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Q.
If $a, b, c$ are $pth, qth$ and $rth$ terms of a GP, then $\begin{vmatrix}\log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1\end{vmatrix}$ is equal to -
Determinants
Solution:
$a = a _0 \cdot r _1{ }^{ p -1} \Rightarrow \log a =( p -1) \log r _1+\log a _0$
$b = a _0 \cdot r _1{ }^{ q -1} \Rightarrow \log b =( q -1) \log r _1+\log a _0 $
$ c = a _0 r _1{ }^{ r -1} \Rightarrow \log c =( r -1) \log r _1+\log a _0$
$\begin{vmatrix}\log a_0+(p-1) \log r_1 & p & 1 \\ \log a_0+(q-1) \log r_1 & q & 1 \\ \log a_0+(r-1) \log r_1 & r & 1\end{vmatrix}$
$=\begin{vmatrix}\log a_0 & p & 1 \\ \log a_0 & q & 1 \\ \log a_0 & r & 1\end{vmatrix}+\log r_1\begin{vmatrix}p-1 & p & 1 \\ q-1 & q & 1 \\ r-1 & r & 1\end{vmatrix}=0$