Matrices and Determinants 2 Question 8
9. Let $a _1, a _2, a _3 \ldots ., a _{10}$ be in GP with $a _i>0$ for $i=1,2, \ldots \ldots, 10$ and $S$ be the set of pairs $(r, k), r, k \in N$ (the set of natural numbers) for which
$$ \left|\begin{array}{llll} \log _e a _1^{r} a _2^{k} & \log _e a _2^{r} a _3^{k} & \log _e a _3^{r} a _4^{k} \\ \log _e a _4^{r} a _5^{k} & \log _e a _5^{r} a _6^{k} & \log _e a _6^{r} a _7^{k} \\ \log _e a _7^{r} a _8^{k} & \log _e a _8^{r} a _9^{k} & \log _e a _9^{r} a _{10}^{k} \end{array}\right|=0 $$
Then, the number of elements in $S$, is (2019 Main, 10 Jan II)
(a) 4
(b) 2
(c) 10
(d) infinitely many
$\begin{array}{lll}1 & b & 2\end{array}$ value of $\frac{\operatorname{det}(A)}{b}$ is
(2019 Main, 10 Jan II)
(a) $-\sqrt{3}$
(b) $-2 \sqrt{3}$
(c) $2 \sqrt{3}$
(d) $\sqrt{3}$
Show Answer
Answer:
Correct Answer: 9. (c)
Solution:
- Given, $\left|\begin{array}{lll}\log _e a _4^{r} a _5^{k} & \log _e a _5^{r} a _6^{k} & \log _e a _6^{r} a _7^{k} \ \log _e a _7^{r} a _8^{k} & \log _e a _8^{r} a _9^{k} & \log _e a _9^{r} a _{10}^{k}\end{array}\right|=0$
On applying elementary operations
$C _2 \rightarrow C _2-C _1$ and $C _3 \rightarrow C _3-C _1$, we get
$\log _e a _1^{r} a _2^{k} \quad \log _e a _2^{r} a _3^{k}-\log _e a _1^{r} a _2^{k}$
$\log _e a _4^{r} a _5^{k} \quad \log _e a _5^{r} a _6^{k}-\log _e a _4^{r} a _5^{k}$
$\log _e a _7^{r} a _8^{k} \quad \log _e a _8^{r} a _9^{k}-\log _e a _7^{r} a _8^{k}$
$$ \begin{aligned} & \log _e a _3^{r} a _4^{k}-\log _e a _1^{r} a _2^{k} \\ & \log _e a _6^{r} a _7^{k}-\log _e a _4^{r} a _5^{k} \\ & \log _e a _9^{r} a _{10}^{k}-\log _e a _7^{r} a _8^{k} \end{aligned} \mid=0 $$
$$ \begin{aligned} & \Rightarrow\left|\begin{array}{lll} \log _e a _1^{r} a _2^{k} & \log _e \frac{a _2^{r} a _3^{k}}{a _1^{r} a _2^{k}} & \log _e \frac{a _3^{r} a _4^{k}}{a _1^{r} a _2^{k}} \\ \log _e a _4^{r} a _5^{k} & \log _e \frac{a _5^{r} a _6^{k}}{a _4^{r} a _5^{k}} & \log _e \frac{a _6^{r} a _7^{k}}{a _4^{r} a _5^{k}} \\ \log _e a _7^{r} a _8^{k} & \log _e \frac{a _8^{r} a _9^{k}}{a _7^{r} a _8^{k}} & \log _e \frac{a _9^{r} a _{10}^{k}}{a _7^{r} a _8^{k}} \end{array}\right|=0 \\ & \because \log _e m-\log _e n=\log _e \frac{m}{n} \end{aligned} $$
$\left[\because a _1, a _2, a _3 \ldots \ldots, a _{10}\right.$ are in GP, therefore put $\left.a _1=a, a _2=a R, a _3=a R^{2}, \ldots, a _{10}=a R^{9}\right]$
$$ \begin{aligned} & \log _e a^{r+k} R^{k} \quad \log _e \frac{a^{r+k} R^{r+2 k}}{a^{r+k} R^{k}} \\ & \Rightarrow \quad \log _e a^{r+k} R^{3 r+4 k} \quad \log _e \frac{a^{r+k} R^{4 r+5 k}}{a^{r+k} R^{3 r+4 k}} \\ & \log _e a^{r+k} R^{6 r+7 k} \quad \log _e \frac{a^{r+k} R^{7 r+8 k}}{a^{r+k} R^{6 r+7 k}} \\ & \log _e \frac{a^{r+k} R^{2 r+3 k}}{a^{r+k} R^{k}} \\ & \log _e \frac{a^{r+k} R^{5+6 k}}{a^{r+k} R^{3 r+4 k}}=0 \\ & \log _e \frac{a^{r+k} R^{8 r+9 k}}{a^{r+k} R^{6 r+7 k}} \end{aligned} $$
$$ \begin{aligned} & \left.\log _e R^{2 r+2 k}=\log _e R^{2(r+k)}=2 \log _e R^{r+k}\right] \end{aligned} $$
So, value of determinant will be zero for any value of $(r, k), r, k \in N$.
$\therefore$ Set ’ $S$ ’ has infinitely many elements.
