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, kÎN (the set of natural numbers) for which rk r k rk 12 23 34 rk rk rk 45 56 67 rk rk rk 7 8 8 9 9 10 log a a log a a log a a log a a log a a log a a 0 log a a log a a log a a ee e ee e eee = Then the number of elements in S, is :

Question:

Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots, \mathrm{a}_{10}$ be in G.P. with $\mathrm{a}_{1}>0$ for $\mathrm{i}=1,2, \ldots, 10$ and $\mathrm{S}$ be the set of pairs $(r, k), r, k \in N$ (the set of natural numbers) for which

$$

\left|\begin{array}{lll}

\log _{e} a_{1}^{\mathrm{r}} \mathrm{a}_{2}^{\mathrm{k}} & \log _{e} \mathrm{a}_{2}^{\mathrm{r}} \mathrm{a}_{3}^{\mathrm{k}} & \log _{e} \mathrm{a}_{3}^{\mathrm{r}} \mathrm{a}_{4}^{\mathrm{k}} \\

\log _{e} \mathrm{a}_{4}^{\mathrm{r}} \mathrm{a}_{5}^{\mathrm{k}} & \log _{e} \mathrm{a}_{5}^{\mathrm{r}} \mathrm{a}_{6}^{\mathrm{k}} & \log _{e} \mathrm{a}_{6}^{\mathrm{r}} \mathrm{a}_{7}^{\mathrm{k}} \\

\log _{e} \mathrm{a}_{7}^{\mathrm{r}} \mathrm{a}_{8}^{\mathrm{k}} & \log _{e} \mathrm{a}_{8}^{\mathrm{r}} \mathrm{a}_{9}^{\mathrm{k}} & \log _{e} \mathrm{a}_{9}^{\mathrm{r}} \mathrm{a}_{10}^{\mathrm{k}}

\end{array}\right|=0

$$

Then the number of elements in S, is : 

  1. 4

  2. infinitely many

  3. 2

  4. 10


Correct Option: 2

JEE Main Previous Year 1 Question of JEE Main from Mathematics Matrices and Determinants chapter.
JEE Main Previous Year Jan. 10, 2019 (II)

Solution:

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