Catalonia PAU Mathematics II (Matemàtiques II - CIC 2026) study plan

Classify and reduce matrices by Gauss-Jordan elimination, stating rank and identifying inconsistent versus compatible systems.

Compute 2x2, 3x3 and 4x4 determinants using cofactor expansion and row reduction, and verify results with properties such as multilinearity and the product rule.

Invert 3x3 matrices by the adjugate method and by Gauss-Jordan, and check the inverse by multiplication.

Solve systems with parameter (Rouché-Frobenius discussion) and express the solution set in each case: none, one, or infinitely many with one free parameter.

Set up and solve matrix equations of the form AX = B, AXB = C and A + XB = C, isolating X symbolically before substituting numbers.

Compute dot and cross products, scalar triple product, and use them to test coplanarity, orthogonality and parallelism of vectors.

Convert between the general, parametric and vector equations of a line and a plane, and pass from one form to another fluently.

Determine relative positions of two lines, line and plane, and three planes by comparing ranks of the coefficient and augmented matrices.

Compute distances point-point, point-line, point-plane, line-line and plane-plane using the appropriate formula, choosing it from the geometry rather than memorising cases.

Compute angles between lines, between planes and between line and plane, and interpret the complement correctly for line-plane angles.

Find projections of a point onto a line and onto a plane, and the symmetric point with respect to each.

Derive the equation of a plane from three conditions (three points, a point and a line, two lines) and verify the result satisfies all conditions.

Compute limits at a point and at infinity, resolving indeterminacies of type 0/0, infinity/infinity, infinity minus infinity, 0 times infinity and 1 to the infinity.

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

Apply the definition of derivative and differentiation rules to obtain derivatives of rational, radical, exponential, logarithmic and trigonometric composites.

Use Rolle's theorem, the mean value theorem and L'Hôpital's rule where applicable, justifying the hypotheses before applying them.

Carry out a complete study of a function: domain, symmetry, asymptotes, monotonicity, extrema, curvature, inflection points and a sketch consistent with all findings.

Solve optimisation problems stated in words by building the objective function, identifying the constraint and verifying the extremum is global on the feasible interval.

Analyse functions defined piecewise or with absolute values, checking differentiability at the join points.

Apply the theorem of Bolzano and the intermediate value theorem to prove existence of roots and locate them approximately.

Compute immediate and near-immediate primitives, including change of variable and integration by parts, and recognise when each method applies.

Evaluate definite integrals using the fundamental theorem of calculus, handling improper integrals that diverge at an endpoint.

Compute areas between curves by finding intersection points and setting up the correct upper-minus-lower integral, splitting where necessary.

Compute volumes of revolution about the x-axis and about the y-axis using the disc and shell methods.

Solve counting problems using permutations, combinations and the multiplication principle, distinguishing ordered from unordered and with or without repetition.

Compute conditional probabilities and test independence of two events using P(A and B) = P(A)P(B).

Apply the total probability theorem and Bayes' theorem to problems given as word scenarios, drawing a tree diagram before calculating.

Model a two-stage experiment with and without replacement and interpret how replacement changes the probabilities.

Define discrete random variables, compute expectation and variance, and recognise the binomial model including when to use it.

Standardise normal variables and compute probabilities and inverse probabilities using the standard normal table, including between two values and tails.

Approximate a binomial by a normal with continuity correction and justify when the approximation is acceptable.

Construct and interpret a confidence interval for the mean with known standard deviation, and explain how sample size and confidence level change its width.