Có bao nhiêu số phức z thỏa mãn \[|z| = 1\;\]và \[\mid {z^3} + 2024z + \overline z \mid - 2\sqrt 3 \mid z + \overline z \mid = 2019\]

Ta có :

\[\mid {z^3} + 2024z + \overline z \mid - 2\sqrt 3 \mid z + \overline z \mid = 2019\]

\[\begin{array}{l} \Leftrightarrow \mid \frac{{\mid {z^3} + 2024z + \overline z \mid }}{{|z|}} - 2\sqrt 3 \mid z + \overline z \mid = 2019\\ \Leftrightarrow \mid {z^2} + 2024 + \frac{{\overline z }}{z}\mid - 2\sqrt 3 \mid z + \overline z \mid = 2019\\ \Leftrightarrow \left| {{z^2} + 2024 + {z^{ - 2}}} \right| - 2\sqrt 3 \mid z + \overline z \mid = 2019\\ \Leftrightarrow \left| {{{\left( {z + \overline z } \right)}^2} - 2z\overline z + 2021} \right| - 2\sqrt 3 \mid z + \overline z \mid = 2019\\ \Leftrightarrow \left| {{{(z + \overline z )}^2} + 2022\mid } \right| - 2\sqrt 3 \mid z + \overline z \mid = 2019\end{array}\]

Đặt\[z = a + bi \Rightarrow \bar z = a - bi \Rightarrow z + \bar z = 2a\]

Khi đó phương trình cuối trở thành

\[\left| {{{\left( {2a} \right)}^2} + 2022} \right| - 2\sqrt 3 .\left| {2a} \right| = 2019 \Leftrightarrow 4{a^2} - 4\sqrt 3 \left| a \right| + 3 = 0\]

\[ \Leftrightarrow {\left( {2\left| a \right| - \sqrt 3 } \right)^2} = 0 \Leftrightarrow \left| a \right| = \frac{{\sqrt 3 }}{2} \Leftrightarrow a = \pm \frac{{\sqrt 3 }}{2}\]

Mà

\[\left| z \right| = 1 \Leftrightarrow {\left| z \right|^2} = 1 \Leftrightarrow {a^2} + {b^2} = 1 \Rightarrow {b^2} = 1 - {a^2} = \frac{1}{4} \Leftrightarrow b = \pm \frac{1}{2}\]

Vậy có bốn số phức thỏa mãn bài toán là

\[{z_1} = \frac{{\sqrt 3 }}{2} + \frac{1}{2}i,\,\,{z_2} = \frac{{\sqrt 3 }}{2} - \frac{1}{2}i,\,\,{z_3} = - \frac{{\sqrt 3 }}{2} - \frac{1}{2}i,\,\,{z_4} = - \frac{{\sqrt 3 }}{2} + \frac{1}{2}i.\]

Đáp án cần chọn là: B