## GATE: Syllabus for Mathematics

## Graduate Aptitude Test in Engineering (GATE)

###
SYLLABUS FOR GENERAL APTITUDE (GA)

(Common to all papers)

**Verbal Ability: **English grammar, sentence completion,
verbal analogies, word groups, instructions, critical reasoning and verbal deduction.

**Numerical Ability: **Numerical computation, numerical
estimation, numerical reasoning and data interpretation.

### Mathematics (MA)

**Linear Algebra: **Finite dimensional vector spaces;
Linear transformations and their matrix representations, rank; systems of linear
equations, eigen values and eigen vectors, minimal polynomial, Cayley-Hamilton Theroem,
diagonalisation, Hermitian, Skew-Hermitian and unitary matrices; Finite dimensional
inner product spaces, Gram-Schmidt orthonormalization process, self-adjoint operators.

**Complex Analysis: **Analytic functions, conformal
mappings, bilinear transformations; complex integration: Cauchy’s integral theorem
and formula; Liouville’s theorem, maximum modulus principle; Taylor and Laurent’s
series; residue theorem and applications for evaluating real integrals.

**Real Analysis: **Sequences and series of functions,
uniform convergence, power series, Fourier series, functions of several variables,
maxima, minima; Riemann integration, multiple integrals, line, surface and volume
integrals, theorems of Green, Stokes and Gauss; metric spaces, completeness, Weierstrass
approximation theorem, compactness; Lebesgue measure, measurable functions; Lebesgue
integral, Fatou’s lemma, dominated convergence theorem.

**Ordinary Differential Equations: **First order
ordinary differential equations, existence and uniqueness theorems, systems of linear
first order ordinary differential equations, linear ordinary differential equations
of higher order with constant coefficients; linear second order ordinary differential
equations with variable coefficients; method of Laplace transforms for solving ordinary
differential equations, series solutions; Legendre and Bessel functions and their
orthogonality.

**Algebra: **Normal subgroups and homomorphism theorems,
automorphisms; Group actions, Sylow’s theorems and their applications; Euclidean
domains, Principle ideal domains and unique factorization domains. Prime ideals
and maximal ideals in commutative rings; Fields, finite fields.

**Functional Analysis: **Banach spaces, Hahn-Banach
extension theorem, open mapping and closed graph theorems, principle of uniform
boundedness; Hilbert spaces, orthonormal bases, Riesz representation theorem, bounded
linear operators.

**Numerical Analysis: **Numerical solution of algebraic
and transcendental equations: bisection, secant method, Newton-Raphson method, fixed
point iteration; interpolation: error of polynomial interpolation, Lagrange, Newton
interpolations; numerical differentiation; numerical integration: Trapezoidal and
Simpson rules, Gauss Legendre quadrature, method of undetermined parameters; least
square polynomial approximation; numerical solution of systems of linear equations:
direct methods (Gauss elimination, LU decomposition); iterative methods (Jacobi
and Gauss-Seidel); matrix eigenvalue problems: power method, numerical solution
of ordinary differential equations: initial value problems: Taylor series methods,
Euler’s method, Runge-Kutta methods.

**Partial Differential Equations: **Linear and quasilinear
first order partial differential equations, method of characteristics; second order
linear equations in two variables and their classification; Cauchy, Dirichlet and
Neumann problems; solutions of Laplace, wave and diffusion equations in two variables;
Fourier series and Fourier transform and Laplace transform methods of solutions
for the above equations.

**Mechanics: **Virtual work, Lagrange’s equations
for holonomic systems, Hamiltonian equations.

**Topology: **Basic concepts of topology, product
topology, connectedness, compactness, countability and separation axioms, Urysohn’s
Lemma.

**Probability and Statistics: **Probability space,
conditional probability, Bayes theorem, independence, Random variables, joint and
conditional distributions, standard probability distributions and their properties,
expectation, conditional expectation, moments; Weak and strong law of large numbers,
central limit theorem; Sampling distributions, UMVU estimators, maximum likelihood
estimators, Testing of hypotheses, standard parametric tests based on normal, X2
, t, F – distributions; Linear regression; Interval estimation.

**Linear programming: **Linear programming problem
and its formulation, convex sets and their properties, graphical method, basic feasible
solution, simplex method, big-M and two phase methods; infeasible and unbounded
LPP’s, alternate optima; Dual problem and duality theorems, dual simplex method
and its application in post optimality analysis; Balanced and unbalanced transportation
problems, u -v method for solving transportation problems; Hungarian method for
solving assignment problems.

**Calculus of Variation and Integral Equations: **
Variation problems with fixed boundaries; sufficient conditions for extremum, linear
integral equations of Fredholm and Volterra type, their iterative solutions.

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