MA is the Mathematics paper, not the Engineering Mathematics section of an engineering paper. Its coverage includes analysis, algebra, topology, differential equations, numerical analysis and linear programming.

Mathematics is tested within a 100-mark paper. The paper also includes General Aptitude. The sections below show the complete subject syllabus.

GATE MA Syllabus and Marks Distribution

There is no optional subject block in this paper. All listed sections are part of the syllabus.

Mathematics Unit-wise Syllabus

Calculus

• Functions of two or more variables, continuity, directional derivatives, partial derivatives, total derivative, Taylor’s theorem, maxima and minima, saddle point, method of Lagrange’s multipliers; Double and Triple integrals, Jacobians and Change of variables, Applications to area, volume and surface area;
• Vector Calculus: gradient, divergence and curl, Line integrals and Surface integrals, Green’s theorem, Stokes’ theorem, and Gauss divergence theorem.

Linear Algebra

• Finite dimensional vector spaces over real or complex fields; Linear transformations and their matrix representations, rank and nullity; systems of linear equations, characteristic polynomial, eigenvalues and eigen - vectors, diagonalization, minimal polynom ial, Cayley -Hamilton Theorem, Finite dimensional inner product spaces, Gram -Schmidt orthonormalization process, symmetric, skew -symmetric, Hermitian, skew -Hermitian, normal, orthogonal and unitary matrices; diagonalization by a unitary matrix, Jordan canonical form; bilinear and quadratic forms.

Real Analysis

• Metric spaces, Baire category theorem, connectedness, compactness, completeness; Continuity and Uniform continuity of functions; Sequences and series of functions, uniform convergence, Ascoli -Arzela theorem; Weierstrass approximation theorem; contraction mapping principle, Power series; Differentiation of functions of several variables, Inverse and Implicit function theorems; Lebesgue measure on the real line, measurable functions; Lebesgue integral, Fatou’s lemma, monotone convergence theorem, dominated convergence theorem, Lp spaces.

Complex Analysis

• Functions of a complex variable: continuity, differentiability, analytic functions, harmonic functions;
• Complex integration: Cauchy’s integral theorem and formula; Liouville’s theorem, maximum modulus principle, Morera’s theorem; zeros and singularities; Power series, radius of convergence, Taylor’s series and Laurent’s series; Residue theorem and applications for evaluating rea l integrals; Rouche’s theorem, Argument principle, Schwarz lemma; Conformal mappings, Mobius transformations.

Ordinary Differential Equations

• First order ordinary differential equations, existence and uniqueness theorems for initial value problems, linear ordinary differential equations of higher order with constant coefficients; Second order linear ordinary differential equations with variable coefficients; Cauchy -Euler equation; Definition and basic properties of Laplace transforms, Applications of Lapalce transform for solving ordinary differential equations, series solutions (power series, Frobenius method); Legendre and Bessel functions and their orthogonal properties; Systems of linear first order ordinary differential equations, Sturm's oscillation and separation theorems, Sturm -Liouville eigenvalue problems, Planar autonomous systems of ordinary differential equations: Stability of stationary points for linear systems with constant coefficients, Linearized stability, Lyapunov functions.

Algebra

• Groups, subgroups, normal subgroups, quotient groups, homomorphisms, automorphisms; cyclic groups, permutation groups, Group action, Finite Abelian groups, Sylow’s theorems and their applications; Rings, ideals, prime and maximal ideals, quotient rings, un ique factorization domains, Principle ideal domains, Euclidean domains, polynomial rings, Eisenstein’s irreducibility criterion; Fields, finite fields, field extensions, algebraic extensions, algebraically closed fields.

Functional Analysis

• Normed linear spaces, Bounded linear operators and compact linear operators, Banach spaces, Separability, Dual spaces, Hahn -Banach theorem, open mapping and closed graph theorems, principle of uniform boundedness; Inner-product spaces, Hilbert spaces, orth onormal bases, projection theorem, Riesz representation theorem, spectral theorem for compact self-adjoint operators.

Numerical Analysis

• Systems of linear equations: Direct methods (Gaussian elimination, LU decomposition, Cholesky factorization), Iterative methods (Gauss-Seidel and Jacobi) and their convergence for diagonally dominant coefficient matrices;
• Numerical solutions of nonlinear equations: bisection method, secant method, Newton -Raphson method, fixed point iteration;
• Interpolation: Lagrange and Newton forms of interpolating polynomial, Error in polynomial interpolation of a function;
• Numerical differentiation and error, Numerical integration: Trapezoidal and Simpson rules, Newton -Cotes integration formulas, composite rules, mathematical errors involved in numerical integration formulae;
• Numerical solution of initial value problems for ordinary differential equations: Methods of Euler, Runge-Kutta method of order 2.

Partial Differential Equations

• Method of characteristics for first order linear and quasilinear partial differential equations; Second order partial differential equations in two independent variables: classification and canonical forms, method of separation of variables for Laplace equation in Cartesian and polar coordinates, Maximum Principle; heat and wave equations in one space variable; Wave equation: Cauchy problem and d'Alembert formula, domains of dependence and influence, non-homogeneous wave equation; Heat equation: Cauchy problem; Laplace and Fourier transform methods.

Topology

• Basic concepts of topology, bases, subbases, subspace topology, order topology, product topology, quotient topology, metric topology, connectedness, path connectedness, compactness, sequentially compact, limit point compact, Tychonoff theorem, countability and separation axioms, Urysohn’s Lemma, Tietze extension theorem.

Linear Programming

• Linear programming models, convex sets, extreme points; Basic feasible solution, graphical method, simplex method, two phase methods, revised simplex method ; Infeasible and unbounded linear programming models, alternate optima; Duality theory, weak dualit y and strong duality; Balanced and unbalanced transportation problems, Initial basic feasible solution of balanced transportation problems (least cost method, north-west corner rule, V ogel’s approximation method); Optimal solution, modified distribution me thod; Solving assignment problems, Hungarian method.

General Aptitude Syllabus

General Aptitude contributes 15 marks within the 100-mark paper. It is not an additional paper that can be left out when choosing the technical sections.

Chapter-wise Weightage and Important Topics

Only the section-wise marks are fixed. IIT Madras has not assigned a guaranteed number of marks to each chapter of Mathematics. A topic may appear in one year and receive fewer questions in another; a past-paper count is not an official 2027 weightage.

For a chapter-weightage comparison, record the year, paper code, session, question number and marks of each matching question. Keep a combined question under one main topic or label the overlap explicitly; counting it twice inflates the total. GATE previous-year question papers can be used for this topic-by-topic practice.

Question Types and Negative Marking

The paper uses one-mark and two-mark questions. The number of MCQ, MSQ and NAT questions is not a fixed chapter-wise quota. In an MSQ, selecting one correct option is not enough when the correct answer contains two options.