$\begin{array}{l} I f λ \in R i s s u c h t h a t t h e s u m o f t h e c u b e s o f t h e r o o t s o f x^{2} + (2 - λ) x + (10 - λ) = 0 i s \min i m u m, t h e n t h e m a g n i t u d e o f t h e d i f f e r e n c e o f t h e r o o t s o f t h i s e q u a t i o n i s \end{array}$

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$2 \sqrt{7}$

$4 \sqrt{2}$

$2 \sqrt{5}$

answer is B.

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Detailed Solution

$\begin{array}{l} L e t α, β a r e r o o t s o f x^{2} + (2 - λ) x + (10 - λ) = 0 \\ T h e n α + β = λ - 2, α β = 10 - λ \\ α^{3} + β^{3} = {(α + β)}^{3} - 3 α β (α + β) \\ = {(λ - 2)}^{3} - 3 (10 - λ) (λ - 2) \\ = (λ - 2) [{(λ - 2)}^{2} - 3 (10 - λ)] \\ = (λ - 2) (λ^{2} - λ - 26) \\ T h e \max i m u m v a l u e a t t a i n a t λ = 4 \\ ∴ | α - β | = | \sqrt{{(α + β)}^{2} - 4 α β} | \\ = | \sqrt{{(λ - 2)}^{2} - 4 (10 - λ)} | \\ = | \sqrt{{(4 - 2)}^{2} - 4 (10 - 4)} | \\ = | \sqrt{4 - 24} | \\ = | i \sqrt{20} | \\ = 2 \sqrt{5} \end{array}$