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KVPY Sample Paper KVPY Stream-SX Model Paper-15

  • question_answer
    If \[\vec{a}\] & \[\vec{b}\] are any two vectors of magnitudes 1 and 2 respectively, and \[{{(1-3\overrightarrow{a}\,\,.\overrightarrow{b})}^{2}}+|2\overrightarrow{a}+\overrightarrow{b}+3(\overrightarrow{a}\,\,\times \,\,\overrightarrow{b}){{|}^{2}}=47,\] then the angle between \[\vec{a}\] and \[\vec{b}\] is -

    A) \[\frac{\pi }{3}\]

    B) \[\pi -{{\cos }^{-1}}\left( \frac{1}{4} \right)\]

    C) \[\frac{2\pi }{3}\]          

    D) \[{{\cos }^{-1}}\left( \frac{1}{4} \right)\]

    Correct Answer: C

    Solution :

    \[1+9\,\,{{(\overrightarrow{a}\,\,.\overrightarrow{b})}^{2}}-6{{(\,\vec{a}.\,\,\vec{b}\,)}^{2}}+4{{\left| \,\overrightarrow{a}\, \right|}^{2}}+{{\left| \,\overrightarrow{b}\, \right|}^{2}}\]\[+\,\,9\,\,{{\left| \overrightarrow{a}\,\,\times \,\,\overrightarrow{b} \right|}^{2}}+4\,\,(\,\overrightarrow{a}.\,\,\overrightarrow{b}\,)=47\]\[\Rightarrow 1+4+4+36-4\cos \theta =47\] \[\Rightarrow \cos \theta =-\frac{1}{2}\] \[\Rightarrow \]Angle between \[\overrightarrow{a}\] & \[\overrightarrow{b}\]  \[\frac{2\pi }{3}\]


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