The number of solutions of the equations $x + 4y - z = 0,$ $3x - 4y - z = 0,\,x - 3y + z = 0$ is
$0$
$1$
$2$
Infinite
The value of $k$ for which the set of equations $x + ky + 3z = 0,$ $3x + ky - 2z = 0,$ $2x + 3y - 4z = 0$ has a non trivial solution over the set of rationals is
$\left| {\,\begin{array}{*{20}{c}}1&5&\pi \\{{{\log }_e}e}&5&{\sqrt 5 }\\{{{\log }_{10}}10}&5&e\end{array}\,} \right| = $
$\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = $
The value of $\left| {\,\begin{array}{*{20}{c}}1&{\cos (\beta - \alpha )}&{\cos (\gamma - \alpha )}\\{\cos (\alpha - \beta )}&1&{\cos (\gamma - \beta )}\\{\cos (\alpha - \gamma )}&{\cos (\beta - \gamma )}&1\end{array}} \right|$ is
The system of equations $kx + y + z =1$ $x + ky + z = k$ and $x + y + zk = k ^{2}$ has no solution if $k$ is equal to