1
JEE Advanced 2025 Paper 2 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

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 $[ -\frac{\pi}{2}, \frac{\pi}{2}]$ and $( -\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)

A

1

B

2

C

3

D

5

2
JEE Advanced 2025 Paper 2 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

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 :

A

$-6\sqrt{2}$

B

$-3\sqrt{2}$

C

$-9\sqrt{2}$

D

$-12\sqrt{2}$

3
JEE Advanced 2025 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
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 $z$, 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?

A

The determinant of $Q - 2I$ is zero

B

The determinant of $Q - 6I$ is 12

C

The determinant of $Q - 3I$ is 15

D

$yz = 2$

4
JEE Advanced 2025 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language

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?

A

$(4, \sqrt{3}) \in S$

B

$(5, \sqrt{2}) \in S$

C

Area of $\mathcal{R}$ is $\frac{14}{3} - 2\sqrt{3}$

D

Area of $\mathcal{R}$ is $\frac{14}{3} - \sqrt{3}$

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