Let $a_{1}, a_{2}, a_{3}, \dots, a_{10}$ be in G.P. with $a_{i} > 0$ for
$i = 1, 2, 3, \dots, 10$ and $S$ be the set of pairs $(r, k), r, k \in N)$ for which
$D = |\begin{array}{lll} \log_{e} a_{1}^{r} a_{2}^{k} \log_{e} a_{2}^{r} a_{3}^{k} \log_{d} a_{3}^{r} a_{4}^{k} \\ \log_{e} a_{4}^{r} a_{5}^{k} \log_{e} a_{5}^{r} a_{6}^{k} \log_{e} a_{6}^{r} a_{7}^{k} \\ \log_{e} a_{7}^{r} a_{8}^{k} \log_{e} a_{8}^{r} a_{9}^{k} \log_{e} a_{9}^{r} a_{10}^{k} \end{array}| = 0$
Then the number of elements in S, is

see full answer

High-Paying Jobs That Even AI Can’t Replace — Through JEE/NEET

🎯 Hear from the experts why preparing for JEE/NEET today sets you up for future-proof, high-income careers tomorrow.

An Intiative by Sri Chaitanya

Infinity many

answer is A.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Let $R$ be the common ratio of the G.P. $a_{1}, a_{2}, \dots, a_{10}$

$D = |\begin{array}{lll} \log_{e} a_{1}^{r} a_{2}^{k} \log_{e} a_{2}^{r} a_{3}^{k} \log_{d} a_{3}^{r} a_{4}^{k} \\ \log_{e} a_{4}^{r} a_{5}^{k} \log_{e} a_{5}^{r} a_{6}^{k} \log_{e} a_{6}^{r} a_{7}^{k} \\ \log_{e} a_{7}^{r} a_{8}^{k} \log_{e} a_{8}^{r} a_{9}^{k} \log_{e} a_{9}^{r} a_{10}^{k} \end{array}| = 0$

Applying $C_{2} \to C_{2} - C_{1} and C_{3} \to C_{3} - C_{2}$ , we obtain

$D = |\begin{array}{lll} \log_{e} a_{1}^{r} a_{2}^{k} \log R^{r + k} \log R^{r + k} \\ \log_{e} a_{4}^{r} a_{5}^{k} \log R^{r + k} \log R^{r + k} \\ \log_{e} a_{7}^{r} a_{8}^{k} \log R^{r + k} \log R^{r + k} \end{array}|$