Matrices & Determinants
Determinants with Parametric Functions
Grade 12
Question:
<p>If \(\Delta = \begin{vmatrix} 1 & 3\cos\phi & 1 \\ \sin\phi & 1 & 3\cos\phi \\ 1 & \sin\phi & 1 \end{vmatrix}\), the maximum value of \(\Delta\) is</p>
<p>(a) \(-10\)</p>
<p>(b) \(-\sqrt{10}\)</p>
<p>(c) \(\sqrt{10}\)</p>
<p>(d) \(10\)</p>
Step-by-Step Solution
Key Concept: The determinant becomes a function of $\sin\phi$ and $\cos\phi$. Its extremum can be found by expressing it in the form $A\sin(\phi + \theta) + B$ and identifying the maximum.
<p>Expand the determinant in terms of $\sin\phi$ and $\cos\phi$ to obtain a linear combination of these functions. Use calculus or trigonometric identities to find the maximum. The maximum value occurs when the linear combination is optimized.</p>
Correct Answer: D