| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with trigonometric functions |
| Difficulty | Challenging +1.2 This is a Further Maths question on De Moivre's theorem requiring multiple steps: expressing powers in trigonometric form, expanding a product, and matching coefficients. While it requires careful algebraic manipulation and understanding of complex numbers, it follows a standard template for this topic with clear signposting through parts (a) and (b). |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | (a) | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | (b) | = 2isin 6θ − 6isin 2θ |
| Answer | Marks |
|---|---|
| ⇒ | B1 |
| Answer | Marks |
|---|---|
| [6] | 2.1 |
| Answer | Marks |
|---|---|
| 2.2a | binomial expansions oe |
| Answer | Marks |
|---|---|
| expression | or etc |
Question 12:
12 | (a) | B1
B1
[2] | 1.1
1.1
12 | (b) | = 2isin 6θ − 6isin 2θ
⇒ | B1
M1
M1
A1
M1
A1
[6] | 2.1
2.1
1.1
1.1
2.1
2.2a | binomial expansions oe
expanding the whole
expression | or etc
award second M1 if changes
into trig and makes some
attempt at using the addition
formulae12
\begin{enumerate}[label=(\alph*)]
\item Given that $z = \cos \theta + \mathrm { i } \sin \theta$, express $z ^ { n } + \frac { 1 } { z ^ { n } }$ and $z ^ { n } - \frac { 1 } { z ^ { n } }$ in simplified trigonometric form.
\item By considering $\left( z + \frac { 1 } { z } \right) ^ { 3 } \left( z - \frac { 1 } { z } \right) ^ { 3 }$, find constants $A$ and $B$ such that $\sin ^ { 3 } \theta \cos ^ { 3 } \theta = A \sin 6 \theta + B \sin 2 \theta$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2020 Q12 [8]}}