Question

If $a,b,c$ are positive and are the $pth,qth$ and $rth$ terms, respectively, of a G.P., then $\Delta =\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|$ is

• A. $0$
• B. $\log \left(abc\right)$
• C. $-(p+q+r)$
• D. none of these

Answer 1

The correct option is A $0$

Given , $a=A{R}^{p-1}$

$\Rightarrow \log a=\log A+(p-1)\log R$
$b=A{R}^{q-1}$

$\Rightarrow \log b=\log A+(q-1)\log R$
$c=A{R}^{r-1}$

$\Rightarrow \log c=\log A+(r-1)\log R$
$\Delta =\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|$
$\Delta =\left|\begin{array}{ccc}\log A+(p-1)\log R& p& 1\\ \log A+(q-1)\log R& q& 1\\ \log A+(r-1)\log R& r& 1\end{array}\right|$
$=\left|\begin{array}{ccc}\log A& p& 1\\ \log A& q& 1\\ \log A& r& 1\end{array}\right|+\left|\begin{array}{ccc}(p-1)\log R& p& 1\\ (q-1)\log R& q& 1\\ (r-1)\log R& r& 1\end{array}\right|$
$=\log A\left|\begin{array}{ccc}1& p& 1\\ 1& q& 1\\ 1& r& 1\end{array}\right|+\log R\left|\begin{array}{ccc}(p-1)& p& 1\\ (q-1)& q& 1\\ (r-1)& r& 1\end{array}\right|$
$=\log R\left|\begin{array}{ccc}(p-1)& p& 1\\ (q-1)& q& 1\\ (r-1)& r& 1\end{array}\right|$
$C_1\to C_1+C_3$
$=\log R\left|\begin{array}{ccc}p& p& 1\\ q& q& 1\\ r& r& 1\end{array}\right|$
$=0$