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 $0$

As $a,b,c$ are $pth,qth,rth$ terms of $G.P.$

Let $A$ is the first term and $R$ is common ratio

Then $a=A{R}^{p-1},b=A{R}^{q-1},c=A{R}^{r-1}$

$\Delta =\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|$

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$

$\Delta =\left|\begin{array}{ccc}\log a& p& 1\\ \log b-\log a& q-p& 0\\ \log c-\log a& r-p& 0\end{array}\right|=(r-p)log\frac{b}{a}-(q-p)log\frac{c}{a}=(r-p)log\left(R^{(q-p)}\right)-(q-p)log\left(R^{(r-p)}\right)=(r-p)(q-p)logR-(r-p)(q-p)logR=0$