1
JEE Main 2025 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $\mathrm{S}=\mathbf{N} \cup\{0\}$. Define a relation R from S to $\mathbf{R}$ by :

$$ \mathrm{R}=\left\{(x, y): \log _{\mathrm{e}} y=x \log _{\mathrm{e}}\left(\frac{2}{5}\right), x \in \mathrm{~S}, y \in \mathbf{R}\right\} . $$

Then, the sum of all the elements in the range of $R$ is equal to :

A
$\frac{3}{2}$
B
$\frac{10}{9}$
C
$\frac{5}{2}$
D
$\frac{5}{3}$
2
JEE Main 2025 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} \in \{0, 1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:

A

$\frac{5}{8}$

B

$\frac{1}{4}$

C

$\frac{3}{4}$

D

$\frac{3}{8}$

3
JEE Main 2025 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let a straight line $L$ pass through the point $P(2, -1, 3)$ and be perpendicular to the lines $ \frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 3}{-2} $ and $ \frac{x - 3}{1} = \frac{y - 2}{3} = \frac{z + 2}{4} $. If the line $L$ intersects the $yz$-plane at the point $Q$, then the distance between the points $P$ and $Q$ is:

A

$\sqrt{10}$

B

$2$

C

$2\sqrt{3}$

D

$3$

4
JEE Main 2025 (Online) 29th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If the domain of the function $ \log_5(18x - x^2 - 77) $ is $ (\alpha, \beta) $ and the domain of the function $ \log_{(x-1)} \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) $ is $(\gamma, \delta)$, then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:

A

186

B

179

C

195

D

174

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