Free Practice Questions for Castilla y León PAU Mathematics II

Algebra and Linear Systems

Matrix operations (addition, scalar multiplication, matrix product, matrix transpose), inverse matrix formula via classical adjoint, determinant properties (Sarrus rule, row operations, product rule), matrix equations of the form AX = B or AX + B = C, and classification and resolution of linear systems with parameters using Rouché-Capelli Theorem and Cramer's Rule.

Differential Calculus

Limits of functions, resolution of indeterminate forms (0/0, infinity/infinity, 1^infinity), L'Hôpital's Rule applications, continuity and Bolzano's Intermediate Value Theorem, differentiation rules, chain rule, implicit differentiation, equations of tangent and normal lines, Rolle's Theorem, Mean Value Theorem, determination of asymptotes, concavity and inflection points, and real-world optimization problems.

Integral Calculus

Indefinite integrals, antiderivatives, integration techniques including u-substitution, integration by parts, and rational function decomposition into partial fractions, Fundamental Theorem of Calculus and Barrow's Rule, and exact calculation of plane areas bounded by single functions, x-axis, and intersecting curves.

3D Analytical Geometry

Vector operations in R3 (addition, scalar multiplication, dot product, cross product, scalar triple product), parametric, continuous, and implicit equations of lines and planes, relative positions of two lines, a line and a plane, and two planes, metric computations including distances (point to plane, point to line, parallel planes, skew lines) and angles (line-line, plane-plane, line-plane).

Probability and Statistics

Sample spaces, union, intersection, conditional probability, independent events, Law of Total Probability, Bayes' Theorem applications, discrete Binomial distribution B(n, p) calculations, and continuous Normal distribution N(mu, sigma) standardization and probability determination.