Question

If $ f(x)= $ $ \left| \begin{matrix} \sin x+\sin 2x+\sin 3x & \sin 2x & \sin 3x \\ 3+4\sin x & 3 & 4\sin x \\ 1+\sin x & \sin x & 1 \\ \end{matrix} \right| $ , then the value of $ \int_{0}^{\pi /2}{f(x)}dx $ is

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

Answer 1

Answer: (C)

We have, $ f(x)=\left| \begin{matrix} \sin x+\sin 2x+\sin 3x & \sin 2x & \sin 3x \\ 3+4\sin x & 3 & 4\sin x \\ 1+\sin x & \sin x & 1 \\ \end{matrix} \right| $ Applying $ {{C}_{1}}\to {{C}_{1}}-{{C}_{2}}-{{C}_{3}} $ ,
we get $ f(x)=\left| \begin{matrix} \sin x & \sin 2x & \sin 3x \\ 0 & 3 & 4\sin x \\ 0 & \sin x & 1 \\ \end{matrix} \right| $
$ =3\sin x-4{{\sin }^{3}}x=\sin 3x $
$ \therefore $ $ \int_{0}^{\pi /2}{f(x)}\,dx=\int_{0}^{\pi /2}{\sin 3x}\,dx $
$ =\left[ -\frac{\cos 3x}{3} \right]_{0}^{\pi /2} $
$ =-\frac{1}{3}\left[ \cos \frac{3\pi }{2}-\cos 0 \right]=\frac{1}{3} $