• # question_answer 9) The values of $'a'$ for which $({{a}^{2}}-1){{x}^{2}}+2(a-1)x+2$ is positive for any $x$ are [UPSEAT 2001] A) $a\ge 1$ B) $a\le 1$ C) $a>-3$ D) $a<-3$or $a>1$

We know that the expression $a{{x}^{2}}+bx+c>0$for all x, if $a>0$and ${{b}^{2}}<4ac$ \ $({{a}^{2}}-1){{x}^{2}}+2(a-1)x+2$is positive for all x if ${{a}^{2}}-1>0$and $4{{(a-1)}^{2}}-8({{a}^{2}}-1)<0$ Þ ${{a}^{2}}-1>0$and $-4(a-1)(a+3)<0$ Þ  ${{a}^{2}}-1>0$and $(a-1)(a+3)>0$ Þ ${{a}^{2}}>1$and $a<-3\,\,\text{or}\,\,a>1$ Þ $a<-3$or $a>1$