Objective mathematics

1245 Pages·2010·21.797 MB·English

by  Sharma, J. K

The Pearson Guide to
OBJECTIVE
MATHEMATICS
For Engineering Entrance Examinations

The Pearson Guide to
OBJECTIVE
MATHEMATICS
For Engineering Entrance Examinations
Third Edition
IIT (Screening Test), AIEEE (CBSE), CEE (Delhi), UPSEAT (UP), CEET
(Haryana), PET (MP), GGS Indraprastha University, Jamia Millia Islamia
University, AMU, PET (Rajasthan), EAMCET (Andhra Pradesh), BCA/BSc
(Hons) Computer Science and other Common Engineering Entrance
Examinations in Orissa, Bihar, Tamil Nadu (TNPCEE), Karnataka
(CET), Assam and West Bengal (WBJEE), MBA and MCA
J K Sharma Anita Khattar Dinesh Khattar
Professor Department of Mathematics Head
Faculty of Management Studies Sarvodaya Vidyalaya Department of Mathematics
University of Delhi Delhi Kirori Mal College
Delhi University of Delhi
Delhi
Chandigarh • Delhi • Chennai
The aim of this publication is to supply information taken from sources believed to be valid and reliable. This is
not an attempt to render any type of professional advice or analysis, nor is it to be treated as such. While much care
has been taken to ensure the veracity and currency of the information presented within, neither the publisher nor
its authors bear any responsibility for any damage arising from inadvertent omissions, negligence or inaccuracies
(typographical or factual) that may have found their way into this book.
Copyright © 2009 Dorling Kindersley (India) Pvt. Ltd.
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owner and the above-mentioned publisher of this book.
ISBN: 978-81-317-2363-0
First Impression
Published by 2009 Dorling Kindersley (India) Pvt. Ltd., licensees of Pearson Education in South Asia.
Head Office: Knowledge Boulevard, a Floor 7th-8(A) Sector 62, Noida, India.
Registered Office: 14 Local Shopping Centre, Panchsheel Park, New Delhi 110 017, India.
Laser typeset by Tantla Compositions Services Pvt. Ltd., Chandigarh
Printed in India by Anand Sons
PREFACE TO THE THIRD EDITION
I take great pleasure in presenting to the readers, the third revised edition of the book. The third
edition of objective Mathematics includes all the basic features of the earlier editions. However, some
typical problems have been added in each chapter. These are designed to test the mental, academic
and creative skills of the students. Questions from recent examination papers of various engineering
entrance examinations have been included in every chapter. Chapters on Mathematical Reasoning and
Mathematical Induction have been included in this new edition.
In preparing this third edition, I am greatly indebted to many teachers as well as students throughout
the country who made constructive criticism and extended valuable suggestions for the improvement of
the book. Any suggestions to ensure further improvement of the book will be greatly acknowledged.
Dinesh Khattar
PREFACE
The last twenty years of preparing/guiding students to success in different engineering entrance examinations has
made us realize that what students really need is a book that imparts the necessary skills to solve questions in
the shortest possible time without compromising on the theoretical aspect of the subject. This book introduces
the concept briefly and then goes straight into solving real-life problems. The graded problems, techniques
to solve them, and huge number of exercises and test papers with solutions will help the student gain the
necessary skills and confidence to take the examinations successfully.
The book is noteworthy in the following aspects:
• Each chapter contains concise definitions and explanations of basic/fundamental principles
and illustrative examples to enable the students to recall the subject matter of the chapter
before attempting to answer the questions.
• Worked out solutions to a large number of problems have been provided. However, we
urge students to attempt the same on their own and only if they fail should they refer to
the solutions provided.
• Problems have been categorised into various types, and working rules to help the students
in solving them have also been provided.
• Large number of problems that have been asked in the competitive examinations in the
recent past have been included in the text.
• To enable the students make a self assessment, practice exercises covering all the topics in
each chapter have been provided at the end.
• In order to help the students decide how much emphasis they should give to various topics
in different competitive examinations, a smart tabLe has been given in the beginning.
This table provides the information about how many questions have been asked from various
topics during 2001–04 in IIT(Screening test), AIEEE (CBSE), CEE (Delhi) and UPSEAT.
• Nine Model Test Papers based on different competitive examinations have been provided
at the end to facilitate students understand the pattern and the type of questions asked in
different examinations.
The answers to almost all unsolved problems have been thoroughly checked. While every effort has
been made to weed out typos, it is possible that a few might have crept in. We will be grateful to the readers
for bringing these errors to our notice. It is earnestly hoped that this book will build a strong foundation for
success in any competitive examination.
We wish to place on record our sincere thanks to our friends and colleagues for their help and suggestions
in planning and preparing the manuscript of this book. We would like to thank Mr. Dinesh Kaushik and Mr.
Pawan Tyagi for their cooperation in typesetting the book.
Suggestions and comments from our esteemed reader to improve the book in content and style are always
welcome and will be greatly appreciated and acknowledged.
thank you for choosing our book.
May you find it stimulating and rewarding!

authors
CONTENTS
Preface to Third Edition v
Preface vi
1. Functions 1
2. Limits 43
3. Continuity and Differentiability 75
4. Differentiation 107
5. Applications of Derivatives 137
6. Indefinite Integration 189
7. Definite Integral and Area 231
8. Differential Equations 305
9. Coordinates and Straight Lines 325
10. Pair of Straight Lines 371
11. Circles 388
12. Conic Sections 443
13. Complex Numbers 515
14. Sequences and Series 565
15. Quadratic Equations and Inequations 621
16. Permutations and Combinations 665
17. Binomial Theorem 695
18. Exponential and Logarithmic Series 729
19. Matrices 739
20. Determinants 763
21. Trigonometric Ratios and Identities 793
22. Trigonometric Equations 829
23. Inverse Trigonometric Functions 847
24. Properties and Solutions Triangles 866
25. Heights and Distances 892
26. Probability 904
27. Statistics 949
28. Vectors 969
29. Three Dimensional Geometry 1021
30. Statics 1047
31. Dynamics 1086
32. Set Theory 1113
33. Numerical Methods 1128
34. Linear Programming 1139
35. Hyperbolic Functions 1146
Model Test Papers 1151
Solutions to Model Test Papers 1187

1
Functions
CHAPTER
xxx Summary of conceptS
function or mapping = set of all image points in Y under the map f.
= f (X) = { f (x) : f (x) ∈ Y; x ∈ X}
Let X and Y be any two non-empty sets and there be corres-
pondence or association between the elements of X and Y such The set Y is also called the co-domain of f. Clearly
that for every element x ∈ X, there exists a unique element f (X) ⊆ Y.
y ∈ Y, written as y = f (x). Then we say that f is a mapping or
function from X to Y, and is written as
f : X → Y such that y = f (x), x ∈ X, y ∈ Y.
Intervals in r
real function
The set of all real numbers lying between two given real num-
If f : X → Y be a function from a non-empty set X to anoth- bers is called an interval in R.
er non-empty set Y, where X, Y ⊆ R (set of all real numbers), Let a and b be any two real numbers such that a < b, then
then we say that f is a real valued function or in short a real we define the following types of intervals.
function.
closed Interval The set of all real numbers x such
features of a mapping f : X → Y that a ≤ x ≤ b is called a closed interval and is denoted by
[a, b]. That is
(i) For each element x ∈ X, there exist is unique element
[a, b] = {x : x ∈ R; a ≤ x ≤ b}
y ∈ Y.
(ii) The element y ∈ Y is called the image of x under the map- open Interval The set of all real numbers x such
ping f. that a < x < b is called an open interval and is denoted by
(iii) If there is an element in X which has more than one image
(a, b) or ]a, b[. That is
in Y, then f : X ∈ Y is not a function. But distinct elements
(a, b) = {x : x ∈ R; a < x < b}
of X may be associated to the same element of Y.
(iv) If there is an element in X which does not have an image
Semi-closed or Semi-open Interval The intervals
in Y, then f : X → Y is not a function.
[a, b) = {x : x ∈ R; a ≤ x < b} and
Throughout this chapter a ‘function’ will mean a ‘real func-
tion’. (a, b] = {x : x ∈ R; a < x ≤ b}
Value of a function are called semi-closed or semi-open intervals. They are also
denoted by [a, b[ and ]a, b] respectively.
The value of a function y = f (x) at x = a is denoted by f (a).
Note that the set R can be thought of as the open interval
It is obtained by putting x = a in f (x).
(– ∞, ∞), so that
Domain and range of a function R = (– ∞, ∞) = {x : x ∈ R; – ∞ < x < ∞}.
Also, the infinite intervals in R can be given by
If f : X → Y be a function, then the set X is said to be the domain
of f and range of f (– ∞, a), (a, ∞), (– ∞, a], [a, ∞).