Với mọi số phức z, mệnh đề nào sau đây luôn đúng? z^2

* Hướng dẫn giải

Đáp án D

$$\begin{gathered} \mathrm{Đặt} \mathrm{z}=\mathrm{a}+\mathrm{bi} (\mathrm{a};\mathrm{b}\in \mathrm{ℝ}) \Rightarrow \overline{\mathrm{z}}=\mathrm{a}-\mathrm{bi} \\ \Rightarrow \left\{\begin{array}{l}{\mathrm{z}}^2={\mathrm{a}}^2-{\mathrm{b}}^2+2\mathrm{abi}\\ {\left|\mathrm{z}\right|}^2={\mathrm{a}}^2+{\mathrm{b}}^2=\left|\overline{\mathrm{z}}\right|\\ {\overline{\mathrm{z}}}^2={\mathrm{a}}^2+{\mathrm{b}}^2-2\mathrm{abi}\end{array}\right.\Rightarrow \mathrm{đáp} \mathrm{án} \mathrm{A},\mathrm{B},\mathrm{C} \mathrm{sai} \\ \mathrm{Mặt} \mathrm{khác}: \left|\overline{{\mathrm{z}}^2}\right|=\left|{\mathrm{a}}^2-{\mathrm{b}}^2-2\mathrm{abi}\right|=\sqrt{{\left({\mathrm{a}}^2-{\mathrm{b}}^2\right)}^2+4{\mathrm{a}}^2{\mathrm{b}}^2} \\ =\sqrt{{\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)}^2}={\mathrm{a}}^2+{\mathrm{b}}^2=\left|{\mathrm{z}}^2\right| \\ \Rightarrow \mathrm{Đáp} \mathrm{án} \mathrm{D} \mathrm{đúng} \end{gathered}$$