AQA A-Level Mathematics Paper 3, June 2025: Question 1

AQA A-Level Mathematics • Pure Maths

Proof by Counter-Example: Rational & Irrational Numbers

What this question tests

Understanding the definition of rational and irrational numbers.
Applying the method of proof (or disproof) by counter-example to refute a mathematical conjecture.
Verifying that all conditions of a counter-example are satisfied: both input conditions must hold, while the concluding statement fails.

Question 1

Multiple Choice • Disproof by Counter-Example • [1 Mark]

Question Statement: A student states that the product of two irrational numbers is always irrational. One of the options is a counter example that shows the student's statement is incorrect. Identify the counter example. Circle your answer.

e × 0 = 0

(1/π) × π = 1 ✓

(1/√2) × √6 = √3

2 × √3 = √6

✅ Correct Answer

The correct option to circle is the second option:

(1/π) × π = 1

Both 1/π and π are irrational numbers, yet their product is 1 , which is a rational number (an integer).

💡 Key Knowledge

Rational Number: Any number that can be expressed in the form a/b , where a and b are integers and b ≠ 0 .
Irrational Number: A real number that cannot be written as a simple fraction (e.g., π, e, √2, √3). The reciprocal of an irrational number is also irrational.
Counter-example Requirements:
1. Number A must be irrational.
2. Number B must be irrational.
3. The product (A × B) must be rational.

🔧 Step-by-Step Analysis of Each Option

Option 1: e × 0 = 0 — Invalid
While e is irrational and the product 0 is rational, the number 0 is rational ( 0 = 0/1 ). The statement requires the product of two irrational numbers.
Option 2: (1/π) × π = 1 — Valid Counter-Example
• First factor: 1/π is irrational.
• Second factor: π is irrational.
• Product: 1 is rational ( 1 = 1/1 ).
Since two irrational numbers produce a rational number, this disproves the student's assertion.
Option 3: (1/√2) × √6 = √3 — Does Not Disprove
Both factors are irrational, but the product √3 is also irrational. This supports the student's statement rather than refuting it.
Option 4: 2 × √3 = √6 — Invalid
The number 2 is rational, not irrational. Furthermore, the product √6 is irrational, and the multiplication itself is incorrect ( 2 × √3 = √12 ≠ √6 ).

🧠 Exam Technique & Examiner Commentary

Target the hypothesis first: Always check whether an example satisfies all premises of the original conditional statement. A counter-example must obey the "if" part to disprove the "then" part.
Assessment Objective (AO 2.3): This question assesses the ability to construct or identify a valid mathematical argument and critique reasoning.
Clean presentation: Clearly circle only one answer in the exam paper. Crossing out and circling multiple answers without clear intent will forfeit the mark.

❌ Common Errors & Traps

Mistaking 0 for an irrational number: A common trap is selecting e × 0 = 0 because it produces a rational result ( 0 ), forgetting that 0 is rational.
Selecting an example that agrees: Choosing (1/√2) × √6 = √3 because both factors are irrational, forgetting that a counter-example must produce a rational output.
Assuming irrational × irrational is always irrational: Many candidates intuitively believe irrational numbers are closed under multiplication. Other classic counter-examples include √2 × √2 = 2 or (3 - √5)(3 + √5) = 4 .

Mark Scheme Breakdown:
• R1 (AO 2.3): Circles the second answer (1/π) × π = 1 . No working needs to be shown for this 1-mark question.