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

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

The total number of real solutions of the equation \( \theta = \tan^{-1}(2 \tan \theta) - \dfrac{1}{2}\sin^{-1}\!\left(\dfrac{6 \tan \theta}{9 + \tan^{2}\theta}\right) \) is
(Here, the inverse trigonometric functions \( \sin^{-1} x \) and \( \tan^{-1} x \) assume values in \( \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] \) and \( \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right) \), 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 \( 4x - \dfrac{3}{\sqrt{2}} y = 0 \).
Then the value of \( pq \) is

Let \(\displaystyle I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \text{ and } P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}. \text{ Let } Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix}\)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 \( Q R = R P \).
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 \( \dfrac{4}{3} \). Let \( R \) 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 \( \dfrac{x^2}{9} + \dfrac{y^2}{4} = 1 \) 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 the line \( x = x_1 \) intersects \( C \) at \( R \), and the line \( x = x_2 \) intersects \( C \) at \( S \), such that the \( y \)-coordinates of \( R \) and \( S \) are positive.
Let \( \angle ROM = \dfrac{\pi}{6} \) and \( \angle SOM = \dfrac{\pi}{3} \), 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

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

Let \( y(x) \) be the solution of the differential equation \( x^2 \dfrac{dy}{dx} + xy = x^2 + y^2, \ x > \dfrac{1}{e} \), satisfying \( y(1) = 0 \).
Then the value of \( 2\dfrac{\,(y(e))^2}{y(e^2)} \) is _____________.

Let \( a_0, a_1, \ldots, a_{23} \) be real numbers such that \( \left(1 + \dfrac{2}{5}x\right)^{23} = \displaystyle\sum_{i=0}^{23} a_i x^i \) for every real number \( x \). Let \( a_r \) be the largest among the numbers \( a_j \) for \( 0 \le j \le 23 \). Then the value of \( r \) is __________.

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 \( 20\% \) of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by \( M_1 \), \( 15\% \) are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by \( M_2 \) is \( \dfrac{2}{5} \).
If a bulb is chosen randomly from the bulbs produced by \( M_3 \), then the probability that it is defective is ___________.

Consider the vectors \( \vec{x}=\hat{i}+2\hat{j}+3\hat{k},\quad \vec{y}=2\hat{i}+3\hat{j}+\hat{k},\; \text{and}\; \vec{z}=3\hat{i}+\hat{j}+2\hat{k}. \)
For two distinct positive real numbers \( \alpha \) and \( \beta \), define \( \vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z},\quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x},\; \text{and}\; \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}. \)
If the vectors \( \vec{X}, \vec{Y} \) and \( \vec{Z} \) lie in a plane, then the value of \( \alpha+\beta-3 \) 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 \( \alpha = \arg\left( \displaystyle\sum_{n=1}^{2025} (-\omega)^n \right) \). Then the value of \( \dfrac{3\alpha}{\pi} \) is __________________.

Let \( \mathbb{R} \) denote the set of all real numbers. Let \( f:\mathbb{R}\to\mathbb{R} \) and \( g:\mathbb{R}\to(0,4) \) be functions defined by \( f(x)=\log_e\!\left(x^2+2x+4\right) \), and \( g(x)=\dfrac{4}{1+e^{-2x}} \). Define the composite function \( f\circ g^{-1} \) by \( (f\circ g^{-1})(x)=f\!\left(g^{-1}(x)\right) \), where \( g^{-1} \) is the inverse of the function \( g \).
Then the value of the derivative of the composite function \( f\circ g^{-1} \) at \( x=2 \) is ___________.

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

If \( \displaystyle \alpha = \int\limits_{\frac{1}{2}}^{2} \frac{\tan^{-1} x}{2x^2 - 3x + 2} \, dx\), then the value of \( \sqrt{7}\,\tan\!\left(\dfrac{2\alpha \sqrt{7}}{\pi}\right) \) is __________.
(Here, the inverse trigonometric function \( \tan^{-1} x \) assumes values in \( \left( -\dfrac{\pi}{2} , \dfrac{\pi}{2} \right) \).)

JEE Advance 2025 Paper-2

Physics Chemistry Mathematics

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

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