| Exam Board | OCR MEI |
|---|---|
| Module | Further Extra Pure (Further Extra Pure) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Challenging +1.2 This is a Further Maths question on group theory requiring systematic verification of four group axioms (closure, associativity, identity, inverses) from a composition table. While conceptually straightforward, it requires careful table reading and checking multiple properties. The closure and identity are immediate from the table, inverses require checking each element, and associativity is typically assumed for finite operations or requires extensive verification. This is harder than standard A-level but a routine exercise for Further Maths students who have learned group axioms. |
| Spec | 8.03c Group definition: recall and use, show structure is/isn't a group |
| \(*\) | \(p\) | \(q\) | \(r\) | \(s\) | \(t\) |
| \(p\) | \(p\) | \(q\) | \(r\) | \(s\) | \(t\) |
| \(q\) | \(q\) | \(p\) | \(s\) | \(t\) | \(r\) |
| \(r\) | \(r\) | \(t\) | \(p\) | \(q\) | \(s\) |
| \(s\) | \(s\) | \(r\) | \(t\) | \(p\) | \(q\) |
| \(t\) | \(t\) | \(s\) | \(q\) | \(r\) | \(p\) |
| Answer | Marks |
|---|---|
| 2 | If it has an identity, then it is p, since pxxpx for all xS. |
| Answer | Marks |
|---|---|
| So not a group | B1 |
| Answer | Marks |
|---|---|
| E1 | 1.1 |
| Answer | Marks |
|---|---|
| 1.2 | Only need to note |
| Answer | Marks |
|---|---|
| Attempt to look at associativity | M1 |
| (q*r)*ts*tq | E1 |
| q*(r*t)q*st | E1 |
| Not associative so not a group | E1 |
| [4] | If not complete |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | 2 | 1 |
| 3i | 4 | 0 |
| 3iiA | 1 | 0 |
| 3iiB | 1 | 1 |
| 3iii | 1 | 1 |
| 3iv | 0 | 1 |
| 4i | 2 | 0 |
| 4ii | 3 | 1 |
| 4iii | 0 | 2 |
| 4iv | 3 | 0 |
| 4v | 1 | 1 |
| 5i | 1 | 1 |
| 5ii | 2 | 0 |
| 5iii | 4 | 1 |
| 5iv | 1 | 3 |
| 5v | 0 | 1 |
| Totals | 31 | 18 |
| Date | Version | Change |
| October 2019 | 2 | Amendments to the front cover rubric instructions to candidates. |
Question 2:
2 | If it has an identity, then it is p, since pxxpx for all xS.
Leading diagonal shows that xxp for all xS, so all elements have order 2.
But 2 is not a factor of 5, the order of S, so this breaks Lagrange.
So not a group | B1
E1
E1
E1 | 1.1
3.1a
2.4
1.2 | Only need to note
this for one element
other than p.
Alternative method
Attempt to look at associativity | M1
(q*r)*ts*tq | E1
q*(r*t)q*st | E1
Not associative so not a group | E1
[4] | If not complete
argument can score
SC1 for attempting
to check 2 axioms,
SC2 for attempting
to check 4
2 | 2 | 1 | 1 | 0 | 4
3i | 4 | 0 | 0 | 0 | 4
3iiA | 1 | 0 | 1 | 0 | 2
3iiB | 1 | 1 | 0 | 0 | 2
3iii | 1 | 1 | 1 | 0 | 3
3iv | 0 | 1 | 0 | 0 | 1
4i | 2 | 0 | 0 | 0 | 2
4ii | 3 | 1 | 0 | 0 | 4
4iii | 0 | 2 | 1 | 0 | 3
4iv | 3 | 0 | 0 | 0 | 3
4v | 1 | 1 | 2 | 0 | 4
5i | 1 | 1 | 0 | 0 | 2
5ii | 2 | 0 | 2 | 0 | 4
5iii | 4 | 1 | 0 | 0 | 5
5iv | 1 | 3 | 2 | 0 | 6
5v | 0 | 1 | 0 | 0 | 1
Totals | 31 | 18 | 11 | 0 | 60
Date | Version | Change
October 2019 | 2 | Amendments to the front cover rubric instructions to candidates.A binary operation $*$ is defined on the set $S = \{p, q, r, s, t\}$ by the following composition table.
\begin{center}
\begin{tabular}{c|ccccc}
$*$ & $p$ & $q$ & $r$ & $s$ & $t$ \\
\hline
$p$ & $p$ & $q$ & $r$ & $s$ & $t$ \\
$q$ & $q$ & $p$ & $s$ & $t$ & $r$ \\
$r$ & $r$ & $t$ & $p$ & $q$ & $s$ \\
$s$ & $s$ & $r$ & $t$ & $p$ & $q$ \\
$t$ & $t$ & $s$ & $q$ & $r$ & $p$
\end{tabular}
\end{center}
Determine whether $(S, *)$ is a group. [4]
\hfill \mbox{\textit{OCR MEI Further Extra Pure Q2 [4]}}