| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Complex conjugate properties and proofs |
| Difficulty | Standard +0.8 This is a two-part proof question requiring students to work with De Moivre's theorem and then manipulate the result algebraically. Part (a) is a standard result that follows directly from |z|=1 and Euler's formula, but part (b) requires non-trivial algebraic manipulation using binomial expansion of (z + 1/z)^4 and substitution. The multi-step reasoning and algebraic dexterity needed place this above average difficulty, though it's a recognizable type for Core Pure 2 students. |
| Spec | 1.05l Double angle formulae: and compound angle formulae4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(z^n + z^{-n} = \cos n\theta + \text{i}\sin n\theta + \cos n\theta - \text{i}\sin n\theta\) | M1 | Identifies correct form for \(z^n\) and \(z^{-n}\) and adds |
| \(= 2\cos n\theta\) | A1* | Achieves printed answer with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \((z+z^{-1})^4 = 16\cos^4\theta\) | B1 | Uses correct index with result from part (a) |
| \((z+z^{-1})^4 = z^4+4z^2+6+4z^{-2}+z^{-4}\) | M1 | Realises need to find expansion of \((z+z^{-1})^4\) |
| \(= z^4+z^{-4}+4(z^2+z^{-2})+6\) | A1 | Terms correctly combined |
| \(= 2\cos4\theta + 4(2\cos2\theta)+6\) | M1 | Links expansion with result in part (a) |
| \(\cos^4\theta = \frac{1}{8}(\cos4\theta+4\cos2\theta+3)\) | A1* | Achieves printed answer with no errors |
## Question 4(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $z^n + z^{-n} = \cos n\theta + \text{i}\sin n\theta + \cos n\theta - \text{i}\sin n\theta$ | M1 | Identifies correct form for $z^n$ and $z^{-n}$ and adds |
| $= 2\cos n\theta$ | A1* | Achieves printed answer with no errors |
## Question 4(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $(z+z^{-1})^4 = 16\cos^4\theta$ | B1 | Uses correct index with result from part (a) |
| $(z+z^{-1})^4 = z^4+4z^2+6+4z^{-2}+z^{-4}$ | M1 | Realises need to find expansion of $(z+z^{-1})^4$ |
| $= z^4+z^{-4}+4(z^2+z^{-2})+6$ | A1 | Terms correctly combined |
| $= 2\cos4\theta + 4(2\cos2\theta)+6$ | M1 | Links expansion with result in part (a) |
| $\cos^4\theta = \frac{1}{8}(\cos4\theta+4\cos2\theta+3)$ | A1* | Achieves printed answer with no errors |
---\begin{enumerate}
\item A complex number $z$ has modulus 1 and argument $\theta$.\\
(a) Show that
\end{enumerate}
$$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta , \quad n \in \mathbb { Z } ^ { + }$$
(b) Hence, show that
$$\cos ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta + 4 \cos 2 \theta + 3 )$$
\hfill \mbox{\textit{Edexcel CP2 Q4 [7]}}