| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Locus with parameter variation |
| Difficulty | Standard +0.3 This is a structured multi-part question guiding students through a standard locus problem. Part (a) is routine complex number division, part (b) is algebraic manipulation following directly from (a), part (c) requires sketching two basic loci (vertical line and circle), and part (d) connects the algebra to geometry. While it requires understanding the relationship between z and 1/z, the heavy scaffolding and standard techniques make this easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Multiply numerator and denominator by \(1-yi\) | M1 | OE |
| Obtain \(\frac{1}{1+y^2}+\frac{-y}{1+y^2}\)i | A1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express \(\left(a-\frac{1}{2}\right)^2+b^2\) in terms of \(y\) and expand the bracket | M1 | \(\left(\frac{1}{1+y^2}-\frac{1}{2}\right)^2+\left(\frac{(-)y}{1+y^2}\right)^2\) |
| Obtain \(\left(\frac{1}{(1+y^2)^2}-\frac{2}{2(1+y^2)}+\frac{1}{4}\right)+\frac{y^2}{(1+y^2)^2}\) | A1 FT | Follow *their* answer from (a) provided it gives an expression in \(y\). |
| Obtain \(\frac{1}{4}\) from full and correct working | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a vertical straight line through \(1+0\)i | B1 | |
| Show a circle centre \(\frac{1}{2}+0\)i | B1 | |
| Show a circle with radius \(\frac{1}{2}\) and centre not at the origin | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Circle centre \(\frac{1}{2}+0\)i with radius \(\frac{1}{2}\) | B1 | OE. Condone inclusion of the origin. |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Multiply numerator and denominator by $1-yi$ | M1 | OE |
| Obtain $\frac{1}{1+y^2}+\frac{-y}{1+y^2}$i | A1 | OE |
**Total: 2 marks**
## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express $\left(a-\frac{1}{2}\right)^2+b^2$ in terms of $y$ and expand the bracket | M1 | $\left(\frac{1}{1+y^2}-\frac{1}{2}\right)^2+\left(\frac{(-)y}{1+y^2}\right)^2$ |
| Obtain $\left(\frac{1}{(1+y^2)^2}-\frac{2}{2(1+y^2)}+\frac{1}{4}\right)+\frac{y^2}{(1+y^2)^2}$ | A1 FT | Follow *their* answer from **(a)** provided it gives an expression in $y$. |
| Obtain $\frac{1}{4}$ from full and correct working | A1 | AG |
**Total: 3 marks**
## Question 8(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a vertical straight line through $1+0$i | B1 | |
| Show a circle centre $\frac{1}{2}+0$i | B1 | |
| Show a circle with radius $\frac{1}{2}$ and centre not at the origin | B1 | |
**Total: 3 marks**
## Question 8(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Circle centre $\frac{1}{2}+0$i with radius $\frac{1}{2}$ | B1 | OE. Condone inclusion of the origin. |
**Total: 1 mark**
---8
\begin{enumerate}[label=(\alph*)]
\item Given that $z = 1 + y \mathrm { i }$ and that $y$ is a real number, express $\frac { 1 } { z }$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are functions of $y$.
\item Show that $\left( a - \frac { 1 } { 2 } \right) ^ { 2 } + b ^ { 2 } = \frac { 1 } { 4 }$, where $a$ and $b$ are the functions of $y$ found in part (a).\\
\includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-14_2716_35_108_2012}
\item On a single Argand diagram, sketch the loci given by the equations $\operatorname { Re } ( z ) = 1$ and $\left| z - \frac { 1 } { 2 } \right| = \frac { 1 } { 2 }$, where $z$ is a complex number.
\item The complex number $z$ is such that $\operatorname { Re } ( z ) = 1$. Use your answer to part (b) to give a geometrical description of the locus of $\frac { 1 } { z }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q8 [9]}}