| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2020 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Standard +0.8 This is a Further Maths group theory question requiring completion of a Cayley table, verification of group axioms (closure, identity, inverses), identification of proper subgroups, and finding generators. While systematic, it demands understanding of abstract algebra concepts beyond standard A-level and involves multiple non-trivial steps including checking all axioms and analyzing subgroup structure. |
| Spec | 8.03a Binary operations: and their properties on given sets8.03b Cayley tables: construct for finite sets under binary operation8.03c Group definition: recall and use, show structure is/isn't a group8.03f Subgroups: definition and tests for proper subgroups8.03g Cyclic groups: meaning of the term |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | × 2 4 6 8 10 12 |
| Answer | Marks |
|---|---|
| 12 10 6 2 12 8 4 | B1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | For any two lines (Rs or Cs) correct |
| Answer | Marks |
|---|---|
| (b) | Closed since no other elements appear in the table |
| Answer | Marks |
|---|---|
| (Hence a group) | B1 |
| Answer | Marks |
|---|---|
| [4] | 2.4 |
| Answer | Marks |
|---|---|
| 2.5 | Don’t accept “closed, from table” only |
| Answer | Marks | Guidance |
|---|---|---|
| (c) | (i) | {8, 6} {8, 2, 4} |
| [2] | 2.2a 1.1 | One correct; both (and no extras). Ignore {8} and G |
| (ii) | 10, 12 | B1 B1 |
| [2] | 1.1 1.1 | One correct; both (and no extras) |
| 4 | 8 | 2 |
Question 4:
4 | (a) | × 2 4 6 8 10 12
14
2 4 8 12 2 6 10
4 8 2 10 4 12 6
6 12 10 8 6 4 2
8 2 4 6 8 10 12
10 6 12 4 10 2 8
12 10 6 2 12 8 4 | B1
B1
B1
B1
[4] | 1.1
1.1
1.1
1.1 | For any two lines (Rs or Cs) correct
For at least two Rs and two Cs correct
For LSP applying to complete table
For symmetry about main diagonal
(Must be fully correct for all 4 marks)
(b) | Closed since no other elements appear in the table
Identity is 8
Inverses: 6 is self-inverse
2 – 1 = 4 and 4 – 1 = 2; 10 – 1 = 12 and 12 – 1 = 10
(Hence a group) | B1
B1
B1
B1
[4] | 2.4
2.2a
1.2
2.5 | Don’t accept “closed, from table” only
Any clear indication of inverses (not just statement they exist)
That is, (2, 4) and (10, 12) are inverse-pairs
Associativity and conclusion not required
(c) | (i) | {8, 6} {8, 2, 4} | B1 B1
[2] | 2.2a 1.1 | One correct; both (and no extras). Ignore {8} and G
(ii) | 10, 12 | B1 B1
[2] | 1.1 1.1 | One correct; both (and no extras)
4 | 8 | 2 | 10 | 4 | 12 | 64
\begin{enumerate}[label=(\alph*)]
\item For the set $S = \{ 2,4,6,8,10,12 \}$, under the operation $\times _ { 14 }$ of multiplication modulo 14, complete the Cayley table given in the Printed Answer Booklet.
\item Show that ( $S , \times _ { 14 }$ ) forms a group, $G$. (You may assume that $\times _ { 14 }$ is associative.)
\item \begin{enumerate}[label=(\roman*)]
\item Write down all the proper subgroups of $G$.
\item Given that $G$ is cyclic, write down all the possible generators of $G$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2020 Q4 [12]}}