| 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 | Complex Numbers Argand & Loci |
| Type | Roots of unity applications |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring understanding of roots of unity, geometric transformations in the complex plane, and connecting midpoints to multiplication by a complex number. Part (a) is routine, but parts (b) and (c) require insight that midpoints correspond to multiplication by a specific complex number and then finding what equation these transformed roots satisfy. The conceptual leap in part (b) and the multi-step reasoning elevate this above average difficulty, though it remains a structured question with clear parts. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (a) | DR |
| Answer | Marks |
|---|---|
| | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 2.5 | |
| 2.5 | modulus 2 | |
| 11 | (b) | DR |
| Answer | Marks |
|---|---|
| So | B1 |
| Answer | Marks |
|---|---|
| [4] | 3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | (c) | DR |
| so | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | taking the 6th power of one of |
Question 11:
11 | (a) | DR
2, 2ei π /3, 2e2i π /3, −2, 2e4i π /3, 2e5i π /3
| M1
A1
[2] | 2.5
2.5 | modulus 2
11 | (b) | DR
modulus of G = √3
modulus of w = √3
2
argument = π/6
So | B1
B1
B1
B1
[4] | 3.1a
1.1
1.1
1.1
11 | (c) | DR
so | M1
A1
[2] | 1.1
1.1 | taking the 6th power of one of
the midpoints
PPMMTT
Y420/01 Mark SchemeNovember 2020
12 (a)
B1 1.1
B1 1.1
[2]
12 (b)
B1 2.1
M1 2.1 binomial expansions oe or etc
M1 1.1 expanding the whole
award second M1 if changes
expression
into trig and makes some
A1 1.1
attempt at using the addition
formulae
= 2isin 6θ − 6isin 2θ M1 2.1
A1 2.2a
⇒
[6]
13 (a)
M1 2.1 substituting
A1 2.2a
= sinh 2x
[2]
13 (b) f(x) = sinh2x SC B1 if other methods used
f′(x) = 2sinh x cosh x [= sinh 2x] B1 2.1
⇒f″(x) = 2cosh 2x* B1 2.2a NB AG
[2]
13 (c) f‴(x) = 4sinh 2x, f(5)( x) = 16 sinh 2x, … M1 2.1
so all odd derivatives are multiples of sinh2x
so f‴(0) = f(5)( 0) = f(7)( 0) = … = 0 E1 2.4
[2]
1211 In this question you must show detailed reasoning.
In Fig. 11, the points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$ and F represent the complex sixth roots of 64 on an Argand diagram. The midpoints of $\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DE } , \mathrm { EF }$ and FA are $\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }$ and L respectively.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[>=latex, scale=1.5]
% Define the radius of the hexagon
\def\r{2}
\def\h{1.732} % r * sin(60)
% Draw the axes
\draw[->] (-2.5, 0) -- (2.5, 0) node[right] {Re};
\draw[->] (0, -2.2) -- (0, 2.2) node[above] {Im};
% Define vertices
\coordinate (A) at (\r, 0);
\coordinate (B) at (\r/2, \h);
\coordinate (C) at (-\r/2, \h);
\coordinate (D) at (-\r, 0);
\coordinate (E) at (-\r/2, -\h);
\coordinate (F) at (\r/2, -\h);
% Draw the hexagon
\draw[thick] (A) -- (B) -- (C) -- (D) -- (E) -- (F) -- cycle;
% Add labels for the vertices
\node[below right] at (A) {A};
\node[above right] at (B) {B};
\node[above left] at (C) {C};
\node[below left] at (D) {D};
\node[below left] at (E) {E};
\node[below right] at (F) {F};
\end{tikzpicture}
\captionsetup{labelformat=empty}
\caption{Fig. 11}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Write down, in exponential ( $r \mathrm { e } ^ { \mathrm { i } \theta }$ ) form, the complex numbers represented by the points $\mathrm { A } , \mathrm { B }$, $\mathrm { C } , \mathrm { D } , \mathrm { E }$ and F .
\item When these complex numbers are multiplied by the complex number $w$, the resulting complex numbers are represented by the points G, H, I, J, K and L.
Find $w$ in exponential form.
\item You are given that $\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }$ and L represent roots of the equation $z ^ { 6 } = p$.
Find $p$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2020 Q11 [8]}}