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:

Match the items of List-$I$ with those of List-$II$. The correct match is:

Let $A$ and $B$ be two $3 \times 3$ non-singular matrices such that $\operatorname{det}(A^T B A) = 27$ and $\operatorname{det}(A B^{-1}) = 8$. Then $\operatorname{det}(B^T A^{-1} B) = $

Let $a, b, c \in \mathbb{R}$ be all non-zero and satisfy $a^{3}+b^{3}+c^{3}=2$. If the matrix $A=\begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$ satisfies $A^{T} A=I$,then a value of $abc$ can be

Let $A = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$,$A^2 = A^T$,then the sum of the diagonal elements of the matrix $(A + I)^3 + (A - I)^3 - 6A$ is equal to . . . . . . .

Let $A, B, C$ be $3 \times 3$ non-singular matrices and $I$ be the identity matrix of order three. If $A B A = B A^2 B$ and $A^3 = I$,then $A B^4 - B^4 A = $