Question

If $f\left(x\right)=$ $\left|\begin{array}{ccc}\sin x& 1& 0\\ 1& 2\sin x& 1\\ 0& 1& 2\sin x\end{array}\right|$ then ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}f\left(x\right)$ equals

• A. $0$
• B. $-1$
• C. $1$
• D. $\frac{3\pi }{2}$

Answer 1

Answer: (A)

$0$

$f\left(x\right)=\left|\begin{array}{ccc}\sin x& 1& 0\\ 1& 2\sin x& 1\\ 0& 1& 2\sin x\end{array}\right|$
Expanding along first column we get,
$f\left(x\right)=\sin x(4{\sin }^{2}x-1)-1(2\sin x-0)=4{\sin }^{3}x-3\sin x$
Clearly $f(-x)=f\left(x\right)\Rightarrow$ f is an odd function
Hence ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}f\left(x\right)=0$