KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    The area of the portion common to \[y={{\sin }^{-\,1}}(\sin x)\] and \[y=[{{\sin }^{-1}}(\sin x)]\] in \[[0,\pi ],\]here \[[\cdot ]\]denotes the greatest integer function is

    A) \[{{(\pi -2)}^{2}}\]                  

    B) \[\frac{{{(\pi -2)}^{2}}}{2}\]

    C) \[\frac{{{(\pi -2)}^{2}}}{4}\]   

    D) None of these

    Correct Answer: C

    Solution :

    We have, \[y={{\sin }^{-\,1}}(\sin x)=x\sin [0,\pi /2]\]\[=\pi -x\sin [\pi /2,\pi ]\]
    \[y=[{{\sin }^{-\,1}}(\sin x)=0\]in \[[0,1)\]\[=1\]in \[[1,\pi -1)\]\[=0\]in \[[\pi -1,\pi ]\]
    Required area \[=\frac{1}{2}(\pi -2)\left( \frac{\pi }{2}-1 \right)\]\[=\frac{{{(\pi -2)}^{2}}}{4}\]

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