• Understand and define complex numbers, including integral powers of 𝑖i.
• Execute addition, subtraction, multiplication, and division of complex numbers.
• Investigate properties of complex numbers, including conjugates and moduli.
• Represent complex numbers using Argand's diagram, express in polar form, and apply De Moivre's theorem

Unit I: 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.

• Learn the concept and conditions of convergence tests for infinite series.
• Use direct and limit comparison tests to determine series convergence or divergence.
• Apply P-series, De Alembert’s ratio, and alternating series tests to analyze series convergence.

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

1. Introduction
2. Convergence test of infinite series (statement only).
3. Direct comparison test, Limit comparison test (statement only).
4. P-series test, De Alembert’s ratio test, and Alternating series test (statement only).

• Learn the concepts, properties, and applications of definite and improper integrals.
• Calculate areas (quadrature) and curve lengths  (rectification)  for  functions
• 𝑦=𝑓(𝑥)y=f(x) using antiderivative techniques
• Study and apply the properties of Beta and Gamma   functions   in   different mathematical contexts.

Unit III: Application of Antiderivative [7 Hrs.]

1. Definite integral
2. Properties of the definite integral
3. Improper Integral
4. Quadrature 
5. Rectification 
6. Beta and Gamma function.

• Learn the concept and importance of partial derivatives
• Use rules of partial differentiation for accurate calculations
• Identify maxima and minima for functions of two variables

Unit IV: 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.

• Understand the order and degree of differential equations
• Solve first-order and first-degree differential equations using     various methods.
• Solve second-order linear differential equations with constant coefficients and address initial and boundary value problem.

Unit V: 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.

• Identify prime numbers and apply them in problem-solving
• Master division algorithm, GCD, and LCM for integer problem-solving.
• Apply modular arithmetic in cryptography for secure message transmission

Unit VI: 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.

• Classify functions as even or odd and analyze their properties.
• Represent periodic functions     using Fourier series and determine coefficients.
• Apply Fourier integrals to represent non-periodic functions

Unit VII: 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 integral

Semester-End examination

50

50