Question

Let $f\left(x\right)=\sin \sqrt{\left[a\right]}x$ ( where $\left[\right]$ denotes the greatest integer function). If $f$ is periodic with fundamental period $\pi$, the $a$ belongs to

• A. $[2,3)$
• B. $\{4,5\}$
• C. $[4,5]$
• D. $[4,5)$

Answer 1

Answer: (D)

$[4,5)$

$f\left(x\right)=\sin \sqrt{\left[a\right]}x$, which has a period $\frac{2\pi }{\sqrt{\left[a\right]}}=\pi$

$\Rightarrow \sqrt{\left[a\right]}=2$

squaring on both sides

$\Rightarrow \left[a\right]=4$
$\Rightarrow 4\le a<5$ $\dots [\because [t]=x\Rightarrow x\le t<x+1]$
$\Rightarrow a\in [4,5)$
Hence, option 'D' is correct.