Jee Main 2024 29 01 2024 Shift 1 - Question2
Question 2
$\lim _{x \rightarrow \frac{\pi^{-}}{2}} \frac{\int _{x^{3}}^{\left(\frac{\pi}{2}\right)^{2}} \cos t^{1 / 3} d t}{\left(x-\frac{\pi}{2}\right)^{2}}$
(1) $\frac{3 \pi^{2}}{4}$
(2) $\frac{3 \pi}{4}$
(3) $\frac{3 \pi^{2}}{8}$
(4) $\frac{3 \pi}{8}$
Show Answer
Answer: (3)
Solution:
$\lim _{h \rightarrow 0} \frac{\int _{\left(\frac{\pi}{2}-h\right)^{3}}^{\left(\frac{\pi}{2}\right.} \cos \left(t^{1 / 3}\right) d t}{h^{2}}$ $=\lim _{h \rightarrow 0} \frac{0+3\left(\frac{\pi}{2}-h\right)^{2} \cos \left(\frac{\pi}{2}-h\right)}{2 h}$
$=\lim _{h \rightarrow 0} \frac{3\left(\frac{\pi}{2}-h\right)^{2} \sin h}{2 h}$
$=\frac{3 \pi^{2}}{8}$
BETA
