Evaluate the determinants

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

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations

$x+y+z=5$    ;    $x+2 y+3 z=\mu$   ;     $x+3 y+\lambda z=1$

is constructed. If $\mathrm{p}$ is the probability that the system has a unique solution and $\mathrm{q}$ is the probability that the system has no solution, then :

Let $k_1$, $k_2$ be the maximum and minimum values of $k$ for which the system of equations given by

$x + ky = 1$ ; $kx + y = 2$;  $x + y = k$  are consistent then $k_1^2 + k_2^2$ is equal to

The values of $\theta, \lambda$ for which the following equations $\sin \theta x – cos\theta y + (\lambda +1)z = 0$; $\cos\theta x + \sin\theta\, y – \lambda z = 0$;$ \lambda x +(\lambda + 1)y + \cos\theta z = 0$ have non trivial solution, is

Let $a ,b ,c $ be such that $b + c \ne 0$  if

$\left| {\begin{array}{*{20}{c}}a&{a + 1}&{a – 1}\\{ – b}&{b + 1}&{b – 1}\\c&{c – 1}&{c + 1}\end{array}} \right| + \left| {\begin{array}{*{20}{c}}{a + 1}&{b + 1}&{c – 1}\\{a – 1}&{b – 1}&{c + 1}\\{{{\left( { – 1} \right)}^{n + 2}} \cdot a}&{{{\left( { – 1} \right)}^{n + 1}} \cdot b}&{{{\left( { – 1} \right)}^n} \cdot c}\end{array}} \right| = 0$ then $n$ equals to