Let $z = 1 – \sqrt{3}\, i$. Find $|z|$ and the principal argument of $z$, taking the argument in $(-\pi, \pi]$.
Question HKC78-AAHL-CPX-01
[Maximum mark: 5]
Let $z = 1 – \sqrt{3}\, i$. Find $|z|$ and the principal argument of $z$, taking the argument in $(-\pi, \pi]$.
Mark Scheme
$|z| = sqrt{1^{2} + (sqrt{3})^{2}} = sqrt{4} = 2$. [2]
The point $(1, -sqrt{3})$ is in the fourth quadrant. The reference angle is $dfrac{pi}{3}$. [2]
Therefore $arg z = -dfrac{pi}{3}$. [1]
Total: 5 marks
Solve $z^{2} – 2z + 5 = 0$, giving each root in the form $a + bi$ with $a, b \in \mathbb{R}$.
Question HKC78-AAHL-CPX-02
[Maximum mark: 4]
Solve $z^{2} – 2z + 5 = 0$, giving each root in the form $a + bi$ with $a, b \in \mathbb{R}$.
Mark Scheme
Discriminant: $(-2)^{2} – 4(1)(5) = -16$. [1]
$z = dfrac{2 pm sqrt{-16}}{2} = dfrac{2 pm 4i}{2}$. [2]
$z = 1 + 2i$ or $z = 1 – 2i$. [1]
Total: 4 marks
