Question

If $a>0,b>0,c>0$ are respectively the $p^{th},q^{th},r^{th}$ terms of a G.P., then the value of the deteminant $\left|\begin{array}{c}\log a\\ \log b\\ \log c\end{array}\begin{array}{c}p\\ q\\ r\end{array}\begin{array}{c}1\\ 1\\ 1\end{array}\right|$ is

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

Answer 1

The correct option is B $0$

$\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|a_n=A{R}^{n-1}$

$\left|\begin{array}{ccc}\log A{R}^{p-1}& p& 1\\ \log A{R}^{q-1}& q& 1\\ \log A{R}^{r-1}& r& 1\end{array}\right|$

Apply to $C_1\log m^n=n\log m$

$C_2\to C_2-C_3$

$\log AR\left|\begin{array}{ccc}p-1& p-1& 1\\ q-1& q-1& 1\\ r-1& r-1& 1\end{array}\right|$

$\Rightarrow △=0$

Option B