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KVPY Sample Paper KVPY Stream-SX Model Paper-6

  • question_answer
    Let \[f\,(x)=min\,(\{x\},\,\,\{-x\})\]\[\forall \]\[x\in R\] (where \[\{\cdot \}\] represents fractional part function) then \[\int\limits_{-100}^{100}{f\,(x)\,\,dx}\]

    A) 50

    B) 100

    C) 200                              

    D) none of these

    Correct Answer: A

    Solution :

    \[I=\int\limits_{-100}^{100}{f\,(x)\,dx=200\int\limits_{0}^{1}{f\,(x)\,dx}}\]
    (\[\because \]1 is the period of f(x))
    \[f\,(x)=\left\{ \begin{matrix}    \{x\} & 0\le x<\frac{1}{2}  \\    \{-x\} & \frac{1}{2}\le x\le 1  \\ \end{matrix} \right.=\left\{ \begin{matrix}    x\,\,\,\,; & 0\le x<\frac{1}{2}  \\    1-x & \frac{1}{2}\le x\le 1  \\ \end{matrix} \right.\]
    \[I=200\,\,\left( \int\limits_{0}^{1/2}{x\,dx+\int\limits_{1/2}^{1}{(1-x)\,dx}} \right)=200\]
    \[\left( \frac{1}{8}+\left( 1-\frac{1}{2} \right)-\frac{1}{2}\left( 1-\frac{1}{4} \right) \right)=200\,\,\left( \frac{1}{8}+\frac{1}{8} \right)\]
    = 50


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