| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Division plus other arithmetic operations |
| Difficulty | Easy -1.2 This is a routine FP1 complex numbers question testing basic operations: addition/scalar multiplication, multiplication with conjugate, and finding a reciprocal. All three parts are standard textbook exercises requiring only direct application of algebraic rules with no problem-solving or insight needed. The calculations are straightforward with minimal steps. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| \(22 - 2i\) | B1B1 | Correct real and imaginary parts |
| Answer | Marks | Guidance |
|---|---|---|
| \(z^* = 2 - 3i\) | B1 | Correct conjugate seen or implied |
| \(\frac{5 - 14i}{}\) | B1B1 | Correct real and imaginary parts |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{4}{17} + \frac{1}{17}i\) | M1 | Attempt to use \(w^*\) |
| A1 | Obtain correct answer in any form |
### (i)
$22 - 2i$ | B1B1 | Correct real and imaginary parts
**Subtotal: 2 marks**
### (ii)
$z^* = 2 - 3i$ | B1 | Correct conjugate seen or implied
$\frac{5 - 14i}{}$ | B1B1 | Correct real and imaginary parts
**Subtotal: 3 marks**
### (iii)
$\frac{4}{17} + \frac{1}{17}i$ | M1 | Attempt to use $w^*$
| A1 | Obtain correct answer in any form
**Subtotal: 2 marks**
**Total: 7 marks**The complex numbers $2 + 3\text{i}$ and $4 - \text{i}$ are denoted by $z$ and $w$ respectively. Express each of the following in the form $x + \text{i}y$, showing clearly how you obtain your answers.
\begin{enumerate}[label=(\roman*)]
\item $z + 5w$, [2]
\item $z^*w$, where $z^*$ is the complex conjugate of $z$, [3]
\item $\frac{1}{w}$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2005 Q3 [7]}}