The rank of the matrix $A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 0 & 1 \\ 4 & 1 & 4 \end{bmatrix}$ 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 $x = 0$

If ${\Delta _1} = \left| {\begin{array}{*{20}{c}} x & b & b \\ a & x & b \\ a & a & x \end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}} x & b \\ a & x \end{array}} \right|$ are the given determinants,then:

Which of the following matrices has rank $3$?

If $\left|\begin{array}{ccc}x^2+3x & x+1 & x-3 \\ x-1 & 2-x & x+4 \\ x-3 & x-3 & 3x\end{array}\right|=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$, then find the value of $(a_1+a_3)+2(a_0+a_2+a_4)$.

Let $A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & x \end{bmatrix}$ and $A^2 = A$. If $r$ is the rank of $A$, then $r + x =$