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Re: CSIR NET Mathematical Sciences Previous Paper
Council of Scientific & Industrial Research (CSIR) is organize Joint CSIR UGC National Eligibility Test (NET) .. here I am giving you question paper for the CSIR NET Mathematical Sciences examination in PDF file attached with it so you can get it easily.. Syllabus for CSIR NET Mathematical Sciences given below : UNIT – 1 Analysis: Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform continuity, differentiability, mean value theorem. Sequences and series of functions, uniform convergence. Riemann sums and Riemann integral, Improper Integrals. Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral. Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation. Metric spaces, compactness, connectedness. Normed Linear Spaces. Spaces of Continuous functions as examples. Linear Algebra: Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations. Algebra of matrices, rank and determinant of matrices, linear equations. Eigenvalues and eigenvectors, CayleyHamilton theorem. Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms. Inner product spaces, orthonormal basis. Quadratic forms, reduction and classification of quadratic forms. UNIT – 2 Complex Analysis: Algebra of complex numbers, the complex plane, polynomials, Power series, transcendental functions such as exponential, trigonometric and hyperbolic functions. Analytic functions, CauchyRiemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem. Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius transformations. Algebra: Permutations, combinations, pigeonhole principle, inclusionexclusion principle, derangements. Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ř function, primitive roots. Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems. Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain. Polynomial rings and irreducibility criteria. Fields, finite fields, field extensions. UNIT – 3 Ordinary Differential Equations (ODEs): Existence and Uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs. General theory of homogenous and nonhomogeneous linear ODEs, variation of parameters, SturmLiouville boundary value problem, Green’s function. Partial Differential Equations (PDEs): Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs. Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat and Wave equations. Numerical Analysis : Numerical solutions of algebraic equations, Method of iteration and NewtonRaphson method, Rate of convergence, Solution of systems of linear algebraic equations using Gauss elimination and GaussSeidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and RungeKutta methods. Calculus of Variations: Variation of a functional, EulerLagrange equation, Necessary and sufficient conditions for extrema. Variational methods for boundary value problems in ordinary and partial differential equations. Linear Integral Equations: Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel. Classical Mechanics: Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s principle and principle of least action, Twodimensional motion of rigid bodies, Euler’s dynamical equations for the motion of a rigid body about an axis, theory of small oscillations. UNIT – 4 Descriptive statistics, exploratory data analysis. Sample space, discrete probability, independent events, Bayes theorem. Random variables and distribution functions (univariate and multivariate); expectation and moments. Independent random variables, marginal and conditional distributions. Characteristic functions. Probability inequalities (Tchebyshef, Markov, Jensen). Modes of convergence, weak and strong laws of large numbers, Central Limit theorems (i.i.d. case). Markov chains with finite and countable state space, classification of states, limiting behaviour of nstep transition probabilities, stationary distribution. Standard discrete and continuous univariate distributions. Sampling distributions. Standard errors and asymptotic distributions, distribution of order statistics and range. Methods of estimation. Properties of estimators. Confidence intervals. Tests of hypotheses: most powerful and uniformly most powerful tests, Likelihood ratio tests. Analysis of discrete data and chisquare test of goodness of fit. Large sample tests. Simple nonparametric tests for one and two sample problems, rank correlation and test for independence. Elementary Bayesian inference. GaussMarkov models, estimability of parameters, Best linear unbiased estimators, tests for linear hypotheses and confidence intervals. Analysis of variance and covariance. Fixed, random and mixed effects models. Simple and multiple linear regression. Elementary regression diagnostics. Logistic regression. Multivariate normal distribution, Wishart distribution and their properties. Distribution of quadratic forms. Inference for parameters, partial and multiple correlation coefficients and related tests. Data reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis, Canonical correlation. Simple random sampling, stratified sampling and systematic sampling. Probability proportional to size sampling. Ratio and regression methods. Completely randomized, randomized blocks and Latinsquare designs. Connected, complete and orthogonal block designs, BIBD. 2K factorial experiments: confounding and construction. Series and parallel systems, hazard function and failure rates, censoring and life testing. Linear programming problem. Simplex methods, duality. Elementary queuing and inventory models. Steadystate solutions of Markovian queuing models: M/M/1, M/M/1 with limited waiting space, M/M/C, M/M/C with limited waiting space, M/G/1.
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#4
 
 
Re: CSIR NET Mathematical Sciences Previous Paper
As you want previous year Mathematical Sciences question paper of CSIR National Eligibility Test (NET) so here I am giving you the same, please have a look……. CSIR UGC  NET: Mathematical Sciences Previous Papers (Solved) and 25 Practice Sets by Dhipragya Dubey 1. An unknown number N of taxis plying in a town are supposed to be erially numbered from 1 to N. If the n different taxis you have come across in the town can be assumed to form a simple random sample with replacement, find an unbiased estimator of the total number of taxis in the town. Also find the variance of your estimator. 2. Show that in a randomized block design the estimates of the elementary block and treatment contrasts are orthogonal observational contrasts. 3. Suppose in a 25–factorial experiment with factors A, B, C, D and E, a replicate is divided into four blocks of size eight each. How many effects will be confounded? Is it possible to confound the effects AB, BC and ABC? Justify your answer. 4. The daily demand for a commodity is approximately 100 units. Every time an order is placed, a fixed cost of Rs.10,000/ is incurred. The daily holding cost per unit inventory is Rs.2/. If the lead time is 15 days, determine the economic lot size and the reorder point. Further suppose that the demand is actually an approximation of a probabilistic distribution in which the daily demand is normal with mean µ = 100 and s.d. σ = 10. How would you determine the size of the buffer stock such that the probability of running out of stock during lead time is at most 0.01? 5. A physiological disorder X always leads to the disorder Y. However, disorder Y may occur by itself. A population shows 4 % incidence of disorder Y. Which of the following inferences is valid? 4% of the population suffers from both X & Y Less then 4% of the population suffers from X At least 4% of the population suffers from X There is no incidence of X in the given population 6. Diabetic patients are advised a low glycaemic index diet. The reason for this is They require less carbohydrate than healthy individuals They cannot assimilate ordinary carbohydrates They need to have slow, but sustained release of glucose in their blood stream They can tolerate lower, but not higher than normal blood sugar levels 7. Standing on a polished stone floor one feels colder than on a rough floor of the same stone. This is because: Thermal conductivity of the stone depends on the surface smoothness Specific heat of the stone changes by polishing it The temperature of the polished floor is lower than that of the rough floor There is greater heat loss from the soles of the feet when in contact with the polished floor than of the rough floor 8. If the atmospheric concentration of carbon dioxide is doubled and there are favorable conditions of water, nutrients, light and temperature, what would happen to water requirement of plants? It decreases initially for a short time and then returns to the original value It increases It decreases It increases initially for a short time and then returns to the original value 9. To examine whether two different skin creams, A and B, have different effect on the human body n randomly chosen persons were enrolled in a clinical trial. Then cream A was applied to one of the randomly chosen arms of each person, cream B to the other. What kind of a design is this? Completely Randomized Design Balanced Incomplete Block Design Randomized Block Design Latin Square Design Previous year Mathematical Sciences question paper of CSIR National Eligibility Test NET
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