1
JEE Main 2024 (Online) 4th April Morning Shift
Numerical
+4
-1

If $$\lim _\limits{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}$$, where $$\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$$, then $$8 \mathrm{~m}+12 \mathrm{n}$$ is equal to _______.

2
JEE Main 2024 (Online) 1st February Morning Shift
Numerical
+4
-1
Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, x \neq 0$. If $\mathrm{L}$ and $\mathrm{R}$ respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x=0$, then $\frac{32}{\pi^2}\left(\mathrm{~L}^2+\mathrm{R}^2\right)$ is equal to ___________.
3
JEE Main 2024 (Online) 31st January Evening Shift
Numerical
+4
-1

If $$\lim _\limits{x \rightarrow 0} \frac{a x^2 e^x-b \log _e(1+x)+c x e^{-x}}{x^2 \sin x}=1$$, then $$16\left(a^2+b^2+c^2\right)$$ is equal to ________.

4
JEE Main 2024 (Online) 30th January Morning Shift
Numerical
+4
-1

If the function

$$f(x)= \begin{cases}\frac{1}{|x|}, & |x| \geqslant 2 \\ \mathrm{a} x^2+2 \mathrm{~b}, & |x|<2\end{cases}$$

is differentiable on $$\mathbf{R}$$, then $$48(a+b)$$ is equal to __________.