100+ Free ENADE Matemática Practice Questions

Cálculo Diferencial e Integral e Análise Real

Encompasses single-variable and multivariable calculus and introductory real analysis: real number axioms, supremum/infimum, Bolzano-Weierstrass theorem, topological properties of $\mathbb{R}^n$, limits, continuity, uniform continuity, intermediate value theorem, differentiability, Mean Value Theorem, Taylor expansions, Riemann-Darboux integration, Fundamental Theorem of Calculus, improper integrals, numerical sequences and series, tests for convergence, sequences and series of functions, pointwise vs. uniform convergence, Weierstrass M-test, power series, radius of convergence, multivariable differential calculus (partial derivatives, directional derivatives, gradient, Jacobian, Hessian, Lagrange multipliers for constrained extrema, Inverse and Implicit Function Theorems), multiple integrals (double and triple integrals, Fubini's theorem, change of variables via Jacobian), vector calculus (conservative vector fields, scalar potentials, line integrals, Green's theorem, surface integrals, flux, Stokes' theorem, and Gauss Divergence theorem), ordinary differential equations (first-order separable and linear ODEs, exact equations and integrating factors, second-order linear ODEs with constant coefficients, method of undetermined coefficients, variation of parameters, Laplace transforms, systems of linear ODEs), and metric space foundations (open and closed balls, compactness, completeness, and Banach fixed-point theorem).

Álgebra Linear e Álgebra Abstrata

Covers linear algebra over real and complex fields and abstract algebraic structures: vector spaces, subspaces, linear span, linear independence, bases, dimension, direct sums, linear transformations, kernel (null space) and image (range), Rank-Nullity Theorem, matrix representations of linear maps, change of basis matrices, determinants, systems of linear equations, eigenvalues, eigenvectors, eigenspaces, characteristic polynomial, minimal polynomial, algebraic and geometric multiplicities, matrix diagonalization, Cayley-Hamilton theorem, Jordan canonical form, inner product spaces, Cauchy-Schwarz inequality, orthogonality, Gram-Schmidt orthogonalization, orthogonal complements, orthogonal projections, adjoint operators, self-adjoint (Hermitian/symmetric) and orthogonal/unitary operators, Spectral Theorem for symmetric operators, bilinear and quadratic forms, Sylvester's law of inertia; Group theory (groups, subgroups, cyclic groups, generators and relations, permutations and symmetric groups $S_n$, alternating groups $A_n$, Lagrange's theorem on cosets and group order, normal subgroups, quotient groups, group homomorphisms, kernel and image, First Isomorphism Theorem, Cayley's theorem, group actions, orbit-stabilizer theorem); Ring and Field theory (rings, subrings, commutative rings, integral domains, division rings, fields, ideals, principal ideal domains (PID), unique factorization domains (UFD), quotient rings, ring homomorphisms, polynomial rings $K[x]$, division algorithm, irreducible polynomials, Eisenstein's criterion, field extensions, algebraic elements, and finite fields $\mathbb{F}_q$).

Geometria Euclidiana, Não-Euclidiana e Topologia

Treats axiomatic Euclidean geometry in the plane and space, triangle congruence and similarity, metric relations in right and general triangles (Law of Cosines, Law of Sines), Ceva's and Menelaus's theorems, power of a point and cyclic quadrilaterals, geometric constructions with ruler and compass, areas and perimeters of plane figures, spatial geometry (prisms, pyramids, cylinders, cones, spheres), Euler's polyhedral formula ($V - E + F = 2$), regular polyhedra (Platonic solids), Cavalieri's principle, analytic geometry in $\mathbb{R}^2$ and $\mathbb{R}^3$ (lines, planes, cross product, scalar triple product, distances, angles between linear subspaces), conic sections (ellipse, parabola, hyperbola: standard equations, foci, directrices, eccentricity, asymptotes), quadric surfaces (ellipsoids, hyperboloids of one and two sheets, elliptic and hyperbolic paraboloids, cones, cylinders, reduction of general second-degree equations to canonical form), foundations of non-Euclidean geometries (Hilbert's axioms, Euclid's fifth postulate and its independence, spherical geometry, hyperbolic geometry and Poincaré disk model), differential geometry of curves and surfaces (regular parametrized curves, arc length parametrization, curvature $\kappa$, torsion $\tau$, Frenet-Serret frame and formulas, regular surfaces, tangent plane, first fundamental form, length and area on surfaces, second fundamental form, normal and principal curvatures, Gaussian curvature $K$, mean curvature $H$, Gauss's Theorema Egregium), and introductory general topology (topological spaces, open and closed sets, interior, closure, boundary, basis and subbasis, subspace topology, continuous functions, homeomorphisms, separation axioms $T_0, T_1, T_2$ (Hausdorff), connectedness, path-connectedness, compactness, and Heine-Borel theorem).

Probabilidade, Estatística e Matemática Discreta

Explores discrete mathematics, combinatorics, graph theory, elementary number theory, probability theory, and mathematical statistics: basic counting principles (multiplication and addition rules), permutations, combinations, combinations with repetition, binomial and multinomial coefficients, Pascal's identity, Inclusion-Exclusion principle, Dirichlet's Pigeonhole Principle, linear homogeneous and non-homogeneous recurrence relations with constant coefficients, generating functions; Graph theory (simple graphs, multigraphs, directed graphs, degree of vertices, Handshaking lemma, paths, cycles, trees, spanning trees, bipartite graphs, Eulerian circuits and paths, Hamiltonian cycles, planar graphs, Kuratowski's theorem, Euler's planar formula $V - E + F = 2$, vertex coloring, chromatic number); Elementary number theory (divisibility, Division Algorithm, greatest common divisor, Euclidean Algorithm, Bézout's identity, prime numbers, Fundamental Theorem of Arithmetic, congruences, modular arithmetic, linear congruences, Chinese Remainder Theorem, Fermat's Little Theorem, Euler's totient function $\phi(n)$, Euler's theorem, Wilson's theorem); Probability and Statistics (sample space, events, Kolmogorov's probability axioms, conditional probability, Law of Total Probability, Bayes' theorem, independence of events, discrete random variables: Bernoulli, Binomial, Poisson, Geometric, Hypergeometric, continuous random variables: Uniform, Exponential, Normal, probability mass/density functions, cumulative distribution function, expected value, variance, standard deviation, moments, covariance, correlation coefficient, joint and marginal distributions, Law of Large Numbers, Central Limit Theorem, point estimation, unbiasedness, confidence intervals for the mean and proportion, and null hypothesis testing).

Educação Matemática e Prática Docente

Focuses on pedagogical knowledge, epistemology, didactics of mathematics, and teacher training for basic and secondary education: epistemological and historical development of mathematics (Babylonian, Egyptian, Greek, Arab, Renaissance, and modern contributions; history of calculus, algebra, and geometry as pedagogical tools); French Didactics of Mathematics: Guy Brousseau's Theory of Didactical Situations (didactical contract, devolutive process, adidactical situations of action, formulation, validation, and institutionalization, obstacles: ontogenic, didactical, epistemological), Yves Chevallard's Anthropological Theory of Didactics (Didactic Transposition: scholarly knowledge, knowledge to be taught, taught knowledge), Gérard Vergnaud's Theory of Conceptual Fields (additive and multiplicative structures, situations, operational invariants: concepts-in-action, theorems-in-action, representations); Brazilian and international perspectives: Ubiratan D'Ambrosio's Ethnomathematics program (sociocultural contexts, ticas, mathema, ethno), Ole Skovsmose's Critical Mathematics Education (formatting power of mathematics, mathemacy, landscape of investigation vs. exercise paradigm), Michele Artigue's Instrumental Genesis (artifacts and instruments, instrumental orchestration); Curriculum guidelines in Brazil: Base Nacional Comum Curricular (BNCC) for Mathematics in Ensino Fundamental and Ensino Médio (development of mathematical thinking, 8 specific competencies, thematic units: Numbers, Algebra, Geometry, Quantities & Measures, Probability & Statistics), Formative and diagnostic assessment in mathematics (Jussara Hoffmann, Charles Hadji), Problem Solving methodologies (George Pólya, Onuchic & Allevato), Mathematical Modeling in education, and digital technologies in the mathematics classroom (GeoGebra, dynamic geometry software, CAS).

Formação Geral e Ética

Evaluates universal competencies required of higher education graduates in Brazil: ethical principles, democratic citizenship, human rights, diversity, affirmative actions and racial equality (Lei nº 10.639/2003 and Lei nº 11.645/2008), socio-environmental sustainability, climate change, circular economy, scientific inquiry, critical reading of media and data, digital citizenship, ethics in artificial intelligence, and public policies for national development.