Kerala SET Mathematics Syllabus 2023 PDF

Kerala State Eligibility Test 2023

Name Of the Examination

State Eligibility Test

Conducting Body

Kerala Pareeksha Bhavan

Official Notification Released

28/03/2023

Kerala set exam 2023 Registration started on

30/03/2023

Exam Mode

Online mode

SET Exam Application Mode

Online mode

SET Exam last date to Apply Online 2023

05/05/2023

Job Name

Teacher in Higher Secondary Level and Non-Vocational Teachers in VHSE

Minimum Percentage of Marks

Main Topics

Sub Topics

UNIT I: FUNDAMENTAL CONCEPTS

Sets and Functions

Module 1

• Sets and functions
• General ideas of sets
• sets of numbers and operations on sets.
• Countable and uncountable sets.
• Properties of relations and functions.
• Domain, codomain and range of functions.
• Injective and surjective functions, bijections and inverses.
• Graphs of polynomials, trigonometric functions and other elementary real valued functions.
• Composition of functions.
• Polynomials and polynomial equations - remainder and factor theorems.
• Relation between roots and coefficients of a polynomial of degree up to three.

Analytic Geometry

Module-2

• Coordinates of points in plane and space.
• Distance in terms of co-ordinates.
• Determination of coordinates of points on a line in terms of two points.
• Slope of a line. Representation of curves in a plane as equations - straight lines, circles and conics.
• Geometric and algebraic properties of their equations. Direction cosines and direction ratios of a line in space.
• Coplanar and non-coplanar lines. Equations of lines, planes and spheres in both cartesian and vector forms.

Elementary Calculus

• Module-3
• Limits of functions, differentiability, and derivative as slope.
• Derivatives of polynomial functions, exponential function and trigonometric functions.
• Derivatives of sums, products, composite and inverse functions.
• Increasing and decreasing functions, local extrema, and simple applications.
• Integration as anti-differentiation, integral as sum.
• Length of curves, area under curves and volume of solids of revolution using integration.

Probability

Module-4

• Basic Combinatorics - Pigeonhole principle, permutations and combinations.
• Random experiment, sample space, events, probability, discrete Probability, conditional probability and independent events.

UNIT II: REAL AND COMPLEX ANALYSIS

Basic Concepts in Real Analysis

Module-1

• Convergence and limits of sequences and series of real numbers. Geometric and harmonic series.
• Sequences and series of real functions, point-wise and uniform convergence.
• The exponential series.
• Limits, continuity, directional and total derivatives of functions of several variables.

Integration and Basic Measure Theory

Module-2

• Functions of bounded variations.
• Riemann integrals and Riemann Stieltjes integrals of real valued functions.
• The concepts of Lebesgue measure and Lebesgue integral of real valued functions.
• Lebesgue measurable sets, measurable functions and simple functions.

Basic concepts in complex Analysis

Module-3

• Basic properties of complex numbers. Absolute value. Polar form of a complex number.
• De Moivre’s theorem and n^th roots of unity and nth roots of complex numbers.
• Properties of complex analytic functions Cauchy Reimann equations, infinite differentiability, Harmonic conjugates.
• Conformal mappings.
• Mobius transformations and cross ratio.

Complex Integration

Module-4

• Power series of functions
• Radius of convergence of power series
• Zeroes, poles and singularities of functions. Liouville’s Theorem
• Open Mapping Theorem
• Maximum Modulus Theorem
• Line integrals of complex valued functions
• Cauchy’s Theorem and Cauchy’s integral formula for complex functions.
• Contour integration and residue theorem

UNIT-III: ABSTRACT ALGEBRA

Groups

Module-1

• Groups and subgroups, Groups of permutations. Abelian and non-abelian groups.
• Cyclic groups. Finite and infinite groups. Normal subgroups and quotients.
• Homomorphisms and isomorphisms. Homomorphic images and quotients.
• Lagrange’s theorem. Order of an element. Groups of prime order and prime-square order.
• Direct products and direct sums of groups.
• Sylows theorem.

Rings

Module-2

• Ring of integers and ring of polynomials. Ring of integers modulus.
• Finite rings.
• Commutative and non-commutative rings
• Ideals, maximal ideals and prime ideals
• Quotient Rings
• Homomorphisms and isomorphisms
• Homomorphic images and quotients
• Zero divisors
• Integral domains
• Euclidean domains
• Units, associates and primes in a ring.
• Groups of units of rings

Fields

Module-3

• Field of rational numbers real numbers and complex numbers. Integers modulo a prime number.
• Finite fields.
• Finite integral domains.
• Polynomials over fields, reducibility and irreducibility.
• Algebraic and transcendental extensions of fields. Degree of extension and irreducible polynomial of an element, Splitting fields, groups of automorphisms and fixed fields.

UNIT IV: LINEAR ALGEBRA AND MATRIX THEORY

Matrix Theory

Module-1

• Algebra of matrices - Types of matrices, nilpotent matrices, invertible matrices.
• Determinants of square matrices.
• Echelon matrices and rank of a matrix.
• Systems of linear equations and solutions and matrix methods for checking consistency.

Vector Spaces

Module-2

• Vector spaces over an arbitrary field, real numbers and complex numbers.
• Linear independence and dependence. Basis and dimension.
• subspaces and quotients.
• Direct sums of vector spaces. Geometry of points, lines and planes in R2 and R3 in terms of subspaces.

Linear Transformation

Module-3

• Linear maps.
• Differentiation and integration as linear maps Range, null space and dimensional relations.
• Invertible transformations.
• Representation of linear maps between finite dimensional vector spaces as matrices and vice-versa.
• Dependence of matrix of a linear trans- formation on the basis and relation with change of basis. Linear functional and dual basis.
• Linear functionals on R2.
• Eigenvalues and eigenvectors.
• Minimal polynomial, characteristic polynomial and Cayley Hamilton theorem, Diagonalization.

UNIT V: NUMBER THEORY AND DIFFERENTIAL EQUATIONS

Number Theory

Module-1

• Principle of Mathematical Induction. Prime factorization.
• Properties of divisibility
• Coprime numbers.
• G C D and L C M of numbers.
• Euler’s totient function and its multiplicative property. Congruences.
• Chinese remainder theorem.
• Fermat’s little theorem. Wilsons theorem

Ordinary Differential Equations

Module-2

• Order and degree of differential equations.
• Formation of differential equation and different types of differential equations of first order.
• Representation of family of curves by differential equations.
• Orthogonal trajectories. Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, Method of variation of parameters, Wronskian.
• Power series solutions of differential equations. Bessel and Legendre polynomials

Partial Differential Equations

Module-3

• Solution of the equation of the form Pdx+Qdy+Rdz=0.
• Charpits and Jacobis method for solving first order PDEs, Classification of higher order PDEs (parabolic, elliptic and hyperbolic types).
• General solution of higher order PDEs with constant coefficients. Metho d of separation of variables.
• Wave equation, Heat equation and Laplace equation

UNIT VI: TOPOLOGY AND FUNCTIONAL ANALYSIS

Metric Topology

• Module-1
• Metric spaces
• Discrete metric
• Euclidean metric on Rn and Cn.
• Supremum metric on the set of real valued functions and complex valued functions on a closed interval
• Limit points and convergence of sequences in a metric space
• Cauchy sequences
• Complete metric spaces
• Completion of a metric space Cantors intersection theorem and Baire’s category theorem
• Compact subsets and connected subsets of R and C
• Heine-Borel theorem for Rn

General Topology

Module-2

• Topological spaces, open sets and closed sets
• Usual topology on R and C
• Base and subbase for a topology
• Closure, interior and boundary of subsets
• Compactness and connectedness
• Convergence of sequences in a topological space. Non-uniqueness of limits
• Hausdor Spaces and separation axioms. Continuity of functions between topological spaces.
• Preservation of compactness and connectedness under continuous functions.
• Homeomorphisms. Product topology.

Normed linear Spaces

Module-3

• Norm on linear spaces
• Euclidean norm on Rn and Cn
• Finite and infinite dimensional p spaces
• Lp spaces
• Closed and non-closed linear subspaces
• Closure and interior of linear subspaces
• Continuous linear maps between normed linear spaces.
• Bounded operators and operator norm. Banach spaces.
• Open mapping theorem and closed graph theorem.

Inner product Spaces

Module-4

• Inner products
• Examples of norms arising from inner products and not arising from any inner product
• Parallelogram law
• Orthogonality in inner product spaces
• Orthonormal bases
• Fourier Expansion
• Bessel’s inequality
• Hilbert spaces
• Parseval’s identity.
• Continuous linear maps between Hilbert spaces.
• Projection map
• Riesz representation theorem.