Question

Let $f(x)=x^2+(a+2)x+a^2-a+2$. Given $a,\alpha ,\beta (\alpha <\beta )$ be real numbers and $\alpha ,\beta$ are the roots of the equation $f(x)=0$.
If both roots of the equation $f(x)=0$ lies in interval $(0,\infty )$ then range of $a$, is

• A. $\left(\frac{2}{3},2\right)$
• B. $(-\infty ,-2)$
• C. $(0,\infty )$
• D. $\varphi$

Answer 1

Answer: (D)

Given, $x^2+(a+2)x+a^2-a+2=0$.
Now, $D>0\Rightarrow \frac{2}{3}<a<2$ .....(i)
Product of roots $>0\Rightarrow a^2-a+2>0$, which is true $\forall a\in R$ .....(ii)
and sum of roots $>0\Rightarrow -(a+2)>0\Rightarrow a<-2$.....(iii)
$\therefore$ (i) $\cap ($ ii $)\cap ($ iii $)\Rightarrow a\in \varphi$.