Question

If $D_{p} = \left| \begin{matrix} p & 15 & 8 \\ p^{2} & 35 & 9 \\ p^{3} & 25 & 10 \end{matrix} \right|$, then $D_{1} + D_{2} + D_{3} + D_{4} + D_{5} =$

• A. 0
• B. 25
• C. 625
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

$\bullet \bullet$ $D_{1} = \left| \begin{matrix} 1 & 15 & 8 \\ 1 & 35 & 9 \\ 1 & 25 & 10 \end{matrix} \right|$, $D_{2} = \left| \begin{matrix} 2 & 15 & 8 \\ 4 & 35 & 9 \\ 8 & 25 & 10 \end{matrix} \right|$, $D_{3} = \left| \begin{matrix} 3 & 15 & 8 \\ 9 & 35 & 9 \\ 27 & 25 & 10 \end{matrix} \right|$,$D_{4} = \left| \begin{matrix} 4 & 15 & 8 \\ 16 & 35 & 9 \\ 64 & 25 & 10 \end{matrix} \right|$, $D_{5} = \left| \begin{matrix} 5 & 15 & 8 \\ 25 & 35 & 9 \\ 125 & 25 & 10 \end{matrix} \right|$

$\Rightarrow$ $D _ { 1 } + D _ { 2 } + D _ { 3 } + D _ { 4 } + D _ { 5 } =$ $\left| \begin{matrix} 15 & 75 & 40 \\ 55 & 175 & 45 \\ 225 & 125 & 50 \end{matrix} \right|$

$= 15(3125) - 75( - 7375) + 40( - 32500) = 46875 + 553125 - 1300000 = - 700000$