Question

If $xyz$ are not equal and $\ne 0,\ne 1$ the value of $\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2x& \log 2y& \log 2z\\ \log 3x& \log 3y& \log 3z\end{array}\right|$ is equal to

• A. $\log (xyz)$
• B. $\log (6xyz)$
• C. $0$
• D. $\log (x+y+z)$

Answer 1

Answer: (C)

We have $\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2x& \log 2y& \log 2z\\ \log 3x& \log 3y& \log 3z\end{array}\right|$
Applying $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$,
$=\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2+\log x& \log 2+\log y& \log 3+\log z\\ \log 3+\log x& \log 3+\log y& \log 3+\log z\end{array}\right|$
$=\log 2\cdot \log 3\left|\begin{array}{ccc}\log x& \log y& \log z\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$
$=0$
$[\because R_2$ and $R_3$ are same]