JEE Main & Advanced Sample Paper JEE Main - Mock Test - 7

  • question_answer
    The number of positive integral solutions of the equation \[{{\tan }^{-1}}x+{{\cot }^{-1}}y={{\tan }^{-1}}3,\] is

    A) two                

    B) one

    C) infinite                        

    D) None of these

    Correct Answer: A

    Solution :

    \[{{\tan }^{-1}}x+{{\tan }^{-1}}\frac{1}{y}={{\tan }^{-1}}3\]
    \[\Rightarrow \,\,{{\tan }^{-1}}\frac{x+\frac{1}{y}}{1-\frac{x}{y}}={{\tan }^{-1}}3\Rightarrow \frac{xy+1}{y-x}=3\]
    \[\Rightarrow \,\,\,y=\frac{1+3x}{3-x}>0\]   [\[\because \]x and y are positive]
    \[\Rightarrow \,\,x-3<0\Rightarrow x<3\] or \[x=1,2\]
    \[\therefore \,\,\,y=2,7\] solution set is \[(x,y)\,\,\in \left\{ (1,2),(2,7) \right\}\]

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