Question

$\text{ Let }f(\theta )=\left|\begin{array}{ccc}{\sin }^2\theta & {\cos }^2\theta & 1+4\sin 4\theta \\ {\sin }^2\theta & 1+{\cos }^2\theta & 4\sin 4\theta \\ 1+{\sin }^2\theta & {\cos }^2\theta & 4\sin 4\theta \end{array}\right|$ 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)

$f(\theta )=\left|\begin{array}{ccc}-1& 0& 1\\ -1& 1& 0\\ 1+{\sin }^2\theta & {\cos }^2\theta & 4\sin 2\theta \end{array}\right|$
$(R_1\to R_1-R_3{R}_2\to R_2-R_3)$
$\left|\begin{array}{ccc}0& 0& 1\\ \cdot 1& 1& 0\\ 1+{\sin }^2\theta +4\sin 4\theta & {\cos }^2\theta & 4sin4\theta \end{array}\right|$
$=-\left({\cos }^2\theta +1+{\sin }^2\theta +4\sin 4\theta \right)$
$=-2(1+2\sin 4\theta )$
which is periodic function with period $\frac{2\pi }{4}=\frac{\pi }{2}$.