📐 Mathematics I BCA 104 | Semester I/I
📖 Course Description
The course covers real numbers, functions and graphs, sequences and series, matrices and determinants, analytical geometry, vector spaces, and permutations & combinations. It provides the essential mathematical foundation for computer applications, programming, and data analysis. Teachers are encouraged to connect mathematical concepts with programming and real-world problem-solving during both theoretical and practical sessions.
🎯 Course Objectives
- Understand real numbers, properties, intervals, complex numbers, and functions
- Solve problems on arithmetic, geometric, harmonic sequences & series
- Apply matrix algebra, determinants, eigenvalues, and transformations for computer graphics
- Analyze conic sections (circle, parabola, ellipse, hyperbola) and polar equations
- Work with vectors, vector spaces, linear dependence/independence, orthogonality
- Compute permutations and combinations using counting principles
| 📌 Major Topic | Core Concepts & Applications |
|---|---|
| Logic, Relations & Functions | Elementary logic, real number axioms, absolute value, complex numbers, Cartesian product, equivalence relation, composite & inverse functions, graphs |
| Sequences & Series | Arithmetic, Geometric, Harmonic progressions, AM, GM, HM, sum of n natural numbers, squares & cubes, Arithmetico-Geometric series |
| Matrices & Determinants | Algebra of matrices, determinants, inverse, rank, transformations (linear/orthogonal), eigenvalues, eigenvectors, application to computer graphics |
| Analytical Geometry | Conic sections: circle, parabola, ellipse, hyperbola, eccentricity, polar equations |
| Vectors & Vector Spaces | Vector operations, scalar/vector product, vector space, subspace, linear combination, independence, norm, orthogonality |
| Permutations & Combinations | Counting principle, permutation of n objects (all distinct / not all distinct), circular permutations, combinations |
📚 Detailed Syllabus
Unit 1: Logic, Relations, Functions & Graphs10 Hrs
- Elementary logic, real number system: field and ordered axioms
- Intervals, rational & irrational numbers, absolute value & properties
- Complex numbers and their properties
- Ordered pairs, Cartesian product, relation, equivalence relation
- Functions: composite functions, domain & range, inverse function
- Types of functions: identity, constant, algebraic, trigonometric, exponential, logarithmic
- Graphs of different functions (polynomial, exponential, trigonometric)
Unit 2: Sequence and Series7 Hrs
- Sequence and Series: Arithmetic, Geometric, Harmonic progressions & properties
- Arithmetic Mean (AM), Geometric Mean (GM), Harmonic Mean (HM) — relation among them
- nth term and sum of arithmetic series; finite & infinite geometric series
- Sum of first n natural numbers, sum of squares, sum of cubes
- Arithmetico-Geometric series
Unit 3: Matrices and Determinants10 Hrs
- Definitions & types of matrices, algebra of matrices
- Determinants, transpose, minors, cofactors, properties of determinants
- Singular, non-singular, adjoint, inverse of a matrix
- Rank of a matrix
- Linear and orthogonal transformation, composite transformation — applications to computer graphics
- Characteristic equations, eigenvalues and eigenvectors
Unit 4: Analytical Geometry7 Hrs
- Defining terms of conic sections
- Standard equations of circle, parabola, ellipse, hyperbola and their graphs
- Conic sections in terms of eccentricity
- Polar equations of circle, ellipse, parabola, and hyperbola
Unit 5: Vectors and Vector Spaces7 Hrs
- Definition of vector and scalar, magnitude, distance, unit vector
- Operations: addition, subtraction, scalar multiplication
- Scalar product (dot) and vector product (cross) of two and three vectors — geometric interpretations
- Vector space, subspace
- Linear combination, linear dependence & independence
- Scalar product, norm, orthogonality
Unit 6: Permutations and Combinations7 Hrs
- Basic counting principle
- Deduction method for formulas of permutations & combinations
- Relation between permutations and combinations
- Permutation of n objects (all different / not all different) — taking all at a time
- Circular permutations
- Combination of different objects and their properties
🧪 Laboratory / Computational Work
Students are expected to implement numerical and algebraic problems using Python, MATLAB, or Mathematica to compare computational results with pen-and-paper solutions. Emphasis on:
- Matrix operations and eigenvalue computation using NumPy
- Plotting functions and conic sections (matplotlib / MATLAB)
- Sequence and series summation, verification of AM, GM, HM relations
- Vector operations and visualization
- Permutation/combination simulations
✔ Practical sessions: bridging math with programming for data science & graphics
📚 Required Readings & References
- Bajracharya, B. C. (2082) – Basic Mathematics, Sukunda Publication
- Boice, W.E., Diprima, R.C. – Elementary Differential Equations, John Wiley & Sons
- Budnick, F. S. (2019) – Applied Mathematics for Business, Economics & Social Sciences, McGraw-Hill
📖 Additional materials: online resources for Python (NumPy/SciPy), lecture notes, and exercise handouts.
📊 Examination Scheme (Indicative): Theory (60% external + internal) evaluates conceptual clarity and problem-solving. Lab component focuses on computational implementations using Python/MATLAB. Internal assessment includes assignments, quizzes, and lab reports.