Question

Which one of the following is not a periodic function?

• A. $\sin 2x+\cos x$
• B. $\cos \sqrt{x}$
• C. $\tan 4x$
• D. $\log \cos 2x$

Answer 1

The correct option is B $\cos \sqrt{x}$

Option A, $f\left(x\right)=\sin 2x+\cos x$

Period of $\sin 2x$ is $\frac{2\pi }{2}=\pi$

Period of $\cos x$ is $2\pi$

LCM of $2\pi$ and $\pi$ is $2\pi$
Option B, $f\left(x\right)=\cos \sqrt{2}$

Period of $f\left(x\right)$ does not exists.
Option C, $f\left(x\right)=\tan 4x$

Period of $\tan 4x$ is $\frac{\pi }{4}$

Option D

Period of $h\left(x\right)=\log \cos 2x$

Here $f\left(x\right)=\log x$ and $g\left(x\right)=\cos 2x$

So, period of $f\left(g\right(x\left)\right)$ is same as period of $g\left(x\right)$

Period of $\cos 2x$ is $\frac{2\pi }{2}=\pi$

Hence, period of $h\left(x\right)$ is $\pi$