1. Paper: Edexcel IGCSE Further Pure Mathematics 4PM1/01, 31 Oct 2023; full coverage of pure algebra, surds, calculus, coordinate geometry, series, and vector transformations.
2. Surds and geometry: triangle side expressions with irrational terms; algebraic manipulation of surd-based area formula to find integer coefficients.
3. Quadratic transformations: completing the square for cubic expressions; minimum value determination through vertex form and comparison with shifted functions.
4. Polynomial construction and factor theorem: derivative-based reconstruction of original function; solving for unknown coefficients and full factorization.
5. Coordinate geometry: line equations with point-slope and perpendicularity conditions; midpoint and ratio division application; standard form of linear equation.
6. Curve intersections and asymptotes: rational function behavior; intercepts and limits; root approximations via algebraic and graphical methods.