Question

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

• A. $2$
• B. $1$
• C. $3$
• D. None of these

Answer 1

Answer: (A)

$2$

Note that the period of $\cos \left(2\pi x\right)$ is $1$ and that of $\sin \left(\pi x\right)$ is $2$.
The period of the function $\left\{x\right\}$ is $1$ as $\left\{x+1\right\}=\left\{x\right\}$. Hence the period of the function $2^{\left\{x\right\}}$ is also $1$.
Similarly the period of the function $\left\{\frac{x}{2}\right\}$ is 2, hence the period of $3^{\left\{x/2\right\}}$ is 2.
therefore the period of the function $2^{\left\{x\right\}}+\sin \left(\pi x\right)+3^{\left\{x/2\right\}}+\cos \left(2\pi x\right)$ is the LCM of the periods of the four functions i.e LCM (1,1,2,2) which is equal to $2$.