Question

Let $12\pi$ be periodic, then p must be.

• A. Rational
• B. Irrational
• C. Positive real number
• D. None of these

Answer 1

Answer: (A)

Rational

Answer 2

Let $f ( x )$ be periodic with period $\lambda$ then $\sin ( x + \lambda ) + \cos p ( x + \lambda ) = \sin x + \cos p x , \forall x \in R$

Putting $x = 0$ and replace $\lambda$ by $- \lambda$, we have $\sin \lambda + \cos p \lambda = 1$ and $- \sin \lambda + \cos p \lambda = 1$

Solving these, we get $\sin \lambda = 0$ so $\lambda = n \pi$ and

$\cos p \lambda = 1$ so $p \lambda = 2 m \pi$ As $\lambda \neq 0 , m$ and $n$ are non-zero integers. Hence $p = \frac { 2 m \pi } { \lambda }$which is rational.