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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 22

  • question_answer
    Let \[\overset{\to }{\mathop{a}}\,,\,\overset{\to }{\mathop{b}}\,and\overset{\to }{\mathop{c}}\,\] be three non-zero vectors such that no two of these are collinear. If the vector \[\overset{\to }{\mathop{a}}\,+2\overset{\to }{\mathop{b}}\,\] is collinear with \[\overset{\to }{\mathop{c}}\,and\,\overset{\to }{\mathop{b}}\,+3\overset{\to }{\mathop{c}}\,\] is collinear with \[\vec{a}\] (\[\lambda \] being some non-zero scalar) then \[\overset{\to }{\mathop{a}}\,+2\overset{\to }{\mathop{b}}\,+6\overset{\to }{\mathop{c}}\,\] equals

    A) 0                     

    B)   \[\lambda \vec{b}\]

    C) \[\lambda \vec{c}\]                 

    D)   \[\lambda \vec{a}\]

    Correct Answer: C

    Solution :

    Let \[\vec{a}+2\vec{b}=\,\,t\vec{c}\,\,and\,\vec{b}\,+3\vec{c}=s\vec{a}\], where t and s are scalars. Adding, we get \[\vec{a}+3\vec{b}\,+3\vec{c}=t\vec{c}+s\vec{a}\Rightarrow \,\,\vec{a}\,+2\vec{b}\,+6\vec{c}=t\vec{c}\,+s\vec{a}-\vec{b}+3\vec{c}\]\[=\,\,t\vec{c}+(\vec{b}+3\vec{c})- \vec{b}+3\vec{c}=\,\,(t\,+6)\vec{c}\] \[\left[ using\,\,s \vec{a}= \vec{b} + 3\vec{c} \right]\]


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