Mathematics Textbook for Class IX PDF

MATHEMATICS Textbook for Class IX 2024-25 0962 – MATHEMATICS ISBN 81-7450-489-3 Textbook for Class 9 First Edition ALL RIGHTS RESERVED February 200...

MATHEMATICS Textbook for Class IX 2024-25 0962 – MATHEMATICS ISBN 81-7450-489-3 Textbook for Class 9 First Edition ALL RIGHTS RESERVED February 2006 Phalguna 1927 q No part of this publication may be reproduced, stored in a retrieval system Reprinted or transmitted, in any form or by any means, electronic, mechanical, October 2006, October 2007, photocopying, recording or otherwise without the prior permission of the publisher. January 2009, January 2010, q This book is sold subject to the condition that it shall not, by way of January 2012, November 2012, trade, be lent, re-sold, hired out or otherwise disposed of without the October 2013, December 2014, publisher’s consent, in any form of binding or cover other than that in December 2015, December 2016, which it is published. December 2017, December 2018, q The correct price of this publication is the price printed on this page, Any revised price indicated by a rubber stamp or by a sticker or by any other August 2019, January 2021, means is incorrect and should be unacceptable. August 2021, November 2021 Revised Edition OFFICES OF THE PUBLICATION October 2022 Ashwina 1944 DIVISION, NCERT NCERT Campus Reprinted Sri Aurobindo Marg March 2024 Chaitra 1946 New Delhi 110 016 Phone : 011-26562708 108, 100 Feet Road Hosdakere Halli Extension PD 525T SU Banashankari III Stage Bangaluru 560 085 Phone : 080-26725740 © National Council of Educational Navjivan Trust Building Research and Training, 2006, 2022 P.O.Navjivan Ahmedabad 380 014 Phone : 079-27541446 CWC Campus Opp. Dhankal Bus Stop Panihati Kolkata 700 114 Phone : 033-25530454 CWC Complex Maligaon Guwahati 781 021 Phone : 0361-2674869 ` 155.00 Publication Team Head, Publication : Anup Kumar Rajput Division Chief Editor : Shveta Uppal Printed on 80 GSM paper with NCERT Chief Production Officer : Arun Chitkara watermark Chief Business : Amitabh Kumar Published at the Publication Division Manager (In charge) by the Secretary, National Council Production Assistant : Prakash Veer Singh of Educational R e s e a r c h a n d Training, Sri Aurobindo Marg, Cover and Illustrations New Delhi 110 016 and printed at Digital Expressions General Offset Printing Press, 42, Industrial Colony, Naini, Allahabad Uttar Pradesh 2024-25  FOREWORD The National Curriculum Framework (NCF) 2005, recommends that children’s life at school must be linked to their life outside the school. This principle marks a departure from the legacy of bookish learning which continues to shape our system and causes a gap between the school, home and community. The syllabi and textbooks developed on the basis of NCF signify an attempt to implement this basic idea. They also attempt to discourage rote learning and the maintenance of sharp boundaries between different subject areas. We hope these measures will take us significantly further in the direction of a child-centred system of education outlined in the national Policy on Education (1986). The success of this effort depends on the steps that school principals and teachers will take to encourage children to reflect on their own learning and to pursue imaginative activities and questions. We must recognize that, given space, time and freedom, children generate new knowledge by engaging with the information passed on to them by adults. Treating the prescribed textbook as the sole basis of examination is one of the key reasons why other resources and sites of learning are ignored. Inculcating creativity and initiative is possible if we perceive and treat children as participants in learning, not as receivers of a fixed body of knowledge. This aims imply considerable change is school routines and mode of functioning. Flexibility in the daily time-table is as necessary as rigour in implementing the annual calendar so that the required number of teaching days are actually devoted to teaching. The methods used for teaching and evaluation will also determine how effective this textbook proves for making children’s life at school a happy experience, rather then a source of stress or boredom. Syllabus designers have tried to address the problem of curricular burden by restructuring and reorienting knowledge at different stages with greater consideration for child psychology and the time available for teaching. The textbook attempts to enhance this endeavour by giving higher priority and space to opportunities for contemplation and wondering, discussion in small groups, and activities requiring hands-on experience. The National Council of Educational Research and Training (NCERT) appreciates the hard work done by the textbook development committee responsible for this book. We wish to thank the Chairperson of the advisory group in science and mathematics, Professor J.V. Narlikar and the Chief Advisor for this book, Professor P. Sinclair of IGNOU, New Delhi for guiding the work of this committee. Several teachers contributed 2024-25 to the development of this textbook; we are grateful to their principals for making this possible. We are indebted to the institutions and organizations which have generously permitted us to draw upon their resources, material and personnel. We are especially grateful to the members of the National Monitoring Committee, appointed by the Department of Secondary and Higher Education, Ministry of Human Resource Development under the Chairpersonship of Professor Mrinal Miri and Professor G.P. Deshpande, for their valuable time and contribution. As an organisation committed to systemic reform and continuous improvement in the quality of its products, NCERT welcomes comments and suggestions which will enable us to undertake further revision and refinement. Director New Delhi National Council of Educational 20 December 2005 Research and Training 2024-25  RATIONALISATION OF CONTENT IN THE TEXTBOOKS In view of the COVID-19 pandemic, it is imperative to reduce content load on students. The National Education Policy 2020, also emphasises reducing the content load and providing opportunities for experiential learning with creative mindset. In this background, the NCERT has undertaken the exercise to rationalise the textbooks across all classes. Learning Outcomes already developed by the NCERT across classes have been taken into consideration in this exercise. Contents of the textbooks have been rationalised in view of the following: Overlapping with similar content included in other subject areas in the same class Similar content included in the lower or higher class in the same subject Difficulty level Content, which is easily accessible to students without much interventions from teachers and can be learned by children through self-learning or peer-learning Content, which is irrelevant in the present context This present edition, is a reformatted version after carrying out the changes given above. 2024-25 2024-25  TEXTBOOK DEVELOPMENT COMMITTEE CHAIRPERSON, ADVISORY GROUP IN SCIENCE AND MATHEMATICS J.V. Narlikar, Emeritus Professor, Chairman, Advisory Committee, Inter University Centre for Astronomy & Astrophysics (IUCAA), Ganeshkhind, Pune University, Pune CHIEF ADVISOR P. Sinclair, Director, NCERT and Professor of Mathematics , IGNOU, New Delhi CHIEF COORDINATOR Hukum Singh, Professor (Retd.), DESM, NCERT MEMBERS A.K. Wazalwar, Professor and Head, DESM, NCERT Anjali Lal, PGT, DAV Public School, Sector-14, Gurgaon Anju Nirula, PGT, DAV Public School, Pushpanjali Enclave, Pitampura, Delhi G.P. Dikshit, Professor (Retd.), Department of Mathematics & Astronomy, Lucknow University, Lucknow K.A.S.S.V. Kameswara Rao, Associate Professor, Regional Institute of Education, Bhubaneswar Mahendra R. Gajare, TGT, Atul Vidyalaya, Atul, Dist. Valsad Mahendra Shanker, Lecturer (S.G.) (Retd.), NCERT Rama Balaji, TGT, K.V., MEG & Centre, ST. John’s Road, Bangalore Sanjay Mudgal, Lecturer, CIET, NCERT Shashidhar Jagadeeshan, Teacher and Member, Governing Council, Centre for Learning, Bangalore S. Venkataraman, Lecturer, School of Sciences, IGNOU, New Delhi Uaday Singh, Lecturer, DESM, NCERT Ved Dudeja, Vice-Principal (Retd.), Govt. Girls Sec. School, Sainik Vihar, Delhi MEMBER-COORDINATOR Ram Avtar, Professor (Retd.), DESM, NCERT (till December 2005) R.P. Maurya, Professor, DESM, NCERT (Since January 2006) 2024-25  ACKNOWLEDGEMENTS The Council gratefully acknowledges the valuable contributions of the following participants of the Textbook Review Workshop: A.K. Saxena, Professor (Retd.), Lucknow University, Lucknow; Sunil Bajaj, HOD, SCERT, Gurgaon; K.L. Arya, Professor (Retd.), DESM, NCERT; Vandita Kalra, Lecturer, Sarvodaya Kanya Vidyalya, Vikas Puri, District Centre, New Delhi; Jagdish Singh, PGT, Sainik School, Kapurthala; P.K. Bagga, TGT, S.B.V. Subhash Nagar, New Delhi; R.C. Mahana, TGT, Kendriya Vidyalya, Sambalpur; D.R. Khandave, TGT, JNV, Dudhnoi, Goalpara; S.S. Chattopadhyay, Assistant Master, Bidhan Nagar Government High School, Kolkata; V.A. Sujatha, TGT, K.V. Vasco No. 1, Goa; Akila Sahadevan, TGT, K.V., Meenambakkam, Chennai; S.C. Rauto, TGT, Central School for Tibetans, Mussoorie; Sunil P. Xavier, TGT, JNV, Neriyamangalam, Ernakulam; Amit Bajaj, TGT, CRPF Public School, Rohini, Delhi; R.K. Pande, TGT, D.M. School, RIE, Bhopal; V. Madhavi, TGT, Sanskriti School, Chanakyapuri, New Delhi; G. Sri Hari Babu, TGT, JNV, Sirpur Kagaznagar, Adilabad; and R.K. Mishra, TGT, A.E.C. School, Narora. Special thanks are due to M. Chandra, Professor and Head (Retd.), DESM, NCERT for her support during the development of this book. The council acknowledges the valuable inputs for analysing syllabi, textbooks and the content, proposed to be rationalised for this edition by Gupreet Bhatnagar, CBSE Resource Person; Rahul Sofat, CBSE Resource Person; Ashutosh K. Wazalwar, Professor, DESM, NCERT and T.P. Sarma, Professor, DESM, NCERT. The Council acknowledges the efforts of Computer Incharge, Deepak Kapoor; D.T.P. Operator, Naresh Kumar; Copy Editor, Pragati Bhardwaj; and Proof Reader, Yogita Sharma. Contribution of APC–Office, administration of DESM, Publication Department and Secretariat of NCERT is also duly acknowledged. 2024-25  CONTENTS FOREWORD iii RATIONALISATION OF CONTENT IN THE TEXTBOOKS v 1. NUMBER SYSTEMS 1 1.1 Introduction 1 1.2 Irrational Numbers 5 1.3 Real Numbers and their Decimal Expansions 8 1.4 Operations on Real Numbers 15 1.5 Laws of Exponents for Real Numbers 21 1.6 Summary 24 2. POLYNOMIALS 25 2.1 Introduction 25 2.2 Polynomials in One Variable 25 2.3 Zeroes of a Polynomial 29 2.4 Factorisation of Polynomials 32 2.5 Algebraic Identities 36 2.6 Summary 42 3. COORDINATE GEOMETRY 43 3.1 Introduction 43 3.2 Cartesian System 46 3.3 Summary 53 4. LINEAR EQUATIONS IN TWO VARIABLES 55 4.1 Introduction 55 4.2 Linear Equations 55 4.3 Solution of a Linear Equation 57 4.4 Summary 59 5. INTRODUCTION TO EUCLID’S GEOMETRY 60 5.1 Introduction 60 5.2 Euclid’s Definitions, Axioms and Postulates 62 5.3 Summary 68 2024-25 6. LINES AND ANGLES 69 6.1 Introduction 69 6.2 Basic Terms and Definitions 70 6.3 Intersecting Lines and Non-intersecting Lines 72 6.4 Pairs of Angles 72 6.5 Lines Parallel to the Same Line 78 6.6 Summary 82 7. TRIANGLES 83 7.1 Introduction 83 7.2 Congruence of Triangles 83 7.3 Criteria for Congruence of Triangles 86 7.4 Some Properties of a Triangle 94 7.5 Some More Criteria for Congruence of Triangles 99 7.6 Summary 103 8. QUADRILATERALS 104 8.1 Properties of a Parallelogram 104 8.2 The Mid-point Theorem 111 8.3 Summary 115 9. CIRCLES 116 9.1 Angle Subtended by a Chord at a Point 116 9.2 Perpendicular from the Centre to a Chord 118 9.3 Equal Chords and their Distances from the Centre 119 9.4 Angle Subtended by an Arc of a Circle 122 9.5 Cyclic Quadrilaterals 125 9.6 Summary 129 10. HERON’S FORMULA 131 10.1 Area of a Triangle – by Heron’s Formula 131 10.2 Summary 136 11. SURFACE AREAS AND VOLUMES 137 11.1 Surface Area of a Right Circular Cone 137 11.2 Surface Area of a Sphere 141 11.3 Volume of a Right Circular Cone 145 2024-25  11.4 Volume of a Sphere 148 11.5 Summary 150 12. STATISTICS 151 12.1 Ggraphical Representation of Data 151 12.2 Summary 166 APPENDIX – 1 PROOFS IN MATHEMATICS 167 A1.1 Introduction 167 A1.2 Mathematically Acceptable Statements 168 A1.3 Deductive Reasoning 171 A1.4 Theorems, Conjectures and Axioms 174 A1.5 What is a Mathematical Proof? 179 A1.6 Summary 186 APPENDIX – 2 INTRODUCTION TO MATHEMATICAL MODELLING 187 A2.1 Introduction 187 A2.2 Review of Word Problems 188 A2.3 Some Mathematical Models 192 A2.4 The Process of Modelling, its Advantages and Limitations 200 A2.5 Summary 203 ANSWERS/HINTS 205-219 2024-25 2024-25