AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 24

Mark scheme

Show the mark scheme

Q Answer Mark Comments

Partially or fully factorises eg 15x(4x3 – x) or x2(60x2 – 15)

numerator M1

or 15x2(2x – 1)(2x + 1)

Factorises at least one eg –(2x + 1) and/or 3(2x – 1)

denominator

or M1

eg –12x2 + (6x – 6x +) 3

correct multiplication of the

denominators may be in a grid

Converts numerators and dep on M1M1

denominators into terms which can 2

be fully cancelled 15x (2x −1)(2x +1)

eg

−(2x +1)3(2x −1)

15x2(2x − 1)(2x + 1) 1

or ×

M1dep −(2x +1) 3(2x −1)

5x2(12x2 − 3)

or 2

−(12x − 3)

factorisation and cancelling may be done

in stages

15x2

or –5x2 with M3 awarded oe with full cancelling of algebraic terms

−3

and A1

2 condone explanation that x2 must be

explanation that x cannot be

negative positive

Additional Guidance

20x4 − 5x2 20x4 − 5x2

or 2 implies M1M1 as 3 has been cancelled

( 2−x −1)(2x −1) −4x +1 M1M1

from both

5x2(2x + 1)

implies complete factorisation of numerator and denominator

−(2x +1) M1M1M1

as 3 and (2x – 1) have been cancelled from both

How to answer it

AQA GCSE Mathematics • Higher Tier

Algebraic Proof: Fractions & Non-Positive Expressions

What this question tests

This question assesses advanced algebraic manipulation at Grade 8/9 level, specifically:

Factorising polynomials completely: Taking out a common algebraic factor and spotting a difference of two squares.
Factorising linear expressions: Factoring out numerical terms and handling negative signs carefully (e.g. taking out −1).
Multiplying and simplifying algebraic fractions: Dividing out common binomial factors between numerators and denominators.
Constructing a formal algebraic proof: Explaining why an expression such as −5x² can never be positive by referencing the property that squares of real numbers cannot be negative (x² ≥ 0).

Question 24 Walkthrough

Prove that (60x⁴ − 15x²) / (−2x − 1) × 1 / (6x − 3) can never be positive. [4 marks]

📐 Step-by-Step Solution

1 Factorise the numerator completely:

60x⁴ − 15x² = 15x²(4x² − 1)
Notice that (4x² − 1) is a difference of two squares:
= 15x²(2x − 1)(2x + 1)

2 Factorise both denominators:

First denominator: −2x − 1 = −(2x + 1)
Second denominator: 6x − 3 = 3(2x − 1)

3 Combine into a single fraction and cancel common terms:

[15x²(2x − 1)(2x + 1)] / [−(2x + 1) × 3(2x − 1)]

Cancel the common brackets (2x − 1) and (2x + 1):
= 15x² / (−1 × 3)
= 15x² / (−3)
= −5x²

4 Complete the proof with a reasoned conclusion:

For all real numbers, x² ≥ 0 (any real number squared cannot be negative).

Therefore, multiplying a non-negative number by −5 gives a value that is less than or equal to 0:

−5x² ≤ 0 → It can never be positive.

✅ Mark Scheme Breakdown

M1: Partially or fully factorises numerator.
15x(4x³ − x) , x²(60x² − 15) , or 15x²(2x − 1)(2x + 1) .
M1: Factorises at least one denominator OR correctly expands product of denominators.
e.g. −(2x + 1) and/or 3(2x − 1) , or expands to −12x² + 3 .
M1 (dep): Converts numerator and denominator into forms that allow full algebraic cancellation.
Dependent on previous M1 M1.
A1: Reaches −5x² (or 15x² / −3 ) AND provides a clear statement that x² cannot be negative (or x² ≥ 0 ).

💡 Key Knowledge

Difference of Two Squares:
a² − b² = (a − b)(a + b)
Recognise 4x² − 1 = (2x − 1)(2x + 1) immediately.
Extracting a Negative Sign:
Expressions like −2x − 1 can be written as −(2x + 1) or −1(2x + 1) to reveal shared factors.
Properties of Squared Numbers:
For any real number x, x² ≥ 0 . Since x² is always 0 or positive, −5 × (non-negative) ≤ 0 . It can be zero, but never positive.

🧠 Exam Technique

Look for matching brackets: When multiplying algebraic fractions, you should anticipate brackets cancelling. Seeing 6x − 3 suggests a factor of (2x − 1) might be lurking in the numerator!
Don't skip the conclusion: Reaching −5x² is only worth 3 marks. You must write a written explanation about x² being non-negative to earn the final 4th mark.
Maintain equivalence: Keep negative signs clearly visible. Don't lose a minus sign when cancelling fractions.

❌ Common Errors

Incomplete Factorisation: Stopping at 15x²(4x² − 1) and failing to spot that 4x² − 1 factors further into (2x − 1)(2x + 1) .
Sign Errors with Negatives: Forgetting the minus sign when dealing with −2x − 1 , incorrectly simplifying to 5x² instead of −5x² .
Missing Explanation: Reaching −5x² and simply stopping without explaining why this proves the expression can never be positive.
Incorrect Reason: Saying "it is negative because it has a minus sign" is incomplete without specifying that x² itself is never negative.

Examiner Insight: This Grade 9 question separated students who merely memorised algebraic steps from those who understood proof. Full-mark candidates showed organised factorisation on both top and bottom before cancelling, clearly arrived at −5x² , and concluded with: "Since x² ≥ 0 for all x, −5x² ≤ 0, which cannot be positive."

Topics

Algebra · 3.2.1 Notation, vocabulary and manipulation