1. Given a complex number $z$ satisfying $|z|=1$, then the maximum value of $\left|z^{3}+3 z+2 \mathrm{i}\right|$ is (where $\mathrm{i}$ is the imaginary unit).
A. $4 \sqrt{2}$
B. $3 \sqrt{3}$
C. $2 \sqrt{5}$
D. $4 \sqrt{3} \quad$
$$
\begin{array}{l}
\text { Let } z=\cos \theta+\mathrm{i} \sin \theta \text {, then }\left|z^{3}+3 z+2 \mathrm{i}\right|^{2}=\left(z^{3}+3 z+2 \mathrm{i}\right)\left(\bar{z}^{3}+3 \bar{z}-2 \mathrm{i}\right) \\
=z^{3} \bar{z}^{3}+3 z^{3} \bar{z}-2 z^{3} \mathrm{i}+3 z \bar{z}^{3}+9 z \bar{z}-6 z \mathrm{i}+2 \bar{z}^{3} \mathrm{i}+6 \bar{z} \mathrm{i}+4 \\
=14+3 z^{2}+3 \bar{z}^{2}-2 \mathrm{i}\left(z^{3}-\bar{z}^{3}+3 z-3 \bar{z}\right)=14+6 \cos 2 \theta+4(\sin 3 \theta+3 \sin \theta) \\
=20-4\left(4 \sin ^{3} \theta+3 \sin ^{2} \theta-6 \sin \theta\right),
\end{array}
$$

When $\sin \theta=\frac{1}{2}$, $\left|z^{3}+3 z+2 \mathrm{i}\right|^{2}$ reaches its maximum value of 27, so the answer is $B$.