<p>If the equation \(|2x + \sin^2 a| + |2x + 3 + 2\sin a| = 0\) has exactly one solution \(x = \lambda\) (where \(a\) is a constant) then find the value of \(4\lambda^2\).</p>
Step-by-Step Solution
Key Concept: A sum of absolute values equals zero if and only if each term equals zero simultaneously. This means both expressions inside the absolute values must equal zero at the same point, creating a system of equations to solve for both a and λ.
<p><strong>Step 1:</strong> Since |2x + sin²a| + |2x + 3 + 2sin a| = 0, and both absolute values are non-negative, each must equal zero:</p><p>2x + sin²a = 0 ... (1)</p><p>2x + 3 + 2sin a = 0 ... (2)</p><p><strong>Step 2:</strong> Subtract equation (1) from equation (2):</p><p>(2x + 3 + 2sin a) - (2x + sin²a) = 0</p><p>3 + 2sin a - sin²a = 0</p><p>sin²a - 2sin a - 3 = 0</p><p><strong>Step 3:</strong> Factor the quadratic in sin a:</p><p>(sin a - 3)(sin a + 1) = 0</p><p>So sin a = 3 or sin a = -1</p><p>Since -1 ≤ sin a ≤ 1, we have sin a = -1</p><p><strong>Step 4:</strong> Substitute sin a = -1 into equation (1):</p><p>2x + (-1)² = 0</p><p>2x + 1 = 0</p><p>x = -1/2</p><p>Therefore λ = -1/2</p><p><strong>Step 5:</strong> Calculate 4λ²:</p><p>4λ² = 4 × (-1/2)² = 4 × 1/4 = 1</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1