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. $\frac{2}{3}$
• C. $\frac{1}{3}$
• D. 0

Answer 1

Answer: (C)

$\frac{1}{3}$

Answer 2

C₂ ® C₂ + C₃

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

C₁ ® C₁ – C₂

f(x) = $\left| \begin{matrix} \sin x & \sin 2x + \sin 3x & \sin 3x \\ 0 & 3 + 4\sin x & 4\sin x \\ 0 & 1 + \sin x & 1 \end{matrix} \right|$

f(x) = sin x (3 – 4 sin²x) = 3sin x – 4sin³x f(x)

= sin 3x

\ $\int_{0}^{\pi/2}{\sin 3xdx}$ = – $\lbrack\cos 3x/3\rbrack_{0}^{\pi/2}$ = 1/3