Let $a_{1}, a_{2}, a_{3}, \dots, a_{10}$ be in G.P. with $a_{i} > 0$ for
$i = 1, 2, 3, \dots, 10$ and $S$ be the set of pairs $(r, k), r, k \in N)$ for which
$D = |\begin{array}{lll} \log_{e} a_{1}^{r} a_{2}^{k} \log_{e} a_{2}^{r} a_{3}^{k} \log_{d} a_{3}^{r} a_{4}^{k} \\ \log_{e} a_{4}^{r} a_{5}^{k} \log_{e} a_{5}^{r} a_{6}^{k} \log_{e} a_{6}^{r} a_{7}^{k} \\ \log_{e} a_{7}^{r} a_{8}^{k} \log_{e} a_{8}^{r} a_{9}^{k} \log_{e} a_{9}^{r} a_{10}^{k} \end{array}| = 0$
Then the number of elements in S, is

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answer is A.

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Detailed Solution

Let $R$ be the common ratio of the G.P. $a_{1}, a_{2}, \dots, a_{10}$

$D = |\begin{array}{lll} \log_{e} a_{1}^{r} a_{2}^{k} \log_{e} a_{2}^{r} a_{3}^{k} \log_{d} a_{3}^{r} a_{4}^{k} \\ \log_{e} a_{4}^{r} a_{5}^{k} \log_{e} a_{5}^{r} a_{6}^{k} \log_{e} a_{6}^{r} a_{7}^{k} \\ \log_{e} a_{7}^{r} a_{8}^{k} \log_{e} a_{8}^{r} a_{9}^{k} \log_{e} a_{9}^{r} a_{10}^{k} \end{array}| = 0$

Applying $C_{2} \to C_{2} - C_{1} and C_{3} \to C_{3} - C_{2}$ , we obtain

$D = |\begin{array}{lll} \log_{e} a_{1}^{r} a_{2}^{k} \log R^{r + k} \log R^{r + k} \\ \log_{e} a_{4}^{r} a_{5}^{k} \log R^{r + k} \log R^{r + k} \\ \log_{e} a_{7}^{r} a_{8}^{k} \log R^{r + k} \log R^{r + k} \end{array}|$