Mathematics - Determinant Question with Solution | TestHub

Let $A$ be the 1 st term and $\mathrm{R}$ the common ratio of G.P., then $\begin{array}{rl}& a=T_p=A{R}^{p-1}\\ & \therefore \log \mathrm{a}=\log \mathrm{A}+(\mathrm{p}-1)\log \mathrm{R}\end{array}$ Similarly, $\log \mathrm{b}=\log \mathrm{A}+(\mathrm{q}-1)\log \mathrm{R}$ and $\log \mathrm{c}=\log \mathrm{A}+(\mathrm{r}-1)\log \mathrm{R}$ $\therefore \Delta =\left|\begin{array}{lll}\log A+(p-1)\log R& p& 1\\ \log A+(q-1)\log R& q& 1\\ \log A+(r-1)\log R& r& 1\end{array}\right|$ Split into two determinants and in the first take $\log A$ common and in the second take $\log \mathrm{R}$ common $\Delta =\log A\left|\begin{array}{lll}1& p& 1\\ 1& q& 1\\ 1& r& 1\end{array}\right|+\log R\left|\begin{array}{lll}p-1& p& 1\\ q-1& q& 1\\ r-1& r& 1\end{array}\right|$ Apply ${\mathrm{C}}_1\to {\mathrm{C}}_2\to {\mathrm{C}}_3$ in the second $\Delta =0+\log R\left|\begin{array}{lll}0& p& 1\\ 0& q& 1\\ 0& r& 1\end{array}\right|=0$