PAU Mathematics II (Galicia) study plan

Compute 2x2 and 3x3 determinants by cofactor expansion and by the rule of Sarrus, and use properties (row swaps, linearity, zero rows) to shortcut calculations.

Classify square matrices by rank using Gaussian elimination, and determine invertibility from rank and determinant.

Invert 3x3 matrices both by the adjugate method (A^-1 = adj(A)/det A) and by Gauss-Jordan elimination, checking the product A·A^-1 = I.

Solve matrix equations of the form AX = B and XA = B, isolating X correctly when A is invertible.

Apply Rouché-Frobenius to decide whether a linear system is incompatible, compatible determined, or compatible indeterminate, expressing the answer in terms of a parameter when the coefficient matrix depends on one.

Solve compatible indeterminate systems by selecting free parameters and writing the general solution in parametric form.

Solve homogeneous systems and describe the solution set as a subspace, giving a basis.

Justify each conclusion with rank arguments rather than by trial and error.

Perform vector operations in R^3 (addition, scalar multiples, dot product, cross product) and interpret each geometrically.

Compute the angle between two vectors and the projection of one vector onto another, and use the scalar triple product to compute volumes of parallelepipeds and tetrahedra.

Determine the equation of a line in parametric, continuous, and implicit form, and convert between the three forms.

Determine the equation of a plane from a point and normal vector, from three points, and from a line and an external point.

Classify the relative position of two lines, of a line and a plane, and of two planes by comparing ranks of the relevant coefficient and augmented matrices.

Compute the angle between lines and between a line and a plane, and the distance from a point to a plane, from a point to a line, and between skew lines.

Find the intersection point of a line and a plane and the projection of a point onto a plane and onto a line.

Solve problems combining a parameter with a position condition (e.g., find k so that two lines intersect).

Compute limits of rational, irrational, exponential, and logarithmic functions, resolving 0/0 and infinity/infinity indeterminacies by factoring, conjugates, and equivalent infinitesimals.

Resolve indeterminacies of the type 1^infinity, 0^0, and infinity^0 using logarithms or the number e, and use L'Hôpital's rule only where its hypotheses hold.

Determine continuity of piecewise functions at a point and classify discontinuities as removable, jump, or essential.

Apply the intermediate value theorem and Bolzano's theorem to prove the existence of roots of a function on an interval.

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

Apply the theorems of Rolle, Lagrange (mean value), and Cauchy, and use derivatives to find monotonicity intervals, extrema, and inflection points.

Compute indefinite integrals using immediate formulas, substitution, and integration by parts, and integrate rational functions by partial fractions.

Compute definite integrals and apply Barrow's rule, including improper integrals on unbounded intervals.

Compute the area between two curves, identifying intersection points and the correct upper/lower function on each subinterval.

Set up and evaluate the volume of a solid of revolution by the disk and washer methods, and by the shell method where more convenient.

Compute the area of a region bounded by a curve given in parametric form or polar coordinates.

Solve combinatorial counting problems using permutations, combinations, and the multiplication principle, distinguishing ordered from unordered selections and with/without replacement.

Apply the axioms of probability, the addition and multiplication rules, conditional probability, total probability, and Bayes' theorem to word problems.

Model a binomial and a normal distribution, compute probabilities, and apply the normal approximation to the binomial with the continuity correction.

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

Construct confidence intervals for the mean with known and unknown variance, and perform a one- or two-tailed hypothesis test for the mean, stating the null and alternative hypotheses, the statistic, the critical region, and the conclusion in context.