Let $z = 5 – 12i$. Find $|z|$ and the real part of $z^{2}$.
Question HKC78-AISL-CPX-01
[Maximum mark: 5]
Let $z = 5 – 12i$. Find $|z|$ and the real part of $z^{2}$.
Mark Scheme
$|z| = sqrt{5^{2} + (-12)^{2}} = sqrt{25 + 144} = sqrt{169} = 13$. [2]
$z^{2} = (5 – 12i)^{2} = 25 – 120i + 144i^{2} = 25 – 120i – 144 = -119 – 120i$. [2]
The real part is $-119$. [1]
Total: 5 marks
The complex number $z$ satisfies $z + \overline{z} = 6$ and $z\overline{z} = 13$. Find $z$ in the form $a + bi$ with $b > 0$.
Question HKC78-AISL-CPX-02
[Maximum mark: 5]
The complex number $z$ satisfies $z + \overline{z} = 6$ and $z\overline{z} = 13$. Find $z$ in the form $a + bi$ with $b > 0$.
Mark Scheme
Write $z = a + bi$. Then $z + overline{z} = 2a = 6$, so $a = 3$. [2]
$zoverline{z} = a^{2} + b^{2} = 13$, so $9 + b^{2} = 13$ and $b^{2} = 4$. [2]
$b > 0$, so $b = 2$ and $z = 3 + 2i$. [1]
Total: 5 marks
