GATE Mathematics Syllabus 2027: Section-wise Topics for the MA Paper
The GATE Mathematics syllabus 2027 is the official topic list for the MA paper of GATE, released by IIT Madras as the organising institute for this cycle. The brochure states that the syllabi of GATE 2027 papers have been revised, so the list you prepare from should be the 2027 version, not one carried over from earlier years.
The MA paper has 11 mathematics sections worth 85 marks and a common General Aptitude section worth 15 marks. This page sets out every section with its topics, a sensible place to start, where sections overlap so you can study a concept once, and a phased plan for the months before the February 2027 exam. IIT Madras does not publish section-wise weightage, so no weightage figures are given here.
Table of Contents
GATE Mathematics Syllabus 2027 at a Glance
Key Changes to Know
Section-wise GATE Mathematics Syllabus
Topic Details by Section
General Aptitude Syllabus
Overlap Between Sections
How to Cover the GATE MA Syllabus
GATE Mathematics Syllabus 2027 at a Glance
Part | Sections | Questions | Marks | Time |
|---|---|---|---|---|
General Aptitude | Verbal, Quantitative, Analytical and Spatial Aptitude | 10 | 15 | Shared 3 hours |
Mathematics | 11 sections, from Calculus to Linear Programming | 55 | 85 | Shared 3 hours |
Total | 12 syllabus areas | 65 | 100 | 3 hours |
Questions come as MCQ, MSQ and NAT, and only MCQs carry negative marking.
Key Changes to Know
Revised for 2027: IIT Madras states that the syllabi for GATE 2027 test papers have been revised. Download the official GATE 2027 MA syllabus PDF and tick topics off against it.
Eleven mathematics sections: Calculus, Linear Algebra, Real Analysis, Complex Analysis, Ordinary Differential Equations, Algebra, Functional Analysis, Numerical Analysis, Partial Differential Equations, Topology and Linear Programming.
No Engineering Mathematics section: Engineering papers carry a separate 13-mark Engineering Mathematics section. MA does not; all 85 subject marks are mathematics from the list below.
No official weightage: Any section-wise percentage you see elsewhere is an estimate from past papers, not an official figure.
Older pages may still show the previous syllabus wording, so follow the 2027 PDF wherever the two differ.
Section-wise GATE Mathematics Syllabus
Section | Topics to cover | Start with |
|---|---|---|
1. Calculus | Functions of several variables, Taylor's theorem, maxima, minima and saddle points, Lagrange multipliers, double and triple integrals, Jacobians, vector calculus | Partial derivatives and Lagrange multipliers |
2. Linear Algebra | Vector spaces, linear transformations, rank and nullity, eigenvalues, minimal polynomial, Cayley-Hamilton, inner product spaces, special matrices, Jordan form, quadratic forms | Rank-nullity and eigenvalues |
3. Real Analysis | Metric spaces, compactness, completeness, uniform convergence, Ascoli-Arzela, contraction mapping, inverse and implicit function theorems, Lebesgue measure and integral | Sequences and series of functions |
4. Complex Analysis | Analytic and harmonic functions, Cauchy's theorem and formula, singularities, Taylor and Laurent series, residues, Rouche's theorem, Mobius transformations | Analytic functions and residues |
5. Ordinary Differential Equations | First-order equations, existence and uniqueness, higher-order linear equations, Cauchy-Euler, Laplace transforms, series solutions, Legendre and Bessel functions, Sturm-Liouville, stability | Linear equations with constant coefficients |
6. Algebra | Groups, normal subgroups, homomorphisms, permutation groups, group actions, Sylow theorems, rings and ideals, UFDs, PIDs, Euclidean domains, fields and extensions | Group basics and cyclic groups |
7. Functional Analysis | Normed and Banach spaces, bounded and compact operators, dual spaces, Hahn-Banach, open mapping, closed graph, uniform boundedness, Hilbert spaces, spectral theorem | Normed spaces and bounded operators |
8. Numerical Analysis | Gaussian elimination, LU and Cholesky, Gauss-Seidel and Jacobi, bisection, secant, Newton-Raphson, interpolation, numerical integration, Euler and Runge-Kutta methods | Root finding and interpolation |
9. Partial Differential Equations | Method of characteristics, classification and canonical forms, separation of variables, maximum principle, heat and wave equations, d'Alembert formula, transform methods | Classification of second-order equations |
10. Topology | Bases, subspace, order, product, quotient and metric topology, connectedness, compactness, Tychonoff theorem, countability and separation axioms, Urysohn's lemma, Tietze extension | Bases and subspace topology |
11. Linear Programming | Convex sets, basic feasible solutions, graphical and simplex methods, two-phase and revised simplex, duality, transportation and assignment problems | Graphical and simplex methods |
IIT Madras does not publish weightage, but the 2026 MA paper drew questions from almost every section, including topology, algebra, analysis, differential equations, numerical methods and linear programming. Plan to cover all 11 sections, and give the most time to Linear Algebra, Real Analysis, Complex Analysis and Algebra, because their ideas feed into several other sections.
Topic Details by Section
Calculus and Linear Algebra
Calculus: Continuity, directional, partial and total derivatives, Taylor's theorem, maxima, minima and saddle points, Lagrange multipliers, double and triple integrals, change of variables, and applications to area, volume and surface area.
Vector calculus: Gradient, divergence and curl, line and surface integrals, and the theorems of Green, Stokes and Gauss.
Linear Algebra: Finite-dimensional vector spaces over real or complex fields, matrix representation of linear maps, systems of equations, characteristic and minimal polynomials, diagonalisation, Gram-Schmidt, Hermitian, normal and unitary matrices, Jordan canonical form, and bilinear and quadratic forms.
Real Analysis and Functional Analysis
Metric space ideas: Baire category theorem, connectedness, compactness, completeness, continuity and uniform continuity.
Function sequences: Uniform convergence, Ascoli-Arzela theorem, Weierstrass approximation theorem, contraction mapping principle and power series.
Measure and integration: Lebesgue measure on the real line, measurable functions, Lebesgue integral, Fatou's lemma, the monotone and dominated convergence theorems, and Lp spaces.
Functional Analysis: Normed linear spaces, Banach spaces, separability, dual spaces, the Hahn-Banach, open mapping and closed graph theorems, uniform boundedness, Hilbert spaces, orthonormal bases, the projection and Riesz representation theorems, and the spectral theorem for compact self-adjoint operators.
Complex Analysis
Continuity and differentiability of complex functions, analytic and harmonic functions, Cauchy's integral theorem and formula, Liouville's theorem, the maximum modulus principle, Morera's theorem, zeros and singularities, power series and radius of convergence, Taylor and Laurent series, the residue theorem for real integrals, Rouche's theorem, the argument principle, the Schwarz lemma, conformal mappings and Mobius transformations.
Algebra
Groups, subgroups, normal subgroups, quotient groups, homomorphisms and automorphisms, cyclic and permutation groups, group actions, finite abelian groups and the Sylow theorems. On the ring side: ideals, prime and maximal ideals, quotient rings, unique factorisation domains, principal ideal domains, Euclidean domains, polynomial rings and Eisenstein's criterion. Fields cover finite fields, field extensions, algebraic extensions and algebraically closed fields.
Ordinary and Partial Differential Equations
ODE: Existence and uniqueness for initial value problems, higher-order linear equations, second-order equations with variable coefficients, Cauchy-Euler equations, Laplace transforms, power series and Frobenius solutions, Legendre and Bessel functions, linear systems, Sturm's theorems, Sturm-Liouville problems, and stability of planar autonomous systems including Lyapunov functions.
PDE: Characteristics for first-order linear and quasilinear equations, classification and canonical forms of second-order equations, separation of variables for the Laplace equation in Cartesian and polar coordinates, the maximum principle, the heat and wave equations, d'Alembert's formula, domains of dependence and influence, and Laplace and Fourier transform methods.
Numerical Analysis
Direct methods (Gaussian elimination, LU decomposition, Cholesky factorisation) and iterative methods (Gauss-Seidel, Jacobi) for linear systems, with convergence for diagonally dominant matrices; bisection, secant, Newton-Raphson and fixed-point iteration; Lagrange and Newton interpolation and its error; numerical differentiation; trapezoidal, Simpson and Newton-Cotes rules with their errors; and Euler's method and the Runge-Kutta method of order 2 for initial value problems.
Topology and Linear Programming
Topology: Bases and subbases, subspace, order, product, quotient and metric topologies, connectedness and path connectedness, compactness in its sequential and limit point forms, Tychonoff theorem, countability and separation axioms, Urysohn's lemma and the Tietze extension theorem.
Linear Programming: Models, convex sets and extreme points, basic feasible solutions, graphical, simplex, two-phase and revised simplex methods, infeasible and unbounded problems, duality, transportation problems (north-west corner, least cost and Vogel's methods, modified distribution) and the Hungarian method for assignment problems.
General Aptitude Syllabus
Section | Topics to cover |
|---|---|
Verbal Aptitude | Basic grammar, vocabulary, idioms and phrases in context, reading comprehension, narrative sequencing |
Quantitative Aptitude | Data interpretation from graphs, maps and tables, ratios, percentages, exponents and logarithms, permutations and combinations, series, mensuration, elementary statistics and probability |
Analytical Aptitude | Deduction and induction, analogy, numerical relations and reasoning |
Spatial Aptitude | Translation, rotation, scaling and mirroring of shapes, paper folding, cutting and patterns in two and three dimensions |
Overlap Between Sections
Area | Where it appears | How to plan |
|---|---|---|
Linear maps and matrices | Linear Algebra, ODE systems, Numerical Analysis, Functional Analysis | Master eigenvalues and diagonalisation first; reuse them for linear systems and operators |
Compactness and connectedness | Real Analysis, Topology, Functional Analysis | Learn the metric space versions, then generalise to topological spaces |
Power series | Real Analysis, Complex Analysis, ODE series solutions | Study convergence once and apply it in all three |
Laplace and Fourier transforms | ODE and PDE | Revise transform rules together and practise mixed problems |
Inner products and orthogonality | Linear Algebra, Functional Analysis, Legendre and Bessel functions | Treat Gram-Schmidt and Hilbert spaces as one thread |
How to Cover the GATE MA Syllabus
Weeks 1 to 5: Linear Algebra, Calculus and Real Analysis, with previous year questions after each topic.
Weeks 6 to 9: Complex Analysis, Algebra and ODE, plus 30 minutes of General Aptitude a day.
Weeks 10 to 12: Topology, Functional Analysis, PDE, Numerical Analysis and Linear Programming, with timed sectional sets.
Final weeks: Full-length timed papers with the on-screen calculator, same-day analysis and error-log revision.
Solve official GATE MA papers topic-wise as you finish each section; IIT Madras hosts past papers and answer keys on the GATE 2027 download page.
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