Sets, Relations & Functions
Sets and their Representations
Grade None
Question:
<p>Let <em>S</em> = {<em>x</em> ∈ ℝ : <em>x</em> ≥ 0 and \(2|\sqrt{x} - 3| + \sqrt{x}(\sqrt{x} - 6) + 6 = 0\)}. Then, <em>S</em></p>
<p>contains exactly one element.</p>
<p>contains exactly two elements.</p>
<p>contains exactly four elements.</p>
<p>is an empty set.</p>
Step-by-Step Solution
Key Concept: Split the absolute value equation into cases based on whether √x ≥ 3 or √x < 3, then solve each case algebraically and check which solutions satisfy the original constraint.
<p><strong>Step 1:</strong> Let u = √x where u ≥ 0. The equation becomes: 2|u - 3| + u(u - 6) + 6 = 0, or 2|u - 3| + u² - 6u + 6 = 0</p><p><strong>Step 2 (Case 1):</strong> If u ≥ 3: 2(u - 3) + u² - 6u + 6 = 0 → 2u - 6 + u² - 6u + 6 = 0 → u² - 4u = 0 → u(u - 4) = 0. Since u ≥ 3, we get u = 4.</p><p><strong>Step 3 (Case 2):</strong> If 0 ≤ u < 3: 2(3 - u) + u² - 6u + 6 = 0 → 6 - 2u + u² - 6u + 6 = 0 → u² - 8u + 12 = 0 → (u - 2)(u - 6) = 0. Since 0 ≤ u < 3, we get u = 2.</p><p><strong>Step 4:</strong> From u = 4: x = 16. From u = 2: x = 4. Verify both in original equation: both satisfy it.</p><p><strong>Step 5:</strong> Therefore S = {4, 16}, which is a non-empty finite set with exactly 2 elements.</p><p>∴ Answer: A</p>
Correct Answer: A