Matrices And Determinants Ques 10
- 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
Then, the number of elements in S, is
(2019 Main, 10 Jan II)
(a) $ 4$
(b) $ 2$
(c) $10$
(d) infinitely many
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Answer:
Correct Answer: 10.(d)
Solution: (d) Given, $\left|\begin{array}{lll}\log_e a_1^r a_2^k & \log_e a_2^r a_3^k & \log_e a_3^r a_4^k \\ \log_e a_4^r a_5^k & \log_e a_5^r a_6^k & \log_e a_6^r a_7^k \\ \log_e a_7^r a_8^k & \log_e a_8^r a_9^k & \log_e a_9^r a_{10}^k\end{array}\right|=0$
On applying elementary operations $C_2 \rightarrow C_2-C_1$ and $C_3 \rightarrow C_3-C_1$, we get
$ \begin{aligned} & \left\lvert, \begin{array}{ll} \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_9^k \end{array}\right. \\ & \left.\begin{array}{l} \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{array} \right\rvert,=0 \\ & \Rightarrow\left|\begin{array}{lll} \log_e a_1^r a_2^k & \log_e\left(\frac{a_2^r a_3^k}{a_1^r a_2^k}\right) & \log_e\left(\frac{a_3^r a_4^k}{a_1^r a_2^k}\right) \\ \log_e a_4^r a_5^k & \log_e\left(\frac{a_5^r a_6^k}{a_4^r a_5^k}\right) & \log_e\left(\frac{a_6^r a_7^k}{a_4^r a_5^k}\right) \\ \log_e a_7^r a_8^k & \log_e\left(\frac{a_8^r a_9^k}{a_7^r a_8^k}\right) & \log_e\left(\frac{a_9^r a_{10}^k}{a_7^r a_8^k}\right) \end{array}\right|=0 \\ & {\left[\because \quad \log_e m-\log_e n=\log_e\left(\frac{m}{n}\right)\right]} \\ \end{aligned} $
$\left[\because \quad a_1, a_2, a_3 \ldots \ldots, a_{10}\right.$ 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] $
$ \Rightarrow\left|\begin{array}{cc} \log_e a^{r+k} R^k & \log_e\left(\frac{a^{r+k} R^{r+2 k}}{a^{r+k} R^k}\right) & \log_e\left(\frac{a^{r+k} R^{2 r+3 k}}{a^{r+k} R^k}\right)\\ \log_e a^{r+k} R^{3 r+4 k} & \log_e\left(\frac{a^{r+k} R^{4 r+5 k}}{a^{r+k} R^{3 r+4 k}}\right) & \log_e\left(\frac{a^{r+k} R^{5 r+6 k}}{a^{r+k} R^{3 r+4 k}}\right) \\ \log_e a^{r+k} R^{6 r+7 k} & \log_e\left(\frac{a^{r+k} R^{7 r+8 k}}{a^{r+k} R^{6 r+7 k}}\right) & \log_e\left(\frac{a^{r+k} R^{8 r+9 k}}{a^{r+k} R^{6 r+7 k}}\right) \end{array}\right|=0 $
$ \Rightarrow\left|\begin{array}{ccc} \log_e\left(a^{r+k} R^k\right) & \log_e R^{r+k} \log_e R^{2 r+2 k} \\ \log_e a^{r+k} R^{3 r+4 k} & \log_e R^{r+k} \log_e R^{2 r+2 k} \\ \log_e a^{r+k} R^{6 r+7 k} & \log_e R^{r+k} \log_e R^{2 r+2 k} \end{array}\right|=0 $
$ \Rightarrow\left|\begin{array}{cc} \log_e\left(a^{r+k} R^k\right) & \log_e R^{r+k} 2 \log_e R^{r+k} \\ \log_e\left(a^{r+k} R^{3 r+4 k}\right) & \log_e R^{r+k} 2 \log_e R^{r+k} \\ \log_e\left(a^{r+k} R^{6 r+7 k}\right) & \log_e R^{r+k} 2 \log_e R^{r+k} \end{array}\right|=0 $
$\left[\because \quad \log m^n=n \log m\right.$ and here
$ \left.\log_e R^{2 r+2 k}=\log_e R^{2(r+k)}=2 \log_e R^{r+k}\right] $
$\because \quad$ Column $\mathrm{C}_2$ and $C_3$ are proportional,
So, value of determinant will be zero for any value of $(r, k), r, k \in N$.
$\therefore \quad $ Set ’ $S$ ’ has infinitely many elements.
BETA
