The maximum value of the determinant of the matrix $\left[\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4 \sin 2 x \\ \sin ^2 x & 1+\cos ^2 x & 4 \sin 2 x \\ \sin ^2 x & \cos ^2 x & 1+4 \sin 2 x \end{array}\right]$ is

Let $A = \begin{bmatrix} 1+i & 1 \\ -i & 0 \end{bmatrix}$ where $i = \sqrt{-1}$. Then,the number of elements in the set $\{n \in \{1, 2, \ldots, 100\} : A^n = A\}$ is

Let $A = [a_{ij}]$,where $a_{ij} \in \mathbb{Z} \cap [0, 4]$ and $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $........$.

The value of $(1+\Delta)(1-\nabla)$ is

The number of integers $x$ satisfying $-3 x^4 + \operatorname{det}\begin{bmatrix} 1 & x & x^2 \\ 1 & x^2 & x^4 \\ 1 & x^3 & x^6 \end{bmatrix} = 0$ is equal to

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]$,$A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=PAP^{T}$. If $P^{T}Q^{2007}P=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$,then $2a+b-3c-4d$ is equal to: