KVPY Sample Paper KVPY Stream-SX Model Paper-7

  • question_answer
    A chord ACB, 5m long is attached at points A and B to the vertical walls 3 m apart.
    A pulley of negligible mass and negligible radius carries 200 N load is free to roll over chord without friction. Dimension x in figure, when pulley is in equilibrium is

    A) \[\frac{9}{8}m\]                        

    B)  \[\frac{7}{5}m\]

    C) \[\frac{4}{3}m\]                        

    D) \[\frac{3}{4}m\]

    Correct Answer: A

    Solution :

    Geometry of given arrangement is as shown below.
    Triangles\[\Delta ADC\], \[\Delta BEC\]and \[\Delta BGF\] are similar triangles. Also, \[\Delta ACD\]is congruent to\[\Delta FCD\]. So, AC = FC.
    Hence,\[FB=5\text{ }m\]
    In \[\Delta BGF\],
    \[\cos \theta =\frac{base}{hypotenuse}=\frac{3}{5}\]
    So, \[\sin \theta =\frac{4}{5}and\tan \theta =\frac{4}{3}\]
    Now, \[\Delta AMC\cong \Delta NMC\]     (by AAS rule)
    Also, \[AM+MN+ND=3m\]
    \[\Rightarrow NP=3-2x\left( \therefore AM=x\,and\,MN=x \right)\]
    Now in \[\Delta BPN\]
    \[\tan \theta =\frac{BP}{NP}\Rightarrow \frac{4}{3}=\frac{1}{3-2x}\Rightarrow x=\frac{9}{8}m\]

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