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. none of these

Answer 1

Answer: (B)

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

$({\lambda }^2+\lambda -2)x^2+(\lambda +2)x<1$

$({\lambda }^2+\lambda -2)x^2+(\lambda +2)x-1<0$

If $({\lambda }^2+\lambda -2)<0$ And $D<0$ then it is always true.
$D={(\lambda +2)}^2+4({\lambda }^2+\lambda -2)<0$
$D=5{\lambda }^2+8\lambda -4<0$
$\lambda \in (-2,\frac{2}{5}).....\left(1\right)$
$({\lambda }^2+\lambda -2)<0$
$\lambda \in (-2,1).....\left(2\right)$
So , $\lambda \in (-2,\frac{2}{5})$.