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# For positive numbers $x,y$ and $z$ the numerical value of the determinant $\begin{vmatrix}1 &\log_xy&\log_xz\\\log_yx&1&\log_yz\\\log_zx&\log_zy&1\end{vmatrix}$ is

$(a)\;1\qquad(b)\;0\qquad(c)\;2\qquad(d)\;3$

Given $x,y,z$ and +ve numbers,then value of
$D=\begin{vmatrix}1 &\log_xy&\log_xz\\\log_yx&1&\log_yz\\\log_zx&\log_zy&1\end{vmatrix}$
$\;\;=\begin{vmatrix}1 &\large\frac{\log y}{\log x}&\large\frac{\log_z}{\log x}\\\large\frac{\log x}{\log y}&1&\large\frac{\log z}{\log y}\\\large\frac{\log x}{\log z}&\large\frac{\log y}{\log z}&1\end{vmatrix}$
$\log_ba=\large\frac{\log a}{\log b}$
Taking $\large\frac{1}{\log x},\frac{1}{\log y}$ and $\large\frac{1}{\log z}$ common from $R_1,R_2$ and $R_3$ respectively.
$D=\large\frac{1}{\log x\log y\log z}$$\begin{vmatrix}\log x&\log y&\log z\\\log x&\log y&\log z\\\log x&\log y&\log z\end{vmatrix}$=0
Using the properties that determinant vanishes if any two rows are identical.
Hence (b) is the correct option.