| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring the quadratic formula with complex coefficients, basic Argand diagram sketching, modulus/argument calculations, and verification of an equilateral triangle using distance formula. While it involves several steps, each part uses standard techniques with no novel insight required. The geometric verification is straightforward once the roots are found. Slightly above average due to the multiple components and complex coefficient handling, but remains a typical Further Maths exercise. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use quadratic formula, completing the square, or substitution \(z = x + iy\) to find a root, using \(i^2 = -1\) | M1 | Allow \((\pm 2 - 2\sqrt{3}\,i)/2\) as final answer. Remaining marks only for roots with \(xy \neq 0\) |
| Obtain a root, e.g. \(1 - \sqrt{3}\,i\) | A1 | |
| Obtain the other root, e.g. \(-1 - \sqrt{3}\,i\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Represent both roots on an Argand diagram in relatively correct positions | B1\(\sqrt{}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State modulus of both roots is \(2\) | B1\(\sqrt{}\) | Treat answers in polar form as misread |
| State argument of \(1 - \sqrt{3}\,i\) is \(-60°\) (or \(300°,\ -\frac{1}{3}\pi,\ -\frac{5}{3}\pi\)) | B1\(\sqrt{}\) | |
| State argument of \(-1 - \sqrt{3}\,i\) is \(-120°\) (or \(240°,\ -\frac{2}{3}\pi,\ -\frac{4}{3}\pi\)) | B1\(\sqrt{}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Give a complete justification of the statement | B1 |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use quadratic formula, completing the square, or substitution $z = x + iy$ to find a root, using $i^2 = -1$ | M1 | Allow $(\pm 2 - 2\sqrt{3}\,i)/2$ as final answer. Remaining marks only for roots with $xy \neq 0$ |
| Obtain a root, e.g. $1 - \sqrt{3}\,i$ | A1 | |
| Obtain the other root, e.g. $-1 - \sqrt{3}\,i$ | A1 | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Represent both roots on an Argand diagram in relatively correct positions | B1$\sqrt{}$ | |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State modulus of both roots is $2$ | B1$\sqrt{}$ | Treat answers in polar form as misread |
| State argument of $1 - \sqrt{3}\,i$ is $-60°$ (or $300°,\ -\frac{1}{3}\pi,\ -\frac{5}{3}\pi$) | B1$\sqrt{}$ | |
| State argument of $-1 - \sqrt{3}\,i$ is $-120°$ (or $240°,\ -\frac{2}{3}\pi,\ -\frac{4}{3}\pi$) | B1$\sqrt{}$ | |
### Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Give a complete justification of the statement | B1 | |
---7 (i) Solve the equation $z ^ { 2 } + ( 2 \sqrt { } 3 ) \mathrm { i } z - 4 = 0$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
(ii) Sketch an Argand diagram showing the points representing the roots.\\
(iii) Find the modulus and argument of each root.\\
(iv) Show that the origin and the points representing the roots are the vertices of an equilateral triangle.
\hfill \mbox{\textit{CAIE P3 2009 Q7 [8]}}