AQA AS Level Mathematics Paper 1, June 2025: Question 5

Mark scheme

Show the mark scheme

$Mark scheme for Question 5: M1 for using one or both of a and b as negative values with a > b (AO 3.1a). R1 for demonstrating a counter example and showing a clear comparison of a simplified fraction with 1, which may be in words (AO 2.1). Typical solution gives a = 1, b = -1, leading to 1 / (-1) = -1 < 1, concluding Jayven is wrong.$

Q Marking instructions AO Marks Typical solution

5 Uses one or both of a and b as 3.1a M1 Let a = 1, b = –1

negative values with a > b 1

= –1 < 1

Demonstrates a counter 2.1 R1 −1

example and shows a clear Therefore Jayven is wrong.

comparison of a simplified

fraction with 1

May be in words

Question 5 Total 2

How to answer it

Disproof by Counter-Example (Inequalities)

📌 What this question tests

This question assesses your understanding of mathematical reasoning and proof, specifically:

Methods of Proof: Disproving a universally quantified mathematical statement by finding a single valid counter-example.
Properties of Real Numbers: Understanding how inequalities behave when dealing with negative values and division.
Clear Mathematical Communication: Demonstrating that chosen values satisfy the premise ( a > b ) while clearly contradicting the conclusion ( a/b > 1 ).

Question 5 Walkthrough

Total Marks: [2 marks]

📐 Step-by-Step Counter-Example

Step 1: Choose values satisfying the initial condition ( a > b ).
Let a = 1 and b = -1 .
Check: 1 > -1 is true.
Step 2: Evaluate the fraction a / b .
Calculate the ratio:
a / b = 1 / (-1) = -1
Step 3: Compare with the claimed condition and conclude.
Since -1 < 1 , the claim a / b > 1 is false.
Conclude clearly: Therefore, Jayven is not correct.

✅ Model Solution & Mark Scheme Breakdown

Full Credit Response:

Let a = 1 and b = -1 .
Here, 1 > -1 .
However, a / b = 1 / (-1) = -1 .
Since -1 < 1 , Jayven's statement is not correct.

M1 (AO 3.1a): Selects one or both of a and b as negative values such that a > b .
R1 (AO 2.1): Demonstrates the counter-example fully with a simplified fraction compared clearly to 1 and a concluding statement.

💡 Key Knowledge

How to Disprove: To prove a general statement false, you only ever need one specific numerical example where the condition holds but the conclusion fails.
The "Negative Trap": When dividing inequalities by a variable, the direction of the inequality depends on whether the divisor is positive or negative. Because b can be negative, dividing by b reverses the inequality sign:
If b < 0 and a > b , then a / b < 1 .
Other Valid Pairs:
- a = 2, b = -3 → 2 / (-3) = -2/3 < 1
- a = -2, b = -5 → -2 / (-5) = 2/5 = 0.4 < 1

🧠 Exam Technique & Examiner Insights

Always State the Comparison: Do not just stop at writing 1 / (-1) = -1 . The mark scheme explicitly demands an explicit comparison: state -1 < 1 or write out in words that -1 is not greater than 1.
Conclude Explicitly: Finish your counter-example with a conclusive sentence: "Hence, Jayven is incorrect" or "This disproves the claim."
Keep Numbers Simple: Always pick small integers like 1 and -1 to avoid arithmetic errors under exam pressure.

❌ Common Errors to Avoid

Picking only positive numbers: For example, choosing a = 4, b = 2 gives 4 / 2 = 2 > 1 , which supports Jayven's claim rather than disproving it!
Violating the initial condition: Picking a = -3, b = 2 . Here a / b = -1.5 < 1 , but a > b is not satisfied ( -3 > 2 is false). A counter-example must satisfy all original premises.
Division by zero: Choosing b = 0 with a = 1 . While 1 > 0 , division by zero is undefined and does not produce a valid real fraction to compare to 1.
Omitting the final conclusion: Leaving the calculation hanging without stating why it disproves the claim can lose the final reasoning mark ( R1 ).

Topics

Pure Mathematics · A: Proof