Question

Let $y=\csc x-\cot x$ such that $\frac{\frac{d^{2}y}{d{x}^2}}{y}=\lambda se{c}^2\frac{x}{2}$, then 1000$\lambda$ is equal to

• A. $400$
• B. $500$
• C. $200$
• D. $100$

Answer 1

The correct option is B $500$

$y=cosecx-\cot x$

$\frac{dy}{dx}=-cosecx\cot x+cose{c}^{2}x$

$\frac{dy}{dx}=cosecx(cosecx-\cot x)$

$\Rightarrow y_1=ycosecx$

$\Rightarrow y_1\sin x=y$

$\Rightarrow y_2\sin x+y_1\cos x=y_1$

$\Rightarrow y_2\sin x=y_1(1-\cos x)$

$y_2\sin x=\frac{y}{\sin x}(1-\cos x)$

$y_2=\frac{y\left(2{\sin }^2\frac{x}{2}\right)}{4{\sin }^2\frac{x}{2}{\cos }^2\frac{x}{2}}$

$\Rightarrow y_2=\frac{1}{2}y{\sec }^2\frac{x}{2}$

$\frac{y_2}{y}=\frac{1}{2}{\sec }^2\frac{x}{2}$

Comparing this with $\frac{\frac{d^{2}y}{d{x}^2}}{y}=\lambda se{c}^2\frac{x}{2}$ (given)

$\Rightarrow \lambda =\frac{1}{2}$

$\Rightarrow 1000\lambda =500$