CBCS PATTERN SYLLABUS
Semester III
Mat/ Sem III/ C 5 – Theory of Real Functions
Instruction: -
Ten questions will be set. Candidates will be required to answer Seven Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions candidates will be required to answer 6 questions selecting at least one from each group. Each question will be of 10 marks.
GROUP - A
Limit of functions:
Limit, algebra of limit of functions. Continuity and discontinuities, algebra of continuous functions. Intermediate value theorem, location of roots theorem, preservation of intervals theorem. Uniform continuity, functions of bounded variations. 3 Questions
GROUP - B
Derivability:
Derivability, relationship with continuity, Rolle’s theorem, Lagrange’s and Cauchy Mean value theorem, Taylor’s theorem, Maclaurin’s theorem, remainder after n terms, power series expansion of (1+x )^n , sin x ,cos x , log x , e^x using suitable remainder after n terms. 3 Questions
GROUP - C
Riemann Integration:
Definition, Darboux theorem I and II, integrability conditions. Particular classes of bounded integrable functions. Primitive, Fundamental theorem, First and Second Mean value theorem. 3
Questions Books Recommended:
1. Introduction to Real Analysis- R Bartle & D R Sherbert
2. Elements of Real Analysis- Shanti Narayan & M D Raisinghania.
CBCS PATTERN SYLLABUS
Semester III
Mat / Sem III / C6 - Group Theory & Matrices
Instruction: -
Ten questions will be set. Candidates will be required to answer Seven Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions candidates will be required to answer 6 questions selecting at least one from each group. Each question will be of 10 marks.
GROUP - A 5 Questions
Groups: Preliminary results, equivalent definitions, sub groups, Cyclic Group and its subgroups, Cosets of a subgroups, Lagrange’s Theorem and it’s applications.
Normal subgroups, Quotient group and homomorphism, Fundamental theorem of homomorphism.
Permutations, Permutation group, Symmetric and Alternating group. Caylay’s Theorem.
GROUP - B 4 Questions
Different types of Matrices, Algebra of Matrices, Adjoint and inverse of a Matrix, different ways of finding inverses.
Elementary row and column operations. Elementary matrices, equivalent matrices, Rank of a matrix, Invariance of rank through elementary row/column operations, rank of sum and product of matrices and related theorems.
Solution of a system of linear equations via matrix methods, Consistency, Inconsistency.
Books Recommended :
1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.
2. Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa Publishing House, New Delhi, 1999.
3. Topics in Algebra : I N Herstein .
4. Basic Abstract Algebra: P B Bhattacharya, Cambridge Univ. Press.
5. Matrices – Shanti Narayan.
6. Matrices – A R Vashishtha.
CBCS PATTERN SYLLABUS
Mat / Sem III / C7 – Differential Equation
Instruction: -
Ten questions will be set. Candidates will be required to answer Seven Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions candidates will be required to answer 6 questions. Each question will be of 10 marks.
Differential equation of first order but not of first degree, Clairaut’s form, singular solutions. Differential equation with constant co-efficients. 2 Questions
Orthogonal trajectories and its simple application in geometrical and mechanical problems. 1 Question
Linear differential equations of higher order with constant coefficients. Differential equations with variable coefficients. 2 Questions
Linear differential equations of second order by method of variation of parameter and by change of independent variable. 2 Questions
Total differential equation in three independent variables. 1 Question Partial differential equation: Lagrange’s linear partial differential equation, Charpit’s method. 1 Question
Books Recommended:
1. Differential Equations – M D Raisinghania.
CBCS PATTERN SYLLABUS
Mat/ Sem III/ GE 3 – MATHEMATICS – III
Instruction for Generic Elective: -
Eleven Questions will be set. Candidates will be required to answer Eight Questions.
Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 3 marks. Out of remaining 10 questions will be required to answer 7 questions selecting at least one from each group. Each question will be of 10 marks.
GROUP - A
REAL ANALYSIS III
Continuity & Derivability of function of one variable, relationship with continuity, Rolle’s theorem, Lagrange’s Mean Value theorem, Taylor’s and Maclaurin's theorem with Rn. 2 Questions
SET THEORY I
Indexed family of sets, Generalised set of operations & Demorgan laws, Set mapping. Equivalence relation and related fundamental theorem of partition. 2 Questions
COMPLEX VARIABLE I
Real functions of two variables: Simultaneous and iterated limits: Continuity, partial derivatives, Differentiability and related necessary and sufficient conditions. 2 Questions
GROUP - B
ABSTRACT ALGEBRA I
Binary operations, Notion of group, Abelian group and non-Abelian group with examples. Uniqueness of identity element and inverse elements in a group, different ways of defining a group, concept of Subgroup and cyclic group, Cosets, Lagrange’s theorem. 2 Questions
DIFFERENTIAL EQUATIONS
Differential equations of first order and higher degree, Clairaut’s form, singular solution, orthogonal trajectories. 2 Questions
SEMESTER 4 (SYLLABUS)
