UPKAR PRAKASHAN, AGRA–2

ByDr. Alok Kumar

Institute of Engineering & Technology,

Dr. B. R. Ambedkar University, Agra

Latest Revised Edition

© Publishers

Publishers

UPKAR PRAKASHAN(An ISO 9001 : 2000 Company)

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● The publishers have taken all possible precautions in publishing this book, yet ifany mistake has crept in, the publishers shall not be responsible for the same.

● This book or any part thereof may not be reproduced in any form byPhotographic, Mechanical, or any other method, for any use, without writtenpermission from the Publishers.

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ISBN : 978-81-7482-380-9

Price : 660·00(Rs. Six Hundred Sixty Only)

Code No. 1587

Printed at : UPKAR PRAKASHAN (Printing Unit) Bye-pass, AGRA

A WORD TO THE READER

The National Eligibility Test (NET) for Junior Research Fellowship and Eligibility for Lectureshipare conducted jointly by Council of Scientific and Industrial Research (CSIR) and University GrantCommission (UGC) twice a year is one of the most competitive examinations in India. Different statesalso now have started conducting this examination for their own examinations under State EligibilityTest (SET).

About the ExaminationThe NET will have a single paper with Multiple Choice Questions, divided in three parts :Part-A : Marks 30Part-B : Marks 75Part-C : Marks 95Part-A : This part shall carry 20 questions pertaining to General aptitude with emphasis on logical

reasoning graphical analysis, analytical and numerical ability, quantitative comparisons, seriesformation, puzzles etc. The candidates shall be required to answer any 15 questions. Each question shallbe of two marks. The total marks allocated to this section shall be 30 out of 200.

Part-B : It will have 40 objective type questions from the subject area. The candidate ofMathematical Sciences will have to attempt any 25 questions out of the given 40 questions, each carry 3marks. Negative marking will be made for wrong answers.

Part-C : It will consist of a number of questions with multiple correct options. Credit in a questionshall be given only on identification of all correct options. (Confined to a maximum of one pagelength). The candidate will be required to answer 20 questions. Each question is of 4·75 marks. Thetotal section is of 95 marks.

The level expected at the examination is that of post-graduate level.The specified number of questions are only evaluated.

Solving the PaperRead the instructions in the question paper carefully. Answer to the multiple choice questions haveto be marked on separately provided OMR Answer Sheets.

How will this book help you ?Primarily meant for CSIR-UGC NET candidates of Mathematical Sciences as per revised

syllabus, this edition thoroughly covers all the topics essential for students of MATHEMATICS,OPERATIONS RESEARCH and STATISTICS stream, as prescribed in the CSIR-UGC : NETSyllabus.

Consistence is maintained and each chapter is kept independent to each other, candidate arerequired to go through the inconsistences too since subject matter differs as per need from chapter tochapter.

We are sure that candidate appearing for the examinations will find this book extremely useful inpreparation of paper and achieving their desired goal.

At last author wish to acknowledge Mr. Mahendra Jain, Publisher UPKAR PRAKASHAN. Theauthor is also grateful to Dr. Madhu Jain and Professor G. C. Sharma for their valuable suggestions anddiscussion on the subject. The author also gratefully acknowledge and express deep appreciation to themany wonderful authors of Mathematical Sciences without consulting their valuable books it isimpossible to write this book. Any suggestions for further improvement and queries regarding this bookis highly welcomed.

With best wishes, —Dr. Alok Kumar

Contents

● Solved Paper

PART–A

General Aptitude……………………....…………………………………….. 1–120

PART–B & Part–C

1. Analysis..……………………....……………………………………… 3–116Part–B…………………………………..……………………………... 33Part–C…………………………………………………………………. 68

2. Linear Algebra…………………………………………….………… 117–191Part–B…………………………………..……………………………... 135Part–C…………………………………………………………………. 158

3. Complex Analysis.…………………………………………………… 192–253Part–B…………………………………..……………………………... 209Part–C…………………………………………………………………. 231

4. Algebra.………………………………………………………………. 254–339Part–B…………………………………..……………………………... 268Part–C…………………………………………………………………. 280

5. Ordinary and Partial Differential Equations..……………..……… 340–401Part–B…………………………………..……………………………... 355Part–C…………………………………………………………………. 382

6. Numerical Analysis..……………………………………..…………… 402–437Part–B…………………………………..………………………….…... 411Part–C……………………………………………………………….…. 420

7. Calculus of Variation.……………………………….…………...…… 438–464Part–B…………………………………..…………………………….... 442Part–C……………………………………………………………….…. 450

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8. Linear Integral Equations..………………………………………...… 465–499Part–B…………………………………..………………………….…... 470Part–C…………………………………………………………….……. 475

9. Classical Mechanics..……………………………...………………..… 500–542Part–B…………………………………..…………………………….... 505Part–C……………………………………………………………….…. 517

10. Probability..…………………………………………..……………..… 543–579Part–B…………………………………..………………………….…... 551Part–C…………………………………………………………….……. 576

11. Statistics..…………………………………………………………..….. 580–699Part–B…………………………………..………………………….…... 611Part–C…………………………………………………………….……. 677

12. Linear Programming & Operations Research.……………….......… 700–792Part–B…………………………………..………………………….…... 730Part–C…………………………………………………………….……. 764

GENERAL INFORMATION

SCHEME OF EXAMINATIONTime : 3 Hrs. Max. Marks : 200

Single Paper Test having Multiple ChoiceQuestions (MCQs) is divided in three parts.

Part ‘A’This part shall carry 20 questions pertaining

to General aptitude with emphasis on logicalreasoning graphical analysis, analytical andnumerical ability, quantitative comparisons, seriesformation, puzzles etc. The candidates shall berequired to answer any 15 questions. Eachquestion shall be of two marks. The total marksallocated to this section shall be 30 out of 200.

Part ‘B’This part shall contain 40 Multiple Choice

Questions (MCQs) generally covering the topicsgiven in the syllabus. A candidate shall berequired to answer any 25 questions. Eachquestion shall be of three marks. The total marksallocated to this section shall be 75 out of 200.

Part ‘C’This part shall contain 60 questions that are

designed to test a candidate's knowledge ofscientific concepts and/or application of thescientific concepts. The questions shall be ofanalytical nature where a candidate is expected toapply the scientific knowledge to arrive at thesolution to the given scientific problem. Thequestions in this part shall have multiple correctoptions. Credit in a question shall be given onlyon identification of ALL the correct options. Nocredit shall be allowed in a question if anyincorrect option is marked as correct answer. Nopartial credit is allowed. A candidate shall berequired to answer any 20 questions. Eachquestion shall be of 4.75 marks. The total marksallocated to this section shall be 95 out of 200.

● For Part ‘A’ and ‘B’ there will be Negativemarking @25% for each wrong answer. NoNegative marking for Part ‘C’.

● To enable the candidates to go through thequestions, the question paper booklet shall bedistributed 15 minute before the scheduledtime of the Exam. The answer sheet (OMRsheet) shall be distributed at the scheduledtime of the Exam.

SYLLABUSPART–A

This part shall carry 20 questions pertainingto General aptitude with emphasis on logicalreasoning graphical analysis, analytical andnumerical ability, quantitative comparisons, seriesformation, puzzles etc. The candidates shall berequired to answer any 15 questions. Eachquestion shall be of two marks. The total marksallocated to this section shall be 30 out of 200.

(Common Syllabus for Part ‘B and C’)UNIT – I

Analysis : Elementary set theory, finite,countable and uncountable sets, Real numbersystem as a complete ordered field, Archimedeanproperty, supremum, infimum.

Sequences and series, convergence, limsup,liminf.

Bolzano Weierstrass theorem, Heine Boreltheorem.

Continuity, uniform continuity, differenti-ability, mean value theorem.

Sequences and series of functions, uniformconvergence.

Riemann sums and Riemann integral,Improper Integrals.

Monotonic functions, types of discontinuity,functions of bounded variation, Lebesguemeasure, Lebesgue integral.

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Functions of several variables, directionalderivative, partial derivative, derivative as a lineartransformation.

Metric spaces, compactness, connectedness.Normed Linear Spaces. Spaces of Continuousfunctions as examples.

Linear Algebra : Vector spaces, subspaces,linear dependence, basis, dimension, algebra oflinear transformations.

Algebra of matrices, rank and determinant ofmatrices, linear equations.

Eigen values and eigen vectors, Cayley-Hamilton theorem.

Matrix representation of linear transfor-mations. Change of basis, canonical forms,diagonal forms, triangular forms, Jordan forms.

Inner product spaces, orthonormal basis.

Quadratic forms, reduction and classificationof quadratic forms.

UNIT – IIComplex Analysis : Algebra of complex

numbers, the complex plane, polynomials, Powerseries, transcendental functions such asexponential, trigonometric and hyperbolicfunctions.

Analytic functions, Cauchy-Riemannequations.

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 ofresidues.

Conformal mappings, Mobius transformations.Algebra : Permutations, combinations,

pigeon-hole principle, inclusion-exclusionprinciple, derangements.

Fundamental theorem of arithmetic,divisibility in Z, congruences, Chinese RemainderTheorem, Euler’s φ- function, primitive roots.

Groups, subgroups, normal subgroups,quotient groups, homomorphisms, cyclic groups,permutation groups, Cayley’s theorem, classequations, 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, GaliosTheory.

Topology : Basis, dense sets, subspace andproduct topology, separation axioms, connected-ness and compactness.

UNIT – IIIOrdinary Differential Equations (ODEs) :

Existence and Uniqueness of solutions of initialvalue problems for first order ordinary differentialequations, singular solutions of first order ODEs,system of first order ODEs.

General theory of homogeneous and non-homogeneous linear ODEs, variation ofparameters, Sturm-Liouville boundary valueproblem, Green’s function.

Partial Differential Equations (PDEs) :Lagrange and Charpit methods for solving firstorder PDEs, Cauchy problem for first order PDEs.

Classification of second order PDEs, Generalsolution of higher order PDEs with constantcoefficients, Method of separation of variables forLaplace, Heat and Wave equations.

Numerical Analysis : Numerical solutions ofalgebraic equations, Method of iteration andNewton-Raphson method, Rate of convergence,Solution of systems of linear algebraic equationsusing Gauss elimination and Gauss-Seidelmethods, Finite differences, Lagrange, Hermiteand spline interpolation, Numerical differentiationand integration, Numerical solutions of ODEsusing Picard, Euler, modified Euler and Runge-Kutta methods.

Calculus of Variations : Variation of afunctional, Euler-Lagrange equation, Necessaryand sufficient conditions for extrema. Variationalmethods for boundary value problems in ordinaryand partial differential equations.

Linear Integral Equations : Linear integralequation of the first and second kind of Fredholmand Volterra type, Solutions with separablekernels. Characteristic numbers and eigenfunctions, resolvent kernel.

Classical Mechanics : Generalized coordi-nates, Lagrange’s equations, Hamilton’s canonicalequations, Hamilton’s principle and principle ofleast action, Two-dimensional motion of rigid

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bodies, Euler’s dynamical equations for themotion of a rigid body about an axis, theory ofsmall oscillations.

UNIT – IVDescriptive Statistics, Exploratory Data

Analysis : Sample space, discrete probability,independent events, Bayes theorem. Randomvariables and distribution functions (univariateand multivariate); expectation and moments.Independent random variables, marginal andconditional distributions. Characteristic functions.Probability inequalities (Tchebyshef, Markov,Jensen). Modes of convergence, weak and stronglaws of large numbers, Central Limit theorems(i.i.d. case).

Markov chains with finite and countable statespace, classification of states, limiting behaviourof n-step transition probabilities, stationary distri-bution, Poisson and birth-and-death processes.

Standard discrete and continuous univariatedistributions. Sampling distributions. Standarderrors and asymptotic distributions, distribution oforder statistics and range.

Methods of estimation. Properties ofestimators. Confidence intervals. Tests ofhypotheses : most powerful and uniformly mostpowerful tests, Likelihood ratio tests. Analysis ofdiscrete data and chi-square test of goodness of fit.Large sample tests.

Simple non-parametric tests for one and twosample problems, rank correlation and test forindependence. Elementary Bayesian inference.

Gauss-Markov models, estimability ofparameters, Best linear unbiased estimators, tests

for linear hypotheses and confidence intervals.Analysis of variance and covariance. Fixed,random and mixed effects models. Simple andmultiple linear regression. Elementary regressiondiagnostics. Logistic regression.

Multivariate normal distribution, Wishartdistribution and their properties. Distribution ofquadratic forms. Inference for parameters, partialand multiple correlation coefficients and relatedtests. Data reduction techniques : Principlecomponent analysis, Discriminant analysis,Cluster analysis, Canonical correlation.

Simple random sampling, stratified samplingand systematic sampling. Probability proportionalto size sampling. Ratio and regression methods.

Completely randomized, randomized blocksand Latin-square designs. Connectedness andorthogonality of block designs, BIBD. 2k factorialexperiments : confounding and construction.

Hazard function and failure rates, censoringand life testing, Series and parallel systems.

Linear programming problem. Simplexmethods, duality. Elementary queuing andinventory models. Steady-state solutions ofMarkovian queuing models : M/M/1, M/M/1 withlimited waiting space, M/M/C, M/M/C withlimited waiting space, M/G/1.

All students are expected to answerquestions from Unit-1. Mathematics studentsare expected to answer additional questionsfrom Unit-II and III. Statistics students areexpected to answer additional questions fromUnit-IV.

Mathematical SciencesCSIR-UGC NET/JRF Exam.

Solved Paper

CSIR-UGC NET/JRF/Set MathematicalSciences

Publisher : Upkar Prakashan ISBN : 9788174823809 Author : Dr. Alok Kumar

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