The values of λ such that sum of the squares of the roots of the quadratic equation, x2 + (3 - λ) x + 2 = λ has the least value is :

Question

The values of λ such that sum of the squares of the roots of the quadratic equation, x2 + (3 - λ) x + 2 = λ has the least value is : (A) 2 (B) (4/9) (C) (15/8) (D) 1

Tardigrade Admin · Accepted Answer

α+β=λ-3 αβ=2-λ α2+β2 =(α+β)2 -2αβ =(λ-3)2 -2(2-λ) =λ2 +9-6λ-4+2λ =λ2 -4λ+5 =(λ-2)2+1 ∴ λ=2 Option (1)

Q. The values of $λ$ such that sum of the squares of the roots of the quadratic equation, $x^{2} + (3 - λ) x + 2 = λ$ has the least value is :

$2$

$\frac{4}{9}$

$\frac{15}{8}$

$1$

$α + β = λ - 3$
$α β = 2 - λ$
$α^{2} + β^{2} = (α + β)^{2} - 2 α β = (λ - 3)^{2} - 2 (2 - λ)$
$= λ^{2} + 9 - 6 λ - 4 + 2 λ$
$= λ^{2} - 4 λ + 5$
$= (λ - 2)^{2} + 1$
$∴ λ = 2$
Option (1)

Q. The values of λ such that sum of the squares of the roots of the quadratic equation, x2+(3−λ)x+2=λ has the least value is :

2

94​

815​

1

α+β=λ−3 αβ=2−λ α2+β2=(α+β)2−2αβ=(λ−3)2−2(2−λ) =λ2+9−6λ−4+2λ =λ2−4λ+5 =(λ−2)2+1 ∴λ=2 Option (1)

Q. The values of $λ$ such that sum of the squares of the roots of the quadratic equation, $x^{2} + (3 - λ) x + 2 = λ$ has the least value is :

$2$

$\frac{4}{9}$

$\frac{15}{8}$

$1$

$α + β = λ - 3$
$α β = 2 - λ$
$α^{2} + β^{2} = (α + β)^{2} - 2 α β = (λ - 3)^{2} - 2 (2 - λ)$
$= λ^{2} + 9 - 6 λ - 4 + 2 λ$
$= λ^{2} - 4 λ + 5$
$= (λ - 2)^{2} + 1$
$∴ λ = 2$
Option (1)