Search any question & find its solution
Question:
Answered & Verified by Expert
If $x, y$ and $z$ are 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
Options:
Solution:
2765 Upvotes
Verified Answer
The correct answer is:
0
Let $\Delta=\left|\begin{array}{ccc}1 & \log _{x}y & \log _{x} z \\ \log _{y} x & 1 & \log _{y} z \\ \log _{z} x & \log _{z} y & 1\end{array}\right|$
$=\left|\begin{array}{lll}\frac{\log x}{\log x} & \frac{\log y}{\log x} & \frac{\log z}{\log x} \\ \frac{\log x}{\log y} & \frac{\log y}{\log y} & \frac{\log z}{\log y} \\ \frac{\log x}{\log z} & \frac{\log y}{\log z} & \frac{\log 2}{\log z}\end{array}\right|$
On taking $\frac{1}{\log x} \frac{1}{\log y} \frac{1}{\log z}$ common from $R_{1}, R_{2}$ and $R_{3}$
we got
$$
=\frac{1}{\log x \cdot \log y \cdot \log z}\left|\begin{array}{lll}
\log x & \log y & \log z \\
\log x & \log y & \log z \\
\log x & \log y & \log z
\end{array}\right|
$$
$\therefore \Delta=0$
$=\left|\begin{array}{lll}\frac{\log x}{\log x} & \frac{\log y}{\log x} & \frac{\log z}{\log x} \\ \frac{\log x}{\log y} & \frac{\log y}{\log y} & \frac{\log z}{\log y} \\ \frac{\log x}{\log z} & \frac{\log y}{\log z} & \frac{\log 2}{\log z}\end{array}\right|$
On taking $\frac{1}{\log x} \frac{1}{\log y} \frac{1}{\log z}$ common from $R_{1}, R_{2}$ and $R_{3}$
we got
$$
=\frac{1}{\log x \cdot \log y \cdot \log z}\left|\begin{array}{lll}
\log x & \log y & \log z \\
\log x & \log y & \log z \\
\log x & \log y & \log z
\end{array}\right|
$$
$\therefore \Delta=0$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.