Castilla-La Mancha PAU Applied Mathematics for Social Sciences II (Matemáticas Aplicadas a las Ciencias Sociales II - UCLM 2026) study plan

Resolve systems of linear equations using Gaussian elimination, identifying inconsistent and indeterminate cases.

Calculate the determinant of 2x2 and 3x3 matrices and apply the Rouché–Frobenius theorem to classify systems by number of solutions.

Compute the inverse of a square matrix using the adjugate method and use it to solve matrix equations of the form AX = B.

Model a word problem (e.g., mixture or production) as a system of linear equations and solve it by matrix methods.

Determine the rank of a matrix by row reduction and relate it to the number of solutions of a system with a parameter.

Solve systems with a real parameter, stating the values of the parameter for which the system is consistent, inconsistent, or indeterminate.

Apply matrix algebra to solve a real-world context (e.g., input-output or network flow) and interpret the solution in context.

Review and correct errors from the week 1 problem set, explaining each mistake in writing.

Formulate a linear programming problem from a word problem by defining decision variables and writing the objective function and constraints.

Solve two-variable linear programming problems graphically, identifying the feasible region and corner points.

Apply the fundamental theorem of linear programming to find optimal values at vertices and distinguish between bounded and unbounded regions.

Interpret the optimal solution in the context of the original problem, including slack and feasibility.

Compute limits of rational, radical, and piecewise functions at a point and at infinity, including indeterminate forms.

Determine continuity of a function at a point and on an interval, and classify discontinuities as removable, jump, or essential.

Apply the definition of derivative and standard rules (product, quotient, chain) to differentiate algebraic and exponential/logarithmic functions.

Solve for parameters that make a piecewise function continuous or differentiable at a given point.

Find intervals of monotonicity and local extrema using the first derivative test and sign chart.

Determine concavity, inflection points, and asymptotes to sketch the graph of a function.

Solve optimisation problems (area, cost, revenue, profit) by setting up a function of one variable and using derivatives.

Apply the second derivative test to classify critical points and justify global extrema on a closed interval.

Compute indefinite integrals using standard formulas and substitution for basic functions.

Apply the fundamental theorem of calculus to evaluate definite integrals.

Calculate the area between two curves and the area under a curve over an interval, including cases where the function crosses the x-axis.

Solve a real-world application of the definite integral (e.g., total cost from marginal cost, or consumer surplus).

Compute measures of central tendency and dispersion for one-variable data and interpret them in context.

Calculate the covariance and Pearson correlation coefficient for two-variable data and interpret the strength and direction of the linear relationship.

Fit a linear regression line by least squares and use it to make predictions, assessing the reliability of the prediction.

Apply conditional probability, independence, and Bayes' theorem to solve problems using tree diagrams or contingency tables.

Identify and compute probabilities for binomial distributions, including mean and variance.

Standardise normal variables and compute probabilities using the standard normal table, including inverse problems.

Approximate a binomial distribution by a normal distribution when conditions are met and apply continuity correction.

Integrate skills from all domains by solving a full mock exam under timed conditions and reviewing errors.