AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 24

📌 What this question tests

This question assesses higher-tier algebraic proportionality and proportional change:

Setting up a direct proportion equation involving a non-linear term: m = k × ∛t .
Understanding how changing an independent variable (doubling t to 2t ) affects the dependent variable.
Calculating a percentage increase using the ratio of new to original values or picking test numbers.

Any value in the range:

25.99% to 26%

A1 Accept any value in the inclusive range [25.9, 26].

Direct proportion: m ∝ ∛t converts to an equation using constant k :
m = k∛t
Scale Factor: Doubling time replaces t with 2t :
m₂ = k∛(2t) = ∛2 × (k∛t) = ∛2 × m₁
Multiplier: ∛2 ≈ 1.25992...
Percentage Increase:
(Multiplier - 1) × 100%

Method 1: Algebraic Multiplier (Most Efficient)

Set up the relationship: Let original mark be m₁ = k∛t .
M1 Awarded for writing m = k∛t or equivalent.
Find the new mark after doubling time: Substitute 2t for t :
m₂ = k∛(2t) = k × ∛2 × ∛t = ∛2 × m₁
Calculate percentage change:
Ratio = m₂ / m₁ = (k∛(2t)) / (k∛t) = ∛2 ≈ 1.25992
M1 dep Dependent on M1: setting up an expression for m₂ / m₁ × 100 or (m₂ - m₁) / m₁ × 100 .
Convert multiplier to percentage increase:
(1.25992 - 1) × 100% = 25.992% ≈ 26%
A1 Correct answer in range [25.9, 26].

Method 2: Choosing Friendly Values (Alternative Full-Mark Method)

Pick values that fit a cube root: Let k = 1 and initial time t₁ = 1 (or 8, or 125).
m₁ = ∛1 = 1
Double the time:
t₂ = 2
m₂ = ∛2 ≈ 1.2599
Work out percentage increase:
Percentage increase = [(1.2599 - 1) / 1] × 100 = 25.99%

Pick smart numbers if stuck: If abstract algebra confuses you, choose concrete numbers. Picking t = 1 and k = 1 gives m₁ = 1 and m₂ = ∛2 immediately.
Always write an equation: The mark scheme specifically notes that just writing m ∝ ∛t earns M0. You must replace the proportionality symbol with an equals sign and a constant k (e.g., m = k∛t ) to secure the first method mark.
Show intermediate values: Write down 1.2599... or 25.99% so you are covered if you round to 26%.

Confusing square root and cube root: Writing √t instead of ∛t leads to calculating √2 ≈ 1.414 (giving 41.4%), which scores zero unless explicitly corrected.
Assuming linear proportionality: Thinking doubling revision time doubles the mark (+100%). Proportionality only means direct multiplying when the power is 1.
Stopping at the multiplier: Giving 126% or 1.26 instead of subtracting 100% to give the increase (26%).
Squaring/cubing the wrong term: Writing 2∛t instead of ∛(2t) . The time doubles, so the 2 belongs inside the cube root!

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