Question

The period of the function \[ f(x) = \begin{vmatrix} \sin x & \cos x & \sin x \cos x \\ \sin x & \cos x & \sin x \\ \cos x & \sin x & \sin x \end{vmatrix} \] is:

• A. \(\pi\)
• B. \(2\pi\)
• C. \(\frac{\pi}{2}\)
• D. None of these

Hint:

If two rows or columns of a determinant are identical, the determinant value is zero.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Evaluate the determinant to find the function.

Step 2: Detailed Explanation:
The matrix has first and third rows identical: \((\sin x, \cos x, \sin x)\) and \((\sin x, \cos x, \sin x)\). Determinant = 0 for all \(x\).
So \(f(x) = 0\), constant function.
Constant function has any period, but typically period is not defined or is said to have no fundamental period. Among options, none specify constant function's period.

Step 3: Final Answer:
Option (D) None of these.