Basque PAU Mathematics II study plan

Classify square matrices by order and compute determinants of orders 2 and 3 using Sarrus' rule and cofactor expansion along a row or column

Compute the inverse of a 3x3 matrix by the adjugate method and verify the result by multiplying back to the identity

Determine the rank of a matrix by Gaussian elimination, stating clearly which elementary row operations preserve rank

Solve Cramer-type 3x3 systems and identify from the determinant value when the system is incompatible or indeterminate

Apply Rouché–Capelli reasoning to decide consistency of systems with a parameter, distinguishing the three cases by comparing rank of coefficient and augmented matrices

Solve parameter-dependent systems and express the solution set in terms of the parameter, including the one-parameter infinite family case

Use matrix equations of the form AX = B and AXB = C to isolate and compute the unknown matrix

Interpret matrix powers and the transpose in short algebraic proofs, such as showing (AB)^T = B^T A^T for 2x2 cases

Perform dot, cross and mixed products on given vectors and state the geometric meaning of each result

Compute the angle between two vectors and the projection of one vector onto another without a calculator

Derive equations of lines and planes in all standard forms (parametric, general, vector) and convert between them

Classify the relative positions of two lines and of a line and a plane by comparing direction vectors and solving the resulting system

Compute distances between points, point-to-line, point-to-plane, and between skew lines using the mixed product formula

Calculate areas of triangles and parallelograms and volumes of tetrahedra and parallelepipeds using cross and mixed products

Find the symmetric point of a point with respect to a line and with respect to a plane

Determine the projection of a point onto a plane and onto a line, and use it to compute the shortest distance

Compute limits of rational, irrational and exponential functions at a point and at infinity, resolving 0/0 and infinity/infinity indeterminacies

Apply L'Hôpital's rule only after verifying the required hypothesis, and identify when it fails

Determine the value of a parameter so that a piecewise function is continuous at a given point

Classify discontinuities as removable, jump or asymptotic and justify each classification with one-sided limits

Differentiate composite, implicit and parametric functions and state the domain where the derivative exists

Use the derivative to find tangent and normal lines at a given abscissa

Apply the mean value theorem conditions and Rolle's theorem to prove existence of a point with a given derivative value

Solve optimisation and monotonicity problems, including finding parameters that make a function have a stationary point of a specified type

Compute indefinite integrals by substitution, by parts and by simple partial fractions, and check each result by differentiating

Apply the fundamental theorem of calculus to definite integrals, handling absolute values and piecewise integrands by splitting the interval

Compute areas between curves by finding intersection points and setting up the correct integral with sign control

Compute volumes of revolution about the x-axis using the disk method

Apply conditional probability, the total probability theorem and Bayes' theorem to multi-stage problems stated in words

Model binomial and normal situations, choose the correct distribution, and apply the standardisation formula with continuity correction

Compute expected value and variance of discrete variables and interpret them in context

Interpret confidence intervals and hypothesis-test language at a conceptual level, identifying the null hypothesis and the meaning of the significance level