GATE Mathematics (MA) Syllabus: Important Topics, Topic-Wise Weightage, Complete Guide

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, eigenvectors, diagonalization, minimal polynomial,
• 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.

Calculus

• Functions of two or more variables
• Continuity
• Directional derivatives
• Partial derivatives
• Total derivative
• Maxima and Minima
• Saddle point
• Method of Lagrange’s Multipliers; Double and Triple integrals and their applications to area
• Volume and surface area; Vector Calculus: gradient, divergence, and curl
• Line integrals and Surface integrals
• Green’s theorem
• Stokes’ theorem
• Gauss divergence theorem

Real Analysis

• Metric spaces
• Connectedness
• Compactness
• Completeness
• 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

Topology

• Basic concepts of topology
• Bases
• Subbases
• Subspace topology
• Order topology
• Product topology
• Quotient topology
• Metric topology
• Connectedness
• Compactness
• Countability and separation axioms
• Urysohn’s Lemma

Complex Analysis

• Functions of a complex variable: continuity, differentiability, analytic functions,
• Radius of convergence
• Taylor’s series and Laurent’s series
• Residue theorem and applications for evaluating real integrals
• Harmonic functions
• Complex integration: Cauchy’s integral theorem and formula
• Liouville’s theorem
• Maximum modulus principle
• Morera’s theorem
• Zeros and singularities
• Power series
• Rouche’s theorem
• Argument principle
• Schwarz lemma
• Conformal mappings
• Mobius transformations

Ordinary Differential Equations

• First order ordinary differential equations
• Method of Laplace transforms 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

• 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

• Planar autonomous systems of ordinary differential equations
• Stability of stationary points for linear systems with constant coefficients
• Sturm's oscillation and separation theorems
• Sturm-Liouville eigenvalue problems
• Linearized stability
• Lyapunov functions

Functional Analysis

• Normed linear spaces
• Banach spaces
• Hahn-Banach theorem
• Open mapping and closed graph theorems
• Principle of uniform boundedness
• Inner-product spaces
• Hilbert spaces
• Orthonormal bases
• Projection theorem
• Riesz representation theorem
• Spectral theorem for compact self-adjoint operators

Algebra

• Groups, subgroups
• Normal subgroups
• Quotient groups
• Homomorphisms
• Automorphisms
• Cyclic groups
• Permutation groups
• Group action
• Sylow’s theorems and their applications
• Rings
• Ideals
• Prime and maximal ideals
• Quotient rings
• Unique factorization domains
• Principle ideal domains
• Euclidean domains
• Polynomial rings
• Eisenstein’s irreducibility criterion Fields
• Finite fields
• Field extensions
• Algebraic extensions
• Algebraically closed fields

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 the 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

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 duality and strong duality
• Balanced and unbalanced transportation problems
• Initial basic feasible solution of balanced transportation problems (least cost method
• North-west corner rule
• Vogel’s approximation method)
• Optimal solution
• Modified distribution method
• Solving assignment problems
• Hungarian method

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
• Heat and wave equations in one space variable
• Wave equation: Cauchy problem and d'Alembert formula
• Domains of dependence and influence
• Nonhomogeneous wave equation
• Heat equation: Cauchy problem
• Laplace and Fourier transform methods

Mode of Examination

Online

Duration

3 hours

Types of Questions

MCQs and NAT

Sections: 2 sections

General Aptitude and Subject-based

Total Questions

65 questions

Total Marks

100 marks

Negative Marking

For MCQs only

MCQ

• 1/3 for 1 mark questions
• 2/3 for 2 marks questions

NAT

No negative marking

MA- Subject-Based

• 25 questions of 1 mark each
• 30 questions of 2 marks each

85 marks

• MCQs
• NATs

GA

• 5 questions of 1 mark each
• 5 questions of 2 marks each

15 marks

MCQs

Vector Calculus

20%

Probability & Statistics

20%

Numerical Methods

20%

Differential Equation

10%

Linear Algebra

10%

Calculus

10%

Complex Variables

10%

Wiley Acing the Gate: Engineering Mathematics and General Aptitude

Anil K. Maini, Wiley

Chapterwise Solved Papers Mathematics GATE

Suraj Singh, Arihant Publication

Higher Engineering Mathematics

B.S. Grewal, Khanna Publishers

GATE: Engineering Mathematics

ME Team, Made Easy Publications

GATE Engineering Mathematics for All Streams

Abhinav Goel, Arihant Publication

Day 1 to 5

Solve all Problems of chapters 1 to 3

Day 6

Revise the 3 chapters done previously and attempt chapter-wise mock tests

Day 7 to 11

Solve all Problems of chapters 1 to 3

Day 12 and 13

Revise all the chapters done between day 7 to day 11 and attempt chapter-wise mock tests

Day 14

Solve as many questions as possible relating to past GATE question papers and test series of any coaching center

Day 15

Rest Day

Day 16 to 20

Learn all concepts of Chapters 4 to 6  and solve available problems.

Day 21

Revise all 3 chapters and attempt chapter-wise mock tests.

Day 22 to 26

Chapters  4 to 6 have to be learned and problems relating to the concept need to be practiced.

Day 27 and 28

Revise all the chapters done between days 22 and 26 and attempt chapter-wise mock tests to analyze your performance

Day 29

Solve questions related to Subject 2 from past GATE previous year question papers and test series.

Day 30

Rest Day

Day 31

Follow the same procedure