A) \[a\ge 1\]
B) \[a\le 1\]
C) \[a>-3\]
D) \[a<-3\]or\[a>1\]
Correct Answer: D
Solution :
We know that the expression\[a{{x}^{2}}+bx+c>0\] for all\[x\], if \[a>0\]and\[{{b}^{2}}<4ac\]. \[\therefore \]\[({{a}^{2}}-1){{x}^{2}}+2(a-1)x+2\]is positive for all\[x\], if \[{{a}^{2}}-1>0\]and \[-4(a-1)(a+3)<0\] \[\Rightarrow \]\[{{a}^{2}}-1>0\]and\[-4(a-1)(a+3)<0\] \[\Rightarrow \]\[{{a}^{2}}-1>0\]and\[(a-1)(a+3)>0\] \[\Rightarrow \] \[{{a}^{2}}>1\]and\[a<-3\]or\[a>1\] \[\Rightarrow \] \[a<-3\]or\[a>1\]
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