JEE Main 2025 Apr 7 Shift 2 — Mathematics

Let $A = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : |\alpha - 1| \leq 4 \text{ and } |\beta - 5| \leq 6\}$ and $B = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \leq 144\}$. Then

If the range of the function $f(x) = \dfrac{5 - x}{x^2 - 3x + 2}$, $x \neq 1, 2$, is $(-\infty, \alpha] \cup [\beta, \infty)$, then $\alpha^2 + \beta^2$ is equal to :

The number of real roots of the equation $x|x - 2| + 3|x - 3| + 1 = 0$ is :

If the locus of $z \in \mathbf{C}$, such that $\mathrm{Re}\left(\dfrac{z - 1}{2z + i}\right) + \mathrm{Re}\left(\dfrac{\bar{z} - 1}{2\bar{z} - i}\right) = 2$, is a circle of radius $r$ and center $(a, b)$, then $\dfrac{15ab}{r^2}$ is equal to :

Let the system of equations $$x + 5y - z = 1$$ $$4x + 3y - 3z = 7$$ $$24x + y + \lambda z = \mu$$ $\lambda, \mu \in \mathbf{R}$, have infinitely many solutions. Then the number of the solutions of this system, if $x, y, z$ are integers and satisfy $7 \leq x + y + z \leq 77$, is :

Let $a_n$ be the $n^{\text{th}}$ term of an A.P. If $S_n = a_1 + a_2 + a_3 + \ldots + a_n = 700$, $a_6 = 7$ and $S_7 = 7$, then $a_n$ is equal to :

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :

Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular polygon $P$ of $n$ sides and $q$ be the number of all quadrilaterals that can be formed by joining the vertices of $P$. If $p + q = 126$, then the eccentricity of the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{n} = 1$ is :

Let a random variable $X$ take values $0, 1, 2, 3$ with $P(X = 0) = P(X = 1) = p$, $P(X = 2) = P(X = 3)$ and $E(X^2) = 2E(X)$. Then the value of $8p - 1$ is :

A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $\dfrac{m}{n}$, $\gcd(m, n) = 1$, then $n^2 - m^2$ is equal to :

If the orthocenter of the triangle formed by the lines $y = x + 1$, $y = 4x - 8$ and $y = mx + c$ is at $(3, -1)$, then $m - c$ is :

Let the length of a latus rectum of an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ be 10. If its eccentricity is the minimum value of the function $f(t) = t^2 + t + \dfrac{11}{12}$, $t \in \mathbf{R}$, then $a^2 + b^2$ is equal to :

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\dfrac{x^2}{b^2} + \dfrac{y^2}{25} = 1$ and the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{b^2} = 1$, respectively. If $b < 5$ and $e_1 e_2 = 1$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

The number of solutions of the equation $\cos 2\theta \cos \dfrac{\theta}{2} + \cos \dfrac{5\theta}{2} = 2 \cos^3 \dfrac{5\theta}{2}$ in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ is :

Let $\vec{a}$ and $\vec{b}$ be the vectors of the same magnitude such that $\dfrac{|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|}{|\vec{a} + \vec{b}| - |\vec{a} - \vec{b}|} = \sqrt{2} + 1$. Then $\dfrac{|\vec{a} + \vec{b}|^2}{|\vec{a}|^2}$ is :

If the equation of the line passing through the point $\left(0, -\dfrac{1}{2}, 0\right)$ and perpendicular to the lines $\vec{r} = \lambda\left(\hat{i} + a\hat{j} + b\hat{k}\right)$ and $\vec{r} = \left(\hat{i} - \hat{j} - 6\hat{k}\right) + \mu\left(-b\hat{i} + a\hat{j} + 5\hat{k}\right)$ is $\dfrac{x - 1}{-2} = \dfrac{y + 4}{d} = \dfrac{z - c}{-4}$, then $a + b + c + d$ is equal to :

Consider the lines $L_1 : x - 1 = y - 2 = z$ and $L_2 : x - 2 = y = z - 1$. Let the feet of the perpendiculars from the point $P(5, 1, -3)$ on the lines $L_1$ and $L_2$ be $Q$ and $R$ respectively. If the area of the triangle $PQR$ is $A$, then $4A^2$ is equal to :

Let $f : \mathbf{R} \to \mathbf{R}$ be a polynomial function of degree four having extreme values at $x = 4$ and $x = 5$. If $\lim\limits_{x \to 0} \dfrac{f(x)}{x^2} = 5$, then $f(2)$ is equal to :

Let $y = y(x)$ be the solution of the differential equation $(x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x$, $y(0) = 1$. Then $\int\limits_{-3}^{3} y(x)\, \mathrm{d}x$ is :

If the area of the region $\{(x, y) : 1 + x^2 \leq y \leq \min\{x + 7, 11 - 3x\}\}$ is $A$, then $3A$ is equal to :

The sum of the series $2 \times 1 \times {}^{20}C_4 - 3 \times 2 \times {}^{20}C_5 + 4 \times 3 \times {}^{20}C_6 - 5 \times 4 \times {}^{20}C_7 + \cdots + 18 \times 17 \times {}^{20}C_{20}$, is equal to _________.

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2a$ and $2b$, respectively, and one focus and the corresponding directrix of this hyperbola be $(-5, 0)$ and $5x + 9 = 0$, respectively. If the product of the focal distances of a point $\left(\alpha, 2\sqrt{5}\right)$ on the hyperbola is $p$, then $4p$ is equal to _________.

For $t > -1$, let $\alpha_t$ and $\beta_t$ be the roots of the equation $\left(\left(t + 2\right)^{1/7} - 1\right)x^2 + \left(\left(t + 2\right)^{1/6} - 1\right)x + \left(\left(t + 2\right)^{1/21} - 1\right) = 0$. If $\lim\limits_{t \to -1^+} \alpha_t = a$ and $\lim\limits_{t \to -1^+} \beta_t = b$, then $72(a + b)^2$ is equal to _________.

If $\int \left(\dfrac{1}{x} + \dfrac{1}{x^3}\right)\left(\sqrt[23]{3x^{-24} + x^{-26}}\right) \mathrm{d}x = -\dfrac{\alpha}{3(\alpha + 1)}\left(3x^{\beta} + x^{\gamma}\right)^{\frac{\alpha + 1}{\alpha}} + C$, $x > 0$, $(\alpha, \beta, \gamma \in \mathbf{Z})$, where $C$ is the constant of integration, then $\alpha + \beta + \gamma$ is equal to _________.

If the function $f(x) = \dfrac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x}$ is continuous at $x = 0$, then $f(0)$ is equal to _________.