If $A$ is a matrix such that $A^2 + A + 2I = O$,then which of the following is $INCORRECT$?

Let $p, q, r$ be three real numbers satisfying $[p \, q \, r] \begin{bmatrix} 2 & p & q \\ -3 & q & -p+r \\ 12 & r & -q+3r \end{bmatrix} = [5 \, b \, c]$. Then the minimum value of $(b+c)$ is:

Let $A = \begin{bmatrix} 1 & 2 & 3 \\ a & 3 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ and $|A| = 2$. If $|2 \operatorname{adj}(2 \operatorname{adj}(2 A))| = 32^n$,then $3n + \alpha$ is equal to:

Let $f(x)=\left|\begin{array}{ccc}1+\sin ^2 x & \cos ^2 x & \sin 2 x \\ \sin ^2 x & 1+\cos ^2 x & \sin 2 x \\ \sin ^2 x & \cos ^2 x & 1+\sin 2 x\end{array}\right|$,where $x \in\left[\frac{\pi}{6}, \frac{\pi}{3}\right]$. If $\alpha$ and $\beta$ are the maximum and minimum values of $f(x)$ respectively,then:

If $p, q, r, s$ are in $A.P.$ and $f(x) = \left| \begin{array}{ccc} p + \sin x & q + \sin x & p - r + \sin x \\ q + \sin x & r + \sin x & -1 + \sin x \\ r + \sin x & s + \sin x & s - q + \sin x \end{array} \right|$ such that $\int_{0}^{\pi} f(x) dx = -4$,then the common difference of the $A.P.$ can be:

Let matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 4 \\ 0 & 3 & 2 \end{bmatrix}$ and $A^3 - 8A^2 + \alpha A + \beta I = O$,then the ordered pair $(\alpha, \beta)$ is