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# $l,m,n$ are the $p^{th},q^{th}$ and $r^{th}$ terms of a GP and all positive then $\begin{vmatrix}\log l&p&1\\\log m&q&1\\\log n&r&1\end{vmatrix}$ equals

$(a)\;3\qquad(b)\;2\qquad(c)\;1\qquad(d)\;zero$

Let $x$ and $y$ are first term and common ratio of GP respectively.
$\therefore l=xy^{p-1}$
$m=xy^{q-1}$
$n=xy^{r-1}$
Now $\begin{vmatrix}\log l&p&1\\\log m&q&1\\\log n&r&1\end{vmatrix}=\begin{vmatrix}\log l/m&p-q&0\\\log m/n&q-r&0\\\log n&r&1\end{vmatrix}$
Apply $R_1\rightarrow R_1-R_2$
$\qquad R_2\rightarrow R_2-R_3$
$\Rightarrow \begin{vmatrix}(p-q)\log y&p-q&0\\(q-r)\log y&q-r&0\\\log n&r&1\end{vmatrix}$
$\Rightarrow (p-q)(q-r)\begin{vmatrix}\log y&1&0\\\log y&1&0\\\log n&r&1\end{vmatrix}$
$1^{st}$ and $2^{nd}$ row are identical.
$\Rightarrow 0$
Hence (d) is the correct option.