JEE Main 2025 Apr 3 Shift 2 — Mathematics

Let $A = \{-2, -1, 0, 1, 2, 3\}$. Let R be a relation on $A$ defined by $x\mathrm{R}y$ if and only if $y = \max\{x, 1\}$. Let $l$ be the number of elements in R. Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l + m + n$ is equal to

If the domain of the function $f(x) = \log_7 (1 - \log_4 (x^2 - 9x + 18))$ is $(\alpha, \beta) \cup (\gamma, \delta)$, then $\alpha + \beta + \gamma + \delta$ is equal to

Let $f$ be a function such that $f(x) + 3f\left(\dfrac{24}{x}\right) = 4x,\ x \neq 0$. Then $f(3) + f(8)$ is equal to

Let the equation $x(x + 2)(12 - k) = 2$ have equal roots. Then the distance of the point $\left(k, \dfrac{k}{2}\right)$ from the line $3x + 4y + 5 = 0$ is

If $z_1, z_2, z_3 \in \mathbb{C}$ are the vertices of an equilateral triangle, whose centroid is $z_0$, then $\displaystyle\sum_{k=1}^{3} (z_k - z_0)^2$ is equal to

The sum $1 + \dfrac{1+3}{2!} + \dfrac{1+3+5}{3!} + \dfrac{1+3+5+7}{4!} + \ldots$ upto $\infty$ terms, is equal to

Line $\mathrm{L}_1$ of slope 2 and line $\mathrm{L}_2$ of slope $\dfrac{1}{2}$ intersect at the origin O. In the first quadrant, $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{12}$ are 12 points on line $\mathrm{L}_1$ and $\mathrm{Q}_1, \mathrm{Q}_2, \ldots, \mathrm{Q}_9$ are 9 points on line $\mathrm{L}_2$. Then the total number of triangles, that can be formed having vertices at three of the 22 points O, $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{12}$, $\mathrm{Q}_1, \mathrm{Q}_2, \ldots, \mathrm{Q}_9$, is:

If the probability that the random variable $X$ takes the value $x$ is given by $P(X = x) = k(x + 1)3^{-x},\ x = 0, 1, 2, 3\ldots$, where $k$ is a constant, then $P(X \geq 3)$ is equal to

Let the Mean and Variance of five observations $x_1 = 1,\ x_2 = 3,\ x_3 = a,\ x_4 = 7$ and $x_5 = \mathrm{b}$, $a > \mathrm{b}$, be 5 and 10 respectively. Then the Variance of the observations $n + x_n,\ n = 1, 2, \ldots, 5$ is

Consider the lines $x(3\lambda + 1) + y(7\lambda + 2) = 17\lambda + 5$, $\lambda$ being a parameter, all passing through a point P. One of these lines (say $L$) is farthest from the origin. If the distance of $L$ from the point $(3, 6)$ is $d$, then the value of $d^2$ is

If the four distinct points $(4, 6)$, $(-1, 5)$, $(0, 0)$ and $(k, 3k)$ lie on a circle of radius $r$, then $10k + r^2$ is equal to

Let $C$ be the circle of minimum area enclosing the ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with eccentricity $\dfrac{1}{2}$ and foci $(\pm 2, 0)$. Let $PQR$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $QR$ of length $2a$ is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle $PQR$ is :

The number of solutions of the equation $\left(4 - \sqrt{3}\right)\sin x - 2\sqrt{3}\cos^2 x = -\dfrac{4}{1 + \sqrt{3}}$, $x \in \left[-2\pi, \dfrac{5\pi}{2}\right]$ is

Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$- and $z$-axes, respectively, is half of the angle that this line makes with the positive $x$-axes. Then the sum of all possible values of the angle $\beta$ is

The distance of the point $(7, 10, 11)$ from the line $\dfrac{x - 4}{1} = \dfrac{y - 4}{0} = \dfrac{z - 2}{3}$ along the line $\dfrac{x - 9}{2} = \dfrac{y - 13}{3} = \dfrac{z - 17}{6}$ is

Let $f : \mathrm{R} \to \mathrm{R}$ be a function defined by $f(x) = \big|\,|x + 2| - 2|x|\,\big|$. If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$, then $m + n$ is

The shortest distance between the curves $y^2 = 8x$ and $x^2 + y^2 + 12y + 35 = 0$ is:

The integral $\displaystyle\int_0^{\pi} \dfrac{8x\,dx}{4\cos^2 x + \sin^2 x}$ is equal to

The area of the region $\left\{(x, y) : |x - y| \leq y \leq 4\sqrt{x}\right\}$ is

Let $y = y(x)$ be the solution of the differential equation $\dfrac{dy}{dx} + 3\left(\tan^2 x\right)y + 3y = \sec^2 x$, $y(0) = \dfrac{1}{3} + e^3$. Then $y\left(\dfrac{\pi}{4}\right)$ is equal to

Let $(1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{20} x^{20}$. If $(a_1 + a_3 + a_5 + \ldots + a_{19}) - 11a_2 = 121k$, then $k$ is equal to ____________.

Let I be the identity matrix of order $3 \times 3$ and for the matrix $\mathrm{A} = \begin{bmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{bmatrix}$, $|\mathrm{A}| = -1$. Let $B$ be the inverse of the matrix $adj\left(A\, adj\left(A^2\right)\right)$. Then $|(\lambda \mathrm{B} + \mathrm{I})|$ is equal to ____________

If the equation of the hyperbola with foci $(4, 2)$ and $(8, 2)$ is $3x^2 - y^2 - \alpha x + \beta y + \gamma = 0$, then $\alpha + \beta + \gamma$ is equal to __________.

Let $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} - 3\hat{j} + 3\hat{k}$, $\vec{c} = 2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{d}$ be a vector such that $\vec{b} \times \vec{d} = \vec{c} \times \vec{d}$ and $\vec{a} \cdot \vec{d} = 4$. Then $\left|\left(\vec{a} \times \vec{d}\right)\right|^2$ is equal to ________.

If $\displaystyle\lim_{x \to 0}\left(\dfrac{\tan x}{x}\right)^{\frac{1}{x^2}} = p$, then $96\log_e p$ is equal to __________