## University of Calcutta

### B.Sc Mathematics Syllabus [CBCS] Hons & General

(w.e.f. 2018)

In this semester, there are two CC papers, each of 100 marks (65+15**+20***=100) & of credits 6 (5+1*=6).

**Core Course-1: Calculus, Geometry & Vector Analysis**

*1 Credit for Tutorial

**15 Marks are reserved for Tutorial

***20 Marks are reserved for Internal Assessmen & Attendance (10 marks for each)

In this semester, there are two CC papers, each of 100 marks (65+15**+20***=100) & of credits 6 (5+1*=6).

**Core Course-3: Real Analysis**

**Core Course-4: Group Theory-I**

*1 Credit for Tutorial

**15 Marks are reserved for Tutorial

***20 Marks are reserved for Internal Assessmen & Attendance (10 marks for each)

In this semester, there are three CC papers each of 100 marks (65+15**+20***=100) & each of credits 6 (5+1*=6) and one SEC-A of marks 100(=80 + 20***) & of credits 2. A student has to opt for any one of the subjects available under each category.

**Core Course-5: Theory of Real Functions**

**Core Course-6: Ring Theory & Linear Algebra-I**

**Core Course-7: Ordinary Diﬀerential Equation & Multivariate Calculus-I**

**SEC-A:**

**C Programming Language****Object Oriented Programming in C++**

*1 Credit for Tutorial

**15 Marks are reserved for Tutorial

***20 Marks are reserved for Internal Assessmen & Attendance (10 marks for each)

In this semester, there are three CC papers each of 100 marks (65+15**+20***=100) & each of credits 6 (5+1*=6) and one SEC-B of marks 100(=80 + 20***) & of credits 2. A student has to opt for any one of the subjects available under each category.

**Core Course-8: Riemann Integration & Series of Functions**

**Core Course-9: Partial diﬀerential equation & Multivariate Calculus-II**

**Core Course-10: Mechanics**

**SEC-B:**

**Mathematical Logic****Scientiﬁc computing with SageMath & R**

*1 Credit for Tutorial

**15 Marks are reserved for Tutorial

***20 Marks are reserved for Internal Assessmen & Attendance (10 marks for each)

In this semester, there are two CC papers each of 100 marks (65+15**+20***=100) & each of credits 6 (5+1*=6) and two DSE of marks 100(65+15**+20***=100) & of credits 6(5+1*=6).

A student has to opt for any one of the subjectsin DSE-A(1) and any one in DSE-B(1) in Semester 5.

**Core Course-11: Probability & Statistics**

**Core Course-12: Group Theory-II & Linear Algebra-II**

**DSE-A(1): **

**Advanced Algebra****Bio Mathematics****Industrial Mathematics**

**DSE-B(1):**

**Discrete Mathematics****Linear Programming**

& Game Theory**Boolean Algebra**

& Automata Theory

*1 Credit for Tutorial

**15 Marks are reserved for Tutorial

***20 Marks are reserved for Internal Assessmen & Attendance (10 marks for each)

In this semester, there are three CC papers each of 100 marks (65+15**+20***=100 for CC 13 , 50+20**=70 for CC 14 and 30 marks for CC 14 Lab) & each of credits 6 (5+1*=6 for CC 13, 4+2=6 for CC 14) and two DSE of marks 100(65+15**+20***=100) & of credits 6.

**Core Course-13: Metric Space & Complex Analysis**

**Core Course-14: Numerical Methods**

**Core Course-14 Practical: Numerical Methods Lab**

**Discipline Speciﬁc Elective- A(2):**

- Diﬀerential Geometry
- Mathematical Modelling
- Fluid Statics & Elementary Fluid Dynamics

**Discipline Speciﬁc Elective-B(2):**

- Point Set Topology
- Astronomy

& Space Science - Advanced Mechanics

*1 Credit for Tutorial

**15 Marks are reserved for Tutorial

***20 Marks are reserved for Internal Assessmen & Attendance (10 marks for each)

A student has to opt for any one of the subjects available under each category.

**SEC-A (for Semester 3)**

- C Programming Language
- Object Oriented Programming in C++

**SEC-B (for Semester 4)**

- Mathematical Logic
- Scientiﬁc computing with SageMath & R

A student has to opt for any one of the subjects in DSE-A(1) and any one in DSE-B(1) in Semester 5. The student has to opt for any one of the subjects in DSE-A(2) and any one in DSE-B(2) in Semester 6.

## Model Question Papers

## Semester: 1

## Calculus, Geometry & Vector Analysis

- Hyperbolic functions, higher order derivatives, Leibnitz rule and its applications to problems of type
*e*sin x ,^{ax}*e*cos x,^{ax}*(ax+b)*sinx,^{n}*(ax+b)*cosx, curvature, concavity and points of inﬂection, envelopes, rectilinear asymptotes (Cartesian & parametric form only), curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves, L’Hospital’s rule, applications in business, economics and life sciences.^{n} - Reduction formulae, derivations and illustrations of reduction formulae of the type ∫
*sin*∫^{n}x dx,*cos*∫^{n}x dx,*tan*∫^{n}x dx,*sec*∫^{n}x d,*(logx)*∫^{n}dx,*sin*∫^{n}x sinmx dx,*sin*^{n}x*cos*^{m}x dx. - Parametric equations, parametrizing a curve, arc length of a curve, arc length of parametric curves, area under a curve, area and volume of surface of revolution.

**Rotation of axes and second degree equations**, classiﬁcation of conics using the discriminant, tangent and normal, polar equations of conics.**Equation of Plane**: General form, Intercept and Normal forms. The sides of a plane. Signed distance of a point from a plane. Equation of a plane passing through the intersection of two planes. Angle between two intersecting planes. Parallelism and perpendicularity of two planes.**Straight lines in 3D**: Equation (Symmetric & Parametric form). Direction ratio and direction cosines. Canonical equation of the line of intersection of two intersecting planes. Angle between two lines. Distance of a point from a line. Condition of coplanarity of two lines. Equation of skew lines. Shortest distance between two skew lines.**Spheres. Cylindrical surfaces**.**Central conicoids**, paraboloids, plane sections of conicoids, generating lines, classiﬁcation of quadrics, illustrations of graphing standard quadric surfaces like cone, ellipsoid. Tangent and normals of conicoids.

Triple product, vector equations, applications to geometry and mechanics — concurrent forces in a plane, theory of couples, system of parallel forces. Introduction to vector functions, operations with vector-valued functions, limits and continuity of vector functions, diﬀerentiation and integration of vector functions of one variable.

## Algebra

- Polar representation of complex numbers, n-th roots of unity, De Moivre’s theorem for rational indices and its applications. Exponential, logarithmic, trigonometric and hyperbolic functions of complex variable.
- Theory of equations : Relation between roots and coeﬃcients, transformation of equation, Descartes rule of signs, Sturm’s theorem, cubic equation (solution by Cardan’s method) and biquadratic equation (solution by Ferrari’s method).
- Inequality : The inequality involving AM ≥ GM ≥ HM, Cauchy-Schwartz inequality.
- Linear diﬀerence equations with constant coeﬃcients (up to 2nd order).

**Relation**: equivalence relation, equivalence classes & partition, partial order relation, poset, linear order relation.**Mapping**: injective, surjective, one to one correspondence, invertible mapping, composition of mappings, relation between composition of mappings and various set theoretic operations. Meaning and properties of f^{-1}(B), for any mapping f : X → Y and B ⊆ Y.- Well-ordering property of positive integers, Principles of Mathematical induction, division algorithm, divisibility and Euclidean algorithm. Prime numbers and their properties, Euclid’s theorem. Congruence relation between integers. Fundamental Theorem of Arithmetic. Chinese remainder theorem. Arithmeticfunctions, some arithmetic functions such as φ, τ, σ and their properties.

- Rank of a matrix, inverse of a matrix, characterizations of invertible matrices.
- Systems of linear equations, row reduction and echelon forms, vector equations, the matrix equation AX=B, solution sets of linear systems, applications of linear systems.