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Manipal Engineering Manipal Engineering Solved Paper-2011

  • question_answer
    The number of solutions of the equation\[2{{\sin }^{-1}}\sqrt{{{x}^{2}}-x-1}+{{\cos }^{-1}}\sqrt{{{x}^{2}}-x}=\frac{3\pi }{2}\]is

    A)  0                                            

    B)  2

    C)  4                            

    D)         \[\infty \]

    Correct Answer: B

    Solution :

    Since,\[{{\sin }^{-1}}\sqrt{x},\,\,{{\cos }^{-1}}\sqrt{x}\]are defined for\[x\le 1\]and\[x\ge 0\]. \[\therefore \]  \[0\le \sqrt{{{x}^{2}}-x+1}\le 1\] and        \[0\le \sqrt{{{x}^{2}}-x}\le 1\] \[\Rightarrow \]               \[-1\le {{x}^{2}}-x\le 0\] and        \[0\le {{x}^{2}}-x\le 0\] \[\Rightarrow \]               \[{{x}^{2}}-x=0\] \[\Rightarrow \]               \[x=0,\,\,1\] Hence, number of solutions are two.


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