Question

If $f(x)=\begin{vmatrix}\cos x & 1 & 0 \\ 1 & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x\end{vmatrix}$, then $\int\limits_{0}^{\pi / 2} f(x) d x=$

• A. $\frac{1}{4}$
• B. $-\frac{1}{3}$
• C. $\frac{1}{2}$
• D. 1

Answer 1

Answer: (B)

$f(x)=\begin{vmatrix}\cos x & 1 & 0 \\ 1 & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x\end{vmatrix}$
$=\cos x\left(4 \cos ^{2} x-1\right)-1\left(2 \cos ^{2} x\right)+0$
$=4 \cos ^{3} x-\cos x-2 \cos x=4 \cos ^{3} x-3 \cos x=\cos 3 x$
$\therefore \int\limits_{0}^{\frac{\pi}{2}} f(x)=\int\limits_{0}^{\frac{\pi}{2}} \cos 3 x=\left(\frac{\sin 3 x}{3}\right)_{0}^{\frac{\pi}{2}}=-\frac{1}{3}$