Question

Match of the following :
List - I List - II
1. Period of $\cos \sqrt{x}$ $a.\frac{\pi }{2}$
2. Period of ${\sin }^{3}x+{\cos }^{3}x$ $b.\pi$
3. Period of $|\sin x|+|\cos x|$ $c.2\pi$
4. Periodof $\sin x\cos x$ d. does not exist

• A. $1-a,2-d,3-b,4-c$
• B. $1-d,2-c,3-a,4-b$
• C. $1-a,2-b,3-c,4-d$
• D. $1-b,2-c,3-d,4-a$

Answer 1

The correct option is B $1-d,2-c,3-a,4-b$

1. Period of $\cos \sqrt{x}$
$\cos \sqrt{x}$ is not a periodic function because domain of $\cos \sqrt{x}$ is $x\ge 0$
But for periodic function domain of $xϵ[-\infty ,\infty ]$

2. Period of ${\sin }^{3}x+{\cos }^{3}x$
${\sin }^{3}x+{\cos }^{3}x=\frac{3\sin x-\sin 3x+\cos 3x+3\cos x}{4}$
$=\frac{3(\cos x+\sin x)+\cos 3x-\sin 3x}{4}$
$=\frac{3\sqrt{2}\cos (x-\frac{\pi }{4})+\sqrt{2}\cos (3x+\frac{\pi }{4})}{4}$
Period of ${\sin }^{3}x+{\cos }^{3}x=LCMof(2\pi ,\frac{2\pi }{3})$
$=2\pi$ [concept -1]

3. Period of $|\sin x|+|\cos x|$
property of periodic function $f(x+T)=f\left(x\right)$
$\therefore$ Put $T=\frac{\pi }{2}$
$|\sin (x+\frac{\pi }{2})|+|\cos (x+\frac{\pi }{2})|=|\cos x|+|\sin x|$
$|\cos x|+|\sin x|=|\cos x|+|\sin x|$
i.e., $\frac{\pi }{2}$ is a period of $|\sin x|+|\cos x|$

4. $\sin x\cos x=\frac{\sin 2x}{2}$

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