# Let ω be a complex number such that where   ifthen k is equal to : Option 1) Option 2) -1 Option 3) 1 Option 4)

We have, $2\omega +1=\sqrt{3}i$

$\Rightarrow \omega =\frac{\sqrt{3}i-1}{2}$

$\Rightarrow \omega^{2}=\frac{-2-2\sqrt{3}i}{4}\Rightarrow \frac{-1-\sqrt{3}i}{2}$

$\Rightarrow \omega ^{3}=-\frac{(\sqrt{3}i+1)(\sqrt{3}i-1)}{4}=1$

In order to find out k, we have to find the value of:

$\begin{vmatrix} 1 &1 & 1\\ 1& -\omega ^{2}-1 &\omega^{2} \\ 1& \omega^{2} & \omega^{7} \end{vmatrix}$

Here, $-\omega^{2}-1=\frac{-1+\sqrt{3}i}{2}=\omega$

So, $\begin{vmatrix} 1 &1 & 1\\ 1& -\omega ^{2}-1 &\omega^{2} \\ 1& \omega^{2} & \omega^{7} \end{vmatrix}=\begin{vmatrix} 1 &1 & 1\\ 1& \omega &\omega^{2} \\ 1& \omega^{2} & \omega \end{vmatrix}$

After expansion of determinant, we get,

$\omega ^2-\omega+\omega ^2-\omega+\omega ^2-\omega=3(\omega ^2-\omega)$=$\omega ^2-\omega+\omega ^2-\omega+\omega ^2-\omega=3(\omega ^2-\omega)=-3\sqrt{3}i$

Therefore, $k=-\sqrt{3}i=-z$

Option 1)

This option has negative sign missing. So, it is wrong.

Option 2)

-1

Not correct.

Option 3)

1

Not correct.

Option 4)

N

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