| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths FP1 question testing basic complex number operations: squaring a complex number and division. Both parts require only direct application of standard algebraic techniques (expanding brackets and multiplying by conjugate) with no problem-solving insight needed. While FP1 content, these are routine computational exercises that are easier than average A-level questions. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z^2 = (5-3i)(5-3i)\) | ||
| \(= 25 - 15i - 15i + 9i^2\) | M1 | Attempt to multiply brackets giving four terms (or four terms implied). \(zw\) is M0 |
| \(= 16 - 30i\) | A1 | Answer only 2/2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{z}{w} = \frac{(5-3i)}{(2+2i)} \times \frac{(2-2i)}{(2-2i)}\) | M1 | Multiplies \(\frac{z}{w}\) by \(\frac{(2-2i)}{(2-2i)}\) |
| \(= \frac{10-10i-6i-6}{4+4}\) | M1 | Simplifies realising real denominator needed; applies \(i^2=-1\) on numerator and denominator |
| \(= \frac{4-16i}{8}\) | ||
| \(= \frac{1}{2} - 2i\) | A1 | \(a=\frac{1}{2}\) and \(b=-2\) or equivalent. Answer as single fraction A0 |
## Question 1:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z^2 = (5-3i)(5-3i)$ | | |
| $= 25 - 15i - 15i + 9i^2$ | M1 | Attempt to multiply brackets giving four terms (or four terms implied). $zw$ is M0 |
| $= 16 - 30i$ | A1 | Answer only 2/2 |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{z}{w} = \frac{(5-3i)}{(2+2i)} \times \frac{(2-2i)}{(2-2i)}$ | M1 | Multiplies $\frac{z}{w}$ by $\frac{(2-2i)}{(2-2i)}$ |
| $= \frac{10-10i-6i-6}{4+4}$ | M1 | Simplifies realising real denominator needed; applies $i^2=-1$ on numerator and denominator |
| $= \frac{4-16i}{8}$ | | |
| $= \frac{1}{2} - 2i$ | A1 | $a=\frac{1}{2}$ and $b=-2$ or equivalent. Answer as single fraction A0 |
---1.
$$z = 5 - 3 \mathrm { i } , \quad w = 2 + 2 \mathrm { i }$$
Express in the form $a + b \mathrm { i }$, where $a$ and $b$ are real constants,
\begin{enumerate}[label=(\alph*)]
\item $z ^ { 2 }$,
\item $\frac { z } { w }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2011 Q1 [5]}}