Syllabus - SAU 2016

M.Sc. in Applied Mathematics

Format of the Entrance Test Paper

The duration of the Entrance Test will be 3 hours and the question paper will consist of 100 multiple choice questions in two parts.

PART A: 40 questions on Basic Mathematics.

PART B: 60 questions on Undergraduate Level Mathematics.

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The combined syllabus for both Part A and Part B is as follows:

Calculus and Analysis: Limit, continuity, uniform continuity and differentiability; Bolzano Weierstrass theorem; mean value theorems; tangents and normal; maxima and minima; theorems of integral calculus; sequences and series of functions; uniform convergence; power series; Riemann sums; Riemann integration; definite and improper integrals; partial derivatives and Leibnitz theorem; total derivatives; Fourier series; functions of several variables; multiple integrals; line; surface and volume integrals; theorems of Green; Stokes and Gauss; curl; divergence and gradient of vectors.

Algebra: Basic theory of matrices and determinants; groups and their elementary properties; subgroups, normal subgroups, cyclic groups, permutation groups; Lagrange's theorem; quotient groups; homomorphism of groups; isomorphism and correspondence theorems; rings; integral domains and fields; ring homomorphism and ideals; vector space, vector subspace, linear independence of vectors, basis and dimension of a vector space.

Differential equations: General and particular solutions of ordinary differential equations (ODEs); formation of ODE; order, degree and classification of ODEs; integrating factor and linear equations; first order and higher degree linear differential equations with constant coefficients; variation of parameter; equation reducible to linear form; linear and quasi-linear first order partial differential equations (PDEs); Lagrange and Charpits methods for first order PDE; general solutions of higher order PDEs with constant coefficients.

Numerical Analysis: Computer arithmetic; machine computation; bisection, secant; Newton-Raphson and fixed point iteration methods for algebraic and transcendental equations; systems of linear equations: Gauss elimination, LU decomposition, Gauss Jacobi and Gauss Siedal methods, condition number; Finite difference operators; Newton and Lagrange interpolation; least square approximation; numerical differentiation; Trapezoidal and Simpsons integration methods.

Probability and Statistics: Mean, median, mode and standard deviation; conditional probability; independent events; total probability and Baye’s theorem; random variables; expectation, moments generating functions; density and distribution functions, conditional expectation.

Linear Programming: Linear programming problem and its formulation; graphical method, simplex method, artificial starting solution, sensitivity analysis, duality and post-optimality analysis.

Note:

·           If the answer given to any of the Multiple Choice Questions is wrong, ¼ of the marks               assigned to that question will be deducted.

·           Calculators will not be allowed. However, Log Tables may be used

  

Syllabus - SAU 2016

M.Phil./ Ph.D. in Applied Mathematics

Format of the Entrance Test Paper

The duration of the Entrance Test will be 3 hours and the question paper will consist of 70 multiple choice questions in two parts: Part A and Part B.

PART A:  30 questions on undergraduate level Mathematics

PART B:  40 questions on Master’s level Mathematics

The areas from which questions may be asked will include the following:

Analysis: Real functions; limit, continuity, differentiability; sequences; series; uniform convergence; functions of complex variables; analytic functions, complex integration; singularities, power and Laurent series; metric spaces; stereographic projection; topology, compactness, connectedness; normed linear spaces, inner product spaces; dual spaces, linear operators; Lebesgue measure and integration; convergence theorems.

Algebra: Basic theory of matrices and determinants; eigen values and eigen vectors; Groups and their elementary properties; subgroups, normal subgroups, cyclic groups, permutation groups; Lagrange's theorem; quotient groups, homomorphism of groups; Cauchy Theorem and p-groups; the structure of groups; Sylow's theorems and their applications; rings, integral domains and fields; ring homomorphism and ideals; polynomial rings and irreducibility criteria; vector space, vector subspace, linear independence of vectors, basis and dimension of a vector space, inner product spaces, orthonormal basis; Gram-Schmidt process, linear transformations.

Differential Equations: First order ordinary differential equations (ODEs); solution of first order initial value problems; singular solution of first order ODEs; system of linear first order ODEs; method of solution of dx/P=dy/Q=dz/R; orthogonal trajectory; solution of Pfaffian differential equations in three variables; linear second order ODEs. Sturm-Liouville problems; Laplace transformation of ODEs; series 
solutions; Cauchy problem for first order partial differential equations (PDEs); method of characteristics; second order linear PDEs in two variables and their classification; separation of variables; solution of Laplace, wave and diffusion equations, Fourier transform and Laplace transform of PDEs.

Numerical Analysis: Numerical solution of algebraic and transcendental equations, direct and iterative methods for system of linear equations; matrix eigenvalue problems; interpolation and approximations; numerical differentiation and integration; composite numerical integration; double numerical integration; numerical solution for initial value problems; finite difference and finite element methods for boundary value problems.

Probability and Statistics: Axiomatic approach of probability; random variables, expectation, moments generating functions, density and distribution functions, conditional expectation.

Linear Programming: Linear programming problem and its formulation; graphical method, simplex method; artificial starting solution; sensitivity analysis; duality and post-optimality analysis.

Note:

·         If the answer given to any of the Multiple Choice Questions is wrong, ¼ of the marks assigned to that question will be deducted.

·         Calculators will not be allowed. However, Log Tables may be used.