IAL 2019 May FP1 Q3

Question

[!problem]

(i) Given that

$z = \frac{2 + 3i}{1 - i}$

find $z$, giving your answer in the form $a + bi$, where $a$ and $b$ are real constants. You must show all your working.

(ii) $w = -4 + 5i$

(a) Find $\arg w$, giving your answer in radians correct to 2 decimal places.

Given that

$\arg(w + k) = \frac{\pi}{2}$

where $k$ is a real constant

(b) write down the value of $k$.

Given that

$|w + ci| = 4\sqrt{5}$

where $c$ is a real constant

(c) find the possible values of $c$.