Question

The set of values of $a$ for which the sum and product of roots of the equation $x^2-(a^2-5a+5)x+(2a^2-3a-4)=0$ is less than $1$, is

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

Answer 1

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

Given equation
$x^2-(a^2-5a+5)x+(2a^2-3a-4)=0$
Let the roots of the equation be $m,n$.
So, $m+n<1,mn<1$

$m+n<1$
$\Rightarrow a^2-5a+5<1\Rightarrow a^2-5a+4<0\Rightarrow (a-4)(a-1)<0\Rightarrow a\in (1,4)\cdots \left(1\right)$

$mn<1$
$\Rightarrow 2a^2-3a-4<1\Rightarrow 2a^2-3a-5<0\Rightarrow (a+1)(2a-5)<0\Rightarrow a\in (-1,\frac{5}{2})\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$,
$a\in (1,\frac{5}{2})$