Madrid PAU Applied Mathematics for Social Sciences II study plan

Define matrix operations (addition, scalar multiplication, transpose, multiplication) and compute them accurately by hand.

Compute determinants of 2x2 and 3x3 matrices using cofactor expansion and Sarrus' rule, and interpret singularity.

Calculate the inverse of a 3x3 matrix using the adjoint method and verify by multiplication.

Determine the rank of a matrix using Gaussian elimination and identify linear dependence of rows/columns.

Classify systems of linear equations as consistent or inconsistent using the Rouché–Capelli theorem, and solve consistent systems by Gaussian elimination.

Solve systems with one, infinite, or no solutions using parameterization, and express the general solution in vector form.

Solve matrix equations of the form AX = B and XA = B where possible, justifying invertibility conditions.

Model a word problem (e.g., mixing, equilibrium) as a linear system and solve it, interpreting the result in context.

Formulate a linear programming problem from a word problem by defining decision variables, objective function, and all constraints (including non-negativity).

Graph the feasible region for a two-variable problem, identify its vertices, and determine the optimal solution by evaluating the objective at each vertex.

Determine whether an LP problem has a unique solution, multiple optimal solutions, unboundedness, or infeasibility from the graphical representation.

Solve a 3-variable LP problem using the simplex method (maximization with ≤ constraints) and interpret the final tableau.

Compute limits of rational, irrational, and exponential functions at finite points and at infinity, including indeterminate forms (0/0, ∞/∞, ∞−∞, 1^∞) using algebraic manipulation and L'Hôpital's rule.

Determine continuity of a piecewise function at a point by checking the three conditions, and find parameters that make a function continuous.

Compute derivatives using the product, quotient, and chain rules, and derive the tangent line equation at a point.

Analyze a function's domain, intercepts, asymptotes, intervals of increase/decrease, extrema, and concavity, and sketch its graph.

Solve optimization word problems (maximize area, minimize cost, maximize profit) by setting up a function, finding critical points, and verifying the extremum.

Compute indefinite integrals using basic rules (power, exponential, logarithmic) and definite integrals using the Fundamental Theorem of Calculus.

Calculate the area between two curves by finding intersection points and setting up the definite integral, including cases where the upper/lower curve changes.

Apply the mean value theorem and Rolle's theorem to verify conditions and find the guaranteed point(s).

Apply the axioms of probability and the addition rule to compute probabilities of unions and complements, including mutually exclusive events.

Compute conditional probabilities and apply Bayes' theorem to invert conditional probabilities in a two-stage experiment.

Determine independence of two events and use the multiplication rule for independent events.

Solve probability problems using tree diagrams and contingency tables, including total probability.

Identify discrete and continuous random variables, and compute the mean (expected value) and variance for a discrete distribution given its probability function.

Recognize and apply the binomial distribution: compute probabilities, mean, and variance for a given n and p.

Use the normal distribution: standardize values (z-scores), find probabilities from the standard normal table, and solve inverse problems (given probability, find the value).

Apply the normal approximation to the binomial distribution with continuity correction and verify conditions (np ≥ 5, n(1−p) ≥ 5).

Distinguish between population and sample, and identify sampling techniques (random, stratified, cluster, systematic) in context.

Construct a confidence interval for the population mean with known population standard deviation, and interpret the confidence level correctly.

Determine the required sample size for a given margin of error and confidence level.

Complete a timed full-length practice exam under exam conditions, and review all errors by domain to target final revision.