| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Discrete (Further Paper 3 Discrete) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Prove group-theoretic identities |
| Difficulty | Standard +0.8 This is a Further Maths discrete mathematics question on binary operations requiring understanding of commutativity, associativity, and identity elements. Part (a)(i) is routine, but (a)(ii) requires careful algebraic manipulation with modular arithmetic, and part (b) involves solving a quadratic congruence condition. The multi-step reasoning and abstract algebraic concepts place it moderately above average difficulty, though it follows standard patterns for Further Maths group theory questions. |
| Spec | 8.02e Finite (modular) arithmetic: integers modulo n8.03a Binary operations: and their properties on given sets8.03b Cayley tables: construct for finite sets under binary operation |
| Answer | Marks |
|---|---|
| 9(a)(i) | Deduces the operation is |
| Answer | Marks | Guidance |
|---|---|---|
| missing | 2.2a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Total | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 9(a)(ii) | Sets up test for associativity by |
| Answer | Marks | Guidance |
|---|---|---|
| missing | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| missing | 2.1 | R1 |
| Total | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 9(b) | Translates problem by applying |
| Answer | Marks | Guidance |
|---|---|---|
| missing | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| the addition must be modulo 11 | 2.2a | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| and no others | 3.2a | A1 |
| Total | 3 | |
| Question total | 6 | |
| Q | Marking instructions | AO |
Question 9:
--- 9(a)(i) ---
9(a)(i) | Deduces the operation is
commutative by considering
both x y and y x and
shows that they are the same
Condone ( m o d k 2 − 1 6 k + 7 4 )
missing | 2.2a | B1 | x y = x + y + 8 ( m o d k 2 − 1 6 k + 7 4 )
y x = y + x + 8 ( m o d k 2 − 1 6 k + 7 4 )
= x + y + 8 ( m o d k 2 − 1 6 k + 7 4 )
Total | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 9(a)(ii) ---
9(a)(ii) | Sets up test for associativity by
considering combinations of the
form ( x y ) z and x ( y z )
and simplifies one combination
to obtain x + y + z + 1 6
Condone ( m o d k 2 − 1 6 k + 7 4 )
missing | 1.1a | M1 | ( ) x y z
= ( ) 8 x + y + + z + 8 ( m o d k 2 − 1 6 k + 7 4 )
( ) x y z
= 1 6 x + y + z + ( m o 2 d 1 k − 6 k + 7 4 )
x ( ) y z
= ( 8 x + y + z + ) + 8 ( m o d k 2 − 1 6 k + 7 4 )
( ) x y z
= 1 6 x + y + z + ( m o 2 d 1 k − 6 k + 7 4 )
( ) x y z = ( x ) y z
Hence, the binary operation is
associative.
Completes a reasoned
argument and concludes that
the binary operation is
associative.
Condone ( m o d k 2 − 1 6 k + 7 4 )
missing | 2.1 | R1
Total | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 9(b) ---
9(b) | Translates problem by applying
binary operation between 3 and
another positive integer
Condone ( m o d k 2 − 1 6 k + 7 4 )
missing | 3.1a | M1 | 3 x = 3 + x + 8 ( m o d k 2 − 1 6 k + 7 4 )
3 x = x + 1 1 ( m o d k 2 − 1 6 k + 7 4 )
3x= x ( mod k2−16k+74 )
k2−16k+74=11
k2−16k+63=0
k =7,9
Uses the condition for 3 to be an
identity element to deduce that
the addition must be modulo 11 | 2.2a | A1
Solves the quadratic equation to
find the two correct values for k
and no others | 3.2a | A1
Total | 3
Question total | 6
Q | Marking instructions | AO | Marks | Typical solutionThe binary operation $\oplus$ acts on the positive integers $x$ and $y$ such that
$$x \oplus y = x + y + 8 \pmod{k^2 - 16k + 74}$$
where $k$ is a positive integer.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that $\oplus$ is commutative.
[1 mark]
\item Determine whether or not $\oplus$ is associative.
Fully justify your answer.
[2 marks]
\end{enumerate}
\item Find the values of $k$ for which 3 is an identity element for the set of positive integers under $\oplus$
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2022 Q9 [6]}}