courses of studies for three- years degree …bhadrakcollege.nic.in/math-science-syllabus.pdf ·...

BHADRAK AUTONOMOUS COLLEGE

BHADRAK- 756100, ODISHA

COURSES OF STUDIES

FOR

THREE- YEARS DEGREE COURSE

IN

SCIENCE First Semester Examination - 2017

Second Semester Examination- 2018

Third Semester Examination- 2018

Fourth Semester Examination - 2019

Fifth Semester Examination - 2019

Six Semester Examination- 2020

SYLLABUS STRUCTURE FOR B.Sc. ( CORE COURSE)

FIRST SEMESTER (WITH PRACTICAL)

SUBJECT PAPER FULL MARK CREDIT MID SEMESTER

END SEMESTER TH PR TH PR TH PR

AECC(ES) 50 2 10 40

CORE COURSE

C.C-1 C.C-1 60 25 5 1 15 60

C.C-2 C.C-2 60 25 5 1 15 60

GENERIC ELECTIVE

G.E-1 G.E-1 60 25 5 1 15 60

TOTAL MARKS

230 75 17 3 55 220

SECOND SEMESTER (WITH PRACTICAL)


END SEMESTER TH PR TH PR TH PR

AECC(ES) (Eng/od/Hn/ur

50 2 10 40

CORE COURSE C.C-3 C.C-3 60 25 5 1 15 60

C.C-4 C.C-4 60 25 5 1 15 60

GENERIC ELECTIVE

G.E-2 G.E-2 60 25 5 1 15 60

TOTAL MARKS 230 75 17 3 55 220

THIRD SEMESTER (WITH PRACTICAL)


MID SEMESTER TH PR TH PR TH PR

CORE COURSE

C.C-5 C.C-5 60 25 5 1 15 60 C.C-6 C.C-6 60 25 5 1 15 60

C.C-7 C.C-7 60 25 5 1 15 60

SEC P-1 50 2 10 40

GENERIC ELECTIVE

G.E-3 G.E-3 60 25 5 1 15 60

TOTAL MARKS

290 100 22 4 70 280

FOURTH SEMESTER (WITH PRACTICAL)



CORE COURSE

C.C-8 C.C-8 50 25 5 1 15 60

C.C-9 C.C-9 60 25 5 1 15 60

C.C-10

C.C-10

60 25 5 1 15 60

SEC P-2 50 2 10 40

GENERIC ELECTIVE

G.E-4 G.E-4 60 25 5 1 15 60

TOTAL MARKS

290 100 22 4 70 280

FIFTH SEMESTER (WITH PRACTICAL)



CORE COURSE

C.C-11

C.C-11

60 25 5 1 15 60

C.C-10

C.C-10

60 25 5 1 15 60

DSE DSE-1 DSE-1 60 25 5 1 15 60

DSE-2 DSE-2 60 25 5 1 15 60 TOTAL MARKS

240 100 22 4 60 240

SIXTH SEMESTER (WITH PRACTICAL)



CORE COURSE

C.C-13

C.C-13

60 25 5 1 15 60

C.C-14

C.C-14

60 25 5 1 15 60

DSE DSE-3 DSE-3 60 25 5 1 15 60

DSE-4 100 6 100

TOTAL MARKS

180 75 15 9 45 280

ENVIRONMENT STUDIES

SEMESTER-1

Ability Enhancement Compulsory Course (AECC)

(2 CREDIT) F.M-50(40+10)

Question must be set from all units with alternatives and each question will be both long

and short answer type

UNIT-1 Introduction to environmental studies and ecosystem

Scope and importance of environmental studies.

What is ecosystem? Structure and function of ecosystem;

Energy flow in an ecosystem; Food webs and ecological succession study of

the ecosystem (forest ecosystem, pond ecosystem)

UNIT-2 Natural Resources: Renewable and Non-Renewable Resources.

Land resources and land use change; Lavel degradation and soil erosion.

Reforestation: causes and impacts due to mining, dam building on

Environment. Forests, biodiversity and tribal population.

Water use and over –exploitation of surface and grand water, draughts.

Energy recourses: Renewable and non-renewable energy sources, use of alternate energy source.

UNIT-3 Biodiversity conservation and Environmental issues.

Level of biology diversity: genetic, species and ecosystem diversity; Bio

geographic zones of India; Biodiversity patterns and global biodiversity hot

sports.

India as a Mega-biodiversity nation; Endangered and endemic species of

India.

Threats to biodiversity. Habital-loss, poaching of wildlife, Man wild life

conflicts; conservation of biodiversity: In sity and Ex-sity conservation of

biodiversity.

Human population growth: Impact on environment, disaster

management(Food , Cyclone, Earthquake) Environmental Movements

(Chipko, Silent-valley, Bis-nois of Rajasthan)

UNIT-4 Environmental pollution and policies.

Environmental pollution: Traps, causes, effects and controls; Air, water, soil

and noise pollution.

Solid waste management control measures of urban and industrial waste.

Climate change, global warming, ozone layer depletion acid main and impacts

on human communities and agriculture.

Environmental laws: Environment pollution Act; Air (prevention and control

of pollution) Act; Forest conservation Act.

Suggested Readings:-

1. Sharma, P.D Ecology & Environmental Biology

2. Dash, M.C. Fundamental of Ecology

3. Rana, S.V.S Environmental Studies

Semester-II

Paper-2. 1: Ability Enhancement Compulsory Course (AECC)

(In English/Odia /Hindi/Urdu)

Duration -2 hour, Mid sem-10, End Sem-49, F.M-50

Question must be set from all units with alternatives and each question will be both long


English

This course at enhancing the English Language proficiency of undergraduate student in

Human and preparing them for the academic, social and professional expectation during

and after the course. The course will help develop academic and social English

competencies in speaking, listening, reading, writing, grammar and usage.

The course will have 2 credits (50 marks), Mid-Sem (Internal Assessment)-10 Marks at End

Sem -40 Marks. These shall be 3 units.

Unit-I : Reading skills, summary, paraphrasing. Analysis, interpretation, knowledge, literary

texts. Candidates shall have to answer 3 questions carrying 5 marks each from the

prescribed text.

Text prescribed: Forms of English Prose (OUP)

Pieces to be studied: 1. The Lament – Anton Chekov

2. The Umbrella- G.D Maupassant

3. The barber’s Trade Union –M.R. Anand

4. The Axe – R.K. Narayan

Unit2: Writing Skill- Report, making Notes, Explain an idea / paragraph/CV/Resume

information Transfer and Business Communication. The candidates shall have to answer

questions carrying 7.5 marks each.

Unit-3 Grammar and usage: Sentence ( Simple, Complex, Compound) , Clause ( Noun

Adjective, Adverb ), Phrasal verb , models, Preposition, Subject- Verb Agreement , Common

Error, Candidates shall have to answer 10 objective questions carrying 1 mark each.

ODIA


Duration -2 Hours, Mid Sem -10, End emester-40, F.M-50

େଯାଗାେଯାଗମଳକ ମାତଭାଷା – ଓଡଆ (AECC)

େଯେକୗଣଶୀ ୨ଟ ପାଠ ବାଛ

ପାଠ� – ୧ : େଯାଗାେଯାଗ ଅନବ�

୧ ମ ଏକକ : େଯାଗାେଯାଗର ଭ� ପରଭାଷା , ଅନବ� ଓ ପରସର

୨ ୟ ଏକକ : େଯାଗାେଯାଗର �କାରେଭଦ : କ�ତ, ଲ�ତ, ବ��ଗତ – ସାମାଜକ –

ସାଂ�ତକ ବ�ବସାୟୀକ- ସାହତ�କ।

୩ ୟ ଏକକ : େଯାଗାେଯାଗର ବାଧକ ଓ ସଫଳ ସାଧନାର ଦଗ ।

୪ ଥ� ଏକକ : େଯାଗାେଯାଗର ସାହତ�ର ଭମକା ।

୫ ମ ଏକକ : ସରଳ େଯାଗାେଯାଗର ଭାଷା ।

Hindi


Duration-2 Hours, Mid Sem-10, End Sem-40, F.M-50

Unit-1

Efnvoer Yee<ee mebcHe£es<eCe

(1) Yee<ee keÀes HeefjYee<ee, He£ke=Àefle Syeb efJeefJeOe ©He ( ceewefKekeÀ, efueefKele,je<ì£Yee<ee,jepeYee<ee )

Unit-2

(2) Efnvoer keÀer JeCe& y³eJemLee-mJej Syeb y³ebpeve-He£keÀejYeso mJej- (nmye,oerIe&,Deewj meb³eg

Yîebpeve- (mye<e&,DeblemLe, G<ce, DeuHeHe^eCe, Iees<e leLee DeIees<e)

Unit-3

Efnvoer yîekeÀjCe Deewj mebcHe^s<eCe

1. He³ee&îeJee®eer Meyo, efyeueesce, DeveskeÀ MeyoeW kesÀ efueS SkeÀ Meyo

Unit-4

(1.)Meyo Megefo, cegneyejW Dewj ueeskeÀesefkele³eBe

Unit-5

Efnefvo mebcHe<eCe keÀer DeJeOeejCee Deewj cenlJe

Mark Distribution:

Unit-1 mes mid Sem Exam. kesÀ meJeeue –(10 Marks)

Unit-2, 3, 4, 5& 6 mes –End Sem Exam 08 x 5=(40 Marks)

Book for Referance:

(1) DeeOegefvekeÀ efnvoer y³ekeÀjCe Deewj j®evee- yemegosJe vebove He^meeo

(2) He^³esefiekeÀ efnvoer – [e.ieg.ce. Keeved –meyeveced yegkedÀ <ìerj, keÀìkeÀ

(3) He^³eespeve cetuekeÀ efnvoer – kewÀueeme ®ebo Yeeefì³ee

URDU

Internal Assessment: Time 1 Hr. Full Marks-10

Semester Assessment: Time-2 Hr. Full Marks-40

Unit-1 PROSE (12)

1. Mujhe mera Dost se Bachao- Sajjad Haider

2. Chema ka Ishq – Pitras Bukhari

3. Haj-e-Akbar-Prem Chand

4. Aakhri Qudam –Zakir Hussain

There shall be one long wuestion with alternative carrying 12 marks.

Unit-2: POETRY (12)

1. Naya Shewala- Mohmmad Iqbal

2. Aasmi Nama-Nazir Akbar Aabadi

3. Kashmir- Durga Sahy Suroor

4. Nasha-re-Ummid – Altaf Hussain Hali

There shall be one long question with alternative carrying 12 marks.

Unit-3: GRAMMAR (8)

Ism, Sifat, Fail, Wahid-o-Jama, Mutazad Alfaz, Mutashabeh Alfaz, Tazkeer-o-

Tanees.

There shall be one question with alternative carrying 8 marks.

Unit-4: RHETORIC (8)

Tashbeeh, Istear, Kenaya, Majaz-e-Mursal, Tazad, ham, Maratun Nazir.

There shall be one question with alternative carrying 8 marks.

SUGGESTED READING:

1. URDU ZABAN-O-QAWAID-PART (I)- SHAFA AHMED SIDDIQI

2. IL MUL BALAGHAT – ABDUL MAJID

SEMESTER-III

SEC-I

Question must be set from all units with alternatives and each question will be both long and short

answer type

Duration- 2 Hrs, Mid Sem -10, End Sem-40, Total Marks-50

Communicative English

There shall be one paper in communicative English of skill enhancement course

of Arts/Science students of +3 2nd Yr. 3rd Semester carrying 40 marks and will

be of 2 hours duration.

Paper-I: Skill Enhancement Course of Arts/Science and Commerce students

This course aims at enhancing the English Language of Arts/ Science /

Commerce proficiency of undergraduate students of ARTS, SCIENCE and

COMMERCE in humanity and preparing them for the academic, social and

professional expectations during and after the course. The course will help to

enhance communicative skill and social English competencies in speaking ,

listening, reading, writing, Grammar ad Usage.

The course will have 2 credits (50 Marks)

Mid semester-10 marks Time, 1 Hour End Sem 40 marks Time 2 hours (there

Shall be 3 units)

Candidates shall have to attempt one long answer type question carrying 4

marks from each until. Alternative questions will be set (from each) against

each question.

Unit-1 Communication: The concept, purpose of communication,

Types of Communication, Verbal Communication,

Non-verbal Communication, Non-verbal Communication: Body

Language

Business Communication, Barriers to communication,

Overcoming communication Barriers

How to sender can overcome communication barriers

How to receive can overcome communication barriers.

Developing effective messages

UNIT-II How can we make communication effect?

Listening

Clarity and Brevity of ideas

The “you” Attitude

Simple and plain English, positive attitude and Bias free language

Computer- Mediated Communication (CMC)

UNIT-III A. How we speak English: The Respiratory system

The Phonatory system, The Articulator System,

International Phonetic Alphabet (IPA), Transcription

Vowels of English, Consonants of English, Varieties of English,

Standard English, American English, Indian English,

Word Stress: Functions of Word stress in English, Intonation

B. Grammar: Aid to communication

Time and Tense: Aspect of Event verb and state verb

Concord. Finite verb and Non-finite verbs,

Interrogatives: Open Questions, Closed Questions and : Rhetorical

Questions

Books prescribed:

Smith L.E. Readings in English as an international Language, Oxford, Pergamon

press (1983)

Banasal. R.K and J.B Harrison- Spoken English – A manual of speech and

phonetics. Madras Orient Longman 1972

Dr. Das Shruti, Contemporary Business Communication New Delhi, S.Chand

Publising, 2008.

O. Conner. J.D Better English pronunciation, 2nd ed. Cambridge, Cup, 1980.

Division of marks:

Unit-I (1) One long answer type question carrying 8 marks- 1x8=08

(2)Two short answer type Questions carrying 4 marks each-2x4=08

UNIT-II (1) One long answer type question carrying 8 marks – 1x8=08

(2)Two Short answer type questions carrying 4 marks -2x4=08

UNIT-III (1) One long answer type question carrying 8 marks -1x8=08

(2)Two short answer type questions carrying 4 marks each -

2x4=08

SEMESTER 1

CORE COURSE – 1

CALCULUS - I

Duration: 3hrs, Mid sem – 15, End Sem – 60, Prac.- 25, Total Mark- 100

Questions must be set from all units with alternatives and each question will be both long


UNIT I

Differential calculus: Reduction formulae,∫ sin ��, ∫ cos ��,∫(log �)n��

∫sinnx��,∫cosnx��, etc. concavity, convexity; Asymptotes, Curve tracing, Curvature of

Cartesian polar and parametric curves.

UNIT II

Integral calculus: Area of plane curves, Volume of curves by axes rotation, Surface area and

rectification (length of) of curves in Cartesian, Polar and parametric curves.

UNIT III

Analytic geometry: Sphere, Cone, Cylinder, Central conicoids.

UNIT IV

Vector Calculus: Triple product, Introduction to vector function, Operation with vector

valued function, Limit and continuity of vector function, Differential and integration of

vector function, Tangent and normal components of acceleration, Divergence curl gradient.

Books Recommended:

1. Text book of calculus, Part II- Shanti Narayan, S. Chand and Co. CH-7,10(Art 33 – Art

38)

2. Text book of calculus, Part III- Shanti Narayan, S. Chand and Co. CH- 1, 3, 4, 5, 6

3. Analytical geometry of quadratic surfaces, B.P. Acharya and D.C. Sahoo, Kalyani

publishers, New Delhi Ludhiana. CH- 2, 3, 4(4.1- )

4. Advanced Higher Calculus (Vidyapuri), G. Samal and others. CH- 20 (20.1-

20.13)(Vector Analysis)

Books for reference:

1. Advanced Higher Calculus (Vidyapuri), G. Samal and others.

2. BSc. Mathematics, Kalayani Publisher, Calculus- 1 (Dhirendra Kumar Dalai)

PART II (PRACTICAL MARKS: 25)

List of Practicals (Using any software)

Practical/lab work to be performed on a computer

1. Plotting the graphs of the function ��, log(�� + �),

1 (�� + �),⁄ sin(�� +�),cos(�� + �), |�� + �| and to illustrate the effect of a and b

on the graph.

2. Sketching parametric curves (eg. Trochoid, Cycloid, Epicycloid, Hypocycloid).

3. Obtaining surface of a revolution of curves.

4. Tracing of conics in Cartesian coordinates/ polar coordinates.

5. Matrix operation (addition, multiplication, inverse, transpose)

BOOKS FOR REFERNCE:

1. Degree practical Mathematics by Dhirendra Kumar Dalai and others, Kalyani

Publishers CH- 3

CORE COURSE 2

ALGEBRA 1

Duration: 3Hrs. MidSem- 20, End Sem-80, Total Mark- 100

UNIT I

Polar representation of complex number, n-th roots of unity, De-Moivers theorem for

rational indices and its applications.

UNIT II

Equivalence relations, Functions, Composition of functions, Invertible function, One to one

correspondence and cardinality of a set, Well ordering property of positive integers,

Division algorithm, Divisibility and Euclidian Algorithm, Congruence relation between

integers, Principles of mathematical induction, Statement of fundamental theorem of

arithmetic.

UNIT III

System of linear equation, Row reduction and Echelon forms, Vector equations, The matrix

equation Ax=b, Solution sets of linear systems, Application of linear systems, Linear

independence.

UNIT IV

Introduction to linear transformations, Matrix of linear transformation, inverse of a matrix,

Characterisation of invertible matrices, Subspaces of Rn, Dimension of subspaces of Rn and

rank of a matrix, Eigen values, Eigen vectors and characteristics equation of a matrix.

Books Recommended:

1. Algebra- 1 , D.K.Dalai, Kalyani Publishers, CH-1 (complex number)

2. (OR) TituAndereescu and DorinAndrica, Complex numbers from A to Z, Birkhauser,

2006 CH- 2.

3. Edger G. Goodaire and Michael M. Permenter, Discrete Mathematics with graph

theory 3rd Edition, Pearson education (Singapore) P. Ltd, Indian reprint, 2005, CH-2

(2.4), 3, 4 (4.1- 4.1.10, 4.2-4.2.11, 4.4 (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 edition, Pearson education Asia,

Indian reprint, 2007, CH- 1 (1.1-1.9), 2 (2.1- 2.3, 2.8, 2.9), 5 (5.1, 5.2)

Books for Reference:

1. Algebra 1 , D.K. Dalai, Kalaynai Publisher (All chapters).

SEMESTER II

CORE COURSE 3

REAL ANALYSIS (ANALYSIS -I)

Duration: 3Hrs. MidSem- 20, End Sem-80, Total Mark- 100



UNIT I

Review of Algebraic ad order properties of R, Neighbourhood of a point in R, Idea of

countable set, uncountable sets and uncountability of R, Bounded above sets, Bounded sets,

Unbounded sets, Suprema and Infima.

UNIT II

The completeness property of R, The Archimedean property, Density of rational (and

irrational) numbers in R, Interval, 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 sequnces, Monotone convergence theorem, Sub sequences, Divergence criteria,

Monotone sub sequence theorem ( statement only), Bolzano Weierstrass Theorem for

sequences, Cauchy sequence, Cauchys convergence criteria.

UNIT IV

Infinite series, Convergence and divergence of infinite series, Cauchy criterion, Test for

convergence: Comparison test, Limit comparison test, ratio test, Cauchys n-th root test,

Integral test, Alternating series, Leibniz test, Absolute and conditional convergence.

Books Recommended:

1. G. Das and S. Pattanayak, Fundamental of Mathematics Analysis, TMH Publishing Co.

CH- 2 (2.1-2.4, 2.5-2.7), 3 (3.1-3.5), 4 (4.1-4.7, 4.10, 4. 11,4. 12, 4.13)


1. R.G. Bartle and D.R. Sherbert, Introduction to Real Analysis, 3rd Edition John Wiley

and sons (Asia) Pvt. Ltd. Singapore 2002

2. Gerald G. Bilodeau, Paul R. Thie, G.E. Keough, An Introduction To Analysis, 2nd Edition

Jones and Bartlett, 2010.

3. Mathematical Analysis, S.C.Mallick&SabitaArora, New Age International Publication.

CORE COURSE 4

DIFFERENTIAL EQUATION

PART I

Duration: 3hrs, Mid sem – 15, End Sem – 60, Prac.- 25, Total Mark- 100



UNIT I

Differential Equations and Mathematical Models.First order and first degree ODE (variable

separable, homogeneous, exact, and linear).Equation of first order but of higher degree.

UNIT II

Second order Linear equation (homogeneous and non-homogeneous) with constant

coefficient, Second order equations with variable coefficients, Variation of parameters,

Method of undetermined coefficients, Equation reducible to Linear equation with constant

coefficient, Euler’s equation.

UNIT III

Power series solutions of second order differential equations.

UNIT IV

Laplace transforms and its application to solutions of differential equations.

Books for Refernce:

1. Degree Practical Mathematics by D.K.Dalai and others, Kalyani Publisher CH- 4.

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 problem

Books Recommended:

1. J. Sinha Roy and S. Padhy, A course of ordinary and partial differential equations,

Kalyani Publishers, New Delhi, CH- 1, 2 (2.1- 2.7), 3, 4 (4.1- 4.8), 5, 7 (7.1- 7.4.2), 9

(9.1,9.2, 9.3, 9.4, 9.5, 9.10, 9.11, 9.13)

Books for references:

1. Martin Braun, Differential Equation and their Application, Springer International

2. M.D. Raisinghania- Advanced Differential Equations, S. Chand and Co. 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 and sons, India, 2004

SEMESTER III

CORE COURSE- 5

THEORY OF REAL FUNCTIONS (ANLYSIS- II)

Duration- 3Hrs, Mid Sem- 20, End Sem- 80, Total Mark- 100

5 Lectures, 1 Tutorial (Per week per student)



UNIT I

Limits of functions (� − � approach), Sequential criterion for limits, Divergence criteria. Limit

theorems, One sided limits. Infinite limits and limit at infinity. Continuous functions,

Sequential criterion for continuity and discontinuity.

UNIT - II

Algebra of continuous functions. Continuous functions on an interval, Intermediate value

theorem, location 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 trigonometric functions.

Books Recommended:

1. G. Das and S. Pattanayak, Fundamentals Of Mathematics Analysis. THM Publishing Co.,

CH- 6 (6.1 – 6.9), 7 (7.1 – 7.7)


1. R. Bartle and D.R. Sherbert, Introduction to Real Analysis, John Wiley and Sons, 2003.

2. Mathematical Analysis by S.C.Mallick&SavitaArora New Age International Publication.

SEMESTER III

CORE COURSE- 6

GROUP THEORY (ALGEBRA- II)




UNIT I

Symmetries of a square, definition and examples of groups including permutation 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

Properties of cycle groups, Classification of subgroupsmof cyclic groups. Cyclic notations for

permutations, Properties of permutations, even and odd permutations, alternating goup,

properties of cosets, Laranges 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, Firsrt, Second and Third isomorphism theorems.

Books Recommended:

1. Joseph A. Gallian, Contemporary Abstract Algebra(4th Edition), Narosa Publishing

House,New Delhi, Part- 2 (1, 2, 3, 4, 5, 6, 7, 8, 9, 10 only)


1. John B. Fraleigh, A First course in Abstract Algebra, 7th Edition, Pearson, 2002.

2. Joseph J. Rotman, An Introduction to the Theory of Groups, 4thEdn., Springer

Verlag, 1995

3. I.N. Herstein, topics in Algebra, Wiley Eastern Limited, India, 1975.

SEMESTER III

CORE COURSE- 7

PARTIAL DIFFERENTIAL EQUATIONS AND SYSTEMS OF ORDINARY DIFFERENTIAL

EQUATIONS

PART- 1

Duration- 3 Hrs., Mid Sem- 15, End Sem- 60, Practical- 25, Total Marks- 100

04 Lectures (per week per student)



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 solutions, Pfaffian differential equations, methods of soulitons 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, Separation of variables, non- linear equations 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.

Books Recommended:

1. J. Sinha Roy and S. Padhy, A course on ordinary and Partial Differential Equations,

Kalyani Publishers, New Delhi, Ludhiana, 2012.

CH- 11, 12, 13 (13.1 – 13.5), 15 (15.1 – 15.5), 16 (16.1 – 16.7), 17 (17.1, 17.2, 17.3)


1. TynMyint-U and LokenathDebnath, Linear Partial Differential Equations for Scientists and

Engineers, 4th Edition, Springer, Indian reprint, 2006.

2. S.L. Ross, Differential Equations, 3rd Edition, John Wiley and Sons, India, 2004.

PART- II (PRACTICAL: MARKS: 25)

List of Practicals (Using any Software)

Practical/Lab work to be performed on a computer.

1. To find the general solution of the non-homogeneous system of the form: ��

��= �1� +b1�+�1(t),

��

��= �2 + b2+ �2(t)

With given conditions. 2. Plotting the integral surfaces of a given first order PDE with initial data.

3. Solution of wave equation ��

��2 – c2�

��

��2 = 0 for the following associated

conditions. (a) �(�, 0) = �(�),ut(�, 0) = �(�), � ∈ ℝ, � > 0. (b)�(�, 0) = �(�),ut(�, 0) = �(�), ux(0, �) = 0, � ∈ (0,∞), t >0 (c) �(�, 0) = �(�),ut(�, 0) = �(�), u(0, �) = 0, � ∈ (0,∞), t >0 (d) �(�, 0) = �(�),ut(�, 0) = �(�), u(1, �) = 0, 0 < � < �, t >0

4. Solution of wave equation ��

��−k2�

��

��2 = 0 for the following associated conditions:

(a) �(0, �) = �(�), �(0, �) = �, �(�, �) = �, 0 < � < �, � > 0.

(b) �(�, 0) = �(�), � ∈ ℝ, 0 < � < �.

(c) �(0, �) = �(�), �(0, �) = �, � ∈ (0,∞), � ≥ 0.


1. Degree practical Mathematics by D.K.Dalai and others, Kalyani Publishers, CH-5.

SEMESTER IV

CORE COURSE- 8

NUMERICAL METHODS

PART- I


04 Lectures (per week per student)

Questions must be set from all units with alternatives and each question will be both long and short answer type.

UNIT I

Algorithms, Convergence, Errors: Relative, Absolute, Round off, Truncation. Transcendental and Polynomial 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 method. Error bounds. Finite difference operators. Newton 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 two or four.

Books Recommended:

1. B.P. Acharya and R.N. Das, A course on Numerical Analysis, Kalyani publishers, New Delhi, Ludhiana, CH- 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).

Books for References:

1. M.K. Jain, S.R.K. Iyengar and R.K. Jain, numerical Metods for Scientific and Engineering Computation, 6th Ed., New Age International Publisher, India, 2007.

2. C.F. Gerald and P.O. Whaetley, 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.

PRAT- II(PARCTICAL: MARKS: 25)

List of Practicals (Using any Software)

Practical/ lab work is to be performed on a computer.

1. Calculate the sum 1/1 + 1/2 + 1/3 + ¼ + --------------------------- + 1/ N. 2. To find the absolute value of an integer. 3. Enter 100 integers into an array and sort them in an ascending order. 4. Bisection Method. 5. Newton Raphson Method. 6. Secant Method. 7. RegulaFalsi Method.

8. LU decomposition Method. 9. Gauss- Jacobi Method. 10. SOR Method or Gauss- Siedel Method 11. Lagrange Interpolation or 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.

SEMESTER- IV

CORE COURSE- 9

RIEMANN INTEGRATION AND SERIES OF FUNCTIONS (ANALYSIS- III)




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; fundamenta theorems of Calculus.

UNIT- II

Improper integrals; Convergence of Beta and Gamma functions.

Unit- III

Pointwise and uniform convergenceof 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 functions 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.

Books Recommended:

1. G. Das and S. Pattanayak- Fundamentals of Mathematics Analysis. TMH Publishing Co., CH- 8, 9.


1. K.A. Ross. Elementary Analysis, The Theory of Calculus, Undergraduate Texts in Mathematics, Springer (SIE), Indian reprint, 2004..

2. S.C. Malik and S. Arora, Mathematical Analysis, New Age International ltd. New Delhi.

3. Shanti Narayan and M.D. Raisinghania, Elements of Real Analysis, S. Chand & Co. Ltd. 4. Riemann integration and Series of functions, Kalayani Publisher, by C. Mallick&S.Mallick.

SEMESTER- IV

CORE COURSE- 10

RING THEORY AND LINEAR ALGEBRA- I (ALGEBRA- III)




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 and 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 spaces,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

representation of a linear transformation, algebra of linear transformations. Isomorphisms,

isomorphism theorems, invertibility and isomorphisms, change of coordinate marix.

Books Recommended:

1. Joseph A. Gallian, Contemporary Abstract Algebra (4th Ed.), Narosa Publishing

House, New Delhi. CH- 12, 13, 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. CH- 1 (1.2 – 1.6), 2 (2.1 – 2.5).


1. I.N.Herstein, Topics in Algebra Will Estern Ltd. Indian publication.

2. V. krishnamurty

SEMESTER- V

CALCULUS

CORE COURSE- 11




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, Extrema of functions

of two variables, method of Lagrange multipliers, constrained optimization problems.

UNIT- II

Homogeneous function of two variables, change of variable. Extrema of functions of two

variables, method of Lagrange multipliers, constrained optimization problems.

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 the

line integrals, conservative vector fields, independence of path. Greens theorem, surface

integrals, Stokes theorem, The Divergence theorem.

Books Recommended:

1. Advanced Higher Calculus (Vidyapuri), by Dr.GhansyamSamal& others, CH- 12, 13,

14, 15,16, 17 & 20 (20.14 – 20.23)

2. S.C.Mallick and SabitaArora, Mathematical Analysis, The New Age International

Publication, CH- 15,17,18(1-5)


1. M.J. Strauss.

SEMESTER- V

CORE COURSE- 12

PROBABILITY AND STATICS




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 distribution: uniform, binomial, Poisson, Continuous distribution: 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, 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,

statement and Central Limit theorem for independent and identically distributed random

variables with finite variance.

Books Recommended:

1. Robert V. Hogg, Joseph W. McKean and Allen T. Craig, Introduction to Mathematical; Statistics,

Pearson Education, Asia, 2007. CH- 1(1.1, 1.3, 1.4 – 1.9), 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. CH- 4, 5(5.1 – 5.5, 5.7), 6(6.2, 6.3, 6.5 – 6.7), 14(14.1, 14.2).


1. S.C. Gupta and V.K. Kapoor- Fundamental of Mathematical Statistics, S. Chand and

Company Pvt. Ltd., New Delhi.

2. PratihariMohanty- Probability and Statistics.

SEMESTER- VI

CORE COURSE- 13

METRIC SPACES AND COMPLEX ANALYSIS (ANALYSIS- IV)




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.

Homeomorphism, Contraction mappings, Banach Fixed point Theorem.

Connectedness, connected subsets ofℝ.

UNIT- II

Properties of complex numbers, regions in the complex lane, functions of the

complex variables, Mappings Derivatives, differentiation formulas, Cauchy- Riemann

equations, sufficient conditions for differentiability.

UNIT- III

Analytic functions, examples of analytic functions, exponential functions, 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.

Books Recommended:

1. P.K. Jain and K. Ahmad, Metric Spaces, Narosa Publishing House, New Delhi, CH- 2(1 -

9), 3(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. CH- 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).


1. SatishShirali and Harikishan L. Vasudeva, Metric Spaces, Springer Verlag, London,

2005.

2. S. Kumaresan, Topology of Metric Spaces, 2nd Ed., Narosa Publishing house, 2011.

3. S. Ponnusamy- Foundation of Complex Analysis, Alpha Science International Ltd.

4. J.B. Conway- Functions of one complex variable, Springer.

5.

6. N. Das- Complex Function Theory, Allied Publishers Pvt. Ltd., Mumbai.

SEMESTER- VI

CORE COURSE- 14

LINEAR PROGRAMMING




UNIT- I

Introduction to Linear programming problems, Theory of simplex method, optimality and

unboundedness, the simplex algorithm, simplex method in tableau format, introduction to

artificial variables, two phase method, Big M method.

UNIT- II

Duality, formulation of the dual problem, primal-dual relationships.

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.

Books Recommended:

1. Mokhtar S. Bazaraa, John J. Jarvis and Hanif D. Sherali, Linear Programming and

Network Flows, 2nd Ed., John Wiley and Sons, India, 2004. CH- 3(3.2 – 3.33, 3.5 – 3.8),

4(4.1 – 4.4), 6(6.1 – 6.3).

2. Hamdy A. Taha, Operations Research, An Introduction, 8th Ed., Prentice Hall India,

2006. CHAPTERS- 5(5.1, 5.3, 5.4).

3. Operation Research, KantiSwarup, P.K.Gupta, Man Mohan, Sultan Chand and Sons,

Ch-(17.1-17.6)

Book for References:

1. P.K.Gupta and and D.S. Hira- Operation Research, S.Chand and Company Pvt. Ltd. ,

New Delhi. CH- 1,2,3 (3.1-3.13), 6,10, 11,12.

2. Operation Research, KantiSwarup, P.K.Gupta, Man Mohan, Sultan Chand and Co.

SEMESTER- V

DSE- 1

PROGRAMMING IN C++ (COMPULSORY)

PART- I (MARKS: 75)



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, expressions, input using the extraction operator >> and

cin, output using the insertion operator << and cout, pre-processor directives, 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, foe, while, do-while loops and continue statement, nested control statement,

value returning functions, value versus reference parameters, local and global variables, one

dimensional array, two dimensional array, pointer data and pointer variables.

Books Recommended:

1. D.S. Malik: C++ Programming Language, Edn. 2009, Course Technology,

CengageLearnning India Edition CH- 2 (Pg. 37 – Pg. 95), CH- 3 (Pg-96 – Pg. 129), CH- 4

(Pg. 134 – Pg. 178), CH- 5 (Pg. 181 – Pg. 236), CH- 6, CH- 7 (Pg. 287 - 304), CH- 9 (Pg.

357 – Pg. 390), CH- 14 (Pg. 594 – Pg. 600).


1. R. Johnsonbaugh and M. Kalin- Applications Programming in ANSI C, Pearson

Education.

2. S.B. Lippman and J. Lajoie, C++ Primer, 3rd Ed., Addison Wesley, 2000.

3. BjarneStroustrup, The C++ Programming Language, 3rd Ed., Addison Wesley.

4. E. Balaguruswami: Object Oriented Programmimg with C++, 5th Ed., Tata McGraw Hill

Education Pvt. Ltd. CH- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

PRAT- II(PARCTICAL: MARKS: 25) List of Practicals (Using any Software)

Practical/ lab work is to be performed on a computer.

1. Calculate the sum of the series �

�+�

�+�

�+ −−−+

�

� for any positive integer N.

2. Write a user defined function to find the absolute value of an integer and use it to evaluate the function (−1)�/|�|, for � = −2,−1, 0, 1, 2.

3. Calculate the factorial of any natural number. 4. Read floating numbers and compute two averages: the average of negative numbers

and the average of positive numbers. 5. Write a program that prompts the user to input a positive integer. It then should

output a message indicating whether the number is a prime number. 6. Write a program that prompts the user to input the value of a, b and c involved in

the equation ��2 + �� + � = �and outputs the type of the roots of the equation. Also the program should outputs all the roots of the equation.

7. Write a program that generates random integer between 0 and 99. Given that first two Fibonacci numbers are 0 and 1, generate all Fibonacci numbers less than or equal to generated number.

8. Write a program that does the following: a. Prompts the user to input five decimal numbers. b. Prints the five decimal numbers. c. Converts each decimal number into the nearest integer. d. Adds these five integers. e. Prints the sum and average of them.

9. Write a program that uses while loops to perform the following steps: a. Prompt the user to input two integers: first Num and second Num (first Num

should be less than second Num). b. Output all odd and even numbers between first Num and second Num. c. Output the sum of all even numbers between first Num and second Num. d. Output the sum of the square of the odd numbers first Num and second

Num. e. Output all uppercase letters corresponding to the numbers between first

Num and second Num if any. 10. Write a program that prompts the user to input five decimal numbers. The program

should then add the five decimal numbers convert the sum to the nearest integer and print the result.

11. Write a program that prompts the user to enter the lengths of three sides of a triangle and then outputs a message indicating whether the triangle is a right triangle or scalene triangle.

12. Write a value returning function smaller to determine the smallest number from a set of numbers. Use this function to determine the smallest number from a set of 10 numbers.

13. Write a function that takes as a parameter an integer (as a long value) and returns the number of odd, even and zero digits. Also write a program to test your function.

14. Enter 100 integers into an array and short them in an ascending/ descending order and print the largest or smallest integer.

15. Enter 10 integers into an array and then search for a particular integer in the array. 16. Multiplication/ Addition of two matrices using two dimensional arrays. 17. Using arrays, read the vectors of the following type: A = (12345678), B = (02340156)

and compute the product and addition of these vectors. 18. Read from a text file and write to a text file. 19. Write a function, reverse Digit, that take an integer as a parameter and returns the

number with its digit reversed. For example, the value of a function reverse Digit 12345 is 54321 and the value of reverse Digit -532 is -235.


1. Degree practical mathematics by Dhirendra Kumar Dalai and Others. 2. Kalyani publishers CH-6

SEMESTER- V

DSE- 2

NUMBER THEORY



UNIT- I

Divisibilty theorem in integers, Primes and their distributions. Fundamental theorem of

arithmetic, Greatest common divisor, Ecledian algorithms, Modular arithmetic, Linear

Diophantine equation, prime counting function, statement of prime number theorem ,

Goldbach conjecture.

UNIT-II

Introduction to congruence,Linearcongruences,Chinese remainder

theorem,polynomialcongruences, system of linear congruences,complete set of

residues,Fermats little theorem,Wilsons theorem.

UNIT-III

Number theoretic function,Sum and number of divisors,Totally multiplicative

functions,definition and properties of the Dirichletproduct,TheMbius inversion formula,The

greatest integer function ,Eulersphifunction,Eulerstheorem,Reduced set of residues,some

properties of Eulers phi-function

UNIT-IV

Orderbof an integer modulo n, Primitive roots for primes ,composite numbers having

primitive roots,Eulerscriterion,Thelegendre symbol and its

propoties,Quadraticreciprocity,quadraticreciprocity,Quadraticcongruences with composite

moduli

Book recommended:

1. D.M. Burton-Elementary Number Theory ,McGraw Hill ,chapters:2(2.1 to 2.4), 3(3.1

to 3.3), 4(4.1 to 4.4), 5(5.1 to 5.4), 6(6.1 to 6.3), 7(7.1 to 7.3), 8(8.1 to 8.2), 9(9.1 to

9.3 )

BOOKS FOR REFERENCES:

1. K.H.Rosen- Elementary Number Theory & Its Applications, Pearson Addition Wesley.

2. Tom M. Apostol- Introduction to Analytic Number Theory, Springer International

Student Edn.

SEMESTER- VI

DSE - 3



DIFFERENTIAL GEOMETRY

UNIT- I

Theory of Space Curves: Space Curves : Space curves ,planer curvature,Torsion and serret –

Frenetformulae.Osulatig circles and spheres,Existence of spacev curves ,Evolutes and

involutes of curves.

UNIT-II

Osculating circle,Osculating circles and spheres,Existence of space curves ,Evolutes and

involutes of curves.

UNIT-III

Developables:Devolapable associated with space curves and curvesonsurfaces,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,Eularstheorem,RodriguesFormula,Conjugate and Asymptotic lines.

Book Recommended:

1. C.E Weatherburn,Differential Geometry Of Three Dimensions,Cambridge University press

2003.

Chapter : 1(1 to4 ,7,8,10) , 2(13,14,16,17),3,4(29-31,35,37,38).


1. T.J Willmore ,An introduction to differential geometry,Dover publications,2012.

2. B.O’Neill, Elementary Differential geometry ,2nd

Ed,academic press,2006.

3. A.N. Pressley-Elementary Differential Geometry, Springer.

4. B.P Acharya and R.N. Das-Fundamentals of Differential Geometry,Kalyani Publishers, Ludhiana, New Delhi

SEMESTER-IV

DSE-4

Project Work (Compulsory)

TOTAL MARK :100( Project :75 Marks+viva-voce:25 marks)

Skill Enhancement courses(SEC)

(Credit:2 eachb, total marks:50)



SEC-2

Logic and Sets

Introduction ,Proposition , Truth table, Nagation ,Conjuction and disjunction .Implication,

Biconditional propositions, converse, contra positive and inverse propositions and

precedence of logical operators. Propositional equivalence,Logicalequivalences,predicates

and quantifiers. Introductionquantifiers, Binding variables and Negations,Sets , Subsets, Set

operations and the laws of set theory and venn diagrams ,Examples of finite and infinite

sets,Finite sets 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, Relation , Product set,composition

of relations,Types of relations,Partitions,Equivalence Relations with examples of congruence

modulo relation , Partial ordering relations, partial ordering relations,Nary relations

Books Reference:

1. kalyani publishers sets and logic

2. R.PGrimaldi-Discrrete Mathematics and combinatorial mathematics, Pearson

education,1998.

3. P.R. Halmos-Naive Set theory, Springer,1974.

4.E.Kamke-Theory of sets dover publishres ,1950.

SEMESTER- I/III

Duration:3hrs., Mid sem-20, End sem-80, Total Marks-100

GE-1/3: Calculus and Ordinary Differrntial Equations



UNIT-I

Curvature, Asymptotes, Tracing of curves (Cartenry, Cycloid ,Folium Of Descartes, Astroid,

Limacon, Cissiod& loops), Rectification, Quardrature, Volume and surface area of solids of

revolution.

UNIT-II

Sphere, cones and cylinders, conicoid

UNIT- III

Limit and continuity of functions of several variable. Partial derivatives, partial derivatives

of higher orders, homogenous functions, change of variables, Expansion of functions of

several variables, maxima and minima.

UNIT-IV

Ordinary differential equations of 1st order and 1st degree(variable separable, homogenous,

exact and linear).equations of 1st order but higher degree.

UNIT-V

Second order llinear equations with constant coefficients , homogeneous forms, second

order equations with variable coefficients, variation of parameters, laplace transforms and

its applications to solutions of differential equations

Books Recommended:

1. B.P.Acharya and D.C Sahu Analytic Geometry of Quadratic surfaces, Kalyani Publishers,

New Delhi, Ludhiana.

2. Advanced higher calculus (Vidyapuri) Dr.Ghanashyamsamal& others ,ch-

6,7,8,9,10,11,12;13.

3. J.Sinharoy and S.Padhy-A Course of ordinary and differential equations, Kalyani

publishers, Chapter: 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).


1. Shanti Narayan and P.K. Mittal-Analytical solid Geometri, 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.

5. Santi Narayan calculus part-III

SEMESTER-II/IV

GE-MATH

General Elective for Computer Science, Physics and Chemistry hnrs.

Duration-3Hrs,Mid sem-20,End sem-80, Total mark100

Linear Algebra and Advanced Algebra

Questions must be set from all units with alternatives and each questions will be both

long and short answer type.

UNIT-I

Vector space, sub space, span of a set, linear dependence and Indepemdence, Dimensions

and Basis, Linear 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 operation, system of linear equations, matrix inversion using row

operations, system of linear equations, matrix inversion using row operation, Determina n t

and rank of matrices, Eigen values, eigen vectors, Quadric forms.

UNIT-III

The integers, Definition of groups with examples & properties sub groups.

UNIT-IV

A counting principle, Normal sub group, Quotient groups, Homomerphism, Isomerphism.

UNIT-V

Definition and examples of rings, some special classes of rings, Homomerphis, Isomerphism,

Ideals and quotient rings, more ideals&Quotients of an integral domain.

Books Recommended:

1. V.Krishnamurty, V.P.Mainra, J.L.Arora-An introduction to linear Algebra, Affiliated

East-West press Pvt. Ltd. New delhi,Chapter:3, 4(4.1------4.7), 5(except 5.3), 6(6.1,

6.2, 6.5, 6.6, 6.8), 7(7.4 only).

2. Topics in algebra, I.N.Herstein (vikas Rub) ch-1(1.3 only), 2(2.1-2.6;2.7 excluiding

Application),3(3.1-3.6)

Books for recommendation:

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 ,Vikash Publishing House Pvt.Ltd.Newdelhi.

4. Gilbert Strang –Linear Algebra & its Applications ,cenggage Learning India Pvt. Ltd.

5. I.N.Herstein-Topic in Algebra, Wiley Eastern Pvt.Ltd .

6. Gallian-contemporary Abstract Algebra, Narosha publishing House.

7. Artian- Algebra, prentice Hall of India.

8. V.K. Khanna and S.K. Bhambri-A cource in Abstract Algebra, Vikas publishing House

Pvt.Ltd

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