Periodic Functions

Period of $\sin \theta -\sqrt{3}\cos \theta$ is

Period of $f\left(x\right)=\left\{x\right\}+\left\{x+\frac{1}{3}\right\}+\left\{x+\frac{2}{3}\right\}$ is equal to (where, $\left\{·\right\}$ represent fractional part function of $x$)

The period of the function $g\left(x\right)=5\cot \left(\frac{\pi }{3}x+\frac{\pi }{6}\right)+2$ is equal to

If $f$ is a function such that $f\left(x+a\right)=\frac{1+f\left(x\right)}{1-f\left(x\right)}\forall x\in R$, then which of the following is correct?

If $f:R\to R$ is a function satisfying the property $f(x+1)+f(x+3)=2$ for all $x\in R$ than $f$ is:

State whether the statement is true or false:

Let $f\left(x\right)$ be a function satisfying $f\left(x\right)+f(x+2)=10, \forall x\in R$, then $f\left(x\right)$ is periodic function.

Find the period of $\tan 3x+\sin \frac{5x}{2}$.

Find the period of the periodic function $\sin \left(4x+5\right)$.

Find the period of the given periodic function $f\left(x\right)=9\sin \left(6x+5\right)$

If $f\left(a+x\right)=f\left(x\right)$, then $\underset{0}{\overset{na}{\int }}f\left(x\right)dx$, where $n\in N$, is equal to

The function $f\left(x\right)=\sin \left(\frac{\pi x}{n!}\right)-\cos \left(\frac{\pi x}{\left(n+1\right)!}\right)$ is

For the following questions, choose the correct answers from the codes $\left(\mathrm{a}\right), \left(\mathrm{b}\right), \left(\mathrm{c}\right)$ and $\left(\mathrm{d}\right)$ defined as follows:
Statement I: The period of $f\left(x\right)=2\mathrm{c}\mathrm{o}\mathrm{s}\frac{1}{3}\left(x-\pi \right)+4\mathrm{s}\mathrm{i}\mathrm{n}\frac{1}{3}\left(x-\pi \right)$ is $3\pi$.
Statement II: If $T$ is the period of $f\left(x\right)$, then the period of $f\left(ax+b\right)$ is $\frac{T}{\left|a\right|}$.

Which of the following functions have period $2\pi$?

Period of $f\left(x\right)=sgn\left(\left[x\right]+[-x]\right)$ is equal to (where $\left[.\right]$ denotes greatest integer function):

Period of $f\left(x\right)=sgn\left(\left[x\right]+[-x]\right)$ is equal to (where $\left[.\right]$ denotes greatest integer function):

The function $f\left(x\right)=\left|sin4x\right|+\left|cos2x\right|$, is a periodic function with a fundamental period 

The function $f\left(x\right)=2\cos 5x+3\sin \sqrt{5}x$ is

The fundamental period of the function $f\left(x\right)={\cos }^{-1}\left(\mathrm{cos\pi }x\right)+\left|\sin \left(\mathrm{\pi }x\right)\right|+sgn\left(x^2-10x+30\right)+2021\mathrm{sin\pi }\left[x\right]$ is

[Note: $\left[x\right]$ denotes greatest integer less than or equal to $x$ and $sgn x$ denotes signum function of $x$.]

Period of $f\left(x\right)=e^{\cot x}+{\sin }^{-1}\left(\sin x\right)+\cos 3x$

Fundamental period of the function $f\left(x\right)={\sin }^{2}4x+\left|\cos 16x\right|$ is