Question

Consider the function $f\left(x\right)=\sin \left(kx\right)+\left\{x\right\}$, where $\left\{x\right\}$ represents the fractional part function
Statement 1: $f\left(x\right)$ is periodic for $k=m\pi$, where $m$ is a rational number
Statement 2: The sum of two periodic functions is always periodic

• A. If both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1
• B. If both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
• C. If STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
• D. If STATEMENT 1 is FALSE and STATEMENT 2 is TRUE

Answer 1

The correct option is C If STATEMENT 1 is TRUE and STATEMENT 2 is FALSE

Period of $sin\left(kx\right)$ is $\frac{2\pi }{k}$
period of $x$ is 1
Hence period of $f\left(x\right)$ is LCM of $\frac{2\pi }{k}$ and 1.
since 1 is a rational number hence $\frac{2\pi }{k}$ should be rational number to get the LCM
Hence $k=m\pi$ where m is a rational number.
Sum of two periodic function is periodic if both the function has either rational period or irrational period of same category.