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

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 equal to

Given the system of equations $a(x + y + z) = x$,$b(x + y + z) = y$,$c(x + y + z) = z$ where $a, b, c$ are non-zero real numbers. If the real numbers $x, y, z$ are such that $xyz \neq 0$,then $(a + b + c)$ is equal to-

$\left|\begin{array}{rr}\sin 35^{\circ} & -\cos 35^{\circ} \\ \sin 55^{\circ} & \cos 55^{\circ}\end{array}\right|=$ . . . . . .

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1$,$2x+Ny+2z=2$,and $3x+3y+Nz=3$ has a unique solution is $\frac{k}{6}$,then the sum of the value of $k$ and all possible values of $N$ is:

If $x, y, z$ are all different and not equal to zero and $\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z\end{array}\right|=0$,then the value of $x^{-1}+y^{-1}+z^{-1}$ is equal to