11. If $a \pm b \mathrm{i}(b \neq 0)$ are the imaginary roots of the equation $x^{3}+q x+r=0$, where $a, b, q$ and $r$ are all real numbers, then $q$ can be expressed in terms of $a, b$ as $\qquad$
11. $q=b^{2}-3 a^{2} \quad a^{3}+3 a^{2} b i+3 a b^{2} i^{2}+b^{3} i^{3}+q a+q b i+r=0$
$$
\left\{\begin{array}{l}
a^{3}-3 a b^{2}+q a+r=0 \\
3 a^{2} b-b^{3}+q b=0
\end{array}\right.
$$

So $q=\frac{b^{3}-3 a^{2} b}{b}=b^{2}-3 a^{2}$.