MATHEMATICS B.A./B.Sc. (GENERAL) THIRD YEAR EXAMINATION, 2012

Paper- I : ANALYSIS

SECTION-A Riemann integral. Integrability of continuous and monotonic functions. The fundamental theorem of integral calculus. Mean value theorems of integral calculus. Improper integrals and their convergence, Comparison tests, Abel’s and Dirichlet’s tests. Beta and Gamma functions. Frullani’s integral. Integral as a function of a parameter. Continuity, derivability and integrability of an integral of a function of a parameter. Double and triple integrals. Fubini’s Theorem without proof, Change of order of integration in double integrals, volume of a region in space, Triple integrals in spherical and cylinderical coordinates, substitution in multiple integrals.

SECTION-B Sequences and series of functions, pointwise and uniform convergence, Cauchy criterion for uniform convergence, Weierstrass M-test, Abel’s and Dirichlet’s tests for uniform convergence, uniform convergence and continuity, uniform convergence and Riemann integration, uniform convergence and differentiation, Weierstrass approximation theorem, Power series, interval of convergence of power series, Abel’s and Taylor’s theorems for power series. Fourier series. Fourier expansion of piecewise monotonic functions.

Paper-II : ABSTRACT ALGEBRA

SECTION A Groups, Subgroups, Cosets, Lagrange’s Theorem, Normal Subgroups and Quotient groups. Homomorphisms, Isomorphism Theorems, Conjugate Elements, Class Equation, Permutation Groups, Alternating Groups, Simplicity of An , n ³ 5 (without proof), Automorphisms of Groups. Rings, Subrings and Ideals, Quotient Rings, Fields and Homomorphisms, Integral Domains, Field of Quotients and Embedding Theorems, Polynomial Rings.

SECTION B Vector Spaces, Subspaces, Linear Dependence, Quotient Spaces, Direct Sums and Complements, Matrices and Change of Bases. Linear Transformation, Algebra of Linear Transformation, Dual Spaces, Matrices and Linear Transformation. Characteristic roots and characteristic vectors of a matrix, nature of characteristic roots of special types of matrices, relation between algebraic and geometric multiplicities of a characteristic root. Minimal polynomial of a matrix. Orthogonal reduction of real symmetric matrices, Unitary reduction of Hermitian matrices, similarity of matrices, diagonalisation of matrices

Paper-III : OPTIONAL

Option (i) : DISCRETE MATHEMATICS

SECTION A The pigeonhole principle : Simple form, strong form and its applications. A Theorem of Ramsey (withoutproof). Permutations and combinations of sets and multisets. Identities involving binomial co- efficients. Pascal’s formula. The multinomial theorem. Newton’s binomial theorem. The inclusion exclusion principle and its applications. Derangements. Permutations with forbidden positions. Recurrence relations Linear Recurrence relations with constant co- efficients. Homogeneous solutions, particular solutions, total solutions. Non homogeneous recurrence relation. Generating functions, Exponential generating functions, solution of recurrence relations using generating functions, Catalan numbers, Difference sequences and stirling numbers, Partition numbers.

SECTION B Combinatorial designs, block designs, Steiner triple system, Latin squares. Graphs : Basic properties, Eulerian trails, Hamilton Chains and Cycles, Bipartite multigraphs. Plane and Planar Graphs. Trees, Spanning trees: Breadth first, Depth first, Dijkstra’s, Kruskal and Prim’s algorithms to generate spanning trees. Directed graphs and networks. Chromatic number.

Option (ii) : PROBABILITY THEORY AND NUMERICAL ANALYSIS

SECTION-A Probability Theory : Notion of probability: Random experiment, sample space, axiom of probability, elementary properties of probability, equally likely outcome problems, Conditional Probability, Bayes’ Theorem. Random Variables: Concept, cumulative distribution function, discrete and continuous random variables, expectations, mean, variance, moment generating function. Discrete random variable: Bernoulli random variable, binomial random variable, geometric random variable, Poisson random variable. Continuous random variables: Uniform random variable, exponential random variable, Gamma random variable, normal random variable. Bivariate random variables: Joint distribution, joint and conditional distributions, Conditional Expectations, Independence, the correlation coefficient.

SECTION-B Numerical Analysis : Solution of Equation: Bisection, Secant, Regula Falsi, Newton’s Method, Roots of Polynomials. Interpolation: Lagrange and Hermite Interpolation, Divided Differences, Difference Schemes, Interpolation Formulas using Difference. Numerical Differentiation. Numerical Quadrature: Newton-Cote’s Formulas, Gauss Quadrature Formulas, Chebychev’s Formulas. Linear Equations: Direct Methods for Solving Systems of Linear Equations (Gauss Elimination, LU Decomposition, Cholesky Decomposition), Iterative Methods (Jacobi, Gauss-Seidel, Relaxation Methods). The Algebraic Eigenvalue problem: Jacobi’s Method, Givens’ Method, Householder’s Method, Power Method, QR Method, Lanczos’ Method. Ordinary Differential Equations: Euler Method, Single step Methods, Runge-Kutta’s Method, Multi-step Methods.