| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Single locus sketching |
| Difficulty | Moderate -0.3 Part (i) requires routine complex number arithmetic (expanding, multiplying by conjugate) with no conceptual difficulty. Part (ii) is a standard circle locus question where students recognize |z-u|=|u| as a circle centered at u with radius |u|, then plot it. This is typical P3/FP1 material requiring only pattern recognition and basic sketching skills, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Either: Expand \((1+2i)^2\) to obtain \(-3 + 4i\) or unsimplified equivalent | B1 | |
| Multiply numerator and denominator by \(2 - i\) | M1 | |
| Obtain correct numerator \(-2 + 11i\) or correct denominator \(5\) | A1 | |
| Obtain \(-\frac{2}{5} + \frac{11}{5}i\) or equivalent | A1 | |
| Or: Expand \((1+2i)^2\) to obtain \(-3+4i\) or unsimplified equivalent | B1 | |
| Obtain two equations in \(x\) and \(y\) and solve for \(x\) or \(y\) | M1 | |
| Obtain final answer \(x = -\frac{2}{5}\) | A1 | |
| Obtain final answer \(y = \frac{11}{5}\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Draw a circle | M1 | |
| Show centre at relatively correct position, following their \(u\) | A1\(\checkmark\) | |
| Draw circle passing through the origin | A1 | [3] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Either:** Expand $(1+2i)^2$ to obtain $-3 + 4i$ or unsimplified equivalent | B1 | |
| Multiply numerator and denominator by $2 - i$ | M1 | |
| Obtain correct numerator $-2 + 11i$ or correct denominator $5$ | A1 | |
| Obtain $-\frac{2}{5} + \frac{11}{5}i$ or equivalent | A1 | |
| **Or:** Expand $(1+2i)^2$ to obtain $-3+4i$ or unsimplified equivalent | B1 | |
| Obtain two equations in $x$ and $y$ and solve for $x$ or $y$ | M1 | |
| Obtain final answer $x = -\frac{2}{5}$ | A1 | |
| Obtain final answer $y = \frac{11}{5}$ | A1 | [4] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Draw a circle | M1 | |
| Show centre at relatively correct position, following their $u$ | A1$\checkmark$ | |
| Draw circle passing through the origin | A1 | [3] |
---4 The complex number $u$ is defined by $u = \frac { ( 1 + 2 \mathrm { i } ) ^ { 2 } } { 2 + \mathrm { i } }$.\\
(i) Without using a calculator and showing your working, express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(ii) Sketch an Argand diagram showing the locus of the complex number $z$ such that $| z - u | = | u |$.
\hfill \mbox{\textit{CAIE P3 2012 Q4 [7]}}