A2 June 2019 Paper 2 Q7
7 In an Argand diagram the points representing the numbers \(2 + 3\mathrm{i}\) and \(1 - \mathrm{i}\) are two adjacent vertices of a square, \(S\).
(a) Find the area of \(S\). [3]
(b) Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \((2 + 3\mathrm{i}) - (1 - \mathrm{i})\ (= \pm(1 + 4\mathrm{i}))\) soi | B1 | 1.1 |
| \(|1 + 4\mathrm{i}| = \sqrt{1^2 + 4^2}\) | M1 | 2.2a |
| 17 | A1 | 1.1 |
| [3] |
Notes
B1: Either way round. Can be implied by vector
M1: Or finding the square of their side
| Scheme | Marks | AO |
|---|---|---|
| \((\pm\mathrm{i}) \times (\pm(1 + 4\mathrm{i}))\) | *M1 | 3.1a |
| \((2 + 3\mathrm{i}) \pm \mathrm{i}(1 + 4\mathrm{i})\) and \((1 - \mathrm{i}) \pm \mathrm{i}(1 + 4\mathrm{i})\) | dep*M1 | 2.2a |
| So vertices at \(-3\) and \(-2 + 4\mathrm{i}\) | A1 | 3.2a |
| Or at \(5 - 2\mathrm{i}\) and \(6 + 2\mathrm{i}\) | A1 | 3.2a |
| [4] |
Notes
*M1: Method to find a complex number representing perpendicular side. Can be implied by \(\pm(4 - \mathrm{i})\). Or vector form if take geometric approach
dep*M1: Method to find both pairs of numbers
A1: Both clearly paired and in complex number form for final A1
If M1M0A0A0 then add SC1 for any two correct vertices