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KVPY Sample Paper KVPY Stream-SX Model Paper-20

  • question_answer
    Let \[\vec{a},\vec{b},\]be three unit vectors, out of which vectors \[\vec{b}\] and \[\vec{c}\] are non-parallel If \[\alpha \] and \[\beta \] are the angles which vector \[\vec{a}\]makes with vectors \[\vec{b}\] and \[\vec{c}\] respectively and \[\vec{a}\times \left( \vec{b}\times \vec{c} \right)=\frac{1}{2}\vec{b},\] then \[\left| a-\beta  \right|\] is Equal to:

    A) \[30{}^\circ \]

    B) \[90{}^\circ \]

    C) \[60{}^\circ \]  

    D) \[45{}^\circ \]

    Correct Answer: A

    Solution :

    \[(\vec{a}.\vec{c})\vec{b}-\left( \vec{a}\vec{b} \right).\vec{c}=\frac{1}{2}\vec{b}\]
    \[\because \]\[\vec{b}\] and \[\vec{c}\] are linearly independent
    \[\because \]\[\vec{a}.\vec{c}=\frac{1}{2}\] and \[\overrightarrow{a}.\overrightarrow{b}=0\]
    (All given vectors are unit vectors)
    \[\because \]\[\vec{a}\wedge \vec{c}=60{}^\circ \]and \[\vec{a}\wedge \vec{b}=90{}^\circ \]
    \[\because \] \[\left| a-\beta  \right|=30{}^\circ .\]


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