1. **Problem Statement:** (a) State four properties of bilinear transformation. (b) Evaluate: i. \((1 - 2i)^4\) ii. \(\frac{2 + i}{2 - i}\) 2. **Properties of Bilinear Transformation:** - It is a rational function of the form \(w = \frac{az + b}{cz + d}\) where \(ad - bc \neq 0\). - It maps circles and lines in the complex plane to circles or lines. - It is conformal, preserving angles at which curves meet. - It is invertible, with the inverse also a bilinear transformation. 3. **Evaluating \((1 - 2i)^4\):** - Use binomial expansion or convert to polar form. - Calculate magnitude: \(|1 - 2i| = \sqrt{1^2 + (-2)^2} = \sqrt{5}\). - Calculate argument: \(\theta = \tan^{-1}(-2/1) = -1.107\) radians. - Raise to power 4: magnitude becomes \(5^2 = 25\), argument becomes \(4 \times -1.107 = -4.428\). - Convert back to rectangular form: $$25(\cos(-4.428) + i\sin(-4.428)) = 25(\cos(1.855) - i\sin(1.855))$$ - Calculate \(\cos(1.855) \approx -0.28\), \(\sin(1.855) \approx 0.96\). - So, \(= 25(-0.28 - 0.96i) = -7 - 24i\). 4. **Evaluating \(\frac{2 + i}{2 - i}\):** - Multiply numerator and denominator by conjugate of denominator: $$\frac{2 + i}{2 - i} \times \frac{2 + i}{2 + i} = \frac{(2 + i)^2}{(2)^2 - (-i)^2}$$ - Calculate numerator: \((2 + i)^2 = 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i\). - Calculate denominator: \(4 - (-1) = 5\). - So, \(\frac{3 + 4i}{5} = \frac{3}{5} + \frac{4}{5}i\). **Final answers:** - \((1 - 2i)^4 = -7 - 24i\) - \(\frac{2 + i}{2 - i} = \frac{3}{5} + \frac{4}{5}i\)