If $α$ is a non-real cube root of -2, then the value of $\begin{vmatrix}1&2α&1\\α^2&1&3α^2\\2&2α&1\end{vmatrix}$, is

-11 -12 -13 0

-13

We have, $α = (−2)^{1/3} w$ or, $α = (− 2)^{1/3} w^2$ $∴α^3=-2$ Now, $\begin{vmatrix}1&2α&1\\α^2&1&3α^2\\2&2α&1\end{vmatrix}$ $=\begin{vmatrix}1&2α&1\\α^2&1&3α^2\\1&0&0\end{vmatrix}$  [Applying $R_3 → R_3 - R_1$] $=6α^3-1$  [On expanding along $R_3$] $=-13$  $[∵ α^3 = -2]$