The number of distinct real roots of the equation $\begin{vmatrix} \cos x & \sin x & \sin x \\ \sin x & \cos x & \sin x \\ \sin x & \sin x & \cos x \end{vmatrix} = 0$ in the interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$ is

If $A = \begin{bmatrix} 5x & 10 \\ 8 & 7 \end{bmatrix}$ and $|A| = 25$,then $x = $ . . . . . . .

If the inverse of the matrix $\begin{bmatrix} \alpha & 14 & -1 \\ 2 & 3 & 1 \\ 6 & 2 & 3 \end{bmatrix}$ does not exist,then the value of $\alpha$ is:

The area of a triangle whose vertices are $(1, -1)$,$(-1, 1)$,and $(-1, -1)$ is given by

If $A, B, C$ are the angles of a triangle,then $\left| \begin{array}{ccc} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{array} \right| = $

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?