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CMC Medical CMC-Medical VELLORE Solved Paper-2009

  • question_answer
    If \[{{\overrightarrow{a}}_{1}}\] and \[{{\overrightarrow{a}}_{2}}\]are two non-collinear unit vectors and if \[\left| {{\overrightarrow{a}}_{1}}-{{\overrightarrow{a}}_{2}} \right|=\sqrt{3},\]then the value of is \[\left( {{\overrightarrow{a}}_{1}}-{{\overrightarrow{a}}_{2}} \right).\]\[\left( 2{{\overrightarrow{a}}_{1}}+{{\overrightarrow{a}}_{2}} \right)\] is

    A)  2                                            

    B)  3/2

    C)  1/2                                       

    D)  1

    Correct Answer: C

    Solution :

                    Since, \[{{\overrightarrow{a}}_{_{1}}}\] and \[{{\overrightarrow{a}}_{_{2}}}\]are non-collinear \[\therefore \]  \[{{a}_{1}}={{a}_{2}}=1\] Now,    \[|\,\,{{\overrightarrow{a}}_{_{1}}}+{{\overrightarrow{a}}_{_{2}}}|\,\,=\sqrt{3}\] \[a_{1}^{2}+a_{2}^{2}+2{{a}_{1}}{{a}_{2}}\cos \theta ={{(\sqrt{3})}^{2}}\] or            \[1+1+2\cos \theta =3\] or                            \[\cos \theta =\frac{1}{2}\] Now,     \[({{\overrightarrow{a}}_{_{1}}}-{{\overrightarrow{a}}_{_{2}}})\cdot (2\,{{\overrightarrow{a}}_{_{1}}}+{{\overrightarrow{a}}_{_{2}}})\] \[=2a_{1}^{2}-a_{2}^{2}-{{a}_{1}}{{a}_{2}}\cos \theta \] \[=2-1-\frac{1}{2}=\frac{1}{2}\]


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