| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Geometric properties in Argand diagram |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: quadratic formula with complex coefficients, plotting on Argand diagram, calculating modulus/argument, and verifying triangle properties. While it involves several steps, each is routine for Further Maths students with no novel insight required. Slightly above average difficulty due to the multi-step nature and the 'hence' proof in part (d), but well within standard FM Pure content. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use quadratic formula, or completing the square \(\left((z-3i)^2 - 3 = 0\right)\) and use \(i^2 = -1\) to find a root | M1 | |
| Obtain a root, e.g. \(\sqrt{3} + 3i\) | A1 | Or exact 2 term equivalent e.g. \(\frac{6i}{2} + \frac{\sqrt{12}}{2}\) ISW |
| Obtain the other root, e.g. \(-\sqrt{3} + 3i\) | A1 | Or exact 2 term equivalent ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show points representing the roots correctly | B1 FT | 2 roots consistent with *their* (a) and with no errors seen on the diagram. B0 if only one root or more than 2 roots. Must match their scale and \(1 < \sqrt{3} < 2\). Linear scales seen or implied. Need some indication of scale (numbers or dashes). Scales along an axis must be approximately consistent but scales may be different on the 2 axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State modulus of either root is \(2\sqrt{3}\), or simplified exact equivalent | B1 FT | ISW if converted to decimal. Ignore modulus of second root if seen. Follow their root(s) not on either axis (from (a) or (b)). |
| Find the argument of one of their roots – get as far as \(\tan^{-1}(...)\) | M1 | SOI but must be correct for their root. |
| Obtain correct arguments \(\frac{1}{3}\pi\) and \(\frac{2}{3}\pi\), or simplified exact equivalents | A1 | Must obtain values. Allow degrees. |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Give a complete justification that the correct triangle is equilateral | B1 | Check *their* diagram in (b). Possible justifications: 3 equal sides, or all angles equal to \(\frac{\pi}{3}\), or isosceles and an angle of \(\frac{\pi}{3}\). |
| 1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use quadratic formula, or completing the square $\left((z-3i)^2 - 3 = 0\right)$ and use $i^2 = -1$ to find a root | M1 | |
| Obtain a root, e.g. $\sqrt{3} + 3i$ | A1 | Or exact 2 term equivalent e.g. $\frac{6i}{2} + \frac{\sqrt{12}}{2}$ ISW |
| Obtain the other root, e.g. $-\sqrt{3} + 3i$ | A1 | Or exact 2 term equivalent ISW |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show points representing the roots correctly | B1 FT | 2 roots consistent with *their* (a) and with no errors seen on the diagram. B0 if only one root or more than 2 roots. Must match their scale and $1 < \sqrt{3} < 2$. Linear scales seen or implied. Need some indication of scale (numbers or dashes). Scales along an axis must be approximately consistent but scales may be different on the 2 axes |
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State modulus of either root is $2\sqrt{3}$, or simplified exact equivalent | B1 FT | ISW if converted to decimal. Ignore modulus of second root if seen. Follow their root(s) not on either axis (from (a) or (b)). |
| Find the argument of one of their roots – get as far as $\tan^{-1}(...)$ | M1 | SOI but must be correct for their root. |
| Obtain correct arguments $\frac{1}{3}\pi$ and $\frac{2}{3}\pi$, or simplified exact equivalents | A1 | Must obtain values. Allow degrees. |
| | **3** | |
## Question 5(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Give a complete justification that the correct triangle is equilateral | B1 | Check *their* diagram in (b). Possible justifications: 3 equal sides, or all angles equal to $\frac{\pi}{3}$, or isosceles and an angle of $\frac{\pi}{3}$. |
| | **1** | |5
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $z ^ { 2 } - 6 \mathrm { i } z - 12 = 0$, giving the answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.
\item On a sketch of an Argand diagram with origin $O$, show points $A$ and $B$ representing the roots of the equation in part (a).
\item Find the exact modulus and argument of each root.
\item Hence show that the triangle $O A B$ is equilateral.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q5 [8]}}