AS June 2019 Paper 1 Q7
7
(a) Sketch on a single Argand diagram
(i) the set of points for which \(|z - 1 - 3\mathrm{i}| = 3\), [3]
(ii) the set of points for which \(\arg(z + 4) = \tfrac{1}{4}\pi\). [3]
(b) Find, in exact form, the two values of \(z\) for which \(|z - 1 - 3\mathrm{i}| = 3\) and \(\arg(z + 4) = \tfrac{1}{4}\pi\). [6]
| Scheme | Marks | AO |
|---|---|---|
 (i) | M1 A1 A1 | 3.1a 1.1 1.1 |
| [3] | ||
| (ii) | B1 B1 B1 | 3.1a 1.1 1.1 |
| [3] |
Notes
(a)(i)
M1: circle
A1: (1st) centre \(1 + 3\mathrm{i}\) indicated
A1: (2nd) touching real axis
(a)(ii)
B1: (1st) line at \(45^\circ\) to real axis
with evidence
B1: (2nd) through \(-4\)
B1: (3rd) correct half-line indicated
| Scheme | Marks | AO |
|---|---|---|
| circle is \((x - 1)^2 + (y - 3)^2 = 9\) | B1ft | 3.1a |
| line is \(y = x + 4\) | B1ft | 3.1a |
| \(\Rightarrow (x - 1)^2 + (x + 1)^2 = 9\) | M1 | 1.1 |
| \(\Rightarrow 2x^2 = 7 \Rightarrow x = \pm\sqrt{\tfrac{7}{2}}\) | A1 | 1.1 |
| \(\Rightarrow y = 4 \pm \sqrt{\tfrac{7}{2}}\) | A1 | 1.1 |
| so \(z = -\sqrt{\tfrac{7}{2}} + \left(4 - \sqrt{\tfrac{7}{2}}\right)\mathrm{i}\) or \(\sqrt{\tfrac{7}{2}} + \left(4 + \sqrt{\tfrac{7}{2}}\right)\mathrm{i}\) | A1 | 3.2a |
| [6] |
Notes
B1ft: (1st) ft their centre
B1ft: (2nd) ft their \(-4\)
M1: eliminating \(y\) (or \(x\))