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

  • question_answer
    If the value of y (greater than 1) satisfying the equation \[\int\limits_{1}^{y}{x\,\ell n\,x\,dx=\frac{1}{4}}\] can be expressed in the form of \[{{e}^{\frac{m}{n}}}\] , where m and n are relative prime then \[(m+n)\] is equal to [Note : e denotes Napier's constant]

    A)  1                    

    B)  2

    C)  3                                

    D)  4

    Correct Answer: C

    Solution :

    \[\int\limits_{1}^{y}{x\,\ln \,xdx\,=\frac{{{y}^{2}}}{2}\,\ln y-\frac{1}{4}\,{{y}^{2}}+\frac{1}{4}}\] \[\left[ \left. \ln \,x.\frac{{{x}^{2}}}{2} \right|_{1}^{y}-\frac{1}{2}\,\int\limits_{1}^{y}{xdx} \right]\] \[\therefore \,\,\frac{{{y}^{2}}}{2}\,\ln y-\frac{1}{4}{{y}^{2}}=0;\,\,{{y}^{2}}\left[ \frac{\ln \,y}{2}\,-\frac{1}{4} \right]=0\] \[\Rightarrow \,\,y={{e}^{1/2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,1+2=3\]

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