1
JEE Main 2025 (Online) 24th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

$\lim _\limits{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right)$ is:

A
$\frac{1}{\sqrt{15}}$
B
$\frac{1}{2 \sqrt{5}}$
C
$0$
D
$-\frac{1}{2 \sqrt{5}}$
2
JEE Main 2025 (Online) 24th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function such that $f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}$. If the $\lim\limits _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in \mathbb{R}$, then $\alpha+2 \beta$ is equal to

A
6
B
5
C
3
D
4
3
JEE Main 2025 (Online) 23rd January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

$\lim \limits_{x \rightarrow \infty} \frac{\left(2 x^2-3 x+5\right)(3 x-1)^{\frac{x}{2}}}{\left(3 x^2+5 x+4\right) \sqrt{(3 x+2)^x}}$ is equal to :

A
$\frac{2 e}{3}$
B
$\frac{2}{3 \sqrt{\mathrm{e}}}$
C
$\frac{2 \mathrm{e}}{\sqrt{3}}$
D
$\frac{2}{\sqrt{3 \mathrm{e}}}$
4
JEE Main 2025 (Online) 23rd January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If the function

$$ f(x)=\left\{\begin{array}{l} \frac{2}{x}\left\{\sin \left(k_1+1\right) x+\sin \left(k_2-1\right) x\right\}, \quad x<0 \\ 4, \quad x=0 \\ \frac{2}{x} \log _e\left(\frac{2+k_1 x}{2+k_2 x}\right), \quad x>0 \end{array}\right. $$

is continuous at $x=0$, then $k_1^2+k_2^2$ is equal to :

A
5
B
10
C
20
D
8
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