Assam SLET Mathematics Previous Year Paper: 2012-2021
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Mathematics SLET Assamese Syllabus 2023
1. Real Analysis : Riemann integrable functions; Improper integrals, their convergence and uniform convergence. Euclidean space R’’, Bolzano-Welerstrass theorem, compact Subsets of R’’, Heine-Borel theorem, Fourier series. Continuity of functions of R’’, Differentiability of F:R’’>Rm, Prop- erties of differential, partial and directional derivatives, continu- ously differentiable functions. Taylor’s series. Inverse function theorem, implict function theorem. Integral functions, line and surface integrants, Green’s theorem, Stoke’s theorem.
2. Complex Analysis : Cauchy’s theorem for convex regions, Power series representation of Analysis function. Liouville’s theorem, Fundamental theorem of algebra, Riemann’s theorem on remov- able singularaties, maximum modulus principle, Schwarz lemma, Open Mapping theorem, Casoratti-Welerstrass-theorem, Welerstrass’s theorem on uniform convergence on compact sets, Bilinear transformations, Multivalued Analytic Functions, Rimann Surfaces.
3. Algebra : Symmetric groups, alternating groups, Simple groups, Rings, Maximal ideals, Prime Ideals, Integral domains, Euclidean domains, principal Ideal domains, Unique Factorisation domains, quotient fields, Finite fields, Algebra of Linear Transformations, Reduction of matrices to Canonical Forms, Inner product Spaces, Orthogonality, quadratic Forms, Reduction of quadration forms. Reduction of Quadritic forms.
4. Advance Analysis : Element of Metric Spaces Convergence, continuity, compactness, Connectedness, Weierstrass’s approximation Theorem, Completeness, Bare category theorem, Labesgue measure, Labesgue integral, Differentiation and Integration.
5. Advanced Algebra : Conjugate elements and class equations of finite groups, Sylow theorem, solvable groups, Jordan Holder Theorem Direct Products, Structure Theorem for finita abellean groups, Chain conditions on Rings; Characteristic of Field, Field extensions, Elements of Galois theory, solvability by Radicals, Ruler and compass construction.
6. Functional Analysis : Banach Spaces, Hahn-Banch Theorem, Open mapping and closed Graph Theorems. Principle of Uniform boundedness, Boundedness and continuity of Linear Transformations. Dual Space, Embedding in the second dual, Hilbert Spaces, Projections. Orthonormal Basis, Riesz-representation theorem, Bessel’s Inequality, persaval’s Identity, self adjoined operators, Normal Operators.
7. Topology : Elements of Topological Spaces, Continuity, Convergence, Homeomorphism, Compactness, Connectedness, Separation Axioms, First and Second Countability, Separability, Subspaces, Product Spaces, quotient spaces, Subspaces, Product Spaces, quotient spaces. Tychonoff’s Theorem, Urysohn’s Metrization theorem, Homotopy and Fundamental Group.
8. Discrete Mathematics : Partially ordered sets, Latices, Complete Latices, Distributive latices, Complements, Boolean Algebra, Boolean Expressions, Application to switching circuits, Elements of Graph Theory, Eulerian and Hamiltonian graphs, planar Graphs, Directed Graphs, Trees, Permutations and Combinations, Pigeonhole principle, principle of Inclusion and Exclusion, Derangements.
9. Ordinary and Partial Differential Equations : Existence and Uniqueness of solution dyxdx-f(x,y) Green’s function, sturm Liouville Boundary Value Problems, Cauchy Problems and Characteristics, Classification of Second Order PDE, Seperation of Variables for heat equation, wave equation and Laplace equation, Special functions.
10. Number Theory : Divisibility : Linear diophantine equations. Congruences. Quadratic residues; Sums of two squares, Arithmatic functions Mu, Tau, Phi and Sigma (and).
11. Mechanics : Generalised coordinates; Lagranges equation; Hamilton’s coronics equations; Variational principles least action; Two dimensional motion of rigid bodies; Euler’s dynamical equations for the motion of rigid body; Motion of a rigid body about an axis; Motion about revolving axes.
12. Elasticity : Analysis of strain and stress, strain and stress tensors; Geomatrical representation; Compatibility conditions; Strain energy function; Constitutive relations; Elastic solids Hookes law; Saint-Venant’s principle, Equations of equilibrium; Plane problemsAiry’s stress function, vibrations of elastic, cylindrical and spherical media.
13. Fluid Mechanics : Equation of continuity in fluid motion; Euler’s equations of motion for perfect fluids; Two dimensional motion complex potential; Motion of sphere in perfect liquid and motion of liquid past a sphere; vorticity; Navier-Stokes’s equations for viscous flows-some exatct solutions.
14. Differential Geometry : Space curves – their curvature and torsion; Serret Frehat Formula; Fundamental theorem of space curves; Curves on surfaces; First and second fundamental form; Gaussian curvatures; Principal directions and principal curvatures; Goedesics, Fundamental equations of surface theory.
15. Calculus of Variations : Linear functionals, minimal functional theorem, general variation of a functional, Euler – Larange equation; Variational methods of boundary value problems in ordinary and partial differential equations.
16. Linear integral Equations : Linear integral Equations of the first and second kind of Fredholm and Volterra type; soluting by successive substitutions and successive approximations; Solution of equations with seperable kernels; The Fredholm Alternative; Holbert-Schmidt theory for symmetric kernels.
17. Numerical analysis : Finite differences, interpolation; Numerical solution of algebric equation; Iteration; Newton-Rephson method; Solutions on linear system; Direct method; Gauss elimination method; Matrix-Inversion, elgenvalue problems; Numerical differentiation and integration. Numerical solution of ordinary differential equation, iteration method, Picard’s method, Euler’s method and improved Euler’s method.
18. Integral Transformal place transform : Transform of elementary
functions, Transform of Derivatives, Inverse Transform, Convolution Theorem, Application, Ordinary and Partial differential equations; Fourier transforms; sine and cosine transform, Inverse Fourier Transform, Application to ordinary and partial differential equations.
19. Mathematical Programming Revised simplex method. Dual simplex method, Sensitivity analysis and perametric linear programming. Kuhn-Tucker conditions of optimality. Quadratic programming; methods due to Beale, Wofle and Vandepanne, Duality in quadratic programming, self duality, Integer programming.
20. Measure Theory : Measurable and measure species; Extension of measure, signed measure, Jordan-Hahn decomposition theorems. Integration, monotone convergence theorem, Fatou’s lemma, dominated convergence theorem. Absolute continuity. Radon Niiodymtheorem, Product measures, Fubinl’s theorem.
21. Probability : Sequences of events and random variables; Zero-one laws of Borel and Kolmogorov. Almost sur convergence, convergence in mean square, Khintchine’s weak law of large numbers; Kologorov’s inequality, strong law of large numbers. Convergence of series of random variables, three-series criterion. Central limit theorems of Liapounov and Lindeberg-Feller. Conditional expectation, martingales.
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