The area of a triangle with vertices $(-3, 0)$, $(3, 0)$, and $(0, k)$ is $9$ sq units. Find the possible values of $k$.

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1$,$2x+Ny+2z=2$,and $3x+3y+Nz=3$ has a unique solution is $\frac{k}{6}$,then the sum of the value of $k$ and all possible values of $N$ is:

Let $\theta \in \left(0, \frac{\pi}{2}\right)$. If the system of linear equations
$(1+\cos^2 \theta) x + \sin^2 \theta y + 4 \sin 3\theta z = 0$
$\cos^2 \theta x + (1+\sin^2 \theta) y + 4 \sin 3\theta z = 0$
$\cos^2 \theta x + \sin^2 \theta y + (1+4 \sin 3\theta) z = 0$
has a non-trivial solution,then the value of $\theta$ is:

$f(x)$ is an $n^{\text{th}}$ degree polynomial satisfying $f(x) = \frac{1}{2} \begin{vmatrix} f(x) & f(\frac{1}{x}) - f(x) \\ 1 & f(\frac{1}{x}) \end{vmatrix}$. If $f(2) = 33$,then the value of $f(3)$ is

$\left| {\begin{array}{*{20}{c}}{{{\log }_3}512}&{{{\log }_4}3}\\{{{\log }_3}8}&{{{\log }_4}9}\end{array}} \right| \times \left| {\begin{array}{*{20}{c}}{{{\log }_2}3}&{{{\log }_8}3}\\{{{\log }_3}4}&{{{\log }_3}4}\end{array}} \right| = $

Evaluate the determinant: $\left| \begin{array}{lll} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{array} \right|$