The values of k for which the system of equations  $x+ky-3z=0, 3x+ky-2z=0, 2x+3y-4z=0$ has a non-trivial solution is

$The condition for a homogeneous system of equations to have a non-trivial solution is \Delta =0$

$\Rightarrow$$\left|\begin{array}{ccc}1& k& -3\\ 3& k& -2\\ 2& 3& -4\end{array}\right|=0$

$\Rightarrow 1\left(-4k+6\right)-k\left(-12+4\right)-3\left(9-2k\right)=0$

$\Rightarrow -4k+6+8k-27+6k=0$

$\Rightarrow 10k-21=0$

$\Rightarrow k=\frac{21}{10}$