KVPY Sample Paper KVPY Stream-SX Model Paper-29

  • question_answer
    A helicopter is flying along the curve given by \[y-{{x}^{3/2}}=7,(x\ge 0).\] A soldier positioned at the point \[\left( \frac{1}{2},7 \right)\] wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:

    A) \[\frac{\sqrt{5}}{6}\]

    B) \[\frac{1}{3}\sqrt{\frac{7}{3}}\]

    C) \[\frac{1}{6}\sqrt{\frac{7}{3}}\]

    D) \[\frac{1}{2}\]

    Correct Answer: C

    Solution :

    \[y={{x}^{3/2}}-2\]     \[\frac{dy}{dx}=\frac{3}{2}\sqrt{x}\]
    Slope of normal \[=-\frac{2}{3\sqrt{x}}\]
    Let point is\[({{x}_{1}},x_{1}^{3/2}-2)\]
    \[\therefore \]Normal\[y-(x_{1}^{3/2}-2)=\frac{-2}{3\sqrt{{{x}_{1}}}}(x-{{x}_{1}})\]
    Now put\[(1,7)\]and solve it. \[\Rightarrow \]           \[{{x}_{1}}=\frac{1}{3}\]
    \[\therefore \]      \[P\Rightarrow \left( \frac{1}{3},7+\frac{1}{3\sqrt{3}} \right),A\Rightarrow (1,7)\]
    \[\therefore \]      \[AD=\frac{1}{6}\sqrt{\frac{7}{3}}.\]

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