(a)

We have, $\left|z^3+z^{-3}\right| \leq 2$

Now, $\left|z^3+\frac{1}{z^3}\right| \leq 2$

$\Rightarrow \quad\left|z^3+\frac{1}{z^3}\right| \leq\left|z^3\right|+\frac{1}{\left|z^3\right|}$

$\left.\Rightarrow \frac{|z|^3+\frac{1}{|z|^3}}{2} \geq \sqrt{|z|^3 \frac{1}{|z|^3}} \quad \because \text { AM } \geq GM \right]$

$\Rightarrow|z|^3+\frac{1}{|z|^3} \geq 2 \Rightarrow|z|^3+\frac{1}{|z|^3}=2$

$\therefore \quad|z|=1$

$\therefore \text { Maximum value of }\left|z+\frac{1}{z}\right| \leq|z|+\frac{1}{|z|}=2$