Let $z_{1} = 2 + i$ and $z_{2} = 1 – 3i$. Find $z_{1}\overline{z_{2}}$ in the form $a + bi$, and find $\left|z_{1}\overline{z_{2}}\right|$.
Question HKC78-AIHL-CPX-01
[Maximum mark: 6]
Let $z_{1} = 2 + i$ and $z_{2} = 1 – 3i$. Find $z_{1}\overline{z_{2}}$ in the form $a + bi$, and find $\left|z_{1}\overline{z_{2}}\right|$.
Mark Scheme
$overline{z_{2}} = 1 + 3i$. [1]
$z_{1}overline{z_{2}} = (2 + i)(1 + 3i) = 2 + 6i + i + 3i^{2} = -1 + 7i$. [3]
$left|-1 + 7iright| = sqrt{1 + 49} = sqrt{50} = 5sqrt{2}$. [2]
Total: 6 marks
Express $z = -1 + i$ in modulus-argument form, with the argument in $(-\pi, \pi]$.
Question HKC78-AIHL-CPX-02
[Maximum mark: 4]
Express $z = -1 + i$ in modulus-argument form, with the argument in $(-\pi, \pi]$.
Mark Scheme
$|z| = sqrt{(-1)^{2} + 1^{2}} = sqrt{2}$. [1]
The point $(-1, 1)$ is in the second quadrant. The reference angle is $dfrac{pi}{4}$, so $arg z = pi – dfrac{pi}{4} = dfrac{3pi}{4}$. [2]
$z = sqrt{2}operatorname{cis}dfrac{3pi}{4}$, or $sqrt{2}left(cosdfrac{3pi}{4} + isindfrac{3pi}{4}right)$. [1]
Total: 4 marks
