Question

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

• A. $(4,5)$
• B. $[4,5)$
• C. $(2,3)$
• D. $[2,3)$

Answer 1

The correct option is B $[4,5)$

As we know that fundamental period of $\sin x$ is $2\pi$
and fundamental period of $\sin ax,(a\in R)$ is $\frac{2\pi }{|a|}$
$\therefore$ fundamental period of
$\sin \left(\sqrt{\left[\lambda \right]}x\right)=\frac{2\pi }{\sqrt{\left[\lambda \right]}}$
But given that fundamental period of
$\sin \left(\sqrt{\left[\lambda \right]}x\right)=\pi$
$\therefore \frac{2\pi }{\sqrt{\left[\lambda \right]}}=\pi$
$\Rightarrow \left[\lambda \right]=4$
$\Rightarrow \lambda \in [4,5)$