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

Statement $1$ : If the system of equations $x + ky + 3z = 0, 3x+ ky – 2z = 0, 2x + 3y – 4z = 0$ has a nontrivial solution, then the value of $k$ is $\frac{31}{2}$

Statement $2$ : A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.

Consider the system of linear equations ${a_1}x + {b_1}y + {c_1}z + {d_1} = 0$, ${a_2}x + {b_2}y + {c_2}z + {d_2} = 0$ and ${a_3}x + {b_3}y + {c_3}z + {d_3} = 0$. Let us denote by $\Delta (a,b,c)$ the determinant $\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ if $\Delta (a,b,c) \ne 0$, then the value of $x$ in the unique solution of the above equations is

If $\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$ and $\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$,then $\frac{a}{\alpha-a}+\frac{b}{\beta-b}+\frac{\gamma}{\gamma-c}$ is equal to :

The solutions of the equation $\left| {\,\begin{array}{*{20}{c}}x&2&{ – 1}\\2&5&x\\{ – 1}&2&x\end{array}\,} \right| = 0$ are