question_answer

Let \[{{a}_{1}},{{a}_{2}},{{a}_{3}},...{{a}_{10}}\] be in G.P. with\[{{a}_{i}}>0\]for\[i=1,\]\[2,....,10\] and S be the set of pairs \[(r,k),r,k\in N\] (the set of natural numbers) for which \[\left| \begin{matrix}    {{\log }_{e}}a_{1}^{r}a_{2}^{k} & {{\log }_{e}}a_{2}^{r}a_{3}^{k} & {{\log }_{e}}a_{3}^{r}a_{4}^{k}  \\    {{\log }_{e}}a_{4}^{r}a_{5}^{k} & {{\log }_{e}}a_{5}^{r}a_{6}^{k} & {{\log }_{e}}a_{6}^{r}a_{7}^{k}  \\    {{\log }_{e}}a_{7}^{r}a_{8}^{k} & {{\log }_{e}}a_{8}^{r}a_{9}^{k} & {{\log }_{e}}a_{9}^{r}a_{10}^{k}  \\ \end{matrix} \right|=0\] Then the number of elements in S, is:

A) \[4\]

B) infinitely many

C) \[2\]          

D) \[10\]