| Exam Board | AQA |
|---|---|
| Module | Paper 1 (Paper 1) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Rational and irrational number properties |
| Difficulty | Standard +0.8 Part (a) requires identifying the exception (zero), which is straightforward. Part (b) demands a formal proof by contradiction or direct argument involving properties of rational/irrational numbers, requiring careful logical reasoning and proof technique beyond routine A-level exercises. The proof structure is non-trivial but accessible to strong students. |
| Spec | 1.01d Proof by contradiction |
| Answer | Marks | Guidance |
|---|---|---|
| 16(a) | Identifies zero as number for which | |
| student’s argument is not true | AO1.2 | B1 |
| (b) | Uses ‘proof by contradiction’ |
| Answer | Marks | Guidance |
|---|---|---|
| argument | AO2.1 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| irrational numbers in symbolic form | AO2.5 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| must be irrational | AO2.2a | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Must define p q c d as integers | AO2.1 | R1 |
| Total | 5 | |
| Q | Marking Instructions | AO |
Question 16:
--- 16(a) ---
16(a) | Identifies zero as number for which
student’s argument is not true | AO1.2 | B1 | 0
(b) | Uses ‘proof by contradiction’
Must see commencement of
argument including stated
assumption and at least two lines of
argument | AO2.1 | M1 | Let a be irrational, and b be a
non-zero rational, so
c
b , where c, d ; c, d ≠ 0
d
Assume ab is rational, so
p
ab , where p, q ; q ≠ 0
q
ac p
d q
pd
a
qc
so a is rational, which is a
contradiction
Hence ab must be irrational
Represents product of rational and
irrational numbers in symbolic form | AO2.5 | M1
Correctly deduces that the product
must be irrational | AO2.2a | A1
Completes a rigorous mathematical
argument, proving that a non-zero
rational multiplied by an irrational is
irrational
Must start with initial assumptions
and prove the result convincingly
Must define p q c d as integers | AO2.1 | R1
Total | 5
Q | Marking Instructions | AO | Marks | Typical SolutionA student argues that when a rational number is multiplied by an irrational number the result will always be an irrational number.
\begin{enumerate}[label=(\alph*)]
\item Identify the rational number for which the student's argument is not true.
[1 mark]
\item Prove that the student is right for all rational numbers other than the one you have identified in part (a).
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 1 Q16 [5]}}