Question

$sinax+cosax$ and $\left|sinx\right|+\left|cosx\right|$ are periodic with the same fundamental period, if $a$ equals to

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (D)

$sinax+cosax=\sqrt{2}sin\left(ax+\frac{\pi }{4}\right)$
So, fundamental period $=\frac{2\pi }{|a|}$
Let, $T$ be the fundamental period of $\left|sinx\right|+\left|cosx\right|$
So $\left|sin(x+T)\right|+\left|cos(x+T)\right|=\left|sinx\right|+\left|cosx\right|.$
Squaring both sides, we get,
$1+\left|sin(2x+2T)\right|=1+\left|sin2x\right|$
Therefore, the fundamental period of $\left|sinx\right|+\left|cosx\right|$ is $\frac{\pi }{2}$
Hence, $\frac{2\pi }{|a|}=\frac{\pi }{2}$
$\Rightarrow a=\pm 4$