Question

If \( \Delta(x)=\begin{vmatrix} 1 & \cos x & 1-\cos x \\ 1+\sin x & \cos x & 1+\sin x-\cos x \\ \sin x & \sin x & 1 \end{vmatrix} \), then \( \int_0^{\pi/2} \Delta(x)\,dx \)

• A. \( -\frac{1}{2} \)
• B. \( \frac{1}{2} \)
• C. \( 1 \)
• D. \( -1 \)
• E. \( 0 \)

Hint:

If \(f(x)+f(a-x)=0\), integral over interval is zero.

Answer 1

Answer: (E)

Concept: Use symmetry property of definite integrals.

Step 1: Let: \[ I = \int_0^{\pi/2} \Delta(x) dx \]

Step 2: Use substitution: \[ x \to \frac{\pi}{2} - x \]

Step 3: Apply trig identities: \[ \sin(\frac{\pi}{2}-x)=\cos x,\quad \cos(\frac{\pi}{2}-x)=\sin x \]

Step 4: Show that: \[ \Delta(x) + \Delta\left(\frac{\pi}{2}-x\right) = 0 \]

Step 5: Hence: \[ I + I = 0 \Rightarrow I = 0 \]

Step 6: Final: \[ \boxed{0} \]