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 nontrivial solution if the determinant of the coefficient matrix is zero.

In a legislative assembly election,a political group hired a public relations firm to promote its candidate in three ways: telephone,house calls,and letters. The cost per contact (in paise) is given in matrix $A$ as $A = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix} \begin{matrix} \text{Telephone} \\ \text{Housecall} \\ \text{Letter} \end{matrix}$. The number of contacts of each type made in two cities $X$ and $Y$ is given by $B = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{bmatrix} \begin{matrix} \text{Telephone} & \text{Housecall} & \text{Letter} \\ \to X \\ \to Y \end{matrix}$. Find the total amount spent by the group in the two cities $X$ and $Y$.

The system of equations $-k x+3 y-14 z=25$,$-15 x+4 y-k z=3$,and $-4 x+y+3 z=4$ is consistent for all $k$ in the set

The system of equations $x+3by+bz=0$,$x+2ay+az=0$ and $x+4cy+cz=0$ has

Let $a, b, c$ be positive real numbers. The following system of equations in $x, y, z$:
$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$
$\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
$-\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
has:

If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$,then the system of three equations $\begin{aligned} & 2x + 3|y| + 5[z] = 0, \\ & x + |y| - 2[z] = 4, \\ & x + |y| + [z] = 1 \end{aligned}$ has