If λ∈R is such that the sum of the cubes of the roots of the equation x²+(2−λ)x+(10−λ)=0 is minimum, then the magnitude of the difference of the roots of this equation is

By quadratic formula, the roots of this equation are:
α, β = (λ−2)±√(2−λ)²−4(10−λ) = (λ−2)±√λ²−4λ+4−40+4λ = (λ−2)±√λ²−36
The magnitude of the difference of the roots is clearly |√λ²−36|
We have,
α³+β³ = ((λ−2)+√λ²−36)³ + ((λ−2)−√λ²−36)³
= 2(λ−2)³ + 6(λ−2)(λ²−36)
= 2(λ−2)( (λ−2)² + 3(λ²−36))
= 2(λ−2)(λ²−4λ+4+3λ²−108)
= 2(λ−2)(4λ²−4λ−104)
= 8(λ−2)(λ²−λ−26)
This function attains its minimum value at λ=4.
Thus, the magnitude of the difference of the roots is clearly |√4²−36| = |√−20| = 2√5.
So the correct answer is option C.