A) \[\frac{\alpha }{\beta }\]is one of the cube roots of unity
B) \[\alpha \]is one of the cube roots of unity
C) \[\beta \]is one of the cube roots of unity
D) \[\alpha \beta \]is one of the cube roots of unity
E) none of the above
Correct Answer: A
Solution :
\[\left| \begin{matrix} \alpha & -\beta & 0 \\ 0 & \alpha & \beta \\ \beta & 0 & \alpha \\ \end{matrix} \right|=0\Rightarrow {{\alpha }^{2}}-{{\beta }^{3}}=0\] \[\Rightarrow \]\[{{\left( \frac{\alpha }{\beta } \right)}^{3}}=1\Rightarrow \frac{\alpha }{\beta }\]is one of the cube roots of unity
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