| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Multiplication and powers of complex numbers |
| Difficulty | Moderate -0.5 This is a straightforward Further Maths question testing basic complex number operations (conjugate multiplication and algebraic manipulation). While it's from Further Maths content, the operations are routine and mechanical with no conceptual challenges—just careful arithmetic. It's easier than an average A-level question overall due to its purely procedural nature, though the Further Maths context prevents it from being rated lower. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z_1^* z_2 = (3-2i)(-1-i) = -3+2i-3i+2i^2\) | M1 (AO 1.1) | Find conjugate, then multiply out brackets |
| \(= -5 - i\) | A1 (AO 1.1) | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\dfrac{z_1 + 2z_2}{z_2} = \dfrac{3+2i-2-2i}{-1-i} = \dfrac{1}{-1-i} \cdot \dfrac{-1+i}{-1+i}\) | M1 (AO 1.1) | Multiply by \(\dfrac{-1+i}{-1+i}\) |
| \(= \dfrac{-1+i}{2} = -\dfrac{1}{2} + \dfrac{1}{2}i\) | A1 (AO 1.1) | |
| [2] |
# Question 2(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z_1^* z_2 = (3-2i)(-1-i) = -3+2i-3i+2i^2$ | **M1** (AO 1.1) | Find conjugate, then multiply out brackets | Working must be seen |
| $= -5 - i$ | **A1** (AO 1.1) | |
| **[2]** | | |
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# Question 2(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\dfrac{z_1 + 2z_2}{z_2} = \dfrac{3+2i-2-2i}{-1-i} = \dfrac{1}{-1-i} \cdot \dfrac{-1+i}{-1+i}$ | **M1** (AO 1.1) | Multiply by $\dfrac{-1+i}{-1+i}$ | Must be seen |
| $= \dfrac{-1+i}{2} = -\dfrac{1}{2} + \dfrac{1}{2}i$ | **A1** (AO 1.1) | |
| **[2]** | | |
---2 In this question you must show detailed reasoning.\\
Given that $z _ { 1 } = 3 + 2 \mathrm { i }$ and $z _ { 2 } = - 1 - \mathrm { i }$, find the following, giving each in the form $a + b \mathrm { i }$.\\
(i) $z _ { 1 } ^ { * } z _ { 2 }$\\
(ii) $\frac { z _ { 1 } + 2 z _ { 2 } } { z _ { 2 } }$
\hfill \mbox{\textit{OCR Further Pure Core AS Q2 [4]}}