EM · GATE CS 2027
16 topics and 27 sheets. Open a topic for its sheets and the topics it depends on.
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.
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.
Connectives and their truth conditions, the equivalence laws (De Morgan, distributive, absorption, implication, contrapositive, biconditional, exportation), how implications split over conjunction and disjunction, converse, inverse and contrapositive, disjunctive and conjunctive normal forms with minterms and maxterms, a conversion routine to CNF, and counts of Boolean functions. Every law is checked by a full truth table in the verification script.
Tautology, contradiction, contingency and satisfiability and how they relate, logical consequence, testing an argument by looking only at rows where the conclusion is false, the standard inference rules, the two classic fallacies, resolution refutation, translating English conditionals (only if, unless, necessary, sufficient), and functional completeness with Post's five classes and NAND constructions. All claims are checked by truth tables or closure computations.
Predicates, domains, free and bound variables, quantifier negation, which distribution laws hold both ways and which hold one way, moving a quantifier past a formula that does not mention its variable (including the implication traps), the order of nested quantifiers, translating all, some, no, only and exactly one, inference rules for quantifiers, finite-domain expansion and the empty-domain convention. Equivalences are checked on every model with up to three elements.
Set identities, power sets and Cartesian products; the six relation properties with matrix and digraph tests; how many relations on an n-element set have each property or combination (brute-forced for n = 3 and 4); the empty and identity relations; composition of relations; reflexive, symmetric and transitive closures with Warshall's algorithm; which properties survive union and intersection; and a solved past question on a relation defined by a formula.
Equivalence classes and the one-to-one match with partitions, Bell numbers and Stirling numbers of the second kind with a small table, counting equivalence relations with k classes or containing a given pair, the equivalence closure; injective, surjective and bijective functions, their counts (including onto maps by inclusion-exclusion), partial functions and involutions, what composition says about each factor, inverses and images, real-valued examples, and countable versus uncountable sets.
Partial and total orders, Hasse diagrams as cover relations, minimal and maximal versus least and greatest elements, upper bounds, least upper bound and greatest lower bound, chains, antichains and linear extensions; lattices with join and meet, bounded, distributive, complemented and Boolean lattices, divisor lattices and the square-free test, complements and their uniqueness, sublattices versus subsets that are lattices, and the M3 and N5 test for distributivity, with a four-panel Hasse diagram.
The ladder from closed operation to semigroup, monoid, group and abelian group with standard examples, uniqueness of identity and inverses, identities and inverses of custom operations, operation-table checks and why a Latin square with an identity can still fail associativity, counts of binary operations, the subgroup test, Lagrange's theorem and its failed converse, element orders, cyclic groups and generators, U(n), S3 and facts that force a group to be abelian.
Walks, trails, paths and cycles, the handshake lemma and its parity consequences, graphic degree sequences and Havel-Hakimi, edge bounds for simple, connected, disconnected, k-component and bipartite graphs with the extremal graph behind each, cut vertices, bridges and blocks with a diagram, vertex and edge connectivity and Whitney's inequality, complements, trees and Cayley's formula, adjacency-matrix walk counts, isomorphism invariants and counts of labelled and unlabelled graphs.
Euler circuits and trails with their degree conditions (undirected and directed), Hamiltonian cycles with Dirac's and Ore's sufficient conditions and quick necessary tests, counts of Hamiltonian cycles in complete graphs, bipartite graphs and the odd-cycle test, planar graphs with Euler's formula for connected and k-component graphs, the 3n - 6 and 2n - 4 edge bounds and why they are only necessary, K5 and K3,3, face degrees, and a diagram of a planar redrawing.
Matchings (maximal, maximum, perfect), vertex covers, independent sets and edge covers with the Gallai identities and Konig's theorem, Hall's marriage condition with an example that fails it, counts of perfect matchings in complete and complete bipartite graphs, vertex colouring with clique, independence and maximum-degree bounds, Brooks and the four-colour theorem, chromatic numbers of standard graphs, edge colouring with Vizing's theorem, and chromatic polynomials, all checked on small graphs.
Sum, product and complement rules, the four selection formulas (order and repetition), arrangements with repeated letters, circular arrangements, integer solutions with lower bounds, distributing distinct or identical objects into distinct or identical boxes (including onto distributions), lattice paths, the pigeonhole principle, binomial identities and the at-least-one overcounting trap, with every count checked by brute force.
Inclusion-exclusion for two, three and n sets with a Venn diagram of region counts, counting integers not divisible by given primes and Euler's phi, solutions with upper bounds, derangements with three equivalent formulas and a past question with repeated letters, permutations with exactly k fixed points, Catalan numbers with the structures they count, Stirling numbers of the second kind and Bell numbers, and a brief note on Stirling numbers of the first kind.
Setting up recurrences from counting problems, the characteristic equation for linear homogeneous recurrences with distinct and repeated roots, particular solutions for constant, polynomial and exponential right-hand sides including the resonance case, worked solutions checked with sympy, standard recurrences (Fibonacci, strings without 00, Hanoi, tilings), ordinary generating functions with a table of standard series, coefficient extraction, counting with restricted choices, and solving a recurrence through its generating function.
Matrix algebra rules that differ from numbers (non-commutativity, transposes and inverses of products, trace), special matrices (symmetric, skew-symmetric, orthogonal, idempotent, nilpotent, involutory, triangular) and what each forces, row operations and the determinant, det(kA), cofactor expansion, the adjugate identities and the 2 x 2 inverse, rank by row reduction with the rank rules for products, sums and transposes, and counting 0/1 matrices, including a solved past question.
The rank test for consistency of Ax = b (unique, infinite or no solution) for square and rectangular systems, homogeneous systems and free variables, Gaussian elimination on the augmented matrix with a worked infinite-solution example, a parameter system classified for every value, Cramer's rule and its limits, the structure of the solution set, geometric pictures, and two solved past questions with a parameter in a 2 x 2 system.
What LU decomposition is and how it stores Gaussian elimination, Doolittle (unit lower) and Crout (unit upper) forms with entry formulas, a full 3 x 3 Doolittle and Crout factorisation of the same matrix, solving Ax = b by forward and back substitution, existence and uniqueness through leading principal minors, a matrix with no LU and the PA = LU fix, a singular matrix that still has an LU, operation counts, and Cholesky in one line. Every factor is checked by exact arithmetic.
The characteristic equation, trace as the sum and determinant as the product of eigenvalues, the 2 x 2 shortcut, eigenvectors, eigenvalues of powers, inverses, shifts, polynomials and transposes, eigenvalues by matrix type, row sums and the all-ones matrix, algebraic and geometric multiplicity, when a matrix is diagonalizable, similar matrices, the Cayley-Hamilton theorem for inverses and powers, and A^k through diagonalization, with every value checked in sympy.
Standard limits, a routine for indeterminate forms, L'Hopital's rule with its exact conditions and a case where it is silent, the 1 to the power infinity shortcut, series expansions for limits, one-sided limits and when a limit exists, limits at infinity of rational functions, continuity and the four kinds of discontinuity, differentiability from one-sided derivatives with the standard counterexamples, fixing constants in piecewise functions, derivative rules and a solved past question on counting discontinuities.
Critical points, the first and second derivative tests and when the second test is silent, global extrema on closed intervals by comparing all candidates, open intervals and endpoints, increasing and decreasing functions, concavity and inflection points, optimisation set-ups, a brief two-variable test, and Rolle's, Lagrange's and Cauchy's mean value theorems with their exact hypotheses, worked values of c and the functions that fail them.
Standard antiderivatives, substitution, integration by parts with the ILATE order, definite-integral properties (reversal, splitting, the a + b - x rule, odd and even integrands, periodicity), the fundamental theorem and Leibniz rule, Wallis and gamma integrals, improper integrals with the p-test and the trap of integrating across a singularity, and areas and averages, with every value computed in sympy.
Kolmogorov's axioms and their consequences, the addition, multiplication and complement rules, conditional probability, independence of two and of several events (pairwise versus mutual), independent versus mutually exclusive, total probability and Bayes theorem with a probability tree and the base-rate effect, classical counting probabilities, the two-children puzzle with its assumptions, and four solved past questions on independence, Bayes, draws without replacement and a Polya urn.
Discrete and continuous random variables, PMF, PDF and CDF and their properties, normalising a density and reading probabilities, median and expectation, expectation of a function, variance and its shortcut, linearity of expectation with indicator variables (fixed points, coupon collector), the tail-sum formula, joint tables with marginals, conditionals and independence, covariance and correlation with the uncorrelated-but-dependent example, and two solved past questions on a randomly broken stick.
The syllabus distributions (uniform, binomial, Poisson, exponential, normal) plus Bernoulli and geometric in one table with PMF or PDF, supports and parameter ranges, mean and variance, a diagram of their shapes, recognising each from the wording, binomial probabilities and mode, the Poisson approximation and sums of Poissons, exponential tails, memorylessness and median, the normal density, standardisation with the 68-95-99.7 values and sums of independent normals, uniform interval probabilities, Chebyshev's bound, and two solved past questions.
Mean, median and mode for raw data, frequency tables and grouped data (with the interpolation formulas for the median and mode), population and sample variance and standard deviation with the shortcut formula, the effect of shifting and scaling data, combined mean and variance of two groups, which centre minimises which error, the order of mean, median and mode under skew, the coefficient of variation and range, with every value computed exactly by script.
33 common mistakes across Engineering Mathematics, each as a wrong belief, the correct statement with a small counterexample or value, and a link to the sheet that explains it: logic translation and quantifiers, relation and function counts, lattices and groups, graph bounds, counting models, recurrences, rank, determinants and eigenvalues, LU, limits and theorem conditions, improper integrals, conditional probability and independence, distribution parameters and variance.
A lookup table of recurring Engineering Mathematics question shapes: the cue in the question, the first step and method, the assumption or boundary to check, and where the method is explained, with the past GATE CS questions solved in this pack listed against their pattern. Covers logic translation, quantifiers, relations, posets, groups, graph bounds, matchings, counting models, recurrences, systems with a parameter, eigenvalues, limits, extrema, integrals, Bayes, expectation and distributions.
The highest-value rules of the pack on one page: logic equivalences and quantifier rules, relation and function counts, lattice and group facts, graph bounds and theorem conditions, counting models and special numbers, recurrence forms, rank, determinant and eigenvalue rules, LU existence, limits, extrema and MVT conditions, integral properties, probability rules, distribution parameters and descriptive statistics, plus the five mistakes that cost the most.
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