The maximum value of $f(x) = \left|\begin{array}{ccc} \sin^{2} x & 1+\cos^{2} x & \cos 2x \\ 1+\sin^{2} x & \cos^{2} x & \cos 2x \\ \sin^{2} x & \cos^{2} x & \sin 2x \end{array}\right|, x \in R$ is:

Let $A = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$ and $B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$ be two $2 \times 1$ matrices with real entries such that $A = XB$,where $X = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & -1 \\ 1 & k \end{bmatrix}$ and $k \in R$. If $a_1^2 + a_2^2 = \frac{2}{3}(b_1^2 + b_2^2)$ and $(k^2 + 1)b_2^2 \neq -2b_1b_2$,then the value of $k$ is ....... .

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

If $A$ and $B$ are both $3 \times 3$ matrices,then which of the following statements are true?
$(i)$ $AB=0 \Rightarrow A=0$ or $B=0$
(ii) $AB=I_3 \Rightarrow A^{-1}=B$
(iii) $(A-B)^2=A^2-2AB+B^2$

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

Let for some real numbers $\alpha$ and $\beta$,$a = \alpha - i \beta$. If the system of equations $4ix + (1 + i)y = 0$ and $8(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3})x + \bar{a}y = 0$ has more than one solution,then $\frac{\alpha}{\beta}$ is equal to