| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of argument on loci |
| Difficulty | Standard +0.8 Part (i) is routine complex division. Part (ii) requires recognizing and sketching a circle locus. Part (iii) is the challenging element: finding maximum argument requires geometric insight to identify the tangent line from the origin to the circle, then using trigonometry to calculate the angle—this optimization on a locus goes beyond standard exercises. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER: Attempt multiplication of numerator and denominator by \(3 + 2i\), or equivalent | M1 | |
| Simplify denominator to 13 or numerator to \(13 + 26i\) | A1 | |
| Obtain answer \(u = 1 + 2i\) | A1 | |
| OR: Using correct processes, find the modulus and argument of \(u\) | M1 | |
| Obtain modulus \(\sqrt{5}\) (or 2.24) or argument \(\tan^{-1}2\) (or \(63.4°\) or 1.11 radians) | A1 | |
| Obtain answer \(u = 1 + 2i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show the point \(U\) on an Argand diagram in a relatively correct position | B1\(\sqrt{}\) | f.t. on the value of \(u\) |
| Show a circle with centre \(U\) | B1\(\sqrt{}\) | |
| Show a circle with radius consistent with 2 | B1\(\sqrt{}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply relevance of the appropriate tangent from \(O\) to the circle | B1\(\sqrt{}\) | Drawing the appropriate tangent is sufficient for B1\(\sqrt{}\) |
| Carry out a complete strategy for finding max arg \(z\) | M1 | A final answer obtained by measurement earns M1 only |
| Obtain final answer \(126.9°\) (2.21 radians) | A1 |
## Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **EITHER:** Attempt multiplication of numerator and denominator by $3 + 2i$, or equivalent | M1 | |
| Simplify denominator to 13 or numerator to $13 + 26i$ | A1 | |
| Obtain answer $u = 1 + 2i$ | A1 | |
| **OR:** Using correct processes, find the modulus and argument of $u$ | M1 | |
| Obtain modulus $\sqrt{5}$ (or 2.24) or argument $\tan^{-1}2$ (or $63.4°$ or 1.11 radians) | A1 | |
| Obtain answer $u = 1 + 2i$ | A1 | |
**Total: [3]**
---
## Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show the point $U$ on an Argand diagram in a relatively correct position | B1$\sqrt{}$ | f.t. on the value of $u$ |
| Show a circle with centre $U$ | B1$\sqrt{}$ | |
| Show a circle with radius consistent with 2 | B1$\sqrt{}$ | |
**Total: [3]**
---
## Question 7(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply relevance of the appropriate tangent from $O$ to the circle | B1$\sqrt{}$ | Drawing the appropriate tangent is sufficient for B1$\sqrt{}$ |
| Carry out a complete strategy for finding max arg $z$ | M1 | A final answer obtained by measurement earns M1 only |
| Obtain final answer $126.9°$ (2.21 radians) | A1 | |
**Total: [3]**
---7 The complex number $u$ is given by $u = \frac { 7 + 4 \mathrm { i } } { 3 - 2 \mathrm { i } }$.\\
(i) Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(ii) Sketch an Argand diagram showing the point representing the complex number $u$. Show on the same diagram the locus of the complex number $z$ such that $| z - u | = 2$.\\
(iii) Find the greatest value of $\arg z$ for points on this locus.
\hfill \mbox{\textit{CAIE P3 2003 Q7 [9]}}