JEE Main 2025 Jan 22 Shift 2 — Mathematics

Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$. Then the number of many-one functions $f : A \to B$ such that $1 \in f(A)$ is equal to :

Let $\alpha_{\theta}$ and $\beta_{\theta}$ be the distinct roots of $2x^2 + (\cos\theta)x - 1 = 0$, $\theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_{\theta}^{4} + \beta_{\theta}^{4}$, then $16(M + m)$ equals :

Let the curve $z(1 + i) + \bar{z}(1 - i) = 4$, $z \in \mathbb{C}$, divide the region $|z - 3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha - \beta|$ equals :

If the system of equations :

$x + y + 2z = 6$,

$2x + 3y + az = a + 1$,

$-x - 3y + bz = 2b$,

where $a, b \in \mathbb{R}$, has infinitely many solutions, then $7a + 3b$ is equal to :

For a $3 \times 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \dfrac{1}{2}$ and trace $(A) = 3$. If $B = \mathrm{adj}(\mathrm{adj}(2A))$, then the value of $|B| + \text{trace}(B)$ equals :

Suppose that the number of terms in an A.P. is $2k$, $k \in \mathbb{N}$. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then $k$ is equal to :

In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is :

Let $\alpha$, $\beta$, $\gamma$ and $\delta$ be the coefficients of $x^7$, $x^5$, $x^3$ and $x$ respectively in the expansion of $\left(x + \sqrt{x^3 - 1}\right)^5 + \left(x - \sqrt{x^3 - 1}\right)^5$, $x > 1$. If $u$ and $v$ satisfy the equations

$\alpha u + \beta v = 18$,

$\gamma u + \delta v = 20$,

then $u + v$ equals :

If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A\,|\,B)$ and $P(B\,|\,A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\dfrac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B})}$ is :

Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral $PQMN$ is equal to :

Let $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ and $H : \dfrac{x^2}{A^2} - \dfrac{y^2}{B^2} = 1$. Let the distance between the foci of $E$ and the foci of $H$ be $2\sqrt{3}$. If $a - A = 2$, and the ratio of the eccentricities of $E$ and $H$ is $\dfrac{1}{3}$, then the sum of the lengths of their latus rectums is equal to :

The sum of all values of $\theta \in [0, 2\pi]$ satisfying $2\sin^2\theta = \cos 2\theta$ and $2\cos^2\theta = 3\sin\theta$ is

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\dfrac{\pi}{3}$. If $\lambda \vec{a} + 2\vec{b}$ and $3\vec{a} - \lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1, 3]$ is :

The perpendicular distance, of the line $\dfrac{x-1}{2} = \dfrac{y+2}{-1} = \dfrac{z+3}{2}$ from the point $P(2, -10, 1)$, is :

Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the square of the area of $\triangle PQR$ is equal to :

If $\lim\limits_{x \to \infty} \left( \left( \dfrac{e}{1-e} \right) \left( \dfrac{1}{e} - \dfrac{x}{1+x} \right) \right)^{x} = \alpha$, then the value of $\dfrac{\log_e \alpha}{1 + \log_e \alpha}$ equals :

Let $f(x) = \displaystyle\int_{0}^{x^2} \dfrac{t^2 - 8t + 15}{e^t} \, dt$, $x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :

The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is :

If $x = f(y)$ is the solution of the differential equation

$$\left(1 + y^2\right) + \left(x - 2e^{\tan^{-1} y}\right) \dfrac{dy}{dx} = 0, \; y \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$$

with $f(0) = 1$, then $f\left(\dfrac{1}{\sqrt{3}}\right)$ is equal to :

If $\displaystyle\int e^x \left( \dfrac{x \sin^{-1} x}{\sqrt{1 - x^2}} + \dfrac{\sin^{-1} x}{\left(1 - x^2\right)^{3/2}} + \dfrac{x}{1 - x^2} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g\left(\dfrac{1}{2}\right)$ equals :

If $\displaystyle\sum_{r=1}^{30} \dfrac{r^2 \left({}^{30}C_r\right)^2}{{}^{30}C_{r-1}} = \alpha \times 2^{29}$, then $\alpha$ is equal to ________.

Let $A(6, 8)$, $B(10\cos\alpha, -10\sin\alpha)$ and $C(-10\sin\alpha, 10\cos\alpha)$, be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ be its orthocenter and centroid respectively, then $(5a - 3h + 6k + 100\sin 2\alpha)$ is equal to ________.

Let $y = f(x)$ be the solution of the differential equation $\dfrac{dy}{dx} + \dfrac{xy}{x^2 - 1} = \dfrac{x^6 + 4x}{\sqrt{1 - x^2}}$, $-1 < x < 1$ such that $f(0) = 0$. If $6\displaystyle\int_{-1/2}^{1/2} f(x)\,dx = 2\pi - \alpha$ then $\alpha^2$ is equal to ________.

Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $(QR)^2$ is equal to ________.

Let $A = \{1, 2, 3\}$. The number of relations on $A$, containing $(1, 2)$ and $(2, 3)$, which are reflexive and transitive but not symmetric, is ________.