Mathematics: COMPLEX NUMBERS Topical Past Paper Questions

Consider the complex numbers $ z_{1}=\cos\frac{11\pi}{12}+i\sin\frac{11\pi}{12} $ and $ z_{2}=\cos\frac{\pi}{6}+i\sin\frac{\pi}{6} $. (a) (i) Find $ \frac{z_{1}}{z_{2}} $ (ii) Find $ \frac{z_{2}}{z_{1}} $ (b) 0, $ \frac{z_{1}}{z_{2}} $ and $ \frac{z_{2}}{z_{1}} $ are represented…

[Maximum mark: 6] Let $P(z) = az^3 - 37z^2 + 66z - 10$, where $z \in \mathbb{C}$ and $a \in \mathbb{Z}$. One of the roots of $P(z)=0$ is $3+i$. Find the value of $a$.

(a) Find the roots of $ z^{24} = 1 $ which satisfy the condition $ 0 < \arg(z) < \frac{\pi}{2} $, expressing your answers in the form $ re^{i\theta} $, where $ r, \theta \in \mathbb{R}^+ $. (b) Let S be the sum of the roots found in part (a). (i) Show that $\mathrm{Re}S=\mathrm{I…