A) \[3\]
B) \[2/3\]
C) \[1/3\]
D) \[0\]
Correct Answer: C
Solution :
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 \[f(x)=\left| \begin{matrix} \sin x & \sin 2x & \sin 3x \\ 0 & 3 & 4\sin x \\ 0 & \sin x & 1 \\ \end{matrix} \right|\] \[=3\sin x-4{{\sin }^{3}}x=\sin 3x\] \[\therefore \] \[\int_{0}^{\pi /2}{f(x)}\,dx=\int_{0}^{\pi /2}{\sin 3x}\,dx\] \[=\left[ -\frac{\cos 3x}{3} \right]_{0}^{\pi /2}\] \[=-\frac{1}{3}\left[ \cos \frac{3\pi }{2}-\cos 0 \right]=\frac{1}{3}\]
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