| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Express roots in trigonometric form |
| Difficulty | Challenging +1.2 This is a standard Further Maths Core Pure question combining de Moivre's theorem with finding roots. Part (a) follows a well-practiced technique (expand (cos θ + i sin θ)^7, equate imaginary parts, express in terms of sin θ only). Part (b) requires recognizing the connection to sin 7θ = -1 and solving for θ, which is methodical rather than insightful. While requiring multiple steps and Further Maths content, this is a textbook application without novel problem-solving demands. |
| Spec | 4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((\cos\theta + \text{i}\sin\theta)^7 = \cos^7\theta + \binom{7}{1}\cos^6\theta(\text{i}\sin\theta) + \binom{7}{2}\cos^5\theta(\text{i}\sin\theta)^2 + \ldots\) | M1 | Attempts to expand \((\cos\theta + \text{i}\sin\theta)^7\) including recognisable attempt at binomial coefficients |
| \(\text{i}\sin 7\theta = {^7C_1}c^6\text{i}s + {^7C_3}c^4\text{i}^3s^3 + {^7C_5}c^2\text{i}^5s^5 + \text{i}^7s^7\) | M1 | Identifies imaginary terms with \(\sin 7\theta\) |
| \(\sin 7\theta = 7c^6s - 35c^4s^3 + 21c^2s^5 - s^7\) | A1 | Correct expression with coefficients evaluated and i's dealt with correctly |
| \(= 7(1-s^2)^3 s - 35(1-s^2)^2 s^3 + 21(1-s^2)s^5 - s^7\) | M1 | Replaces \(\cos^2\theta\) with \(1 - \sin^2\theta\) and applies expansions of \((1-\sin^2\theta)^2\) and \((1-\sin^2\theta)^3\) |
| \(\sin 7\theta = 7\sin\theta - 56\sin^3\theta + 112\sin^5\theta - 64\sin^7\theta\) | A1* | Reaches printed answer with no errors and expansion of brackets seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1 + \sin 7\theta = 0 \Rightarrow \sin 7\theta = -1\) | M1 | Makes connection with part (a) and realises need to solve \(\sin 7\theta = -1\) |
| \(7\theta = -450, -90, 270, 630, \ldots\) or \(7\theta = -\frac{5\pi}{2}, -\frac{\pi}{2}, \frac{3\pi}{2}, \frac{7\pi}{2}, \ldots\) | A1 | At least one correct value for \(7\theta\) |
| \(\theta = -\frac{450}{7}, -\frac{90}{7}, \frac{270}{7}, \frac{630}{7}, \ldots \Rightarrow \sin\theta = \ldots\) or \(\theta = -\frac{5\pi}{14}, -\frac{\pi}{14}, \frac{3\pi}{14}, \frac{7\pi}{14}, \ldots \Rightarrow \sin\theta = \ldots\) | M1 | Divides by 7 and deduces \(x\) values found by finding at least one value for \(\sin\theta\) |
| \(x = \sin\theta = -0.901, -0.223, 0.623, 1\) | A1 | Awrt 2 correct values for \(x\) |
| A1 | Awrt all 4 \(x\) values correct and no extras |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(\cos\theta + \text{i}\sin\theta)^7 = \cos^7\theta + \binom{7}{1}\cos^6\theta(\text{i}\sin\theta) + \binom{7}{2}\cos^5\theta(\text{i}\sin\theta)^2 + \ldots$ | M1 | Attempts to expand $(\cos\theta + \text{i}\sin\theta)^7$ including recognisable attempt at binomial coefficients |
| $\text{i}\sin 7\theta = {^7C_1}c^6\text{i}s + {^7C_3}c^4\text{i}^3s^3 + {^7C_5}c^2\text{i}^5s^5 + \text{i}^7s^7$ | M1 | Identifies imaginary terms with $\sin 7\theta$ |
| $\sin 7\theta = 7c^6s - 35c^4s^3 + 21c^2s^5 - s^7$ | A1 | Correct expression with coefficients evaluated and i's dealt with correctly |
| $= 7(1-s^2)^3 s - 35(1-s^2)^2 s^3 + 21(1-s^2)s^5 - s^7$ | M1 | Replaces $\cos^2\theta$ with $1 - \sin^2\theta$ and applies expansions of $(1-\sin^2\theta)^2$ and $(1-\sin^2\theta)^3$ |
| $\sin 7\theta = 7\sin\theta - 56\sin^3\theta + 112\sin^5\theta - 64\sin^7\theta$ | A1* | Reaches printed answer with no errors and expansion of brackets seen |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1 + \sin 7\theta = 0 \Rightarrow \sin 7\theta = -1$ | M1 | Makes connection with part (a) and realises need to solve $\sin 7\theta = -1$ |
| $7\theta = -450, -90, 270, 630, \ldots$ or $7\theta = -\frac{5\pi}{2}, -\frac{\pi}{2}, \frac{3\pi}{2}, \frac{7\pi}{2}, \ldots$ | A1 | At least one correct value for $7\theta$ |
| $\theta = -\frac{450}{7}, -\frac{90}{7}, \frac{270}{7}, \frac{630}{7}, \ldots \Rightarrow \sin\theta = \ldots$ or $\theta = -\frac{5\pi}{14}, -\frac{\pi}{14}, \frac{3\pi}{14}, \frac{7\pi}{14}, \ldots \Rightarrow \sin\theta = \ldots$ | M1 | Divides by 7 and deduces $x$ values found by finding at least one value for $\sin\theta$ |
| $x = \sin\theta = -0.901, -0.223, 0.623, 1$ | A1 | Awrt 2 correct values for $x$ |
| | A1 | Awrt all 4 $x$ values correct and no extras |
---\begin{enumerate}
\item (a) Use de Moivre's theorem to prove that
\end{enumerate}
$$\sin 7 \theta = 7 \sin \theta - 56 \sin ^ { 3 } \theta + 112 \sin ^ { 5 } \theta - 64 \sin ^ { 7 } \theta$$
(b) Hence find the distinct roots of the equation
$$1 + 7 x - 56 x ^ { 3 } + 112 x ^ { 5 } - 64 x ^ { 7 } = 0$$
giving your answer to 3 decimal places where appropriate.
\hfill \mbox{\textit{Edexcel CP2 2020 Q4 [10]}}