Define and use set notation of the number system for complex numbers (ℂ).

Consider the set \(S = \{z \in \mathbb{C} : |z - 2 + 3i| \leq 5\}\). (a) State whether each of the following complex numbers belongs to the set \(S\). Show evidence of your reasoning. (i) \(z_1 = 4 + i\) (ii) \(z_2 = -2 - 2i\) (b) Determine the maximum value of \(|z|\) for \(z \in S\).

Consider the complex number \( z = -3 + 4i \). (a) State which set(s) this number belongs to by using correct set notation from: \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\). (b) Determine the values of \(a\) and \(b\) such that \( z^2 = a + bi \) where \( a, b \in \mathbb{R} \). (c) Show that \( z^2 \in \mathbb{C} \setminus \mathbb{R} \).

The shaded region in the Argand diagram represents which set of complex numbers \(z\)?