### Course Outcome

#### B.Sc. Mathematics

 Semester Course code Course title Course outcomes I (core) MM 1141 Methods of Mathematics CO 1: Understanding the concepts of fundamental methods of solving problems like limit, continuity and differentiation CO 2: Finding absolute maximum and minimum of functions CO 3: Understanding application of extrema problems to Economics CO 4: Understanding various Integration Techniques CO 5: Finding Area under a curve through integration, work done, Pappu‟s Theorem and understanding the concept of hyperbolic functions and their applications I (compl) MM 1131.1 Calculus with applications in Physics-I CO 1: Understands the special points of a function, curvature and applies Rolle‟s Theorem and Mean value theorem on functions CO 2: Understands integration by parts and reduction formula CO 3: Understands the concept of infinite and improper integrals CO 4: Applies the integration techniques to evaluate the area, volume etc CO 5: Understands various types of Series such as arithmetic series, geometric series, the difference method, series involving natural numbers and transformation of series CO 6: Understands Convergence of infinite series (Absolute and conditional convergence) and series containing only real positive terms; alternating series test CO 7: Understands Operations with series (Sum and product) Convergence of power series and Taylor series CO 9: Understands Scalars and vectors, Addition and subtraction of vectors, Multiplication by a scalar, Basis vectors and components, Magnitude of a vector, Multiplication of vectors CO 10: Understands Equations of lines, planes and spheres, using vectors to find distances from Point to line; point to plane; line to line and line to plane I (compl) MM 1131.2 Calculus with applications in Chemistry I CO 1: Understands the special points of a function, curvature and applies Rolle‟s Theorem and Mean value theorem on functions CO 2: Understands the Basic operations of complex numbers, modulus and argument; multiplication; complex conjugate, Polar representation of complex numbers and de Moivers theorem CO 3: Understands the trigonometric identities and finding the nth roots of unity; solving polynomial equations, Complex logarithms and complex powers, CO 4: Applies the complex numbers to differentiation and integration, Definition of hyperbolic and trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions CO 5: Understands Scalars and vectors, Addition and subtraction of vectors, Multiplication by a scalar, Basis vectors and components, Magnitude of a vector, Multiplication of vectors CO 6: Understands Equations of lines, planes and spheres, using vectors to find distances from Point to line; point to plane; line to line and line to plane CO 7: Understands integration by parts and reduction formula CO 8: Understands the concept of infinite and improper integrals CO 9: Applies the integration techniques to evaluate the area, volume etc I (compl) MM 1131.3 Algebra, Geometry and Trigonometry CO 1: Understands preliminary algebra including calculators, algebraic expressions, quadratic equations, etc. CO 2: Understands plane geometry – lines and angles, circles, triangles, straight line, ellipse, etc. CO 3: Understands basic trigonometric functions, vectors and oblique triangles, law of sines, etc. II (core) Mm 1221 Foundations of Mathematics CO 1: Understanding the concepts of sets, functions and the way in which a mathematician formally makes statements and proves or disproves it CO 2: Visualize some of the properties of graphs of elementary functions CO 3: Understanding foundations of co-ordinate geometry CO 4: Understand the application of polar coordinates in Astronomy CO 5: Understanding three-dimensional rectangular co-ordinate system and basic operations on vectors II (compl) MM 1231.1 Calculus with applications in Physics-II CO 1: Apply Integral calculus and vectors to problems in chemistry CO 2: Use integration to find the area and volume of a surface of revolution CO 3: Evaluate multiple integrals CO 4: Solving first order and second order linear differential equations CO 5: Identify the Equations of different types of conics in Cartesian and polar coordinates and sketch them II (compl) MM 1231.2 Calculus with applications in Chemistry II CO 1: Understands the total differential and total derivative, Exact and inexact differentials, theorems of partial differentiation, CO 2: Understands the chain rule, Change of variables, Taylors theorem for many-variable functions CO 3: Understands the Stationary values of many-variable functions, Stationary values under constraints CO 4: Understands various types of Series such as arithmetic series, geometric series, the difference method, series involving natural numbers and transformation of series CO 5: Understands Convergence of infinite series (Absolute and conditional convergence) and series containing only real positive terms; alternating series test CO 6: Understands Operations with series (Sum and product) Convergence of power series and Taylor series CO 7: Understands the Differentiation of vectors, Integration of vectors, Space curves, Vector functions of several arguments, Surfaces, Scalar and vector fields CO 8: Gets the knowledge of Vector operators like Gradient, divergence and curl, Cylindrical and spherical polar coordinates CO 9: Understands Double integrals, Triple integrals, Applications of multiple integrals (Areas and volumes), Change of variables in multiple integrals and properties of Jacobians II (compl) MM 1231.3 Calculus and Linear Algebra CO 1: Solve special types of first order equations CO 2: Solve second order linear differential equation, homogeneous and non-homogeneous equation. CO 3: Solve second order equations by operator method. CO 4: Solve Euler, Cauchy and Legender equations CO 5: Solve system of linear equations CO 6: Compute the rank of a matrix III (core) MM 1341 Elementary Number Theory and calculus I CO 1: Understanding the fundamental facts in elementary Number Theory CO 2: Understand the physical and geometrical interpretations of vectors. CO 3: Explain more properties of curves in three-dimension space using the concepts of differentiability CO 4: Visualizing functions of more than one variable, sketching, contours and level surface plotting CO 5: Understanding limits and continuity of multivariable functions, partial derivatives and its geometrical interpretation CO 6: Solving extremum problems with constraints using Lagrange multipliers III (compl) MM 1331.1 Calculus and Linear algebra CO 1: Solve special types of first order equations CO 2: Solve second order linear differential equation, homogeneous and non-homogeneous equation. CO 3: Solve second order equations by operator method. CO 4: Solve Euler, Cauchy and Legender equations CO 5: Solve system of linear equations CO 6: Compute the rank of a matrix CO 7: Determine whether a square matrix is diagonalizable and compute its diagonalization if it is CO 8: Understand the relation between roots and coefficients of a polynomial and apply these relations to solve polynomial Equations CO 9: Characterise roots of a polynomial. CO 10: Calculate approximate roots of a polynomial equation using bisection and Newton Raphson method III (compl) MM 1331.2 Linear Algebra, Probability Theory and Numerical Methods CO 1: Understands row reduction of Matrices, Determinants, Cramer's rule for solving system of equations CO 2: Understands vectors, lines and planes, linear combinations, linear functions, linear operators, linear dependence and independence, special matrices like Hermitian matrices and formulas CO 3: Understands linear vector spaces, eigen values and eigen vectors, diagonalizing matrices and applications of diagonalization CO 4: Understands the Basics of statistics such as Sample Space, Probability Theorems, Methods of Counting Random Variables CO 5: Understands the Continuous Distributions, Binomial Distribution, The Normal or Gaussian Distribution and the Poisson Distribution CO 6: Understanding the Algebraic and transcendental equations Convergence of iteration schemes CO 7: Solves the Simultaneous linear equations using Gaussian elimination, Gauss-Seidel iteration CO 8: Evaluates integrals using Numerical integration techniques such as Trapezoidal rule Simpsons rule; Gaussian integration; Monte Carlo methods CO 9: Understands Finite differences, Differential equations; Taylor series solutions; prediction and correction; Runge-Kutta methods III (compl) MM 1331.3 Complex Numbers, Algebra and Calculus CO 1: Understands the algebra of Complex numbers, point representation and its vector and polar form CO 2: Understands the concept of limit and continuity of functions of complex variable Prove the Cauchy-Riemann equations IV (core) MM 1441 Elementary Number Theory and calculus Ii CO 1: Defining the congruence relation and the congruence classes in integers CO 2: Understanding Chinese remainder theorem and its applications CO 3: Finding double and triple integrals and their applications CO 4: Evaluating the integrals of vector valued functions CO 5: Understanding the concept of Divergence Theorem, Gauss Law, Stoke‟s Theorem and its applications IV (compl) MM 1431.1 Complex Analysis, Special Functions and Probability Theory CO 1: Demonstrate accurate and efficient use of complex analysis techniques CO 2: Apply problem-solving using complex analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts CO 3: Evaluate integrals using Cauchy‟s Residue integration method CO 4: Understands the Factorial Function, the Gamma Function; Recursion Relation, the Gamma Function of Negative Numbers, some Important Formulas Involving Gamma Functions, Beta Functions, Beta Functions in Terms of Gamma Functions CO 5: Understands the Basics of statistics such as Sample Space, Probability Theorems, Methods of Counting Random Variables CO 6: Understands the Continuous Distributions, Binomial Distribution, The Normal or Gaussian Distribution and the Poisson Distribution IV (compl) MM 1431.2 Differential Equations, Vector Calculus and abstract Algebra CO 1: Understands the General form First-degree first order equations and solving using Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoullis equation; solves Higher-degree first-order Clairaut‟s equation CO 2: Solving Linear equations with constant coefficients; linear recurrence relations; Laplace transform method, Linear equations with variable coefficients such as The Legendre and Euler linear equations; CO 3: Solves exact equations using partially known complementary function; variation of parameters; Green's functions; canonical form for second-order equations CO 4: Solves general ordinary differential equations; non-linear exact equations; isobaric or homogeneous equations and solves equations homogeneous in x or y alone and equations having y = Aex as a solution CO 5: Evaluate line, surface and volume integrals CO 6: Acquire fundamental concept of Group theory CO 7: Enhance capacity for mathematical reasoning CO 8: Develop problem solving skill IV (compl) MM 1431.2 Basic Statistics and Differential Equations CO 1: Understands basic aspects of Statistics including probability and sample spaces, graphical representation of data, frequency distribution, measures of central tendency and dispersion CO 2: Understands the General form First-degree first order equations and solving using Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoullis equation; solves Higher-degree first-order Clairaut‟s equation V (core) MM 1541 Real Analysis-I CO 1: Understands the existence of irrational numbers CO 2: States the completeness axiom of the reals and do simple calculations with suprema and infima of bounded sets CO 3: Proving the uncountability of R CO 4: calculate limits of sequences using the algebra of limits for sequences and the standard list of basic sequences, limits of sequences and to prove Bolzano Weierstrass theorem CO 5: state various convergence tests for series (e.g. comparison test or the ratio test) and use them to detect convergence or divergence of series CO 6: Understands abstract metric spaces CO 7: Understands the construction of Cantor set CO 8: Understands the open and closed sets in R and their complements CO 9: Understands the compactness, open covers, perfect and connected sets in R CO 10: Proves the Baire‟s Theorem V (core) MM 1542 Complex Analysis I CO 1: Understands the algebra of Complex numbers, point representation and its vector and polar form CO 2: Understands the concept of limit and continuity of functions of complex variable Prove the Cauchy-Riemann equations CO 3: Understanding polynomials and rational functions, the exponential, trigonometric, hyperbolic, the logarithmic functions and inverse trigonometric functions CO 4: Gets the knowledge of contour integrals and proves Cauchy‟s Integral formula. Also discusses about its applications in evaluating integrals CO 5: Understands the Bounds of Analytic functions V (core) MM 1543 Abstract Algebra – Group Theory CO 1: Acquire fundamental concept of Group theory CO 2: Enhance capacity for mathematical reasoning CO 3: Develop problem solving skill CO 4: Students can connect the theory of groups to problems in other discipline CO 5: Defining and analysing various permutation groups CO 6: Understanding Cosets, Lagrange‟s theorem and fundamental theorem of Isomorphism CO 7: Solve boundary value problem V (core) MM 1544 Differential Equations CO 1: Understands first order differential equations and various methods to solve them CO 2: Understanding the existence and uniqueness of solutions theorem CO 3: Understands second order differential equations and various methods to solve them V (core) MM 1545 Mathematics software- LATEX & Sage Math CO1: Enables to prepare a project report in Mathematics using LATEX CO 2: Typesets a simple article, prepares a table, inserts figures in the document and adds bibliography CO 3: Understands to start Sage Math, use Sage Math cloud CO 4: Do simple calculations using Sage Math calculator and by basic functions CO 5: Plots the graphs of simple functions CO 6: Understands matrix algebra, defining functions, operations on polynomials, complex number arithmetic, differentiation of functions CO7: Understands the concepts of combinatorics and number theory, vector calculus V (open course) MM 1551 Business Mathematics CO 1: Provides basic mathematics for finance – interest and discount calculations, depreciation, etc. CO 2: Differentiation and their applications to Business and Economics CO 3: Provides basic knowledge on index numbers – their types, applications and limitations VI (core) MM 1641 Real Analysis-II CO 1: State the definition of continuous functions and verify or disprove this in easy examples, formulate characterizations of continuity in terms of convergent sequences. CO 2: State the intermediate value theorem and the boundedness theorem and apply them to solve equations. CO 3: State the definition of differentiable functions and to verify or disprove this in easy examples. CO 4: Calculate derivatives using the chain rule, the algebra of differentiable functions and the rule on derivatives of compositional inverses. CO 5: State Rolle's theorem, the Mean Value Theorem and L'Hospital's Rule and to apply them to recognise the shape of functions (e.g. existence of local extrema, subjectivity of the derivative) and to calculate limits. CO 6: State the definition of Riemann Integrability and derive the Cachy criteria. CO 7: Establish the integrability using various results, like squeeze theorem, integrability of monotone functions etc. CO 8: Derive the relation between integration and differentiation via fundamental theorem of calculus. VI (core) MM 1642 Complex Analysis II CO 1: Compute the Taylor and Laurent expansions of simple functions, determining the nature of the singularities CO 2: Understands about the point at infinity CO 3: Prove the Cauchy Residue Theorem and use it to evaluate improper integrals CO 4: Understands the geometric considerations of conformal mapping CO 5: Gets the knowledge of Mobius Transformations VI (core) MM 1643 Abstract Algebra – Ring Theory CO 1: Explain fundamental concepts of homomorphism of Groups CO 2: Develop the notion of Ring theory CO 3: Handle Factor ring CO 4: Use the theory of rings to problems in other discipline VI (core) MM 1644 Linear Algebra CO 1: Understands the basics of Linear Algebra and matrix theory through geometry CO 2: Demonstrate understanding of linear independence, span, and basis. CO 3: Determine eigenvalues and eigenvectors and solve eigenvalue problems CO 4: Apply principles of matrix algebra to linear transformations VI (core) MM 1645 Integral Transforms CO 1: Understands Laplace Transforms and its properties CO 2: Understands its applications to Non- homogeneous Linear ODE CO 3: Understands the Fourier series representation of periodic functions, odd and even functions, Half range expansions CO 4: Understands Fourier integrals and its properties CO 5: Understands Fourier Transform and its properties VI (core) MM 1661.1 Graph Theory CO 1: Understands the Fundamental Concepts of graph CO 2: Understands the trees and Connectedness of graphs CO 3: Understands Euler tours and Hamiltonian cycles CO 4: Understands the concept of Chinese postman problem, Travelling salesman problem CO 5: Understands the idea of planar graphs CO 6: Gets the knowledge of Platonic bodies and Kuratowski‟s Theorem VI (core) MM 1646 Project CO 1: Computational understanding of mathematics to a broad understanding encompassing logical reasoning, generalization, abstraction, and formal proof. CO 2: Create and verify their own conjectures, rather than simply using provided formulas, rules and theorems in multiple courses throughout the mathematics curriculum. CO 3: Construct clear and well-supported mathematical arguments to explain mathematical problems, topics, and ideas in writing.