Periodic Functions

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

If range of the function $f\left(x\right)={\cos }^2\left(\cos x\right)+{\sin }^2\left(\sin x\right),$ is $[a, b]$ then.

Which of the following functions is/are periodic ?

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 $f\left(x\right)=\tan \left(4x-1\right)$

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

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

If the graph of the curve$y=f\left(x\right)$ is symmetric about the lines $x=1$ and $x=2$ then which of the followings is/are true?

Fundamental period of the function $f\left(x\right)=\frac{1}{\left|\right|\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{ }4x|-|\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{ }4x\left|\right|}+\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{ }\left(\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{ }6x\right)$ is

The period of the function $f\left(x\right)=\sin \left(\sin \frac{x}{5}\right)$ is