| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Challenging +1.2 This is a multi-part loci question requiring modulus/argument calculation, sketching perpendicular bisectors and circles on an Argand diagram, and optimization using geometric reasoning. While it involves several steps and geometric insight to find the minimum argument, the techniques are standard for Further Maths and the optimization is guided by the diagram rather than requiring algebraic manipulation. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain modulus \(\sqrt{8}\) | B1 | |
| Obtain argument \(\frac{1}{4}\pi\) or \(45°\) | B1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show 1, i and \(u\) in relatively correct positions on an Argand diagram | B1 | |
| Show the perpendicular bisector of the line joining 1 and i | B1 | |
| Show a circle with centre \(u\) and radius 1 | B1 | |
| Shade the correct region | B1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply relevance of the appropriate tangent from \(O\) to the circle | B1\(\sqrt{}\) | |
| Carry out complete strategy for finding \( | z | \) for the critical point |
| Obtain answer \(\sqrt{7}\) | A1 | [3] |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain modulus $\sqrt{8}$ | B1 | |
| Obtain argument $\frac{1}{4}\pi$ or $45°$ | B1 | [2] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show 1, i and $u$ in relatively correct positions on an Argand diagram | B1 | |
| Show the perpendicular bisector of the line joining 1 and i | B1 | |
| Show a circle with centre $u$ and radius 1 | B1 | |
| Shade the correct region | B1 | [4] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply relevance of the appropriate tangent from $O$ to the circle | B1$\sqrt{}$ | |
| Carry out complete strategy for finding $|z|$ for the critical point | M1 | |
| Obtain answer $\sqrt{7}$ | A1 | [3] |
---7 The complex number $2 + 2 \mathrm { i }$ is denoted by $u$.\\
(i) Find the modulus and argument of $u$.\\
(ii) Sketch an Argand diagram showing the points representing the complex numbers 1, i and $u$. Shade the region whose points represent the complex numbers $z$ which satisfy both the inequalities $| z - 1 | \leqslant | z - \mathrm { i } |$ and $| z - u | \leqslant 1$.\\
(iii) Using your diagram, calculate the value of $| z |$ for the point in this region for which $\arg z$ is least.
\hfill \mbox{\textit{CAIE P3 2010 Q7 [9]}}