JEE Main2025Jan 23, Shift 2

JEE Main 2025 — Jan 23, Shift 2

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

Mathematics

Q1If in the expansion of $(1 + x)^{p} (1 - x)^{q}$ , the coefficients of $x$ and $x^{2}$ are $1$ and $- 2$ , respectively, then $p^{2} + q^{2}$ is equal to :
Q2Let $A = {(x, y) \in R \times R : ∣ x + y ∣ \geq 3}$ and $B = {(x, y) \in R \times R : ∣ x ∣ + ∣ y ∣ \leq 3}$ . If $C = {(x, y) \in A \cap B : x = 0 or y = 0}$ , then $(x, y) \in C \sum ∣ x + y ∣$ is :
Q3The system of equations $x + y + z = 6,$ $x + 2 y + 5 z = 9,$ $x + 5 y + λ z = μ,$ has no solution if
Q4Let $\int x^{3} sin x d x = g (x) + C$ , where $C$ is the constant of integration. If $8 (g (\frac{π}{2}) + g^{'} (\frac{π}{2})) = α π^{3} + β π^{2} + γ$ , where $α, β, γ \in Z$ , then $α + β - γ$ equals :
Q5 A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x - y + \dots$
Q6The distance of the line $\frac{x - 2}{2} = \frac{y - 6}{3} = \frac{z - 3}{4}$ from the point $(1, 4, 0)$ along the line $\frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3}$ is :
Q7 Let the point $A$ divide the line segment joining the points $P (- 1, - 1, 2)$ and $Q (5 \dots$
Q8If the area of the region ${(x, y) : - 1 \leq x \leq 1, 0 \leq y \leq a + e^{∣ x ∣} - e^{- x}, a > 0}$ is $\frac{e ^{2} + 8 e + 1}{e}$ , then the value of $a$ is :
Q9 A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is $1$ cm, the ice-cream melts at the rate of $81$ cm $^{3}$ /min and the thicknes…
Q10 A board has $16$ squares as shown in the figure : [ADD IMAGE: 4 by 4 grid of squares (16 unit squares arranged in a 4x4 array)] Out of these $16$ squares, two squares are chosen at random. The probability that they have no side in common is : …
Q11Let $x = x (y)$ be the solution of the differential equation $y = (x - y \frac{d x}{d y}) sin (\frac{x}{y}), y > 0, x (1) = \frac{π}{2} .$ Then $cos (x (2))$ is equal to :
Q12 Let the range of the function $f (x) = 6 + 16 cos x \dots$
Q13 Let the shortest distance from $(a, 0)$ , $a > 0$ , to the parabola $\dots$
Q14 Let $X = R \times R$ . Define a relation $R$ on $\dots$
Q15The length of the chord of the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ , whose mid-point is $(1, \frac{1}{2})$ , is :
Q16Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A 010 = 001, A 413 = 010 and A 212 = 100,$ then $a_{23}$ equals :
Q17The number of complex numbers $z$ , satisfying $∣ z ∣ = 1$ and $\frac{z}{z ˉ} + \frac{z ˉ}{z} = 1$ , is :
Q18 If the square of the shortest distance between the lines $\frac{x - 2 \dots}{1}$
Q19If $I = \int_{0}^{π /2} \frac{sin ^{3/2} x}{sin ^{3/2} x + cos ^{3/2} x} d x$ , then $\int_{0}^{2 I} \frac{x sin x cos x}{sin ^{4} x + cos ^{4} x} d x$ equals :
Q20 $x \to \infty lim \frac{( 2 x ^{2} - 3 x + 5 ) ( 3 x - 1 ) ^{x /2}}{( 3 x ^{2} + 5 x + 4 ) ( 3 x + 2 ) ^{x}}$ is equal to :
Q21The number of ways, $5$ boys and $4$ girls can sit in a row so that either all the boys sit together or no two boys sit together, is ________.
Q22 Let $α, β$ be the roots of the equation $x^{2} - \dots$
Q23 The focus of the parabola $y^{2} = 4 x + \dots$
Q24The variance of the numbers $8, 21, 34, 47, \dots, 320$ , is ________.
Q25 The roots of the quadratic equation $3 x^{2} - p x +$

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