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Skills in Mathematics Algebra for IIT JEE Main and Advanced Arihant Dr. S K Goyal PDF

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Preview Skills in Mathematics Algebra for IIT JEE Main and Advanced Arihant Dr. S K Goyal

Algebra With Sessionwise Theory & Exercises Algebra With Sessionwise Theory & Exercises Dr. SK Goyal ARIHANT PRAKASHAN (Series), MEERUT All Rights Reserved © AUTHOR No part of this publication may be re-produced, stored in a retrieval system or by any means, electronic mechanical, photocopying, recording, scanning, web or otherwise without the written permission of the publisher. Arihant has obtained all the information in this book from the sources believed to be reliable and true. However, Arihant or its editors or authors or illustrators don’t take any responsibility for the absolute accuracy of any information published, and the damages or loss suffered thereupon. All disputes subject to Meerut (UP) jurisdiction only. Administrative & Production Offices Regd. Office ‘Ramchhaya’ 4577/15, Agarwal Road, Darya Ganj, New Delhi -110002 Tele: 011- 47630600, 43518550 Head Office Kalindi, TP Nagar, Meerut (UP) - 250002 Tel: 0121-7156203, 7156204 Sales & Support Offices Agra, Ahmedabad, Bengaluru, Bareilly, Chennai, Delhi, Guwahati, Hyderabad, Jaipur, Jhansi, Kolkata, Lucknow, Nagpur & Pune. ISBN : 978-93-25298-63-7 PO No : TXT-XX-XXXXXXX-X-XX Published by Arihant Publications (India) Ltd. For further information about the books published by Arihant, log on to www.arihantbooks.com or e-mail at [email protected] Follow us on P R E FAC E ‘‘THE ALGEBRAIC SUM OF ALL THE TRANSFORMATIONS OCCURRING IN A CYCLICAL PROCESS CAN ONLY BE POSITIVE, OR, AS AN EXTREME CASE EQUAL TO NOTHING’’ MEANS IF YOU CONTINUOUSLY PUT YOUR EFFORTS ON AN ASPECT YOU HAVE VERY GOOD CHANCE OF POSITIVE OUTCOME i.e. SUCCESS It is a matter of great pride and honour for me to have received such an overwhelming response to the previous editions of this book from the readers. In a way, this has inspired me to revise this book thoroughly as per the changed pattern of JEE Main & Advanced. I have tried to make the contents more relevant as per the needs of students, many topics have been re-written, a lot of new problems of new types have been added in etcetc. All possible efforts are made to remove all the printing errors that had crept in previous editions. The book is now in such a shape that the students would feel at ease while going through the problems, which will in turn clear their concepts too. A Summary of changes that have been made in Revised & Enlarged Edition — Theory has been completely updated so as to accommodate all the changes made in JEE Syllabus & Pattern in recent years. — The most important point about this new edition is, now the whole text matter of each chapter has been divided into small sessions with exercise in each session. In this way the reader will be able to go through the whole chapter in a systematic way. — Just after completion of theory, Solved Examples of all JEE types have been given, providing the students a complete understanding of all the formats of JEE questions & the level of difficulty of questions generally asked in JEE. — Along with exercises given with each session, a complete cumulative exercises have been given at the end of each chapter so as to give the students complete practice for JEE along with the assessment of knowledge that they have gained with the study of the chapter. — Last 13 Years questions asked in JEE Main & Adv, IIT-JEE & AIEEE have been covered in all the chapters. However I have made the best efforts and put my all Algebra teaching experience in revising this book. Still I am looking forward to get the valuable suggestions and criticism from my own fraternity i.e. the fraternity of JEE teachers. I would also like to motivate the students to send their suggestions or the changes that they want to be incorporated in this book. All the suggestions given by you all will be kept in prime focus at the time of next revision of the book. Dr. SK Goyal CONTENTS 1. COMPLEX NUMBERS 1-102 LEARNING PART Session 4 Session 1 — nth Root of Unity — Integral Powers of Iota (i) — Vector Representation of Complex Numbers — Switch System Theory — Geometrical Representation of Algebraic Operation on Complex Numbers Session 2 — Rotation Theorem (Coni Method) — Definition of Complex Number — Shifting the Origin in Case of Complex — Conjugate Complex Numbers Numbers — Representation of a Complex Number in — Inverse Points Various Forms — Dot and Cross Product Session 3 — Use of Complex Numbers in Coordinate — amp (z)– amp (–z)=± p, According as amp (z) Geometry is Positive or Negative — Square Root of a Complex Number PRACTICE PART — Solution of Complex Equations — JEE Type Examples — De-Moivre’s Theorem — Chapter Exercises — Cube Roots of Unity 2. THEORY OF EQUATIONS 103-206 LEARNING PART Session 4 Session 1 — Equations of Higher Degree — Polynomial in One Variable — Rational Algebraic Inequalities — Identity — Roots of Equation with the — Linear Equation Help of Graphs — Quadratic Equations Session 5 — Standard Quadratic Equation — Irrational Equations Session 2 — Irrational Inequations — Transformation of Quadratic Equations — Exponential Equations — Condition for Common Roots — Exponential Inequations — Logarithmic Equations Session 3 — Logarithmic Inequations — Quadratic Expression — Wavy Curve Method PRACTICE PART — Condition for Resolution into Linear Factors — JEE Type Examples — Location of Roots (Interval in which Roots Lie) — Chapter Exercises 3. SEQUENCES AND SERIES 207-312 LEARNING PART Session 5 Session 1 — Mean — Sequence Session 6 — Series — Arithmetico-Geometric — Progression Series (AGS) — Sigma (S) Notation Session 2 — Natural Numbers — Arithmetic Progression Session 7 Session 3 — Application to Problems of Maxima and — Geometric Sequence or Geometric Minima Progression Session 4 PRACTICE PART — Harmonic Sequence or Harmonic Progression — JEE Type Examples — Chapter Exercises 4. LOGARITHMS AND THEIR PROPERTIES 313-358 LEARNING PART Session 3 Session 1 — Properties of Monotonocity of Logarithm — Definition — Graphs of Logarithmic Functions — Characteristic and Mantissa PRACTICE PART Session 2 — JEE Type Examples — Principle Properties of Logarithm — Chapter Exercises 5. PERMUTATIONS AND COMBINATIONS 359-436 LEARNING PART Session 5 Session 1 — Combinations from Identical Objects — Fundamental Principle of Counting Session 6 — Factorial Notation — Arrangement in Groups Session 2 — Multinomial Theorem — Divisibility Test — Multiplying Synthetically — Principle of Inclusion and Exclusion Session 7 — Permutation — Rank in a Dictionary Session 3 — Gap Method — Number of Permutations Under Certain [when particular objects are never together] Conditions PRACTICE PART — Circular Permutations — JEE Type Examples — Restricted Circular Permutations — Chapter Exercises Session 4 — Combination — Restricted Combinations 6. BINOMIAL THEOREM 437-518 LEARNING PART Session 4 Session 1 — Use of Complex Numbers in Binomial Theorem — Binomial Theorem for Positive Integral Index — Multinomial Theorem — Pascal’s Triangle — Use of Differentiation — Use of Integration Session 2 — When Each Term is Summation Contains the — General Term Product of Two Binomial Coefficients or — Middle Terms Square of Binomial Coefficients — Greatest Term — Binomial Inside Binomial — Trinomial Expansion — Sum of the Series Session 3 PRACTICE PART — Two Important Theorems — Divisibility Problems — JEE Type Examples — Chapter Exercises 7. DETERMINANTS 519-604 LEARNING PART — System of Linear Equations Session 1 — Cramer’s Rule — Definition of Determinants — Nature of Solutions of System of Linear — Expansion of Determinant Equations — Sarrus Rule for Expansion — System of Homogeneous Linear Equations — Window Rule for Expansion Session 4 Session 2 — Differentiation of Determinant — Minors and Cofactors — Integration of a Determinant — Use of Determinants in Coordinate Geometry — Walli’s Formula — Properties of Determinants — Use of S in Determinant Session 3 PRACTICE PART — Examples on Largest Value of a — JEE Type Examples Third Order Determinant — Chapter Exercises — Multiplication of Two Determinants of the Same Order 8. MATRICES 605-690 LEARNING PART Session 2 Session 1 — Transpose of a Matrix — Definition — Symmetric Matrix — Types of Matrices — Orthogonal Matrix — Difference Between a Matrix and a — Complex Conjugate (or Conjugate) of a Matrix Determinant — Hermitian Matrix — Equal Matrices — Unitary Matrix — Operations of Matrices — Determinant of a Matrix — Various Kinds of Matrices — Singular and Non-Singular Matrices Session 3 Session 4 — Adjoint of a Matrix — Solutions of Linear Simultaneous Equations — Inverse of a Matrix Using Matrix Method — Elementary Row Operations PRACTICE PART — Equivalent Matrices — JEE Type Examples — Matrix Polynomial — Chapter Exercises — Use of Mathematical Induction 9. PROBABILITY 691-760 LEARNING PART Session 4 Session 1 — Binomial Theorem on Probability — Some Basic Definitions — Poisson Distribution — Mathematical or Priori or Classical Definition — Expectation of Probability — Multinomial Theorem — Odds in Favours and Odds Against the Event — Uncountable Uniform Spaces Session 2 — Some Important Symbols PRACTICE PART — Conditional Probability — JEE Type Examples Session 3 — Chapter Exercises — Total Probability Theorem — Baye’s Theorem or Inverse Probability 10. MATHEMATICAL INDUCTION 761-784 LEARNING PART PRACTICE PART — Introduction — JEE Type Examples — Statement — Chapter Exercises — Mathematical Statement 11. SETS, RELATIONS AND FUNCTIONS 785-836 LEARNING PART Session 3 Session 1 — Definition of Function — Definition of Sets — Domain, Codomain and Range — Representation of a Set — Composition of Mapping — Different Types of Sets — Equivalence Classes — Laws and Theorems — Partition of Set — Venn Diagrams (Euler-Venn Diagrams) — Congruences Session 2 PRACTICE PART — Ordered Pair — JEE Type Examples — Definition of Relation — Chapter Exercises — Ordered Relation — Composition of Two Relations SYLLABUS JEE MAIN numbers. Relation between AM and GM Sum upto n terms of special series: ∑ n, ∑ n2, ∑n3. Arithmetico - Unit I Sets, Relations and Functions Geometric progression. Sets and their representation, Union, intersection and Unit VIII Probability complement of sets and their algebraic properties, Power Probability of an event, addition and multiplication set, Relation, Types of relations, equivalence relations, theorems of probability, Baye’s theorem, probability functions, one-one, into and onto functions, composition distribution of a random variate, Bernoulli and Binomial of functions. distribution. Unit II Complex Numbers Complex numbers as ordered pairs of reals, JEE ADVANCED Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra Algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a Algebra of complex numbers, addition, multiplication, complex number, triangle inequality. conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of Unit III Matrices and Determinants unity, geometric interpretations. Matrices, algebra of matrices, types of matrices, Quadratic equations with real coefficients, relations determinants and matrices of order two and three. between roots and coefficients, formation of quadratic Properties of determinants, evaluation of deter-minants, equations with given roots, symmetric functions of roots. area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using Arithmetic, geometric and harmonic progressions, determinants and elementary transformations, Test of arithmetic, geometric and harmonic means, sums of finite consistency and solution of simultaneous linear arithmetic and geometric progressions, infinite geometric equations in two or three variables using determinants series, sums of squares and cubes of the first n natural and matrices. numbers. Unit IV Permutations and Combinations Logarithms and their Properties Fundamental principle of counting, permutation as an Permutations and combinations, Binomial theorem for a arrangement and combination as selection, Meaning of positive integral index, properties of binomial P(n,r) and C (n,r), simple applications. coefficients. Unit V Mathematical Induction Matrices as a rectangular array of real numbers, equality Principle of Mathematical Induction and its simple of matrices, addition, multiplication by a scalar and applications. product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square Unit VI Binomial Theorem and its matrix of order up to three, properties of these matrix Simple Applications operations, diagonal, symmetric and skew-symmetric Binomial theorem for a positive integral index, general matrices and their properties, solutions of simultaneous term and middle term, properties of Binomial coefficients linear equations in two or three variables. and simple applications. Addition and multiplication rules of probability, Unit VII Sequences and Series conditional probability, independence of events, Arithmetic and Geometric progressions, insertion of computation of probability of events using permutations arithmetic, geometric means between two given and combinations.

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