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JEE Main & Advanced Mathematics Vector Algebra Question Bank Critical Thinking

  • question_answer
    The value of 'a' so that the volume of parallelopiped formed by \[\mathbf{i}+a\mathbf{j}+\mathbf{k},\mathbf{j}+a\,\mathbf{k}\] and \[a\,\mathbf{i}+\mathbf{k}\] becomes minimum is [IIT Screening 2003]

    A) - 3

    B) 3

    C) \[\frac{1}{\sqrt{3}}\]

    D) \[\sqrt{3}\]

    Correct Answer: C

    Solution :

    • \[V=\left| \,\begin{matrix}    1 & a & 1  \\    0 & 1 & a  \\    a & 0 & 1  \\ \end{matrix}\, \right|=1+{{a}^{3}}-a\,\,\Rightarrow \,\,\frac{dV}{da}=3{{a}^{2}}-1\]       \[=3\,\left( a+\frac{1}{\sqrt{3}} \right)\,\left( a-\frac{1}{\sqrt{3}} \right)\]                   
    • \[\therefore \]  Minimum at \[\frac{1}{\sqrt{3}}\].


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