Mathematics I BCA TU First Semester
Course Title: Mathematics I (3 Cr.)
Course Code: CACS101
Year/Semester: I/I
Class Load: 4 Hrs. / Week (Theory: 3 Hrs., Tutorial: 1 Hr.)
Course Description
This course provides a fundamental understanding of algebra and analytical geometry, which are essential for students pursuing a career in computer applications. Topics covered include set theory, real and complex numbers, relations, functions, sequences and series, matrices, determinants, conic sections, vectors in space, and permutation and combination. These mathematical concepts form the foundation for further study in computing and help students develop critical problem-solving skills required for technical courses. Additionally, the course integrates practical applications using software tools like Matlab and Mathematica to enhance learning and demonstrate the real-world use of mathematical techniques.
Course Objectives
The primary objective of this course is to equip students with the essential mathematical skills needed to understand and excel in computer application courses. Specifically, students will:
- Gain proficiency in basic mathematical concepts that are pivotal to computing.
- Develop the ability to solve problems involving sets, functions, sequences, series, and matrices.
- Understand the geometric properties of conic sections and vectors in space, including their application in computer science.
- Apply mathematical reasoning to real-world problems, with a focus on computational tools like Matlab and Mathematica.
Course Contents
Unit 1: Set Theory and Real & Complex Numbers (7 Hrs.)
- Set Theory:
- Concepts, notations, and specification of sets
- Types of sets: finite, infinite, equal, etc.
- Operations on sets: union, intersection, difference, complement
- Venn diagrams and laws of algebra of sets (without proof)
- Cardinal number of sets and problems related to sets
- Real and Complex Numbers:
- Real number system, intervals, and absolute value
- Introduction to complex numbers and their geometric representation
- Algebraic properties of complex numbers: addition, multiplication, inverse, and absolute value
Unit 2: Relation, Functions, and Graphs (8 Hrs.)
- Relations:
- Ordered pairs, Cartesian product
- Domain, range, and inverse of a relation
- Types of relations: reflective, symmetric, transitive, equivalence
- Functions:
- Definition and properties of functions
- Domain and range of functions, inverse functions
- Special functions: identity, constant, algebraic (linear, quadratic, cubic), trigonometric
- Graphs of algebraic and trigonometric functions
- Exponential and logarithmic functions
- Composite functions and their mathematical properties
Unit 3: Sequence and Series (7 Hrs.)
- Sequence and Series:
- Types of sequences: arithmetic, geometric, harmonic
- Properties of sequences and relations among arithmetic mean (A.M.), geometric mean (G.M.), and harmonic mean (H.M.)
- Sum of infinite geometric series
- Taylor’s Theorem (without proof), Taylor’s series, and exponential series
Unit 4: Matrices and Determinants (8 Hrs.)
- Matrices:
- Introduction to matrices and types of matrices
- Matrix operations: addition, multiplication, and equality
- Determinants, transpose, minors, and cofactors
- Properties of determinants (without proof)
- Singular and non-singular matrices
- Adjoint and inverse of matrices
- Linear Transformations:
- Understanding orthogonal transformations and rank of matrices
- Practical application of matrices in computation (Matlab)
Unit 5: Analytical Geometry (8 Hrs.)
- Conic Sections:
- Definitions and properties of conic sections: circle, ellipse, parabola, hyperbola
- Equations and graphs of conic sections
- Classifying conic sections by eccentricity and solving related problems
- Polar equations of lines, circles, ellipses, parabolas, and hyperbolas (Mathematica)
- Vectors in Space:
- Introduction to vectors in space
- Vector operations: length, distance between points, unit vector, null vector
- Scalar product of two and three vectors and their geometric interpretations
- Practical applications in computing (Matlab)
Unit 6: Permutation and Combination (7 Hrs.)
- Permutation and Combination:
- Basic principle of counting
- Permutation of distinct objects, objects not all distinct, circular arrangements, and repeated objects
- Combination of distinct objects and properties of combinations
- Applications in solving combinatorial problems
Teaching Methods
The course will be taught using a combination of lectures, tutorials, and practical exercises. Instructors will employ problem-solving techniques and use software tools like Matlab and Mathematica to demonstrate the real-world application of the mathematical concepts discussed. The emphasis will be on interactive learning, with students encouraged to participate in class discussions, solve problems, and engage in hands-on activities to strengthen their understanding.
Evaluation
- Internal Evaluation (40%)
- Attendance: 5%
- Participation and presentation in class: 5%
- Writing assignments and problem sets: 15%
- Mid-term test: 15%
- Final Evaluation (60%)
- Comprehensive understanding of course material
- Application of mathematical concepts to solve problems
- Written exam testing comprehension, vocabulary, and problem-solving skills
- Practical application of matrices, determinants, and vectors using Matlab and Mathematica
Text and Reference Books
Textbook
- Thomas, G. B., Finney, R. S. Calculus with Analytic Geometry, Addison-Wesley, 9th Edition.
Reference Books
- Bajracharya, D. R., Shreshtha, R. M. & et al, Basic Mathematics I, II, Sukunda Pustak Bhawan, Nepal
- Budnick, F. S., Applied Mathematics for Business, Economics, and the Social Sciences, McGraw-Hill Ryerson Limited.
This syllabus ensures that students acquire the essential mathematical knowledge required for their computer science studies, helping them build a solid foundation in mathematics that will support their understanding of various computing concepts and techniques.