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Let $\alpha $, $\beta$ $\gamma$, $\delta$ are distinct imaginary roots of
$z^5=1$ then value of $\left| {\begin{array}{*{20}{c}}
{{e^\alpha }}&{{e^{2\alpha }}}&{{e^{3\alpha + 1}}}&{ - {e^{ - \delta }}} \\
{{e^\beta }}&{{e^{2\beta }}}&{{e^{3\beta + 1}}}&{ - {e^{ - \delta }}} \\
{{e^\gamma }}&{{e^{2\gamma }}}&{{e^{3\gamma + 1}}}&{ - {e^{ - \delta }}}
\end{array}} \right|$
$0$
$e$
$1$
${e^5}$
Solution
$\left|\begin{array}{lll}{e^{\alpha}} & {e^{2 \alpha}} & {e^{3 \alpha+1}} \\ {e^{\beta}} & {e^{2 \beta}} & {e^{3 \beta+1}} \\ {e^{\gamma}} & {e^{2 \gamma}} & {e^{3 \gamma+1}}\end{array}\right|-\left|\begin{array}{ccc}{e^{\alpha}} & {e^{2 \alpha}} & {e^{-\delta}} \\ {e^{\beta}} & {e^{2 \beta}} & {e^{-\delta}} \\ {e^{\gamma}} & {e^{2 \gamma}} & {e^{-\delta}}\end{array}\right|$
$e^{\alpha} \cdot e^{\beta} \cdot e^{\gamma} \cdot e^{1}\left|\begin{array}{ccc}{1} & {e^{\alpha}} & {e^{2 \alpha}} \\ {1} & {e^{\beta}} & {e^{2 \beta}} \\ {1} & {e^{\gamma}} & {e^{2 \gamma}}\end{array}\right|-e^{-\delta}\left|\begin{array}{ccc}{1} & {e^{\alpha}} & {e^{2 \alpha}} \\ {1} & {e^{\beta}} & {e^{2 \beta}} \\ {1} & {e^{\gamma}} & {e^{2 \gamma}}\end{array}\right|$
$\left(e^{\alpha+\beta+\gamma+1}-e^{-\delta}\right)\left|\begin{array}{ccc}{1} & {e^{\alpha}} & {e^{2 \alpha}} \\ {1} & {e^{\beta}} & {2^{2 \beta}} \\ {1} & {e^{\gamma}} & {e^{2 \gamma}}\end{array}\right|=0$
$\because \alpha+\beta+\gamma+\delta+1=0$
