If $\alpha$ is a real root of the equation $x^3+6 x^2+5 x-42=0$, then the determinant of the matrix

$\left[\begin{array}{lll}\alpha-1 & \alpha+1 & \alpha+2 \\ \alpha-2 & \alpha+3 & \alpha-3 \\ \alpha+4 & \alpha-4 & \alpha+5\end{array}\right]$ is

The rank of the matrix $\left[\begin{array}{cccc}2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4\end{array}\right]$ is

If the least positive integer $n$ satisfying the equation $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^n=-1$ is $p$ and the least positive integer $m$ satisfying the equation $\left(\frac{1-\sqrt{3 i}}{1+\sqrt{3} i}\right)^m=\operatorname{cis} \frac{2 \pi}{3}$ is $q$, then $\sqrt{p^2+q^2}=$