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Manipal Engineering Manipal Engineering Solved Paper-2013

  • question_answer
    Let\[A=\left[ \begin{matrix}    0 & \alpha   \\    0 & 0  \\ \end{matrix} \right]\]and\[{{(A+I)}^{50}}-50A=\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]\]then the value of\[a+b+c+d\]is

    A)  2                                            

    B)  1

    C)  4                            

    D)         None of these

    Correct Answer: A

    Solution :

    As,\[{{A}^{2}}=O,\,\,{{A}^{K}}=O\]          \[\forall \,\,K\ge 2\] Thus,     \[{{(A+I)}^{50}}=I+50\,\,A\] \[\Rightarrow \]               \[{{(A+I)}^{50}}-50A=I\] \[\therefore \]  \[I=\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]\] \[\Rightarrow \]               \[a=1,\,\,b=0,\,\,c=0,\,\,d=1\] \[\therefore \]  \[a+b+c+d=1+0+0+1=2\]


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