A-Math 2025 Paper 1 Worked Solutions

Exponential-decay model

5 marks · algebra (exponential decay, logarithms)

Always-negative condition

6 marks · algebra (quadratic always-negative condition)

A^3 b^3 factorisation

5 marks: (a)(i) 1 + (a)(ii) 1 + (b) 3 · algebra (difference of cubes, divisibility proof)

Product rule (with x)

5 marks: 2 + 3 · calculus (product rule, integration by reversing a derivative)

Linear law (ln-ln)

6 marks: 2 + 4 · coordinate geometry (linear law / ln-ln transformation)

Stationary points

7 marks: 4 + 3 · calculus (stationary point with a surd term, second-derivative test)

Binomial expansion (general term

7 marks: 4 + 3 · algebra (binomial expansion, coefficient comparison)

Line-and-curve intersection

7 marks: 5 + 2 · algebra (line and curve, discriminant for intersection and tangency)

Double-angle identities

7 marks: 4 + 3 · trigonometry (double-angle identity proof, then solve)

Tangents & normals

10 marks: 7 + 3 · calculus (normal to a trig curve, triangle area)

Connected rates of change

8 marks: 5 + 3 · calculus (connected rates of change)

Graphs of trig functions (amplitude

8 marks: (a)(i) 2 + (a)(ii) 3 + (b) 3 · trigonometry (cosine graph, equilateral-triangle geometry)

Equation of a circle

9 marks: 7 + 2 · coordinate geometry (equation of a circle, point inside)