Q83 / 225 · JEE Main 2004 · Mathematics

If $a_1, a_2, a_3, \ldots, a_n, \ldots$. are in G.P., then the value of the determinant $$ \left|\begin{array}{ccc} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array}\right| \text {, is } $$

  • Correct answer
    A
    0
  • B
    $-2$
  • C
    2
  • D
    1

Solution

$$ \begin{aligned} & \left|\begin{array}{lll} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array}\right| \\ & C_3 \rightarrow C_3-C_2, C_2 \rightarrow C_3-C_1 \\ & =\left|\begin{array}{lll} \log a_n & \log r & \log r \\ \log a_{n+3} & \log r & \log r \\ \log a_{n+6} & \log r & \log r \end{array}\right|=0 \end{aligned} $$ (where $r$ is a common ratio).

Hence, the correct answer is option: 'A'.

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