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 :

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 : (A) 4 √2 (B) 2 √5 (C) 2 √7 (D) 20

Tardigrade Admin · Accepted Answer

By quadratic formula, the roots of this equation are:

α, β = \frac{λ - 2 \pm 4 - 4 λ + λ ^{2} - 40 + 4 λ}{2} = \frac{λ - 2 \pm λ ^{2} - 36}{2}

.
The magnitude of the difference of the roots is clearly

∣ ∣ λ^{2} - 36 ∣ ∣

We have,

α^{3} + β^{3} = \frac{( λ - 2 ) ^{3}}{4} + \frac{3 ( λ - 2 ) ( λ ^{2} - 36 )}{4} = \frac{( λ - 2 ) ( 4 λ ^{2} - 4 λ - 104 )}{4} = (λ -

2) (λ^{2} - λ - 26)

This function attains its minimum value at

λ = 4

.
Thus, the magnitude of the difference of the roots is clearly

∣ i 20 ∣ = 25

.

Q. If $λ \in R$ is such that the sum of the cubes of the roots of the equation, $x^{2} + (2 - λ) x + (10 - λ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is :

$42$

$25$

$27$

$20$

Q. 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 :

42​

25​

27​

20

Q. If $λ \in R$ is such that the sum of the cubes of the roots of the equation, $x^{2} + (2 - λ) x + (10 - λ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is :

$42$

$25$

$27$

$20$