A2 June 2025 Paper 1 Q8
8 In this question you must show detailed reasoning.
In this question, arguments of complex numbers are in the interval \([0, 2\pi)\).
| Scheme | Marks | AO |
|---|---|---|
| DR \(|z| = \sqrt{5^2 + 5^2} \quad \left(= 5\sqrt{2}\right)\) or \(\arg(z) = \pi - \tan^{-1}\left(\dfrac{5}{5}\right) \quad \left(= \dfrac{3}{4}\pi\right)\) or \(5\sqrt{2}\mathrm{e}^{\mathrm{i}\frac{3}{4}\pi}\) | B1* | 1.1 |
| \(5\sqrt{2}\mathrm{e}^{\mathrm{i}\frac{3}{4}\pi}\) from complete working | B1dep* | 1.1 |
| [2] |
Notes
B1*: For completely correct working for either \(|z|\) or \(\arg(z)\) or for the correct answer of \(5\sqrt{2}\mathrm{e}^{\mathrm{i}\frac{3}{4}\pi}\) (accept \(\sqrt{50}\) and any exact equivalent for \(\frac{3}{4}\pi\))
For \(|z|\) must see either \(\sqrt{25 + 25}\) or \(\sqrt{(-5)^2 + 5^2}\) or \(\sqrt{5^2 + 5^2}\) but not just \(\sqrt{50}\) or \(5\sqrt{2}\)
For \(\arg(z)\) must see either \(\pi - \tan^{-1}\left(\frac{5}{5}\right)\) or \(\frac{1}{2}\pi + \tan^{-1}\left(\frac{5}{5}\right)\) or \(\pi + \tan^{-1}\left(-\frac{5}{5}\right)\) or equivalent with \(\tan\theta = \frac{5}{5}\)
B1dep*: www must have shown complete working for both \(|z|\) and \(\arg(z)\) as DR – accept \(\sqrt{50}\) and any exact equivalent for \(\frac{3}{4}\pi\)
| Scheme | Marks | AO |
|---|---|---|
| \(|w| = \dfrac{36}{27} \quad \left(= \dfrac{4}{3}\right)\) | B1 | 3.1a |
| \(\cos\left(\frac{1}{7}\pi\right) - \mathrm{i}\sin\left(\frac{1}{7}\pi\right) = \cos\left(-\frac{1}{7}\pi\right) + \mathrm{i}\sin\left(-\frac{1}{7}\pi\right)\) or \(\mathrm{e}^{-\mathrm{i}\frac{1}{7}\pi}\) | M1 | 2.1 |
| \(\sin\left(\frac{3}{7}\pi\right) + \mathrm{i}\cos\left(\frac{3}{7}\pi\right) = \mathrm{i}\left(\cos\left(\frac{3}{7}\pi\right) - \mathrm{i}\sin\left(\frac{3}{7}\pi\right)\right)\) or \(\mathrm{i}\mathrm{e}^{-\mathrm{i}\frac{3}{7}\pi}\) or \(\mathrm{e}^{\mathrm{i}\frac{1}{2}\pi}\mathrm{e}^{-\mathrm{i}\frac{3}{7}\pi}\) or \(\dfrac{\mathrm{i}}{-\cos\left(\frac{3}{7}\pi\right) + \mathrm{i}\sin\left(\frac{3}{7}\pi\right)}\) | M1 | 3.1a |
| \(\dfrac{\mathrm{e}^{-\mathrm{i}\frac{1}{7}\pi}}{\mathrm{e}^{\mathrm{i}\frac{1}{2}\pi} \times \mathrm{e}^{-\mathrm{i}\frac{3}{7}\pi}} = \dfrac{\mathrm{e}^{-\mathrm{i}\frac{1}{7}\pi}}{\mathrm{e}^{\mathrm{i}\frac{1}{14}\pi}} = \mathrm{e}^{-\mathrm{i}\frac{3}{14}\pi}\) or \(\dfrac{\mathrm{e}^{\mathrm{i}\frac{13}{7}\pi}}{\mathrm{e}^{\mathrm{i}\frac{1}{2}\pi} \times \mathrm{e}^{-\mathrm{i}\frac{3}{7}\pi}} = \dfrac{\mathrm{e}^{\mathrm{i}\frac{13}{7}\pi}}{\mathrm{e}^{\mathrm{i}\frac{1}{14}\pi}} = \mathrm{e}^{\mathrm{i}\frac{25}{14}\pi}\) | A1 | 1.1 |
| \(w = \dfrac{4}{3}\mathrm{e}^{\mathrm{i}\frac{25}{14}\pi}\) | A1 | 2.2a |
| [5] |
Notes
B1: DR
Correct modulus of \(|w|\) seen anywhere (check final answer)
M1: Correctly re-writes \(\cos\left(\frac{1}{7}\pi\right) - \mathrm{i}\sin\left(\frac{1}{7}\pi\right)\) in either modulus-argument or exponential form, or states that the argument of the numerator is \(-\frac{1}{7}\pi\) or \(\frac{13}{7}\pi\)
M1: Correct first step to re-write \(\sin\left(\frac{3}{7}\pi\right) + \mathrm{i}\cos\left(\frac{3}{7}\pi\right)\) in either modulus-argument (so in terms of c + is) or exponential form e.g. \(\cos\left(\frac{1}{2}\pi - \frac{3}{7}\pi\right) + \mathrm{i}\sin\left(\frac{1}{2}\pi - \frac{3}{7}\pi\right)\) or \(-\mathrm{i}\left(-\cos\left(\frac{3}{7}\pi\right) + \mathrm{i}\sin\left(\frac{3}{7}\pi\right)\right)\)
If argument of \(\sin\left(\frac{3}{7}\pi\right) + \mathrm{i}\cos\left(\frac{3}{7}\pi\right)\) stated as \(\frac{1}{14}\pi\) without one step of correct intermediate working, then M0
A1: Correct simplified \(\arg w\) which follows directly from their working – allow as a minimum \(\dfrac{\mathrm{e}^{-\mathrm{i}\frac{1}{7}\pi}}{\mathrm{e}^{\mathrm{i}\frac{1}{14}\pi}} = \mathrm{e}^{-\mathrm{i}\frac{3}{14}\pi}\) (oe) provided \(\frac{1}{14}\pi\) in denominator derived convincingly (see previous M mark). Note that \(\arg w = -\frac{3}{14}\pi\) stated with no working is A0
A1: Dependent on all previous marks – allow \(\dfrac{36}{27}\mathrm{e}^{\mathrm{i}\frac{25}{14}\pi}\) and any positive multiple of \(\frac{25}{14}\). Common incorrect answers are \(\frac{4}{3}\mathrm{e}^{-\mathrm{i}\frac{4}{7}\pi}\) or \(\frac{4}{3}\mathrm{e}^{\mathrm{i}\frac{10}{7}\pi}\) - these score B1 M1 only
Alternative method
| Scheme | Marks |
|---|---|
| \(|w| = \dfrac{36}{27} \quad \left(= \dfrac{4}{3}\right)\) | B1 |
| \(\dfrac{\cos\left(\frac{1}{7}\pi\right) - \mathrm{i}\sin\left(\frac{1}{7}\pi\right)}{\sin\left(\frac{3}{7}\pi\right) + \mathrm{i}\cos\left(\frac{3}{7}\pi\right)} \times \dfrac{\sin\left(\frac{3}{7}\pi\right) - \mathrm{i}\cos\left(\frac{3}{7}\pi\right)}{\sin\left(\frac{3}{7}\pi\right) - \mathrm{i}\cos\left(\frac{3}{7}\pi\right)}\) | M1 |
| \(= \sin\left(\frac{3}{7}\pi - \frac{1}{7}\pi\right) - \mathrm{i}\cos\left(\frac{3}{7}\pi - \frac{1}{7}\pi\right)\) | M1 |
| \(= \cos\left(\frac{1}{2}\pi - \frac{2}{7}\pi\right) - \mathrm{i}\sin\left(\frac{1}{2}\pi - \frac{2}{7}\pi\right)\) \(= \cos\left(-\frac{3}{14}\pi\right) + \mathrm{i}\sin\left(-\frac{3}{14}\pi\right)\) | A1 |
| \(w = \dfrac{4}{3}\mathrm{e}^{\mathrm{i}\frac{25}{14}\pi}\) | A1 |
B1: Correct modulus of \(|w|\) seen anywhere (check final answer)
M1: Multiplying numerator and denominator by a correct suitable conjugate
For reference:
\(\dfrac{\cos\left(\frac{1}{7}\pi\right)\sin\left(\frac{3}{7}\pi\right) - \sin\left(\frac{1}{7}\pi\right)\cos\left(\frac{3}{7}\pi\right) - \mathrm{i}\left(\sin\left(\frac{1}{7}\pi\right)\sin\left(\frac{3}{7}\pi\right) + \cos\left(\frac{1}{7}\pi\right)\cos\left(\frac{3}{7}\pi\right)\right)}{\sin^2\left(\frac{3}{7}\pi\right) + \cos^2\left(\frac{3}{7}\pi\right)}\)
M1: For correctly writing the numerator as a single term in sine and a single term in cosine – the angle must be seen as two separate terms e.g. \(-\sin\left(\frac{1}{7}\pi - \frac{3}{7}\pi\right) - \mathrm{i}\cos\left(\frac{1}{7}\pi - \frac{3}{7}\pi\right)\)
A1: Correct simplified \(\arg w\) with at least two terms shown e.g. \(\frac{1}{2}\pi - \frac{2}{7}\pi\). Note that \(\arg w = -\frac{3}{14}\pi\) stated or implied with no working is A0
A1: Dependent on all previous marks – allow \(\dfrac{36}{27}\mathrm{e}^{\mathrm{i}\frac{25}{14}\pi}\) and any positive multiple of \(\frac{25}{14}\). A common incorrect answer is \(\frac{4}{3}\mathrm{e}^{-\mathrm{i}\frac{4}{7}\pi}\) or \(\frac{4}{3}\mathrm{e}^{\mathrm{i}\frac{10}{7}\pi}\) - these scores B1 M1 only