Let S be the set of all complex numbers z satisfying | z ^ 2 + z +...

${z}^{2} + z + 1 = {e}^{i\theta}\theta \in ( - \pi, \pi ]$

$⇒ {z}^{2} + z + 1 - {e}^{i\theta} = 0$

By quadratic formula, we get

$z = \dfrac{- 1 \pm \sqrt{4{e}^{i\theta} - 3}}{2} . . . . . . . . . . . . . \left(1\right)$

$⇒ z + \dfrac{1}{2} = \pm \sqrt{\left(4cos\theta - 3\right) + i4sin\theta}$
$⇒ \left|z + \dfrac{1}{2}\right| = \left[{\left(4cos\theta - 3\right)}^{2} + {\left(4sin\theta\right)}^{2}\right]\dfrac{1}{4}$

$⇒ \left|z + \dfrac{1}{2}\right| \in \left[1, \sqrt{7}\right]$

Option (C) is correct
By equation \left(1\right)$,</p><p>\left|2z\right| \le 1 + \sqrt{\left|4{e}^{i\theta} - 3\right|}$

$⇒ \left|2Z\right| \le 1 + ( 25 - 24cos\theta )\dfrac{1}{4}$

$⇒ \left|2Z\right| \le 1 + \sqrt{7} < 4$

$⇒ \left|Z\right| \le 2$

Option (B) is correct.