Mathematics Textbook for Class XI PDF

MATHEMATICS Textbook for Class XI 2024-25  11076 – MATHEMATICS ISBN 81-7450-486-9 Textbook for Class XI First Edition ALL RIGHTS RESERVED February 2006 Phalguna 1927...

MATHEMATICS Textbook for Class XI 2024-25  11076 – MATHEMATICS ISBN 81-7450-486-9 Textbook for Class XI First Edition ALL RIGHTS RESERVED February 2006 Phalguna 1927 q No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, Reprinted mechanical, photocopying, recording or otherwise without the prior permission of the publisher. October 2006, November 2007, q This book is sold subject to the condition that it shall not, by way of December 2008, December 2009, trade, be lent, re-sold, hired out or otherwise disposed of without the January 2011, February 2012, publisher’s consent, in any form of binding or cover other than that in December 2012, November 2013, which it is published. December 2014, May 2016, q The correct price of this publication is the price printed on this page, December 2016, December 2017, Any revised price indicated by a rubber stamp or by a sticker or by any other means is incorrect and should be unacceptable. January 2019, August 2019, January 2021 and November 2021 OFFICES OF THE PUBLICATION DIVISION, NCERT Revised Edition November 2022 Agrahayan 1944 NCERT Campus Sri Aurobindo Marg Reprinted New Delhi 110 016 Phone : 011-26562708 March 2024 Chaitra 1946 108, 100 Feet Road Hosdakere Halli Extension Banashankari III Stage PD 375T SU Bengaluru 560 085 Phone : 080-26725740 © National Council of Educational Navjivan Trust Building P.O.Navjivan Research and Training, 2006, 2022 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 ` 230.00 Publication Team Head, Publication : Anup Kumar Rajput Division Chief Editor : Shveta Uppal Chief Production : Arun Chitkara Printed on 80 GSM paper with NCERT Officer watermark Chief Business : Amitabh Kumar Manager (In charge) Published at the Publication Division by the Secretary, National Council of Assistant Production : Om Prakash Educational Research and Training, Officer Sri Aurobindo Marg, New Delhi 110 016 and printed at Amit Printing Press, Cover and Layout D-12 and 13, Industrial Area, Site-A, Arvinder Chawla Mathura- 281 001 (U.P.) 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 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. The National Council of Educational Research and Training (NCERT) appreciates the hard work done by the Textbook Development Committee responsible for this 2024-25 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.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. 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 the 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 iv 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. v 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 (IUCCA), Ganeshkhind, Pune University, Pune CHIEF ADVISOR P.K. Jain, Professor, Department of Mathematics, University of Delhi, Delhi CHIEF COORDINATOR Hukum Singh, Professor, DESM, NCERT, New Delhi MEMBERS A.K. Rajput, Associate Professor, RIE Bhopal, M.P. A.K. Wazalwar, Associate Professor, DESM NCERT, New Delhi B.S.P. Raju, Professor, RIE Mysore, Karnataka C.R. Pradeep, Assistant Professor, Department of Mathematics, Indian Institute of Science, Bangalore, Karnataka. Pradeepto Hore, Sr. Maths Master, Sarla Birla Academy Bangalore, Karnataka. S.B. Tripathy, Lecturer, Rajkiya Pratibha Vikas Vidyalaya, Surajmal Vihar, Delhi. S.K.S. Gautam, Professor, DESM, NCERT, New Delhi Sanjay Kumar Sinha, P.G.T., Sanskriti School Chanakyapuri, New Delhi. Sanjay Mudgal, Lecturer, CIET, New Delhi Sneha Titus, Maths Teacher, Aditi Mallya School Yelaharika, Bangalore, Karnataka Sujatha Verma, Reader in Mathematics, IGNOU, New Delhi. Uaday Singh, Lecturer, DESM, NCERT, New Delhi. MEMBER-COORDINATOR V.P. Singh, Associate Professor, DESM, NCERT, New Delhi 2024-25  Acknowledgements The Council gratefully acknowledges the valuable contributions of the following participants of the Textbook Review Workshop: P. Bhaskar Kumar, P.G.T., Jawahar Navodaya Vidyalaya, Ananthpur, (A.P.); Vinayak Bujade, Lecturer, Vidarbha Buniyadi Junior College, Sakkardara Chowk Nagpur, Maharashtra; Vandita Kalra, Lecturer, Sarvodaya Kanya Vidyalaya Vikashpuri District Centre, New Delhi; P.L. Sachdeva Deptt. of Mathematics, Indian Institute of Science, Bangalore, Karnataka; P.K.Tiwari Assistant Commissioner (Retd.), Kendriya Vidyalaya Sangathan; Jagdish Saran, Department of Statistics, University of Delhi; Quddus Khan, Lecturer, Shibli National P.G. College Azamgarh (U.P.); Sumat Kumar Jain, Lecturer, K.L. Jain Inter College Sasni Hathras (U.P.); R.P. Gihare, Lecturer (BRC), Janpad Shiksha Kendra Chicholi Distt. Betul (M.P.); Sangeeta Arora, P.G.T., A.P.J. School Saket, New Delhi; P.N. Malhotra, ADE (Sc.), Directorate of Education, Delhi; D.R. Sharma, P.G.T., J.N.V. Mungespur, Delhi; Saroj, P.G.T. Government Girls Sr. Secondary School, No. 1, Roop Nagar, Delhi, Manoj Kumar Thakur, P.G.T., D.A.V. Public School, Rajender Nagar, Sahibabad, Ghaziabad (U.P.) and R.P. Maurya, Reader, DESM, NCERT, New Delhi. Acknowledgements are due to Professor M. Chandra, Head, Department of Education in Science and Mathematics for her support. The Council acknowledges the efforts of the Computer Incharge, Deepak Kapoor; Rakesh Kumar, Kamlesh Rao and Sajjad Haider Ansari, D.T.P. Operators; Kushal Pal Singh Yadav, Copy Editor and Proof Readers, Mukhtar Hussain and Kanwar Singh. The contribution of APC–Office, administration of DESM and Publication Department is also duly acknowledged. 2024-25  Contents Foreword iii Rationalisation of Content in the Textbooks v 1. Sets 1 1.1 Introduction 1 1.2 Sets and their Representations 1 1.3 The Empty Set 5 1.4 Finite and Infinite Sets 6 1.5 Equal Sets 7 1.6 Subsets 9 1.7 Universal Set 12 1.8 Venn Diagrams 13 1.9 Operations on Sets 13 1.10 Complement of a Set 18 2. Relations and Functions 24 2.1 Introduction 24 2.2 Cartesian Product of Sets 24 2.3 Relations 28 2.4 Functions 30 3. Trigonometric Functions 43 3.1 Introduction 43 3.2 Angles 43 3.3 Trigonometric Functions 49 3.4 Trigonometric Functions of Sum and Difference of Two Angles 57 4. Complex Numbers and Quadratic Equations 76 4.1 Introduction 76 4.2 Complex Numbers 76 2024-25  4.3 Algebra of Complex Numbers 77 4.4 The Modulus and the Conjugate of a Complex Number 81 4.5 Argand Plane and Polar Representation 83 5. Linear Inequalities 89 5.1 Introduction 89 5.2 Inequalities 89 5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation 91 6. Permutations and Combinations 100 6.1 Introduction 100 6.2 Fundamental Principle of Counting 100 6.3 Permutations 104 6.4 Combinations 114 7. Binomial Theorem 126 7.1 Introduction 126 7.2 Binomial Theorem for Positive Integral Indices 126 8. Sequences and Series 135 8.1 Introduction 135 8.2 Sequences 135 8.3 Series 137 8.4 Geometric Progression (G.P.) 139 8.5 Relationship Between A.M. and G.M. 144 9. Straight Lines 151 9.1 Introduction 151 9.2 Slope of a Line 152 9.3 Various Forms of the Equation of a Line 159 9.4 Distance of a Point From a Line 164 10. Conic Sections 176 10.1 Introduction 176 10.2 Sections of a Cone 176 10.3 Circle 179 x 2024-25  10.4 Parabola 182 10.5 Ellipse 187 10.6 Hyperbola 195 11. Introduction to Three Dimensional Geometry 208 11.1 Introduction 208 11.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space 209 11.3 Coordinates of a Point in Space 209 11.4 Distance between Two Points 211 12. Limits and Derivatives 217 12.1 Introduction 217 12.2 Intuitive Idea of Derivatives 217 12.3 Limits 220 12.4 Limits of Trigonometric Functions 234 12.5 Derivatives 239 13. Statistics 257 13.1 Introduction 257 13.2 Measures of Dispersion 259 13.3 Range 259 13.4 Mean Deviation 259 13.5 Variance and Standard Deviation 271 14. Probability 289 14.1 Event 289 14.2 Axiomatic Approach to Probability 295 Appendix 1: Infinite Series 314 A.1.1 Introduction 314 A.1.2 Binomial Theorem for any Index 314 A.1.3 Infinite Geometric Series 316 A.1.4 Exponential Series 318 A.1.5 Logarithmic Series 321 xi 2024-25 Appendix 2: Mathematical Modelling 323 A.2.1 Introduction 323 A.2.2 Preliminaries 323 A.2.3 What is Mathematical Modelling 327 Answers 335 Supplementary Material 357 xii 2024-25