Question

Which of the following is not periodic

• A. $\left|\sin 3x\right|+{\sin }^{2}x$
• B. $\cos \sqrt{x}+{\cos }^{2}x$
• C. $\cos 4x+{\tan }^{2}x$
• D. $\cos 2x+\sin x$

Answer 1

The correct option is B $\cos \sqrt{x}+{\cos }^{2}x$

Period of $\left|\sin 3x\right|$ is $\frac{\pi }{3}$ and periode of ${\sin }^{2}x$ is $\pi$

(1) Same as the periode of $\left|\sin 3x\right|$ or
$\frac{1-\cos 2x}{2}$ whose period is $\pi$
Now period of $\left|\sin 3x\right|+{\sin }^{2}x$ is the L.C.M of their periods
$\therefore$ L.C.M of $(\frac{\pi }{3},\pi )=\frac{L.C.M(\pi ,\pi )}{H.C.F(3,1)}=\pi$

(2) Similarly we can say that $\cos 4x+{\tan }^{2}x$ and $\cos 2x+\sin x$ are periodic function.

(3)Now ${\cos }^{2}x$ is periodic with periode $\pi$ and for periode of $\cos \sqrt{x}$
Let $f\left(x\right)=\cos \sqrt{x}$
Let $f(x+T)=f\left(x\right)$
$\Rightarrow \cos \sqrt{T+x}=\cos \sqrt{x}\Rightarrow \sqrt{T+x}=2n\pi \pm \sqrt{x}$
which gives no value of $T$ independent of $x$
$\therefore f\left(x\right)$ can not be periodic

now
say $g\left(x\right)={\cos }^{2}x+\cos \sqrt{x}$ which is sum of a
periodic and non-periodic function and such function have no period.
So, $\cos \sqrt{x}+{\cos }^{2}x$ is non periodic function.