Let a₁,a₂,a₃,....,a_{10} be in G.P. with aᵢ > 0 for i = 1, 2,....,10 and S be set of pair

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3 and 4 .Determinants and Matrices

hard

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

$\left| {\begin{array}{*{20}{c}}
{{{\log }_e}\,a_1^ra_2^k}&{{{\log }_e}\,a_2^ra_3^k}&{{{\log }_e}\,a_3^ra_4^k} \\
{{{\log }_e}\,a_4^ra_5^k}&{{{\log }_e}\,a_5^ra_6^k}&{{{\log }_e}\,a_6^ra_7^k} \\
{{{\log }_e}\,a_7^ra_8^k}&{{{\log }_e}\,a_8^ra_9^k}&{{{\log }_e}\,a_9^ra_{10}^k}
\end{array}} \right| = 0$

Then the number of elements in $S$, is

A

$4$

B

infinitely many

C

$2$

D

$10$

(JEE MAIN-2019)

Solution

For any value of $r$ determinant is zero.

Standard 12

Mathematics

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