GATE Mathematics Syllabus: Check Exam Pattern and Section-wise Weightage

Total Duration

180 minutes (3 hours)

Total Marks

100 marks

Total Questions

65 questions

Sections

- General Aptitude – 15 marks - Mathematics (Core/Subject) – 85 marks

Question Distribution

- General Aptitude: 10 questions (5 × 1 mark + 5 × 2 marks) - Mathematics Section: 55 questions (approx. 25 × 1 mark + 30 × 2 marks)

Question Types

- MCQ: Multiple Choice Questions - MSQ: Multiple Select Questions - NAT: Numerical Answer Type

Negative Marking

- 1-mark MCQ → –⅓ mark for wrong answer - 2-mark MCQ → –⅔ mark for wrong answer - No negative marking for MSQ & NAT

Exam Mode

Computer-Based Test (CBT)

Conducting Body (2026)

IIT Guwahati

Calculus

- Functions of two or more variables - Continuity, directional & partial derivatives - Total derivative, maxima & minima, saddle point, method of Lagrange multipliers - Double & triple integrals and their applications (area, volume, surface area) - Vector calculus: gradient, divergence, curl - Line & surface integrals - Green’s theorem, Stokes’ theorem, Gauss divergence theorem

Linear Algebra

- Finite dimensional vector spaces (ℝ or ℂ) - Linear transformations & matrix representation - Rank, nullity & systems of linear equations - Characteristic polynomial, eigenvalues & eigenvectors - Diagonalization, minimal polynomial, Cayley–Hamilton theorem - Inner product spaces, Gram–Schmidt orthonormalization - Symmetric, skew-symmetric, Hermitian, unitary, normal matrices - Jordan canonical form, bilinear & quadratic forms

Real Analysis

- Metric spaces: definitions, topology, completeness, compactness, connectedness - Sequences & series of functions, uniform convergence, Ascoli–Arzelà theorem - Power series, Weierstrass approximation theorem - Differentiation in several variables, inverse & implicit function theorems - Lebesgue measure & integration - Measurable functions, Lebesgue integral, Fatou’s lemma, monotone & dominated convergence theorems

Complex Analysis

- Analytic & harmonic functions, Cauchy–Riemann equations - Complex integration: Cauchy’s theorem & integral formula - Power series, Taylor & Laurent expansions - Singularities, residues & applications to real integrals - Rouche’s theorem, Argument Principle, Schwarz lemma - Conformal mappings, Möbius transformations - Liouville’s theorem, Maximum Modulus Principle, Morera’s theorem

Ordinary Differential Equations (ODE)

- First-order ODEs, existence & uniqueness theorems - Linear ODEs of higher order with constant coefficients - Variable coefficient second order ODE, Cauchy–Euler equations - Laplace transform & Frobenius series method - Special functions: Legendre, Bessel & orthogonality - Systems of first-order ODEs, phase plane analysis - Sturm–Liouville problems, eigenvalue problems, stability

Partial Differential Equations (PDE)

- Basic classification (first-order, second-order) - Method of separation of variables - Heat, wave & Laplace equations - Boundary value problems - Fourier series & transforms

Algebra

- Group theory: subgroups, cosets, normal subgroups, quotient groups, homomorphism, automorphism - Cyclic groups, permutation groups, Sylow theorems, group actions - Ring theory: ideals, prime & maximal ideals, quotient rings - Integral domains, unique factorization & Euclidean domains - Polynomial rings, irreducibility criteria (Eisenstein) - Field theory: extension fields, algebraic extensions, splitting fields, Galois theory basics

Functional Analysis

- Normed linear & Banach spaces - Inner product & Hilbert spaces - Hahn–Banach theorem, open mapping & closed graph theorems - Uniform boundedness principle (Banach–Steinhaus) - Riesz representation theorem, orthonormal bases, projection theorem - Spectral theorem for compact self-adjoint operators

Numerical Analysis

- Nonlinear equations: bisection, Newton–Raphson, fixed-point iteration - Interpolation: Lagrange & Newton forms, error analysis - Numerical differentiation & integration: trapezoidal, Simpson’s rules - ODE initial value problems: Euler’s, Runge–Kutta methods - Systems of linear equations: Gaussian elimination, LU decomposition, Jacobi & Gauss–Seidel methods

Topology

- Open & closed sets, basis, subspace topology, product topology - Order & metric topology, quotient topology - Connectedness, compactness, countability axioms - Separation axioms (T0, T1, T2, etc.), Urysohn’s Lemma

Linear Programming

- Formulation of linear programming problems - Simplex method - Duality theory & primal-dual relationships - Sensitivity analysis & dual simplex method

Calculus

10%

Functions of several variables, maxima-minima, vector calculus, Green’s & Stokes’ theorems

Revise core calculus formulas daily; solve past-year numerical problems and conceptual theorems.

Linear Algebra

10%

Eigenvalues, eigenvectors, diagonalization, matrix operations

Master theory first; practice solving system equations & matrix transformations quickly.

Real Analysis

8%

Sequences, series, continuity, convergence, Lebesgue integration

Focus on theorems; make flashcards for definitions and practice proof-based questions.

Complex Analysis

7%

Cauchy’s theorem, residues, contour integration, conformal mapping

Solve previous GATE questions on residues and contour integrals; memorize key theorems.

Ordinary Differential Equations (ODE)

10%

First & higher order ODEs, special functions, Laplace transforms

Learn standard forms; solve numerical and analytical problems from each category.

Partial Differential Equations (PDE)

8%

Heat, wave, Laplace equations, separation of variables

Practice boundary value problems and Fourier method derivations.

Algebra

8%

Group, ring, and field theory, homomorphisms, Sylow theorems

Prepare short notes on important theorems and group classifications.

Functional Analysis

6%

Banach & Hilbert spaces, Riesz theorem, spectral theory

Focus on conceptual clarity and relationships between function spaces.

Numerical Analysis

12%

Interpolation, differentiation, integration, ODE solvers

Practice problems using formulas; focus on iterative & error-based questions.

Topology

6%

Compactness, connectedness, separation axioms

Study definitions with examples; attempt theory-based questions.

Linear Programming

5%

Simplex method, duality, sensitivity analysis

Practice stepwise simplex problems; revise standard formulations.

General Aptitude (GA)

10%

Verbal & numerical ability, logical reasoning

Practice 1–2 aptitude questions daily; focus on accuracy and speed.