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5th November 2014, 04:01 PM
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Re: Kerala SET Mathematics Solved Question Paper

In Kerala SET Mathematics paper, questions asked from following topics:

ALGEBRA

Sets, Relations, Functions, Elementary Number Theory Including the relation of congruence Modulo, Simultaneous linear / quadratic equations, Indices, Logarithms, Arithmetic, Geometric and Harmonic progressions, Binomial theorem, Surds, Complex numbers, Demoivre’s theorem and its simple application

MATRICES AND DETERMINANTS

Matrix operations, Definition and properties of determinats, Cofactors, Adjoint, Elementry Transformations, Rank and inverse of a Matrix, Matrix Polynomial, Characteristic Equations,Eigen Values, Latent Vectors, Caylay Hamilton theorem, Linear system of Equations

THEORY OF EQUATIONS

Polynomials and their charcteristics, Roots of an equation, Relations between Roots and Coefficients,Transformation of Equations, Symmetric function etc.

ABSTRACT ALGEBRA

Groups, Cyclic groups, Subgroups, Normal Groups, Lagrange’s theorem, Homomorphism, Isomorphism, Ring, Field, Vector Spaces, Linear Independence of Vectors, Basis, Dimension, Linear Transformation and diagonalisation of Matrices etc

VECTOR ALGEBRA

Scalar and Vector quantities their representation, Addition and subtraction of vectors, Scalar and vector product of two vectors, Scalar triple product, Expansion formula of Vector triple product

TRIGONOMETRY

Simple identities, trignometric equations, Properties of triangles, Solution of triangle, Height and distance, Inverse function

CALCULAS AND REAL ANALYSIS

Real number system, Concept of neighborhood and limit points, Continuity and limits, Indeterminate forms, properties of continuous functions in closed interval, Differentiation, Successive Differentiation, Maxima Minima, Roll's theorem, Mean value theorems, Maclaurine's series and Taylor's series, Integration, definite integral, Evaluation of length, Area and volume of curves, Curvature, Asymptotes, Tracing of curves, Partial Differentiation, functions of two variables, definition of partial derivates, Total differentiation, Sequence, Sequence of real numbers, Convergent Sequences, Cauchy Sequences, Monotonic Sequences, Infinite series of positive terms and their different test of convergence,Alternating infinite series and Leibnitz's test of convergence, Absolute convergence, Conditional convergence, Uniform Convergence, Multiple integration, Change of order and change of variables, Application to evaluation of Area, Surface and volume.

DIFFERENTIAL EQUATIONS

Differential equations of first order and their solutions, Linear differential equations with constant coefficients, Homogenous linear differential equations, Orthogonal Trajectories, Singular solutions.

COORDINATE GEOMETRY

2-d :

Pair of straight lines, Transformation of coordinate system, Circles, Parabola, Ellipse and Hyperbola, Pair of tangents from a point, Chord of contact, Equation of chord in terms of middle point, Diameter of Conic, Conjugate diameter, Classification of curves of second degree.

3-d :

Introduction to lines and planes, Spheres, Quadric Cones and Cylinders.

LINEAR PROGRAMMING

Linear inequalities with two variables, Mathematical formulation of L.P.P, Basic concepts of graphical and Simplex method.

NUMERICAL ANALYSIS

Interpolation, Extrapolation, Quadrature formula, Simpson's 1/3rd rule, Trapezoidal rule, Solution of non-linear equations using iterative methods, Numerical differentiation.

STATISTICS

Classifications of Data and frequency distribution, Calculation of measures of Central tendency and measures of dispersion, Skewness and Kurtosis, Permutation and Combination, Probability, Random variables and distribution functions, Mathematical expectations and generating functions, Binomial, Poisson, Geometric, Exponential and Normal distributions, Curve fitting and principal of least square, Correlation and Regression,Index numbers and their importance,Simple Time Series analysis, Sampling and large sample tests,test of significance based on t, Chi-square and F distributions


COMPUTER AWARENESS


COMPUTER BASICS

Organization of a computer, Central Processing Unit (CPU), Structure of instructions in CPU, input / output devices, computer Memory, memory organization, back-up devices.

DATA REPRESENTATION

Representation of characters, integers, and fractions, binary and Hexadecimal representations

BINARY ARITHMETIC

Addition, subtraction, division, multiplication, single arithmetic and two complement arithmetic, floating point representation of numbers, normalized floating point representation, Boolean Algebra, truth tables, Venn diagrams

COMPUTER ARCHITECTURE

Block structure of computers, communication between processor and I / O devices, interrupts

COMPUTER LANGUAGE

Assembly language and high level language, Multiprogramming and time sharing operating systems, Computer Programming in C

MATHEMATICAL LOGIC

Venn Diagrams in logic, Logical operators, Negations, Logical Equivalence and Tautology

FLOW CHART AND ALGORITHMS
Kerala SET Maths paper
21. MATHEMATICS DETAILS OF SYLLABUS UNIT I - BASIC MATH 1. ALGEBRA Basic properties of real and complex numbers. Absolute value. Polar form of a complex number. De Moivre’s Theorem and complex nth roots. Polynomials and polynomial equations, remainder and factor theorems. Quadratic equations. Systems of linear equations and their consistency. Matrix methods of checking consistency and solving systems of linear equations. Algebra of matrices. Basic combinatorics involving permutations and combinations. 2. ANALYTIC GEOMETRY Coordinates of points in plane or space. Distance in terms of coordinates. Coordinates of points on line determined by two specified points. Slope of a line. Representation of curves in a plane as equations and vice versa, especially straight lines, circles and conics. Inferring geometric properties of plane curves from the algebraic properties of their equations and vice versa. Direction cosines and direction ratios of a line in space. Equations of lines in space. Coplanar and non-coplanar lines. Equations of planes and spheres. 3. SETS AND FUNCTIONS General ideas of sets, especially sets of numbers and operations on sets. Real valued functions as transformations, domain and range of such functions. Injective and surjective functions, bijections and inverses. Graphs of real valued functions, especially polynomials and trigonometric functions. Composition of functions 4. CALCULUS Intuitive idea of limits of functions, differentiability, derivative as slope. Derivatives of polynomial functions, exponential function and trigonometric functions; derivatives of sums and products, composite and inverse functions. Increasing and decreasing functions, local extrema, simple applications.
Integration as anti-differentiation, integral as sum. Areas under curves and volumes of solids of revolution, using integration. UNIT II - REAL AND COMPLEX ANALYSIS Limits Convergence sequences and series of real and complex numbers. Geometric and harmonic series. Sequences and series of real and complex functions, point-wise and uniform convergence. Power series, radius of convergence. The exponential series. Limits and continuity of real and complex functions Differentiation Differentiation of real and complex functions. Analytic functions. Power series as analytic functions, extension of exponential and trigonometric functions to complex numbers. Branches of logarithm. Isolated singularities of complex function. Integration Riemann integrals and Riemann Stieltjes integrals of real valued functions. The concepts of Lebesgue measure and Lebesgue integral of real valued functions. Line integrals of complex valued functions, Cauchy’s Theorem and Integral Formula for complex functions. Contour integration. Analytic functions Properties of complex analytic functions, such as infinite differentiability, power series expansion, isolated zeros. Liouvilles’ Theorem. Open Mapping Theorem. Maximum Modulus Theorem. Cauchy-Riemann Equations, harmonic conjugates. Conformal mappings, Mobius transformations.
UNIT III - ABSTRACT ALGEBRA Rings Ring of integers and ring of polynomials over real numbers. Integers modulo n. Finite rings. Commutative and non-commutative rings. Ideals, maximal ideals, prime ideals. Quotients. Homomorphisms and isomorphisms. Homomorphic images as quotients. Divisors of zeros. Integral domains. Euclidean domains. Factorization, units, associates, primes. Primitive polynomials. Fields Rational numbers, real numbers and complex numbers as fields. Integers modulo a prime number. Finite fields. Finite integral domains are fields. Polynomials over fields, reducibility and irreducibility. Algebraic and transcendental extensions of fields. Splitting fields. Vector spaces Vector spaces over a field, especially over real numbers and complex numbers. Linear independence and dependence. Basis. Dimension. Linear subspaces and quotients. Geometry of R2 and R3Linear maps. Representation of linear maps between finite dimensional vector spaces as matrices and vice-versa. Change of basis. Eigenvalues and eigenvectors. Function spaces as linear spaces. Differentiation and integration as linear maps. Groups Groups of permutations. Groups of units of rings. Abelian and non-abelian groups. Cyclic groups. Finite and infinite groups. Subgroups. 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. Sylow’s Theorem as a partial converse of Lagrange’s Theorem.
UNIT IV - ABSTRACT ANALYSIS Topology Metric as an abstractions of the absolute value of real and complex numbers. The Euclidean metric on Rn and Cn. The supremum metric and the integral metric on the set of real or complex valued functions on a closed interval. Limit points in a metric space. Convergence of sequences in a metric space. Cauchy sequences. Complete metric spaces. Completion of a metric space. Cantor’s Theorem and Baire’s Theorem. Topological spaces as generalizations of metric spaces. Non-metrizable spaces. Usual topology on R and C. Basis for a topology. Closure, interior and boundary of subsets. Compactness and connectedness. Compact subsets and connected subsets of R and C. Heine-Borel Theorem for RnConvergence of sequences in a topological space. Non-uniqueness of limits. Hausdorff spaces. Continuity of functions between topological spaces. Preservation of compactness and connectedness. Non-continuity of inverse functions. Homeomorphisms. Product topology and the topology of point-wise convergence in function spaces. Functional Analysis The norm on a vector space as a generalization of the length of a vector. Euclidean norm on Rn and Cn. Then ℓnp and ℓp spaces. Supremum norm and integral norm on the space of continuous complex valued functions on a closed interval. The Lp spaces. Closed and non-closed linear subspaces. Closure and interior of linear subspaces. Continuous linear maps between normed linear spaces. Non-continuity of the inverse. Boundedness and continuity. Banach spaces. Open Mapping Theorem and the Bounded Inverse Theorem. Quotients as images for Banach spaces. Inner products as generalization of the dot product. Examples of norms arising from inner products and not arising from any inner product. Parallelogram Law. Orthogonality in inner product spaces. Orthogonal bases. Bessel’s Inequality. Hilbert Spaces. Parseval’s Identity. Fourier Expansion. Continuous linear maps between Hilbert spaces. Adjoint of a linear map.


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