Question

If $\Delta(x) = \left| \begin{matrix} 1 & \cos x & 1 - \cos x \\ 1 + \sin x & \cos x & 1 + \sin x - \cos x \\ \sin x & \sin x & 1 \end{matrix} \right|$, then $\int_{0}^{\pi/2}{\Delta(x)}dx$ is

equal to

• A. ¼
• B. ½
• C. 0
• D. –1/2

Answer 1

Answer: (D)

–1/2

Answer 2

Applying $C_{3} \rightarrow C_{3} + C_{2} - C_{1}$

1 & \cos x & 0 \\ 1 + \sin x & \cos x & 0 \\ \sin x & \sin x & 1 \end{matrix} \right| = \cos x - \cos x(1 + \sin x) = - \sin x\cos x$$ $$\therefore\int_{0}^{\pi/2}{\Delta(x)dx = - \frac{1}{2}}\int_{0}^{\pi/2}{\sin 2xdx} = - \frac{1}{2}\left\lbrack - \frac{\cos 2x}{2} \right\rbrack_{0}^{\pi/2} = \frac{1}{4}(\cos\pi - \cos 0) = - \frac{1}{2}$$