| Exam Board | AQA |
|---|---|
| Module | Paper 2 (Paper 2) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Identifying errors in proofs |
| Difficulty | Standard +0.3 Part (a) requires recognizing that checking examples isn't a proof (basic understanding of proof methodology). Part (b) requires representing consecutive odd numbers algebraically as (2n+1) and (2n+3), then showing their sum equals (2n+2)² - (2n)² using difference of squares—a standard algebraic proof with straightforward manipulation. This is easier than average A-level proof questions. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01b Logical connectives: congruence, if-then, if and only if |
| Answer | Marks | Guidance |
|---|---|---|
| 7(a) | States any correct reason | AO2.3 |
| Answer | Marks |
|---|---|
| (b) | Commences an argument, writing |
| Answer | Marks | Guidance |
|---|---|---|
| (at least two lines of argument) | AO2.1 | R1 |
| Answer | Marks | Guidance |
|---|---|---|
| algebraically | AO2.5 | R1 |
| Answer | Marks | Guidance |
|---|---|---|
| correct working | AO2.2a | R1 |
| Total | 4 | |
| Q | Marking Instructions | AO |
Question 7:
--- 7(a) ---
7(a) | States any correct reason | AO2.3 | B1 | Just checking a few cases only
proves it for those cases
(b) | Commences an argument, writing
the sum of two consecutive odd
numbers algebraically
(at least two lines of argument) | AO2.1 | R1 | Two consecutive odd numbers
can be written as
2n1 and 2n3
Their sum is
2n12n34n4
n2and (n2)2 are two square
numbers
Their difference is
(n2)2 n2
n2 4n4n2
4n4
Therefore the sum of two
consecutive odd numbers can
always be written as the
difference of two square numbers
At some point in the argument
correctly writes the difference of
two appropriate square numbers
algebraically | AO2.5 | R1
Correctly deduces the result from
correct working | AO2.2a | R1
Total | 4
Q | Marking Instructions | AO | Marks | Typical SolutionA student notices that when he adds two consecutive odd numbers together the answer always seems to be the difference between two square numbers.
He claims that this will always be true.
He attempts to prove his claim as follows:
Step 1: Check first few cases
$3 + 5 = 8$ and $8 = 3^2 - 1^2$
$5 + 7 = 12$ and $12 = 4^2 - 2^2$
$7 + 9 = 16$ and $16 = 5^2 - 3^2$
Step 2: Use pattern to predict and check a large example
$101 + 103 = 204$
subtract 1 and divide by 2 for the first number
Add 1 and divide by two for the second number
$52^2 - 50^2 = 204$ it works!
Step 3: Conclusion
The first few cases work and there is a pattern, which can be used to predict larger numbers.
Therefore, it must be true for all consecutive odd numbers.
\begin{enumerate}[label=(\alph*)]
\item Explain what is wrong with the student's "proof".
[1 mark]
\item Prove that the student's claim is correct.
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 2 Q7 [4]}}