<p>The number of distinct real roots of the equation \(\begin{vmatrix} \cos x & \sin x & \sin x \\ \sin x & \cos x & \sin x \\ \sin x & \sin x & \cos x \end{vmatrix} = 0\) in the interval \(\left[-\dfrac{\pi}{4}, \dfrac{\pi}{4}\right]\) is</p>
Step-by-Step Solution
Key Concept: Factor the determinant by adding all rows to the first row, yielding (cos x + 2sin x) as a common factor. Then solve (cos x + 2sin x)(cos x - sin x)² = 0 within the given interval.
<p><strong>Step 1:</strong> Add all three rows and factor from the first row:</p><p>R₁ → R₁ + R₂ + R₃ gives first row as (cos x + 2sin x, cos x + 2sin x, cos x + 2sin x)</p><p>Factor out (cos x + 2sin x): determinant = (cos x + 2sin x) · M</p><p><strong>Step 2:</strong> The remaining 2×2 minor M simplifies to (cos x - sin x)²</p><p>So: (cos x + 2sin x)(cos x - sin x)² = 0</p><p><strong>Step 3:</strong> Case 1 – cos x - sin x = 0 ⟹ tan x = 1 ⟹ x = π/4 ✓ (in interval)</p><p><strong>Step 4:</strong> Case 2 – cos x + 2sin x = 0 ⟹ tan x = -1/2</p><p>Since tan(-π/4) = -1 and tan(0) = 0, and -1/2 lies between them, there exists unique x₀ ∈ (-π/4, 0) ✓</p><p><strong>Step 5:</strong> Total distinct real roots in [-π/4, π/4]: x = π/4 and x = arctan(-1/2)</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: A