| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Sum geometric series with complex terms |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring: (a) recognizing C+jS as a geometric series with complex terms, applying the sum formula, and separating real/imaginary parts; (b) working with regular hexagons in the Argand diagram, finding vertices using rotations, squaring complex numbers, and calculating area. While the techniques are standard for FP2, the question requires fluency with multiple concepts (geometric series, complex exponentials, modulus-argument form, geometric transformations) and careful algebraic manipulation across several steps. It's moderately challenging for Further Maths but not exceptionally difficult. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| and \(C = \frac{a\cos\theta - a^2}{1-2a\cos\theta + a^2}\) | M1, M1, A1, M1*, M1, A1(ag), A1 | Forming \(C + jS\) as a series of powers |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sqrt{3} - j\) | B1, B1, B1, B1, B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Area \(= \frac{1}{2} \times 4\sqrt{3} \times 6 = 12\sqrt{3}\) | M1, A2, B1 | Attempt to square at least one of their vertices in (i) |
### 2(a)
$C + jS = ae^{i\theta} + a^2e^{2j\theta} + ...$
This is a geometric series with $r = ae^{i\theta}$
Sum to infinity $= \frac{ae^{i\theta}}{1-ae^{i\theta}}$
$\frac{ae^{i\theta}}{1-ae^{i\theta}} \times \frac{1-ae^{-i\theta}}{1-ae^{-i\theta}} = \frac{ae^{i\theta} - a^2}{1-ae^{i\theta}-ae^{-i\theta}+a^2}$
$= \frac{a\cos\theta + aj\sin\theta - a^2}{1-2a\cos\theta + a^2}$
$= \frac{a\cos\theta - a^2}{1-2a\cos\theta + a^2} + \frac{aj\sin\theta}{1-2a\cos\theta + a^2}$
$\Rightarrow S = \frac{a\sin\theta}{1-2a\cos\theta + a^2}$
and $C = \frac{a\cos\theta - a^2}{1-2a\cos\theta + a^2}$ | M1, M1, A1, M1*, M1, A1(ag), A1 | Forming $C + jS$ as a series of powers | Identifying G.P. and attempting sum. Dependent on first M1 | Multiplying numerator and denominator by $1-ae^{-i\theta}$ o.e. | Strictly this, or trig equivalent; Multiplying out denominator. Dependent on M1*; Use of FOIL with powers combined correctly (allow one slip); Introducing trig functions. Dependent on M1 | If trig used throughout award last M1 for using $\cos^2\theta + \sin^2\theta = 1$; Answer given. www which leads to S, e.g. condone sign error in num. | ...a^2(cos 2 sin2θ) insufficient. Powers must be correct; Allow M1 for sum to n terms
### 2(b)(i)
$\sqrt{3} + j = 2e^{i\frac{\pi}{6}}$; need to rotate by $\frac{\pi}{3}$ so vertices are
$2j$
$-\sqrt{3} + j$
$-\sqrt{3} - j$
$-2j$
$\sqrt{3} - j$ | B1, B1, B1, B1, B1 | | NB answer space continued (BP) | If vertices not given in form x + jy: B1 for $2e^{i\frac{\pi}{3}}$; B1 for $2e^{i\frac{\pi}{2}}$ and $2e^{i\frac{3\pi}{2}}$; B1 for $2e^{i\frac{5\pi}{6}}$ and $2e^{i\frac{11\pi}{6}}$ i.e. maximum of 3/5. If B0 scored give SC B2 for five vertices in form x + jy obtained by repeatedly rotating their P by $\frac{\pi}{3}$
### 2(b)(ii)
Vertices are $4e^{i\frac{\pi}{3}} = 2 + 2\sqrt{3}j$
$4e^{i\frac{\pi}{3}} = -4$
and $4e^{i\frac{5\pi}{3}} = 2 - 2\sqrt{3}j$
Area $= \frac{1}{2} \times 4\sqrt{3} \times 6 = 12\sqrt{3}$ | M1, A2, B1 | Attempt to square at least one of their vertices in (i) | Three correct in form x + jy (and simplified) and no more | Give A1 for any two of these, or all three and no extras in polar form; awrt 20.8; Dependent on A2 above
---2
\begin{enumerate}[label=(\alph*)]
\item The infinite series $C$ and $S$ are defined as follows.
$$\begin{gathered}
C = a \cos \theta + a ^ { 2 } \cos 2 \theta + a ^ { 3 } \cos 3 \theta + \ldots \\
S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots
\end{gathered}$$
where $a$ is a real number and $| a | < 1$.\\
By considering $C + \mathrm { j } S$, show that
$$S = \frac { a \sin \theta } { 1 - 2 a \cos \theta + a ^ { 2 } }$$
Find a corresponding expression for $C$.
\item P is one vertex of a regular hexagon in an Argand diagram. The centre of the hexagon is at the origin. P corresponds to the complex number $\sqrt { 3 } + \mathrm { j }$.
\begin{enumerate}[label=(\roman*)]
\item Find, in the form $x + \mathrm { j } y$, the complex numbers corresponding to the other vertices of the hexagon.
\item The six complex numbers corresponding to the vertices of the hexagon are squared to form the vertices of a new figure. Find, in the form $x + \mathrm { j } y$, the vertices of the new figure. Find the area of the new figure.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2014 Q2 [17]}}