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(D) Let $A$ be the first term and $R$ be the common ratio of the $G$.$P$. Then,$l = A R^{p-1} \Rightarrow \log l = \log A + (p-1) \log R$ ... $(i)$$m

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$0$

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(D) Let $A$ be the first term and $R$ be the common ratio of the $G$.$P$. Then,$l = A R^{p-1} \Rightarrow \log l = \log A + (p-1) \log R$ ... $(i)$$m = A R^{q-1} \Rightarrow \log m = \log A + (q-1) \log R$ ... $(ii)$$n = A R^{r-1} \Rightarrow \log n = \log A + (r-1) \log R$ ... $(iii)$Let the determinant be $\Delta = \left| \begin{array}{ccc} \log l & p & 1 \ \log m & q & 1 \ \log n & r & 1 \end{array} ight|$.Applying the column operations $C_1 	o C_1 - (\log A) C_3$ and then $C_1 	o C_1 / (\log R)$,we observe that the first column becomes proportional to the second column shifted by a constant. More simply,applying $C_1 	o C_1 - (\log A) C_3$,we get $\log l - \log A = (p-1) \log R$,etc. Since the columns become linearly dependent (specifically,$C_1$ is a linear combination of $C_2$ and $C_3$),the value of the determinant is $0$.

$l, m, n$ are the $p^{th}, q^{th}$ and $r^{th}$ terms of a $G$.$P$.,all positive,then $\left| \begin{array}{ccc} \log l & p & 1 \\ \log m & q & 1 \\ \log n & r & 1 \end{array} \right|$ equals

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