<p>Let the roots of the equation \(x^2 + (2-\lambda)x + (10-\lambda) = 0\) be \(\alpha\) and \(\beta\). The minimum value of \(\alpha^3 + \beta^3\) occurs at \(\lambda = 4\). What is \(|\alpha - \beta|\)?</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to express α³ + β³ in terms of λ, then minimize using calculus. The condition that minimum occurs at λ = 4 is self-consistent (verification), and |α - β| = √(discriminant) evaluated at that λ.
<p><strong>Step 1:</strong> From Vieta's formulas: α + β = λ - 2 and αβ = 10 - λ</p><p><strong>Step 2:</strong> Express α³ + β³ = (α + β)³ - 3αβ(α + β)</p><p>= (λ - 2)³ - 3(10 - λ)(λ - 2)</p><p>= (λ - 2)[(λ - 2)² - 3(10 - λ)]</p><p>= (λ - 2)[λ² - 4λ + 4 - 30 + 3λ]</p><p>= (λ - 2)[λ² - λ - 26]</p><p><strong>Step 3:</strong> To find extremum: d/dλ[α³ + β³] = (λ² - λ - 26) + (λ - 2)(2λ - 1) = 0</p><p>= λ² - λ - 26 + 2λ² - 5λ + 2 = 0</p><p>= 3λ² - 6λ - 24 = 0 ⟹ λ² - 2λ - 8 = 0</p><p>⟹ (λ - 4)(λ + 2) = 0 ⟹ λ = 4 or λ = -2</p><p><strong>Step 4:</strong> At λ = 4 (minimum): Discriminant = (λ - 2)² - 4(10 - λ)</p><p>= (4 - 2)² - 4(10 - 4) = 4 - 24 = -20</p><p><strong>Step 5:</strong> |α - β| = √(Discriminant) = √|−20| = √20 = <strong>2√5</strong></p><p>∴ Answer: A</p>
Correct Answer: A