| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring students to translate geometric features (a half-line and circle) on an Argand diagram into complex number equations and inequalities. Part (i) involves standard recall of arg(z) for a half-line and |z - centre| for a circle. Part (ii) requires combining these with inequality signs, which is routine practice for FP1 students. The circle passes through the origin, making the radius calculation trivial (radius = 3). No problem-solving or novel insight required—just direct application of standard complex number loci formulas. |
| Spec | 4.02o Loci in Argand diagram: circles, half-lines4.02p Set notation: for loci |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\arg(z-3i) = \frac{1}{4}\pi\) | M1 | Use \(\arg(z-a) = \theta\) in equation for \(l\), condone missing brackets |
| A1 | Obtain correct answer | |
| \( | z - 3i | = 3\) |
| A1 | Obtain correct answer | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | z-3i | \leq 3\) or e.g. \(x^2 + (y-3)^2 \leq 9\) |
| \(\frac{1}{4}\pi \leq \arg(z-3i) \leq \frac{1}{2}\pi\) or \(y \geq x+3,\ x \geq 0\) | B1B1 | Each correct single inequality, or answer consistent with sensible (i) |
| [3] | SC | if \(<\) used consistently, but otherwise all correct, B2 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\arg(z-3i) = \frac{1}{4}\pi$ | M1 | Use $\arg(z-a) = \theta$ in equation for $l$, condone missing brackets |
| | A1 | Obtain correct answer |
| $|z - 3i| = 3$ | M1 | Use $|z-a| = k$ in equation for $C$, $k$ must be real |
| | A1 | Obtain correct answer |
| **[4]** | | |
---
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|z-3i| \leq 3$ or e.g. $x^2 + (y-3)^2 \leq 9$ | B1 | Obtain correct inequality, or answer consistent with sensible (i) |
| $\frac{1}{4}\pi \leq \arg(z-3i) \leq \frac{1}{2}\pi$ or $y \geq x+3,\ x \geq 0$ | B1B1 | Each correct single inequality, or answer consistent with sensible (i) |
| **[3]** | **SC** | if $<$ used consistently, but otherwise all correct, B2 |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{2ba2e0bf-d20a-41ab-a77c-86a08e700b40-2_885_803_1425_630}
The Argand diagram above shows a half-line $l$ and a circle $C$. The circle has centre 3 i and passes through the origin.\\
(i) Write down, in complex number form, the equations of $l$ and $C$.\\[0pt]
(ii) Write down inequalities that define the region shaded in the diagram. [The shaded region includes the boundaries.]
\hfill \mbox{\textit{OCR FP1 2013 Q6 [7]}}