KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    The solution set of the inequality \[\left| {{9}^{x}}-{{3}^{x+1}}-15 \right|<{{2.9}^{x}}-{{3}^{x}}\]is

    A) \[(-\infty ,1)\]

    B) \[(1,\infty )\]

    C) \[(-\infty ,\text{ }1]\]

    D) None of these

    Correct Answer: B

    Solution :

    Let \[{{3}^{x}}=y,\] then the inequality is \[\left| {{y}^{2}}-3y-15 \right|<2{{y}^{2}}-y\]                                 ? (1)
    The inequality holds if \[2{{y}^{2}}-y>0\Rightarrow y<0\,\operatorname{or}\,y>\frac{1}{2}\]
    \[\because \]\[y={{3}^{x}}0\Rightarrow y>\frac{1}{2}\]
    Now the inequality on solving,
    \[-(2{{y}^{2}}-y)<{{y}^{2}}-3y-15<2{{y}^{2}}-y\] \[\Rightarrow \]\[3{{y}^{2}}-4y-15>0\]and\[{{y}^{2}}+2y+15>0\]
    Solution of first inequality
    Solution of second inequality
    \[{{y}^{2}}+2y+15>0\] is \[y\in R\]
    The common solution is
    \[y>3\Rightarrow {{3}^{x}}>x\Rightarrow x>1\Rightarrow x\in (1,\infty )\]

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