11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer 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 \[a>1\]

    Correct Answer: D

    Solution :

    We know that the expression \[a{{x}^{2}}+bx+c>0\]for all x, if \[a>0\]and \[{{b}^{2}}<4ac\] \ \[({{a}^{2}}-1){{x}^{2}}+2(a-1)x+2\]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 \[a>1\]

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