Question

If $f\left(x\right)=\left|\begin{array}{ccc}\cos x& 1& 0\\ 1& 2\cos x& 1\\ 0& 1& 2\cos x\end{array}\right|$, then ${\int }_0^{\pi /2}f\left(x\right)dx$ is

• A. $1/4$
• B. $-1/3$
• C. $1/2$
• D. $0$

Answer 1

The correct option is B $-1/3$

$f\left(x\right)=\left|\begin{array}{ccc}\cos x& 1& 0\\ 1& 2\cos x& 1\\ 0& 1& 2\cos x\end{array}\right|$

$=\cos x(4{\cos }^{2}x–1)–1\left(2\cos x\right)$

$=4{\cos }^{3}x–3\cos x$

$=\cos 3x$

${\int }_0^{\frac{\pi }{2}}\cos 3x\cdot dx={\left[\frac{\sin 3x}{3}\right]}_0^{\frac{\pi }{2}}$

$=\frac{\sin 3\left(\frac{\pi }{2}\right)}{3}-\frac{\sin 3\left(0\right)}{3}$

$=-\frac{1}{3}$.