Tìm phần ảo của số phức $z = {\left(1+i\sqrt{3}\right)}^3$

Đáp án A

$$\begin{gathered} \mathrm{Xét} \mathrm{đáp} \mathrm{án} \mathrm{A}: \\ \mathrm{z}=1+\cos \frac{\mathrm{\pi }}{7}+\mathrm{i}.\sin \frac{\mathrm{\pi }}{7} \\ \Rightarrow \left|\mathrm{z}\right|={\left(1+\cos \frac{\mathrm{\pi }}{7}\right)}^2+{\left(\sin \frac{\mathrm{\pi }}{7}\right)}^2=1+2\cos \frac{\mathrm{\pi }}{7}+{\left(\cos \frac{\mathrm{\pi }}{7}\right)}^2+{\left(\sin \frac{\mathrm{\pi }}{7}\right)}^2 \\ =2+2\cos \frac{\mathrm{\pi }}{7}\ne 1 \\ \Rightarrow \mathrm{Đáp} \mathrm{án} \mathrm{A} \mathrm{đúng} \\ \mathrm{Xét} \mathrm{đáp} \mathrm{án} \mathrm{B}: \\ \mathrm{z}=\cos \frac{2000\mathrm{\pi }}{3}+\mathrm{i}.\sin \frac{2000\mathrm{\pi }}{3} \\ \Rightarrow \left|\mathrm{z}\right|=\sqrt{{\left(\cos \frac{2000\mathrm{\pi }}{3}\right)}^2+{\left(\sin \frac{2000\mathrm{\pi }}{3}\right)}^2}=1 \Rightarrow \mathrm{Đáp} \mathrm{án} \mathrm{B} \mathrm{loại} \\ \mathrm{Xét} \mathrm{đáp} \mathrm{án} \mathrm{C}: \\ \mathrm{z}=\frac{1-2\mathrm{i}\sqrt{2}}{3} \\ \Rightarrow \left|\mathrm{z}\right|=\sqrt{{\left(\frac{1}{3}\right)}^2+{\left(\frac{-2\sqrt{2}}{3}\right)}^2}=\sqrt{\frac{1}{9}+\frac{8}{9}}=1 \Rightarrow \mathrm{Đáp} \mathrm{án} \mathrm{C} \mathrm{loại} \\ \mathrm{Xét} \mathrm{đáp} \mathrm{án} \mathrm{D}: \\ \mathrm{z}=\frac{4-\mathrm{i}}{4+\mathrm{i}}=\frac{15}{17}+\frac{8}{17}\mathrm{i} \\ \Rightarrow \left|\mathrm{z}\right|=\sqrt{{\left(\frac{15}{17}\right)}^2+{\left(\frac{8}{17}\right)}^2}=\sqrt{\frac{289}{289}}=1 \Rightarrow \mathrm{Đáp} \mathrm{án} \mathrm{D} \mathrm{loại} \end{gathered}$$

Đáp án B

$$\begin{gathered} \mathrm{Gọi} \mathrm{z}=\mathrm{a}+\mathrm{bi} (\mathrm{a};\mathrm{b}\in \mathrm{ℝ} \mathrm{và} \mathrm{khác} 0) \mathrm{có} \mathrm{điểm} \mathrm{biểu} \mathrm{diễn} \mathrm{là} \mathrm{M}(\mathrm{a};\mathrm{b}) \\ \Rightarrow \mathrm{w}=\overline{-\mathrm{z}}=\overline{-(\mathrm{a}+\mathrm{bi})}=\overline{-\mathrm{a}-\mathrm{bi}}=-\mathrm{a}+\mathrm{bi} \\ \Rightarrow \mathrm{Điểm} \mathrm{biểu} \mathrm{diễn} \mathrm{số} \mathrm{phức} \mathrm{w} \mathrm{là} \mathrm{N}(-\mathrm{a};\mathrm{b}) \\ \mathrm{Ta} \mathrm{thấy} \mathrm{M} \mathrm{và} \mathrm{N} \mathrm{đối} \mathrm{xứng} \mathrm{nhau} \mathrm{qua} \mathrm{trục} \mathrm{Oy} \end{gathered}$$