1
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

$-12\sqrt{2}$

2
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

$yz = 2$

3
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 $\frac{14}{3} - \sqrt{3}$

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

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 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 $\angle R O M=\frac{\pi}{6}$ and $\angle S O M=\frac{\pi}{3}$, where $O$ denotes the origin $(0,0)$. Let $|X Y|$ denote the length of the line segment $X Y$.

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

A

The equation of the line joining P and Q is $2x + 3y = 3(1 + \sqrt{3})$

B

The equation of the line joining P and Q is $2x + y = 3(1 + \sqrt{3})$

C

If $N_2 = (x_2, 0)$, then $3|N_2Q| = 2|N_2S|$

D

If $N_1 = (x_1, 0)$, then $9|N_1P| = 4|N_1R|$

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