| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question testing basic complex number arithmetic operations (linear combination, squaring, and division). All three parts are routine calculations requiring only direct application of standard techniques with no problem-solving insight needed. While FP1 content, these are foundational operations that are more computational than conceptually challenging. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) \(-7i\) | B1, B1, 2 | Real part correct; Imaginary part correct |
| Part (ii) \(2 + 3i\) | B1, B1, B1, 3 | \(iz\) stated or implied or \(i^2 = -1\) seen; Real part correct; Imaginary part correct |
| Result: \(-5 + 12i\) | ||
| Part (iii) \(\frac{1}{5}(4 - 7i)\) or equivalent | M1, A1, A1, 3 | Multiply by conjugate; Real part correct; Imaginary part correct |
| N.B. Working must be shown |
**Part (i)** $-7i$ | B1, B1, 2 | Real part correct; Imaginary part correct
**Part (ii)** $2 + 3i$ | B1, B1, B1, 3 | $iz$ stated or implied or $i^2 = -1$ seen; Real part correct; Imaginary part correct
Result: $-5 + 12i$ |
**Part (iii)** $\frac{1}{5}(4 - 7i)$ or equivalent | M1, A1, A1, 3 | Multiply by conjugate; Real part correct; Imaginary part correct
| N.B. Working must be shown5 The complex numbers $3 - 2 \mathrm { i }$ and $2 + \mathrm { i }$ are denoted by $z$ and $w$ respectively. Find, giving your answers in the form $x + \mathrm { i } y$ and showing clearly how you obtain these answers,\\
(i) $2 z - 3 w$,\\
(ii) $( \mathrm { i } z ) ^ { 2 }$,\\
(iii) $\frac { z } { w }$.
\hfill \mbox{\textit{OCR FP1 2006 Q5 [8]}}