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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank System of linear equations, Some special determinants, differentiation and integration of determinants

  • question_answer
    If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},........,{{a}_{n}},......\] are in G.P. and \[{{a}_{i}}>0\]for each i, then the value of the determinant \[\Delta =\left| \,\begin{matrix}    \log {{a}_{n}} & \log {{a}_{n+2}} & \log {{a}_{n+4}}  \\    \log {{a}_{n+6}} & \log {{a}_{n+8}} & \log {{a}_{n+10}}  \\    \log {{a}_{n+12}} & \log {{a}_{n+14}} & \log {{a}_{n+16}}  \\ \end{matrix} \right|\] is equal to

    A) 1

    B) 2

    C) 0

    D) None of these

    Correct Answer: C

    Solution :

    If r is the common ratio, then \[{{a}_{n}}={{a}_{1}}{{r}^{n-1}}\]for all \[n\ge 1\Rightarrow \log {{a}_{n}}=\log {{a}_{1}}+(n-1)\log r\]        = \[A+(n-1)R\], where \[\log {{a}_{1}}=A\]and \[\log r=R\]. Thus in\[\Delta \], on applying \[{{C}_{2}}\to {{C}_{2}}-{{C}_{1}}\] and \[{{C}_{3}}\to {{C}_{3}}-{{C}_{2}}\], we obtain \[{{C}_{2}}\]and \[{{C}_{3}}\]are identical. Thus\[\Delta =0\].


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