VITMEE Syllabus for Mathematics.
Vellore Institute of Technology Master’s entrance examination (VITMEE) For M.Tech and MCA Admission 2012 VITMEE Syllabus for Mathematics
MA-MATHEMATICS
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.
VITMEE Syllabus for Mathematics
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:Normalsubgroups 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 -u method for solving transportation problems; Hungarian method for solving assignment problems.
Calculus ofVariation and Integral Equations: Variation problems with fixed boundaries; sufficient conditions for extremum, linear integral equations of Fredholm and Volterra type, their iterative solutions.
MB – Research Aptitude & Quantitative Ability
Verbal Comprehension(for research
This section aims to test the candidate’s comprehension of and interpretative abilities in English as a language of business. Given the potential manager’s decision-making roles, this section seeks to examine the candidate’s felicity with common forms of English expression, grammar and usage in business that would enable him/her to extract essential information from a variety of data, and arrive at an informed decision. Regular analysis of business articles and non-fiction prose, besides a firm grasp of communicative English grammar would be helpful in preparing for this section.
Logical Reasoning
This section consists of analytical reasoning, argument analysis, and analysis of explanation questions
Quantitative Ability
Basic Mathematics (Numbers; Operations; HCF and LCM; Fractions, Decimals and Percentages; Ratio and Proportion; Roots and Power; Progressions; Elementary Geometry and Mensuration; Introductory Set Theory), Linear Algebra (Equations and Inequalities; Matrices; Determinants; Simultaneous equations and solutions; Elementary Linear Programming; Elementary differential candidates only)
calculus involving functions of one variable; Elementary integral calculus), and Probability and Statistics
(Types of Data; Frequency Distributions; Measures of Central Tendency and Dispersion; Probability Concepts: Basic Outcomes, Events, Sample Spaces; Probability Calculations: Counting Rules using Permutations and Combinations, Unions and Intersections, Complementary Events, Mutually Exclusive Events, Conditional Probability and Independent Events; Correlation and Simple Linear Regression) for their use in business applications such as Partnership and Shareholding;
Present Worth and Discounts; Depreciation; Demand and Supply; Cost and Revenue, and common applications such as Banking Transactions; Inventories; Mixtures; Time and Work; Time and Distance; Pipes and Tanks; Estimation of time, distance, area, volume, effort, etc.
Data Interpretation
Assess the ability of the examinee to make valid interpretations from a given data set. The section also assesses the ability of the examinee to understand data in different representative forms such as simple tables, histograms, pie charts, graphs, scatter diagrams, etc. Although involved calculations are not expected, simple data manipulations would be required.
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