Question

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 ______.

विकल्प

• 20

• `2sqrt(5)`

• `2sqrt(7)`

• `4sqrt(2)`

Answer 1

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 `underlinebb(2sqrt(5))`.

Explanation:

Let, the roots of the equation,

x² + (2 – λ)x + (10 – λ) = 0 are α and β.

Also roots of the given equation are

`(λ - 2 ± sqrt(4 - 4λ + λ^2 - 40 + 4λ))/2 = (λ - 2 ± sqrt(λ^2 - 36))/2`

The magnitude of the difference of the roots is `|sqrt(λ^2 - 36)|`

So, α³ + β³ = (α + β)³ – 3αβ(α + β)

= (λ – 2)² – 3(10 – λ)(λ – 2)

= (λ – 2)(λ² – λ – 26)

= f(λ)

As f(λ) attains its minimum value at λ = 4.

Therefore, the magintude of the difference of the roots is `|"i" sqrt(20)| = 2sqrt(5)`