## M.Sc. Operational Research Entrance Delhi University (DU) Syllabus

#### Mathematics:

**Algebra:**

Linear Algebra. Vector space, subspace and its properties, linear independence and dependence of vector, matrices, rank of matrix, reduction to normal forms, linear homogeneous and non-homogeneous equations, Cayley-Hamilton theorem, characteristic roots and vectors.

Theory of equation. De Moivere’s theorem, relation between roots and coefficient of nth degree equation, solution to cubic and biquadratic equation, transformation of equation.

**Calculus:**

Diff. calculus. Limit and continuity, differentiability of functions, successive differentiation, leibnitz’s theorem, partial differentiation, Euler’s theorem on homogeneous functions, tangents and normals, asymoptotes, singular points, curve tracing.

Integral calculus. Reduction formulae, integration and properties of definite integrals, quadrature, rectification of curves, volumes and surfaces of solids of revolution.

**Differential Equation:**

Linear, homogeneous equation, first order higher degree equations, algebraic properties of solutions, linear homogeneous equation with constant coefficients, solution of second order differential equation, linear non-homogeneous differential equations.

**Real Analysis:**

Neighbourhood, open and sets, limit point and Bolzano weirstrass theorem, continuous functions, sequences and their properties, limit superior and limit inferior of a sequence, infinite series and their convergence, Rolle’s theorem, mean value theorem, Taylor’s theorem, Taylor’s series, Maclaurin’s series, maxima and minima, indeterminate forms.

**Statistics:**

Measures of central tendency and dispersion and their properties, skewness and kurtosis, introduction to probability, theorem of total and compound probability, Bayes theorem, random variables, probability

## DU M.A./M.Sc Mathematics Entrance Syllabus

- Elementary set theory, Finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum.

- Sequence and series, Covergencelimsup, liminf.

- Bolzano Weierstrass theorem, Heine Borel theorem.

- Continuity, Uniform continuity, Intermediate value theorem, Differentiability, Mean value theorem, Maclaurin’s theorem and series, Taylor’s series.

- Sequences and series of functions, Uniform convergence.

- Riemann 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.

- Eigenvalues and eigenvectors of matrices, Cayley-Hamilton theorem.

- Divisibility in Z, congruences, Chinese remainder theorem, Euler’s φ- 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, Subspaces, Linear dependence, Basis, Dimension, Algebra of linear transformations, Matrix representation of linear transformations, Change of basis, Inner product spaces, Orthonormal basis.