Question

If $({\lambda }^2+\lambda -2)x^2+(\lambda +2)x<1$ for all $x\in R$, then $\lambda$ belongs to the interval

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

Answer 1

Answer: (B)

$(-2,\frac{2}{5})$

We have, $({\lambda }^2+\lambda -2)x^2+(\lambda +2)x<1$
$\Rightarrow ({\lambda }^2+\lambda -2)x^2+(\lambda +2)x-1<0$
We know, for $a{x}^2+bx+c<0$ $a<0,D=b^2-4ac<0$
$\Rightarrow {\lambda }^2+\lambda -2<0\Rightarrow (\lambda -1)(\lambda +2)<0$
$\Rightarrow \lambda \in (-2,1)\Rightarrow \left(1\right)$
And $(\lambda +2{)}^2+4({\lambda }^2+\lambda -2)<0$
$\Rightarrow (\lambda +2{)}^2+4(\lambda +2\left)\right(\lambda -1)<0$
$\Rightarrow (\lambda +2)(5\lambda -2)<0$
$\Rightarrow \lambda \in (-2,\frac{2}{5})\Rightarrow \left(2\right)$
Hence from (1) and (2) $\lambda \in (-2,\frac{2}{5})$