<p>The number of distinct real roots of \(\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0\) in the interval \(-\pi/4 \leq x \leq \pi/4\) is</p>
Step-by-Step Solution
Key Concept: Factor the determinant by adding all rows to the first row to get a common factor, then use the eigenvalue structure of this symmetric matrix (two equal diagonal elements imply a repeated eigenvalue).
<p><strong>Step 1:</strong> Add all rows to the first row:</p><p>R₁ → R₁ + R₂ + R₃</p><p>First row becomes: (sin x + 2cos x, sin x + 2cos x, sin x + 2cos x)</p><p><strong>Step 2:</strong> Factor out (sin x + 2cos x):</p><p>Δ = (sin x + 2cos x) · <begin>vmatrix</begin> 1 & 1 & 1 \\ cos x & sin x & cos x \\ cos x & cos x & sin x \end{vmatrix}</p><p><strong>Step 3:</strong> Simplify the remaining determinant by R₂ → R₂ - R₁ and R₃ → R₃ - R₁:</p><p>= (sin x + 2cos x) · <begin>vmatrix</begin> 1 & 1 & 1 \\ cos x - 1 & sin x - 1 & cos x - 1 \\ cos x - 1 & cos x - 1 & sin x - 1 \end{vmatrix}</p><p><strong>Step 4:</strong> Further simplify to get:</p><p>Δ = (sin x + 2cos x)(sin x - cos x)² = 0</p><p><strong>Step 5:</strong> Solve in [-π/4, π/4]:</p><p>• sin x + 2cos x = 0 ⟹ tan x = -2 (no solution in interval since tan x ∈ [-1, 1])</p><p>• sin x - cos x = 0 ⟹ tan x = 1 ⟹ x = π/4 ✓</p><p>∴ <strong>Answer: 1</strong></p>
Correct Answer: C