Question

Match of the following :
List - I List - II
1. Period of $\cos x^2$ $a.$ $\frac{\pi }{3}$
2. Period of $\cos (e^{\tan x}+e^{\cot x})$ $b.$ $2\pi$
3. Period of $\cos (3x+2x+x)$ $c.$ does not exist
4. Period of $\sin \left(\right(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...\infty \left)x\right)$ $d.$ $\pi$

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

Answer 1

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

1. $\cos \left(x^2\right),x^2\ge 0$
Quantity inside $\cos$ is always positive.
It does not contain all conditions of periodic period.
Period of $\cos x^2$ does not exist.
2. $\cos (e^{\tan x}+e^{\cot x})$
Period of $\tan x$ or $\cot x=\pi$
So, by property of periodic function,
$\to$ If period of $g\left(x\right)$ is T, then period of $fo\left(g\right)$ is also T.
$\therefore$ period of $e^{\tan x}=\pi$
period of $\cos (e^{\tan x}+e^{\cot x})=\pi$
3. $\cos (3x+2x+x)=\cos \left(6x\right)$
period of $\cos \left(6x\right)=\frac{2\pi }{6}=\frac{\pi }{3}$
4. $\sin \left(\right(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...\infty \left)x\right)=\sin x$
$(s=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}....\infty )\Rightarrow s=1$, period of $\sin x=2\pi$