| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex arithmetic operations |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question testing standard complex number techniques: converting to modulus-argument form (routine calculation), multiplying complex numbers (basic algebra), and applying simple geometric conditions. All parts follow textbook procedures with no novel insight required, making it slightly easier than average for A-level Further Maths. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | z_1 | =\sqrt{13}\) |
| \(\arg(z_1)=\arctan(-3/2)\) | M1 | Allow \(\arctan(3/2)\) for M1. M0 if no working i.e. just an angle of \(-0.983\) or \(-56.3°\). Need to see some evidence of use of arctan or \(\tan^{-1}\). \(\tan\theta=\frac{3}{2}\) and \(\theta=0.983\) is enough. |
| \(\sqrt{13}\,\text{cis}(-0.983)\) or \(\sqrt{13}\,\text{cis}(5.30)\) | A1 | Any equivalent mod-arg form (including exponential). Condone \(-56.3°\) or \(304°\) if degree symbol shown. Must not have \(\pi\) attached to \(-0.983\). A1 cannot be awarded if M0 awarded. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2a+8\text{i}-3a\text{i}-12\text{i}^2\) | M1 | Expanding brackets; allow one error |
| \(2a+12+(8-3a)\text{i}\) | A1 | i terms must be collected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2a+12=8-3a\) | M1 | Equating real and imaginary parts. No i. |
| \(a=-\frac{4}{5}\) | A1ft | Or \(-0.8\). Follow through provided \(a\) appears in both real and imaginary parts. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8-3a=0\) | M1 | Or \(2a+12+(8-3a)\text{i}=2a+12-(8-3a)\text{i}\). Setting imaginary part of (ii) to 0. |
| \(a=\frac{8}{3}\) | A1ft | Or awrt \(2.67\) |
# Question 3:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|z_1|=\sqrt{13}$ | B1 | |
| $\arg(z_1)=\arctan(-3/2)$ | M1 | Allow $\arctan(3/2)$ for M1. M0 if no working i.e. just an angle of $-0.983$ or $-56.3°$. Need to see some evidence of use of arctan or $\tan^{-1}$. $\tan\theta=\frac{3}{2}$ and $\theta=0.983$ is enough. |
| $\sqrt{13}\,\text{cis}(-0.983)$ or $\sqrt{13}\,\text{cis}(5.30)$ | A1 | Any equivalent mod-arg form (including exponential). Condone $-56.3°$ or $304°$ if degree symbol shown. Must not have $\pi$ attached to $-0.983$. A1 cannot be awarded if M0 awarded. |
**[3 marks total]**
---
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2a+8\text{i}-3a\text{i}-12\text{i}^2$ | M1 | Expanding brackets; allow one error |
| $2a+12+(8-3a)\text{i}$ | A1 | i terms must be collected |
**[2 marks total]**
---
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2a+12=8-3a$ | M1 | Equating real and imaginary parts. No i. |
| $a=-\frac{4}{5}$ | A1ft | Or $-0.8$. Follow through provided $a$ appears in both real and imaginary parts. |
**[2 marks total]**
---
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8-3a=0$ | M1 | Or $2a+12+(8-3a)\text{i}=2a+12-(8-3a)\text{i}$. Setting imaginary part of (ii) to 0. |
| $a=\frac{8}{3}$ | A1ft | Or awrt $2.67$ |
**[2 marks total]**
---3 In this question you must show detailed reasoning.\\
The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by $z _ { 1 } = 2 - 3 i$ and $z _ { 2 } = a + 4 i$ where $a$ is a real number.\\
(i) Express $z _ { 1 }$ in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.\\
(ii) Find $z _ { 1 } z _ { 2 }$ in terms of $a$, writing your answer in the form $c + \mathrm { id }$.\\
(iii) The real and imaginary parts of a complex number on an Argand diagram are $x$ and $y$ respectively. Given that the point representing $z _ { 1 } z _ { 2 }$ lies on the line $y = x$, find the value of $a$.\\
(iv) Given instead that $z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }$ find the value of $a$.
\hfill \mbox{\textit{OCR Further Pure Core AS 2018 Q3 [9]}}