KVPY Sample Paper KVPY Stream-SX Model Paper-19

  • question_answer
    The range of \['\alpha '\] for which the point \[(\alpha ,\alpha )\] lies inside the region bounded by the Curves \[y=\sqrt{1-{{x}^{2}}}\]and \[x+y=1\] is -

    A) \[\frac{1}{2}<\alpha <\frac{1}{\sqrt{2}}\]

    B) \[-\frac{1}{\sqrt{2}}<\alpha <\frac{1}{\sqrt{2}}\]

    C) \[\alpha >\frac{1}{\sqrt{2}}\]

    D) \[0<\alpha <\frac{1}{2}\]

    Correct Answer: A

    Solution :

    \[\Rightarrow 2{{\alpha }^{2}}-1<0\]\[\Rightarrow -\frac{1}{\sqrt{2}}<\alpha <\frac{1}{2}\]        ??..(1)
    and \[\alpha +\alpha >1\]           \[\Rightarrow \alpha >\frac{1}{2}\]                ???(2)
    \[\therefore \]Common solution of (1) and (2) is:
    \[\frac{1}{2}<\alpha <\frac{1}{\sqrt{2}}\]

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