Let $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $P = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$. Let $Q = \begin{bmatrix} x & y \\ z & 4 \end{bmatrix}$ for some non-zero real numbers $x, y$,and $z$,for which there exists a $2 \times 2$ matrix $R$ with all entries being non-zero real numbers,such that $QR = RP$. Then which of the following statements is (are) true?

Let $A$ be a square matrix of order $2$ such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x, y)$ satisfying $A^2+xA+yI=0$ lie on a hyperbola,whose transverse axis is parallel to the $x$-axis,eccentricity is $e$ and the length of the latus rectum is $\ell$,then $e^4+\ell^4$ is equal to?

Among the statements:
$I$: If $\begin{vmatrix} 1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} = \begin{vmatrix} 0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0 \end{vmatrix}$, then $\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=\frac{3}{2}$
$II$: If $\begin{vmatrix} x^{2}+x & x+1 & x-2 \\ 2x^{2}+3x-1 & 3x & 3x-3 \\ x^{2}+2x+3 & 2x-1 & 2x-1 \end{vmatrix} = px+q$, then $p^{2}=196q^{2}$

Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If $\left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & -\omega^2 - 1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{array} \right| = 3k$,then $k$ is equal to:

If ${\Delta _1} = \left| {\begin{array}{*{20}{c}} x & {\sin \theta } & {\cos \theta } \\ {\sin \theta } & { - x} & 1 \\ {\cos \theta } & 1 & x \end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}} x & {\sin 2\theta } & {\cos 2\theta } \\ {\sin 2\theta } & { - x} & 1 \\ {\cos 2\theta } & 1 & x \end{array}} \right|$,$x \ne 0$; then for all $\theta \in \left( {0, \frac{\pi }{2}} \right)$:

$A$ and $B$ are two square matrices such that $A^2B = BA$. If $(AB)^{10} = A^K B^{10}$,then $k$ is: