Question

Let $f\left(x\right)=\sin \sqrt{p}x,$ where $p=\left[a\right]=$ the greatest integer less than or equal to $a$. If the period of $f\left(x\right)$ is $\pi$ then

• A. $a\in [4,5]$
• B. $a=4,5$
• C. $a\in [4,5)$
• D. none of these

Answer 1

The correct option is C $a\in [4,5)$

Note that period of $\sin \left(kx\right)$ is $\frac{2\pi }{k}$.

Hence the period of $\sin \left(\sqrt{p}x\right)$ is $\frac{2\pi }{\sqrt{p}}$.

$\therefore \frac{2\pi }{\sqrt{p}}=\pi$

$\Rightarrow p=4$

Since $p=\left[a\right]$, we have $a\in [4,5)$.