Subject 3. Let $a, b \in \mathbb{C}$ and the function $f: \mathbb{C} \rightarrow \mathbb{C}$, defined by $f(z)=z^{2}+a|z|+b$.

i) Determine $a, b \in \mathbb{C}$ such that $f(1)=f(2)=0$.

ii) For $a, b$ determined in the previous point, find all complex numbers $z$ such that $f(z)=0$.

Mathematical Gazette 1980
Subject 3. i) $1+a+b=0$ (1p); $4+2 a+b=0$ (1p); $a=-3, b=2$ (1p). ii) substitute $z=x+i y$ into $z^{2}-3|z|+2=0$ : $(x+i y)^{2}-3 \sqrt{x^{2}+y^{2}}+2=0$ (1p);

![](https://cdn.mathpix.com/cropped/2024_06_07_a049303b8b5804c00308g-2.jpg?height=136&width=1404&top_left_y=905&top_left_x=339)