Question

Let $f$ be a quadratic polynomial such that $f(-\pi )=f\left(\pi \right)=0$ and $f\left(\frac{\pi }{2}\right)=\frac{3{\pi }^2}{4}$. If $\underset{x\to \pi }{lim}f\left(x\right)\cdot \cos \left(\sin x\right)\cdot \text{cosec}\left(\sin x\right)=m\pi ,$ then the value of $m$ is

Answer 1

$f\left(x\right)=a(x-\pi )(x+\pi )$
Given, $f\left(\frac{\pi }{2}\right)=\frac{3{\pi }^2}{4}$
$\therefore a=-1$
$\therefore f\left(x\right)=(\pi +x)(\pi -x)$

Now, $\underset{x\to \pi }{lim}f\left(x\right)\cdot \cos \left(\sin x\right)\cdot \text{cosec}\left(\sin x\right)=m\pi$
$\Rightarrow \underset{x\to \pi }{lim}\frac{(\pi +x)(\pi -x)}{\tan \left(\sin x\right)}=m\pi$
$\Rightarrow \underset{x\to \pi }{lim}\frac{(\pi +x)(\pi -x)}{\frac{\tan \left(\sin x\right)}{\sin x}\cdot \sin x}=m\pi$
$\Rightarrow \underset{x\to \pi }{lim}\frac{(\pi +x)(\pi -x)}{\sin x}=m\pi$
$\Rightarrow \underset{x\to \pi }{lim}\frac{(\pi +x)(\pi -x)}{\frac{\sin (\pi -x)}{(\pi -x)}\cdot (\pi -x)}=m\pi$
$\Rightarrow 2\pi =m\pi$
$\Rightarrow m=2$