### SYLLABUS FOR ENTRANCE EXAMINATION FOR DU(M.A./M.Sc.) MATHEMATICS

### SECTION 1

- Elementary set theory, Finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum.
- Sequence and series, Covergence limsup, liminf
- Bolzano Weierstrass theorem, Heine Borel theorem
- Continuity, Uniform continuity, Intermediate value theorem, Differentiability, Mean value theorem, Maclaurin’s,br> theorem and series, Taylor’s series
- Sequences and series of functions, Uniform convergence
- Reimann sums and Riemann integral, Improper integrals.
- Monotonic functions, Types of discontinuity
- Functions of several variables,Directional derivative, Partial derivative
- Metric spaces, Completeness, Total boundedness, Separability, Compactness, Connectedness

### SECTION 2

- Eigenvalues and eigenvectors of matrices, Cayley-Hamilton theorem.
- Divisibility in Z, congruences, Chinese remainder theorem, Euler’s f- function.
- Groups, Subgroups, Normal subgroups, Quotient groups, Homomorphisms, Cyclic groups, Cayley’s theorem,Class equations, Sylow theorems.
- Rings,fields, Ideals, Prime and Maximal ideals, Quotient rings, Unique factorization domain, Principal ideal domain, Euclidean domain, Polynomial rings and irreducibility criteria.
- Vector spaces, Subsaces, Linear dependence, Basis, Dimension, Algebra of linear transformations, Matrix representation of linear transformations, Change of basis, Inner product spaces, Orthonormal

### SECTION 3

- Existence and Uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ordinary differential equations
- System of first order ordinary differential equations, General theory of homogeneous and non- homogeneous linear ordinary differential equations, Variation of parameters, Sturm Liouville boundary value problem, Green’s function
- Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for firstorder PDEs, Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for laplace. Heat and Wave equation
- Numerical solutions of algebraic equation, Method of iteration and Newton-Raphson method, Rate of convergence, Solution of systems of linear algebraic equations using Guass elimination and Guass-Seidel method, Finite differences, Lagrange, Hermite and Spline interpolation, Numerical integration, Numerical solutions of ODEs using Picard,
- Euler, modified Euler and second order Runge- Kutta methods.