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. pqr

Answer 1

Answer: (A)

$a=a_0\cdot r_1{}^{p-1}\Rightarrow \log a=(p-1)\log r_1+\log a_0$
$b=a_0\cdot r_1{}^{q-1}\Rightarrow \log b=(q-1)\log r_1+\log a_0$
$c=a_0{r}_1{}^{r-1}\Rightarrow \log c=(r-1)\log r_1+\log a_0$
$\left|\begin{array}{ccc}\log a_0+(p-1)\log r_1& p& 1\\ \log a_0+(q-1)\log r_1& q& 1\\ \log a_0+(r-1)\log r_1& r& 1\end{array}\right|$
$=\left|\begin{array}{ccc}\log a_0& p& 1\\ \log a_0& q& 1\\ \log a_0& r& 1\end{array}\right|+\log r_1\left|\begin{array}{ccc}p-1& p& 1\\ q-1& q& 1\\ r-1& r& 1\end{array}\right|=0$