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

$$g(x) = \frac{3xe^x + 3x - ae^x - ax}{3(e^x + 1)}$$

for all real numbers $$x$$.

Then which one of the following statements is TRUE?

Let $$\mathbb{R}$$ 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

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 $$\tan^{-1} x$$ assume values in $$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and $$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, respectively.)

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 $$(p, 0)$$ and $$(0, q)$$, $$q > 0$$, and parallel to the line $$4x - \frac{3}{\sqrt{2}} y = 0$$.

Then the value of $$pq$$ is

Let $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ and $$P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$$. Let $$Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix}$$ for some non-zero real numbers $$x$$, $$y$$, and $$z$$, for which there is a $$2 \times 2$$ matrix $$R$$ with all entries being non-zero real numbers, such that $$QR = RP$$.

Then which of the following statements is (are) TRUE?

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 quadrant, enclosed by the parabola $$y^2 = x$$, the curve $$S$$, and the lines $$x = 1$$ and $$x = 4$$.

Then which of the following statements is (are) TRUE?

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 that $$y_1 > 0$$, and $$y_2 > 0$$. Let $$\mathcal{C}$$ denote the circle $$x^2 + y^2 = 9$$, and $$M$$ be the point $$(3, 0)$$.

Suppose the line $$x = x_1$$ intersects $$\mathcal{C}$$ at $$R$$, and the line $$x = x_2$$ intersects $$\mathcal{C}$$ at $$S$$, such that the $$y$$-coordinates of $$R$$ and $$S$$ are positive. Let $$\angle ROM = \frac{\pi}{6}$$ and $$\angle SOM = \frac{\pi}{3}$$, where $$O$$ denotes the origin $$(0, 0)$$. Let $$|XY|$$ denote the length of the line segment $$XY$$.

Then which of the following statements is (are) TRUE?

Let $$\mathbb{R}$$ denote the set of all real numbers. Let $$f: \mathbb{R} \to \mathbb{R}$$ be defined by

$$f(x) = \begin{cases} \frac{6x + \sin x}{2x + \sin x} & \text{if } x \neq 0, \\ \frac{7}{3} & \text{if } x = 0. \end{cases}$$

Then which of the following statements is (are) TRUE?

Let $$y(x)$$ be the solution of the differential equation

$$x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e},$$

satisfying $$y(1) = 0$$. Then the value of $$2\frac{(y(e))^2}{y(e^2)}$$ is ______.

Let $$a_0, a_1, \ldots, a_{23}$$ be real numbers such that

$$\left(1 + \frac{2}{5}x\right)^{23} = \sum_{i=0}^{23} a_i x^i$$

for every real number $$x$$. Let $$a_r$$ be the largest among the numbers $$a_j$$ for $$0 \leq j \leq 23$$. Then the value of $$r$$ is ______.