| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Prove group-theoretic identities |
| Difficulty | Standard +0.3 This is a straightforward abstract algebra question testing basic properties of binary operations. Part (a) requires simple algebraic manipulation to show commutativity (a∇b = b∇a is immediate from commutativity of addition and multiplication). Part (b) requires expanding (a∇b)∇c and a∇(b∇c) and showing they're equal—mechanical algebra with no conceptual difficulty. While this is Further Maths content, it's a standard textbook exercise requiring only careful algebraic manipulation, making it slightly easier than average overall. |
| Spec | 8.03a Binary operations: and their properties on given sets |
| Answer | Marks | Guidance |
|---|---|---|
| 7(a) | Obtains b∇a = b+a+ba | 3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| conclude that ∇ is commutative | 2.1 | R1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 7(b) | Writes down both (a∇b) ∇c | |
| and a∇(b∇c) | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a∇b) ∇c or a∇(b∇c) correctly | 1.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| associative | 2.1 | R1 |
| Subtotal | 3 | |
| Question total | 5 | |
| Q | Marking instructions | AO |
Question 7:
--- 7(a) ---
7(a) | Obtains b∇a = b+a+ba | 3.1a | M1 | b∇a = b+a+ba
= a+b+ab
= a∇b
Hence∇ is commutative on Ρ
Completes reasoned argument to
conclude that ∇ is commutative | 2.1 | R1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 7(b) ---
7(b) | Writes down both (a∇b) ∇c
and a∇(b∇c) | 3.1a | M1 | (a∇b) ∇c = (a+b+ab) ∇c
= a+b+ab +c+ac+bc
+abc
a∇(b ∇c) = a ∇(b+c+bc)
= a+b+c +bc+ab+ac
+abc
As (a∇b) ∇c = a ∇(b∇c)
then ∇ is associative on Ρ
Fully expands at least one of
(a∇b) ∇c or a∇(b∇c) correctly | 1.1b | M1
Completes reasoned argument,
expanding both expressions
correctly to prove that ∇ is
associative | 2.1 | R1
Subtotal | 3
Question total | 5
Q | Marking instructions | AO | Marks | Typical solutionThe binary operation $\nabla$ is defined as
$a \nabla b = a + b + ab$ where $a, b \in \mathbb{R}$
\begin{enumerate}[label=7 (\alph*)]
\item Determine if $\nabla$ is commutative on $\mathbb{R}$
Fully justify your answer.
[2 marks]
\item Prove that $\nabla$ is associative on $\mathbb{R}$
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2024 Q7 [5]}}