Question

If \[ f(x) = \begin{vmatrix} 2 \cos^4 x & 2 \sin^4 x & 3 + \sin^2 2x \\ 3 + 2 \cos^4 x & 2 \sin^4 x & \sin^2 2x \\ 2 \cos^4 x & 3 + 2 \sin^4 x & \sin^2 2x \end{vmatrix} \] then \( \frac{1}{5} f'(0) \) is equal to

• A. 0
• B. 1
• C. 2
• D. 6

Answer 1

The Correct Option is A

By simplifying the determinant using row operations:

\( R_2 \rightarrow R_2 - R_1 \),    \( R_3 \rightarrow R_3 - R_1 \)

we find that \( f(x) \) is constant. Therefore, \( f'(x) = 0 \).

Thus,

\[ \frac{1}{5} f'(0) = 0 \]