JEE Main2025Apr 2, Shift 1

JEE Main 2025 — Apr 2, Shift 1

75 questions with step-by-step solutions. All free.

Mathematics

Q1The largest $n \in N$ such that $3^{n}$ divides $50!$ is:
Q2 Let one focus of the hyperbola $H : a ^{2}$
Q3The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to
Q4 Let $f : R \to R$ be a twice differentiable function such that $(sin x\dots$
Q5Let $A = [α - 1 6 β]$ , $α > 0$ , such that $det (A) = 0$ and $α + β = 1$ . If $I$ denotes $2 \times 2$ identity matrix, then the matrix $(I + A)^{8}$ is:
Q6The term independent of $x$ in the expansion of x2/3+1−x1/3(x+1)−x−x1/2(x+1))10,x>1 is:
Q7If $θ \in [- 2 π, 2 π]$ , then the number of solutions of $22 cos^{2} θ + (2 - 6) cos θ - 3 = 0$ , is equal to:
Q8Let $a_{1}, a_{2}, a_{3}, \dots$ be in an A.P. such that $\sum_{k = 1}^{12} a_{2 k - 1} = - \frac{72}{5} a_{1}$ , $a_{1} \neq = 0$ . If $\sum_{k = 1}^{n} a_{k} = 0$ , then $n$ is:
Q9 If the function $f (x) = 2 x^{3} - \dots$
Q10Let $z$ be a complex number such that $∣ z ∣ = 1$ . If $\frac{2 + k ^{2} z}{k + z ˉ} = k z$ , $k \in R$ , then the maximum distance of $k + i k^{2}$ from the circle $∣ z - (1 + 2 i) ∣ = 1$ is:
Q11If $a$ is nonzero vector such that its projections on the vectors $2 \hat{i} - \hat{j} + 2 \hat{k}$ , $\hat{i} + 2 \hat{j} - 2 \hat{k}$ and $\hat{k}$ are equal, then a unit vector along $a$ is:
Q12Let $A$ be the set of all functions $f : Z \to Z$ and $R$ be a relation on $A$ such that $R = {(f, g) : f (0) = g (1) and f (1) = g (0)}$ . Then $R$ is:
Q13For $α, β, γ, \in R$ , if $x \to 0 lim \frac{x ^{2} sin ( α x ) + ( γ - 1 ) e ^{x^{2}}}{sin 2 x - β x} = 3$ , then $β + γ - α$ is equal to:
Q14If the system of linear equations $3 x + y + β z = 3$ $2 x + α y - z = - 3$ $x + 2 y + z = 4$ has infinitely many solutions, then the value of $22 β - 9 α$ is :
Q15Let $P_{n} = α^{n} + β^{n}$ , $n \in N$ . If $P_{10} = 123$ , $P_{9} = 76$ , $P_{8} = 47$ and $P_{1} = 1$ , then the quadratic equation having roots $\frac{1}{α}$ and $\frac{1}{β}$ is :
Q16If $S$ and $S^{'}$ are the foci of the ellipse $\frac{x ^{2}}{18} + \frac{y ^{2}}{9} = 1$ and $P$ be a point on the ellipse, then $min (S P \cdot S^{'} P) + max (S P \cdot S^{'} P)$ is equal to :
Q17 Let the vertices $Q$ and $R$ of the triangle $P QR$ lie on the line $\dots$
Q18 Let $A B C D$ be a tetrahedron such that the edges $A B$ , $A C$ and $\dots$
Q19 Let $a \in R$ and $A$ be a matrix of order $3 \times \dots$
Q20 Let the focal chord $P Q$ of the parabola $y^{2} = \dots$
Q21Let $[\cdot]$ denote the greatest integer function. If $\int_{0}^{e^{3}} ⌊ \frac{1}{e ^{x - 1}} ⌋ d x = α - lo g_{e} 2$ , then $α^{3}$ is equal to _____.
Q22Let $f : R \to R$ be a thrice differentiable odd function satisfying $f^{'} (x) \geq 0$ , $f^{''} (x) = f (x)$ , $f (0) = 0$ , $f^{'} (0) = 3$ . Then $9 f (lo g_{e} 3)$ is equal to _____.
Q23If the area of the region ${(x, y) : ∣4 - x^{2} ∣ \leq y \leq x^{2}, y \leq 4, x \geq 0}$ is $(\frac{80 2}{α} - β)$ , $α, β \in N$ , then $α + β$ is equal to _____.
Q24 Three distinct numbers are selected randomly from the set ${1, 2, 3, \dots, 40}$ . If the probability that the selected numbers are in an increasing G.P. with integer common ratio is $\dots$
Q25 The absolute difference between the squares of the radii of the two circles passing through the point $(- 9, 4)$ and touching the lines $x + y =\dots$

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