Question

If $l,m,n$ are $p^{th},q^{th},r^{th}$ terms of $G.P$. all positive, then $\left|\begin{array}{ccc}\log l& p& 1\\ \log m& q& 1\\ \log n& r& 1\end{array}\right|$ equals

• A. -1
• B. 2
• C. 1
• D. 0

Answer 1

The correct option is D 0

Let $A$ be the first term and $R$ be the common ratio of G.P

$\therefore l=T_p=A{R}^{p-1}$

$\Rightarrow \log l=\log A+(p-1)\log R$

Similarly,

$\log m=\log A+(q-1)\log R$

$\log n=\log A+(r-1)\log R$
$\therefore \left|\begin{array}{ccc}\log l& p& 1\\ \log m& q& 1\\ \log n& r& 1\end{array}\right|=\left|\begin{array}{ccc}\log A+(p-1)\log R& p& 1\\ \log A+(q-1)\log R& q& 1\\ \log A+(r-1)\log R& r& 1\end{array}\right|$
$=\left|\begin{array}{ccc}\log A-\log R& p& 1\\ \log A-\log R& q& 1\\ \log A-\log R& r& 1\end{array}\right|+\left|\begin{array}{ccc}p\log R& p& 1\\ q\log R& q& 1\\ r\log R& r& 1\end{array}\right|$
$=(\log A-\log R)\left|\begin{array}{ccc}1& p& 1\\ 1& q& 1\\ 1& r& 1\end{array}\right|+\log R\left|\begin{array}{ccc}p& p& 1\\ q& q& 1\\ r& r& 1\end{array}\right|$
$=0+0=0$