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KVPY Sample Paper KVPY Stream-SX Model Paper-18

  • question_answer
    Find the value of \[\int\limits_{-1}^{3/2}{\left| x\sin \,\pi x \right|}\,dx\]

    A) \[\frac{3}{\pi }+\frac{2}{{{\pi }^{2}}}\]

    B) \[\frac{3}{{{\pi }^{2}}}+\frac{1}{\pi }\]

    C) \[\frac{3}{{{\pi }^{{}}}}+\frac{1}{{{\pi }^{2}}}\]

    D) \[\frac{3}{{{\pi }^{{}}}}-\frac{1}{{{\pi }^{2}}}\]

    Correct Answer: C

    Solution :

    \[\int\limits_{-1}^{3/2}{\left| x\sin \pi x \right|dx}\]
    \[For\,-1\le x<0\Rightarrow -\pi <px<0\]
    \[\Rightarrow \sin \,\pi x<0\] \[\Rightarrow x\sin \,\pi x>0\]
    For \[1<x<3/2\Rightarrow \pi <\pi x<3\pi /2\]
    \[\Rightarrow \sin \,\pi x<0\] \[\Rightarrow x\sin \,\pi x<0\]
    \[\therefore \int\limits_{-1}^{3/2}{\left| x\sin \pi x \right|}\,dx\] \[=\int\limits_{-1}^{1}{x\,\sin \pi xdx+\int\limits_{1}^{3/2}{\left( -x\sin \,\pi x\, \right)ax}}\]\[=2\int\limits_{0}^{1}{x\,\sin \pi x\,dx-\int\limits_{1}^{3/2}{x\sin \pi x\,dx}}\]\[=2\left[ \frac{-x\cos \pi x}{\pi }+\frac{\sin \pi x}{{{\pi }^{2}}} \right]_{0}^{1}\]\[-\left[ \frac{-x\cos \pi x}{x}+\frac{\sin \pi x}{{{\pi }^{2}}} \right]_{1}^{3/2}\] \[=2\left[ \left( \frac{-\cos \pi }{\pi }+0 \right)-\left( 0+0 \right) \right]\]\[-\left[ \left( \frac{-3/2\cos 3\pi /2}{\pi } \right)+\frac{\sin 3\pi /2}{{{\pi }^{2}}} \right]\]\[-\left( \frac{-\cos \pi }{\pi }+\frac{\sin \pi }{{{\pi }^{2}}} \right)\] \[=2\left[ \frac{1}{\pi } \right]-\left[ -\frac{1}{\pi }-\frac{1}{\pi } \right]=\frac{2}{\pi }+\frac{1}{{{\pi }^{2}}}+\frac{1}{\pi }\]\[=\frac{3}{\pi }+\frac{1}{{{\pi }^{2}}}\]


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