If $\mathrm{a} 0, \mathrm{~b} 0, \mathrm{c} 0$ are respectively the $\mathrm{pth}, \mathrm{q}$ th, rth terms of GP., then the value of the determinant $\left|\begin{array}{lll}\log \mathrm{a} & \mathrm{p} & 1 \ \log \mathrm{b} & \mathrm{q} & 1 \ \log \mathrm{c} & \mathrm{r} & 1\end{array}\right|$ is

Question

If $\mathrm{a} > 0, \mathrm{~b} > 0, \mathrm{c} > 0$ are respectively the $\mathrm{pth}, \mathrm{q}$ th, rth terms of GP., then the value of the determinant $\left|\begin{array}{lll}\log \mathrm{a} & \mathrm{p} & 1 \ \log \mathrm{b} & \mathrm{q} & 1 \ \log \mathrm{c} & \mathrm{r} & 1\end{array}\right|$ is, 0, 1, -1, None of these

MARKS App · Accepted Answer

Let $A$ be the 1 st term and $\mathrm{R}$ the common ratio of G.P., then $\begin{aligned} & a=T_p=A R^{p-1} \ & \therefore \log \mathrm{a}=\log \mathrm{A}+(\mathrm{p}-1) \log \mathrm{R} \end{aligned}$ 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 $\mathrm{C}_1 \rightarrow \mathrm{C}_2 \rightarrow \mathrm{C}_3$ in the second $\Delta=0+\log R\left|\begin{array}{lll} 0 & p & 1 \ 0 & q & 1 \ 0 & r & 1 \end{array}\right|=0$

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