Question

If $\sin x\cdot \cos y=\frac{1}{2}$, then the value of $\frac{d^{2}y}{d{x}^2}$ at $\left(\frac{\pi }{4},\frac{\pi }{4}\right)$ is equal to

• A. 1
• B. -1
• C. 4
• D. -4

Answer 1

Answer: (D)

$\cos x\cos y-\sin y\sin x\cdot \frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=\cot x\cdot \cot y$
$\text{ and }\frac{d^{2}y}{d{x}^2}=-{\mathrm{cosec}}^{2}x\cot y-{\mathrm{cosec}}^{2}y\cdot \cot x\cdot \frac{dy}{dx}$
$\because {\left(\frac{dy}{dx}\right)}_{\left(\frac{\pi }{4},\frac{\pi }{4}\right)}=1$
$\therefore {\left(\frac{d^{2}y}{d{x}^2}\right)}_{\left(\frac{\pi }{4},\frac{\pi }{4}\right)}=-4$