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JEE Main & Advanced Sample Paper JEE Main Sample Paper-32

  • question_answer
    Let \[y'(x)+\frac{g'(x)}{g(x)}y(x)=\frac{g'(x)}{1+{{g}^{2}}(x)}\]where f(x) denotes \[\frac{df(x)}{dx}\]and g(x) is a given non-constant differentiable function an R If \[g\left( 1 \right)=y\left( 1 \right)=1\]and \[g(e)=\sqrt{(2e-1)}\] then y(e) equals (Here e denotes napier's constant)

    A) \[\frac{3}{2g(e)}\]  

    B)                    \[\frac{1}{2g(e)}\]

    C) \[\frac{2}{3g(e)}\]          

    D) \[\frac{1}{3g(e)}\]

    Correct Answer: A

    Solution :

    \[I.F.={{e}^{\int{\frac{g'(x)}{g(x)}dx}}}={{e}^{In\,g(x)}}=g(x).\]                 \[\Rightarrow \,y.g(x)=\frac{1}{2}\int{\frac{2g(x)g'(x)}{1+{{g}^{2}}(x)}}dx=\frac{1}{2}\ell n(1+{{g}^{2}}(x))+C\]                 \[C=1-\frac{\ell n\,\,2}{2}\] \[\tan ({{90}^{o}}\,-{{60}^{o}})\,=\tan {{30}^{o}}=\frac{1}{\sqrt{3}}\]


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