Module Code and Title:       MAT101          Mathematics

Programme:                          BCA

Credit Value:                         12

Module Tutor:                       Somnath Chaudhuri

General Objective: This module aims to build on the mathematical knowledge the students have from their Class XII mathematics, especially on the branches of Algebra, Calculus and Numerical Methods. Students will be trained to apply the knowledge gained in problem solving, with a view to support the kind of mathematical thinking necessary for further modules in mathematics and programming.

Learning Outcomes – On completion of the module, learners will be able to:

1. Solve cubic and bi-quadratic equations under certain conditions.
2. Analyse different types of matrices.
3. Solve the Eigen roots and Eigen vectors of a given square matrix.
4. Solve problems related to maxima and minima for functions up to two independent variables.
5. Solve problems on limit, differential and integral calculus.
6. Evaluate and determine the value of a given definite integral.
7. Solve a given differential equation up to second order.
8. Describe the Regula-Falsi and Newton – Raphson’s method.
9. Calculate numerical integrals using trapezoidal and Simpson’s rule.

Learning and Teaching Approach:

Assessment Approach:

A. Individual Assignment: Portion of Final Mark: 20%

Students should submit two assignments relate to Matrices, Calculus and Numerical Analysis to obtain this 20%. The first one will be before the midterm and it constitutes half of the total 20% allocated. Students will be submitting an assignment based on theory of equation and matrix. The next assignment for the other half of the total marks will be done after the midterm, which will be on topics from calculus and numerical analysis. For each, 40% will be awarded for solving the problem, 40% for analysing the problem and 20% for the overall report. 3-4 numerical problems will be assigned to them to solve.

Activity: Different problems relate to relate to Matrices, Calculus and Numerical Analysis will be assigned to the students. Students have to analyse those and solve them. Solutions should be submitted in hardcopy format with proper logical expressions.

B. Class Test: Portion of Final Mark: 10%

This is a written test conducted within the class for duration of 30-40 minutes. There will two such tests, one before midterm comprising of topics from the beginning to the quarter point of the subject matter and the other after the midterm comprising of topics from after the midterm to quarter pointer after midterm. Each class test will consist of 3-4 numerical problems. The students have to solve those problems in the class within predefined time.

C. Individual Presentation: Portion of Final Mark: 5%

The presentation will be conducted within the class hours. Students are expected present the topic assigned to them respectively. The presentation will be approximately 10-12 minutes, and include power points slides. 30% will be awarded for content of the presentation, 30% for preparedness, 10% for timing, 15% for handling of Q&A session and 15% for presentation skill.

Activity: A few topics will be taught in the class. Then, when students have adequately grasped the concept of the topic, they have to prepare a presentation on its real-time application. Topics will be on numerical analysis. The presentation will be evaluated by the contents of the presentation and individual presentation skill.

D. Class Participation: Portion of Final Mark: 5%

This component assesses the student’s overall performance in class throughout the semester. This portion is awarded for the active participation in class activities like discussion and question-answer sessions.

Activity: Questions based on previous class are asked to the whole class at the beginning of every lecture session. Students who give answers are marked (similarly discussion participation is also marked), and these are counted at the end of the semester to assign class participation marks for every student.

E. Midterm Exam: Portion of Final Marks: 20%

This a college wide examination conducted at the half-way into the semester. This examination is conducted for 1 hour and 30 Minutes and it includes all topics till the half-way point in the subject matter.

Prerequisites:

Subject Matter:

1. Theory of equations
   • Relation between the roots and coefficients
   • Solving cubic and bi quadratic equations when some condition on the roots is given
   • Reciprocal equations
   • Transformation of equations
2. Matrices
   • Types of Matrices – Symmetric, Skew symmetric, Hermitian, Skew Hermitian, Orthogonal and Unitary matrices
   • Determinants
   • Characteristic Equation and Cayley-Hamilton theorem (Proof not required)
   • Rank of a matrix
   • Eigen values and Eigen vectors
   • Solution of a system of linear equations
3. Calculus:
   • Limits of function and continuity, fundamental properties of continuous functions (without proof)
   • Geometric meaning of derivative and differential, rules of differentiation, successive differentiation
   • Rolle’s theorem, mean value theorem
   • Taylor’s and Maclaurin’s theorems with Cauchy’s and Lagrange’s forms of reminder
   • Taylor’s series, function of several variables, partial derivatives, total differential
   • Euler’s theorem on homogeneous functions of two variables. Maxima and Minima of functions of two variables
   • Introduction to: Application to plane curves
   • Integral calculus: Rules of integration of indefinite integrals, solution of definite integrals and their elementary properties, idea of improper integral
4. Numerical Analysis
   • Solution of numerical algebraic and transcendental equations using Regula – Falsi method and Newton – Raphson’s method
   • Solution of simultaneous linear algebraic equations using Gaussian elimination and Gauss – Seidal methods

Reading List:

1. Essential Reading
   • Advanced Engineering Mathematics, (2011), Erwin Kreyszig, Wiley India Edition.
   • Atkinson & Kendell E. (2005). An Introduction to Numerical Analysis, John Wiley and Sons, Inc.
   • Marjorie Darrah and Edgar Fuller.(2009).Introduction to Calculus
   • Hari Kishan. (2013).Theory of Equations
2. Additional Reading
   • Apostol, T. M. (2005). Calculus – Volume I. John Wiley and Sons, Inc.
   • Lang, S. (1988). Introduction to Linear Algebra, (2nd Edition). Springer
   • Pillai, M.T.K., Natarajan S, (1986), Analytical Geometry of Two Dimensions. Chennai
   • James Victor Uspensky, Theory of Equations, McGraw-Hill
   • Leonard Eugene Dickson, Elementary Theory of Equations, BiblioBazaar
   • Joel N. Franklin, Matrix Theory, Dover Publications

Date: May 30, 2015