question_answer

Let \[{{a}_{1}},\text{ }{{a}_{2}},{{a}_{3}},.....\text{ }{{a}_{10}}\] be in G.P. with \[{{a}_{i}}>0\] for \[i=1,\text{ }2,\text{ }......10\] and S be the set of pairs (r, k), r, k e 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 -

  [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

A) 10                                            

B) 4

C) 2                     

D)                  infinitely many