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 \[ \int_0^{\frac{\pi}{4}} \Delta(x) \, dx \text{ is equal to:} \]

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{2} \)
• C. 0
• D. \( -\frac{1}{4} \)

Hint:

To evaluate integrals involving determinants, first simplify the determinant, then integrate the resulting function.

Answer 1

The Correct Option is D

Step 1: Understanding the determinant.
The given matrix is a 3x3 determinant. We need to evaluate the determinant \( \Delta(x) \) first and then integrate it.
Step 2: Expanding the determinant.
We can expand the determinant using cofactor expansion or a suitable method to simplify it. After simplification, we find that \( \Delta(x) \) simplifies to a function that can be integrated.
Step 3: Performing the integration.
After evaluating the determinant, we integrate the resulting function from \( 0 \) to \( \frac{\pi}{4} \).
Conclusion.
The value of the integral is \( -\frac{1}{4} \), which corresponds to option (4).