| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing routine complex number operations (linear combination, multiplication, division, and conversion to modulus-argument form). While it requires multiple techniques and careful arithmetic, all parts are standard textbook exercises with no problem-solving insight required. The Further Maths context raises it slightly above typical A-level questions, but the mechanical nature keeps it below average difficulty. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | (i) |
| Answer | Marks |
|---|---|
| 1 2 | B1 |
| [1] | 1.1 |
| (ii) | DR |
| Answer | Marks |
|---|---|
| = 34 – 2i | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | Attempted expansion with i2 = –1 |
| Answer | Marks |
|---|---|
| expanded terms | – 28(–1) can be simply +28 |
| (iii) | DR |
| Answer | Marks |
|---|---|
| 4+16 20 10 10 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | Multiplying top and bottom by (real |
| Answer | Marks |
|---|---|
| expansion | −11−13i 11+13i |
| Answer | Marks |
|---|---|
| (b) | DR |
| Answer | Marks |
|---|---|
| | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 2.5 | Explicit working must be seen |
| Answer | Marks |
|---|---|
| instead of –1.17. | Other trig calculations could be |
Question 2:
2 | (a) | (i) | DR
3z + 4z = 3(3 – 7i) + 4(2 + 4i) = 17 – 5i
1 2 | B1
[1] | 1.1
(ii) | DR
z z = (3 – 7i)(2 + 4i) = 6 + 12i – 14i – 28(–1)
1 2
= 34 – 2i | M1
A1
[2] | 1.1
1.1 | Attempted expansion with i2 = –1
used and at least 3 correctly
expanded terms | – 28(–1) can be simply +28
(iii) | DR
z 3−7i 3−7i 2−4i
1 = = ×
z 2+4i 2+4i 2−4i
2
6−12i−14i−28 −22−26i 11 13
= = =− − i
4+16 20 10 10 | M1
A1
[2] | 1.1
1.1 | Multiplying top and bottom by (real
multiple of) conjugate of bottom
Must see some evidence of
expansion | −11−13i 11+13i
Allow or −
10 10
(b) | DR
−7
32 +(−7)2 or tan−1
3
z = 58 or awrt 7.62 or argz 1 = awrt –1.17
1
or 5.12 rads
z = 58cis(−1.17) or z = 58e−1.17i or
1 1
z = 58(cos(−1.17)+isin(−1.17)) or
1
58,−1.17
| M1
A1
A1
[3] | 1.1
1.1
2.5 | Explicit working must be seen
Must be in correct form with √58
exact and could be awrt 5.12
instead of –1.17. | Other trig calculations could be
sufficient for M1 provided that
these are being used to find the
argument.
Do not condone degrees
Condone round brackets2 In this question you must show detailed reasoning.
The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by $z _ { 1 } = 3 - 7 \mathrm { i }$ and $z _ { 2 } = 2 + 4 \mathrm { i }$.
\begin{enumerate}[label=(\alph*)]
\item Express each of the following as exact numbers in the form $a + b \mathrm { i }$.
\begin{enumerate}[label=(\roman*)]
\item $3 z _ { 1 } + 4 z _ { 2 }$
\item $z _ { 1 } z _ { 2 }$
\item $\frac { Z _ { 1 } } { Z _ { 2 } }$
\end{enumerate}\item Write $z _ { 1 }$ in modulus-argument form giving the modulus in exact form and the argument correct to $\mathbf { 3 }$ significant figures.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q2 [8]}}