JEE Main 2025 Apr 4 Shift 1 — Mathematics

Consider the sets $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$, $B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$, $C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \le 4\}$ and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:

Let $f, g : (1, \infty) \to \mathbb{R}$ be defined as $f(x) = \frac{2x+3}{5x+2}$ and $g(x) = \frac{2-3x}{1-x}$. If the range of the function $fog : [2, 4] \to \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta - \alpha}$ is equal to

Consider the equation $x^2 + 4x - n = 0$, where $n \in [20, 100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

Let $A = \{1, 6, 11, 16, \ldots\}$ and $B = \{9, 16, 23, 30, \ldots\}$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $n(A \cup B)$ is

$1 + 3 + 5^2 + 7 + 9^2 + \ldots$ upto 40 terms is equal to

In the expansion of $\left(\sqrt[3]{2} + \frac{1}{\sqrt[3]{3}}\right)^n$, $n \in \mathbb{N}$, if the ratio of $15^{\text{th}}$ term from the beginning to the $15^{\text{th}}$ term from the end is $\frac{1}{6}$, then the value of $^{n}\mathrm{C}_3$ is

For an integer $n \ge 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2n-3}$ is 16, then the distance of the point $\mathrm{P}(2n - 1, n^2 - 4n)$ from the line $x + y = 8$ is

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is

Let the three sides of a triangle are on the lines $4x - 7y + 10 = 0$, $x + y = 5$ and $7x + 4y = 15$. Then the distance of its orthocentre from the orthocentre of the tringle formed by the lines $x = 0$, $y = 0$ and $x + y = 1$ is

The length of the latus-rectum of the ellipse, whose foci are $(2, 5)$ and $(2, -3)$ and eccentricity is $\frac{4}{5}$, is

If $10\sin^4\theta + 15\cos^4\theta = 6$, then the value of $\frac{27\,\mathrm{cosec}^6\theta + 8\sec^6\theta}{16\sec^8\theta}$ is

Considering the principal values of the inverse trigonometric functions, $\sin^{-1}\left(\frac{\sqrt{3}}{2}x + \frac{1}{2}\sqrt{1 - x^2}\right)$, $-\frac{1}{2} < x < \frac{1}{\sqrt{2}}$, is equal to

Let the shortest distance between the lines $\frac{x-3}{3} = \frac{y-\alpha}{-1} = \frac{z-3}{1}$ and $\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-\beta}{4}$ be $3\sqrt{30}$. Then the positive value of $5\alpha + \beta$ is

Let $A$ and $B$ be two distinct points on the line $L : \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2\sqrt{17}$ from the foot of perpendicular drawn from the point $(1, 2, 3)$ on the line $L$. If $O$ is the origin, then $\overrightarrow{OA} \cdot \overrightarrow{OB}$ is equal to

Consider two vectors $\vec{u} = 3\hat{i} - \hat{j}$ and $\vec{v} = 2\hat{i} + \hat{j} - \lambda\hat{k}$, $\lambda > 0$. The angle between them is given by $\cos^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\vec{v} = \vec{v_1} + \vec{v_2}$, where $\vec{v_1}$ is parallel to $\vec{u}$ and $\vec{v_2}$ is perpendicular to $\vec{u}$. Then the value $|\vec{v_1}|^2 + |\vec{v_2}|^2$ is equal to

If $\lim\limits_{x \to 1^{+}} \frac{(x-1)(6 + \lambda\cos(x-1)) + \mu\sin(1-x)}{(x-1)^3} = -1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda + \mu$ is equal to

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying $f(0) = 1$ and $f(2x) - f(x) = x$ for all $x \in \mathbb{R}$. If $\lim\limits_{n \to \infty}\left\{f(x) - f\left(\frac{x}{2^n}\right)\right\} = G(x)$, then $\sum\limits_{r=1}^{10} G(r^2)$ is equal to

The value of $\int\limits_{-1}^{1} \frac{\left(1 + \sqrt{|x| - x}\right)e^x + \left(\sqrt{|x| - x}\right)e^{-x}}{e^x + e^{-x}}\, dx$ is equal to

Let $f : [0, \infty) \to \mathbb{R}$ be a differentiable function such that $f(x) = 1 - 2x + \int\limits_{0}^{x} e^{x-t} f(t)\, dt$ for all $x \in [0, \infty)$. Then the area of the region bounded by $y = f(x)$ and the coordinate axes is

Let $\mathrm{A} = \{z \in \mathbb{C} : |z - 2 - i| = 3\}$, $\mathrm{B} = \{z \in \mathbb{C} : \mathrm{Re}(z - iz) = 2\}$ and $\mathrm{S} = \mathrm{A} \cap \mathrm{B}$. Then $\sum\limits_{z \in S} |z|^2$ is equal to ________

Let $\mathrm{A} = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$, $\mathrm{A}^2 = \mathrm{A}^{\mathrm{T}}$, then the sum of the diagonal elements of the matrix $(\mathrm{A} + \mathrm{I})^3 + (\mathrm{A} - \mathrm{I})^3 - 6\mathrm{A}$ is equal to ________.

Let $C$ be the circle $x^2 + (y-1)^2 = 2$, $E_1$ and $E_2$ be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line $x + y = 3$ touch the curves $C$, $E_1$ and $E_2$ at $P(x_1, y_1)$, $Q(x_2, y_2)$ and $R(x_3, y_3)$ respectively. Given that $P$ is the mid point of the line segment $QR$ and $PQ = \frac{2\sqrt{2}}{3}$, the value of $9(x_1 y_1 + x_2 y_2 + x_3 y_3)$ is equal to ________

Let $m$ and $n$ be the number of points at which the function $f(x) = \max\{x, x^3, x^5, \ldots, x^{21}\}$, $x \in \mathbb{R}$, is not differentiable and not continuous, respectively. Then $m + n$ is equal to ________

If the area of the region $\{(x, y) : |x - 5| \le y \le 4\sqrt{x}\}$ is $A$, then $3A$ is equal to ________