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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Critical Thinking

  • question_answer
    If \[y={{\cot }^{-1}}{{(\cos 2x)}^{1/2}}\], then the value of \[\frac{dy}{dx}\]at \[x=\frac{\pi }{6}\]will be                                    [IIT 1992]

    A) \[{{\left( \frac{2}{3} \right)}^{1/2}}\]

    B) \[{{\left( \frac{1}{3} \right)}^{1/2}}\]

    C) \[{{(3)}^{1/2}}\]

    D) \[{{(6)}^{1/2}}\]

    Correct Answer: A

    Solution :

    • \[\frac{dy}{dx}=\frac{-1}{1+\cos 2x}\times \frac{1}{2\sqrt{\cos 2x}}\times -2\sin 2x=\frac{\tan x}{\sqrt{\cos 2x}}\]           
    • \[\Rightarrow {{\left( \frac{dy}{dx} \right)}_{x=\pi /6}}=\frac{1/\sqrt{3}}{\sqrt{1/2}}=\sqrt{\frac{2}{3}}\].


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