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JEE Main & Advanced Sample Paper JEE Main Sample Paper-41

  • question_answer
    Let \[\overrightarrow{a},\overrightarrow{b}\] and \[\overrightarrow{c}\] be three non-zero vectors, no two of which are collinear. If the vector \[\overrightarrow{a}+2\overrightarrow{b}\]is collinear with \[\overrightarrow{c}\] and \[\overrightarrow{b}+3\overrightarrow{c}\] is collinear with\[\overrightarrow{a}\], then \[\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}\]is equal to

    A)  \[\lambda \overrightarrow{a}\]                                  

    B)  \[\lambda \overrightarrow{b}\]

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

    D)  \[\overrightarrow{0}\]

    Correct Answer: D

    Solution :

    Since, \[\overrightarrow{a}+2\overrightarrow{b}\] is collinear with \[\overrightarrow{c}\] \[\therefore \]\[\overrightarrow{a}+2\overrightarrow{b}=\lambda \overrightarrow{c},\forall \lambda \in R\] and \[\overrightarrow{b}+3\overrightarrow{c}\] is collinear with \[\overrightarrow{a}.\] \[\therefore \]\[\overrightarrow{b}+3\overrightarrow{c}=\mu \overrightarrow{a},\forall \mu \in R\] \[\Rightarrow \]\[\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}=(\lambda +6)\overrightarrow{c}\]and\[\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}=(2\mu +1)\overrightarrow{a}\] \[\Rightarrow \]\[(\lambda +6)\overrightarrow{c}=(2\mu +1)\overrightarrow{a}\] \[\Rightarrow \]\[\lambda +6=0\]and\[2\mu +1=0\] \[\Rightarrow \]\[\lambda =-6,\lambda =-\frac{1}{2}\] Hence,   \[\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}=\overrightarrow{0}\]


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