Let $A = \begin{bmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{bmatrix}$,where $a, c \in \mathbb{R}$. If $A^3 = A$ and the positive value of $a$ belongs to the interval $(n-1, n]$,where $n \in \mathbb{N}$,then $n$ is equal to $..........$.

Let $A = \begin{pmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{pmatrix}$ and $\det(A - \alpha I) = 0$,where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$,then the circle $(x - p)^2 + (y - 2p)^2 = 320$ intersects the coordinate axes at

The number of cubic polynomials $P(x)$ satisfying $P(1)=2, P(2)=4, P(3)=6, P(4)=8$ is

Let $P=\begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{bmatrix}$ and $I$ be the identity matrix of order $3$. If $Q=[q_{ij}]$ is a matrix such that $P^{50}-Q=I$,then $\frac{q_{31}+q_{32}}{q_{21}}$ equals

If $A = \begin{bmatrix} 0 & \sin \alpha \\ \sin \alpha & 0 \end{bmatrix}$ and $\det\left(A^{2} - \frac{1}{2} I\right) = 0$,then a possible value of $\alpha$ is

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix,where
$a_{ij} = 1$,if $i = j$
$a_{ij} = -x$,if $|i - j| = 1$
$a_{ij} = 2x + 1$,otherwise
Let a function $f: R \rightarrow R$ be defined as $f(x) = \det(A)$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to: