| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Session | Specimen |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | De Moivre to derive trigonometric identities |
| Difficulty | Challenging +1.2 This is a standard Further Maths question on De Moivre's theorem and complex exponentials. Parts (i) are bookwork proofs requiring straightforward manipulation. Parts (ii) and (iii) involve binomial expansion and algebraic manipulation to find coefficients, which is methodical but lengthy. Part (iv) requires substituting a specific angle and simplifying, which is computational rather than conceptually demanding. While this covers multiple techniques and requires careful algebra, it follows well-established patterns for this topic with no novel insights needed. |
| 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 |
|---|---|---|
| 14 | (i) | (A) |
| (cid:32)ein(cid:84)(cid:32)cosn(cid:84)(cid:14)isinn(cid:84) AG | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 2.1 | n |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (i) | (B) |
| Hence obtain given result | M1 |
| Answer | Marks |
|---|---|
| [2] | m |
| Answer | Marks |
|---|---|
| i2.1 | Answer given so working |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (ii) | e |
| Answer | Marks |
|---|---|
| (cid:32) 32c6 – 48c4 (cid:14) 18c2 – 1 | c |
| Answer | Marks |
|---|---|
| [6] | 3.1a |
| Answer | Marks |
|---|---|
| 2.2a 1.1 | A1 any two terms correct, A2 |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (iii) | 6 6 |
| Answer | Marks |
|---|---|
| 32 32 32 16 | M1 |
| Answer | Marks |
|---|---|
| [5] | 3.1a |
| Answer | Marks |
|---|---|
| 2.2a 1.1 | set up expansion |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (iv) | Use result in part (iii): |
| Answer | Marks |
|---|---|
| (cid:169) (cid:185) | M1 |
| Answer | Marks |
|---|---|
| [3] | m |
| Answer | Marks |
|---|---|
| 2.1 | Putting correct trig values |
Question 14:
14 | (i) | (A) | (cid:11)cos(cid:84)(cid:14) i sin(cid:84)(cid:12)n (cid:32) (cid:11) ei(cid:84)(cid:12)n
(cid:32)ein(cid:84)(cid:32)cosn(cid:84)(cid:14)isinn(cid:84) AG | M1
A1
[2] | 1.1
2.1 | n
e
Answer given so working
must be convincing
14 | (i) | (B) | e(cid:16)i(cid:84) (cid:32) cos(cid:84)– isin(cid:84) AG
Hence obtain given result | M1
A1
[2] | m
1.1
i2.1 | Answer given so working
must be convincing
14 | (ii) | e
(cid:11) is(cid:12)6(cid:12)
cos6(cid:84)(cid:32) Re (cid:11)c (cid:14)
p
(cid:32) c6 –15 c4s2 (cid:14)15 c2s4 –s6
(cid:32) c6 –15 c4(cid:11) 1–c2(cid:12) (cid:14)15 c2(cid:11) 1S–c2(cid:12)2 – (cid:11) 1–c2(cid:12)3
(cid:32) 32c6 – 48c4 (cid:14) 18c2 – 1 | c
M1
A1
M1A1
A1A1
[6] | 3.1a
2.1
2.11.1
2.2a 1.1 | A1 any two terms correct, A2
all four
a, b, c, d implicit or explicit
14 | (iii) | 6 6
(cid:167)1(cid:183) (cid:167) 1(cid:183)
cos6(cid:84)(cid:32)(cid:168) (cid:184) (cid:168)z(cid:14) (cid:184) where z(cid:32)ei(cid:84)
(cid:169)2(cid:185) (cid:169) z(cid:185)
1 (cid:167) 1 (cid:167) 1 (cid:183) (cid:167) 1 (cid:183) (cid:183)
(cid:32) (cid:168)z6 (cid:14) (cid:14)6 z4 (cid:14) (cid:14)15 z2 (cid:14) (cid:14)20(cid:184)
(cid:168) (cid:184) (cid:168) (cid:184)
64(cid:169) z6 (cid:169) z4 (cid:185) (cid:169) z2 (cid:185) (cid:185)
1 6 15 5
(cid:32) cos6(cid:84)(cid:14) cos4(cid:84)(cid:14) cos2(cid:84)(cid:14)
32 32 32 16 | M1
A1
M1
A2
[5] | 3.1a
2.1
1.1
2.2a 1.1 | set up expansion
result of expansion
collecting terms
n
A1 for any two coefficients
ecorrect,
A2 all four P, Q, R, S
implicit or explicit
14 | (iv) | Use result in part (iii):
(cid:83) 1 (cid:83) 6 (cid:83) 15 (cid:83) 5
cos6 (cid:32) cos (cid:14) cos (cid:14) cos (cid:14)
12 32 2 32 3 32 6 16
e
1 6 1 15 3 5
(cid:32) (cid:117)0(cid:14) (cid:117) (cid:14) (cid:14) p
32 32 2 32 2 16
S
1
(cid:83) (cid:167)26(cid:14)15 3(cid:183)6
cos (cid:32)(cid:168) (cid:184) AG
(cid:168) (cid:184)
12 64
(cid:169) (cid:185) | M1
c
A1
A1
[3] | m
3.1a
i
1.1
2.1 | Putting correct trig values
into their (iii)\begin{enumerate}[label=(\roman*)]
\item Starting with the result
$$e^{i\theta} = \cos \theta + i \sin \theta,$$
show that
\begin{enumerate}[label=(\Alph*)]
\item $(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta$ [2]
\item $\cos \theta = \frac{1}{2}(e^{i\theta} + e^{-i\theta})$. [2]
\end{enumerate}
\item Using the result in part (i) (A), obtain the values of the constants $a$, $b$, $c$ and $d$ in the identity
$$\cos 6\theta = a \cos^6 \theta + b \cos^4 \theta + c \cos^2 \theta + d.$$ [6]
\item Using the result in part (i) (B), obtain the values of the constants $P$, $Q$, $R$ and $S$ in the identity
$$\cos^6 \theta = P \cos 6\theta + Q \cos 4\theta + R \cos 2\theta + S.$$ [5]
\item Show that $\cos \frac{\pi}{12} = \left(\frac{26 + 15\sqrt{3}}{64}\right)^{\frac{1}{4}}$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core Q14 [18]}}