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$l,m,n$ are the ${p^{th}},{q^{th}}$and ${r^{th}}$term of a G.P., all positive, then $\left| {\,\begin{array}{*{20}{c}}{\log l}&{p\,\,\,\,\,\begin{array}{*{20}{c}}1\end{array}}\\{\log m}&{q\,\,\,\,\,\begin{array}{*{20}{c}}1\end{array}}\\{\log n}&{r\,\,\,\,\,\begin{array}{*{20}{c}}1\end{array}}\end{array}\,} \right|$ equals
$-1$
$2$
$1$
$0$
Solution
(d) Let $A$ be the first term and $ R$ be the common ratio of the $ G.P$ then,
$l = A{R^{p – 1}} \Rightarrow \log l = \log A + (p – 1)\log R$…..$(i)$
$m = A{R^{q – 1}} \Rightarrow \log m = \log A + (q – 1)\log R$…..$(ii)$
$n = A{R^{r – 1}} \Rightarrow \log n = \log A + (r – 1)\log R$…..$(iii)$
Multiplying $(i), (ii)$ and $(iii)$ by
$(q – r),\,(r – p)\,$ and $(p – q)$respectively and adding we get, $\log l\,(q – r) + \log m(r – p) + \log n(p – q) = 0$
$\therefore \,\Delta = 0$.
