| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Modulus-argument form conversion |
| Difficulty | Easy -1.2 This is a straightforward recall question testing basic definitions: finding modulus using Pythagoras, argument using arctan, and complex conjugate by negating the imaginary part. Part (b) requires only knowing that conjugates reflect in the real axis. No problem-solving or multi-step reasoning required. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | z | = 5\) |
| \(\arg z = -0.927\) rads or \(-53.1°\) | B1 | Or 5.36 rads or 307° (or 306.8698...); From \(\tan^{-1}(-\frac{4}{3})\), \(\tan\theta = \frac{4}{3}\) or BC |
| \(z^* = 3 + 4i\) | B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(A\) and \(B\) are reflections of each other... | M1 | Allow references to \(w\) and \(w^*\) (or \(z\) and \(z^*\)); do not allow "mirrored" unless also a reference to "reflection" |
| ...in the real (or horizontal or \(x\)) axis | A1 [2] | Correct mirror line \(y=0\) ok; do not allow "positive real axis"; diagram only is no marks; diagram with accompanying description for a general case is fine |
**Question 1:**
**(a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|z| = 5$ | B1 | From $\sqrt{3^2 + 4^2}$ or BC |
| $\arg z = -0.927$ rads or $-53.1°$ | B1 | Or 5.36 rads or 307° (or 306.8698...); From $\tan^{-1}(-\frac{4}{3})$, $\tan\theta = \frac{4}{3}$ or BC |
| $z^* = 3 + 4i$ | B1 [3] | |
**(b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $A$ and $B$ are reflections of each other... | M1 | Allow references to $w$ and $w^*$ (or $z$ and $z^*$); do not allow "mirrored" unless also a reference to "reflection" |
| ...in the real (or horizontal or $x$) axis | A1 [2] | Correct mirror line $y=0$ ok; do not allow "positive real axis"; diagram only is no marks; diagram with accompanying description for a general case is fine |
---1 You are given that $z = 3 - 4 \mathrm { i }$.
\begin{enumerate}[label=(\alph*)]
\item Find
\begin{itemize}
\item $| z |$,
\item $\arg ( z )$,
\item $Z ^ { * }$.
\end{itemize}
On an Argand diagram the complex number $w$ is represented by the point $A$ and $w ^ { * }$ is represented by the point $B$.
\item Describe the geometrical relationship between the points $A$ and $B$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core AS 2019 Q1 [5]}}