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

Let $\alpha ,\beta$ be the roots of the given equation, then
$\alpha +\beta ={\lambda }^2-5\lambda +5<1$,and $\alpha \beta ={2\lambda }^2-3\lambda -4<1$
$\Rightarrow {\lambda }^2-5\lambda +4<0$ and ,${2\lambda }^2-3\lambda -5<0$
$\therefore (\lambda -4)(\lambda -1)<0$ and $(2\lambda -5)(\lambda +1)<0$

$1<\lambda <4$ and $-1<\lambda <\frac{5}{2}$
$\therefore$ Required set $=(-1,\frac{5}{2})\cap (1,4)=(1,\frac{5}{2}).$