Question

If $\left({\lambda }^2+\lambda -2\right)x^2+(\lambda +2)x<1\forall x\in R$ then

• A. $\lambda \in (-2,1)$
• B. $\lambda \in (-2,2/5)$
• C. $\lambda \in (2/5,1)$
• D. $\lambda \in [-2,2/5)$

Answer 1

Answer: (D)

$\left({\lambda }^2+\lambda -2\right)x^2+(\lambda +2)x-1<0\forall x\in R$
$\Rightarrow {\lambda }^2+\lambda -2<0\text{ and }D<0$
$\Rightarrow (\lambda -1)(\lambda +2)<0\text{ and }(\lambda +2{)}^2+4(\lambda +2)(\lambda -1)<0$
$\Rightarrow (\lambda -1)(\lambda +2)<0\text{ and }(\lambda +2)(5\lambda -2)<0$
$\Rightarrow \lambda \in (-2,1)\text{ and }-2<\lambda <2/5$
$\Rightarrow \lambda \in (-2,2/5)$
But when $\lambda =-2$
$\Rightarrow 0<1$ which is true
Hence $\lambda \in [-2,2/5)$