Mathematics AA: COMPLEX NUMBERS Topical Past Paper Questions

[Maximum mark: 20] (a) It is given that $ w = 1 + i\sqrt{3} $. (i) Express $w$ in the form $w = re^{\mathrm{i}\alpha}$, where $r > 0$ and $-\pi [6][6] It is given that $ (1 + i \tan \theta)^n \equiv \frac{\cos n\theta + i \sin n\theta}{\cos^n \theta} $ for $ \cos \theta \neq 0 $,…

[Maximum mark:6] (a) Given that $ w \in \mathbb{C} $, prove that $ ww^* = |w|^2 $. [2] [2] Two complex numbers z and w satisfy the following equations: $$ 5w^{*}=(1-2\mathrm{i})z^{2} $$ $$ zw\equiv10+10i. $$ (b) Given that $ |w| = 2\sqrt{5} $, find z expressing your answer in the…

[Maximum mark: 17] (a) Find the binomial expansion of $ (\cos\theta + i\sin\theta)^{5} $. Give your answer in the form $ a + bi $ where a and b are expressed in terms of $ \sin\theta $ and $ \cos\theta $. (b) By using De Moivre's theorem and your answer to part (a), show that $ \…