**DU MA/MSc Mathematics Entrance Exam Syllabus**

DU (MA/M.Sc) SYLLABUS FOR ENTRANCE EXAMINATION FOR 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 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 basis.

**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.

Joint Admission Test

**Msc (OR) Syllabus for the Common Entrance Test**

**PART-1**

This part is intended to test the candidate’s vocabulary and analytical skills at a level essential for accurate comprehension and presentation of material appropriate for the degree. The language background expected will be of the level of English at Senior Secondary Examination. The paper will include passages for comprehension, test of vocabulary (synonyms and antonyms).

**PART-II**

Mathematics : Vector Space, subspace and its properties, linear independence and dependence of vectors, matrices, rank of matrix, reduction to normal forms, linear homogenous and non-homogenous equations, Cayley-Hamilton theorem, characteristic roots and vectors. De Moivre’s theorem, relation between roots and coefficient of nth degree equation, solution to cubic and biquadratic equation, transformation of equations.

**Calculus :** Limits and continuity, differentiability of functions, successive differentiation, Leibnitz’s theorem, partial differentiation, Euler’s theorem on homogenous functions, tangents and normals, asymptotes, singular points, curve tracing, reduction formulae, integration and properties of definite integrals, quadrature, rectification of curves, volumes and surfaces of solids of revolution.

**Differential Equations :** Linear, homogenous, separable equations, first order higher degree equations, algebraic properties of solutions, linear homogenous equation with constant coefficients, solution of second order differential equations. Linear nonhomogenous differential equations.

**Real Analysis : Neighbourhoods, open and closed sets, limit points and Bolzano Weiestrass theorem, continuous functions, **sequences and their properties, limit superior and limit interior of a sequence, infinite series and their convergence, Rolle’s theorem, mean value theorem, Taylor’s theorem, Taylor’s theorem, Taylor’s series, Maclaurin’s series, maximum and minima, indeterminate forms.

**Statistics :** Measures of central tendency and dispersion and their properties, skewness and kurtosis, introduction to probability, theorems of total and compound probability, Bayes theorem, random variables, probability mass and density functions, mathematical expectation, moment generating functions, Binomial, Possion, Geometric, Exponential and Normal distribution and their properties, method of least squares, correlation and regression, introduction to sampling, sampling distributions and tests of significance based on t, Chisquare and F-distributions.

**Operation Research :**

Definition & scope of Operation Research, Formulation of simple Linear Programming Problems, Simplex method and basics of Duality.

Characteristics of Inventory System, Simple Economic Lot Size Inventory models, Recorder Level, Simple single period Stochastic Inventory Model.

Definition of Queues and their characteristics, Queueing Models with Markovian Input and Markovian Service, M/M/1 & M/M/C Queueing Models.

Definitions of Reliability, Avalability, Reliability of multicomponents systems, failure time distributions : exponential and Weibull.

**Computer Science :**

Flowcharts and algorithms, Number system : binary, octal, hexadecimal; Truth values, Logical operations, Logic functions and their evaluation.

Computer basics, Computer generation and classific

Mathematical Statistics (MS) Certificate Courses

The Mathematical Statistics (MS) test paper comprisesof Mathematics (40% weightage) and Statistics (60%weightage).

Sequences and Series: Convergence of sequences of real numbers, Comparison, root and ratio tests for convergence of series of real numbers.

**Differential Calculus:** Limits, continuity and differentiability of functions of one and two variables. Rolle’s theorem, mean value theorems, Taylor’s theorem, indeterminate forms, maxima and minima of functions of one and two variables.

**Integral Calculus:** Fundamental theorems of integral calculus. Double and triple integrals, applications of definite integrals, arc lengths, areas and volumes. Matrices: Rank, inverse of a matrix. systems of linear equations. Linear transformations, eigenvalues and eigenvectors. Cayley-Hamilton theorem, symmetric, skewsymmetric and orthogonal matrices.

**Differential Equations**: Ordinary differential equations of the first order of the form y’ = f(x,y). Linear differential equations of the second order with constant coefficients.

**Statistics Probability:** Axiomatic definition of probability and properties, conditional probability, multiplication rule. Theorem of total probability. Bayes’ theorem and independence of events.

**Random Variables:** Probability mass function, probability density function and cumulative distribution functions, distribution of a function of a random variable. Mathematical expectation, moments and moment generating function. Chebyshev’s inequality.

**Standard Distributions: **Binomial, negative binomial, geometric, Poisson, hypergeometric, uniform, exponential, gamma, beta and normal distributions. Poisson and normal approximations of a binomial distribution.

**Joint Distributions:** Joint, marginal and conditional distributions. Distribution of functions of random variables. Product moments, correlation, simple linear regression. Independence of random variables.

**Sampling distributions:** Chi-square, t and F distributions, and their properties.

Limit Theorems: Weak law of large numbers. Central limit theorem (i.i.d.with finite variance case only).

**Estimation:** Un-biasedness, consistency and efficiency of estimators, method of moments and method of maximum likelihood. Sufficiency, factorization theorem. Completeness, Rao-Blackwell and Lehmann-Scheffe theorems, uniformly minimum variance unbiased estimators. Rao-Cramer inequality. Confidence intervals for the parameters of univariate normal, two independent normal, and one parameter exponential distributions.

Testing of Hypotheses: Basic concepts, applications of Neyman-Pearson Lemma for testing simple and composite hypotheses. Likelihood ratio tests for parameters of univariate normal distribution

Mathematics (MA) Certificate Courses

**SEQUENCES, SERIES AND DIFFERENTIAL CALCULUS:**

Sequences and Series of real numbers: Sequences and series of real numbers. Convergent and divergent sequences, bounded and monotone sequences, Convergence criteria for sequences of real numbers, Cauchy sequences, absolute and conditional convergence; Tests of convergence for series of positive terms – comparison test, ratio test, root test, Leibnitz test for convergence of alternating series.

Functions of one variable: limit, continuity, differentiation, Rolle’s Theorem, Mean value theorem. Taylor’s theorem. Maxima and minima.

Functions of two real variable: limit, continuity, partial derivatives, differentiability, maxima and minima. Method of Lagrange multipliers, Homogeneous functions including Euler’s theorem.

egral Calculus: Integration as the inverse process of differentiation, definite integrals and their properties, Fundamental theorem of integral calculus. Double and triple integrals, change of order of integration. Calculating surface areas and volumes using double integrals and applications. Calculating volumes using triple integrals and applications.

Differential Equations: Ordinary differential equations of the first order of the form y’=f(x,y). Bernoulli’s equation, exact differential equations, integrating factor, Orthogonal trajectories, Homogeneous differential equations-separable solutions, Linear differential equations of second and higher order with constant coefficients, method of variation of parameters. Cauchy- Euler equation.

Vector Calculus: Scalar and vector fields, gradient, divergence, curl and Laplacian. Scalar line integrals and vector line integrals, scalar surface integrals and vector surface integrals, Green’s, Stokes and Gauss theorems and their applications.

Group Theory: Groups, subgroups, Abelian groups, non-abelian groups, cyclic groups, permutation groups; Normal subgroups, Lagrange’s Theorem for finite groups, group homomorphisms and basic concepts of quotient groups (only group theory).

Linear Algebra: Vector spaces, Linear dependence of vectors, basis, dimension, linear transformations, matrix representation with respect to an ordered basis, Range space and null space, rank-nullity theorem; Rank and inverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions. Eigenvalues and eigenvectors. Cayley-Hamilton theorem. Symmetric, skewsymmetric, hermitian, skew-hermitian, orthogonal and unitary atrices.

Real Analysis: Interior points, limit points, open sets, closed sets, bounded sets, connected sets, compact sets; completeness of R, Power series (of real variable) including Taylor’s and Maclaurin’s, domain of convergence, term-wise differentiation and integration of power series.

**DU Exam Test Pattern**

It includes 3 sections A,B,C having 7 questions in each section.

There are total 21 questions and out of these questions you have to attempt only 15questions in every sections(A, B C).

It is compulsory to attempt any 4 questions and rest 3 questions you can attempt from any section or any question

**IIT JAM Test Pattern**

**IIT JAM TEST PATTERN 2015 **

The Jam 2015 examination for all test papers will be carried out as ONLINE COMPUTER BASED TEST (CBT) where the candidates will be shown the questions in a random sequence on a computer screen. The duration of the examination will be 3 hours. The medium of the test papers will be English only. There will be total 60 question carrying 100 marks. The entire paper will be divided into 3 sections A, B and C. All sections are compulsory.

**Question in each section are of different types as given below :**

**Section –** A contains a total of 30 Multiple Choice Questions (MCQ) carrying one or two marks each. Each MCQ type question has four choices out of which only one choice is the correct answer. Candidate can mark the answer by clicking the choice.

**Section –** B contains a total 10 Multiple Select Questions (MSQ) carrying two marks each. Each MSQ type question is similar to MCQ but with a difference that there may be one or than one choice(s) that are correct out of the four given choices. The candidate get full credit if he/she selects all the correct answers only and no wrong answers. Candidate can mark the answers by clicking the choices.

**Section –** C cantains a total of 20 Numerical Answer Type (NAT) questions carrying one or two marks each. For these NAT type questions, the answer is a signed real number which needs to be entered using the virtual keyboard on the monitor. No choices will be shown for these types of questions. Candidates have to enter the answer by using a virtual numeric keypad.

In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will result in negative marks. For all mark questions, 1/3 marks will be deducted for each wrong answer. For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ) there is no negative and no partial marking provisions. There is no negative marking in Section – C (NAT) as well.

An-on – screen virtual scientific calculator will be available for the candidates to do the calculations. Physical calculators, charts, graph sheets, tables, cellular phones or any other electronic gadgets are NOT allowed in the examination hall.

A scribble pad will be provided for rough work and this has to be returned back at the end of examination.

The candidates are required to select the answer for MCQ & MSQ type questions, and to enter the answer for NAT questions using only a mouse on a virtual keypad (The keyboard of the computer will be disabled). At the end of the 3 hours window, the computer will automatically will close the screen from further actions.

Use of unfair means by a candidate in Jam 2015, whether detected at the time of examination, or at any stage, will lead to cancellation of his/her candidature as well as disqualification of the candidate from appearing in Jam in future.

The candidates are advised to visit the Jam 2015 website for more details on the patterns of questions for Jam 2015, including examples of the questions. Candidate will also be able to take a mock examination through a ‘MOCK TEST’ link that will be made available on the website closer to the examination dates.

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