| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Square roots of complex numbers |
| Difficulty | Standard +0.3 Part (a) is a standard algebraic exercise finding square roots by equating real/imaginary parts—routine for P3 students. Part (b)(i) requires sketching a circle and half-line, which is straightforward. Part (b)(ii) involves finding the perpendicular distance from a point to a line, a common geometric application. Overall slightly easier than average due to standard techniques throughout. |
| Spec | 4.02h Square roots: of complex numbers4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Square \(x + \mathrm{i}y\) and equate real and imaginary parts to \(7\) and \(-6\sqrt{2}\) respectively | M1 | |
| Obtain equations \(x^2 - y^2 = 7\) and \(2xy = -6\sqrt{2}\) | A1 | |
| Eliminate one variable and find an equation in the other | M1 | |
| Obtain \(x^4 - 7x^2 - 18 = 0\) or \(y^4 + 7y^2 - 18 = 0\), or 3-term equivalent | A1 | |
| Obtain answers \(\pm(3 - \mathrm{i}\sqrt{2})\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show point representing \(1 + 2\mathrm{i}\) | B1 | |
| Show circle with radius \(1\) and centre \(1 + 2\mathrm{i}\) | B1\(\checkmark\) | |
| Show a half line from the point representing \(1\) | B1 | |
| Show line making the correct angle with the real axis | B1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply the relevance of the perpendicular from \(1 + 2\mathrm{i}\) to the line | M1 | |
| Obtain answer \(\sqrt{2} - 1\) (or \(0.414\)) | A1 | [2] |
# Question 10:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Square $x + \mathrm{i}y$ and equate real and imaginary parts to $7$ and $-6\sqrt{2}$ respectively | M1 | |
| Obtain equations $x^2 - y^2 = 7$ and $2xy = -6\sqrt{2}$ | A1 | |
| Eliminate one variable and find an equation in the other | M1 | |
| Obtain $x^4 - 7x^2 - 18 = 0$ or $y^4 + 7y^2 - 18 = 0$, or 3-term equivalent | A1 | |
| Obtain answers $\pm(3 - \mathrm{i}\sqrt{2})$ | A1 | [5] |
## Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show point representing $1 + 2\mathrm{i}$ | B1 | |
| Show circle with radius $1$ and centre $1 + 2\mathrm{i}$ | B1$\checkmark$ | |
| Show a half line from the point representing $1$ | B1 | |
| Show line making the correct angle with the real axis | B1 | [4] |
## Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply the relevance of the perpendicular from $1 + 2\mathrm{i}$ to the line | M1 | |
| Obtain answer $\sqrt{2} - 1$ (or $0.414$) | A1 | [2] |10
\begin{enumerate}[label=(\alph*)]
\item Showing all your working and without the use of a calculator, find the square roots of the complex number $7 - ( 6 \sqrt { } 2 ) \mathrm { i }$. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.
\item \begin{enumerate}[label=(\roman*)]
\item On an Argand diagram, sketch the loci of points representing complex numbers $w$ and $z$ such that $| w - 1 - 2 \mathrm { i } | = 1$ and $\arg ( z - 1 ) = \frac { 3 } { 4 } \pi$.
\item Calculate the least value of $| w - z |$ for points on these loci.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2016 Q10 [11]}}