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KVPY Sample Paper KVPY Stream-SX Model Paper-18

  • question_answer
    Let \[f:\left[ -2,3 \right]\to \left[ 0,\infty  \right]\] be a continuous function such that \[f\left( 1-x \right)=f\left( x \right)\] for all  \[x\in \left[ -2,3 \right]\] If \[{{R}_{1}}\]is the numerical value of area of the region bound by \[y=f\left( x \right),x=-2,x=3\] and the axis of x and \[{{R}_{2}}\]\[=\int\limits_{-2}^{3}{xf\left( x \right)dx\,then:}\]

    A) \[3{{R}_{1}}=2{{R}_{2}}\]

    B) \[2{{R}_{1}}=3{{R}_{2}}\]

    C) \[{{R}_{1}}={{R}_{2}}\]

    D) \[{{R}_{1}}=2{{R}_{2}}\]

    Correct Answer: D

    Solution :

    We have \[{{\operatorname{R}}_{2}}=\int\limits_{-2}^{3}{x\,f\left( x \right)dx}=\int\limits_{a}^{b}{\left( 1-x \right)f\left( 1-x \right)dx}\]\[\left[ using\int\limits_{a}^{b}{f\left( x \right)dx=\int\limits_{a}^{b}{f\left( a+b-x \right)dx}} \right]\]\[\Rightarrow {{\operatorname{R}}_{2}}=\int\limits_{-2}^{3}{\left( 1-x \right)f\left( x \right)dx}\]                                           \[\left( \because f\left( x \right)=f\left( 1-x \right)\operatorname{on}\left[ -2,3 \right] \right)\]\[\therefore {{R}_{2}}+{{R}_{2}}=\int\limits_{-2}^{3}{x\,f\left( x \right)dx+\int\limits_{-2}^{3}{\left( 1-x \right)f\left( x \right)dx}}\] \[=\int\limits_{-2}^{3}{f\left( x \right)dx={{R}_{1}}}\]\[=2{{R}_{2}}={{R}_{1}}\]


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