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

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

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 \( z^2 \in \mathbb{C} \setminus \mathbb{R} \).

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 \(z \in S\).