Question

$\left[x\right]$ denotes the greatest integer less than or equal to $x$. If $f\left(x\right)=\sin \left[{\pi }^{2}x\right]+\sin [-{\pi }^{2}x]$ then $f\left(\frac{\pi }{2}\right)=$

• A. $-1$
• B. $\sin \left(15\right)-\sin \left(16\right)$
• C. $0$
• D. $\sin \left(15\right)+\sin \left(16\right)$

Answer 1

The correct option is B $\sin \left(15\right)-\sin \left(16\right)$

$f\left(x\right)=\sin \left[{\pi }^{2}x\right]+\sin [-{\pi }^{2}x]$
$\therefore f\left(\frac{\pi }{2}\right)=\sin \left[{\pi }^3/2\right]+\sin [-{\pi }^3/2]$
Now $15\le {\pi }^3/2\le 16$ and $-16\le {\pi }^3/2\le -15$
$\therefore f\left(\frac{\pi }{2}\right)=\sin 15-\sin 16$