Question

If x, y, z are all positive and are the pth, qth and rth terms of a geometric progression respectively, then the value of the determinant $\left|\begin{array}{ccc}\log x& p& 1\\ \log y& q& 1\\ \log z& r& 1\end{array}\right|$ is equal to

• A. $\log xyz$
• B. $(p-1)(q-1)(r-1)$
• C. $pqr$
• D. $0$

Answer 1

Answer: (D)

$0$

$\left|\begin{array}{ccc}\log x& p& 1\\ \log y& q& 1\\ \log z& r& 1\end{array}\right|$

A-First term

R-difference

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

$\log m^n=n\log m$

$C_2\to C_2-C_1$

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

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

$\Rightarrow △=0$

Option D