| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with single inequality |
| Difficulty | Moderate -0.3 Part (i) requires straightforward expansion of (2+i)² and calculating modulus using standard formulas—pure recall. Part (ii) involves recognizing the inequality represents a disc centered at w² with radius |w²|, then shading it—a standard locus question requiring minimal problem-solving beyond pattern recognition. This is slightly easier than average due to its routine nature and clear structure. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt multiplication and use \(i^2 = -1\) | M1 | |
| Obtain \(3 + 4i\) | A1 | |
| Obtain \(5\) for modulus | B1 | [3] |
| (ii) Draw complete circle with centre corresponding to their \(w^2\) | B1 | |
| \(\ldots\) and radius corresponding to their \( | w | \) |
| Shade the correct region | B1 | cwo |
**(i)** Attempt multiplication and use $i^2 = -1$ | M1 |
Obtain $3 + 4i$ | A1 |
Obtain $5$ for modulus | B1 | [3]
**(ii)** Draw complete circle with centre corresponding to their $w^2$ | B1 |
$\ldots$ and radius corresponding to their $|w|$ | B1 |
Shade the correct region | B1 | cwo | [3]3 The complex number $w$ is defined by $w = 2 + \mathrm { i }$.\\
(i) Showing your working, express $w ^ { 2 }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real. Find the modulus of $w ^ { 2 }$.\\
(ii) Shade on an Argand diagram the region whose points represent the complex numbers $z$ which satisfy
$$\left| z - w ^ { 2 } \right| \leqslant \left| w ^ { 2 } \right|$$
\hfill \mbox{\textit{CAIE P3 2010 Q3 [6]}}