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Let a1, a2,..., a10 be a G.P. with a_i > 0 for all i. Let S be the set of pairs (r, k), r, k in N, for which the 3x3 determinant with entries logₑ(a_r^? aₖ^?) (as defined) equals 0. The number of elements in S is:

1. Infinitely many
2. 4
3. 10
4. 2

Correct answer: Infinitely many

Solution

Since log of GP terms is linear (arithmetic progression), the rows become linearly dependent for all (r,k), so the determinant is always 0 and S is infinite.

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