SATHEE: Matrices and Determinants 2 Question 8

Matrices and Determinants 2 Question 8

9. Let $a_{1}, a_{2}, a_{3} \dots ., a_{10}$ be in GP with $a_{i} > 0$ for $i = 1, 2, \dots \dots, 10$ and $S$ be the set of pairs $(r, k), r, k \in N$ (the set of natural numbers) for which

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

Then, the number of elements in $S$ , is (2019 Main, 10 Jan II)

(a) 4

(b) 2

(d) infinitely many

$\begin{array}{lll} 1 & b & 2 \end{array}$ value of $\frac{\det (A)}{b}$ is

(2019 Main, 10 Jan II)

(a) $- \sqrt{3}$

(b) $- 2 \sqrt{3}$

(d) $\sqrt{3}$

Show Answer

Answer:

Correct Answer: 9. (c)

Solution:

Given, $| \begin{array}{lll} \log_{e} a_{4}^{r} a_{5}^{k} & \log_{e} a_{5}^{r} a_{6}^{k} & \log_{e} a_{6}^{r} a_{7}^{k} \log_{e} a_{7}^{r} a_{8}^{k} & \log_{e} a_{8}^{r} a_{9}^{k} & \log_{e} a_{9}^{r} a_{10}^{k} \end{array} | = 0$

On applying elementary operations

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

$\log_{e} a_{1}^{r} a_{2}^{k} \log_{e} a_{2}^{r} a_{3}^{k} - \log_{e} a_{1}^{r} a_{2}^{k}$

$\log_{e} a_{4}^{r} a_{5}^{k} \log_{e} a_{5}^{r} a_{6}^{k} - \log_{e} a_{4}^{r} a_{5}^{k}$

$\log_{e} a_{7}^{r} a_{8}^{k} \log_{e} a_{8}^{r} a_{9}^{k} - \log_{e} a_{7}^{r} a_{8}^{k}$

$\begin{aligned} \log_{e} a_{3}^{r} a_{4}^{k} - \log_{e} a_{1}^{r} a_{2}^{k} \\ \log_{e} a_{6}^{r} a_{7}^{k} - \log_{e} a_{4}^{r} a_{5}^{k} \\ \log_{e} a_{9}^{r} a_{10}^{k} - \log_{e} a_{7}^{r} a_{8}^{k} \end{aligned} ∣= 0$

$\begin{aligned} \Rightarrow | \begin{array}{lll} \log_{e} a_{1}^{r} a_{2}^{k} & \log_{e} \frac{a_{2}^{r} a_{3}^{k}}{a_{1}^{r} a_{2}^{k}} & \log_{e} \frac{a_{3}^{r} a_{4}^{k}}{a_{1}^{r} a_{2}^{k}} \\ \log_{e} a_{4}^{r} a_{5}^{k} & \log_{e} \frac{a_{5}^{r} a_{6}^{k}}{a_{4}^{r} a_{5}^{k}} & \log_{e} \frac{a_{6}^{r} a_{7}^{k}}{a_{4}^{r} a_{5}^{k}} \\ \log_{e} a_{7}^{r} a_{8}^{k} & \log_{e} \frac{a_{8}^{r} a_{9}^{k}}{a_{7}^{r} a_{8}^{k}} & \log_{e} \frac{a_{9}^{r} a_{10}^{k}}{a_{7}^{r} a_{8}^{k}} \end{array} | = 0 \\ ∵ \log_{e} m - \log_{e} n = \log_{e} \frac{m}{n} \end{aligned}$

$[∵ a_{1}, a_{2}, a_{3} \dots \dots, a_{10}$ are in GP, therefore put $a_{1} = a, a_{2} = a R, a_{3} = a R^{2}, \dots, a_{10} = a R^{9}]$

$\begin{aligned} \log_{e} a^{r + k} R^{k} \log_{e} \frac{a^{r + k} R^{r + 2 k}}{a^{r + k} R^{k}} \\ \Rightarrow \log_{e} a^{r + k} R^{3 r + 4 k} \log_{e} \frac{a^{r + k} R^{4 r + 5 k}}{a^{r + k} R^{3 r + 4 k}} \\ \log_{e} a^{r + k} R^{6 r + 7 k} \log_{e} \frac{a^{r + k} R^{7 r + 8 k}}{a^{r + k} R^{6 r + 7 k}} \\ \log_{e} \frac{a^{r + k} R^{2 r + 3 k}}{a^{r + k} R^{k}} \\ \log_{e} \frac{a^{r + k} R^{5 + 6 k}}{a^{r + k} R^{3 r + 4 k}} = 0 \\ \log_{e} \frac{a^{r + k} R^{8 r + 9 k}}{a^{r + k} R^{6 r + 7 k}} \end{aligned}$