题目内容
12.已知复数z的共轭复数为$\overline z$,且z•$\overline z-3iz=\frac{10}{1-3i}$,求复数z.分析 设z=a+bi(a,b∈R),则$\overline{z}$=a-bi,代入化简$\left.\begin{array}{l}{z•\overline{z}-3iz}\end{array}\right.$,再代入z•$\overline z-3iz=\frac{10}{1-3i}$化简,利用复数相等的条件列出方程组,求出a、b的值即可求出复数z.
解答 解:设z=a+bi(a,b∈R),则$\overline{z}$=a-bi,
∴$\left.\begin{array}{l}{z•\overline{z}-3iz={a}^{2}+{b}^{2}+3b-3ai}\end{array}\right.$,
∵z•$\overline z-3iz=\frac{10}{1-3i}$,
∴$\left.\begin{array}{l}{{a}^{2}+{b}^{2}+3b-3ai=\frac{10(1+3i)}{(1-3i)(1+3i)}=1+3i}\end{array}\right.$,
则$\left.\begin{array}{l}{\left\{\begin{array}{l}{{a}^{2}+{b}^{2}+3b=1}\\{-3a=3}\end{array}\right.}\end{array}\right.$,解得$\left.\begin{array}{l}{\left\{\begin{array}{l}{a=-1}\\{b=0}\end{array}\right.}\end{array}\right.$或$\left.\begin{array}{l}{\left\{\begin{array}{l}{a=-1}\\{b=-3}\end{array}\right.}\end{array}\right.$
∴z=-1或$\left.\begin{array}{l}{z=-1-3i}\end{array}\right.$.
点评 本题考查复数代数形式的乘除运算,共轭复数和复数相等的定义,以及化简、计算能力.
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(2)能否在犯错误的概率不超过0.001的前提下,认为该市外来务工人员春节买票回家是否需要交通部门提供帮助与性别有关?
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