JEE MAIN 2025 (Online) 6th April Morning Shift

Question 1:

Let [t] denote the greatest integer function. If the domain of the function f(x) = sin⁻¹((x + [x])/3) is [α, β), then α² + β² is equal to:

Question 2:

Let one root of the quadratic equation in x: (k² – 15k + 27)x² + 9(k – 1)x + 18 = 0 be twice the other. Then the length of the latus rectum of the parabola y² = 6kx is equal to:

Question 3:

Let e₁ and e₂ be two distinct roots of the equation x² – ax + 2 = 0. Let the sets {a ∈ R : e₁ and e₂ are the eccentricities of hyperbolas} = (α, β), and {a ∈ R : e₁ and e₂ are the eccentricities of an ellipse and a hyperbola, respectively} = (γ, ∞]. Then α² + β² + γ² is equal to:

Question 4:

Let the set of all values of k ∈ R such that the equation z(z̄ + 2 + i) + k(2 + 3i) = 0, z ∈ C, has at least one solution, be the interval [α, β]. Then 9(α + β) is equal to:

Question 5:

The value of 1³ – 2³ + 3³ – … + 15³ is:

Question 6:

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

Question 7:

The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

Question 8:

If the coefficients of the middle terms in the binomial expansions of (1 + αx)²⁶ and (1 – αx)²⁸, α ≠ 0, are equal, then the value of α is:

Question 9:

A data consists of 20 observations x₁, x₂, …, x₂₀. If ∑ᵢ₌₁²⁰ (xᵢ + 5)² = 2500 and ∑ᵢ₌₁²⁰ (xᵢ – 5)² = 100, then the ratio of mean to standard deviation of this data is:

Question 10:

A bag contains (N + 1) coins – N fair coins, and one coin with ‘Head’ on both sides. A coin is selected at random and tossed. If the probability of getting ‘Head’ is 9/16, then N is equal to:

Question 11:

If the eccentricity e of the hyperbola x²/a² – y²/b² = 1 passing through (6, 4√3) satisfies 15(e² + 1) = 34e, then the length of the latus rectum of the hyperbola x²/b² – y²/(2(a² + 1)) = 1 is:

Question 12:

Let chord PQ of length 3√13 of the parabola y² = 12x be such that the ordinates of points P and Q are in the ratio 1:2. If the chord PQ subtends an angle α at the focus of the parabola, then sin α is equal to:

Question 13:

Let 0 < α < 1, β = 1/(3α) and tan⁻¹(1 – α) + tan⁻¹(1 – β) = π/4. Then 6(α + β) is equal to:

Question 14:

Let S = {θ ∈ (-2π, 2π) : cos θ + 1 = √3 sin θ}. Then ∑_{θ∈S} θ is equal to:

Question 15:

Let the image of the point P(1, 6, a) in the line L : x/1 = (y-1)/2 = (z-a+1)/b, b > 0 be (a/3, 0, a+c). If S(α, β, γ), α > 0 is the point on L such that the distance of S from the foot of perpendicular from the point P on L is 2√14, then α + β + γ is equal to:

Question 16:

Let a line L be perpendicular to both the lines L₁ : (x+1)/3 = (y+3)/5 = (z+5)/7 and L₂ : (x-2)/1 = (y-4)/4 = (z-6)/7. If θ is the acute angle between the lines L and L₃ : (x-8/7)/2 = (y-4/7)/1 = z/2, then tan θ is equal to:

Question 17:

The value of lim(x→0) (x² sin²x / (x² – sin²x)) is:

Question 18:

The value of the integral ∫₍₋π/₄₎^(π/4) (32cos⁴x / (1 + e^(sin x))) dx is:

Question 19:

The area of the region {(x, y) : 0 ≤ y ≤ 6 – x, y² ≥ 4x – 3, x ≥ 0} is:

Question 20:

Let e be the base of natural logarithm and let f : {1, 2, 3, 4} → {1, e, e², e³} and g : {1, e, e², e³} → {1, 1/2, 1/3, 1/4} be two bijective functions such that f is strictly decreasing and g is strictly increasing. If φ(x) = [f⁻¹{g⁻¹(1/2)}]ˣ, then the area of the region R = {(x, y) : x² ≤ y ≤ φ(x), 0 ≤ x ≤ 1} is:

Question 21:

Let A = [-1 1 -1; 1 0 1; 0 0 1], satisfy A² + α(adj(adj(A))) + β(adj(A)(adj(adj(A)))) = [2 -2 2; -2 0 -1; 0 0 -1] for some α, β ∈ R. Then (α – β)² is equal to

Question 22:

Let the centre of the circle x² + y² + 2gx + 2fy + 25 = 0 be in the first quadrant and lie on the line 2x – y = 4. Let the area of an equilateral triangle inscribed in the circle be 27√3. Then the square of the length of the chord of the circle on the line x = 1 is

Question 23:

If a⃗ = î + ĵ + k̂, b⃗ = ĵ – k̂ and c⃗ be three vectors such that a⃗ × c⃗ = b⃗ and a⃗ · c⃗ = 3, then c⃗ · (a⃗ – 2b⃗) is equal to

Question 24:

For the functions f(θ) = αtan²θ + βcot²θ and g(θ) = αsin²θ + βcos²θ, α > β > 0, let min(0<θ<π/2) f(θ) = max(0<θ<π) g(θ). If the first term of a G.P. is (α/2β), its common ratio is (2β/α) and the sum of its first 10 terms is m/n where gcd(m, n) = 1, then m + n is equal to

Question 25:

Let y = y(x) be the solution of the differential equation (x² – x√(x² – 1))dy + (y(x – √(x² – 1)) – x)dx = 0, x ≥ 1. If y(1) = 1, then the greatest integer less than y(√5) is