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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Mock Test - Determinants

  • question_answer
    If \[{{\Delta }_{1}}=\left| \begin{matrix}    x & b & b  \\    a & x & b  \\    a & a & x  \\ \end{matrix} \right|\] and \[{{\Delta }_{2}}=\left| \begin{matrix}    x & b  \\    a & x  \\ \end{matrix} \right|\]are the given determinants, then

    A)  \[{{\Delta }_{1}}=3{{({{\Delta }_{2}})}^{2}}\]

    B)  \[\frac{d}{dx}({{\Delta }_{1}})=3({{\Delta }_{2}})\]

    C)  \[\frac{d}{dx}({{\Delta }_{1}})=3{{({{\Delta }_{2}})}^{2}}\]

    D)  \[{{\Delta }_{1}}=3{{\Delta }_{2}}^{3/2}\]

    Correct Answer: B

    Solution :

    [b] \[{{\Delta }_{1}}=x({{x}^{2}}-ab)-b(ax-ab)+b({{a}^{2}}-ax)\] \[={{x}^{3}}-3abx+a{{b}^{2}}+{{a}^{2}}b\] \[\frac{d}{dx}({{\Delta }_{1}})=3{{x}^{2}}-3ab=3({{x}^{2}}-ab)=3{{\Delta }_{2}}\]


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