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ABOUT PEARSON:


Pearson is the world’s learning company, with presence across 70 countries worldwide. Our unique insights and world-class expertise comes from a long history of working closely with renowned teachers, authors and thought leaders, as a result of which, we have emerged as the preferred choice for millions of teachers and learners across the world. We believe learning opens up opportunities, creates fulfilling careers and hence better lives. 




We hence collaborate with the best of minds to deliver you class-leading products, spread across the Higher Education and K12 spectrum. Superior learning experience and improved outcomes are at the heart of everything we do. This product is the result of one such effort. 


 CONTENTS:


  1. POLYNOMIALS
  2. INEQUALITIES
  3. MATHEMATICAL INDUCTION
  4. RECURRENCE RELATION
  5. FUNCTIONAL EQUATIONS
  6. NUMBER THEORY
  7. COMBINATORICS
  8. GEOMETRY

BRIEF CONTENTS:

POLYNOMIALS:



  • Polynomial Functions
  • Division in Polynomials
  • Remainder Theorem and factor Theorem
  • Fundamental Theorem for Algebra
  • Polynomial Equations
  • Vieta's Relation
  • Symmetric Functions
  • Common Roots of Polynomials Equations
  • Irreducibility of Polynomials

INEQUALITIES:


  • Basic Rules
  • Weirststras's Inequality
  • Modulus Inequalities
  • Sum of Squares (SOS)
  • Arithmetic Mean and Geometric Mean
  • Weighted Means
  • Power Mean Inequality
  • Rearrangement Inequality
  • Chebyshev's Inequality
  • Cauchy-Schwarz Inequality
  • Holders Inequality
  • Some Geometrical inequalities
  • Jensen's Inequalities

MATHEMATICAL INDUCTION:


  • Introduction
  • First (or Weak) Principle of Mathematical Induction
  • Second (or strong) Principle of Mathematical Induction

RECURRENCE RELATION:


  • Introduction
  • Classification
  • First Order Linear Recurrence Relation
  • First Order Non Linear
  • Linear Homogeneous Recurrence Relation with constant coefficient
  • General Form of Homogeneous Recurrence Relation with constant coefficient
  • General Method For Non-Homogeneous

FUNCTIONAL EQUATIONS:


  • Functions
  • Functional Equations

NUMBER THEORY:


  • Divisibility of Integers
  • Euclid's Division Division  (GCD)
  • Primes
  • Fundamental Theorem of Arithmetic
  • Modular Arithmetic
  • Complete Residue System
  • Some Important Functions Theorem
  • Scales of Notation
  • Greatest Integer Functions
  • Diophantine Equations

COMBINATORICS:


  • Definition of Factorial
  • Basic Counting Principles
  • Combinations
  • The Bijective Principle
  • Combinations with Repetitions Allowed
  • Definition of permutation
  • Division and Distribution of Non-identical items
  • Number of Integral Solution
  • Binomial, Multinomial and Generating functions
  • Applications of recurrence Relations
  • principle of Inclusion and exclusion (PIE)
  • Dearrangement
  • Classical Occupancy Problems
  • Dirichlet's (or Pigeon Hole)  Principle (PHP)

GEOMETRY:


  • Angle
  • Congruent Triangles
  • Triangle Inequality
  • Ratio and Proportion Theorem
  • Mid-point Theorem
  • Basic Proportionality Theorem
  • Similar Triangles
  • Baudhayana (Pythagoras) Theorem
  • Quadrilaterals


PREFACE:

 “For another hundred years, School will teach children ‘to do’ rather than ‘to think’” observed Bertrand Russell. This statement is still seen to be true without being even remotely contradicted. NCF 2005 (National Curriculum Framework) provides a vision for perspective planning of school education in scholastic and non-scholastic domains. It also emphasizes on ‘mathematisation’ of the child’s thought and processes by recognizing mathematics as an integral part of development of the human potential. The higher aim of teaching mathematics is to enhance the ability to visualize, logically understand, build arguments, prove statements and in a sense, handle abstraction. 




For motivated and talented students, there is a need to widen the horizon as these students love challenges and always look beyond the curriculum at school. Hence, we created this book to cater to the needs of these students. With numerous problems designed to develop thinking and reasoning, the book contains statements, definitions, postulates, formulae, theorems, axioms, and propositions, which normally do not appear in school textbooks. These are spelt out and interpreted to improve the student’s conceptual knowledge. The book also presents ‘non-routine problems’ and detailed, step-by-step solutions to these problems to enable the reader to acquire a better understanding of the concepts as well as to develop analytical and reasoning (logical) abilities. Thus,  readers get the ‘feel’ of problem-solving as an activity which, in turn, reveals the innate pleasure of successfully solving a challenging problem. This ‘pleasure’ is permanent and helps to build-in them a positive attitude towards the subject. Developing ability for critical analysis and problem solving is an essential requirement if one wants to become successful in life. No one has yet discovered a way of learning mathematics better than by  solving problems in the subject. 




This book helps students to face competitive examinations such as the Olympiads (RMO, INMO, IMO), KVPY and IIT-JEE confidently  without being befuddled by the intricacies of the subject. It has been designed to enable  students and all lovers of mathematics to master the subject at their own pace. We have made efforts to provide solutions along with the problems in an error-free and unambiguous manner as far as possible. However, if any error is detected by the reader, it may please be brought to our notice, so that we may make necessary corrections in the future editions of the book. We look forward to your suggestions and shall be grateful for them. Lastly, we share the observation made by Pundit Jawaharlal Nehru: “Giving opportunity to potential creativity is a matter of life and death for an enlightened society because the contributions of a few creative individuals are the mankind’s ultimate capital asset.” We wish best of luck at all times to all those using this book. 



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