| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Modulus and argument calculations |
| Difficulty | Standard +0.8 This is a Further Maths question requiring multiple complex number techniques: converting between forms, finding modulus/argument, division, cube roots with multiple values, and proving an algebraic relationship. While each individual step uses standard FP2 methods, the multi-part structure, exact value calculations (involving π/12 angles), and especially part (iv) requiring insight to match cube roots to given forms elevates this above routine exercises. The question demands careful work across several techniques but doesn't require exceptional creativity. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02r nth roots: of complex numbers |
2 You are given the complex numbers $z = \sqrt { 32 } ( 1 + \mathrm { j } )$ and $w = 8 \left( \cos \frac { 7 } { 12 } \pi + \mathrm { j } \sin \frac { 7 } { 12 } \pi \right)$.\\
(i) Find the modulus and argument of each of the complex numbers $z , z ^ { * } , z w$ and $\frac { z } { w }$.\\
(ii) Express $\frac { z } { w }$ in the form $a + b \mathrm { j }$, giving the exact values of $a$ and $b$.\\
(iii) Find the cube roots of $z$, in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.\\
(iv) Show that the cube roots of $z$ can be written as
$$k _ { 1 } w ^ { * } , \quad k _ { 2 } z ^ { * } \quad \text { and } \quad k _ { 3 } \mathrm { j } w ,$$
where $k _ { 1 } , k _ { 2 }$ and $k _ { 3 }$ are real numbers. State the values of $k _ { 1 } , k _ { 2 }$ and $k _ { 3 }$.
\hfill \mbox{\textit{OCR MEI FP2 2008 Q2 [18]}}