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

846 Pages·2022·11.107 MB·English

by  S K Goyal

Algebra
With Sessionwise Theory & Exercises
Algebra
With Sessionwise Theory & Exercises
Dr. SK Goyal
ARIHANT PRAKASHAN (Series), MEERUT
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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.