AQA Further AS Paper 2 Discrete 2019 June — Question 5 5 marks

Exam BoardAQA
ModuleFurther AS Paper 2 Discrete (Further AS Paper 2 Discrete)
Year2019
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGroups
TypeComplete or analyse Cayley table
DifficultyStandard +0.3 This is a straightforward Further Maths question on group theory basics. Part (a) requires simple modular arithmetic (multiplication mod 4). Part (b)(i) uses commutativity to fill in a small Cayley table by symmetry. Part (b)(ii) requires checking associativity, which is systematic but tedious for a 4-element set. While this is Further Maths content, it's mostly mechanical application of definitions with minimal conceptual insight required, making it slightly easier than average overall.
Spec8.02e Finite (modular) arithmetic: integers modulo n8.03a 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 group
5
  1. Complete the Cayley table in Figure 1 for multiplication modulo 4 \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-08_761_1017_434_493}
    \end{figure} 5
  2. The set \(S\) is defined as $$S = \{ a , b , c , d \}$$ Figure 2 shows an incomplete Cayley table for \(S\) under the commutative binary operation • \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2}
    •\(a\)\(b\)\(c\)\(d\)
    \(a\)\(b\)\(a\)\(a\)\(c\)
    \(b\)\(c\)\(c\)
    \(c\)\(d\)\(d\)
    \(d\)\(d\)\(d\)
    \end{table} 5 (b) (i) Complete the Cayley table in Figure 2. 5 (b) (ii) Determine whether the binary operation • is associative when acting on the elements of \(S\). Fully justify your answer.
Question 5:
Part 5(a):
AnswerMarks
Partially completes Cayley table (\(\times_4\)) with 3 correct rows or columnsM1
Fully correct table: rows 0,1,2,3 giving \((0,0,0,0)\), \((0,1,2,3)\), \((0,2,0,2)\), \((0,3,2,1)\)A1
Part 5(b)(i):
AnswerMarks Guidance
Completes operation table for \(S = \{*, a, b, c, d\}\) correctlyB1 Full table required
Part 5(b)(ii):
AnswerMarks Guidance
Sets up test for associativity with at least two different elements of \(S\): \(a \bullet (b \bullet c) = a \bullet d = c\) and \((a \bullet b) \bullet c = a \bullet c = a\)M1 Must use at least two different elements
\(a \bullet (b \bullet c) \neq (a \bullet b) \bullet c\), therefore \(\bullet\) is not associativeR1 Complete mathematical argument required 
# Question 5:

## Part 5(a):
| Partially completes Cayley table ($\times_4$) with 3 correct rows or columns | M1 | |
| Fully correct table: rows 0,1,2,3 giving $(0,0,0,0)$, $(0,1,2,3)$, $(0,2,0,2)$, $(0,3,2,1)$ | A1 | |

## Part 5(b)(i):
| Completes operation table for $S = \{*, a, b, c, d\}$ correctly | B1 | Full table required |

## Part 5(b)(ii):
| Sets up test for associativity with at least two different elements of $S$: $a \bullet (b \bullet c) = a \bullet d = c$ and $(a \bullet b) \bullet c = a \bullet c = a$ | M1 | Must use at least two different elements |
| $a \bullet (b \bullet c) \neq (a \bullet b) \bullet c$, therefore $\bullet$ is not associative | R1 | Complete mathematical argument required |

---
5
\begin{enumerate}[label=(\alph*)]
\item Complete the Cayley table in Figure 1 for multiplication modulo 4

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
  \includegraphics[alt={},max width=\textwidth]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-08_761_1017_434_493}
\end{center}
\end{figure}

5
\item The set $S$ is defined as

$$S = \{ a , b , c , d \}$$

Figure 2 shows an incomplete Cayley table for $S$ under the commutative binary operation •

\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\begin{tabular}{|l|l|l|l|l|}
\hline
• & $a$ & $b$ & $c$ & $d$ \\
\hline
$a$ & $b$ & $a$ & $a$ & $c$ \\
\hline
$b$ &  & $c$ &  & $c$ \\
\hline
$c$ &  & $d$ & $d$ &  \\
\hline
$d$ &  &  & $d$ & $d$ \\
\hline
\end{tabular}
\end{center}
\end{table}

5 (b) (i) Complete the Cayley table in Figure 2.

5 (b) (ii) Determine whether the binary operation • is associative when acting on the elements of $S$.

Fully justify your answer.
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2019 Q5 [5]}}
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