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:

If $\left[\begin{array}{ccc}1 & -1 & x \\ 1 & x & 1 \\ x & -1 & 1\end{array}\right]$ has no inverse,then the real value of $x$ is

Let ${a_2},{a_3} \in R$ such that $\left| {{a_2} - {a_3}} \right| = 6$ and $f\left( x \right) = \left| \begin{array}{ccc} 1 & {a_3} & {a_2} \\ 1 & {a_3} & {2{a_2} - x} \\ 1 & {2{a_3} - x} & {a_2} \end{array} \right|, x \in R.$ Then the greatest value of $f(x)$ is

If $D = \left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{array} \right|$ for $x \neq 0, y \neq 0$,then $D$ is

If $a + b + c = 0$,then the solution of the equation $\left| \begin{array}{ccc} a - x & c & b \\ c & b - x & a \\ b & a & c - x \end{array} \right| = 0$ is

If $w = \frac{-1-i \sqrt{3}}{2}$ where $i = \sqrt{-1}$,then the value of $\left|\begin{array}{ccc}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{array}\right|$ is