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(D) Given that $a_1, a_2, \ldots, a_9$ are in $G.P.$Let $r$ be the common ratio,then $\frac{a_{n+1}}{a_n} = r$ for all $n$.Let $\Delta = \begin{vmatri

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(D) Given that $a_1, a_2, \ldots, a_9$ are in $G.P.$Let $r$ be the common ratio,then $\frac{a_{n+1}}{a_n} = r$ for all $n$.Let $\Delta = \begin{vmatrix} \log a_1 & \log a_2 & \log a_3 \ \log a_4 & \log a_5 & \log a_6 \ \log a_7 & \log a_8 & \log a_9 \end{vmatrix}$.Applying column operations $C_2 	o C_2 - C_1$ and $C_3 	o C_3 - C_2$,we get:$\Delta = \begin{vmatrix} \log a_1 & \log a_2 - \log a_1 & \log a_3 - \log a_2 \ \log a_4 & \log a_5 - \log a_4 & \log a_6 - \log a_5 \ \log a_7 & \log a_8 - \log a_7 & \log a_9 - \log a_8 \end{vmatrix}$.Using the property $\log m - \log n = \log(\frac{m}{n})$,we have:$\Delta = \begin{vmatrix} \log a_1 & \log(\frac{a_2}{a_1}) & \log(\frac{a_3}{a_2}) \ \log a_4 & \log(\frac{a_5}{a_4}) & \log(\frac{a_6}{a_5}) \ \log a_7 & \log(\frac{a_8}{a_7}) & \log(\frac{a_9}{a_8}) \end{vmatrix} = \begin{vmatrix} \log a_1 & \log r & \log r \ \log a_4 & \log r & \log r \ \log a_7 & \log r & \log r \end{vmatrix}$.Since column $C_2$ and column $C_3$ are identical,the value of the determinant is $0$.

If $a_1, a_2, \ldots, a_9$ are in $G.P.$,then $\left|\begin{array}{lll}\log a_1 & \log a_2 & \log a_3 \\ \log a_4 & \log a_5 & \log a_6 \\ \log a_7 & \log a_8 & \log a_9\end{array}\right|$ is equal to

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