MATHEMATICS

Textbook for Class XII

PART I

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12079 – MATHEMATICS
ISBN 81-7450-629-2
Textbook for Class XII

First Edition ALL RIGHTS RESERVED

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Reprinted mechanical, photocopying, recording or otherwise without the prior
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December 2009, January 2012,
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January 2019, August 2019, Any revised price indicated by a rubber stamp or by a sticker or by any
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January 2021, July 2021 and
November 2021 OFFICES OF THE PUBLICATION
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 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.

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 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.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 December 2005 Research and Training

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 Rationalisation of Content in the Textbook

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.

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 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, the Advisory 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 contain 13 main chapters and two appendices. Each Chapter contain 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/sub
concepts.
§ Motivating and introducing the concepts/sub concepts. Illustrations have been
provided wherever possible.

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 viii

§ Proofs/problem solving involving deductive or inductive reasoning, multiplicity

of approaches wherever possible have been inducted.
§ Geometric viewing / visualisation of concepts have been emphasised 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 sub concepts 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 endeavor 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 the
Advisory Group in Science and Mathematics, for his specific suggestions and advice
towards the improvement of the book from time to time. I, also, thank Prof. 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
Textbook Development Committee

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 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
Arun Pal Singh, Sr. Lecturer, Department of Mathematics, Dayal Singh College,
University of Delhi, Delhi
A.K. Rajput, Reader, RIE, Bhopal, M.P.
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., JNV-Mungeshpur, Delhi
Ram Avtar, Professor (Retd.) and Consultant, DESM, NCERT, New Delhi
R.P. Maurya, Reader, DESM, NCERT, New Delhi
S.S. Khare, Pro-Vice-Chancellor, NEHU, Tura Campus, Meghalaya
S.K.S. Gautam, Professor, DESM, NCERT, New Delhi
S.K. Kaushik, Reader, Department of Mathematics, Kirori Mal College, University
of Delhi, Delhi
Sangeeta Arora, P.G.T., Apeejay School Saket, New Delhi-110017
Shailja Tewari, P.G.T., Kendriya Vidyalaya, Barkakana, Hazaribagh, Jharkhand
Vinayak Bujade, Lecturer, Vidarbha Buniyadi Junior College, Sakkardara Chowk,
Nagpur, Maharashtra
Sunil Bajaj, Sr. Specialist, SCERT, Gurgaon, Haryana
MEMBER – COORDINATOR
V.P. Singh, Reader, DESM, NCERT, New Delhi

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 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, PGT, Jawahar Navodaya Vidyalaya,
Lepakshi, Anantapur, (A.P.); Mrs. S. Kalpagam, PGT, K.V. NAL Campus, 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.

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 Contents
PART I
Foreword iii
Rationalisation of Content in the Textbook v
Preface vii
1. Relations and Functions 1
1.1 Introduction 1
1.2 Types of Relations 2
1.3 Types of Functions 7
1.4 Composition of Functions and Invertible Function 12
2. Inverse Trigonometric Functions 18
2.1 Introduction 18
2.2 Basic Concepts 18
2.3 Properties of Inverse Trigonometric Functions 27
3. Matrices 34
3.1 Introduction 34
3.2 Matrix 34
3.3 Types of Matrices 39
3.4 Operations on Matrices 43
3.5 Transpose of a Matrix 61
3.6 Symmetric and Skew Symmetric Matrices 63
3.7 Invertible Matrices 68
4. Determinants 76
4.1 Introduction 76
4.2 Determinant 76
4.3 Area of a Triangle 82
4.4 Minors and Cofactors 84
4.5 Adjoint and Inverse of a Matrix 87
4.6 Applications of Determinants and Matrices 93

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5. Continuity and Differentiability 104
5.1 Introduction 104
5.2 Continuity 104
5.3 Differentiability 118
5.4 Exponential and Logarithmic Functions 125
5.5 Logarithmic Differentiation 130
5.6 Derivatives of Functions in Parametric Forms 134
5.7 Second Order Derivative 137
6. Application of Derivatives 147
6.1 Introduction 147
6.2 Rate of Change of Quantities 147
6.3 Increasing and Decreasing Functions 152
6.4 Maxima and Minima 159
Appendix 1: Proofs in Mathematics 187
A.1.1 Introduction 187
A.1.2 What is a Proof? 187
Appendix 2: Mathematical Modelling 196
A.2.1 Introduction 196
A.2.2 Why Mathematical Modelling? 196
A.2.3 Principles of Mathematical Modelling 197
Answers 208
Supplementary Material 223

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