Let a1, a2, a3,…. , a10 be in G.P. with ai > 0 for I = 1, 2, …, 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which\(\left| {\begin{array}{*{20}{c}}{{\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}&{{\rm{lo}}{{\rm{g}}_e}a_2^ra_3^k}&{{\rm{lo}}{{\rm{g}}_e}a_3^ra_4^k}\\{{\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}&{{\rm{lo}}{{\rm{g}}_e}a_5^\pi a_6^k}&{{\rm{lo}}{{\rm{g}}_e}a_6^ra_7^k}\\{{\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k}&{{\rm{lo}}{{\rm{g}}_e}a_8^\pi a_9^k}&{{\rm{lo}}{{\rm{g}}_e}a_9^ra_{10}^k}\end{array}} \right| = 0\)Then the number of elements in S, is:

Let a1, a2, a3,…. , a10 be in G.P. with ai > 0 for I = 1, 2, …, 10

1.	Let a1, a2, a3,…. , a10 be in G.P. with ai > 0 for I = 1, 2, …, 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which\(\left\| {\begin{array}{*{20}{c}}{{\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}&{{\rm{lo}}{{\rm{g}}_e}a_2^ra_3^k}&{{\rm{lo}}{{\rm{g}}_e}a_3^ra_4^k}\\{{\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}&{{\rm{lo}}{{\rm{g}}_e}a_5^\pi a_6^k}&{{\rm{lo}}{{\rm{g}}_e}a_6^ra_7^k}\\{{\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k}&{{\rm{lo}}{{\rm{g}}_e}a_8^\pi a_9^k}&{{\rm{lo}}{{\rm{g}}_e}a_9^ra_{10}^k}\end{array}} \right\| = 0\)Then the number of elements in S, is:
A.	4
B.	Infinitely many
C.	2
D.	10
Answer» C. 2

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Let a1, a2, a3,…. , a10 be in G.P. with ai > 0 for I = 1, 2, …, 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which\(\left| {\begin{array}{*{20}{c}}{{\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}&{{\rm{lo}}{{\rm{g}}_e}a_2^ra_3^k}&{{\rm{lo}}{{\rm{g}}_e}a_3^ra_4^k}\\{{\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}&{{\rm{lo}}{{\rm{g}}_e}a_5^\pi a_6^k}&{{\rm{lo}}{{\rm{g}}_e}a_6^ra_7^k}\\{{\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k}&{{\rm{lo}}{{\rm{g}}_e}a_8^\pi a_9^k}&{{\rm{lo}}{{\rm{g}}_e}a_9^ra_{10}^k}\end{array}} \right| = 0\)Then the number of elements in S, is:

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