Question

If $a,b,c$ are $p^{th},q^{th}$ and $r^{th}$ terms respectively of a geometric progression then the value of the determinant
$\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|$ is equal to

• A. $0$
• B. $1$
• C. $-1$
• D. None of these

Answer 1

The correct option is A $0$

If $a,b,c$ are $p^{th},q^{th}$ and $r^{th}$ terms of a G.P.
Let the first term be $\alpha$ and common ratio $\beta$.
$a=\alpha {\beta }^{p-1}$
$b=\alpha {\beta }^{q-1}$
$c=\alpha {\beta }^{r-1}$

Substituting in determinant
$\left|\begin{array}{ccc}\log \alpha {\beta }^{p-1}& p& 1\\ \log \alpha {\beta }^{q-1}& q& 1\\ \log \alpha {\beta }^{r-1}& r& 1\end{array}\right|$

Using properties of log
$\left|\begin{array}{ccc}\log \alpha +(p-1)\log \beta & p& 1\\ \log \alpha +(q-1)\log \beta & q& 1\\ \log \alpha +(r-1)\log \beta & r& 1\end{array}\right|$

Using properties of determinant
$\left|\begin{array}{ccc}\log \alpha & p& 1\\ \log \alpha & q& 1\\ \log \alpha & r& 1\end{array}\right|+\left|\begin{array}{ccc}(p-1)\log \beta & p& 1\\ (q-1)\log \beta & q& 1\\ (r-1)\log \beta & r& 1\end{array}\right|$
For 1st determinant $C_1\to C_1-\left(\log \alpha \right)C_3$
For 2nd determinant $C_1\to C_1-\left(\log \beta \right)C_2+\left(\log \beta \right)C_3$
$\left|\begin{array}{ccc}0& p& 1\\ 0& q& 1\\ 0& r& 1\end{array}\right|+\left|\begin{array}{ccc}0& p& 1\\ 0& q& 1\\ 0& r& 1\end{array}\right|=0$