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

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

If $f\left(x\right)$ be a function such that $f\left(x-1\right)+f\left(x+1\right)=\sqrt{3}f\left(x\right)$ and $f\left(5\right)=10,$ then the sum of digits of the value of $\sum _{r=0}^{19}f\left(5+12r\right)$ is

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 $x$.]

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

If period of $\frac{\cos \left(sinnx\right)}{\tan \left(x/n\right)}\left(n\in N\right)$ is $6\pi$, then find value of $n$ is

Find period of function $2^{\left\{x\right\}}+\mathrm{sin\pi }x+3^{\left\{x/2\right\}}+\cos 2\pi x$ is (where $\left\{\right\}$ represent fractional part of $x$)

Find the period of the function satisfying the relation $f\left(x\right)+f\left(x+3\right)=0 \forall x\in R.$

Let $f\left(x\right)$ be a periodic differentiable function, $g\left(x\right)=\int f\left(x\right)dx;h\left(x\right)={\int }_a^{b}f\left(x\right)dx,$ where $a,b$ are constants. Consider the statements
(i) $g\left(x\right)$ may be a periodic function
(ii) $h\left(x\right)$ may be a periodic function
(iii) $g\left(x\right)$ may not be a periodic function
(iv) $h\left(x\right)$ may not be a periodic function

Then number of statements(s) which is/are true is

Which of the following functions is periodic?

The fundamental period of trigonometric function $f\left(x\right)=\frac{\sin x +\mathrm{ }\sin 3x\mathrm{ }+\sin 5x+.....+\sin 2017x}{\cos x\mathrm{ }+\mathrm{ cos}3x\mathrm{ }+\mathrm{ cos}5x\mathrm{ }+.....+\cos 2017x},$ is: