If $abc \neq 0$ and the system of equations $x+7ay+2az=0$,$x+6by+2bz=0$,$x+5cy+2cz=0$ has a non-trivial solution,then $a, b, c$ are in

$\left| {\begin{array}{ccc} 19 & 17 & 15 \\ 9 & 8 & 7 \\ 1 & 1 & 1 \end{array}} \right| = $

The characteristic equation of the matrix $A = \begin{bmatrix} 2 & 3 & 0 \\ 1 & 2 & 5 \\ 3 & -1 & 2 \end{bmatrix}$ is:

Let $f(\theta) = \left| \begin{array}{ccc} 1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1 \end{array} \right|$. Suppose $A$ and $B$ are respectively the maximum and minimum values of $f(\theta)$. Then $(A, B)$ is equal to

The area of a triangle whose vertices are $(1, -1)$,$(-1, 1)$,and $(-1, -1)$ is given by

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