Question

Choose the correct statement:

• A. x+sin2x is a periodic function
• B. x+sin2x is not a periodic function
• C. cos(√x +1) is a periodic function
• D. cos(√x +1) is not a periodic function

Hint:

To check periodicity, analyze each term of the function separately. If any term is nonperiodic, the entire function cannot be periodic.

Answer 1

The Correct Option is B, D

1. Check periodicity of \(x + \sin 2x\):

- The term \(\sin 2x\) is periodic with a period of \(\pi\).

- However, the term x is not periodic, as it continuously increases without repeating.

- Since the sum of a periodic function (\(\sin 2x\)) and a non-periodic function (x) cannot be periodic, \(x + \sin 2x\) is not periodic.

2. Check periodicity of \(\cos(\sqrt{x} + 1)\):

- The term \(\sqrt{x}\) is not periodic, as it is a continuously increasing function.

- Adding 1 to \(\sqrt{x}\) does not change its non-periodic nature.

- Since \(\cos(\sqrt{x} + 1)\) depends on a non-periodic term, it is also not periodic.

Thus, the correct answers are (B) and (D).