Question

Let $a_1,a_2,a_3,\cdots ,a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,\cdots ,10$ and $S$ be the set of pairs $(r,k),r,k\in N$ (the set of natural numbers) for which
$\left|\begin{array}{ccc}{\log }_e{a}_1^r{a}_2^k& {\log }_e{a}_2^r{a}_3^k& {\log }_e{a}_3^r{a}_4^k\\ {\log }_e{a}_4^r{a}_5^k& {\log }_e{a}_5^r{a}_6^k& {\log }_e{a}_6^r{a}_7^k\\ {\log }_e{a}_7^r{a}_8^k& {\log }_e{a}_8^r{a}_9^k& {\log }_e{a}_9^r{a}_{10}^k\end{array}\right|$

Then the number of elements in $S$, is :

• A. $2$
• B. $4$
• C. $10$
• D. infinitely many

Answer 1

The correct option is D infinitely many

$a_1,a_2,a_3,\cdots ,a_{10}$ are in G.P.
Let $m$ be the common ration of the G.P.
$\therefore m=\frac{a_2}{a_1}=\frac{a_2}{a_3}=\cdots =\frac{a_9}{a_{10}}$
Let, $|A|=\left|\begin{array}{ccc}{\log }_e{a}_1^r{a}_2^k& {\log }_e{a}_2^r{a}_3^k& {\log }_e{a}_3^r{a}_4^k\\ {\log }_e{a}_4^r{a}_5^k& {\log }_e{a}_5^r{a}_6^k& {\log }_e{a}_6^r{a}_7^k\\ {\log }_e{a}_7^r{a}_8^k& {\log }_e{a}_8^r{a}_9^k& {\log }_e{a}_9^r{a}_{10}^k\end{array}\right|$

Apply $c_1\to c_2-c_1$ then apply $c_2\to c_3-c_2$

$\Rightarrow |A|=\left|\begin{array}{ccc}{\log }_e{\left(\frac{a_2}{a_1}\right)}^r{\left(\frac{a_3}{a_2}\right)}^k& {\log }_e{\left(\frac{a_3}{a_2}\right)}^r{\left(\frac{a_4}{a_3}\right)}^k& {\log }_e{a}_3^r{a}_4^k\\ {\log }_e{\left(\frac{a_5}{a_4}\right)}^r{\left(\frac{a_6}{a_5}\right)}^k& {\log }_e{\left(\frac{a_6}{a_5}\right)}^r{\left(\frac{a_7}{a_6}\right)}^k& {\log }_e{a}_6^r{a}_7^k\\ {\log }_e{\left(\frac{a_8}{a_7}\right)}^r{\left(\frac{a_9}{a_8}\right)}^k& {\log }_e{\left(\frac{a_9}{a_8}\right)}^r{\left(\frac{a_{10}}{a_9}\right)}^k& {\log }_e{a}_9^r{a}_{10}^k\end{array}\right|$
$\Rightarrow |A|=\left|\begin{array}{ccc}{\log }_e{m}^{r+k}& {\log }_e{m}^{r+k}& {\log }_e{a}_3^r{a}_4^k\\ {\log }_e{m}^{r+k}& {\log }_e{m}^{r+k}& {\log }_e{a}_6^r{a}_7^k\\ {\log }_e{m}^{r+k}& {\log }_e{m}^{r+k}& {\log }_e{a}_9^r{a}_{10}^k\end{array}\right|$
$\therefore |A|=0\forall r,k$
Hence, the number of elements in $S$ is infinitely many.