| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Square roots of complex numbers |
| Difficulty | Moderate -0.3 This question tests standard A-level techniques: finding square roots of complex numbers by equating real/imaginary parts, simplifying complex fractions by multiplying by conjugate, plotting on Argand diagrams, and applying the modulus property |z₁/z₂| = |z₁|/|z₂|. All parts are routine procedural exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02h Square roots: of complex numbers4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Mark | Guidance |
| (a) EITHER: Square \(x + iy\) and equate real and/or imaginary parts to \(-3\) and/or \(4\) respectively | M1 | |
| Obtain \(x^2 - y^2 = -3\) and \(2xy = 4\) | A1 | |
| Eliminate one variable and obtain an equation in the other variable | M1 | |
| Obtain \(x^4 + 3x^2 - 4 = 0\), or \(y^4 - 3y^2 - 4 = 0\), or 3-term equivalent | A1 | |
| Obtain final answers \(\pm(1+2i)\) and no others | A1 | [Accept \(\pm 1 \pm 2i\), or \(x = 1, y = 2\) and \(x = -1, y = -2\) as final answers, but not \(\pm \pm 1, \pm \pm 2\).] |
| OR: Convert \(-3 + 4i\) to polar form \((R, \theta)\) | M1 | |
| Use fact that a square root has polar form \((\sqrt{R}, \frac{1}{2}\theta)\) | M1 | |
| Obtain one root in polar form e.g. \((\sqrt{5}, 63.4°)\) (allow \(63.5°\); argument is \(1.11\) radians) | A1 | |
| Obtain answer \(1 + 2i\) | A1 | |
| Obtain answer \(-1 - 2i\) and no others | A1 | Max 5 marks |
| (b) (i) Carry out multiplication of numerator and denominator by \(2 - i\) | M1 | |
| Obtain answer \(\frac{1}{5} + \frac{7}{5}i\) or \(0.2 + 1.4i\) | A1 | Max 2 marks |
| (ii) Show all three points on an Argand diagram in relatively correct positions | B1✓ | [Accept answers on separate diagrams.] |
| (iii) State that \(OC = \frac{OA}{OB}\), or equivalent | B1 | [Accept the answer \(OA.OC = 2OB\), or equivalent.] |
| Content | Mark | Guidance |
|---------|------|----------|
| **(a)** **EITHER:** Square $x + iy$ and equate real and/or imaginary parts to $-3$ and/or $4$ respectively | M1 | |
| Obtain $x^2 - y^2 = -3$ and $2xy = 4$ | A1 | |
| Eliminate one variable and obtain an equation in the other variable | M1 | |
| Obtain $x^4 + 3x^2 - 4 = 0$, or $y^4 - 3y^2 - 4 = 0$, or 3-term equivalent | A1 | |
| Obtain final answers $\pm(1+2i)$ and no others | A1 | [Accept $\pm 1 \pm 2i$, or $x = 1, y = 2$ and $x = -1, y = -2$ as final answers, but not $\pm \pm 1, \pm \pm 2$.] | Max 5 marks |
| | | |
| **OR:** Convert $-3 + 4i$ to polar form $(R, \theta)$ | M1 | |
| Use fact that a square root has polar form $(\sqrt{R}, \frac{1}{2}\theta)$ | M1 | |
| Obtain one root in polar form e.g. $(\sqrt{5}, 63.4°)$ (allow $63.5°$; argument is $1.11$ radians) | A1 | |
| Obtain answer $1 + 2i$ | A1 | |
| Obtain answer $-1 - 2i$ and no others | A1 | Max 5 marks |
| | | |
| **(b)** **(i)** Carry out multiplication of numerator and denominator by $2 - i$ | M1 | |
| Obtain answer $\frac{1}{5} + \frac{7}{5}i$ or $0.2 + 1.4i$ | A1 | Max 2 marks |
| **(ii)** Show all three points on an Argand diagram in relatively correct positions | B1✓ | [Accept answers on separate diagrams.] |
| **(iii)** State that $OC = \frac{OA}{OB}$, or equivalent | B1 | [Accept the answer $OA.OC = 2OB$, or equivalent.] | Max 1 mark |
---8
\begin{enumerate}[label=(\alph*)]
\item Find the two square roots of the complex number $- 3 + 4 \mathrm { i }$, giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item The complex number $z$ is given by
$$z = \frac { - 1 + 3 \mathrm { i } } { 2 + \mathrm { i } } .$$
\begin{enumerate}[label=(\roman*)]
\item Express $z$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
\item Show on a sketch of an Argand diagram, with origin $O$, the points $A , B$ and $C$ representing the complex numbers $- 1 + 3 \mathrm { i } , 2 + \mathrm { i }$ and $z$ respectively.
\item State an equation relating the lengths $O A , O B$ and $O C$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2002 Q8 [9]}}