TS CPGET 2022 Syllabus M.Sc Mathematics

CPGET-2022 Syllabus 68. MATHEMATICS Differential and Integral Calculus (20 Marks) Partial Differentiation : Introduction - Functions of two variables - Neighborhood of a point (a, b) - Continuity of a Function of two variables, Continuity at a point - Limit of a Function of two variables - Partial Derivatives - Geometrical representation of a Function of two Variables - Homogeneous Functions. Theorem on Total Differentials - Composite Functions - Differentiation of Composite Functions - Implicit Functions - Equality of fxy(a, b) and fyx(a, b) - Taylor’s theorem for a function of two Variables - Maxima and Minima of functions of two variables – Lagrange’s Method of undetermined - multipliers. Curvature and Evolutes : Introduction - Definition of Curvature - Radius of Curvature - Length of Arc as a Function, Derivative of arc - Radius of Curvature - Cartesian Equations - Newtonian Method - Centre of Curvature - Chord of Curvature. Evolutes: Evolutes and Involutes - Properties of the evolute. Envelopes : One Parameter Family of Curves - the family of straight lines - Definition - Determination of Envelope. Lengths of Plane Curves : Introduction - Expression for the lengths of curves y = f (x) - Expressions for the length of arcs x = f (y); x = f (t), y = ϕ(t); r = f (θ) Volumes and Surfaces of Revolution : Introduction - Expression for the volume obtained by revolving about either axes - Expression for the volume obtained by revolving about any line - Area of the surface of the frustum of a cone - Expression for the surface of revolution - Pappus Theorems - Surface of revolution. Differential Equations (20 Marks) Differential Equations of first order and first degree: Introduction - Equations in which Variables are Separable - Homogeneous Differential Equations - Differential Equations Reducible to Homogeneous Form - Linear Differential Equations - Differential Equations Reducible to Linear Form - Exact differential equations - Integrating Factors - Change in variables - Total Differential Equations - Simultaneous Total Differential Equations - Equations of the form = = Differential Equations first order but not of first degree: Equations Solvable for p - Equations Solvable for y - Equations Solvable for x - Equations that do not contain x (or y)- Equations Homogeneous in x and y - Equations of the First Degree in x and y - Clairaut’s equation. Applications of First Order Differential Equations : Growth and Decay - Dynamics of Tumour Growth - Radioactivity and Carbon Dating - Compound Interest - Orthogonal Trajectories. Higher order Linear Differential Equations: Solution of homogeneous linear differential equations with constant coefficients - Solution of non-homogeneous differential equations P (D)y = Q(x) with constant coefficients by means of polynomial operators when Q(x) =eax, b sin ax/b cos ax,bxk , V eax - Method of undetermined coefficients. Method of variation of parameters - Linear differential equations with non constant coefficients - The Cauchy - Euler Equation - Legendre’s Linear Equations - Miscellaneous Differential Equations. Partial Differential Equations: Formation and solution- Equations easily integrable - Linear equations of first order. 1

CPGET-2022 Syllabus Real Analysis (20 Marks) Sequences: Limits of Sequences- A Discussion about Proofs-Limit Theorems for Sequences- Monotone Sequences and Cauchy Sequences -Subsequences-Lim superiors and Lim inferior- Series-Alternating Series and Integral Test . Continuity: Continuous Functions -Properties of Continuous Functions -Uniform Continuity - Limits of Functions Differentiation: Basic Properties of the Derivative - Mean Value Theorems - L ’Hospital Rule – Taylor ’s Theorem. Integration : The Riemann Integral - Properties of Riemann Integral-Fundamental Theorem of Integral Calculus. Algebra (20 Marks) Groups: Definition and Examples of Groups- Elementary Properties of Groups-Finite Groups - Subgroups -Terminology and Notation -Subgroup Tests - Examples of Subgroups. Cyclic Groups: Properties of Cyclic Groups - Classification of Subgroups - Cyclic Groups. Permutation Groups: Definition and Notation -Cycle Notation-Properties of Permutations -A Check Digit Scheme Based on D5. Isomorphisms ; Motivation- Definition and Examples - Cayley’s Theorem Properties of Isomorphisms -Automorphisms-Cosets and Lagrange’s Theorem Properties of Cosets - Lagrange’s Theorem and Consequences-An Application of Cosets to Permutation Groups -The Rotation Group of a Cube and a Soccer Ball. Normal Subgroups and Factor Groups: Normal Subgroups-Factor Groups -Applications of Factor Groups -Group Homomorphisms - Definition and Examples -Properties of Homomorphisms -The First Isomorphism Theorem. Introduction to Rings: Motivation and Definition -Examples of Rings -Properties of Rings - Subrings. Integral Domains: Definition and Examples - Fields –Characteristics of a Ring. Ideals and Factor Rings: Ideals -Factor Rings -Prime Ideals and Maximal Ideals. Ring Homomorphisms: Definition and Examples-Properties of Ring- Homomorphisms. Linear Algebra (20 Marks) Vector Spaces: Vector Spaces and Subspaces -Null Spaces, Column Spaces, and Linear Transformations -Linearly Independent Sets; Bases -Coordinate Systems -The Dimension of a Vector Space- Rank-Change of Basis - Eigenvalues and Eigenvectors - The Characteristic Equation - Diagonalization -Eigenvectors and Linear Transformations -Complex Eigenvalues - Applications to Differential Equations. Orthogonality and Least Squares : Inner Product, Length, and Orthogonality -Orthogonal Sets -Orthogonal Projections - The Gram-Schmidt othogonalization Process. 2