1. The complex number $z$ satisfies $|z|=5$, and $(3+4 \mathrm{i}) z$ is a pure imaginary number. Then $\bar{z}=$ $\qquad$
$$
-1 \cdot \bar{z}= \pm(4-3 \mathrm{i})
$$

From the given information, we have
$$
\begin{array}{l}
|(3+4 \mathrm{i}) z|=|3+4 \mathrm{i}||z|=25 \\
\Rightarrow(3+4 \mathrm{i}) z= \pm 25 \mathrm{i} \\
\Rightarrow z=\frac{ \pm 25 \mathrm{i}}{3+4 \mathrm{i}}= \pm(4+3 \mathrm{i}) \\
\Rightarrow \bar{z}= \pm(4-3 \mathrm{i}) .
\end{array}
$$