1. Given the complex number $z$ satisfies $|z|-z=3-\mathrm{i}$. Then $z=$
$\qquad$ .
\begin{array}{l}\text {1. }-\frac{4}{3}+\mathrm{i} . \\ \text { From }|z|-3=z-\mathrm{i} \in \mathbf{R} \\ \Rightarrow z=a+\mathrm{i} \Rightarrow \sqrt{a^{2}+1}-3=a \\ \Rightarrow a=-\frac{4}{3} \Rightarrow z=-\frac{4}{3}+\mathrm{i} .\end{array}