Specific Objectives 

Content

• To describe the need for extending the set of real numbers to the set of complex numbers;

• Identify the real and complex part of the complex number

• Define and find the modulus and conjugate of complex numbers;

• Represent complex numbers in the polar form.

Unit 1: Complex Numbers     [8 hrs.]

1. Definition of a complex number, integral powers of i.

2. Algebra of complex numbers (sum, difference, multiplication, division). 

3. Properties of complex numbers (without proof), conjugate of a complex number and its properties.

4. Modulus of a complex number and its properties (without proof), representation of a complex number by a point in a plane (Argand's diagram). 

5. Polar representation of a complex number.

6. Square roots of a complex number (only Cartesian form)

7.  De Moivre's theorem (statement only) and its application.

• Identify the convergence and divergence of an infinite series

• Explain the concept of convergence and divergence and use it in real-life problems.

Unit 2: Infinite Sequence and Series    [7 hrs.]

2.1. Introduction

2.2. Convergence test of infinite series (statement only).

2.3. Direct comparison test, Limit comparison test (statement only).

2.4. P-series test, De Alembert’s ratio test, and Alternating series test (statement only).

Specific Objectives

Contents

• To calculate the area with the help of definite integral

• Find the arc length by using integration.

• Calculate the value of integrals with the help of Beta and Gamma function.

Unit 3: Application of Antiderivative [7 Hrs.]

1.  Definite integral

2. Properties of the definite integral

3. Improper Integral

4. Quadrature [y=f(x)]

5.  Rectification [y=f(x)]

6.  Beta and Gamma function.

Specific Objectives

Contents

• To calculate the partial derivatives of multivariable functions.

• To know about the basic application of partial derivatives.

Unit 4: Optimization: Functions of several variables [6 hrs]

1.  Introduction 

2.  Partial derivative

3.  Rules of partial differentiation

4.   Maxima and minima for the function of two variables.

Specific Objectives

Contents

• Identify the general and particular solution of the differential equation.

• Formulate a differential equation that describes how the system changes in time.

Unit 5: Ordinary Differential Equation

                                                             [7 Hrs.]

1.  Introduction.

2.  Order and degree of differential equation.

3.  Solution of the first order and first-degree differential equation.

4. Variable separation, homogeneous, linear differential equation.

5.  Second-order linear differential equation with constant coefficients.

6.  Initial and boundary value problems.

Specific Objectives

Contents

• Foundation of Computer Programming

• To the study of the integers and their properties.

Unit 6:  Integers and Division          [6 Hrs.]

1.  Introduction 

2.  Division, primes, the fundamental theorem of arithmetic (statement only)

3.  The infinitude of primes

4.  The division algorithm, GCD and LCM

5.  Modular arithmetic

6.  Application of congruence’s Cryptology.

Specific Objectives

Contents

• Learn the required conditions for deriving the Fourier series

• Understand the meaning of Fourier sine and cosine series

• Able to calculate the Fourier integral of a function.

Unit 7:  Fourier Series and Integrals [7 Hrs.]

1.  Introduction 

2.  Even and odd function 

3.  Periodic function

4.  Fourier series and Fourier coefficients(without proof)

5.  Fourier sine and cosine series

6.  Fourier integral

7.  Fourier sine and cosine integrals

External Evaluation

Marks

Internal Evaluation

Weight

Marks

Semester-End examination

50

Theory

50

Attendance & Class Participation

10%

Assignments

20%

Presentations/Quizzes

10%

Internal Assessment

60%

Total External

50

Total Internal

50

Full Marks: 50 + 50 = 100