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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
    The value of \[\sum\limits_{n=1}^{N}{{{U}_{n}},}\] if \[{{U}_{n}}=\left| \,\begin{matrix}    n & 1 & 5  \\    {{n}^{2}} & 2N+1 & 2N+1  \\    {{n}^{3}} & 3{{N}^{2}} & 3N  \\ \end{matrix}\, \right|\] is  [MNR 1994]

    A) 0

    B) 1

    C) - 1

    D) None of these

    Correct Answer: A

    Solution :

    \[\sum\limits_{n=1}^{N}{{{U}_{n}}=}\left| \,\begin{matrix}    \frac{N(N+1)}{2} & 1 & 5  \\    \frac{N(N+1)(2N+1)}{6} & 2N+1 & 2N+1  \\    {{\left\{ \frac{N(N+1)}{2} \right\}}^{2}} & 3{{N}^{2}} & 3N  \\ \end{matrix}\, \right|\] \[=\frac{N(N+1)}{12}\left| \,\begin{matrix}    6 & 1 & 5  \\    4N+2 & 2N+1 & 2N+1  \\    3N(N+1) & 3{{N}^{2}} & 3N  \\ \end{matrix}\, \right|\] \[=\left| \,\begin{matrix}    6 & 1 & 6  \\    4N+2 & 2N+1 & 4N+2  \\    3N(N+1) & 3{{N}^{2}} & 3N(N+1)  \\ \end{matrix}\, \right|=0\],                                       {Applying\[{{C}_{3}}\to {{C}_{3}}+{{C}_{2}}\}\].


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