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JEE Main & Advanced Sample Paper JEE Main Sample Paper-21

  • question_answer
    Let \[f(x)={{\cot }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)\] and  \[g(x)={{\cot }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\]then \[\underset{x\to c}{\mathop{\lim }}\,\frac{f(x)-f(c)}{g(x)-g(c)}\]  where \[c\in \left( 0,\frac{1}{2} \right)\] is

    A)  \[\frac{3}{2}\]                                     

    B)  \[\frac{-3}{2}\]

    C)  \[\frac{1}{2}\]                                     

    D)    \[\frac{-1}{2}\]

    Correct Answer: B

    Solution :

    \[\underset{x\to c}{\mathop{Lim}}\,\,\frac{f(x)-f(c)}{g(x)-g(c)}\,=\frac{f'(c)}{g'(c)}\] Now, \[f(x)\,=\frac{\pi }{2}-3{{\tan }^{-1}}x,\,\,\] in\[\left( 0,\,\frac{1}{2} \right)\] \[g(x)={{\tan }^{-1}}\,x,\,\] in \[\left( 0,\,\frac{1}{2} \right)\] \[\therefore \,\,\frac{f'(c)}{g'(c)}\,=\frac{-3}{2}\]


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