A-Math 2025 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)

Rationalising the denominator

6 marks · algebra (surds; no calculator)

Completing the square (max/min)

7 marks: 5 + 2 · algebra (completing the square, maximum)

Integrate e^ax and 2x

7 marks · calculus (integration of exponential and trig, constant from a point)

Factor theorem

9 marks: 6 + 3 · algebra (factor and remainder theorems; no calculator)

Quotient rule

6 marks · calculus (quotient rule, increasing function)

Pythagorean identity cosec^2=1+ ^2

7 marks · trigonometry (identity to quadratic, solve over a range)

Integrate acceleration to velocity

7 marks · calculus (kinematics: integrate acceleration)

Solve an exponential equation with logarithms

9 marks: 4 + 5 · algebra (logarithms; simultaneous exponential equations)

Area between a curve and a line

10 marks: 7 + 3 · calculus (area between a curve and a line; equal areas)

Alternate segment theorem

10 marks: (a) 3 + (b)(i) 3 + (b)(ii) 4 · geometry (alternate segment theorem, cyclic quadrilateral)

R-formula R ( - )

12 marks: (a) 1 + (b) 3 + (c)(i) 2 + (c)(ii) 6 · trigonometry (r-formula, kinematics modelling)