Mathematics AI: COMPLEX NUMBERS Topical Past Paper Questions

[Maximum mark:5] Two complex numbers are $ z = 2 + ai $ and $ w = b + 4i $, where $ a, b \in \mathbb{R} $. (a) Find an expression for the real part of $ z^{2} $, in terms of a. (b) Find the value of $a$ and of $b$, given that $z^{2}+2w=3+6i$.

[Maximum mark: 8] Consider the complex number $ z = -1 + i $. (a) Express $z$ in the form $r e^{\mathrm{i}\theta}$ where $-\pi [2] A is mapped onto B by the composition of a rotation and an enlargement. (b) (i) Describe fully this mapping of A onto B, stating the scale factor of…

[Maximum mark: 6] Given $ z = \sqrt{3} - i $. (a) Write $ z $ in the form $ z = r e^{\theta i} $, where $ r \in \mathbb{R}^+ $, $ -\pi 0 $, $ t \in \mathbb{R} $ and $ -\pi \leq q \leq \pi $. [4]