VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE. CBCS … Calculus Maths Original... · VIKRAMA SIMHAPURI UNIVERSITY:: NELLORE. CBCS B.A./B.Sc. Mathematics (Revised in April, 201Course Structure

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE. CBCS B.A./B.Sc. Mathematics Course Structure w.e.f. 2015-16 (Revised in April, 2017)

Year Seme- Paper Subject Hrs. Credits IA EA Total

ster

Differential Equations &

1 I I Differential Equations 6 5 25 75 100

Problem Solving Sessions

Solid Geometry &

II II Solid Geometry 6 5 25 75 100


Abstract Algebra &

2 III III Abstract Algebra 6 5 25 75 100


Real Analysis &

IV IV Real Analysis 6 5 25 75 100


3 V Ring Theory & Matrices 5 5 25 75 100

V Problem Solving Sessions

Linear Algebra &

VI Linear Algebra 5 5 25 75 100


Electives: (any one)

VII-(A) Vector Calculus

VII-(B) Operations Research

VI VII VII-(C) Number Theory & Elective 5 5 25 75 100


Cluster Electives:

VIII-A-1: Laplace Transforms 5 5 25 75 100

VIII-A-2: Integral Transforms

VIII-A-3: Project work 5 5 25 75 100

or

VIII VIII-B-1: Principles of Mechanics 5 5 25 75 100

VIII-B-2: Fluid Mechanics

VIII-B-3: Project work

or

VIII-C-1: Graph Theory

VIII-C-2: Applied Graph

VIII-C-3: Project work

Theory

or

VIII-D-1: Numerical Analysis

VIII-D-2: Advanced Numerical

Analysis

VIII-D-3: Project work

VIKRAMA SIMHAPURI UNIVERSITY::NELLORE B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS

SEMESTER – V, PAPER- 5

RING THEORY & MATRICES 60 Hrs

UNIT – I (12 hrs) : Rings-I :-

Definition of Ring and basic properties, Boolean Rings, Zero Divisors of Ring - Cancellation laws in a

Rings - Integral Domain Division Ring – Fields Examples.

UNIT –II (12 hrs) : Rings-II :-

Characteristic of Ring – Characteristic of an Integral Domain – Characteristic of Field

Characteristic of Boo Loan Ring.

Sub Ring Definition – Sub ring test – Union and Intersection of sub rings – Ideal Right and left

Ideals – Union and Intersection of Ideals. Excluding Principal prime and maximal Ideals.

UNIT –III (12 hrs) : Rings-III :-

Definition of Homomorphism – Homorphic Image – Elementary Properties of Homomorphism –

Kernel of a Homomorphism – Fundamental theorem of Homomorhphism.

UNIT – IV (12 hrs) Matrix-I :-

Rank of a Matrix – Elementary operations – Normal form of a matrix Echecon from of a Matrix -

Solutions of Linear Equations System of homogenous Linear equations – System of non Homogenous

Linear Equations method of consistency.

UNIT – V (12 hrs) Matrix-II :-

Characteristic Roots, Characteristic Values & Vectors of square Matrix, Cayley – Hamilton Theorem.

Reference Books :- 1. Abstract Algebra by J. Fralieh, Published by Narosa Publishing house. 2. A text Book of B.Sc., Mathematics by B.V.S.S.Sarma and others, published by S. Chand & Company Pvt. Ltd., New Delhi. 3. Rings and Linear Algebra by Pundir & Pundir, Published by Pragathi Prakashan.

4. Matrices by Shanti Narayana, published by S.Chand Publications.

Suggested Activities: Seminar/ Quiz/ Assignments/ Project on Ring theory and its applications

B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS SEMESTER – V, PAPER -6

LINEAR ALGEBRA

60 Hrs UNIT – I (12 hrs) : Vector Spaces-I : Vector Spaces, General properties of vector spaces, n-dimensional Vectors, addition and scalar

multiplication of Vectors, internal and external composition, Null space, Vector subspaces, Algebra of

subspaces, Linear Sum of two subspaces, linear combination of Vectors, Linear span Linear

independence and Linear dependence of Vectors.

UNIT –II (12 hrs) : Vector Spaces-II : Basis of Vector space, Finite dimensional Vector spaces, basis extension, co-ordinates, Dimension of a Vector space, Dimension of a subspace, Quotient space and Dimension of Quotientspace.

UNIT –III (12 hrs) : Linear Transformations : Linear transformations, linear operators, Properties of L.T, sum and product of LTs, Algebra of Linear

Operators, Range and null space of linear transformation, Rank and Nullity of linear transformations –

Rank – Nullity Theorem.

UNIT –IV (12 hrs) : (Inner product space-I) : Inner product spaces, Euclidean and unitary spaces, Norm or length of a Vector, Schwartz inequality,

Triangle in Inequality, Parallelogram law.

UNIT –V (12 hrs) : (Inner product space-II) : Orthogonal and Orthonormal Vectors, Orthogonal and Orthonormal Sets of Inner product Space,

Phythagoras theorem, The Diagonals are perpendicular in a rhombus, orthogonal set of non-zero vectos is

linearly independent, orthonormal set of vectors is liner independent, Gram-schmidt Orthogonalisation

process, Bessel’s Inequality and parseval’s Identity.

Reference Books : 1. Linear Algebra by J.N. Sharma and A.R. Vasista, published by Krishna Prakashan Mandir, Meerut-

250002. 2. Linear Algebra by Kenneth Hoffman and Ray Kunze, published by Pearson Education

(low priced edition), New Delhi.

3. Linear Algebra by Stephen H. Friedberg et al published by Prentice Hall of India Pvt. Ltd. 4th

Edition 2007.

Suggested Activities : Seminar/ Quiz/ Assignments/ Project on “Applications of Linear algebra Through Computer Sciences”

B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS SEMESTER – VI, PAPER – VII-(A)

VECTOR CALCULUS 60 Hrs

UNIT – I (12 hrs) : Vector Differentiation – I :-

Vector Function of Scalar Variable continuity of a vector function partial differentiation scalar

point Faction vector point faction – Gradient of a scalar point Function – Unit normal – Directional

Derivative at a Point – Angle between two surfaces.

UNIT – II (12 hrs) : Vector Differentiation – II :-

Vector differential Operator – Scalar Differential Operator – Divergence of a vector – Solenoidal

vector – Laplacian operator – curl of a vector – Ir rotational Vector – Vector identities.

UNIT – III (12 hrs) : Vector Integration - I :-

Definition – Integration of a vector – simple problems – smooth curve – Line integral –

Tangential Integral – circulation Problems on line Integral. Surface Integral – Flux Problems on Surface

Integral.

UNIT – IV(12 hrs) : Vector Integration - II :-

Volume Integrals – Gauss Divergence Theorem statement and proof – Applications of Gauss

Divergence theorem.

UNIT – V (12 hrs) : Vector Integration - III :-

Green’s Theorem in a plane Statement and proof – Application of Green’s Theorem.

Statement and Proof of Stoke Theorem – Application of stoke Theorem.

Reference Books :-

1. Vector Calculus by Santhi Narayana, Published by S. Chand & Company Pvt. Ltd., New Delhi.

2. Vector Calculus by R. Gupta, Published by Laxmi Publications.

3. Vector Calculus by P.C. Matthews, Published by Springer Verlag publicattions.

Suggested Activities: Seminar/ Quiz/ Assignments/ Project on Vector Calculus and its applications

B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS

SEMESTER – VI, PAPER – VII-(B)

ELECTIVE–VII-(B); OPERATIONS RESEARCH

60 Hrs

UNIT-I (12 hrs):

Introduction to Operations Research, Origin and Development of OR, Definition of OR,

Applications of OR, Models and their classifications, Advantages and Limitations of OR

UNIT-II (12 hrs):

Linear programming problem (LPP), Formulation of LPP, Solution of LPP using graphical

method and simplex method ( inequality only).

UNIT-III (12 hrs):

Transportation problem: Mathematical formulation, IBFS of transportation problem using north-

west corner rule, least-cost rule and Vogel’s approximation method, Simple problems.

UNIT-IV (12 hrs):

Assignment problem, definition, mathematical formulation of assignment problem, solution of

assignment problem using Hungarian algorithm, unbalanced assignment problem, simple problems.

UNIT-V (12 hrs):

Introduction – Definition – Terminology and Notations Principal Assumptions,

Problems with n Jobs through Two Machines

Problems with n Jobs through Three Machines

Prescribed Text Book:

Operations Research (2nd Edition) by S.Kalavathi, Vikas Publications Towers Pvt. Ltd.

Scope:

UNIT-I: 1.1, 1.2, 1.3, 1.5, 1.6, 1.7

UNIT-II: 2.1, 2.2, 2.2.1, 2.2.2, 3.1, 3.1.1, 4.1, 4.2, 4.3

UNIT-III: 8.1, 8.2, 8.3, 8.4.1, 8.4.2, 8.4.3

UNIT-IV: 9.1, 9.2, 9.2.1, 9.2.2, 9.3, 9.4

UNIT-V: 12.1, 12.2, 12.2.1, 12.2.2, 12.3, 12.4

Reference books:

1. Operations Research by Kanthiswaroop, P.K.Gupta, Manmohan by Sultan Chand & Sons

2. Operations Research by Paneerselvam by Prentice Hall of India

INSTRUCTIONS TO PAPER SETTER:-

1. The Paper setter is instructed to set theory questions from the first unit and problems from the

remaining units. The importance of applications to emphasis.

2. Number of constraints in LPP should be less than or equal to 3.

3. The order of transportation and assignment matrix should be less than or equal to 5.


SEMESTER – VI, PAPER – VII-(C)

ELECTIVE– VII-(C) : NUMBER THEORY

60 Hrs

UNIT-I (12 hours) Divisibility – Greatest Common Divisor – Euclidean Algorithm – The Fundamental Theorem of Arithmetic

UNIT-II (12 hours)

Congruences – Special Divisibility Tests - Chinese Remainder Theorem- Fermat‟s Little Theorem –

Wilson‟s Theorem – Residue Classes and Reduced Residue Classes – Solutions of Congruences

UNIT-III (12 hours)

Number Theory from an Algebraic Viewpoint – Multiplicative Groups, Rings and Fields

UNIT-IV (12 hours)

Quadratic Residues - Quadratic Reciprocity – The Jacobi Symbol

UNIT-V (12 hours)

Greatest Integer Function – Arithmetic Functions – The Moebius Inversion Formula

Reference Books:

1. “Introduction to the Theory of Numbers” by Niven, Zuckerman & Montgomery (John Wiley &

Sons) 2. “Elementary Number Theory” by David M. Burton.

3. Elementary Number Theory, by David, M. Burton published by 2nd

Edition (UBS Publishers).

4. Introduction to Theory of Numbers, by Davenport H., Higher Arithmetic published by 5th

Edition (John Wiley & Sons) Niven,Zuckerman & Montgomery.(Camb, Univ, Press)

5. Number Theory by Hardy & Wright published by Oxford Univ, Press. 6. Elements of the Theory of Numbers by Dence, J. B & Dence T.P published by Academic Press.

B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS SEMESTER – VI, PAPER – VIII-A-1

Cluster Elective –VIII-A-1; LAPLACE TRANSFORMS

60 Hrs UNIT – 1 (12 hrs) Laplace Transform I : -

Definition of - Integral Transform – Laplace Transform Linearity, Property, Piecewise continuous

Functions, Existence of Laplace Transform, Functions of Exponential order, and of Class A. Linear

property, First Shifting Theorem.

UNIT – 2 (12 hrs) Laplace Transform II : -

Second Shifting Theorem, Change of Scale Property, Laplace Transform of the derivative of f(t),

Initial Value theorem and Final Value theorem.

UNIT – 3 (12 hrs) Laplace Transform III : -

Laplace Transform of Integrals – Multiplication by t, Multiplication by tn – Division by t. Laplace

transform of Bessel Function Only.

UNIT –4 (12 hrs) Inverse Laplace Transform I : -

Definition of Inverse Laplace Transform. Linearity, Property, First Shifting Theorem, Second

Shifting Theorem, Change of Scale property, use of partial fractions, Examples.

UNIT –5 (12 hrs) Inverse Laplace Transform II : -

Inverse Laplace transforms of Derivatives–Inverse Laplace Transforms of Integrals –

Multiplication by Powers of „P’– Division by powers of „P’– Convolution Definition – Convolution

Theorem – proof and Applications – Heaviside’s Expansion theorem and its Applications.

Prescribed Text Books :-

Integral Transforms by A.R. Vasistha and Dr. R.K. Gupta Published by Krishna Prakashan Media Pvt. Ltd. Meerut.

Reference Books :-

1. Laplace Transforms by A.R. Vasistha and Dr. R.K. Gupta Published by Krishna Prakashan Media Pvt. Ltd. Meerut.

2. Fourier Series and Integral Transforms by Dr. S. Sreenadh Published by S.Chand and Co., Pvt.

Ltd., New Delhi.

3. Laplace and Fourier Transforms by Dr. J.K. Goyal and K.P. Gupta, Published by Pragathi Prakashan, Meerut.

4. Integral Transforms by M.D. Raising hania, - H.C. Saxsena and H.K. Dass Published by S. Chand

and Co., Pvt.Ltd., New Delhi.

Suggested Activities: Seminar/ Quiz/ Assignments

B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS, SEMESTER – VI, CLUSTER – A, PAPER – VIII-A-2

Cluster Elective- VIII-A-2: INTEGRAL TRANSFORMS

60 Hrs UNIT – 1 (12 hrs) Application of Laplace Transform to solutions of Differential Equations : -

Solutions of ordinary Differential Equations.

Solutions of Differential Equations with constants co-efficient

Solutions of Differential Equations with Variable co-efficient

UNIT – 2 (12 hrs) Application of Laplace Transform : - Solutions of partial Differential Equations.

UNIT – 3 (12 hrs) Application of Laplace Transforms to Integral Equations : -

Definitions : Integral Equations-Abel’s, Integral Equation-Integral Equation of Convolution Type, Integro Differential Equations. Application of L.T. to Integral Equations.

UNIT –4 (12 hrs) Fourier Transforms-I : -

Definition of Fourier Transform – Fourier’s in Transform – Fourier cosine Transform – Linear

Property of Fourier Transform – Change of Scale Property for Fourier Transform – sine Transform and

cosine transform shifting property – modulation theorem.

UNIT – 5 (12 hrs) Fourier Transform-II : -

Convolution Definition – Convolution Theorem for Fourier transform – parseval’s Indentify – Relationship between Fourier and Laplace transforms – problems related to Integral Equations.

Prescribed Text Books :-

Integral Transforms by A.R. Vasistha and Dr. R.K. Gupta Published by Krishna Prakashan Media Pvt. Ltd. Meerut.

Reference Books :-

1. Laplace Transforms by A.R. Vasistha and Dr. R.K. Gupta Published by Krishna Prakashan Media Pvt. Ltd. Meerut.

2. Fourier Series and Integral Transforms by Dr. S. Sreenadh Published by S.Chand and Co.,

Pvt. Ltd., New Delhi. 3. Laplace and Fourier Transforms by Dr. J.K. Goyal and K.P. Gupta, Published by

Pragathi Prakashan, Meerut. 4. Integral Transforms by M.D. Raising hania, - H.C. Saxsena and H.K. Dass Published by S. Chand

and Co., Pvt.Ltd., New Delhi.

Suggested Activities: Seminar/ Quiz/ Assignments

B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS SEMESTER – VI, CLUSTER-B, PAPER – VIII-B-1

Cluster Elective – VIII-B-1 : PRINCIPLES OF MECHANICS

60 Hrs

Unit – I : (10 hours) D’ Alembert’s Principle and Lagrange’s Equations : some definitions – Lagrange’s equations for a Holonomic system – Lagrange’s Equations of motion for conservative, nonholonomic system. Unit – II: (10 hours) Variational Principle and Lagrange’s Equations: Variatonal Principle – Hamilton’s Principle – Derivation

of Hamilton’s Principle from Lagrange’s Equations – Derivation of Lagrange’s Equations from

Hamilton’s Principle – Extension of Hamilton’s Principle – Hamilton’s Principle for Non-conservative,

Non-holonomic system – Generalised Force in Dynamic System – Hamilton’s Principle for Conservative,

Non-holonomic system – Lagrange’s Equations for Non-conservative, Holonomic system - Cyclic or

Ignorable Coordinates. Unit –III: (15 hours) Conservation Theorem, Conservation of Linear Momentum in Lagrangian Formulation – Conservation of angular Momentum – conservation of Energy in Lagrangian formulation. Unit – IV: (15 hours) Hamilton’s Equations of Motion: Derivation of Hamilton’s Equations of motion – Routh’s procedure –

equations of motion – Derivation of Hamilton’s equations from Hamilton’s Principle – Principle of Least

Action – Distinction between Hamilton’s Principle and Principle of Least Action. Unit – V: (10 hours) Canonical Transformation: Canonical coordinates and canonical transformations – The necessary and

sufficient condition for a transformation to be canonical – examples of canonical transformations –

properties of canonical transformation – Lagrange’s bracket is canonical invariant – poisson’s bracket is

canonical invariant - poisson’s bracket is invariant under canonical transformation – Hamilton’s

Equations of motion in poisson’s bracket – Jacobi’s identity for poisson’s brackets. Reference Text Books : 1. Classical Mechanics by C.R.Mondal Published by Prentice Hall of India, New Delhi. 2. A Text Book of Fluid Dynamics by F. Charlton Published by CBS Publications, New Delhi. 3. Classical Mechanics by Herbert Goldstein, published by Narosa Publications, New Delhi. 4. Fluid Mechanics by T. Allen and I.L. Ditsworth Published by (McGraw Hill, 1972)

5. Fundamentals of Mechanics of fluids by I.G. Currie Published by (CRC, 2002)

6. Fluid Mechanics : An Introduction to the theory, by Chia-shun Yeh Published by (McGraw Hill,

1974)

7. Introduction to Fluid Mechanics by R.W Fox, A.T Mc Donald and P.J. Pritchard Published by (John

Wiley and Sons Pvt. Ltd., 2003)

B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS SEMESTER – VI, CLUSTER-B, PAPER – VIII-B-2

Cluster Elective–VIII-B-2 : FLUID MECHANICS

60 Hrs

Unit – I : (10 hours) Kinematics of Fluids in Motion Real fluids and Ideal fluids – Velocity of a Fluid at a point – Streamlines and pthlines – steady and

Unsteady flows – the velocity potential – The Vorticity vector – Local and Particle Rates of Change –

The equation of Continuity – Acceleration of a fluid – Conditions at a rigid boundary – General Analysis

of fluid motion. Unit – II : (10 hours)

Equations of motion of a fluid- Pressure at a point in fluid at rest – Pressure at a point in a moving fluid –

Conditions at a boundary of two inviscid immiscible fluids – Euler’s equations of motion – Bernoulli’s

equation – Worked examples. Unit – III : (10 hours) Discussion of the case of steady motion under conservative body forces - Some flows involving axial

symmetry – Some special two-dimensional flows – Impulsive motion – Some further aspects of vortex

motion. Unit – IV : (15 hours) Some Two – dimensional Flows, Meaning of two-dimensional flow – Use of Cylindrical polar coordinates

– The stream function – The complex potential for two-dimensional, Irrotational, Incompressible flow –

Uniform Stream – The Milne-Thomson Circle theorem – the theorem of Blasius.

Unit – V : (15 hours)

Viscous flow, Stress components in a real fluid – Relations between Cartesian components of stress –

Translational motion of fluid element – The rate of strain quadric and principal stresses – Some further

properties of the rate of strain quadric – Stress analysis in fluid motion – Relations between stress and

rate of strain – the coefficient of viscosity and laminar flow - The Navier-Stokes equations of motion of a

viscous fluid. Reference Text Books : 1. A Text Book of Fluid Dynamics by F. Charlton Published by CBS Publications, New Delhi. 2. Classical Mechanics by Herbert Goldstein, published by Narosa Publications, New Delhi. 3. Fluid Mechanics by T. Allen and I.L. Ditsworth published by (McGraw Hill, 1972) 4. Fundamentals of Mechanics of fluids by I.G. Currie published by (CRC, 2002) 5. Fluid Mechanics, An Introduction to the theory by Chia-shun Yeh published by (McGraw Hill, 1974) 6. Fluids Mechanics by F.M White published by (McGraw Hill, 2003) 7. Introduction to Fluid Mechanics by R.W Fox, A.T Mc Donald and P.J. Pritchard published by (John

Wiley and Sons Pvt. Ltd., 2003


SEMESTER – VI, CLUSTER-C, PAPER – VIII-C-1 Cluster Elective–VIII-C-1: GRAPH THEORY

60 Hrs

UNIT – I (12 hrs) Graphs and Sub Graphs : Graphs , Simple graph, graph isomorphism, the incidence and adjacency matrices, sub graphs, vertex degree, Hand shaking theorem, paths and connection, cycles.

UNIT – II (12 hrs) Applications, the shortest path problem, Sperner‟s lemma. Trees : Trees, cut edges and Bonds, cut vertices, Cayley‟s formula.

UNIT – III (12 hrs) : Applications of Trees - the connector problem. Connectivity Connectivity, Blocks and Applications, construction of reliable communication Networks,

UNIT – IV (12 hrs): Euler tours and Hamilton cycles Euler tours, Euler Trail, Hamilton path, Hamilton cycles , dodecahedron graph, Petersen graph, hamiltonian graph, closure of a graph.

UNIT – V (12 hrs) Applications of Eulerian graphs, the Chinese postman problem, Fleury‟s algorithm - the travelling

salesman problem.

Reference Books : 1. Graph theory with Applications by J.A. Bondy and U.S.R. Murthy published by Mac. Millan Press 2. Introduction to Graph theory by S. Arumugham and S. Ramachandran, published by

scitech Publications, Chennai-17. 3. A Text Book of Discrete Mathamatics by Dr. Swapan Kumar Sankar, published by S.Chand & Co.

Publishers, New Delhi. 4. Graph theory and combinations by H.S. Govinda Rao published by Galgotia Publications.


SEMESTER – VI, CLUSTER-C, PAPER – VIII-C-2 Cluster Elective -VIII-C-2: APPLIED GRAPH THEORY

60 Hrs

UNIT – I (12 hrs) : Matchings Matchings – Alternating Path, Augmenting Path - Matchings and coverings in Bipartite graphs, Marriage Theorem, Minimum Coverings.

UNIT –II (12 hrs) : Perfect matchings, Tutte‟s Theorem, Applications, The personal Assignment problem -The optimal

Assignment problem, Kuhn-Munkres Theorem. UNIT –III (12 hrs) : Edge Colorings Edge Chromatic Number, Edge Coloring in Bipartite Graphs - Vizing‟s theorem.

UNIT –IV (12 hrs) : Applications of Matchings, The timetabling problem. Independent sets and Cliques Independent sets, Covering number , Edge Independence Number, Edge Covering Number - Ramsey‟s

theorem.

UNIT –V (12 hrs) : Determination of Ramsey‟s Numbers – Erdos Theorem, Turan‟s theorem and Applications, Sehur‟s

theorem. A Geometry problem.

Reference Books :- 1. Graph theory with Applications by J.A. Bondy and U.S.R. Murthy, published by Mac. Millan Press. 2. Introduction to graph theory by S. Arumugham and S. Ramachandran published by SciTech

publications, Chennai-17. 3. A text book of Discrete Mathematics by Dr. Swapan Kumar Sarkar, published by S. Chand Publishers. 4. Graph theory and combinations by H.S. Govinda Rao, published by Galgotia Publications.


SEMESTER – VI, PAPER – VIII-(D)-1

Cluster Elective –VIII-(D)-1; NUMERICAL ANALYSIS

60 Hrs

UNIT- I: (10 hours) Errors in Numerical computations : Errors and their Accuracy, Mathematical Preliminaries, Errors and

their Analysis, Absolute, Relative and Percentage Errors, A general error formula, Error in a series

approximation.

UNIT – II: (12 hours) Solution of Algebraic and Transcendental Equations: The bisection method, The iteration method,

The method of false position, Newton Raphson method, Generalized Newton Raphson method. Muller‟s

Method UNIT – III: (12 hours) Interpolation - I Interpolation : Errors in polynomial interpolation, Finite Differences, Forward differences, Backward

differences, Central Differences, Symbolic relations, Detection of errors by use of Differences Tables,

Differences of a polynomial UNIT – IV: (12 hours) Interpolation - II Newton‟s formulae for interpolation. Central Difference Interpolation Formulae, Gauss‟s central

difference formulae, Stirling‟s central difference formula, Bessel‟s Formula, Everett‟s Formula. UNIT – V : (14 hours) Interpolation - III

Interpolation with unevenly spaced points, Lagrange‟s formula, Error in Lagrange‟s formula, Divided

differences and their properties, Relation between divided differences and forward differences, Relation

between divided differences and backward differences Relation between divided differences and central

differences, Newton‟s general interpolation Formula, Inverse interpolation. Reference Books : 1. Numerical Analysis by S.S.Sastry, published by Prentice Hall of India Pvt. Ltd., New Delhi. (Latest

Edition) 2. Numerical Analysis by G. Sankar Rao published by New Age International Publishers, New –

Hyderabad. 3. Finite Differences and Numerical Analysis by H.C Saxena published by S. Chand and Company, Pvt.

Ltd., New Delhi. 4. Numerical methods for scientific and engineering computation by M.K.Jain, S.R.K.Iyengar, R.K. Jain. Suggested Activities: Seminar/ Quiz/ Assignments

B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS SEMESTER – VI: PAPER – VIII-D-2

Cluster Elective –VIII-D-2: ADVANCED NUMERICAL ANALYSIS

60 Hrs

Unit – I (10 Hours) Curve Fitting: Least – Squares curve fitting procedures, fitting a straight line, nonlinear curve fitting, Curve fitting by a sum of exponentials. UNIT- II : (12 hours) Numerical Differentiation: Derivatives using Newton’s forward difference formula, Newton’s backward difference formula, Derivatives using central difference formula, stirling’s interpolation formula, Newton’s divided difference formula, Maximum and minimum values of a tabulated function. UNIT- III : (12 hours) Numerical Integration: General quadrature formula on errors, Trapozoidal rule, Simpson’s 1/3 – rule, Simpson’s 3/8 – rule, and Weddle’s rules, Euler – Maclaurin Formula of summation and quadrature, The Euler transformation. UNIT – IV: (14 hours) Solutions of simultaneous Linear Systems of Equations: Solution of linear systems – Direct methods, Matrix inversion method, Gaussian elimination methods, Gauss-Jordan Method ,Method of factorization,

Solution of Tridiagonal Systems,. Iterative methods. Jacobi‟s method, Gauss-siedal method. UNIT – V (12 Hours) Numerical solution of ordinary differential equations: Introduction, Solution by Taylor’s Series, Picard’s method of successive approximations, Euler’s method, Modified Euler’s method, Runge – Kutta methods. Reference Books : 1. Numerical Analysis by S.S.Sastry, published by Prentice Hall India (Latest Edition). 2. Numerical Analysis by G. Sankar Rao, published by New Age International Publishers, New –

Hyderabad. 3. Finite Differences and Numerical Analysis by H.C Saxena published by S. Chand and Company,

Pvt. Ltd., New Delhi. 4. Numerical methods for scientific and engineering computation by M.K.Jain, S.R.K.Iyengar, R.K. Jain. Suggested Activities: Seminar/ Quiz/ Assignments

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE.

I, II, III, IV, V AND VI SEMESTER

CBCS B.A. / B.Sc. MATHEMATICS

w.e.f. 2015-16 (Revised in March, 2017)

MODEL QUESTION PAPER

Time: 3 Hours Max. Marks : 75

PART-A

Answer any FIVE of the following Questions: (5 x 5= 25 Marks)

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

PART - B

Answer any FIVE of the following Questions.

Choosing at least ONE Question from Each Section. (5 10 =50 Marks)

SECTION – A

UNIT- I 11. 12.

UNIT- II 13. 14.

UNIT- III 15. 16.

SECTION - B

UNIT- IV 17. 18.

UNIT- V 19. 20. Instruction to Paper Setter:

Two questions must be given from each unit in Part-A and Part-B

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE. (w.e.f. 2016-17)

B.A./B.Sc. FIRST YEAR MATHEMATICS SEMESTER-I MODEL QUESTION PAPER-1

(DIFFERENTIAL EQUATIONS)

TIME : 3 Hours Max.Marks : 75

PART – A

I. Answer any FIVE Questions : 5 X 5 = 25M

1. Solve 2

2dy xxy edx

.

2. Find Integrating factor of 3 2 2 42 0xy y dx x y x y dy .

3. Find the Orthogonal trajectories of the family of curves

2

3x

2 2 2

3 3 3x y a where

‘a’ is a parameter.

4. Solve 2 42y xP x P .

5. Solve 4 28 16 0D D y .

6. Solve 2 45 6 xD D y e .

7. Solve 2 4 sinD y x x .

8. Solve 2 34 4D D y x .

9. Solve 2 2 1 logx D xD y x .

10. Find the complementary function yc

of 2 2 23 5 sin logx D xD y x x .

PART - B Answer any FIVE of the following Questions.


SECTION - A UNIT - I

11. Solve 2 3 4 1dy

x y xdx

.

12. Solve 2 3 3 0x ydx x y dy .

UNIT - II

13. Find the orthogonal Trajectories of the families of Curves 2

1 cos

ar

when “a” is

Parameter.

14. 2 22 cotP Py x y .

UNIT - III

15. Solve 2

3 1 1xD y e .

16. Solve 2 3 2 cos3 .cos 2D D y x x .

SECTION - B UNIT - IV

17. Solve 2

36 13 8 sin 22

d y dy xy e xdxdx

.

18. Solve 2 2 21 cosxD y x e x x .

UNIT - V

19. Solve by the method of variation of parameters 2 1 cosD y ecx .

20. Solve 2 21 1 1 4cos log 1x D x D y x

.

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE B.A./B.Sc. FIRST YEAR MATHEMATICS

MODEL QUESTION PAPER SEMESTER-II

(SOLID GEOMETRY)


PART-A I. Answer any FIVE of the following Questions : (5 X 5= 25 Marks)

1. Find the Equation of the plane through the point (-1,3,2) and perpendicular to the planes

2 2 5x y z and 3 3 2 8x y z .

2. Find the angles between the planes 2 3 5,x y z 3 3 9x y z .

3. Show that the line 1 2 5

1 3 5

x y z

lies in the plane x+2y-z=0.

4. Find the point of intersection with the plane 3 4 5 5x y z and the line 1 3 2

1 3 2

x y z .

5. Find the centre and radius of the sphere 2 2 22 2 2 2 4 2 1 0x y z x y z .

6. Find the equation of the sphere through the circle 2 2 2 9x y z , 2 3 4 5x y z and the

point (1,2,3)

7. Find the equation of the tangent plane to the sphere 2 2 23 3 3 2 3 4 22 0x y z x y z

at the point (1,2,3)

8. Show that the spheres are orthogonal 2 2 2 6 2 8 0;x y z y z

2 2 2 6 8 4 20 0x y z x y z .

9. Find the equation of the cone which passes through the three co-ordinate axis and the lines

1 2 3

x y z

and 2 1 1

x y z .

10. Find the equation of the cylinder whose generators are parallel to 1 2 3

x y z and which

Passes through the curve 2 2 16, 0x y z .

PART - B



SECTION – A UNIT - I

11. Find the equation of the plane passing through the intersection of the planes

2 3 4,2 5 0x y z x y z and perpendicular to the plane 6 5 3 8 0z x y .

12. Prove that Equation 2 2 22 6 12 18 2 0x y z yz zx xy represents a pair of planes

and find the angle between them.

UNIT - II

13. Find the image of the point (2,-1,3) in the plane 3x-2y+z=9.

14. Find the length and equation to the line of shortest distance between the lines 2 3 1

,3 4 2

x y z

4 5 2

4 5 3

x y z .

UNIT - III

15. Find the equation of the sphere through the circle 2 2 2 2 3 6 0x y z x y ,

2 4 9 0x y z and the centre of the sphere 2 2 2 2 4 6 5 0x y z x y z .

16. Find whether the following circle is a great circle or small circle 2 2 2 4 6 8 4 0,x y z x y z 3x y z .

SECTION – B UNIT - IV

17. Find the equation of the sphere which touches the plane 3x+2y-z+2=0 at (1,-2,1) and

cuts orthogonally the sphere 2 2 2 4 6 4 0x y z x y .

18. Find limiting points of the co axial system of spheres

2 2 2 20 30 40 29x y z x y z 2 3 4 0x y z .

UNIT - V

19. Find the vertex of the cone 2 2 27 2 2 10 10 26 2 2 17 0x y z zx xy x y z .

20. Find the equation to the right circular cylinder whose guiding circle 2 2 2 9,x y z 3x y z .

VIKRAMA SIMHAPURI UNIVERSITY::NELLORE

(w.e.f. 2016-17) B.A./B.Sc. SECOND YEAR MATHEMATICS

MODEL QUESTION PAPER SEMESTER – III

(ABSTRACT ALGEBRA)


PART - A

I. Answer any FIVE of the following Questions : (5 X 5= 25 Marks)

1. Prove that in a group G Inverse of any Element is unique.

2. 1,2,3,4,5,6G Prepare composition table and prove that G is a finite abelian group of order

6 with respect to 7

X .

3. If H is any subgroups of G then prove that 1H H .

4. Prove that any two left cosets of a subgroups are either disjoint or identical.

5. Prove that intersection of any two normal subgroup is again a normal subgroup.

6. Define the following :

(a) Normal subgroups (b) Simple Groups.

7. Prove that the homomorphic image of a group is a group.

8. If for a group ,G :F G G is given by 2,f x x x G is a homomorphism then prove

that G is abelian.

9. If 1 2 3 1 2 3

,2 3 1 3 1 2

A B

find AB and BA.

10. Find the inverse of the permutation: 1 2 3 4 5 6

3 4 5 6 1 2

PART - B




11. Define abelian group. Prove that the set of thn roots of unity under multiplication form a finite

abelian group.

12. Show that the set of all positive rational numbers form on abelian group under the composition ‘0’

defined by 2

abaob .

UNIT - II

13. Prove that a non-empty finite subset of a group which is closed under multiplication is a

subgroup of G.

14. Prove that the union of two subgroups of a group is a subgroup if f one is contained in the other.

UNIT - III

15. Prove that a subgroup H of a group G is a normal subgroup of G if f each left coset of H in G

is a right coset of H in G.

16. If G is a group and H is a subgroup of index 2 in G then prove that H is a normal subgroup of G.

SECTION - B

UNIT - IV

17. ,G and 1,G be two groups 1:f G G is an into homomorphism then prove

(i) 1f e e (ii) 11f a f a

Where e , 1e are then identity elements in G and 1G respectively.

18. State and prove fundamental theorem on Homomorphism of Groups.

UNIT - V

19. Examine the following permutation are even (or) odd

(i) 1 2 3 4 5 6 7

3 2 4 5 6 7 1f

(ii) 1 2 3 4 5 6 7 8

7 3 1 8 5 6 2 4g

20. Define cyclic group. Prove that every cyclic group is an abelian group.

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE. B.A./B.Sc. SECOND YEAR MATHEMATICS

SEMESTER – IV MODEL QUESTION PAPER

(REAL ANALYSIS)


PART - A I. Answer any FIVE of the following Questions : (5 X 5= 25 Marks)

1. Test for convergence 1

2 1n

.

2. State cauchy’s root test and test for convergence

21

1n

n

.

3. Discuss various types of discontinuity.

4. Examine for continuity of a function 1f n x x at x=0.

5. If 1

1

xf x

xe

if 0x and 0f x if x=0 show that f is not derivable at x = 0.

6. Prove that 12 sin , 0f x x xx

and 0 0f is derivable at the origin.

7. State cauchy’s Mean value theorem.

8. Find ‘C’ of the Lagrange’s mean value theorem for 1 2 3f x x x x on 0,4 .

9. If 2f x x on 0,1 and 1 2 3

0, , , ,14 4 4

P

compute ,L P f and ,U P f .

10. Prove that a constant function is Reiman integrable on ,a b .

PART - B




11. Prove that a monotonic Increasing sequence which is bounded above is convergent.

12. State and prove P-test.

UNIT - II

13. Discuss the continuity of

1 1

1 1

x xx e e

f x

x xe e

for 0x and 0 0f at x = 0.

14. If f is continuous on ,a b and ,f a f b having opposite sign then prove that there

exit , 0C a b f c .

UNIT - III

15. Show that 1

sin , 0, 0f x x x f xx

when x=0 is continuous but not derivable at x=0.

16. Show that

1

1

1

1

xx e

f x

xe

if 0x and 0 0f is continuous at x=0 but not

derivable at x=0.


17. State and prove Rolle’s theorem.

18. Using Lagrange’s theorem show that log 11

xx n

x

if log 1f x x .

UNIT - V

19. If : ,f a b R is monotonic on ,a b then f is integrable on ,a b .

20. If ,f R a b and m, M are the infimum and supremum of f on ,a b , then

b

m b a f x dx M b a

a

.

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE. B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS

SEMESTER – V, PAPER - 5 MODEL QUESTION PAPER

RING THEORY & MATRICES



1. Define Types of rings and give one example for each.

2. If R is a Boolean ring then prove that 0a a a R .

3. If the characteristic of a ring is 2 and ab ba then prove that

2 22 2 ,a b a b a b a b R .

4. State and prove “Sub ring test”.

5. If 1:f R R be a homomorphism of a ring R into a ring 1R and 0 R , 1 10 R be

the zero elements then prove (1) 10 0f (2) f a f a a R .

6. Prove that the Homorphic image of a Commutative ring is Commutative.

7. Obtain the rank of the matrix

1 2 0

3 7 1

5 9 3

A

.

8. Show that the system 2 3 0,x y z 7 13 9 0,x y z 2 3 4 0x y z has trial

solution only.

9. Find the characteristic equation of the matrix

0 1 2

1 0 1

2 1 0

A

.

10. Find the Eigen values of 5 4

1 2A

.

PART - B




11. Prove that A division ring has no zero divisors.

12. Prove that A finite integral domain is a field.

UNIT - II

13. Prove that characteristic of Booliean Ring is 2.

14. Prove that A field has no proper ideals.

UNIT - III

15. If ‘f ’ is a homomorphism of a ring ‘R’ in to the ring 1R then prove that ‘f ’ is an

into isomorphism iff test = 0 .

16. Prove that every quotient ring of a ring is a homomorphic image of the ring.


17. Reduce the Matrix

1 2 3 0

2 4 3 2

3 2 1 3

6 8 7 5

A

into echelon form and hence find its rank.

18. Show that the equations 3 0,x y z 3 5 2 8 0,x y z 5 3 4 14 0x y z are

consistent and solve them.

UNIT - V

19. If

2 1 2

5 3 3

1 0 2

A

verfy cayley – Hamilton theorem. Hence find 1A .

20. Find the characteristic roots and vectors to the matrix

2 1 0

0 2 1

0 0 2

A

.

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE B.A./B.Sc. THIRD YEAR MATHEMATICS SYLLABUS

SEMESTER – V, PAPER - 6 MODEL QUESTION PAPER

LINEAR ALGEBRA



1. F is field and , ,p q r F . Show that , , / 0w x y z px qy rz form a vector

subspace of 3

v F .

2. Show that the system of vector 1,3,2 , 1, 7, 8 , 2,1, 1 of 3

V R is Linearly dependent.

3. State and prove “Invariance theorem”.

4. Show that the vectors 1,1,2 , 1,2,5 , 5,3,4 of 3R R do not form a basis set of 3R R .

5. Show that the mapping :3 2

T V R V R is defined by : , , ,T x y z x y x z is a

Linear Transformation.

6. :3 2

T V R V R and :3 2

H V R V R be two Linear Transformations

, , ,T x y z x y y z and , , 2 , 3H x y z x y Find (i) H+T (ii) aH.

7. State and prove Triangle Inequality.

8. If , are two vectors in Euclidean space V R such that prove that

, 0 .

9. In an inner product space prove that 1,u v u v are orthogonal if u v .

10. State and prove Pythagoras Theorem.

PART - B




11. If V F be a vector space. V . Prove that the necessary and sufficient conditions

for to be a subspace of V are

(i) ,

(ii) ,a F a .

12. If show that are the sub sets of a vector space v F then prove that

L S T L S L T .

UNIT - II

13. State and prove Basis Existence theorem.

14. 1

and 2

be two subspaces of 4R .

, , , : 2 01

a b c d b c d

, , , : , 22

a b c d a d b c

Find dim 1 3

UNIT - III

15. Find , ,T x y z where 3:T R R is defined by 1,1,1 3,T

0,1, 2 1,T 0,0,1 2T .

16. Define Null space. Prove that Null space N T is subspace of U F where

:T U V is a Linear Transformation.


17. State and prove parallelogram Law.

18. If , and two vectors in an I.P.S. then prove that , are Linear

Independent iff , .

UNIT - V

19. Prove that in an I.P.S. any orthonormal set of vectors in Linear independent.

20. State and prove Bessel’s inequality.

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VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE. CBCS … Calculus Maths Original... · VIKRAMA SIMHAPURI UNIVERSITY:: NELLORE. CBCS B.A./B.Sc. Mathematics (Revised in April, 201Course Structure