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Old January 26th, 2014, 07:05 PM
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Here I am looking for the Mathematical Science syllabus of State Level Eligibility Test (SLET) - North East Region of India, can you please provide me the same??

As you are looking for the Mathematical Science syllabus of State Level Eligibility Test (SLET) - North East Region of India so here I am sharing the same with you

1. Basic concepts of Real and Complex analysis-sequences and series, continuity, uniform continuity, Differentiability, Mean Value Theorem, sequences and series of functions, uniform convergence, Riemann integral-definition and simple properties. Algebra of Complex numbers, Analytic functions, Cauchy’s Theorem and integral formula, Power series, Taylor’s and Laurent’s series, Residues,
contour integration.

2. Basic Concepts of Linear Algebra-Space of n-vectors, Linear dependence, Basic, Linear transformation, Algebra of matrices, Rank of a matrix, Determinants, Linear equations, Quadratic forms, Characteristic roots and vectors.

3. Basic concepts of probability-Sample space, discrete probability, simple theorem on probability, independence of events, Bayes Theorem. Discrete and continuous random variables, Binomial, Poisson and Normal distributions; Expectation and moments, independence of random variables, Chebyshev’s inequality.

4. Linear Programming Basic Concepts-Convex sets, Linear Programming Problem (LPP). Examples of LPP. Huperplane, Open and closed half-spaces. Feasible, basic feasible and optimal solutions. Extreme point and graphical method.

5. Real Analysis-finite, countable and uncountable sets, Bounded and unbounded sets, Archimedean property, ordered field, completeness of R, Extended real number system, lumsup and limit of a sequence, the epsilon-delta definition of continuity and convergence, the algebra of continuous functions, monotonic functions,





Rest of the detailed syllabus is attached in below file which is free of cost
Attached Files Available for Download
File Type: pdf SLET North East Region Mathematical Science Syllabus.pdf (45.5 KB, 69 views)

Last edited by Aakashd; October 15th, 2019 at 03:09 PM.
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Old September 29th, 2015, 06:24 PM
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Can you provide me the syllabus of Mathematics subject of SLET or State Level Eligibility Test Commission exam as I will be giving the exam this session for need it for preparation?
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Old September 29th, 2015, 06:25 PM
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Default Re: Syllabus of Mathematics SLET exam

The syllabus of Mathematics subject of SLET or State Level Eligibility Test Commission exam as you will be giving the exam this session for need it for preparation is as follows:

SYLLABUS

MATHEMATICAL SCIENCE

PAPER -I
Subject : General Paper on Teaching & Research Aptitude
The test is intended to assess the teaching/research aptitude of the candidate. They are supposed to possess and exhibit cognitive abilities like comprehension, analysis, evaluation, understanding the structure of arguments, evaluating and distinguishing deductive and inductive reasoning, weighing the evidence with special reference to analogical arguments and inductive generalization, evaluating, classification and definition, avoiding logical inconsistency arising out of failure to see logical relevance due to ambiguity and vagueness in language. The candidates are also supposed to have a general acquaintance with the nature of a concept, meaning and criteria of truth, and the source of knowledge. There will be 60 questions, out of which the candidates can attempt any 50. In the event of the candidate attempting more than 50 questions, the first 50 questions attempted by the candidate will only be evaluated.

1. The Test will be conducted in objective mode from SET 2012 onwards. The Test will consist of three papers. All the three papers will consists of only objective type questions and will be held on the day of Test in two separate sessions as under :
Session Paper Number of Marks Duration
Questions First I 60 out of which 50%2=100 1¼ Hours 50 questions are to be attempted
First II 50 questions all 50%2=100 1¼ Hours of which are compulsory Second III
75 questions all 75%2=150 2½ Hours of which are compulsory

2. The candidates are required to obtain minimum marks separately in Paper-II and Paper -III as given below Minimum marks (%) to be obtained Category Paper-I Paper-II Paper-III General 40 (40%) 40 (40%) 75 (50%) OBC 35 (35%) 35 (35%) 67.5 (45%) rounded off to 68 PH/VH/ 35 (35%) 35 (35%) 60 (40%) SC/ST Only such candidates who obtain the minimum required marks in each Paper, separately, as mentioned above, will be considered for final preparation of result. However, the final qualifying criteria for eligibility for Lectureship shall be decided by Steering Committee before declaring of result.
3. The syllabus of Paper-I, Paper-II and Paper-III will remain the same.
(1) (2) Syllabus/Mathematical Science Syllabus/Mathematical Science

MATHEMATICAL SCIENCE

SECTION – B

PAPER-II
General Information : Units 1, 2, 3 and 4 are compulsory for all candidates.
Candidates with Mathematics background my omit units 10 -14
and units 17, 18. Candidates with Statistics background may omit units
6, 7 and 9. Adequate alternatives would be given for candidates with O.
R. background.
1. Basic concepts of Real and Complex analysis-sequences and series,
continuity, uniform continuity, Differentiability, Mean Value
Theorem, sequences and series of functions, uniform convergence,
Riemann integral-definition and simple properties. Algebra of Complex
numbers, Analytic functions, Cauchy’s Theorem and integral
formula, Power series, Taylor’s and Laurent’s series, Residues,
contour integration.
2. Basic Concepts of Linear Algebra-Space of n-vectors, Linear dependence,
Basic, Linear transformation, Algebra of matrices, Rank
of a matrix, Determinants, Linear equations, Quadratic forms,
Characteristic roots and vectors.
3. Basic concepts of probability-Sample space, discrete probability,
simple theorem on probability, independence of events, Bayes
Theorem. Discrete and continuous random variables, Binomial,
Poisson and Normal distributions; Expectation and moments, independence
of random variables, Chebyshev’s inequality.
4. Linear Programming Basic Concepts-Convex sets, Linear Programming
Problem (LPP). Examples of LPP. Huperplane, Open and
closed half-spaces. Feasible, basic feasible and optimal solutions.
Extreme point and graphical method.
5. Real Analysis-finite, countable and uncountable sets, Bounded and
unbounded sets, Archimedean property, ordered field, completeness
of R, Extended real number system, lumsup and limit of a
sequence, the epsilon-delta definition of continuity and convergence,
the algebra of continuous functions, monotonic functions,
types of discontinuities, Infinite limits and limits at infinity, functions
of bounded variation, elements of metric spaces.
6. Complex Analysis-Riemann Sphere and Stereographic projection.
Lines, Circles crossratio. Mobius transformations, Analytic functions,
Cauchy-Riemann equations, line integrals, Cauchy’s theorem,
Morera’s theorem, Liouville’s theorem, integral formula, zerosets
of analytic functions, exponential, sine and cosine functions,
Power series representation, Classification of singularities, Conformal
Mapping.
7. Algebra-Group, subgroups, Normal subgroups, Quotient Groups,
Homomorphisms, Cyclic Groups, Permutation Groups, Cayley’s
Theorem, Rings, Ideals, integral Domains, Fields, Polynomial
Rings.
8. Linear-Algebra-Vector spaces, subspaces, quotient spaces, Linear
independence, Bases, Dimension. The algebra of linear Transformations,
kernel, range, isomorphism, Matrix Representation of a
linear transformation, change of bases, Linear functionals, dual
space, projection, determinant function, eigen values and eigen
vectors, Cayley-Hamittion Theorem, invarient Sub-spaces, Canonical
Forms; diagonal form, Triangular form, Jordan form, inner product
spaces.
9. Differential Equations-First order ODE, singular solutions, initial
value Problems of First Order ODE, General theory of homogenous
and non-homogeneous Linear ODE, Variation of Parameters,
Lagrange’s and Charpit’s methods of solving First order Partel Differential
Equations. PDE’s of higher order with constant coefficients.
10. Data Analysis Basic Concepts-Graphical representation, measures
of central tendency and dispersion. Bivariate data, correlation and
regression, Least squares-polynomial regression, Application of
normal distribution.
11. Probability - Axiomatic definition of probability. Random variables
and distribution functions (univeriate and multivariate); expectation
and moments; independent events and independent random
variables; Bayes’ theorem; marginal and conditional distribution
in the multivariate case, coveriance matrix and correlation coefficients
(product moment, partial and multiple), regression.
(3) (4) Syllabus/Mathematical Science Syllabus/Mathematical Science
Moment generating functions; characteristic functions; probability
inequalities (Tchebyshef, Markov, Jenson). Convergence in
probability and in distribution; weak law of large numbers and
central limit theorem for independent identically distributed random
variables with finite variance.
12. Probability Distribution-Berhount, Binomial, Multinomial,
Hypergeomatric, Poisson, Geometric and Negative binomial distribution,
Uniform, exponential, Cauchy, Beta, Gamma, and normal
(univariate and multivariate) distributions Transformations of
random variables; sampling distributions, t, F and chi-square distributions
as sampling distributions, Standard errors and large
sample distributions. Distribution of order statistics and range.
13. Theory of Statistics : Methods of estimation : maximum likelihood
method, method of moments, minimum chi-square method,
least-squares method. Unblasedness, efficiency, consistency.
Cramer-Rao linequality. Sufficient Stastics. Rao-Blackwell Theorem.
Uniformly minimum variance unblased estimators. Estimation
by confidence intervals. Tests of hypothesis : Simple and composite
hypotheses, two types of errors, critical region, randomized
test, power function, most powerful and unifirmly most powerful
tests, Likelihood-ratio tests. Wald’s sequential probability ratio test.
14. Stastical methods and Data Anlysis- Tests for mean and variance
in the normal distribution : one-population and two-population
cases; related confidence intervals. Tests for product moment, partial
and multiple correlation coefficients; comparison of k linear
regressions. Fitting polynomial regression; related test. Analysis
of discrete data : chi-square test of goodness of fit, contingency
tables. Analysis of varience : one-way and two-way classification
(equal number of observations per cell). Large-sample tests through
normal approximation.
Nonparametric tests : sign test, median test, Mann-Whitney test,
Wilcoxom test for one and two-samples, rank correlation and test
of independence.
15. Operational Research Modelling - Definition and scope of Operational
Research. Different types of models. Replacement models
and sequencing theory, inventory problems and their analytical
structure. Simple deterministic of queueing system, different performance
measures. Steady state solution of Markovian queueing
models : M/M/1, M/M/1 with limited waiting space M/M/C,
M/M/C with limited waiting space.
16. Linear Programming - Linear Programming. Simplex method,
Duality in linear programming. Transformation and assignment
problems. Two person-zero sum games. Equivalence of rectangular
game and linear programming.
17. Finite Population : Sampling Techniques and Estimation : Simple
random sampling with and without replacement. Stratified sampling;
allocation problem; systematic sampling. Two stage sampling.
Related estimation problems in the above cases.
18. Design of Experiments : Basic principles of experimental design.
Randomisation structure and analysis of completely randomised,
randomised blocks and Latinsquare designs. Factorial experiments.
Analysis of 2n factorial experiments in randomised blocks.

PAPER-III
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, Properties
of differential, partial and directional derivatives, continuously
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 removable
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.
(5) (6) Syllabus/Mathematical Science Syllabus/Mathematical Science
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 problems-
Airy’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 suc-
(7) (8) Syllabus/Mathematical Science Syllabus/Mathematical Science
cessive 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.
22. Distribution Theory : Properties of distribution functions and characteristic
functions; continuity theorem, inversion formula, Representation
of distribution function as a mixture of discrete and
continuous distribution functions, Convolutions, marginal and conditional
distributions of bivariate discrete and continuous distributions.
Relations between characteristic functions and moments: Moment
inequalities of Holder and Minkowski.
23. Statistical inference and Decision Theory : Statistical Decision
problem; non-randomized, mixed and randomized decision rules;
risk function, admissibility, Bayes’ rules, minimax rules, least
favourable distributions, complete class and minimal complete
class. Decision problem for finite perameter space. Convex loss
function. Role of sufficiency.
Families of distributions with monotone likelihood property, exponential
family of distributions. Test of simple hypothesis against
a simple alternative from decision, theoretic viewpoint. Tests with
Neyman structure. Uniformly most powerful unbiased tests. Locally
most powerful tests, inference on location and scale parameters;
estimation and tests. Equivariant estimators, invarience in
hypothesis testing.
24. Large sample statistical methods : Various modes of convergence.
Op and op, CLT, Sheffe’s theorem, Polya’s theorem and Slutsky’s
theorem, Transformation and varience stabilizing formula. asymptotic
distribution of function of sample moments, Sample quantiles,
Order statistics and their functions, Tests on correlations, coefficients
of variation, skewness and kurtosis, Pearson Chi-square,
contingency Chi-square and childhood ratio statistics, U-statistics,
Consistency of Tests, Asymptotic relative efficiency.
25. Multivariate statistical Analysis : Singular and non-singular multivariate
distributions. Characteristics functions Multivariate normal
distribution; marginal and conditional distribution, distribution
of linear forms, and quadratic forms, Cochran’s theorem.
inference on parameters of multivariate normal distributions; one
population and two-population cases. Wishart distribution, Handlings
T2, Mahalenobils D2, Discrimina-Analysis, Principal components,
Canonical correlations, Cluster analysis.
26. Linear Models and Regression : Standard Gauss Markov models;
Estimability of parameters; best linear, unbised estimates (Bell..);
(9) (10) Syllabus/Mathematical Science Syllabus/Mathematical Science
Method of least squares and Gauss-Markov theorem; Variancecoveriance
matrix of BLUES. Test of linear hypothesis, One-way
and two-way classifications. Fixed, random and mixed effects
models (two-way classifications only); variance components, Bivariable
and multiple linear regression; Poly-normal regression;
use of orthogonal poly-normals. Analysis of coveriance. Linear
and nonlear regression, Outliers.
27. Sample Surveys : Sampling with varying probability of selection,
Hurwitz-Thompson estimator; PPS sampling; Double sampling.
Cluster sampling. Non-sampling errors : Interpenetrating samples.
Multiphase sampling. Ratio and regression methods of estimation.
28. Design of Experiments : Factorial experiments, confounding and
fractional replication. Split and strip plot designs; Quasi-Latin
square designs; Youden square. Design for study of response surfaces;
first and second order designs. Incomplete block designs;
Balanced, connectedness and orthogonality, BIBD with recovery
of inter-block information; PBIBD with 2 associate classes. Analysis
of series of experiments, estimation of residual effects. Construction
of orthogonal-Latin squares, BIB designs and confounded factorial
designs. Optimality criteria for experimental designs.
29. Time-Series Analysis : Discrete-parameter stochastic processes;
strong and weak stationary; autocovariance and autocorrelation.
Moving average, autoregressive, autoregressive moving average
and autoregressive integrated moving average processes. Box-
Jenkins models. Estimation of the parameters in ARIMA models;
forecasting. Periodogram and correlogram analysis.
30. Stochastic Process : Markov chains with finite and countable state
space, classification of states, limiting behaviour of n-step transition
probabilities, stationary distribution; branching processes;
Random walk; Gambler’s ruin. Markov processes in continuous
time; Poisson processes, birth and death processes, Wiener process.
31. Demography and Vital Statistics : Measures of fertility and mortality,
period and Cohort measures. Life tables and its applications;
Methods of construction of abridged life tables. Application of
stable population theory to estimate vital rates. Population projections.
Stochastic models of fertility and reproduction.
32. Industrial Statistics : Control charts for variables and attributes;
Acceptance sampling by attributes; single, double and sequential
sampling plans; OC and ASN functions, AOQL and ATI; Acceptance
sampling by varieties. Tolerance limit. Reliability analysis :
Hazard function, distribution with DFR and IFR; Series and parallel
systems. Life testing experiments.
33. Inventory and Queueing theory : Inventory (S,s) policy, periodic
review models with stochesitic demand. Dynamic inventory models.
Probabilistics re-order point, lox size inventory system with
and without lead time. Distribution free analysis. Solution of inventory
problem with unknown density function. Warehousing
problem. Queues : Imbeded markov chain method to obtain steady
state solution of M/G/1,G/M/1 and M/D/C, Network models. Machine
maintenance models. Design and control of queing systems.
34. Dynamic Programming and Marketing : Nature of dynamic programming,
Determinstic processes, Non-sequential discrete
optimisation-allocation problems, assortment problems. Sequential
discrete optimisation long-term planning problems, multistage
production processes. Functional approximations. Marketing systems,
application of dynamic programming to marketing problems.
Introduction of new product, objective in setting market price and
its policies, purchasing under fluctuating prices. Advertising and
promotional decisions. Brands switching analysis, Distribution, decisions.
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