| Exam Board | AQA |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Identifying errors in proofs |
| Difficulty | Standard +0.3 This is a proof question testing understanding of rational/irrational numbers and basic algebraic manipulation. Part (a)(i) requires spotting an algebraic error in adding fractions (1/a + 1/b ≠ 2/(a+b)), which is straightforward. Part (a)(ii) needs the correct formula (a+b)/(ab) and recognizing rationals are closed under division. Part (b) is a standard proof by contradiction following a template structure. While it involves proof writing, the concepts are elementary and the steps are routine for A-level students who have practiced proof techniques. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01d Proof by contradiction |
| Answer | Marks | Guidance |
|---|---|---|
| 7(a)(i) | Identifies the error lies in step 1 | |
| without contradiction. | 2.3 | E1 |
| Answer | Marks |
|---|---|
| Subtotal | 1 |
| Answer | Marks |
|---|---|
| 7(a)(ii) | Recalls correct addition |
| Answer | Marks | Guidance |
|---|---|---|
| ab ab | 1.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| r ational 𝑎𝑎𝑏𝑏 | 2.1 | R1 |
| Subtotal | 2 |
| Answer | Marks |
|---|---|
| 7(b) | States assumption to begin |
| Answer | Marks | Guidance |
|---|---|---|
| b d b d | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone missing b,d ≠0 | 2.5 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| bd | 1.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| (including b,d ≠0) | 2.1 | R1 |
| Subtotal | 4 | |
| Question Total | 7 | |
| Q | Marking instructions | AO |
Question 7:
--- 7(a)(i) ---
7(a)(i) | Identifies the error lies in step 1
without contradiction. | 2.3 | E1 | Mistake is
1 1 2
𝑎𝑎+ 𝑏𝑏 = 𝑎𝑎+𝑏𝑏
Subtotal | 1
--- 7(a)(ii) ---
7(a)(ii) | Recalls correct addition
b a
Accept +
ab ab | 1.1b | M1 | 1 1 𝑎𝑎+𝑏𝑏
a + b is rational and ab is rational
𝑎𝑎+ 𝑏𝑏 = 𝑎𝑎𝑏𝑏
and therefore
1 1
+ is rational.
a b
Completes rigorous argument
to complete proof. Must state
that abis rational (and non-
a+bis
zero) and rational and
1 1
conclude that + or is
a b
𝑎𝑎+𝑏𝑏
r ational 𝑎𝑎𝑏𝑏 | 2.1 | R1
Subtotal | 2
--- 7(b) ---
7(b) | States assumption to begin
proof by contradiction may
PI by
a c a c
−x= or x− =
b d b d | 3.1a | M1 | Assume that the difference
between a rational and an irrational
number is rational.
a c
−x =
b d
Where a,b,cand d are integers,
b,d ≠0 and x is irrational
a c
x= −
b d
ad cb
= −
bd bd
ad −cb
=
bd
Hence x is rational. This is a
contradiction hence the difference
of any rational number and any
irrational number is irrational.
Uses language and notation
correctly to state initial
assumptions:
States their a,b,cand d are
integers
and
x is irrational do not accept the
irrational written as a fraction
Condone missing b,d ≠0 | 2.5 | A1
Demonstrates that x can be
expressed as a rational number
ad −cb
by obtaining x= OE
bd | 1.1b | M1
Completes rigorous argument
to prove the required result,
clearly explaining where the
contradiction lies with ALL
assumptions correct at the start
(including b,d ≠0) | 2.1 | R1
Subtotal | 4
Question Total | 7
Q | Marking instructions | AO | Mark | Typical solution$a$ and $b$ are two positive irrational numbers.
The sum of $a$ and $b$ is rational.
The product of $a$ and $b$ is rational.
Caroline is trying to prove $\frac{1}{a} + \frac{1}{b}$ is rational.
Here is her proof:
Step 1 \quad $\frac{1}{a} + \frac{1}{b} = \frac{2}{a + b}$
Step 2 \quad $2$ is rational and $a + b$ is non-zero and rational.
Step 3 \quad Therefore $\frac{2}{a + b}$ is rational.
Step 4 \quad Hence $\frac{1}{a} + \frac{1}{b}$ is rational.
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Identify Caroline's mistake.
[1 mark]
\item Write down a correct version of the proof.
[2 marks]
\end{enumerate}
\item Prove by contradiction that the difference of any rational number and any irrational number is irrational.
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 2 2020 Q7 [7]}}