If \(x, y\) and \(z\) be greater than 1 , then the value of \(\left|\begin{array}{ccc}1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 1 & \log _{y} z \\ \log _{z} x & \log _{z} y & 1\end{array}\right|\) is

Question

If \(x, y\) and \(z\) be greater than 1 , then the value of \(\left|\begin{array}{ccc}1 & \log _{x} x & \log _{x} z \\ \log _{z} x & \log _{z} y & 1\end{array}\right|\) is
(A) \(\log x \cdot \log y \cdot \log z\)
(B) \(\log x+\log y+\log z\)
(C) 0
(D) \(1-\{(\log x) \cdot(\log y) \cdot(\log z)\}\)

Answer 1

Ans: (C)
 \(\left|\begin{array}{lll}\frac{\log x}{\log x} & \frac{\log y}{\log x} & \frac{\log z}{\log x} \\ \frac{\log y}{\log y} & \frac{\log z}{\log y} \\ \mid \frac{\log x}{\log z} & \frac{\log y}{\log z} & \frac{\log z}{\log z}\end{array}\right|\)
Taking \(\frac{1}{\log x}, \frac{1}{\log y}, \frac{1}{\log z}\) common from \(R_{1}, R_{2}, R_{3}\) all rows are identical. So \(\Delta=0\)

Answer 2

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