JEE PYQ: Sequence And Series Question 62
Question 62 - 2019 (10 Jan Shift 2)
Let $a_1, a_2, a_3, \ldots, a_{10}$ be in G.P. with $a_i > 0$ for $i = 1, 2, \ldots, 10$ and $S$ be the set of pairs $(r, k)$, $r, k \in N$ (the set of natural numbers) for which
$\begin{vmatrix} \log_e a_1^r a_2^k & \log_e a_2^r a_3^k & \log_e a_3^r a_4^k \ \log_e a_4^r a_5^k & \log_e a_5^r a_6^k & \log_e a_6^r a_7^k \ \log_e a_7^r a_8^k & \log_e a_8^r a_9^k & \log_e a_9^r a_{10}^k \end{vmatrix} = 0$
Then the number of elements in $S$, is:
(1) 4
(2) infinitely many
(3) 2
(4) 10
Show Answer
Answer: (2)
Solution
Let common ratio of G.P. be $R$
$\Rightarrow a_2 = a_1R, a_3 = a_1R^2, \ldots, a_{10} = a_1R^9$
Each column becomes a linear function of $r$ and $k$ with the same structure, so $\Delta = 0$ for all $r, K \in N$
Hence, number of elements in $S$ is infinitely many.
BETA
