JEE Main 2024 — 9 April, Shift 2 — Question 14

$\quad \ln (y)=3 \sin ^{-1} x$

$\frac{1}{y} \cdot y^{\prime}=3\left(\frac{1}{\sqrt{1-x^{2}}}\right)$

$\Rightarrow y^{\prime}=\frac{3 y}{\sqrt{1-x^{2}}}$

at $x=\frac{1}{2}$

$\Rightarrow y^{\prime}=\frac{3 e^{3\left(\frac{\pi}{6}\right)}}{\frac{\sqrt{3}}{2}}=2 \sqrt{3} \mathrm{e}^{\frac{\pi}{2}}$

$\Rightarrow y^{\prime \prime}=3\left(\frac{\sqrt{1-x^{2}} y^{\prime}-y \frac{1}{2 \sqrt{1-x^{2}}}(-2 x)}{\left(1-x^{2}\right)}\right)$

$\Rightarrow\left(1-x^{2}\right) y "=3\left(3 y+\frac{x y}{\sqrt{1-x^{2}}}\right)$

$\downarrow$

at $\mathrm{x}=\frac{1}{2}, \mathrm{y}=\mathrm{e}^{3 \sin ^{-1}\left(\frac{1}{2}\right)}=\mathrm{e}^{3\left(\frac{\pi}{6}\right)}=\mathrm{e}^{\frac{\pi}{2}}$

$$\begin{aligned} & \left.\left(1-\mathrm{x}^{2}\right) y^{\prime \prime}\right|_{\mathrm{atx}=\frac{1}{2}}=3\left(3 \mathrm{e}^{\frac{\pi}{2}}+\frac{\frac{1}{2}\left(\mathrm{e}^{\frac{\pi}{2}}\right)}{\frac{\sqrt{3}}{2}}\right) & =3 \mathrm{e}^{\frac{\pi}{2}}\left(3+\frac{1}{\sqrt{3}}\right) & \left(1-\mathrm{x}^{2}\right) \mathrm{y}^{\prime \prime}-\left.\mathrm{xy}\right|_{\text {at } x=\frac{1}{2}} \end{aligned}$$ $$\begin{aligned} & =3 \mathrm{e}^{\frac{\pi}{2}}\left(3+\frac{1}{\sqrt{3}}\right)-\frac{1}{2}\left(2 \sqrt{3} \mathrm{e}^{\frac{\pi}{2}}\right)=9 \mathrm{e}^{\frac{\pi}{2}} \end{aligned}$$