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VIKRAM DEB AUTONOMOUS COLLEGE

JEYPORE, KORAPUT, ODISHA - 764001

COURSE OF STUDIES

OF

BACHELOR DEGREE SCIENCE CBCS

ADMISSION BATCH 2015 ONWARD

MATHEMATICS

Published by VIKRAM DEB AUTONOMOUS COLLEGE

JEYPORE, KORAPUT, ODISHA www.vikramdebcollege.org

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Three Year Degree Course Bachelor of Science – Coures Structure

2015 Onwards MATHEMATICS

1st Year 1st Semester C.C. – I 100 C.C. – II 100 AECC 50 G.E – I (Paper-I) 100 Total 350

2nd Semester C.C. – III 100 C.C. – IV 100 AECC – EVS 50 G.E. – II (Paper-I) 100 Total 350 2nd Year 3rd Semester C.C. – V 100 C.C. – VI 100 C.C. – VII 100 SEC – I 50 G.E. – I (Paper II) 100 Total 450 4th Semester C.C. – VIII 100 C.C. – IX 100 C.C. – X 100 SEC – II 50 G.E. – II (Paper II) 100 Total 450 3rd Year 5th Semester C.C. – XI 100 C.C. – XII 100 DSE - I 100 DSE - II 100 Total 400 6th Semester C.C. – XIII 100 C.C. – XIV 100 DSE - III 100 DSE - II 100 Total 400

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VIKRAM DEB AUTONOMOUS COLLEGE, JEYPORE

Three Years Degree Course Bachelor of Science C.B.C.S. Course Structure

1st Year 1St emester Credit

Total Marks Mid Sem

End Sem

Core - 1 4 75 15 60 Core - 1 Practical 2 25 - 25 Core - 2 (Except Mathematics) 4 75 15 60 Core - 2 Practical 2 25 - 25 Core - 2 Mathematics 6 100 20 80 AECC - MIL/Eng 2 50 10 40 Generic Elective - 1 Paper – 1 (Except Mathematics) 4 75 15 60 Generic Elective - 1 Paper - 1 Practical 2 25 - 25 Generic Elective – 1 Paper – 1 Mathematics 6 100 20 80

20 350

2nd emester Credit

Total Marks Mid Sem

End Sem

Core - 3 (Except Mathematics) 4 75 15 60 Core - 3 Practical 2 25 - 25 Core - 3 Mathematics 6 100 20 80 Core - 4 4 75 15 60 Core - 4 Practical 2 25 - 25 EVS 2 50 10 40 Generic Elective - 2 Paper – 1 (Except Mathematics) 4 75 15 60 Generic Elective – 2 Paper – 1 Mathematics 6 100 20 80 Generic Elective - 2 Paper - 1 Practical 2 50 10 40

20 350

2nd Year 3rd Semister Credit

Total Marks Mid Sem

End Sem

Core - 5 (Except Mathematics) 4 75 15 60 Core - 5 Practical 2 25 - 25 Core - 5 Mathematics 6 100 20 80 Core - 6 (Except Mathematics) 4 75 15 60 Core - 6 Practical 2 25 - 25 Core - 6 Mathematics 6 100 20 80 Core - 7 4 75 15 60 Core - 7 Practical 2 25 - 25 SEC - I Subject Specific Skills 2 50 10 40 Generic Elective - 1 Paper - 2 (Except Mathematics) 4 75 15 60 Generic Elective - 1 Paper - 2 Practical 2 25 - 25 Generic Elective - 1 Paper - 2 Mathematics 6 100 20 80

26 450

4th Semister Credit

Total Marks Mid Sem

End Sem

Core - 8 4 75 15 60 Core - 8 Practical 2 25 - 25 Core - 9 (Except Mathematics) 4 75 15 60 Core - 9 Practical 2 25 - 25

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Core - 9 Mathematics 6 100 20 80 Core - 10 (Except Mathematics) 4 75 15 60 Core - 10 Practical 2 25 - 25 Core - 10 Mathematics 6 100 20 80 SEC - 2 Subject Specific Skills 2 50 10 40 Generic Elective - 2 Paper - 2 (Except Mathematics) 4 75 15 60 Generic Elective - 2 Paper - 2 Practical 2 25 - 25 Generic Elective - 2 Paper - 2 Mathematics 6 100 20 80

26 450

3RD Year 5th Semister Credit

Total Marks Mid Sem

End Sem

Core - 11 (Except Mathematics) 4 75 15 60 Core - 11 Practical 2 25 - 25 Coere - 11 Mathematics 6 100 20 80 Core – 12 (Except Mathematics) 4 75 15 60 Core - 12 Practical 2 25 - 25 Core - 12 Mathematics 6 100 20 80 DSE - 1 6 100 20 80 DSE - 2 6 100 20 80

24 400

6th Semister Credit

Total Marks Mid Sem

End Sem

Core - 13 (Except Mathematics) 4 75 15 60 Core - 13 Practical 2 25 - 25 Core - 13 Mathematics 6 100 20 80 Core – 14 (Except Mathematics) 4 75 15 60 Core - 14 Practical 2 25 - 25 Core - 14 Mathematics 6 100 20 80 DSE - 3 6 100 20 80

DSE - 4 (Project) 6 100

Viva-20 Presentation-

20 Disserteation

– 60

24 400

Total Credit = 140

Total Marks = 2400 * AECC - Ability Enhancement Compulsory Course SEC - Skill Enhancement Courses DSE - Discipline Specific Elective GE - Generic Elective

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MATHEMATICS COURSE STRUCTURE Semester Paper Subject Marks Duration of

Exam 1st Year

1st sem. C.C. I CALCULUS - I 75 3 PRACTICAL 25 3 C.C. II ALGEBRA – I 100 3

2nd Sem. C.C. III REAL ANALYSIS (ANALYSIS – I) 100 3 C.C. IV DIFFERENTIAL EQUATIONS 75 3 PRACTICAL 25 3

2nd Year 3rd Sem. C.C. V THEORY OF REAL FUNCTIONS

(ANALYSIS-II) 100 3

C.C. VI GROUP THEORY (ALGEBRA-II) 100 3 C.C. VII DIFFERENTIAL EQUATIONS AND

SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

75 3

PRACTICAL 25 3 4th Sem. C.C. VIII NUMERICAL METHODS 75 3

PRACTIVAL 25 3 C.C. IX RIEMANN INTEGRATION AND SERIES OF

FUNCTIONS (ANALYSIS – III) 100 3

C.C. X RING THEORY AND LINEAR ALGEBRA – I (ALGEBRA-III)

100 3

3rd Year 5th Sem. C.C. XI MULTIVARIATE CALCULUS (CALCULUS-

II) 100 3

C.C. XII PROBABILITY AND STATISTICS 100 3 6th Sem. C.C. XIII METRIC SPACES AND COMPLEX

ANALYSIS (ANALYSIS – IV) 100 3

C.C. XIV LIEAR PROGRAMMING 75 3 SKILL ENHANCEMENT COURSE

2nd Year 3rd Sem. SEC I THEORY OF EQUATIONS 50 3 4th Sem. SEC II LOGIC AND SETS 50 3

DISCIPLINE SPECIFIC ELECTIVE 3rd Year

5th Sem. DSE I PROGRAMMING IN C++ 100 3 DSE II MATHEMATICAL MODELING 100 3

6th Sem. DSE III DIFFERENTIAL GEOMETRY 100 3 DSE IV PROJECT 100 3

N. B.: 1) 20 % marks in each paper is allocated to Mid Semester Examination and 80 % marks allocated to term End Examination. 2) There shall be only one Mid Semester Examination in each Semester.

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Combination for Honours PHYSICAL SCIENCE

The Candidate has to select one of the following combination.

Core AECC Compulsory AECC Skill Based DSE GE

Physics

MIL/Alt. English EVS

Subject Specific

Physics Chemistry Mathematics

Chemistry Chemistry Physics Mathematics

Mathematics Mathematics Physics Chemistry

Computer Science *

Computer Science

Physics Chemistry Mathematics

Computer Science (Self-Financing) with a course fee of Rs. 30,000/- (Thirty Thousand) only @ Rs.

10,000/- per year. Students alloted computer science hons. shall have to pay Rs. 10,000/- every year along with admission fees. if a student discontinues in mid course, he/she shall forfeit the amount already depositied and in addition shall have to deposit the balance amout of the course fee Rs. 30,000/- before taking C.L.C.

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Semester - I AECC – MIL (Odia)

Full Marks – 50 `ÐWÔ : jcèÐ], bþÐiÐ J jÐkÞþ[Ô

1c HLL - jcèÐ]eþ `eÞþbþÐiÐ, `eÞþjeþ J `õLÐeþ

2 Æ HLL - jcèÐ] jÕNõkþeþ `w[Þ, õªÊÆ[Þ, `eÞþÒaiZ

3 Æ HLL - jcèÐ]eþ bþÐiÐ, jcèÐ] J jÐkÞþ[Ô

jkþЯÆL Nõx :

1 - jcèÐ] J jÐcèÐ]ÞL[Ð - QtõÒhMeþ ckþÐ`Ð[ö, JXÞA `ÐWÔ`ʪÆL õLÐh_ J `õZ¯Æ_ jÕyÐ, bËþaÒ_hèeþ¼¼Ð

2 - jÐeþjè[ jÐlÐ[LÐeþ - aÞS¯ÆÐ_t jÞÕkþ, aÞ]ÔÐ`ÊeþÑ, LVL

3 - jÕÒdÐN @_ÊaÞ Þ / aÔÐakþÐeÞþL JXÞA bþÐiÐ J õÒ¯ÆÐNÐ[ëL aÔÐLeþZ - jÒ«ÆÐi [öÞ ÐWÑ, _ÐftÐ, LVL

4 - mþM_LfÐ J õÐÒ¯ÆÐNÞL aÔÐLeþZ - @S Æ LÊcÐeþ cÞhõ, LmþÔÐZÑ `aîÞhjà, LVL

Mid Sem - 10

Term End - 40

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Semester - I Ability Enhancement Compulsory Course for Science

ENGLISH Full Marks - 50

This course aims at enhancing the English language proficiency of undergraduate students in humanity, science and commerce streams to prepare them for the academic, social and professional expectations during and after the course. The course will help develop academic and social English competencies in speaking, listening, pronunciation, reading and writing, grammar and usage, vocabulary, syntax, and rhetorical patterns. Students, at the end of the course, should be able to use English appropriately and effectively for further studies or for work where English is used as the language of communication. Unit I Reading Comprehension [15] " Locate and remember the most important points in the reading " Interpret and evaluate events, ideas, and information " Read "between the lines" to understand underlying meanings " Connect information to what they already know Unit II Writing [15] 1. Expanding an Idea 2. Writing a Memo 3. Report Writing 4. Creative Writing 5. News Story 6. Setting in Creative Writing 7. Writing a Business Letter 8. Letters to the Editor 9. Précis Writing 10. CV & Resume Writing 11. Dialog writing 12. Covering Letter 13. Writing Formal Email 14. Elements of Story Writing 15. Note Making 16. Information Transfer 17. Interviewing for news papers Unit III Language functions in listening and conversation [06] 1. Discussion on a given topic in pairs 2. Speaking on a given topic individually (Practice to be given using speaking activities from the prescribed textbooks) Grammar and Usage [14] 1. Simple and Compound Sentences 2. Complex Sentences 3. Noun Clause 4. Adjective Clause 5. Adverb Clause 6. The Conditionals in English 7. The Second Conditional 8. The Third Conditional 9. Words and their features 10. Phrasal Verbs 11. Collocation 12. Using Modals 13. Use of Passives 14. Use of Prepositions 15. Subject-verb Agreement 16. Sentence as a system 17. Common Errors in English Usage

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Book Prescribed Vistas and Visions: An Anthology of Prose and Poetry. (Ed.)Kalyani Samantray, Himansu S. Mohapatra, Jatindra K. Nayak, Gopa Ranjan Mishra, Arun Kumar Mohanty. OBS Texts to be studied (For Science) Prose " Pleasures of Ignorance " Ecology and Society " Lifestyle English " Playing the English gentleman Poetry " Lover's Infiniteness " Last Sonnet " The Felling of the Banyan Tree All grammar and writing activities in the textbook

Mid Sem - 10

Term End - 40

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Semester - II Ability Enhancement Compulsory Course (AECC)

ENVIRONMENTAL STUDIES Full Marks - 50

Unit - I Basic Concepts of Environment Ecological Factors; Air, Water, Soil, Light, Temperature, Biotic factors Ecosystem: Concepts, Structure and Function, Food Chain, Food Web, Ecological Pyramid, Energy Flow, Biogeochemical Cycle (Carbon Cycle, Nitrogen Cycle) Unit - II Environmental Pollution Air Pollution; Water Pollution, Industrial Pollution, Noise Pollution, Types of Pllutants, Sources and fate of Pollutants in the environment. Sewage treatment, Greenhouse Effect, Global Warming, Acid Rain. Unit - III Conservation of Natural Resourses Renewal and Non-Renewable resourcs, soil erosion and conservation, Forest Conservation, Wildlife Conservation, Pollution control Board and its function, Environmental education and awarness. Suggested Readings :

1. Fundamental of Ecology by E.P. Odum 2. Fundamental of Ecology by M.C. Dash 3. Ecology and Environment by P.D. Sharma 4. Concepts of Ecology by E.J. Kormondy 5. Fundamentals of Enviromental studies by N.K. Tripathy 6. Environmental studies by Dr. A.K. Panigrahi and Alaka Sahu 7. Parivesh Bigyana : Dr. Ranganath Mishra

Mid Sem - 10

Term End - 40

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2nd Year Semester – III

MATHEMATICS Skill Enhancement Courses(SEC)

SEC-I THEORY OF EQUATIONS

Full Marks : 100 Unit-l Preliminary,properties of equations, Descartes' Rule of Signs,relations between roots and coefficients, application to the solution of an equation ,symmetric functions of the roots. Unit-II Transformation of equations.transformation in general,multiple roots,sum of powers of roots,reciprocal equations. Unit-III Cubic Eqcations-Algebraic solutions,solution of the cubic, nature of the roots of cubic, nature of the roots from Cardan's solution.application of Cardan's method to numerical examples expressing the cubic as a difference of two cubes,solution by symmetric functions of roots. Unit-IV Biquadratic equations:-Roots of biquadratic in terms of M,solutions by radicals Suggested Readings : A TEXT BOOKON ALGEBRAAND THEORYOF EQUATIONS CHANDRIKA PRASAD.

POTHISHALA PRIVATE LIMITED, 2, LAJPAT ROAD, ALLAHABAD – 211002 Mid Sem. - 10 Term End - 40

Semester – III MATHEMATICS

Skill Enhancement Courses(SEC) SEC-II

LOGIC AND SETS Full Marks : 100

Unit-I Introduction, propositions, truth table, negation, conjunction and disjunction. Implications, biconditional propositions, converse, contra positive and inverse propositions and precedence of logical operators. Propositional equivalence: logical equivalence. Unit-II Predicates and quantifiers: introduction, quantifiers, binding variables and negations. Sets, subsets, set operations and the laws of set theory and Venn diagrams. Unit-III Example of finite and infinite sets. Finite set and counting principle. Empty set, properties of empty set. Standard set operations. Classes of sets. Power set of a set. Difference and symmetric difference of two sets. Set identities. Generalized union and intersections. Unit-IV Relation, product set, composition of relations, types of relations, partitions, equivalence relations with examples of congruence modulo relation, partial orders relations, and nary relations. Suggested Readings : 1. R.P.Grimaldi-Discrete mathematics and combinatorial mathematics, Pearson education.199B. 2. P.R Halmos-Naive set theory, springer, 1974. 3. E.kamke-theory of sets, dover publishers, 1950.

Mid Sem. - 10 Term End - 40

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1st Year Semester – I

Core Course : MATHEMATICS Core Course (C.C.) - I

CALCULUS-I Full Marks : 100

Unit-I Hyperbolic functions, higher order derivatives, Leibniz rule and its applications to problems of the type e ax+b x.e ax+b cos x:,(ax + b)nsin x, (ax + b)n cos x, concavity and inflection points, asymptotes, curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves, LHospitals rule, applications in bu. siness, economics and life sciences. Unit-II Reduction formulae, derivations and illustrations of reduction formulae of the type ∫ sin nxdx, ∫ cos nxdx, ∫ tannx dx, ∫ sec nx dx. ∫ (log x)n dx, ∫ sinnx cosn xdx, volumes by slicing, disks and washers methods, volumes by cylindrical shells, parametric equations, parameterizing ~ curve, arc length, arc length of parametric curves, area of surface of revolution. Unit-III Techniques of sketching conics, reflection properties of conics, rotation of axes and second degree equations, classification into conics using the discriminant, polar equations of conics. Sphere, Cone, Cylinder, Central Conicoids. Unit-IV Triple product, introduction to vector functions, operations with vector-valued functions. limits and continuity of vector functions, differentiation and integration of vector functions, tangent and nOrITiClI com ponents of acceleration. Part-II (Practical, Marks:25) List of Practicals (Using any software) Practical/Lab work to be performed on a Computer. 1. Plotting the graphs ofthe functions, eax+b.Iog(ax + b). 1/(ax + b). sin(ax+b), cost(ax+b). lax + bl and to illustrate the effect of a and b on the graph. 2. Plotting the graphs of the polynomial of degree <1 and 5. the derivative graph, the second derivative graph and comparing them. 3. Sketching parametric curves (Eg. Trochoid, cycloid, epicycloids, hypocycloid). 4. Obtaining surface of revolution of curves. 5. Tracing of conics in cartesian coordinates/polar coordinates. 6. Sketching ellipsoid, hyperboloid of one and two sheets, elliptic cone, elliptic, paraboloid, hyperbolic paraboloid using cartesian coordinates. 1. Matrix operation (addition, multiplication, inverse, transpose Suggested Readings : 1. M.J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) P. Ltd. (Pearson Education), Delhi, 2007. Chapters:4(4.3,4.4,4.5 & 4.7), 9(9.4), 10(10.1-10.4). 2. H. Anton, I. Bivens and S. Davis, Calculus, 7th Ed., John Wiley and Sons (Asia) P. Ltd., Singapore, 2002. Chapters: 6, (6.2-6.5),7(7.8), 8(8.2-8.3, Pages:532-538), 11(11.1), 13(13.5) 3. Analytical Geometry of Quadratic Surfaces, B.P. Acharya and D.C. Sahu, Kalyani Publishers, New Delhi, Ludhiana.-878 Books for Reference: 1. G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005. 2. R. Courant and F. John, Introduction to Calculus and Analysis (Volumes I & II), Springer-Verlag, New York, Inc., 1989. 3. Text Book of Calculus, Part-II- Shantinarayan, S. Chand & Co., 4. Text Book of Calculus, Part-III-Shantinarayan, S. Chand & Co., 5. Shanti Narayan and P.K. Mittal-Analytical Solid Geometry, S. Chand & Company Pvt. Ltd., New Delhi.

Mid Sem. - 15 Term End - 60 Prctical - 25

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Semester – I Core Course : MATHEMATICS

Core Course (C.C.) - II ALGEBRA-I

Full Marks : 100 Unit-I Polar representation of complex numbers, n-th roots of unity, De Moivres theorem for rational indices and its applications. Unit-II Equivalence relations, Functions, Composition of functions, Invertible functions, One to one correspondence and cardinality of a set, Well-ordering property of positive integers, Division algorithm, Divisibility and Euclidean algorithm, Congruence relation between integers, Principles of Mathematical Induction, statement of Fundamental Theorem of Arithmetic. Unit-III Systems of linear equations, row reduction and echelon forms, vector equations, the matrix equation Ax = U, solution sets of linear systems, applications of linear systems, linear independence. Unit-IV Introduction to linear transformations, matrix of a linear transformation, inverse of a matrix, characterizations of invertible matrices. Subspaces of IRn, dimension of subspaces of IRn and rank of a matrix, Eigen values, Eigen Vectors and Characteristic Equation of a matrix. Suggested Readings : 1. L.V. Ahlfors, Complex Analysis, McGraw-Hill(lnternational Student Edn.) 2. Titu Andreescu and Dorin Andrica, Complex Numbers from A to Z, Birkhauser, 2006. Chapter:2 3. Edgar G. Goodaire and Michael M. Parmenter, Discrete Mathematics with Graph Theory, 3rd Ed., Pearson Education (Singapore) P. Ltd., Indian Reprint, 2005. Chapters:2(2.4), 3,4(4.1-4.1.6, 4.2-4.2.11, 4.4(4.1-4.4.8) ,4.3-4.3.9, 5(5.1-5.1.4). 4. David C. Lay, Linear Algebra and its Applications, 3rd Ed., Pearson Education Asia, Indian Reprint, 2007. Chapters:1(1.1-1.9), 2(2.1-2.3, 2.8, 2.9), 5(5.1,5.2

Mid Sem. - 20 Term End - 80

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Semester – II Core Course : MATHEMATICS

Core Course (C.C.) - III REAL ANALYSIS (ANALYSIS-I)

Full Marks : 100 Unit-I eview of Algebraic and Order Properties of lR. Neighborhood of a point in lR. Idea of countable sets, ncountable sets and uncountability of JR. Bounded above sets, Bounded below sets, Bounded Sets, Unbounded sets, Suprema and Infima. Unit-II The Completeness Property of JR, The Archimedean Property, Density of Rational (and Irrational) numbers in JR, Intervals. Limit points of a set, Isolated points, Illustrations of Bolzano-Weierstrass theorem for sets. Unit-III Sequences, Bounded sequence, Convergent sequence, Limit of a sequence. Limit Theorems, Monotone Sequences, Monotone Convergence Theorem. Subsequences, Divergence Criteria, Monotone Subsequence Theorem (statement only), Bolzano Weierstrass Theorem for Sequences. Cauchy sequence, Cauchys Convergence Criterion. Unit-IV Infinite series, convergence and divergence of infinite series, Cauchy Criterion, Tests for convergence: Comparison test, Limit Comparison test, Ratio Test, Cauchys n-th root test, Integral test, Alternating series, Leibniz test, Absolute and Conditional convergence. Suggested Readings : 1. G. Das and S. Pattanayak, Fundamentals of Mathematics Analysis, TMH Pub-lishing Co. ,Chapters: 2(2.1 to 2.4, ~, 3(3.1-3.5), 4(4.1 to 4.7,4.10,4.11,4.12,4.13). Books for References: 1. R.G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002. 2. Gerald G. Bilodeau, Paul R. Thie, G.E. Keough, An Introduction to Analysis, 2nd Ed., Jones & Bartlett, 2010. 3. Brian S. Thomson, Andrew. M. Bruckner and Judith B. Bruckner, Elementary Real Analysis, Prentice Hall, 2001. 4. S.K. Berberian, A First Course in Real Analysis, Springer Verlag, New York, 1004. 5. S.c. Mallik and S. Arora-Mathematical Analysis, New Age International Publications. 6. D. Smasundaram and B. Choudhury-A First Course in Mathematical Analysis, Narosa Publishing House. 7. S.L. Gupta and Nisha Rani-Real Analysis, Vikas Publishing House Pvt. l.td., New Delhi

Mid Sem. - 20 Term End - 80

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Semester – II Core Course : MATHEMATICS

Core Course (C.C.) - IV DIFFERENTIAL EQUATIONS

Full Marks : 100

Unit-I Differential equations and mathematical models. First order and first degree ODE (variables separable, homogeneous, exact, and linear). Equations of first order but of higher degree. Applications of first order differential equations(Growth, Decay and Chemical Reactions, Heat flow, Oxygen debt, Economics). Unit-II Second order linear equations(homogeneous and non-homogeneous) with constant coefficients, second order equations with variable coefficients, variation of parameters, method of undetermined coefficients, equations reducible to linear equations with constant coefficients, Euler's equation. Applications of second order differential equations. Unit-III Power series solutions of second order differential equations. Unit-IV Laplace transforms and its applications to solutions of differential equations. Part-II (Practical: Marks:25) List of Practicals (Using any Software) Practical/Lab work to be performed on a Computer. 1. Plotting of second order solution of family of differential equations. 2. Plotting of third order solution of family of differential equations. 3. Growth model (exponential case only). 4. Decay model (exponential case only). 5. Oxygen debt model. 6. Economic model. 7. Vibration problems. Suggested Readings : 1. J. Sinha Roy and S. Padhy, A Course of Ordinary and Partial Differential Equations, Kalyani Publishers, New Delhi. Chapters: 1,2(2.1 to 2.7),3,4(4.1 to 4.7),5,7(7.1-7.4),9(9.1,9.2,9.3, 9.4, 9.5, 9.10, 9.11, 9.13). Books for References: 1. Martin Braun, Differential Equations and their Applications, Springer International. 2. M.D. Raisinghania-Advanced Differential Equations, S. Chand & Company Ltd., New Delhi. 3. G. Dennis Zill-A First Course in Differential Equations with Modelling Applications, Cengage Learning India Pvt. Ltd. 4 S.L. Ross, Differential Equations, John Wiley & Sons, India, 2004

Mid Sem. - 15 Term End - 60 Prctical - 25

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2nd Year Semester – III

Core Course : MATHEMATICS Core Course (C.C.) - V

THEORY OF REAL FUNCTIONS (ANALYSIS-II) Full Marks : 100

Unit-I Limits of functions (휖- 훿 approach), sequential criterion for limits, divergence criteria. Limit theorems, one sided limits. Infinite limits and limits at infinity. Continuous functions, sequential criterion for continuity and discontinuity. Unit-II Igebra of continuous functions. Continuous functions on an interval, intermediate value theorem, ocation of roots theorem, preservation of intervals theorem. Uniform continuity, non-uniform continuity criteria, uniform continuity theorem. Differentiability of a function at a point and in an interval, Caratheodorys theorem, algebra of differentiable functions. Unit-III Relative extrema, interior extremum theorem. Rolles theorem, Mean value theorem, intermediate value property of derivatives, Darbouxs theorem. Applications of mean value theorem to inequalities and approximation of polynomials, Taylors theorem to inequalities. Unit-IV Cauchys mean value theorem. Taylors theorem with Lagranges form of remainder, Taylors theorem with Cauchys form of remainder, application of Taylors theorem to convex functions, relative extrema. Taylors series and Maclaurins series expansions of exponential and trigcnometric functions, In(l + x). l/(ax + b) and (1 + z)". Suggested Readings : 1. G. Das and S. Pattanayak, Fundamentals of Mathematics Analysis, TMH Pub-lishing Co., Chapters:6(6.1-6.8), 7(7.1-7.7), Books for References: 1. R. Bartle and D.R. Sherbert, Introduction to Real Analysis, John Wiley and Sons, 2003. 2. K.A. Ross, Elementary Analysis: The Theory of Calculus, Springer, 2004. 3. A. Mattuck, Introduction to Analysis, Prentice Hall, 1999. 4. S.R. Ghorpade and B.v. Limaye, A Course in Calculus and Real Analysis, Springer, 2006.

Mid Sem. - 20 Term End - 80

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Semester – III Core Course : MATHEMATICS

Core Course (C.C.) - VI ALGEBRA-II

Full Marks : 100

Unit-I Summetries of a square, Dihedral groups, definition and examples of groups including permutation groups " d quaternion groups (illustration through matrices), elementary properties of groups. Subgroups and examples of subgroups, centralizer, normalizer, center of a group, product of two subgroups Unit-II yo erties of cyclic groups, classification of subgroups of cyclic groups. Cycle notation for permutation, properties of permutations, even and odd permutations, alternating group, properties of cosets, ....a ranges theorem and consequences including Fermats Little theorem. Unit-III External direct product of a finite number of groups, normal subgroups, factor groups, Cauchys theorem for finite abelian groups. Unit-IV Group homomorphisms, properties of homomorphisms, Cayleys theorem, properties of isomorphisms, First, Second and Third isomorphism theorems. Suggested Readings : 1. Joseph A .. Gallian, Contemporary Abstract Algebra(4th Edn.), Narosa Publishing House, New Delhi. Books for References: 1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. 2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. 3. Joseph J. Rotman, An Introduction to the Theory of Groups, 4th Ed., Springer Verlag, 1995. 4. I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.

Mid Sem. - 20 Term End - 80

19

Semester – III Core Course : MATHEMATICS

Core Course (C.C.) - VII PARTIAL DIFFERENTIAL EQUATIONS AND SYSTEMS OF ORDINARY

DIFFERENTIAL EQUATIONS Full Marks : 100

Unit-I Systems of linear differential equations, types of linear systems, differential operators, an operator method for linear systems with constant coefficients, Basic Theory of linear systems in normal form, homogeneous linear systems with constant coefficients(Two Equations in two unknown functions). Simultaneous linear first order equations in three variables, methods of solution, Pfaffian differential equations, methods of solutions of Pfaffian differential equations in three variables. Unit-II Formation of first order partial differential equations, Linear and non-linear partial difFerential equations of first order, special types of first-order equations, Solutions of partial differential equations of first order satisfying given conditions. Unit-III Linear partial differential equations with constant coefficients, Equations reducible to linear partial differential equations with constant coefficients, Partial differential equations with variable coefficients, eparation of variables, Non-linear equation of the second order Unit-IV Laplace equation, Solution of laplace equation by separation of variables, One dimensional wave equation, Solution of the wave equation (method of separation of variables), Diffusion equation, Solution of one-dimensional diffusion equation, method of separation of variables. Part-II (Practical: Marks:25) List of Practicals (Using any Software) Practical/Lab work to be performed on a Computer. To find the general solution of the non-homogeneous system of the form:

푑푥푑푡

= 푎 푥 + 푏 푦 + 푓 (푡),푑푦푑푡

= 푎 푥 + 푏 푦 + 푓 (푡) with given conditions. Plotting the integral surfaces of a given first order PDE with initial data. 3. 0 ution 0 wave equation − 푐 = 0 for the following associated conditions : (a) 푢(푥, 0) = ∅(푥),푢 (푥, 0) = 휑(푥), 푥휖푅, (b) 푢(푥, 0) = ∅(푥),푢 (푥, 0) = 휑(푥),푢 (0, 푡) = 0, 푥 ∈ (0,∝)푦 > 0 (c) 푢(푥, 0) = ∅(푥),푢 (푥, 표) = 휑(푥),푢(0, 푡) = 0, 푥 ∈ (0,∝),푦 > 0 (d) 푢(푥, 0) = ∅(푥),푢 (푥, 0) = 휑(푥),푢(0, 푡) =0,푢(1, 푡) = ∅(푥),푢 (푥, 0) = 휑(푥), 푢(0, 푡) = 0,푢(1, 푡) = 0,0 < 푥 < 푙, 푡 > 0. Solution of wave equation − 푘 = 0 for the following associated conditions:

(a) u(x,0) = ∅(푥),푢 (0,푦) = 푎,푢(푙, 푡) = 푏, 0 < 푥 < 푙, 푡) > 0 (b) 푢(푥, 0) = ∅(푥), 푥휖푅, 0 < 푡 < 푇 (c) 푢(푥, 0) = ∅(푥),푢(0, 푡) = 푎, 푥휖(0,∞), 푡 ≥ 0

Suggested Readings : 1. J.Sinha Roy and S, Padhy, A Course on Ordinary and Partial Differential Equations, Kalyani Publishers, New Delhi, Ludhiana, 2012, Chapters:l1, 12, 13(13.1-13.5), 15(15.1,15,5), 16(16,1, 16.1.1), 17(17.1, 17.2, 17.3). Books for References: 1. Tyn Myint-U and Lokenath Debnath, Linear Partial DifFerential Equations for Scientists and Engineers, 4th edition, Springer, Indian reprint, 2006, 2. S.L. Ross, DifFerential equations, 3rd Ed., John Wiley and Sons, India, 2004

Mid Sem. - 15 Term End - 60 Prctical - 25

20

Semester – IV Core Course : MATHEMATICS

Core Course (C.C.) - VIII NUMERICAL METHODS

Full Marks : 100 Unit-I Algorithms, Convergence, Errors: Relative, Absolute, Round off, Truncation. Transcendental and Polyial equations: Bisection method, Newtons method, Secant method. Rate of convergence of these Methods Unit-II System of linear algebraic equations: Gaussian Elimination and Gauss Jordan methods. Gauss Jacobi Method, Gauss Seidel method and their convergence analysis. Unit-III Interpolation: Lagrange and Newtons methods. Error bounds. Finite difFerence operators. Gregory Forward and backward difference interpolation. Unit-IV Numerical Integration: Trapezoidal rule, Simpsons rule, Simpsons 3/8th rule, Booles Rule. Midpoint rule, Composite Trapezoidal rule, Composite Simpsons rule. Ordinary DifFerential Equations: Eulers method. Runge-Kutta methods of orders two and four. Part-II (Practical: Marks:25) List of Practicals (Using any Software) Practical/Lab work to be performed on a Computer. 1. Calculate the sum 1/1 + 1/2 + 1/3 + 1/4 + - - - - - - - - - - +1/N. 2. Enter n integers into an array and sort them in an ascending order. 3. Bisection Method. 4. Newton Raphson Method. 5. Secant Method. 6. Regulai Falsi Method. 7. LU decomposition Method. 8.Gauss-Jacobi Method. 9. Gauss-Siedel Method. 10. SOR Method 11. Lagrange Interpolation / Newton Interpolation. 12. Simpsons rule. Note: For any of the CAS (Computer aided software) Data types-simple data types, floating data types, character data types, arithmetic operators and operator precedence, variables and constant declarations, expressions, input/output, relational operators, logical operators and logical expressions, control statements and loop statements, Arrays should be introduced to the students. Suggested Readings :

1. B.P. Acharya and R.N. Das, A Course on Numerical Analysis, Kalyani Publishers, New Delhi, Ludhiana. Chapters: 1, 2(2.1 to 2.4, 2.6, 2.8, 2.9), 3(3.1 to 3.4, 3.6 to 3.8, 3.10), 4(4.1, 4.2), 5(5.1, 5.2, 5.3), 6(6.1, 6.2, 6.3, 6.10, 6.11), 7(7.1, 7.2, 7.3, 7.4 &7.7).

2. Brian Bradie, A Friendly Introduction to Numerical Analysis, Pearson Education, India, 2007. Books for References: 1. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, 6th Ed., New age International Publisher, India, 2007. 2. C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Pearson Education, India, 2008. 3. Uri M. Ascher and Chen Greif, A First Course in Numerical Methods, 7th Ed., PHI Learning Private Limited, 2013. 4. John H. Mathews and Kurtis D. Fink, Numerical Methods using Matlab, 4th Ed., PHI Learning Private Limited, 2012.

Mid Sem. - 15 Term End - 60 Prctical - 25

21

Semester – IV Core Course : MATHEMATICS

Core Course (C.C.) - IX RIEMANN INTEGRATION AND SERIES OF FUNCTIONS (ANALYSIS-III)

Full Marks : 100 Unit-I Riemann integration; inequalities of upper and lower sums; Riemann conditions of integrability. Riemann sum and definition of Riemann integral through Riemann sums; equivalence of two definitions; Riemann integrability of monotone and continuous functions, Properties of the Riemann integral; definition and integrability of piecewise continuous and monotone functions. Intermediate Value theorem for Integrals; Fundamental theorems of Calculus. Unit-II Improper integrals; Convergence of Beta and Gamma functions. Unit-III Pointwise and uniform convergence of sequence of functions. Theorems on continuity, derivability and integrability of the limit function of a sequence of functions. Series of functions; Theorems on the continuity and derivability of the sum function of a series of functions; Cauchy criterion for uniform convergence and Weierstrass M- Test. Unit-IV Limit superior and Limit inferior. Power series, radius of convergence, Cauchy Hadamard Theorem, Differentiation and integration of power series; Abels Theorem; Weierstrass Approximation Theorem. Suggested Readings :

1. G. Das and S. Pattanayak-Fundamentals of Mathematics Analysis, TMH Publishing Co., Chapters 8,9. Books for References: l. K.A. Ross, Elementary Analysis, The Theory of Calculus, Undergraduate Texts in Mathematics, Springer (SIE), Indian reprint, 2004. 2. R.G. Bartle D.R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002. 3. Charles G. Denlinger, Elements of Real Analysis, Jones & Bartlett (Student Edition), 2011 . 4. S.c. Mallik and S. Arora-Mathematical Analysis, New Age International Ltd., New Delhi. 5. Shanti Narayan and M.D. Raisinghania-Elements of Real Analysis, S. Chand & Co. Pvt. Ltd.

Mid Sem. - 20 Term End - 80

22

Semester – IV Core Course : MATHEMATICS

Core Course (C.C.) - X RING THEORY AND LINEAR ALGEBRA-I (ALGEBRA-III)

Full Marks : 100 Unit-I Definition and examples of rings, properties of rings, subrings, integral domains and fields, characteristic of a ring. Ideal, ideal generated by a subset of a ring, factor rings, operations on ideals, prime ar.d maximal ideals. Unit-II Ring homomorphisms, properties of ring homomorphisms, Isomorphism theorems I, II and III, field of quotients. Unit-III Vector spaces, subspaces, algebra of subspaces, quotient spaces, linear combination of vectors, linear span, linear independence, basis and dimension, dimension of subspaces. Unit-IV Linear transformations, null space, range, rank and nullity of a linear transformation, matrix representationof a linear transformation, algebra of linear transformations. Isomorphisms, Isomorphism theorems, invertibility and isomorphisms, change of coordinate matrix. Suggested Readings : l. Joseph A. Gallian, Contemporary Abstract Algebra(4th Edn.), Narosa Publishing House, New Delhi. Chapters:12, l3, 14, 15. 2. Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra, 4th Ed., Prentice- Hall of India Pvt. Ltd., New Delhi, 2004. Chapters: 1 (l.2-l.6), 2(2.1-2.5). Books for References: l. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002. 2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011. 3. S. Lang, Introduction to Linear Algebra, 2nd Ed., Springer, 2005. 4. Gilbert Strang, Linear Algebra and its Applications, Cengage Learning India Pvt. Ltd. 5. S. Kumaresan, Linear Algebra- A Geometric Approach, Prentice Hall of India,1999. 6. Kenneth Hoffman, Ray Alden Kunze, Linear Algebra, 2nd Ed., Prentice-Hall of India Pvt. Ltd., 1971. 7. I.N. Herstein-Topics in Algebra, Wiley Eastern Pvt. Ltd.

Mid Sem. - 20 Term End - 80

23

3rd Year Semester – V

Core Course : MATHEMATICS Core Course (C.C.) - XI

MULTIVARIATE CALCULUS (CALCULUS-II) Full Marks : 100

Unit-I Functions of several variables, limit and continuity of functions of two variables Partial differentiation, total differentiability and differentiability, sufficient condition for differentiability. Chain rule for one and two independent parameters, directional derivatives, the gradient, maximal and normal property of the gradient, tangent planes. Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems, Definition of vector field, divergence and curl Unit-II Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems, Definition of vector field, divergence and curl. Unit-III Double integration over rectangular region, double integration over non-rectangular region, Double integrals in polar co-ordinates, Triple integrals, Triple integral over a parallelepiped and solid regions. Volume by triple integrals, cylindrical and spherical co-ordinates. Change of variables in double integrals and triple integrals. Unit-IV Line integrals, Applications of line integrals: Mass and Work. Fundamental theorem for line integrals, conservative vector fields, independence of path. Greens theorem, surface integrals, integrals over parametrically defined surfaces. Stokes theorem, The Divergence theorem. Suggested Readings :

1. M.J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) Pvt. Ltd.(Pearson Education), Delhi, 2007. Chapters: 11(11.1(Pages: 541-543),11.2-11.6, 1l.7(Pages:598- 605), 11.8(Pages:610-614)), 12 (12.1, -12.3, 12.4(Pages:652-660), 12.5, 12.6), 13 (13.2, 13.3, 13.4(Pages:712-716), 13.5(Pages:723-726; 729-730),13.6 (Pages:733-737), 13.7(Pages:742 ..745)) .

Books for Reference: 1. G.B. Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005. 2. E. Marsden, A.J. Tromba and A. Weinstein, Basic Multivariable Calculus, Springer (SIE), Indian reprint, 2005. 3. Santosh K. Sengar-Advanced Calculus, Cengage Learning India Pvt. Ltd.

Mid Sem. - 20 Term End - 80

24

Semester – V Core Course : MATHEMATICS

Core Course (C.C.) - XII PROBABILITY AND STATISTICS

Full Marks : 100 Unit-I Sample space, probability axioms, real random variables (discrete and continuous), cumulative distribution function, probability mass/density functions, mathematical expectation, moments, moment generating function, characteristic function. Unit-II Discrete distributions: uniform, binomial, Poisson, geometric, negative binomial, continuous distributions: uniform, normal, exponential. Joint cumulative distribution function and its properties, joint probability density functions, marginal and conditional distributions. Unit-III Expectation of function of two random variables, conditional expectations, independent random variables, bivariate normal distribution, correlation coefficient, joint moment generating function (jmgf) and calculation of covariance (from jmgf), linear regression for two variables. Unit-IV Chebyshevs inequality, statement and interpretation of (weak) law of large numbers and strong law of large numbers, Central Limit theorem for independent and identically distributed random variables with finite variance, Markov Chains, Chapman-Kolmogorov equations, classification of states. Suggested Readings : l. Robert V. Hogg, Joseph W. McKean and Allen T. Craig, Introduction to Mathematical Statistics, Pearson Education, Asia, 2007. Chapters: 1(l.1, l.3. l.5-l.9), 2(2.1, 2.3-2.5). 2. Irwin Miller and Marylees Miller, John E. Freund, Mathematical Statistics with Applications, 7th Ed., Pearson Education, Asia, 2006. Chapters: 4, 5(5.1-5.5, 5.7), 6(6.2,6.3, 6.5-6.7), 14(14.1, 14.2) 3. Sheldon Ross, Introduction to Probability Models, 9th Ed., Academic Press, Indian Reprint, 2007. Chapters:2(2.7),4(4.1-4.3). Books for References: l. Alexander M. Mood, Franklin A. Graybill and Duane C. Boes, Introduction to the Theory of Statistics, 3rd Ed., Tata McGraw- Hill, Reprint 2007. 2. S.c. Gupta and V.K. Kapoor-Fundamentals of Mathematical Statistics, S. Chand and Company Pvt. Ltd., New Delhi. 3. S. Ross-A First Course in Probability, Pearson Education.

Mid Sem. - 20 Term End - 80

25

Semester – VI Core Course : MATHEMATICS

Core Course (C.C.) - XIII METRIC SPACES AND COMPLEX ANALYSIS (ANALYSIS-IV)

Full Marks : 100 Unit-I Metric spaces: definition and examples. Sequences in metric spaces, Cauchy sequences. Complete Metric Spaces. Open and closed balls, neighbourhood, open set, interior of a set. Limit point of a set, closed set, diameter of a set, Cantors theorem. Subspaces, dense sets, separable spaces. Continuous mappings, sequential criterion and other characterizations of continuity. Uniform continuity. Horneomorphism, Contraction mappings, Banach Fixed point Theorem. Connectedness, connected subsets of R. Unit-II Properties of complex numbers, regions in the complex plane, functions of complex variable, mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability. Unit-III Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function, derivatives of functions, definite integrals of functions. Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals. Cauchy- Goursat theorem, Cauchy integral formula. Unit-IV Liouvilles theorem and the fundamental theorem of algebra. Convergence of sequences and series, Taylor series and its examples. Laurent series and its examples, absolute and uniform convergence of power series. Suggested Readings : 1. P.K. Jain and K. Ahmad, Metric Spaces, Narosa Publishing House, New Delhi. Chapters: 2(1-9), 3(1-4),4(1-4),6(1-2), 7(1 only). 2. James Ward Brown and Ruel V. Churchill, Complex Variables and Applications, 8th Ed., McGraw Hill International Edition, 2009. Chapters: 1(11 only), 2(12,13), 2(15-22, 24, 25), 3(29, 30, 34) 4(37-41, 43-46, 50-53), 5(55-60, 62,63,66). Books for References: 1. Satish Shirali and Harikishan L. Vasudeva, Metric Spaces, Springer Verlag, London, 2006. 2. S. Kumaresan, Topology of Metric Spaces, 2nd Ed., Narosa Publishing House, 2011. 3. S. Ponnusamy-Foundations of Complex Analysis, Alpha Science International Ltd. 4. J.B. Conway-Functions of one complex variable, Springer. 5. N. Das- Complex Function Theory, Allied Publishers Pvt. Ltd., Mumbai.

Mid Sem. - 20 Term End - 80

26

Semester – VI Core Course : MATHEMATICS

Core Course (C.C.) - XIV LINEAR PROGRAMMING

Full Marks : 100 Unit-I Introduction to linear programming problem, Theory of simplex method, optimality and unboundedness, the simplex algorithm, simplex method in tableau format, introduction to artificial variables, twophase method, BigM method and their comparison. Unit-II Duality, formulation of the dual problem, primal-dual relationships, economic interpretation of the dual. Unit-III Transportation problem and its mathematical formulation, northwestcorner method least cost method and Vogel approximation method for determination of starting basic solution, algorithm for solving transportation problem, assignment problem and its mathematical formulation, Hungarian method for solving assignment problem. Unit-IV Game theory: formulation of two person zero sum games, solving two person zero sum games. Games with mixed strategies, graphical solution procedure, linear programming solution of games. Suggested Readings : 1. Mokhtar S. Bazaraa, John J. Jarvis and Hanif D. Sherali, Linear Programming and Network Flows, 2nd Ed., John Wiley and Sons, India, 2004. Chapters:3(3.2-3.3, 3.5-3.8), 4(4.1-4.4), 6(6.1-6.3). 2. F.5. Hillier and G.J. Lieberman, Introduction to Operations Research, 9th Ed., Tata McGraw Hill, Singapore, 2009. Chapter:14 3. Hamdy A. Taha, Operations Research, An Introduction, 8th Ed., PrenticeHall India, 2006. Chapter: 5(5.1, 5.3, 5.4). Books for Reference: 1. G. Hadley, Linear Programming, Narosa Publishing House, New Delhi, 2002. 2. Kanti Swarup, P.K. Gupta and Man Mohan-Operations Research, S. Chand and Co. Pvt. Ltd. 3. N.v.R. Naidu, G. Rajendra and T. Krishna Rao-Operations Research, I.K. lnternational Publishing House Pvt. Ltd., New Delhi, Bangalore. 4. R. Veerachamy and V. Ravi Kumar-Operations Research- I.K. International Publishing House Pvt. Ltd., New Delhi, Bangalore. 5. P.K. Gupta and D.5. Hira-Operations Research, S. Chand and Company Pvt. Ltd., New Delhi.

Mid Sem. - 20 Term End - 80

27

3rd Year Semester – V

MATHEMATICS Discipline Specific Electives (DES)

DSE-I PROGRAMMING IN C++ (COMPULSORY)

Full Marks : 100

Unit-I Introduction to structured programming: data types- simple data types, floating data types, character data types, string data types, arithmetic operators and operators precedence, variables and constant declarations Unit-II Expressions, input using the extraction operator U. and cin, output using the insertion operator i i and cout, preprocessor directives Unit-III increment(++) and decrement( -) operations, creating a C++ program, input, output, relational operators, logical operators and logical expressions, if and if-else statement, switch and break statements. for, while and do-while loops and continue statement, nested control statement Unit-IV Value returning functions, value versus reference parameters, local and global variables, one dimensional array, two dimensional array, pointer data and pointer variables. Suggested Readings : 1. D. S. Malik: C++ Programming Language, Edition-2009, Course Technology, Cengage Learning, India Edition. Chapters: 2(Pages:37-95), 3(Pages:96-129), 4(Pages: 134-178), 5(Pages: 181-2:i6), 6, 7(Pages:287-304), 9 (pages: 357-390), 14(Pages:594-600). Books for References: 1. E. Balaguruswami: Object oriented programming with C++, fifth edition, T3t3 McGraw Hiil Education Pvt. Ltd. 2. R. Johnsonbaugh and M. Kalin-Applications Programming in ANSI C, Pearson Education. 3. S. B. Lippman and J. Lajoie, C++ Primer, 3rd Ed., Addison Wesley, 2000. 4. Bjame Stroustrup , The C++ Programming Language, 3rd Ed., Addison Welsley.

Mid Sem. - 20 Term End - 80

28

Semester – V MATHEMATICS

DSE-II MATHEMATICAL MODELLING

Full Marks : 100 Unit-I Simple situations requiring Mathematical modelling. The technique of Mathematical modelling, Mathematical modelling through differential equations, linear growth and decay models, non-linear growth and decay models, compartment models, Mathematical modelling of geometrical problems through ordinary differential equations Unit-II Mathematical modelling in population dynamics, Mathematical modelling of epidemics through systems of ordinary differential equations of first order, compartment models through systems of ordinary differential equations, Mathematical modelling in economics through systems of ordinary differential equations of first order. Unit-III Mathematical models in medicine, arms race, battles and international trade in terms of systems of ordinary differential equations, Mathematical modelling of planetary motions, Mathematical modeling of circular motion and motion of satellites, mathematical modelling through linear differential equations of second order. Unit-IV Situation giving rise to partial differential equations models, mass balance equations: First method of getting PDE models, momentum balance equations. The second method of obtaining partial differential models, variational principles, third function, fourth method of obtaining partial differential equation models, models for traffic flow of a highway. Situation that can be modelled through graphs, mathematical models in terms of directed graphs, optimization principles and techniques, Mathematical modeling through calculus of variations. Suggested Readings :

1. J.N. Kapur-Mathematical Modelling, Chapters: 1(1.1 and 1.2), 2(2.1 to 2.4, 2.6), 3(3.1 to 3.5), 4(4.1 to 4.3),6 (6.1 to 6.6), 7(7.1 to 7.2), 9(9.1 and 9.2).

Mid Sem. - 20 Term End - 80

29

Semester – VI MATHEMATICS

DSE-III DIFFERENTIAL GEOMETRY

Full Marks : 100 Unit-I Theory of Space Curves: Space curves, Planer curves, Curvature, torsion and Serret-Frenet formulae. Osculating circles, Osculating circles and spheres. Existence of space curves. Evolutes and involutes of curves. Unit-II Oscuiating circles, Osculating circles and spheres. Existence of space curves. Evolutes and involutes of curves. Unit-III Developables: Developable associated with space curves and curveson surfaces, Minimal surfaces. Unit-IV Theory of Surfaces: Parametric curves on surfaces. Direction coefficients. First and second Fundamental forms. Principal and Gaussian curvatures. Lines of curvature, Eulers theorem. Rodrigues formula, Conjugate and Asymptotic lines. Suggested Readings : 1. C.E. Weatherburn, Differential Geometry of Three Dimensions, Cambridge University Press 2003. Chapters:1(1-4, 7,8.10), 2(13,14,16,17),3,4(29-31,35,37,38). Books for References 1. T.J. Willmore, An Introduction to Differential Geometry, Dover Publications, 2012. 2. S. Lang, Fundamentals of Differential Geometry, Springer, 1999. 3. B. O'Neill, Elementary Differential Geometry, 2nd Ed., Academic Press, 2006. 4. A.N. Pressley-Elementary Differential Geometry, Springer. 5. B.P. Acharya and R.N. Das-Fundamentals of Differential Geometry, Kalyani Publishers, Ludhiana. New Delhi.

Mid Sem. - 20 Term End - 80

30

Semester – VI MATHEMATICS

DSE- IV

PROJECT Full Marks : 100

Dissertation 60 Marks Presentation 20 Marks Viva 20 Marks

31

GENERIC ELECTIVE A Student has to choose two Generic Elective (G.E.) Subjects

other than the Core Course

32

CHEMISTRY PAPER – I

CHEMISTRY – I Theory : 60 Lecturers

Full Marks : 100

Unit - I (A) Atomic Structure: Bohr's Theory, Sommerfelds Model of atom, de-broglie's concept of matter wave. Helsenbergs uncertainity principle. Concept of orbital. Shape of S,P,D & F orbital. Quantam number and its significance. Rules for filling electron in different orbitals electronic configuration of atoms. Stability of half filled and fully filled subshell. Concept of exchange energy. Relative energy of atomic. Unit - I (B) Modern perodic table: Classification of elements into S,P,D & F blocks. Atomic Size (Covalent radii, ionic radii and vander waals radii) lonisation enthalpy, elecron affinity and electronegativity as periodic properties.

(15 Lecturers) Unit - II (A) Ionic bonding, energy consideration, lattice energy and solvation energy. Born Haber cycle and estimation of lattice energy. Valence bond approach, Hybridisation of atomic orbitals SP, SP2 & SP3 hybridisation. VSEPR theory to predict shape of cavalent molecules. Molecular orbital theory, Bonding and antibonding MO's application to simple diatomic molecules of H,O,N & F. Unit - II (B) Co-ordination chemistry : Isomerism, EAN Rule, Werner's theory, Valence bond approach and crystal field theory for octahedral and tetrahedral complex. (15 Lecturers) Unit - III (A) Kinetic theory of gases, expression for pressure, Derivation of gas laws (Boy)'s law. Charl's law, Avogadro's law, Dalton's law of partial pressure and grahm's of diffusion). Maxwell Boltzman distribution of molecular speed root mean square speed and most probable speed semple calculations. Unit - III (B) First law of thermodynamics enthalpy and enthalpy change, laws of thermochemistry, semple calculations second law of thermodynamics and entropy. Concept of free energy, Gibbs Helmholtz equation and its significance.

(15 Lecturers) Unit - IV (A) Reversible reaction, chemical equilibrium. Equilibrium constant, Le-chatliers principle & its applications in haber's method of synthesis of ammonia. Ionic equilibrium. Theories of acid and base, pH scale, common ion effect. solubility product. Buffer solution, Hydrolysis of salt. Unit - IV (B) Phase equilibra : Phase, degrees of freedom and components. Phase rule. Phase diagram of water and sulphur system. Distribution law, Nernst's distribution law and application to solvent extraction. Dilute solution and colligative properties. Lawering of vapaur pressure, Elevation of boiling point, Depression of freezing point and osmotic pressure as colligative properties. Abnormal Colligative properties and vant Hoff factor.

(15 Lecturers) PRACTICAL Generic Elective - I 1. Estimation of sodium corbonate and sodium hydrogen carbonate present in a mixture. 2. Estimation of oxalic acid by titrating with KmnO4

3. Estimation of fe(II) ions by titrating with K2Cr2O7

4. Estimation of Cu(II)ion iodometrically using Na2S2O3

Mid Sem. - 15 Term End - 60 Prctical - 25

33

CHEMISTRY PAPER – II

CHEMISTRY-II Theory : 60 Lecturers

Full Marks : 100 Unit-I Structure and Bonding in Organic molecules: Concept of hybridization and application to explain structure and bonding in organic molecules. Distribution of electrons in organic molecules, Inductive Effect, Resonance and Hyper conjugation. Organic Reaction Mechanism. Homolysis and heterolysis. Generation ,Structure, Stability and Reactions of Carbocation, Carbanion, and Carbon free radicals. Attacking agents: Electrophile, Nucleophile and Free radicals. SN

1, SN2,E1,E2AdE and, Ad N reactions with suitable examples. (15 Lecturers)

Unit-II Aliphatic hydrocarbons: Preperation of alkanes, alkenes and alkynes.Saytzeff and Hoffman elimination. Maronokoffs addition. Cis- trans isomerism in alkenes, Acidic properties of acetylene. Cycloalkanes. Preperation and Properties. Relative stability of cycloalkanes and Bayers strain theory. Preperation of Cycloalkenes. Aromaticity. Structure and bonding in benzene. Electrophilic substitution reaction in benzene.

(15 Lecturers) Unit-III Alkylhalides ,preparation and properties .Chlorination of Benzene.lnertness of halogen atom of Chlorobenzene towards SNreaction. Preperation and properties of alcohol. Distinction between primery secondary and tertiary alcohol. Preperation of Phenol and acidic character. Reimer Tiemann Reaction. Preperation and properties of aliphatic and aromatic carbonyl compounds. Benzoin condensation, Aldol condensation, and Cannizzaro reaction.

(15 Lecturers) Unit-IV Carboxylic acid: Preparation and properties(Aliphatic and Aromatic) .Benzoic acid . Effect of substitution on acid strength. Active methylene compounds. Preperation and synthetic applications of Aceto acetic ester. Amines. Preperation and properties (Aliphatic and aromatic ). Basic properties of different types of amines. Seperation of Primery , secondary and tertiary amines. Diazonium salt . Preparation and synthetic applications.

(15 Lecturers) GENERICELECTIVECHEMISTRY-II(LAB) Qualitative analysis of organic compounds: Test for Unsaturation, Aromatic character, Element Detection and functional group and determination of melting point / boiling point. Suggested Readings : • Garland, C. W.; Nibler, J. W. & Shoemaker, D. P. Experiments in Physical Chemistry 8th Ed.; McGraw-Hill: New York (2003). • Halpern, A. M. & McBane, G. C. Experimental Physical Chemistry 3rd Ed.; W.H. Freeman & Co.: New York (2003).

Mid Sem. - 15 Term End - 60 Prctical - 25

34

MATHEMATICS PAPER – I

CALCULUS AND ORDINARY DIFFERENTIAL EQUATIONS Full Marks : 100

Unit-I Curvature, Asymptotes, Tracing of Curves (Cartenary, Cycloid, Folium of Descartes, Astroid, Limacon, Cissoid & loops), Rectification, Quardrature, Volume and Surface area of solids of revolution. Unit-II Explicit and Implicit functions, Limit and Continuity of functions of several variables, Partial derivatives, Partial derivatives of higher orders, Homogeneous functions, Change of variables, Mean value theorem, Taylors theorem and Maclaurins theorem for functions of two variables. Maxima and Minima of functions of two and three variables, Implicit functions, Lagranges multipliers. Multiple integrals. Unit-III Ordinary Differential Equations of Ist order and Ist degree (Variables separable, homogenous, exact and linear). Equations of 1st order but higher degree. Unit-IV Second order linear equations with constant coefficients, homogeneous forms, Second order equations with ariable coefficients, Variation of parameters. Laplace transforms and its applications to solutions of differential equations. Suggested Readings : 1. Shantinarayan-Text Book of Calculus, Part-II, S. Chand and Co., Chapter-8 (Art. 24,25,26) 2. Shantinarayan-Text Book of Calculus, Part-III, S. Chand and Co., Chapter-l (Art 1,2),3, 4(Art. 10 to 12 ommitting Simpsons Rule), 5(Art-13) and 6(Art-15). 3. Santosh K. Sengar-Advanced Calculus, Chapters: 2,4,5,6, 7, II,12,13. 4. J. Sinharoy and S. Padhy-A Course of Ordinary and Partial Differential Equations, Kalyani Publishers. Chapters: 2(2.1 to 2.7).3,4(4.1 to 4.7),5,9 (9.1, 9.2, 9.3, 9.4, 9.5, 9.10, 9.11, 9.13). Books for References: 1. Shanti Narayan and P.K. Mittal-Analytical Solid Geometry, S. Chand & Company Pvt Ltd., New Delhi. 2. David V. Weider-Advanced Calculus, Dover Publications. 3. Martin Braun-Differential Equations and their Applications-Martin Braun, Springer International. 4. M.D. Raisinghania-Advanced Differential Equations, S. Chand & Company ltd., New Delhi G. Dennis Zill-A First Course In Differential Equations with Modelling Applications, Cengage Learning India Pvt. Ltd.

Mid Sem. - 20 Term End - 80

35

MATHEMATICS PAPER – II

LINEAR ALGEBRA AND ADVANCED ALGEBRA Full Marks : 100

Unit-I Vector space, Subspace, Span of a set, Linear dependence and Independence, Dimensions and Basis. Near transformations, Range, Kernel, Rank, Nullity, Inverse of a linear map, Rank-Nullity theorem. Unit-II Matrices and linear maps, Rank and Nullity of a matrix, Transpose of a matrix, Types of matrices. Elementary row operations, System of linear equations, Matrix inversion using row operations, Determinant .d Rank of matrices, Eigen values, Eigen vectors, Quadratic forms. Unit-III Group Theory: Definition and examples, Subgroups, Normal subgroups, Cyclic groups, Cosets, Quotient groups, Permutation groups, Homomorphism. Unit-IV Ring Theory: Definition and examples, Some special classes of Rings, Ideals, Quotient rings, Ring homomorphism. Isomorphism theorems. Zero divisions integral domain. Suggested Readings : 1. V Krishnarnurty, V I) Maima, J L Arora-An introduction to Linear Algebra, Affiliated East-West Pless Pvt. Ltd, New Delhi, Chapters: 3, 4(41 to 47), 5(except 53).6(61,62,6.5,66,68), 7(74 only) 2. I.N. Henstrain-Topics in Algebra. Prentice Hall of India Pvt. Ltd, New Delhi. Chapters 13, 14, 15, 16, 17, 18, 19,20 Books for Rofercnces: 1 S Kumaresan-Linear Algebra: A Geometric Approach, Prentice Hall of India. 2 Rao and Bhimasankaran-Linear Algebra, Hindustan Publishing House. 3. S Singh-Linear Algebra, Vikas Publishing House Pvt. Ltd, New Delhi 4. Gilbert Strang-Linear Algebra & its Applications, Cengage Learning lndia Pvt. Ltd 5 I.N. Herstein-Topics in Algebra. Wiley Eastern Pvt Ltd 6 Gallian-Conteruporary Abstract Algebra, Narosa publishing House 7. Artill-Algebra. Prentice Hall of India. 8 V.K. Khanna and 5 K. Bhambri-A Course in Abstract Algebra, Vikas Publishing House Pvt. Ltd, New Delhi

Mid Sem. - 20 Term End - 80

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PHYSICS PAPER – I

MECHANICS AND ELECTROMAGNETISM Full Marks : 100

UNIT-I: Gravitation, Rotational motion and Oscillation Rotational Motion: Angular velocity and angular momentum. Torque.Conservationof angular momentum. Gravitation: Newton's Law of Gravitation. Motion of a particle in a central forcefield (motion is in a plane, angular momentum is conserved, areal velocity isconstant). Kepler's Laws (statement only).Satellite in circular orbit and applications. Geosynchronous orbits. Basic idea of global positioning system (GPS).Weightlessness.Oscillations: Simple harmonic motion. Differential equation of SHM and itssolutions. Kinetic and Potential Energy, Total Energy and their time averages. UNIT-II: Elasticity and Special Theory of Relativity Elasticity: Hooke's law - Stress-strain diagram, Elastic moduli, Relation between elastic constants - Poisson's Ratio-Expression for Poisson's ratio in terms of elastic constants - Work done in stretching and work done in twisting a wire, Twisting couple on a cylinder ,Determination of Rigidity modulus by static torsion, Torsional pendulum ,Determination of Rigidity modulus and moment of inertia by Searle's method. Special Theory of Relativity: Constancy of speed of light. Postulates of Special Theory of Relativity. Lorentz Transformation, Length contraction .Time dilation. UNIT-III: Vector analysis and Electrostatics Vector Analysis: Review of vector algebra (Scalar and Vector product), gradient,divergence, Curl and their significance, ector Integration, Line, surface and volumeintegrals of Vector fields, Gauss-divergence theorem and Stoke's theorem of vectors(statement only).Electrostatics: Electrostatic Field, electric flux, Gauss's theorem of electrostatics.Applications of Gauss theorem- Electric field due to point charge, infinite line ofcharge, plane charged sheet,charged conductor. Electric potential as line integral of electric field, potential due t~a point charge, electric dipole, uniformly charged spherical shell and solid sphere. Calculation of electric field from potential.Capacitance of an isolateds sphericalconductor. Parallel plate, spherical and cylindrical condenser with and without dielectric. Energy per unit volumejn electrostatic field. Dielectric medium, Polarisation, Displacement vector.Gauss'stheorem in dielectrics. UNIT-IV: Magnetism and Electromagnetic Induction: Biot-Savart's law and its applications- straight conductor, circularcoil, solenoid carrying current. Divergence and curl of magnetic field.Magneticvector potential.Ampere's circuital law. Application to straight wire and solenoids, Magnetic properties of materials: Magnetic intensity, magnetic induction; Permeability, magnetic susceptibility and relation between them. Electromagnetic Induction: Faraday's laws of electromagnetic induction, Lenz'slaw, self and mutual inductance, L of single coil, M of two coils. Energy stored il!.ffiagnetic field. Suggested Readings : 1. University Physics. F.W. Sears, M.W. Zemansky and H.D. Young, 13/e, 1986.Addison-esley 2. Mechanics Berkeley Physics, v.1: Charles Kittel, et. al. 2007, Tata McGraw-Hill. 3. Physics - Resnick, Halliday& Walker 9/e, 2010, Wiley 4. Engineering Mechanics, Basudeb Bhattacharya, 2 ndedn., 2015, OxfordUniversity Press. 5. Electricity and Magnetism, Edward M. Purcell, 1986, McGraw-Hill Education 6.Electricity & Magnetism, J.H. Fewkes&J.Yarwood.Vol. 1,1991, Oxford Univ. Press 7.Electricity and Magnetism, D C Tayal, 1988, Himalaya Publishing House. University Physics, Ronald Lane Reese, 2003, Thomson Brooks/Cole. 9.DJ .Griffiths, Introduction to Electrodynamics, 3rd Edn, 1998, Benjamin Cummings. PRACTICAL MECHANICS and ELECTROMAGNETISM 1. Measurements oflength (or diameter) using vernier caliper, screw gauge andtravelling microscope. 2. To determine the Height of a Building using a Sextant. 3. To determine the Elastic Constants of a Wire by Searle's method. 4. To determine g by Bar Pendulum. 5. To determine g by Kater's Pendulum 6.To use a Multimeter for measuring (a) Resistances, (b) AC and DC Voltages,(c) DC Current, and (d) checking electrical fuses.7.To determine a Low Resistance by Carey Foster's Bridge. Reference Books: 1. Advanced Practical Physics for students, B.L. Flint and H.T. Worsnop, 1971,Asia Publishing House.

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2. Advanced level Physics Practicals, Michael Nelson and Jon M. Ogborn, 4 thEdition, reprinted 1985, Heinemann Educational Publishers. 3.Engineering Practical Physics, S.Panigrahi& B.Mallick,201S, CengageLearning India Pvt. Ltd. 4. A Text Book of Practical Physics, InduPrakash and Ramakrishna, 11 thEdition, 2011, KitabMahal, ew Delhi.

Mid Sem. - 15 Term End - 60 Prctical - 25

38

PHYSICS PAPER – II

MODERN PHYSICS AND OPTICS Full Marks : 100

(Credits: Theory-04, Practicals-02) UNIT-I: Thermodynamics and thermodynamic potentials Laws of Thermodynamics: Thermodynamic Description of system: Zeroth Law of thermodynamics and temperature. First law and internal energy, conversion of heat into work, Various Thermo dynamical Processes, Applications of First Law: General Relation between C, and Cv, Work Done during Isothermal and Adiabatic Processes Compressibility and Expansion Coefficient, Reversible and irreversible processes, Second law and Entropy, Carnot's cycle & theorem, Entropy changes in reversible & irreversible processes, Entropy-temperature diagrams, Third law of thermodynamics, Thermodynamical Potentials: Enthalpy, Gibbs, Helmholtz and Internal Energy functions, Maxwell's relations and applications - Joule-Thompson Effect, Clausius- Clapeyron Equation, Expression for (C, - C v), C p ICv , TdS equations. UNIT-II: Wave optics, interference, diffraction and polarization Wave Optics: Electromagnetic nature of light. Definition and Properties of wave front. Huygens Principle. Interference: Division of amplitude and division of wavefront. Young's Double Slit experiment. Lloyd's Mirror and Fresnel's Biprism.: Stokes' treatment Phase change on reflection. Interference in Thin Films: parallel and wedge-shaped films. Fringes of equal inclination (Haidinger Fringes); Fringes of equal thickness (Fizeau Fringes). Newton's Rings: measurement of wavelength and refractive index Diffraction: Fraunhofer diffraction- Single slit; Double Slit. Multiple slits and Diffraction grating. Fresnel Diffraction: Half-period zones. Zone plate. Fresnel Diffraction pattern of a straight edge, a single slit and a wire using half-period zone analysis. Polarization: Transverse nature oflight waves. Plane polarized light - production and analysis. Circular and elliptical polarization. UNIT-III: Modern physics Planck's quantum theory, Planck's constant and light as a collection of photons; Photo- electric effect and Compton scattering. De Broglie wavelength and matter waves; Davisson-Germer experiment. Problems with Rutherford model- instability of atoms and observation of discrete atomic spectra; Bohr's quantization rule and atomic stability; calculation of energy levels for hydrogen like atoms and their spectra. Wave-particle duality, Heisenberg uncertainty principlegamma ray microscope; Estimating minimum energy of a confined particle using uncertainty principle; Energy-time uncertainty principle. Two slit interference experiment with photons, atoms & particles; linear superposition principle as a consequence; Matter waves and wave amplitude. UNIT-IV: Solid state physics Crystal Structure of Solids: Amorphous and Crystalline Materials. Lattice Translation Vectors. Lattice with a Basis - Central and Non-Central Elements. Unit Cell. Miller Indices. Reciprocal Lattice. Types of Lattices. Brillouin Zones. Diffraction of X-rays by Crystals. Bragg's Law. Elementary band theory:Band Gaps. Conductors, Semiconductors and insulators. P and N type Semiconductors. Conductivity of Semiconductors, mobility, Hall Effect, Hall coefficient. LAB -II: MODERN PHYSICS and OPTICS (Credits: Practicals-02) 1. To determine the Frequency of an Electrically Maintained Tuning Fork byMelde's Experiment 2. To determine the Refractive Index of the Material ofa Prism using Sodium Light. 3. To determine Dispersive Power of the Material of a Prism using Mercury Light. 4. To determine the value of Cauchy Constants. 5. To determine the Resolving Power of a Prism. 6. To determine wavelength of sodium light using Fresnel Biprism. 7. To determine wavelength of sodium light using Newton's Rings. 8. To determine Mechanical Equivalent of Heat. Suggested Readings : 1. Thermal Physics, S. Garg, R. Bansal and C. Ghosh, 1993, Tata McGraw-Hill. 2. Heat and Thermodynamics, M.W.Zemasky and R. Dittman, 1981, McGraw Hill. 3. Thermodynamics, Kinetic theory & Statistical thermodynamics, F.W.Sears and G.L. Salinger. 1988, Narosa. 4. Thermal Physics, A. Kumar and S.P. Taneja, 2014, R. chand Publications. 5.Fundamentals of Optics, F.A Jenkins and H.E White, 1976, McGraw-Hill 6. Principles of Optics, B.K. Mathur, 1995, Gopal Printing

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7. Fundamentals of Optics, H.R. Gulati and D.R. Khanna, 1991, R. Chand Publications 8. University Physics. F.W. Sears, M.W. Zemansky and H.D. Young. 13/e, 1986.Addison-Wesley 9. Concepts of Modem Physics, Arthur Beiser, 2009, McGraw-Hill 10. Introduction to Solid State Physics, Charles Kittel, 8 th Ed., 2004, Wiley India Pvt. Ltd. 11. Elements of Solid State Physics, J.P. Srivastava, 2 nd Ed., 2006, Prentice-Hall oflndia 12.0ptics, B.S.Agarwal, Kedamath-Ramnath publications

Mid Sem. - 15 Term End - 60 Prctical - 25