| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Modulus and argument with operations |
| Difficulty | Moderate -0.5 This is a straightforward FP1 complex numbers question testing basic operations: finding an argument of a simple complex number, multiplication, and division using conjugate method. All three parts are routine textbook exercises requiring standard techniques with no problem-solving insight. While it's Further Maths content, these are foundational FP1 skills, making it slightly easier than an average A-level question overall. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(\arg z_1 = -\arctan(1)\) | M1 | \(-\arctan(1)\) or \(\arctan(1)\) or \(\arctan(-1)\) |
| \(= -\dfrac{\pi}{4}\) | A1 | or \(-45°\) or awrt \(-0.785\) (oe e.g. \(\dfrac{7\pi}{4}\)) |
| Correct answer only 2/2 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(z_1 z_2 = (1-i)(3+4i) = 3 - 3i + 4i - 4i^2\) | M1 | At least 3 correct terms (unsimplified) |
| \(= 7 + i\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(\dfrac{z_2}{z_1} = \dfrac{(3+4i)}{(1-i)} = \dfrac{(3+4i)(1+i)}{(1-i)(1+i)}\) | M1 | Multiply top and bottom by \((1+i)\) |
| \(= \dfrac{(3+4i)(1+i)}{2}\) | A1 | \((1+i)(1-i) = 2\) |
| \(= -\dfrac{1}{2} + \dfrac{7}{2}i\) | A1 | or \(\dfrac{-1+7i}{2}\) |
| Answer | Marks |
|---|---|
| Correct answers only in (b) and (c) scores no marks | Total 7 |
# Question 1:
## Part (a)
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $\arg z_1 = -\arctan(1)$ | M1 | $-\arctan(1)$ or $\arctan(1)$ or $\arctan(-1)$ |
| $= -\dfrac{\pi}{4}$ | A1 | or $-45°$ or awrt $-0.785$ (oe e.g. $\dfrac{7\pi}{4}$) |
**Correct answer only 2/2** | **(2)**
---
## Part (b)
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $z_1 z_2 = (1-i)(3+4i) = 3 - 3i + 4i - 4i^2$ | M1 | At least 3 correct terms (unsimplified) |
| $= 7 + i$ | A1 | cao |
**(2)**
---
## Part (c)
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $\dfrac{z_2}{z_1} = \dfrac{(3+4i)}{(1-i)} = \dfrac{(3+4i)(1+i)}{(1-i)(1+i)}$ | M1 | Multiply top and bottom by $(1+i)$ |
| $= \dfrac{(3+4i)(1+i)}{2}$ | A1 | $(1+i)(1-i) = 2$ |
| $= -\dfrac{1}{2} + \dfrac{7}{2}i$ | A1 | or $\dfrac{-1+7i}{2}$ |
**Special case:** $\dfrac{z_1}{z_2} = \dfrac{(1-i)}{(3+4i)} = \dfrac{(1-i)(3-4i)}{(3+4i)(3-4i)}$ — Allow M1A0A0
**(3)**
**Correct answers only in (b) and (c) scores no marks** | **Total 7**\begin{enumerate}
\item Given that $z _ { 1 } = 1 - \mathrm { i }$,\\
(a) find $\arg \left( z _ { 1 } \right)$.
\end{enumerate}
Given also that $z _ { 2 } = 3 + 4 \mathrm { i }$, find, in the form $a + \mathrm { i } b , a , b \in \mathbb { R }$,\\
(b) $z _ { 1 } z _ { 2 }$,\\
(c) $\frac { z _ { 2 } } { z _ { 1 } }$.
In part (b) and part (c) you must show all your working clearly.\\
\hfill \mbox{\textit{Edexcel FP1 2012 Q1 [7]}}