AQA Further AS Paper 2 Discrete 2021 June — Question 4 5 marks

Exam BoardAQA
ModuleFurther AS Paper 2 Discrete (Further AS Paper 2 Discrete)
Year2021
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGroups
TypeProve group-theoretic identities
DifficultyStandard +0.3 This is a straightforward abstract algebra question requiring basic proof techniques. Part (a) involves showing ab+1 = ba+1 (trivial commutativity of multiplication), and part (b) requires computing (a*b)*c vs a*(b*c) and showing they differ with a simple counterexample. Both parts are routine applications of definitions with no conceptual depth or problem-solving required beyond substitution and algebraic manipulation.
Spec8.03a Binary operations: and their properties on given sets
The binary operation \(*\) is defined as $$a * b = ab + 1 \quad \text{where } a, b \in \mathbb{R}$$
  1. Prove that \(*\) is commutative on \(\mathbb{R}\) [2 marks]
  2. Prove that \(*\) is not associative on \(\mathbb{R}\) [3 marks]
Question 4:
AnswerMarks
4(a)Sets up a test for commutativity
using 2 distinct elements by
AnswerMarks Guidance
considering1.1a M1
𝑏𝑏Asβˆ— π‘Žπ‘Ž = π‘π‘π‘Žπ‘Ž+1
then
π‘Žπ‘Žπ‘π‘+1 = π‘π‘π‘Žπ‘Ž+1
Thereπ‘Žπ‘Žfoβˆ—re𝑏𝑏 = is𝑏𝑏 cβˆ—omπ‘Žπ‘Žmutative
βˆ—
π‘π‘βˆ—π‘Žπ‘Ž
Constructs a rigorous
mathematical argument to prove
AnswerMarks Guidance
that is commutative2.1 R1
βˆ—
AnswerMarks Guidance
Total2
QMarking instructions AO
AnswerMarks Guidance
4(b)Sets up a test for associativity
using 3 elements1.1a M1
= 3βˆ—3
= 10
1βˆ—( 2βˆ—3) = 1βˆ—(2Γ—3+1)
= 1βˆ—7
=As 8 β‰ 
then is not associative
(1βˆ—2)βˆ—3 1βˆ—(2βˆ—3)
βˆ—
Finds two correct values for a
proof by counter example
or
Finds two correct simplified
AnswerMarks Guidance
algebraic expressions1.1b A1
Constructs a rigorous
mathematical argument to prove
AnswerMarks Guidance
that is not associative2.1 R1
βˆ—
AnswerMarks Guidance
Total3
Question total5
QMarking instructions AO 
Question 4:
--- 4(a) ---
4(a) | Sets up a test for commutativity
using 2 distinct elements by
considering | 1.1a | M1 | π‘Žπ‘Ž βˆ—π‘π‘ = π‘Žπ‘Žπ‘π‘+1
𝑏𝑏Asβˆ— π‘Žπ‘Ž = π‘π‘π‘Žπ‘Ž+1
then
π‘Žπ‘Žπ‘π‘+1 = π‘π‘π‘Žπ‘Ž+1
Thereπ‘Žπ‘Žfoβˆ—re𝑏𝑏 = is𝑏𝑏 cβˆ—omπ‘Žπ‘Žmutative
βˆ—
π‘π‘βˆ—π‘Žπ‘Ž
Constructs a rigorous
mathematical argument to prove
that is commutative | 2.1 | R1
βˆ—
Total | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 4(b) ---
4(b) | Sets up a test for associativity
using 3 elements | 1.1a | M1 | (1βˆ—2 )βˆ—3 = (1Γ—2+1)βˆ—3
= 3βˆ—3
= 10
1βˆ—( 2βˆ—3) = 1βˆ—(2Γ—3+1)
= 1βˆ—7
=As 8 β‰ 
then is not associative
(1βˆ—2)βˆ—3 1βˆ—(2βˆ—3)
βˆ—
Finds two correct values for a
proof by counter example
or
Finds two correct simplified
algebraic expressions | 1.1b | A1
Constructs a rigorous
mathematical argument to prove
that is not associative | 2.1 | R1
βˆ—
Total | 3
Question total | 5
Q | Marking instructions | AO | Marks | Typical solution
The binary operation $*$ is defined as
$$a * b = ab + 1 \quad \text{where } a, b \in \mathbb{R}$$

\begin{enumerate}[label=(\alph*)]
\item Prove that $*$ is commutative on $\mathbb{R}$
[2 marks]

\item Prove that $*$ is not associative on $\mathbb{R}$
[3 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2021 Q4 [5]}}
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