Castilla y León PAU Mathematics II study plan

Compute 2×2 and 3×3 determinants by cofactor expansion and by the rule of Sarrus, and state when a determinant vanishes

Classify square matrices by rank using Gaussian elimination, and invert a 3×3 matrix by the adjugate method, verifying A·A⁻¹ = I

Solve matrix equations of the form AX = B and XA = B by isolating X, including cases where A is singular and the equation has no unique solution

Apply matrix powers to a recurrence-type problem by computing A², A³ and inferring Aⁿ for a diagonalizable 2×2 case

Apply Rouché–Capelli to decide whether a linear system is inconsistent, determinate or indeterminate, and express the solution set with parameters

Solve homogeneous systems and describe the solution set as a linear span of independent vectors

Discuss a system with a real parameter a, identifying the critical values of a where the rank changes

Translate a short word problem (mixing, equilibrium, network flow) into a linear system and solve it

Perform vector operations in ℝ³ (dot product, cross product, mixed product) and use them to test coplanarity and orthogonality

Convert fluently among the general, vector and parametric equations of a line and of a plane

Determine the relative position of two lines, of a line and a plane, and of three planes, stating the geometric interpretation of each algebraic outcome

Compute the angle between two lines, two planes, and a line and a plane using dot and cross products

Compute the distance from a point to a plane, from a point to a line, and between two skew lines using the appropriate formula

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

Obtain the projection of a point onto a plane and onto a line, and construct the perpendicular line or plane involved

Solve an area and volume problem using cross products and the mixed product, including the volume of a tetrahedron

Evaluate limits of rational, irrational and exponential functions, resolving 0/0 and ∞/∞ indeterminacies by factoring, conjugates and L'Hôpital's rule

Use equivalent infinitesimals and notable limits to resolve trigonometric and logarithmic indeterminacies

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 identify vertical, horizontal and oblique asymptotes

Differentiate composite, implicit and parametrically defined functions, and compute higher-order derivatives

Apply the tangent-line equation and the mean value theorems (Rolle, Lagrange) to locate points satisfying the hypothesis

Determine parameters so that a piecewise function is differentiable at a join point, checking continuity first

Optimize a geometric or economic quantity by setting the derivative to zero and verifying the extremum with the second derivative or a sign test

Sketch a curve from its domain, asymptotes, monotonicity and concavity intervals

Compute indefinite integrals by substitution and by parts, recognizing which method applies from the integrand's structure

Integrate rational functions using partial fractions for distinct linear and repeated linear factors

Apply Barrow's rule to definite integrals and use symmetry to simplify integrals over symmetric intervals

Compute the area between two curves and the volume of a solid of revolution about the x-axis

State and apply the mean value theorem for integrals to find the point where the function attains its average value

Solve counting problems with permutations, combinations and variations, distinguishing ordered from unordered selections and with versus without repetition

Apply conditional probability, the total probability theorem and Bayes' theorem to a two-stage problem stated in words

Model a discrete random variable, compute its mean and variance, and recognize the binomial distribution in context

Use the normal distribution with standardization and the symmetry of the table to compute probabilities and inverse values

Complete a full past-paper under exam timing and self-mark against the official solution criteria