<p>If \(|2x + \sin^2 a| + |2x + 3 + 2\sin a| = 0\) and \(4\lambda^2 = 1\), find \(4\lambda^2\).</p>
Step-by-Step Solution
Key Concept: Sum of absolute values equals zero if and only if each absolute value term equals zero individually. This creates a system of two equations that must be satisfied simultaneously.
<p><strong>Step 1:</strong> For |2x + sin²a| + |2x + 3 + 2sin a| = 0, since sum of absolute values equals zero, each term must be zero:</p><p>2x + sin²a = 0 ... (1)</p><p>2x + 3 + 2sin a = 0 ... (2)</p><p><strong>Step 2:</strong> From (1): 2x = -sin²a</p><p>From (2): 2x = -3 - 2sin a</p><p><strong>Step 3:</strong> Equating both expressions for 2x:</p><p>-sin²a = -3 - 2sin a</p><p>sin²a - 2sin a - 3 = 0</p><p><strong>Step 4:</strong> Factoring: (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 5:</strong> From equation (1): 2x = -(-1)² = -1, so x = -1/2</p><p><strong>Step 6:</strong> The problem states 4λ² = 1 and asks to find 4λ² (which appears to be a transcription issue in the problem statement). The answer is: <strong>4λ² = 1</strong></p>
Correct Answer: 1