The value of $x,$ if $\left| {\,\begin{array}{*{20}{c}}{ – x}&1&0\\1&{ – x}&1\\0&1&{ – x}\end{array}\,} \right| = 0 $ is equal to

If the system of equations $\alpha x+y+z=5, x+2 y+$ $3 z=4, x+3 y+5 z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha, \beta)$ is equal to:

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

$\left| {\,\begin{array}{*{20}{c}}{bc}&{bc' + b'c}&{b'c'}\\{ca}&{ca' + c'a}&{c'a'}\\{ab}&{ab' + a'b}&{a'b'}\end{array}\,} \right|$ is equal to

If the system of equations

$x+y+z=2$

$2 x+4 y-z=6$

$3 x+2 y+\lambda z=\mu$ has infinitely many solutions, then