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JEE Main & Advanced Sample Paper JEE Main Sample Paper-24

  • question_answer
    The least value of the volume of parallelepiped formed by the vectors \[{{\vec{V}}_{1}}=\hat{i}+\hat{j}\], \[{{\vec{V}}_{2}}=\hat{i}+(2\cos ec\alpha )\hat{j}+\hat{k}\] and \[{{\vec{V}}_{3}}=\hat{j}+(2\cos ec\alpha )\,\hat{k}\] where \[\alpha \in (0,\pi )\], is

    A)  16                               

    B)  9

    C)  5                                

    D)  1

    Correct Answer: D

    Solution :

    We have \[V=\left| \begin{matrix}    1 & 1 & 0  \\    1 & 2\cos ec\alpha  & 1  \\    0 & 1 & 2\cos ec\alpha   \\ \end{matrix} \right|\] \[=4\cos e{{c}^{2}}\alpha \,-1-2\cos ec\alpha \] \[=4\left[ \cos e{{c}^{2}}\alpha -\frac{1}{2}\cos ec\,\alpha  \right]\,-1=4{{\left( \cos ec\alpha =\frac{1}{4} \right)}^{2}}=\frac{5}{4}\] \[\therefore \,\,{{V}_{least}}\left( \alpha =\frac{\pi }{2} \right)=4\times \frac{9}{16}-\frac{5}{4}=\frac{4}{4}=1\]


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