Question

If $a,b,c$ are $pth$,$qth$and $rth$ terms of a GP, then $\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|$ is equal to

• A. $0$
• B. $1$
• C. $\log abc$
• D. none of these

Answer 1

The correct option is A $0$

It is given that the given terms are in G.P

$\therefore p^{th}$ term$=a,q^{th}$ term$=b$ and $r^{th}$ term$=c$

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

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

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

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

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

$C_2\to C_2-C_3$

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

$C_1\to C_1-\left(\log A\right)C_3-\left(\log R\right)C_2$

$=\left|\begin{array}{ccc}0& p-1& 1\\ 0& q-1& 1\\ 0& r-1& 1\end{array}\right|$

$=0$