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JEE Main & Advanced Sample Paper JEE Main Sample Paper-18

  • question_answer
    \[{{\Delta }_{1}}={{\Delta }_{1}}=\left| \begin{matrix}    x & b & b  \\    a & x & b  \\    a & a & x  \\ \end{matrix} \right|\] 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 :

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


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