If a, b and c are cube roots of unity,

\(\begin{array}{l}\left|\begin{array}{ccc} e^{a} & e^{2 a} & e^{3 a} \\ e^{b} & e^{2 b} & e^{3 b} \\ e^{c} & e^{2 c} & e^{3 c} \end{array}\right|-\left|\begin{array}{ccc} e^{a} & e^{2 a} & 1 \\ e^{b} & e^{2 b} & 1 \\ e^{c} & e^{2 c} & 1 \end{array}\right|\end{array} \)

1) 0

2) e

3) e

4) e

Solution: (1) 0

\(\begin{array}{l}\begin{aligned} \Delta &=\left|\begin{array}{ccc} e^{a} & e^{2 a} & e^{3 a} \\ e^{b} & e^{2 b} & e^{3 b} \\ e^{c} & e^{2 c} & e^{3 c} \end{array}\right|-\left|\begin{array}{ccc} e^{a} & e^{2 a} & 1 \\ e^{b} & e^{2 b} & 1 \\ e^{c} & e^{2 c} & 1 \end{array}\right| \\ &=e^{a} e^{b} e^{c}\left|\begin{array}{ccc} 1 & e^{a} & e^{2 a} \\ 1 & e^{b} & e^{2 b} \\ 1 & e^{c} & e^{2 c} \end{array}\right|-\left|\begin{array}{ccc} 1 & e^{a} & e^{2 a} \\ 1 & e^{b} & e^{2 b} \\ 1 & e^{c} & e^{2 c} \end{array}\right| \\ &=e^{a+b+c-1}\left|\begin{array}{ccc} 1 & e^{a} & e^{2 a} \\ 1 & e^{b} & e^{2 b} \\ 1 & e^{c} & e^{2 c} \end{array}\right| \\ &=0 \end{aligned}\end{array} \)

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