| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Moderate -0.3 This is a standard Further Pure 1 question testing basic complex number operations: squaring a complex number, simplifying expressions, and rationalizing a complex fraction. All parts use routine techniques with no novel insight required. While it's Further Maths content, these are foundational FP1 skills, making it slightly easier than an average A-level question overall. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(z^2 = (2+i)^2 = 4 + 4i + i^2 = 3 + 4i\) | M1, A1 | For showing 3-term or 4-term expansion / For correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(zz^* = (2+i)(2-i) = 5\) | B1, B1, B1 | For correct value 5 / For stating or using \(z^* = 2-i\) / For correct verification of given result |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) \(\frac{z+1}{z-1} = \frac{3+i}{1+i} = \frac{(3+i)(1-i)}{(1+i)(1-i)} = \frac{4-2i}{2} = 2-i\) | B1, M1, A1 | For correct initial form / For multiplying top and bottom by \(1-i\) / For correct answer \(2-i\) |
**(i)** $z^2 = (2+i)^2 = 4 + 4i + i^2 = 3 + 4i$ | M1, A1 | For showing 3-term or 4-term expansion / For correct answer
**Total: 2 marks**
**(ii)** $4z - z^2 = 8 + 4i - 3 - 4i = 5$
$zz^* = (2+i)(2-i) = 5$ | B1, B1, B1 | For correct value 5 / For stating or using $z^* = 2-i$ / For correct verification of given result
**Total: 3 marks**
**(iii)** $\frac{z+1}{z-1} = \frac{3+i}{1+i} = \frac{(3+i)(1-i)}{(1+i)(1-i)} = \frac{4-2i}{2} = 2-i$ | B1, M1, A1 | For correct initial form / For multiplying top and bottom by $1-i$ / For correct answer $2-i$
**Total: 3 marks**
**Question 3 Total: 8 marks**
---3 The complex number $2 + \mathrm { i }$ is denoted by $z$, and the complex conjugate of $z$ is denoted by $z ^ { * }$.\\
(i) Express $z ^ { 2 }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real, showing clearly how you obtain your answer.\\
(ii) Show that $4 z - z ^ { 2 }$ simplifies to a real number, and verify that this real number is equal to $z z ^ { * }$.\\
(iii) Express $\frac { z + 1 } { z - 1 }$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real, showing clearly how you obtain your answer.
\hfill \mbox{\textit{OCR FP1 Q3 [8]}}