| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2018 |
| Session | March |
| Marks | 5 |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex arithmetic operations |
| Difficulty | Moderate -0.8 This is a straightforward Further Maths question testing basic complex number operations and properties. Part (i) is routine algebraic expansion, parts (ii)-(iv) test standard recall of modulus/argument relationships (|z²|=|z|² and arg(z²)=2arg(z)). While it's Further Maths content, it requires no problem-solving or insight—just direct application of definitions and well-known properties. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| \(2^2 + 2 \times 2 \times i + i^2 = 4 + 4i - 1 = 3 + 4i\) | M1 | |
| A1 [2] | AG | \(i^2 = -1\) must be shown |
| Answer | Marks | Guidance |
|---|---|---|
| Both \(z\) and \(z^*\) correctly located | B1 [1] | If clearly labelled then award mark unless wildly inaccurate |
| Answer | Marks | Guidance |
|---|---|---|
| \( | z^2 | = |
| Answer | Marks |
|---|---|
| \(\text{arg}(z^2) = 2\text{arg}(z)\) | B1 [1] |
## 2(i)
$2^2 + 2 \times 2 \times i + i^2 = 4 + 4i - 1 = 3 + 4i$ | M1 |
| A1 [2] | AG | $i^2 = -1$ must be shown
## 2(ii)
Both $z$ and $z^*$ correctly located | B1 [1] | If clearly labelled then award mark unless wildly inaccurate
## 2(iii)
$|z^2| = |z|^2$ | B1 [1]
## 2(iv)
$\text{arg}(z^2) = 2\text{arg}(z)$ | B1 [1]
---The complex number $2 + i$ is denoted by $z$.
\begin{enumerate}[label=(\roman*)]
\item Show that $z^2 = 3 + 4i$. [2]
\item Plot the following on the Argand diagram in the Printed Answer Booklet.
\begin{itemize}
\item $z$
\item $z^2$
\end{itemize} [1]
\item State the relationship between $|z^2|$ and $|z|$. [1]
\item State the relationship between $\arg(z^2)$ and $\arg(z)$. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2018 Q2 [5]}}