CMI Entrance Exam Syllabus 2027 PDF: Download Subject, Section wise Syllabus

Algebra

Geometry

Calculus

Number Theory

Relation and Function

Coordinate Geometry

Vector

Matrices (upto 3 dimension)

Counting

Probability

All the NCERT topics for Mathematics from class 9 to 12

Topics

Sub topics

Discrete Mathematics

Sets and relations, elementary counting techniques, pigeon hole principle, partial orders,

Elementary probability theory

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Automata Theory

Regular expressions, non deterministic and deterministic finite automata, subset construction, regular languages, non regularity (pumping lemma), context free grammars, basic ideas about computable and uncomputable functions.

Algorithms

O notation, recurrence relations, time complexity of algorithms, sorting and searching (bubble sort, quick sort, merge sort, heap sort).

Data structures

Lists, queues, stacks, binary search trees, heaps.

Graphs

Basic definitions, trees, bipartite graphs, matchings in bipartite graphs, breadth first search, depth first search, minimum spanning trees, shortest paths.

Algorithmic techniques

Dynamic programming, divide and conquer, greedy.

Logic

Boolean logic, truth tables, boolean circuits — and, or, not, and, and gates.

Topics

Sub topics

Algebra

(a) Groups, homomorphisms, cosets, Lagrange’s Theorem, group actions, Sylow Theorems, symmetric group Sn, conjugacy class, rings, ideals, quotient by ideals, maximal and prime ideals, fields, algebraic extensions, finite fields

(b) Matrices, determinants, vector spaces, linear transformations, span, linear independence, basis, dimension, rank of a matrix, characteristic polynomial, eigenvalues, eigenvectors, upper triangulation, diagonalization, nilpotent matrices, scalar (dot) products, angle, rotations, orthogonal matrices, GLn, SLn, On, SO2, SO3.

Complex Analysis

Holomorphic functions, Cauchy-Riemann equations, integration, zeros of analytic functions, Cauchy formulas, maximum modulus theorem, open mapping theorem, Louville’s theorem, poles and singularities, residues and contour integration, conformal maps, Rouche’s theorem, Morera’s theorem

Calculus and Real Analysis

(a) Real Line: Limits, continuity, differentiability, Reimann integration, sequences, series, limsup, liminf, pointwise and uniform convergence, uniform continuity, Taylor expansions

(b) Multivariable: Limits, continuity, partial derivatives, chain rule, directional derivatives, total derivative, Jacobian, gradient, line integrals, surface integrals, vector fields, curl, divergence, Stoke’s theorem

(c) General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass approximation.

Topology

Topological spaces, base of open sets, product topology, accumulation points, boundary, continuity, connectedness, path connectedness, compactness, Hausdorff spaces, normal spaces, Urysohn’s lemma, Tietze extension, Tychonoff’s theorem,

Topics

Sub topics

School Level Mathematics

Arithmetic and geometric progressions; arithmetic, geometric and harmonic mean; polynomials, matrices (basic operations, inverse, transpose), determinants, solving linear equations, prime numbers and divisibility, GCD, LCM, modular arithmetic, logarithms, basic properties of functions (domain, range, injective, bijective, surjective), elementary calculus (differentiation, maxima-minima, integration and its applications)

Discrete Mathematics

Sets and relations, combinations and permutations, elementary counting techniques, pigeonhole principle, binomial theorem, mathematical induction, boolean logic and truth tables

Probability Theory

Elementary probability theory, conditional probability, and Bayes theorem; random variables, density functions, distribution functions; standard distributions (Gaussian etc.); expectation and variance; data interpretation; summary statistics

Programming

Ability to read and interpret algorithms written in simple pseudocode (variables, conditionals, loops)