department of mathematicsdrvskrishnagdc.edu.in/syllabus/mathematics_2014_15.pdf · 2016-10-27 ·...

MATHEMATICS – SYLLABUS & QUESTION

PAPERS WITH EFFECT FROM 2014-2015

CHOICE BASED CREDIT SYSTEM

DEPARTMENT OF MATHEMATICS

Phone: 0891 2 55 32 62 // 2 71 44 65 E.Mail:[email protected] Fax:0891 2 55 81 23

Dr.V.S.KRISHNA GOVERNMENT DEGREE COLLEGE (A) Affiliated to Andhra University

NAAC accredited B++ Institution by NAAC

VISAKHAPATNAM 530 013, A.P.INDIA

Dr.V.S.KRISHNA GOVERNMENT DEGREE COLLEGE (A), VISAKHAPATNAM

I B.Sc., MATHEMATICS, COURSE-I

SYLLABUS FOR THE FIRST SEMESTER

DIFFERENTIAL EQUATIONS & SOLID GEOMETRY

W.E.F. 2014-2015

DIFFERENTIAL EQUATIONS (30 hours)

MODULE - I: Differential equations of first order and first degree:

Variables and separable, Homogeneous and reducible homogeneous, Linear differential equations,

Differential equations reducible to linear form, Exact differential equations, Integrating factors.

MODULE - II: Differential equations of the first order but not of the first degree:

Equations solvable for p, Equations solvable for y, Equations solvable for x, Equations that do not

contain x (or y), Equations of the first degree in x and y - Clairaut's equation, Orthogonal

trajectories.

SOLID GEOMETRY (30 hours)

MODULE - III: The Plane & Straight line:

ii)) TThhee ppllaannee:: DDeetteerrmmiinnaattiioonn ooff aa ppllaannee uunnddeerr ggiivveenn ccoonnddiittiioonnss,, EEqquuaattiioonnss ooff aa ppllaannee iinn

tteerrmmss ooff iittss iinntteerrsseeccttss oonn tthhee aaxxeess,, Equations of the plane through the given points,

system of the planes, Two sides of a plane, Bisectors of angles between two planes,

joint equation of two planes.

iiii)) SSttrraaiigghhtt lliinnee:: Equations of a line, Angle between a line and a plane, the condition that a

given line may lie in a given plane, the condition that two given lines are coplanar,

Number of arbitrary constants in the equations of a straight line. Sets of conditions

which determine a line, The shortest distance between two lines. The length and

equations of the line of shortest distance between two straight lines, Length of the

perpendicular from a given point to a given line, Intersection of three planes, Triangular

Prism.

MODULE - IV: The Sphere:

Definition and equation of the sphere, Equation of the sphere through four given points, Plane

sections of a sphere. Intersection of two spheres. Equation of a circle. Sphere through a given circle;

Intersection of a sphere and a line. Power of a point, Tangent plane, Plane of contact. Polar plane,

Pole of a plane, Conjugate points, Conjugate planes; Angle of intersection of two spheres.

Conditions for two spheres to be orthogonal; Radical plane. Coaxial system of spheres; Simplified

from of the equation of two spheres.

Prescribed Text Books: 1. A Text book of Mathematics-volume –I published by S.Chand & company

2. Scope as in Analytical Solid Geometry by Shanti Narayan and P.K.

Mittal, Published by S. Chand & Company Ltd. Seventeenth edition

Reference books:

1. Scope as in differential equations and their application by zafar ahson.

22.. P.K. Jain and Khaleel Ahmed, “A Text Book of Analytical Geometry of

Three Dimensions”, Wiley Eastern Ltd., 1999.

33.. Differential equations-M.L.Khanna-S.V.Nath parteners, Jai prakash

nath&co (1954).

Question paper pattern and scheme(Blue Print)

Section - A : Very Short Answer type 5×2=10 Marks

Section-B: Short Answer type:

5 questions out of 8 questions 5×5 = 25 Marks

Section-C: Essay Answer type:

4 questions with internal choice 4×10 = 40 Marks

______________

Total = 75 Marks

Time allotted: 3 hours

IInntteerrnnaall AAsssseessssmmeenntt:: 2255 MMaarrkkss

Guidelines to the paper setters:

Paper setter is requested to cover all the modules and also requested to

give question based on remembrance and understanding (80%) and application

or creativity (20%)

TOPIC V.S.A.Q S.A.Q E.Q Marks

allowed to

the module

Module - I 2 2 2 34

Module - II 1 2 2 32

Module - III 1 2 2 32

Module - IV 1 2 2 32

Total Marks Including choice 130


I B.Sc., MATHEMATICS, COURSE- II

SYLLABUS FOR THE SECOND SEMESTER

DIFFERENTIAL EQUATIONS & SOLID GEOMETRY

W.E.F. 2014-2015

DIFFERENTIAL EQUATIONS (Higher order linear differential equations) (30 hours)

MODULE - I:

Solution of homogeneous linear differential equations of order n with constant coefficients.

Solution of the non-homogeneous linear differential equations with constant coefficients by means

of polynomial operators.

MODULE - II:

Method of undetermined coefficients, Method of variation of parameters. Linear differential

equations with non-constant coefficients, The Cauchy-Euler equation, solution of a system of

linear equations with constant cofficients an equivalent triangular- system, Degenerate case:

1 4 2 3( ) ( ) ( ) ( ) 0− =P D P D P D P D

SOLID GEOMETRY (30 hours)

MODULE - III:

CCoonneess:: Definitions of a cone, vertex, guiding curve, generators. Equation of the cone with a given

vertex and guiding curve. Enveloping cone of a sphere. Equations of cones with vertex at origin are

homogeneous. Condition that the general equation of the second degree should represent a cone.

Condition that a cone may have three mutually perpendicular generators Intersection of a line and a

quadric cone. Tangent lines and tangent plane at a point.Condition that a plane may touch a cone.

Reciprocal cones. Intersection of two cones with a common vertex. Right circular cone. Equation of

the right circular cone with a given vertex, axis and semi-vertical angle.

CCyylliinnddeerr::

Definition of a cylinder, Equation to the cylinder whose generators intersect a given cone

and are parallel to a given line, Enveloping cylinder of a sphere. The right circular cylinder.

Equation of the right circular cylinder with a given axis and radius.

MODULE - IV:

The conicoid:

The general equation of the second degree and the various surfaces represented by it; Shapes

of some surfaces. Nature of Ellipsoid. Nature of Hyperboloid, enveloping cone, enveloping

cylinder.

Prescribed Text Books:

1. A Text book of Mathematics-volume –I published by S.Chand &

company

2. Scope as in Analytical Solid Geometry by Shanti Narayan and P.K.

Mittal, Published by S. Chand & Company Ltd. Seventeenth edition

Reference books: 1. Scope as in differential equations and their application by zafar ahson.

22.. P.K. Jain and Khaleel Ahmed, “A Text Book of Analytical Geometry of

Three Dimensions”, Wiley Eastern Ltd., 1999.







______________

Total = 75 Marks






or creativity (20%)


allowed to

the module

Module - I 2 2 2 34






II B.Sc., MATHEMATICS, COURSE-1I1

SYLLABUS FOR THE THIRD SEMESTER

ABSTRACT ALGEBRA & REAL ANALYSIS W.E.F. 2014-2015

ABSTRACT ALGEBRA (30 hours)

MODULE - I:

Groups: Binary operations- Definitions and properties, Groups--Definition and elementary

properties, Finite group and group composition tables

Subgroups: CompleX definition, subgroups, their intersection and union.

Cosets: Groups of cosets and applications, Legrange’s theorem, normalizer of an element of a

group, self conjugate element of a group.

Normal subgroups:

Definition of a normal sub group, properties, simple groups, quotient groups center of a group.

Homomorphisms(Homomorphism, isomorphism of groups) Definition and elementary properties, fundamental theorem on homomorphisms of groups, Inner

and outer automorphism of a group.

MODULE - II:

Permutation groups: Functions and permutations, groups of permutations, cycles and cyclic notation, even and odd

permutations, the alternating groups, cayley’s theorem.

Cyclic groups:

Elementary properties, The classification of cyclic groups , sub groups of finite cyclic groups

REAL ANALYSIS (30 hours)

MODULE - III:

Real analysis: The Completeness Properties of R, Applications of the Supremum Property

(No question is to be set from this part)

Sequence: Sequences and their limits, limit theorems, Monotonic Sequences, Sub-sequences and the Bolzano-

Weirstrass theorem,The Cauchy’s Criterion, Properly divergent sequences.

Infinite series:

Introduction to a series, Absolute convergence, test for absolute convergence, test for non-absolute

convergence.

MODULE - IV: LIMITS AND CONTINUITY

Limits: Limits of functions- limit theorem-some extension of the limit concept

Continuous functions: Continuous functions, continuous functions on intervals, Uniform continuity.


1. A Text book of Mathematics-volume –II published by S.Chand &

company

2. A Text book of Mathematics-volume –II published by Deepti

publications

3. Introduction to Real analysis, by Robert G. Bartle and Donald R.

Sherbert , John Wiley

Reference books:

1. Topic in Algebra, by I.N.Herstein, Wilay eastern.

2. Contemporary abstract algebra by Joseph a Gallian, Narosa pubilications .

3. A course of Mathematical Analysis, Shanthi Narayan and P.K.Mittal,

S.Chand & Company.

4. Mathematical analysis by S.C.Malik and Savita Arora, Wiley Eastern Ltd.







______________

Total = 75 Marks






or creativity (20%)


allowed to

the module

Module - I 2 2 2 34






II B.Sc., MATHEMATICS, COURSE- IV

SYLLABUS FOR THE FOURTH SEMESTER

ABSTRACT ALGEBRA & REAL ANALYSIS W.E.F. 2014-2015

ABSTRACT ALGEBRA (30 hours)

MODULE - I:

Rings, Integral domain and Fields:

Definition and basic properties, divisors of zero and Cancellation laws, Integral domains, fields the

characteristic of a ring, some non – commutative rings, Examples.

Subrings:

Definition of subrings, ideals, quotient rings, Euclidean rings.

MODULE - II:

Homomorphism of rings:

Definition and elementary properties, Maximal and Prime ideals, Prime fields.

REAL ANALYSIS (25 hours)

MODULE - III:

Differentiation:

The derivative, Mean value theorems, L’Hospital Rule, Taylor’s Theorem.

MODULE - IV:

Riemann Integartion:

Riemann integration - Riemann integral, Riemann integrable functions, Fundamental theorem.


1. A Text book of Mathematics-volume –II published by S.Chand &

company .

2. A Text book of Mathematics-volume –II published by Deepti

publications.

Reference books:

1. Modern algebra Goyal and Gupta

2. Linear algebra- A Ramachandra rao & P.Bhima sankaram, Tata MC graw

hill publishing company, New Delhi


S.Chand & Company

4. Real analysis- J.N.Sharma, A.R.Vasista (1975)-Krishna prakashan media

(P) Ltd. New Delhi.







______________

Total = 75 Marks






or creativity (20%)


allowed to

the module

Module - I 2 2 2 34






III B.Sc., MATHEMATICS, COURSE-V

SYLLABUS FOR THE FIFTH SEMESTER

LINEAR ALGEBRA & VECTOR CALCULUS

W.E.F. 2014-2015

LINEAR ALGEBRA (30 hours)

MODULE - I:

Vector spaces, General properties of vector spaces, Vector subspaces, Algebra of subspaces, linear

combination of vectors. Linear span, linear sum of two subspaces, Linear independence and

dependence of vectors, Basis of vector spaces, Finite dimensional vector spaces, Dimension of a

vector space, Dimension of a subspace.

MODULE - II:

Linear transformations, linear operators, Range and null space of linear transformation, Rank and

nullity of linear transformations, Linear transformation as vectors, Product of linear

transformations, Invertible linear transformation.

Multiple Integrals: (25 hours)

Introduction, the concept of a plane, Curve, line integral- Sufficient condition for the existence of

the integral. The area of a subset of 2R , Calculation of double integrals, Jordan curve, Area, Change

of the order of integration.

MODULE - IV:

Double Integral as a limit:

Change of variable in a double integration, length of curves, surface areas.


1. A Text book of Mathematics-volume –III published by S.Chand &

company.

2. Linear algebra by J.N.Sarma & A.R.Vasista, Krishna prakasham Mandir

meerut-250002.


S.Chand & Company.

Reference books:

1. Linear Algebra by Kenneth Hoffman and Ray Kunze, Pearson Education

(low priced edition), New Delhi.







______________

Total = 75 Marks






or creativity (20%)


allowed to

the module

Module - I 2 2 2 34



Module - IV 1 2 2 32 Total Marks Including choice 130


III B.Sc., MATHEMATICS, COURSE-VI

SYLLABUS FOR THE SIXTH SEMESTER

LINEAR ALGEBRA & VECTOR CALCULUS W.E.F. 2014-2015

LINEAR ALGEBRA (35 hours)

MODULE - I:

The adjoint or transpose of a linear transformation, Sylvester’s law of nullity, characteristic values

and characteristic vectors , Cayley- Hamilton theorem, Diagonalizable operators.

MODULE - II:

Inner product spaces, Euclidean and unitary spaces, Norm or length of a vector, Schwartz

inequality, Orthogonality, Orthonormal set, complete orthonormal set, Gram - Schmidt

orthogonalisation process.

VECTOR CALCULUS: (30 hours)

MODULE - III:

Vector differentiation. Ordinary derivatives of vectors, Space curves, Continuity, Differentiability,

Gradient, Divergence, Curl operators, Formulas involving these operators.

MODULE - IV:

Vector integration, Theorems of Gauss and Stokes, Green’s theorem in plane and applications of

these theorems.



meerut-250002.

2. Vector Analysis by Murray. R.Spiegel, Schaum series Publishing

Company.

Reference books:


meerut-250002.

4. Text book of vector Analysis by Shanti Narayana and P. K. Mittal,

S. Chand & Company Ltd, New Delhi.







______________

Total = 75 Marks






or creativity (20%)


allowed to

the module

Module - I 2 2 2 34






III B.Sc., MATHEMATICS, COURSE-V (A)

SYLLABUS FOR THE FIFTH SEMESTER

NUMERICAL ANALYSIS W.E.F. 2014-2015

MODULE - I: (30 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, Ramanujan’s method, Muller’s method.

MODULE - II:

Errors in Numerical computations:

Numbers and their Accuracy, Errors and their Computation, Absolute, Relative and percentage

errors, A general error formula, Error in a series approximation.

Finite Differences:

Errors in polynomial interpolation, Forward differences, Backward differences, Central Differences,

Symbolic relations, Detection of errors by use of D.Tables, Differences of a polynomial.

MODULE - III: (30 hours)

Interpolation with equal intervals-central diffrence, interpolation formulae:

Newton’s formulae for interpolation, Gauss’s central difference formula, Stirling’s central

difference formula, Bessel’s diffrence formula, Everetts difference formula.

Interpolation with unequal intervals(unevenly spaced points)

Lagrange’s formula, Error in Lagrange’s formula, Derivation of governing equations, End

conditions, Divided differences and their properties, Newton’s general interpolation.

MODULE - IV:

Curve Fitting:

Least-Squares curve fitting procedures, fitting a straight line, nonlinear curve fitting, Curve fitting

by a sum of exponentials.


1. Numerical analysis by S.Ranganatham, M.V.S.S.N.Prasad, Dr.V.ramesh

babu. S.Chand & co.

2. Numerical analysis by Dr. A.Anjaneyulu, Deepthi Publications.

Reference books:

1. Numrical analysis by S.S.Sastry prentice Hall New Delhi.

2. Numerical analysis by Kamali suryanarayana S.Chand and co, New

Delhi.

3. Numrical Analsyis by Gupta & Malik, Krishna prakasham Mandir

meerut.







______________

Total = 75 Marks





give question based on remembrance and understanding (80%) and

application or creativity (20%)


allowed to

the module

Module - I 2 2 2 34






III B.Sc., MATHEMATICS, COURSE-V1 (A)

SYLLABUS FOR THE SIXTH SEMESTER

NUMERICAL ANALYSIS W.E.F. 2014-2015

MODULE - I: (10 hours)

Solution of simulations linear system of equations:

Direct methods, Matrix inversion method, Gaussian elimination method, Method of factorization,

Ill-conditioned linear systems. Iterative methods: Jacobi’s method, Gauss-siedal method.

MODULE - II: (15 hours)

Numerical diffeenciation:

Numrical differentiation using newton forward and backward formula, derivatives using central

diffrence formula, Maximum and minimum

MODULE - III: (15 hours)

Numerical Integration:

General quadrature formula, Trapezoidal rule, Simpson’s 1/3 –rule, Simpson’s 3/8 –rule, Boole’s

and Weddle’s rule, the eular and maclaurins formula of summation and quadrature, the eular

transormaion

MODULE - IV: (15 hours)

Numerical solution of ordinary differential equations:

Taylor’s Series method, Picard’s method of successive approximations, Euler’s method, Modified

Euler’s method, Runge – Kutta method.


1. Numerical analysis by S.Ranganatham, M.V.S.S.N.Prasad, Dr.V.ramesh

babu. S.Chand & co

2. Numerical analysis by Dr. A.Anjaneyulu, Deepthi Publications.

Reference books:

3. Numrical analysis by S.S.Sastry prentice Hall New Delhi.

4. Numerical analysis by Kamali suryanarayana S.Chand and co, New

Delhi.

5. Numrical Analsyis by Gupta & Malik, Krishna prakasham Mandir

meerut.







______________

Total = 75 Marks






or creativity (20%)


allowed to

the module

Module - I 2 2 2 34





department of mathematicsdrvskrishnagdc.edu.in/syllabus/mathematics_2014_15.pdf · 2016-10-27 ·...

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