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 $\lambda \in[-2,2 / 5)$