Question

If the function $f ( x )=\lambda|\sin x |+\lambda^2|\cos x |+ g (\lambda), \lambda \in R$ ( $g$ 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

Answer: (B)

$f \left(\frac{\pi}{2}+ x \right)= f ( x ) \forall x \in R$
$\lambda|\cos x |+\lambda^2|\sin x |+ g (\lambda)=\lambda|\sin x |+\lambda^2|\cos x |+ g (\lambda) $
$\left(\lambda-\lambda^2\right)|\cos x |+\left(\lambda^2-\lambda\right)|\sin x |=0 \forall x \in R$
$\lambda-\lambda^2=0 \Rightarrow \lambda=0,1 \text { but } \lambda=0 \text { (rejected) } $
$\Rightarrow \lambda=1 $