AS June 2018 Paper 1 Q14
14
(a) Sketch, on the Argand diagram below, the locus of points satisfying the equation\[|z - 3| = 2\]
[1 mark]
(b) There is a unique complex number \(w\) that satisfies both\[|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha\]
where \(\alpha\) is a constant such that \(0 \lt \alpha \lt \pi\)
(i) Find the value of \(\alpha\). [2 marks]
(ii) Express \(w\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\).
Give each of \(r\) and \(\theta\) to two significant figures. [4 marks]
Give each of \(r\) and \(\theta\) to two significant figures. [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Draws a circle with centre \((3, 0)\) and radius 2. Accept freehand circle. Ignore any straight lines drawn on the diagram. | B1 | 1.1b |
Typical solution
| Scheme | Marks | AO |
|---|---|---|
| (i) Uses fully correct method for \(\sin\alpha\) or \(\cos\alpha\) or \(\tan\alpha\) \(\sin\alpha = \frac{2}{4} \qquad \cos\alpha = \frac{\sqrt{4^2 - 2^2}}{4} \qquad \tan\alpha = \frac{2}{\sqrt{4^2 - 2^2}}\) | M1 | 3.1a |
| Obtains correct value for \(\alpha\) Accept \(0.52(35987756)\) Condone \(30^\circ\) | A1 | 1.1b |
| (ii) Forms an equation in \(r\) using cosine rule or equivalent. Follow through their \(\alpha\). Or forms a correct equation in \(x\) and \(y\). | M1 | 3.1a |
| Forms an equation in \(\theta\) using sine rule or equivalent. Follow through their \(\alpha\). Or forms a second correct equation in \(x\) and \(y\). | M1 | 1.1a |
| Obtains correct value for \(r\) or \(\theta\). Or obtains correct values for \(x\) and \(y\). | A1 | 1.1b |
| Expresses \(w\) in the required form. Accept \(2.6[45751311]\) or \(\sqrt{7}\) for \(r\) Accept \(0.71[37243789]\) or \(\sin^{-1}\left(\frac{\sqrt{21}}{7}\right)\) or \(\sin^{-1}\left(\sqrt{\frac{3}{7}}\right)\) for \(\theta\) | A1 | 1.1b |
| (7 marks) |
Typical solution
(i)
(ii)
