Question

If $x y z$ are not equal and $\neq 0, \neq 1 $ the value of $\begin{vmatrix} \log x& \log y& \log z\\ \log 2x& \log 2y& \log 2z\\ \log 3x& \log 3y& \log 3z\end{vmatrix}$ is equal to

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

Answer 1

Answer: (C)

We have $\begin{vmatrix}\log x&\log y&\log z\\ \log 2x& \log2y&\log2z\\ \log3x&\log 3y&\log3z\end{vmatrix}$
Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$,
$ = \begin{vmatrix}\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{vmatrix}$
$=\log \,2 \cdot \log \,3\begin{vmatrix} \log x & \log y & \log z \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{vmatrix}$
$= 0$
$\left[\because R_{2}\right.$ and $R_{3}$ are same]