First slide

Theory of equations

Question

$The set of all real values of λ for which the quadratic equations.$ $(λ^{2} + 1) x^{2} - 4 λ x + 2 = 0$ always have exactly one root in the interval (0,1) is :

Moderate

A

(2,4]
B

(-3, -1)
C

(1,3]
D

(0,2)

Solution

$Let f (x) = (λ^{2} + 1) x^{2} - 4 λ x + 2$
If f(x) = 0 has exactly one root in (0,1) then
$\begin{matrix} f (0) \cdot f (1) \leq 0 \\ 2 (λ^{2} + 1 - 4 λ + 2) \leq 0 \\ (λ - 1) (λ - 3) \leq 0 \end{matrix}$
$But for λ = 1 \Rightarrow n = 1 is a repeat root,$
$Therefore, λ \in (1, 3]$

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