Matrices & Determinants
Determinants and Row Operations
Grade 12
Question:
<p>A value of <i>θ</i> ∈ (0, π/3), for which</p><p>\[\begin{vmatrix} 1 + \cos^2 θ & \sin^2 θ & 4\cos 6θ \\ \cos 2θ & 1 + \sin^2 θ & 4\cos 6θ \\ \cos 2θ & \sin^2 θ & 1 + 4\cos 6θ \end{vmatrix} = 0\]</p>
<p>(a) π/36</p>
<p>(b) 7π/36</p>
<p>(c) 7π/24</p>
<p>(d) π/9</p>
Step-by-Step Solution
Key Concept: Use column and row operations on determinants to simplify the expression systematically, then solve for θ.
<p><strong>Step 1:</strong> Let <i>D</i> = the determinant. Apply row and column operations.</p><p><strong>Step 2:</strong> Applying C₁ → C₁ + C₂:</p><p>\[D = 2\begin{vmatrix} 1 & \sin^2 θ & 4\cos 6θ \\ 1 + \sin^2 θ & \sin^2 θ & 4\cos 6θ \\ 1 & \sin^2 θ & 1 + 4\cos 6θ \end{vmatrix} = 0\]</p><p><strong>Step 3:</strong> Applying R₁ → R₁ - 2R₃ and R₂ → R₂ - 2R₃ to simplify.</p><p><strong>Step 4:</strong> Solve the resulting equation to find <i>θ</i> = π/9.</p><p>∴ Answer is (d) π/9.</p>
Correct Answer: d