Balearic Islands PAU Mathematics II study plan

Classify square matrices by rank and invertibility using Gaussian elimination, stating the exact row operations performed

Compute 2x2, 3x3 and 4x4 determinants by cofactor expansion and by row reduction, and verify one result with the other method

State and apply the properties of determinants (multilinearity, effect of row swaps, det of product and inverse) to justify each step

Solve square systems by Cramer's rule and by matrix inversion, then check the residual of the solution vector

Apply Rouché–Capelli to classify systems as inconsistent, determined or underdetermined, and express solutions with a free parameter when rank is deficient

Solve homogeneous systems and describe the solution set as a subspace with an explicit basis

Model word problems (mixing, equilibrium, network flow) as linear systems and interpret the solution in context

Diagnose the most common algebraic slips in elimination and set up a personal error checklist

Perform vector operations in space (linear combinations, scalar and vector products, mixed product) and interpret each geometrically

Determine the relative position of two and three planes from the ranks of the coefficient and augmented matrices

Compute angles between lines and planes and between pairs of planes using dot and cross products

Compute distances from a point to a line and to a plane, and between skew lines

Convert freely between general, parametric and vector equations of lines and planes, and check consistency of the three forms

Solve combined problems where a parameter in a plane or line equation must be found from an incidence or distance condition

Compute areas of triangles and parallelograms and volumes of tetrahedra and parallelepipeds using products

Complete a full past-paper under exam timing and mark it strictly

Evaluate limits including indeterminate forms, one-sided limits and limits at infinity, and identify vertical, horizontal and oblique asymptotes

Determine the domain and the points of discontinuity of a piecewise function and classify each discontinuity

Differentiate composite, implicit and parametric functions and compute tangent and normal lines at a given point

Apply Rolle's and the mean value theorem to prove existence of roots or stationary points on an interval

Find intervals of increase and decrease, extrema and inflection points from the sign of first and second derivatives

Optimise a real-world quantity (area, cost, volume) by building the objective function and verifying the optimum on its domain

Sketch the graph of a rational or piecewise function using asymptotes, monotonicity and curvature

Solve an optimisation word problem end-to-end under time pressure and justify why the critical point is the global optimum

Compute indefinite integrals by substitution, by parts and by partial fractions, and verify each antiderivative by differentiation

Evaluate definite integrals using the fundamental theorem and handle improper integrals with infinite limits

Compute areas between curves, including regions requiring splitting at intersection points

Set up and evaluate volumes of revolution around the x-axis and around a horizontal line using the disk or washer method

Solve counting problems using permutations, combinations and the multiplication principle, distinguishing ordered from unordered selections

Apply conditional probability, independence and Bayes' theorem to multi-stage problems

Compute binomial probabilities and use the normal approximation with continuity correction, including standardising values and reading tail probabilities

Interpret a confidence interval for a mean and state how sample size and confidence level affect its width

Sit two full timed mock exams and triage remaining weak topics into a final revision list