Question

If both sum of roots and product of roots of $x^2-({\lambda }^2-5\lambda +5)x+(2{\lambda }^2-3\lambda -4)=0$ are less than $1,$ then the range of $\lambda$ is

• A. $(-1,\frac{5}{2})$
• B. $(1,4)$
• C. $[1,\frac{5}{2}]$
• D. $(1,\frac{5}{2})$

Answer 1

The correct option is D $(1,\frac{5}{2})$

$x^2-({\lambda }^2-5\lambda +5)x+(2{\lambda }^2-3\lambda -4)=0$

Sum of roots $<1$
$\Rightarrow {\lambda }^2-5\lambda +5<1\Rightarrow {\lambda }^2-5\lambda +4<0\Rightarrow (\lambda -1)(\lambda -4)<0\Rightarrow \lambda \in (1,4)\cdots \left(1\right)$

Product of roots $<1$
$\Rightarrow 2{\lambda }^2-3\lambda -4<1\Rightarrow 2{\lambda }^2-3\lambda -5<0\Rightarrow (2\lambda -5)(\lambda +1)<0\Rightarrow \lambda \in (-1,\frac{5}{2})\cdots \left(2\right)$

From equations $\left(1\right)$ and $\left(2\right),$ we get
$\lambda \in (1,\frac{5}{2})$