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Mathematics X, Second Edition Mathematics X Based on the syllabus prescribed by the BSEM For Class X A J Sanasam N C Cogent Nirtish Laishram This is a preview copy containing some selected pages only. Some pages have been intentionally left blank. LCLC Lousing Chaphu A J Sanasam, Assistant Professor, Manipur Technical University, Imphal Email: ajsanasam[at]lousingchaphu.com Ningombam Cha Cogent Email: ncogent[at]lousingchaphu.com Nirtish Laishram, M.Sc. Mathematics, University of Delhi, SLET, B.Ed. Email: nklaishram[at]lousingchaphu.com Editorial Board A J Sanasam, N C Cogent, Nirtish Laishram, Hidam Gobind, Shyamson Laishram Cover Picture: Fractals are any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size. Front Cover: (a) 282589933 1 is the largest known prime (as of January 2019) with − 24862048 digits. It is a Mersenne prime (a prime of the form 2p 1, where p is a prime). − (b) The diagram shows the construction of a pair of tangents to a circle from an external point. Back Cover: RSA-768 is a semiprime (a number with exactly two prime factors). It was a part of the RSA Factoring Challenge. Large semiprimes play an important role in cryptography. Mathematics X c 2019 Authors This work is subject to copyright. All rights are reserved. This work may not be translated or copied or stored as soft copy in whole or in part without the prior written permission of the authors. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Price: | 400/- First Edition 2015 Second Edition 2019 This edition is published by Lousing Chaphu, Thoubal. info[at]lousingchaphu.com https://www.lousingchaphu.com Printed in Manipur by International Printers, Thoubal Melaground, Thoubal 795138. To teachers who inspire the young minds of tomorrow Preface This edition is equipped with more than a hundred pages of new materials. Unlike the first edition, this edition includes complete proofs of almost all theorems. We have added numerous examples and exercises. The purpose of these examples and exercises is to develop problem solving skills. It is through these examples and exercises that the students can as- sess progress and understanding. Hints are given to several of the exercises. Our general approach on this book is simple, and we hope that the students will be equally interested in all parts of this book. The main emphasis is given on understanding the underlying fundamental principles. This book can be used as a main text, a reference text or a supplementary text. It can also be used for self study. This new edition gave us the opportunity to streamline some arguments to correct errors and misprints and to ratify proofs and solutions. We would like to thank all those who gave comments and suggestions to the first edition. We welcome remarks and suggestions from our readers. Thoubal A J Sanasam February 2019 N C Cogent Nirtish Laishram Acknowledgements This work has been made possible with the support and encouragement from several persons. First of all, we are grateful to Hidam Gourashyam for his encouragement and mental support. The publication of this book would have been delayed if it were not for the support given by Laishram Aboy Singh, Laishram Wanglen Meitei, Phaln Maibam, Rabinath Akham, Md Gayaz Ahamed, Puyam Ajit Singh and Mayanglambam Stalin Singh. Finally, we would like to thank Jasoba Singh Konthoujam, Arjun Laishram, Oinam Premdas Singh and Jocyline Thokchom for their endless support. Contents Prefacev Notations and Abbreviationsx 1 Number System1 Divisibility .............................. 1 Euclid’s Division Lemma...................... 3 Euclid’s Algorithm......................... 5 Exercise 1.1............................. 8 Fundamental Theorem of Arithmetic............... 14 Exercise 1.2............................. 18 Field Properties of Real Numbers................. 27 Absolute Value or Modulus of a Real Number........... 31 Exercise 1.3............................. 35 2 Polynomials 39 Division Algorithm for Polynomials................ 39 Exercise 2.1............................. 43 Remainder Theorem......................... 51 Exercise 2.2............................. 53 Factorisation of Polynomials.................... 59 Exercise 2.3............................. 60 3 Factorisation 68 Factorisation of Cyclic Expressions................ 68 Exercise 3.1............................. 74 4 Pair of Linear Equations in Two Variables 83 Graphical Method for Solving a Pair of Linear Equations . 83 Exercise 4.1............................. 87 Algebraic Methods of Solving a Pair of Linear Equations . 108 Exercise 4.2............................. 112 Problems Involving a Pair of Linear Equations . 137 viii Contents | Exercise 4.3............................. 138 5 Quadratic Equations 150 Solution of Quadratic Equations . 150 Exercise 5.1............................. 158 Word Problems based on Quadratic Equation . 172 Exercise 5.2............................. 174 6 Arithmetic Progression (AP) 183 General Terms of an AP...................... 183 Exercise 6.1............................. 187 Sum of the First n Terms of an AP . 198 Exercise 6.2............................. 200 7 Triangles 213 Similar Figures........................... 213 Exercise 7.1............................. 214 Similarity of Triangles....................... 216 Exercise 7.2............................. 219 Internal and External Bisectors of an Angle of a Triangle . 226 Exercise 7.3............................. 228 Criteria for Similarity of Triangles . 231 Exercise 7.4............................. 238 Areas of Similar Triangles..................... 248 Exercise 7.5............................. 250 Pythagoras Theorem........................ 254 Exercise 7.6............................. 257 8 Circles 265 Tangent to a Circle......................... 265 Exercise 8.1............................. 270 9 Construction 278 Division of a Line Segment in a given Ratio . 278 Construction of a Triangle Similar to a given Triangle . 281 Exercise 9.1............................. 284 Construction of Tangents to a Circle . 294 Exercise 9.2............................. 297 10 Trigonometry 301 Trigonometric Ratios of an Acute Angle . 301 Exercise 10.1 ............................ 308 Contents ix | Relationships between the Trigonometric Ratios . 310 Exercise 10.2 ............................ 313 Trigonometric Ratios of Complementary Angles . 324 Exercise 10.3 ............................ 326 Heights and Distances........................331 Exercise 10.4 ............................ 334 11 Coordinate Geometry 346 Section Formula .......................... 346 Exercise 11.1 ............................ 350 Area of a Triangle ......................... 356 Exercise 11.2 ............................ 359 12 Mensuration 366 Perimeter and Area of a Circle .................. 366 Exercise 12.1 ............................ 369 Areas of Sectors and Segments of a Circle . 375 Exercise 12.2 ............................ 379 Area of Combinations of Plane Figures . 388 Exercise 12.3 ............................ 389 Surface Area and Volume of Solids . 400 Exercise 12.4 ............................ 403 Conversion of Solid from One Shape to Another . 417 Exercise 12.5 ............................ 418 Frustum of a Right Circular Cone . 430 Exercise 12.6 ............................ 434 13 Statistics 443 Measures of Central Tendency................... 443 Methods for Finding Arithmetic Mean . 445 Measures of Location........................ 449 Cumulative Frequency Curve or Ogive . 452 Exercise 13.1 ............................ 458 14 Probability 474 Classical Definition of Probability . 474 Independent Events and Independent Experiments . 482 Exercise 14.1 ............................ 489 Bibliography 496 Index 497 Notations and Abbreviations N the natural numbers Z the integers Q the rational numbers R the real numbers C the complex numbers if and only if ⇐⇒ = implies ⇒ for all ∀ there exists ∃ triangle 4 ∠ angle x A the element x belongs to the set A ∈ A B the union of A and B ∪ A B the intersection of A and B ∩ n! the product of first n natural numbers x the greatest integer less than or equal to x b c x the absolute value of x | | (a, b) the gcd of a and b (see Chapter 1) [a, b] the lcm of a and b (x, y) the coordinates of a point (see Chapter 11) min a, b the minimum of a and b { } max a, b the maximum of a and b { } Q.E.D. (quod erat demonstrandum), that which was to be demonstrated CMO Canadian Mathematical Olympiad IMO International Mathematical Olympiad INMO Indian National Mathematical Olympiad USAMO United States of America Mathematical Olympiad Chapter 1 Number System “Begin at the beginning,” the King said gravely, “and go on till you come to the end: then stop.” — Lewis Carroll, Alice in Wonderland A number is an abstract (mathematical) object used to count, measure and label. A number system is a writing system to represent numbers. The set of natural numbers, the set of integers, the set of rational numbers, the set of real numbers, etc., are some commonly used number systems. Divisibility Notation: In this chapter, the
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