| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle of Apollonius locus |
| Difficulty | Standard +0.3 This is a standard locus question requiring routine manipulation of modulus equations and complex conjugates. Part (i) involves straightforward verification of basic properties, part (ii) is algebraic manipulation following a clear hint ('by squaring both sides'), and part (iii) requires recognizing a circle. The question guides students through each step with minimal problem-solving required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Show that \(a^2 + b^2 = (a + ib)(a - ib)\) | B1 | |
| Show that \((a + ib - ki)^* = a - ib + ki\) | B1 | [2 marks] |
| (ii) Square both sides and express the given equation in terms of \(z\) and \(z^*\) | M1 | |
| Obtain a correct equation in any form, e.g. \((z - 10i)(z^* + 10i) = 4(z - 4i)(z^* + 4i)\) | A1 | |
| Either express \( | z - 2i | = 4\) in terms of \(z\) and \(z^*\) or reduce the given equation to the form \( |
| Obtain the given answer correctly | A1 | [5 marks] |
| (iii) State that the locus is a circle with centre \(2i\) and radius \(5\) | B1 | [1 mark] |
**(i)** Show that $a^2 + b^2 = (a + ib)(a - ib)$ | B1 |
Show that $(a + ib - ki)^* = a - ib + ki$ | B1 | [2 marks]
**(ii)** Square both sides and express the given equation in terms of $z$ and $z^*$ | M1 |
Obtain a correct equation in any form, e.g. $(z - 10i)(z^* + 10i) = 4(z - 4i)(z^* + 4i)$ | A1 |
Either express $|z - 2i| = 4$ in terms of $z$ and $z^*$ or reduce the given equation to the form $|z - u| = r$ | M1 |
Obtain the given answer correctly | A1 | [5 marks]
**(iii)** State that the locus is a circle with centre $2i$ and radius $5$ | B1 | [1 mark]
---7 The complex number $z$ is defined by $z = a + \mathrm { i } b$, where $a$ and $b$ are real. The complex conjugate of $z$ is denoted by $z ^ { * }$.\\
(i) Show that $| z | ^ { 2 } = z z ^ { * }$ and that $( z - k \mathrm { i } ) ^ { * } = z ^ { * } + k \mathrm { i }$, where $k$ is real.
In an Argand diagram a set of points representing complex numbers $z$ is defined by the equation $| z - 10 \mathrm { i } | = 2 | z - 4 \mathrm { i } |$.\\
(ii) Show, by squaring both sides, that
$$z z ^ { * } - 2 \mathrm { i } z ^ { * } + 2 \mathrm { i } z - 12 = 0$$
Hence show that $| z - 2 i | = 4$.\\
(iii) Describe the set of points geometrically.
\hfill \mbox{\textit{CAIE P3 2013 Q7 [8]}}