| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Order of elements |
| Difficulty | Standard +0.8 This is a Further Maths group theory question requiring understanding of modular arithmetic with complex numbers. Part (i) is straightforward recall, part (ii) requires basic calculation, but part (iii) demands proof that all non-zero elements have order 5, requiring systematic reasoning about the structure of Z₅ × Z₅. The combination of abstract algebra concepts with modular arithmetic places this above average difficulty, though it's a standard FP3 exercise rather than requiring deep insight. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03e Order of elements: and order of groups |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| identity \(0+0i\) | B1 | Or '\(0\)' |
| order \(25\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3+i\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(5(a+bi)=5a+5bi=0+0i\) | M1 | Shows 5 times any element equals \(e\) |
| every non-zero element has order 5 or 25 | M1 | Attempt to show that order \(\neq 2,3,4\) |
| So order is 5 | A1 | Argument is convincing, exhaustive and conclusive; must consider all non-zero elements |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| identity $0+0i$ | B1 | Or '$0$' |
| order $25$ | B1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3+i$ | B1 | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5(a+bi)=5a+5bi=0+0i$ | M1 | Shows 5 times any element equals $e$ |
| every non-zero element has order 5 or 25 | M1 | Attempt to show that order $\neq 2,3,4$ |
| So order is 5 | A1 | Argument is convincing, exhaustive and conclusive; must consider all non-zero elements |
---2 The elements of a group $G$ are the complex numbers $a + b \mathrm { i }$ where $a , b \in \{ 0,1,2,3,4 \}$. These elements are combined under the operation of addition modulo 5 .\\
(i) State the identity element and the order of $G$.\\
(ii) Write down the inverse of $2 + 4 \mathrm { i }$.\\
(iii) Show that every non-zero element of $G$ has order 5 .
\hfill \mbox{\textit{OCR FP3 2013 Q2 [6]}}