Question

$\text { Let } f(\theta)=\begin{vmatrix} \sin ^2 \theta & \cos ^2 \theta & 1+4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ 1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \end{vmatrix}$ then $f$ is

• A. a non periodic function
• B. periodic with period $\pi$
• C. periodic with period $\pi / 2$
• D. odd function

Answer 1

Answer: (C)

$f(\theta)=\begin{vmatrix}-1 & 0 & 1 \\ -1 & 1 & 0 \\ 1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 2 \theta\end{vmatrix}$
$\left(R_1 \rightarrow R_1-R_3\right. \left.R_2 \rightarrow R_2-R_3\right)$
$\begin{vmatrix}0&0&1\\ \cdot1&1&0\\ 1+\sin^{2}\theta +4 \sin 4\theta&\cos^{2}\theta&4 sin 4\theta\end{vmatrix}$
$=-\left(\cos ^2 \theta+1+\sin ^2 \theta+4 \sin 4 \theta\right) $
$=-2(1+2 \sin 4 \theta)$
which is periodic function with period $\frac{2 \pi}{4}=\frac{\pi}{2}$.