| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle equations in complex form |
| Difficulty | Standard +0.3 This question requires computing powers of a complex number (standard technique), finding modulus/argument (routine), and recognizing that the midpoint of a diameter is the center with radius half the distance between endpoints. All steps are straightforward applications of well-practiced techniques with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) Use correct method for finding modulus of \(w^2\) or \(w^3\) or both | M1 | |
| Obtain \(\ | w^2\ | = 2\) and \(\ |
| Use correct method for finding argument of \(w^2\) or \(w^3\) or both | M1 | |
| Obtain \(\arg(w^2) = -\frac{1}{2}\pi\) or \(\frac{3}{2}\pi\) and \(\arg(w^3) = \frac{1}{4}\pi\) | A1ft | [4] |
| (ii) Obtain centre \(-\frac{1}{2} - \frac{1}{2}i\) | B1ft | (their \(w^2\)) |
| Calculate the diameter or radius using \(\ | w - w^2\ | \) w21 or right-angled triangle or cosine rule or equivalent |
| Obtain radius \(\frac{1}{2}\sqrt{10}\) or equivalent | A1 | |
| Obtain \(\left\ | z + \frac{1}{2} + \frac{1}{2}i\right\ | = \frac{1}{2}\sqrt{10}\) or equivalent |
## Question 6:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** Use correct method for finding modulus of $w^2$ or $w^3$ or both | M1 | |
| Obtain $\|w^2\| = 2$ and $\|w^3\| = 2\sqrt{2}$ or equivalent | A1 | |
| Use correct method for finding argument of $w^2$ or $w^3$ or both | M1 | |
| Obtain $\arg(w^2) = -\frac{1}{2}\pi$ or $\frac{3}{2}\pi$ and $\arg(w^3) = \frac{1}{4}\pi$ | A1ft | [4] |
| **(ii)** Obtain centre $-\frac{1}{2} - \frac{1}{2}i$ | B1ft | (their $w^2$) |
| Calculate the diameter or radius using $\|w - w^2\|$ w21 or right-angled triangle or cosine rule or equivalent | M1 | |
| Obtain radius $\frac{1}{2}\sqrt{10}$ or equivalent | A1 | |
| Obtain $\left\|z + \frac{1}{2} + \frac{1}{2}i\right\| = \frac{1}{2}\sqrt{10}$ or equivalent | A1ft | [4] |
---6 The complex number $w$ is defined by $w = - 1 + \mathrm { i }$.\\
(i) Find the modulus and argument of $w ^ { 2 }$ and $w ^ { 3 }$, showing your working.\\
(ii) The points in an Argand diagram representing $w$ and $w ^ { 2 }$ are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form $| z - ( a + b \mathrm { i } ) | = k$.
\hfill \mbox{\textit{CAIE P3 2011 Q6 [8]}}