| Exam Board | AQA |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Contradiction proof about integers |
| Difficulty | Standard +0.8 Part (a) is trivial (1 mark for stating 6-8-10), but part (b) requires a modular arithmetic proof that odd squares are ≡1 (mod 4), leading to a contradiction in a²+b²=c². This is a non-routine proof requiring number theory insight beyond standard A-level content, though the technique is accessible once recognized. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01d Proof by contradiction |
| Answer | Marks | Guidance |
|---|---|---|
| 6(a) | States an appropriate even | |
| Pythagorean triple | 2.2a | B1 |
| Answer | Marks |
|---|---|
| 6(b) | Begins an appropriate method of |
| Answer | Marks | Guidance |
|---|---|---|
| unknowns | 3.1a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| words or algebraically | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| CSO | 2.1 | R1 |
| Total | 4 | |
| Q | Marking instructions | AO |
Question 6:
--- 6(a) ---
6(a) | States an appropriate even
Pythagorean triple | 2.2a | B1 | a=6
b=8
c=10
--- 6(b) ---
6(b) | Begins an appropriate method of
proof assuming at least two sides
are odd eg states ‘assume a, b
odd’ or defines a, b (or c)
algebraically with different
unknowns | 3.1a | B1 | Assume a and b are odd
so a = 2m + 1 and b = 2n + 1
+
= + 24m + 1 + 2 + 4n + 1
(2𝑚𝑚+1) (2𝑛𝑛+1)
= 2 2
4𝑚𝑚 4 𝑛𝑛
2 2
wh2i(c2h𝑚𝑚 is e+ve2n𝑚𝑚, +so 2𝑛𝑛 is+ e2v𝑛𝑛en+, 1s)o c
is even. Therefore i2t is not possible
for all three to be ocdd.
Uses Pythagoras’ theorem with
at least two odd sides either in
words or algebraically | 1.1a | M1
Completes rigorous argument to
prove the required result
CSO | 2.1 | R1
Total | 4
Q | Marking instructions | AO | Mark | Typical solutionThe three sides of a right-angled triangle have lengths $a$, $b$ and $c$, where $a, b, c \in \mathbb{Z}$
\includegraphics{figure_6}
\begin{enumerate}[label=(\alph*)]
\item State an example where $a$, $b$ and $c$ are all even. [1 mark]
\item Prove that it is not possible for all of $a$, $b$ and $c$ to be odd. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 2019 Q6 [4]}}