Periodic Functions

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

Find the fundamental period of the function $f(x)=3 \sin \frac{\pi x}{3}+4 \cos \frac{\pi x}{4}$
 

Find the fundamental period of the function $f(x)=4 \cos (2 x+3)$

The period of $f(x)=\sin 2 \sqrt{x+1}$ is:-

The fundamental period of $f\left(x\right)=\sin x$ is $2\mathrm{\pi }$. So the fundamental period of $g\left(x\right)=\sin \frac{x}{6}$ is:-

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

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

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 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 ?