| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Simultaneous equations with complex numbers |
| Difficulty | Standard +0.3 Part (a) is a straightforward simultaneous equations problem with complex numbers requiring substitution and algebraic manipulation. Part (b)(i) is standard recognition of a circle on an Argand diagram, and (b)(ii) requires finding the minimum argument using basic geometry and inverse tangent. All techniques are routine for P3 level with no novel insight required, making this slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve for \(v\) or \(w\) | M1 | |
| Use \(i^2=-1\) | M1 | |
| Obtain \(v=-\frac{2i}{1+i}\) or \(w=\frac{5+7i}{-1+i}\) | A1 | |
| Multiply numerator and denominator by the conjugate of the denominator | M1 | |
| Obtain \(v=-1-i\) | A1 | |
| Obtain \(w=1-6i\) | A1 | |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre \(2+3i\) | B1 | |
| Show a circle with radius 1 and centre not at the origin | B1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a complete method for finding the least value of \(\arg z\) | M1 | |
| Obtain answer \(40.2°\) or \(0.702\) radians | A1 | |
| Total | 2 |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve for $v$ or $w$ | M1 | |
| Use $i^2=-1$ | M1 | |
| Obtain $v=-\frac{2i}{1+i}$ or $w=\frac{5+7i}{-1+i}$ | A1 | |
| Multiply numerator and denominator by the conjugate of the denominator | M1 | |
| Obtain $v=-1-i$ | A1 | |
| Obtain $w=1-6i$ | A1 | |
| **Total** | **6** | |
## Question 10(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $2+3i$ | B1 | |
| Show a circle with radius 1 and centre not at the origin | B1 | |
| **Total** | **2** | |
## Question 10(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a complete method for finding the least value of $\arg z$ | M1 | |
| Obtain answer $40.2°$ or $0.702$ radians | A1 | |
| **Total** | **2** | |10
\begin{enumerate}[label=(\alph*)]
\item The complex numbers $v$ and $w$ satisfy the equations
$$v + \mathrm { i } w = 5 \quad \text { and } \quad ( 1 + 2 \mathrm { i } ) v - w = 3 \mathrm { i } .$$
Solve the equations for $v$ and $w$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item \begin{enumerate}[label=(\roman*)]
\item On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying $| z - 2 - 3 \mathrm { i } | = 1$.
\item Calculate the least value of $\arg z$ for points on this locus.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q10 [10]}}