Periodic Functions

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

Given is the partial graph of an even periodic function $f$ whose period is $8$. If $\left[*\right]$ denotes greatest integer function then find the value of the expression $f\left(-3\right)+2\left|f\left(-1\right)\right|+\left[f\left(\frac{7}{8}\right)\right]+f\left(0\right)+\arccos \left(f\left(-2\right)\right)+f\left(-7\right)+f\left(20\right).$

Let $\text{'}f\text{'}$ be an even periodic function with period $\text{'}4\text{'}$ such that $f\left(x\right)=2^x-1, 0\le x\le 2$. The number of solutions of the equation $f\left(x\right)=1$ in $\left[-10,20\right]$ are

Let $f$ be a periodic function with period $\frac{1}{2}$. If $f\left(2\right)=5$ and $f\left(\frac{9}{4}\right)=2$, then find $f\left(-3\right)$ and $f\left(\frac{1}{4}\right)$.

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 $x$)

An odd function is symmetric about the vertical line $x=a \left(a>0\right)$ and if $\sum _{r=0}^{\infty }{\left[f\right(1+4ar\left)\right]}^r=8$, then find the numerical value of $8f\left(1\right)$ $($where $0<f\left(1\right)<1)$.

The period of the function $f\left(x\right)$, which satisfies the relation $f\left(x\right)+f\left(x+4\right)=f\left(x+2\right)+f\left(x+6\right)$ is

Let $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{3x}{10}$ for all real $x.$ The least natural number $n$ such that $f\left(10n\pi +x\right)=f\left(x\right)$ for all real values of $x$ 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 $x$.]

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

Which of the following functions is periodic?

The period of $f\left(x\right)=\sin \left(\frac{2x}{3}\right)+\sin \left(\frac{3x}{2}\right)$ will be

If the period of the function $f\left(x\right)=\tan \left(\sqrt{\left[k\right]}x\right)$ is $\mathrm{\pi }$, where $\left[.\right]$ denotes the greatest integer function, then which of the following is correct$?$

A function is given by $f\left(x\right)=a^{\left\{\tan \left(\mathrm{\pi }x\right)+x-\left[x\right]\right\}}$, where $a>0, x\in R$ and $\left[·\right]$ denotes the greatest integer function. Then the period of $f\left(x\right)$ is

If the periods of the periodic functions $\mathrm{s}\mathrm{i}\mathrm{n}ax+\mathrm{c}\mathrm{o}\mathrm{s}ax$ and $\left|\mathrm{s}\mathrm{i}\mathrm{n}x\right|+\left|\mathrm{c}\mathrm{o}\mathrm{s}x\right|$ are equal, then $a$ is equal to

The period of the function $f\left(x\right)=\frac{\sin 8x\cos x-\sin 6x\cos 3x}{\cos 2x\cos x-\sin 3x\sin 4x}$ is

If the period of the function $f\left(x\right)=\frac{\sin \left(\sin \left(nx\right)\right)}{\tan \left(\frac{x}{n}\right)}, n\in N,$ is $6 \pi ,$ then $n$ is equal to

The period of the function $f\left(x\right)=\left|\sin x \right|-\left|\cos x\right|$ , is

The period of $f\left(x\right)=\sin \frac{2\pi x}{3}+\cos \frac{\pi x}{2}$ is

Let $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{3x}{10}$ for all real $x.$ The least natural number $n$ such that $f\left(10n\pi +x\right)=f\left(x\right)$ for all real values of $x$ is