| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: purely real or imaginary RHS |
| Difficulty | Moderate -0.3 Finding sixth roots of a real number is a standard Further Maths exercise requiring routine application of De Moivre's theorem. While it involves multiple steps (converting -64 to polar form, dividing argument by 6, finding all 6 roots), this is a textbook procedure with no novel insight required. The 'show detailed reasoning' instruction and Argand diagram are typical for this topic, making it slightly easier than an average A-level question overall due to its mechanical nature. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | DR |
| Answer | Marks |
|---|---|
| z = 2ei/6, 2ei/2, 2e5i/6, 2e–i/6, 2e–i/2, 2e–5i/6 | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | condone −𝜋 for 𝜋 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (b) | 2 |
| i | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | six roots lying on approximate circle centre O |
Question 5:
5 | (a) | DR
Let 𝑧 = 𝑟(cos𝜃+isin𝜃) or 𝑧 = 𝑟e(cid:3036)(cid:3087)
𝑧(cid:2874) = 64(cos𝜋+isin𝜋) or 64e(cid:2919)(cid:3095)
r = 2
z = 2ei/6, 2ei/2, 2e5i/6, 2e–i/6, 2e–i/2, 2e–5i/6 | M1
B1
A1
A1
[4] | 1.1
1.1
2.5
1.1 | condone −𝜋 for 𝜋
soi
1 correct
all correct or 2e7i/6, 2e3i/2, 2e11i/6
5 | (b) | 2
i | M1
A1
A1
[3] | 1.1
1.1
1.1 | six roots lying on approximate circle centre O
forming an approximate regular hexagon
root at 2i or −2i oe indicated. If part (a) contains incorrect
roots do not award final A1.5
\begin{enumerate}[label=(\alph*)]
\item In this question you must show detailed reasoning.\\
Determine the sixth roots of - 64 , expressed in $r \mathrm { e } ^ { \mathrm { i } \theta }$ form.
\item Represent the roots on an Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2023 Q5 [7]}}