EM · Section 1: Engineering Mathematics
Engineering Mathematics
Discrete mathematics, linear algebra, calculus, and probability and statistics.
Topics
15 topics in syllabus order, each with study material.
Propositional laws, normal forms, tautology and satisfiability, valid arguments and inference rules, functional completeness; first-order logic with quantifier negation, nested quantifier order, distribution laws and translation of English statements.
Set identities and power sets; relation properties with matrix tests and counts; reflexive, symmetric and transitive closures; equivalence relations, classes and partitions with Bell and Stirling numbers; injective, surjective and bijective functions with counts, composition, inverses and countability.
Partial orders, Hasse diagrams, extremal elements and bounds, chains, antichains and linear extensions; lattices with join and meet, bounded, distributive, complemented and Boolean lattices, sublattices, the M3 and N5 test and divisor lattices.
Closure, associativity, identity and inverse in semigroups, monoids and groups; operation tables and counts of binary operations; subgroups and the subgroup test, Lagrange's theorem, element orders, cyclic groups and generators, abelian groups and small examples such as Zn, U(n) and S3.
Degrees and the handshake lemma, connectivity, components, cut vertices and bridges, edge bounds; Euler and Hamilton graphs, bipartite graphs, planar graphs and Euler's formula; matching and Hall's theorem, independence and covering numbers, vertex and edge colouring and chromatic number bounds.
Sum and product rules, permutations and combinations with and without repetition, distribution models, pigeonhole; inclusion-exclusion, derangements, Catalan and Stirling numbers; linear recurrences, homogeneous and non-homogeneous solution forms; ordinary generating functions and coefficient extraction.
Matrix operations and special matrices (symmetric, skew-symmetric, orthogonal, idempotent, nilpotent, triangular); determinant properties, cofactors, adjugate and inverse; rank by row reduction and the rank rules for products, sums and transposes.
Consistency of Ax = b by comparing rank A with rank [A | b], unique, infinite or no solution, homogeneous systems and free variables, Gaussian elimination and Cramer's rule; LU decomposition (Doolittle and Crout), when it exists, PA = LU and solving by forward and back substitution.
Characteristic equation, trace and determinant as sum and product of eigenvalues, eigenvalues of powers, inverses, shifts and special matrices, eigenvectors, algebraic and geometric multiplicity, Cayley-Hamilton theorem and its uses, diagonalization and powers of matrices.
Standard limits, indeterminate forms and L'Hopital's rule with its conditions, the 1 to the power infinity form, one-sided limits, continuity and types of discontinuity, differentiability and the relation between continuity and differentiability.
Critical points, first and second derivative tests, global extrema on closed intervals, increasing and decreasing functions, concavity and inflection; Rolle's theorem, Lagrange's mean value theorem and Cauchy's form with their exact hypotheses.
Standard integrals, substitution, integration by parts, definite-integral properties (linearity, symmetry, the a + b - x rule, odd and even functions, periodicity), the fundamental theorem and Leibniz rule, and convergence of simple improper integrals.
Axioms and the addition rule, conditional probability, multiplication rule, independence of two and of several events, mutually exclusive versus independent, total probability, Bayes theorem with trees, and classical counting probability.
Discrete and continuous random variables, PMF, PDF and CDF, expectation, variance, linearity of expectation and indicators, covariance; uniform, binomial, Poisson, exponential and normal distributions with mean, variance, memorylessness and standardization.
Mean, median and mode for raw, frequency and grouped data, standard deviation and variance (population and sample), effect of shifting and scaling, combined mean and variance, and the mean-median-mode order for skewed data.
Engineering Mathematics revision: traps, patterns and last day
Pack-wide revision: the traps that cost marks with the correct rule, question patterns with a first step and a check, and a last-day master sheet of the highest-value rules and conditions.