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KVPY Sample Paper KVPY Stream-SX Model Paper-18

  • question_answer
    If the product  of \[n\] matrices  \[\left[ \begin{matrix}    1 & 1  \\    0 & 1  \\ \end{matrix} \right]\,\,\left[ \begin{matrix}    1 & 2  \\    0 & 1  \\ \end{matrix} \right]\,\,\left[ \begin{matrix}    1 & 3  \\    0 & 1  \\ \end{matrix} \right]\,......\left[ \begin{matrix}    1 & n  \\    0 & 1  \\ \end{matrix} \right]\] is equal to the matrix \[\left[ \begin{align}   & 1\,\,378 \\  & 0\,\,\,\,\,1 \\ \end{align} \right]\], then the value of \[n\]is equal to

    A) 26

    B) 27

    C) 337      

    D) 378

    Correct Answer: B

    Solution :

    observe \[{{A}_{1}}{{A}_{2}}\] \[=\left[ \begin{align}   & 1\,\,\,1 \\  & 0\,\,1 \\ \end{align} \right]\left[ \begin{align}   & 1\,\,2 \\  & 0\,\,1 \\ \end{align} \right]=\left[ \begin{align}   & 1\,\,3 \\  & 0\,\,1 \\ \end{align} \right]=\left[ \begin{align}   & 1\left( 2+1 \right) \\  & 0\,\,\,\,\,\,\,1 \\ \end{align} \right]\]
    And \[{{A}_{1}}{{A}_{2}}{{A}_{3}}=\left[ \begin{align}   & 1\,\,3 \\  & 0\,\,1 \\ \end{align} \right]\left[ \begin{align}   & 1\,\,3 \\  & 0\,\,1 \\ \end{align} \right]=\left[ \begin{align}   & 1\left( 3+2+1 \right) \\  & 0\,\,\,\,\,\,\,\,\,\,\,\,1 \\ \end{align} \right]\]
    So, in general \[{{A}_{1}}{{A}_{2}}{{A}_{3}}...{{A}_{n}}\]\[=\left[ \begin{align}   & 1n+\left( n-1 \right)+\left( n-2 \right)+...+3+2+1 \\  & 01 \\ \end{align} \right]\]
    \[So\frac{n\left( n+1 \right)}{2}=378\Rightarrow n=27\]


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