| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Sum geometric series with complex terms |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on geometric series with complex terms. Part (a) tests basic understanding of modulus and argument (straightforward recall). Part (b)(i) is algebraic manipulation with complex exponentials (routine). Part (b)(ii) requires summing a geometric series and equating imaginary parts, which is a standard technique in Further Maths. While it involves multiple steps and careful algebra, it follows a well-established method with clear signposting. Slightly above average difficulty due to the multi-step nature and Further Maths content, but not requiring novel insight. |
| Spec | 1.04j Sum to infinity: convergent geometric series |r|<14.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| 13 | (a) | (i) |
| [1] | 1.1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 13 | (a) | (ii) |
| [1] | 1.1 | |
| 13 | (b) | (i) |
| Answer | Marks |
|---|---|
| = 10β6cosπ | M1 |
| Answer | Marks |
|---|---|
| [2] | 2.1 |
| 2.1 | Expanding correctly to give at least three terms. Condone |
| Answer | Marks | Guidance |
|---|---|---|
| 13 | (b) | (ii) |
| Answer | Marks |
|---|---|
| 3 9 10β6cosπ | M1 |
| Answer | Marks |
|---|---|
| [6] | 2.1 |
| Answer | Marks |
|---|---|
| 2.2a | At least two terms of series in exponential form soi by |
Question 13:
13 | (a) | (i) | 1:3 | B1
[1] | 1.1 | 1
or :1 only
3
13 | (a) | (ii) | π | B1
[1] | 1.1
13 | (b) | (i) | (3βeiπ)(3βeβiπ) = 9β3eiπ β3eβiπ +1
= 10β3(eiπ +eβiπ)
= 10β6cosπ | M1
A1
[2] | 2.1
2.1 | Expanding correctly to give at least three terms. Condone
π0 = 1.
www. Condone only incorrect values quoted for π and π.
Intermediate step not required here.
13 | (b) | (ii) | [π§+π§2+...=] 1 eiπ+ 1 e2iπ+β¦
3 9
13 e i ο± eiπ
= [= ]
1 13 e i ο± β 3βeiπ
eiπ(3βeβiπ)
=
(3βeiπ)(3βeβiπ)
3eiπ β1
=
10β6cosπ
3(cosπ+isinπ)β1
=
10β6cosπ
1 1 3sinπ
[ sinπ+ sin2π...=]
3 9 10β6cosπ | M1
A1
M1*
A1
M1dep
A1
[6] | 2.1
2.1
3.1a
2.1
2.1
2.2a | At least two terms of series in exponential form soi by
π§
correct GP formula or seen. Condone modulus-
1βπ§
argument form.
Using sum to infinity formula correctly
Multiplying their numerator and denominator by a
multiple of 3βeβiπ. Must be a clear attempt at a sum to
infinity.
oe
eiΞΈ = cosπ+isinπ used when denominator has been
simplified to a real expression. No errors allowed.
AG www13 The complex number $z$ is defined as $z = \frac { 1 } { 3 } \mathrm { e } ^ { \mathrm { i } \theta }$ where $0 < \theta < \frac { 1 } { 2 } \pi$.\\
On an Argand diagram, the point O represents the complex number 0 , and the points $P _ { 1 } , P _ { 2 } , P _ { 3 } , \ldots$ represent the complex numbers $z , z ^ { 2 } , z ^ { 3 } , \ldots$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Write down each of the following.
\begin{enumerate}[label=(\roman*)]
\item The ratio of the lengths $\mathrm { OP } _ { n + 1 } : \mathrm { OP } _ { n }$
\item The angle $\mathrm { P } _ { n + 1 } \mathrm { OP } _ { n }$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that $\left( 3 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = \mathrm { a } + \mathrm { b } \cos \theta$, where $a$ and $b$ are integers to be determined.
\item By considering the sum to infinity of the series $z + z ^ { 2 } + z ^ { 3 } + \ldots$, show that
$$\frac { 1 } { 3 } \sin \theta + \frac { 1 } { 9 } \sin 2 \theta + \frac { 1 } { 27 } \sin 3 \theta + \ldots = \frac { 3 \sin \theta } { 10 - 6 \cos \theta } .$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2024 Q13 [10]}}