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KVPY Sample Paper KVPY Stream-SX Model Paper-20

  • question_answer
    \[\underset{x+\infty}{\mathop{\lim}}\,\left( \frac{n}{{{n}^{2}}+{{1}^{2}}}+\frac{n}{{{n}^{2}}+{{n}^{2}}}+\frac{n}{{{n}^{2}}+{{3}^{2}}}+...\frac{1}{5n} \right)\] is equal to:

    A) \[\frac{\pi }{4}\]

    B) \[{{\tan }^{-1}}(3)\]

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

    D) \[{{\tan }^{-1}}(2)\]

    Correct Answer: D

    Solution :

    \[\underset{x\to \infty }{\mathop{\lim }}\,\sum\limits_{r\,=\,1}^{2n}{\frac{n}{{{n}^{2}}+{{r}^{2}}}}\]
    \[\underset{x\to \infty }{\mathop{\lim }}\,\sum\limits_{r\,=\,1}^{2n}{\frac{1}{n\left( 1+\frac{{{r}^{2}}}{{{n}^{2}}} \right)}}\] Using D.I as limit of sum, we get
    \[=\int\limits_{0}^{2}{\frac{dx}{1+{{x}^{2}}}={{\tan }^{-1}}2.}\]


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