1. Paper: Edexcel International GCSE Further Pure Mathematics 4PM1/01, 26 May 2023, 100 marks; coverage: series summation, trigonometric differentiation, graph roots estimation, kinematics, optimization via calculus.
2. Series and summation: manipulation of summation expressions, evaluation of cubic polynomial series using algebraic identities and term-by-term simplification.
3. Calculus and trigonometry: application of product and chain rule for differentiation of composite trig-polynomial functions, integer coefficient extraction.
4. Root estimation: graphical method involving curve-sketching and straight-line intersections to approximate roots of nonlinear cubic equations.
5. Kinematics: velocity-time polynomial modeling, critical points analysis, integration of absolute velocity to calculate exact travel distance between rest positions.
6. Surface area optimization: variable-dependent surface formula derivation from fixed volume constraint, use of calculus to identify and justify surface area minimum.