Question

The set of possible values of $\lambda$ for which $x^2-({\lambda }^2-5\lambda +5)x+(2{\lambda }^2-3\lambda -4)=0$ has roots whose sum and product are both less than $1$ are

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

Answer 1

Answer: (D)

$(1,\frac{5}{2})$

Given,

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

Sum of roots $\alpha +\beta ={\lambda }^2-5\lambda +5$
Product of roots $\alpha \beta =(2{\lambda }^2-3\lambda -4)$

Given, $\alpha +\beta <1$ and $\alpha \beta <1$
$\Rightarrow {\lambda }^2-5\lambda +4<0$ and $2{\lambda }^2-3\lambda -5<0$
$\Rightarrow (\lambda -4)(\lambda -1)<0$ and $(\lambda +1)(2\lambda -5)<0$
$\Rightarrow \lambda \in (1,4)$ and $\lambda \in (-1,\frac{5}{2})$
$\Rightarrow \lambda \in (1,\frac{5}{2})$.