| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Non-group structures |
| Difficulty | Standard +0.3 This is a straightforward group theory question requiring recall of group axioms and checking them systematically. Part (i) only needs identifying that multiplicative inverses don't exist in integers (e.g., 2 has no integer inverse). Part (ii) requires recognizing {1, -1} as the answer and verifying the four axioms, which is routine bookwork for Further Maths students. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group8.03d Latin square property: for group tables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\forall n, 1n = n1 = n\) so 1 is identity | M1 | Identify identity (can be implicit) |
| But not all integers have an inverse, e.g. \(2n \neq 1\) for any \(n\) | A1 | Complete argument (example or general) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\{-1, 1\}\) | B1* | |
| Demonstrates closure, references associativity, references identity | B1 | Any two of these, without contradiction; dep on 1st B1 |
| \((-1)^{-1} = -1\) (and \(1^{-1}=1\)) so inverses | *B2 | B1 for any two properties; dep on 1st B1 |
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\forall n, 1n = n1 = n$ so 1 is identity | M1 | Identify identity (can be implicit) |
| But not all integers have an inverse, e.g. $2n \neq 1$ for any $n$ | A1 | Complete argument (example or general) |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\{-1, 1\}$ | B1* | |
| Demonstrates closure, references associativity, references identity | B1 | Any two of these, without contradiction; dep on 1st B1 |
| $(-1)^{-1} = -1$ (and $1^{-1}=1$) so inverses | *B2 | B1 for any two properties; dep on 1st B1 |
---4 Let $A$ be the set of non-zero integers.\\
(i) Show that $A$ does not form a group under multiplication.\\
(ii) State the largest subset of $A$ which forms a group under multiplication. Show that this is a group.
\hfill \mbox{\textit{OCR FP3 2016 Q4 [5]}}