1
JEE Main 2025 (Online) 23rd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $\left(\log _e 2, k\right)$. If the curve satisfies the differential equation $2(3+y) e^{2 x} d x-\left(7+e^{2 x}\right) d y=0$, then $k$ is equal to

A
32
B
8
C
4
D
16
2
JEE Main 2025 (Online) 22nd January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $x=f(y)$ is the solution of the differential equation $\left(1+y^2\right)+\left(x-2 \mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $f(0)=1$, then $f\left(\frac{1}{\sqrt{3}}\right)$ is equal to :

A
$\mathrm{e}^{\pi / 4}$
B
$e^{\pi / 12}$
C
$\mathrm{e}^{\pi / 6}$
D
$e^{\pi / 3}$
3
JEE Main 2025 (Online) 22nd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $x=x(y)$ be the solution of the differential equation $y^2 \mathrm{~d} x+\left(x-\frac{1}{y}\right) \mathrm{d} y=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

A
$\frac{3}{2}+\mathrm{e}$
B
$\frac{1}{2}+\mathrm{e}$
C
$3+e$
D
$3-e$
4
JEE Main 2025 (Online) 22nd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in \mathbf{R}$. Then $\sum_\limits{n=1}^{100} \log _e f(n)$ is equal to :

A
2406
B
5220
C
2525
D
2384
JEE Main Subjects
EXAM MAP