Let \( \omega \) be a complex number such that \( 2 \omega+1=\mathr...

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Let \( \omega \) be a complex number such that \( 2 \omega+1=\mathrm{z} \) where \( \mathrm{z}=\sqrt{-3} \). If \( \left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & -\omega^{2}-1 & \omega^{2} \\ 1 & \omega^{2} & \omega^{7}\end{array}\right|=3 \mathrm{k} \), then \( \mathrm{k} \)
\( \mathrm{P} \) is equal to
(1) \( -\mathrm{z} \)
(2) z
(3) \( -1 \)
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