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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 26

  • question_answer
    The number of real solutions of \[{{\cos }^{-1}}\sqrt{{{x}^{2}}+x+1}-{{\cot }^{-1}}\sqrt{x(x+1)}=\frac{\pi }{2}\] is                          

    A)  \[0\]                

    B)        \[1\]

    C) \[2\]                 

    D)        \[3\]

    Correct Answer: A

    Solution :

    [a] We have \[{{\cos }^{-1}}\sqrt{{{x}^{2}}+x+1}-{{\cot }^{-1}}\sqrt{x(x+1)}=\frac{\pi }{2}\] We must have \[x(x+1)\ge 0\] \[\therefore \,\,\,\,\,x(x+1)+1\ge 1\] \[\therefore \,\,\,\,\,x(x+1)+1=1\] (\[\because \,\,\,{{\cos }^{-1}}\] is defined only on \[[-1,1]\]) \[\Rightarrow \,\,\,\,\,x(x+1)=0\] \[\Rightarrow \,\,\,\,\,x\in \{0,\,-1\}\] But, none of them satisfies the equation.


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