For a real number $\alpha$, if the system

$\left[\begin{array}{ccc}1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right]$

of linear equations, has infinitely many solutions, then $1+\alpha+\alpha^2=$

If $\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2\end{array}\right|=\frac{9}{8}(103 x+81)$, then $\lambda$, $\frac{\lambda}{3}$ are the roots of the equation

If $\omega $ is an imaginary root of unity, then the value of $\left| {\,\begin{array}{*{20}{c}}a&{b{\omega ^2}}&{a\omega }\\{b\omega }&c&{b{\omega ^2}}\\{c{\omega ^2}}&{a\omega }&c\end{array}\,} \right|$ is

If the sum of squares of all real values of $\alpha$, for which the lines $2 x-y+3=0,6 x+3 y+1=0$ and $\alpha x+2 y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $\mathrm{p}$ is $.........$

If ${\Delta _1} = \left| {\begin{array}{*{20}{c}}
  x&{\sin \,\theta }&{\cos \,\theta } \\ 
  {\sin \,\theta }&{ - x}&1 \\ 
  {\cos \,\theta }&1&x 
\end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}}
  x&{\sin \,2\theta }&{\cos \,\,2\theta } \\ 
  {\sin \,2\theta }&{ - x}&1 \\ 
  {\cos \,\,2\theta }&1&x 
\end{array}} \right|$, $x \ne 0$ ; then for all $\theta  \in \left( {0,\frac{\pi }{2}} \right)$

If ${x^a}{y^b} = {e^m},{x^c}{y^d} = {e^n},{\Delta _1} = \left| {\,\begin{array}{*{20}{c}}m&b\\n&d\end{array}\,} \right|\,\,{\Delta _2} = \left| {\,\begin{array}{*{20}{c}}a&m\\c&n\end{array}\,} \right|$ and ${\Delta _3} = \left| {\,\begin{array}{*{20}{c}}a&b\\c&d\end{array}\,} \right|$, then the values of $x$  and $y$  are respectively