Question

If $f\left(x\right)=\cos \sqrt{p}x$ has a fundamental period of $\pi$ where $p=\left[a\right]$ where $\left[x\right]$ is the integral part of $x$. then $a$ belongs to

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

Answer 1

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

We know fundamental period of $\cos x$ is $2\pi$
Thus fundamental period of $\cos \sqrt{p}x$ is $\frac{2\pi }{\sqrt{p}}=\pi$ (given)
$\Rightarrow \sqrt{p}=2$

$\Rightarrow p=4=\left[a\right]$

$\Rightarrow 4\le a<5$ $\dots [\because \left[t\right]=x\Rightarrow x\le t<x+1]$

$\therefore a\in [4,5)$