Question

If $a _{1}, a _{2}, a _{3}, \ldots \ldots \ldots a _{ n }$ are in G.P then $\begin{vmatrix}\log a_{1} & \log a_{2} & \log a_{3} \\ \log a_{4} & \log a_{5} & \log a_{6} \\ \log a_{7} & \log a_{8} & \log a_{9}\end{vmatrix}$

• A. 2
• B. 1
• C. 0
• D. 5

Answer 1

Answer: (C)

If $a_{1}, a_{2}, a_{3} \ldots \ldots \ldots a_{n}$ are in G.P.
then $\log a_{1}, \log a_{2}, \log a_{3} \ldots \ldots . . \log a_{n}$ are in A.P.
Let $\log a_{1}=a, \log a_{2}=a+d$ etc then
G.E. $=\begin{vmatrix}a & a+d & a+2 d \\ a+3 d & a+4 d & a+5 d \\ a+6 d & a+7 d & a+8 d \end{vmatrix}=\begin{vmatrix}a & d & 2 d \\ a+3 d & d & 2 d \\ a+6 d & d & 2 d \end{vmatrix}=0$
by c $_{2}- c _{1}$ and $c _{3}- c _{1}$