PANJAB UNIVERSITY, CHANDIGARH

OUTLINES OF TESTS, SYLLABI AND COURSES OF READING FOR B.Sc. MATHEMATICS (SEMESTER SYSTEM)

UNDER THE FRAMEWORK OF NEP-2020

ACADEMIC SESSION 2025-2026

PREAMBLE

To teach the fundamental concepts of Mathematics and their applications. The syllabus pertaining to B.Sc. in the subject of Mathematics has been designed as per the provision of the NATIONAL EDUCATION POLICY-2020.

The syllabus contents are duly arranged unit-wise and contents are included in such a manner so that due importance is given to requisite intellectual skills and the demand of the academic environment.

PROGRAMME OUTCOME

The Department of Mathematics, Panjab University, Chandigarh offers a rigorous UG programme in Mathematics to provide an orientation of a wide range of essential courses in Mathematics.

The programme aims to achieve the following outcomes:

1. Develop a solid foundation in Mathematics, including Calculus, Algebra, Real Analysis, Numerical Analysis, Discrete Mathematics, and Statistics.
2. Students will cultivate critical thinking skills, enabling them to evaluate mathematical arguments, identify assumptions, and construct logical proofs.
3. Cultivate the integration of mathematical concepts and tools to solve real-world problems in various fields such as Finance, Data Analysis, Cryptography, and Scientific Research.
4. Enhance students’ analytical thinking and problem-solving skills, empowering them to tackle complex mathematical and computational challenges.
5. Students will develop logical reasoning skills and the ability to apply mathematical reasoning to solve complex problems and prove mathematical statements.
6. Prepare graduates for diverse career paths in fields such as Data Science, Actuarial Sciences, Financial Analysis, Research, and Academia.

The B.Sc. Mathematics programme at Panjab University aims to produce graduates who are well-equipped with a rigorous mathematical foundation and a wide variety of mathematical reasoning at an early stage.

As a result, every year a large number of UG students get enrolled into institutes of national and international repute.

The UG courses also equip students for careers in various fields besides Mathematics, including:

• Finance
• Banking
• Bureaucracy
• Actuarial Sciences
• Defence Services
• Research and Academia

Important Notes

1. The minor and major subjects opted by a student will remain the same for two consecutive semesters (i.e. Semester I & II, Semester III & IV, Semester V & VI, and Semester VII & VIII). No change in these subjects will be allowed during the running session.
2. The contact hours of AEC courses are doubled in order to meet the syllabus requirements for teaching and improving writing skills.
3. Only those students who have more than 75% CGPA till the 6th Semester will be allowed to undertake research work.
4. This paper is meant for those students who have less than 75% CGPA till the 6th Semester and are not eligible to opt for a research project.

Criteria for the Award of Certificate / Degree

1. Students exiting the programme after securing 48 credits will be awarded a UG Certificate in the relevant discipline/subject, provided they secure:
   • 4 credits in work-based vocational courses offered during summer term or internship/apprenticeship
   • 6 credits from skill-based courses earned during the first and second semesters
2. Students exiting the programme after securing 96 credits will be awarded a UG Diploma in the relevant discipline/subject, provided they secure an additional 4 credits in skill-based vocational courses offered during the second year or summer term.
3. Students completing the 3-year UG programme and securing 144 credits will be awarded a UG Degree in the relevant discipline/subject.
4. Students securing 192 credits and fulfilling the minimum credit requirement in the respective subject will be awarded a UG Degree (Honours) with Research.

Evaluation

1. There shall be one Mid-Term Examination carrying 20% marks in each semester.
2. There shall be continuous internal assessment for practicals carrying 20% marks.
3. Each practical examination shall be of 3 hours duration.
4. The End-Semester Examination shall carry 80% marks.

Pattern of End-Semester Question Paper

1. The question paper will consist of nine questions of equal weightage. Candidates will be required to attempt five questions.
2. There will be one compulsory question consisting of short-answer-type questions covering the entire syllabus, with no choice in this question.
3. The remaining eight questions will be divided into four units, with two questions from each unit.
4. Candidates are required to attempt one question from each unit along with the compulsory question.

Discipline Specific Core Courses

Semester	Course Code	Name of Course	Credits
I	MAT-DSC-1	Calculus	6
II	MAT-DSC-2	Algebra and Ordinary Differential Equations	6
III	MAT-DSC-3	Metric Spaces	6
III	MAT-DSC-4	Number Theory	6
IV	MAT-DSC-5	Linear Algebra	6
IV	MAT-DSC-6	Numerical Analysis	6
IV	MAT-DSC-7	Partial Differential Equations	6
V	MAT-DSC-8	Riemann Integration and Series of Functions	6
V	MAT-DSC-9	Group Theory	6
V	MAT-DSC-10	Multivariate Calculus	6
VI	MAT-DSC-11	Complex Analysis	6
VI	MAT-DSC-12	Rings and Modules	6
VI	MAT-DSC-13	Mechanics	6
VII	MAT-DSC-14	Topology	6
VII	MAT-DSC-15	Advanced Group Theory	6
VII	MAT-DSC-16	Classical Mechanics	6
VII	MAT-DSC-17	Probability and Statistics	6
VIII	MAT-DSC-18	Field Theory	6
VIII	MAT-DSC-19	Lebesgue Integration	6
VIII	MAT-DSC-20	Advanced Differential Equations	6
VIII	MAT-DSC-21	Operations Research	6

Minor Courses

(To be offered to the students of other departments)

Semester	Course Code	Name of Course	Credits
I	MAT-M-1	Calculus	6
II	MAT-M-2	Algebra and Ordinary Differential Equations	6
III	MAT-M-3	Partial Differential Equations and Numerical Analysis	6
IV	MAT-M-4	Number Theory and Group Theory	6
V	MAT-M-5	Mathematical Modelling	4
VI	MAT-M-6	Discrete Mathematics	4
VII	MAT-M-7	Special Functions and Integral Transformations	4
VIII	MAT-M-8	Advanced Calculus / Linear Algebra	4

The above Minor Courses will be offered only to those students who studied Mathematics in 10+2.

Skill Enhancement Compulsory Courses

(3 Credits)

Semester	Course Code	Name of Course
I	MAT-SEC-1	Discrete Mathematics
II	MAT-SEC-2	Working with Mathematical Softwares
IV	MAT-SEC-3	Programming in C

Multidisciplinary Courses

(3 Credits)

(To be offered to the students of other departments)

Semester	Course Code	Name of Course
I, II	MAT-IDC	Algebra and Geometry
III	MAT-IDC-3	Matrices

These IDC courses will be offered only to those students who did not study Mathematics after matriculation.

Students are not permitted to repeat any course they have already completed for credit.

Ability Enhancement Compulsory Courses

A student is required to take two Ability Enhancement Courses (Language) of 2 credits per semester in Semesters I and II from the pool of AECC courses offered by the University.

Value Added Courses

One Value Added Course of 2 credits per semester in Semesters I, II and V shall be chosen from the pool of VAC courses offered by the University.

The department may also offer some of the following Value Added Courses (VACs). These courses are not open to students of B.Sc. (Mathematics) or B.Sc. (Mathematics and Computing). However, the VAC course on Number Theory may be opted by students of B.Sc. (Mathematics and Computing).

Course Code	Name of Course
MAT-VAC-1	Logic and Sets
MAT-VAC-2	LaTeX and HTML Theory-I
MAT-VAC-3	Discrete Mathematics
MAT-VAC-4	Matrices and Their Applications
MAT-VAC-5	Graph Theory
MAT-VAC-6	Probability and Statistics
MAT-VAC-7	Number Theory

Students are not permitted to repeat any course they have already completed for credit.

MAT-DSC-1: Calculus

Credits: 6 (L=5; T=1; P=0)

Total Marks: 150 (Including Internal Assessment = 30)

Time Allowed for Examination: 3 Hours

Instructions for the Candidates and Paper Setters

Candidates will be required to attempt five questions out of nine questions carrying equal marks.

Question No. 1, covering the entire syllabus, will be compulsory.

There will be two questions from each unit, and students will be required to attempt one question from each unit.

Learning Outcome

The main goal of this course is to provide students with the fundamentals of sequences and series, differential calculus, and integral calculus for both real-valued and multivariable functions.

The course is designed to help students develop analytical and proof-writing skills, especially in Unit I and Unit III, rather than relying only on formulas and computations.

Unit-I

Differential Calculus

• Precise definition of limit
• Continuity
• One-sided limit
• Limits involving infinity
• Asymptotes of graphs
• Tangents and derivative at a point
• Derivative of a function
• Extreme values of functions
• Mean Value Theorem
• Monotone functions and first derivative test
• Test for concavity
• Tracing of curves

Scope: Sections 2.3 – 2.6, 3.1, 3.2, 4.1 – 4.4 of Textbook (A)

Unit-II

Integral Calculus

• Riemann sums
• Definite integrals
• Area between curves
• Volumes using cross sections
• Volumes using cylindrical shells
• Arc length
• Areas of surfaces of revolution

Scope: Sections 5.1, 5.6, 6.1 – 6.4 of Textbook (A)

Unit-III

Infinite Sequences and Series

• Sequences
• Infinite series
• Integral test
• Comparison test
• Root test
• Ratio test
• Alternating series
• Absolute and conditional convergence
• Power series
• Taylor and Maclaurin series
• Convergence of Taylor series
• Applications of Taylor series

Scope: Sections 10.1 – 10.10 of Textbook (A)

Unit-IV

Polar Coordinates

• Polar coordinates
• Graphing in polar coordinates
• Areas in polar coordinates
• Lengths in polar coordinates

Multivariable Functions

• Limits and continuity for functions of several variables
• Partial derivatives
• The chain rule
• Directional derivatives
• Gradient vectors
• Tangent planes
• Extreme values and saddle points

Scope: Sections 11.3 – 11.5, 14.2 – 14.7 of Textbook (A)

Essential Textbook

(A) G. B. Thomas, M. D. Weir and J. R. Hass, Thomas’ Calculus, 12th Edition, Pearson Education, 2014.

Further Readings

1. J. L. Taylor, Foundations of Analysis, American Mathematical Society, 2012.
2. S. Narayan, Integral Calculus, S. Chand and Company Ltd., 2001.
3. M. J. Strauss, G. L. Bradley and K. J. Smith, Calculus, 3rd Edition, Pearson Education, 2007.
4. H. Anton, I. Bivens and S. Davis, Calculus, 7th Edition, John Wiley and Sons, 2002.
5. R. Courant and F. John, Introduction to Calculus and Analysis (Volumes I & II), Springer-Verlag, Inc., 1989.

MAT-DSC-2: Algebra and Ordinary Differential Equations

Credits: 6 (L=5; T=1; P=0)

Total Marks: 150 (Including Internal Assessment = 30)

Time Allowed for Examination: 3 Hours

Instructions for the Candidates and Paper Setters

Candidates will be required to attempt five questions out of nine questions carrying equal marks.

Question No. 1, covering the entire syllabus, will be compulsory.

There will be two questions from each unit, and students will be required to attempt one question from each unit.

Learning Outcome

The objectives of this course are to introduce systems of linear equations and their solutions, polynomials and their roots, and matrix theory.

This course familiarizes students with algebra before introducing advanced abstract concepts such as groups, rings, and fields.

The course also introduces techniques for obtaining solutions to ordinary differential equations along with the theoretical concepts behind those techniques.

The emphasis is on combining abstract concepts with practical examples to strengthen conceptual understanding.

Unit-I

Linear Equations and Matrices

• Review of systems of linear equations
• General theory of systems of linear equations
• Rank of a matrix
• Row rank and column rank of a matrix
• Homogeneous and non-homogeneous systems of linear equations
• Matrices
• Inverse of a matrix
• Elementary linear transformations
• Determinants and their properties
• Cramer’s Rule
• Cayley-Hamilton Theorem

Unit-II

Complex Numbers and Polynomials

• Complex numbers and their properties
• Taking roots of complex numbers
• Operations on polynomials
• Divisors and greatest common divisor
• Roots of polynomials
• Fundamental Theorem of Algebra (without proof)
• Corollaries of the Fundamental Theorem
• Roots of third and fourth degree polynomials
• Bounds of roots
• Sturm’s Theorem
• Descartes’ Rule of Signs
• Approximation of roots

Unit-III

First Order Differential Equations

• Origin of Differential Equations
• Basic definitions
• Family of solutions
• Geometric interpretation
• Isoclines
• Initial and boundary value problems
• Basic Existence Theorem (Statement only)
• Equations of order one
• Separation of variables
• Exact equations
• Linear equations
• Integrating factors
• Bernoulli’s Equation

Unit-IV

Higher Order Differential Equations

• General linear equations
• General solutions
• Linear independence of solutions
• Differential operators
• Linear equations with constant coefficients
• Auxiliary equation
• Non-homogeneous equations
• Method of Undetermined Coefficients
• Variation of Parameters Method
• Nonlinear equations

Scope: Chapters 2, 3, 4, 5 and 9 of Textbook (A), and Chapter 1, Sections 2.1–2.6, 5.1–5.5, 6.1–6.9, 7.1–7.3, 7.5, 8.3, 8.5, 9.1–9.4 and Chapter 16 of Textbook (B).

Essential Textbooks

(A) A. Kurosh, Higher Algebra, MIR Moscow, 1982.

(B) E. D. Rainville, P. E. Bedient and R. E. Bedient, Elementary Differential Equations, 8th Edition, Pearson India, 2016.

Further Readings

1. Shanti Narayan and P. K. Mittal, A Textbook of Matrices, S. Chand and Company Limited, 2019.
2. S. L. Ross, Differential Equations, 3rd Edition, John Wiley and Sons, India, 2004.
3. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 7th Edition, John Wiley and Sons, Inc., 2001.
4. E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice Hall.