Maths new edition

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CLASS 9 Pearson IIT Foundation Series Mathematics Seventh Edition

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CLASS 9 Pearson IIT Foundation Series Mathematics Seventh Edition Trishna Knowledge Systems

Photo Credits All Chapter Openers: 123rf.com Icons of Practice Questions: graphixmania. Shutterstock Icons of Answer Keys: Viktor88. Shutterstock Icons of Hints and Explanation: graphixmania. Shutterstock Copyright © 2018 Pearson India Education Services Pvt. Ltd Published by Pearson India Education Services Pvt. Ltd, CIN: U72200TN2005PTC057128. No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent. This eBook may or may not include all assets that were part of the print version. The publisher reserves the right to remove any material in this eBook at any time. ISBN: 9789352866830 eISBN: 9789353061876 Head Office: 15th Floor, Tower-B, World Trade Tower, Plot No. 1, Block-C, Sector-16, Noida 201 301, Uttar Pradesh, India. Registered Office: 4th Floor, Software Block, Elnet Software City, TS-140, Block 2 & 9, Rajiv Gandhi Salai, Taramani, Chennai 600 113, Tamil Nadu, India. Fax: 080-30461003, Phone: 080-30461060 Website: in.pearson.com, Email: [email protected]

Brief Contents Prefacexiii Chapter Insights xiv Series Chapter Flow xvi Chapter 1   Number Systems 1.1 Chapter 2  Logarithms 2.1 Chapter 3   Polynomials and Square Roots of Algebraic Expressions 3.1 Chapter 4   Linear Equations and Inequations 4.1 Chapter 5   Quadratic Expressions and Equations 5.1 Chapter 6   Sets and Relations 6.1 Chapter 7  Matrices 7.1 Chapter 8   Significant Figures 8.1 Chapter 9  Statistics 9.1 Chapter 10 Probability 10.1 Chapter 11  Banking and Computing 11.1 Banking (Part I) 11.13 Computing (Part II) 12.1 Chapter 12 Geometry 13.1 Chapter 13 Mensuration 14.1 Chapter 14  Coordinate Geometry 15.1 Chapter 15 Locus 16.1 Chapter 16 Trigonometry 17.1 Chapter 17 Percentages, Profit and Loss, Discount and Partnership 18.1 Chapter 18 Sales Tax and Cost of Living Index 19.1 Chapter 19 Simple Interest and Compound Interest 20.1 Chapter 20 Ratio, Proportion and Variation 21.1 Chapter 21  Shares and Dividends 22.1 Chapter 22  Time and Work 23.1 Chapter 23  Time and Distance

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Contents Prefacexiii CHAPTER 2  LOGARITHMS 2.1 Chapter Insights xiv Introduction2.2 Series Chapter Flow xvi Definition2.2 System of Logarithms 2.2 CHAPTER 1  NUMBER SYSTEMS 1.1 Properties2.2 Introduction1.2 Laws2.3 Natural Numbers 1.2 Variation of log ax with x2.3 Whole Numbers 1.2 Sign of log ax for Different Values of x and a2.4 Integers1.2 Antilog2.7 Rational Numbers 1.2 To Find the Antilog 2.7 Irrational Numbers 1.2 Practice Questions 2.9 Real Numbers 1.2 Answer Keys 2.15 Number Line 1.3 Hints and Explanation 2.17 Properties of Real Numbers 1.4 CHAPTER 3 POLYNOMIALS AND SQUARE ROOTS Laws of Indices for Real Numbers 1.5 OF ALGEBRAIC EXPRESSIONS3.1 Exponential Equation 1.7 Radicals1.7 Surds1.8 Introduction3.2 Types of Surds 1.8 Constant3.2 Laws of Radicals 1.9 Variable3.2 Order of a Surd 1.10 Algebraic Expression 3.2 Comparison of Monomial Surds 1.10 Terms3.2 Addition and Subtraction of Surds 1.11 Coefficient of a Term 3.2 Multiplication and Division of Surds 1.11 Rationalizing Factor 1.12 Polynomial3.3 Rationalization of Mixed Surds  1.14 Degree of a Polynomial in One Variable 3.3 Conjugate of the Surd of the Form a + b 1.14 Addition of Polynomials 3.4 Comparison of Compound Surds 1.15 Subtraction of Polynomials 3.4 Rationalizing the Numerator  1.16 Multiplication of Two Polynomials 3.4 Square root of a Quadric Surd 1.18 Division of a Polynomial by a Monomial 3.5 Square Root of a Trinomial Quadratic Surd 1.18 Division of a Polynomial by a Polynomial 3.5 Practice Questions 1.20 Factorization3.7 Answer Keys 1.27 HCF of Given Polynomials 3.11 Hints and Explanation 1.29 LCM of the Given Polynomials 3.12

viii Contents Relation between the HCF, the LCM 3.13 Quadratic Expression 5.2 and the Product of Polynomials  3.13 3.18 Zeroes of a Quadratic Expression 5.2 Concept of Square Roots 3.21 Rational Integral Function of x 3.21 Quadratic Equation 5.2 Homogeneous Expression 3.22 Symmetric Expressions Solutions or Roots of a Quadratic Equation 5.2 Cyclic Expressions 3.24 Finding the Solutions or Roots Practice Questions 3.31 of a Quadratic Equation 5.2 Answer Keys 3.33 Relation Between the Roots and the Hints and Explanation Coefficients of a Quadratic Equation 5.4 Nature of the Roots of a Quadratic Equation 5.5 Signs of the Roots of a Quadratic Equation 5.5 Constructing the Quadratic CHAPTER 4 LINEAR EQUATIONS Equation when its Roots are Given 5.7 AND INEQUATIONS 4.1 Equations which can be Reduced Introduction4.2 to Quadratic Form  5.10 Algebraic Expressions 4.2 Reciprocal Equation 5.11 Open Sentences 4.2 Constructing a New Quadratic Equation by Changing the Roots of a Equation4.2 Given Quadratic Equation 5.12 Linear Equation 4.2 Maximum or Minimum Value Simple Equation 4.2 of a Quadratic Expression 5.13 Solving an Equation in One Variable 4.2 Practice Questions 5.15 Simultaneous Linear Equations 4.3 Answer Keys 5.22 Solving Two Simultaneous Equations 4.4 Hints and Explanation 5.24 4.13 Nature of Solutions CHAPTER 6  SETS AND RELATIONS 6.1 4.14 Word Problems and Application of Simultaneous Equations Linear Inequations 4.16 Introduction6.2 Introduction4.16 Set6.2 Definition4.17 Elements of a Set 6.2 Inequation4.17 Representation of Sets 6.3 Continued Inequation 4.17 Some Simple Definitions of Sets 6.3 Linear Inequation 4.17 Operations on Sets 6.6 Solving Linear Inequation in One Variable 4.17 Dual of an Identity 6.8 Solving Linear Inequations in Two Variables 4.20 Venn Diagrams 6.8 System of Inequations 4.21 Some Formulae on the Cardinality of Sets 6.9 Absolute Value 4.23 Ordered Pair 6.11 Properties of Modulus 4.23 Cartesian Product of Sets 6.11 Interval Notation 4.23 Some Results on Cartesian Product 6.11 Practice Questions 4.26 Answer Keys 4.34 Relation6.13 Hints and Explanation 4.36 Definition6.13 Domain and Range of a Relation 6.13 CHAPTER 5 QUADRATIC EXPRESSIONS Representation of Relations 6.14 AND EQUATIONS 5.1 Inverse of a Relation 6.14 Introduction5.2 Types of Relations 6.15 Properties of Relations 6.16

Contents ix Function6.17 Some Basic Definitions 9.2 Tabulation or Presentation of Data 9.2 Practice Questions 6.19 Class Interval 9.4 Class Boundaries 9.4 Answer Keys 6.26 Hints and Explanation 6.28 Statistical Graphs 9.5 CHAPTER 7  MATRICES 7.1 Bar Graph 9.5 Histograms9.6 Introduction7.2 Frequency Polygon 9.7 Order of a Matrix 7.2 Frequency Curve 9.8 Types of Matrices 7.3 Cumulative Frequency Curves 9.9 Comparable Matrices 7.5 Measures of Central Tendencies Equality of Two Matrices 7.5 for Ungrouped Data 9.10 Addition of Matrices 7.6 Arithmetic Mean or Mean 9.10 Properties of Matrix Addition 7.6 Median9.13 Mode9.14 Matrix Subtraction 7.6 Measure of Central Tendencies for Grouped Data Transpose of a Matrix 7.7 9.15 Mean of Grouped Data Symmetric Matrix 7.8 9.15 Median of Grouped Data Skew-symmetric Matrix 7.8 9.16 Mode of Grouped Data 9.18 Multiplication of Matrices 7.8 Range9.19 Multiplication of a Matrix by Scalar 7.8 Quartiles9.20 Multiplication of Two Matrices 7.9 Properties of Matrix Multiplication 7.10 First Quartile or Lower Quartile 9.20 Practice Questions 7.13 Third Quartile or Upper Quartile 9.20 Answer Keys 7.22 Semi-inter Quartile Range or Hints and Explanation 7.24 Quartile Deviation 9.21 Estimation of Median and Quartiles from Ogive 9.22 CHAPTER 8  SIGNIFICANT FIGURES 8.1 Mean Deviation 9.24 Introduction8.2 Mean Deviation for Ungrouped 9.24 or Raw Data 9.25 Addition and Subtraction  8.4 9.25 Mean Deviation for Discrete Data 9.27 Multiplication and Division 8.4 Mean Deviation for Grouped Data 9.37 Practice Questions 9.39 Absolute Error and Relative Error 8.5 Answer Keys Hints and Explanation 10.1 Practice Questions 8.7 Answer Keys 8.11 CHAPTER 10  PROBABILITY Hints and Explanation 8.12 CHAPTER 9  STATISTICS 9.1 Introduction10.2 Observations10.2 Introduction9.2 Practice Questions 10.7 Data9.2 Answer Keys 10.12 Types of Data 9.2 Hints and Explanation 10.13

x Contents CHAPTER 11 BANKING 11.1 Practice Questions 11.21 AND COMPUTING 11.1 Answer Keys 11.32 Hints and Explanation 11.34 Banking (Part I) Introduction  11.2 CHAPTER 12  GEOMETRY 12.1 Remittance of Funds  11.2 Introduction12.2 Safe Deposit Lockers  11.2 Public Utility Services  11.2 Line12.2 Deposit Accounts  11.2 Line Segment 12.2 Savings Bank Account  11.2 Ray12.2 Types of Cheques  11.3 Angle12.3 Bearer Cheque  11.3 Types of Angles 12.3 Crossed Cheque  11.4 Bouncing of Cheques  11.4 Parallel Lines 12.5 Parties Dealing with a Cheque  11.4 Current Account 11.7 Properties of Parallel Lines 12.5 Term Deposit Accounts 11.7 Transversal12.5 Loans  11.9 Intercepts12.6 Demand Loans  11.9 Term Loans  11.9 Triangles12.7 Overdrafts  11.9 Types of Triangles 12.7 Compound Interest 11.10 Important Properties of Triangles 12.7 Hire Purchase and 11.10 Instalment Scheme  Congruence of Triangles 12.8 11.11 Hire Purchase Scheme  11.11 Quadrilaterals12.9 Instalment Scheme  Different Types of Quadrilaterals 12.9 Geometrical Results on Areas 12.11 Some Theorems on Triangles 12.13 Mid-point Theorem 12.13 Basic Proportionality Theorem  12.13 Similarity12.15 Pythagorean Theorem  12.18 Computing (Part II) 11.13 Polygons12.19 Introduction  11.14 Convex Polygon and Concave Polygon  12.20 Some Important Results on Polygons  12.20 Architecture of a Computer 11.14 Construction of Triangles  12.21 Input Device 11.14 Construction of Quadrilaterals 12.26 Central Processing Unit 11.14 Output Device 11.14 Circles  12.32 Block Diagram of a Computer 11.14 Arc of a Circle 12.32 Hardware11.15 Semi-circle  12.32 Software11.15 Segment of a Circle 12.32 Algorithm11.15 Congruence of Circles  12.32 Flowchart11.16 Arcs–Chords12.32 Operators11.18 Constructions Related to Circles  12.36 Shift Operators 11.19 Practice Questions 12.40 Logical Operators 11.19 Answer Keys 12.51 Relational Operators 11.19 Arithmetical Operators 11.19 Hints and Explanation 12.53

Contents xi CHAPTER 13  MENSURATION 13.1 Intercepts of a Straight Line 14.12 Equation of a Line in General Form 14.13 Introduction13.2 Equations of Some Standard Lines 14.13 Oblique Line 14.14 Plane Figures 13.2 Practice Questions 14.21 Units of Measurement 13.2 Answer Keys 14.27 Estimating Areas 13.2 Hints and Explanation 14.29 Area of a Rectangle 13.3 Area of a Parallelogram 13.3 CHAPTER 15  LOCUS 15.1 Area of a Triangle 13.3 Area of a Trapezium 13.4 Introduction15.2 Area of a Rhombus 13.4 Area of a Square 13.5 Equation of a Locus 15.4 Polygon13.5 Steps to Find the Equation of a Locus 15.5 Area of a Polygon 13.5 Concurrency—Geometric Centres of a Triangle Area of an Equilateral Triangle 13.6 15.5 Geometric Centres of a Triangle Area of an Isosceles Triangle 13.6 15.5 Practice Questions Area of Right Triangle 13.7 15.8 Answer Keys Circles13.8 15.13 Hints and Explanation Circumference of a Circle 13.8 15.14 Area of a Ring 13.9 Sector of a Circle 13.9 Solids13.10 CHAPTER 16  TRIGONOMETRY 16.1 Volume of a Solid 13.10 Introduction16.2 Cubes and Cuboids 13.12 Right Circular Cylinder 13.13 Angle16.2 Pyramid13.14 Systems of Measurement of Angle 16.2 Circular Cone 13.16 Trigonometric Ratios 16.5 Torus (Solid Ring) 13.19 Pythagorean Triplets 16.6 Sphere13.20 Trigonometric Identities 16.6 Polyhedrons13.22 Trigonometric Ratios of Compound Angles 16.8 Practice Questions 13.24 Practice Questions 16.14 Answer Keys 13.31 Answer Keys 16.21 Hints and Explanation 13.33 Hints and Explanation 16.23 CHAPTER 14 COORDINATE CHAPTER 17 PERCENTAGES, GEOMETRY14.1 PROFIT AND LOSS, DISCOUNT AND Introduction14.2 PARTNERSHIP17.1 Coordinates of a Point 14.2 Introduction17.2 Convention of Signs 14.2 Points on the Plane 14.3 Percentage17.2 Point on X-axis and Y-axis14.3 Expressing x% as a Fraction 17.2 Distance Between Two Points 14.3 Expressing a Fraction a/b as a Decimal Straight Lines 14.8 and as a Percentage 17.2 Inclination of a Line 14.8 Percentage: A Relative Value 17.3 Slope or Gradient of a Line 14.8 Comparison of Percentages 17.4

xii Contents Profit and Loss 17.7 Proportion20.7 Cost price 17.7 Properties of Proportion 20.7 Selling price 17.7 Continued Proportion 20.7 Profit17.8 Types of Variation 20.8 Loss17.8 Direct Variation 20.8 Overheads17.8 Indirect Variation 20.8 Partnership17.11 Joint Variation 20.8 Types of Partnership 17.12 Practice Questions 20.15 Practice Questions 17.15 Answer Keys 20.22 Answer Keys 17.23 Hints and Explanation 20.24 Hints and Explanation 17.25 CHAPTER 18 SALES TAX AND COST CHAPTER 21 SHARES AND OF LIVING INDEX 18.1 DIVIDENDS21.1 Introduction21.2 Introduction18.2 Nominal Value of a Share 21.2 Sales Tax 18.2 Market Value of a Share 21.2 Value-Added Tax  18.2 Dividend21.2 Cost of Living Index 18.3 Practice Questions 21.8 Practice Questions 18.6 Answer Keys 21.13 Answer Keys 18.12 Hints and Explanation 21.14 Hints and Explanation 18.13 CHAPTER 22  TIME AND WORK 22.1 CHAPTER 19 SIMPLE INTEREST AND Introduction22.2 COMPOUND INTEREST 19.1 Sharing of the Money Earned 22.9 Interest  19.2 Pipes and Cisterns  22.10 Simple Interest 19.2 Practice Questions 22.12 Compound Interest 19.2 Answer Keys 22.21 Practice Questions 19.10 Hints and Explanation 22.23 Answer Keys 19.17 CHAPTER 23  TIME AND DISTANCE 23.1 Hints and Explanation 19.19 Introduction23.2 CHAPTER 20 RATIO, PROPORTION Speed23.2 AND VARIATION 20.1 Average Speed 23.2 Ratio20.2 Relative Speed 23.5 Terms of a Ratio 20.2 Practice Questions 23.14 Properties of a Ratio 20.2 Answer Keys 23.23 Types of Ratios 20.3 Hints and Explanation 23.25

Preface Pearson IIT Foundation Series has evolved into a trusted resource for students who aspire to be a part of the elite undergraduate institutions of India. As a result, it has become one of the best-selling series, providing authentic and class-tested content for effective preparation—strong foundation, and better scoring. The structure of the content is not only student-friendly but also designed in such a manner that it motivates students to go beyond the usual school curriculum, and acts as a source of higher learning to strengthen the fundamental concepts of Physics, Chemistry, and Mathematics. The core objective of the series is to be a one-stop solution for students preparing for various competitive examinations. Irrespective of the field of study that the student may choose to take up later, it is important to understand that Mathematics and Science form the basis for most modern-day activities. Hence, utmost effort has been made to develop student interest in these basic blocks through real-life examples and application-based problems. Ultimately, the aim is to ingrain the art of problem-solving in the mind of the reader. To ensure high level of accuracy and practicality, this series has been authored by a team of highly qualified teachers with a rich experience, and are actively involved in grooming young minds. That said, we believe that there is always scope for doing things better and hence invite you to provide us with your feedback and suggestions on how this series can be improved further.

2112NSKyuinsmteembmNSKeaystruiinscmtesembmeastrics Banking and Computing 11.9 Chapter Insights n(n + 1) 1 R SI = P × 2 × 12 × 100 5.2 Chapter 5 Re= m600e×m24(2b25e) ×R112 × 5 = 750 100 INTRODUCRTIeOmNembeR \\BeTfootarleambeogunint n=i(n2g4 ×th6i0s0)c+ha7p5t0er, you should be able to: = 14,400 + 750 = `15,150. Remember section In previous topicB, weefohraevebleeagrnint lninienagr etxhpirsescsihona,p•litneeRra,reyepqorueuasetisnohntonaunuldmdlibbneeerasraoibnneleqnutuaotmi:obne. rInlitnheiss will help them to memorize and review topic, we shall learn•quaRdreaptirceesexnprtenssuiomn banedrLsqOounaAdnr•aNutimcOSebbqeturaailtniionnLe.sCM and HCF of numbers neAQxuppmUroeblAsyesnrioDsonamRn. idTaAlhaoTe≠fIgs0Cee.nc••oeErnaXdlOAanfdPpobderptgRmaliryrieEnreoaSofitLnipaSoCeqoInrMuOnaaaetldiNvanoraanunrtdiimscabHoBbelxeneaCp231nrirss...kFientsel•sOTDotorieeofmaevgrnnAamnemeesnrudirpadncdsnmlrpaao,adxnsliafbyrftnalibrsreossaaqero(aactOusnxipctaosli2Deadonrs+rsnsaaa)ittslbfiii,xcenoddun+peomsicnclytwobionmnehotreamhsrileesni,atfaloe,rolgalbroet,iwqrocsuin,anaradegflrrrataehtciarctleieoncas,tegdoerciiems: atlhso,enrpaatrieopvnaiaroltiucsulleaarrtnoipnigc Example: 2x2 + 3x, x2 − 2, 3x2 + 11x − 108 DareesmomaenqduadLroataicnesxpressions. thKeesyetNuxppodorteeiesnnsiottssnTswth.oKeielilexdphYerenesltspIiiDofynes xA2 S+ x12 − 3, x3Tp26rho+AKemmbf3oieostnserxoYtrrhro+yswc3Ifnoer,Dorom2tmhxeeap2sitAnl−hteoeftx1Saridven+opagtau4eyrtoatohhrffeiedstnlhioscoebahtunbqarsasuponeankmtde,derrpaen,rtmtoiycmoaonfiudstih.nTesghhltooehauarnetl.pdhTaeybhmweeeobnuaotlbdrolreforetwhpteeoary:lohtahanse has to be done within to execute a demand loan unconditionally thcZIizsfeehercaoaareqelpolsuse.stadeeedsanrrazttoiiecafroleaxoppfAoQretihfsunt••seieoatersnSDxd,cptireaanroaxetfeim2santsi+iteocphnbtlexEehaxptex+ir2pnnoc+grupbeeembtcxrshotbsi+mieeisroseTrTacs.escsenohpAaphzrfeaneeaymnrqbrdmopuuo•••ttharefrmLoeedrneoUrDSorprrtbaasw,,trxtaetdeaneniieyprtcfi=sdunreiscouenmeeaesαlnutxarelnhb,sttsptitetweserotahrehrhnnhpsseoeossedriifmronnoouientnfutplohαdsotemctheaarnaienlbbtnslibmabeeeaahaargsenrasnsarenviouekabcaeteflm.snl,mannecdetbuouetettmcnrnmhore.tcebpT:blweeimrhernpei,etroteshtsseshrtenteohptntwfeatyαohbrmaednemnikctaropelnsegraaiarodndnidnuigsmtghthebeeneelrpareawlrillniysoedmoofofrelotahnanan3d6 mode of months. aEQTSFxqohigUauelumauredAepqtr1iluaDeo.ta:1itcnRi2oexsnAq2 uoo+TafrtI3itCoxhRne+o••••,Efooo5QrrOrDSUct=masaotUdenunb0masoi,ddtecxcAabxyeqfa2ri2irnuilT+narsbsa−atIaetsabtOiqt5Qoxinitouonh=d+Nunaneor0acatefslah=aiudtnzrnehro0idrdedn,oab,sg3cwtaettOATmTICxocsyinfho2incohhaaoptcevfc−ecneejrreluuErqeesstedcces4eerogquuorrsax•••tdaratellh,adluoonentna+da(obgiaacdraRtnDOcS,otnrnnreiaoaetcatnossopFetubetlcima5fhosaairno.stc)dtrtcrrdecsaeaosbeeyni=nreiousnaroinisaaurnemr(nb0fvvfaaOisrotaesaasIulodatqtnriifihrcnlluitsrDaoeueetothoadundeddhnalssnlesmctidroreac)bbueeeamsblaayylsivursanelzureatstrptroeidierrdsnluqoadodaa,augrdctbnn,tnectaooelhdirydfeodrntnsLaasprsfoa.cisjaitdenueonqttha≠asirotgpcueishonalra0eayieonstsrlda(qiifadapRnsoaurwmtrcaknaiooaFctotnsagsdiiro)cnocodruuohewnisecsnnief,snturn.mt,esrboaudreabfsynssru,itdtsthb.swueuOipilstdntuhocrhetdaoh,iuemnistbeasesaxv,onieempkrtyecuw.rmqahutwaiiaclroTlirhumeetneelcpislrxet,iexstdtfrhit:mpxremeCulidtlaoscoaiabnhtnnnyuiemncartdmheeeterdpooinutbdasnarnantakwdis. The values of x for which the equation ax2 i+ncbrxea+sedc =by0thisatsaatmisfoieudnta.re called the roots of the quadratic equation. A quadratic equation canDnoatilyhapvreomduocrte=thBanalatwncoer×ooNtsu. mber of days it has remained as balance. Interest = Sum of daily products × Rate . 100 × 365 1.14 ChaEpXteAr m1 PlE 5.1 NVoetreifybwohxeethsearrxe=s2o,mis ea solution of 2x2 + x − 1N0o=te0. If the loan is fully repaid, the date on which it is repaid is not counted for calculation HNA_78982.indd 1 adredOSlR-aOontaelnstduUiobiTtsnntIoiOfatpouNltiriicnzmsgaaxtti=oion2nionof2fxm2 +ixxe−d10S,uwredgset of interest. If the loan is partially repaid, the day of repayment is also counted for calculating the interest. a+ b issotlhuetioranti(oonra)lirzoiontgoffac2txo22r (+o2f)x2a−+−120−b=,E10Gw0x.ah=Ane1meres0hPa−tLaa1nEk0de1s=b1aa0.rl.e5oarnatioofna`l2. 0,000 on 1-4-2005. He repays `2000 on the 10th of every month, ∴2 is a exAmPLe 1.11 beginning from May, 2005. If the rate of interest is 15% per annuEmx,acmalcpulleasteatrhee ginitvereenst till top2/i7c/20-1w8 1i1s:4e8:09tAoM apply FindRinatgiontahliezeSthoeludetnioomnisnaotorr oRfo22o+−ts 55o.f 30-6-2005. Equation 2/7/2018 11:48:09 AM the concepts learned There are two ways of finding the a Quadratic in a particular chapter 1. SFOacLtoUrTizIaOtiNon method roots of a quadratic equation. 2. Application of formula M11_TRIS22HN+−A_7895582_=P01.in22dd+−9 5  2 + 5 2/7/2018 1:18:2    5   2 + 5  = (2 + ( 5 )2 (2)2 − 5 )2 NA_78982.indd 2 2/7/2018 12:43:39 PM = 4 +4 5+ 5 Illustrative examples 4−5 solved in a logical and step-wise manner = 9 +4 5 = −(9 + 4 5) −1 exAmPLe 1.12

ChaptNeurmIbnesriSgyhsttesms 1x.2v1 1.20 Chapter 1 41. Find the positive square root of the following: 44. Gi=ven 2 1=.414, 3 1.732, 5 = 2.236 , find 10 + 2 6 + 60 + 2 10 the value, correct to three decimals, of the TeST YOUR CONCePTS 42. If x = 11 5 , find the value of x2 – 8x + 11. following: 1− 3 4− 45. If x = 3 3 + Very Short Answer Type Questio43n.s If both a and b are rational numbers, find the val- 5 −Dif2ferent levels 1. 5 × 125 = _______. ues of a and1b6.inMthueltifpolyll:ow3 5inbgye4q2u.ation: 26 , fibondef etqhnue eivnasltcuielouondfse21hdaxv+e 1  . 2 ++EWx35hpirce55hss=itshager+esuabtredr,52 x  20 174. 2 or 3 3 ? 2. 320 = _______. 18. 3 as a pure surd. in the Test Your 3. 3 6 × 3 6 × 3 6 = _______. Essay Type Q1u9e. sMtiuoltnipsly: 14 by 8 . Concepts as well 4. 3 √ 4 12 = _______. as on Concept 20. ( 3 3 )4 = _______. Chapter567892..... 3TnT_wd1x3_iuw0hhs+_ms7eooi3_m3sbs_e,mueyia_lirmnainsixsrusid/us)ermadrad_d1tip_fis1ufoqa_.enrun_re3deaa_dlns_urdpca_areerrd_toeni,_ocdoi_f_ufms.a_cuxit_rrn=d_aaats_tri,_ooe_a_rn__r+_aia_sl_t__iao_bn44_snu_d_67a_r_a..dla.,n_ns(IR5d_.zafxi_er(rae2.rsartxoiaim–−c/ot=iaoion7llnalnaxbe7reald/yi),l+z–e142222t5h2143y3....e2.EREE,3d33233xxxeya6+83ppp3nt21=i1rrroo5eee6nmsss27sssa1ilttt−−inhhhz1eeeai43ntfffogooor33lllflllaooooc,wwwftofiiitinnnrhngggoedfiaifnnso5ta1httl/hhloep3eeuw+ssvriiei5mman−lsgupp1u/:ller3eedss:ittosfff_oo_rrmm_44_::98_..__iRG.nigavteisounnrda,li3zceo=33rtr1he++.ec711t3dt2+3eo1n2ot,7htAwpm33f+ori+sren+i−piko3ned5ldp11laibdltlte5+olhhei3lvcecreea44iemomlvt+−lpfiaoao-ltluhnpsspe33eotluwatoflochvfdelhiletseohn.iwnecgithfnosgll:ow- 23.28 1‘T0.eCstonYjuogautersurd of 5 − 3 is _______. 25. 3 3 + 2 is a surd. (True/False) L∴Q(PP(ievvQRRt))TL∴ir==t=aesv(t(2sPe8s8prlRr4s0e−s))e=pd–e26ptCsfAdta66resbPohh,e00031113311p‘direSQp=C6ocv32351214rotee+rhh0........s=tpikne3boce6oi=decEtIxFFIE(kml4ponhTcff0lrliiimxxhnneexaann8e67tr5cetan668ppnsunh=ddd8sacmd1Apo++rrra00−8pdkeesee0pheettla8/ntkwmsrsvhht4yaasopssiFt0potmssec8ee54o7ksritaatewh’5ifokh8=thoorlspmf=tsn7ermdeoapmeooeeh16nmuip=)4rtrfs’sasrfe6aiucohltta+tlt8Trhri_leit.vldoy=he_osoketn_wepfmty8sra_wqeatil318_n,w)ctuo1it_goQkoaoh_inrmnwsieue.njsanuaueiplrttgiorx_hshzhdaot_ie+tsnrite_aogsi_toyisinm_fofs=auas_npc6r_a_rtlde−ola_CostdrL_21tifooe_oeO..ft2fnno_fhv0oar_W(E(A(a(e5elNmmacacn_.x5.o))))2nl.6di:hp.tn8uA(B1Ch030iwr+a7mcLLLIee..,431.tl446556tehhaorsbeeeCs.+ns78e?r(ltnte7P.sr8old7qr13+0e.pgtia,88+f3thT4q.nteBs31qhe0ted147shB3233222dA4pol5=eD0978768e+.0ef=.......neP=)71f0Pg4odWEIWIEiD3213.=sfft3P3lxxqih1lab=0hhvx4ppop32Le1miuiirrwo0dccl==ee5((((n−Iahheqfss/bddb6iiCsss:222n1qQs2))))imotteCsaug5++16−A+hhAAcf00es1/1.seebm..f,i1s4tTs66rean2fshe13u533iBa04aoun88sg5Icel3r7cl0lrl79dd,Oglledatbaomet.ifrrnnywwoiC,e(nddm.33dNaTnoide6tn65rpseDc2u4gq=treihl?Dexmw−.iiie/ens45p.Fi1arvtatlaebhl1ah.o7lmsrle7sesuarreitatoe)wsitionimooen3sfn,epa−anlxlelns+d1ut152e1mfx1n?o−obr. mmer1:,i65n0t..ahteoW(v(((W((anacacacran.))))))ldhqhu6131(9iaec34023xtsh4?/i23s(,ow)tx2fh/h4e3texh,vr/e(e3a7lf4xuo)elx∈l/oo1wR2f ((((i)4bdbdn))))2_gx_5242−p_51532a_62,i_ris((f_bd.i())s16((h32)a265vx26i+n))324g//=7xt,,w(((664o433))e)x24q+//37uxa?l Hints and Explanation51. 52. PRACTICE QUESTIONS PRACTICE QUESTIONS ( ) ( )T6i0sm0e−task6+e0n5002Lc=o333=;e4656mva2...0sen0SoAIpflifdrhmxlrm1o aepLn=au;lxgig rfecneLsyitvi:ttbeyhue2ved+:le ef14:o32,lllof−iwn3din3gx2s+u+rx1d2s. 48 . as3c.en((Id5acfi6)n) .g26552R=LLon5reede7tt=eq2htru1hisi2seer5ce34uuod90s,s..unutahdiSoAamlesilfrm.tnrmseiapmpna5e((l=gigbdeefnney))d6lbi4tt1t+1heu2ho5112ed1e5=qf1t+e5f4Hf5h:oo_0olla_ll6oour_swwir−_esbiic3_.nneo_4gg6n3.,u:sd+4uksr35dm,s3pin8−h.an1a53sc+e23ndi2n78g..orT((pIdcaflhe))imfreisceiomaednmxpbdppleeronecusosausnmriroedden.pssuoarsdit.5iv((e−db))inbm3tienixgoeemdrs7i,sau−tlhrsdeu.nr2df.orwahepnossiitmive- in an 600(s + 50 − s) = 23 9, 9 5, 3 7 s(s + 50) If x_4_D_+_is_1t_a=n. c1e2(9in7 kamnd) y=4 1+ =5)(2t4–001,) then y2 – x2 mn = number a, m n a = _____. { ( )}30000 = 2(s2 + 50s) 4. (-u s2 + 50s – 15000 = 0 = (u − 2.5)  t + 48  = ut 60  M01_TRISHNA_78982.indd 20 2/7/2018 11:48:48 AM (u + 5)(t – 1) = ut (s + 150)(s – 100) = 0 s>0 M01_TRISHNA_78982.indd 21 ∴ ut + 5t – u – 5 = ut Hints and 2/7/2018 11:48:52 AM s = 100. 5t – 5 = u (1) Explanation for key questions along with 53. Let the speeds of the faster and the slower cars be (u − 2.5)  t + 48  = ut highlights on the f kmph and s kmph respectively. 60  common mistakes that students usually make 360 = 9(f – s) = 3(f + s) ut − 2.5t + 4 u − 2 = ut f – s = 40 and f + s = 120 5 (2) in the examination Adding these and then simplifying, f = 80. 54. Let the distance be d km 4 u = 2.5t + 2 5 Let his usual speed be u kmph. 4t – 4 = 2.5t + 2 (From Eqs. (1) and (2)) 1.5t = 6 Let his usual time be t hours. t=4 d = t ∴ u = 15 u d = t + 9 ⇒ t = t + 9 ∴ut = 60 60 4 60  4  u 57. Let the speeds of Ram, Shyam and Tarun be r m/  5  5 sec, s m/sec and t m/sec respectively.

Series Chapter Flow Class 7 1 Expressions and 3 Indices 5 Number Systems Special Products 4 Geometry Ratio and 2 Its applications 10 Statistics 8 Equations and their Applications 6 Set Theory 9 Formulae 7 Mensuration Class 9 Logarithms 3 Linear Equations 5 2 and Inequations 1 Polynomials and Square Roots Quadratic Expressions of Algebraic Expressions 4 and Equations Number Systems Geometry Probability 9 Significant Figures Sets and Relations 12 11 Statistics 8 7 10 6 Banking and Matrices Computing Mensuration Locus Percentages, Profit and Loss, 18 13 15 Discount and Partnership 14 16 Coordinate Geometry 17 Sales Tax and Cost Trigonometry of Living Index 23 Time and Work 21 Ratio, Proportion and Variation 19 Time and Distance 22 20 Shares and Simple Interest and Dividends Compound Interest

Series Chapter Flow xvii Class 8 1 Squares and Square 3 Polynomials and 5 Roots and Cubes and Indices LCM and HCF of Formulae Real Numbers and LCM and HCF Cube Roots Polynomials 2 4 10 Simple Interest and 8 Percentages 6 Compound Interest 7 Time and Work, Profit and Loss, Ratio, Proportion Pipes and Cisterns 9 Discount and Partnership and Variation 11 Linear Equations 13 Statistics 15 Geometry 17 Time and Distance and Inequations Sets 14 Matrices 16 Mensuration 12 Class 10 Polynomials and Quadratic Equations Sets, Relations Rational Expressions and Inequalities and Functions 13 4 5 6 Statements 2 Number Linear Equations Systems in Two Variables Mensuration Statistics Matrices Trigonometry 12 10 13 11 97 14 Remainder and 8 Geometry Factor Theorems Limits Progressions Mathematical Induction Linear Permutations and and Binomial Theorem Programming Combinations 15 17 19 16 18 20 Coordinate Modular Arithmetic Computing Geometry 27 Partial Fractions Instalments 23 Banking 21 Logarithms 26 25 Taxation 22 Probability 24 Shares and Dividends

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112CChhaapptteerr NKuinmembeatrics Systems Figure 1.1 RemembeR Before beginning this chapter, you should be able to: • Represent numbers on number lines • Obtain LCM and HCF of numbers • Apply operations on integers, fractions, decimals, rational and irrational numbers KeY IDeAS After completing this chapter, you should be able to: • Define the numbers and represent them on a number line • State the properties of numbers • Understand the basic concepts of radicals and the laws of radicals • Describe the surd, types and order of the surd, its operations, combination and conjugations • Study rationalizing factor (RF) in surds • Obtain square root of quadratic surds

1.2 Chapter 1 INTRODUCTION In this chapter, we shall discuss types of numbers, laws of indices and then about the surds, types of surds, laws of surds, the four basic operations on surds, their comparison etc. Basically, a surd is an irrational number. Hence, let us first review the types of numbers and recall the definitions. Natural Numbers All counting numbers are natural numbers. If N is the set of natural numbers, then N = {1, 2, 3, …} Whole Numbers Natural numbers including zero represent the set of whole numbers. It is denoted by the symbol W. For example: W = {0, 1, 2, 3,…} Integers The set of integers, Z = {…, -3, -2, -1, 0, 1, 2, 3, …} Rational Numbers p q Any number that can be expressed in the form (where q ≠ 0 and p, q are integers) is called a rational number. All integers, all recurring decimals and all terminating decimals are rational numbers. Any integer can be expressed in the form p (q ≠ 0). q τ Example: 2 , 3 , 4 , etc. 111 p Any recurring decimal can be expressed in the form q (q ≠ 0). τ Example: 0.333... = 1 3 0.1666... = 1 6 p Any terminating decimal can be expressed in the form q (q ≠ 0). τ E=xample: 0.5 15=0 or 12 ,0.25 25 or 1 100 4 Irrational Numbers p q A number that cannot be expressed in the form (where q ≠ 0), p and q are integers, is an irrational number or any non-terminating and non-recurring decimal is an irrational number. Example: 2, 3, 3 5, 4 25 cannot be expressed in the form p (q ≠ 0), where p and q are integers. q τ \\ These numbers are irrational numbers. Real Numbers A number that is either rational or irrational is a real number.

Number Systems 1.3 Number Line A straight line on which points are identified with real numbers is called a number line. Successive integers are placed on the number line at regular intervals. The following is an illustration of the number line: −2 −1 0 12 Representation of Numbers on the Number Line We now learn the procedure of representing real numbers on the number line. Representation of natural numbers: Draw a line. Mark a point on it which represents 0 (zero). Now on the right hand side of zero (0), mark points at equal intervals of length, as shown below: ••• 0 1 2 34 These points represent natural numbers 1, 2, 3, … respectively. The three dots on the number line indicate the continuation of these numbers indefinitely. Representation of Whole Numbers This is similar as above, but with the inclusion of 0 in the number line, it is as follows: ••• 0 1 2 34 Representation of Integers Draw a line. Mark a point on it which represents 0 (zero). ••• ••• −3 −2 −1 0 1 2 3 Three dots on either side show the continuation of integers indefinitely on each side. Representation of Rational Numbers Rational numbers can be represented by some points on the number line. Draw a line. Mark a point on it which represents 0 (zero). Set equal distances on both sides of 0. Each point on the division represents an integer as shown below. ••• • • ••• x −5 −4 −3 −2 −1 0 1 2 3 4 5 6 The length between two successive integers is called unit length. Let us consider a rational number 2 . 7 • −2 −1 0 P 1 2

1.4 Chapter 1 Divide unit length between 0 and 1 into 7 equal parts; call them sub-divisions. The point at the line indicating the second sub-division from 0 which represents 2 . 7 In this way any rational number can be represented on the number line. Representation of Irrational Numbers on the Number Line We use the Pythagoras property of a right-angled triangle, according to which, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of other two sides. Consider the number line l and a perpendicular line l1 l1 to it. Let OA = 1 unit and OP = 1 unit. Let OAXP be a square. OX = OA2 + AX 2 = 12 + 12 = 2 P XY Taking O as the centre and OX as the radius, cut the 2 number line at the point A′. l 3 That is, OX = OA′ = 2. O 1 A′ B′ 2 3 A B C Representing 3 on the Number Line Let OA′YP be a rectangle. OA′ = 2 units and A′Y = 1 unit. OY = ( 2 )2 + (1)2 = 3 units. Taking O as centre and OY as radius, cut the number line with an arc at the point B ′. That is, OB ′ = 3 units. In the same way, 5, 6,… can be represented on the number line. Properties of Real Numbers Addition 1. Closure The sum of two real numbers is also a real number. Example: 1 + 3 = 3 12 . 2 2. Commutative If a and b are the real numbers, then a + b = b + a. Example: 1 + 2 = 2 + 1. 3. Associative If a, b and c are three real numbers, then (a + b) + c = a + (b + c). Example: (1 + 2) + 3 = 1 + (2 + 3). 4. Existence of identity element Zero is the additive identity element for all real numbers. For every real number, there exists a number 0 ∈ R such that a + 0 = 0 + a = a. 5. Existence of inverse element If a is a real number, then there exists a real number (-a) such that a + (-a) = (-a) + a = 0.

Number Systems 1.5 Multiplication 1. Closure If a and b are real numbers, then a × b is also a real number. Example: 2 × 3 = 2 3 . 2. Commutative If a and b are real numbers, then a × b = b × a. Example: 3 × 5 = 5 × 3. 3. Associative If a, b and c are real numbers, then (a × b) × c = a × (b × c). Example: (3 × 5) × 3 = 3 (5 × 3 ). 4. Existence of identity element One is the identity element for all real numbers. That is, ∀ a ∈ R, a × l = l × a = a. (∀ means for all) Example: 2 × 1 = 2 = 1 × 2. 5. Existence of inverse element 1 such that a × 1 = 1 × a = 1. For every non-zero real number, there exists a real number, a a a Distribution of Multiplication over Addition If a, b and c are real numbers, a × (b + c) = a × b + a × c. Example: 13 × (10 + 7) = 13 × l0 + 13 × 7. Laws of Indices for Real Numbers 1. am × an = am+n (Product of powers) Examples: (i) 23 × 26 = 23+6 = 29 (ii)  5 4 ×  5 5 =  5 4+5 =  5 9  6  6  6  6 (iii) 23 × 24 × 25 × 28 = 2(3 + 4 + 5 + 8) = 220 (iv) ( 7)3 × ( 7)5 = ( )7 3+5 = ( 7)8  Note   am1 × am2 × am3 × … × amn = aml + m2 + m3 +…+ mn 2. am ÷ an = am–n, a ≠ 0 (Quotient of powers) Examples: (i) 78 ÷ 73 = 78–3 = 75 7 9  75 7 9−5  74 (ii)  3  ×  3 =  3  =  3    Note   We now consider, what meaning we can assign to a°. If we want these laws to be true for all values of m and n, i.e., even for n = m, from (2) above we get am = am−m or am 1 = a°. We see that if we define a° as 1, this law will be true even for n = m. Therefore, we define a° as 1, provided a ≠ 0.

1.6 Chapter 1 When a = 0, an −n = an = 0 , which is not defined. an 0 \\ 0° is not defined.  Note   We shall now consider, what meaning we can assign to an, where n is a negative integer. We have am × an = am+n. Consider an × a–n = an+(–n) (if we want the law to be true) = a° = 1 \\ an × a–n = 1 ⇒ a−n 1 and 1 = an. If we define a-n as 1 , this law is true even for negative = an a−n an value of n. Example: 2−4 = 1 , 5−1 = 1 , a−n = 1 (provided a ≠ 0). 24 5 an  Note   a −1 = 1 = b  b  a a b 3. (am)n = am×n (Power of a power) Examples: (i) (52)3 = 52×3 = 56  2  4 5  2 4×5  2 20 3    3   3 (ii)  = =  Note   [(am)n]p = amnp. 4. (ab)n = a n × b n (power in the product) Examples: (i) (20)5 = (4 × 5)5 = 45 × 55. (ii) (42)7 = (2 × 3 × 7)7 = 27 × 37 × 77. In problems, we may often want to write an × bn as (ab)n. Examples: (i) 8 × 27 = 23 × 33 = (2 × 3)3 = 63 (ii) 125 × 729 =  5 9 3 =  45 3 343 8  7   2   14   Note   (a b c d … z)n = an bn cn dn … zn. 5.  a n = an (Power in a quotient)  b  bn Examples: (i)  4 7 = 47  5 57 (ii)  6 8 = (28 )(38 )  8 88 n In problems, we may want to write down an as  a  . bn  b

Number Systems 1.7 Examples: (i) 48 =  48 58  5  25  25  16  5  3 3  27 (ii)  95 =  9 =  8  8   6.  a −n =  bn  b   a Examples:  5 −3  9 3  9  5 (i) =  1 −1  51  5  1 (ii) = = 5  Note   1 −1 =  a 1 = a.  a   1  Exponential Equation 1. If am = an, then m = n, if a ≠ 0 and a ≠ –1. Examples: ∴ (i) If 5p = 53 ⇒ p = 3 (ii) If 4p = 256 4p = 44 ⇒ p = 4 2. If an = bn, then a = b (when n is odd). Examples: (i) If 57 = p7, then p = 5. then 5 = 3p or p = 5 . (ii) If (5)2n-1 = (3 × p)2n-1, 3 3. If an = bn, n ≠ 0, then a = ±b (when n is even). Example: 24 = x4 ⇒ x = ±2. RADICALS A radical expression (or simply a radical) is an expression of the type n x. The sign ‘ n ’ is called the radical sign. The number under this sign, i.e., ‘x’ is called the radicand and the number in the angular part of the sign, i.e., ‘n’ is the order of the radical. At present, we shall deal only with cases where x is a real number. Depending on the values of x, n can have certain corresponding values. Initially we shall consider only positive integral values of n. But remember that, n x = x1/n . Last year, we studied such exponential expressions and we know the values that the index can have for different values of the base. From those results, we get the following results for radicals. If x > 0, n can have any real value except zero. If x = 0, n can have any positive real value. If x < 0, n can have any real value except zero. But, we shall consider only certain rational values. Later, we shall examine what these values are for other values of n, (some rational and all irrational) the radical does not have a real value. We shall study such expressions in higher classes.

1.8 Chapter 1 Examples: 1. 2, 3 3, 4 7, 2 8, are some radicals. 2. 0, 3 0 are also radicals. But 0 0 and −1 0 are undefined. 3. 3 −2, 5 −64, 7 −128 are radicals and 2 −4, 4 −16 are also radicals, but they are not real numbers, we shall study them only in higher classes. In all these examples, the value of n, i.e., the order of the radical is a positive integer. But as stated above, it can have other values. SURDS If a is a positive rational number, which is not the nth power (n is any natural number) of any rational number, then the irrational numbers ±n a are called simple surds or monomial surds. Every surd is an irrational number (but every irrational number is not a surd). So, the representation of monomial surd on a number line is same as that of irrational numbers. Examples: 1. 3 is a surd and 3 is an irrational number. 2. 3 5 is a surd and 3 5 is an irrational number. 3. π is an irrational number, but it is not a surd. 4. 3 3 + 2 is an irrational number. It is not a surd, because 3 + 2 is not a rational number. Example 1.1 Which of the following are surds? (a) 9 (b) 3 13 (c) 4 25 (d) 6 32 Solution (b) 3 13 is a surd. (a) 9 = 3 is not a surd. (d) 6 32 is a surd. (c) 4 25 is a surd. Types of Surds 1. U nit surds and multiples of surds: If n a is a surd, it is also referred to as a unit surd. If k is a rational number, kn a is a multiple of a surd.   Note    All multiples of surds can be expressed as unit surds as kn a = n kn ⋅ a. 2. Mixed surds: If a is a rational number (not equal to 0) and n b is a surd, then a + n b, a − n b, are called mixed surds. If a = 0, then they are called pure surds. Example: 2 + 3, 5 − 3 6 are mixed surds, while 3, 3 6 are pure surds. 3. Compound surd: A surd which is the sum or difference of two or more surds is called a compound surd. Example: 2 + 3 3, 3 + 5 7 − 3 2 and 7 + 2 − 3 are compound surds. 4. Binomial surd: A compound surd consisting of two surds is called a binomial surd. Example: 3, 6 + 4 5, 8 − 3 7.

Number Systems 1.9 5. S imilar surds: If two surds are different multiples of the same surd, they are called similar surds. Otherwise they are called dissimilar surds. Example: 2 2,5 2,7 2 are similar surds. ( 2 + 3 3 ),(2 2 + 6 3 ) are similar surds and 1 + 2, 2 + 2 2 are similar surds. 3 3 and 6 5 are dissimilar surds. Equality of Two Mixed Surds of the Form a + c b and d + e b Two mixed surds a + c b and d + e b are equal, if and only if their respective rational parts and the irrational parts are equal, i.e., a = d and c = e. Example 1.2 Identify the following types of surds: (a) 6 + 5 3 (b) 15 + 8 − 11 (c) 5 (d) 5 + 7 Solution (a) 6 + 5 3. It is the sum of two surds. ∴ It is a compound surd of two surds, i.e., a binomial surd. (b) 15 + 8 − 11. It is the combination of three surds. ∴ It is a compound surd. (c) 5. It is a monomial surd or a simple surd. (d) 5 + 7. It is the sum of a rational number and a surd. ∴ It is a mixed surd. Example 1.3 Which of the following surds are similar? (a) 2 5 (b) 33 5 (c) 4 5 (d) 5 4 5 Solution 5. 2 5 and 4 5 are multiples of the same surd, ∴ They are similar. Laws of Radicals If a > 0, b > 0 and n is a positive rational number, then 1. (n a )(n b ) = n ab 2. n a = n a n b b 3. m =n a m=n a n m a =4. n a p a=p/n and n a p n m (a p )m

1.10 Chapter 1 The application of laws of radicals are as follows: 1. Convert the multiple of a surd into a unit surd. 2. Convert certain unit surds into multiples of other unit surds. 3. Express surds of different orders as surds of the same order. Example 1.4 Express the following surds in their simplest form as multiples of smaller surds: (a) 3 1458 (b) 3 144  (c) 4 1024 Solution (a) 3=1458 3=2(93 ) 93 2. (b)= 3 144 3=24(32 ) 3 23(2)(32 ) = 23 18. (c) 4 1=024 4=210 4 =(28 )(22 ) 4=28 4 22 4 2. Order of a Surd 2, 3 3, 4 5 are 2, 3 and 4 In the surd n a, n is called the order of the surd. Thus, the orders of respectively.  Note   The orders of the radicals 4 9, 6 16, 6 27 are 4, 6 and 6 respectively. We note that =4 9 4=32 32 4 = 3 and the order of 3 is 2. Thus the order of a surd is not a property of the surd itself, but of the way in which it is expressed. Comparison of Monomial Surds If two simple surds are of the same order, then they can easily be compared. If a < b, n a < n b for all positive integral values of n. Example: 4 2 < 4 7, 3 3 < 3 5, 5 10 < 5 13, etc. If two simple surds of different orders have to be compared, they have to be expressed as radicals of the same order. Thus to compare 4 6 and 3 5, we express both as the radicals of 12th (LCM of 3, 4) order. 4 6 = 12 63 and 3 5 = 12 54 As 63 < 54 , 4 6 < 3 5. Example 1.5 Arrange the following in ascending or descending order of magnitude: 4 6, 3 7, 5 Solution =4 6 6=1/4 , 3 7 71/3, 5 = 51/2 LCM of the denominators of the exponents of these three terms, 4, 3 and 2 is 12.

Number Systems 1.11 Now express the exponent of each term, as a fraction in which the denominator is 12. 13 1 6=4 6=12 (63 )12 = 12 216 14 1 7=3 7=12 (74 )12 = 12 2401 16 1 5=2 5=12 (56 )12 = 12 15625 =Now 4 6 1=2 216, 3 7 12 2401, 5 = 12 15625 5, 3 7, 4 6. Hence, their ascending order is 12 216,12 2401,12 15625, i.e., 4 6, 3 7, 5 ∴ The descending order of magnitude of the given radicals is Addition and Subtraction of Surds Addition and subtraction of similar surds can be done using the distributive law, a c ± b c = (a ± b) c. Example 1.6 Simplify the following by combining similar surds: (a) 2 5 + 5 5 (b) 3 6 + 216 (c) 2 3 − 5 12 + 3 48 Solution (a) 2 5 + 5 5 = (2 + 5) 5 = 7 5. (b) 3 6 + 216 = 3 6 + 62(6) = 3 6 + 6 6 = 9 6. (c) 2 3 − 5 12 + 3 48 = 2 3 − 5 22(3) + 3 42(3) = 2 3 − 5(2) 3 + 3(4) 3 = (2 − 10 + 12) 3 = 4 3. Multiplication and Division of Surds Surds of the same order can be multiplied according to the law, (n x )(n y ) = n xy.  Note   When the surds to be multiplied or divided are not of the same order, they have to be necessarily brought into the same order before the operation is performed. Example 1.7 (a) 15 × 35. (b) 2 3 ÷3 27. τ (c) Multiply 3 3 by 4 2. (d) Divide 6 5 by 3 10.

1.12 Chapter 1 Solution (a) ( 15)( 35) = (15)(35) = =(5)(3)(5)(7) 5 21. (b) 2 3 ÷3 27 = 2 3 3 27 = 23 = 23 3 = 2 . (3) 32(3) (3)(3) 9 =(c) 3 3 3=1/3 and 4 2 21/4 The LCM of 3 and 4 is 12. ∴ 31/3 = 34/12 = 12 34 21/4 = 23/12 = 12 23 ( 3 3 )( 4 2 ) = (12 34 )(12 23 ) = 12 (34 )(23 ) = 12 (81)(8) = 12 648. (d) 6 5 = 51/6 3 10 = 101/3 LCM of 3 and 6 is 6. 3 10 = 101/3 = 102/6 = 6 102 = 6 100 ∴ 65 = 65 3 10 6 100 = 6 5 = 6 1 . 100 20 Rationalizing Factor (RF) If the product of two surds is a rational number, then each of the two surds is a RF of the other. 1. RF is not unique. 2. If one RF of a surd is known, then the product of this factor and any non-zero rational number is also the RF of the given surd. 3. It is convenient to use the simplest of all RFs of the given surd to convert it to a rational number. Examples: 1. (3 3 )=( 3 ) (=3)(3) 9,  a rational number. ∴ 3 is a RF of 3 3. 2. ( 3 + 2 )( 3 − 2 ) = ( 3 )2 − ( 2 )2 = 3 – 2 = 1, a rational number ∴ 3 − 2 is a RF of 3 + 2 and 3 + 2 is a RF of 3 − 2

Number Systems 1.13 Example 1.8 Find the simplest RF of: (a) 4 216  and (b) 5 16 Solution (a) 4 216 = 4 (23 )(33 ) = 23/4 × 33/4 So RF is 21/4 × 31/4 = (2 × 3)1/4 = 4 6. ∴ 4 6 is the simplest RF of 4 216. (b) 5=16 5=24 24/5 ∴ RF is 21/5 ⇒ (24/5 )(21/5 ) = 25/5 = 1. ∴ 5 2 is the simplest RF of 5 16  Notes  1. n a is a RF of n an−1 and vice-versa. 2. n am is a RF of n an−m and vice-versa. 3. a + b is a RF of a − b and vice-versa. 4. 3 a + 3 b is a RF of a2/3 − a1/3 ⋅ b1/3 − b2/3 and vice-versa. 5. 3 a − 3 b is a RF of a2/3 + a1/3 ⋅ b1/3 + b2/3 and vice-versa. Example 1.9 Express the following surds with rational denominators: (a) 2 (b)  23 3 14 3 25 Solution (a) 2 14 2 14 14 . 14 × 14 = 14 = 7 (b) 23 3 = 23 3 × 3 5 = 23 3 × 5 = 23 15 . 3 25 3 25 3 5 3 53 5 Example 1.10 Given that 2 = 1.414, find the value of 3 up to three decimal places. 2 Solution 3 3 2 2 2 2 = × = 32 = 1.5 2 2 = 1.5 (1.414) = 2.121.

1.14 Chapter 1 Rationalization of Mixed Surds a + b is the rationalizing factor of a − b , where a and b are rational. Example 1.11 Rationalize the denominator of 2+ 5 . 2− 5 Solution 2+ 5 = 2 + 5  2 + 5 2− 5  −     2 5   2 + 5  = (2 + ( 5 )2 (2)2 − 5 )2 = 4 +4 5+ 5 4−5 = 9+4 5 = −(9 + 4 5) −1 Example 1.12 Given 2 = 1.414, 3 = 1.732, 5 = 2.236, 6 = 2.449 and 10 = 3.162. Find the value of 2 −1 upto three decimal places. 3− 5 Solution We have to rationalize the denominator. The RF of 3 − 5 is 3 + 5 . ∴ 2 −1 =  2 −1  ( 3 + 5) 3− 5   ( 5 )   3− 5  3+ = 6 − 3 + 10 − 5= 10 + 6 − 3 − 5 ( 3 )2 − ( 5)2 3−5 = 3.162 + 2.449 − 1.732 − 2.236 = −0.822. −2 Conjugate of the Surd of the Form a + b Two binomial quadratic surds a + b and a − b , are called conjugate surds. The product of conjugate surds is rational.

Number Systems 1.15 Example 1.13 Write the conjugate of: (a) 3 + 5 (b) 5 + 3 7 Solution (a) 3 − 5 is the conjugate of 3 + 5 and 5 − 3 is also the conjugate of 3 + 5. (b) 5 − 3 7 is conjugate of 5 + 3 49. The following formulae are helpful in finding the rationalizing factors of mixed quadratic and cubic surds: 1. (a + b) (a – b) = a2 – b2 2. (a + b) (a2 – ab + b2) = a3 + b3 3. (a – b) (a2 + ab + b2) = a3 – b3 Example 1.14 (a) Find the RF of 21/3 + 2−1/3 (b) Find the RF of 51/3 − 5−1/3 Solution (a) 21/3 + 2−1/3 Let a = 21/3 and b = 2−1/3 a3 = (21/3 )3 = 2 b3 = (2−1/3 )3 = 2−1 = 1 2 But a3 + b3 = (a + b)(a2 − ab + b2 ) ∴ a2 − ab + b2 = (21/3 )2 − (21/3 ⋅ 2−1/3 ) + (2−1/3 )2 = 22/3 − 1 + 2−2/3 ∴ RF of 21/3 + 2−1/3 is 22/3 − 1 + 2−2/3. (b) 51/3 − 5−1/3 We have a3 − b3 = (a − b)(a2 + ab + b2 ) ∴ RF of 51/3 − 5−1/3 is [51/3 ]2 + [51/3 ⋅ 5−1/3 ] + [5−1/3 ]2 = 52/3 + 1 + 5−2/3. Comparison of Compound Surds Example 1.15 Among 7 − 3 and 11 − 7, which is greater?

1.16 Chapter 1 Solution By rationalizing, ( )( )7 − 3 = 7− 3 7+ 3 4 = 7+ 3 7+ 3 ( )( )11 − 7 = 11 − 7 11 + 7 4 = 11 + 11 + 7 7 The numerator of each of the irrational number is 4. But 11 + 7 > 7 + 3 ∴4 > 4 7 + 3 11 − 7 7 − 3 > 11 − 7. Example 1.16 Compare the surds A = 8 + 7 and B = 10 + 5 . Solution Since there is a positive sign, by squaring both the surds, we get, ( )A2 = 8 + 7 2 = 8 + 7 + 2 56 = 15 + 2 56 ( )B2 = 10 + 5 2 = 10 + 5 + 2 50 = 15 + 2 50 As 56 > 50,15 + 2 56 > 15 + 2 50 ⇒ A > B. i.e., 8 + 7 > 10 + 5. Rationalizing the Numerator Example 1.17 Rationalize the numerator of 2− 3+x . x −1 Solution Rationalizing factor of 2 − 3 + x is 2 + 3 + x. ∴ 2 − 3+x = 2 − 3 + x 2 + 3+x x −1      x −1   2 + 3 + x  ( )(2)2 − 3 + x 2 4 − (3 + x) == ( ) ( )(x − 1) 2 + 3 + x (x − 1) 2 + 3 + x ( )=(x − 1) 1−x 3+ x = 2+ −1 . 2+ 3+x

Number Systems 1.17 Example 1.18 Express E = 1 with a rational denominator. 5+ 3− 8 Solution The denominator is a trinomial surd, an expression having all the three terms as surds. We group any two of the three terms, say 5 and 3 . Thus, 5 + 3 − 8 =( 5 + 3 ) − 8 Consider the product, 5+ 3 2 − 82 ( ) ( ) ( ) ( )5+3− 8   5+ 3+ 8  =     = 5 + 3 + 2 5 3 − 8 = 2 15 5+ 3+ 8 ( )( )∴ 1 = 5+ 3− 8 5+ 3− 8 5+ 3+ 8 = ( 5 + 3 + 8). 2 15 Rationalizing the denominator, E= 5+ 3+ 8  15  2 15  15  5 3+3 5+2 30 . = 30 Example 1.19 If both a and b are rational numbers, find the value of a and b in each of the following: 3+ 5 =a+b 5     (b) 3 + 2 3 =a+b 3 (a) 3 − 5 5 − 2 3 Solution 3+ 5 (a) 3 − 5 3 + 5 is the rationalizing factor of 3 − 5. ( )∴ 3 + 52 ( )3 − 52 5 = 3+ 5 ×3+ 5 = 3+ 5 3− 5 3+ 5 (3)2 − = 9 +5+6 5 = 14 +6 5 = 14 + 6 5 = 7 + 3 5 =a+b 5 9−5 4 4 4 2 2 ∴a = 7 and b = 3 . 2 2

1.18 Chapter 1 3+2 3 (b)  5 − 2 3 5 + 2 3 is the RF of 5 − 2 3. ( )∴3 + 2 3 ×5+2 3 15 +10 3 +12 + 6 3 5−2 3 5+2 3 (5)2 − 2 32 3 =3+2 = 3 5−2 = 27 + 16 3 = 27 + 16 3 =a+b 3 25 − 12 13 ⇒ a = 27 and b = 16 . 13 13 Square root of a Quadratic Surd Consider the real number a + b , where a and b are rational numbers and b is a surd. Equate the square root of a + b to x + y , where x and y are rational numbers, i.e., a + b = x + y. Squaring both sides, a + b = x + y + 2 xy Equating the rational numbers on the two sides of the above equation, we get a = x + y (1) and equating the irrational numbers, we get b = 2 xy (2) By solving (1) and (2) we get the values of x and y. Similarly, a − b = x − y. Square Root of a Trinomial Quadratic Surd Consider the real number a + b + c + d , where a is a rational number and b, c and d are surds. Let, a + b + c + d = x + y + z . By squaring both sides, and comparing rational and irrational parts on either sides, we get, x + y + z = a. =x 21=bcd , y 1 bc and z = 1 cd . 2 d 2 b Example 1.20 (a) Find the square root of 7 + 4 3. Solution Let 7 + 4 3 = x + y

Number Systems 1.19 Squaring both the sides, 7 + 4 3 = x + y + 2 xy ⇒ x + y = 7 and xy = 2 3 ⇒ xy =12. By solving, we get x = 4 and y = 3 x + y = 4 + 3 = 2 + 3. (b) Find the square root of 10 + 24 + 60 + 40 . Solution Let the given expression be equal to a + x + y + z. As per the method discussed=, a 1=0,=b 24, c 6=0 and d 40. x = 1 bd = 1 24 × 40 = 2 2 c 2 60 y = 1 bc 1 24 × 60 = 3 2 d =2 40 z = 1 cd 1 60 × 40 =5 2 b =2 24 ∴ x + y + z = 2 + 3 + 5. Alternative method: = 10 + 24 + 60 + 40 = 10 + 2 6 + 2 15 + 2 10 = (2 + 3 + 5) + 2 2(3) + 2 3(5) + 2 2(5) = ( 2 + 3 + 5)2 = 2 + 3 + 5. Example 1.21 If x = 2 + 21/3 + 22/3, then find the value of x3 - 6x2 + 6x - 2. Hints Cubing on both sides for (x – 2) = (21/3 + 22/3 ).

1.20 Chapter 1 TEST YOUR CONCEPTS Very Short Answer Type Questions 1. 5 × 125 = _______. 1 6. Multiply: 3 5 by 4 2 . 1 7. Which is greater, 2 or 3 3 ? 2. 20 = _______. 320 18. Express the surd 2 3 as a pure surd. 3. 3 6 × 3 6 × 3 6 = _______. 1 9. Multiply: 14 by 8 . 20. ( 3 3 )4 = _______. 4. 3 √ 4 12 = _______. 5. The sum/difference of a rational and an irrational 2 1. Rationalizing factor of 51/3 + 5−1/3 is _______. number is __________. 2 2. Express the following in the simplest form: 6. 3 7 , in rational denominator is _______. 33831 3 7. Two mixed quadratic surds, a + b and a − b , 23. Express the following in the simplest form: whose sum and product are rational, are called 3 625 ______ surds. 8. 10 3 and 11 3 are _________ surds. (similar/ 24. Express the following as a pure surd: dissimilar) 32 3 16 9. x + y is a pure surd, if x = ________. (zero/one) 25. 3 3 + 2 is a surd. (True/False) 1 0. Conjugate surd of 5 − 3 is _______. 35 1 1. If x + 5 = 4 + y , then x + y = _______. (where 2 6. Express the surd 36 with rational denominator. x and y are rational) PRACTICE QUESTIONS 12. If the product of two surds is a rational number, 2 7. If p = 2 + 3 and pq is a rational number, then q then each of the two is a _______ of the other. is a unique surd. (True/False) 1 3. 6 − 7 is the conjugate surd of 6 + 7. 28. Divide: 6 144 by 6 4 . (True/False) 3 2 9. Express the following in the simplest form: 11 3 15625 1 4. Express the surd  with rational denominator. 15. Find the smallest rationalizing factor of 28 . 30. Which is smaller, 2 − 1 or 3 − 2? Short Answer Type Questions 31. 7 + 48 = _______. 37. If x = 2− 3 , find the value of x+ 1 . 2+ 3 x 32. Find the positive square root of 6 − 20 . 33. Express the following in the simplest form: 38. Which of the two expressions, 11 − 10 and 12 − 11 is greater? 4 5 1048576 39. Simplify the following: 34. Simplify: 2 12 − 3 32 + 2 48 . 55 33 32 3 5. If x = c b + 4, find x + 1 . 11 + 6− 6+ 3 − 15 + 3 2 x 36. Arrange the following surds in an ascending order 40. Arrange the following surds in an ascending order of magnitude: of magnitude: 3 9, 9 5, 3 7 3 4, 4 5, 8

Number Systems 1.21 41. Find the positive square root of the following: 44. Gi=ven 2 1=.414, 3 1.732, 5 = 2.236 , find the value, correct to three decimals, of the 10 + 2 6 + 60 + 2 10 following: 42. If x = 4 11 5 , find the value of x2 – 8x + 11. 1− 3 − 43. If both a and b are rational numbers, find the val- 5− 2 ues of a and b in the following equation: 45. If x = 3 3 + 26 , find the value of 1  x 1 . 2+3 5 2  + x  4+5 5 a +b 5 = Essay Type Questions 46. If x = 7 1 3 , y = 7 1 3 , find the value of 48. Rationalize the denominator of the following: +4 −4 2+3 5 5x2 – 7xy – 5y2. 3 7+5 3 47. Rationalize the denominator of the following: 49. Given 3 = 1.7321 , find the value of the follow- ing surd, correct to three decimal places. 1 3+ 2−3 3 3 + 1 3 − 1 4 + 3 3 + 1 3 + 1 4 − 3 + + CONCEPT APPLICATION Level 1 1. Which of the following fractions lie between 1 (a) 10 (b) 25 PRACTICE QUESTIONS 5 (c) 13 (d) 43 1 and 4 ? 13 7 5. What is the value of 42x−2 , if (16)2x+3 = (64)x+3 ? 57 17 A. 7   B. 4   C.   D. (a) 64 (b) 256 33 11 (c) 32 (d) 512 (a) A and B (b) A and C 6. Which of the following pairs is having two equal values? (where x ∈ R) ______. (c) B, C and D (d) A, B and D 2. Express 0.34 + 0.34 as a single decimal. (a) 0.6788 (b) 0.689 (a) 9x/2, 24x/3 (b) (256)4/x , (43 )4/x (c) 0.6878 (d) 0.687 (c) (343)x/3, (74 )x/12 (d) (362 )2/7, (63 )2/7 3. If 5n = 125 , then 5n 64 = ______. ( )( ) 7. The expression 5 − 3 7 − 2 when sim- plified becomes a (a) 25 (b) 1 (a) simple surd. (b) mixed surd. 125 (c) compound surd. (d) binomial surd. (c) 625 (d) 1 8. If m and n are positive integers, then for a positive 25 { ( )}mn 4. If x4 + 1 = 1297 and y4 - 1 = 2400, then y2 – x2 = ______. number a, m n a = _____.

1.22 Chapter 1 (a) amn (b) a 1 8. Which of the following surd is the smallest? 10 − 5, 19 − 14, 22 − 17 and 8 − 3 (c) am/n (d) 1 (a) 10 − 5 (b) 19 − 14 9. If 2−m × 1 = 1 , then 1 (4m )1/2 +  1 −1  = (c) 22 − 17 (d) 8 − 3 ____ 2m 4 14   5m    (a) 1 (b) 2 19. If m = a + c and m , a and c are three 2 surds, then −1 (c) 4 (d) 4 10. The surds 2, 3 3 and 5 5 , in their descending (a) m is dissimilar to a and c . order are (b) a and c are similar to m. (a) 3 3, 5 5, 2 (b) 2, 3 3, 5 5 (c) only a is similar to m. (d) None of these (c) 2, 5 5, 3 3 (d) 3 3, 2, 5 5 2 0. The surd obtained after rationalizing the numera- 11. 2[(16 − 15)−1 + 25 (13 − 8)−2 ]−1 + (1024)0 = _____. 4 − 25 − a (a) 2 (b) 3 tor of a−9 is equal to (c) 1 (d) 5 a−9 4 − 25 − a  x x−y  y  y −x (a)  y   x  12. If x =2 and y = 4, then + = _____. (a) 4 (b) 8 (b) 4 − 1 25 − a (c) 12 (d) 2 (c) 1 13. In which of the following pairs of surds are the (a − 9)(4 + 25 − a ) given two surds similar? (d) 4 + 1 (a) 5, 7 5 (b) 3 7, 2 7 25 − a PRACTICE QUESTIONS (c) 7, 28 (d) Both (a) and (c) 21. If 13 − x 10 = 8 + 5 , then what is the value of x? 1 4. Which of the following is the greatest? (a) 72 (b) (49)3/2 (a) -5 (b) -6 (c)  1 −1/3 (d) (2401)−1/4 (c) -4 (d) -2  343  22. If the surds 4 4, 6 5, 8 6 and 12 8 are arranged in ( ) 15. ( 6 5)( 3 2 ) 3 (12 6 ) = ascending order from left to right, then the third surd from the left is (a) 12 1749600 (b) 3 2 × 12 109350 (a) 12 8 (b) 4 4 (c) 12 177960 (d) Both (a) and (b) (c) 8 6 (d) 6 5 16. If p = 3 and q = 2, then (3p - 4q)q-p ÷ (4p - 3q)2q-p 2 3. 11 11 11 … 4 terms = = ______. (a) 1 (b) 6 (a) 16 115 (b) 16 11 (c) 1 (d) 2 (c) 16 1114 (d) 16 1115 6 3 5− 3 17.  (32)0.2 + (81)0.25  = _______. 24. If 2+ 3 = x + y 3 , then (x, y) is (256)0.5 − (121)0.5  (a) 2 (b) 5 (a) (13, -7) (b) (-13, 7) (c) (-13, -7) (d) (13, 7) (c) 1 (d) 11

Number Systems 1.23 25. The simplified form of 125 + 125 − 845 is 28. If 35x × (81)2 × 6561 = 37 , then x =_______ 32x (a) 15 (b) 2 5 (c) − 5 (d) −2 5 (a) 3 (b) -3 26. Which of the following statements is true? (c) 1 (d) −1 3 3 I. If x is a conjugate surd of y, then x can be a RF of y. 2 9. If 2n = 1024, then  n −4 = _______. 4 II. If x is a RF of y, then x need not be the conju- 3 gate of y. (a) 3 (b) -3 (a) Only I (b) Only II (c) 27 (d) 81 (c) Both I and II (d) Neither I nor II 2 7. If 3−2 5 =a+b 5 where a and b are rational 30. If  1 −2 −1/3 1/4 = 7m , then m = _______. 6− 5 72    numbers, then what are the values of a and b?    (a) 8 , −9 (b) 8 , −9 (a) −1 (b) 1 35 35 31 31 3 4 (c) −8 , 9 (d) −8 , 9 (c) -3 (d) 2 31 31 35 35 Level 2 a +b 1 (a) 4 2 (b) 32   1 1  (a +b)2 (c) 8 2 (d) 64 −b  31.  a  = _______.  xa2 −b2    35 If a = 6 − 3, b = 3 − 2 and c = 2 − 6,  then find the value of a3 + b3 + c3 − 2abc . PRACTICE QUESTIONS (a) x2 (b) 1 (a) 3 2 − 5 3 − 6 x (c) 73 (d) 1 (b) 3 2 − 5 3 − 6 x2 (c) 3 2 − 4 3 + 6 32. If 2m+n = 16 and 1 then (a2m+n− p )2 = (d) 3 2 + 4 3 + 6 2m−n (am −2n + 2 p )−1 a = 210 _______. (b) 1 36. 81 81 81 81 ∞= (a) 2 4 64 64 64 64 (c) 9 (d) 1 (a) 81 (b) 9 8 64 8 33. Simplify (c) 3 (d) 3 2 22 ( p−1 + q−1 )( p−1 − q−1 ) ÷  1 − 1  1 + 1   ( pq)2 .   p −1 q−1   p −1 q−1   37. If a=p b=q cr = abc , then pqr = _______.  (a) (pq)2 (b) -1 (a) p2q + q2r (b) pq + qr + pr (c) -(pq)-2 (d) 1 3 4. If x = 2 8,y = 2 , then (x − y)2 = (c) ( pq + qr + rp)2 10 − 10 + 2 2 (d) pq(qr + rp)

1.24 Chapter 1 38. The value of (23 + 22 )2/3 + (140 − 29)1/2 2 is (A) 13 − 2 40 = x−y 40 ___________. 8−5 (a) 196 (b) 289 (B) ( 8 )2 + ( 5)2 − 2( 8 )( 5) = x − y 40 ( 8 )2 − ( 5)2 (c) 324 (d) 400 3 9. If x = 6+ 5, then x2 + 1 −2= (C) x −y = 11 x2 3 (a) 2 6 (b) 2 5 = (D) x 1=33 and y 2 (c) 24 (d) 20 3 40. 6 + 6 + 6 + …∞ is equal to _________. (E) ( 8− 5 )( 8− 5) = x − y 40 ( 8+ 5 )( 8− 5) (a) -3 (b) 3 (a) EABDC (c) 6 (d) 2 (b) EBADC (c) ABDEC 41. Simplify (d) DEBAC 1 − 1+ 2= 19 − 360 21 − 440 20 + 396 47. The following are the steps involved in finding the _________. least among 3, 3 4 and 6 15. Arrange them in sequential order. (a) 1 (b) 2 (c) 0 (d) None of these (A) ∴ 6 15 is the smallest. 4 2. If a = 17 − 16 and b = 16 − 15 then (B) ∴ 31/2 = 33/6, 41/3 = 42/6, 151/6 = 151/6 (a) a < b (b) a > b (C) The LCM of the denominators of the expo- nents is 6. (c) a = b (d) None of these (=D) 3 3=1/2, 3 4 41/3, 6 15 = 151/6 ( ) ( 43. 6 15 − 2 56 ⋅ 3 )7 + 2 2 = _________. PRACTICE QUESTIONS (E) ∴ 3 = 6 27, 3 4 = 6 16, 6 15 = 6 15 (a) 0 (b) 1 (c) -1 (d) 2 (a) DCABE (b) DABEB (c) DCBEA (d) DBCAE 44. 63 + 56 = 48. If y = 3 − 8 , then  y − 1 2 = _________. (a) 4 7( 3 + 5) (b) 4 7( 3 + 1)  y  (c) 4 7( 3 + 5) (d) 4 7( 2 + 1) (a) 9 (b) 81 45. If 7 + 2 3 c + p+ q+ r ( p < q < r ), where (c) 4 (d) 32 2 7− 5 23 = 49. The following are the steps involved in finding the p, q, r are rational numbers, then q + r – p = 2+ 3 =a+b 2− 3 (a) 361 (b) 302 value of a + b from 3. Arrange (c) 418 (d) 426 them in sequential order. 46. The following are the steps involved in finding the (A) 22 + ( 3 )2 + 2 × 2 × 3 =a+b 3 22 − ( 3 )2 value of x – y from 8 − 5 = x − y 40 . Arrange 8+ 5 (B) a + b = 7 + 4 = 11 them in sequential order.

Number Systems 1.25 (C) (2 + 3 )(2 + 3 ) = a + b 3 1  1  2 (2 − 3 )(2 + 3 ) 3+  x  5 1. If x = 2 , then x + = ______. 7+4 3 (a) 16 (b) 3 4−3 (D) =a+b 3 (c) 12 (d) 6 (E) a = 7 and b = 4 5 2. If x 3 × y 5 = 10125 , then 12xy = ______. (a) CDAEB (b) CAEBD (a) 1 (b) 1 (c) CADEB (d) CEDAB 3 1 5 0. The following are the steps involved in finding the (c) 2 (d) 2 greatest among 3 2, 6 3 and 6 . Arrange them 53. If x = 1 , then x2 − 10x + 1 = ______. in sequential order. 5+2 6 (A) The LCM of the denominators of the expo- (a) 1 (b) -1 nents is 6. (c) 0 (d) 10 (B) ∴ 6 216 i.e., 6 is the greatest. 5 4. If x = 2 5 and y = 2 , then x + y = 3− 3+ 5 ________. (=C) 3 2 2=1/3, 6 3 31/6, 6 = 61/2 =(D) 21/3 2=2/6, 31/6 31/6, 61/2 = 63/6 (a) 3 (b) 4 3 (c) −2 3 (d) 6 3 (E=) 3 2 6=4 6 3 6 3, 6 = 6 216 5 5. 7 lies between the fractions ______. (a) CADEB (b) CDABE (a) 4 , 5 (b) 43 , 4 9 9 99 9 (c) DCAEB (d) DACBE 42 4 41 42 (c) 99 , 9 (d) 99 , 99 Level 3 ∑143 k +1k+1 =a− b , then a and b respec- (a) ABC 81 (b) 2 PRACTICE QUESTIONS (c) ABC 27 (d) ABC 9 56. If k=4 tively are 1 2− (a) 10 and 0 (b) -10 and 4 59. If x= 3 , the value of x3 − 2x2 − 7x + 10 is (c) 10 and 4 (d) -10 and 0 equal to 5 7. The surd 12 2 , after rationalizing the (a) 2 + 3 (b) 10 5+2 3+ (c) 7 + 2 3 (d) 8 denominator becomes 60. If x = 1 + 51/3 + 52/3 , then find the value of x3 - 3x2 - (a) 5 − 2 + 10 + 1 12x + 6. (b) 5 + 10 + 2 + 1 (a) 22 (b) 20 (c) 10 + 2 + 5 + 1 (c) 16 (d) 14 (d) 5 − 10 − 2 − 1 61. 4 = _____. 10 − 2 21 5 8. If A1/A = B1/B = C1/C , ABC + B AC + C AB = 729 . (a) 1 ( 7+ 3 ) (b) 1 ( 7− 3) 4 4 Which of the following equals A1/A ? (c) 7 + 3 (d) 7 − 3

1.26 Chapter 1 62. If y = 31/3 + 3 , then y3 − 9y2 + 27y = ____. 66. 2 3 6 9x x x2 x3 x3 x6 x4 x10 = (a) 27 (b) -27 (a) 18 (b) 54 (c) -30 (d) 30 (c) 24 (d) 36 6 3. 1 = ______. 8 + 2 + 15 6 7. 7 + 2 6 + 7 − 2 6 = ______. 1 1 (a) 2 ( 5+ 3 ) (b) 2 ( 5− 3) (a) 14 (b) 6 (c) 1 ( 5 + 1) (d) 1 ( 5 − 1) (c) 2 6 (d) 7 2 2 64. If x = 21/3 − 2 , then x3 + 6x2 + 12x = ______. 6 8. 32 92 (81)2 1616 = ______. (a) 6 (b) -6 (a) 6 × 24 (b) 33 × 2 (d) –8 (c) 63 × 23 (d) 63 × 2 (c) 8 8 = ______. 14 + 2 33 65. 3 69. 6 15 − 2 56 3 7 + 2 2 = ______. − 19 − 2 88 (a) 0 (b) 2 (a) 19 + 2 33 (c) 1 (d) 6 2 (b) 14 − 2 88 p2 + 1 70. If p=7−4 3 , then 7 p = ______. (c) 11 + 2 24 (d) 11 − 2 55 (a) 2 (b) 1 (c) 7 (d) 3 PRACTICE QUESTIONS

Number Systems 1.27 TEST YOUR CONCEPTS Very Short Answer Type Questions 1. 25 1 6. 12 5000 17. 3 3 2. 1 4 18. 12 3. 6 4. 1 19. 4 7 8 20. 3 × 3 3 5. irrational 6. 21 2 1. 52/3 − 1 + 5−2/3 9 22. 3 7. conjugate 23. 53 5 8. similar 3 128 27 9. zero 24. 10. −3 − 5 25. False 11. 9 26. 3 180 6 1 2. rationalizing factor 1 3. True 27. False 1 4. 3 11 2 8. 6 36 11 29. 5 15. 7 3 0. 3 − 2 Shot Answer Type Questions 31. 2 + 3 39. 55 − 3 2 − 3 32. 5 − 1 40. 4 5, 3 4, 8 33. 2 41. 2 + 3 + 5 3 4. 12 3 − 12 2 42. 0 ANSWER KEYS 67 −2 35. 5 + 3 3 43. a = 109 , b = 109 2 3 6. 9 5, 6 9, 3 7 44. -0.891 3 7. 14 45. 3 3 38. 11 − 10 Essay Type Questions 46. −7(1 + 80 3 ) 48. 3 14 + 9 35 − 5 6 − 15 15 −12 4 7. −7 2 − 24 3 − 9 6 − 30 92 49. 6.527

1.28 Chapter 1 CONCEPT APPLICATION Level 1 1. (b) 2. (d) 3. (a) 4. (c) 5. (b) 6. (a) 7. (c) 8. (b) 9. (a) 10. (d) 11. (a) 12. (b) 13. (d) 14. (b) 15. (d) 16. (c) 17. (c) 18. (c) 19. (b) 20. (d) 21. (c) 22. (d) 23. (d) 24. (a) 25. (c) 26. (c) 27. (b) 28. (b) 29. (b) 30. (a) Level 2 31. (b) 32. (a) 33. (b) 34. (b) 35. (c) 36. (a) 37. (b) 38. (d) 39. (d) 40. (b) 41. (c) 42. (a) 43. (b) 44. (d) 45. (a) 46. (b) 47. (c) 48. (d) 49. (c) 50. (a) 51. (a) 52. (a) 53. (c) 54. (c) 55. (c) Level 3 58. (b) 59. (d) 60. (a) 61. (c) 62. (d) 63. (b) 64. (b) 65. (c) 68. (d) 69. (c) 70. (a) 56. (a) 57. (b) 66. (a) 67. (c) ANSWER KEYS

Number Systems 1.29 CONCEPT APPLICATION Level 1 1. Convert them into decimal form. 17. Apply laws of indices. 1 8. Rationalize each binomial. 2. Express them in p form. 19. Recall the definitions of similar and dissimilar q surds. 3. Find n. 2 0. RF of a − b is a + b . 2 1. Taking squares on both the sides. 4. Find x and y. 22. Convert the surds into similar surds. 5. Find the value of x. 2n −1 6. Simplify the numbers given in options. 23. a a a… n terms = a 2n . 24. Rationalize the denominator of (5 − 3 ) . 7. Find the product and recall the types of surds. (2+ 3) 8. Recall the laws of radicals. 2 5. Factorize each rational part in the compound surd. 9. Find the value of m. 2 6. Recall the definitions of RF and conjugate. 1 0. Convert the given surds into similar surds 27. Rationalize the denominator of (3 − 2 5) . (6 − 5) 1 1. Simplify the expression. 28. Apply laws of indices. 12. Substitute the values of x and y. 2 9. Find n. 13. Recall the definition of similar surds. 30. Apply laws of indices. 14. Convert the base into same number. 1 5. Convert all factors into similar surds. Hints and Explanation 1 6. Substitute the values of p and q. Level 2 31. (i) (am )n = amn . (ii) Let y = 81 81 6841 … ∞ 64 64 ( ) (ii) Use (am )n s = amns . 32. (i) Apply laws of indices (ii) 22mm + n = 16 ⇒ 22m = 24 . Find the value of (iii) y2 = 81 81 6841 … ∞ − n 64 64 m and apply in the given expression. y2 81 y. 64 1 1 ⇒ = a−m am 3 3. (i) = am and a−m = . (ii) p −1 + q−1 = 1 + 1 and 1 + 1 = p + q. 3 7. (i) Assume each of the power as k then find pqr. p q p −1 q−1 (ii) U sing, a=p b=q cr = abc , find the values of a, 34. (i) Rationalize the denominators of x and y b and c in terms of (abc). (ii) Rationalize the denominators of both x and y. (iii) Multiply the obtained a, b and c values and 35. (i) If a + b + c = 0 , then a3 + b3 + c3 = 3abc. compare the exponents. (ii) If a + b + c = 0 , then a3 + b3 + c3 − 3abc = 0 . (iii) a3 + b3 + c3 − 2abc = abc. 3 8. (i) S implify the numbers in the brackets first and then write into exponential forms. 36. (i) Let the given expression be y and square on (ii) 23 + 22 = 27 = 33; 140 − 19 both sides. = 121 = 112.

1.30 Chapter 1 39. (i) Rationalize the denominator of 1 . 4 9. CADEB is the required sequential order. 1 x 5 0. CADEB is the required sequential order. x (ii) x = 6+ 5 ⇒ = 6− 5. 1 3+2  2 1 − 2  1  2 51. Given that x =  x2  x  (iii) x + = x − . 40. (i) Say, the given expression is y and square on ⇒ x = (2 + 2− 3 3) both sides. 3 )(2 − (ii) Let y = 6 6 6 + …∞ . ⇒ x = 2− 3 4− 3 (iii) y2 = 6 + 6 + 6 + …∞ ⇒ y2 = 6 + y. ⇒ x = 2 − 3. 4 1. (i) Express every denominator in the form And 1 3+2 x= a − b or a + b by finding the square root and rationalize them.  1 2 3 + 2)2  x  (ii) Simplify the surds in each denominator and Now x + = (2 − 3+ then rationalize. =(4)2 =16. 4 2. (i) If a > b, then 1 < 1 . a b (ii) Rationalize the numerators of given surds and 5 2. x 3 × y 5 =10125 then compare. 31/x ⋅ 511/y = 34 × 53 43. (i) Use ‘ 6 x = 3 x ’ and simplify by finding the ⇒ 1 = 4, 1 = 3 square root wherever necessary. x y Hints and Explanation (ii) 6 x = 3 2 x ⇒ 4x = 1, 3y = 1 (iii) 3 x + y . 3 x − y = 3 (x2 − y2 ). ⇒ 12xy = 1. 4 4. (i) Bring a possible term out of the root. 5 3. x = 1 (ii) 63 + 56 = 4 7 9 + 8 5+2 6 (iii) Let 3 + 2 2 = x + y. x = 5−2 6 ⇒ x =5−2 6 25 − 24 45. (i) Rationalize the denominator of the LHS of the equation. 1 =5+2 6 ⇒ x + 1 = 10 x x (ii) R ationalize the denominator of LHS and com- pare it with RHS. x2 + 1 = 10x ⇒ x2 − 10x + 1 = 0. 46. EBADC is the required sequential order. 54. x= 2 ,y= 2 3− 5 3+ 5 4 7. DCBEA is the required sequential order. x+y= 2 + 2 48. Given that, y = 3 − 8 3− 5 3+ 5 1 =1 8 = (3 − 3+ 8 8) =2 3+2 5+2 3 −2 5 43 y 3− 8 )(3 + 3−5 −2 = = 3+ 8 =3 + 8. (x + y)= −2 3. 9−8 Now,  y − 1 2 =(3 − 8 −3− 8 )2 5 5. 3 = 0.428571  y  7   (a) 4 = 0.444…   5 = 0.555… =(−2 8 )2 = 32. 9 9

Number Systems 1.31 (b) 43 = 0.434343…   4 = 0.4444… (d) 41 = 0.414141…  42 = 0.424242… 99 9 99 99 (c) 4 = 0.4444…  42 = 0.424242… 1= 1 9 99 8 + 2 15 ( 5)2 + ( 3 )2 + 2 5× 3 Level 3 =1 5+ 3 56. (i) Substitute the values of k and rationalize every 5+ 5− 3 3) = 1 ( 5− 3 ). term of LHS. =( 3)( 5− 2 (ii) ∑143 1 64. Given x = 21/3 − 2 k+ k +1 k=4 ⇒ x + 2 = 21/3 = 1 4+ 1 5+ + 1. Taking the cubes of the terms on both the sides, 5+ 6+ 144 + 143 ⇒ (x + 2)3 = (21/3 )3 (iii) Rationalize the denominator of each term and ⇒ x3 + 6x2 + 12x + 8 = 2 simplify. ⇒ x3 + 6x2 + 12x = −6. 57. (i) Rationalize the denominator two times. (ii) RF of denominator is (3 + 5 − 2 2 ) . 65. 3 − 8 (iii) R ationalize the denominator twice and simplify. 19 − 2 88 14 + 2 33 61. 4 = 4 3 8 11 − 11 + 10 − 2 21 ( 7 )2 + ( 3 )2 − 2 7×3 = 8− 3 44 = 3( 11 + 8 ) − 8( 11 − 3) Hints and Explanation == 11− 8 11− 3 ( 7 − 3 )2 7 − 3 = 4( 7 + 3 ) = 11 + 8 − 11 + 3 ( 7 − 3)( 7 + 3) = 8 + 3 = 11+ 2 24 . = 4( 7+ 3) = 7+ 3. 66. 2 3 6 9x x x2 x3 x3 x6 x4 x10 4 = 2x x ⋅ 3x3 x3 ⋅ 6x6 x6 ⋅ 9x10 x10 = 18. 6 2. Given, y = 31/3 + 3 ⇒ y − 3 = 31/3 Taking the cubes on both sides 6 7. 7 + 2 6 + 7 − 2 6 ⇒ (y − 3)3 = (31/3 )3 = 6+1+2 6 + 6+1− 2 6 = 6 +1+ 6 −1= 2 6. ⇒ y3 − 9y2 + 27y − 27 = 3 ⇒ y3 − 9y2 + 27y = 30. 63. 1 = 1 6 8. 32 92 (81)2 1616 8 + 2 15 ( 5)2 + ( 3 )2 + 2 5× 3 = (32 )1 2 × (92 )1 4 × (81)2 1/8 × (16)16 1/16 = 3 × 3 × 3 × 16 = 63 × 2. =1 5+ 3 =( 5+ 5− 3 3) = 1 ( 5− 3 ). 3)( 5− 2