Let $A$ and $B$ be two non-singular matrices of order $3$ such that $A + B = I$ and $A^{-1} + B^{-1} = 2I$. Then $|adj(4AB)|$ is equal to (where $adj(A)$ is the adjoint of matrix $A$):

If $A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$,then $\operatorname{det}\left(A^6+B^6\right)=$

If $a, b, c, d, e, f$ are in $G.P.$,then the value of $\left| \begin{array}{ccc} a^2 & d^2 & x \\ b^2 & e^2 & y \\ c^2 & f^2 & z \end{array} \right|$ depends on

Let $S = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4 \} \text{ and } A^2 - 4A + 3I = 0 \right\}$ be a set of $2 \times 2$ matrices. Then the number of matrices in $S$, for which the sum of the diagonal elements is equal to $4$, is:

If $\Delta_{r}=\left|\begin{array}{cc}\frac{1}{3r-2} & \frac{2}{3r-5} \\ 0 & \frac{3}{3r+1}\end{array}\right|$, then $\sum_{r=1}^{33} \Delta_{r}=$

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: