Matrices & Determinants
Determinant with Trigonometry
Grade 12
Question:
<p>If \(f(x) = \begin{vmatrix} \sec^2 x & 1 & 1 \\ \cos^2 x & \cos^2 x & \csc^2 x \\ 1 & \cos^2 x & \cot^2 x \end{vmatrix}\), then which statement is correct?</p>
<p>(a) \(\int_0^{\pi/4} f(x)\,dx = \frac{1}{6}(3\pi - 8)\)</p>
<p>(b) \(f'(3\pi/4) = 0\)</p>
<p>(c) The maximum value of \(f(x)\) is \(1\)</p>
<p>(d) The minimum value of \(f(x)\) is \(0\)</p>
Step-by-Step Solution
Key Concept: Use trigonometric identities to simplify the determinant before differentiation.
<p>Expand the determinant using trigonometric identities. Differentiate and evaluate at $x = 3\pi/4$ to verify the critical point.</p>
Correct Answer: B