If $A, B$ are two non-singular matrices of order $3$ and $|B|=k$, where $k$ is a positive integer, then match the items of List-$I$ with the items of List-$II$.

List-$I$	List-$II$
$A$. $|k^{-1} A^{-1}|$	$I$. $BA^k + A^kB$
$B$. $|\text{Adj}(A^{-1})|$	$II$. $\frac{B\text{Adj}(B)}{|B|}$
$C$. $BAB^{-1} = I \Rightarrow BA^kB^{-1} =$	$III$. $\frac{1}{|B|^3|A|}$
$D$. $\text{Adj}(\text{Adj}(A^{-1})) =$	$IV$. $\frac{1}{|A|}(A^{-1})$
	$V$. $\frac{1}{|A|^2}$

Let $|M|$ denote the determinant of a square matrix $M$. Let $g:\left[0, \frac{\pi}{2}\right] \rightarrow R$ be the function defined by $g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$,where $f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _e\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _e\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right|$. Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$,and $p(2)=2-\sqrt{2}$. Then,which of the following is/are $TRUE$?
$(A) \ p \left(\frac{3+\sqrt{2}}{4}\right) < 0$
$(B) \ p \left(\frac{1+3 \sqrt{2}}{4}\right)>0$
$(C) \ p \left(\frac{5 \sqrt{2}-1}{4}\right)>0$
$(D) \ p \left(\frac{5-\sqrt{2}}{4}\right) < 0$

Let $A_1, A_2, A_3$ be three $A$.$P$.s with the same common difference $d$ and having their first terms as $A, A+1, A+2$,respectively. Let $a, b, c$ be the $7^{\text{th}}, 9^{\text{th}}, 17^{\text{th}}$ terms of $A_1, A_2, A_3$,respectively,such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$. If $a=29$,then the sum of the first $20$ terms of an $A$.$P$. whose first term is $c-a-b$ and common difference is $\frac{d}{12}$,is equal to $........$.

If $A = \begin{bmatrix} -8 & 5 \\ 2 & 4 \end{bmatrix}$ satisfies the equation $x^2 + 4x - p = 0$, then $p$ is equal to

$\left|\begin{array}{ll}2 & 1 \\ 3 & 1\end{array}\right|+\left|\begin{array}{cc}1 & 1/3 \\ 3 & 1\end{array}\right|+\left|\begin{array}{cc}1/2 & 1/9 \\ 3 & 1\end{array}\right|+\left|\begin{array}{cc}1/4 & 1/27 \\ 3 & 1\end{array}\right|+\ldots \infty=$

Let the minimum $m$ $(m \in Z^+)$ be defined as the power of a square matrix $A$ such that $A^m = I$. If $A^5 = I$ and $ABA^{-1} = B^2$,then the power of matrix $B$ such that $B^k = I$ is between: