Canary Islands PAU Mathematics II / Matemáticas II COPAU ULPGC ULL study plan

Classify and compute determinants of 2x2, 3x3 and 4x4 matrices using row operations, Laplace expansion and the properties of determinants (row swaps, linearity, zero rows).

Compute the inverse of a square matrix by the adjugate method and by Gauss-Jordan elimination, and verify the result by multiplying back to the identity.

Determine the rank of a matrix by Gaussian elimination, identifying the pivot columns and stating the rank from the number of nonzero rows.

Set up and solve homogeneous and non-homogeneous linear systems, and classify them as incompatible, compatible determined or compatible indeterminate before solving.

Apply Rouché-Frobenius to decide the compatibility of systems depending on one or two parameters, and express the solution set in terms of the parameter.

Solve systems with infinitely many solutions by writing the general solution with free parameters and checking two particular cases numerically.

Use Cramer's rule where it applies and justify when it does not.

Solve matrix equations of the form AX = B and AX + B = X, isolating X correctly using invertibility arguments.

Compute dot, cross and scalar triple products and interpret each geometrically (angle, area of a parallelogram, volume of a parallelepiped).

Convert fluently between the general, vector, parametric and point-normal equations of a line and a plane in space.

Determine the relative position of two lines, a line and a plane, and two planes (intersecting, parallel, coincident, skew) using rank arguments on the coefficient and augmented matrices.

Compute the distance from a point to a plane, from a point to a line, and between skew lines using the cross-product formula.

Compute angles between lines, between planes and between a line and a plane, using absolute values correctly for the line-plane case.

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

Compute areas of triangles and volumes of tetrahedra defined by three or four points in space.

Solve locus problems: find the plane or line satisfying two or three geometric conditions simultaneously.

Evaluate limits, including indeterminate forms 0/0, infinity/infinity, infinity minus infinity and 1 to the power infinity, using L'Hôpital's rule and algebraic manipulation.

Determine the value of a parameter that makes a piecewise function continuous at a given point.

Determine the value of a parameter that makes a piecewise function differentiable at a point, checking continuity first and then matching one-sided derivatives.

State and apply the theorems of Bolzano, Weierstrass and Rolle/mean value to prove existence of roots or of points with a given derivative.

Compute derivatives of composite, implicit and parametrically defined functions, and write the tangent line at a given point.

Find intervals of increase and decrease, extrema and inflection points from the signs of the first and second derivatives, and sketch the graph.

Solve optimisation problems with a constraint (maximise area, volume or profit; minimise cost or distance) and justify that the extremum is global on the stated domain.

Solve related-rates problems by differentiating a geometric relation with respect to time.

Compute primitives using immediate integrals, substitution, integration by parts and simple partial fractions.

Apply the fundamental theorem of calculus and Barrow's rule to definite integrals, handling absolute values and piecewise integrands by splitting the interval.

Compute the area between two curves, finding the intersection points first and deciding which function is on top on each subinterval.

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

Compute conditional probabilities using the definition and the multiplication rule, and build a tree diagram for two-stage experiments.

Apply the law of total probability and Bayes' theorem to reverse a conditional probability.

Model binomial experiments (fixed n, independent trials, constant p) and compute probabilities, mean and variance.

Standardise a normal variable, use the standard normal table for both tails, and apply the normal approximation to the binomial with continuity correction.

Solve a full past-paper-style mock under exam timing and allocate remaining revision hours by weakest domain.