| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.3 This question involves standard complex number manipulations (multiplying by conjugate), sketching familiar loci (perpendicular bisector and circle), and finding arguments at intersection points. While it requires multiple techniques across three parts, each step is routine for A-level Further Maths students with no novel problem-solving insight needed. The geometric interpretation is straightforward, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: Substitute for \(u\) in \(\frac{i}{u}\) and multiply numerator and denominator by \(1 + i\) | M1 | |
| Obtain final answer \(-\frac{1}{2} + \frac{1}{2}i\), or equivalent | A1 | |
| OR: Substitute for \(u\), obtain two equations in \(x\) and \(y\) and solve for \(x\) or for \(y\) | M1 | |
| Obtain final answer \(-\frac{1}{2} + \frac{1}{2}i\), or equivalent | A1 | 2 |
| (ii) Show a point representing \(u\) in a relatively correct position | B1 | |
| Show the bisector of the line segment joining \(u\) to the origin | B1 | |
| Show a circle with centre at the point representing \(i\) | B1 | |
| Show a circle with radius 2 | B1 | 4 |
| (iii) State argument \(-\frac{1}{2}\pi\), or equivalent, e.g. \(270°\) | B1 | |
| State or imply the intersection in the first quadrant represents \(2 + i\) | B1 | |
| State argument 0.464, (0.4636)or equivalent, e.g. \(26.6°\) (26.5625) | B1 | 3 |
**(i)** **EITHER:** Substitute for $u$ in $\frac{i}{u}$ and multiply numerator and denominator by $1 + i$ | M1 |
Obtain final answer $-\frac{1}{2} + \frac{1}{2}i$, or equivalent | A1 |
**OR:** Substitute for $u$, obtain two equations in $x$ and $y$ and solve for $x$ or for $y$ | M1 |
Obtain final answer $-\frac{1}{2} + \frac{1}{2}i$, or equivalent | A1 | 2 |
**(ii)** Show a point representing $u$ in a relatively correct position | B1 |
Show the bisector of the line segment joining $u$ to the origin | B1 |
Show a circle with centre at the point representing $i$ | B1 |
Show a circle with radius 2 | B1 | 4 |
**(iii)** State argument $-\frac{1}{2}\pi$, or equivalent, e.g. $270°$ | B1 |
State or imply the intersection in the first quadrant represents $2 + i$ | B1 |
State argument 0.464, (0.4636)or equivalent, e.g. $26.6°$ (26.5625) | B1 | 3 |8 The complex number 1 - i is denoted by $u$.\\
(i) Showing your working and without using a calculator, express
$$\frac { \mathrm { i } } { u }$$
in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(ii) On an Argand diagram, sketch the loci representing complex numbers $z$ satisfying the equations $| z - u | = | z |$ and $| z - \mathrm { i } | = 2$.\\
(iii) Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).
\hfill \mbox{\textit{CAIE P3 2015 Q8 [9]}}