A) 0
B) 1
C) 4 loge
D) 5 loge
Correct Answer: A
Solution :
\[\left| \begin{matrix} \log e & \log {{e}^{2}} & \log {{e}^{3}} \\ \log {{e}^{2}} & \log {{e}^{3}} & \log {{e}^{4}} \\ \log {{e}^{3}} & \log {{e}^{4}} & \log {{e}^{5}} \\ \end{matrix} \right|\] \[=\left| \begin{matrix} \log e & 2\log e & 3\log e \\ 2\log e & 3\log e & 4\log e \\ 3\log e & 4\log e & 5\log e \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \\ \end{matrix} \right|\] Applying \[{{C}_{2}}\to {{C}_{2}}-{{C}_{1}},{{C}_{3}}\to {{C}_{3}}-{{C}_{2}}\] \[=\left| \begin{matrix} 1 & 1 & 1 \\ 2 & 1 & 1 \\ 3 & 1 & 1 \\ \end{matrix} \right|\] (\[\because \]Two columns are identical)
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