Let be a complex number such that 2 + 1 = z where z = - 3 . If | … - Vidyadip

MCQ

Let $\omega $ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt { - 3} $ . If $\left| {\begin{array}{*{20}{c}}1&1&1\\1&{ - {\omega ^2} - 1}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^7}}\end{array}} \right| = 3k$ then $k$ is equal to :

A
$1$
✓
$-z$
C
$z$
D
$-1$

✓

Answer

Correct option: B.

$-z$

Given $2\omega + 1 = z;$$z = \sqrt {3i} $
$ \Rightarrow \omega = \frac{{\sqrt {3i} - 1}}{2}$
$ \Rightarrow \omega $ is complex cube root of unity
Applying ${R_1} \to {R_1} + {R_2} + {R_3}$
$ = \left| \begin{array}{l}
3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\
1\,\,\,\,\, - {\omega ^2} - 1\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^2}\\
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\omega
\end{array} \right|\,$
$ = 3\left( { - 1 - \omega - \omega } \right) $
$= - 3\left( {1 + 2\omega } \right)\, = - 3z$
$ \Rightarrow k = - z$

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