| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Complex number loci on Argand diagrams |
| Difficulty | Standard +0.3 This question involves standard complex number techniques: rationalizing denominators to find Cartesian form, using argument properties with conjugates, and interpreting geometric loci (perpendicular bisector and circle) on an Argand diagram. While multi-part, each component uses routine A-level methods without requiring novel insight or particularly challenging problem-solving. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Multiply numerator and denominator by \(a - 2i\), or equivalent | M1 | |
| Use \(i^2 = -1\) at least once | A1 | |
| Obtain answer \(\frac{6}{a^2+4} + \frac{3ai}{a^2+4}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Either state that \(\arg u = -\frac{1}{3}\pi\) or express \(u^*\) in terms of \(a\) (FT on \(u\)) | B1 | |
| Use correct method to form an equation in \(a\) | M1 | |
| Obtain answer \(a = -2\sqrt{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show the perpendicular bisector of points representing \(2i\) and \(1+i\) | B1 | |
| Show the point representing \(2+i\) | B1 | |
| Show a circle with radius 2 and centre \(2+i\) | B1FT | FT on the position of the point for \(2+i\) |
| Shade the correct region | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the critical point \(2+3i\) | B1 | |
| Obtain answer \(56.3°\) or \(0.983\) radians | B1 |
# Question 10(a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Multiply numerator and denominator by $a - 2i$, or equivalent | M1 | |
| Use $i^2 = -1$ at least once | A1 | |
| Obtain answer $\frac{6}{a^2+4} + \frac{3ai}{a^2+4}$ | A1 | |
---
# Question 10(a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Either state that $\arg u = -\frac{1}{3}\pi$ or express $u^*$ in terms of $a$ (FT on $u$) | B1 | |
| Use correct method to form an equation in $a$ | M1 | |
| Obtain answer $a = -2\sqrt{3}$ | A1 | |
---
# Question 10(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show the perpendicular bisector of points representing $2i$ and $1+i$ | B1 | |
| Show the point representing $2+i$ | B1 | |
| Show a circle with radius 2 and centre $2+i$ | B1FT | FT on the position of the point for $2+i$ |
| Shade the correct region | B1 | |
---
# Question 10(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the critical point $2+3i$ | B1 | |
| Obtain answer $56.3°$ or $0.983$ radians | B1 | |10
\begin{enumerate}[label=(\alph*)]
\item The complex number $u$ is defined by $u = \frac { 3 \mathrm { i } } { a + 2 \mathrm { i } }$, where $a$ is real.
\begin{enumerate}[label=(\roman*)]
\item Express $u$ in the Cartesian form $x + \mathrm { i } y$, where $x$ and $y$ are in terms of $a$.
\item Find the exact value of $a$ for which $\arg u ^ { * } = \frac { 1 } { 3 } \pi$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z - 2 \mathbf { i } | \leqslant | z - 1 - \mathbf { i } |$ and $| z - 2 - \mathbf { i } | \leqslant 2$.
\item Calculate the least value of $\arg z$ for points in this region.\\
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 [12]}}