Analyze complex problem conditions, formulate multi-step strategies, verify solutions using multiple representations, and justify mathematical reasoning with formal precision.

Simplify complex fractions, rewrite improper rational expressions into quotient-remainder form, and solve applied rational modeling problems.

Represent arithmetic and geometric sequences recursively and explicitly, link to linear/exponential growth, and derive and evaluate geometric series.

Analyze bivariate data, calculate and interpret Pearson's r, fit linear, quadratic, and exponential models, and evaluate predictions.

Analyze how mapping a 3D spherical surface onto 2D planes induces metric distortions in angles, scale, and surface area across cylindrical, conic, and azimuthal projections.

Use Cavalieri's principle and solid decomposition methods to derive and calculate surface areas and volumes of prisms, cylinders, pyramids, cones, spheres, and composite solids.

Derive and calculate arc length, sector area, and circular segment area in degrees and radians, and solve composite geometric problems.

Prove circle theorems involving chords, central and inscribed angles, angles in the same segment, and cyclic quadrilaterals.