AS October 2021 Paper 1 Q6
6 In this question you must show detailed reasoning.
| Scheme | Marks | AO |
|---|---|---|
| \(z = \dfrac{-(-10) \pm \sqrt{(-10)^2 - 4 \times 2 \times 25}}{2 \times 2}\) | M1 | 2.1 |
| \(z = \dfrac{5}{2} \pm \dfrac{5}{2}\mathrm{i}\) | A1 | 1.1 |
| [2] |
Notes
M1: Correct substitution into formula. If formula quoted allow one slip.
Or completing the square – one slip allowed.
A1: Allow \(z = \dfrac{5 \pm 5\mathrm{i}}{2}\) or equivalent fractions
NB: This question required detailed reasoning
| Scheme | Marks | AO |
|---|---|---|
| \(3\omega - 2 = 5\mathrm{i} + 2\mathrm{i}\omega \Rightarrow 3\omega - 2\mathrm{i}\omega = 2 + 5\mathrm{i}\) | M1 | 1.1 |
| \((3 - 2\mathrm{i})\omega = 2 + 5\mathrm{i} \Rightarrow \omega = \dfrac{2 + 5\mathrm{i}}{3 - 2\mathrm{i}}\) | M1 | 1.1 |
| \(\omega = \dfrac{2 + 5\mathrm{i}}{3 - 2\mathrm{i}} \times \dfrac{3 + 2\mathrm{i}}{3 + 2\mathrm{i}} = \dfrac{6 + 4\mathrm{i} + 15\mathrm{i} - 10}{9 + 4}\) | M1 | 2.1 |
| \(\omega = -\dfrac{4}{13} + \dfrac{19}{13}\mathrm{i}\) | A1 | 1.1 |
| [4] |
Notes
M1: (1st) Expanding and rearranging
Must rearrange to isolate \(\omega\) terms on one side and other terms on other side
M1: (2nd) Factorising and dividing by two term complex number
NB: This question required detailed reasoning
M1: (3rd) Multiplying top and bottom by conjugate of bottom
Alternative method
| Scheme | Marks |
|---|---|
| \(\omega = a + b\mathrm{i} \Rightarrow 3a + 3b\mathrm{i} - 2 = 5\mathrm{i} + 2a\mathrm{i} - 2b\) | M1 |
| \(3a - 2 = -2b\) and \(3b = 5 + 2a\) | M1 |
| \(9a - 6 + 10 + 4a = 0 \Rightarrow a = -\dfrac{4}{13}\) | M1 |
| \(\Rightarrow b = \dfrac{19}{13} \Rightarrow \omega = -\dfrac{4}{13} + \dfrac{19}{13}\mathrm{i}\) | A1 |
M1: (1st) Assigning real and imaginary parts, to \(\omega\) expanding and rearranging
M1: (2nd) Comparing real and imaginary parts
M1: (3rd) Using valid algebra to eliminate one unknown and finding the other