Tìm số phức z thỏa mãn z + (1-2i) z ngang = 5i; 2z - (1-2i) z ngang

* Hướng dẫn giải

Đáp án D

$$\begin{gathered} \mathrm{Đặt} \mathrm{z}=\mathrm{a}+\mathrm{bi} (\mathrm{a};\mathrm{b}\in \mathrm{ℝ}) \Rightarrow \overline{\mathrm{z}}=\mathrm{a}-\mathrm{bi} \\ \mathrm{Ta} \mathrm{có}:\left\{\begin{array}{l}\mathrm{z}+\left(1-2\mathrm{i}\right)\overline{\mathrm{z}}=5\mathrm{i} \left(1\right)\\ 2\mathrm{z}-(1-2\mathrm{i})\overline{\mathrm{z}}=1+4\mathrm{i} \left(2\right)\end{array}\right. \\ \mathrm{Lấy} \left(1\right)+\left(2\right)\Rightarrow 3\mathrm{z}=1+9\mathrm{i}\iff \mathrm{z}=\frac{1}{3}+3\mathrm{i}. \mathrm{Thay} \mathrm{vào} \left(1\right) \\ \left(\frac{1}{3}+3\mathrm{i}\right)+\left(1-2\mathrm{i}\right)\overline{\mathrm{z}}=5\mathrm{i} \\ \iff \left(1-2\mathrm{i}\right)\overline{\mathrm{z}}=\frac{-1}{3}+2\mathrm{i} \\ \iff \overline{\mathrm{z}}=\frac{{\displaystyle \frac{-1}{3}}+2\mathrm{i}}{1-2\mathrm{i}}=\frac{-13}{15}+\frac{4}{15}\mathrm{i} \\ \mathrm{Ta} \mathrm{thấy} \overline{\mathrm{z}} \mathrm{không} \mathrm{phải} \mathrm{số} \mathrm{phức} \mathrm{liên} \mathrm{hợp} \mathrm{của} \mathrm{z} \\ \Rightarrow \mathrm{không} \mathrm{có} \mathrm{số} \mathrm{phức} \mathrm{z} \mathrm{thỏa} \mathrm{mãn} \mathrm{bài} \mathrm{toán} \end{gathered}$$