Question

If $f\left(x\right)=\cos \left[\frac{1}{2}{\pi }^2\right]x+\sin \left[\frac{1}{2}{\pi }^2\right]x$ , $\left[x\right]$ denoting the greatest integer function, then

• A. $f\left(0\right)=0$
• B. $f\left(\frac{\pi }{3}\right)=\frac{1}{4}$
• C. $f\left(\frac{\pi }{2}\right)=1$
• D. $f\left(\pi \right)=0$

Answer 1

Answer: (C)

$f\left(\frac{\pi }{2}\right)=1$

$f\left(x\right)=\cos \left[\frac{1}{2}{\pi }^2\right]x+\sin \left[\frac{1}{2}{\pi }^2\right]$

We know, $9<{\pi }^2<10\Rightarrow 4.5<\frac{1}{2}{\pi }^2<5\therefore \left[\frac{1}{2}{\pi }^2\right]=4$
Thus,
$f\left(x\right)=\cos 4x+\sin 4x$

$\therefore f\left(\frac{\pi }{2}\right)=\cos 2\pi +\sin 2\pi =1+0=1$