Books Recommended



DEENBANDHU CHHOTU RAM UNIVERSITY OF SCIENCE & TECHNOLOGY, MURTHALDEPARTMENT OF MATHEMATICS(Accredited ‘A’ Grade by NAAC)Scheme of Study and Examination ofDual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics (w.e.f. Session July 2018-2019)Semester IPaper CodeNaturePaper titleTeachingSchemeExamination SchemeDuration of Exam.CreditLTPInternal MarksExternalMarksTotalDMT 211BCAlgebra 41025751003 Hours5DMT 213BCCalculus41025751003 Hours5DMT 215BC Ordinary Differential Equations 41025751003 Hours5EElective -I41025751003 Hours5DMT 219BGComputer Fundamentals and MS- OFFICE41025751003 Hours5DMT 221BGPractical based on Paper DMT 219B--42525503 Hours2ENG 225 BAECEnglish-I30075753 HoursQualifyingTotal235440055027Aberrations used in Nature C –Core Paper, E – Elective Paper, OE – Open Elective Paper G- Generic , AEC – Ability Enhancement Course.Elective Papers:Students are required to take Elective-I from anyone of the following: DMT 217B Discrete Mathematics-IDMT223B Operations Research-IDMT225B Descriptive StatisticsNote: Electives can be offered subject to availability of requisite resources/ faculty in the departmentDEENBANDHU CHHOTU RAM UNIVERSITY OF SCIENCE & TECHNOLOGY, MURTHALDEPARTMENT OF MATHEMATICS(Accredited ‘A’ Grade by NAAC)Scheme of Study and Examination ofDual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics Under Choice Based Credit System (w.e.f. Session July 2018-2019)Semester IIPaper CodeNaturePaper titleTeachingSchemeExamination SchemeDuration of Exam.CreditLTPInternal MarksExternalMarksTotalDMT 212BCNumber Theory and Trigonometry 41025751003 Hours5DMT 214BCVector Calculus 41025751003 Hours5DMT 216BCSolid Geometry41025751003 Hours5EElective -II41025751003 Hours5DMT 220BG Programming in Visual Basic41025751003 Hours5DMT 222BGPractical based on Paper DMT 220B--42525503 Hours2ENG 226BAECEnglish-II30075753 HoursQualifying Total235415040055027Aberrations used in Nature C –Core Paper, E – Elective Paper, G- Generic, and AEC – Ability Enhancement course.Elective Papers:Students are required to take Elective-II from anyone of the following: DMT 218B Discrete Mathematics-IIDMT224B Operations Research-IIDMT226B Numerical MethodsNote: Electives can be offered subject to availability of requisite resources/ faculty in the departmentDual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics (Semester-I)DMT 211B: ALGEBRA(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives: To familiarize the students with the concept of rank of a matrix, eigen values and eigen vectors, applications of matrices to a system of linear equations, relations between the roots and coefficients of general equation in one variable, nature of the roots of equation, solution of cubic and biquadratic equations.Course outcomes: After the successful completion of the course the student would be able to find rank, eigen values and eigen vectors, understand applications of matrix to a system of linear equations, solve equations using relation between roots and coefficients of the equations, describe the nature of the roots of an equations, solve cubic and biquadratic equations.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit– ISymmetric, skew-symmetric, Hermitian and skew-Hermitian matrices. Elementary operations on matrices. Rank of a matrix. Inverse of a matrix. Linear dependence and independence of rows and columns of matrices. Eigenvalues, eigenvectors and the characteristic equation of a matrix. Minimal polynomial of a matrix. Cayley Hamilton theorem and its use in finding the inverse of a matrix. Unit – IIApplications of matrices to a system of linear (both homogeneous and non–homogeneous) equations. Theorems on consistency of a system of linear equations. Unitary and orthogonal matrices. Bilinear and quadratic forms.Unit – III Relations between the roots and coefficients of general polynomial equation in one variable. Solutions of polynomial equations having conditions on roots. Common roots and multiple roots. Transformation of equations.Unit – IVNature of the roots of an equation. Descarte’s rule of signs. Solutions of cubic equations by Cardan’s method. Biquadratic equations and their solutions.Books Recommended: H.S. Hall and S.R. Knight, Higher Algebra, H.M. Publications, 1994.Shanti Narayan and P.K.Mittal, A Text Book of Matrices, S. Chand & Co., New Delhi.Erwin Kreyszig, Advanced Engineering Mathematics, Wiley India.Chandrika Prasad, Text Book of Algebra & Theory of Equations, Pothishala Pvt. Ltd., AllahabadDual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-I)DMT 213B: CALCULUS(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives : Calculus is primarily concerned with developing the students’ understanding of the concepts of calculus and providing experience with its methods and applications. The course emphasizes a multi-representational approach to calculus, with concepts, results, and problems being expressed geometrically, numerically, analytically, and verbally.Course outcomes: Students will be able to know the basic rules of differentiation and use them to find derivatives of products and quotients and they will be able to find tangents and normals to graphs of functions given in explicit, implicit and parametric forms.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit – IDefinition of the limit of a function. Basic properties of limits, Indeterminate forms, Continuous functions and various types of discontinuities. Differentiability. Successive differentiation. Leibnitz theorem. Maclaurin and Taylor series expansions. Unit – IITangents and normals, sub-tangents and sub-normals, Asymptotes in Cartesian coordinates and polar coordinates. Curvature, radius of curvature for Cartesian, parametric, pedal and polar forms. Newton’s method, Centre of curvature. Circle of curvature. Chord of curvature, Evolutes. Unit – IIITests for concavity and convexity. Points of inflexion. Multiple points. Cusps, nodes & conjugate points. Type of cusps, Tracing of curves in Cartesian, parametric and polar co-ordinates. Reduction formulae. Unit – IVRectification.,Quadrature, Area bounded by closed curves. Volumes and surfaces of solids of revolution. Theorems of Pappu’s and Guilden. Books Recommended : Shanti Narayan: Differential and Integral Calculus.Murray R. Spiegel: Theory and Problems of Advanced Calculus. Schaun’s Outline series. Schaum Publishing Co., New York.N. Piskunov: Differential and integral Calculus. Peace Publishers, Moscow.M.J.Strauss, G.L. Bradley, K.J.Smith: Calculus (3rd edition), Darling Kindasley (INDIA) Pvt. Limited ( Pearson Education) Delhi, 2007.Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-I)DMT 215B: ORDINARY DIFFERENTIAL EQUATIONS(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Objectives and outcomes: This course has been devised to make the students learn, the theory of ordinary differential Equations. Exact differential equations and their integrating Factors along with equations of first order but of higher degree are solved. To taught the students Orthogonal Trajectories and Linear Differential of Various orders, transformation of equations to normal form, change of dependent and independent variable, solutions of simultaneous and total differential Equations.Course outcomes: By the end of this course the students would be well versed with various kinds of differential Equations and their solutions. They would be able to translate a Physical or Engineering phenomenon into a differential Equation. It would also help in Mathematical modeling.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit – IGeometrical meaning of a differential equation. Exact differential equations, integrating factors. First order higher degree equations solvable for x,y,p Lagrange’s equations, Clairaut’s equations. Equation reducible to Clairaut’s form. Singular solutions. Unit – IIOrthogonal trajectories: in Cartesian coordinates and polar coordinates. Self orthogonal family of curves.. Linear differential equations with constant coefficients. Homogeneous linear ordinary differential equations. Equations reducible to homogeneous form.Unit – IIILinear differential equations of second order: Reduction to normal form. Transformation of the equation by changing the dependent variable/ the independent variable. Solution by operators of non-homogeneous linear differential equations. Reduction of order of a differential equation. Method of variations of parameters. Method of undetermined coefficients.Unit – IVOrdinary simultaneous differential equations. Solution of simultaneous differential equations. Simultaneous equation of the form dx/P = dy/Q = dz/R. Total differential equations. Condition for Pdx + Qdy +Rdz = 0 to be exact. General method of solving Pdx + Qdy + Rdz = 0 by taking one variable constant. Method of auxiliary equations. Books Recommended : E.A. Codington, Introduction to Differential Equations.S.L.Ross, Differential Equations, John Wiley & SonsB.Rai & D.P. Chaudhary, Ordinary Differential Equations, Narosa Publishing House Pvt. Ltd.Spiegal, Differential Equations, Schaum Outline Series.Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-I)DMT217B: DISCRETE MATHEMATICS-I(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives : This course has been devised to learn the students to solve counting problems by applying elementary counting techniques using the product and sum rules, permutations ,combinations the pigeon-hole principle, the concepts of countability and equivalent sets, equivalence and partial order relations, simplify and evaluate basic logic statements, implications ,inverse , converse and contrapositive using truth tables and properties of logic , solve problems using recurrence relations such as finding Fibonacci numbers. Course outcomes: Students after completing this course would be able to express a logic sentence in terms of predicates, quantifiers and logical connectives, apply rule of inference and methods of proof including direct and indirect proofs forms.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks. Unit – ISets, principle of inclusion and exclusion, relations, equivalence relations and partition, denumerable sets, Cardinality of sets, Posets and their properties, Pigeon Hole Principle and its applications. Unit – II Propositions, logical operations, logical equivalence, conditional propositions, Tautologies and contradictions. Quantifier, Predicates and Validity. Types of proof: Direct proof , indirect proof, proof by mathematical induction. Unit – III Permutations and combinations, Functions: injective, surjective, bijective, many to one, Range, domain, Invertible and composite. Examples of functions including hash function, floor function, ceiling function, characteristic function.Unit -IV Discrete numeric functions, Generating functions, recurrence relations with constant coefficients. Complementary function, particular solution, total solution, Solution of recurrence relation by the method of generating functions. Books Recommended : Johnsonbaugh: Discrete mathematics, Pearson EducationKolman, Busby and Rose: Discrete mathematics, Pearson Education.C.L. Liu: Elements of Discrete Mathematics, McGraw-Hilll Book CoBabu Ram: Discrete Mathematics, Pearson’s Publishers, 2009. Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-I)DMT 219B: Computer Fundamentals and MS-OFFICE(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives : To help students know the basic components of computer and working of each device and to understand the basic concept of software and its types. Student will understand the functions of Operating System in general and Windows operating system in particular. Provide hands-on use of Microsoft Office applications Word, Excel and PowerPoint. Course outcomes: After the completion of the course the students would be able in MS Office applications, knowledge and skills.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit -IFundamentals of Computer: Model of a digital computer, Functioning of a digital computer, Historical evolution of computers, classification of computers, Human being vs computer, Input / Output devices, Storage devices, Memory and mass storage devices, characteristics of memory systems, types of memory, RAM, ROM, concepts of Virtual and Cache memory, Types of software, Application and system software and its functions, time sharing, multiprocessing, Applications of Computer. Unit -IIIntroduction to Windows: Types of windows, windows as an operating system, windows explorer, using clipboard, using paintbrush, control panel, installing a printer.MS Power Point: Introduction, Power point slide creation, Slide-show, Adding graphics, Formatting Customizing and Printing. Unit -IIIMS-Word: Introduction to MS-Word, Standard Toolbar, Word Wrap, Text formatting, Indents, Tabs, Formatting paragraphs, Applying Effects to text, Applying animation to text. Unit -IVMS Excel: Introduction to MS Excel, Working with Toolbars, Formatting, Formulas, Data management, Graphs and Charts, Macros and other additional functions.Books Recommended:Donald Sanders, Computers Today, McGraw-Hill Publishers.Davis, Introduction to Computers, McGraw-Hill Publishers.V. Rajaraman, Fundamental of Computers, Prentice-Hall India Ltd., New Delhi.Balagurusamy, Computing Fundamentals & C Programming, McGraw-Hill Publishers.Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-I)DMT 221B: PRACTICAL(Based on paper DMT 219B)(w.e.f. Session July 2018- 19) LTPMarks for External Exam : 25--4 (2 Credits)Marks for Internal Exam : 25 Total : 50 Duration of Exam : 3 Hours Course objectives : The objective of this lab students are come to know why computers are essential components in business, education and society particularly with respect to personal use of computer hardware and software, the Internet, networking and mobile computing. It will provide hands-on use of Microsoft Office like applications Word, Excel, Access and PowerPoint. Course outcomes: The student would be able to apply technical knowledge and perform specific technical skills, including usage of computers and why computers are essential components in business and society. This Utilize the Internet Web resources and evaluate on-line e-business system. [ List of ExperimentIntroduction of Ms-Word.Prepare Time-table in Ms-Word.Prepare Document using by applying Formatting attribute. Introduction of MS-ExcelPrepare Mark sheet in MS-Excel. Prepare Bill in MS-Excel. Introduction of MS-Power Point. Prepare Presentation by applying Formatting Tools.Prepare Presentation using by applying Formatting Tools.Introduction to HTML & formatting with HTMLDual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-I)DMT 223B: OPERATIONS RESEARCH-I(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3Hours Course objectives : This course introduces the students to the fundamentals of operations research models including linear programming and applications. OR is an interdisciplinary subject drawing from game theory, statistics, and computer science as well as applied mathematics, and we will show some of these connections. the course focusses on linear optimisation problems involving both continuous and integer variable, because these are used in a vast range of real situations.Course outcomes: On successful completion of this course the student would be able to define and formulate linear programming problems and appreciate their limitations; conduct and interpret post-optimal and sensitivity analysis; and?explain the primal-dual relationship.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit - IDefinition, scope, methodology and applications of OR. Types of OR models, Concept of optimization, Linear Programming: Introduction, Formulation of a Linear Programming Problem (LPP), Requirements for an LPP, Advantages and limitations of LP. Graphical solution: Multiple, unbounded and infeasible solutions.Unit -IIPrinciple of simplex method: standard form, basic solution, basic feasible putational Aspect of Simplex Method: Cases of unique feasible solution, no feasible solution, multiple solution and unbounded solution and degeneracy. Two Phase and Big- M methods. Unit -IIIDuality in LPP, primal-dual relationship. Dual Simplex Method, Transportation Problem: Methods for finding basic feasible solution of a transportation problem, Modified distribution method for finding the optimum solution, Unbalanced and degenerate transportation problems, transshipment problem, maximization in a transportation problem. Unit -IVAssignment Problem:Solution by Hungarian method, Unbalanced assignment problem, maximization in an assignment problem, Crew assignment and Travelling salesman problem, Game Theory: Two person zero sum game, Game with saddle points, the rule of dominance; Algebraic, graphical and linear programming methods for solving mixed strategy games.Books Recommended J.K. Sharma, Mathematical Model in Operations Research, Tata McGraw Hill. H.A. Taha, Operations Research – An Introduction.Kanti Swarup, P.K. Gupta, and Manmohan, Operations Research, S.Chand Publishers.P.K. Gupta and D.S. Hira, Operations Research, S. Chand & Co.Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-I)DMT 225B: DESCRIPTIVE STATISTICS(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objective: This course is designed to aware the students about some basics of statistics, measures of central tendency, moments, skewness and kurtosis and theory of attributes.Course outcomes: After the completion of course, the students can present the data in different forms, they can find Mean, median, mode, geometric mean, harmonic mean and partition. Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit – IIntroduction of Statistics, Basic knowledge of various types of data, Collection, classification and tabulation of data. Presentation of data: histograms, frequency polygon, frequency curve and ogives. Stem- and- Leaf and Box plots.Unit – II Measures of Central Tendency and Location: Mean, median, mode, geometric mean, harmonic mean, partition values. Types of errors and propagation in statistical data.Measures of Dispersion: Absolute and relative measures of range, quartile deviation, mean deviation, standard deviation (σ), coefficient of variation.Unit – III Moments, Skewness and Kurtosis: Moments about mean and about any point and derivation of their relationships, effect of change of origin and scale on moments, Sheppard’s correction for moments (without derivation), Charlier’s checks, Concepts of Skewness and Kurtosis.Unit – IV Theory of Attributes: Symbolic notation, dichotomy of data, class frequencies, order of class frequencies, consistency of data, independence and association of attributes, Yule’s coefficient of association and coefficient of colligation.Correlation for Bivariate Data: Concept and types of correlation, Scatter diagram, Karl Pearson Coefficient (r) of correlation and rank correlation coefficient.Books Recommended: A.M. Goon, M.K. Gupta, and B. Das Gupta: Fundamentals of Statistics, Vol-I.S. Bernstein and R. Bernstein, Elements of Statistics, Schaum’s outline series, McGraw-Hill.S.C. Gupta and V.K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, 2002.E.Tanis: Statistics1: Descriptive statistics and Probability, Harcourt Brace, College Outline Series.Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-I)ENG 225B: English - I(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 753 - -Total : 75 Duration of Exam : 3 HoursCourse Objective: To equip the students with language skills in English needed in academic and professional world and to inculcate human values in them.Course Outcome: The students will be able to enhance their writing and speaking skills and develop an ability to critically read the literary texts so as to develop proficiency in reading along with sensitivity to the impact literary texts can have on their minds/ lives.Course Contents:Unit IReading Skills: Language through Literature (15 Marks)“The Responsibility of Young Men” by Lal Bahadur Shastri#“The Mark of Vishnu” by Khushwant Singh# # iii) “What is Duty?” by Swami Vivekananda* “The Cherry Tree” by Ruskin Bond**Unit IIVocabulary Building (15 Marks) i) One word substitution (List attached)*** ii) Commonly used Idioms (List attached)*** iii) Common Phrasal verbs (List attached)*** iv) Common Foreign Expressions (List attached)***Unit IIIWriting Skills: Basics of Grammar (15 Marks)Correct usage of nouns, pronouns Verb-subject agreement Use of Conjunctions Tag Questions Unit IVProfessional Communication (15 Marks)Letters (Social and Official)NoticesCircularsWriting emailsScheme of End Semester Examinations (Major Test):The duration of examinations will be 3 hours.Nine questions of 15 marks each will be set out of which the students will have to attempt five questions.First question of 15 marks will be compulsory. It will cover all the four units of the syllabus. The nature of the questions in each unit will depend upon the nature of content therein. The questions may have sub-parts with marks assigned against each. Question No 02 to 09 will be set from the four units of the syllabus --- two from each unit of 15 marks each. Students will have to attempt four more questions, selecting one from each unit. Recommended pattern of questions in each unit will be as follows:Unit IOne question having sub-parts from the literary texts has to be answered in about 200 words each or may be set on vocabulary items from the texts themselves. The second question will either be in the form of a comprehension passage from the texts or explanation with reference to context of the lines / passage from the prescribed texts.Unit IITwo questions with or without parts will be set from this unit. Questions may be in the form of supplying one word substitution for the phrases/ sentences, providing meanings of phrasal verbs/idioms by using them in sentences, choosing the correct alternative or in the form of matching exercises or on writing the meaning of commonly used foreign expressions.Unit IIITwo questions will be set from this unit also out of which one is to be attempted. Questions will be in the form of correcting the errors in the sentences, picking up the right alternative, filling in the blanks or completing the sentences.Unit IVTwo questions from this unit will be in the form of writing letter / notice/ circular/ email related to day to day social or professional life. Approved by Board of UG Studies, Department of Humanities on 19 March 2018 Recommended Readings (Online sources have been underlined):# Mohan, Loveleen, Randeep Rana and Jaibir S. Hooda. Eds. Literature and Language-I. New Delhi: Orient Blackswan Pvt. Ltd.,2015.(Chapter 9)# # # #Murthy, M.G. Narasimha. Famous Indian Stories. Mumbai: Orient BlackSwan Pvt. Ltd.,2009.(Chapter 5)**Vivekananda, Swami. Karma Yoga. New Delhi: Sahityashila Prakashan, 2015*****Bhatnagar, Nitin and Mamta Bhatnagar. Communicative English for Engineers and Professionals. New Delhi: Pearson Education, 2016.Konar, Nira. Communication Skills for Professionals. New Delhi: PHI Learning Pvt. Ltd., 2009 Sinha, R.P. Current English Grammar and Usage with Composition. New Delhi: OUP, 2016.Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)DMT 212B: NUMBER THEORY AND TRIGONOMETRY(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives: The objective of this course is to familiarize the students with basic concept of elementary number theory such as results on divisibility, congruence, solution of linear congruence equations. Further some basic results on trigonometric functions are also taught.Course outcomes: After successful completion of the syllabus, the students will be able to find the solutions of simple number theoretic problems related to the applications of Fermat Theorem, Wilson Theorem, Chinese Remainder Theorem and Mobius Inversion formula etc. Students will also be able to solve problems related to trigonometric functions, series, De Moivre’s Theorem etc.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks. Unit – IDivisibility. Division (Euclidean) algorithm. Greatest common divisors. Least common multiple. Prime numbers. Well ordering property. Division algorithm. Fundamental Theorem of Arithemetic. Linear Congruences. Fermat’s theorem. Wilson’s theorem and its converse. Linear Diophanatine equations in two variables.Unit – II Complete residue system and reduced residue system modulo m. Euler’s ? function and Euler’s generalization of Fermat’s theorem. Chinese Remainder Theorem. Quadratic residues. Legendre symbols. Lemma of Gauss; Gauss reciprocity law. Greatest integer function. The number of divisors and the sum of divisors of a natural number n (The functions d(n) and (n)). Mobius function and Mobius inversion formula. Unit - III De Moivre’s Theorem and its Applications. Determination of roots of some polynomials. Expansion of trigonometrical functions. Direct circular and hyperbolic functions and their properties.Unit – IV Inverse circular and hyperbolic functions and their properties. Logarithm of a complex quantity. Gregory’s series. Summation of Trigonometry series. Books Recommended : S.L. Loney, Plane Trigonometry Part – II, Macmillan and Company, London.Ivan Ninen and H.S. Zuckerman, An Introduction to the Theory of Numbers.D.M. Burton, Elementary Number Theory, McGraw Hill Education, 2006.C. V. Durell and A. Robson, Advanced Trigonometry, Dover Publications, Inc, Mineola, New York, 2003.Thomas Koshy, Elementary Number Theory with Applications, Student Solutions Manual, 2nd edition, Academic press, USA 2007Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)DMT 214B: VECTOR CALCULUS(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives: To familiarize the students with the concept of scalar and vector product of three and four vectors, vector differentiation, gradient, divergence and curl, orthogonal curvilinear coordinates, vector integration, line integral, surface integral and volume integral.Course outcomes: After the successful completion of the course the student would be able to solve problems on scalar and vector product of three and four vectors, understand the concept of vector differentiation, gradient, divergence and curl, understand various coordinate systems, solve vector integration problem.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit – IScalar and vector product of three vectors, product of four vectors. Reciprocal vectors. Vector differentiation. Scalar Valued point functions, vector valued point functions, derivative along a curve, directional derivatives.Unit – IIGradient of a scalar point function, geometrical interpretation of grad , characteristics of gradient as a point function. Divergence and curl of vector point function, characteristics of Div and Curl as point function, geometrical interpretation of Div and Curl. Gradient, divergence and curl of sums and product and their related vector identities. Laplacian operator.Unit – III Orthogonal curvilinear coordinates Conditions for orthogonality fundamental triad of mutually orthogonal unit vectors. Gradient, Divergence, Curl and Laplacian operators in terms of orthogonal curvilinear coordinates, Cylindrical co-ordinates and Spherical co-ordinates.Unit – IVVector integration; Line integral, Surface integral, Volume integral.Theorems of Gauss, Green & Stokes and problems based on these theorms.Books Recommended: Murrary R. Spiegal, Theory and Problems of Advanced Calculus, Schaum Publishing Company, New York.Murrary R. Spiegal, Vector Analysis, Schaum Publisghing Company, New York.Shanti Narayan, A Text Book of Vector Calculus. S. Chand & Co., New Delhi.N.Saran and S.N. Nigam, Introduction to Vector Analysis, Pothishala Pvt. Ltd., Allahabad.Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)DMT 216B: SOLID GEOMETRY(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives : The course is aimed to give information about tracing of conics and also aimed to give information about tangents and normals to the conics. Here we provide the knowledge of central conicoids and their generating lines. Course outcomes: After successful of this course, the students would be able to know the basic concepts of conics i.e. circle , ellipse, and hyperbola. And they would also be able to understand the concepts of 3-dimensions conicoids. Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit – IGeneral equation of second degree. Tracing of conics. Tangent at any point to the conic, chord of contact, pole of line to the conic, director circle of conic. System of conics. Confocal conics. Polar equation of a conic, tangent and normal to the conic.Unit – IISpheres: Plane section of a sphere. Sphere through a given circle. Intersection of two spheres, radical plane of two spheres. Co-axial system of spheres. Cones: Right circular cone, enveloping cone and reciprocal cone. Cylinders: Right circular cylinder and enveloping cylinder.Unit – IIICentral Conicoids: Equation of tangent plane. Director sphere. Normal to the conicoids. Polar plane of a point. Enveloping cone of a coincoid. Enveloping cylinder of a coincoid. Unit – IVParaboloids: Circular section, Plane sections of conicoids. Generating lines. Confocal conicoid. Reduction of second degree equations.Books Recommended: 1.R.J.T. Bill, Elementary Treatise on Coordinary Geometry of Three Dimensions, MacMillan India Ltd. 1994. 2.P.K. Jain and Khalil Ahmad : A Textbook of Analytical Geometry of Three Dimensions, Wiley Eastern Ltd. 1999. 3.Shanti Narayan and P.K. Mittal, analytical Solid Geometry, S.Chand Publishers, 2007.4.H.E. Slaught and N.J. Lennes, Solid Geometry and Applications, Create Space Independent Publishing Plateform, 2014Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)DMT 218B: DISCRETE MATHEMATICS-II(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives: This course has been devised to make the students learn, about Lattices and their properties , types of lattices , evaluate Boolean functions and simplifying expression using the properties of Boolean algebra apply Boolean algebra to circuits and gating networks ,determine if a graph has an Euler or a Hamilton path or circuit perform tree traversal. Course outcomes: Students after completing this course would be able to evaluate Boolean functions and simplify expressions using the properties of Boolean algebra , use tree and graph algorithms to solve problems .Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks. Unit -ILattices and their properties, lattice as algebraic system, Bounded, Complement and distributive lattices. Unit -IIBoolean algebra, definition and examples, properties, duality, distributive and complemented Calculus. Design and implementation of digital networks, switching circuits, simplification of Boolean expression using Algebraic methods and Karnaugh map. Unit -IIIGraphs: Definition, Types of graphs, paths and circuits, Koingsberg seven brige problem, Eulerian and Hamiltonion paths and circuits. Seven bridges problem, shortest path traveling salesman problems. Planar graph and Euler’s formula. Non-planarity of K5 and K3,3. Matrix of a graph. Unit -IVDirected Graphs, Trees, Isomorphism of Trees, Representation of Algebraic Expressions by Binary Trees, Spanning Tree of a Graph, Shortest Path Problem, Minimal spanning Trees: Prim’s and Kruskal’s Algorithms. Shortest Route Problems: Dijkastra’s Algorithm. Cut Sets, Tree Searching. Books Recommended: Johnsonbaugh: Discrete mathematics, Pearson Education.Kolman, Busby and Rose: Discrete mathematics, Pearson Education.C.L. Liu, Elements of Discrete Mathematics, McGraw-Hilll Book Co.Babu Ram, Discrete Mathematics, Pearson Publications, 2009. Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)DMT 220B: PROGRAMMING IN VISUAL BASIC(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3 Hours Course objectives : To help students understand the benefits of using Microsoft Visual Basic 6.0 for Windows as an application tool and make them learn event-driven programming concepts, terminology and available tools. Next is to learn how to use the Visual Basic toolbox, object methods and to modify the object properties. Course outcomes: successfully completing this course the student would learn to use the menu, design window and study various procedures and functions and understand proper debugging and error-handling procedures. Gain a basic understanding of database access and management, ActiveX controls and library functions.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit -IVisual Basic: Introduction, Analyzing, Controls and Properties, Coding, Control structures: Decision & Loops, Control Array, ArraysUnit -IIText Boxes, Command Buttons, Additional Controls – List Box, Option Buttons, Frames, Check Boxes, Scroll Bars, Timer Control, Unit -IIIMenus: Menu Editor, Menu controls, Dialog Boxes, Procedures and Functions, Using Debugging Windows, Database Programming. Unit -IVCrystal Reports, Simple Active X controls. Library Functions: String, Numeric, Time-related & Misc. FunctionsBooks Recommended Reselman & Other, Using Visual Basic 6, Prentice Hall of India. Donald & Oancea, Visual Basic 6 from Scratch, Prentice- Hall of India. Noel Jerke, Visual Basic 6, Tata Mc-Graw HillDays Maver, Teach Yourself More VB in 21 days, Techmedia. Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)DMT 222B: PRACTICAL(Based on paper DMT220B)(w.e.f. Session July 2018- 19) LTPMarks for External Exam : 25--4 (2 Credits)Marks for Internal Exam : 25 Total : 50 Duration of Exam : 3 HoursCourse objectives : The objective of the course is to cover visual programming skills needed for modern software development. This course introduces computer programming using the Visual BASIC programming language with object-oriented programming principles. Emphasis is on event-driven programming methods, including creating and manipulating objects, classes, and using object-oriented tools such as the class debugger. Course outcomes: The students would come to know fundamental skills in utilizing the tools of a visual environment in terms of the set of available command menus and toolbars and explain and use of delegates and events for producing event-driven applications.List of experimentDesign calculator using various controls Design an application to make a bill using Combo Box & List Box Design an application using Image Control & Picture Box Design an application using scrollbar & timer Design an application to draw Line & Shape Design an application using File System Control & Common Dialog box Control Design an application that explore OLE ctl Design an application using Standard & Form Module 9Design an application using Class Module Develop a menu with MDI Form ObjectDual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)DMT224B: OPERATIONS RESEARCH-II(w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7550- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3Hours Course objectives : The objectives of this course are to introduce students to the techniques of operations research in mining operations, provide students with basic skills and knowledge of operations research and its application in mineral industry, able to Model a dynamic system as a queuing model and compute important performance measures.Course outcomes: Upon successful completion of this course, the student will be able to explain the meaning of operations research, know the various techniques of operations research; ,eliminate customers / clients waiting period for service delivery ,determine critical path analysis to solve real life project scheduling time and timely delivery ,use critical path analysis and programming evaluation production and review techniques for timely project scheduling.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.Unit - IInventory Control: introduction of inventory, factors affecting inventory, Inventory models, Deterministic models: Economic order quantity model when shortages are allowed/not allowed, price discounts model, multi-item inventory models. Unit -IIQueuing Theory : Basic characteristics of queuing system, Birth-death equations, Steady state solution of Markovian queuing models with single and multiple servers (M/M/1 and M/M/c), with limited capacity (M/M/1/K and M/M/c/K). Unit -IIISequencing problems: Processing of n jobs through 2 machines, n jobs through 3 machines, 2 jobs through m machines, n jobs through m machines.Replacement problems: Replacement of items whose running cost increases with time, Replacement policies for the items that fail completely - Individual and the group replacement policies. Unit -IVPERT and CPM: Introduction of PERT and CPM, Earliest and latest times, Determination of critical path and various types of floats, Probablistic and cost considerations in project schedulingBooks Recommended:J.K. Sharma, Mathematical Model in Operations Research, Tata McGraw Hill. H.A. Taha, Operations Research – An Introduction.Kanti Swarup, Gupta, P.K. and Manmohan. Operations Research.P.K. Gupta and D.S Hira, Operations Research, S. Chand & Co. Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)DMT226B NUMERICAL METHODS (w.e.f. Session July 2018- 19)LTPMarks for External Exam : 7541- (5 Credits)Marks for Internal Exam : 25 Total : 100 Duration of Exam : 3Hours Course objectives : To provide the student with numerical methods of solving the non-linear equations, interpolation, differentiation, and integration. - To improve the student’s skills in numerical methods by using the numerical analysis software and computer facilities.Course outcomes: On completion of this course, the student should be able to solve interpolation and curve fitting, non-linear Equations, simultaneous linear equations, integration, numerical solution of ordinary differential equations, numerical solution of partial differential Equations.Note:The question paper will consist of four Units and one compulsory question containing four to five parts distributed equally all over the syllabus. Each unit will contain two questions and the students shall be asked to attempt five questions, selecting one question from each unit and the compulsory question. All questions carry equal marks.UNIT IInterpolation and Curve Fitting: Interpolation problem, Lagrangian polynomials, Divided differences, Interpolating with a cubic spline, Bezier curves and B-spline curves, Least Square Approximations. Non-Linear Equations: Bisection Method, Linear Interpolation methods, Newton’s method, Muller’s method, fixed point method. UNIT IISimultaneous Linear Equations: Elimination Method, Gauss and Gauss-Jordan method, Jacobi’s method, Gauss-Seidal method, Relaxation Method. Numerical Differentiation and Integration: Derivatives from differences tables, Higher order derivatives, Extrapolation techniques, Newton-cotes integration formula, Trapezoidal rule, Simpson’s rule, Boo;e’s rule and Weedle’s rule, Romberg’s integrationUNIT III Numerical Solution of Ordinary Differential Equations: Taylor series method, Euler and modified Euler method, Runge-Kutta method, Milne’s method, Adams-Moulton method, Power method for Eigen values by interation .UNIT IVNumerical Solution of Partial Differential Equations: Finite difference approximations of partial derivatives, solution of Laplace equation (Standard 5-point formula only), one- dimensional heat equation( Schmidt method, Crank-Nicolson method, Dufort and Frankel method) and wave equation. Books Recommended: Applied Numerical Analysis: Curtis F. Gerald and Patrik G. Wheatley-Pearson, Education Ltd. Numerical Method: E. Balaguruswamy T.M.H REFERENCE BOOKS: 1. Numerical Methods for scientific and Engg. Computations: MK Jain, SRK Iyenger and R.K. Jain-Wiley Eastern Ltd. 3.Introductory Methods of Numerical Analysis S.S Sastry, PHI 4. Numerical Methods in Engineering and Science: BS Grewal Dual degree-B.Sc.(Hons.) Mathematics-M.Sc. Mathematics(Semester-II)ENG 226B: English - II (w.e.f. Session July 2018- 19)LTPMarks for External Exam : 753 - -Total : 75 Duration of Exam : 3 HoursCourse Objective: To make the students proficient in comprehension, writing and speaking skills in English with correct pronunciation so that they enter the professional world with ease and confidenceCourse Outcome: The students will be able to enhance their writing and speaking skills and develop an ability to critically read the literary texts so as to develop proficiency in reading along with sensitivity to the impact literary texts can have on their minds/ lives.Course Contents:Unit IReading Skills: Language through Literature (15 Marks) “ Playing the English Gentleman” by M. K. Gandhi#“The Home Coming” by Rabindranath Tagore # # “The Ideal of Karma-Yoga” by Swami Vivekananda*“The Last Leaf” by O. Henry**Unit IIGrammar (15 Marks)Correct usage of TensesUse of Conditional sentences Active and Passive VoiceDirect and Indirect Speech Unit IIISpeaking skills (15 Marks)IPA Symbols of ‘Received Pronunciation’ Identifying the symbols / sounds in the wordsTranscription of monosyllabic and disyllabic words Dialogue Writing: Mechanics of writing good dialogues; common phrases and responses used in routine conversations, conversations in different situations of daily life Unit IVWriting Composition (15 Marks)Developing the outline into meaningful paragraphParagraph writing on current / social issues or given situationsBusiness letters: Constituents of formal letter Writing formal letters in complete / semi blocked style on hypothetical business issues (Calling quotations, placing an order, making an enquiry or complaint) Scheme of End Semester Examinations (Major Test):The duration of examinations will be 3 hours.Nine questions of 15 marks each will set, out of which the students will have to attempt five questions.First question of 15 marks will be compulsory. It will cover all the four units of the syllabus. The nature of the questions in each unit will depend upon the contents therein. The question will have sub-parts with marks assigned against each. Question No 02 to 09 will be set from the four units of the syllabus --- two from each unit of 15 marks each. Students will have to attempt four more questions, selecting one from each unit.Recommended pattern of questions in each unit will be as follows:Unit IOne question having subparts from the literary texts has to be answered in about 200 words each or may be set on vocabulary items from the texts themselves. The second question will be on comprehension of a passage from the text or on explanation with reference to context of the lines / passage from the prescribed texts. Examinees will attempt one question from this unit.Unit IITwo questions will be set from this unit also out of which one needs to be attempted. Questions will be in the form of correcting the errors in the sentences; picking up the right alternative, filling in the blanks or improving the sentence for desired results or completing the sentences/ changing mode of narration / providing tags to the statements.Unit IIIOne question will be on recognizing the sounds / phonemes in underlined alphabet/s of given words or on transcription of given words in IPA symbols and the second question will be on dialogue writing on imaginary situations or writing the short responses to given questions. Examinees will attempt one out of the two given questions. Unit IVTwo questions in the form of writing business letter / paragraph/ developing a given outline into a meaningful paragraph will be set from this unit. The questions may have parts if required. Examinees will attempt one out of the two given questions. Recommended readings (Online sources have been underlined): # Mohan, Loveleen, Randeep Rana and Jaibir S. Hooda. Eds. Literature and Language-I. New Delhi: Orient Blackswan Pvt. Ltd.,2015 (Chapter 7)# # Tagore,Rabindranath. Stories from Tagore. New York: The Macmillan Company,1918.*.*Vivekananda, Swami. Karma Yoga. New Delhi: Sahityashila Prakashan, 2015.** Bhatnagar, Nitin and Mamta Bhatnagar. Communicative English for Engineers and Professionals. New Delhi: Pearson Education, 2016. 9. Konar, Nira. Communication Skills for Professionals. New Delhi: PHI Learning Pvt. Ltd., 2009. 10. Sinha, R P. Current English Grammar and Usage with Composition. New Delhi: OUP, 2016.Bansal, R. K. and J. B. Harrison. Spoken English?for India: A Manual of Speech and Phonetics. Orient Longman Ltd., 1988.12Hill, L. A. A Guide to Correct English. Oxford: OUP, 1968. Approved by Board of UG Studies, Department of Humanities on 19 March 2018 ................
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