JEE Advanced 2025 Paper 2 Online

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be defined by
$f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x}, & \text{if } x \neq 0, \\ \dfrac{7}{3}, & \text{if } x = 0. \end{cases}$
Then which of the following statements is ...

Let ℝ denote the set of all real numbers. Then the area of the region
$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x}, 5x - 4y - 1 > 0, 4x + 4y - 17 < 0 \right\} $
is

For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument of $z$, with $-\pi<\arg (z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0<\arg (\omega)<\pi$. Let
$$ \alpha=\arg \left(\sum\limits_{n=1}^{2025}...

If
\(\alpha=\int\limits_{\frac{1}{2}}^2 \frac{\tan ^{-1} x}{2 x^2-3 x+2} d x\)
then the value of $\sqrt{7} \tan \left(\frac{2 \alpha \sqrt{7}}{\pi}\right)$ is _________.
(Here, the inverse trigonometric function $\tan ^{-1} x$ assumes val...

Let $y(x)$ be the solution of the differential equation
\(x^2 \frac{d y}{d x}+x y=x^2+y^2, \quad x>\frac{1}{e}\)
satisfying $y(1)=0$. Then the value of $2 \frac{(y(e))^2}{y\left(e^2\right)}$ is ____________.

Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$ be two distinct points on the ellipse
\(\frac{x^2}{9}+\frac{y^2}{4}=1\)
such that $y_1>0$, and $y_2>0$. Let $C$ denote the circle $x^2+y^2=9$, and $M$ be the point $(3,0)$.
Suppose...

Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(0,4)$ be functions defined by
\(f(x)=\log _e\left(x^2+2 x+4\right), \text { and } g(x)=\frac{4}{1+e^{-2 x}}\)
D...

The total number of real solutions of the equation
$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1}\left(\frac{6 \tan \theta}{9 + \tan^2 \theta}\right) $
is
(Here, the inverse trigonometric functions $\sin^{-1} x$ and $\tan^{-1} ...

Let $x_0$ be the real number such that $e^{x_0} + x_0 = 0$. For a given real number $\alpha$, define \(g(x) = \frac{3x e^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)}\) for all real numbers $x$. Then which one of the following statements is T...

Let $a_0, a_1, \ldots, a_{23}$ be real numbers such that
\(\left(1+\frac{2}{5} x\right)^{23}=\sum\limits_{i=0}^{23} a_i x^i\)
for every real number $x$. Let $a_r$ be the largest among the numbers $a_j$ for $0 \leq j \leq 23$. Then the val...

Let $I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$ and $P=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)$. Let $Q=\left(\begin{array}{ll}x & y \\ z & 4\end{array}\right)$ for some non-zero real numbers $x, y$, and $z$,...

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 enclosed between the parabola and the chord is $\frac{4}{3}$. Let $\mathcal{R}$ denote the region lying in the first quadra...

A factory has a total of three manufacturing units, $M_1, M_2$, and $M_3$, which produce bulbs independent of each other. The units $M_1, M_2$, and $M_3$ produce bulbs in the proportions of $2: 2: 1$, respectively. It is known that $20 \%$ ...

Let S denote the locus of the point of intersection of the pair of lines$4x - 3y = 12\alpha$,$4\alpha x + 3\alpha y = 12$,where $\alpha$ varies over the set of non-zero real numbers. Let T be the tangent to S passing through the points $(p,...

Let
\(\alpha=\frac{1}{\sin 60^{\circ} \sin 61^{\circ}}+\frac{1}{\sin 62^{\circ} \sin 63^{\circ}}+\cdots+\frac{1}{\sin 118^{\circ} \sin 119^{\circ}}\)
Then the value of
\(\left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2\)
is ...

Consider the vectors
\(\vec{x}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \quad \vec{y}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k}, \quad \text { and } \quad \vec{z}=3 \hat{\imath}+\hat{\jmath}+2 \hat{k}\)
For two distinct positive real numbers $\...