Let's solve each part by separating the real and imaginary parts.

Problem Analysis

The task is to separate the real and imaginary components of complex expressions given in three parts:

1. $\left(\frac{3 - 2i}{2 - 3i}\right)^{-2}$
2. $Z = i - 4zi - 3$
3. $\frac{3z + i}{2 + 4}$ and $z = 3 + 2i$

I'll go through each part in detail.

Solution

Part (i) $\left(\frac{3 - 2i}{2 - 3i}\right)^{-2}$

1. First, simplify $\frac{3 - 2i}{2 - 3i}$:

   • Multiply numerator and denominator by the conjugate of the denominator: $2 + 3i$.
   • This gives: $\frac{(3 - 2i)(2 + 3i)}{(2 - 3i)(2 + 3i)} = \frac{6 + 9i - 4i - 6i^2}{4 + 9} = \frac{6 + 5i + 6}{13} = \frac{12 + 5i}{13}$
   • So, $\frac{3 - 2i}{2 - 3i} = \frac{12}{13} + \frac{5}{13}i$.

2. Now, raise this result to the power of -2.

Part (ii) $Z = i - 4zi - 3$

Separate the real and imaginary parts directly by rearranging terms.

Part (iii) $\frac{3z + i}{2 + 4}$ where $z = 3 + 2i$

1. Substitute $z = 3 + 2i$ into $3z + i$:
   • Compute