Mathematics and Physics And Chemistry with your mind properly and then practice
- 1. MATHEMATICS PART II Textbook for Class XII Reprint 2024-25
- 2. First Edition First Edition First Edition First Edition First Edition January 2007 Magha 1928 Reprinted Reprinted Reprinted Reprinted Reprinted October 2007, January 2009, December 2009, January 2012, November 2012, November 2013, December 2014, December 2015, December 2016, December 2017, December 2018, August 2019, January 2021, July 2021, November 2021 Revised Revised Revised Revised Revised Edition Edition Edition Edition Edition January 2023 Pausa 1944 Reprinted Reprinted Reprinted Reprinted Reprinted March 2024 Chaitra 1946 PD 215T SU PD 215T SU PD 215T SU PD 215T SU PD 215T SU © National Council of Educational © National Council of Educational © National Council of Educational © National Council of Educational © National Council of Educational Research and Training, 2007, Research and Training, 2007, Research and Training, 2007, Research and Training, 2007, Research and Training, 2007, 2022 2022 2022 2022 2022 ` 155.00 155.00 155.00 155.00 155.00 Printed on 80 GSM paper with NCERT watermark OFFICES OF THE PUBLICATION DIVISION, NCERT NCERT Campus Sri Aurobindo Marg New Delhi 110 016 Phone : 011-26562708 108, 100 Feet Road Hosdakere Halli Extension Banashankari III Stage Bangaluru 560 085 Phone : 080-26725740 Navjivan Trust Building 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 Publication Team Publication Team Publication Team Publication Team Publication Team Head, Publication : Anup Kumar Rajput Division Chief Editor : Shveta Uppal Chief Production : Arun Chitkara Officer Chief Business : Amitabh Kumar Manager (In charge) Assistant Production : Sunil Kumar Officer Cover and Illustration Cover and Illustration Cover and Illustration Cover and Illustration Cover and Illustration Arvinder Chawla ALL RIGHTS RESERVED q No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. q This book is sold subject to the condition that it shall not, by way of trade, be lent, re-sold, hired out or otherwise disposed of without the publisher’s consent, in any form of binding or cover other than that in which it is published. 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 means is incorrect and should be unacceptable. Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi 110 016 and printed at Laxmi Offset Printers, Plot No-1, Saiyad Nagar, chawad ka madh, Nai ki thadi, Jaipur -302027 12080 12080 12080 12080 12080 – M – M – M – M – MATHEMATICS ATHEMATICS ATHEMATICS ATHEMATICS ATHEMATICS P P P P PART ART ART ART ART II II II II II Textbook for Class XII ISBN ISBN ISBN ISBN ISBN 81-7450-629-2 (Part-I) 81-7450-629-2 (Part-I) 81-7450-629-2 (Part-I) 81-7450-629-2 (Part-I) 81-7450-629-2 (Part-I) 81-7450-653-5 (Part-II) 81-7450-653-5 (Part-II) 81-7450-653-5 (Part-II) 81-7450-653-5 (Part-II) 81-7450-653-5 (Part-II) Reprint 2024-25
- 3. Foreword The National Curriculum Framework, 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 recognise 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. These aims imply considerable change in 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 than 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. Reprint 2024-25
- 4. 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 ChiefAdvisor for this book, Professor P.K. Jain for guiding the work of this committee. Several teachers contributed to the development of this textbook; we are grateful to their principals for making this possible. We are indebted to the institutions and organisations which have generously permitted us to draw upon their resources, material and personnel. 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 November 2006 Research and Training iv Reprint 2024-25
- 5. 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. Reprint 2024-25
- 7. Preface The National Council of Educational Research and Training (NCERT) had constituted 21 Focus Groups on Teaching of various subjects related to School Education, to review the National Curriculum Framework for School Education - 2000 (NCFSE - 2000) in face of new emerging challenges and transformations occurring in the fields of content and pedagogy under the contexts of National and International spectrum of school education. These Focus Groups made general and specific comments in their respective areas. Consequently, based on these reports of Focus Groups, National Curriculum Framework (NCF)-2005 was developed. NCERT designed the new syllabi and constituted Textbook Development Teams for Classes XI and XII to prepare textbooks in Mathematics under the new guidelines and new syllabi. The textbook for Class XI is already in use, which was brought in 2005. The first draft of the present book (Class XII) was prepared by the team consisting of NCERT faculty, experts and practicing teachers. The draft was refined by the development team in different meetings. This draft of the book was exposed to a group of practicing teachers teaching Mathematics at higher secondary stage in different parts of the country, in a review workshop organised by the NCERT at Delhi. The teachers made useful comments and suggestions which were incorporated in the draft textbook. The draft textbook was finalised by an editorial board constituted out of the development team. Finally, theAdvisory Group in Science and Mathematics and the Monitoring Committee constituted by the HRD Ministry, Government of India have approved the draft of the textbook. In the fitness of things, let us cite some of the essential features dominating the textbook. These characteristics have reflections in almost all the chapters. The existing textbook contains thirteen main chapters and two appendices. Each chapter contains the followings : § Introduction: Highlighting the importance of the topic; connection with earlier studied topics; brief mention about the new concepts to be discussed in the chapter. § Organisation of chapter into sections comprising one or more concepts/ subconcepts. § Motivating and introducing the concepts/subconcepts. Illustrations have been provided wherever possible. Reprint 2024-25
- 8. § Proofs/problem solving involving deductive or inductive reasoning, multiplicity of approaches wherever possible have been inducted. § Geometric viewing / visualisation of concepts have been emphasized whenever needed. § Applications of mathematical concepts have also been integrated with allied subjects like Science and Social Sciences. § Adequate and variety of examples/exercises have been given in each section. § For refocusing and strengthening the understanding and skill of problem solving and applicabilities, miscellaneous types of examples/exercises have been provided involving two or more subconcepts at a time at the end of the chapter. The scope of challenging problems to talented minority have been reflected conducive to the recommendation as reflected in NCF-2005. § For more motivational purpose, brief historical background of topics have been provided at the end of the chapter and at the beginning of each chapter, relevant quotation and photograph of eminent mathematician who have contributed significantly in the development of the topic undertaken, are also provided. § Lastly, for direct recapitulation of main concepts, formulas and results, brief summary of the chapter has also been provided. I am thankful to Professor Krishan Kumar, Director, NCERT who constituted the team and invited me to join this national endeavour for the improvement of Mathematics education. He has provided us with an enlightened perspective and a very conducive environment. This made the task of preparing the book much more enjoyable and rewarding. I express my gratitude to Professor J.V. Narlikar, Chairperson of theAdvisory Group in Science and Mathematics, for his specific suggestions and advice towards the improvement of the book from time to time. I, also, thank Professor G. Ravindra, Joint Director, NCERT for his help from time to time. I express my sincere thanks to Professor Hukum Singh, Chief Coordinator and Head, DESM, Dr. V. P. Singh, Coordinator and Professor, S. K. Singh Gautam who have been helping for the success of this project academically as well as administratively. Also, I would like to place on records my appreciation and thanks to all the members of the team and the teachers who have been associated with this noble cause in one or the other form. PAWAN K. JAIN Chief Advisor viii Reprint 2024-25
- 9. Textbook Development Committee CHAIRPERSON, ADVISORY GROUP IN SCIENCE AND MATHEMATICS J.V. Narlikar, Emeritus Professor, Inter-University Centre for Astronomy and Astrophysics (IUCAA), Ganeshkhind, Pune University, Pune CHIEF ADVISOR P.K. Jain, Professor, Department of Mathematics, University of Delhi, Delhi CHIEF COORDINATOR Hukum Singh, Professor and Head, DESM, NCERT, New Delhi MEMBERS A.K. Rajput, Reader, RIE, Bhopal, M.P. Arun Pal Singh, Sr. Lecturer, Department of Mathematics, Dayal Singh College, University of Delhi, Delhi B.S.P. Raju, Professor, RIE Mysore, Karnataka C.R. Pradeep, Assistant Professor, Department of Mathematics, Indian Institute of Science, Bangalore, Karnataka D.R. Sharma, P.G.T., Jawahar Navodaya Vidyalaya, Mungeshpur, Delhi R.P. Maurya, Reader, DESM, NCERT, New Delhi Ram Avtar, Professor (Retd.) and Consultant, DESM, NCERT, New Delhi S.K. Kaushik, Reader, Department of Mathematics, Kirori Mal College, University of Delhi, Delhi S.K.S. Gautam, Professor, DESM, NCERT, New Delhi S.S. Khare, Pro-Vice-Chancellor, NEHU, Tura Campus, Meghalaya Sangeeta Arora, P.G.T., Apeejay School, Saket, New Delhi Shailja Tewari, P.G.T., Kendriya Vidyalaya, Barkakana, Hazaribagh, Jharkhand Sunil Bajaj, Sr. Specialist, SCERT, Gurgaon, Haryana Vinayak Bujade, Lecturer, Vidarbha Buniyadi Junior College, Sakkardara Chowk, Nagpur, Maharashtra MEMBER-COORDINATOR V.P. Singh, Reader, DESM, NCERT, New Delhi Reprint 2024-25
- 11. Acknowledgements The Council gratefully acknowledges the valuable contributions of the following participants of the Textbook Review Workshop: Jagdish Saran, Professor, Deptt. of Statistics, University of Delhi; Quddus Khan, Lecturer, Shibli National P.G. College, Azamgarh (U.P.); P.K. Tewari, Assistant Commissioner (Retd.), Kendriya Vidyalaya Sangathan; S.B. Tripathi, Lecturer, R.P.V.V., Surajmal Vihar, Delhi; O.N. Singh, Reader, RIE, Bhubaneswar, Orissa; Miss Saroj, Lecturer, Govt. Girls Senior Secondary School No.1, Roop Nagar, Delhi; P. Bhaskar Kumar, P.G.T., Jawahar Navodaya Vidyalaya, Lepakshi, Anantapur, (A.P.); Mrs. S. Kalpagam, P.G.T., K.V. NALCampus, Bangalore; Rahul Sofat, Lecturer,Air Force Golden Jubilee Institute, Subroto Park, New Delhi; Vandita Kalra, Lecturer, Sarvodaya Kanya Vidyalaya, Vikaspuri, District Centre, New Delhi; Janardan Tripathi, Lecturer, Govt. R.H.S.S., Aizawl, Mizoram and Ms. Sushma Jaireth, Reader, DWS, NCERT, New Delhi. The Council acknowledges the efforts of Deepak Kapoor, Incharge,Computer Station; Sajjad Haider Ansari, Rakesh Kumar and Nargis Islam, D.T.P. Operators; Monika Saxena, Copy Editor; and Abhimanu Mohanty, Proof Reader. The contribution of APC-Office, administration of DESM and Publication Department is also duly acknowledged. Reprint 2024-25
- 12. Contents of MATHEMATICS PART I For Class XII Chapter 1 Relations and Functions 1 - 17 Chapter 2 Inverse Trigonometric Functions 18 - 33 Chapter 3 Matrices 34 - 75 Chapter 4 Determinants 76 - 103 Chapter 5 Continuity and Differentiability 104 - 146 Chapter 6 Application of Derivatives 147 - 186 Appendix 1: Proofs in Mathematics 187 - 195 Appendix 2: Mathematical Modelling 196 - 207 Answers 208 - 224 Reprint 2024-25
- 13. Contents PART II Foreword iii Rationalisation of Content in the Textbooks v Preface vii 7. Integrals 225 7.1 Introduction 225 7.2 Integration as an Inverse Process of Differentiation 226 7.3 Methods of Integration 235 7.4 Integrals of Some Particular Functions 243 7.5 Integration by Partial Fractions 252 7.6 Integration by Parts 259 7.7 Definite Integral 267 7.8 Fundamental Theorem of Calculus 267 7.9 Evaluation of Definite Integrals by Substitution 271 7.10 Some Properties of Definite Integrals 273 8. Application of Integrals 292 8.1 Introduction 292 8.2 Area under Simple Curves 292 9. Differential Equations 300 9.1 Introduction 300 9.2 Basic Concepts 300 9.3 General and Particular Solutions of a 304 Differential Equation 9.4 Methods of Solving First Order, First Degree 306 Differential Equations 10. Vector Algebra 338 10.1 Introduction 338 10.2 Some Basic Concepts 338 10.3 Types of Vectors 341 10.4 Addition of Vectors 343 10.5 Multiplication of a Vector by a Scalar 346 10.6 Product of Two Vectors 355 Reprint 2024-25
- 14. 11. Three Dimensional Geometry 377 11.1 Introduction 377 11.2 Direction Cosines and Direction Ratios of a Line 377 11.3 Equation of a Line in Space 381 11.4 Angle between Two Lines 383 11.5 Shortest Distance between Two Lines 385 12. Linear Programming 394 12.1 Introduction 394 12.2 Linear Programming Problem and its Mathematical Formulation 395 13. Probability 406 13.1 Introduction 406 13.2 ConditionalProbability 406 13.3 Multiplication Theorem on Probability 415 13.4 Independent Events 417 13.5 Bayes' Theorem 423 Answers 439 xiv Reprint 2024-25