If the system of equations $2x + 3y - z = 0$,$x + ky - 2z = 0$ and $2x - y + z = 0$ has a non-trivial solution $(x, y, z)$,then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$ is equal to

$\left|\begin{array}{ccc} \log e & \log e^2 & \log e^3 \\ \log e^2 & \log e^3 & \log e^4 \\ \log e^3 & \log e^4 & \log e^5 \end{array}\right| \text{ is equal to: }$

$\left| {\begin{array}{ccc} bc & bc' + b'c & b'c' \\ ca & ca' + c'a & c'a' \\ ab & ab' + a'b & a'b' \end{array}} \right|$ is equal to

If $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{array}\right]$,then show that $|3 A|=27|A|$.

If $1$,$\log_{10}(4^{x}-2)$ and $\log_{10}(4^{x}+\frac{18}{5})$ are in arithmetic progression for a real number $x$,then the value of the determinant $\left|\begin{array}{ccc} 2(x-\frac{1}{2}) & x-1 & x^{2} \\ 1 & 0 & x \\ x & 1 & 0 \end{array}\right|$ is equal to ...... .

Solve the equation $\left|\begin{array}{ccc}x+a & x & x \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0$,where $a \neq 0$.