Question

Let $f\left(x\right)=\{\begin{array}{cc}-1+\sin K_1\pi x,& \text{x is rational}\\ 1+\cos K_2\pi x,& \text{x is irrational}\end{array}$

If $f\left(x\right)$ is a periodic function, then

• A. either $K_1,K_2\in$ rational or $K_1,K_2\in$ irrational
• B. $K_1,K_2\in$ rational only
• C. $K_1,K_2\in$ irrational only
• D. $K_1,K_2\in$ irrational such that $\frac{K_1}{K_2}$ is rational

Answer 1

The correct option is B $K_1,K_2\in$ rational only

Range of $-1+sin{K}_1\pi x$ is [-2, 0] and range of $1+cos{K}_2\pi x$ is [0,2].
$\Rightarrow$ If $g\left(x\right)=-1+sin{K}_1\pi x$
$\Rightarrow g(x+T_1)=-1+sin{K}_1\pi x$, where $T_1$ is a period of $1+sin{K}_1\pi x$
$\Rightarrow T_1$ is rational $\Rightarrow$ period of f(x) is rational.
$\Rightarrow K_1$ and $K_2$ are rational.
Hence,option 'B' is correct.