| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Divisibility proof for all integers |
| Difficulty | Moderate -0.3 Part (a) is a straightforward algebraic proof requiring factorization (3n(3n+2)) and recognizing that consecutive even/odd numbers give a factor of 2, combined with the factor of 3. Part (b) is immediate once (a) is done. This is a standard AS-level proof question with clear structure, requiring only basic factorization and parity arguments—slightly easier than average due to its routine nature and limited conceptual demand. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps |
| Answer | Marks |
|---|---|
| 12(a) | Begins to construct a rigorous |
| Answer | Marks | Guidance |
|---|---|---|
| the given expression | AO2.1 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| extracts 12 as a common factor | AO1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| when n is an even number | AO2.1 | R1 |
| (b) | Uses a counter example by |
| Answer | Marks | Guidance |
|---|---|---|
| multiple of 12 | AO2.2a | R1 |
| Answer | Marks | Guidance |
|---|---|---|
| Total | 4 | |
| Q | Marking Instructions | AO |
Question 12:
--- 12(a) ---
12(a) | Begins to construct a rigorous
mathematical proof by
generalising the form of an even
number and substituting it into
the given expression | AO2.1 | M1 | Let n = 2m
9n2 + 6n = 9(2m)2 + 6(2m)
= 36m2 + 12m
= 12(3m2 + m)
Hence 12 is a factor of the
expression 9n2 + 6n when n is any
even number
Simplifies expression and
extracts 12 as a common factor | AO1.1b | A1
Completes rigorous proof – well
explained. A statement is
required that links the factor of
12 to the expression 9n + 6n
when n is an even number | AO2.1 | R1
(b) | Uses a counter example by
substituting any odd number into
the expression and shows that
the resulting value is not a
multiple of 12 | AO2.2a | R1 | Let n = 1
9(1)2 + 6(1) = 15
12 is not a factor of 15 and hence
statement is not true for all
integers n
Total | 4
Q | Marking Instructions | AO | Marks | Typical Solution\begin{enumerate}[label=(\alph*)]
\item Given that $n$ is an even number, prove that $9n^2 + 6n$ has a factor of 12 [3 marks]
\item Determine if $9n^2 + 6n$ has a factor of 12 for any integer $n$. [1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 Q12 [4]}}