The rank of the matrix $\left[ {\begin{array}{*{20}{c}}4&1&0&0\\3&0&1&0\\6&0&2&0\end{array}} \right]$ is:

If $f(x) = \left| \begin{array}{ccc} \cos(2x) & \cos(2x) & \sin(2x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x \end{array} \right|$,then:
$A$. $f'(x) = 0$ at exactly three points in $(-\pi, \pi)$
$B$. $f'(x) = 0$ at more than three points in $(-\pi, \pi)$
$C$. $f(x)$ attains its maximum at $x = 0$
$D$. $f(x)$ attains its minimum at $x = 0$

The rank of the matrix $\begin{bmatrix} 2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4 \end{bmatrix}$ is

If $f(x) = \left| \begin{array}{ccc} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{array} \right|$,then find $\lim_{x \to 0} \frac{f'(x)}{x}$.

If $f(x) = \left| \begin{array}{ccc} \cos(x+a+b) & \sin(x+a+b) & 10 \\ \cos(x+b+c) & \sin(x+b+c) & 10 \\ \cos(x+c+a) & \sin(x+c+a) & 10 \end{array} \right|$,then find the value of $f(2019)^{f(2020)} - f(2020)^{f(2019)}$.

The rank of the matrix $\begin{bmatrix} -1 & 2 & 5 \\ 2 & -4 & a - 4 \\ 1 & -2 & a + 1 \end{bmatrix}$ is