Periodic Functions

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

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

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

The period of $f\left(x\right)=x-\left[x\right],$ if it is periodic, is

If $f\left(x\right)=2^{\mathrm{s}\mathrm{i}{\mathrm{n}}^3\pi x+x-\left[x\right]}$ , where $[*]$denotes the integral part of $x$ is a periodic function with period

Let $f:R\to R$ be a periodic function such that $f\left(T+x\right)=1+{\left[1-3f\left(x\right)+3{\left(f\left(x\right)\right)}^2-{\left(f\left(x\right)\right)}^3\right]}^{\frac{1}{3}}$, where $T$ is a fixed positive number, then period of $f\left(x\right)$ is

If $f\left(x\right)=2^{\mathrm{s}\mathrm{i}{\mathrm{n}}^3\pi x+x-\left[x\right]}$, where $\left[·\right]$ denotes the integral part of $x$ is a periodic function with period

Let $f:R\to R$ be a periodic function such that $f\left(T+x\right)=1+{\left[1-3f\left(x\right)+3{\left(f\left(x\right)\right)}^2-{\left(f\left(x\right)\right)}^3\right]}^{\frac{1}{3}}$ where $T$ is a fixed positive number, then period of $f\left(x\right)$ is