Question

Let
$( {λ} ^{2} + 1) {x}^{2} - 4λx + 2 = 0$
be a quadratic equation then set of values of λ is exactly one root of quadratic equation lies in (0, 1) is ​

Answer 1

it has given that (λ² + 1)x² - 4λx + 2 = 0 be a quadratic equation.

To find : The set of values of λ, if exactly one root of quadratic equation lies in (0, 1) is .....

solution : concept : if α and β are the roots of quadratic function, y = f(x) such that one root α lies in (k₁, k₂).

then, f(k₁)f(k₂) < 0 and D ≥ 0

let's use this condition here

f(x) = (λ² + 1)x² - 4λx + 2

here k₁ = 0, and k₂ = 1

so, f(0)f(1) = [(λ² + 1)(0)² - 4λ(0) + 2][(λ² + 1) - 4λ + 2] < 0

⇒(2)(λ² + 1 - 4λ + 2) < 0

⇒λ² - 4λ + 3 < 0

⇒(λ - 1)(λ - 3) < 0

⇒1 < λ < 3

Therefore the set of values is (1, 3)