JEE Main 2025 Jan 24 Shift 2 — Mathematics

The function $f:(-\infty, \infty) \to (-\infty, 1)$, defined by $f(x) = \dfrac{2^x - 2^{-x}}{2^x + 2^{-x}}$ is :

Let $A = \left\{x \in (0, \pi) - \left\{\dfrac{\pi}{2}\right\} : \log_{(2/\pi)}|\sin x| + \log_{(2/\pi)}|\cos x| = 2\right\}$ and

$B = \left\{x \geq 0 : \sqrt{x}\left(\sqrt{x} - 4\right) - 3\left|\sqrt{x} - 2\right| + 6 = 0\right\}.$ Then $\mathrm{n(A \cup B)}$ is equal to :

The number of real solution(s) of the equation $x^2 + 3x + 2 = \min\{|x - 3|, |x + 2|\}$ is :

For some a, b, let $f(x) = \begin{vmatrix} a + \dfrac{\sin x}{x} & 1 & b \\ a & 1 + \dfrac{\sin x}{x} & b \\ a & 1 & b + \dfrac{\sin x}{x} \end{vmatrix}$, $x \neq 0$, $\lim\limits_{x \to 0} f(x) = \lambda + \mu a + \nu b$. Then $(\lambda + \mu + \nu)^2$ is equal to :

If the system of equations

$x + 2y - 3z = 2$

$2x + \lambda y + 5z = 5$

$14x + 3y + \mu z = 33$

has infinitely many solutions, then $\lambda + \mu$ is equal to :

If $7 = 5 + \dfrac{1}{7}(5+\alpha) + \dfrac{1}{7^2}(5 + 2\alpha) + \dfrac{1}{7^3}(5 + 3\alpha) + \ldots\ldots \infty$, then the value of $\alpha$ is :

In an arithmetic progression, if $\mathrm{S_{40}} = 1030$ and $\mathrm{S_{12}} = 57$, then $\mathrm{S_{30}} - \mathrm{S_{10}}$ is equal to :

Let $\mathrm{A} = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is :

Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1+x)^{2n-1}$. If $2\mathrm{A} = 5\mathrm{B}$, then $n$ is equal to :

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to :

Let the points $\left(\dfrac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to :

If the equation of the parabola with vertex $\mathrm{V}\left(\dfrac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to :

The equation of the chord, of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$, whose mid-point is $(3, 1)$ is :

If $\alpha > \beta > \gamma > 0$, then the expression

$\cot^{-1}\left\{\beta + \dfrac{(1 + \beta^2)}{(\alpha - \beta)}\right\} + \cot^{-1}\left\{\gamma + \dfrac{(1 + \gamma^2)}{(\beta - \gamma)}\right\} + \cot^{-1}\left\{\alpha + \dfrac{(1 + \alpha^2)}{(\gamma - \alpha)}\right\}$ is equal to :

Let $\vec{a} = 3\hat{i} - \hat{j} + 2\hat{k}$, $\vec{b} = \vec{a} \times \left(\hat{i} - 2\hat{k}\right)$ and $\vec{c} = \vec{b} \times \hat{k}$. Then the projection of $\vec{c} - 2\hat{j}$ on $\vec{a}$ is :

Let the position vectors of three vertices of a triangle be $4\vec{p} + \vec{q} - 3\vec{r}$, $-5\vec{p} + \vec{q} + 2\vec{r}$ and $2\vec{p} - \vec{q} + 2\vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\dfrac{\vec{p} + \vec{q} + \vec{r}}{4}$ and $\alpha\vec{p} + \beta\vec{q} + \gamma\vec{r}$ respectively, then $\alpha + 2\beta + 5\gamma$ is equal to :

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $\mathrm{m + n}$ is equal to :

Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2\log_e(x - 2) - x^2 + \mathrm{a}x + 1$ is strictly increasing and $(\mathrm{b}, \mathrm{c})$ be the largest open interval, in which the function $\mathrm{g}(x) = (x-1)^3(x + 2 - \mathrm{a})^2$ is strictly decreasing. Then $100(\mathrm{a} + \mathrm{b} - \mathrm{c})$ is equal to :

The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is :

Let $f : (0, \infty) \to \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2xf(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to :

Number of functions $f : \{1, 2, \ldots, 100\} \to \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to __________.

Let P be the image of the point $\mathrm{Q}(7, -2, 5)$ in the line $\mathrm{L} : \dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z}{4}$ and $\mathrm{R}(5, \mathrm{p}, \mathrm{q})$ be a point on L. Then the square of the area of $\Delta\mathrm{PQR}$ is __________.

If $\displaystyle\int \dfrac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}}\, \mathrm{d}x = x\sqrt{x^2 + x + 1} + \alpha\sqrt{x^2 + x + 1} + \beta \log_e\left|x + \dfrac{1}{2} + \sqrt{x^2 + x + 1}\right| + \mathrm{C}$,

where C is the constant of integration, then $\alpha + 2\beta$ is equal to __________.

Let $y = y(x)$ be the solution of the differential equation $2\cos x \dfrac{\mathrm{d}y}{\mathrm{d}x} = \sin 2x - 4y\sin x$, $x \in \left(0, \dfrac{\pi}{2}\right)$. If $y\left(\dfrac{\pi}{3}\right) = 0$, then $y'\left(\dfrac{\pi}{4}\right) + y\left(\dfrac{\pi}{4}\right)$ is equal to __________.

Let $\mathrm{H_1} : \dfrac{x^2}{\mathrm{a}^2} - \dfrac{y^2}{\mathrm{b}^2} = 1$ and $\mathrm{H_2} : -\dfrac{x^2}{\mathrm{A}^2} + \dfrac{y^2}{\mathrm{B}^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their ecentricities be $\mathrm{e_1} = \sqrt{\dfrac{5}{2}}$ and $\mathrm{e_2}$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25\mathrm{e_2^2}$ is equal to __________.