 14, α < β, be 8 and 16 respectively. Then the quadratic equation whose roots are 3α + 2 and 2β + 1 is:

Question 9:

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is:

Question 10:

Let C be a circle having centre in the first quadrant and touching the x-axis at a distance of 3 units from the origin. If the circle C has an intercept of length 6√3 on y-axis, then the length of the chord of the circle C on the line x – y = 3 is:

Question 11:

The eccentricity of an ellipse E with centre at the origin O is √3/2 and its directrices are x = ±4√6/3. Let H : x²/a² – y²/b² = 1 be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of E. Then the distance between the foci of H is:

Question 12:

Let x = 9 be a directrix of an ellipse E, whose centre is at the origin and eccentricity is 1/3. Let P(a, 0), α > 0, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is:

Question 13:

If sin(tan⁻¹(x√2)) = cot(sin⁻¹√(1 – x²)), x ∈ (0, 1), then the value of x is:

Question 14:

The shortest distance between the lines (x-4)/1 = (y-3)/2 = (z-2)/(-3) and (x+2)/2 = (y-6)/4 = (z-5)/(-5) is:

Question 15:

Let a⃗ = 2î + 3ĵ + 3k̂ and b⃗ = 6î + 3ĵ + 3k̂. Then the square of the area of the triangle with adjacent sides determined by the vectors (2a⃗ + 3b⃗) and (a⃗ – b⃗) is:

Question 16:

Let lim(x→2) [(tan(x-2))(rx² + (p-2)x – 2p)]/(x-2)² = 5 for some r, p ∈ R. If the set of all possible values of q, such that the roots of the equation rx² – px + q = 0 lie in (0, 2), be the interval (α, β], then 4(α + β) equals:

Question 17:

Let A = [1 3 -1; 2 1 α; 0 1 -1] be a singular matrix. Let f(x) = ∫₀ˣ (t² + 2t + 3)dt, x ∈ [1, α]. If M and m are respectively the maximum and the minimum values of f in [1, α], then 3(M – m) is equal to:

Question 18:

Let f : R → R be such that f(xy) = f(x)f(y), for all x, y ∈ R and f(0) ≠ 0. Let g : [1, ∞) → R be a differentiable function such that x²g(x) = ∫₁ˣ (t²f(t) – tg(t))dt. Then g(2) is equal to:

Question 19:

The area of the region {(x, y) : x² – 8x ≤ y ≤ -x} is:

Question 20:

The value of the integral ∫₋₁¹ (x³ + |x| + 1)/(x² + 2|x| + 1) dx is equal to:

Question 21:

Let R = {(x, y) ∈ N × N : logₑ(x + y) ≤ 2}. Then the minimum number of elements, required to be added in R to make it a transitive relation, is

Question 22:

If (1 – x³)¹⁰ = ∑ᵣ₌₀¹⁰ aᵣxʳ(1 – x)³⁰⁻²ʳ, then 9a₉/a₁₀ is equal to

Question 23:

Let the line x – y = 4 intersect the circle C : (x – 4)² + (y + 3)² = 9 at the points Q and R. If P(α, β) is a point on C such that PQ = PR, then (6α + 8β)² is equal to

Question 24:

Let the image of the point P(0, -5, 0) in the line (x-1)/2 = y/1 = (z+1)/(-2) be the point R and the image of the point Q(0, -1/2, 0) in the line (x-1)/(-1) = (y+9)/4 = (z+1)/1 be the point S. Then the square of the area of the parallelogram PQRS is

Question 25:

Let f(x) = {x³ + 8; x < 0, x² – 4; x ≥ 0} and g(x) = {(x – 8)^(1/3); x < 0, (x + 4)^(1/2); x ≥ 0}. Then the number of points, where the function g ∘ f is discontinuous, is