Question

Let a₁, a₂, a₃,…. , a₁₀ be in G.P. with a_i > 0 for I = 1, 2, …, 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which

\(\left| {\begin{array}{*{20}{c}} {{\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}&{{\rm{lo}}{{\rm{g}}_e}a_2^ra_3^k}&{{\rm{lo}}{{\rm{g}}_e}a_3^ra_4^k}\\ {{\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}&{{\rm{lo}}{{\rm{g}}_e}a_5^\pi a_6^k}&{{\rm{lo}}{{\rm{g}}_e}a_6^ra_7^k}\\ {{\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k}&{{\rm{lo}}{{\rm{g}}_e}a_8^\pi a_9^k}&{{\rm{lo}}{{\rm{g}}_e}a_9^ra_{10}^k} \end{array}} \right| = 0\)

Then the number of elements in S, is:
1. 4
2. Infinitely many
3. 2
4. 10

Answer 1

Correct Answer - Option 2 : Infinitely many

From question, the numbers given is:

\(\left| {\begin{array}{*{20}{c}} {{\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}&{{\rm{lo}}{{\rm{g}}_e}a_2^ra_3^k}&{{\rm{lo}}{{\rm{g}}_e}a_3^ra_4^k}\\ {{\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}&{{\rm{lo}}{{\rm{g}}_e}a_5^\pi a_6^k}&{{\rm{lo}}{{\rm{g}}_e}a_6^ra_7^k}\\ {{\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k}&{{\rm{lo}}{{\rm{g}}_e}a_8^\pi a_9^k}&{{\rm{lo}}{{\rm{g}}_e}a_9^ra_{10}^k} \end{array}} \right| = 0\)

On applying, C₂ → C₂ - C₁ and C₃ → C₃ - C₁,

\(\Rightarrow \left| {\begin{array}{*{20}{c}} {{\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}&{{\rm{lo}}{{\rm{g}}_e}a_2^ra_3^k - {\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}&{{\rm{lo}}{{\rm{g}}_e}a_3^ra_4^k - {\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}\\ {{\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}&{{\rm{lo}}{{\rm{g}}_e}a_5^ra_6^k - {\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}&{{\rm{lo}}{{\rm{g}}_e}a_6^ra_7^k - {\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}\\ {{\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k}&{{\rm{lo}}{{\rm{g}}_e}a_8^ra_9^k - {\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k}&{{\rm{lo}}{{\rm{g}}_e}a_9^ra_{10}^k - {\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k} \end{array}} \right| = 0\)

\(\because \left[ {{\rm{lo}}{{\rm{g}}_e}m - {\rm{lo}}{{\rm{g}}_e}n = {\rm{lo}}{{\rm{g}}_e}\left( {\frac{m}{n}} \right)} \right]\) 

\(\Rightarrow \left| {\begin{array}{*{20}{c}} {{\rm{lo}}{{\rm{g}}_e}a_1^ra_2^k}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{a_2^ra_3^k}}{{a_1^ra_2^k}}} \right)}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{a_3^ra_4^k}}{{a_1^ra_2^k}}} \right)}\\ {{\rm{lo}}{{\rm{g}}_e}a_4^ra_5^k}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{a_5^ra_6^k}}{{a_4^ra_5^k}}} \right)}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{a_6^ra_7^k}}{{a_4^ra_5^k}}} \right)}\\ {{\rm{lo}}{{\rm{g}}_e}a_7^ra_8^k}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{a_8^ra_9^k}}{{a_7^ra_8^k}}} \right)}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{a_9^ra_{10}^k}}{{a_7^ra_8^k}}} \right)} \end{array}} \right| = 0\)

a₁, a₂, a₃,……,a₁₀ are in GP, therefore, we can put, a₁ = a; a₂ = aR; a₃ = aR²; …. a₁₀ = aR⁹

\(\Rightarrow \left| {\begin{array}{*{20}{c}} {{\rm{lo}}{{\rm{g}}_e}{a^{r + k}}{R^k}}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{{a^{r + k}}{R^{r + 2k}}}}{{{a^{r + k}}{R^k}}}} \right)}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{{a^{r + k}}{R^{2r + 3k}}}}{{{a^{r + k}}{R^k}}}} \right)}\\ {{\rm{lo}}{{\rm{g}}_e}{a^{r + k}}{R^{3r + 4k}}}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{{a^{r + k}}{R^{4r + 5k}}}}{{{a^{r + k}}{R^{3r + 4k}}}}} \right)}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{{a^{r + k}}{R^{5r + 6k}}}}{{{a^{r + k}}{R^{3r + 4k}}}}} \right)}\\ {{\rm{lo}}{{\rm{g}}_e}{a^{r + k}}{R^{6r + 7k}}}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{{a^{r + k}}{R^{7r + 8k}}}}{{{a^{r + k}}{R^{6r + 7k}}}}} \right)}&{{\rm{lo}}{{\rm{g}}_e}\left( {\frac{{{a^{r + k}}{R^{8r + 9k}}}}{{{a^{r + k}}{R^{6r + 7k}}}}} \right)} \end{array}} \right| = 0\)

\(\Rightarrow \left| {\begin{array}{*{20}{c}} {{\rm{lo}}{{\rm{g}}_e}\left( {{a^{r + k}}{R^k}} \right)}&{{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}}&{{\rm{lo}}{{\rm{g}}_e}{R^{2r + 2k}}}\\ {{\rm{lo}}{{\rm{g}}_e}{a^{r + k}}{R^{3r + 4k}}}&{{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}}&{{\rm{lo}}{{\rm{g}}_e}{R^{2r + 2k}}}\\ {{\rm{lo}}{{\rm{g}}_e}{a^{r + k}}{R^{6r + 7k}}}&{{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}}&{{\rm{lo}}{{\rm{g}}_e}{R^{2r + 2k}}} \end{array}} \right| = 0\)

\(\because \left[ {\begin{array}{*{20}{c}} {{\rm{log}}{m^n} = n{\rm{log}}m}\\ {{\rm{lo}}{{\rm{g}}_e}{R^{2r + 2k}} = {\rm{lo}}{{\rm{g}}_e}{R^{2\left( {r + k} \right)}} = 2{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}} \end{array}} \right]\) 

\(\Rightarrow \left| {\begin{array}{*{20}{c}} {{\rm{lo}}{{\rm{g}}_e}\left( {{a^{r + k}}{R^k}} \right)}&{{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}}&{2{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}}\\ {{\rm{lo}}{{\rm{g}}_e}\left( {{a^{r + k}}{R^{3r + 4k}}} \right)}&{{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}}&{2{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}}\\ {{\rm{lo}}{{\rm{g}}_e}\left( {{a^{r + k}}{R^{6r + 7k}}} \right)}&{{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}}&{2{\rm{lo}}{{\rm{g}}_e}{R^{r + k}}} \end{array}} \right| = 0\)

Now, column C₂ and C₃ are proportional.

So, value of determinant will be zero for any value of (r, k), r, k ∈ N.

Thus, Set ‘S’ has infinitely many elements.