| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2018 |
| Session | December |
| Marks | 4 |
| Topic | Complex numbers 2 |
| Type | Express roots in trigonometric form |
| Difficulty | Standard +0.3 This is a straightforward application of De Moivre's theorem requiring students to convert to polar form (r=2, θ=π/6), then find when nθ is a multiple of 2π (giving n=12) and calculate 2^12=4096. While it requires multiple steps, each is routine for Further Maths students and the question structure clearly guides the solution path. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\arg(z) = \frac{\pi}{6}\); \(\Rightarrow \arg(z^n) = \frac{n\pi}{6} = 2\pi\); \(\Rightarrow n = 12\) | B1, M1, A1 | Equating arg to \(2\pi\) |
| (b) \(4096\) or \(2^{12}\) | B1 |
**(a)** $\arg(z) = \frac{\pi}{6}$; $\Rightarrow \arg(z^n) = \frac{n\pi}{6} = 2\pi$; $\Rightarrow n = 12$ | B1, M1, A1 | Equating arg to $2\pi$
**(b)** $4096$ or $2^{12}$ | B1 |
---4 In this question you must show detailed reasoning.\\
You are given that $z = \sqrt { 3 } + \mathrm { i }$.\\
$n$ is the smallest positive whole number such that $z ^ { n }$ is a positive whole number.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $n$.
\item Find the value of $z ^ { n }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q4 [4]}}