PAU Applied Math ACS II (Aragón) study plan

Compute determinants of 2×2, 3×3 and 4×4 matrices by cofactor expansion and by row reduction to triangular form, and state which method is faster for each case

Classify a square matrix as regular or singular from its determinant and justify the conclusion in one written sentence

Find the inverse of a 3×3 matrix using the adjoint method and verify the result by multiplying A·A⁻¹

Determine the rank of a parameter-dependent matrix by Gaussian elimination, identifying the values of the parameter that change the rank

Solve square systems by Cramer's rule and by matrix inversion, and compare the two results for consistency

Apply Rouché–Fröbenius to classify a system as incompatible, compatible determined or compatible indeterminate

Solve compatible indeterminate systems by Gaussian elimination and express the solution set with a free parameter

Discuss a system whose coefficients depend on a real parameter, stating for which parameter values each case occurs

Write the matrix equation of a system and interpret the solution set geometrically as intersection of planes or lines

Formulate a linear programming problem from a word problem: define decision variables, write the objective function and all constraints including non-negativity

Graph the feasible region for two-variable problems and identify vertices algebraically by solving pairs of binding constraints

Evaluate the objective function at every vertex and state the optimal solution with its value

Detect unbounded and infeasible regions and explain in writing why no finite optimum exists in each case

Convert a standard maximisation problem to the slack-variable form used by the simplex tableau

Solve maximisation problems with the simplex method, performing pivot operations correctly and reading the optimum from the final tableau

Interpret the dual of a minimisation problem and solve it, checking that the optimal value matches the primal

Identify integer-programming-style constraints in a statement (whole units, indivisible items) and round or adjust the continuous solution appropriately

Re-solve two Domain 1 system problems under exam time pressure to confirm retention

Compute limits of rational, irrational and exponential functions at a point and at infinity, resolving 0/0 and ∞/∞ indeterminacies by factoring, conjugates and L'Hôpital's rule

Apply the definition of continuity at a point and classify discontinuities as removable, jump or essential

Determine the value of a parameter that makes a piecewise function continuous at the join point

Compute derivatives using the product, quotient and chain rules, and differentiate implicit and logarithmic functions

Find the equation of the tangent line to a curve at a given point

Locate and classify critical points using the first and second derivative tests, and identify intervals of increase, decrease and concavity

State and apply the hypotheses of Rolle's theorem and the mean value theorem to a given function on a given interval

Solve optimisation word problems by building the objective function from the constraint and checking the endpoint and boundary cases

Compute definite integrals using Barrow's rule and apply integration by parts and by substitution

Calculate the area between two curves, finding the intersection points first and splitting the integral where the curves cross

Apply the addition and multiplication rules, conditional probability and independence to multi-stage experiments, using tree diagrams or contingency tables

Apply Bayes' theorem to reverse a conditional probability and interpret the result in context

Model a discrete random variable with its probability function, compute the mean and variance, and recognise the binomial setting

Use the normal distribution with standardisation and read probabilities from the table, including the inverse problem of finding a value from a given probability

Re-do two area problems from week 6 to confirm the integration techniques are retained

Distinguish population and sample, and identify the sampling distribution of the sample mean and its standard error

Construct a confidence interval for the mean with known or large-sample standard deviation and state the meaning of the confidence level

Carry out a one- and two-tailed hypothesis test for the mean, stating the null and alternative hypotheses, the statistic, the critical region and the conclusion in context

Compute and interpret the p-value and relate it to the significance level

Complete two full past-paper-style exams under timing and review every error against the domain weighting