Question:
The set of all real values of $\lambda$ for which the quadratic equations, $\left(\lambda^{2}+1\right) x^{2}-4 \lambda x+2=0$ always have exactly one root in the interval $(0,1)$ is :
Correct Option: , 3
Solution:
The given quadratic equation is
$\left(\lambda^{2}+1\right) x^{2}-4 \lambda x+2=0$
$\because$ One root is in the interval $(0,1)$
$\therefore f(0) f(1) \leq 0$
$\Rightarrow 2\left(\lambda^{2}+1-4 \lambda+2\right) \leq 0$
$\Rightarrow 2\left(\lambda^{2}-4 \lambda+3\right) \leq 0$
$(\lambda-1)(\lambda-3) \leq 0 \Rightarrow \lambda \in[1,3]$
But at $\lambda=1$, both roots are 1 so $\lambda \neq 1$
$\therefore \lambda \in(1,3]$