1
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
2
JEE Main 2025 (Online) 22nd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbf{R}$. If $f^{\prime}(0)=4 \mathrm{a}$ and $f$ satisfies $f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x)=0, \mathrm{a}>0$, then the area of the region $\mathrm{R}=\{(x, y) \mid 0 \leq y \leq f(a x), 0 \leq x \leq 2\}$ is :

A
$\mathrm{e}^2-1$
B
$e^4+1$
C
$\mathrm{e}^2+1$
D
$e^4-1$
3
JEE Main 2024 (Online) 9th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$\int_\limits0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0$$. Then at $$x=2, y^{\prime \prime}+y+1$$ is equal to

A
$$\sqrt2$$
B
2
C
1/2
D
1
4
JEE Main 2024 (Online) 9th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The solution of the differential equation $$(x^2+y^2) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0$$, is :

A
$$\left|x^2-4 y^2\right|^5=x^2$$
B
$$\left|x^2-2 y^2\right|^6=x$$
C
$$\left|x^2-2 y^2\right|^5=x^2$$
D
$$\left|x^2-4 y^2\right|^6=x$$
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