Question

If the function $f\left(x\right)=\lambda |\sin x|+{\lambda }^2|\cos x|+g\left(\lambda \right),\lambda \in R,$ where $g$ is a function of $\lambda ,$ is periodic with fundamental period $\frac{\pi }{2},$ then

• A. $\lambda =0,1$
• B. $\lambda =1$
• C. $\lambda =0$
• D. $\lambda =-1$

Answer 1

The correct option is B $\lambda =1$

Given $f\left(x\right)$ is periodic with fundamental period $\frac{\pi }{2}.$
So, $f(\frac{\pi }{2}+x)=f\left(x\right)\forall x\in R$
$\Rightarrow \lambda |\cos x|+{\lambda }^2|\sin x|+g\left(\lambda \right)=\lambda |\sin x|+{\lambda }^2|\cos x|+g\left(\lambda \right)$
$\Rightarrow (\lambda -{\lambda }^2)|\cos x|+({\lambda }^2-\lambda )|\sin x|=0\forall x\in R$
$\Rightarrow \lambda -{\lambda }^2=0\Rightarrow \lambda =0,1$
But $\lambda =0$ is rejected as if $\lambda =0,$ then $f\left(x\right)$ becomes a constant function.
Hence, $\lambda =1$