JEE Main 2025 Apr 3 Shift 1 — Mathematics

Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$. Let R be a relation on A defined by $x\mathrm{R}y$ if and only if $0 \le x^2 + 2y \le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l + m$ is equal to

If the domain of the function $f(x) = \log_e\left(\frac{2x-3}{5+4x}\right) + \sin^{-1}\left(\frac{4+3x}{2-x}\right)$ is $[\alpha, \beta)$, then $\alpha^2 + 4\beta$ is equal to

Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then $\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}}$ is equal to

Let $z \in \mathbb{C}$ be such that $\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$. Then the sum of all possible values of $z^2$ is

Let $A$ be a matrix of order $3 \times 3$ and $|A| = 5$. If $\left|2\,adj\left(3A\,adj\left(2A\right)\right)\right| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha + \beta + \gamma$ is equal to

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$, then $24\,(a_1 + a_2 + a_3)$ is equal to

The sum $1 + 3 + 11 + 25 + 45 + 71 + \ldots$ upto 20 terms, is equal to

If $\sum_{r=1}^{9}\left(\frac{r+3}{2^r}\right) \cdot {}^{9}C_r = \alpha\left(\frac{3}{2}\right)^{9} - \beta$, $\alpha, \beta \in \mathbb{N}$, then $(\alpha + \beta)^2$ is equal to

The sum of all rational terms in the expansion of $\left(2 + \sqrt{3}\right)^{8}$ is

The number of solutions of the equation $2x + 3\tan x = \pi$, $x \in [-2\pi, 2\pi] - \left\{\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}\right\}$ is:

The radius of the smallest circle which touches the parabolas $y = x^2 + 2$ and $x = y^2 + 2$ is

A line passing through the point $P\left(\sqrt{5}, \sqrt{5}\right)$ intersects the ellipse $\frac{x^2}{36} + \frac{y^2}{25} = 1$ at $A$ and $B$ such that $(PA) \cdot (PB)$ is maximum. Then $5(PA^2 + PB^2)$ is equal to :

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $\mathrm{L}_1 : 2x + y + 6 = 0$ and $\mathrm{L}_2 : 4x + 2y - p = 0$, $p > 0$, at the points A and B, respectively. If $\mathrm{AB} = \frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point A on the line $\mathrm{L}_2$ is M, then $\frac{\mathrm{AM}}{\mathrm{BM}}$ is equal to

Line $\mathrm{L}_1$ passes through the point $(1, 2, 3)$ and is parallel to $z$-axis. Line $\mathrm{L}_2$ passes through the point $(\lambda, 5, 6)$ and is parallel to $y$-axis. Let for $\lambda = \lambda_1, \lambda_2$, $\lambda_2 < \lambda_1$, the shortest distance between the two lines be 3. Then the square of the distance of the point $(\lambda_1, \lambda_2, 7)$ from the line $\mathrm{L}_1$ is

Let a line passing through the point $(4, 1, 0)$ intersect the line $\mathrm{L}_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ at the point $A(\alpha, \beta, \gamma)$ and the line $\mathrm{L}_2 : x - 6 = y = -z + 4$ at the point $B(a, b, c)$. Then $\begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix}$ is equal to

If $y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix}$, $x \in \mathbb{R}$, then $\frac{d^2y}{dx^2} + y$ is equal to

Let $f(x) = \begin{cases} (1+ax)^{1/x} & , \ x < 0 \\ 1+b & , \ x = 0 \\ \dfrac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2} & , \ x > 0 \end{cases}$

be continuous at $x = 0$. Then $e^{a}bc$ is equal to:

Let $f(x) = \int x^3 \sqrt{3 - x^2}\, dx$. If $5f\left(\sqrt{2}\right) = -4$, then $f(1)$ is equal to

Let the domain of the function $f(x) = \log_2 \log_4 \log_6(3 + 4x - x^2)$ be $(a, b)$. If $\int_0^{b-a}\left[x^2\right] dx = p - \sqrt{q} - \sqrt{r}$, $p, q, r \in \mathbb{N}$, $\gcd(p, q, r) = 1$, where $[\cdot]$ is the greatest integer function, then $p + q + r$ is equal to

Let $g$ be a differentiable function such that $\int_0^x g(t)\, dt = x - \int_0^x t\, g(t)\, dt$, $x \ge 0$ and let $y = y(x)$ satisfy the differential equation $\frac{dy}{dx} - y\tan x = 2(x+1)\sec x \ g(x)$, $x \in \left[0, \frac{\pi}{2}\right)$. If $y(0) = 0$, then $y\left(\frac{\pi}{3}\right)$ is equal to

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^n$; $m, n \in \{1, 2, 3, \ldots, 9\}$, then $m + n$ is equal to __________

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by $\mathrm{W_n}$. Let the probability $\mathrm{P(W_n)}$ of choosing the word $\mathrm{W_n}$ satisfy $\mathrm{P(W_n)} = 2\mathrm{P(W_{n-1})}$, $n > 1$.

If $\mathrm{P(CDBEA)} = \frac{2^{\alpha}}{2^{\beta} - 1}$, $\alpha, \beta \in \mathbb{N}$, then $\alpha + \beta$ is equal to : __________

Let the product of the focal distances of the point $\mathrm{P}\left(4, 2\sqrt{3}\right)$ on the hyperbola $\mathrm{H}: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be 32. Let the length of the conjugate axis of H be $p$ and the length of its latus rectum be $q$. Then $p^2 + q^2$ is equal to __________

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}$, $\vec{c} = \lambda\hat{j} + \mu\hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a} \times \hat{d} = \vec{b} \times \hat{d}$ and $\vec{c} \cdot \hat{d} = 1$. If $\vec{c}$ is perpendicular to $\vec{a}$, then $\left|3\lambda\hat{d} + \mu\vec{c}\right|^2$ is equal to __________

The area of the region bounded by the curve $y = \max\{|x|,\ x|x-2|\}$, the $x$-axis and the lines $x = -2$ and $x = 4$ is equal to __________