| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.3 This is a straightforward loci question requiring students to sketch a circle centered at -2 with radius 3 and a half-line from i at angle -π/4, then identify their intersection point. The geometric interpretation is standard and the intersection can be found by inspection or simple calculation, making it slightly easier than average. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \( | z+2 | = 3\) Circle, Centre \(-2 + 0\)i, radius 3 M1 A1 |
(i) $|z+2| = 3$ Circle, Centre $-2 + 0$i, radius 3 **M1 A1**
$\arg(z-i) = -\frac{1}{4}\pi$ Half-line, From $0 + i$ at $45°$ downwards **M1 A1** [4]
[Allow full line through i for M1 A0]
(ii) $1 (+ 0$i$)$ **B1** [1]3 (i) On a single copy of an Argand diagram, sketch the loci defined by
$$| z + 2 | = 3 \quad \text { and } \quad \arg ( z - \mathrm { i } ) = - \frac { 1 } { 4 } \pi$$
(ii) State the complex number $z$ which corresponds to the point of intersection of these two loci.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2014 Q3 [5]}}