The value of $\left|\alpha \right|$ for which the system of equation 

$\alpha x+y+z=\alpha -1$

$x+\alpha y+z=\alpha -1$

$x+y+\alpha z=\alpha -1$

has no solution, is _____.

Let $\alpha , \lambda , \mu \in R$. Consider the system of linear equations $$\begin{gathered} \alpha x+2y=\lambda \\ 3x-2y=\mu \end{gathered}$$

Which of the following statement(s) is(are) correct?

Let $\omega$ be the complex number $\cos \frac{2\mathrm{\pi }}{3}+i \sin \frac{2\mathrm{\pi }}{3}$.Then the number of distinct complex numbers $z$ satisfying

$\left|\begin{array}{ccc}z+1& \omega & {\omega }^2\\ \omega & z+{\omega }^2& 1\\ {\omega }^2& 1& z+\omega \end{array}\right|=0$ is equal to_.

The total number of distinct $x \in R$ for which

$\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10 is \_\_\_\_\_.$

For a real number $\alpha$, if the system $\left[\begin{array}{ccc}1& \alpha & {\alpha }^2\\ \alpha & 1& \alpha \\ {\alpha }^2& \alpha & 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$ of linear equations, has infinitely many solutions, then $1+\alpha +{\alpha }^2=\_.$