If $a, b, c$ are the sides of a triangle $ABC$ and $2(\cos A + \cos B + \cos C) = \left|\begin{array}{lll}b & 1 & a \\ a & 1 & c \\ c & 1 & b\end{array}\right| = 0$, then find the value of the expression.

If $A_\alpha = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}$, then the determinant of $A_{\pi / 5} A_{\pi / 4} A_{3 \pi / 10}$ is:

The value of $x,$ if $\left| \begin{matrix} -x & 1 & 0 \\ 1 & -x & 1 \\ 0 & 1 & -x \end{matrix} \right| = 0$ is equal to

$\left|\begin{array}{lll}24 & 25 & 26 \\ 25 & 26 & 27 \\ 26 & 27 & 27\end{array}\right|$ is equal to

If $\left|\begin{array}{ccc}x & 4 & 6 \\ 2 & 3 & -9 \\ 5 & 6 & 1\end{array}\right|+\left|\begin{array}{ccc}5 & 6 & 1 \\ 6 & 4 & 5 \\ 2 & 3 & -9\end{array}\right|=\left|\begin{array}{ccc}2 & 3 & -9 \\ 1-2 x & -8 & -11 \\ 5 & 6 & 1\end{array}\right|$,then $x=$ . . . . . .

Let $\sigma_1, \sigma_2, \sigma_3$ be planes passing through the origin. Assume that $\sigma_1$ is perpendicular to the vector $(1, 1, 1)$,$\sigma_2$ is perpendicular to a vector $(a, b, c)$,and $\sigma_3$ is perpendicular to the vector $(a^2, b^2, c^2)$. What are all the positive values of $a, b$,and $c$ so that $\sigma_1 \cap \sigma_2 \cap \sigma_3$ is a single point?