Jee Advanced 2025 - Paper 2

Let $x_0$ be the real number such that $e^{x_0} + x_0 = 0$. For a given real number $\alpha$, define $$g(x) = \frac{3xe^

Let $\mathbb{R}$ denote the set of all real numbers. Then the area of the region

The total number of real solutions of the equation $\theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \fr

Let $S$ denote the locus of the point of intersection of the pair of lines $4x - 3y = 12\alpha,$

Let $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$. Let $Q =

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2 = x$, such that the area of the region e

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ such tha

Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \begin{cas

Let $y(x)$ be the solution of the differential equation $x^2 \frac{dy}{dx} + xy = x^2 + y^2, x > \frac{1}{e}$ satisfyi

Let $a_0, a_1, \dots, a_{23}$ be real numbers such that $$\left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i$

A factory has a total of three manufacturing units, $M_1, M_2$, and $M_3$, which produce bulbs independent of each other

Consider the vectors $\vec{x} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{y} = 2\hat{i} + 3\hat{j} + \hat{k}$, and $\vec{z}

For a non-zero complex number $z$, let $\arg(z)$ denote the principal argument of $z$, with $-\pi < \arg(z) \le \pi$. Le

Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to (0, 4)$ b

Let $$\alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \dots + \frac{1}{\sin 118

If $\alpha = \int_{1/2}^2 \frac{\tan^{-1} x}{2x^2 - 3x + 2} dx$, then the value of $\sqrt{7} \tan \left( \frac{2\alpha