AS June 2018 Q5
5. A complex number \(z\) is represented by the point \(P\) on an Argand diagram.
Given that \(\arg\left(\dfrac{z - 6\mathrm{i}}{z - 3\mathrm{i}}\right) = \dfrac{\pi}{3}\)
(a) sketch the locus of \(P\) as \(z\) varies, (3)
(b) find the exact maximum possible value of \(|z|\) (5)
| Scheme | Marks | AO |
|---|---|---|
 | M1 A1 A1 | 3.1a 1.1b 1.1b |
| (3) |
Notes
M1: Interprets the locus correctly as a circle or as an arc of a circle
A1: A circle or an arc of a circle passing through or touching at 3 and 6 on the positive imaginary axis.
A1: Correct diagram – a major arc that is wholly to the left of the imaginary axis and wholly above the real axis with 3 and 6 marked on the imaginary axis
| Scheme | Marks | AO |
|---|---|---|
| \(y\)-coordinate of centre of circle is 4.5 | B1 | 1.1b |
| \(x\)-coordinate of centre of circle is \(-\dfrac{1.5}{\tan\frac{\pi}{3}}\left(= -\dfrac{\sqrt{3}}{2}\right)\) | M1 | 3.1a |
| Radius of circle is \(\dfrac{1.5}{\sin\frac{\pi}{3}}\) or \(\sqrt{1.5^2 + \left(\dfrac{1.5}{\tan\frac{\pi}{3}}\right)^2}\) | M1 | 1.1b |
| \(d = \sqrt{4.5^2 + 0.75} + \sqrt{3}\) | M1 | 3.1a |
| \(d = \sqrt{21} + \sqrt{3}\) | A1 | 1.1b |
| (5) | ||
| (8 marks) |
Notes
B1: Correct \(y\)-coordinate of the centre
M1: Correct strategy for finding the \(x\)-coordinate of the centre
M1: Correct strategy for finding the radius of the circle
M1: Fully correct method for the maximum using their values
A1: Correct value