Question

If $f\left(x\right)=\left|\begin{array}{ccc}\sin 2x(1+2\cos x)& \sin 2x& \sin 3x\\ 3+4\sin x& 3& 4\sin x\\ 1+\sin x& \sin x& 1\end{array}\right|$
then the value of ${\int }_0^{\pi /2}f\left(x\right)$ is

• A. 3
• B. 2/3
• C. 1/3
• D. 0

Answer 1

Answer: (C)

1/3

$f\left(x\right)=\left|\begin{array}{ccc}\sin 2x(1+2\cos x)& \sin 2x& \sin 3x\\ 3+4\sin x& 3& 4\sin x\\ 1+\sin x& \sin x& 1\end{array}\right|$

Applying $C_1\to C_1-C_2-C_3$

$f\left(x\right)=\left|\begin{array}{ccc}2\cos x\sin 2x-\sin 3x& \sin 2x& \sin 3x\\ 0& 3& 4\sin x\\ 0& \sin x& 1\end{array}\right|$

$f\left(x\right)=(2\cos x\sin 2x-\sin 3x)(3-4{\sin }^{2}x)-\sin 2x(0-0)+\sin 3x(0-0)$

$f\left(x\right)=(2\cos x\sin 2x-\sin 3x)(3-4{\sin }^{2}x)$

$f\left(x\right)=\sin 3x$

Now,

${\int }_0^{\pi /2}f\left(x\right)$$={\int }_0^{\pi /2}\sin 3xdx$

$=\frac{1}{3}{[-\cos 3x]}_0^{\pi /2}$

$=\frac{1}{3}$