mathematics for iit jee main and advanced differential calculus algebra trigonometry sanjiva dayal...
TRANSCRIPT
Mathematics for IIT-JEE
Differential Calculus
Algebra
Trigonometry
Volume-1
Second Edition 2015
This book contains theory and a large collection of about 7500 questions and is useful for
students and learners of IIT-JEE, Higher and Technical Mathematics; and also for the students
who are preparing for Standardized Tests, Achievement Tests, Aptitude Tests and other
competitive examinations all over the world
Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
First Edition 2014
Second Edition 2015
Sanjiva Dayal Classes For IIT-JEE Mathematics
Head Office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA.
Phone: +91-512-2581426.
Mobile: +91-9415134052.
Email: [email protected].
Website: www.amazon.com/author/sanjivadayal.
Website: www.sanjivadayal-iitjee.blogspot.com.
ISBN-13: 978-1507743584
ISBN-10: 1507743580
Copyright © Sanjiva Dayal, Author
All rights reserved. No part of this publication may be reproduced, distributed or transmitted
in any form or by any means electronic, mechanical, photocopying and recording or
otherwise, or stored in a database or retrieval system, including, but not limited to, in any
network or other electronic storage or transmission, or broadcast for distance learning
without the prior written permission of the Author.
i
ABOUT THE AUTHOR
Indian Institute of Technology Joint Entrance Examination (IIT-JEE) is considered to be one of
the toughest competitive entrance examinations in the world with success ratio well below 1%. The author of
this book Sanjiva Dayal was selected in IIT-JEE in his first attempt. After completing B.Tech. from IIT Kanpur,
since the last more than 25 years, he has been teaching IIT-JEE Mathematics to the students aspiring for
success in IIT-JEE and other engineering entrance examinations with an objective to provide training of the
highest order by way of specialized and personalized oral coaching, teaching, training, guidance, educational
facilities including tutorials, tests, assignments, reading materials etc. and all such facilities which are helpful to
the students seeking admission in the IIT and other Engineering Institutes of India.
His excellent teaching and result-oriented methods have combined to produce outstanding
performance by his students in the past years. A large number of his students were selected in their First
Attempt with top ranks. Most of his students have secured admission in the IIT and other Engineering
Institutes of India and they are doing well in their engineering education and professional career also.
Conceptual understanding of Mathematics provided by him not only helps his students to do
well in the IIT-JEE and other engineering entrance examinations but also helps them to do well in their
engineering studies where they are pitted against the best students of the country and competition is much
more intense.
ii
ACKNOWLEDGEMENTS
I thank the Almighty God for my very existence and for providing me the opportunity and
resources to write this book.
Mother is the first teacher and father is the second teacher. My respects and deepest gratitude
to my mother Late Pushpa Dayal, my father Late Chandra Mauleshwar Dayal, my grandfather Late
Lakshmeshwar Dayal, my grandmother Late Sarju Dayal, my maternal grandfather Late Ranbir Jang Bahadur
and my maternal grandmother Late Chandra Kishori Bahadur who are no longer in this world but their
blessings are always with me.
I am extremely thankful to all my school teachers and all my IIT Kanpur Professors who have
taught and trained me.
I am also extremely thankful to all my batch mates in IIT Kanpur because studying and training
with such brilliant persons was an amazing experience for me which will be cherished by me throughout my
life.
I am also extremely thankful to all my students because teaching and interacting with such
brilliant minds has been an amazing learning experience for me which will be cherished by me throughout my
life.
I am also extremely thankful to my wife Dr. Vidya Dayal, my sister Ms. Smita Prakash and my
niece Ms. Somya Prakash for their love, cooperation and moral support.
I am also extremely thankful to my childhood friends Mr. Navneet Butan, Mr. Pankaj Agarwal,
Mr. Akhil Srivastava and Mr. Subir Nigam for their love, cooperation and moral support.
I am also thankful to all other persons who have ever helped me directly or indirectly.
I am also thankful to my student Mr. Ashutosh Verma for his help and efforts in publishing this
book.
Place: Kanpur, India.
Date: 27th January, 2015. Sanjiva Dayal
B.Tech. (I.I.T. Kanpur)
Author
iii
PREFACE TO THE SECOND EDITION 2015
"If there is a God, he's a great mathematician."~Paul Dirac
"How is an error possible in mathematics?"~Henri Poincare
"Go down deep enough into anything and you will find mathematics."~Dean Schlicter
About Indian Institute of Technology Joint Entrance Examination (IIT-JEE)
Indian Institute of Technology (IIT) are the premier Engineering Institutes of India.
Career in engineering calls for high level of confidence, motivation and capacity to do hard work besides
intelligence and analytical & deductive thinking. Therefore, in order to select the brightest students for
admission in the Indian Institute of Technology (IIT), National Institute of Technology (NIT) and other
Engineering Institutes of India, the competitive entrance examination Indian Institute of Technology Joint
Entrance Examination (IIT-JEE) is conducted on all India basis every year, along with other state level
engineering entrance examinations. IIT-JEE is considered to be one of the toughest competitive entrance
examinations in the world with success ratio well below 1%. The number of candidates appearing in IIT-
JEE as well as the level of competition has grown tremendously in the past years thus creating a need for
highly specialized course content and result-oriented coaching and training.
About the student
An engineering aspirant should possess sharp mental reflexes, acumen and skill to
understand and master the fundamental concepts of Science and Mathematics; should have the basic
qualities of an ideal student; should be capable of going through the rigorous and tough training schedule
to achieve the standards set by the IIT-JEE and other engineering entrance examinations; and must be
hungry for achievement. With specialized coaching and training, such talented students can achieve
success in IIT-JEE and other engineering entrance examinations.
iv
Purpose of the book
This book is useful for students and learners of IIT-JEE, Higher and Technical
Mathematics; and also for the students who are preparing for Standardized Tests, Achievement Tests,
Aptitude Tests and other competitive examinations all over the world.
Organization of the book
This book is a part of book series which is divided into two volumes, seven parts and
twenty eight chapters as under:-
Volume-1
Part-I: Differential Calculus Chapter-1: Real Functions, Domain, Range Chapter-2: Limit Chapter-3: Continuity Chapter-4: Derivatives Chapter-5: Applications Of Derivatives Chapter-6: Investigation Of Functions And Their Graphs
Part-II: Algebra Chapter-7: Equations And Inequalities Chapter-8: Quadratic Expressions Chapter-9: Progressions Chapter-10: Determinants And Matrices
Part-III: Trigonometry Chapter-11: Trigonometric Identities And Expressions Chapter-12: Trigonometric And Inverse Trigonometric Functions, Equations And Inequalities Chapter-13: Properties Of Triangles
Volume-2
Part-IV: Two Dimensional Coordinate Geometry Chapter-14: Point And Straight Line Chapter-15: Circle Chapter-16: Parabola Chapter-17: Ellipse Chapter-18: Hyperbola
Part-V: Vector And Three Dimensional Geometry Chapter-19: Vector Chapter-20: Three Dimensional Coordinate Geometry
Part-VI: Integral Calculus Chapter-21: Indefinite Integrals Chapter-22: Definite Integral Chapter-23: Differential Equations Chapter-24: Applications Of Calculus
Part-VII: Algebra Chapter-25: Complex Numbers Chapter-26: Binomial Theorem
v
Chapter-27: Permutations and Combinations Chapter-28: Probability
This book contains theory and a large collection of about 7500 questions. In each
chapter, theory is divided into Sections and questions are divided into Question Categories. For each
Section there are one or more corresponding Question Category/ Categories in order to make this book
more readable and more useful for the readers and students.
Using the book
Each Section and its corresponding Question Category/ Categories is the prerequisite of
its following Sections and their corresponding Question Category/ Categories. It is recommended that
readers and students should read a Section and then solve the questions of its corresponding Question
Category/ Categories; and thereafter go to the next Section and so on. This approach is necessary for
proper conceptual development and problem solving skills.
Reader's feedback
All readers and students are requested and welcome to send their feedback, comments,
corrections, suggestions, improvements, mathematical theory and questions on my email ID
[email protected]. This will help in continuous improvement and updation of this book. All readers
and students are invited to join me as friends on Facebook at URL www.facebook.com/sanjiva.dayal and
they are also invited to join my Facebook Group "Mathematics By Sanjiva Dayal".
Place: Kanpur, India.
Date: 27th January, 2015. Sanjiva Dayal
B.Tech. (I.I.T. Kanpur)
Author
vi
CONTENTS
About The Author i
Acknowledgements ii
Preface iii
Part-I: Differential Calculus
Chapter-1: Real Functions, Domain, Range 1.1
Chapter-2: Limit 2.1
Chapter-3: Continuity 3.1
Chapter-4: Derivatives 4.1
Chapter-5: Applications Of Derivatives 5.1
Chapter-6: Investigation Of Functions And Their Graphs 6.1
Part-II: Algebra
Chapter-7: Equations And Inequalities 7.1
Chapter-8: Quadratic Expressions 8.1
Chapter-9: Progressions 9.1
Chapter-10: Determinants And Matrices 10.1
Part-III: Trigonometry
Chapter-11: Trigonometric Identities And Expressions 11.1
Chapter-12: Trigonometric And Inverse Trigonometric Functions, Equations And Inequalities 12.1
Chapter-13: Properties Of Triangles 13.1
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-I
DIFFERENTIAL CALCULUS
CHAPTER-1
REAL FUNCTIONS, DOMAIN, RANGE
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.2
CHAPTER-1
REAL FUNCTIONS, DOMAIN, RANGE
LIST OF THEORY SECTIONS
1.1. Introduction To Mathematics 1.2. Real Number System 1.3. Real Functions 1.4. Defining And Representing Real Functions 1.5. Domain And Range 1.6. Mathematical Operations On Functions
LIST OF QUESTION CATEGORIES
1.1. Defining Analytical Functions 1.2. Domain And Range 1.3. Equivalent Functions 1.4. Simplifying Expressions Containing Powers And Logarithms 1.5. Miscellaneous Questions On Power And Logarithm 1.6. Mathematical Operations On Functions 1.7. Mathematical Operations On Piecewise Functions 1.8. Additional Questions
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.3
CHAPTER-1
REAL FUNCTIONS, DOMAIN, RANGE
SECTION-1.1. INTRODUCTION TO MATHEMATICS
1. The subject of Mathematics Mathematics is the study of relationships among quantities, magnitudes and properties and of logical operations by which unknown quantities, magnitudes and properties may be deduced.
2. The language of Mathematics Mathematics can be regarded as a language which uses symbols to provide an exact theory of logical deduction and inference based on definitions, axioms, postulates and rules for combining and transforming primitive elements into more complex relations and theorems.
3. Definition Definition is a brief precise statement describing or stating clear and unambiguous meaning of a word or an expression. A word or an expression which has a definition is said to be defined otherwise is said to be not defined.
4. Symbol Symbols are various signs and abbreviations used in mathematics to indicate entities, relations or operations.
5. Notation Notation is any system of marks or symbols used to represent entities, processes, facts or relationships in an abbreviated or nonverbal form.
6. Proof Proof is an argument that is used to show the truth of a mathematical assertion.
7. Axiom and postulate Axiom is a basic principle that is assumed to be true without proof. The terms axiom and postulate are often used synonymously.
8. Rule and law Rule is a statement or relationship that is assumed or proved to hold under given conditions. The terms rule and law are often used synonymously.
9. Formula Formula is a set of symbols and numbers that expresses a fact or rule.
10. Theorem Theorem is a proposition or formula that is derivable or provable from a set of axioms and basic assumptions.
11. Standard result Standard result is a result that can be used without giving reason or proof.
12. Method Method is a mathematical procedure that can be used to solve problems of a particular type.
13. Quantity Quantity is anything that is completely characterized by its numerical value.
14. Constant and variable quantities A constant quantity (or constant) is a quantity which has one and only one value within the framework of a given problem. A variable quantity (or variable) is a quantity which can take different values within the framework of a given problem. A quantity can be a constant in one problem and a variable in the other.
15. Parameter
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.4
A parameter is a quantity which can take different constant values within the framework of a given problem. 16. Expression
An expression is a mathematical phrase constructed with numbers, variables and mathematical operations (addition, substraction, multiplication, division, power) formed according to the rules.
SECTION-1.2. REAL NUMBER SYSTEM
1. Natural numbers and Integers i. Set of natural numbers, denoted by N, is
,5,4,3,2,1N . ii. Set of integers, denoted by Z or I, is
,5,4,3,2,1,0 I . iii. IN . iv. The three basic rules i.e. rules of comparison, addition and multiplication are defined for numbers.
Subtraction and division are defined as inverse operations to addition and multiplication respectively. v. Division by zero is not defined. vi. In decimal notation symbol 0 denotes zero, symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 denotes the first nine natural
numbers, ten is designated as 10 and each integer is represented as 01
22
11 10101010 aaaaa n
nn
n
which is written as 0121 aaaaa nn where na
is one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and each of 0a , 1a , 2a , ........., 1na is one of the numbers 0,
1, 2, 3, 4, 5, 6, 7, 8, 9. 2. Rational numbers
i. Concept of fraction and reciprocal. ii. Set of rational numbers, denoted by Q, is
0,, qIqpq
pQ .
iii. QIN .
iv. Any rational number q
p, where the natural number q does not have any prime divisors other than 2 and
5 can be uniquely represented as a terminating decimal fraction which is written as naaaaa ........321
where a is a non-negative integer and each of 1a , 2a , ........., na is one of the numbers 0, 1, 2, 3, 4, 5, 6,
7, 8, 9.
v. Any rational number q
p, where the natural number q contains at least one prime factor different from 2
and 5, can be uniquely represented as a non-terminating repeating decimal fraction which is written as ........321 aaaa where a is a non-negative integer and each of 1a , 2a , ......... is one of the numbers 0, 1,
2, 3, 4, 5, 6, 7, 8, 9 and one or several digits (a block) are repeated in an unchanged order. vi. Any rational number can be represented as a terminating decimal fraction or as a non-terminating
repeating decimal fraction and conversely, any terminating decimal fraction or non-terminating repeating decimal fraction represents a definite rational number.
vii. Sum, difference, product and ratio of two rational numbers is always a rational number. 3. Irrational numbers
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.5
i. Irrational numbers are those numbers which are defined such that they cannot be expressed in q
p form
and can be contained between two rational numbers a and b ab , whose difference ab can be made smaller than any positive number.
ii. Any irrational number x is contained between two rational numbers naaaaa ........321 and
nnaaaaa10
1........321 such that
nnn aaaaaxaaaaa10
1................ 321321 for any number n
whose difference n10
1 can be made smaller than any positive number. The rational number
naaaaa ........321 is called its approximate value by defect and the rational number
nnaaaaa10
1........321 is called its approximate value by excess. Therefore any irrational number can
be approximated by rational numbers with any degree of accuracy. iii. Therefore for every irrational number x, there exists rational numbers id as its approximate values by
defect and there exists rational numbers ie as its approximate values by excess such that
01233210 ................ eeeexdddd .
iv. An irrational number has non-terminating non-repeating decimal fraction form. v. Irrational numbers cannot be expressed as fraction or decimal fraction and so they are denoted by
symbols. vi. Types of irrational numbers:- roots, , e.
4. Real numbers i. Set of real numbers, denoted by R, is the set of all rational numbers and irrational numbers. ii. Sum of a rational number and an irrational number is a irrational number. iii. Product of a non-zero rational number and an irrational number is an irrational number. iv. Between any two real numbers, there are innumerable rational as well as irrational numbers.
5. Geometric representation of Real numbers on number line i. On a given a horizontal straight line, an arbitrary chosen point corresponds to number 0. The portion of
the line to the right of 0 is taken the positive side and to the left of 0 is taken the negative side. This line is known as number line.
ii. There is a one-to-one correspondence between the set of all points of the number line and the set of all real numbers. Thus, every point of the number line is associated with one and only one real number, different points of the number line are associated with different numbers, there is not a single number which would not be associated with some point of the number line.
iii. "A real number" is also called "a point". 6. Definition of power
i. Powers with 0 or 1 as base or exponent a. 00 is not defined b. 0,00 xx
c. x0 is not defined for 0x d. 0,10 aa
e. 11 x
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.6
f. aa 1 ii. Negative exponent is defined as
0,1
xa
ax
x .
iii. Positive integer exponent is defined as timesnaaaaan .
iv. Positive non-integer rational exponent
a. 21
na n is defined as the number such that aa
n
n
1
. na1
is also written as n a and is called the
nth root of a.
b. By definition, qpq
p
aa1
. c. Even roots of negative numbers are not defined but odd roots of negative numbers are defined
v. Positive irrational exponent a. If x is a positive irrational number and rational numbers id approximate the number x by defect and
rational numbers ie approximate the number x by excess, i.e.
01233210 ................ eeeexdddd ; then xa is the number which is contained
between any number ida and iea and such a number exists and is unique. b. Irrational exponent power of negative numbers are not defined
vi. Binomial theorem, Binomial series, Exponential series 7. Logarithm
i. Definition of natural logarithm If 0x , then the real number denoted by xelog is the logarithm of the number x to the base e if
xe xe log . xelog is also called the natural logarithm of the number x .
a. xelog is also denoted as xln .
b. 01ln and 1ln e . c. Log of negative numbers and 0 are not defined. d. Logarithmic series.
e. axxax eea lnln .
f. Series of xa . ii. Definition of xalog
If 0x , 0a and 1a , then the real number denoted by xalog is the logarithm of the number x to
the base a if xa xa log .
a. Base of log must be positive and not 1, i.e. 0a , 1a b. 01log a and 1log aa .
c. a
xxa ln
lnlog
d. x10log is also denoted as xlog .
8. Concept of 'Plus infinity (+)' and 'Minus infinity (-)' i. If a quantity 'increases without bound' then it is said that it 'approaches '. If a quantity 'decreases
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.7
without bound' then it is said that it 'approaches '. is also denoted as . ii. and are not numbers and hence mathematical operations cannot be used with them.
9. Set notation for representing sub-sets of real numbers i. ba, denotes the set of all real number x satisfying the condition bxa .
ii. ba, denotes the set of all real number x satisfying the condition bxa .
iii. ba, denotes the set of all real number x satisfying the condition bxa .
iv. ba, denotes the set of all real number x satisfying the condition bxa .
v. ,a denotes the set of all real number x satisfying the condition ax .
vi. ,a denotes the set of all real number x satisfying the condition ax .
vii. b, denotes the set of all real number x satisfying the condition bx .
viii. b, denotes the set of all real number x satisfying the condition bx .
ix. , or R denotes the set of all real numbers.
x. a denotes the set of real number a. xi. Any sub-set of real number can be written using above mentioned sets alongwith set union and
subtraction 10. Interval
i. An interval is any of the set of real numbers ba, , ba, , ba, , ba, , ,a , ,a , b, ,
b, and , .
ii. Intervals ba, are called closed intervals.
iii. Intervals ba, , ba, , ,a and b, are called semi-open or semi-closed intervals.
iv. Intervals ba, , ,a , b, and , are called open intervals. SECTION-1.3. REAL FUNCTIONS
1. Calculus of real quantities i. For the study of calculus, all quantities are considered within the set of real numbers.
2. Concept and Definition of real function i. A real function, denoted by f, is a rule which relates a real number x to another real number )(xf such
that:- a. for at least one value of x, )(xf should have a real value; b. for each value of x for which )(xf has a value, )(xf should have only one value.
3. One-one (injective) and many-one functions i. A function is said to be a One-one (injective) function if for any two different values of x, )(xf has
different value; i.e. )()(,, 212121 xfxfxxxx . ii. A function is said to be a Many-one function if there exists two different values of x for which )(xf has
same value; i.e. )()(,, 212121 xfxfxxxx . SECTION-1.4. DEFINING AND REPRESENTING REAL FUNCTIONS
1. Methods to define and represent real functions i. Graphical method
a. The graph of a function )(xf is the set of points on a rectangular coordinate axes having coordinates
)(, xfx .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.8
b. Conventions in graph drawing Graph should contain all the properties of a function End-point marking Single point exclusion Solid and dotted lines
ii. Tabular method a. Values of x and corresponding values of )(xf are tabulated in horizontal or vertical table.
iii. Analytical method a. )(xf is expressed as an expression containing x.
2. Basic functions and their graphs
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.9
i. Constant functions axf )(
ii. Identity function xxf )(
iii. Power functions
a. axxf )( , 1a , q
pa (p even, q odd); eg. 2x , 4x , 3
4
x , etc.
b. axxf )( , 1a , q
pa (p odd, q odd) ; eg. 3x , 5x , 3
5
x , etc.
a
0 x
0 x
0 x
0 x
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.10
c. axxf )( , 1a , q
pa (p odd, q even) or a is irrational; eg. 2
3
x , 2x , etc.
d. axxf )( , 10 a , q
pa (p even, q odd) ; eg. 3
2
x , etc.
e. axxf )( , 10 a , q
pa (p odd, q odd) ; eg. 3 x , 5
3
x , etc.
f. axxf )( , 10 a , q
pa (p odd, q even) or a is irrational; eg. x , 2
1
x , etc.
0 x
0 x
0 x
0 x
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.11
g. axxf )( , 0a , q
pa (p even, q odd) ; eg.
2
1
x, etc.
h. axxf )( , 0a , q
pa (p odd, q odd) ; eg.
x
1,
3
1
x, etc.
i. axxf )( , 0a , q
pa (p odd, q even) or a is irrational; eg. 2
1
x , 2x , etc.
iv. Exponential functions a. xaxf )( , 1a ; eg. xe , x2 , x10 , etc.
b. xaxf )( , 10 a ; eg. x5.0 , etc.
0 x
0 x
0 x
0 x
1
ax
0 x
1
ax
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.12
v. Logarithmic functions a. xxf alog)( , 1a ; eg. xln , x2log , xlog , etc.
b. xxf alog)( , 10 a ; eg. x5.0log , etc.
vi. Trigonometric functions a. xxf sin)(
b. xxf cos)(
0 x 1
logax
0 x 1
logax
2
2 x
1
0
1
23
2 2
x 0
1
1
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.13
c. xxf tan)(
d. ecxxf cos)(
e. xxf sec)(
f. xxf cot)(
2
2 2
3 x 0
2
2 2
3
x
2 0
1
1
23
2
23 2
x 0
1
1
2
2 2
3
x 0 2
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.14
vii. Inverse trigonometric functions a. xxf 1sin)(
b. xxf 1cos)(
c. xxf 1tan)(
d. xecxf 1cos)(
2
2
x 0 1
1
2
x 0 1 1
2
0 x 1 1
2
2
2
x 0
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.15
e. xxf 1sec)(
f. xxf 1cot)(
viii. Hyperbolic functions
a. 2
sinh)(xx ee
xxf
b. 2
cosh)(xx ee
xxf
2
0 x 1 1
2
x 0
x 0
0 x
1
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.16
c. xx
xx
ee
eexxf
tanh)(
3. Methods to define functions from basic/predefined functions i. Defining analytical functions by taking part (or parts) of basic/ predefined functions (Piecewise
functions) a. If xf1 is a basic/predefined function and a set RX 1 then a function xf may be defined as
11 , Xxxfxf .
b. If xf1 , xf2 , xf3 , ……….., xf n are basic/predefined functions and sets
RXXXX n ...,..........,,, 321 then a function xf may be defined as
11 , Xxxfxf
22 , Xxxf
33 , Xxxf
nn Xxxf , .
ii. Defining analytical functions by adding/ multiplying two basic/ predefined functions a. If xf1 and xf2 are basic/predefined functions then a function xf may be defined as
xfxfxf 21 .
b. If xf1 and xf2 are basic/predefined functions then a function xf may be defined as
xfxfxf 21 . c. Negative of a function d. Subtraction of two functions
iii. Defining analytical functions as Composite function of two basic/ predefined functions a. If xf1 and xf2 are basic/predefined functions then a function xf may be defined as
xffxf 21 .
b. xff 21 is also written as xoff 21 .
c. Defining xf
1 type of functions
d. Division of two functions
e. Defining axf and xfa type of functions
f. Defining xgxf type of expo-logarithmic functions
x 0
1
1
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.17
4. Explicit functions i. If x and y are related as xy , where x denotes an expression containing x, then y is said to be an
explicit function of x. 5. Implicit functions
i. If x and y are related by an equation 0, yx , where yx, denotes an expression containing x and y, then y is said to be an implicit function of x.
6. Parametric functions i. If x and y are related as ty , tx where t and t denote expressions containing t, then y is
said to be a parametric function of x and t is called parameter. 7. Other ways of defining functions
i. Height of falling body as a function of time ii. Rotation of fan as a function of voltage
8. Standard functions x and xsgn and their graphs
i. 0, xxx
0, xx
ii. 0,1 xxsgn 0,0 x 0,1 x
9. Graphs of linear functions i. 0,)( abaxxf
0 x
x 0
1
1
0 x
b
ab
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.18
ii. 0,)( abaxxf
10. How function is different from it’s ways of representations i. Every graph is not a function ii. A function may not be represented graphically iii. Every tabular data is not a function iv. A function may not be represented completely by tables v. Every expression is not a function vi. A function may not have an expression vii. Two (or more) different expressions may represent the same function viii. A function may involve more than one expression
11. Graphical representation shows all the values and properties of a function i. Checking whether a graph is of a function or not- Vertical line test
a. If a vertical line cuts the curve at two (or more) points then the graph is not a function, otherwise the graph is a function.
ii. Checking whether a graph is of one-one (injective) or many-one function- Horizontal line test a. If a horizontal line cuts the curve at two (or more) points then the function is a many-one function,
otherwise the function is a one-one (injective) function. iii. Checking basic functions and standard functions for being one-one (injective) or many-one functions
axf )( ; many-one function.
xxf )( ; one-one (injective) function.
axxf )( , 1a , q
pa (p even, q odd) ; many-one function.
axxf )( , 1a , q
pa (p odd, q odd) ; one-one (injective) function.
axxf )( , 1a , q
pa (p odd, q even) or a is irrational; one-one (injective) function.
axxf )( , 10 a , q
pa (p even, q odd) ; many-one function.
axxf )( , 10 a , q
pa (p odd, q odd) ; one-one (injective) function.
axxf )( , 10 a , q
pa (p odd, q even) or a is irrational; one-one (injective) function.
axxf )( , 0a , q
pa (p even, q odd) ; many-one function.
axxf )( , 0a , q
pa (p odd, q odd) ; one-one (injective) function.
0 x ab
b
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.19
axxf )( , 0a , q
pa (p odd, q even) or a is irrational; one-one (injective) function.
xaxf )( ; one-one (injective) function.
xxf alog)( ; one-one (injective) function.
xxf sin)( ; many-one function.
xxf cos)( ; many-one function.
xxf tan)( ; many-one function.
ecxxf cos)( ; many-one function.
xxf sec)( ; many-one function.
xxf cot)( ; many-one function.
xxf 1sin)( ; one-one (injective) function.
xxf 1cos)( ; one-one (injective) function.
xxf 1tan)( ; one-one (injective) function.
xecxf 1cos)( ; one-one (injective) function.
xxf 1sec)( ; one-one (injective) function.
xxf 1cot)( ; one-one (injective) function.
2
sinh)(xx ee
xxf
; one-one (injective) function.
2
cosh)(xx ee
xxf
; many-one function.
xx
xx
ee
eexxf
tanh)( ; one-one (injective) function.
xxf )( ; many-one function.
xsgnxf )( ; many-one function. SECTION-1.5. DOMAIN AND RANGE
1. Properties of functions: Domain, Range, Limit, Continuity, Derivatives 2. Definition of Domain of a function
Domain is the set of all real values of x for which )(xf attains real value. Domain and Domain R . 3. Definition of Range of a function
Range is the set of all real values attained by )(xf . Range and Range R . 4. Reading Domain and Range of a function from it’s graph 5. Reading Domain and Range of a function from it’s table 6. Domain and Range of Basic functions and standard functions
i. axf )( ; Domain: R ; Range: a . ii. xxf )( ; Domain: R ; Range: R .
iii. axxf )( , 1a , q
pa (p even, q odd) ; Domain: R ; Range: ,0 .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.20
iv. axxf )( , 1a , q
pa (p odd, q odd) ; Domain: R ; Range: R .
v. axxf )( , 1a , q
pa (p odd, q even) or a is irrational; Domain: ,0 ; Range: ,0 .
vi. axxf )( , 10 a , q
pa (p even, q odd) ; Domain: R ; Range: ,0 .
vii. axxf )( , 10 a , q
pa (p odd, q odd) ; Domain: R ; Range: R .
viii. axxf )( , 10 a , q
pa (p odd, q even) or a is irrational; Domain: ,0 ; Range: ,0 .
ix. axxf )( , 0a , q
pa (p even, q odd) ; Domain: ]0[R ; Range: ),0( .
x. axxf )( , 0a , q
pa (p odd, q odd) ; Domain: ]0[R ; Range: ]0[R .
xi. axxf )( , 0a , q
pa (p odd, q even) or a is irrational; Domain: ),0( ; Range: ),0( .
xii. xaxf )( ; Domain: R ; Range: ),0( .
xiii. xxf alog)( ; Domain: ),0( ; Range: R .
xiv. xxf sin)( ; Domain: R ; Range: ]1,1[ . xv. xxf cos)( ; Domain: R ; Range: ]1,1[ .
xvi. xxf tan)( ; Domain: InnR
,
2)12(
; Range: R .
xvii. ecxxf cos)( ; Domain: InnR ],[ ; Range: ,11, .
xviii. xxf sec)( ; Domain: InnR
,
2)12(
; Range: ,11, .
xix. xxf cot)( ; Domain: InnR ],[ ; Range: R .
xx. xxf 1sin)( ; Domain: ]1,1[ ; Range:
2,
2
.
xxi. xxf 1cos)( ; Domain: ]1,1[ ; Range: ],0[ .
xxii. xxf 1tan)( ; Domain: R ; Range:
2,
2
.
xxiii. xecxf 1cos)( ; Domain: ,11, ; Range:
2,00,
2
.
xxiv. xxf 1sec)( ; Domain: ,11, ; Range:
,
22,0 .
xxv. xxf 1cot)( ; Domain: R ; Range: ),0( .
xxvi. 2
sinh)(xx ee
xxf
; Domain: R ; Range: R .
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.21
xxvii. 2
cosh)(xx ee
xxf
; Domain: R ; Range: ,1 .
xxviii. xx
xx
ee
eexxf
tanh)( ; Domain: R ; Range: )1,1( .
xxix. xxf )( ; Domain: R ; Range: ,0 .
xxx. xsgnxf )( ; Domain: R ; Range: 101 . 7. Domain of defined analytical functions
i. Theorem: If 1D is the domain of the function 1f and 2D is the domain of the function 2f and the
function xfxfxf 21)( , then the domain of the function 21, DxDxxf , i.e. domain of
)(xf is 21 DD .
ii. Theorem: If 1D is the domain of the function 1f and 2D is the domain of the function 2f and the
function xfxfxf 21)( , then the domain of the function 21, DxDxxf , i.e. domain of
)(xf is 21 DD .
iii. Theorem: If 1D is the domain of the function 1f and 2D is the domain of the function 2f and the
function xffxf 21)( , then the domain of the function 122 ,| DxfDxxf . iv. Domain conditions
8. Domain of piecewise functions If 1D is the domain of the function 1f and 2D is the domain of the function 2f and the function
11 ,)( Xxxfxf
22 , Xxxf ,
then the Domain of 2211 XDXDf . 9. Applications of Domain 10. Range of defined analytical functions 11. Applications of Range 12. Equivalent functions
Two functions )(xf and )(xg are said to be equivalent functions, written as xgxf )( , if they are
defined in different ways but they have the same domain and xgxf )( for all values of x in their domain;
otherwise )(xf and )(xg are said to be non-equivalent functions, written as )(xf ≢ )(xg .
SECTION-1.6. MATHEMATICAL OPERATIONS ON FUNCTIONS
1. Properties of modulus i. 0, xxx
0, xx
ii. xx
iii. 22xx
iv. 2xx
v. 0x
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.22
vi. yxxy
vii. y
x
y
x
viii. yxyx , if 0,0 yx or 0,0 yx
ix. yxyx
x. yxyxyx ~
2. Properties of power i. 0,10 aa
ii. 0,00 xx
iii. aa 1 iv. 11 a
v. x
x
aa
1
vi. xxx baab
vii. x
xx
b
a
b
a
viii. yxyx aaa
ix. yxy
x
aa
a
x. xyyx aa 3. Properties of logarithm
i. 0,1,0,log xaaxa xa
ii. 1,0,01log aaa
iii. 1,0,1log aaaa
iv. 0,0,1,0,logloglog yxaaxyyx aaa
v. 0,1,0,logloglog xyaayxxy aaa
vi. 0,0,1,0,logloglog yxaay
xyx aaa
vii. 0,1,0,logloglog y
xaayx
y
xaaa
viii. 0,1,0,loglog xaaxbx ab
a
ix. 0,1,0,log2log 2 xaaxx aa
x. 1,0,0,1,0,log
loglog ccbaa
a
bb
c
ca
xi. 1,0,1,0,log
1log bbaa
ab
ba
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.23
xii. 0,0,1,0,loglog bxaaxx b
aa b
4. Mathematical operations on functions i. To find the value of a given function )(xf at a given point a.
ii. Given two functions )(xf and )(xg , to determine the functions xgxf )( , xgxf )( , xgxf )( ,
xg
xf )( and xgf .
5. Mathematical operations on graphical functions 6. Mathematical operations on tabular functions 7. Mathematical operations on single expression analytical functions 8. Mathematical Operations On Piecewise Functions
i. To find the value of a given piecewise function )(xf at a given point a. ii. Given a single expression function )(xf and a piecewise function )(xg , to determine the functions
xgxf )( , xgxf )( , xgxf )( , xg
xf )( and xgf .
iii. Given two piecewise functions )(xf and )(xg , to determine the functions xgxf )( , xgxf )( ,
xgxf )( , xg
xf )( and xgf .
iv. Given a single expression function )(xf and a piecewise function )(xg , to determine the function
xgf . v. Given a single expression function )(xf and a piecewise function )(xg , to determine the function
xfg .
vi. Given two piecewise functions )(xf and )(xg , to determine the function xgf .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.24
EXERCISE-1
CATEGORY-1.1. DEFINING ANALYTICAL FUNCTIONS
1. Define the following analytical functions using basic functions:- i. 1,lnsin xxxxf
ii. 0,cos2 xxxxf
iii. x
iv. xsgn
v. 0,sin xxexf x
0,3 xx
vi. xx
xxf
coshln
sin
vii. 0,1 2
xx
exf
x
0,ln
ln2
1
x
xe
xxx
viii. xxxf lnsin)ln(sin
ix. 3
sin1
x
xxf
x. xxxf cos2
xi. xxxf
xii. xxxf cossin
xiii. x
x
xxf
1
1
xiv. xxxf 11 sinlnsin
xv.
xxxxf 11 tanlncos
xvi. xxf x coslogsin
CATEGORY-1.2. DOMAIN AND RANGE
2. Find domain (write conditions only):-
i. 2)1ln(
1)(
x
xxf
ii. )(logsin)( 31 xxf
iii. xxxf lnsinsinln)( 1
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.25
iv.
xxf
sin24
3cos)( 1
v. )(loglog)( 32 xxf
vi. ))(lnsec1ln(
)(1 x
xxf
vii. xxxxf 1524)(
viii. )}nln{cosh(si
1)(
xxf
ix. )(sin2
1)(
1 xcosecxxf
x. )2(cos2)( 1cos
11 xxxf
xi.
xexf
1tan1
1tan)(
xii. 43 cossin)( xxxf
xiii. xxxf )(
xiv. xxxf1cossin)(
xv. x
x
xxf
1
1)(
xvi. xxf x coslog)( sin
3. Find domain of the function
2,3
1)(
x
xxf
11,1
xx
2,1
1
x
x. {Ans. ,33,21,00,12, }
4. If 3,1
1)(
2
x
x
xxf
3,3
sin
x
x
x,
for what values of x is the function not defined? {Ans. 1, 3} CATEGORY-1.3. EQUIVALENT FUNCTIONS
5. Are the following functions equivalent?
i. x
xxf )( and 1)( x {Ans. No}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.26
ii.
x
xf11
)( and xx )( {Ans. No}
iii. xx
xf11
)( and 0)( x {Ans. No}
iv. 2)( xxf and xx )( {Ans. No}
v. 2)( xxf and xx )( {Ans. No}
vi. 2)( xxf and xx )( {Ans. Yes}
vii. 2log)( xxf and xx log2)( {Ans. No} viii. )3log()2log()( xxxf and )3)(2log()( xxx {Ans. No}
ix. xxf cot)( and x
xtan
1)( {Ans. No}
x. xxxf 22 cossin)( and 1)( x {Ans. Yes}
xi. ecx
xfcos
1)( and xx sin)( {Ans. No}
CATEGORY-1.4. SIMPLIFYING EXPRESSIONS CONTAINING POWERS AND LOGARITHMS
6. 9log
2
13
3
. {Ans. 3
1}
7. 5log2 22 . {Ans. 5
4}
8. nm loglog10 . {Ans. mn }
9. 15log 222 . {Ans. 3 225 }
10. 110log 8.58.5 . {Ans. 58}
11. 3
1121log 3
2
8
. {Ans. 242}
12. 4log25.0 . {Ans. 2}
13. 25.0tanlog . {Ans. 0}
14. 81loglog 32 . {Ans. 2}
15. 9log
4
36log3log
1
795 32781
. {Ans. 890} 16. 2log5log 33 52 . {Ans. 0}
17. 27log5log 253 . {Ans. 2
3}
18. 9log27log 279 . {Ans. 6
5}
19. 9log8log7log6log5log4log 876543 . {Ans. 2}
20. 15log14log4log3log2log 1615543 . {Ans. 4
1}
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.27
21. 6216log6 . {Ans. 2
7}
22. 2log2256logloglog2422 . {Ans. 5}
23.
9log2
13log2
5
5
27
1
. {Ans.
2
3
16
3
3
13}
24. 8081
log32425
log51516
log7 . {Ans. 2log }
25. 55log327log2log16log 534
83
2 . {Ans. 6
17 }
26.
37
3
4
3
3249
7log
2128
8log
27
33log
44
1log . {Ans.
12
103 }
27. 3
2
14
8
1
3
13
1 2128log32log9log9log3
. {Ans. 12
139 }
28.
27
64log27log27log27log
2
33
133 . {Ans. 6}
29. 3
3
13
2
1
53
2 39log2
4log
2
1624log
. {Ans.
10
31 }
30.
5
22log
5
15log50
5
1log 32.06.0
34.0 . {Ans.
3
4}
31. 324log55log5log135
2
5 35
. {Ans. 4
15}
32. 4
2222log22log . {Ans.
2
3}
33.
42
42
1
3 2
2log
3
3log 4 . {Ans.
22
1}
34. 55log55log5
1log
55
15
. {Ans. 6 }
35. 25log25log2
15log2 2
554
5 . {Ans. 4
1}
36. 22log392
12
4
12log2
15log 1625
. {Ans. 9}
37. 16logloglog 283 . {Ans. 12log3 }
38. 64logloglog 248 . {Ans.
3
113loglog 22 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.28
39. 81logloglog 324 . {Ans. 2
1}
40.
52log62
1loglog 2
223 . {Ans. 2}
41.
3
2.0
5
1255
25log5log25log125log . {Ans. 8
9}
42. 52
16
1
4
1
2
1 8log4
1log2
2
1log6
4
1log
. {Ans.
2
5}
43. 23log4log1 23 23 . {Ans.
4
51}
44. 13log2log32
34 5.14 . {Ans. 126}
45. 12log23log3 74 72 . {Ans.
3
828 }
46. 5log33log
2
15log1 82
8 416
. {Ans.
3
4
5
1675 }
47. 16log
2
122log42log2 3
813 39
. {Ans. 384}
48.
20log2
3
8log1.0log5.11.0log2 1.01.0 . {Ans.
104
3}
49.
45
77 log6log9log2
1
54972 . {Ans. 2
45}
50. 676 log2log1
3
3 49369log
81log . {Ans.
18
325}
51.
2log28log2
14
24
2 10022log2log
16log. {Ans. 96}
52. 27
33log2log5log9log2
1
710
. {Ans. 5
294}
53. 81log8log36log3log 4336 . {Ans. 16}
54.
2
8log102log
5
1log72
5
23
252 . {Ans. 10 }
55. 10log64log
3
132log
5
2333
3
. {Ans. 10}
56. 4log32log92
12.02.02.0 . {Ans. 22 }
57. 5log210log25log25log3 22222
. {Ans. 10
1}
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.29
58. 3log5
1
533log300log5log2log . {Ans. 5
1
53 }
59.
3log4log
3log12log
31
log27log108
3
36
35.08 . {Ans. 0}
60. 37
37
3
37
22
32
3
2log
3
2log
7log32
log2
7log32
log6 . {Ans. 0}
61. 81log2log31
2 49
5.2log13log
5
5
. {Ans. 1}
62. 7log6log5log4log3log2log 876543 . {Ans. 3
1}
CATEGORY-1.5. MISCELLANEOUS QUESTIONS ON POWER AND LOGARITHM
63. Given that a2log6 , find 72log24 in terms of a. {Ans. 12
2
a
a}
64. Given that a8log36 , find 9log36 in terms of a. {Ans. 3
23 a}
65. Given that a125log4 , find 64log in terms of a. {Ans. a23
18
}
66. Given that a3log100 and b2log100 , find 6log5 in terms of a and b. {Ans.
b
ba
21
2
}
67. Given that a15log6 and b18log12 , find 24log25 in terms of a and b. {Ans. baba
b
212
5
}
68. If 10ln16log625log2ln a , then find the value of a. {Ans. 5}
69. Prove that 1logloglog cba acb .
70. Prove that 1loglogloglog dcba adcb .
71. If xaba log , then evaluate abblog in terms of x . {Ans. 1x
x}
72. Prove that bn
na
ab
a log1loglog
.
73. Prove that xx
xxx
ba
baab loglog
logloglog
.
74. If abba 722 , prove that baba loglog21
31
log .
75. Show that nnnn !434332 log
1
log
1
log
1
log
1 .
76. If !1983n , compute the sum nnnn 1983432 log
1
log
1
log
1
log
1 . {Ans. 1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.30
77. If xaay log1
1
and yaaz log1
1
, prove that zaax log1
1
.
78. If ba
c
ac
b
cb
a
logloglog, prove that 1 cba cba .
79. If qp
z
pr
y
rq
x
logloglog, prove that rqpqpprrq zyxzyx .
80. Prove that n
nnnnnnnnn
abc
cbaaccbba log
logloglogloglogloglogloglog .
81. If 0a , 0c , acb , 1a , 1c , 1ac and 0n , prove that nn
nn
n
n
cb
ba
c
a
loglog
loglog
log
log
.
82. Prove that if cbx bc loglog , acy ca loglog , baz ab loglog then 4222 zyxxyz .
CATEGORY-1.6. MATHEMATICAL OPERATIONS ON FUNCTIONS
83. If cbxaxxf 2)( , find )0(f , )1(f , )1(f , )(af and )(bf . {Ans. c , cba , cba ,
caba 3 , cbab 22 }
84. If x
xxf
1)(
and 1)( 2 xxg , find )1(fg and )1(gf . {Ans.
2
1,1 }
85. If 132 xxxf and 32 xxg . Find fog, gof, fof and gog. {Ans. 164 2 xxxfog ;
162 2 xxxgof ; 515146 234 xxxxxfof ; 94 xxgog }
86. Let 2xxf and 12 xxg . Find fog and gof. Also show that goffog . {Ans.
144 2 xxxfog ; 12 2 xxgof }
87. If 22 xxf and 1
x
xxg . Find fog and gof. {Ans.
22
1
243
x
xxxfog ;
1
22
2
x
xxgof }
88. If 232 xxxf , find xff . {Ans. xxxxxff 3106 234 }
89. Find )(x and )(x if 1)( 2 xx and .3)( xx {Ans. 12 2
3,13 xx }
90. If xxf sin and 2xxg , then find xfog and xgof . {Ans. 2sin xxfog , 2sin xxgof }
91. If xxxf cos)( and 21
)(x
xxg
, then find xfog and xgof . {Ans.
22 1cos
1 x
x
x
xxfog
;
xx
xxxgof
22 cos1
cos
}
92. If 2
2 1)(
xxxf , prove that
xfxf
1)( .
93. If x
xxf1
, prove that
xfxfxf
1333 .
94. Given the function )0(,2
)(
aaa
xfxx
, show that )()(2)()( yfxfyxfyxf .
95. Given the functions xf and xg .
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.31
x xf x xg
1 3 1 2 2 1 2 4 3 4 3 1 4 2 4 3
i. Determine the function xgxfxgxfxh 22)( ;
ii. Determine the function xgfx )( ;
iii. Determine the function xfgx )( . iv. Which of the above functions are injective? {Ans. i.
x xh
1 7 2 13 3 13 4 7
ii. x x
1 1 2 2 3 3 4 4
iii. x x
1 1 2 2 3 3 4 4
iv. x and x are injective} CATEGORY-1.7. MATHEMATICAL OPERATIONS ON PIECEWISE FUNCTIONS
96. Given the function 01,13)( xxf x
xx
0,2
tan
6,22
xx
x .
Find:- i. )1(f ; {Ans. 2 }
ii.
2
f ; {Ans. 1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.32
iii.
3
2f ; {Ans. 3 }
iv. )4(f ; {Ans. 7
2}
v. )6(f . {Ans. 17
3}
97. Let 1)( xf , if x is a rational number = 0, if x is a irrational number. Find:-
i.
3
1f ; {Ans. 1}
ii. )7(f ; {Ans. 0}
iii.
7
22f ; {Ans. 1}
iv. f ; {Ans. 0}
v. ef ; {Ans. 0}
vi. )4327.1(ff ; {Ans. 1}
vii. )3(ff . {Ans. 1}
98. Given 0,)( 2 xxxf 0, xx
and 1,1
)( xx
xg
1,1 x . Determine the following functions:- i. xgxxh )( ;
ii. xgxfx )( ;
iii. xgxfx )( . {Ans. i. 1,1)( xxh 1, xx ;
ii. 1,1
)( 2 xx
xx
10,12 xx 0,1 xx ; iii. 1,)( xxx
10,2 xx 0, xx }
99. Given 12,1)( xxxf 1,1 xx
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.33
and 0,)( xxxg 20,1 xx .
i. Determine the function xgxfx )( ;
ii. Determine the function xgxx )( ;
iii. Plot graph of xf and write its domain and range;
iv. Plot graph of xg and write its domain and range; v. Plot graph of )(x and write its domain and range; vi. Plot graph of )(x and write its domain and range; vii. Which of the above functions are injective? {Ans. i. 02,1)( xx 10,0 x 21,2 xx ii. 02,1)( xxx 10,1 xx 21,13 xx
vii. xg and )(x are injective}
100. Determine the function xsgnxxf )( . {Ans. xxf }
101. Determine the following functions:- i. xxf sin1 ;
ii. xxf sgnsin 12
;
iii. 2
3 xxf ;
iv. 24 sgn xxf ;
v. xexf sgn5 .
{Ans. i. 0,sin1 xxxf 0,sin xx
ii. 0,22 xxf
0,0 x
0,2
x
iii. 23 xxf
iv. 0,14 xxf 0,0 x
v. 0,5 xexf
0,1 x
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.34
0,1
xe
}
102. Determine the following functions:- i. 11 xxf ;
ii. 122 xxf ;
iii. xxf ln3 ;
iv. xexf 4 ;
v. 3sgn5 xxf ;
vi. 23sgn6 xxf ;
vii. xxf lnsgn7 ;
viii. xexf sgn8 .
{Ans. i. 1,11 xxxf 1,1 xx
ii. 2
1,122 xxxf
2
1,12 xx
iii. 1,ln3 xxxf
10,ln xx
iv. xexf 4
v. 3,15 xxf
3,0 x 3,1 x
vi. 3
2,16 xxf
3
2,0 x
3
2,1 x
vii. 1,17 xxf
1,0 x 10,1 x
viii. 18 xf }
103. Given 02,2)( xxxf 40,1 xx .
i. Determine the function
2
xfx and plot its graph and write its domain and range.
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.35
ii. Determine the function xfx 2 and plot its graph and write its domain and range.
iii. Determine the function xfxh and plot its graph and write its domain and range.
iv. Determine the function 11 xfxf and plot its graph and write its domain and range.
v. Determine the function 122 xfxf and plot its graph and write its domain and range. vi. Which of the above functions are one-one functions? {Ans.
i. 04,22
xx
x
80,12
xx
ii. 01,22 xxx 20,12 xx
iii. 04,1 xxxh 20,2 xx
iv. 13,31 xxxf 31, xx
v. 2
1
2
3,322 xxxf
2
3
2
1,2 xx
vi. None of these functions are one-one function} 104. Given 1,)( 2 xxxf
21,1
xx
.
Determine the functions 2xfxg . {Ans.
12,1
2 x
xxg
11,4 xx
21,1
2 x
x}
105. Given 01,1)( xxxf
10, xx 01,1)( xxxg
10,1 xx .
i. Determine the function xgfxf )(1 and plot its graph and write its domain and range.
ii. Determine the function xfgxf )(2 and plot its graph and write its domain and range.
iii. Determine the function xffxf 213 )( and plot its graph and write its domain and range.
iv. Determine the function xffxf 124 )( and plot its graph and write its domain and range.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.36
v. Determine the function xfxf )(5 and plot its graph and write its domain and range.
vi. Determine the function xfxf )(6 and plot its graph and write its domain and range.
vii. Determine the function xfsgnxf )(7 and plot its graph and write its domain and range.
viii. Determine the function xsgnfxf )(8 and plot its graph and write its domain and range.
ix. Which of the above functions are injective functions? {Ans. i. 01,1 xxxf 10,1 xx 1,1 x
ii. 01,2 xxxf 10,1 xx
iii. 1,13 xxf
01,1 xx 10,1 xx 1,1 x
iv. 01,14 xxxf 10,1 xx
v. 01,5 xxxf
0,1 x 10, xx
vi. 01,16 xxxf
10, xx
vii. 1,07 xxf
01,1 x 10,1 x
viii. 0,08 xxf
0,1 x 0,1 x
ix. xf is injective function.} 106. Given 20,1)( xxxf
32,3 xx .
i. Determine the function ))(( xffxg and plot its graph and write its domain and range.
ii. Determine the function xgfxh and plot its graph and write its domain and range.
iii. Determine the function xhxfx and plot its graph and write its domain and range.
iv. Determine the function xfx and plot its graph and write its domain and range. {Ans. i. 10,2 xxxg 21,2 xx
PART-I/CHAPTER-1
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.37
32,4 xx
ii. 0,3 xxh 10,1 xx 21,3 xx 32,5 xx
iii. 0,4 xx 10,2 x 21,4 x 32,28 xx
iv. 0,2 xx 10,4 x 21,26 xx 32,2 x 3,4 x }
107. Given 02,1)( xxf 20,1 xx .
i. Determine the function xfxfxg and plot its graph and write its domain and range.
ii. Determine the function xgxgx and plot its graph and write its domain and range.
{Ans. i. 02, xxxg 10,0 x 21,22 xx
ii. 12,23 xxx 01, xx 10,0 x 21,44 xx }
108. Given 02,1)( xxxf
0,0 x 20,1 xx ,
))(()( xffx and
22)(
xxF . Determine the function ))(( xFxg and plot its graph and write
its domain and range. {Ans. 34,4 xxxg 3,0 x 13,2 xx 1,0 x 11, xx 1,0 x
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
1.38
31,2 xx 3,0 x 43,4 xx }
109. Determine the function xxxf 1)( and plot its graph and write its domain and range.
{Ans. 1,1 xxf
2
11,12 xx
02
1,12 xx
0,1 x } CATEGORY-1.8. ADDITIONAL QUESTIONS
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-I
DIFFERENTIAL CALCULUS
CHAPTER-2
LIMIT
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.2
CHAPTER-2
LIMIT
LIST OF THEORY SECTIONS
2.1. Definition Of Limit 2.2. Existence Of Limit 2.3. Finding Limit Of A Function By Theorems 2.4. Finding Limit Of A Function By Methods 2.5. Limits With Parameters
LIST OF QUESTION CATEGORIES
2.1. Limit Of Defined Analytical Functions By Theorems 2.2. Limit Of Piecewise Functions 2.3. Limit Of Rational Functions At A Finite Point 2.4. Limit Of Irrational Functions At A Finite Point 2.5. Limits Involving 2.6. Substitution
2.7. Use Of Standard Limits 1sin
lim0
x
xx
And 1tan
lim0
x
xx
2.8. Use Of Standard Limits 1sin
lim1
0
x
xx
And 1tan
lim1
0
x
xx
2.9. Use Of Standard Limit 1lim
nnn
axna
ax
ax
2.10. Use Of Standard Limits kx
xekx
1
01lim And k
x
xe
x
k
1lim
2.11. Use Of Standard Limit ax
a x
xln
1lim
0
2.12. Use Of Series 2.13. Application Of Methods 2.14. L’ Hospital’s Rule 2.15. Application Of Methods And L’ Hospital’s Rule 2.16. Sandwich Theorem 2.17. Finding Limit Of A Function Containing Parameters 2.18. Finding Values Of The Parameters Given The Value Of Limit 2.19. Finding The Limit As Function Of Parameter 2.20. Additional Questions
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.3
CHAPTER-2
LIMIT
SECTION-2.1. DEFINITION OF LIMIT
1. Definition of Neighbourhood (nbd) of a point, Deleted neighbourhood, δ-neighbourhood, Right hand and Left hand neighbourhood i. Any interval containing a point a as its interior point is called the neighbourhood (nbd) of point a. ii. Any neighbourhood of point a which does not contain the point a is called the deleted neighbourhood
(nbd) of point a. iii. The interval aa , is called the δ-neighbourhood of point a. iv. Any interval whose left hand end-point is a is called right hand neighbourhood of point a. v. Any interval whose right hand end-point is a is called left hand neighbourhood of point a.
2. Meaning of tends to () and its use i. Tends to, denoted by symbol , means "approaching without bound but not equal to". ii. ax means "x approaches a without bound but ax ". iii. ax means ax and ax . iv. ax means ax and ax . v. x means ‘x approaches without bound’ or ‘x is increasing without bound’. vi. x means ‘x approaches without bound’ or ‘x is decreasing without bound’.
3. Definition of Right hand limit of a function at a finite point i. A function xf is said to have a right hand limit at a finite point a if xf is either tending to or equal
to b (a real number or or ) as ax and b is said to be the right hand limit of xf at the
point a, denoted by symbol bxfax
lim ; otherwise xf has no right hand limit at the point a.
4. Definition of Left hand limit of a function at a finite point i. A function xf is said to have a left hand limit at a finite point a if xf is either tending to or equal to
b (a real number or or ) as ax and b is said to be the left hand limit of xf at the point a,
denoted by symbol bxfax
lim ; otherwise xf has no left hand limit at the point a.
5. Definition of limit of a function at a finite point i. A function xf is said to have a limit at a finite point a if xf is either tending to or equal to b (a real
number or or ) as ax and b is said to be the limit of xf at the point a, denoted by symbol
bxfax
lim ; otherwise xf has no limit at the point a.
6. Definition of limit of a function at i. A function xf is said to have a limit at if xf is either tending to or equal to b (a real
number or or ) as x x and b is said to be the limit of xf at ,
denoted by symbol bxfx
lim bxfx
lim ; otherwise xf has no limit at .
7. A function may not have a limit at a point or SECTION-2.2. EXISTENCE OF LIMIT
1. Existence of limit of a function at a finite point
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.4
i. The limit of a function xf is said to exist at a finite point a iff:-
a. xfax
lim is finite,
b. xfax
lim is finite,
c. xfax
lim and xfax
lim are both equal,
otherwise the limit of the function xf does not exist at a. 2. Existence of limit at the end points of the Domain
i. If xf is not defined in a left hand nbd of the point a and xfax
lim is finite, then the limit of the
function xf is said to exist at the point a, otherwise the limit of the function xf does not exist at a.
ii. If xf is not defined in a right hand nbd of the point a and xfax
lim is finite, then the limit of the
function xf is said to exist at the point a, otherwise the limit of the function xf does not exist at a.
3. Existence of limit at i. If xf
x lim is finite, then the limit of the function xf is said to exist at , otherwise the limit of the
function xf does not exist at .
ii. If xfx lim is finite, then the limit of the function xf is said to exist at , otherwise the limit of the
function xf does not exist at . 4. Cases when limit does not exist SECTION-2.3. FINDING LIMIT OF A FUNCTION BY THEOREMS
1. Reading the limit of a function from it’s graph 2. Limit of analytical functions
i. Limit of Basic functions a. From values/graphs b. Theorem of limit for Basic functions
If xf is a basic function and a finite point a belongs to its domain, then xfax
lim exists and is
equal to af . ii. Limit of defined analytical functions
a. Meaningless (Indeterminate) forms Certain forms of limits are said to be meaningless when merely knowing the limiting behavior of individual parts of the expression is not sufficient to actually determine the overall limit since the value of the overall limit actually depends on the limiting behavior of the combination of the two
expressions. There are seven meaningless forms 00 form,
form, 0 form, form, 00
form, 0 form, 1 form. b. Non-meaningless forms and Properties of mathematical operations
All other forms of limits except the seven meaningless forms are non-meaningless forms and merely knowing the limiting behavior of individual parts of the expression is sufficient to actually determine the overall limit by Properties of mathematical operations.
c. Theorems of limit & its applications If a is a finite point or or and xf
axlim and xg
axlim both exist and lxf
ax
lim and
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.5
mxgax
lim then:-
xgxfax
lim must exist and is equal to ml .
xgxfax
lim must exist and is equal to ml .
xg
xfax
lim must exist and is equal to m
l provided that 0m .
xg
axxf
lim must exist and is equal to ml provided that ml is defined.
d. Finding limit by values, theorems and properties in cases of non-meaningless forms e. Limit of piecewise functions
SECTION-2.4. FINDING LIMIT OF A FUNCTION BY METHODS
1. Methods to calculate limit i. There are methods to calculate limits in cases of meaningless forms and in other cases where
theorems/properties are not applicable. ii. Limit of rational functions at a finite point iii. Limit of irrational functions at a finite point iv. Limit involving v. Substitution vi. Use of Standard limits
a. 1sin
lim0
x
xx
b. 1tan
lim0
x
xx
c. 1sin
lim1
0
x
xx
d. 1tan
lim1
0
x
xx
e. 1lim
nnn
axna
ax
ax (a, n are constants)
f. kx
xekx
1
01lim (k is a non-zero constant); ex x
x
1
01lim
g. kx
xe
x
k
1lim (k is a non-zero constant); e
x
x
x
11lim
h. ax
a x
xln
1lim
0
(a is a positive constant); 1
1lim
0
x
e x
x
vii. Use of Binomial theorem & Series. a. Binomial Theorem
If Nn and a, b and x are numbers, then
nn
nnn
nnnnnnnn bCabCbaCbaCaCba
11
222
110 ....................
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.6
nnnnn bnabbann
bnaa
1221 ....................!2
1;
nn
nnn
nnnnn xCxCxCxCCx
11
2210 ....................1
nn xnxxnn
nx
12 ....................!2
11 ;
where
1;!!
!0
n
nnr
n CCrrn
nC .
b. Sine series
...............!7!5!3
sin753
xxx
xx
c. Cosine series
...............!6!4!2
1cos642
xxx
x
d. Exponential series
...............!3!2!1
132
xxx
e x
...............!3
ln
!2
ln
!1
ln1
3322
xaxaxa
a x
e. Logarithmic series If 11 x , then
...............432
1ln432
xxx
xx
f. Binomial series If Na and x is a number such that 11 x , then
...............!4
321
!3
21
!2
111 432
x
aaaax
aaax
aaaxx a
g. Proving standard limits viii. Application of methods ix. L’ Hospital’s rule & it’s restricted use
If a is a finite point or or ; and if as ax xf and xg either both tends to 0 or both tends
to ; and if xg
xfax
lim and xg
xfax
lim both exist or both are infinity, then they must be equal.
x. Sandwich theorem a. Let xgxfxh in a neighbourhood of a except probably at a and xh , xf and xg have
limits at a then xgxfxhaxaxax
limlimlim .
b. Let xgxfxh in a neighbourhood of a except probably at a and let xh and xg have the
same limit L at a, then Lxfax
lim .
SECTION-2.5. LIMITS WITH PARAMETERS
1. Finding limit of a function containing parameters
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.7
2. Finding values of the parameters given the value of limit 3. Finding the limit as function of parameter
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.8
EXERCISE-2
CATEGORY-2.1. LIMIT OF DEFINED ANALYTICAL FUNCTIONS BY THEOREMS
1. limtan
lnx
x
x
1
{Ans. 0}
2. limtanh
x
x
x3
{Ans. 0}
3. 2
0sgnlim x
x {Ans. 1}
4. 3
0sgnlim x
x {Ans. 1, 1}
5. lim(sgn )x
x0
2 {Ans. 1}
6. x
xx sgn
0)(sgnlim
. {Ans. 1, 1 }
7. limsgn(ln )x
x0
{Ans. -1}
8. x
xesgn
0lim
{Ans. e , e
1}
9. x
xe sgn
0lim
{Ans. e }
10. xe
xxsgnlim
0 {Ans. 1}
11. x
xx sgnln
0tanlim
{Ans. 1}
12. limln
x
x
x
0
11
{Ans. }
13. 543lim 2
1
xx
x. {Ans. 12}
14. limx
x x
x x
1
5
6 3
4 9 73 1
{Ans. 4}
15. x
xx sin1
coslim
0 . {Ans. 1}
16. limsin
cos tanx
x x
x x
0 1 1
2 {Ans. 2
}
17. lim lnx e
x x
2 {Ans. e2 }
18. limcos lnx
x xx x
1
12
1 1 {Ans. 2}
19. limln(sin cos )x
x x
1
{Ans. 0}
20. limsin ln(cos )x
x
0
1 {Ans. 0}
21. )ln(cossgnlim0
xx
{Ans. 1}
22. )1ln(sgnlim0
xx
{Ans. 1, 1}
23. )ln(sgnsgnlim0
xx
{Ans. 0}
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.9
24.
x
xx
sgnlnlim
0 {Ans. 0}
25. xxx
sgnlnlim
{Ans. 0}
26. x
xxsgnlnlim
0 {Ans. 0}
27. x
xx sgnlnlim
{Ans. 1}
28. x
xxsgnlim
{Ans. 1}
29. lim( )sin
x
xx
0
1 {Ans. 1}
30. lim(cos )cos
x
xx
0
1 {Ans. 2
}
31. limx
x
x 1 1 {Ans. +, -}
32. limlnx
xx e
0
{Ans. -}
33. limtan tanx
x x
2
2 {Ans. -, +}
34. limcos cot tanx
x x x
0
1 1 {Ans. +, -}
35. limx
xe
x0 2 {Ans. }
36. lim lnx
x x
{Ans. }
37. 31 7
3lim
x
xx
{Ans. 1}
38. limx
xx e
{Ans. }
39. limln
x
x
x0 {Ans. -}
40. xx sinln
1lim
2
{Ans. -}
41. limtanx
x
x
2
2 {Ans. 0}
42. limx
x x
0
1
{Ans. 0}
43. limx
xx
{Ans. }
44. lim(sin )cot
x
xx0
{Ans. 0}
45. lim ln
x
xx
{Ans. }
46. limx x
x
e {Ans. -}
47. limx
x
0
221
{Ans. 0, 1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.10
48. lim tanx xe 2
11
{Ans. 1, 0}
49. x
xx sin
lnlim
0 {Ans. -, }
50. xxx
sinloglim cos0
{Ans. }
CATEGORY-2.2. LIMIT OF PIECEWISE FUNCTIONS
51. Given f x x x
x x
( ) ,
,
2 3 1
3 5 1
Find lim ( )x
f x1
. {Ans. 2, 1}
52. Given 1,2)( xxxf
31,14 xx
3,52 xx . Find:- i. lim ( )
xf x
1. {Ans. 3}
ii. )(lim3
xfx
. {Ans. 14, 11}
53. A function is defined as
0,2
0,1)(
x
xxf
Does the limit )(lim0
xfx
exists? {Ans. Yes}
54. Draw the graph of function x
xxf )( . Is )0(f defined? Does )(lim
0xf
x exist? {Ans. No, No}
55. Given 0,)( xxxf
0,1 x
0,2 xx
Does )(lim0
xfx
exist? {Ans. Yes}
56. Evaluate the limit of the function
4,4
4
x
x
xxf
4,0 x at 4x . Whether the limit exists or not. {Ans. 1, -1, does not exist}
57. Evaluate the limit of the function 10,1 2 xxxf
1,2 xx at 1x . Whether the limit exists or not. {Ans. 1, 2, does not exist}
58. If
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.11
0,
xx
xxxf
0,2 x ,
show that xfx 0lim
does not exist.
59. If 10,45 xxxf
21,34 3 xxx ,
show that xfx 1lim
exists.
60. Let 0,cos xxxf
0, xkx .
Find the value of constant k, given that xfx 0lim
exists. {Ans. 1k }
61. Given 10, xxxf
21,24 xx
10,1 xxxg 21,2 xx . Find:- i. xgxf
x
0lim ; {Ans. 1}
ii. xgxfx
1
lim ; {Ans. 3}
iii. xgxfx
2
lim ; {Ans. 0}
iv. xgfx 0lim
; {Ans. 2}
v. xgfx 1lim
; {Ans. 1, 0}
vi. xgfx 2lim
; {Ans. 0}
vii. xfgx 0lim
; {Ans. 1}
viii. xfgx 1lim
; {Ans. 0, 2}
ix. xfgx 2lim
; {Ans. 1}
x. xfffx 1lim
; {Ans. 0, 1}
xi. xgggx 1lim
. {Ans. 0, 1}
CATEGORY-2.3. LIMIT OF RATIONAL FUNCTIONS AT A FINITE POINT
62. limx
x x
x
2
2 5 62
{Ans. 1}
63. limx
x x
x x
5
2
2
2 11 54 16 20
{Ans. 3
8}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.12
64. limx
x x
x x
3
2
2
6 92 21
{Ans. 0}
65. limx
x
x x
3
3
2
272 3 27
{Ans. 9
5}
66. limx
x
x x
12
8 1
6 5 1
3
2 {Ans. 6}
67. 86
6116lim
2
23
2
xx
xxxx
. {Ans. 2
1}
68. limx
x x x
x x
2
3 2
3 2
5 8 43 4
{Ans. 1
3}
69. limx x x
3 2
13
69
{Ans. 1
6}
70. limx
x x x
x x x
1
3 2
3 2
11
{Ans. +, }
71. limx
x x
x x
4
2
2
2 4 2416
14
{Ans. 13
8}
72. limx
x x x
x x
2
3 2
3
3 9 26
{Ans. 15
11}
73. limx
x x
x x
1
3
4
3 24 3
{Ans. 1
2}
74. limx x x
1 3
11
31
{Ans. -1}
75. limx
x x x x
x x x x
2
5 4 2
4 3 2
2 3 22 3 5 22
{Ans. 3
5}
76. lim( )x x x x x
2 2 2
12
13 2
{Ans. }
77. limx x x x x
2 2 3
22
12 4
248
{Ans. 11
12}
CATEGORY-2.4. LIMIT OF IRRATIONAL FUNCTIONS AT A FINITE POINT
78. limx
x
x
1 2
11
{Ans. 1
4}
79. limx
x
x
4
3 51 5
{Ans. 1
3}
80. limx
x x
x x
1
8 8 15 7 3
{Ans. 7
12}
81. limx
x x
x
0
4 44
{Ans. 1
8}
82. limx
x
x
7 2
2 349
{Ans. 1
56}
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.13
83. limx
x x
x
0
1 1 {Ans. 1}
84. limx
x x x x
x x
3
2 2
2
2 6 2 64 3
{Ans. 1
3}
85. limx
x
x x
1 2
1
2 4 {Ans. 4}
86. limx
x
x
0
2
2
1 1
16 4 {Ans. 4}
87. 39
11lim
2
2
0
x
xx
. {Ans. 3}
88. limx
x
x
2
3 10 22
{Ans. 112 }
89. 326
1lim
31
x
xx
{Ans. 27}
90. 2
3 2
0
11lim
x
xx
{Ans. 1
3 }
91. limx
x x
x
0
3 31 1 {Ans. 2
3 }
CATEGORY-2.5. LIMITS INVOLVING
92. limx
x x
x x
2 4 14 6 5
2
2 {Ans. 1
2}
93. 1
)1(lim
3
3
x
xx
{Ans. 1}
94. limx
x
x
x
x
3
2
2
3 4 3 2 {Ans. 2
9}
95. limx
x x
x x
3
4 23 1 {Ans. 0}
96. limx
x
x
3 21
2
4 {Ans. 0}
97. xx
xx
1
lim2
3
{Ans. 0}
98. 13
5lim
2
4
xx
xxx
{Ans. -}
99. limx
x
x
2 22
{Ans. 1}
100. limx
x x
x
1 42 2
{Ans. 0}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.14
101. x
xxx 2
24 11lim
. {Ans. 0}
102. limx
x x x
32 3 31 1 {Ans. 1}
103. limx
x x
2 21 1 {Ans. 0}
104. limx
x x
x x
1 9 1
1 9 1
2 2
2 2 {Ans. 4
3}
105. x
xx ln2
1lnlim
0
{Ans. –1}
106. 11
lim
x
x
x e
e {Ans. 1}
107. 3ln
1lnlnlim
2
x
xxx
{Ans. }
108. 32
1lim
23
2
xxx
xx
x eee
ee {Ans. 0}
CATEGORY-2.6. SUBSTITUTION
109. lim( )x
x x
x
1
23 3
2
2 11
{Ans. 1
9}
110. limx
x
x
1
3
4
11
{Ans. 4
3}
111. limx
x
x
1
3
5
11
{Ans. 5
3}
112. limx
x
x
1 4
117 2
{Ans. 32}
113. limsin sin
sin sinx
x x
x x
6
2 12 3 1
2
2 {Ans. -3}
CATEGORY-2.7. USE OF STANDARD LIMITS 1sin
lim0
x
xx
AND 1tan
lim0
x
xx
114. limsin
x
x
x0
3 {Ans. 3}
115. limtan
x
kx
x0 {Ans. k}
116. limsinsinx
x
x0
{Ans.
}
117. limtansinx
x
x0
25
{Ans. 2
5}
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.15
118.
x
xx ln
lnsinlim
1 {Ans. 1}
119. x
x
x e
etanlim
{Ans. 1}
120. x
xx
2
0
sinlim
{Ans. 0}
121. 2
2
0
2sinlim
x
xx
{Ans. 4}
122. limcos
x
x
x
0 2
1 {Ans. 1
2}
123.
limcos
x
x
x 1
2
1
{Ans. 2
}
124. lim tanx
x x
2 2 {Ans. 1}
125. lim( )tanx
xx
11
2
{Ans. 2
}
CATEGORY-2.8. USE OF STANDARD LIMITS 1sin
lim1
0
x
xx
AND 1tan
lim1
0
x
xx
126. x
xx 3
2sinlim
1
0
{Ans. 2
3}
127. x
xx 2tanlim
10 . {Ans.
2
1}
128.
21
21
0 tan
sinlim
x
xx
{Ans. 1}
129. limtan ( )x
x
x
3
2
1
93
{Ans. -6}
130. limsintanx
x x
x x
0
1
1
22
{Ans. 13 }
CATEGORY-2.9. USE OF STANDARD LIMIT 1lim
nnn
axna
ax
ax
131. 1
1lim
10
1
x
xx
{Ans. 10}
132. 1
1lim
17
13
1
x
xx
{Ans. 17
13}
133. limx
m
n
x
x
1
11
{Ans. m
n}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.16
CATEGORY-2.10. USE OF STANDARD LIMITS kx
xekx
1
01lim AND k
x
xe
x
k
1lim
134. x
xx
1
0)21(lim
{Ans. 2e }
135. x
x x
21lim . {Ans. 2e }
136. xx
x1
2
01lim
{Ans. 1}
137. x
xx 3
1
0)1(lim
{Ans. e
13 }
138. x
x x
11lim {Ans. 1
e }
139. limx
x
x
1
1 7
{Ans. e7 }
140. limx
mxk
x
1 {Ans. emk }
141. x
x x
2
11lim {Ans. 1}
142. limx
xx
x
1 {Ans. e1 }
143. 3
1
3lim
x
x x
x. {Ans. 4e }
144. lim sin cos
x
ecxx
0
1 {Ans. e }
145. lim( tan )cot
x
xx
0
21 32
{Ans. e3 }
146. limln( )
x
kx
x
0
1 {Ans. k}
147. limsin
ln( sin )x x
a x
0
11 {Ans. a}
148. limln
x e
x
x e
1 {Ans. 1
e}
CATEGORY-2.11. USE OF STANDARD LIMIT ax
a x
xln
1lim
0
149. limx
xe
x
0
1 {Ans. -1}
150. limx
xe
x
0
2 1
3 {Ans. 2
3 }
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.17
151. limx
xa
x
0
2 1 {Ans. 2ln a }
152. x
x
x
12lim
2
0
{Ans. 0}
153. lim ( )x
x e x
1
1 {Ans. 1}
154. lim( ) ( )x
a x ax
1
1 0 {Ans. lna}
155. limx
ax bxe e
x
0 {Ans. a - b}
156. limsinx
xe
x
0
1 {Ans. 1}
157. limln( )
xx
x
0
1
3 1 {Ans. 1
3ln }
158. limtanx
xe
x
0
4 1 {Ans. 4}
159. limx
xe e
x
1 1 {Ans. e }
160. 20
110lim
x
x
x
{Ans. +, -}
CATEGORY-2.12. USE OF SERIES
161. limsin
x
x x
x
0 3 {Ans. 1
6}
162. 20
1lim
x
xe x
x
{Ans.
2
1}
163. 5
3
0
6sin
limx
xxx
x
. {Ans.
120
1}
164. limx
x xe e
x
02
2 {Ans. 1}
165. limsinx
x xe e
x
0 {Ans. 2}
166. limsin
x
x xe e x x
x
0 5
2 4 {Ans. 1
30}
167. limcos sin
x
x x x
x
0 3 {Ans. 1
3}
168. limcosx
x x x
x
e x
x
0
6 2
2
3 2
2
11
{Ans. 1}
169. limln( )
sinx
x x x x x
x x x
0
4 2 43
3 4
3
1 4 26 6
{Ans. 16}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.18
170.
x
xxx
57
0
31)21(lim
{Ans. 1}
171.
2
48
0
21)1(lim
x
xxx
{Ans. 4}
172. x
xxx
43
0
211lim
{Ans. 1
6}
173. 2
3
0
6141lim
x
xxx
{Ans. 2}
CATEGORY-2.13. APPLICATION OF METHODS
174. limx a
x a
x a
3 3
{Ans. 236 a
}
175. limx
x x
x x
2 3 3
7 3 2 3
6 2 3 5 {Ans. 34
23 }
176. 36
3loglim
3
x
xa
x {Ans. loga 6 }
177. xxx
319lim 2
{Ans. 0, +}
178. limx
x x x
x x
2 3 5
3 2 2 3
3 5
3 {Ans. 23
}
179. xxx
532lim 2
{Ans. +}
180. limx
x x x
2 1 {Ans. 12 , }
181. limx
x
x
2 3
4 2
2
{Ans. 12 2
12 2
, }
182. limx
xx
52
3 {Ans. 25}
183. limx
k x
xk
0
1 1 ( is positive integer) {Ans. 1
k }
184. limsin( )
cosx
x
x
6
6
3 2 {Ans. 1}
185. limln( ) ln
x
a x a
x
0 {Ans. 1
a }
186. CBxAx
cbxaxx
2
2
lim . {Ans. A
a}
187. xxa
xxaax 23
32lim
. {Ans.
33
2}
188. 34
1213lim
22
x
xxx
. {Ans. 4
23 }
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.19
189. 11lim 22
xxxx
. {Ans. 2
1}
190. xxx
xx
x sincos4
sincoslim
4
. {Ans. 1}
191. ax
axax
coscoslim . {Ans.
a
a
2
sin }
192.
x
xxx
5ln5lnlim
0. {Ans.
5
2}
193. x
xx cot2
0sin1lim
. {Ans. 2e }
194. limx
x
x
xx
1
1
2
11
{Ans. 23 }
195. x
x xx
xx
52
34lim
2
2
. {Ans. 6e }
196. limx
x x
x x
xx
2
2
2 1
2 3 2
2 11
{Ans. 14 }
197. limx
xx
x
2 3
2 5
2
2
8 32
{Ans. e8 }
198. 2
1
2
2
0 31
51lim
x
x x
x
. {Ans. 2e }
199.
x
x x
x cos1
15lim
0
. {Ans. 5ln2 }
200.
xx
xxx 3sin
5sin2cos1lim
20
. {Ans.
3
10}
201. x
xx 11
4sinlim
0. {Ans. 8}
202. x
x x
x
1
1lim . {Ans. 2e }
203. x
x x
x
6
6lim
2
2
. {Ans. 1}
204. 23
23
2
2
52
32lim
x
x
x xx
xx. {Ans.
2
1}
205. 20
sinlim
x
xx
. {Ans. 1, -1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.20
206. x
b
xax
1lim
0. {Ans. abe }
207. 3
3lim
3
x
xx
. {Ans. 1, -1}
208. limtansin
sin
x
x
x
x
0
11
1
{Ans. 1}
209. lim( sin )cot
x
xx
1
1 {Ans. 1e }
210. limcos tan
x
x x
x x
04
4
2 2 {Ans. 1
2 }
211. 2
281
21lim
3 3
x
x x
x
{Ans. -1}
212. limln( )x
ax axe e
x
0
2
1 {Ans. 3a}
213. limx
x
x x
1
3 1 2 1
2 {Ans. 4
9 }
214. limsinx
x xe e x
x x
0
2 {Ans. 2}
215. limsinx
x
x
0
4 25
{Ans. 120 }
216. x
x xx
xx
2
35lim
2
2
. {Ans. 4e }
217. limsin( )
x
x
x
1
1
1 {Ans. -2}
218. limsin
12
2
{Ans. 2 }
219. lim tan tan( )x
x x
4
2 4 {Ans. 12 }
220. limtan tan
cos( )x
x x
x
3
3
6
3 {Ans. -24}
221. 3
4lim
x
x x
x {Ans. e4 }
222. limx
xx
x
2 12 1
{Ans. 1e }
223. limx
x x
x x
xx
3 2 1
2
2
2
6 13 2
{Ans. 9}
224. limx
x
x
xx
1 3
2 3
11
{Ans. 1}
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.21
225. limln(tan )
cotx
x
x 4 1
{Ans. 1}
226. limx
x
x x
3
2
4 2
3
1 {Ans. 0}
227. limx
x x
x x x
1
3
3 2
2
1 {Ans. +, -}
228. )23(3
4
45
2lim
221
xx
x
xx
xx
{Ans. 0}
229. 1212
lim2
2
3
x
x
x
xx
{Ans. 14 }
230. 2
22
4
)23)(12(
12
3lim
x
xxx
x
xx
{Ans. 1
2 }
231. lim( ) ( ) ... ( )
x
x x x
x
1 2 100
10
10 10 10
10 10 {Ans. 100}
232. limx
x x
x x x
2
34
1 {Ans. -1}
233. limx
x x
x x
2 23
44 45
1 1
1 1 {Ans. 1}
234. limx
x x
x x x
75 34
8 76
3 2 1
1 {Ans. }
235. limx
x x
x
43 35
73
3 4
1 {Ans. 0}
236. limx
x
x
02
1 1 {Ans. +, -}
237. lim ( )x a
x b a b
x aa b
2 2 {Ans. 1
4a a b}
238. 1
1lim
1
m
n
x x
x {Ans. m
n }
239. limx
x x
x x
0
23 4
2
1 1 2 {Ans. 1
2 }
240. xaxx
lim {Ans. 0}
241. xxx
1lim 2 {Ans. 0, +}
242. xbxaxx
))((lim {Ans. a b 2 , }
243. 3712lim 22
xxxxx
{Ans. 52
52, }
244. 3 23 2 )1()1(lim
xxx
{Ans. 0}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.22
245. limcos
sinx
x
x x
0
31
2 {Ans. 3
4 }
246. limtan
( cos )
0 23 1 {Ans. +, -}
247. limsin cos
sin cosx
x x
x x
0
1
1 {Ans. -1}
248. limtan sin
03 {Ans. 1
2 }
249. lim( cos )
tan sin
0
2
3 3
1 {Ans. +, -}
250. xxx tan
1
sin
1lim
0
{Ans. 0}
251. limsin
x
x
x
2
1
2
2 {Ans. 12 }
252. limcos
( sin )x
x
x 2 1 23
{Ans. -, +}
253. limsin
sinx
x
x
3
2 {Ans. 3
2 }
254. a
yayay 2
tan2
sinlim
{Ans. a
}
255. limcos sin
cosx
x x
x
4 2
{Ans. 12
}
256. limsin
cos cos sinx
x
x x x
12
2 4 4
{Ans. 12
}
257. x
xxx cos
tan2lim2
{Ans. -2}
258. limcos( ) cos( )
x
a x a x
x
0 {Ans. 2sina }
259. limcos cos
x
x x
x
02
{Ans. 2 2
2 }
260. limsin( ) sin( )tan( ) tan( )x
a x a x
a x a x
0 {Ans. cos3 a}
261. limsin sin
2 2
2 2 {Ans. sin22
}
262. limsin( ) sin( ) sin
h
a h a h a
h
02
2 2 {Ans. sina }
263. limtan( ) tan( ) tan
h
a h a h a
h
02
2 2 {Ans. 2
3sin
cosaa
}
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.23
264. limcos
sinx
x
x
02
2 1 {Ans. 1
4 2}
265. limsin sin
tanx
x x
x
0
1 1 {Ans. 1}
266. limsin cos
tanx
x x xx
0 2
1 2
2
{Ans. 6}
267. 20
cos2cos1lim
x
xxx
{Ans. 3
2 }
268. limx a
m m
n n
x a
x a
{Ans. m
nm na }
269. limcosx
xe
x
0
2
1
1 {Ans. -2}
270. ex
exx
1
)(lnlim . {Ans. ee1
}
271. limsin cosx
x xe e
x x
0 {Ans. 2}
272. limx x
xx
11
1
{Ans. 1}
273. limx
xx
x
1
2
2 1
{Ans. e6 }
274. limx
x
x
x
3 4
3 2
13
{Ans. e 23 }
275. limx
xx
x
2
2
1
1
2
{Ans. e2 }
276. limx
xx
x
1
2 1 {Ans. 0, +}
277. limx
xx
x
2 1
1 {Ans. +, 0}
278. limx
x
x
1
12
{Ans. +, 0}
279. limx
xx x
x x
2
2
2 1
4 2 {Ans. e2 }
280. lim tanx
xx
0
211
2 {Ans. e }
281. xaxxx
ln)ln(lim
{Ans. a}
282. limcos
x
xe x
x
02
2
{Ans. 32 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.24
283. limsin sin
x
x xe e
x
0
2
{Ans. 1}
284. limcos
x
x e
x
x
0 4
2
2
{Ans. 112 }
285. xxx
sinhcoshlim
{Ans. 0, +}
286. lim tanhx
x
{Ans. 1, -1}
287.
21lim 42 xxxx
x {Ans. 0, -}
288. limsin
x
x
x {Ans. 0}
289. limtan
x
x
x
1
{Ans. 0}
290. limsin
cosx
x x
x x
{Ans. 1}
291. limsin
tanx
xx
1
1
2
{Ans. 0}
292. 2sin6sin4sin3sin2tanlim 222
2
xxxxxx
{Ans. 1
12 }
293. limlnx
xx x
x x
1 1 {Ans. 2}
294. limcos( cos )
x
x
x
04
1 1 {Ans. 1
8 }
295. lim cosx
xx
2 1
1 {Ans. 1
2 }
296. lim(cos ) sin
xx x
0
1
{Ans. 1}
297. limsin
x
x x
x
0
1
3 {Ans. 6
1 }
298. limlncos
x
x
x02 {Ans. 1
2 }
299. limsin
sinsin
x
x
x
xx x
0 {Ans. 1
e }
300. lim(cos sin )x
x x x
0
1
{Ans. e}
301. lim(cos sin )x
x a bx x
0
1
{Ans. eab }
302. xxx
cos1coslim
{Ans. 0}
303. 20
ln1lim
x
axa x
x
{Ans.
2
ln 2 a}
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.25
304. xx
xx
x 65
43lim
{Ans. 0}
305. limcos ( )
x
x
x
0
1 1 {Ans. 2 }
306. limcos
x
x
x
1
1
1
{Ans. 1
2}
307. limtanx
e
x
x
12 1
2 1 2 {Ans. 1
2 }
308. limcos
x
x
x
0
1 2 {Ans. 2 2, }
309. limx
x
0
1
1
2 2 {Ans. 0, 1
2 }
310. limx
x
1
1
1
31
1 7 {Ans. 4, 3}
311. 20
limx
xx
{Ans. }
312. If
0,1
sin)( xx
xxf
0,1 x
then find )(lim0
xfx
. {Ans. 0}
313. 2
321lim
2
x
xx
. {Ans. 38
1}
314. x
xx cos1
cos1lim
2
0
. {Ans. 2 }
315. 30
tanlim
x
xxx
. {Ans.
3
1}
316. xxxxx
lim . {Ans. 2
1}
317. xx
xxx 57
3lim
0
. {Ans. 2,
6
1}
318.
1
51
sin23lim
23
324
xxx
xx
xx
x. {Ans. 2 }
319. axx
axxaax
2
sinsinlim 0a . {Ans.
a
aa
sincos }
CATEGORY-2.14. L’ HOSPITAL’S RULE
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.26
320. x
xx
lnlim
{Ans. 0}
321. xxx
lnlim0
{Ans. 0}
322. limx
xx0
{Ans. 1}
323. x
xx
1
lim
{Ans. 1}
324. Show that the following limits cannot be evaluated by L’ Hospital’s rule:-
i. limsin
sinx
xx
x0
2 1
{Ans. 0}
ii. xx exx
xxsin)2sin2(
2sin22lim
{Ans. no limit}
iii. limsin
sinx
x x
x x
{Ans. 1}
iv. x
xx sec
tanlim
2
{Ans. 1}
CATEGORY-2.15. APPLICATION OF METHODS AND L’ HOSPITAL’S RULE
325. limln( )
cosxx
x
x e
0
213
{Ans. 0}
326. limsin
lncos( )x
x
x x 0
2
2
32
{Ans. -6}
327. limlog
x
ak
x
x (k > 0) {Ans. 0}
328. 1
1
ln
1lim
1
xxx {Ans. 1
2 }
329. x
xx
1cotlim
0
{Ans. 0}
330. 1
11lim
0
xx ex {Ans. 1
2 }
331. lim ln ( )x
nx x n
0
0 {Ans. 0}
332. )1(lncot)sin1ln(lim 22
0xx
x
{Ans. 1}
333. limsin
x
x
x
0
1 {Ans. 1}
334. xx
x1
0sinlim
{Ans. 0}
335. lim ln( )
xx ex
0
1
1 {Ans. e}
336. lim(tan )cot
x
xx
2
{Ans. 1}
337. limln( )
x
x
x x
2
2
2
3
3 10 {Ans. 4
7 }
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.27
338. limln
ln
x
xa x
x
1 {Ans. 1ln a }
339. limsin ( )
x
x
x
1
26
2
1 4
1
{Ans. 36 }
340. limsin cot( )x a
x a
ax a
1 {Ans. 1
a }
341. lim ( tan ) lnx
x x
2 1 {Ans. 0}
342. lim(cos )x
mxn
x
0
2 {Ans. em n
2
2 }
343. xxx
220
cot1
lim
{Ans. 23 }
344.
xxx
x
11lnlim 2 {Ans. 1
2 }
345. lim coshx
xa
x
2 1 {Ans. a2
2 }
346. limsin
x x
x
0
5
2 9
1
{Ans. e 130 }
347. lim(lncot )tan
x
xx0
{Ans. 1}
348. limcos
tanx
x x
x
0
2
4
1 1 {Ans. 1
3 }
349. limx
x x
x
0
3
2
1 3 1 2 {Ans. 1
2 }
350. limsin ( )
x
xe x x x
x
03
1 {Ans. 1
3 }
351. limcos
cosx
x
x
e x
e x
0
{Ans.
}
352. limtan
x
x x
x
0
1
3 {Ans. 13 }
353. limsinx
a xe
bx
0
1 {Ans. a
b}
354. limsin
tanx
x x
x x
0 {Ans. 1
2 }
355. limtan
lnx
x
x
2
11
1
{Ans. 2}
356. limx
x x
x x
a b
c d
0 {Ans. ln
ln
abcd
}
357. limx
x xa b
x x
0 21 {Ans. ln a
b }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.28
358. limcos ln( )
ln( )x ax a
x x a
e e
{Ans. cosa}
359. limtan
tan
x
x xe e
x x
0 {Ans. 1}
360. limsinx
xe x
x
0
3
6
3
1
2 {Ans. 1
128 }
361. limtan sin
sin tanx
x x
x x
0
13 13
1 1
1 3 1 3
1 2 1 2 {Ans. -1}
362. limcos sin
x
xx e x x
x
0
2 2 3
4
2 2 232
{Ans. -1}
363. limlnsin
lnsinx
x
x0
2 {Ans. 1}
364. limln
lnsinx
x
x0 {Ans. 1}
365. limln( ) tan
cotx
xx
x
1
12
{Ans. -2}
366. xn
xex
lim (n > 0) {Ans. 0}
367. lim sinx
xa
x {Ans. a}
368. a
aa 2
tanlim 22
{Ans. 4 2a
}
369. limcos ln( )
x xx
1
1
21
{Ans. -}
370. xxcxbxax
3 ))()((lim {Ans. a b c 3 }
371. 21
2
0lim xexx
{Ans. }
372. lim(tan )x
xx
2
2 {Ans. 1}
373. limx
xe x x
0
1
{Ans. e2 }
374. lim( )x
xe x
1
1
{Ans. e }
375. xx
x
a1
0 2
1lim
{Ans. a }
376. limtan
x a
x
a
xa
2
2
{Ans. e2 }
377. xx
x x
x
1)1ln(lim
2
1
0
{Ans. 1
2 }
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.29
378.
xx
xx x
x sinlnsin1
sinsinlim
sin
2
{Ans. 2}
379. limx
x x
x
1
33 27 3
1 {Ans. 1
4 }
380. lim( )
x
x e
x
x
0
11
{Ans. 2
e }
381. lim(sin ) tan
x
xx
4
2
2 2 {Ans. e 12 }
382. lim(sin ) tan
x
xx
2
{Ans. 1}
383. limtan
x
x
x
x
0
12
{Ans. 31
e }
384. lim tanx
x x
04
1
{Ans. 2e }
385. 0,2
lim
11
ba
bax
x
xx
{Ans. ab }
386. xxxx
x
cba1
0 3lim
. {Ans. 3 abc }
387. integer)an is , (sin
sinlim
1
kkaa
x ax
ax
{Ans. e acot }
388. lim( )
x
xex
e
x
x
0 2
12
1
{Ans. 24
11e}
389. limlog cos
log cos
sin
sin
x
x xx
x
0
2 2
{Ans. 4}
390.
nx
n
x n
aaa xxx
111
..........lim 21
{Ans. naaa ......21 }
391. lim( ... )x
nna x a x a x x
01 2
211
{Ans. 1ae }
392. lim( )x
xx x
x
1 21 {Ans. 1}
393. limcos( cos )
x
x
x0
22
{Ans. 4
}
394. x
x x
x1
0
1lnlim
{Ans. 2
1
e }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.30
395. xx
xx
x
1
3
2 22
622lim {Ans. 8}
396. x
xx
x 2sin1
)sin(cos22lim
3
4
{Ans. 2
3}
397. )0(lim
a
ax
axax
xa
ax {Ans.
a
a
ln1
ln1
}
398. 2
1)(cos)(sinlim
2
x
xx
x
{Ans. coslncossinlnsin 22 }
399.
xex
xx
x
x
11ln
11lim 32 {Ans.
8
e}
400.
a
x
axaxaxax
coslnlim
222
0 {Ans. a }
401. 6
422
0
3sin
limx
xxx
x
{Ans.
45
2 }
402. 3
2
0
21lim
x
xex x
x
{Ans.
3
1}
403.
30
tansinlim
x
xxx
{Ans.
6
1}
404.
xa
xaxa
ax 2cotlim 22
{Ans.
a4}
405. n
n
x
x bxx
bxxb
xx
a
coscos
sinsin
sin
1lim
0 {Ans. a
bn
ln3
}
406. axbx
axbbxa xx
x tantan
sinsinlim
0
{Ans. 1}
407. limsin( ) sin( ) sin( ) sin
h
a h a h a h a
h
03
3 3 2 3 {Ans. cosa }
408.
6
2
0
sinsinsinlim
x
xxxx
{Ans.
18
1}
409. x
x x
x2
sin
2cos
)(sin1lim
{Ans. 2
1}
410. 1sinlim 2
nn
(n is an integer) {Ans. 0}
411. lim( ... ) ( ... )x
n nn
n nnx a x a x b x bn n
11
111 1
{Ans. n
ba 11 }
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.31
412. ax x
e x1
0lim
(a > 0) {Ans. 0}
413. lim ln(sec )x
xx0
{Ans. 1}
414. lim tann
nn
4
41 1 {Ans. 2e }
415. limlog cos
log cos
sec
sec
x
x
x x
x0
2
2
{Ans. 16}
416. x
a
x
a
x
a
x
a
x
ee
ee
0lim (a > 0). {Ans. 1, –1}
417. dxc
x bxa
11lim (a, b, c, d are positive). {Ans. b
d
e }
418. 101
!coslim
K
n
nnK
n. {Ans. 0}
419. nnn
n
1
32lim
. {Ans. 3}
420. Find m
m m
x
coslim without using L’ Hospital’s rule. {Ans. 1}
421. Find x
xx
x
x
1
1sin
lim
2
. {Ans. 0}
422. 3
24
1
1sin
limx
xx
x
x
{Ans. 1}
423. If 1,)( 24 xxxf
1 , 2 xx .
Discuss the existence of limit at x = 1 and x = 1. {Ans. exists, does not exist} 424. If ,2,1,0,,sin)( nnxxxf
nx ,2
and 2,0,1)( 2 xxxg 0,4 x 2,5 x
then find )]([lim0
xfgx
. {Ans. 1}
425. limcos cos ... cos
x
x x nx
n
x
0
212
{Ans.
12
121
nn
e }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.32
CATEGORY-2.16. SANDWICH THEOREM
426. If ecxxfx cossin in the neighbourhood of 2
, find xf
x2
lim
. {Ans. 1}
427. If xxfx 2 in the neighbourhood of 0, find xfx 0lim
. {Ans. 0}
428. If xxxf , find xfx 0lim
. {Ans. 0}
429. If 0ln xxxf , find xfx lim . {Ans. }
CATEGORY-2.17. FINDING LIMIT OF A FUNCTION CONTAINING PARAMETERS
430. limx
x
x
a
aa
10
Ans
a
a
a
a
a
a
.
,
,
,
,
,
,
,
1 1
0 1
1
0 1
1 1
112
12
431. limx
x x
x x
a a
a aa
0
Ans
a
a
a
a
a
a
.
,
,
,
,
,
,
,
1 1
1 1
0 1
1 1
1 1
0 1
432. 11lim 2222
xaaxxax
.
0,2
1
0,0
0,2
1
.
a
a
a
Ans
433. x
ax x
11lim .
1,
1,
1,1
.
a
ae
a
Ans
434. limh
x h x
h
0
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.33
0,2
1
0,
.
xx
x
Ans
435.
m
n
x x
x
sin
sinlim
0 (m and n are positive integers)
oddismnmn
evenismnmn
mn
mn
Ans
&0,
&0,
,1
0,0
.
436. How do the roots of the equation ax bx c2 0 change when b and c retain constant values b 0 and a
tends to zero. {Ans. One root tends to b
c and the other root tends to }
437. x
b
x a
x
lim
10&0,
1&0,0
1&0,1
10&0,
1&0,0
.
ab
ab
ab
ab
ab
Ans
438.
limx
n
n
ax
x A
1. Consider separately the cases when n is (1) positive integer, (2) negative integer, (3) zero.
0,1function,anot
0,1,)3(
0,0,0,
0,0,
0,0,0,
0,0,0,0)2(
0,)1(
.
11
1
nA
nA
naA
naA
naAa
naA
na
Ans
A
A
n
n
CATEGORY-2.18. FINDING VALUES OF THE PARAMETERS GIVEN THE VALUE OF LIMIT
439. Find the constants a and b such that 01
1lim
2
bax
x
xx
{Ans. a = 1, b = -1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.34
440. If
bax
x
xx 1
1lim
2
, find the value of a and b. {Ans. Rba ,1 }
441. If 21
1lim
2
bax
x
xx
, find the values of a and b. {Ans. 3,1 ba }
442. If 21
1lim
2
3
bax
x
xx
, then find a and b. {Ans. 2,1 ba }
443. Find the constants a and b such that 01lim 2
baxxxx
{Ans. a = -1, b = 12 }
444. If 1lim30
x
cxbeae xx
x, find the values of a, b, c. {Ans. 6,3,3 cba }
445. Find the values of a, b, c so that 2sin
coslim
0
xx
cexbae xx
x. {Ans. 1,2,1 cba }
446. If 30
)1ln(sinlim
x
xcbeaex xx
x
is finite, find a, b, c and the value of the limit. {Ans.
a b c 12
12
130, , , }
447. If 40
2cos4coslim
x
bxaxx
is finite, find a, b and the value of the limit. {Ans. a = -4, b = 3, 8}
448. If 2
2
1
sinlnlim
x
axxxxe x
x
is finite, find a and the value of the limit. {Ans. a = 0, 0}
449. If 3
2
0
sin)1ln(lim
x
xbaxxx
is finite, find a, b and the value of the limit. {Ans. a b 2 5 2
6, ,
a b 2 5 26, }
450. If 50
sinh2sinh3sinhlim
x
xbxaxx
is finite, find a, b and the value of the limit. {Ans. a = -4, b = 5, 1}
451. Given
0,sin
xx
xxf
0, xbax .
If xfx 0lim
exists, find a, b and the value of the limit. {Ans. Ra , b = 1, 1}
452. If
0sintan
lim20
x
xxnxnax
, where n is non-zero real no., then find the value of a. {Ans. n
n1
}
CATEGORY-2.19. FINDING THE LIMIT AS FUNCTION OF PARAMETER
453. Sketch the graph of the function
0,1lim)(
xxxf n n
n.
Ans
f x x
x x
.
( ) ,
,
1 0 1
1
454. Sketch the graph of the function f x xn
n( ) limsin
2 .
PART-I/CHAPTER-2
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.35
2)12(,0
2)12(,1)(
.
mx
mxxf
Ans
455. Determine the function
nn
xxxxf
2cos
4cos
2coslim .
0,1
0,sin
)(Ans.
x
xx
xxf
456. Determine the function n
mnxmxf !coslimlim)(
.
irrational is ,0
rational is ,1)(
Ans.
x
xxf
CATEGORY-2.20. ADDITIONAL QUESTIONS
457. If )()(lim xgxfax
exists, then both )(lim xfax
and )(lim xgax
exists. {Ans. False}
458. Find the value of k if 22
334
1lim
1
1lim
kx
kx
x
xkxx
. {Ans.
3
8}
459. If 9lim99
ax
axax
, find the value of a. {Ans. 1 }
460. Let 218
1
xxf
. Find the value of
3
3lim
3
x
fxfx
. {Ans. 9
1}
461. Find the polynomial function xf of least degree satisfying 2
1
301lim e
x
xf x
x
. {Ans. 42x }
462. Find a polynomial xf of least degree such that 2
1
202lim e
x
xf x
x
. {Ans. 232 xxxf }
463. Evaluate 30
sinlim
x
xxx
without using series and L’ Hospital’s rule. {Ans.
6
1}
464. The function xf satisfies the condition
2
51
3
1
xfxfxf . Find xf
x lim . {Ans.
2
5 }
465. Find
nnnn
n
1cos1
1cos1
1cos1lim 2 . {Ans.
2
1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
2.36
466. Find
x
x
xx 24
sin
cos1coscos1cos1cos1coslim
222222
0
. {Ans. 2 }
467. 1
2)12(4321lim
2
n
nnn
. {Ans. 1}
468. n
xxxxn
242 1111lim
, where 1x . {Ans. x1
1}
469. Given nxxfn
1tanlim
; xxg n
n
2sinlim
and xfxgxh 2coscos2
1 . Find the function
xh . {Ans.
2
12,1
nxxh
0,2
12,0
nx
0,1 x }
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-I
DIFFERENTIAL CALCULUS
CHAPTER-3
CONTINUITY
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.2
CHAPTER-3
CONTINUITY
LIST OF THEORY SECTIONS
3.1. Continuity At A Point 3.2. Continuity In A Set/ Domain
LIST OF QUESTION CATEGORIES
3.1. Continuity At A Point By First Principles 3.2. Right Hand And Left Hand Continuity 3.3. Removable And Non-Removable Discontinuity 3.4. Continuity In A Set/ Domain By First Principles 3.5. Continuity By Theorems 3.6. Intermediate Value Theorem 3.7. Additional Questions
PART-I/CHAPTER-3
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.3
CHAPTER-3
CONTINUITY
SECTION-3.1. CONTINUITY AT A POINT
1. Continuity of a function at a finite point i. A function xf is said to be continuous at a finite point a iff:-
a. xfax
lim exists,
b. af is defined,
c. xfax
lim and af are both equal, i.e. afxfax
lim ,
otherwise xf is said to be discontinuous at the point a and the point a is called a point of discontinuity.
2. Right hand and Left hand continuity at a finite point i. A function xf is said to be continuous from the right at a finite point a iff:-
a. xfax
lim is finite,
b. af is defined,
c. xfax
lim and af are both equal, i.e. afxfax
lim ,
otherwise xf is said to be discontinuous from the right at the point a.
ii. A function xf is said to be continuous from the left at a finite point a iff:-
a. xfax
lim is finite,
b. af is defined,
c. xfax
lim and af are both equal, i.e. afxfax
lim ,
otherwise xf is said to be discontinuous from the left at the point a. 3. Continuity of a function at the end points of the Domain
i. If a function xf is not defined in a left (right) hand neighbourhood of a finite point a and is continuous
from the right (left) at the point a then the function xf is said to be continuous at the point a,
otherwise xf is said to be discontinuous at the point a. 4. Checking continuity of a function at a point by first principles
i. Graphical a. If the curve of the function xf is unbroken at the point a then xf is continuous at a and vice-
versa. b. If the curve of the function xf is broken at the point a then xf is discontinuous at a and vice-
versa. ii. Analytical
a. To check continuity at the point a, check whether af is defined or not, xfax
lim exists or not and
afxfax
lim or not.
5. Removable and non-removable discontinuities
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.4
i. A point a is called removable discontinuity of the function xf if xfax
lim exists but either af is not
defined or af is different from xfax
lim . This removable discontinuity can be removed and the
function can be made continuous at a by defining the value of af equal to the value of xfax
lim .
ii. A point a is called non-removable discontinuity of the function xf if xfax
lim does not exist.
SECTION-3.2. CONTINUITY IN A SET/ DOMAIN
1. Continuity of a function in a set i. A function xf is said to be continuous on a set A RA if xf is continuous at every point of set
A, otherwise xf is said to be discontinuous on the set A.
ii. If Axxfhxfh
0lim0
then xf is continuous on set A.
2. Continuity of a function in it’s Domain i. A function xf is said to be a continuous function if xf is continuous at every point in its domain,
otherwise xf is said to be a discontinuous function.
ii. If fh
Dxxfhxf
0lim0
then xf is a continuous function.
3. Checking continuity of a function in a set/Domain by first principles i. Graphical
a. If the curve of the function xf is unbroken in a set/domain then xf is continuous in the set/domain and vice-versa.
b. If the curve of the function xf is broken in a set/domain then xf is discontinuous in the set/domain and vice-versa.
ii. Analytical a. Check whether Axxfhxf
h
0lim
0 or not without using Theorem of limit of basic
functions and L’ Hospital’s rule. 4. Theorem of continuity of Basic functions
i. All Basic functions are continuous functions, i.e. all basic functions are continuous at every point in their domain..
5. Theorems of continuity and its converse i. If the functions xf and xg are continuous at a point/set then:-
a. the function xgxf must be continuous at the point/set;
b. the function xgxf must be continuous at the point/set;
c. the function xgxf must be continuous at the point/set;
d. the function xg
xf must be continuous at the point/set provided 0xg at the point/set.
ii. If xf is continuous at the point a and xg is continuous at the point afx then the composite
function xfg is continuous at the point a.
iii. If xf is continuous in the set A and the values of xf is the set B and xg is continuous in the set B
then the composite function xfg is continuous in the set A. 6. Applications of Theorems of continuity
PART-I/CHAPTER-3
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.5
7. Cases where theorems of continuity are not applicable 8. Checking continuity by theorems
i. For checking the continuity of a function in a set, identify the points at which theorems of continuity are applicable and the points at which theorems of continuity are not applicable. At the points where theorems of continuity are applicable, the function is continuous. At the points where theorems of continuity are not applicable, check each point for continuity by first principles.
9. Intermediate value theorem i. If xf is continuous on a closed interval ba, then it assumes all intermediate values between af
and bf within the interval, i.e. if xf be continuous on a closed interval ba, and let Aaf and
Bbf and if C is any value between A and B then there exists at least one point ba, such that
Cf .
ii. If xf be continuous on a closed interval ba, and if af and bf be of opposite signs, then there
exists at least one point in the open interval ba, at which the value of xf is zero. iii. If the value of a function is zero at a point then that point is called a zero of the function.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.6
EXERCISE-3
CATEGORY-3.1. CONTINUITY AT A POINT BY FIRST PRINCIPLES
1. Given
0,)1()(1
xxxf x 0, xe . Check continuity at x = 0. {Ans. continuous}
2. Given
0,1
sin)(
x
xxxf
0,0 x . Check continuity at x = 0. {Ans. continuous}
3. Given
0,1
)(
xx
exf
x
0, xe . Check continuity at x = 0. {Ans. discontinuous}
4. Given
0,3sin
)( xx
xxf
0,1 x . Check continuity at x = 0. {Ans. discontinuous}
5. Given
1),32(5
1)( 2 xxxf
31,56 xx 3,3 xx . Check continuity at x = 1 and x = 3. {Ans. continuous, discontinuous}
6. Given
4,
2cos
sincos)(
x
x
xxxf
4
,2
1 x .
Check continuity at 4
x . {Ans. continuous}
7. Check the function
2
,
2
cos
xx
xxf
2
,1
x
PART-I/CHAPTER-3
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.7
for continuity at 2
x . {Ans. Continuous}
8. Given
0,11
)(3
xx
xxxf
0,6
1 x .
Check continuity at x = 0. {Ans. continuous} 9. Given
0,0)( xxf 10, xx
31,242 xxx 3,4 xx . Check continuity at x = 0, 1 and 3. {Ans. continuous}
10. A function )(xf is defined as
1,1
34)(
2
2
x
x
xxxf
1,2 x . Test the continuity of the function at x = 1. {Ans. discontinuous}
11. Construct the graph of the function given below 0,1)( xxxf
0,4
1 x
0,2 xx .
Find )(lim0
xfx
and )(lim0
xfx
discuss the continuity of )(xf at x = 0. {Ans. 0, 1, discontinuous}
12. Let 1,112
1)(
2
2
x
xx
xxf
1,2
1 x .
Discuss the continuity of function at x = 1. {Ans. discontinuous} 13. For what value of a will the function f (x) be continuous at x = 1
1,1)( xxxf
1,3 2 xax . {Ans. a = 1}
14. Let 2,)2(
2016)(
2
23
x
x
xxxxf
,k 2x . If )(xf is continuous for all x, then find k. {Ans. 7}
15. Let 0,4
sin2
1
xxxf
x
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.8
= K, 0x . If xf is continuous at 0x , then find K. {Ans. 0}
16. 0,4cos1
)(2
xx
xxf
0, xa
.0,416
xx
x
If possible find the values of a so that the function may be continuous at x = 0. {Ans. 8} 17. For what value of a, the function
0,1
sin xx
xxf a
0,0 x is continuous at 0x . {Ans. 0a }
18. Choose A and B so as to make the function f (x) continuous at 2
x
2,sin2)(
xxxf
22
,sin
xBxA
2
,cos
xx . {Ans. A = -1, B = 1}
19. Let 06
, sin 1)( sin xxxf x
a
0, xb
6
0,3tan
2tan
xe x
x
.
Determine a and b such that )(xf is continuous at x = 0. {Ans. 3
2
,3
2e }
20. Given the function
2
0,5
6 5tan
6tan
xxf
x
x
2
,2
xb
xx b
xa
2,cos1
tan
.
Determine the values of a and b, if xf is continuous at 2
x . {Ans. 0a , 1b }
21. Determine a, b and c for which the function
0,sin)1sin(
)(
xx
xxaxf
PART-I/CHAPTER-3
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.9
0, xc
0,)(23
21
21
2
xbx
xbxx
is continuous at x = 0. {Ans. 21
23 ,0, cRba }
22. Given the function
0,sintan
tansin
x
xx
aaxf
xx
0,cossec
1ln1ln 22
x
xx
xxxx.
If xf is continuous at 0x , find the value of a. Now, axaxa
xxg
,cot2ln . If xg is
continuous at ax , show that ee
g
1. {Ans.
ea
1 }
23. Given 0,1)( xxf
10, xx 1,0 x 21,3 xx 2,2 x
0,0)( xxg 10,1 xx 1,0 x 21,2 xx 2,1 x . Check the continuity of the following functions at the indicated points:- i. xgxf )( at 0x {Ans. Continuous}
ii. xgxf )( at 1x {Ans. Discontinuous}
iii. xgxf )( at 2x {Ans. Discontinuous}
iv. xgxf )( at 0x {Ans. Continuous}
v. xgxf )( at 1x {Ans. Discontinuous}
vi. xgxf )( at 2x {Ans. Discontinuous}
vii. )( xgf at 0x {Ans. Discontinuous}
viii. )( xgf at 1x {Ans. Continuous}
ix. )( xgf at 2x {Ans. Continuous}
x. )( xfg at 0x {Ans. Discontinuous}
xi. )( xfg at 1x {Ans. Discontinuous}
xii. )( xfg at 2x {Ans. Continuous}
xiii. )( xff at 0x {Ans. Continuous}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.10
xiv. )( xff at 1x {Ans. Continuous}
xv. )( xff at 2x {Ans. Continuous}
xvi. )( xgg at 0x {Ans. Discontinuous}
xvii. )( xgg at 1x {Ans. Discontinuous}
xviii. )( xgg at 2x {Ans. Discontinuous}
CATEGORY-3.2. RIGHT HAND AND LEFT HAND CONTINUITY
24. Given
0,
1)(
11
xe
xxf
xx
0,1
xe
0,sincosln11
xx x . Check the left hand and right hand continuity at x = 0. {Ans. Continuous from the left but discontinuous from the right}
25. Find the constant A such that the function
0,tan
1)(
4
xx
exf
x
0,1672 xAA 0, xA is continuous from the left but discontinuous from the right at x = 0. {Ans. A = 3}
CATEGORY-3.3. REMOVABLE AND NON-REMOVABLE DISCONTINUITY
26. Define the following functions at x = 0 so as to make them continuous:-
i. x
xxxf
2
35)(
2 {Ans. 2
3 }
ii. x
xxf
cos1)(
{Ans. Non-removable discontinuity}
iii. x
xxxf
)1ln()1ln()(
{Ans. 2 }
iv. x
xxf
2sin
42)(
{Ans. 8
1 }
v. 11
1sincos)(
2
22
x
xxxf {Ans. 4 }
vi. x
xxf
2sin
cos1)(
{Ans. 2
1 }
vii. xxxf cot)1()( . {Ans. e}
PART-I/CHAPTER-3
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.11
CATEGORY-3.4. CONTINUITY IN A SET/ DOMAIN BY FIRST PRINCIPLES
27. Prove the continuity of the following functions by first principles:- i. nxxf )(
ii. x
xf1
)(
iii. xexf )( iv. xxf ln)( v. xxf sin)( vi. xxxf sin)( vii. xxxf lncos)(
28. If yxyfxfyxf ,)()()( and )(xf is continuous at x = 0, then show that )(xf is continuous .x 29. If f x y f x f y x y( ) ( ) ( ) , and f x( ) is continuous at x = 0, then show that f x( ) is continuous x. 30. If 0,)()()( yxyfxfxyf and f x( ) is continuous at x = 1, then check the continuity of f x( ) .
{Ans. Continuous for all x except x = 0} 31. If yxyfxfxyf ,)()()( and )(xf is continuous at x = 1, then show that )(xf is continuous for all x
except x = 0. 32. If f x y f x f y x y( ) ( ) ( ) , and f x( ) is continuous at a point x = a, then show that f x( ) is continuous
x.
33. If yxyfxfyx
f ,2
)()(
2
and f x( ) is continuous at x = 0, then check the continuity of f x( ) .
{Ans. Continuous for all x} 34. If yxfyfxfyxf ,0222 and f x( ) is continuous at x = 0, then check the continuity of
f x( ) . {Ans. Continuous for all x}
35. If
yxfyfxfyx
f ,3
0
3
and f x( ) is continuous at x = 0, then check the continuity of
f x( ) . {Ans. Continuous for all x}
36. If
yxyfxfyx
f ,3
2
3
2
and f x( ) is continuous at x = 0, then check the continuity of f x( ) .
{Ans. Continuous for all x} CATEGORY-3.5. CONTINUITY BY THEOREMS
37. Show that a polynomial function 011
1 ...... axaxaxaxP nn
nnn
is a continuous function.
38. Show that a rational function is a continuous function.
39. Test the following functions for continuity
xe
xxxxf
x cosh
tansinlncos)(
13
. {Ans. continuous}
40. Check the continuity of x and xsgn . {Ans. Continuous, Discontinuous at x = 0}
41. Test the following functions for continuity 0,0)( xx
2
10,
2
1 xx
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.12
2
1,
2
1 x
12
1,
2
3 xx
1,1 x .
Also draw the graph of the function. {Ans. Discontinuous at 1,2
1,0 }
42. Find the values of a and b so that the function
40,sin2)(
xxaxxf
24
,cot2
xbxx
xxbxa2
, sin2cos
is continuous for x0 . {Ans. 12
,6
ba }
43. Given the function x
xf
1
1)( . Find the points of discontinuity of the function )(),( xffxf &
)(xfff . {Ans. 1; 0 & 1; 0 & 1}
44. Let 1
1
xxf &
2
12
xx
xg . Find the points where xfg is discontinuous. {Ans. 2
1, 1, 2}
45. Given 0,1)( xxxf
0,1 xx .
Test the function 2)()( xfx for continuity. {Ans. continuous}
46. Investigate the functions xgf and xfg for continuity if xxf sgn)( and ).1()( 2xxxg {Ans. discontinuous at 1,0 x , continuous}
47. Given 20,1 xxxf
32,3 xx .
Test the function xff for continuity. {Ans. Continuous 2,1x & discontinuous at 2,1x } 48. Show that
rationalis,1)( xxf irrationalis,1 x is discontinuous for all x.
49. Given rationalis,)( xxxf
irrationalis, xx . Show that )(xf is continuous at x = 0 only.
50. Test the function for continuity
PART-I/CHAPTER-3
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.13
2,1,02
1
2
1,
2
1)(
1
nxxf
nnn
0,0 x . {Ans. Discontinuous at ,.......3,2,1,2
1 nx
n}
51. Discuss the continuity of the function
1sin1
1sin1lim)(
n
n
n x
xxf
at the point x = 1. {Ans. Discontinuous}
52. Check the continuity of the function
n
n
n x
xxxxf
2
2
1
sin2loglim
. {Ans. Discontinuous at 1x }
53. If )(xf and )(xg are two functions continuous everywhere and
n
n
n x
xgxxfxh
2
2
1lim)(
, then prove
that )(xh is continuous everywhere except at 1,1 x . Find the condition on )(xf and )(xg which
makes )(xh continuous everywhere. {Ans. 1)1( gf & 1)1( gf }
54. If yxyfxfyxf ,2 2 and f x( ) is continuous at x = 0, then check the continuity of f x( ) .
{Ans. Continuous for all x} 55. Can one assert that the square of a discontinuous function is also a discontinuous function? Give an
example of a function discontinuous everywhere whose square is a continuous function. {Ans. No} 56. Let )(xf be a continuous and )(xg be a discontinuous function. Prove that )(xgxf is a
discontinuous function. CATEGORY-3.6. INTERMEDIATE VALUE THEOREM
57. Does the function 3sin4
)(3
xx
xf take the value 3
12 within the interval 2,2 ? {Ans. Yes}
58. Given 1)( 3 xxxf , show that )(xf has a zero in the interval 0,1 .
59. Show that the equation 0135 xx has at least one root lying between 1 and 2. 60. Show that the equation 12 xx has at least one positive root not exceeding 1. 61. Show that 133 xxxf has a zero in the interval 2,1 .
62. Does the equation 02185 xx has a root in the interval 1,1 ? {Ans. Yes} 63. Does the equation 01sin xx has a root? {Ans. Yes} 64. Show that the equation bxax sin , where 10 a , 0b , has at least one positive root which does
not exceed ba . 65. Show that 0ln xx has a solution in the interval 1,0 .
66. Show that the equation 03
3
2
2
1
1
x
a
x
a
x
a, where 01 a , 02 a , 03 a and 321 , has
two real roots lying in the intervals 21, and 32 , .
67. Given 02,1)( 2 xxxf
20,12 xx .
Is there a point in the interval 2,2 at which 0)( xf ? {Ans. No}
68. The function )(xf is continuous in the interval ba, and has values of the same sign on its end-points.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
3.14
Can one assert that there is no point in ba, at which the function becomes zero? {Ans. No}
69. Show that the function 145 xxxf has at least two zeros in the interval 2,0 .
70. If xf be continuous in ba, and the equation 0xf has only two roots and ( ) in the
interval ba, , then prove that xf retains the same sign in the interval , . 71. Let )(xf be a continuous function defined for 31 x . If )(xf takes rational values for all x and
10)2( f , then find )5.1(f . {Ans. 10}
72. Let the function )(xf be continuous in the interval ba, and baxbxfa ,)( . Prove that in this close interval there exists at least one point at which xxf )( . Explain this geometrically.
73. If xf is continuous and 10 ff then prove that there exists
2
1,0c such that
2
1cfcf .
74. Let the function )(xf be continuous in the interval ba, . Prove that in this close interval there exists at
least one point at which
2)(
bfafxf
.
75. Prove that if the function )(xf is continuous in the interval ba, and 1x , 2x , ………, nx are any values
in this open interval, then we can always find a real number c in this open interval such that
n
xfxfxfcf n
.........21 .
CATEGORY-3.7. ADDITIONAL QUESTIONS
76. If xf is continuous and 9
2
2
9
f then find
20
3cos1lim
x
xf
x. {Ans.
9
2}
77. If xf is continuous in 1,0 and 13
1
f then find
19lim
2n
nf
n. {Ans. 1}
78. If xf and xh are continuous functions and
1,332
1lim
x
xx
xhxfxxg
m
m
m and x
xexg ln
2
1lnlim1
and xg is continuous at 1x , then find the value of 11212 hfg . {Ans. 1}
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-I
DIFFERENTIAL CALCULUS
CHAPTER-4
DERIVATIVES
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.2
CHAPTER-4
DERIVATIVES
LIST OF THEORY SECTIONS
4.1. Differentiability And First Derivative At A Point 4.2. Differentiability In A Set/Domain And First Derivative Function 4.3. Higher Order Derivatives
LIST OF QUESTION CATEGORIES
4.1. First Derivative Of A Function At A Point By First Principles 4.2. First Derivative Function Of A Function By First Principles 4.3. Differentiating Explicit Functions 4.4. Logarithmic Differentiation 4.5. Differentiating Implicit Functions 4.6. Differentiating Parametric Functions 4.7. Differentiating A Function W.R.T. Another Function 4.8. First Derivative By Differentiation And By First Principles 4.9. Higher Order Derivatives Of Explicit Functions By Differentiation 4.10. Higher Order Derivatives Of Implicit Functions By Differentiation 4.11. Higher Order Derivatives Of Parametric Functions By Differentiation 4.12. Leibniz Theorem 4.13. Miscellaneous Questions On Differentiation 4.14. Higher Order Derivatives By Differentiation And By First Principles 4.15. Additional Questions
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.3
CHAPTER-4
DERIVATIVES
SECTION-4.1. DIFFERENTIABILITY AND FIRST DERIVATIVE AT A POINT
1. Differentiability and First derivative of a function at a point i. The first derivative of the function xf at a finite point a, denoted by af , is defined as
h
afhaf
ax
afxfaf
hax
0limlim if this limit exists. If this limit exists then it is said that the
function xf is differentiable at the point a and the value of this limit is the value of af . If this limit
does not exist then it is said that the function is not differentiable at the point a and af does not exist. 2. Right hand & left hand derivative at a point
i. The right hand derivative of the function xf at a finite point a, denoted by af , is defined as
h
afhaf
ax
afxfaf
hax
0limlim if this limit is finite. If this limit is not finite then
af has no value.
ii. The left hand derivative of the function xf at a finite point a, denoted by af , is defined as
h
hafaf
ax
afxfaf
hax
0limlim if this limit is finite. If this limit is not finite then
af has no value. 3. Differentiability and First derivative at the end points of the domain
i. If a function xf is not defined in a left hand neighbourhood of a finite point a and af is finite,
then it is said that the function xf is differentiable at the point a and the value of af is the value
of af , otherwise it is said that the function is not differentiable at the point a and af does not exist.
ii. If a function xf is not defined in a right hand neighbourhood of a finite point a and af is finite,
then it is said that the function xf is differentiable at the point a and the value of af is the value
of af , otherwise it is said that the function is not differentiable at the point a and af does not exist.
4. Continuity is a necessary condition for differentiability i. Continuity is a necessary but not sufficient condition for differentiability. ii. If a function is discontinuous at a point then it cannot be differentiable at that point. iii. If a function is differentiable at a point then it must be continuous at that point.
5. Checking differentiability and finding first derivative at a point by first principles i. Graphical
a. If the curve of the function xf is unbroken and smooth at the point a and the tangent at the point a
is not vertical (not perpendicular to x-axis) then the function xf is differentiable at a and af = slope of tangent at a.
b. If the curve of the function xf is unbroken and smooth at the point a and the tangent at the point a
is vertical (perpendicular to x-axis) then the function xf is continuous but not differentiable at a.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.4
c. If the curve of the function xf is unbroken but not smooth at the point a then the function xf is continuous but not differentiable at a.
d. If the curve of the function xf is broken at the point a then the function xf is discontinuous and not differentiable at a.
ii. Analytical a. Find af by finding the limit given in the definition without using L’ Hospital’s rule.
SECTION-4.2. DIFFERENTIABILITY IN A SET/DOMAIN AND FIRST DERIVATIVE FUNCTION
1. Differentiability in a set i. A function xf is said to be differentiable on a set A RA if xf is differentiable at every point of
set A, otherwise it is said that xf is not differentiable on the set A. 2. Differentiability in the domain
i. A function xf is said to be a differentiable function if xf is differentiable at every point in its
domain, otherwise it is said that xf is not a differentiable function. 3. First derivative function
i. The First derivative function of a function xf , denoted by xf , is defined as
xh
xfhxfxf
h
0lim at which this limit exists.
ii. The value of xf at a point is the value of first derivative of xf at that point. iii. Domain of First derivative function is the set of all the points at which the function is differentiable. iv. If a function xf is a differentiable function then the domain of xf is same as the domain of the
function xf , otherwise domain of xf is a subset of the domain of the function xf . 4. Symbolic notations of first derivative
i. First derivative function is usually denoted by xf or xfdx
d.
ii. First derivative at a point a is usually denoted by af or ax
xfdx
d
.
5. Checking differentiability in a set/domain and finding first derivative function by first principles i. Graphical
a. If the curve of the function xf is unbroken and smooth and tangent is not vertical in a set/domain
then xf is differentiable in the set/domain, otherwise xf is not differentiable in the set/domain.
b. Plot the graph of xf by plotting slope of tangent at the points on the curve of the function xf . ii. Analytical
a. Find xf by finding the limit given in the definition without using L’ Hospital’s rule. 6. Process of differentiation
i. Differentiability of Basic functions a. All basic functions are differentiable functions except the following basic functions at the indicated
points:- 10 axa at 0x ;
x1sin at 1x ; x1cos at 1x ;
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.5
xec 1cos at 1x ; x1sec at 1x .
ii. First derivative function of basic functions a. cxf , 0 xf
b. axxf , 1 aaxxf
c. xexf , xexf
d. xaxf , aaxf x ln
e. xxf ln , 0,1
xx
xf
f. xxf alog , 0,ln
1 x
axxf
g. xxf sin , xxf cos
h. xxf cos , xxf sin
i. xxf tan , xxf 2sec
j. ecxxf cos , xecxxf cotcos
k. xxf sec , xxxf tansec
l. xxf cot , xecxf 2cos
m. xxf 1sin , 21
1
xxf
n. xxf 1cos , 21
1
xxf
o. xxf 1tan , 21
1
xxf
p. xecxf 1cos , 1
12
xx
xf
q. xxf 1sec , 1
12
xx
xf
r. xxf 1cot , 21
1
xxf
s. xxf sinh , xxf cosh
t. xxf cosh , xxf sinh
u. xxf tanh , xxf 2sech
v. xxf cosech , xxxf cothcosech
w. xxf sech , xxxf tanhsech
x. xxf coth , xxf 2cosech iii. Theorems (Rules) of differentiation
If the functions xf and xg are differentiable at a point/set then:-
a. the function xgxfx must be differentiable at the point/set and xgxfx ;
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.6
b. the function xgxfx must be differentiable at the point/set and
xgxfxgxfx ;
c. the function xg
xfx must be differentiable at the point/set and
2xg
xgxfxgxfx
if 0xg . iv. Theorems (Rules) of differentiation for composite functions (Chain rule)
a. If xf is differentiable at the point a and xg is differentiable at the point afx then the
composite function xfgx must be differentiable at the point a and afafga .
b. If xf is differentiable in the set A and the values of xf is the set B and xg is differentiable in
the set B then the composite function xfgx must be differentiable in the set A and
xfxfgx or dx
duu
du
dx
dx
d , where xfu .
7. Differentiating functions by Process of differentiation i. Explicit functions ii. Logarithmic differentiation iii. Implicit function iv. Parametric function v. Differentiation of one function w.r.t. another function
8. Checking differentiability and finding first derivative by differentiation and by first principles i. For checking the differentiability of a function in a set, identify the points at which theorems of
differentiation are applicable and the points at which theorems of differentiation are not applicable. At the points where theorems of differentiation are applicable, find first derivative by differentiation. At the points where theorems of differentiation are not applicable, find first derivative by first principles.
SECTION-4.3. HIGHER ORDER DERIVATIVES
1. Definition of higher order derivatives of a function i. For 2n , the nth order derivative of xf , denoted by xf n , is the derivative of the 1n th order
derivative of xf , i.e. xfdx
dxf nn 1 .
2. Symbolic notations of higher order derivatives i. Higher order derivative functions of xf are usually denoted by
xf , xf , xf , xf IV , ................
or xf 2 , xf 3 , xf 4 , ...................
or xfdx
d, xf
dx
d2
2
, xfdx
d3
3
, ..................
ii. Higher order derivative functions of xy are usually denoted by
xy1 , xy2 , xy3 , ................
iii. Higher order derivatives of xf at a point a are usually denoted by
af , af , af , af IV , ................
or af 2 , af 3 , af 4 , ...................
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.7
or ax
xfdx
d
, ax
xfdx
d
2
2
, ax
xfdx
d
3
3
, ..................
iv. Higher order derivatives of xy at a point a are usually denoted by
ay1 , ay2 , ay3 , ................
3. Finding higher order derivatives by differentiation i. Explicit functions ii. Implicit functions iii. Parametric functions
4. Leibniz theorem
i.
n
n
nn
n
n
nn
n
nn
n
nn
n
nn
n
n
dx
vduC
dx
vd
dx
duC
dx
vd
dx
udC
dx
dv
dx
udCv
dx
udCuv
dx
d1
1
12
2
2
2
21
1
10
n
n
n
n
n
n
n
n
n
n
dx
vdu
dx
vd
dx
dun
dx
vd
dx
udnn
dx
dv
dx
udnv
dx
ud1
1
2
2
2
2
1
1
!2
1
where
1;!!
!0
n
nnr
n CCrrn
nC .
ii. 0cdx
dn
n
.
iii. mnxnm
mx
dx
d nmmn
n
,)!(
!
mn ,0 .
iv. xxn
n
eedx
d .
v. xnxn
n
aaadx
dln .
vi. nn
n
n
xnxdx
d !11ln 1 .
vii.
2sinsin
nxx
dx
dn
n
.
viii.
2coscos
nxx
dx
dn
n
.
5. Finding higher order derivatives of a function by differentiation and by first principles i. Graphical ii. Analytical
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.8
EXERCISE-4
CATEGORY-4.1. FIRST DERIVATIVE OF A FUNCTION AT A POINT BY FIRST PRINCIPLES
1. Given ,)( 2xxf find )0(f , )1(f & )1(f by first principles. {Ans. 2,2,0 }
2. Given ,)( xxf find )0(f & )1(f by first principles. {Ans. does not exist, 21 }
3. Given ,)( 3 xxf find )0(f by first principles. {Ans. does not exist}
4. Given ,)( axxf show that )0(f does not exist if 0 < a < 1.
5. Given ,)( xexf find )0(f , )1(f & )1(f by first principles. {Ans. ee 1,,1 }
6. Given ,ln)( xxf find )1(f & )(ef by first principles. {Ans. e1,1 }
7. Given ,sin)( xxf find )0(f , )( 2f & )(f by first principles. {Ans. 1,0,1 }
8. Given ,tan)( xxf find )0(f & )( 4f by first principles. {Ans. 2,1 }
9. Given ,sin)( 1 xxf find )0(f , )1(f & )1(f by first principles. {Ans. 1 , does not exist, does not exist}
10. Given ,cos)( 1 xxf find )0(f , )1(f & )1(f by first principles. {Ans. 1 , does not exist, does not exist}
11. Given ,)( 2 xexxf find )0(f & )1(f by first principles. {Ans. e3,0 } 12. Given ,ln)( xxxf find )1(f by first principles. {Ans. 1}
13. Given ,sin)( xexf x find )0(f & )(f by first principles. {Ans. e,1 } 14. Given ),sin(ln)( xxf find )1(f by first principles. {Ans. 1}
15. Given ,)( xxf find )0(f by first principles. {Ans. does not exist}
16. Prove that xxf ln)( is continuous but not differentiable at x = 1.
17. Given 0,)( 2 xxxf
0, xx . Find ).0(f {Ans. does not exist}
18. Given 0,)( 2 xxxf
0,0 x Find ).0(f {Ans. 0 }
19. Given 0,)( 3 xxxf
0,2 xx . Find ).0(f {Ans. 0 }
20. Given 1,13 xxxf
1,1 xx ,
find 1f . {Ans. does not exist}
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.9
21. Given 0,)( xexf x
0,1 xx . Find ).0(f {Ans. 1}
22. Given
0,2
xx
xxf
0,0 x ,
find 0f . {Ans. does not exist}
23. If 0,1 xxf
2
0,sin1
xx ,
find 0f . {Ans. does not exist} 24. Given
1,ln)( xxxf 1),1sin( xx . Find ).1(f {Ans. 1}
25. Given
2,32
21,2
1 ,)(
2
xxx
xx
xxxf
Find )1(f & )2(f . {Ans. does not exist, -1} 26. Show that the function
0 ,0
0,1
sin)(
x
xx
xxf
is continuous but not differentiable at x = 0. 27. Given
0 ,0
0,)(11
11
x
x
ee
eexxf
xx
xx
Show that )(xf is not differentiable at x = 0. 28. Given
0 ,0
0,)(
11
x
xxexfxx
Check continuity and differentiability at x = 0. {Ans. continuous but not differentiable at x = 0} 29. Discuss the limit, continuity and differentiability of the function
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.10
0 ,0
0,
2
43)(
1
1
x
x
e
exxf
x
x
at x = 0. {Ans. limit exists, continuous but not differentiable at x = 0}
30. If 1)( xxf and xxg sin)( then calculate x)fog( and x)gof( and discuss differentiability of
x)gof( at x = 1. {Ans. not differentiable at x = 1 } 31. If
,22,12)(
31,11)(
xxxg
xxxf
then calculate x)fog( and x(gof) . Draw their graphs. Discuss the continuity of x)fog( at x = 1 and differentiability of 1 )gof( at xx . {Ans. continuous, not differentiable}
32. Given 1,)( 3 xxxf
1, xbax . Find the constants a & b such that )1(f exists. {Ans. 2,3 ba }
33. If 1,2 xbaxxf
1,2 xcaxbx ,
where 0b . Find a and c such that xf is continuous and differentiable at 1x . {Ans. ba 2 , 0c } 34. Given
1,ln)( xxxf 1, xbax . Find the constants a & b such that )1(f exists. {Ans. 1,1 ba }
35. Given
0 ,0
0,1
cos)(
x
xx
xxf p
What conditions should be imposed on p so that (i) xf may be continuous at x = 0. {Ans. 0p }
(ii) xf may be differentiable at x = 0. {Ans. 1p } 36. Given
3 ,2
31 ,1
1,1)(
2
xqxpx
xx
xbxaxxf
Find the constants a, b, p and q so that xf is differentiable at x = 1 & x = 3. {Ans. 1a , 0b ,
3
1p , 1q }
37. Given
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.11
02
1,
2sin)( 1
x
cxbxf
0 ,2
1 x
2
10 ,
12
xx
eax
.
If xf is differentiable at x = 0 and 2
1c , then find the value of ‘a’ and prove that 22 464 cb .
{Ans. 1a }
CATEGORY-4.2. FIRST DERIVATIVE FUNCTION OF A FUNCTION BY FIRST PRINCIPLES
38. Find )(xf by first principles:-
i. 2)( xxf {Ans. x2 }
ii. x
xf1
)( {Ans. 21x
}
iii. xxf )( {Ans. x2
1 }
iv. axxf )( {Ans. 1aax }
v. xexf )( {Ans. xe }
vi. xaxf )( {Ans. aax ln }
vii. xxf ln)( {Ans. 0,1 xx }
viii. xxf alog)( {Ans. 0,ln1 xax }
ix. xxf sin)( {Ans. xcos } x. xxf cos)( {Ans. xsin }
xi. xxf tan)( {Ans. x2sec } xii. xcosecxf )( {Ans. xcosecxcot } xiii. xxf sec)( {Ans. tanxxsec }
xiv. xxf cot)( {Ans. xcosec2 }
xv. xxexf )( {Ans. xex 1 } xvi. xxxf ln)( {Ans. xln1 } xvii. xxxf sin)( {Ans. xxx sincos }
xviii. xxf 2sin)( {Ans. xxcossin2 }
xix. xxf ln)( {Ans. xx ln2
1}
xx. )sin(ln)( xxf {Ans. x
xlncos}
xxi. xexf sin)( {Ans. xeSinx cos }
xxii. xxxf )( {Ans. xx x ln1 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.12
xxiii. xxxf1
. {Ans.
2
1
ln1
x
xx x }
39. If 20&, fyxyfxfyxf , then test the differentiability of xf . {Ans. Differentiable x }
40. If 10&, fyxyfxfyxf , then test the differentiability of xf . {Ans. Differentiable x }
41. If 31&, fyxyfxfyxf , then test the differentiability of xf . {Ans. Differentiable x }
42. If 31&0, fyxyfxfxyf , then test the differentiability of xf . {Ans. Differentiable 0x }
43. If 21&, fyxyfxfxyf , then test the differentiability of xf . {Ans. Differentiable 0x }
44. If yfxfyxf for all 30,25,, ffRyx , then find 5f . {Ans. 6} 45. Suppose the function f satisfies the conditions:
(i) )()()( yfxfyxf for all the x and y
(ii) )(1)( xxgxf where 1)(lim0
xgx
.
Show that the derivative )(xf exists and )()( xfxf for all x.
46. Let yfxfyxf and xgxxf 2 for all ,, Ryx where xg is continuous function. Then
find xf . {Ans. 0} CATEGORY-4.3. DIFFERENTIATING EXPLICIT FUNCTIONS
47. 33 )1( xy {Ans.
3 2
231
x
x}
48.
b
k
xay tan {Ans.
b
k
xk
a
2cos}
49. pxy 21 {Ans. pxpx
p
2212 }
50. If )23(tan 21 xxy , find dx
dy,
0
xdx
dy,
1
xdx
dy. {Ans.
22 )23(1
32
xx
x;
5
3 ; 1 }
51. )cos(log10 xxy . {Ans. 10ln)cos(
sin1
xx
x
}
52. xxy 32 coscos3 {Ans. )2(cos2sin2
3xx }
53. 8
tan5
tan5
x
y {Ans. 5
sec2 x}
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.13
54. 3
1
xxy
{Ans.
3 46
21
xxx
x
}
55. xx
y 2sin2
sin {Ans. xx
xx
2sin2
cos2
12cos
2sin2 }
56. xexy cossin {Ans. )sin(cos 2cos xxe x }
57. 3 65 8 xxy {Ans. 3 26
64
)8(
)407(
x
xx}
58. If xey x ln2 , find
dx
dy,
1
xdx
dy. {Ans.
xx
xe x ln2
12
; e
1}
59. 10
1
xxy {Ans.
91)1(5
xx
xx
x}
60. 1
1tan 1
x
xy {Ans.
21
1
x }
61. If
2
1232 xxey x , find dx
dy,
0
xdx
dy. {Ans. 3222 xex ; 0}
62. x
xy
2cos
sin2 2
{Ans. x
x
2cos
2sin22
}
63. 2
1
1
3tan
3
1
x
xy
{Ans.
42
2
1
1
xx
x
}
64. x
yxx22 cottan
{Ans. xx
xxx22 sin
)sincos(2 }
65. 2
cot3
sin 2 xxy {Ans.
2cosec
3sin
2
1
3
2sin
2cot
3
1 22 xxxx }
66. 4
9 5
3
24
x
xy
{Ans.
9 855
5
)24(27
)1831(4
xx
x}
67. )ln( 22 xaxy {Ans. 22
1
ax }
68. xxy 1tan {Ans. )1(2
tan 1
x
xx
}
69. xxy 42 tantan1 {Ans. xxx
xx422
2
tantan1cos
)tan21(tan
}
70. xxy ln2cos {Ans. xxx
xln2sin2
2cos }
71. 2
11
1tan
3
1tan
3
2
x
xxy
{Ans.
6
4
1
1
x
x
}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.14
72. )sin(sin 1 xny {Ans. xn
xn22 sin1
cos
}
73. xy sinsin 1 {Ans. xx
x2sinsin2
cos
}
74. xxy 3sin24
13sin
18
1 86 {Ans. xx 3cos3sin 35 }
75. xxxy 12 sin1 {Ans. 2
1
1
sin
x
xx
}
76. 2
sincos
1 xy
{Ans. 2
1
1
1
2
sinsin
2
1
x
x
}
77. xxxy {Ans. xxxxxx
xxxx
8
421}
78.
x
xy
sin1
costan 1 . {Ans.
2
1 }
79.
4
3
2
2ln
x
xey x . {Ans.
4
12
2
x
x}
80. xy 31cos 1 {Ans. 2932
3
xx }
81.
x
xy
ln1sin 2 {Ans.
x
x
x
x ln12sin
2ln2
}
82. )sin(log 23 xxy {Ans.
3ln)sin(
cos22 xx
xx
}
83. x
xy
1
1tan 1 {Ans.
212
1
x }
84.
2
21
1
1sin
x
xy . {Ans.
21
2
x }
85. x
xxy
21ln
{Ans.
)1(1
122 xxxx
}
86. )(lnsin 1 xxy {Ans. x
x2
1
ln1
1)(lnsin
}
87. x
x
e
ey
1
1tan {Ans.
x
x
x
x
e
e
e
e
1
1sec
)1(
2 22
}
88. xxy 2sin1cos {Ans. x
x2
3
sin1
sin2
}
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.15
89. 2
8.0sin2
12cos4.0
x
xy {Ans.
x
xx
x8.0cos8.0
2
12sin8.0sin
2
12cos8.0 }
90. xxy 10 {Ans.
10ln
2110
xx }
91. x
y2tan
12
{Ans. xx 2sin2tan
42
}
92. x
y
1
1tanln 1 {Ans.
xxx
11
tan)22(
1
12
}
93. 1
1ln
2
xxy {Ans.
1
12
x
}
94. 3 31 xxy {Ans. 3 2)31(32
2
xxx
x }
95. xxy 12 {Ans. x
xx
14
)98(}
96. x
y2sin1
1
{Ans.
32 )sin1(2
2sin
x
x
}
97. 313 tan xxy {Ans. 6
5312
1
3tan3
x
xxx
}
98. xy x sinlogcos {Ans. x
xxxx
cosln
sinlntancoslncot2
}
99. If 21 1sin xxy , find dx
dy,
0
xdx
dy. {Ans.
x
x
1
1; 1}
100. x
xy
41
4sin 1
{Ans.
xx
x
x4sin
41
41
)41(
4 12
}
101. xey ln1
{Ans. xx
e x
2
ln
1
ln }
102.
x
x
e
ey
1ln {Ans.
1
1
xe}
103. xxy tan10 {Ans.
x
xxxx
2tan
costan10ln10 }
104. xecay1
10 coslog
. {Ans.
axxxec
a xec
1021
coslog
log1
1
cos
110
}
105. 22 sinsin xxy {Ans. )sincoscossin(sin2 22 xxxxxx }
106. x
xy
2cos
cos2 {Ans.
xx
x
2cos2cos
sin2}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.16
107. 21
1
x
xxy
{Ans.
22
3
1
1
)1)(1(2
32
x
x
xx
xx
}
108. xx
xy 1tan
2
1
1
1ln
4
1
{Ans.
4
2
1 x
x
}
109. xx
y ln2 {Ans. 2lnln
1ln2
2ln
x
xx
x
}
110. bx
xababxxay
1tan)())(( {Ans.
bx
xa
}
111. xx
xy
cossin2
3sin2
{Ans. x
x
2sin
)1cos2(22
2 }
112. xx
ey
11
{Ans. x
x
exx
1
1
21)1(
1}
113. a
xaxay 122 cos {Ans.
xa
xa
}
114.
22 1
11
ln1xx
xy {Ans.x
x 12 }
115. x
x
x
xy
tan1
cos
cot1
sin 22
{Ans. x2cos }
116. 1
)1ln(2
2
x
xxxy {Ans.
32
2
)1( x
x}
117. )cossin( xxaey ax {Ans. axxea sin)1( 2 }
118. xxey cos1 {Ans. )sin1(cos1 xxe x }
119. xe
y21tan
1
{Ans. 2214
2
))(tan1(
2xx
x
ee
e
}
120. )3cos33(sin xxey x {Ans. xex 3sin10 }
121. 2213 1)2(sin3 xxxxy {Ans. xx 12 sin9 }
122. xe
y
1
1 {Ans.
314 x
x
ex
e
}
123. 21 426
2sin2 xx
xy
{Ans.
242 xx
x
}
124. )sincosln( xexey xx {Ans. xexe
eexxxx
xx
sincos
))(sin(cos
}
125. 2
1
1
tan1
x
xxy
{Ans. 32
1
)1(
tan
x
x
}
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.17
126. )coscos(
1
xxy
{Ans.
)cos(cos
)sin1)(cossin(2 xx
xxx
}
127. xxey x 3cossin {Ans. )tan3cot1(cossin 3 xxxxex }
128. 11 5 969 xy {Ans. 11 105 9
5 4
)69(55
54
x
x
}
129. )1412ln( 2 xxx eeexy {Ans. 14
12 xx ee
}
130. )32ln(1tan 1
xey {Ans.
)32ln(1)]32ln(2)[32(
)32ln(1tan 1
xxx
e x
}
131. xx
x
ee
ey
2
{Ans. )]()(2[)( 2
2
xxxxxx
x
eeeexee
e
}
132. xxxx
y )sin1ln(cot2
tanln {Ans. x
x2sin
)sin1ln( }
133. xxxy 2sin6)4132ln(2 12 {Ans. 24132
40
xx }
134. xxx
xy 12
3
2
tan1ln3
13
{Ans. )1(
124
5
xx
x}
135. 2
1sin221)3(
2
1 12 x
xxxy {Ans. 2
2
21 xx
x
}
136. )1sinln( 2xxxy {Ans. xx
x
xcot
1
12
}
137. xxxy sin1 2 {Ans. 2
22
1
cos)1(sin)21(
x
xxxxx
}
138. 5
4
)1(
)3(2
x
xxy {Ans.
2)1(2
)3)(7332(6
32
xx
xxx }
139. 5 3)1( xxey {Ans. 5 2)1(10
)2(3x
x
xe
xe
}
140. 1lntan
21121
xxx
ex
y {Ans. x
e
xx
xxx 1ln2
1tan
2
12
1
12
}
141. 2
224 tan1
tan1ln
8
3
cos8
sin3
cos4
sinx
x
x
x
x
xy
{Ans.
x5cos
1}
142. x
xxey
x
5
1
ln
tan
{Ans.
xxxx
x
xex
ln
5
tan)1(
11
ln
tan125
1
}
143. 31
132
)(cos
cos)1(
x
xexy
x
{Ans.
xxx
x
xx
x
xex x
122
2
31
132
cos1
3tan
1
323
)(cos
cos)1( }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.18
144. )ln(2
3
2
3)( 22
422
2322 axx
aax
xaaxxy {Ans. 322 )(4 ax }
145. xxxxxy 1221 sin122)(sin {Ans. 21 )(sin x }
146.
2tancosln 1
xx eey {Ans.
xx
xx
ee
ee
}
147.
b
ae
abmy mx1tan
1 {Ans.
mxmx beae
1}
148. 3
12tan
3
1
1
1ln
3
1 1
2
x
xx
xy {Ans.
1
13 x
}
149. x
x
xx
xxy
1
1tan2
11
11ln 1 {Ans.
x
x
x
1
11}
150. 35 2 4
5
x
xy {Ans.
3 5 222
2
4)5()4(15
20103
xxx
xx}
151.
3
12tan
3
12tan
32
1
1
1ln 114
2
2 xx
xx
xxy {Ans.
1
124 xx
}
152. 1
1cos
2
21
n
n
x
xy {Ans.
1
22
1
n
n
x
nx if n is even and ,
)1(
22
n
n
xx
nx if n is odd.}
153. 3
14tan
6
3
421
)21(ln
12
1
811
2
2
3
x
xx
x
x
xy {Ans.
23
3
)81(
24
x
x
}
CATEGORY-4.4. LOGARITHMIC DIFFERENTIATION
154. 2xxy {Ans. )1ln2(12
xxx }
155. xxxy {Ans. )ln(ln. 12
xxx xxxx
x
}
156. xxy cos)(sin {Ans.
xx
x
xx x sinlnsin
sin
cos)(sin
2cos }
157. 2cot)2(tanx
xy {Ans.
2sin2
2tanln
4sin2
cot4)2(tan
2
2cot
xx
x
x
xx
}
158. xxy )(ln {Ans.
x
xx x lnln
ln
1)(ln }
159. xxy2
)1( {Ans.
22 )1ln(
)1(
1)1(2
x
x
xxxx }
160. xexy x 2sin23 {Ans. )2cot223(2sin 22 2
xxxxex x }
161. 3
32
)5(
1)2(
x
xxy {Ans.
3 24
2
)1()5(3
)111)(2(2
xx
xxx}
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.19
162. xxy ln {Ans. xx x ln2 1ln }
163. 5 2
43
)3(
2)1(
x
xxy {Ans.
5 2
422
)3(
2)1(.
)3)(2(20
36130257
x
xx
xx
xx}
164. xexxy 1sin {Ans.
x
xx
e
ex
xexx
1.
2
1cot
11sin
2
1}
165. x
xy
1
1
sin1
sin1
{Ans.
x
x
xx1
1
212 sin1
sin1
]1)[(sin1
1
}
166. xxy1
{Ans. )ln1(2
1
xx x
}
167. xxy sin {Ans.
x
xxxx x sin
lncossin }
168. x
x
xy
1 {Ans.
1ln
1
1
1 x
x
xx
xx
169. xxy 2 {Ans. )ln2(2
1
xxx
}
170. xxy sin2 )1( {Ans.
)1ln(cos
1
sin2)1( 2
2sin2 xx
x
xxx x }
171. 322
2
)1(
)1(
x
xxy {Ans. 3
22
2
4
24
)1(
)1(
)1(3
16
x
xx
xx
xx}
CATEGORY-4.5. DIFFERENTIATING IMPLICIT FUNCTIONS
172. 0223 yyxx {Ans. yx
xyx
2
232
2
}
173. cex x
y
ln {Ans. x
y
ex
y }
174. 0410422 yxyx {Ans. 5
2
y
x}
175. 32
32
32
ayx {Ans. 3
x
y }
176. 0tan 1 xyy {Ans. 2
11
y }
177. xyee yx {Ans. y
x
e
e
1
1}
178. yxeyx {Ans. 1
1
yx
yx
e
e}
179. 0552 223 yxyxx {Ans. yx
xxy2
22
41
534
}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.20
180. ayxyx {Ans. ayx
yxa
22
22
}
181. 221 lntan yxx
y
{Ans. yx
yx
}
182. 0cossin xeye yx {Ans. xeye
xeyeyx
yx
coscos
sinsin
}
183. exye y {Ans. yex
y
}
184. 0625 22 yxyxyx {Ans. 125
522
yx
yx}
185. If 5cos4sin3 xyxy , then show that x
y
dx
dy .
186. If yxy sin , then find 10
yxdx
dy. {Ans. 1}
187. If yaxy sinsin , then find 0
yxdx
dy. {Ans. asin }
188. If yxyx 222 , then find the value of 1
yxdx
dy. {Ans. -1}
CATEGORY-4.6. DIFFERENTIATING PARAMETRIC FUNCTIONS
189. ),sin( ttax )cos1( tay {Ans. 2
cott
}
190. ,sinsin kttkx kttky coscos {Ans. tk
2
1cot }
191. ,cotln2 tx tty cottan {Ans. t2cot }
192. ,ctex ctey {Ans. cte 2 }
193.
tby
tax3
3
sin
cos {Ans. t
a
btan }
194.
13
133
3
tty
ttx {Ans.
1
12
2
t
t}
195.
)cos(sin
)sin(cos
tttay
tttax {Ans. ttan }
196.
tey
text
t
sin
cos {Ans.
tt
tt
sincos
sincos
}
197. 3, tyex t {Ans. tet 23 } 198. tytx tan ,sec {Ans. tcosec }
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.21
199. tb
tcy
tb
tax
cos1
cos,
cos1
sin
{Ans.
)cos(
sin
tba
tc
}
200. ttytx 12 tan),1ln( {Ans. 2
t}
201. tt
ytx 3
2,+3
2 {Ans. t
t
2
12 }
202. )12(tan, 12
tyex t {Ans. )122(2 2
2
ttt
et
}
203. tbtayt
x cossin,2
tan4 2 {Ans.
2sin4
2cos)sincos( 3
t
ttbta
}
204. Given tytx 2cos1),-(sin 121 , find
4
1
tdx
dy {Ans.
23
3
2
1 }
205. Given 21 1,sin tytx , find
2
1
tdx
dy {Ans.
2
1 }
CATEGORY-4.7. DIFFERENTIATING A FUNCTION W.R.T. ANOTHER FUNCTION
206. Differentiate x
x 11tan
21 w.r.t. x1tan {Ans.
2
1}
207. Differentiate 2
1
1
2tan
x
x
w.r.t. 2
1
1
2sin
x
x
{Ans. 1}
208. Differentiate 12cos 21 x with respect to x1cos . {Ans. 2}
209. Differentiate x
x
1
1sin 1 w.r.t. x {Ans.
x
1
2}
210. Differentiate xx1sin
w.r.t. x1sin {Ans. )1
sin(ln2
1sin 1
x
xxxx x
}
211. Differentiate 12
1sec
2
1
x w.r.t. 21 x {Ans.
x
2}
212. Differentiate x
x1
1
tan1
tan
w.r.t. x1tan . {Ans.
21 )tan1(
1
x}
213. Let U
21
1
2sin
x
x and V
21
1
2tan
x
x, then find
dV
dU. {Ans.
2
2
1
1
x
x
}
CATEGORY-4.8. FIRST DERIVATIVE BY DIFFERENTIATION AND BY FIRST PRINCIPLES
214. Let 1,0,1 2
2
xx
xxf , then find 2f . {Ans.
9
4}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.22
215. If xexxf 21tan1 , then find 0f . {Ans. 14
}
216. If xxfx
lnlog 2 , then find ef . {Ans. e2
1}
217. Find the derivative of the function x2coscot 1 at 6
x . {Ans.
3
2}
218. If 2
0,sin1
sin1tan 1
x
x
xxf , then find
6
f . {Ans.
2
1}
219. Find the derivative w.r.t. x of the function 2
11sincos 1
2sin)cos)(logsin(log
x
xxx xx
at
4
x . {Ans.
2ln
1
16
48
2}
220. Given 31 xxxf , find 2f . {Ans. 0}
221. ,cosln)( 1 xxxf find )1(f . {Ans. 0}
222. ,cos)1()( 12 xxxf find )1(f & )1(f . {Ans. 0&2 }
223. Given xxf )( , find ).(xf
{Ans. 0,1 xxf 0,1 x }
224. Given xxf sgn)( , find ).(xf
{Ans. 0,0 xxf }
225. Find the derivative of xln . {Ans. x
1}
226. Check the differentiability of the function & find f x .
f x x x
x x
( ) ,
,
2 0
0
{Ans. 0,2 xxxf 0,1 x }
227. Check the differentiability of the function & find f x . 0,sin)( xxxf
0,1 xe x
{Ans. 0,cos xxxf
0, xex } 228. Given
0 ,0
0 ,1
sin)( 2
x
xx
xxf
find ).(xf
{Ans. 0,cossin2 11 xxxf xx
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.23
0,0 x } 229. Given
0,sin 2
xx
xxf
0,0 x ,
find xf .
{Ans. 0,sincos2
2
222
xx
xxxxf
0,1 x }
230. Given ,sin)( 3 xxxf find ).(xf {Ans.
0,
0,cos1sin
3
32
61
31
x
xxxxxf}
231. Given ,1)(4
xexf find ).(xf {Ans.
0,0
0,1
24
43
x
xe
exxf
x
x
}
232. Given ,sin)1()( 12 xxxf find ).(xf {Ans.
1,
11,1sin2 21
x
xxxxxf
}
233. Given ,sinln)( 1 xxxf find ).(xf {Ans.
1,
10,1
lnsin
2
2
1
x
xx
x
x
xxf
}
234. Given ,sin)( xxxf find ).(xf {Ans.
0,0
0,2
sincos
x
xx
xxxxf
}
235. If 122 xxxf , then find xf . {Ans. 1,1 xxf
1,1 x }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.24
236. Given 1)( xxxf , find ).(xf
{Ans. 0,2 xxf
10,0 x 1,2 x }
237. Given
1,1
1,1
1sin1)( 2
x
xxx
xxf
Find the set of points where xf is not differentiable. {Ans. x = 0} 238. Discuss the continuity and differentiability of the function
1,1
1,1
)(
xx
x
xx
xxf
{Ans. Discontinuous at 1x , not differentiable at 1x } 239. Let
0, 0
0,1
sin1
0,1
sin1)(
x
xx
xx
xx
xxxf
Show that )(xf exists everywhere and is finite except at x = 0.
240. Draw the graph of the function 21 xxy in the interval [0, 3] and discuss the continuity and
differentiability of the function in this interval. {Ans. continuous, not differentiable at x = 1 & 2} 241. Let )(xf be defined in the interval [2, 2] such that
20,1
02,1)(
xx
xxf
and )()( xfxfxg . Test the differentiability of )(xg in (2, 2). {Ans. not differentiable at x = 0,
1} 242. Find a & b such that the function
1 ,1
1 ,)( 2
xx
xbaxxf
is continuous and differentiable function. {Ans. 23
21 , ba }
243. Test for continuity and differentiability of the function & find f x rational is ,)( xxxf
irrational is , xx .
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.25
{Ans. Continuous at x 0 only, not differentiable x } 244. Check continuity and differentiability of the function & find xf
irrational is ,
rational is ,)(3
2
xx
xxxf
{Ans. Continuous at x = 0 & 1 only, 0,0 xxf }
245. If xgexf x , 20 g , 10 g , then find 0f . {Ans. 3}
246. Discuss continuity & differentiability of n
n xaxaxaaxf 2
210 , where 0a , 1a , ........., na
are constants. {Ans. Continuous x ; differentiable 0x and not differentiable at 0x if 01 a ;
differentiable x if 01 a ) 247. If the derivative of the function
1,2 xbaxxf
1,42 xaxbx is everywhere continuous, then find a and b. {Ans. 3,2 ba }
248. Let R be the set of real numbers and RRf : such that for all x and y in R, 3
)()( yxyfxf .
Prove that )(xf is a constant.
249. Suppose nn xaxaxaaxp ..............2
210 . If 011 xexp x , then prove that
1.............2 21 nnaaa .
CATEGORY-4.9. HIGHER ORDER DERIVATIVES OF EXPLICIT FUNCTIONS BY
DIFFERENTIATION
250. If 232 xxy , find 2
2
dx
yd. {Ans. 2}
251. If 32 1 xy , find 2
2
dx
yd. {Ans. 1656 24 xx }
252. If nx
ay , find
2
2
dx
yd. {Ans.
2
1
nx
nan}
253. If 2xxey , find
2
2
dx
yd. {Ans. 3232
2
xxex }
254. If 31
1
xy
, find
2
2
dx
yd. {Ans.
33
3
1
126
x
xx}
255. If xxy 12 tan1 , find 2
2
dx
yd. {Ans. x
x
x 12
tan21
2
}
256. If 22 xay , find 2
2
dx
yd. {Ans.
322
2
xa
a
}
257. If 21ln xxy , find 2
2
dx
yd. {Ans.
321 x
x
}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.26
258. If xa
y
1
, find 2
2
dx
yd. {Ans.
34
3
xaxx
xa
}
259. If xey , find 2
2
dx
yd. {Ans.
xx
xe x
4
1}
260. If xxy 12 sin1 , find 2
2
dx
yd. {Ans.
32
21
1
1sin
x
xxx
}
261. If xay sinsin 1 , find 2
2
dx
yd. {Ans.
322
2
sin1
sin1
xa
xaa
}
262. If xxy , find 2
2
dx
yd. {Ans.
xxx x 1
1ln 2 }
263. If 12 xey , find 0
2
2
xdx
yd. {Ans.
e
4}
264. If xy 1tan , find 1
2
2
xdx
yd. {Ans.
2
1 }
265. If 421 xxy , find 3
3
dx
yd. {Ans. x24 }
266. If xy 2cos , find 3
3
dx
yd. {Ans. x2sin4 }
267. If 610 xy , find 2
3
3
xdx
yd. {Ans. 207360}
268. If xxf lnln , find ex
dx
yd
3
3
. {Ans. 3
7
e}
269. If xxy ln3 , find 4
4
dx
yd. {Ans.
x
6}
270. If xay 2sin , find 4
4
dx
yd. {Ans. xa 2sin16 }
271. If 44 36 xxy , find 1
4
4
xdx
yd. {Ans. 360}
272. If x
y
1
1, find
5
5
dx
yd. {Ans.
6)1(
120
x}
273. If x
xy
1
1, find
15
5
xdx
yd. {Ans.
4
15 }
CATEGORY-4.10. HIGHER ORDER DERIVATIVES OF IMPLICIT FUNCTIONS BY
DIFFERENTIATION
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.27
274. If 12 22 byhxyax , find 2
2
dx
yd. {Ans.
32
byhx
abh
}
275. If 222222 bayaxb , find 2
2
dx
yd. {Ans.
32
4
ya
b }
276. If stes 1 , find 2
2
dt
sd. {Ans.
3
2
2
3
s
es s
}
277. If 0333 axyxy , find 2
2
dx
yd. {Ans.
32
32
axy
xya
}
278. If yxy sin , find 2
2
dx
yd. {Ans.
3cos1 yx
y
}
279. If xye yx , find 2
2
dx
yd. {Ans.
32
22
1
11
yx
yxy}
280. If exye y , find 0
2
2
xdx
yd. {Ans.
2
1
e}
281. If yxy tan , find 3
3
dx
yd. {Ans.
8
24 5832
y
yy }
282. If 222 ryx , find 0
3
3
xdx
yd. {Ans. 0}
CATEGORY-4.11. HIGHER ORDER DERIVATIVES OF PARAMETRIC FUNCTIONS BY
DIFFERENTIATION
283. If tytx , , then find 2
2
dx
yd. {Ans.
3t
tttt
}
284. If atyatx 2,2 , find 2
2
dx
yd. {Ans.
32
1
at }
285. If 2atx , 3bty , find 2
2
dx
yd. {Ans.
ta
b24
3}
286. If tax cos , tay sin , find 2
2
dx
yd. {Ans.
ta 3sin
1 }
287. If ttax sin , tay cos1 , find 2
2
dx
yd. {Ans.
2cos1
1
ta }
288. If tax 2cos , tay 2sin , find 2
2
dx
yd. {Ans. 0}
289. If tx ln , 12 ty , find 2
2
dx
yd. {Ans. 24t }
290. If tx 1sin , 21ln ty , find 2
2
dx
yd. {Ans.
1
22 t
}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.28
291. If tatx cos , taty sin , find 0
2
2
tdx
yd. {Ans.
a
2}
292. If tax 3cos , tay 3sin , find 3
3
dx
yd. {Ans.
tta
tt372
22
sincos9
sin4cos }
293. If tax cos , tby sin , find
2
3
3
tdx
yd. {Ans. 0}
CATEGORY-4.12. LEIBNIZ THEOREM
294. If xxy sin2 , find 25
25
dx
yd. {Ans. xxxx sin50cos6002 }
295. If 12 xey x , find 24
24
dx
yd. {Ans. 551482 xxex }
296. If xxy sin3 , find 20y . {Ans. xxxxxxx cos6840sin1140cos60sin 23 }
297. If 43 2 xey x , find ny . {Ans. 43363 22 nnnxxex }
298. If xxy cos1 2 , find ny2 . {Ans. xnxxxnnn sin4cos1241 22 }
299. If 52
232
xx
xy , prove that
0
5
10
5
20 21
nnn y
nnnyy for 2n .
CATEGORY-4.13. MISCELLANEOUS QUESTIONS ON DIFFERENTIATION
300. If 2
3 xexf , then find the value of 003
12 ffxxfxf . {Ans. 1}
301. If ayx
yxlnsin
22
221
, then show that x
y
dx
dy .
302. If ax
x
cey , then show that 2ax
ay
dx
dy
.
303. If yxy ex , then show that 2ln1
ln
x
x
dx
dy
.
304. If xexexey
, prove that y
y
dx
dy
1
305. If xxxy , prove that
xxy
y
dx
dy
)ln1(
2
306. If
xaxay , show that
yxyx
yy
dx
dy
lnln1
ln2
.
307. If cossec x , nny cossec then prove that 44 2222 ynyx .
308. If 2bxay , where a and b are arbitrary constants, then prove that dx
dy
dx
ydx
2
2
.
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.29
309. If qpqp yxyx )( , prove that 0&2
2
dx
yd
x
y
dx
dy
310. If ntbntax sincos , then show that 022
2
xndt
xd.
311. If nn bxaxy 1 , then show that ynndx
ydx 1
2
22 .
312. If
bxa
xxy ln , show that
2
2
23
y
dx
dyx
dx
ydx
313. If )sincos( xbxaey x prove that 0222
2
ydx
dy
dx
yd
314. If n
n
x
b
y
lncos 1 , prove that 02
2
22 yn
dx
dyx
dx
ydx
315. If )sin(ln)cos(ln xbxaxy n , then prove that 0)1()21( 22
22 yn
dx
dyxn
dx
ydx
316. Prove that 3
2
2
2
2
dx
dy
dx
yd
dy
xd.
317. If xexy sin , find 2
2
dy
xd in terms of x. {Ans.
3cos
sinx
x
ex
ex
}
318. If xexy , find 2
2
dy
xd in terms of x. {Ans.
31 x
x
e
e
}
CATEGORY-4.14. HIGHER ORDER DERIVATIVES BY DIFFERENTIATION AND BY FIRST
PRINCIPLES
319. Given 0,sin)( xxxf 0, xx ,
find ).(&)(),(),( xfxfxfxf IV {Ans. 0,cos xxxf
0,1 x ;
0,sin xxxf 0,0 x ;
0,cos xxxf 0,0 x ;
0,sin xxxf IV 0,0 x }
320. Given 0,)( 4 xxxf
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.30
0,3 xx
find ).(&)(),(),( xfxfxfxf IV {Ans. 0,4 3 xxxf
0,3 2 xx ;
0,12 2 xxxf 0,6 xx ;
0,24 xxxf 0,6 x ;
0,24 xxf IV 0,0 x }
321. Given 0,)( xexf x
0,2
12
xx
x ,
find ).(&)(),(),( xfxfxfxf IV {Ans. 0, xexf x
0,1 xx ;
0, xexf x 0,1 x ;
0, xexf x 0,0 x ;
0, xexf xIV 0,0 x }
322. Given xxxf )( , find ).(&)(),("),(' xfxfxfxf IV
{Ans. 0,2 xxxf
0,2 xx ;
0,2 xxf
0,2 x ;
0,0 xxf ;
0,0 xxf IV } 323. Given 1,ln)( xxxf
1,1 xx , find ).(&)(),( xfxfxf {Ans.
1,1
xx
xf
PART-I/CHAPTER-4
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.31
1,1 x ;
1,1
2 x
xxf
1,0 x ;
1,2
3 x
xxf
1,0 x }
324. Let 3xxf , then find 0f . {Ans. does not exist}
325. Given 1,ln)( xxxf
1,2 xcbxax ,
find the constants a, b, c such that 1f exists. {Ans. 21a , b = 2, 2
3c }
CATEGORY-4.15. ADDITIONAL QUESTIONS
326. Let xxf sin , 2xxg and xxh ln . If xhogofxF , then find xF . {Ans. xec 2cos2 }
327. If 92 xxf , then find
4
4lim
4
x
fxfx
. {Ans. 5
4}
328. Let 42 f and 42 f . Then find
2
22lim
2
x
xfxfx
. {Ans. -4}
329. If 99 f , 49 f , then find
3
3lim
9
x
xfx
. {Ans. 4}
330. If 22 f , 12 f , then find
2
2lim
2
2
x
xfxx
. {Ans. 2}
331. If 3cosax , 3sinay then find 2
1
dx
dy. {Ans. sec }
332. If xey1sin
and xu ln , then find
du
dy
ydx
d 1. {Ans.
2
321
1
x
}
333. Find the derivative of
12
1sec
21
x w.r.t. x31 at
3
1x . {Ans. does not exist}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
4.32
THIS PAGE IS INTENTIONALLY LEFT BLANK
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-I
DIFFERENTIAL CALCULUS
CHAPTER-5
APPLICATIONS OF DERIVATIVES
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.2
CHAPTER-5
APPLICATIONS OF DERIVATIVES
LIST OF THEORY SECTIONS
5.1. Rate Of Change Of A Function 5.2. Monotonicity Of A Function 5.3. Extremum (Maxima/ Minima) Of A Function 5.4. Greatest And Least Value Of A Function In An Interval 5.5. Concavity Of A Function 5.6. Point Of Inflection 5.7. Rolle’s Theorem And Lagrange’s Mean Value Theorem
LIST OF QUESTION CATEGORIES
5.1. Rate Of Change Of A Function 5.2. Monotonicity Of A Function 5.3. Extremum Points Of A Function 5.4. Greatest And Least Value Of A Function 5.5. Concavity Of A Function 5.6. Point Of Inflection 5.7. Rolle’s Theorem 5.8. Lagrange’s Mean Value Theorem 5.9. Additional Questions
PART-I/CHAPTER-5
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.3
CHAPTER-5
APPLICATIONS OF DERIVATIVES
SECTION-5.1. RATE OF CHANGE OF A FUNCTION
1. Mean (Average) rate of change of a function in an interval i. The mean (average) rate of change of a function xf with respect to x in the interval ba, is given by
ab
afbf
, which is also equal to the slope of the chord joining the end points of the curve of the
function in the interval ba, . 2. Instantaneous rate of change of a function at a point
i. The instantaneous rate of change of a function xf with respect to x at a point ax is given by
ax
afxfax
lim , which is af , which is also equal to the slope of tangent of the curve of the function
at ax . 3. Stationary point
i. Stationary points of a function are those points where it's first derivative is zero. ii. The instantaneous rate of change of a function xf with respect to x at the Stationary point is zero. iii. Stationary points of graphical functions:- Points at which tangent to the curve of the function is parallel
to x-axis are Stationary points. iv. Stationary points of Basic functions.
4. Approximate value of a function i. If ax then afaxafxf .
SECTION-5.2. MONOTONICITY OF A FUNCTION
1. Function increasing, decreasing, non-decreasing, non-increasing, constant in an open interval i. A function xf is said to be strictly increasing (or increasing) in an interval ba, if
baxxxfxfxx ,, 212121 .
ii. A function xf is said to be strictly decreasing (or decreasing) in an interval ba, if
baxxxfxfxx ,, 212121 .
iii. A function xf is said to be non-decreasing in an interval ba, if
baxxxfxfxx ,, 212121 .
iv. A function xf is said to be non-increasing in an interval ba, if
baxxxfxfxx ,, 212121 .
v. A function xf is said to be constant in an interval ba, if baxxxfxfxx ,, 212121 .
vi. A function xf may be neither increasing, decreasing or constant in an interval ba, . 2. Function increasing and decreasing at a point
i. A function xf is said to be increasing (decreasing) at a point a if there is a neighbourhood of point a
in which xf is increasing (decreasing), otherwise xf is not increasing (decreasing) at the point a. 3. Function increasing and decreasing in a closed interval and semi-closed interval
i. A function xf is said to be increasing (decreasing) in a closed interval ba, if it is increasing
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.4
(decreasing) ba, and also increasing (decreasing) at points a and b.
ii. A function xf is said to be increasing (decreasing) in a semi-closed interval ba, if it is increasing
(decreasing) ba, and also increasing (decreasing) at point b.
iii. A function xf is said to be increasing (decreasing) in a semi-closed interval ba, if it is increasing
(decreasing) ba, and also increasing (decreasing) at point a. 4. Intervals of monotonicity of a function
i. Intervals in which a function increases or decreases are called intervals of monotonicity of the function. ii. A function increasing (decreasing) in its domain is called a monotonous function.
5. Intervals of monotonicity of a graphical function i. A function is increasing in an interval if its curve moves upwards as x increases. ii. A function is decreasing in an interval if its curve moves downwards as x increases.
6. Intervals of monotonicity of basic functions 7. Testing a defined analytical function for monotonicity
i. Theorem:- If xf is continuous in an interval ba, and 0 xf in the interval, may be zero or non-
existent at finite number of isolated points, then xf is increasing in the interval.
ii. Theorem:- If xf is continuous in an interval ba, and 0 xf in the interval, may be zero or non-
existent at finite number of isolated points, then xf is decreasing in the interval.
iii. Theorem:- If xf is continuous in an interval ba, and 0 xf in the interval, then xf is constant in the interval.
SECTION-5.3. EXTREMUM (MAXIMA/ MINIMA) OF A FUNCTION
1. Definition of extremum (maxima/ minima or local maxima/ local minima) of a function i. A point a belonging to the domain of the function xf is called a point of maxima (or local maxima) if
there is a deleted neighbourhood of point a in which afxf , otherwise the point a is not a point of maxima.
ii. A point a belonging to the domain of the function xf is called a point of minima (or local minima) if
there is a deleted neighbourhood of point a in which afxf , otherwise the point a is not a point of minima.
iii. A point may be neither maxima nor minima. iv. The points of maxima and minima are called the extremum points of the function.
2. Extremum of a graphical function 3. Extremum of basic functions 4. Testing a point for extremum
i. By first principles ii. First derivative sign change test iii. Second derivative test iv. Higher order derivative test
5. Testing a defined analytical function for extremum i. Theorem:- If point a is an extremum of the function xf then af is either zero or does not exist. ii. Critical point:- Critical points of a function are those points in its domain at which its first derivative is
either zero or does not exist. iii. Critical points of graphical function
PART-I/CHAPTER-5
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.5
iv. Critical points of Basic functions v. To test a defined analytical function for extremum, find all critical points and test each critical point for
extremum. SECTION-5.4. GREATEST AND LEAST VALUE OF A FUNCTION IN AN INTERVAL
1. Definition of Greatest (largest or maximum or global maximum) value & least (smallest or minimum or global minimum) value of a function in an interval i. A function xf has a greatest value G in an interval A if the function attains the value G within the
interval A and AxGxf .
ii. A function xf has a least value L in an interval A if the function attains the value L within the interval
A and AxLxf . iii. A function may not have a greatest or least value in an interval. iv. A function may have same greatest and least value in an interval.
2. Greatest & least value of a graphical function 3. Greatest & least value of basic functions 4. Greatest & least value of defined analytical functions
i. Theorem:- If a function xf is continuous in a closed interval ba, then xf must have greatest value
and least value in the interval ba, and the greatest value and least value are attained either at the critical points within the interval or at the end-points of the interval.
5. Intermediate value theorem i. If xf is continuous in a interval and has greatest value G and least value L in the interval then it
assumes all intermediate values between L and G in the interval, i.e. if C is any value between L and G then there exists at least one point in the interval such that Cf .
SECTION-5.5. CONCAVITY OF A FUNCTION
1. Concavity of a function in an interval, concave upward, concave downward i. The curve of the function xf is said to be concave upwards in the interval ba, if the function xf
has non-vertical tangent at every point of the interval; and the curve of the function lies not below than any of its tangents within the interval.
ii. The curve of the function xf is said to be concave downwards in the interval ba, if the function
xf has non-vertical tangent at every point of the interval; and the curve of the function lies not above than any of its tangents within the interval.
iii. A function may be neither concave upwards nor concave downwards in an interval. 2. Concavity of a function at a point
i. A function xf is said to be concave upwards (concave downwards) at a point a if there is a
neighbourhood of point a in which xf is concave upwards (concave downwards), otherwise xf is not concave upwards (concave downwards) at the point a.
3. Intervals of concavity of a function i. Intervals in which a function is concave upwards or concave downwards are called intervals of concavity
of the function. 4. Concavity of a graphical function 5. Concavity of basic functions 6. Testing a defined analytical function for concavity
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.6
i. Theorem:- If xf is differentiable in an interval ba, and 0 xf in the interval, may be zero or
non-existent at finite number of isolated points, then xf is concave upwards in the interval.
ii. Theorem:- If xf is differentiable in an interval ba, and 0 xf in the interval, may be zero or
non-existent at finite number of isolated points, then xf is concave downwards in the interval. SECTION-5.6. POINT OF INFLECTION
1. Definition of point of inflection of a function i. A point a is said to be a point of inflection of the function xf if there is tangent at the point a and the
function has different concavity on the left and on the right of the point a. 2. Point of infection of a graphical function
i. The point on the curve of the function at which tangent cuts through the curve, is a point of inflection. 3. Point of infection of basic functions 4. Testing a point for Point of inflection
i. If there is tangent at point a and xf has different signs on the left and on the right of the point a, then a is a point of inflection.
5. Testing a defined analytical function for Point of inflection i. Theorem:- If point a is a point of inflection of the function xf then af is either zero or does not
exist. ii. To test a defined analytical function for Point of inflection, find all points where xf is either zero or
does not exist and test each such point for Point of inflection. SECTION-5.7. ROLLE’S THEOREM AND LAGRANGE’S MEAN VALUE THEOREM
1. Rolle's theorem i. If xf is continuous in ba, and differential in ba, and if bfaf , then there exists a point
bac , such that 0 cf . 2. Lagrange’s Mean Value theorem
i. If xf is continuous in ba, and differential in ba, , then there exists a point bac , such that
ab
afbfcf
.
PART-I/CHAPTER-5
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.7
EXERCISE-5
CATEGORY-5.1. RATE OF CHANGE OF A FUNCTION
1. Find the mean rate of change of the following functions in the interval 2,1 :-
i. 2)( xxf . {Ans. 3}
ii. 3)( xxf . {Ans. 7}
iii. xxf )( . {Ans. 0.414}
iv. x
xf1
)( . {Ans. –0.5}
v. xexf )( . {Ans. 4.67} vi. xxf ln)( . {Ans. 0.693} vii. xxf sin)( . {Ans. 0.068} viii. xxf cos)( . {Ans. –0.956}
2. For what value of a, the mean rate of change of the function 2xxf in the interval 1, aa is 2? {Ans.
2
1}
3. Find the instantaneous rate of change of the following functions at 1x and also find Stationary points:- i. 2)( xxf . {Ans. 2; 0}
ii. 3)( xxf . {Ans. 3; 0}
iii. xxf )( . {Ans. 2
1; no Stationary point}
iv. x
xf1
)( . {Ans. -1; no Stationary point}
v. xexf )( . {Ans. 2.72; no Stationary point} vi. xxf ln)( . {Ans. 1; no Stationary point}
vii. xxf sin)( . {Ans. 0.54; 2
12
n }
viii. xxf cos)( . {Ans. –0.84; n }
4. For what value of a, the mean rate of change of the function 3)( xxf in the interval a,1 is equal to
the instantaneous rate of change at a? {Ans. 2
1}
5. Find the approximate value of the following:- i. 31cos . {Ans. 0.851} ii. 21.10log . {Ans. 1.009}
iii. 5 33 . {Ans. 2.012} iv. 0145cot . {Ans. 0.994}
CATEGORY-5.2. MONOTONICITY OF A FUNCTION
6. Test the following functions for monotonicity and find stationary points:-
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.8
i. 291292 23 xxxxf . {Ans. 1, increasing, 2,1 decreasing, ,2 increasing; 1, 2}
ii. 1763 23 xxxy . {Ans. , increasing}
iii. xxxf 33 . {Ans. 1, increasing, 1,1 decreasing, ,1 increasing; -1, 1}
iv. 22 3 xxy . {Ans. 0, decreasing,
2
3,0 increasing,
3,2
3 decreasing, ,3 increasing; 0,
2
3, 3}
v. 643 79 xxxf . {Ans. , increasing; 0}
vi. xxxf ln)( . {Ans. e1,0 decreasing, ,1
e increasing; e
1}
vii. x
xxf
ln)( . {Ans. e,0 increasing, ,e decreasing; e}
viii. x
xxf
ln . {Ans. 1,0 decreasing, e,1 decreasing, ,e increasing; e}
ix. xxxf )( . {Ans. e1,0 decreasing, ,1
e increasing; e
1}
x. xxxf1
)( . {Ans. e,0 increasing, ,e decreasing; e}
xi. xxexf )( . {Ans. 1, decreasing, ,1 increasing; -1}
xii. xxexf )( . {Ans. 1, increasing, ,1 decreasing; 1}
xiii. xexf1
)( . {Ans. 0, decreasing, ,0 decreasing}
xiv. xxexf1
)( . {Ans. 0, increasing, 1,0 decreasing, ,1 increasing; 1}
xv. )(lnlog)( xxf x . {Ans. ee,1 increasing, ,ee decreasing; ee }
xvi. x
xxxf
2
21ln)( . {Ans. ,1 increasing; 0}
xvii. xxxf sin)( . {Ans. increasing for all x; Inn ,12 }
xviii. 20,cos2
cos2sin4)(
x
x
xxxxxf . {Ans.
2,0
increasing,
2
3,
2
decreasing,
2,
2
3
increasing; 2
,
2
3}
xix. xxxf ln2
1tan)( 1 . {Ans. 0, increasing, ,0 decreasing, 1}
xx. 1lncot 21 xxxxy . {Ans. , increasing}
xxi. xxxf ln2 2 . {Ans.
2
1, decreasing,
0,
2
1 increasing,
2
1,0 decreasing,
,
2
1
increasing; 2
1 ,
2
1}
7. Show that the function no.rationalais,sin xxxf
PART-I/CHAPTER-5
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.9
no.irrationalanis, xx
has positive derivative at 0x but xf is not increasing at 0x . 8. Show that the function
0,1
sin2
2 xx
xx
xf
0,0 x
is continuous and differentiable in any neighbourhood of 0x and 0f is positive but xf is not increasing at 0x .
9. For what values of b the function cbxxxf sin is decreasing in the interval , ? {Ans. 1b }
10. If xfxfxg 1)( and 10;0 xxf , show that )(xg increases in 2
10 x and decreases in
12
1 x .
11. Given 22 1410 xxfxxfxg and 0 xf for all real x, except at finite no. of real values
of x for which 0 xf . Discuss the monotonicity of the function xg . {Ans. 3, decreasing,
2
1,3 increasing,
4,2
1 decreasing, ,4 increasing}
CATEGORY-5.3. EXTREMUM POINTS OF A FUNCTION
12. Investigate the following functions for extremum at x = 0:- i. xxxf sin)( . {Ans. no extremum}
ii. !3
sin)(3x
xxxf . {Ans. no extremum}
iii. !4!3
sin)(43 xx
xxxf . {Ans. maxima}
iv. 0,)(1
xexf x 0,0 x . {Ans. minima}
v. xxxf coscosh)( . {Ans. minima}
vi. !3!2
1cos)(32 xx
xxf . {Ans. no extremum}
vii. 2
1cos)(2x
xxf . {Ans. minima}
viii. 3
2
)( xxxf . {Ans. minima}
ix. 3
12)( xxxf . {Ans. no extremum}
x. 2)( 3
4
xxf . {Ans. minima}
xi. 3)( 3
5
xxf . {Ans. no extremum}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.10
xii. 0,1
sin)( xx
xf
0,2 x . {Ans. maxima}
xiii. 0,1
sin2)(
xx
xxf
0,0 x . {Ans. minima}
xiv. 0,21
cos)( 2
xx
xxf
0,0 x . {Ans. maxima}
xv. no.rationalais,2 xxxf
no.irrationalanis,4 xx {Ans. minima} 13. Test the following functions for extremum:-
i. 884152)( 23 xxxxf . {Ans. maxima at 2 , minima at 7 }
ii. 896 23 xxxxf . {Ans. maxima at 1, minima at 3}
iii. 1052
458
4
3 234 xxxxf . {Ans. maxima at 0, -5, minima at -3}
iv. 794
3)( 234 xxxxf . {Ans. maxima at 0, minima at -2, 3}
v. 1224228)( 234 xxxxxf . {Ans. maxima at 2, minima at 1, 3}
vi. 23 )3()1()( xxxxf . {Ans. maxima at 4173 , minima at 3,4
173 }
vii. 12
23)(
2
2
xx
xxxf . {Ans. minima at 5
7 }
viii. 23 23)( xxxf . {Ans. maxima at 1 , minima at 0 }
ix. 3 23 2 )1()1()( xxxf . {Ans. maxima at 0 , minima at 1 }
x. 0,2)( xxxf 0,53 xx . {Ans. no extremum}
xi. 0,32)( 2 xxxf 0,4 x . {Ans. maxima at 0 }
xii. 601883
50)(
234
xxxxf . {Ans. maxima at 1,3 , minima at 0 }
xiii. 1)(2
xexf . {Ans. minima at 0 }
xiv. xxexf . {Ans. minima at 1 }
xv. 24)( xexxf . {Ans. maxima at 2 , minima at 0 }
xvi. xexxf 2)( . {Ans. maxima at 2 , minima at 0 }
xvii. 4
4)(
2
x
xxf . {Ans. maxima at 2 , minima at 2 }
xviii. 5 22 )2()( xxxf . {Ans. maxima at 0 , 2, minima at 35 }
PART-I/CHAPTER-5
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.11
xix. 28
14)(
24
xxxf . {Ans. maxima at 2 , minima at 0 }
xx. 3 23 3632)( xxxxf . {Ans. maxima at 3 , minima at 2 }
xxi. xxxf ln)( 2 . {Ans. minima at 21e }
xxii. xxxf 2ln)( . {Ans. maxima at 2e , minima at 1}
xxiii. 2
1)(
x
xxf
. {Ans. maxima at 2, minima at 1}
xxiv. 21)( xxxxf . {Ans. minima at 1}
xxv. 2
0,cossin 44 xxxxf . {Ans. minima at
4
}
xxvi. 22
,2sin
xxxxf . {Ans. maxima at 6
, minima at
6
}
xxvii. 2
0,2cos2
1sin
xxxxf . {Ans. maxima at
6
, minima at
2
}
14. The function xxaxf 3sin3
1sin has maximum value at
3
x . Find the value of a . {Ans. 2}
15. Given 2,1ln2 2 xaxxf
2,53 xx .
Find values of a for which xf has local minima at 2x . {Ans. ,11, 1111 eea }
16. Find the polynomial of degree 6 which satisfies 2
1
301lim e
x
xf x
x
and has local maxima at 1x and
local minima at 0x & 2x . {Ans. 456 25
12
3
2xxx }
CATEGORY-5.4. GREATEST AND LEAST VALUE OF A FUNCTION
17. Find greatest & least value of the following functions in the indicated intervals:- i. xxxf 33 in 2,0 . {Ans. 2, 2}
ii. 11232)( 23 xxxxf in
2
5,2 . {Ans. 19,8 }
iii. 107242 3 xxxf in 3,1 . {Ans. 89, 75}
iv. xxxf ln)( 2 in e,1 . {Ans. 0,2e }
v. )21)(1()( 22 xxxf in 1,1 . {Ans. 0,22
3 }
vi. )(cos)( 21 xxf in
2
1,
2
1. {Ans. 32 , }
vii. xxxf )( in 4,0 . {Ans. 0,6 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.12
viii. 22
3
3
2
4)(
234
xxx
xf in 4,2 . {Ans. 437
316 , }
ix. 24)( xxf in 2,2 . {Ans. 0,2 }
x. xxxf ln2
1tan)( 1 in
3,
3
1. {Ans. 4
3ln34
3ln6 , }
xi. xxxf 2sinsin2)( in
2
3,0 . {Ans. 2,2
33 }
xii. xxxf ln2)( in e,1 . {Ans. 2ln22,1 }
xiii. 863 2 xxxf . {Ans. No greatest value, 5}
xiv. 51 xxf . {Ans. 5, No least value}
xv. 43sin xxf . {Ans. 5, 3}
xvi. 13 xxf . {Ans. No greatest value, no least value}
xvii. xxf sinsin . {Ans. 1sin , 1sin }
xviii. 3 xxf . {Ans. No greatest value, 0}
xix. 2,00,2,2
2)( 22 x
xxxf
0,1 x
in 2,2 . {Ans. no greatest value, 1} 18. Prove that:-
i. 0,1 xxe x .
ii. 0,sin6
3
xxxx
x .
iii. 1,1ln1
xxxx
x.
iv. 0,tan1
12
xxxx
x.
v.
1,1
12ln
x
x
xx .
vi. 21 1lntan2 xxx .
vii. 0,1
tan1ln
1
xx
xx .
viii. 0,1206
sin53
xxx
xx .
ix. 2
0,2tansin
xxxx .
x. 0,2
1cosh2
xx
x .
xi. 22 1)1ln(1 xxxx .
PART-I/CHAPTER-5
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.13
xii. xxx 3tansin2 for 2
0
x .
19. The function 962 24 axxxxf attains its maximum value on the interval 2,0 at 1x . Find the value of a . {Ans. 120}
20. Let
10,23
1)(
2
233
x
bb
bbbxxf
31,32 xx . Find all possible real values of b such that )(xf has the smallest value at x = 1. {Ans. ),1[)1,2( b }
21. Use the function 0,)(1
xxxf x to determine the bigger of the two numbers e & e . {Ans. e } CATEGORY-5.5. CONCAVITY OF A FUNCTION
22. Find the intervals of concavity of the following functions:- i. 122418)( 234 xxxxxf . {Ans. concave upward in 2, , concave downward in
2
3,2 , concave upward in
,
2
3}
ii. 2353)( 45 xxxxf . {Ans. concave downward in 1, , concave upward in ,1 }
iii. 46 10)( xxxf . {Ans. concave upward in 2, ; concave downward in 2,2 ; concave
upward in ,2 }
iv. 1ln)( 2 xxf . {Ans. concave downward in 1, and ,1 }
v. xexxf 41)( . {Ans. concave upward in , }
vi. xxxf ln)( 2 . {Ans. concave downward in
2
3
,0 e ; concave upward in
,2
3
e }
vii. 3
4
)( xxxf . {Ans. concave upward in , }
viii. 1)( 3
5
xxxf . {Ans. concave downward in 0, ; concave upward in ,0 }
ix. 3
2
)( xxxf . {Ans. concave downward in 0, ; concave downward in ,0 }
x. 0,2 xxxf
0,3 xx . {Ans. concave upward in , }
xi. 0,3 xxxf
0,2 xx . {Ans. concave downward in 0, ; concave upward in ,0 }
xii. 1,3 xxxf
1,2 xx . {Ans. concave downward in 0, ; concave upward in 1,0 ; concave upward in
,1 } CATEGORY-5.6. POINT OF INFLECTION
23. Test the indicated points for point of inflection:-
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.14
i. 535)( 23 xxxxf at 2,3
5,1x . {Ans.
3
5 is point of inflection; 1, 2 are not points of
inflection} ii. 234 4812)( xxxxf at 4,3,2,1x . {Ans. 2, 4 are points of inflection; 1, 3 are not points of
inflection}
iii. 2)( 3
5
xxxf at 0x . {Ans. 0 is point of inflection}
iv. 1)( 3
42 xxxf at 0x . {Ans. 0 is not point of inflection}
v. 4)( 3
2
xxxf at 0x . {Ans. 0 is not point of inflection}
vi. 3)( 5
3
xxxf at 0x . {Ans. 0 is point of inflection}
vii. !5!3
sin)(53 xx
xxf at 0x . {Ans. 0 is point of inflection}
viii. 62
)(32 xx
exf x at 0x . {Ans. 0 is not point of inflection}
ix. 0,sin xxxf
0,6
3
xx
x
at 0x . {Ans. 0 is point of inflection} 24. Test the following functions for concavity & find points of inflection:-
i. 432 236)( xxxxxf . {Ans. concave downward in 3, , concave upward in 2,3 ,
concave downward in ,2 ; 3, 2}
ii. 12683)( 234 xxxxf . {Ans. concave upward in
3
1, , concave downward in
1,3
1,
concave upward in ,1 ; 3
1, 1}
iii. 222)( 6 xxxf . {Ans. concave upward in , }
iv. 21
)(x
xxf
. {Ans. concave downward in 3, , concave upward in 0,3 , concave
downward in 3,0 , concave upward in ,3 ; 0, 3 }
v. 21ln)( xxf . {Ans. concave downward in 1, , concave upward in 1,1 , concave
downward in ,1 ; 1, 1}
vi. 7ln12)( 4 xxxf . {Ans. concave downward in 1,0 , concave upward in ,1 ; 1}
vii. xxxf ln)( . {Ans. concave upward in ,0 }
viii. x
xxf
ln)( . {Ans. concave downward in 2
3
,0 e , concave upward in ,23
e ; 23
e }
ix. xxxf )( . {Ans. concave upward in ,0 }
x. xxexf )( . {Ans. concave downward in 2, , concave upward in ,2 ; 2}
PART-I/CHAPTER-5
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.15
xi. xxexf )( . {Ans. concave downward in 2, , concave upward in ,2 ; 2}
xii. xexf1
)( . {Ans. concave downward in 21, , concave upward in 0,2
1 , concave upward in
,0 ; 21 }
xiii. xxexf1
)( . {Ans. concave downward in 0, , concave upward in ,0 }
xiv. 3
5
)( xxxf . {Ans. concave downward in 0, , concave upward in ,0 ; 0}
xv. 35 151)( xxxf . {Ans. concave upward in ,1 }
xvi. 12)( 5 xxf . {Ans. concave downward in 0, , concave upward in 1,0 , concave
downward in ,1 ; 0}
xvii. 5 23)( xxxf . {Ans. concave upward in 3, and ,3 }
xviii. 52)( 3
4
xxxf . {Ans. concave upward in , }
xix. 23)( 5
3
xxxf . {Ans. concave upward in 0, , concave downward in ,0 ; 0}
xx. 0,123 xxxxxf
0,12 xxx . {Ans. concave downward in
3
1, , concave upward in
0,
3
1,
concave downward in ,0 ; 3
1 , 0}
25. Given 0, xexf x
0,23 xdcxbxax .
Find the constants a, b, c, d if 0f exists and xf has a point of inflection at 1x .
{Ans. 6
1a ,
2
1b , 1 dc }
CATEGORY-5.7. ROLLE’S THEOREM
26. Verify Rolle’s theorem for the following functions:-
i. 242 23 xxxxf in 2,2 .
ii. xxf sin in ,0 .
iii. xxf tan in ,0 .
iv. x
xf1
cos in 1,1 .
v. 23x
exxxf
in 0,3 .
vi. xxexf x cossin in
4
5,
4
.
vii. xxf in 1,1 .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.16
viii. 3
2
23 xxf in 3,1 .
ix.
xba
abxxf
2
ln in ba, , a>0.
x. nm bxaxxf in ba, , where m and n are positive integers.
27. Show that the equation 033 cxx cannot have two different roots in the interval 1,0 .
28. Prove that the equation 08153 5 xx has only one real solution. 29. Prove that the polynomial 144 xx has exactly two different real roots. 30. Show that the equation 2xxe has only one solution which lies in the interval 1,0 .
31. Show that the equation 0224 xx has exactly one real solution in the interval 1,0 .
32. If the equation 0...... 11
1
xaxaxa nn
nn has a positive solution a, then prove that the equation
0......1 12
11
axanxna nn
nn also has a positive solution which is smaller than a.
33. If 0632 cba , then show that the equation 02 cbxax has at least one real root between 0 and 1.
34. Let 012
..........11
1210
nn aa
n
a
n
a
n
a. Show that there exists at least one real x between 0 and 1
such that 0...... 11
10
nnnn axaxaxa .
CATEGORY-5.8. LAGRANGE’S MEAN VALUE THEOREM
35. Show that 2xxf satisfies Lagrange’s Mean value theorem in the interval 1,0 and find the value of c.
36. Prove the validity of Lagrange’s theorem for the function xy ln in the interval e,1 and find the value of c.
37. With the aid of Lagrange’s theorem prove the inequalities b
ba
b
a
a
ba
ln , for the condition
ab 0 .
38. With the aid of Lagrange’s theorem prove the inequalities a
baba
b
ba22 cos
tantancos
, for the
condition 2
0
ab .
39. Using Mean value theorem, show that baba coscos .
40. Use Lagrange’s theorem to prove that 011 xxeex xx .
41. If xf exists for all points in ba, and
cb
cfbf
ac
afcf
, where bca , then there is a
number such that ba and 0 f .
42. If xf is differentiable and xfx lim is finite and xf
x
lim is finite, then show that 0lim
xf
x.
43. If baxxf ,0 , show that
222121 xfxfxx
f
for baxx ,, 21 .
CATEGORY-5.9. ADDITIONAL QUESTIONS
PART-I/CHAPTER-5
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.17
44. Suppose that xf and xg are non-constant differentiable real valued functions on R. If for every
Ryx , , ygxgyfxfyxf and ygxfyfxgyxg and 00 f , then prove
that maximum and minimum values of the function xgxf 22 are same for all real x.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
5.18
THIS PAGE IS INTENTIONALLY LEFT BLANK
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-I
DIFFERENTIAL CALCULUS
CHAPTER-6
INVESTIGATION OF FUNCTIONS AND THEIR GRAPHS
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.2
CHAPTER-6
INVESTIGATION OF FUNCTIONS AND THEIR GRAPHS
LIST OF THEORY SECTIONS
6.1. Calculation Of Domain Of Analytical Functions 6.2. Plotting The Graphs Of Analytical Functions By Properties 6.3. Simple Transformations On Graphs 6.4. Even And Odd Functions 6.5. Periodic Functions 6.6. Inverse Functions 6.7. Greatest Integer Function And Fractional Part Function 6.8. Finding Range Of Analytical Functions 6.9. Function (Mapping)
LIST OF QUESTION CATEGORIES
6.1. Calculation Of Domain Of Analytical Functions 6.2. Plotting Graph By Properties 6.3. Plotting Graphs By Transformation 6.4. Even And Odd Functions 6.5. Periodic Functions 6.6. Miscellaneous Questions On Graph Plotting 6.7. Inverse Functions 6.8. Greatest Integer Function 6.9. Finding Range Of Analytical Functions 6.10. Miscellaneous Questions On Functions 6.11. Additional Questions
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.3
CHAPTER-6
INVESTIGATION OF FUNCTIONS AND THEIR GRAPHS
SECTION-6.1. CALCULATION OF DOMAIN OF ANALYTICAL FUNCTIONS
1. Calculation of Domain of analytical functions i. Write domain conditions and solve to get domain.
SECTION-6.2. PLOTTING THE GRAPHS OF ANALYTICAL FUNCTIONS BY PROPERTIES
1. For investigating a function and sketching its graph, determine the following properties of the function:- i. Find Domain of the function. ii. Check continuity and find discontinuities if any. iii. Check differentiability and find first derivative function. Find the points where first derivative function
is positive, negative, zero and does not exist. Find the following properties:- a. Intervals of monotonicity of the function. b. Stationary points of the function. c. Critical points of the function. d. Maxima and minima of the function.
iv. Find second derivative function. Find the points where second derivative function is positive, negative, zero and does not exist. Find the following properties:- a. Intervals of concavity of the function. b. Points of inflection of the function.
v. Find value/limit of the function at important points by finding the following: properties:- a. Value/limit of the function at the end points of the domain. b. Right hand limit, left hand limit and value of the function at the points of discontinuity. c. Value of the function at critical points and points of inflection. d. Zeros of the function if any, i.e. points where the function cuts x-axis. e. Value of the function at 0x if any, i.e. point where the graph cuts y-axis.
vi. Find slope of the tangent at important points by finding the following: properties:- a. Value/limit of the first derivative function at the end points of the domain. b. Right hand limit and left hand limit of the first derivative function at the points of discontinuity. c. Right hand limit and left hand limit of the first derivative function at the points where function is
not differentiable. d. Value of the first derivative function at points of inflection. e. Value of the first derivative function at zeros of the function if any, i.e. slope of tangent at the points
where the function cuts x-axis. f. Value of the first derivative function at 0x if any, i.e. slope of tangent at the point where the
graph cuts y-axis. vii. Based on these properties, plot the graph of the function. viii. Write the Range of the function. ix. Write the greatest and least value of the function.
2. Graph of the function xaxf )( 3. Graph of the function xxf alog)(
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.4
4. Graph of the function axxf )( 5. Graphs of trigonometric functions 6. Graphs of inverse trigonometric functions 7. Graphs of hyperbolic functions SECTION-6.3. SIMPLE TRANSFORMATIONS ON GRAPHS
1. Plotting graphs by simple transformations i. If graph of the function )(xf is known then graphs of the functions )( xf , )(xf , xf , xf ,
)(xf , )( xf , )(xf , )( xf can be plotted by simple transformations on the graph of the function )(xf without finding its properties.
2. Graphs of )( xf , )(xf , )( xf
3. Graphs of xf , xf , xf
4. Graphs of )(xf , )( xf 5. Graphs of )(xf , )( xf 6. Applications of transformations in graph plotting SECTION-6.4. EVEN AND ODD FUNCTIONS
1. Definition of even and odd functions i. A function )(xf is said to be an even function if xfxf )( for all x in the domain of )(xf ; is said
to be an odd function if xfxf )( for all x in the domain of )(xf ; otherwise )(xf is neither even nor odd function.
ii. A function may be both even and odd. iii. Domain of an even/odd function must be symmetrical about origin.
2. Checking a graphical function for being even or odd or neither even nor odd i. If graph of the function is in itself symmetrical about y-axis then the function is even; if graph of the
function is in itself symmetrical about origin then the function is odd; otherwise the function is neither even nor odd.
3. Basic functions which are even or odd or neither even nor odd 4. Testing an analytical function for being even or odd or neither even nor odd
i. Check by definition. ii. If xfxfxf 2)( then the function is even; if 0)( xfxf then the function is odd;
otherwise the function is neither even nor odd. 5. Applications of even/odd property in graph plotting
i. If the graph of the function is to be plotted by properties and if the function is even or odd, then first plot the graph for positive values of x only and then plot the graph for negative values of x by symmetry to get the complete graph.
SECTION-6.5. PERIODIC FUNCTIONS
1. Definition of periodic functions i. A function )(xf is called periodic if there exists a non-zero constant real number P such that
xfPxf )( for all values of x within the domain of )(xf and the number P is called a period of )(xf . All other functions are non-periodic.
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.5
ii. If P is a period then nP (n is any non-zero integer) is also a period. Hence, a periodic function has infinite periods.
iii. The least positive period is called the Fundamental period, denoted by T. iv. If T is the Fundamental period then nT (n is any non-zero integer) is also a period of the function. v. A periodic function may not have fundamental period. vi. Domain of periodic functions must extend to and to .
2. Checking a graphical function for being periodic 3. Basic functions which are periodic/non-periodic 4. Testing an analytical function for periodicity
i. By definition ii. By method iii. By Theorems
5. Applications of periodic/non-periodic property in graph plotting SECTION-6.6. INVERSE FUNCTIONS
1. Definition of inverse functions i. The function )(xf has an inverse function, denoted by )(1 xf , if aabfbaf 1)( domain
of )(xf and b domain of )(1 xf , i.e. xxxff 1 domain of )(xf and
xxxff 1 domain of )(1 xf . 2. Properties of inverse functions
i. Domain of )(1 xf = Range of )(xf and Range of )(1 xf = Domain of )(xf . ii. A function has an inverse iff the function is one-one function (invertible function). iii. Many-one functions may have different inverse functions for its different one-one parts. iv. A function may be inverse to itself.
3. Finding inverse of graphical function i. Graphs of inverse functions are symmetric about xy line.
4. Inverse functions of basic functions 5. Finding inverse of an explicit analytical function
i. Given an explicit analytical function )(xf , to find its inverse function y, solve the equation xyf )( . 6. Inverse of implicit and parametric functions 7. Inverse of piecewise functions. 8. Derivative of inverse function
i. xff
xfdx
d1
1 1
.
9. If xf is monotonously increasing (decreasing) then xf 1 is also monotonously increasing (decreasing)
10. Applications of inverse function in graph plotting SECTION-6.7. GREATEST INTEGER FUNCTION AND FRACTIONAL PART FUNCTION
1. Definition of Greatest Integer Function [x] i. Greatest integer function, denoted by [x], is the greatest integer less than or equal to x for any real
number x. 2. Graph and properties of Greatest Integer Function
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.6
3. Definition of Fractional Part Function {x} i. Fractional part function, denoted by {x}, is the fractional part of x for any real number x, i.e.
xxx . 4. Graph and properties of Fractional Part Function 5. Graphs of )(xf and xf 6. Graphs of xf and xf SECTION-6.8. FINDING RANGE OF ANALYTICAL FUNCTIONS
1. By graph, if known or easily drawn by transformation 2. Range of Composite functions 3. If xf is continuous & domain is closed interval, find least value (L) and greatest value (G), range is
[L,G] 4. Find inverse function of xf and find its domain, which is range of xf 5. Find values of y for which equation yxf has solution, then these values of y is the range of xf 6. If xf is periodic with period T then find range in any interval of length T 7. By properties (concavity, slope of tangent not required) 8. If xf is piecewise function, find range of each part and take their union 9. Find an equivalent function of xf and find its range SECTION-6.9. FUNCTION (MAPPING)
1. Definition of function (mapping) i. If A and B are two non-empty sets, then a function f from set A to set B is a rule which associates each
element of set A to a unique element of set B. ii. f is a function from set A to set B is denoted by BAf : . iii. If RA and RB then f is a real function.
2. Domain and Co-domain i. Set A is known as the domain of f and set B is known as the co-domain of f.
3. Value (Image) and pre-image i. If and element Aa is associated to an element Bb then it is written as baf and b is called
value or image of a and a is called the pre-image of b. 4. Range of a function
i. The set of all values (images) of elements of set A is called the range of f or image set of A and is denoted by Af , i.e. AxxfAf | .
ii. BAf . 5. Ways of representing functions
i. By diagram; ii. By notation yxf ; iii. By sets of ordered pairs; iv. By formula.
6. One-one (injective) function and many-one function i. A function BAf : is said to be a one-one function or injective function if different elements of set A
have different images in set B, i.e. Ababfafba , .
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.7
ii. A function BAf : is said to be a many-one function if two or more elements of set A have same
image in set B, i.e. there exists Aba , such that ba but bfaf . 7. Into and Onto (Surjective) function
i. A function BAf : is said to be an into function if there exists an element in set B having no pre-
image in set A, i.e. BAf . ii. A function BAf : is said to be an onto function or surjective function if every element of set B is the
image of some element of set A, i.e. BAf . 8. One-one and into functions; one-one and onto (Bijective) functions; many-one and into functions;
many-one and onto functions
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.8
EXERCISE-6
CATEGORY-6.1. CALCULATION OF DOMAIN OF ANALYTICAL FUNCTIONS
1. 23 22 xxxxxf . {Ans. 1 }
2.
50
50
42
421
x
xxf . {Ans. ,00,4 }
3. 3log3
1
3
xxf . {Ans. ,3030,3 }
4. x
xxf
9log
5. {Ans. 9,88,5 }
5.
2
2log2 x
xxf . {Ans. ,22, }
6. 212log xxf . {Ans. ,11, }
7. x
xxf
3log . {Ans. 2
3,0 }
8. xx
xf
1
. {Ans. 0, }
9. 2log4 xxxxf . {Ans. ,0 }
10.
3logsin 3
1 xxf . {Ans. 9,1 }
11. xx
xf
4log
2
3sin 1 . {Ans. 4,1 }
12. 21sin 1 xxf . {Ans. 4,20,2 }
13.
2
21
2
1logsin xxf . {Ans. 2,11,2 }
14. 36
1
5
1log
24.0
xx
xxf . {Ans. ,66,1 }
15. 2
3
2 16log xxfx
x
. {Ans. 4,23,4 }
16. 11 3log4
2cos
x
xxf . {Ans. 3,22,6 }
17.
4
5log
2
10
xxxf . {Ans. 4,1 }
18.
183
1log2
3.0
xx
xxf . {Ans. 6,2 }
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.9
19. 3loglog4logloglog xxxf . {Ans. 43 10,10 }
20. xxxf 64log . {Ans. 6,4 }
21. 6log5loglog 2 xxxf . {Ans. ,1000100,0 }
22. 165log1log 2 xxxf . {Ans. 3,2 }
23.
23
13loglog 25.0 x
xxf . {Ans.
,
3
1}
24. 103log 252 xxxf x . {Ans. ,5 }
25. Find the set of values of x for which the function 2
12
1 1sin
xxxf x is defined. {Ans. }
26. If a function xf is defined for 1,0x , then find the domain of the function 32 xf . {Ans.
1,
2
3}
27. If 20,1)( xxxf , find the domain of )(xffx . {Ans. 2,1 } CATEGORY-6.2. PLOTTING GRAPH BY PROPERTIES
28. Plot graph of the following functions and write their range, greatest value and least value:- i. xxxf ln)( .
ii. x
xxf
ln)( .
iii. xxxf )( .
iv. xxexf )( .
v. xxexf )( .
vi. xexf1
)( .
vii. xxexf1
)( .
viii. 1ln)( xxxf . CATEGORY-6.3. PLOTTING GRAPHS BY TRANSFORMATION
29. Plot graph:-
i. 1)( xxf .
ii. 22)( xxf .
iii. 21)( xxf .
iv. x
xf1
1)( .
v. 2
sin)(
xxf .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.10
vi. 4
sin)(
xxf .
vii. 1)(
xexf .
viii. 1)(
xexf .
ix. 11)(
xexf .
x. 1)(
xexf .
xi. 2)( xexf .
xii. xxf 1ln)( .
xiii. xxf 1ln)( .
xiv. 1ln)( xxf .
xv. xxf 1ln)( .
xvi. 1ln)( xxf .
xvii. 1ln)( xxf .
xviii. 1ln)( xxf .
xix. 1sgn)( xxf .
xx. 1sgn)( xxf .
xxi. 1sgn)( xxf .
xxii. 1sgn)( xxf .
xxiii. 21tanh)( xxf .
xxiv. 1
)(
x
xxf .
xxv. 1cos)( 1 xxf .
xxvi. 1)( xxf .
xxvii. 1)( xxf .
xxviii. x
xf
2
1)( .
xxix. 11
1)(
xxf .
CATEGORY-6.4. EVEN AND ODD FUNCTIONS
30. Test the following functions for even, odd or neither:-
i. 21log)( xxxf . {Ans. odd}
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.11
ii. x
xxf
1
1log)( . {Ans. odd}
iii. 12)( 3 xxxf . {Ans. neither}
iv. 24 2)( xxxf . {Ans. even}
v. 2)( xxxf . {Ans. neither} vi. xxxf cossin)( .{Ans. neither}
vii. 2
2)( xxf . {Ans. even}
viii. 2
)(xx aa
xf
. {Ans. even}
ix. 2
)(xx aa
xf
. {Ans. odd}
x. 1
)(
xa
xxf . {Ans. neither}
xi. 4
2)( xxxf . {Ans. neither}
xii. 1
1)(
x
x
a
axxf . {Ans. even}
xiii. xxxf 24 sin24)( . {Ans. even}
xiv. 22 11)( xxxxxf . {Ans. odd}
xv. kx
kx
a
axf
1
1)( . {Ans. odd}
xvi. xxxf cossin)( . {Ans. neither}
xvii. 3 23 2 11)( xxxf . {Ans. odd}
xviii. xxxf 2)( . {Ans. even}
xix. 32sin)( xxxxf . {Ans. odd}
xx.
x
x
xf2
21)(
2
. {Ans. even}
xxi. xx
xxxf
cottan
sincos)(
. {Ans. even}
xxii. xxxf 53 tan2sin)( . {Ans. odd}
xxiii. xxx
xxxf
tan
cossin2
44
. {Ans. odd}
xxiv. xxx
xecxxf
cot
cossec43
44
. {Ans. odd}
xxv. 121
x
e
xxf
x. {Ans. even}
31. For what values of a, the function 322 22 aaxaaxf is (a) even, (b) odd? {Ans. Even for 2,1 a ; odd for 3,1 a }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.12
32. Prove that the product of two even or two odd functions is an even function, whereas the product of an even and an odd function is an odd function.
33. Prove that if the domain of the function f(x) is symmetrical with respect to x = 0, then xfxf is an
even function and xfxf is an odd function. 34. Prove that any function f(x), whose domain is symmetrical about origin, can be presented as a sum of an
even and an odd function. Rewrite the following functions in the form of a sum of an even and an odd function:-
i. 21
2)(
x
xxf
. {Ans. 22 11
2xx
xxf
}
ii. xaxf )( . {Ans. 22
xxxx aaaaxf }
iii. 23)( 2 xxxf . {Ans. xxxf 322 }
iv. 543 21)( xxxxf . {Ans. 534 21 xxxxf }
v. xxxf x tancos2sin)( 2 . {Ans. xxxf x tan2sincos 2 }
vi. 1001)( xxf . {Ans. 2
112
11 100100100100 xxxxxf }
35. Extend the function xxxf 2)( defined on the interval [0, 3] onto the interval [-3, 3] in an even and an odd way. {Ans. 30,2 xxxxf
03,2 xxx ;
30,2 xxxxf
03,2 xxx }
36. Let the function xxxxxf cossin2 be defined on the interval 1,0 . Find the odd and even
extensions of xf in the interval 1,1 . {Ans. 10,cossin2 xxxxxxf
01,cossin2 xxxxx ;
10,cossin2 xxxxxxf
01,cossin2 xxxxx }
37. If xf is an odd function and if xfx 0lim
exists, prove that this limit must be zero.
38. Show that derivative of an even function is an odd function and derivative of an odd function is an even function.
39. Let )(xf be an even function and if )0(f exists, find it’s value. {Ans. 0}
40. If )(xf is an even function defined on the interval 5,5 then find the real values of x satisfying the
equation
2
1)(
x
xfxf . {Ans.
2
51,
2
51,
2
53,
2
53}
41. If Ryxyfxfyxfyxf ,2)()( and 00 f , then determine that )(xf is an even function or odd function or neither. {Ans. even}
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.13
CATEGORY-6.5. PERIODIC FUNCTIONS
42. Which of the following functions are periodic:- i. 2cos)( xxf . {Ans. non-periodic} ii. xxxf sin)( . {Ans. non-periodic}
iii. xxf cos)( . {Ans. non-periodic} 43. Find the period of the following functions:-
i. xxf 4sin5)( . {Ans. 2 }
ii. 43sin4)( xxf . {Ans. 32 }
iii. xxf 2tan)( . {Ans. 2 }
iv. 2cot)( xxf . {Ans. 2 }
v. xxf 2sin)( . {Ans. 1}
vi. xxf 2sin)( . {Ans. }
vii.
6
32sin
xxf . {Ans. 26 }
viii. xxxf 44 cossin)( . {Ans. 2
}
ix. xxf cos)( . {Ans. }
x. xxxf 3cos2sin)( . {Ans. 2 }
xi. 22 cos4sin3)( xxxf . {Ans. 4 }
xii. xxf tantan)( 1 . {Ans. }
xiii. 3
cos2)(
x
xf . {Ans. 6 }
xiv.
2cos
2sin
xxxf
. {Ans. 4}
xv. 2
cos3
2sin)(
xxxf
. {Ans. 12}
xvi. 4
sin3
sinxx
xf
. {Ans. 24}
xvii. xxxxf
5sin34
3sin23
2sin
. {Ans. 2}
xviii. xxf sincos)( . {Ans. }
xix. xxxf coscossincos)( . {Ans. 2
}
xx. ecxx
xxxf
cos1cos1
sec1sin1)(
. {Ans. }
xxi. xxxf cossin)( . {Ans. 2
}
xxii. 4
tan3
cos22
sinxxx
xf
. {Ans. 12}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.14
44. Plot the graph of a periodic function f(x) with fundamental period T = 1 defined in the interval (0,1] if i. xxf )( .
ii. 2)( xxf . iii. xxf ln)( .
45. Prove that the function
.,0
.,1)(
noirrationalaisx
norationalaisxxf
is periodic but has no fundamental period.
46. Prove that if f(x) is a periodic function with period T, then the function f(ax+b) is periodic with period a
T.
47. Prove that if the function axxxf cossin)( is periodic, then a is a rational number. 48. Let )(xf be a function and k be a positive real number such that Rxxfkxf 0)()( . Prove that
)(xf is a periodic function with period 2k. 49. Let f be a real valued function defined for all real numbers x such that for some fixed 0a
xxfxfaxf 2
2
1. Show that the function xf is periodic with period 2a.
50. Let )(xf be a real valued function with domain R such that
31
323321)( xfxfxfpxf holds good for all Rx and some positive constant p, then prove that )(xf is a periodic function.
51. For what integral value of n, the function n
xnxxf
5sincos is periodic with period 3 ? {Ans.
15,5,3,1 n } CATEGORY-6.6. MISCELLANEOUS QUESTIONS ON GRAPH PLOTTING
52. Plot graph of the following functions:-
i. x
xxf1
)( .
ii. 1ln)( 2 xxf .
iii. xxf sinsin)( 1 .
iv. xxf coscos)( 1 .
v. xxf tantan)( 1 . 53. Plot graph of the following functions:-
i. xey ln .
ii. 2logxxy .
iii. xy sinsgn .
iv. xxy sgn .
v. xy 1tansgn .
vi. xxy sgnsgn .
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.15
vii. xy sgnsin 1 .
viii. xy sgncos 1 .
ix. xxxy 21 .
x. x
xx
y
2 .
xi. 2log xy x .
xii. xxy cotlntanln .
54. Plot the graphs of the functions )(
)(&
2
)()(,
2
)()(
xf
xfxfxfxfxf from the graph of f(x) for the
following functions:- i. xxf ln)( .
ii. 3)( xxf . iii. xxf sin)( . iv. xxf tan)( .
v. xxf 1sin)( .
vi. xxf 1tan)( . 55. Plot the graphs of the functions
i. 2,max xxxf .
ii. 2,min xxxf .
iii. xxxf cos,sinmax .
iv. xxxf cos,sinmin .
56. Determine the function 2,1,1max xxxf . {Ans. 1,1 xxxf
11,2 x 1,1 xx }
57. Plot the graphs of the functions i. 0,0:sinmax xxttxf
0,0:sinmax xtxt .
ii. 0,0:cosmin xxttxf
0,0:cosmin xtxt .
iii. 0,0:sinmin xxttxf
0,0:sinmin xtxt .
iv. 0,0:cosmax xxttxf
0,0:cosmax xtxt .
v. xttxf 0:tanmax .
vi. xttxf 0:tanmin .
58. If 12,2,min xxxxf , then draw the graph of xf and also discuss its continuity and
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.16
differentiability. {Ans. Continuous x ; not differentiable at 2
5,2,1,0,
2
1x }
59. Let 1)( 23 xxxxf and 10},0:)(max{)( xxttfxg
21,3 xx .
Discuss the continuity and differentiability of the function )(xg in the interval (0, 2). {Ans. not differentiable at x = 1}
60. Plot graph of the function
1,1)( 1
12
xexxf x
1,0 x . Write i. all the points of discontinuity of xf ;
ii. all the points where xf is not differentiable;
iii. all the stationary points of xf ;
iv. intervals of monotonicity of xf ;
v. all the critical points of xf ;
vi. all the points of maxima of xf ;
vii. all the points of minima of xf ;
viii. intervals of concavity of xf ;
ix. all the points of inflection of xf ;
x. range of xf ;
xi. greatest and least value of xf ; CATEGORY-6.7. INVERSE FUNCTIONS
61. Find the inverse of the following functions:- i. xxf . {Ans. xxf 1 }
ii. 2xxf . {Ans. xxf 1
1 ; xxf 1
2 }
iii. 3xxf . {Ans. 31 xxf }
iv. xxf . {Ans. 0,21 xxxf }
v. x
xf1
. {Ans. x
xf11 }
vi. 2
1
xxf . {Ans.
xxf
111 ;
xxf
112 }
vii. xexf . {Ans. xxf ln1 }
viii. xaxf . {Ans. xxf alog1 }
ix. 22
,sin
xxxf . {Ans. xxf 11 sin }
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.17
x. 2
3
2,sin
xxxf . {Ans. xxf 11 sin }
xi. xxxf 0,cos . {Ans. xxf 11 cos }
xii. 22
,tan
xxxf . {Ans. xxf 11 tan }
xiii. xxf sinh . {Ans. 1ln 21 xxxf }
xiv. xxf cosh . {Ans. 1ln 211 xxxf , 1ln 21
2 xxxf }
xv. xxf tanh . {Ans.
x
xxf
1
1ln
2
11 }
xvi. xxf 2 . {Ans. 2
1 xxf }
xvii. xxf 31 . {Ans. 3
11 xxf
}
xviii. 53 xxf . {Ans. 3
51 x
xf }
xix. 12 xxf . {Ans. 111 xxf , 11
2 xxf }
xx. 12 xxxf . {Ans. 2
34111
xxf ,
2
34112
xxf }
xxi. xxxf 22 . {Ans. xxf 111
1 , xxf 111
2 }
xxii. 53 xxf . {Ans. 3
11 5 xxf }
xxiii. x
xf
1
1. {Ans.
x
xxf
11 }
xxiv. 3 2 1 xxf . {Ans. 1311 xxf , 131
2 xxf }
xxv. 110 xxf . {Ans. 1log1 xxf }
xxvi. 12 xxxf . {Ans. 2
log411 211
xxf
, 2
log411 212
xxf
}
xxvii. xxf log5 . {Ans. xxf 5log1 10 }
xxviii. )2ln(1 xxf . {Ans. 211 xexf }
xxix. 2logxxf . {Ans. xxf1
1 2 }
xxx. 1log 2 xxxf a . {Ans. 2
1xx aa
xf
}
xxxi. 12
2
x
x
xf . {Ans.
x
xxf
1log2
1 }
xxxii. xx
xx
xf
1010
1010 {Ans.
x
xxf
1
1log
2
11 }
xxxiii. 51
3)5(1 xxf . {Ans. 31
51 15 xxf }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.18
xxxiv. xxf 3sin2 . {Ans. xy 3sin2 }
xxxv. 1
1sin21
x
xxf . {Ans.
2
1
1
1sin
x
y
y}
xxxvi. 21 1sin4 xxf . {Ans. 20,4
cos11 x
xxf ; 20,
4cos1
2 x
xxf }
xxxvii. 0)1(log1 22 xy . {Ans. 0,21
2111 xxf x ; 0,21
2112 xxf x }
xxxviii.
63
1,
63
1,13sin
xxxf . {Ans.
3
sin1 11 x
xf
}
xxxix. 3,3,3
sin 1 xx
xf . {Ans. 22
,sin31 xxxf }
xl. 3,1,13ln 2 xxxxf . {Ans. 19ln5ln,2
4531
xe
xfx
}
xli. 1, xxxf
41,2 xx
4,2 xx . {Ans.
1,1 xxxf
161, xx
16,log2 xx }
xlii. 0, xexf x 10,1 xx
1,2 xx . {Ans.
10,ln1 xxxf 21,1 xx
2,4
2
xx
}
xliii. 0,13 xxxf
0,12 xx . {Ans.
1,131 xxxf
1,1 xx }
62. Prove that the function x
ky is inverse to itself.
63. Prove that the function x
xy
1
1 is inverse to itself.
64. Prove that the function 1
2
x
xy is inverse to itself.
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.19
65. Find the value of the parameter , for which the function 0,1 xxf is the inverse of itself. {Ans. -1}
66. Let 1,1
xx
xxf
. For what value of , xf is the inverse of itself? {Ans. -1}
67. Prove that the inverse of the linear-fractional function 0
bcad
dcx
baxy is also a linear-fractional
function. Under what conditions does this function coincides with it’s inverse? {Ans. da }
68. Show that the function 0,)( xxaxf n n is inverse to itself.
69. If 32 xxf and 53 xxg , then find xfog 1 . {Ans. 3
1
1
2
7
xxfog }
70. Find the set of all values of a for which 532 23 axxaxxf is invertible. {Ans. 4,1a }
71. If 13 xxxf , then find 1
1
x
xfdx
d. {Ans. 1}
72. If xexxf and xg be its inverse function, then find 1g . {Ans. 2
1}
73. If xxxf cos and xg be its inverse function, then find
2
3g and
2
1
4
g . {Ans.
2
1,
12
2
}
74. Show that the function 2
1,12 xxxxf and
4
3
2
1 xxg are mutually inverse, and solve
the equation 4
3
2
112 xxx . {Ans. 1x }
CATEGORY-6.8. GREATEST INTEGER FUNCTION
75. Find the domain of the function 671
1
xxxf ([ ] denotes greatest integer function). {Ans.
,87,66,55,44,33,22,10, }
76. xx lim . {Ans. no limit}
77. 2
2lim xxxx
. {Ans. 3}
78. xxxx
111lim1
. {Ans. -1}
79. 4
lim4
x
xxx
. {Ans. , }
80. xx
xxx
0lim . {Ans. 0}
81.
x
xx sgn
sgnsinlim
0. {Ans. 0}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.20
82.
xx
x
1lim
0. {Ans. 1}
83.
xx
x
1lim . {Ans. 0}
84. x
nx1lim
. {Ans. n1 , 11
n }
85. xx 0lim
. {Ans. 0, 1}
86.
n
nxn lim . {Ans. x}
87.
2
.........2lim
n
nxxxn
. {Ans. 2
x }
88.
3
2222 ......321lim
n
xnxxxn
. {Ans.
3
x}
89.
x
xxx 2
2222
0 sin
2tan2tanlim
. {Ans. 2020tan }
90.
3
222
0
sin....sin2sin1limlim
n
xnxx xxx
nx
. {Ans.
3
1}
91.
xx
xxx
12
1cos1sinlim
11
0. {Ans.
22,
2
}
92.
xx
x e
x
11
0
1lim
. {Ans.
ee
2,2
1
}
93. Show that xxxx 221 .
94. Show that xxxxx 332
31 .
95. Show that xxx 121
21 .
96. Plot graphs:- i. xxf sin)( .
ii. xxf 1sin)( .
iii. xy sgn .
iv. xy sinln .
v. 2)( xexf .
vi. xxxf 22)( .
vii. xxxf )( .
viii. 21)( xxxf .
ix. xxf 1)( .
x. x
xf1
)( .
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.21
xi. xxxf )( .
xii. 21
21)( xxxf .
97. If xxxf 1 , then find the function xfxg sgn . {Ans. 1xg }
98. Show that the function xxxf has removable discontinuity for integral values of x. 99. Draw the graphs and discuss continuity and differentiability of the following functions:-
i. 22,32 xxxxf . {Ans. Discontinuous & not differentiable at 2,1,0,1x }
ii. 22, xxxxf . {Ans. Discontinuous at 2,1,1x & not differentiable at 2,1,0,1x }
iii. 2
3
2
3,2 xxxf . {Ans. Discontinuous & not differentiable at 2,1,1,2 x }
iv. 2
3
2
3,2 xxxxf . {Ans. Discontinuous & not differentiable at 2,1,1,2 x }
v. 22
,sin
xxxf . {Ans. Differentiable everywhere}
vi. 22
,sin
xxxf . {Ans. Continuous x ; not differentiable at 1,0 }
vii. 31,1][)( xxxxf . {Ans. Discontinuous & not differentiable at 0, 1, 2, 3}
viii. 20, xxxxf
32,1 xxx . {Ans. Discontinuous at 1x & not differentiable at 2,1x }
ix. 22,2
1
xxxxf . {Ans. Discontinuous at 2,1,0,1x ; not differentiable at
2,1,0,2
1,1 x }
x.
21
4sin
x
xxf
. {Ans. Continuous and differentiable for all x)
100. If Ixxf ,0
Ixx ,2 .
Discuss the continuity and differentiability of xf and xf . {Ans. Discontinuous & not differentiable
at Nnnx , . Continuous & differentiable x .}
101. Prove that
32
2
tansin2
x
xxxxf is an odd function.
102. Check the function
1,1
1 22
2
x
x
xxf
1,0 2 x for continuity at 1x . {Ans. Discontinuous}
103. For what values of a and b, the function
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.22
0,cos32
xx
xaxf
0,3
tan
xx
b
is continuous at 0x . {Ans. 3a , 2
3b }
104. Discuss the continuity of the function
0, xxxxf
0,sin xx . {Ans. Continuous x } 105. If
1,1 xxxxf
1,0 x . Test the differentiability at 1x . {Ans. Not differentiable}
106. Draw the graph of the function
2212, 2
1
122,][)(
nxn
nxnxxxF
where n is an integer. Is the function periodic? If periodic, what is its period? What are the points of discontinuity of )(xF ? {Ans. T = 2, discontinuous for all integer x}
107. Find the period of the function xnxxxxexf cos.......3cos2coscos)( . {Ans. 1}
108. For what values of a & b the function bxaxxf )( is periodic? Find it’s period. {Ans.
aTab ,1 }
109. If xxxf )( , then find its inverse function. {Ans. 122,2
11 IxIxxxf }
110. For what values of the constant a, the function axxxf )( is inverse to itself and plot it’s graph. {Ans. -2, 0}
111. For what values of a, .constant xaxax {Ans. ,....,1,,0 23
21
2 Ia }
112. If f be function defined on set of non-negative integers and taking values in the same set. Given that:-
i.
9090
1919
xfxxfx for all non-negative integers;
ii. 200019901900 f .
Find the possible values of 1990f . ([ ] denotes greatest integer function). {Ans. 1904, 1994} CATEGORY-6.9. FINDING RANGE OF ANALYTICAL FUNCTIONS
113. x
xxf . {Ans. 11 }
114. 2xxxf . {Ans.
2
1,0 }
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.23
115. 543 2 xxxf . {Ans.
,
3
11}
116. 543log 2 xxxf . {Ans.
,
3
11log }
117. 485log 2 xxxf . {Ans.
,
5
4log }
118. xxxf 321 . {Ans. 10,2 }
119. 2
23cossinlog2
xxxf . {Ans. 2,1 }
120. xxxf 12 . {Ans. 6,3 }
121. 1
x
xxf . {Ans. 1R }
122. x
xf3sin2
1
. {Ans.
1,3
1}
123. 106log 2 xxxf . {Ans. ,0 }
124.
21
2
1sin xxf ([ ] denotes Greatest integer function). {Ans.
20
}
125. xx
xx
ee
eexf
. {Ans.
2
1,0 }
126. x
xfcos34
1
. {Ans.
1,
7
1}
127. x
xf1
. {Ans. ,1 }
128. 22
16sin3 xxf
. {Ans.
2
3,0 }
129. Find domain and range of
x
xxf
1
4logsin)(
2
. {Ans. Domain: 1,2 ; Range: 1,1 }
CATEGORY-6.10. MISCELLANEOUS QUESTIONS ON FUNCTIONS
130. Let ,3,2,1A , 4,3,2B then which of the following is function from A to B :-
i. 3,3,3,2,3,1,2,11 f . {Ans. not a function}
ii. 4,2,3,12 f . {Ans. not a function}
iii. 3,3,2,2,3,13 f . {Ans. function}
iv. 4,3,2,3,3,2,2,14 f . {Ans. not a function}
131. Let 2,1,0,1,2 A and 6,5,4,3,2,1,0B and a rule 2xxf . Whether BAf : is a function
or not? If yes, find range of f. {Ans. Yes; 4,1,0Af }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.24
132. Consider a rule 32 xxf . Whether NNf : is a function or not?. {Ans. No}
133. Let 2,1,0,1,2 A and IAf : given by 322 xxxf . Find range of f. Also find pre-
images of 6, -3 and 5. {Ans. 5,0,3,4 Af ; no pre-image of 6; 0 and 2 are pre-images of –3; -2 is the pre-image of 5}
134. Given 11,6,5,2,0,1A and 108,28,18,0,1,2 B and 22 xxxf . Find Af . Whether
BAf or not. {Ans. 108,28,18,0,2Af ; BAf }
135. Let RRf : be given by 32 xxf . Find 28| xfx . Also find the pre-images of 39 and 2
under f. {Ans. 5,5 ; pre-images of 39 are –6 and 6; no pre-image of 2} 136. Express the following functions as sets of ordered pairs and determine their ranges:-
i. RAf : , 12 xxf , where 4,2,0,1A . {Ans. 17,4,5,2,1,0,2,1f ;
17,5,2,1Af }
ii. NAg : , xxg 2 , where 5,| xNxxA . {Ans. 10,5,8,4,6,3,4,2,2,1g ;
10,8,6,4,2Ag }
137. Whether 7,4,5,3,3,2,1,1g a function or not? If g is defined by the rule baxxg , then what values should be assigned to a and b? {Ans. 1,2 ba }
138. If the function f and g are given by 1,4,5,3,2,1f and 3,1,1,5,3,2g . Write fog and gof
as set of ordered pairs. {Ans. 5,1,2,5,5,2fog ; 3,4,1,3,3,1gof }
139. If 4,3,2,1A and 8,6,4,2B and BAf : is given by xxf 2 , then write 1f as a set of
ordered pairs. {Ans. 4,8,3,6,2,4,1,21 f }
140. Let 3xxf be a function with domain 3,2,1,0 . Find the domain of 1f . {Ans. 27,8,1,0 } 141. Which of the following functions are one-one, many-one, into, onto & bijective:-
i. xxfRRf sin,: . {Ans. many-one, into}
ii. xxfRf sin,1,1: . {Ans. many-one, onto}
iii. xxfRf sin,2
,2
:
. {Ans. one-one, into}
iv. xxff sin,1,12
,2
:
. {Ans. one-one, onto, bijective}
v. xxff cos,1,1,0: . {Ans. one-one, onto, bijective}
vi. 12,: xxfRRf . {Ans. one-one, into}
vii. RRf : , 33 xxf . {Ans. one-one, onto, bijective}
viii. xxf ln . {Ans. one-one, onto, bijective}
ix. xxfRf ln,1,0: . {Ans. one-one, into}
x. xxfIRf ,: . {Ans. many-one, onto}
xi. xxfNNf ,: . {Ans. one-one, onto, bijective}
xii. xxfIRf sgn,: . {Ans. many-one, into}
xiii. xxfRf ,1,0: . {Ans. many-one, onto}
xiv. xxff ,1,01,0: . {Ans. one-one, onto, bijective}
PART-I/CHAPTER-6
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.25
xv. NNf : , 32 xxf . {Ans. injective, into} 142. Which of the following are functions?
i. Ryxxyyx ,,:, 2 . {Ans. not a function}
ii. Ryxxyyx ,,:, . {Ans. function}
iii. Ryxyxyx ,,1:, 22 . {Ans. not a function}
iv. Ryxyxyx ,,1:, 22 . {Ans. not a function}
v. yxRyxyx 2,,, . {Ans. function}
vi. xyRyxyx 2,,, . {Ans. not a function}
vii. 3,,, yxRyxyx . {Ans. function}
viii. 3,,, xyRyxyx . {Ans. function}
143. If a function Bf ,2: defined by 542 xxxf is a bijection, then find B. {Ans. ,1 } CATEGORY-6.11. ADDITIONAL QUESTIONS
144. Find the minimum value of
33
3
66
6
11
211
xx
xx
xx
xx
for 0x . {Ans. 6}
145. If xxf and xxg , then find the function x satisfying 022 xgxxfx . {Ans.
,0, xxx }
146. If 32)( baxxf , then find the function g such that xfgxgf . {Ans.
2
1
3
1
a
bxxg }
147. Draw a graph of the function 11,)( 2 xxxxxf and discuss its continuity or discontinuity
in the interval 11 x . {Ans. continuous} 148. The function f is defined by )(xfy where Rttttyttx ,,2 2 . Draw the graph of f for the
interval 11 x . Also discuss it’s continuity and differentiability at x = 0. {Ans. continuous and differentiable}
149. Given 511
xxbfxaf , 0x , ba . Find xf . {Ans.
baba
bxx
a
xf
5
22}
150. If 1b and x
xf1
, then show that the conditions of L. M. V. theorem are not satisfied in the interval
b,1 but the conclusion of the theorem is true iff 21b .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
6.26
THIS PAGE IS INTENTIONALLY LEFT BLANK
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-II
ALGEBRA
CHAPTER-7
EQUATIONS AND INEQUALITIES
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.2
CHAPTER-7
EQUATIONS AND INEQUALITIES
LIST OF THEORY SECTIONS
7.1. Equation 7.2. Polynomial Functions 7.3. Polynomial Equations 7.4. Rational Equations 7.5. Irrational Equations 7.6. Equations Containing Modulus Function 7.7. Exponential Equations 7.8. Logarithmic Equations 7.9. Expo-Logarithmic Equations 7.10. Inequality 7.11. Polynomial Inequalities 7.12. Rational Inequalities 7.13. Irrational Inequalities 7.14. Inequalities Containing Modulus Function 7.15. Exponential Inequalities 7.16. Logarithmic Inequalities 7.17. Expo-Logarithmic Inequalities 7.18. Parametric Equations And Inequalities 7.19. Proving Inequalities 7.20. System Of Non-Linear Equations In More Than One Variables
LIST OF QUESTION CATEGORIES
7.1. Roots Of Polynomial Functions 7.2. Descartes’ Rule Of Signs 7.3. Polynomial Equations 7.4. Rational Equations 7.5. Irrational Equations 7.6. Equations Containing Modulus Function 7.7. Exponential Equations 7.8. Logarithmic Equations 7.9. Expo-Logarithmic Equations 7.10. Quadratic Inequalities 7.11. Polynomial Inequalities 7.12. Rational Inequalities 7.13. Irrational Inequalities 7.14. Inequalities Containing Modulus Function 7.15. Exponential Inequalities 7.16. Logarithmic Inequalities
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.3
7.17. Expo-Logarithmic Inequalities 7.18. Parametric Equations And Inequalities 7.19. Proving Inequalities 7.20. System Of Non-Linear Equations In Two Variables 7.21. System Of Non-Linear Equations In Three Or More Variables 7.22. Solving Equations And Inequalities Graphically 7.23. Equations And Inequalities Containing Greatest Integer Function 7.24. Additional Questions
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.4
CHAPTER-7
EQUATIONS AND INEQUALITIES
SECTION-7.1. EQUATION
1. Equation, Solution (root) of equation, Solution set of equation, Domain of equation i. If xf and xg are functions, then xgxf is said to be an equation.
ii. A real number a is said to be a solution (root) of the equation xgxf iff
a. af is defined;
b. ag is defined;
c. af and ag are numerically equal, i.e. agaf ;
otherwise a is not a solution of the equation xgxf .
iii. The set of all the solutions of an equation is called its solution set, denoted by eS . RSe .
iv. Domain of the equation xgxf , denoted by eD , is defined as gfe DDD . RDe .
v. ee DS .
2. Identity & contradiction i. If ee DS , then the equation is said to be an identity.
ii. If eS , then the equation is said to be a contradiction.
iii. If eS and ee DS , then the equation is neither an identity nor a contradiction.
3. Solving an equation i. Solving an equation means finding its solution set.
4. Solving an equation graphically i. Solutions of the equation xgxf are the x-coordinates of the points of intersection of the curves of
xf and xg in their graphical representation. The number of solutions is the number of points of
intersection of their curves. From the graphical sketch of the curves of xf and xg , number of
solutions of the equation xgxf and approximate values of solutions can be determined. 5. Simplest algebraic equations
i. If a is constant real number, then equations of type ax are said to be simplest algebraic equations having solution set aSe .
6. Equivalent equations i. Two (or more) equations are said to be equivalent equations iff they have the same solution set,
otherwise they are said to be non- equivalent equation. ii. If the equations xgxf 11 and xgxf 22 are equivalent then they are written as
xgxfxgxf 2211 and if they are non- equivalent then they are written as xgxf 11 ≢
xgxf 22 . 7. Equivalent transformations in equations
i. Theorem: Consider the equations xgxf and xxgxxf . If a is a solution of the
equation xgxf and a does not belong to the domain of x , then a is not a solution of the
equation xxgxxf and hence, both the equations are non-equivalent.
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.5
ii. Theorem: Consider the equations xgxf and xxgxxf . If a is a solution of the
equation xgxf and a does not belong to the domain of x , then a is not a solution of the
equation xxgxxf and hence, both the equations are non-equivalent.
iii. Theorem: Consider the equations xgxf and xxgxxf . If a is not a solution of the
equation xgxf , a belongs to the domain of this equation and 0a , then a is a solution of the
equation xxgxxf and hence, both the equations are non-equivalent. 8. System & Collection of equations/ inequalities/ systems/ collections
i. A system of two or more equations/inequalities is written as
xgxf
xgxf
22
11
or ,, 2211 xgxfxgxf (separated by comma). The solution set of a
system is defined as intersection of the solution sets of the equations/inequalities in the system, i.e every solution of the system satisfies each of the equations/inequalities in the system.
ii. A collection of two or more equations/inequalities is written as ;; 2211 xgxfxgxf (separated by semicolon). The solution set of a collection is defined as
union of the solution sets of the equations/inequalities in the collection, i.e every solution of the collection satisfies at least one of the equations/inequalities in the collection.
iii. A system or collection may contain systems and collections. 9. Equivalent equations/ inequalities/ systems/ collections
i. Two or more equations/ inequalities/ systems/ collections are said to be equivalent if they have the same solution set.
10. Solving an equation analytically i. The process of solving an equation analytically consists of certain analytical transformation leading to
equivalent simplest algebraic equations. 11. Types of equations
i. Polynomial equations ii. Rational equations iii. Irrational equations iv. Equations containing modulus function v. Exponential equations vi. Logarithmic equations vii. Expo-logarithmic equations
SECTION-7.2. POLYNOMIAL FUNCTIONS
1. Polynomial function, degree, coefficients, real polynomial, complex polynomial i. The function 01
11 axaxaxaxP n
nn
nn
is called a polynomial function where Nn ; na ,
1na , .............., 1a , 0a are real/complex constants; 0na .
ii. n is called the degree of the polynomial. iii. na , 1na , .............., 1a , 0a are called coefficients.
iv. If all the coefficients are real numbers then the polynomial is called a real polynomial. If at least one coefficient is non-real complex number then the polynomial is called a complex polynomial.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.6
2. Theorem of roots i. Any polynomial in the set of complex numbers can always be factored into a product of n linear factors,
i.e. 01
11 axaxaxaxP n
nn
nn
nnn xxxxa 121 .
ii. 1 , 2 , .........., 1n , n are real/complex numbers and are called the n roots of the polynomial. A
polynomial of degree n always has n roots and the set of roots are unique. iii. Some of the roots may be equal to one another. If m roots are equal to , then it is said that the number
is a root of order (multiplicity) m of the polynomial. iv. If is a root of P(x), then x is a factor of P(x).
v. If is a root of order m of P(x), then mx is a factor of P(x).
vi. If is a root of P(x), then 0P . vii. Polynomials P(x) and kP(x) have same roots.
3. Theorem of equivalence i. Two polynomials xP and xQ are equivalent functions if and only if their degrees are equal and the
coefficients of same powers of x are also equal, otherwise they are non-equivalent. Thus, if 01
11 axaxaxaxP n
nn
nn
011
1 bxbxbxbxQ mm
mmm
then xQxP mn iff mn and niba ii ,,2,1,0 .
4. Theorem of conjugate complex roots i. If a real polynomial has a complex root iba then it is necessary that its conjugate iba is also a root
of the polynomial. ii. A real polynomial function either has no non-real complex roots or has even number of non-real
complex roots which are in conjugate pairs. iii. If a real polynomial has a complex roots iba and its conjugate iba , then the real quadratic
222 2 baaxx is a factor of the polynomial. iv. If a real polynomial has p real roots and q non-real complex roots then the polynomial has p real linear
factors and 2
q real quadratic factors.
5. Theorem of roots of type qp NqIp ,
i. If a real polynomial has rational coefficients and if has a root qp NqIp , then it is necessary
that its conjugate qp NqIp , is also a root of the polynomial.
6. Theorem of rational roots i. If the real polynomial 01
11 axaxaxaxP n
nn
nn
has integer coefficients and if a rational
number q
p is a root of the polynomial then p must be a divisor of 0a and q must be a divisor of na .
7. Descartes’ rule of signs i. A real polynomial xP cannot have more positive roots than there are changes in sign in xP and
cannot have more negative roots than there are changes of sign in xP . 8. To find the roots of a real polynomial function
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.7
i. Polynomial of degree one (Linear function)
a. The root of linear function baxxP 1 is a
b1 .
ii. Polynomial of degree two (Quadratic function) a. Given a quadratic function cbxaxxP 2
2 , the number acb 42 is called the discriminant.
b. The two roots of the quadratic function are a
acbb
2
42
1
and
a
acbb
2
42
2
.
c. If 0 both the roots are real and different 0 both the roots are real and equal 0 both the roots are non-real complex and different.
iii. Higher degree polynomial function a. There is no formula for the roots of polynomials of degree greater than two. b. Find roots using Theorem of rational roots. c. Find the roots by substitution. d. Find the roots by factorization.
SECTION-7.3. POLYNOMIAL EQUATIONS
1. Linear equation i. If 0a then 0bax is a linear equation.
ii. a
bxbax 0 .
2. Quadratic equation i. If 0a then 02 cbxax is a quadratic equation. ii. Type-I: If 042 acb
02 cbxax
a
acbbx
a
acbbx
2
4;
2
4 22
.
iii. Type-II: If 042 acb 02 cbxax
a
bx
2
.
iv. Type-III: If 042 acb 02 cbxax
x . 3. Higher order polynomial equation
i. Type-I: Write the equation as 0xP . Find real distinct roots of the polynomial xP using Theorem of rational roots and these are the real solutions of the equation.
ii. Type-II: Substitution. iii. Type-III: Factorization. iv. Type-IV: Using Descartes' rule of signs and other theorems, if it can be shown that xP has no real root,
then the equation has no solution.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.8
SECTION-7.4. RATIONAL EQUATIONS
1. Rational equations i. An equation containing rational functions of x is called a rational equation.
2. Solving rational equations i. Type-I:
0xQ
xP
0
0
xP
xQ.
ii. Type-II: Substitution. SECTION-7.5. IRRATIONAL EQUATIONS
1. Irrational equations i. An equation containing irrational functions of x is called an irrational equation.
2. Solving irrational equations i. Type-I: For equations containing square root, repeatedly square the equation to obtain a polynomial
equation. Solve the polynomial equation. Back check each solution of the polynomial equation in the original equation to discard extraneous solutions.
ii. Type-II:
1xhxgxf
xgxfxhxgxfxgxf
2
xh
xgxfxgxf
.
Add equations (1) and (2) and solve. iii. Type-III:
xhxgxf 33
333333 xhxgxfxgxfxgxf
3333 xhxhxgxfxgxf
xgxfxhxhxgxf 3333 .
Cube both sides to get polynomial equation and solve. Back check each solution of the polynomial equation in the original equation to discard extraneous solutions.
iv. Type-IV: Substitution. SECTION-7.6. EQUATIONS CONTAINING MODULUS FUNCTION
1. Equations containing modulus function i. An equation containing modulus of expressions of x is called an equation containing modulus function.
2. Solving equations containing modulus function i. Type-I: Using definition of modulus. ii. Type-II: Method of interval. iii. Type-III: Substitution.
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.9
SECTION-7.7. EXPONENTIAL EQUATIONS
1. Exponential equations i. An equation containing exponential functions of x is called an exponential equation.
2. Solving exponential equations i. Type-I: Simplest exponential equations
a. 0, bxba x .
b. 0,log bbxba ax .
ii. Type-II: a. 0, bxba xf .
b. 0,log bbxfba axf .
iii. Type-III: xgxfaa xgxf . iv. Type-IV: Substitution
0
/
y
yaa xfx
.
v. Type-V: Homogeneous equation in two exponential functions u and v 00
11
222
11
nnnn
nn
nn vauvavuavuaua
001
1
1
av
ua
v
ua
v
ua
n
n
n
n
0011
1 ayayaya
yv
u
nn
nn
.
SECTION-7.8. LOGARITHMIC EQUATIONS
1. Logarithmic equations i. An equation containing logarithmic functions of x is called a logarithmic equation.
2. Solving logarithmic equations i. Type-I: Simplest logarithmic equation
ba axbx log .
ii. Type-II: b
a axfbxf log .
iii. Type-III: xgxf aa loglog
xgxf
xg
xf
0
0
.
iv. Type-IV:
xgxf xaxa loglog
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.10
xgxf
xa
xa
xg
xf
1
0
0
0
.
v. Type-V: Substitution
0
log/log/log
y
yxfxfx xaaa
.
vi. Type-V: Homogeneous equation in two logarithmic functions u and v 00
11
222
11
nnnn
nn
nn vauvavuavuaua
0
0
;0
0
01
1
1 av
ua
v
ua
v
ua
v
u
vn
n
n
n
n
n
n
0
0
;0
0
011
1 ayayaya
yv
u
v
u
v
nn
nn
.
SECTION-7.9. EXPO-LOGARITHMIC EQUATIONS
1. Expo-logarithmic equations i. An equation containing expo-logarithmic functions of x is called an expo-logarithmic equation.
2. Solving expo-logarithmic equations i. Type-I:
xxg xxf
xxxfxg aa loglog .
Take log (of a suitable base) both sides and solve as logarithmic equation.
ii. Type-II: Write xfxgxg aaxf log for a suitable base a and solve as exponential equation.
iii. Type-III: Substitution
0y
yxf xg
.
SECTION-7.10. INEQUALITY
1. Inequality, Solution of inequality, Solution set of inequality, Domain of inequality i. If xf and xg are functions, then xgxf or xgxf is said to be an inequality.
ii. A real number a is said to be a solution of the inequality xgxf iff
a. af is defined;
b. ag is defined;
c. af is numerically greater than ag , i.e. agaf ;
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.11
otherwise a is not a solution of the inequality xgxf .
iii. The set of all the solutions of the inequality is called its solution set, denoted by iS . RSi .
iv. Domain of the inequality xgxf , denoted by iD , is defined as gfi DDD . RDi .
v. ii DS .
2. Strict & non-strict inequalities i. xgxf or xgxf are called strict inequality and xgxf or xgxf are called non-
strict inequality. ii. xgxfxgxfxgxf ; . xgxfxgxfxgxf ; .
3. Identity & contradiction i. If ii DS , then the inequality is said to be an identity.
ii. If iS , then the inequality is said to be a contradiction.
iii. If iS and ii DS , then the inequality is neither an identity nor a contradiction.
4. Solving an inequality i. Solving an inequality means finding its solution set.
5. Solving an inequality graphically i. Inequalities can be solved from the graphical sketch of the curves of the functions contained in the
inequality. 6. Simplest algebraic inequalities
i. If a is constant real number, then the following inequalities are said to be the simplest algebraic inequalities:- a. ax having solution set ,aSi .
b. ax having solution set aSi , .
c. ax having solution set ,aSi .
d. ax having solution set aSi , .
7. Equivalent inequalities i. Two (or more) inequalities are said to be equivalent inequalities iff they have the same solution set,
otherwise they are said to be non- equivalent inequalities. ii. If the inequalities xgxf 11 and xgxf 22 are equivalent then they are written as
xgxfxgxf 2211 and if they are non- equivalent then they are written as xgxf 11 ≢
xgxf 22 . 8. Equivalent transformations in inequalities
i. aa , where a is a constant. ii. aa , if 0a
aa , if 0a .
iii. nn , if 0 , 0 ; n is even nn , if 0 , 0 ; n is even.
iv. nn , if n is odd.
v. nn , if 0 , 0 ; n is even.
vi. nn , if n is odd.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.12
vii.
11
, if 0 , 0
11 , if 0 , 0
11 , if 0 , 0 .
viii. aa , if 0,0,0 a aa , if 0,0,0 a .
ix. aa , if 1a aa , if 10 a .
x. aa loglog , if 0 , 0 , 1a
aa loglog , if 0 , 0 , 10 a .
9. System & Collection of equations/ inequalities/ systems/ collections i. A system of two or more equations/inequalities is written as
xgxf
xgxf
22
11
or ,, 2211 xgxfxgxf (separated by comma). The solution set of a
system is defined as intersection of the solution sets of the equations/inequalities in the system, i.e every solution of the system satisfies each of the equations/inequalities in the system.
ii. A collection of two or more equations/inequalities is written as ;; 2211 xgxfxgxf (separated by semicolon). The solution set of a collection is defined as
union of the solution sets of the equations/inequalities in the collection, i.e every solution of the collection satisfies at least one of the equations/inequalities in the collection.
iii. A system or collection may contain systems and collections. 10. Equivalent equations/ inequalities/ systems/ collections
i. Two or more equations/ inequalities/ systems/ collections are said to be equivalent if they have the same solution set.
11. Solving an inequality analytically i. The process of solving an inequality analytically consists of certain analytical transformation leading to
equivalent simplest algebraic inequalities. 12. Types of inequalities
i. Polynomial inequalities ii. Rational inequalities iii. Irrational inequalities iv. Inequalities containing modulus function v. Exponential inequalities vi. Logarithmic inequalities vii. Expo-logarithmic inequalities
SECTION-7.11. POLYNOMIAL INEQUALITIES
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.13
1. Linear inequalities i. Type-I: 0bax . ii. Type-II: 0bax . iii. Type-III: 0bax . iv. Type-IV: 0bax .
2. Quadratic inequalities i. Type-I: 02 cbxax . ii. Type-II: 02 cbxax . iii. Type-III: 02 cbxax . iv. Type-IV: 02 cbxax . v. Case-I: 0,0 a . vi. Case-II: 0,0 a . vii. Case-III: 0,0 a . viii. Case-IV: 0,0 a . ix. Case-V: 0,0 a . x. Case-VI: 0,0 a .
3. Higher order polynomial inequalities i. Type-I: Method of interval
a. 0xP
b. 0xP
c. 0xP
d. 0xP . ii. Type-II: Substitution. iii. Type-III: Factorization.
SECTION-7.12. RATIONAL INEQUALITIES
1. Solving rational inequalities i. Type-I: Method of interval
a.
0xQ
xP
b.
0xQ
xP
c.
0xQ
xP
d.
0xQ
xP.
ii. Type-II: Substitution. iii. Type-III: Factorization.
SECTION-7.13. IRRATIONAL INEQUALITIES
1. Solving irrational inequalities i. Type-I:
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.14
a. xgxf
xgxf
xg
xf
0
0
.
b. xgxf
xgxf
xg
xf
0
0
.
c. xgxf
xgxf
xg
xf
0
0
.
d. xgxf
xgxf
xg
xf
0
0
.
ii. Type-II:
a. xhxgxf
xgxhxf
2
0
0
xhxgxf
xg
xf
.
b. xhxgxf
xgxhxf
2
0
0
xhxgxf
xg
xf
.
c. xhxgxf
xgxhxf
2
0
0
xhxgxf
xg
xf
.
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.15
d. xhxgxf
xgxhxf
2
0
0
xhxgxf
xg
xf
.
iii. Type-III:
a. xgxf
0
0;0
0
2xg
xf
xgxf
xg
xf
.
b. xgxf
0
0;0
0
2xg
xf
xgxf
xg
xf
.
iv. Type-IV:
a. xgxf
2
0
0
xgxf
xg
xf
b. xgxf
2
0
0
xgxf
xg
xf
v. Type-V: a. xgxf
33 xgxf
b. xgxf
33 xgxf
c. xgxf
33 xgxf
d. xgxf
33 xgxf vi. Type-VI: Substitution.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.16
SECTION-7.14. INEQUALITIES CONTAINING MODULUS FUNCTION
1. Solving inequalities containing modulus function i. Type-I: Using definition of modulus. ii. Type-II: Method of interval. iii. Type-III: Substitution.
SECTION-7.15. EXPONENTIAL INEQUALITIES
1. Solving exponential inequalities i. Type-I: Simplest exponential inequalities
a. 0, bRxba x
1,0,log abbx a
10,0,log abbx a .
b. 0, bRxba x
1,0,log abbx a
10,0,log abbx a .
c. 0, bxba x
1,0,log abbx a
10,0,log abbx a .
d. 0, bxba x
1,0,log abbx a
10,0,log abbx a .
ii. Type-II: a. 0, bDxba f
xf
1,0,log abbxf a
10,0,log abbxf a .
b. 0, bDxba fxf
1,0,log abbxf a
10,0,log abbxf a .
c. 0, bxba xf
1,0,log abbxf a
10,0,log abbxf a .
d. 0, bxba xf
1,0,log abbxf a
10,0,log abbxf a .
iii. Type-III: a. 1, axgxfaa xgxf
10, axgxf .
b. 1, axgxfaa xgxf
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.17
10, axgxf .
c. 1, axgxfaa xgxf
10, axgxf .
d. 1, axgxfaa xgxf
10, axgxf . iv. Type-IV: Substitution v. Type-V: Homogeneous inequality in two exponential functions u and v
a. 001
122
21
1
nnnn
nn
nn vauvavuavuaua
001
1
1
av
ua
v
ua
v
ua
n
n
n
n
0011
1 ayayaya
yv
u
nn
nn
.
b. 001
122
21
1
nnnn
nn
nn vauvavuavuaua
001
1
1
av
ua
v
ua
v
ua
n
n
n
n
0011
1 ayayaya
yv
u
nn
nn
.
c. 001
122
21
1
nnnn
nn
nn vauvavuavuaua
001
1
1
av
ua
v
ua
v
ua
n
n
n
n
0011
1 ayayaya
yv
u
nn
nn
.
d. 001
122
21
1
nnnn
nn
nn vauvavuavuaua
001
1
1
av
ua
v
ua
v
ua
n
n
n
n
0011
1 ayayaya
yv
u
nn
nn
.
SECTION-7.16. LOGARITHMIC INEQUALITIES
1. Solving logarithmic inequalities i. Type-I: Simplest logarithmic inequalities
a. 1,log aaxbx ba
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.18
10,0
a
ax
xb
.
b. 1,log aaxbx ba
10,0
a
ax
xb
.
c. 1,0
log
a
ax
xbx
ba
10, aax b .
d. 1,0
log
a
ax
xbx
ba
10, aax b . ii. Type-II:
a. 1,log aaxfbxf ba
10,
0
a
axf
xfb
.
b. 1,log aaxfbxf ba
10,
0
a
axf
xfb
.
c.
1,
0log
a
axf
xfbxf
ba
10, aaxf b .
d.
1,
0log
a
axf
xfbxf
ba
10, aaxf b . iii. Type-III:
a.
1,0
0
loglog
a
xgxf
xg
xf
xgxf aa
10,0
0
a
xgxf
xg
xf
.
b.
1,0
0
loglog
a
xgxf
xg
xf
xgxf aa
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.19
10,0
0
a
xgxf
xg
xf
.
iv. Type-IV: a. xgxf xaxa loglog
xgxf
xa
xg
xf
xgxf
xa
xa
xg
xf
1
0
0
;
1
0
0
0
b. xgxf xaxa loglog
xgxf
xa
xg
xf
xgxf
xa
xa
xg
xf
1
0
0
;
1
0
0
0
c. xgxf xaxa loglog
xgxf
xa
xg
xf
xgxf
xa
xa
xg
xf
1
0
0
;
1
0
0
0
d. xgxf xaxa loglog
xgxf
xa
xg
xf
xgxf
xa
xa
xg
xf
1
0
0
;
1
0
0
0
v. Type-V: Substitution vi. Type-VI: Homogeneous inequality in two logarithmic functions u and v
a. 001
122
21
1
nnnn
nn
nn vauvavuavuaua
0
0
;0
0;
0
0
0101 av
ua
v
ua
v
ua
v
av
ua
v
ua
vn
n
n
nn
n
n
n
n
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.20
0
0
;0
0;
0
0
0101 ayaya
yv
u
v
ua
v
ayaya
yv
u
v
nn
n
nn
n
nn
n
.
b. 001
122
21
1
nnnn
nn
nn vauvavuavuaua
0
0
;0
0;
0
0
0101 av
ua
v
ua
v
ua
v
av
ua
v
ua
vn
n
n
nn
n
n
n
n
0
0
;0
0;
0
0
0101 ayaya
yv
u
v
ua
v
ayaya
yv
u
v
nn
n
nn
n
nn
n
.
c. 001
122
21
1
nnnn
nn
nn vauvavuavuaua
0
0
;0
0;
0
0
0101 av
ua
v
ua
v
ua
v
av
ua
v
ua
vn
n
n
nn
n
n
n
n
0
0
;0
0;
0
0
0101 ayaya
yv
u
v
ua
v
ayaya
yv
u
v
nn
n
nn
n
nn
n
.
d. 001
122
21
1
nnnn
nn
nn vauvavuavuaua
0
0
;0
0;
0
0
0101 av
ua
v
ua
v
ua
v
av
ua
v
ua
vn
n
n
nn
n
n
n
n
0
0
;0
0;
0
0
0101 ayaya
yv
u
v
ua
v
ayaya
yv
u
v
nn
n
nn
n
nn
n
.
SECTION-7.17. EXPO-LOGARITHMIC INEQUALITIES
1. Solving expo-logarithmic inequalities i. Type-I:
a. xxg xxf .
b. xxg xxf .
c. xxg xxf .
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.21
d. xxg xxf .
Take log (of a suitable base) both sides and solve as logarithmic inequality.
ii. Type-II: Write xfxgxg aaxf log for a suitable base a and solve as exponential inequality.
iii. Type-III: Substitution. SECTION-7.18. PARAMETRIC EQUATIONS AND INEQUALITIES
1. Parametric equations and inequalities i. An equation/inequality containing parameter is to be solved for each value of the parameter.
SECTION-7.19. PROVING INEQUALITIES
1. Property-1 If a > b & b > c then a > c.
2. Property-2 If nn bababa ....,,........., 2211 then nn bbbaaa ........................... 2121 .
3. Property-3 For 0ia , 0ib , if nn bababa ...,..........,, 2211 then nn bbbaaa ........................... 2121 .
4. Theorem-1 For a, b > 0,
HGA or ba
abba
11
2
2
equality holds only when a = b. 5. Theorem-2
For 0,......,, 21 naaa ,
HGA or naaa
nn
n naaa
n
aaa11121
21
........................
...........
21
equality holds only when naaa ..............21 .
6. Theorem-3 If 0ia ,
i. 10.................. 2121
mormif
n
aaa
n
aaam
nm
nmm
ii. 10.................. 2121
mif
n
aaa
n
aaam
nm
nmm
equality holds only when naaa ............21 .
SECTION-7.20. SYSTEM OF NON-LINEAR EQUATIONS IN MORE THAN ONE VARIABLES
1. Linear and non-linear equations i. A linear equation in n variables 1x , 2x , ..........., nx is
002211 axaxaxa nn , where 0a , 1a , 2a , ..........., na are constants.
All other equations in n variables 1x , 2x , ..........., nx are non-linear equations.
2. System of non-linear equations in two variables
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.22
i. Equations in two variables x and y are of type yxgyxf ,, or 0, yxf . ii. Solution and solution set
a. An ordered pair of numbers 11, yx corresponding to the variables x and y respectively, which satisfies
the equation is called a solution of the equation, written as 11,, yxyx . b. All the solutions of the equation is called solution set of the equation, written as
,,,,, 2211 yxyxyx . iii. Solving a system of non-linear equations in two variables
A system containing at least one non-linear equation in two variables x and y is called a system of non-linear equation in two variables. Solving a system of non-linear equations in two variables means finding its solution set.
iv. Methods of solving a system of non-linear equations in two variables a. Elimination b. Substitution c. Symmetric equations
3. System of non-linear equations in three or more variables i. Equations in three variables x, y and z are of type zyxgzyxf ,,,, or 0,, zyxf . ii. Solution and solution set
a. An ordered triplet of numbers 111 ,, zyx corresponding to the variables x, y and z respectively, which
satisfies the equation is called a solution of the equation, written as 111 ,,,, zyxzyx . b. All the solutions of the equation is called solution set of the equation, written as
,,,,,,,, 222111 zyxzyxzyx . iii. Solving a system of non-linear equations in three variables
A system containing at least one non-linear equation in three variables x, y and z is called a system of non-linear equation in three variables. Solving a system of non-linear equations in three variables means finding its solution set.
iv. Methods of solving a system of non-linear equations in three variables a. Elimination b. Substitution
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.23
EXERCISE-7
CATEGORY-7.1. ROOTS OF POLYNOMIAL FUNCTIONS
1. x x x3 25 3 9 . {Ans. 13 3, , }
2. 2 18 1083 2x x . {Ans. 3 3 3 3 3 3 3 , , }
3. 8
1
4
1
2
1 23 xxx . {Ans. 2
1,
2
1,
2
1 }
4. 13 xx . {Ans. No rational root} 5. x x x x4 3 212 54 108 81 . {Ans. 3 3 3 3, , , }
6. 253 234 xxxx . {Ans. 1,1,1,2 }
7. 2 3 3 14 3 2x x x x . {Ans. 11
211, , , }
8. x x x x4 3 214 71 154 120 . {Ans. 5 4 3 2, , , }
9. 43
4911
3
7 234 xxxx . {Ans.
32 7
3
2 7
34, , , }
10. 144 234 xxx . {Ans. 21,21,1,1 }
11. 1510105 2345 xxxxx . {Ans. 1,1,1,1,1 }
12. 27822 2345 xxxxx . {Ans. 1,1,1,1,2 }
13. 4 5 11 23 13 25 4 3 2x x x x x . {Ans. 21
4111, , , , }
14. 2 9 8 15 28 125 4 3 2x x x x x . {Ans. 3
211, , ,2,2 }
15. 3 19 9 71 84 205 4 3 2x x x x x . {Ans. 21
31 5, , ,2, }
16. 1716167 2345 xxxxx . {Ans. 32,32,1,1,1 } CATEGORY-7.2. DESCARTES’ RULE OF SIGNS
17. Show that the polynomial 1235 xxxxP cannot have a positive real root.
18. Show that the polynomial 5732 23457 xxxxxxxP cannot have a negative real root.
19. Find the nature of roots of the polynomial 13 xxxP . {Ans. One real root, two imaginary roots}
20. Find the nature of roots of the polynomial 45123 24 xxxxP . {Ans. One positive real root, one negative real root, two imaginary roots}
21. Find the nature of roots of the polynomial 352 24 xxxP . {Ans. Four imaginary roots}
22. Find the nature of roots of the polynomial 732 248 xxxxP . {Ans. No real root, eight imaginary roots}
23. Find the nature of roots of the polynomial xxxxxP 359 32 . {Ans. One root is 0, eight imaginary roots}
24. Show that the polynomial 542 347 xxxxP has at least four imaginary roots.
25. Find the least possible number of imaginary roots of the polynomial 12459 xxxxxP . {Ans.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.24
Six} 26. Show that the polynomial 275 389 xxxxxP has at least four imaginary roots.
27. Show that the polynomial 324 4610 xxxxxP has at least four imaginary roots. CATEGORY-7.3. POLYNOMIAL EQUATIONS
28. x x x3 24 6 3 0 {Ans. 1 }
29. x x x 1 2 3 27 83 3 3 {Ans. 12
23 3 }
30. 2 9 13 5 04 3 2x x x x {Ans. 1 52 }
31. 8 6 13 3 04 3 2x x x x {Ans. 12
34
1 52
1 52 }
32. x x x x4 3 24 19 106 120 0 {Ans. 2 3 4 5 }
33. x x x x 4 5 6 7 1680 {Ans. 1 12 }
34. 6 5 3 2 1 352x x x {Ans. 5 216
5 216 }
35. x x x4 32 132 0 {Ans. 3 4 }
36. x x x x 1 1 2 24 {Ans. 2 3 }
37. x x x x 4 2 8 14 304 {Ans. 5 85 5 85 5 5 5 5 }
38. x x x x x x2 2 21 2 2 3 3 1 {Ans. 0 1 }
CATEGORY-7.4. RATIONAL EQUATIONS
39. x
x x
1
3
4
71 {Ans. 1 }
40. 6 5
4 3
3 3
2 5
x
x
x
x
{Ans. 16
17 }
41. 1
1
4
2
3
x x x
{Ans. }
42. x x
x
x
5
2
2 1
2 3
5 1
10
7
5 {Ans. 3 }
43. 3 5
22
11
4
x
x
x
x {Ans. 5 61
45 61
4 }
44. 3
1
7
2
6
1x x x
{Ans. 5 5
4 }
45. 7
12
6
2 80
2 2x x x x
{Ans. }
46. x x
x x
x x
x x
2
2
2
2
7 10
7 12
3 10
3 8
{Ans. 2 }
47. x
x x
x
x x
3
3 4
1
22 2 {Ans. }
48. x xx x
22
47
4 51
{Ans. 2 6 2 6 }
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.25
49. 53
6
43
2
33
1222
xxxxxx
. {Ans. ]2[]1[ }
50. 1
8
1
6
1
6
1
80
x x x x
{Ans. 5 2 0 5 2 }
51. 2
14
5
13
2
9
5
11x x x x
{Ans. 17 }
52. 1
1
4
2
4
3
1
4
1
30x x x x
{Ans. 2 1 6 7 }
CATEGORY-7.5. IRRATIONAL EQUATIONS
53. x x 1 8 3 1 {Ans. 8 }
54. 17 17 2 x x {Ans. 8 }
55. 3 7 1 2x x {Ans. 1 3 }
56. 25 2 9 x x {Ans. }
57. x x2 4 1 0 {Ans. 1 2 }
58. 4 3 5 1 15 4x x x {Ans. 3 }
59. x x x 5 3 2 7 {Ans. }
60. 4 5 3 x x {Ans. 5 4 }
61. 4 2 4 2 4x x {Ans. 1716 }
62. 2 5 3 5 2x x {Ans. 2 }
63. x x x x2 25 8 4 5 {Ans. 2 }
64. x x x x2 21 1 1 {Ans. }
65. x x x x 11 11 4 {Ans. 5 }
66. x x x x 2 2 2 {Ans. }
67. x x 34 3 13 3 {Ans. 61 30 }
68. x x x 5 6 2 113 3 3 {Ans. 6 5 112 }
69. x x x 1 3 1 13 3 3 {Ans. 1 }
70. x x x 1 2 3 03 3 3 {Ans. 2 }
71. 1 1 23 3 x x {Ans. 0 } CATEGORY-7.6. EQUATIONS CONTAINING MODULUS FUNCTION
72. 31 xx {Ans. 1 }
73. x x 1 1 {Ans. 1 0, }
74. x x 1 2 2 {Ans. 12
52 }
75. x x 1 2 1 {Ans. 2, }
76. x x 2 4 3 {Ans. 32
92 }
77. x x 1 2 1 {Ans. 1,2 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.26
78. x x x 2 3 2 4 9 {Ans. 1 112 }
79. 2 1 3 4x x x {Ans. 32 }
80. x x x 1 1 2 2 {Ans. 2 25 }
81. x x x 2 1 3 2 0 {Ans. 2 }
82. x x x x x 1 3 1 2 2 2 {Ans. , ,2 2 }
83. x x x 2 1 3 3 0 {Ans. 76 }
84. 0233 23 xxx . {Ans. 22 }
85. 0452 xx . {Ans. }
86. x x2 9 2 5 {Ans. 3 2 1 652 }
87. x x2 1 1 0 {Ans. 1 }
88. x x2 24 9 5 {Ans. , ,3 3 }
89. x x2 29 4 5 {Ans. 3 2 2 3, , }
90. x x x x x2 22 2 {Ans. 1 52 }
91. 052342 xxx . {Ans. 314 }
92. 2 1 1 x {Ans. }
93. 3 2 2 x {Ans. 7 3 1 3 }
94. 101 xxx {Ans. 39 }
95. 3 2 1 2 x x {Ans. 12 }
96. 542 xxx {Ans.
4
1
2
9 }
97. 15
342
2
xx
xx {Ans. 2
2
1
3
2
}
98. 0232
xx . {Ans. 2112 }
CATEGORY-7.7. EXPONENTIAL EQUATIONS
99. 93 772 2
xx . {Ans.
2
51 }
100. 3 5 225x x {Ans. [4]}
101. 2 5 16003x x {Ans. [2]}
102. 9 7 13 5 5 3 x x {Ans. 3
5
}
103. 3 595
5 32 1 3 2 2 3x x x x {Ans. [-3]}
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.27
104. 3 413
9 6 412
92 1 1 x x x x {Ans.
1
2}
105. 7 22 5 9
23 7
2
2x x
log
{Ans.
3
24 }
106. 4 3 5 3 7 3 402 1 x x x {Ans. log3 2 }
107. 16 512 645
7
17
3
x
x
x
x
{Ans.
5
93
11 }
108. 5 254 6 3 4x x {Ans. 7
5
}
109. 3 313
811
2
2 1
x
x
x x
x {Ans. [81]}
110. 35
259
27125
2 12 3
x x
{Ans.
5
23 }
111. 2 3 2 761 3x x {Ans. [5]} 112. 3 81 10 9 3 0 x x {Ans. 2 2 }
113. 2log228343 25284 xx . {Ans.
2
31 }
114. 2 5 6 2 03 3 3 3x x {Ans. 4
3
3 3
32
log}
115. 3 3 9 9 61 1 x x x x {Ans. log log3 3
5 1
22 5
}
116. 64 2 12 01
33
x x
{Ans. log6 8 3 }
117. 4 6 2 2 09 9 3 27log log logx x {Ans. 9 81 }
118. 4 2 9 23 2 1 3 22 2x x x x {Ans.
1
31 }
119. 7 4 9 14 2 49 02 2 2
x x x {Ans. 1 0 1 }
120. 3 16 36 2 81 x x x {Ans. 1
2
}
121. 8 18 2 27x x x {Ans. 0 }
122. 6 9 13 6 6 4 0 x x x {Ans. 1 1 }
123. 16 5 8 6 4 0x x x {Ans. 1 32 log }
124. 27 12 2 8x x x {Ans. 0 }
125. 4 15 4 15 62 x x
{Ans. 2 2 }
126. 5 2 6 5 2 6 10 x x
{Ans. 2 2 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.28
127. a a a a ax x
2 21 1 2 {Ans. 2 2 }
128. 5 5 241 13 3 x x {Ans. 1 }
129. 5 515
2612
xx
{Ans. 1 3 }
130. 10 25174
502 1 1
x x x {Ans.
1
2
1
2 }
CATEGORY-7.8. LOGARITHMIC EQUATIONS
131. 17 54log 27
xxx . {Ans. 32 }
132. 2124log 23 xx . {Ans. 13 }
133. log log log3 9 27
11
2x x x {Ans. 27 }
134. log log2 23 1 3 x x {Ans. 1 }
135. log logx x 3 6 1 {Ans. 4 }
136. log log logx x x 4 3 5 4 {Ans. 8 }
137. ln ln lnx x x3 211
22 1 3 {Ans. 2 }
138. log log log5 53
512 2 2 2 4x x x {Ans. 3 }
139. 2 2 4 03 32log logx x {Ans. 3 3 2 }
140. log log log22
22
22 10 4 3x x {Ans. 11 6 7 1 6 7 }
141. log log2 2
2
11
3 7
3 1
x
x
x
x
{Ans. 3 }
142. 27
1
1
112 2log log
x
x
x
x
{Ans. 17 }
143. log log log3 3 35 2 2 3 1 1 4x x {Ans. 1 }
144. log log log3 2 21
22 50x x {Ans. 2 }
145. log log log2 2 214
14
42
2
11
x x x {Ans. 2 6 }
146. log log24
22 1x x {Ans. 2 2
16
16 }
147. log log10 12x x {Ans. 110 10 }
148. log
loglog log2
2
2 221
2
2 3x
xx x
{Ans. 1
2 4 }
149. 2 9 3 109log logx x {Ans. 3 39 }
150. log logx x x125 1252 {Ans. 1
625 5 }
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.29
151. log log logx x xx5 594
52 {Ans. 5 515 }
152. log log log logx x 3 2 0 {Ans. 10 }
153. log log3 72
2 329 12 4 4 6 23 21x xx x x x {Ans. 1
4 }
154. log log log log2 24 412
212
0
x x x x {Ans. 0
3 2 6
274
}
155. log logx x276
04 {Ans. 4 813 }
156. log log log0 52
163
414 40 0 x x xx x x {Ans. 11
24
}
157. 4 2 32
42
23log log logx x x xx x {Ans. 1
8 1 4 }
158. log log3 323
1x xx
{Ans. 1
9 1 3 }
159. 1 3 10 04 0 2 log log. .x x {Ans. 25 }
160. 2 912
3
loglogx x
{Ans. 19 }
161. log logx xx x x2 2 21 1 4 {Ans.
1 5
2
}
162. log . log2 20 5x x {Ans. 2 }
163. log log log2 4 9 1 2 12 2 x x {Ans. 2 4 }
164. log log6 5 25 20 5 x x x {Ans. }
165. xx 44 log31log1 {Ans. 8 }
CATEGORY-7.9. EXPO-LOGARITHMIC EQUATIONS
166. x xx
1 100 11log {Ans. 910 99 }
167. 2log 1000xx x . {Ans. 100010
1
}
168. xx
xlog
log
5
3 510 {Ans. 10 103 5 }
169. x xx x
1 12 2 3log log {Ans. 1
10 2 1000 }
170. x xx x
3 31
4
2
3 {Ans. 2 4 11 }
171. x x x
3 13 10 32
{Ans. 13 2 4 }
172. xxlog5 1
5
{Ans. 15 25 }
173. x x xlog log 7 4 110 {Ans. 10 104 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.30
174. 3log
1log3
10
xx
x . {Ans.
3
2
3
2
1010 }
CATEGORY-7.10. QUADRATIC INEQUALITIES
175. 0473 2 xx {Ans. 34,1 }
176. 0673 2 xx {Ans. } 177. 0673 2 xx {Ans. 3,3
2 }
178. 0532 xx {Ans. R } 179. 015142 xx {Ans. ,151, }
180. 02 2 xx {Ans. 1,2 }
181. 0252 x {Ans. 5,5 }
182. xxx 7102 {Ans. 0,3 }
183. 372 xx {Ans. 2617
2617 , }
184. 08162 xx {Ans. 4 }
185. 0852 xx {Ans. R }
186. 37115 2 xxx . {Ans. 4,2 }
187. 5232 22 xxx . {Ans. } CATEGORY-7.11. POLYNOMIAL INEQUALITIES
188. 01 2xx {Ans. ,11,0 }
189. 01 2xx {Ans. 10, }
190. 032132 xxx {Ans. 2,, 23
31 }
191. 021323 433 xxxx {Ans. 3,1,22, 3
2 }
192. 0643 xx {Ans. ,80,8 }
193. 0128 234 xxx {Ans. ,26, }
194. 0831 2 xxx {Ans. 1, }
195. 01111 432 xxxx {Ans. 11 }
196. 031416 2222 xxxxxx {Ans. ,4,4, 2131
2131 }
197. 04486444 2222 xxxxxxx {Ans. 4,22,2 }
198. 03952 222 xxxxx {Ans. 3,0,3 4411
4411 }
199. 024103 23 xxx {Ans. ,42,3 }
200. 0254 23 xxx {Ans. 12, }
201. 03732 23 xxx {Ans. ,21 }
202. 0233 234 xxxx {Ans. ,211, }
203. 05666 234 xxxx {Ans. 1,5 }
204. 03103 24 xx {Ans. ,3,3,3
13
1 }
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.31
CATEGORY-7.12. RATIONAL INEQUALITIES
205. 332
23
x
x {Ans. ,, 3
723 }
206. 12
47
x
x {Ans. ,12, }
207. 3
11
x {Ans. ,30, }
208.
0
25
231
x
xx {Ans. 2
532 ,1, }
209. 11
432
x
xx. {Ans. ,31,1 }
210. 632
722
x
xx. {Ans. 11,1
2
3,
}
211. 03512
652
2
xx
xx {Ans. ,75,32, }
212. 01
652
2
xx
xx. {Ans. ,32, }
213. 09
242
2
x
xx {Ans. ,623,623, }
214. 289
41822
2
xx
xx {Ans. 1,8 }
215. 832
3417102
2
xx
xx. {Ans. 2,1
2
5,3
}
216.
3432
51168 2
xx
xx. {Ans.
2
5,
2
33,4 }
217. 2
1
2
3
2
1
xx
x {Ans. ,22, }
218. 31
1
1
2
xx {Ans. 6
7316
731 ,11, }
219. 21
11
x
x
x
x {Ans. ,1,01, 2
1 }
220. 14
32
2
2
x
x
x
x {Ans. ,41,2, 4
1 }
221. 1
8
1
2
1 2
xxx
x {Ans. ,31,12, }
222.
03412
321
xxx
xxx {Ans. 3,1,23,4 2
1 }
223. 0259 2
23
x
xxx {Ans. ,0, 3
535 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.32
224. 08
123
x
xxx {Ans. 1,8 }
225.
034
1222
2
xx
xxx. {Ans. 4,12, }
226. 2
3
3
2
1
1
xxx {Ans. 1,12,3 }
227. xxx
1
1
1
2
1
{Ans. ,22,10,2 }
228. 1
6
2
7
1
3
xxx {Ans. 5,11,2, 4
5 }
229. 1
74
2
13
1
12
x
x
x
x
x
x {Ans. ,51,1,2 4
5 }
230. 054
12
24
xx
xx {Ans. 5,1 }
231. 01
822
24
xx
xx {Ans. 2,,2 2
512
51 }
232. 22 347
3
23
16
xxxx
{Ans. 2,,1 3
44558 }
233. 06
1
8
1
6
1
8
1
xxxx {Ans. ,825,60,625,8 }
234. 30
1
4
1
3
4
2
4
1
1
xxxx {Ans. ,76,43,21,12, }
CATEGORY-7.13. IRRATIONAL INEQUALITIES
235. 512 x {Ans. 12,21 }
236. 123 x {Ans. ,1 }
237. 53102 xx {Ans. ,3 }
238. 1313 xxx {Ans. 1, }
239. 42124 xxx {Ans. ,21 }
240. xxx 852 {Ans. 1374,52, }
241. xxx 122 {Ans. ,4 }
242. xx 209 {Ans. ,54,920 }
243. 342 xxx {Ans. ,0, 29 }
244. 72223 2 xxx {Ans. 0, }
245. 4652 xxx {Ans. ,32,1310 }
246. 35072 2 xxx {Ans. , }
247. 121 xx {Ans. ,3 }
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.33
248. 243 xx {Ans. 1673,4 }
249. 121 xx {Ans. }
250. 054413 xxx {Ans. 5,4 }
251. 32112 xxx {Ans. 15151615,3 }
252. 5813 xxx {Ans. }
253. 1135417 xxx {Ans. }
254. 5216 xxx {Ans. 21495
25 , }
255. 0232 xxx {Ans. }
256. 33 325 xx {Ans. , }
257. 33 121 xx {Ans. ,0 }
CATEGORY-7.14. INEQUALITIES CONTAINING MODULUS FUNCTION
258. 132 x . {Ans. 2,1 }
259. xx 11 . {Ans. 1, }
260. 51 xx {Ans. 3,2 }
261. 411 xx . {Ans. 2,2 }
262. 521 xx {Ans. ,32, }
263. 12512 xx {Ans. 32
72 , }
264. 253213 xxx {Ans. 411
21 , }
265. xxx 321 {Ans. ,60, }
266. 6321 xxx . {Ans. ,40, }
267. 0322 xx {Ans. 1,1 }
268. 02452 xx {Ans. ,33, }
269. xxxx 22 2153 {Ans. ,3, 35 }
270. 27310 22 xxxx {Ans. ,14, }
271. 65112 22 xxxx {Ans. ,15, }
272. 3694 22 xxxx {Ans. , }
273. 956 2 xxx {Ans. 3,1 }
274. 4512 xx {Ans.
,80,2
3
10, }
275. 111 x . {Ans. 3,1 }
276. 213 x {Ans. ,42, }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.34
277. 21 xxx {Ans. ,11, }
278. xxx 32 {Ans.
,
3
5}
279. 2232 xxx {Ans.
,
2
293
2
293, }
CATEGORY-7.15. EXPONENTIAL INEQUALITIES
280. 12 63 x {Ans. 21, }
281. 21663 x {Ans. ,0 }
282. 2162 252 6
xx {Ans. ,71, }
283. 813
12
x
{Ans. ,26, }
284. xx
3
3
12
{Ans. ,2 }
285. 425
5256
5
2
xx
{Ans. ,2, 52 }
286. 162373 5854 xx {Ans. 66, }
287. 3088 121 xx {Ans. 60log, 832 }
288. 0624 1 xx {Ans. 17log, 2 }
289. 50525 1 xx {Ans. 1, }
290. 0123422
xx {Ans.
,loglog, 2
5322
532 }
291. 212
121
1
x
x
{Ans.
7
2log,1 2 }
292. 13
1
53
11
xx
{Ans. 1,1 }
293. 09818236 xxx {Ans. ,2 }
294. 495485 22
49798 xxxx {Ans. 5,10 }
295. xxx 10725245 {Ans. 1,0 }
296. 51312132513 xxx {Ans. 1,5log13 }
297. xxx 39239 {Ans. ,log 1983
3 }
298. xx 25174 1 {Ans. ,2 }
299. 75752452 xxx {Ans. 2,7log5 }
CATEGORY-7.16. LOGARITHMIC INEQUALITIES
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.35
300. 032log21 x {Ans. 1,2
3 }
301. 02
3log2
x
x {Ans. ,3 }
302. 12log 832
85 xx {Ans. 1,, 4
341
21 }
303. 02loglog 222 xx {Ans. 2,4
1 }
304. 6log32log2 22
24 xx {Ans. 3,3 }
305. 01log2log 22
41 xx {Ans. 4,2
1 }
306. 21loglog4log xx {Ans. 7,4 }
307. 27
1loglog7 xx {Ans. ,1 }
308. 2loglog 210
2100 xx {Ans. 10,100
1 }
309. 9log4
1log
16
97log 7
2
223 xx {Ans. 7,6,2 320 }
310. 4log2
3loglog
22
1233 xx {Ans. ,9,0 3
1 }
311. 2loglog 52
5 xx . {Ans.
5,25
1}
312. 2
1
log1
log1
2
4
x
x {Ans. ,2,0 2
1 }
313. 21log xx {Ans. }
314. 416
3log 2
xx {Ans. 1,, 2
321
43 }
315. 1616log 2 xxx {Ans. 2,1,0 21137 }
316. 32log 23 xxxx {Ans. ,2 }
317.
11
3log
x
xx {Ans. 3,1 }
318. 1421log xx {Ans. 3,1 }
319. 165log 22 xxx {Ans. 6,32,1,0 2
1 }
320. 244 4log205log xx xx {Ans. ,13,4 }
321. 1log 232 xx {Ans. 3,00,11,2
3 }
322. 02
3log
4
2 2
x
xx {Ans. 2,,11, 2
323
21 }
323. 010
3log
51
xx {Ans. ,1 }
324.
1
51
22log 1
xx
xx
{Ans. 2,1 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.36
325. 141
log2
21
xx
xx
{Ans. ,,0 253
253 }
326. 013log 2
11
xx
xx {Ans. 3,2
53 }
CATEGORY-7.17. EXPO-LOGARITHMIC INEQUALITIES
327. 10001log3log2
xxx {Ans. ,1000 }
328. 10010
2log
x
x {Ans. 1000,1 }
329. 310 loglog xx xx {Ans. ,1010,0 5log5log }
330. 1112
2
x
xx {Ans. 2,251 }
CATEGORY-7.18. PARAMETRIC EQUATIONS AND INEQUALITIES
331. Solve the equation 222 axaa . {Ans. xa ,0 Rxa ,2
axa
2
1,2,0 }
332. Solve the equation 0341221 2 axaxa .
{Ans. xa ,5
4
6
7,1 xa
3
1,
5
4xa
1
4512
1
4512,1,
5
4
a
aa
a
aaxaa }
333. Solve the equation a
x
axaxa
x
2
1
2
12
2
.
{Ans. xa ,0
1,1 xa
3
1,2 xa
1
11,2,1,0
a
axa }
334. Solve the inequality 4
313
117 xa
a
x
.
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.37
{Ans. xa ,3
5,3
,
25103
44,
3
53,5
2 aaxaa
25103
44,,
3
5,35
2 aaxaa
Rxa ,5 }
335. Solve the inequality 02 aaxx .
{Ans. ,,,,40, 24
24 22 aaaaaaxa
Rxa ,4,0
,,,40 22aaxa }
336. Solve the inequality 0422 xax .
{Ans. Rxa ,4
1
,
411411,,
4
10
a
a
a
axa
a
a
a
axa
411,
411,0
2,,0 xa }
337. Find all values of a for which the inequality 33 11 axax has no less than four different integral
solutions. {Ans. 3 2,a } CATEGORY-7.19. PROVING INEQUALITIES
338. 21
xx if 0x .
339. 21
x
x if 0x .
340. 2cottan 22 xx . 341. 21010 xx . 342. 2seccos xx .
343. cabcabcba 222 . 344. abxybyaxxyab 4))(( . 345. abcbaaccb 8))()(( . 346. abcabcabccba 9))(( .
347. 9111
)(
cbacba .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.38
348. 9
c
g
b
f
a
e
g
c
f
b
e
a.
349. )(222222 cbaabcbaaccb .
350. cbac
ab
b
ca
a
bc .
351. cbac
ab
b
ca
a
bc 111333
.
352. abcabccba
111111 .
353. If 1222 cba then show that 12
1 cabcab .
354. )()()()(2 333 baabaccacbbccba
355.
333
222333 cbacbacba.
356. If nixi ,2,1,0 , prove 2
2121
111)( n
xxxxxx
nn
.
357. If 1 cba then prove that 811
11
11
27
8
cbaabc.
358. If azyx then prove that 3
27
8))()((8 azayaxaxyz .
359. If a, b, c are positive real numbers such that 1 cba , prove that
6111
bc
aa
ab
cc
ac
bb.
360. Prove that ))()()((81 dscsbsasabcd where dcbas 3 . 361. Prove that abcddScSbSaS 81))()()(( where dcbaS .
362. If 1222 cba and 1222 zyx , show that 1 czbyax .
363. If x and y are positive real numbers and m and n are any positive integers, then 4
1
11 22
mn
mn
yx
yx.
(True/False) {Ans. False}
364. For 0a , 0b and 0 yx , prove that yyyxxx baba11
. CATEGORY-7.20. SYSTEM OF NON-LINEAR EQUATIONS IN TWO VARIABLES
365. 35 yx , 256 22 xy . {Ans. 7,2 ,
19
97,
19
8}
366. 70644 yx , 8 yx . {Ans. 3,5 , 5,3 }
367. 1843 yx , 6
511
yx. {Ans. 3,2 ,
5
9,
5
18}
368. 1822
x
y
y
x, 12 yx . {Ans. 4,8 , 8,4 }
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.39
369. 826
36
322
yx
y
yx
y, 2673 yx . {Ans. 2,4 ,
17
52,
17
26, 26,52 ,
11
52,
11
26}
370. 2
5
x
y
y
x, 10 yx . {Ans. 2,8 , 8,2 }
371. 31
223
2
3
2
32 yxyxyx , 1323 yx . {Ans. 7,9 ,
0,3
13}
372. 1712 xyx , 1312 yxy . {Ans. 2,3 ,
7
19,
7
23}
373. 0743 2 yxyxy , 01222 2 yxyxy . {Ans. Rx , 1y ; 3,2 }
374. 153 22 yxyx , 455331 22 yxxy . {Ans. 3,1 , 1,2 }
375. 5023 22 yx , 13 2 yxy . {Ans. 1,4 ,
87
3,
87
38}
376. 11 xyyx , 3022 xyyx . {Ans. 1,5 , 5,1 , 3,2 , 2,3 }
377. 12733 yx , 4222 xyyx . {Ans. 6,7 , 7,6 }
378. 26134224 yyxx , 6722 yxyx . {Ans. 2,7 , 7,2 }
379. 9111
33
yx, 1
11
yx. {Ans.
6
1,
5
1,
5
1,
6
1}
380. 4log3log 43 yx , yx loglog 34 . {Ans.
4
1,
3
1}
381. 2022 yx , 6 yx . {Ans. 2,4 , 4,2 }
382. 02438233
26
yxyxy
x , 2xy . {Ans. 1,2 , 2,1 , 2,1 }
383. Find the positive solutions of the system of equations nyx yx , nnyx yxy 2 , where 0n . {Ans. 1,1 ,
2
8114,
2
181 nnn}
CATEGORY-7.21. SYSTEM OF NON-LINEAR EQUATIONS IN THREE OR MORE VARIABLES
384. 022 zyx , 0967 zyx , 1728333 zyx . {Ans. 10,8,6 }
385. 089 zyx , 0784 zyx , 47 xyzxyz . {Ans. 4,5,3 }
386. 4 zyxx , 9 zyxy , 12 zyxz . {Ans.
5
12,
5
9,
5
4}
387. 3 zyx , 4 xzy , 5 yxz . {Ans.
6,
2
6,
3
6}
388. yxxy , zyyz 2 , xzzx 3 . {Ans. 0,0,0 ,
12,
7
12,
5
12}
389. 922 zyx , 1522
xzy , 322 yxz . {Ans. 3,1,1 , 3,1,1 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.40
390. 23 yxxy , 41 xzxz , 27 zyyz . {Ans. 6,3,5 , 8,5,7 }
391. 2 yx , 12 zxy . {Ans. 0,1,1 }
392. 84222 zyx , 14 zyx , 2yxz . {Ans. 2,4,8 , 8,4,2 }
393. 7 zyx , 2 yzxzxy , 21222 zyx . {Ans. 4,2,1 , 2,4,1 }
394. 24132 222 zyx , 63 zxyzxy , 3032 zyx . {Ans. 3,5,6 ,
3
13,
3
19,
3
10}
395. 04 xyyx , 06 yzzy , 08 zxxz . {Ans. 0,0,0 ,
5
1,1,
3
1}
396. zyxz 7 , zxyz 8 , 12 zyx . {Ans.
6,
7
66,
7
60, 2,6,4 }
CATEGORY-7.22. SOLVING EQUATIONS AND INEQUALITIES GRAPHICALLY
397. Equation xx ln has how many solutions? {Ans. No solution} 398. Equation 1ln xx has how many solutions? {Ans. One}
399. xe x . {Ans. No solution}
400. Solve xxx 543 . {Ans. 2 }
401. Solve 12 xx . {Ans. 1,0 }
402. Solve 743 xx . {Ans. ,1 } CATEGORY-7.23. EQUATIONS AND INEQUALITIES CONTAINING GREATEST INTEGER
FUNCTION
403. Solve ][Sgn xx . {Ans. 2,10,1 }
404. Solve 212 xx ([ ] denotes Greatest Integer Function). {Ans.
1,
2
51
2
51,2 }
405. Let x and x denote the fractional and integral parts of a real number x respectively. Solve
xxx 4 . {Ans.
3
50 }
406. Solve 32 xy , 523 xy ([ ] denotes Greatest Integer Function). {Ans. 11,5,4 yx }
407. Solve xxx 222 , where [x] is the greatest integer x and (x) is the least integer x . {Ans.
2
10 I }
408. Solve 20,0322 xxxx ([ ] denotes Greatest Integer Function). {Ans. 10 }
409. Solve 24 xy , 6 yy ([ ] denotes Greatest Integer Function). {Ans. 3,2,10,1 yx }
410. Solve the equation 5
36
2
13
4
13
4
13
xxxx ([ ] denotes Greatest integer function). {Ans.
6
7}
411. Solve the system of equations:-
PART-II/CHAPTER-7
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.41
1.1 zyx
2.2 zyx
3.3 zyx . ([ ] denotes Greatest integer function, { } denotes Fractional part function). {Ans. 1.0x , 2.1y , 2z }
412. For every +ve integer n, prove that 24114 nnnn . Hence, prove that
141 nnn , where [ ] denotes greatest integer function. CATEGORY-7.24. ADDITIONAL QUESTIONS
413. Find out whether the following numerical expressions are defined or not.
i. 7.0log4.1log 22 . {Ans. Not defined}
ii. 07.0log15log . {Ans. Defined}
iii. 11logloglog . {Ans. Defined}
414. Without using tables prove 2log
1log
1
43
.
415. Without using tables prove that 251
log2log17log 3
5
12 .
416. Which is greater? i. 3log2 or 5log
2
1 . {Ans. 3log2 }
ii. 5log4 or
251
log16
1 . {Ans. equal}
iii. 11log7 or 5log8 . {Ans. 11log7 }
iv. 3log2 or 11log3 . {Ans. 11log3 }
v. 21
log3
1 or 31
log2
1 . {Ans. 31
log2
1 }
417. Find all pairs yx, of real numbers such that 1161622
yxyx . {Ans. 2
1x ;
2
1y }
418. Given abcbbcba 20112012201220112011201220122011 and cba 0 and
xxx bbccbac , then find the interval of solution of x. {Ans.
2
1,0x }
419. For each integer 1n , define
n
nan , where [ ] denotes the Greatest Integer Function. Find the
number of all n in the set 2010,,3,2,1 for which 1 nn aa . {Ans. 43}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
7.42
THIS PAGE IS INTENTIONALLY LEFT BLANK
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-II
ALGEBRA
CHAPTER-8
QUADRATIC EXPRESSIONS
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.2
CHAPTER-8
QUADRATIC EXPRESSIONS
LIST OF THEORY SECTIONS
8.1. Relationship Between The Roots And Coefficients 8.2. Formation Of Polynomial Equation From Given Roots 8.3. Formation Of Polynomial Equation From Transformation 8.4. Nature Of Roots And Sign Of Quadratic Function 8.5. Common Roots 8.6. Position Of Roots 8.7. Range Of Rational Functions
LIST OF QUESTION CATEGORIES
8.1. Relationship Between The Roots And Coefficients 8.2. Formation Of Polynomial Equation From Given Roots 8.3. Formation Of Polynomial Equation From Transformation 8.4. Nature Of Roots And Sign Of Quadratic Functions 8.5. Common Roots 8.6. Position Of Roots 8.7. Miscellaneous Questions On Quadratic Functions 8.8. Range Of Rational Functions 8.9. Additional Questions
PART-II/CHAPTER-8
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.3
CHAPTER-8
QUADRATIC EXPRESSIONS
SECTION-8.1. RELATIONSHIP BETWEEN THE ROOTS AND COEFFICIENTS
1. Vieta’s Theorem i. If the polynomial 01
11 axaxaxaxP n
nn
nn
has roots 1 , 2 , .........., 1n , n then the
following equalities hold true:-
Sum of the roots = n
nn a
a 1321
;
Sum of the product of the roots taken two at a time = n
n
a
a 221
;
Sum of the product of the roots taken three at a time = n
n
a
a 3321
;
Product of the roots = n
nn a
a0321 1 .
2. Application of Vieta’s Theorem in quadratic function i. If the quadratic function cbxaxxP 2
2 has roots and then the following equalities hold true:-
a
b ;
a
c .
3. Application of Vieta’s Theorem in cubic function i. If the cubic function dcxbxaxxP 23
3 has roots , and then the following equalities hold
true:-
a
b ;
a
c ;
a
d .
4. Application of Vieta’s Theorem in bi-quadratic function i. If the bi-quadratic function edxcxbxaxxP 234
4 has roots , , and then the following equalities hold true:-
a
b ;
a
c ;
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.4
a
d ;
a
e .
SECTION-8.2. FORMATION OF POLYNOMIAL EQUATION FROM GIVEN ROOTS
1. To find a polynomial from given roots i. Given n numbers 1 , 2 , .........., 1n , n , then a polynomial of degree n having these numbers as roots
is
nnnnnn
n SxSxSxSxxP 133
22
11 ;
where
1S = Sum of the roots = n 321 ;
2S = Sum of the product of the roots taken two at a time = 21 ;
Sum of the product of the roots taken three at a time = 321 ;
Sum of the product of the roots taken k at a time = k 321 ;
Product of the roots = n 321 .
2. To find a quadratic function from given two roots i. Given two numbers and , then a quadratic function having these numbers as roots is
212
2 SxSxxP , i.e.
xxxP 22 .
3. To find a cubic function from given three roots i. Given three numbers , and , then a cubic function having these numbers as roots is
322
13
3 SxSxSxxP , i.e.
xxxxP 233 .
4. To find a bi-quadratic function from given four roots i. Given three numbers , , and , then a bi-quadratic function having these numbers as roots is
432
23
14
4 SxSxSxSxxP , i.e.
xxxxxP 2344 .
SECTION-8.3. FORMATION OF POLYNOMIAL EQUATION FROM TRANSFORMATION
1. Formation of polynomial equation from transformation i. Sometimes a given polynomial equation can be transformed into another polynomial equation whose
roots bears a certain assigned relation with those of the given equation. 2. To transform a given polynomial equation into another polynomial equation whose roots are negative
of the roots of the given polynomial equation i. Substitute xy .
3. To transform a given polynomial equation into another polynomial equation whose roots are a given
PART-II/CHAPTER-8
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.5
multiple of the roots of the given polynomial equation i. Substitute kxy .
4. To transform a given polynomial equation into another polynomial equation whose roots exceed the roots of the given polynomial equation by a given quantity i. Substitute kxy .
5. To transform a given polynomial equation into another polynomial equation whose roots are reciprocal of the roots of the given polynomial equation
i. Substitute x
y1
.
6. To transform a given polynomial equation into another polynomial equation whose roots are square of the roots of the given polynomial equation i. Substitute 2xy .
7. To transform a given polynomial equation into another polynomial equation whose roots are cube of the roots of the given polynomial equation i. Substitute 3xy .
SECTION-8.4. NATURE OF ROOTS AND SIGN OF QUADRATIC FUNCTION
1. Nature of roots of quadratic function i. To determine whether the roots are real, imaginary, rational, integer, even, odd etc.
2. Sign of quadratic function i. To determine the intervals where the quadratic function is positive or negative.
SECTION-8.5. COMMON ROOTS
1. Common roots of quadratic functions i. Two quadratic functions 11
21 cxbxa and 22
22 cxbxa have one root common if
2122112211221 cacababacbcb and common root is 1221
2112
baba
caca
or
2112
1221
caca
cbcb
.
ii. Two quadratic functions 112
1 cxbxa and 222
2 cxbxa have both roots common if 2
1
2
1
2
1
c
c
b
b
a
a .
2. Common roots of polynomial functions i. If all roots of xPm are roots of xQn mn then xPm is a factor of xQn , i.e.
xPxxQ mmnn .
SECTION-8.6. POSITION OF ROOTS
1. Given a quadratic function cbxaxxf 2 and given a number k, to find the condition that one root is less than k and the other root is greater than k.
2. Given a quadratic function cbxaxxf 2 and given a number k, to find the condition that both the roots are less than k.
3. Given a quadratic function cbxaxxf 2 and given a number k, to find the condition that both the roots are greater than k.
4. Given a quadratic function cbxaxxf 2 and given a two numbers k and l, to find the condition
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.6
that both the roots lie between k and l. 5. Given a quadratic function cbxaxxf 2 and given a two numbers k and l, to find the condition
that both the roots do not lie between k and l. 6. Given a quadratic function cbxaxxf 2 and given a two numbers k and l, to find the condition
that exactly one of the roots lie between k and l. SECTION-8.7. RANGE OF RATIONAL FUNCTIONS
1. To find the range of rational functions of type 22
22
112
1
cxbxa
cxbxaxf
i. Find the values of for which the equation
222
2
112
1
cxbxa
cxbxa has a solution. These values of are
the range of the function.
PART-II/CHAPTER-8
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.7
EXERCISE-8
CATEGORY-8.1. RELATIONSHIP BETWEEN THE ROOTS AND COEFFICIENTS
1. If both the roots of cbxax 2 are zero, then find the value of a, b, c. {Ans. 0,0 cba }
2. If 8, 2 are the roots of axx2 and 3, 3 are the roots of bxx 2 , then find the roots of baxx 2 . {Ans. 1, 9}
3. If the product of the roots of the equation 01262 mxmx is –1, then find m. {Ans. 3
1}
4. If the sum of the roots of the equation 043321 2 axaxa is 1 , then find the product of the roots. {Ans. 2}
5. If a, b are the roots of the equation 012 xx , then find the value of 22 ba . {Ans. -1}
6. If , are the roots of the equation 0734 2 xx , then find the value of
11 . {Ans.
7
3 }
7. If and are the roots of 02 cbxax , find the values of the following:-
i. baba
11. {Ans.
ac
b}
ii. baba
. {Ans.
a
2 }
iii. 33 baba . {Ans.
33
3 3
ca
abcb }
iv. 22 baba . {Ans.
22
2 2
ca
acb }
8. If , are roots of the equation 0166 2 xx , then find the value of
3232
2
1
2
1 dcbadcba in terms of a, b, c, d. {Ans. dcba 34612
12
1 }
9. If and are the roots of 02 baxx , then prove that
is a root of the equation
02 22 bxabbx .
10. If and are the roots of 012 cxpx , show that c 111 . Hence prove that
12
12
2
122
2
2
2
cc
.
11. If the roots of the equation 0 kbxax be c and d, then prove that the roots of the equation
0 kdxcx are a and b.
12. If the roots of the equation 02 qpxx are obtained –2 and –15 when the coefficient of x was misread 17 in place of 13, then find the correct roots of the equation. {Ans. –10, –3}
13. Two candidates attempt to solve a quadratic of the form 02 qpxx . One starts with a wrong value of p and finds the roots to be 2 and 6. The other starts with a wrong value of q and finds the roots to be 2 and –9. Find the correct roots. {Ans. –3, -4}
14. If be a root of 0124 2 xx , prove that 34 3 is the other root.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.8
15. Let a, b, c be real numbers with 0a and let , be the roots of the equation 02 cbxax . Express the roots of 0323 cabcxxa in terms of , . {Ans. 22 , }
16. If 3 and 733 , then show that and are the roots of 020279 2 xx .
17. If , be the roots of 022 cbxax and , be those of 022 CBxAx , then prove that 2
2
2
A
a
ACB
acb.
18. The ratio of the roots of the equation 02 cbxax is same as the ratio of the roots of the equation 02 CBxAx . If 1D and 2D are the discriminants of 02 cbxax and 02 CBxAx
respectively, then show that 2221 :: BbDD .
19. If and be the roots of 02 qpxx and , the roots of 02 rpxx , prove that
rq .
20. If and are the roots of 012 pxx and , the roots of 012 qxx , show that
22 pq .
21. If sin and cos are roots of the equation 02 rqxpx , then show that 0222 prqp .
22. If one root of the equation 0135 2 kxx is reciprocal of other, then find the value of k. {Ans. 5} 23. If the roots of 022 qxpx are reciprocal of each other, then find p. {Ans. 2}
24. Find the condition that the roots of the equation 02 cbxax be such that
i. One root is n times the other. {Ans. 22 1 nacnb }
ii. One root is three times the other. {Ans. acb 163 2 }
25. If the roots of the equation 02 cbxax are of the form k
k 1 and
1
2
k
k, prove that
acbcba 422 .
26. If one root of the equation 02 cbxax be the square of the other, then prove that abccaacb 3223 .
27. Find the relation between a, b, c such that one root of the equation 02 cbxax may be double of the other. {Ans. acb 92 2 }
28. If one root of the quadratic equation 02 cbxax is equal to the nth power of the other root, then show
that 01
1
1
1
bcaac nnnn .
29. If the equation 1
bx
b
ax
a has roots equal in magnitude but opposite in sign, then find the value of
ba . {Ans. 0}
30. If the roots of the equation rqxpx
111
are equal in magnitude but opposite in sign, show that
rqp 2 and the product of the roots is equal to 22
2
1qp .
31. If the equation 042365 2222 kxkkxkk is satisfied by more than two values of x, then determine the value of k. {Ans. 2}
32. If the sum of the roots of 02 cbxax be equal to sum of their squares, prove that 22 babac .
PART-II/CHAPTER-8
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.9
33. , are the roots of the equation 052 xxx . If 1 and 2 be the two values of for which
and are connected by the relation 5
4
, then find the value of
1
2
2
1
. {Ans. 254}
34. If the ratio of the roots of the equation 02 qpxx be equal to the ratio of the roots of 02 mlxx ,
then prove that qlmp 22 .
35. Find the value of p for which 1x is a factor of 692533 234 xpxpxpx . Find the remaining factors for this value of p. {Ans. p = 4, 3,2,1 xxx }
36. If 232 xx is a factor of qpxx 24 , prove p = 5, q = 4.
37. If the difference of the roots of the equation 0122 pxx is 1, find the value of p. {Ans. 7 }
38. Find the value of p for which the difference between the roots of the equation 082 pxx is 2. {Ans. 6 }
39. If the roots of the equation 02 qpxx differ by unity, then prove that 142 qp .
40. If ba, are roots of the equation xaabx 12 , then find the value of b. {Ans. 1}
41. Knowing that 2 and 3 are the roots of the equation 0132 23 nxmxx , determine m and n and find
the third root of the equation. {Ans. 2
5,30,5 nm }
42. Find the value of m for which the equation 1
mbx
b
max
a has roots equal in magnitude but
opposite in sign. {Ans. 0} 43. If p and q are roots of the quadratic equation 022 ammxx , then find the value of pqqp 22 .
{Ans. a }
44. If , are the roots of the equation 012 xx and
, are roots of the equation 02 qpxx ,
then find the value of p . {Ans. 1}
45. If and are the roots of 02 qpxx and also of 02 nnnn qxpx and if
,
are the roots of
011 nn xx , then prove that n must be a even integer.
CATEGORY-8.2. FORMATION OF POLYNOMIAL EQUATION FROM GIVEN ROOTS
46. Find the polynomial whose roots are 1, 2 and 3. {Ans. 6116 23 xxx }
47. Form a quadratic equation whose roots are baa
a
. {Ans. 02 22 axaabx }
48. If and are the roots of 0632 2 xx , find the equation whose roots are 2,2 22 . {Ans.
0118494 2 xx }
49. Find the equation whose roots are 2 and 2
, where , are the roots of
022 222 nmxnmx . {Ans. 042222 nmmnxx }
50. If and are the roots of 0322 xx , find the equation whose roots are:- i. 2,2 . {Ans. 01162 xx }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.10
ii. 1
1,
1
1
. {Ans. 0123 2 xx }
iii. 5,253 2323 . {Ans. 0232 xx }
51. If and are the roots of 02 cbxax , find the equation whose roots are given below:-
i.
11,
1
. {Ans. 022 abxacbbcx }
ii.
, . {Ans. 0222 acxacbacx }
iii.
1
,1
. {Ans. 022 caxcabacx }
iv. 22
22 11,
. {Ans. 022
22222222 acbxcaacbxca }
v. baba
1,
1. {Ans. 012 bxacx }
52. If , are the roots of 02 cbxax , ,1 are the roots of 0112
1 cxbxa , show that 1, are
the roots of 01
1
1
1
1
2
c
b
c
bx
a
b
a
bx
.
CATEGORY-8.3. FORMATION OF POLYNOMIAL EQUATION FROM TRANSFORMATION
53. Form an equation whose roots are negative of the roots of the equation 0375 23 xxx . {Ans. 0375 23 xxx }
54. Form an equation whose roots are double the roots of the equation 013 xx . {Ans. 0843 xx } 55. Form an equation whose roots exceed the roots of the equation 013 xx by 2. {Ans.
09136 23 xxx } 56. If , are the roots of 02 cbxax then find the equation whose roots are 2,2 . {Ans.
02442 cbaabxax }
57. Form an equation whose roots are reciprocal of the roots of the equation 013 xx . {Ans. 0123 xx }
58. Find the quadratic equation whose roots are reciprocal of the roots of the equation 02 cbxax . {Ans. 02 abxcx }
59. Form an equation whose roots are squares of the roots of the equation 06116 23 xxx . {Ans. 0364914 23 xxx }
60. Form an equation whose roots are cubes of the roots of the equation 023 dcxbxax . {Ans. 03333 33223233 dxcbcdadxbabcdaxa }
61. If , , are roots of 013 xx , then find the polynomial whose roots are 1
32
, 1
32
, 1
32
.
{Ans. 02793 xx }
PART-II/CHAPTER-8
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.11
CATEGORY-8.4. NATURE OF ROOTS AND SIGN OF QUADRATIC FUNCTIONS
62. For what values of a, the roots of the equation axax 6822 are real. {Ans. 8,2 }
63. For what values of a the function 12 axx has no real roots? {Ans. 2,2a }
64. If one root of the equation 0122 pxx is 4, while the equation 02 qpxx has equal roots, then
find the value of q . {Ans. 4
49}
65. For what values of k , 22 kkxx has equal roots? {Ans. 122 } 66. Find the values of k for which the quadratic 1311 2 xkxk has real and equal roots. {Ans.
7,5 }
67. If a and b are integers and 02 baxx has discriminant as a perfect square then prove that its roots are integers.
68. Show that the expression cbxax 2 has always the same sign as c if 24 bac . 69. Find the values of a for which 2121 22 xaxa is positive for any x . {Ans. ,13, a }
70. If the graph of the function 575816 2 axaxy is strictly above the x -axis, then find a. {Ans. 215 a }
71. For what values of m the function 1592 mmxmx is positive for all x? {Ans. 614,0m }
72. For what values of m the function 1592 mmxmx is negative for all x? {Ans. m } 73. Prove that the roots of 0 axcxcxbxbxax are always real and they will be
equal if and only if a = b = c.
74. Examine the nature of the roots of the quadratic 042 xcxaxb where a, b, c are real. {Ans.
real} 75. Show that the equation 02 cbxax where a, b, c are real numbers connected by the relation
024 cba and 0ab has real roots. 76. If the roots of the equation 02 baxx are real and differ by a quantity which is less than c (c > 0),
then b lies between 4
22 ca and
4
2a.
77. If the two roots of the equation 01111 2422 xxaxxa are real and distinct then prove
that a lies in the interval ,22, . 78. Show that a polynomial of an odd degree has at least one real root. 79. Show that a polynomial of an even degree has at least two real roots if it attains at least one value opposite
in sign to the coefficient of its highest degree term. CATEGORY-8.5. COMMON ROOTS
80. For what value of a , 0112 axx and 02142 axx have a common root? {Ans. 0, 24} 81. If 0212 axx and 03532 axx have a common root, then find the value of a. {Ans. 4 } 82. Find k if the equations 02114 2 kxx and 032 kxx have a common root and obtain the
common root for this value of k. {Ans. 36
170 ork , common root
6
170 or }
83. If the equations 0322 xx and 0532 2 xx have a non-zero common root, then find the value of . {Ans. -1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.12
84. If the equations 02 baxx and 02 abxx have a common root, then find the numerical value of ba . {Ans. -1}
85. If the equations 02 qpxx and 02 qxpx have a common root, then find the common root.
{Ans.
pp
qq}
86. If , are the roots of 02 qpxx and , are the roots of 02 srxx , evaluate
in terms of p, q, r and s. Hence deduce the condition that the equations have a
common root. {Ans. pqrqsprsqrspsq 22222 , 022222 pqrqsprsqrspsq }
87. If the equations 02 cabxx and 02 abcxx have a common root, then show that their other roots are the roots of the equation 02 bcaxx .
88. Find the condition that a root of the equation 02 cbxax be reciprocal of a root of the equation
02 cxbxa . {Ans. cbbabcabaacc 2 }
89. If each pair of the three equations 0112 qxpx , 022
2 qxpx and 0332 qxpx have a
common root, then prove that 13322132123
22
21 24 ppppppqqqppp
90. If every pair of equations 02 bcaxx , 02 cabxx , 02 abcxx have a common root, then
find the sum and product of these common roots. {Ans. abccba
,2
}
91. If the three equations 0122 axx , 0152 bxx and 0362 xbax have a common positive root, then find a and b and the roots. {Ans. 12,3;5,3;4,3,8,7 ba }
92. If the equations 02 cbxax and 0233 23 xxx have two common roots, then show that cba .
CATEGORY-8.6. POSITION OF ROOTS
93. If the roots of the equation 0523 2 mmxxm are real and positive then prove that
4
15,3m .
94. For what values of a do both roots of the function 22 9226 aaaxx exceed 3? {Ans. ,911a }
95. For what values of a, the roots of the equation 032 22 aaaxx are real and less than 3? {Ans. a < 2}
96. Find all the values of m for which both roots of the function 52 22 mmxx
i. are less than 1. {Ans. 740
2131
2131
740 ,, m }
ii. exceed –1. {Ans. 740
2131
2131
740 ,, m }
97. For what values of a does the function 59122 axax has
i. no real roots. {Ans. 6,1a }
ii. only negative roots. {Ans. ,61,95 a }
iii. only positive roots. {Ans. a } 98. If 0, cb , then show that roots of the equation 02 cbxx are of opposite sign.
99. Find the set of values of p for which the roots of the equation 0123 2 ppxx are of opposite
signs. {Ans. 1,0 }
PART-II/CHAPTER-8
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.13
100. If the roots of the equation 023123 222 kkxkx be of opposite signs, then prove that 21 k .
101. For what values of a does the function aaxaax 4182 232 has the roots of opposite signs? {Ans.
4,0a }
102. For what values of a does the function 2722 322 aaxxaa has the roots of opposite signs?
{Ans. 3,21, a }
103. For what values of a do both roots of the function 22 axx belong to the interval 3,0 ? {Ans.
311,22a }
104. For what values of k do both roots of the function 9322 xkx belong to the interval 1,6 ? {Ans.
427,6k }
105. Find all the values of k for which one root of the function 81 22 kkxkx exceeds 2 and the other
root is less than 2? {Ans. 3,2k }
106. For what values of k, one root of the function 425 2 kkxxk is smaller than 1 and the other root
exceeds 2? {Ans. 24,5k }
107. For what values of a does the function 5122 axax has at least one positive root? {Ans.
1,a }
108. Let a, b, c be real. If 02 cbxax has two real roots and , where < -1 and > 1, then show that
01 a
b
a
c.
109. Prove that the value of a for which 011222 2 aaxax may have one root less than a and the other root greater than a, are given by a > 0 or a < -1.
110. Find all values of the parameter a for which the roots of the function axx 2 are real and exceed a? {Ans. 2,a }
111. a, b, c are real numbers, 0a . If is a root of 022 cbxxa , is a root of 022 cbxxa and 0 , then show that the equation 02222 cbxxa has a root that always lies between and .
CATEGORY-8.7. MISCELLANEOUS QUESTIONS ON QUADRATIC FUNCTIONS
112. For what values of k, the function 399 23 xxkxxf is monotonically increasing in every interval? {Ans. 3k }
113. For what values of k, the function 30279 23 xkxxxf is increasing on R? {Ans. 11 k }
114. Find the minimum value of bxax . {Ans.
4
2ba }
115. If the function 52 kxxxf is increasing on 4,2 , then find the value of k. {Ans. 4,k }
116. Find all numbers a for which the least value of the quadratic function 2244 22 aaaxx in the
interval 2,0 is equal to 3? {Ans. 10521 a }
117. Find all real values of m for which the inequality 01342 mxmx is satisfied for all positive x? {Ans. ,0m }
118. For what values of a does the inequality 0324 aa xx has at least one solution? {Ans. ,2a }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.14
119. For what values of a does the equation 0loglog2 323 axx possess
i. four solutions. {Ans. 81,0a }
ii. three solutions. {Ans. 0a }
iii. two solutions. {Ans. 810, a }
iv. one solution. {Ans. a } v. no solution. {Ans. ,8
1a }
120. Solve the equation 02 axx for every real number a.
121. For what values of a does the equation 522 xxa possess i. two solutions. {Ans. 4
25,0a }
ii. one solution. {Ans. 4250, a }
iii. no solution. {Ans. ,425a }
CATEGORY-8.8. RANGE OF RATIONAL FUNCTIONS
122. Find the range of the function 1
12
2
xx
xxxf . {Ans. 3,3
1 }
123. Find the range of the function 1
12
xx
xxf . {Ans. 1,3
1 }
124. Find the range of the function 42
422
2
xx
xx. {Ans.
3,3
1}
125. For real values of x, prove that the value of the expression 24
612112
2
xx
xx cannot lie between -5 and 3.
126. If x be real, prove that the expression 632
22
xx
x takes all values in the interval
3
1,
13
1.
127. Prove that for real values of x the expression 42
31
xx
xx cannot lie between
9
4 and 1.
128. If x is real, show that the expression 3
1122
x
xx takes all values which do not lie between 4 and 12.
129. If x is real, find the maximum and minimum values of 32
9142
2
xx
xx. {Ans. 4, -5}
130. Show that 43
432
2
xx
xx can never be greater than 7 nor less than
7
1 for real values of x.
131. Show that the expression 2
2
43
43
xxp
xpx
will be capable of all real values when x is real, provided that p
has any value between 1 and 7.
132. For what values of a is the inequality 1322
12
2
xx
axx fulfilled for all x? {Ans. 2,6a }
133. For what values of a is the inequality 41
426
2
2
xx
axx fulfilled for all x? {Ans. 4,2a }
PART-II/CHAPTER-8
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.15
CATEGORY-8.9. ADDITIONAL QUESTIONS
134. Find the value of 222 . {Ans. 2}
135. If xf is a polynomial function satisfying
xfxf
xfxf
11 and 283 f , then find 4f .
{Ans. 65} 136. If Rnl , , 0, qp and the roots of the equation 02 nnxlx be real and the ratio of the roots be
qp : , then prove that 0l
n
p
q
q
p.
137. A function RRf : is defined by 2
2
86
86
xx
xxxf
, find the interval of values of for which f is
onto. Is the function one-to-one for 3 ? Justify your answer. {Ans. 14,2 , xf is not one-to-one for 3 }
138. Let c be a fixed real number. Show that a root of the polynomial cxxxxxP 200921 can have multiplicity at most 2.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
8.16
THIS PAGE IS INTENTIONALLY LEFT BLANK
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-II
ALGEBRA
CHAPTER-9
PROGRESSIONS
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.2
CHAPTER-9
PROGRESSIONS
LIST OF THEORY SECTIONS
9.1. Sequence And Series 9.2. Arithmetic Progressions 9.3. Geometric Progressions 9.4. Harmonic Progressions 9.5. Sum Of Standard Series
LIST OF QUESTION CATEGORIES
9.1. Arithmetic Progression 9.2. Arithmetic Series 9.3. Arithmetic Mean 9.4. Geometric Progression 9.5. Geometric Series 9.6. Geometric Mean 9.7. Harmonic Progression And Harmonic Series 9.8. Harmonic Mean 9.9. Polynomial Series 9.10. Difference Of Consecutive Terms In A.P. 9.11. Difference Of Consecutive Terms In G.P. 9.12. Arithmetic-Geometric Series 9.13. Method Of Difference 9.14. Exponential And Logarithmic Series 9.15. Additional Questions
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.3
CHAPTER-9
PROGRESSIONS SECTION-9.1. SEQUENCE AND SERIES
1. Sequence i. A sequence is an ordered arrangement of numbers (real/complex) according to a definite rule. ii. Each number in the sequence is called a term.
2. Notation and representation of a sequence i. A sequence is written as ,,,,, 321 naaaa , which is also denoted as na .
ii. The number na is called the nth term of the sequence.
3. Ways of defining sequences i. By means of formula for its general term, i.e. nfan .
ii. By a recursive relationship, i.e. a formula which expresses na in terms of some preceding terms of the
sequence. iii. By other ways.
4. Series i. If ,,,,, 321 naaaa is a sequence, then naaaa 321 is a series, generally
denoted by S. 5. Partial sum of a series
i. If there is a series naaaaS 321 , then 11 aS , 212 aaS , 3213 aaaS ,
............, nn aaaaS 321 , ........... are called the partial sums of the series S.
6. Convergent and divergent series i. Given a series S and its partial sum nS , if LSn
n
lim exists, then the series is said to be convergent and
L is its sum, otherwise the series is said to be divergent. 7. Sigma notation, Pi notation and their properties
i. If m and n mn are given integers and if is an expression containing i, then the symbol
n
mi
if
means nfmfmfmf 21 .
ii.
n
mi
n
mi
n
mi
igifigif .
iii.
n
mi
n
mi
ifkikf .
iv. kmnkn
mi
1
.
v. If m and n mn are given integers and if is an expression containing i, then the symbol
n
mi
if
means nfmfmfmf 21 .
vi.
n
mi
n
mi
n
mi
igifigif .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.4
vii.
n
mi
mnn
mi
ifkikf 1 .
viii. 1
mn
n
mi
kk .
SECTION-9.2. ARITHMETIC PROGRESSIONS
1. Definition of Arithmetic Progression (A.P.) and General Term of an A.P. i. A sequence is called an Arithmetic Progression if the difference of its two consecutive terms is a
constant. ii. Terms of an A.P. are of the form ,3,2,, dadadaa .
iii. If na is an A.P., then dnaan 1 , where a and d are parameters and a is called the first term and
d is called the common difference. 2. Properties of A.P.
i. If a, b, c are in A.P. then cab 2 . ii. If a constant k is added to all the terms of a given A.P. with first term a and common difference d, then
the resulting sequence is also an A.P. with the first term ka and the same common difference d. iii. If a constant k is multiplied to all the terms of a given A.P. with first term a and common difference d,
then the resulting sequence is also an A.P. with the first term ak and the common difference dk . 3. Arithmetic series, partial sum of Arithmetic series and its convergence/ divergence
i. The sum of the terms of an A.P. is called an Arithmetic series. ii. If na is an A.P. with first term a and common difference d and the Arithmetic series
naaaaS 321 , then
dnan
Sn 122
naan
12.
iii. All Arithmetic series diverge except when 0 da . 4. Arithmetic mean(s) between two numbers
i. When three numbers are in A.P., then the middle number is the arithmetic mean of the other two numbers.
ii. If a and b are two numbers and A is their arithmetic mean then a, A, b are in A.P., hence
2
baAaAAb
.
iii. A given number of arithmetic means can be inserted between two given numbers. Given two numbers a and b and n the number of means, let 1A , 2A , ........, nA be the n arithmetic means such that
bAAAa n ,,,,, 21 are in A.P.. Then nin
abiaAi ,,2,1,
1
.
5. Arithmetic mean of two or more numbers
i. If naaaa ,,,, 321 are n numbers then their arithmetic mean is n
aaaa n 321 .
SECTION-9.3. GEOMETRIC PROGRESSIONS
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.5
1. Definition of Geometric Progression (G.P.) and General Term of a G.P. i. A sequence of non-zero numbers is called a Geometric Progression if the ratio of its two consecutive
terms is a constant. ii. Terms of a G.P. are of the form ,,,, 32 ararara .
iii. If na is a G.P., then 1 nn ara , where a and r are non-zero parameters and a is called the first term
and r is called the common ratio. 2. Properties of G.P.
i. If a, b, c are in G.P. then acb 2 . ii. If a constant k is multiplied to all the terms of a given G.P. with first term a and common ratio r, then the
resulting sequence is also a G.P. with the first term ak and the same common ratio r. iii. If all the terms of a given G.P. with first term a and common ratio r, are raised to a constant power k then
the resulting sequence is also a G.P. with the first term ka and the common ratio kr . 3. Geometric series, partial sum of Geometric series and its convergence/ divergence
i. The sum of the terms of an G.P. is called an Geometric series. ii. If na is an G.P. with first term a and common r and the Geometric series
naaaaS 321 , then
1,1
1
1
1
r
r
ra
r
raS
nn
n
1, rna .
iii. When 1r then the Geometric series converges and r
aS
1. When 1r then the Geometric series
diverges. 4. Geometric mean(s) between two positive numbers
i. When three positive numbers are in G.P., then the middle number is the geometric mean of the other two numbers.
ii. If a and b are two numbers and G is their geometric mean then a, G, b are in G.P., hence
abGa
G
G
b .
iii. A given number of geometric means can be inserted between two given numbers. Given two numbers a and b and n the number of means, let 1G , 2G , ........, nG be the n arithmetic means such that
bGGGa n ,,,,, 21 are in G.P.. Then nia
baG
n
i
i ,,2,1,1
.
5. Geometric mean of two or more positive numbers i. If naaaa ,,,, 321 are n numbers then their geometric mean is n
naaaa 321 .
SECTION-9.4. HARMONIC PROGRESSIONS
1. Definition of Harmonic Progression (H.P.) and General Term of a H.P. i. A sequence of is called a Harmonic Progression if the reciprocal of its terms are in A.P.
ii. Terms of a H.P. are of the form ,3
1,
2
1,
1,
1
dadadaa .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.6
iii. If na is a H.P., then dna
an 1
1
, where a and d are parameters such that Nndna 01 .
2. Properties of H.P.
i. If a, b, c are in H.P. then ca
acb
2.
ii. If a constant k is multiplied to all the terms of a given H.P. then the resulting sequence is also a H.P.. 3. Harmonic series, partial sum of Harmonic series and its convergence/ divergence
i. The sum of the terms of a H.P. is called a Harmonic series. ii. If na is a H.P. and the Harmonic series naaaaS 321 , then there is no formula
for nS .
iii. All Harmonic series diverge. 4. Harmonic mean(s) between two numbers
i. When three numbers are in H.P., then the middle number is the harmonic mean of the other two numbers.
ii. If a and b are two numbers and H is their harmonic mean then a, H, b are in H.P., hence bHa
1,
1,
1 are in
A.P. therefore ba
abH
2.
iii. A given number of harmonic means can be inserted between two given numbers. Given two numbers a and b and n the number of means, let 1H , 2H , ........, nH be the n harmonic means such that
bHHHa n ,,,,, 21 are in H.P.. Then
niabn
bai
aH i
,,2,1,1
11
.
5. Harmonic mean of two or more numbers
i. If naaaa ,,,, 321 are n numbers then their harmonic mean is
naaa
n111
21
.
ii. For two positive numbers, 2GAH . iii. For two or more positive numbers, HGA .
SECTION-9.5. SUM OF STANDARD SERIES
1. Polynomial series
i. Sum of first n natural numbers =
2
1
1
nni
n
i
.
ii. Sum of the squares of first n natural numbers =
6
121
1
2
nnni
n
i
.
iii. Sum of the cubes of first n natural numbers =
4
1 22
1
3
nni
n
i
.
2. Difference of consecutive terms is in A.P. 3. Difference of consecutive terms is in G.P. 4. Arithmetic-Geometric series 5. Method of difference
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.7
6. Exponential and logarithmic series i. Standard exponential series:-
a.
0
32
!!!3!2!11
n
nnx
n
x
n
xxxxe .
b.
0
2242
!22
!2!4!212
n
nnxx
n
x
n
xxxee .
c.
0
121253
!122
!12!5!3!12
n
nnxx
n
x
n
xxxxee .
d.
0
3322
!
ln
!
ln
!3
ln
!2
ln
!1
ln1
n
nnnnx
n
ax
n
axaxaxaxa .
ii. Standard logarithmic series:-
a. 11,11432
1ln1
11432
xn
x
n
xxxxxx
n
nn
nn
.
b. 11,12
21253
21
1ln
1
121253
xn
x
n
xxxx
x
x
n
nn
.
iii. Sum of certain infinite series can be obtained by expressing it in terms of standard exponential and logarithmic series.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.8
EXERCISE-9
CATEGORY-9.1. ARITHMETIC PROGRESSION
1. If 7th and 13th terms of an AP be 34 and 64 respectively, then find its 18th term. {Ans. 89} 2. Divide 28 into four parts in A.P. so that ratio of the product of first and third with the product of second
and fourth is 8:15. {Ans. 4, 6, 8, 10}
3. If cbax ,,, are real and 022 cbxbax , then show that cba ,, are in AP.
4. Prove that if p, q, r (pq) are terms (not necessarily consecutive) of an A.P., than there exists a rational
number k such that
kpq
qr
.
5. Prove that the numbers 2 3 5, , cannot be the terms of a single A.P. with non-zero common difference. 6. Find the number of terms common to the two A.P.’s 3, 7, 11, ..., 407 and 2, 9, 16, ..., 709. {Ans. 14} 7. Prove that there are 17 identical terms in the two A.P.’s 2, 5, 8, 11, ...60 terms and 3, 5, 7, 9, ...50 terms. 8. If the roots of the equation 0283912 23 xxx are in A.P., then find their common difference. {Ans.
3 } 9. The mth term of an A. P. is n and its nth term is m. Prove that its pth term is pnm . Also show that its
nm th term is zero. 10. If the pth , qth and rth terms of an A.P. be a, b and c respectively, then prove that
a q r b r p c p q 0.
11. If a, b, c are in A.P., then prove that acbca 22 4 . 12. If a, b, c are in A.P., prove that the following are also in A.P.:-
i. 1 1 1
bc ca ab, , .
ii. b +c, c + a, a + b. iii. a b c b c a c a b2 2 2 , , .
iv. ab c
bc a
ca b
1 1 1 1 1 1
, , .
v. 1 1 1
b c c a a b , , .
13. If 2a , 2b , 2c are in A.P., then the following are also in A.P.:-
i. 1 1 1
b c c a a b , , .
ii. a
b c
b
c a
c
a b , , .
14. If c
cba
b
bac
a
acb ,, are in A.P., then
cba
1,
1,
1 are also in A.P.
15. If b c c a a b 2 2 2, , are in A.P. then prove that
1 1 1
b c c a a b , , are also in A.P.
16. If 2log , 12log x and 32log x be three consecutive terms of an A.P., then find x. {Ans. log2 5 }
17. For what values of the parameter a are there values of x such that xx 11 55 , 2
a, xx 2525 are three
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.9
consecutive terms of an A.P.? {Ans. 12, }
18. If 282118 zyx , prove that 3, xylog3 , yzlog3 , zxlog7 form an A.P.
19. Each of the two triplets of numbers alog , blog , clog and ba 2loglog , cb 3log2log , ac log3log are in A.P. Can the numbers a, b, c be the lengths of the sides of a triangle? {Ans. yes}
CATEGORY-9.2. ARITHMETIC SERIES
20. The fifth term of an A.P. is 1 whereas its 31st term is 77 . Find its 20th term and sum of its first fifteen terms. Also find which term of the series will be 17 and sum of how many terms will be 20. {Ans. -44, -120, 11, 8}
21. The third term of an A.P. is 7 and its 7th term is 2 more than thrice of its 3rd term. Find the first term, common difference and the sum of its first 20 terms. {Ans. -1, 4, 740}
22. Find the number of terms in the series 3
218
3
11920 of which the sum is 300. Explain the double
answer. Also find the maximum sum of the series. {Ans. 25 or 36; 310} 23. How many terms of the sequence 54, 51, 48, ............. be taken so that their sum is 513? Explain the double
answer. {Ans. 18 or 19} 24. If the sum of three consecutive terms of an increasing AP is 51 and the product of the first and third of
these terms is 273, then find the third term. {Ans. 21} 25. Find the sum of all 2 digit odd numbers. {Ans. 2475} 26. If the sum of the sequence 2, 5, 8, 11, .......... is 60100, then find the number of terms. {Ans. 200}
27. The nth term of a series is given to be 3
4
n, find the sum of 105 terms of this series. {Ans. 1470}
28. Find the sum of first 24 terms of the A.P. ,,, 321 aaa if it is known that
a a a a a a1 5 10 15 20 24 225 . {Ans. 900}
29. Find a a a a1 6 11 16 if it is known that a a1 2, , is an A.P. and a a a a1 4 7 16 147 . {Ans. 98}
30. The first and last terms of an AP are 1 and 11. If the sum of its terms is 36, then find the number of terms. {Ans. 6}
31. If the sum of first 8 and 19 terms of an A.P. are 64 and 361 respectively, find the common difference and sum of its n terms. {Ans. 2, n2}
32. A man arranges to pay off a debt of Rs. 3600 in 40 annual installments, which form an arithmetic series. When 30 of the installments are paid he dies leaving one-third of the debt unpaid. Find the value of the first installment. {Ans. Rs. 51}
33. A class consists of a number of boys whose ages are in A.P., the common difference being 4 months. If the youngest boy is just eight years old and if the sum of the ages is 168 years, find the number of boys. {Ans. 16}
34. The sum of n term of a series is nn 43 2 . Show that the series is an A.P. and find the first term and common difference. What will be its nth term? {Ans. 7, 6, 6n+1}
35. If the sum of n terms of an AP be nn 23 , then find its first term and common difference. {Ans. 2, 6} 36. If the sum of n terms of an AP is nn 52 2 , then show that its nth term is 34 n . 37. Find the sum of all natural numbers between 250 and 1000 which are exactly divisible by 3. {Ans.
156375}
38. If 7101185
753
terms
termsn
, then find the value of n . {Ans. 35}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.10
39. Show that the sum of all odd numbers between 1 and 1000, which are divisible by 3, is 83667. 40. Find the sum of first n odd natural numbers. {Ans. n2} 41. Find the sum of all odd integers between 2 and 100 divisible by 3. {Ans. 867} 42. Find the sum of all two digit numbers which when divided by 4, yield unity as remainder. {Ans. 1210} 43. Find the sum of integers from 1 to 100 which are divisible by 2 or 5. {Ans. 3050} 44. The sum of n terms of the two series 17103 and 676563 are equal, then find the
value of n . {Ans. 25} 45. The series of natural numbers is divided into groups (1), (2, 3, 4), (3, 4, 5, 6, 7), (4, 5, 6, 7, 8, 9, 10), .....
Find the sum of the numbers in nth group. {Ans. 212 n } 46. The series of natural numbers is dividend into groups (1); (2, 3, 4); (5, 6, 7, 8, 9); ...... and so on . Show
that the sum of the numbers in the nth group is 331 nn . 47. N, the set of natural numbers, is partitioned into subsets S1 = {1}, S2 ={2, 3}, S3 = {4, 5, 6}, S4 = {7, 8, 9,
10}. Find the sum of the elements in the subset S50. {Ans. 62525} 48. Show that sum of the terms in the nth bracket (1); (3, 5); (7, 9, 11); .... is n3. 49. The sum of three numbers in A.P. is 15 whereas sum of their squares is 83. Find the numbers. {Ans. 3, 5, 7
or 7, 5, 3} 50. The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers. {Ans. 2, 4, 6
or 6, 4, 2} 51. Find four numbers in A.P. whose sum is 20 and sum of their squares is 120. {Ans. 2, 4, 6, 8 or 8, 6, 4, 2} 52. Find four numbers in A.P. whose sum is 32 and sum of squares is 276. {Ans. 5, 7, 9, 11 or 11, 9, 7, 5} 53. The number of terms of an A.P. is even; the sum of the odd terms is 24, of the even terms 30, and the last
term exceeds the first by 1012
, find the number of terms and the series. {Ans. 8 terms, ...,29,3,
23 }
54. Find an A.P. in which sum of any number of terms is always three times the squared number of these terms. {Ans. 3, 9, 15, ....}
55. If bnanSn 2 , prove that series is an A.P.
56. Sum of certain consecutive odd positive integers is 22 1357 . Find them. {Ans. 1539, 1541 767, 769, 771, 773 299, 301,......10 numbers 207, 209, .....14 numbers 135, 137, .....20 numbers 119, 121, .....22 numbers 83, 85, .........28 numbers 27, 29, .........44 numbers.}
57. 150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped the second day, four more workers dropped the third day and so on. It takes 8 more days to finish the work now. Find the number of days in which the work was completed. {Ans. 25}
58. Along a road lie an odd number of stones placed at intervals of 10 meters. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried the job with one of the end stones by carrying them in succession. In carrying all the stones he covered a distance of 3 km. Find the number of stones. {Ans. 25}
59. Balls are arranged in rows to from an equilateral triangle. The first row consists of one ball, the second row
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.11
of two balls and so on. If 669 more balls are added then all the ball can be arranged in the shape of a square and each of the sides then contains 8 balls less then each side of the triangle did. Determine the initial numbers of balls. {Ans. 1540}
60. Certain numbers appear in both arithmetic progressions 17, 21, 25, ... and 16, 21, 26, ... Find the sum of first hundred numbers appearing in both progressions. {Ans. 101100}
61. Let nS denote the sum of first n terms of an A.P. If nn SS 32 , then find the ratio n
n
S
S3 . {Ans. 6}
62. If PnQnSn 2 , where nS denotes the sum of the first n terms of an A.P., then show that the third term
is QP 5 . 63. The first and last term of an A.P. are a and l respectively. If S be the sum of all the term of the A.P., show
that the common difference is
l a
S l a
2 2
2
.
64. Show that the sum of an A.P. whose first term is a, second term is b and the last term is c is equal to
a c b c a
b a
2
2.
65. If the ratio of the sum of m term and n terms of an A.P. be 22 : nm , prove that the ratio of its mth and nth terms will be 12:12 nm .
66. The ratio between the sum of n term of two A.P.’s is 157:83 nn . Find the ratio between their 12th
terms. Also find the ratio of their common difference. {Ans. 167 ,
73 }
67. The ratio between the sum of n term of two A.P.’s is 274:17 nn . Find the ratio between their nth
terms. {Ans. 238614
nn }
68. The sum of the first n term of two A.P.’s are as 95:53 nn . Prove that their 4th terms are equal.
69. The pth term of an A.P. is a and qth term is b. Prove that sum of its qp terms isp q
a ba b
p q
2.
70. If the sums of p, q and r terms of an A.P. be a, b and c respectively then prove that
a
pq r
b
qr p
c
rp q 0.
71. If in an A.P the sum of p terms is equal to sum of q terms, then prove that the sum of qp terms is zero. 72. In an A.P., of which a is the first term, if the sum of the first p terms is zero, show that the sum of the next
q terms is
a p q q
p 1.
73. The sum of first p terms of an A.P. is q and the sum of the first q terms is p. Find the sum of the first qp
terms. {Ans. qp } 74. Prove that the sum of the latter half of 2n terms of any A.P. is one-third the sum of 3n terms of the same
A.P. 75. The sums of n terms of three arithmetical progressions are S1, S2 and S3. The first term of each is unity and
the common differences are 1, 2 and 3 respectively. Prove that 231 2SSS .
76. The sums of n, 2n, 3n terms of an A.P. are S1, S2, S3 respectively. Prove that 123 3 SSS .
77. If pnSn2 and pmSm
2 , nm , in an A.P., prove that 3pS p .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.12
78. There are n A.P.’s whose common difference are 1, 2, 3, ... n respectively, the first term of each being
unity. Prove that sum of their nth terms is 1
212n n .
79. If there be m A.P.’s beginning with unity whose common differences are 1, 2, 3, ... m respectively, show
that the sum of their nth terms is 1
21m mn m n .
80. The sum of n terms of m arithmetical progressions are S1, S2, S3, .... Sm. The first term and common
differences are 1, 2, 3, ..., m respectively. Prove that S S S S mn m nm1 2 3
1
41 1 .
81. If S1, S2, S3,...., Sm are the sums of n term of m A.P.’s whose first terms are 1, 2, 3, ..., m and common
differences are 1, 3, 5, ...., 12 m respectively. Show that S S S S mn mnm1 2 3
1
21 .
82. If the sum of m terms of an A.P. is equal to sum of either the next n terms or the next p terms, prove that
m nm p
m pm n
1 1 1 1.
83. Show that in arithmetical progression 1a , 2a , 3a , ...., .12
22
21
22
24
23
22
21 kk aa
k
kaaaaa
84. If 1a , 2a , 3a , ...., na be an A.P. of non-zero terms prove that
i.
nnnnnn aaaaaaaaaaaaa
11121111
211123121
;
ii. 1 1 1 1
1 2 2 3 1 1a a a a a a
n
a an n n
.
85. If 1a , 2a , 3a , ...., na are in A.P. where 0ia for all i, show that
.1111
113221 nnn aa
n
aaaaaa
86. Let the sequence 1a , 2a , 3a , ...., na from an A.P. and let 01 a , prove that
2322
14
5
3
4
2
3 111
nn
n
aaaa
a
a
a
a
a
a
a
a .
1
2
2
1
n
n
a
a
a
a
CATEGORY-9.3. ARITHMETIC MEAN
87. Find fifth of the ten arithmetic means inserted between 1 and 100. {Ans. 46} 88. If fedcba ,,,,, are AM’s between 2 and 12, then find the value of fedcba . {Ans. 42}
89. The sum of two numbers is 6
12 . An even number of arithmetic means are being inserted between them
and their sum exceeds their number by 1. Find the number of means inserted. {Ans. 12} 90. Between 1 and 31 are inserted m arithmetic means so that the ratio of the 7th and 1m th means is 5:9.
Find the value of m. {Ans. 14} 91. Prove that the sum of the n arithmetic means inserted between two quantities is n times the single
arithmetic mean between them.
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.13
92. For what value of nn
nn
ba
ban
11
, is the arithmetic mean of a and b? {Ans. 0}
CATEGORY-9.4. GEOMETRIC PROGRESSION
93. If 33,22, xxx are in GP, then find the fourth term. {Ans. –13.5} 94. Find three numbers in G.P. whose sum is 65 and whose product is 3375. {Ans. 45, 15, 5 or 5, 15, 45}
95. The product of three numbers in G.P. is 125 and sum of their products taken in pairs is 8712
. Find them.
{Ans. 10, 5, 25 or
25 , 5, 10}
96. If the product of three numbers in G.P. is 216 and sum of the products taken in pairs is 156, find the numbers. {Ans. 18, 6, 2 or 2, 6, 18}
97. Three numbers are in G.P. whose sum is 70. If the extremes be each multiplied by 4 and the mean by 5, they will be in A.P. Find the numbers. {Ans. 10, 20, 40 or 40, 20, 10}
98. The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers. {Ans. 2, 4, 8 or 8, 4, 2}
99. Three numbers whose sum is 15 are in A.P. If 1, 4, 19 be added to them respectively, then they are in G.P. Find the numbers. {Ans. 26, 5, -16 or 2, 5, 8}
100. In a set of four numbers the first three are in G.P. and the last three in A.P. with common difference 6. If the first number is the same as the fourth, find the four numbers. {Ans. 8, -4, 2, 8}
101. Find four numbers in G.P. whose sum is 85 and product is 4096. {Ans. 1, 4, 16, 64 or 64, 16, 4, 1} 102. Does there exist a geometric progression containing 27, 8 and 12 as three of its terms? If it exists, how
many such progressions are possible? {Ans. yes, infinite} 103. Show that the numbers 10, 11, 12 cannot be the terms of a single G.P. with common ratio not equal to 1. 104. The third term of a G.P. is 4. Find the product of first five terms. {Ans. 45 } 105. In a G.P. the first, third and fifth terms may be considered as the first, fourth and sixteenth terms of an A.P.
Determine the fourth term of the A.P., knowing that its first term is 5 and determine T1, T3, T5 of G.P. {Ans. 20, 5, 20, 80}
106. The sum of first ten terms of an A.P. is equal to 155, and sum of the first two terms of a G.P. is 9, find these progressions if the first term of A.P. is equal to common ratio of G.P. and the first term of G.P. is
equal to common difference of A.P. {Ans. 2, 5, 8, 11, ..... & 3, 6, 12, 24,..... or ,...683,
679,
225 &
,...6
625,325,
32 }
107. Find the three numbers constituting a G.P. if it is known that the sum of the numbers is equal to 26 and that when 1, 6 and 3 are added to them respectively, the new numbers are obtained which from an A.P. {Ans. 2, 6, 18 or 18, 6, 2}
108. Three numbers from a G.P. If the 3rd term is decreased by 64, then the three numbers thus obtained will constitute an A.P. If the second term of this A.P. is decreased by 8, a G.P. will be formed again. Determine
the numbers. {Ans. 4, 20, 100 or 9
676,952,
94 }
109. Find the numbers a, b, c between 2 and 18 such that (i) their sum is 25 (ii) the numbers 2, a, b are consecutive terms of an A.P and (iii) the numbers b, c, 18 are consecutive terms of a G.P. {Ans. 5, 8, 12}
110. In a G.P. if the nm th term be p and nm th term be q, then prove that its mth term is pq.
111. The first and second terms of a GP are 4x and nx respectively. If 52x is the eight term of the same
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.14
progression, then find n . {Ans. 4} 112. If rqp ,, are in AP, then show that qthpth, and rth terms of any GP are in GP. 113. If cba ,, are in AP, ab , bc and a are in GP, then find cba :: . {Ans. 1:2:3}
114. If the roots of 22222 2 cbxcabxba are equal then show that cba ,, are in GP.
115. Let na be a GP such that 4
1
6
4 a
a and 21652 aa . Then find 1a . {Ans. 12 or
7
108}
116. The rth, sth and tth terms of a certain G.P. are R, S and T respectively. Prove that .1 srrtts TSR 117. The fourth, seventh and tenth terms of a GP are rqp ,, respectively, then show that prq 2 . 118. If x, y, z be respectively the pth, qth and rth terms of a G.P., then prove that
q r x r p y p q z log log log .0
119. If the pth, qth, rth terms of an A.P. are in G.P. show that common ratio of the G.P. is q r
p q
.
120. If a, b, c, d be in G.P., prove that
i. 22222 cdbcabdbdbcaca ;
ii. a b c b c d ab bc cd2 2 2 2 2 2 2 .
121. If a, b, c be in G.P., then prove that a ab b
bc ca ab
b a
c b
2 2
.
122. If a, b, c are three distinct real numbers in G.P. and xbcba , then prove that either 1x or 3x .
123. Find all the numbers x and y and such that x, yx 2 , yx 2 from an A.P. while the numbers 21y ,
5xy , 21x form a G.P. Write down the progressions. {Ans. x = 3, y = 1; 3, 5, 7, & 4, 8, 16}
124. If a, b, c, x are all real numbers, and a b x b a c x b c2 2 2 2 22 0 then show that a, b, c are in
G.P., and x is their common ratio.
125. If zyx cba111
and a, b, c be in G.P. then prove that x, y, z are in A.P. 126. If a, b, c be in G.P. then prove that nnn cba log,log,log are in A.P. 127. If the mth, nth, and pth terms of an A.P. and G.P. be equal and be respectively x, y and z, then prove that
.1 yxxzzy zyx 128. If a, b, c are in G.P. and x, y respectively be arithmetic means between a, b and b, c, then prove that
2y
c
x
a and .
211
byx
129. If a, b, c be distinct positive and in G.P. and bca abc log,log,log be in A.P. then show that the common
difference of this A.P. is 2
3.
130. Prove that the three successive terms of a G.P. will form the sides of a triangle if the common ratio r
satisfies the inequality 12
5 112
5 1 r .
CATEGORY-9.5. GEOMETRIC SERIES
131. The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.15
{Ans. 16, 24, 36, .......; 8
6305 }
132. The sum of first three of a G.P. is to the sum of the first six terms as 125:152. Find the common ratio of
the G.P. {Ans. 53 }
133. For a sequence 2, 1 aan and 3
11
n
n
a
a. Then find the value of
20
1rra . {Ans.
203
113 }
134. Find the value of 27
1
9
1
3
1
999 upto . {Ans. 3}
135. Find the value of 16
1
8
1
4
1
2
1
xxxx to . {Ans. x} 136. If S 1 2 4 8 16 32 , then S is a positive number. Multiply both sides by 2, then it is found
that 12 SS which leads to conclusion 1S which is certainly negative. Do you agree with the conclusion? Your answer should be supported by explanation.
137. The first term of an infinite G.P. is 1 and any term is equal to the sum of all the succeeding terms. Find the
series. {Ans. ,41,
21,1 }
138. The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of succeeding terms. Find the series. {Ans. ,,1,4 4
1 }
139. Sum of a certain number of terms of the series 2
1
3
1
9
2 is
72
55. Find the number. {Ans. 5}
140. How many terms of the series 1, 4, 16, .... must be taken to have their sum equal to 341 ? {Ans. 5} 141. In a G.P. sum of n terms is 255, the last term is 128 and common ratio is 2. Find n. {Ans. 8} 142. In an increasing G.P., the sum of the first and the last term is 66, the product of the second and the last but
one term is 128, and the sum of all the terms is 126. How many terms are there in the progression? {Ans. 6}
143. In a G.P. sum of n terms is 364, first term is 1 and the common ratio is 3. Find n. {Ans. 6}
144. Express the recurring decimal 0125125125. as a rational number. {Ans. 999125 }
145. Find the value of 0123. regarding it as a geometric series. {Ans. 49561 }
146. Find the value of 0 23.4 . {Ans. 990419 }
147. Find the value of 357.2 . {Ans. 999
2355}
148. After striking a floor a certain ball rebounds 5
4th of the height from which it has fallen. Find the total
distance that it travels before coming to rest, if it is gently dropped from a height of 120 meters. {Ans. 1080m}
149. A ball is dropped from a height of 48 ft. and rebounds two-third of the distance it falls. If it continues to fall and rebound in this way, how far will it travel before coming to rest ? {Ans. 240 ft.}
150. A GP consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then find the common ratio. {Ans. 4}
151. If in an infinite GP, first term is equal to 10 times the sum of all successive terms, then find its common
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.16
ratio. {Ans. 11
1}
152. If second term of a GP is 2 and the sum of its infinite terms is 8, then find its first term. {Ans. 4} 153. Find the sum of the term of an infinitely decreasing G.P. in which all the terms are positive, the first term
is 4, and the difference between the third and fifth term is equal to 81
32. {Ans.
22312,6
}
154. The sum of an infinite geometric progression is 2 and the sum of the geometric progression made from the
cube of the terms of this infinite series is 24. Then find the series. {Ans. 83
43
233 }
155. If the sum of an infinite GP be 3 and the sum of squares of its term is also 3, then find its first term and
common ratio. {Ans. 2
1,
2
3}
156. The length of a side of a square is a meters. A second square is formed by joining the middle points of this square. Then a third square is formed by joining the middle points of the second square and so on. The process is carried on ad-infinitum. Find the sum of the areas of the squares. {Ans. 2a2}
157. Show that the sum of n terms of a G.P. of common ratio r beginning with the pth term is qpr times the sum of an equal number of terms of the same series beginning with qth term.
158. Find the sum of 2n term of a series of which every even term is a times the term before it, and every odd
term c times the term before it, the first term being unity. {Ans. nncaaca
1
11 }
159. If S is the sum to infinity of a GP, whose first term is a , then find the sum of the first n terms. {Ans.
n
S
aS 11 }
160. If S be the sum, P the product and R the sum of the reciprocals of n terms of a G.P., prove that S
RP
n
2 .
161. If 21 aaS to 1 a , then show that S
Sa
1 .
162. If the sum of the series 32
27931
xxx to is a finite number, then show that 3x .
163. Prove that
1
1
11
log2
logn
nn
rr
r
b
an
b
a.
164. If to1 2 aa rrA and to1 2 bb rrB , prove that ba
B
B
A
Ar
/1/111
.
165. If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that
i. S S S S S1 3 2 2 1
2
ii. S S S S S12
22
1 2 3 .
166. If S1, S2, ...., Sn are the sums of infinite geometric series whose first terms are 1, 2, 3, ....n and common
ratios are 1
1,,
4
1,
3
1,
2
1
n respectively then prove that S S S S n nn1 2 3
12
3 .
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.17
167. If Sp denotes the sum of series 1 2 r rp p to and sp the sum of the series 1 to r rp p2 , prove that ppp SsS 22 .
168. If Sn represents the sum of n terms of a G.P. whose first term and common ratio are a and r respectively, then prove that
i.
S S S S
na
r
ar r
rn
n
1 2 3 21
1
1
;
ii.
S S S S
an
r
ar r
r rn
n
1 3 5 2 1
2
21
1
1 1
.
169. The sum of the squares of three distinct real numbers, which are in G.P. is 2S . If their sum is S , show
that 3,11,3
12
.
170. Prove the equality 666 6 888 8 444 4
2
2
n n n digits digits digits .
171. Find the natural number a for which 12161
nn
k
kaf , where the function f satisfies the relation
f x y f x f y . for all natural numbers x, y and further 21 f . {Ans. 3}
172. Sum the series a b a b a b 2 32 3 to n terms. {Ans. 2
111
nn
ba
naa }
173. Sum the series xx
xx
xx
xx
nn
1 1 1 122
2
23
3
2 2
. {Ans. nnx
nx
x
nx 22
122
1212
}
174. Sum the series 322 1111 xxxxxx to n terms. {Ans.
nxxxnx
1121
1 }
175. Sum the series 333222 yxxyxxyxx to n terms. {Ans.
xy
nynxxy
x
nxx1
121
212 }
176. If 1 to1 32 aaaax and 1 to1 32 bbbby . Prove that
1 to +1 3322
yx
xybaba+ab .
177. Sum the series
144
21217
324
725
12
223
32
121 . {Ans.
123232
}
178. Find the sum of the first n terms of the series 16
15
8
7
4
3
2
1. {Ans. 12 nn }
179. Let r be the common ratio of the G.P. ,, 21 aa , show that
)(
1111
1
1
13221mmm
nm
mn
mn
mmmm rra
r
aaaaaa
.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.18
180. Find the sum of the infinite series 1 1 1 12 2 2 3 3 a r a a r a a a r ......., r and a being proper
fractions. {Ans.
arr 11
1 }
181. Find 1
........lim
2
1
x
nxxx n
x. {Ans.
2
1nn}
182. Find n
nn
nnnnnn
12 2
1
2
1
2
11lim
2
. {Ans. 2e }
CATEGORY-9.6. GEOMETRIC MEAN
183. Insert five geometric means between 486 and 3
2. {Ans. 162, 54, 18, 6, 2}
184. If A and G be the A.M. and G.M. between two numbers, prove that the numbers are A A G A G .
185. Construct a quadratic in x such that A.M. of its roots is A and G.M. is G. {Ans. x Ax G2 22 0 } 186. If one G.M. G and two arithmetic means p and q be inserted between any two given numbers then show
that G p q q p2 2 2 .
187. If one A.M. A and two geometric means p and q be inserted between any two given numbers then show that p q Apq3 3 2 .
188. The A.M. between m and n and the G.M. between a and b are each equal to nmnbma . Find m and
n in terms of a and b. {Ans. ba
banba
abm
2,2 }
189. If n geometric means be inserted between a and b, then prove that their product is 2n
ab .
190. For what value of na b
a b
n n
n n,
1 1
is the geometric mean of a and b? {Ans. 21 }
191. If A is the AM of the roots of the equation 022 baxx and G is the GM of the roots of the equation 02 22 abxx , then show that GA .
192. If 21, AA be two AM’s and 21,GG be two GM’s between a and b , then prove that ab
ba
GG
AA
21
21 .
193. If the A.M. between a and b is twice as great as their G.M. show that a b: : . 2 3 2 3
194. The A.M. of a and b is to their G.M as m to n, show a b m m n m m n: : . 2 2 2 2 195. If A be the arithmetic mean of b and c and G1, G2, be the two geometric means between them, then prove
that AGGGG 213
23
1 2 . CATEGORY-9.7. HARMONIC PROGRESSION AND HARMONIC SERIES
196. The 7th term of a H.P. is 101 and 12th term is
251 , find the 20th term. {Ans.
491 }
197. Solve the equation 6 11 6 1 03 2x x x if its roots are in harmonic progression. {Ans. 31,
21,1 }
198. The value of zyx is 15 if a, x, y, z, b are in A.P. while the value of 1 1 1 5
3x y z is if a, x, y, z, b are
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.19
in H.P., find a and b. {Ans. 9 & 1} 199. If x, y, z are in H.P., prove that log log log .x z x z y x z 2 2
200. Show that 2log,2log,2log 1263 are in HP.
201. Prove that a, b, c are in A.P., G.P or H.P. according as the value of a b
b c
is equal to
c
a
b
a
a
aor ,
respectively. 202. If 1a , 2a , 3a , ...., na are in harmonic progression, prove that a a a a a a n a an n n1 2 2 3 1 11 .
203. If a, b, c be respectively the pth, qth and rth terms of an H.P., then prove that bc q r ca r p ab p q 0 .
204. If pth term of an H.P. is qr and qth term is rp, prove that rth term is pq. 205. If the roots of the equation 02 bacxacbxcba be equal, then prove that a, b, c are in H.P.
206. If the mth term of an H.P. is n and nth term be m, then prove that nm th term is mn
m n.
207. If a, b, c be in H.P., prove that
i. c
a
cb
ba
;
ii. 1 1 2
b a b c b
;
iii. b a
b a
b c
b c
2. ;
iv. 2
34111111
bacacbcba
;
v. bc ca ab ca ab bc ac b ac 4 32 .
208. If a, b, c be in H.P. prove that
i. a
b c
b
c a
c
a b , , are in H.P.;
ii. a
b c a
b
c a b
c
a b c , , are in H.P.;
iii. 1 1 1 1 1 1a b c b c a c a b
, , are in H.P.
209. If
1 1 1a b c b c a c a b
, , be in H.P. then a, b, c are also in H.P.
210. If cb , ac , ba are in H.P. then prove that
i. ba
c
ac
b
cb
a
,, are in A.P.;
ii. 222 ,, cba are in A.P.
211. If a, b, c are in A.P., prove that cabc
ab
abbc
ca
abca
bc
,, are in H.P.
212. First three of the four numbers are in A.P. & the last three in H.P. Prove that the four numbers are proportional.
213. If a, b, c be in A.P. b, c, a be in H.P. then prove that c, a, b are in G.P.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.20
214. If x, y, z are in A.P., ax, by, cz in G.P. and a, b, c in H.P. prove that x
z
z
x
a
c
c
a .
215. If a b cx y z and a, b, c be in G.P. then prove that x, y, z are in H.P. 216. If a, b, c be in G.P. then prove that nnn cba log,log,log are in H.P.
217. Given a b c dx y z u and a, b, c, d are in G.P., show that x, y, z, u are in H.P. 218. If a, b, c, d, e be five numbers such that a, b, c are in A.P., b, c, d are in G.P. and c, d, e are in H.P., prove
that a, c, e are in G.P. and
eb a
a
2 2
. If a = 2 and e = 18, find all possible values of b, c and d. {Ans. 4,
6, 9 or -2, -6, -18}
219. a, b, c are in H.P., b, c, d are in G.P. and c, d, e are in A.P. show that
.2 2
2
ba
abe
220. If a x
px
a y
qy
a z
rz
and p, q, r be in A.P. then prove that x, y, z are in H.P.
221. If cba ,, are all positive and in H.P., then show that the roots of cbxax 322 are imaginary.
222. If a, b, c be in A.P. and a2, b2, c2 in H.P. then prove that either 2
a , b, c are in G.P. or a=b=c.
223. p, q, r are three numbers in G.P. Prove that the first term of an A.P., whose pth, qth and rth terms are in H.P., is to the common difference as 1:1q .
224. A G.P. and a H.P. have the same pth, qth and rth terms as a, b, c respectively. Show that a b c a b c a b c a b c log log log .0
225. An A.P., a G.P and a H.P. have a and b for their first two terms. Show that their 2n th terms will be in
G.P. if
b a
ab b a
n
n
n n
n n
2 2 2 2
2 2
1
.
226. An A.P. and a H.P., have the same first term, the same last term, and the same number of terms; prove that the product of the rth term from the beginning in one series and the rth term from the end in the other is independent of r.
227. , , are the geometric means between ca, ab; ab, bc; bc, ca respectively. Prove that if a, b, c are in A.P., then 2 , 2 , 2 are also in A.P., and , , are in H.P.
228. If the ththth rnm 1 and 1,1 terms of an A.P. are in G.P., m, n, r are in H.P. show that the ratio of
the common difference to the first term in the A.P. is n
2 .
229. If S1, S2, S3 denote the sums of n terms of three A.P.’s whose first terms are unity and common differences
in H.P., prove that nS S S S S S
S S S
2
23 1 1 2 2 3
1 2 3
.
230. If a, b, c are in A.P., , , in H.P., a, b, c in G.P., (with common ratio not equal to 1.), then prove
that a b c: : : : .1 1 1
CATEGORY-9.8. HARMONIC MEAN
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.21
231. Insert six harmonic means between 3 and 236 . {Ans.
103,
176,
73,
116,
43,
56 }
232. If the harmonic mean of two numbers is to their geometric mean as 12:13, prove that the numbers are the ratio of 4:9.
233. Let the harmonic mean and geometric mean of two positive numbers be in the ratio 4:5, then find the ratio of the two numbers. {Ans. 1:4}
234. If A.M between two numbers is 5 and their GM is 4, then find their HM. {Ans. 5
16}
235. A.M and H.M. between two quantities are 27 and 12 respectively, find their G.M. {Ans. 18} 236. If zx ,1, are in AP and zx ,2, are in GP, then find the harmonic mean of x and z. {Ans. 4}
237. If H be the HM between a and b , then show that 2b
H
a
H.
238. The harmonic mean of two numbers is 4. Their A.M., A, and G.M., G, satisfy the relation 272 2 GA . Find the two numbers. {Ans. 6 & 3}
239. The A.M. of two numbers exceeds their G.M. by 15 and H.M. by 27, find the numbers. {Ans. 120 & 30}
240. If the A.M. between two numbers exceeds their G.M. by 2 and the G.M. exceeds their H.M. by 5
8; find the
numbers. {Ans. 4 & 16} 241. If the A.M., the G.M and the H.M. of first and last terms of the sequence 25, 26, 27, ..., 1N , N are the
term of this sequence, find the value of N. {Ans. 225 & 1225}
242. If 9 arithmetic and harmonic means be inserted between 2 and 3, prove that AH
6
5 where A is any of
the A.M.’s and H the corresponding H.M.
243. If H be the harmonic mean between a and b then prove that 1 1 1 1
H a H b a b
.
244. If A, G, H be respectively the A.M., G.M. and H.M. between two given quantities a and b, then prove that A, G, H are in G.P.
245. If A be the A.M. and H the H.M. between two numbers a and b then a A
a H
b A
b H
A
H
.
246. If A1, A2; G1, G2; and H1, H2 be two A.M.’s and G.M.’s and H.M.’s between two quantities then prove that G G
H H
A A
H H1 2
1 2
1 2
1 2
.
247. If n harmonic means are inserted between 1 and r then show that 1st meanth meann
n r
nr
1.
248. If H1, H2, ... Hn be n harmonic means between a and b show that H a
H a
H b
H bnn
n
1
1
2
. If n be a root of
the equation x ab x a b ab2 2 21 1 0 , prove that baabHH n 1 .
249. For what value of na b
a b
n n
n n,
1 1
is the harmonic mean of a and b? {Ans. -1}
250. If a be A.M. of b and c, b the G.M. of c and a, then prove that c is the H.M. of a and b. 251. If ay 2 is the H.M. between y-x and y-z, then show that ax , ay , az are in G.P. 252. If p be the first of n arithmetic means between two numbers and q be the first of n harmonic means
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.22
between the same two numbers, prove that the value of q cannot be between p and n
np
11
2
.
CATEGORY-9.9. POLYNOMIAL SERIES
253. Sum the series terms. to433221 222 n {Ans. 12
5321
nnnn
}
254. Sum the series terms.20 to735231 222 {Ans. 188090}
255. Sum the series terms. to543432321 n {Ans. 4
321
nnnn
}
256. Sum the series terms. to743632521 n {Ans. 12
17321
nnnn
}
257. Sum the series terms to4321321211 n . {Ans. 6
21
nnn
}
258. Sum the series terms to321211 222222 n . {Ans. 12
221
nnn
}
259. Sum the series terms.16 to531
321
31
21
1
1 333333
{Ans. 446}
260. Find the sum of the series .503231 333 {Ans. 1409400}
261. Sum to n terms the series 222222 654321 . {Ans. even ,2
1n
nn
; odd ,2
1n
nn
}
262. Sum the series .132211 nnnn {Ans. 6
21
nnn
}
263. Find the sum of all possible products of the first n natural numbers taken two by two. {Ans.
242311
nnnn
}
264. Find the coefficient of xn-2 in the polynomial .321 nxxxx {Ans. 24
2311
nnnn
}
265. Find the coefficient of x99 in the polynomial 100321 xxxx {Ans. -5050} 266. If s and t are respectively the sum and the sum of the squares of n successive positive integers beginning
with a then show that 2snt is independent of a. 267. If S1, S2, S3, ..., Sn are the sums of infinite geometric series whose first terms are 1, 2, 3, ….., n and whose
common ratios are 1
1,,
4
1,
3
1,
2
1
n respectively, then find the value of .2
122
32
22
1 nSSSS
{Ans.
13
1412
nnn}
268. Consider n A.P.s whose first terms are 1, 2, 3, ………, n and their common differences are 1, 2, 3, ………, n respectively. If iiS , denotes sum of i terms of ith A.P., then find nnSSSS ,3,32,21,1 . {Ans.
24
1321 nnnn}
269. On the ground are placed n stones, the distance between the first and second is one yard, between the 2nd and 3rd is 3 yards, between the 3rd and 4th, 5 yards, and so on. How far will a person have to travel who
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.23
shall bring them one by one to a basket placed at the first stone? {Ans. 3
121
nnn
yards}
270. Find
2222 11
3
1
2
1
1lim
n
n
nnnn . {Ans.
2
1 }
271. Find 3
2222 321lim
n
nn
. {Ans.
3
1}
CATEGORY-9.10. DIFFERENCE OF CONSECUTIVE TERMS IN A.P.
272. Sum the series 1 3 6 10 15 to terms.n {Ans. 6
21
nnn
}
273. Sum the series 4 6 9 13 18 to terms.n {Ans.
6
2032 nnn}
274. Sum the series 2 4 7 11 16 ...... to terms.n {Ans.
6
832 nnn}
CATEGORY-9.11. DIFFERENCE OF CONSECUTIVE TERMS IN G.P.
275. Sum the series 1 3 7 15 31 to terms.n {Ans. 2 21n n }
276. Sum up to n terms the series 0 7 0 77 0 777. . . . {Ans.
n
n10
11817
97 }
277. Sum up to n terms the series 6 66 666 . {Ans. 10910816 1 nn }
278. Sum up to n terms the series 8 88 888 . {Ans. 10910818 1 nn }
CATEGORY-9.12. ARITHMETIC-GEOMETRIC SERIES
279. Sum the series 1 2 2 3 2 4 2 100 22 3 99 . . . . . {Ans. 99 2 1100 }
280. Find the sum of n terms of the series the rth term of which is (2r + 1) 3r. {Ans. 13 nn }
281. Sum the series 132
54
78
terms.n {Ans. 12326
n
n }
282. Sum the series 145
75
1052 3 to terms and to .n {Ans.
1635,
516712
1635
1
n
n }
283. Sum the series . to3
5
3
3
3
11
32 {Ans. 2}
CATEGORY-9.13. METHOD OF DIFFERENCE
284. Sum to n terms the series and sum of infinite series .......43
1
32
1
21
1
{Ans.
1n
n, 1}
285. Sum to n terms the series and sum of infinite series .......97
1
75
1
53
1
{Ans.
323 n
n,
6
1}
286. Sum to n terms the series and sum of infinite series .......321
1
21
11
{Ans.
1
2
n
n, 2}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.24
287. Sum to n terms the series and sum of infinite series .......975
1
753
1
531
1
{Ans.
32123
2
nn
nn,
12
1}
288. Sum to n terms the series and sum of infinite series .......43
7
32
5
21
3222222
{Ans.
21
2
n
nn, 1}
289. Sum to n terms the series and sum of infinite series .......7531
3
531
2
31
1
{Ans.
12............531
11
2
1
n,
2
1}
290. Sum to n terms the series and sum of infinite series .......34
1
23
1
12
1
{Ans.
11n , diverging}
291. Sum to n terms the series and sum of infinite series .......4
11ln
3
11ln
2
11ln
222
{Ans.
12
2ln
n
n, 2ln }
292. Find
n
rn rr1 !2
1lim . {Ans.
2
1}
293. Prove that nnn
12
1
3
1
2
1
1
1
1
11
2222
.
CATEGORY-9.14. EXPONENTIAL AND LOGARITHMIC SERIES
294. Find
1 !1
1
n n. {Ans. 2e }
295. Find
1 !2
1
n n. {Ans.
2
5e }
296. Find
1 !12
1
n n. {Ans.
2
1 ee}
297. Find
1 !12
1
n n. {Ans.
2
21 ee}
298. Find
1 !1
1
n n. {Ans. e }
299. Find
2 !2
1
n n. {Ans. e }
300. Sum the infinite series !3
7
!2
5
!1
31 . {Ans. e3 }
301. Sum the infinite series !4
4
!3
3
!2
21
222
. {Ans. e2 }
PART-II/CHAPTER-9
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.25
302. Sum the infinite series !4
4
!3
3
!2
21
333
. {Ans. e5 }
303. Sum the infinite series !9
8
!7
6
!5
4
!3
2. {Ans.
e
1}
304. Sum the infinite series !7
8
!5
6
!3
4
!1
2. {Ans. e }
305. Sum the infinite series
!4
1
!3
1
!2
11
322 aaaaaa. {Ans.
1
a
eea
}
306. Find
1
2
!1n n
n. {Ans. 1e }
307. Sum the infinite series
!4
4321
!3
321
!2
211 . {Ans.
2
3e}
308. Sum the infinite series
!4
321
!3
21
!2
1. {Ans.
2
e}
309. Sum the infinite series
!3
43
!2
32
!1
21 222
. {Ans. e7 }
310. Sum the infinite series !4
10
!3
6
!2
31 . {Ans.
2
3e}
311. Sum the infinite series !5
78
!4
50
!3
28
!2
12. {Ans. e5 }
312. Sum the infinite series
753 37
1
35
1
33
1
3
1. {Ans. 2ln
2
1}
313. Sum the infinite series
642 27
1
25
1
23
11 . {Ans. 3ln }
314. Sum the infinite series
65
1
43
1
21
1. {Ans. 2ln }
315. Sum the infinite series
76
1
54
1
32
1. {Ans. 2ln1 }
316. Sum the infinite series
94
1
73
1
52
1
31
1. {Ans. 2ln2 }
317. Sum the infinite series
765
9
543
7
321
5. {Ans. 12ln3 }
318. Sum the infinite series
765
1
543
1
321
1. {Ans.
2
12ln }
CATEGORY-9.15. ADDITIONAL QUESTIONS
319. If the sum of first n natural numbers is 5
1 times the sum of their squares, then find the value of n . {Ans.
7}
320. If 11 a and 1,23
341
n
a
aa
n
nn and if ,lim aan
n
then find the value of a. {Ans. 2 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
9.26
321. Prove that each number is the square of an odd integer in the sequence 49, 4489, 444889, ............... in which every number is formed by inserting 48 in the middle of the previous number as indicated.
322. Find n for which the sum of the product of integers n ,,3,2,1,0 taken two at a time, lies between 50 to 100 . {Ans. 5, 6}
323. Let 11321 ,,,, aaaa be real numbers satisfying 151 a , 0227 2 a and 212 kkk aaa for
11,,4,3 k . If 9011
211
22
21
aaa , then find the value of
111121 aaa
. {Ans. 0}
324. If 0 ba and Nn , prove that 2
1
n
nn abbanba .
325. Given 1x , sum to infinite terms
75
4
53
2
3 111111
1
xx
x
xx
x
xx. {Ans.
211
1
xx }
326. Let 1031 x
xf for all Rx and xffxf nn 11 for 2n . Then find xf n . {Ans.
153
15
nn
xxf }
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-II
ALGEBRA
CHAPTER-10
DETERMINANTS AND MATRICES
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.2
CHAPTER-10
DETERMINANTS AND MATRICES
LIST OF THEORY SECTIONS
10.1. Determinant And Its Properties 10.2. System Of Linear Equations In More Than One Variables 10.3. Matrix And Its Properties 10.4. Mathematical Operations On Matrices 10.5. Solving System Of Linear Equations By Matrix
LIST OF QUESTION CATEGORIES
10.1. Evaluating Determinants 10.2. Proving Identities By Determinants 10.3. Equations Containing Determinants 10.4. Miscellaneous Questions On Determinants 10.5. Solving System Of Linear Equations By Determinants 10.6. System Of Linear Equations With Parameters 10.7. Matrix And Its Properties 10.8. Equality Of Matrices 10.9. Multiplication By Scalar, Addition, Subtraction Of Matrices 10.10. Multiplication Of Matrices, Power Of A Matrix 10.11. Transpose Of A Matrix 10.12. Symmetric And Skew-Symmetric Matrix 10.13. Determinant Of A Square Marix, Singular And Non-Singular Matrix 10.14. Adjoint Of A Square Marix 10.15. Inverse Of A Square Matrix 10.16. Orthogonal Matrix 10.17. Indempotent, Involutary, Periodic, Nilpotent Matrix 10.18. Rank Of A Matrix 10.19. Solving System Of Linear Equations By Matrix 10.20. Additional Questions
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.3
CHAPTER-10
DETERMINANTS AND MATRICES
SECTION-10.1. DETERMINANT AND ITS PROPERTIES
1. Definition of Determinant of order two
i. 2221
1211
aa
aa represents a determinant of order two.
ii. A determinant of order two has two horizontal rows denoted by 1R & 2R and has two vertical columns
denoted by 1C & 2C .
iii. ija (i, j = 1, 2) are real/ complex expressions and they are called the elements of the determinant.
iv. ija denotes element of thi row and thj column.
v. Value of the determinant is 12212211 aaaa . 2. Higher order determinants
i.
333231
232221
131211
aaa
aaa
aaa
represents a determinant of order 3. It has 9 elements arranged in 3 rows and 3 columns.
ii. Likewise, a determinant of order n has 2n elements arranged in n rows and n columns. 3. Minors
i. The determinant obtained by deleting the row and column of the element ija is called the minor of ija
and is denoted by ijM .
ii. Order of the determinant ijM is one less than the order of the original determinant.
4. Cofactor i. ij
ji M1 is called the cofactor of ija and is denoted by ijC .
5. Value of the determinant of order 3 or more The value of a determinant of order 3 or more is the sum of the products of elements of any row (column) with their corresponding cofactors.
For example, let
333231
232221
131211
aaa
aaa
aaa
, then
131312121111
3
111 CaCaCaCa
jjj
3
122
jjjCa
3
133
jjjCa
313121211111
3
111 CaCaCaCa
iii
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.4
3
122
iii Ca
3
133
iii Ca
6. Properties of Determinants i. If rows and columns are interchanged, then the value of the determinant remains unchanged
db
ca
dc
ba
ii. If two rows (columns) are identical then the value of the determinant is zero
0ba
ba
iii. If two rows (columns) are interchanged then the sign of the determinant changes
ba
dc
dc
ba
iv. If all the elements of a row (column) are multiplied by a factor then the value of the determinant gets multiplied by that factor
dc
ba
dc
ba
v. If all the elements of a row (column) are written as the sum of two factors, then the determinant can be written as the sum of two determinants
d
b
dc
ba
dc
ba
vi. If to the elements of a row (column), any multiple of the elements of any other row (column) is added, then the value of the determinant remains unchanged
dc
dbca
dc
ba
vii. Multiplication of two determinants of the same order
dcdc
baba
dc
ba
viii. Derivative of a function expressed in determinant form
xgxg
xfxf
xgxg
xfxf
xgxg
xfxf
dx
d
21
21
21
21
21
21
ix. Sum of the products of the elements of any row (column) with the cofactor of the corresponding elements of any other row (column) is zero.
qpCaCan
iqjpj
n
jqjpj
011
x. If is a determinant of order n and c denotes the determinant of cofactors then
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.5
1
00
00
00
ncnc
.
For example, if
333231
232221
131211
aaa
aaa
aaa
, then 2
333231
232221
131211
CCC
CCC
CCCc .
SECTION-10.2. SYSTEM OF LINEAR EQUATIONS IN MORE THAN ONE VARIABLES
1. A linear equation in more than one variable i. Solution ii. Solution set
2. System of linear equations in more than one variable i. Solution ii. Solution set iii. Consistent and determinate iv. Consistent and indeterminate v. Inconsistent
3. No. of equations = no. of variables i. Unique solution set: Cramer’s rule ii. No solution set iii. Infinite solution set: parametric representation iv. Consider the system of linear equations
1111 dzcybxa
2222 dzcybxa
3333 dzcybxa
We define
333
222
111
cba
cba
cba
,
333
222
111
cbd
cbd
cbd
x ,
333
222
111
cda
cda
cda
y ,
333
222
111
dba
dba
dba
z
Case-1: If 0 The system has unique solution given by
z
y
x
z
y
x
(Cramer’s rule)
The system is said to be consistent and determinate. Case-2: If 0 but at least one 0i The system has no solution.
The system is said to be inconsistent Case-3: If zyx0 The system has infinite solutions; represented parametrically.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.6
The system is said to be consistent and indeterminate. v. When 0 zyx , there may be no solution in some cases.
4. Homogeneous equations, trivial and non trivial solutions i. A linear equation whose free term is 0, is called a homogeneous linear equation. ii. All variables simultaneously equal to 0 is always a solution of any homogeneous linear equation and this
solution is called trivial solution. iii. Consider the system of homogeneous linear equations
0111 zcybxa
0222 zcybxa
0333 zcybxa
For the system of homogeneous equations 0 zyx .
Case-1: If 0 The system has unique solution given by 0,0,0 , i.e. trivial solution is the only solution. Case-2: If 0 The system has infinite solutions, trivial solution 0,0,0 is a solution and other solutions are called non-trivial solutions.
5. No. of equations = no. of variables + 1 i. Consider the system of 3 linear equations in 2 unknowns
111 cybxa
222 cybxa
333 cybxa
This system is consistent if 0
333
222
111
cba
cba
cba
and inconsistent if 0
333
222
111
cba
cba
cba
(Consistency
condition). 6. No. of equations < no. of variables
i. Unique solution not possible. SECTION-10.3. MATRIX AND ITS PROPERTIES
1. Definition of Matrix i. Matrix is a rectangular arrangement of real/ complex expressions called elements. ii. A matrix is written as
mnmjmm
inijii
nj
nj
aaaa
aaaa
aaaa
aaaa
21
21
222221
111211
.
iii. Matrix having m rows and n columns is said to be of the order nm 1, nm .
iv. A matrix is represented by nmijnm aA
, where ija denotes element of thi row & thj column, m
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.7
denotes no. of rows and n denotes no. of columns. 2. Row matrix
A matrix having only one row is called a row matrix. 3. Column matrix
A matrix having only one column is called a column matrix. 4. Square matrix
A matrix in which number of rows and columns are equal, is called a square matrix. 5. Principal diagonal of a square matrix
The elements iia lying on the diagonal constitute the principal diagonal.
6. Trace of a square matrix Sum of the diagonal elements of a square matrix A is called trace of A, i.e. Trace of nnaaaA 2211 .
7. Diagonal matrix A square matrix in which all the elements, except those in the principal diagonal, are zero, is called a diagonal matrix, i.e. in a diagonal matrix jiaij 0 . A diagonal matrix of order nn having 1d , 2d ,
…….., nd as diagonal elements is also denoted by nddddiag ,........,, 21 .
8. Scalar matrix A diagonal matrix in which all the elements in the principal diagonal are equal, is called a scalar matrix.
9. Unit (Identity) matrix A diagonal matrix in which all the elements in the principal diagonal are 1, is called a unit or identity matrix. A unit matrix of order n is denoted by nI . Thus for an identity matrix 1iia and jiaij 0 .
10. Null (Zero) matrix A matrix in which all the elements are zero, is called a null or zero matrix and is denoted by O or nmO .
11. Upper triangular matrix A square matrix
nmijaA
is called an upper triangular matrix if jiaij 0 .
12. Lower triangular matrix A square matrix
nmijaA
is called a lower triangular matrix if jiaij 0 .
SECTION-10.4. MATHEMATICAL OPERATIONS ON MATRICES
1. Equality of matrices Two matrices
nmijaA
and srijbB
are said to be equal, denoted by A = B, iff m=r, n=s and
jiba ijij , , otherwise they are not equal, denoted by BA .
2. Multiplication of a matrix by a scalar The product of a scalar k and a matrix A, denoted by kA, is a matrix defined as ijkakA .
Properties:- i. AA 1 ii. nmnm OA 0
iii. kAllAkAkl 3. Negative of a matrix
A1 , denoted by A , is said to be the negative of matrix A, i.e. ijaA .
4. Addition of two matrices
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.8
Two matrices A and B are conformable for addition if they are of the same order. Sum of two matrices A and B, denoted by BA , is defined as a matrix obtained by adding the corresponding elements of A and B and their sum matrix is of the same order as A and B, i.e.
nmijijnmnm baBA .
Properties:- i. ABBA ii. CBACBACBA iii. AOAOA iv. BACBCA v. kBkABAk
vi. lAkAAlk 5. Subtraction of two matrices
Two matrices A and B are conformable for subtraction if they are of the same order. Subtraction of two matrices A and B, denoted by BA , is a matrix defined as BABA . Therefore, the difference of two matrices A and B is a matrix obtained by subtracting the corresponding elements of A and B and their difference matrix is of the same order as A and B, i.e.
nmijijnmnm baBA .
Properties:- i. OAA ii. AOA iii. AAO
6. Multiplication of two matrices Two matrices A and B are conformable for multiplication if A is of the order nm and B is of the order
pn . Their product matrix, denoted by AB, is of the order pm and is defined as ijCAB where
n
kkjikij baC
1
. The matrix A is called pre-multiplier and matrix B is called post-multiplier.
Properties:- i. In general, BAAB . ii. AB and BA are both defined iff A is of the order nm and B is of the order mn . iii. AB and BA are both defined and are of the same order iff A and B are square matrices. iv. If AB and BA are both defined and are of the same order, even then AB and BA may not be equal. v. If BAAB , then A and B must be square matrices and A and B are said to be commuting matrices. vi. AA is defined iff A is a square matrix. vii. kBABkAABk
viii. BCACAB
ix. ACABCBA
x. nnmnmnmm IAAAI
xi. mmm III
xii. pmpnnm OOA
xiii. npnmmp OAO
xiv. If AB = O, then it does not imply that either A = O or B = O. xv. If AB = O, then it does not imply that BA = O. xvi. If AB = AC, then it does not imply that B = C.
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.9
7. Natural powers of a square matrix If A is a square matrix, then its natural power is defined as i. AA 1 ii. AAA nn 1 iii. timesnAAAAn Properties:- i. nmnm AAA
ii. mnnm AA
iii. mn
m II
8. Transpose of a matrix A matrix obtained by interchanging rows and columns of a matrix A is called transpose of A and is denoted by TA or A . If A is a nm matrix then TA is a mn matrix.
mnijT bA
, jiij ab .
Properties:-
i. AATT
ii. TT kAkA
iii. TTT BABA
iv. TTT ABAB
v. TTTT ABCABC
vi. nT
n II
9. Symmetric matrix and Skew symmetric matrix i. A square matrix is said to be symmetric if jiij aa or AAT .
ii. nI is a symmetric matrix.
iii. A square matrix is said to be skew symmetric if jiij aa or AAT .
iv. The diagonal elements of a skew symmetric matrix are zero. 10. Determinant of a square matrix
Determinant of a square matrix A, denoted by det A or A , is the determinant formed by the elements of
matrix A. If 11A then A is defined as 11aA .
Properties:- i. AkkA n
nn
ii. If A & B are square matrices then BAAB
iii. If A is a square matrix then nn AA
iv. If A is a square matrix then AAT
v. 0nnO
vi. 1nI
11. Singular matrix and Non-singular matrix i. A square matrix A is said to be singular if 0A .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.10
ii. A square matrix A is said to be non-singular if 0A .
12. Adjoint of a square matrix Adjoint of a square matrix
nnijaA
is the matrix TnnijC
, where ijC denotes cofactor of ija . Adjoint of
A is denoted by adj A. Properties:- i. AAadjIAAadjA
ii. If A is non-singular square matrix of order n then 1
n
AAadj
iii. If A & B are non-singular square matrices of same order then AadjBadjABadj
iv. If A is non-singular square matrix then TT AadjAadj
v. If A is non-singular square matrix of order n then AAAadjadjn 2
vi. nn IIadj
13. Inverse of a square matrix A square matrix A is invertible if there exists a square matrix B of the same order such that BAIAB , then B is said to be the inverse matrix of matrix A, denoted by 1A , otherwise matrix A is not invertible. Properties:- i. A square matrix is invertible iff it is non-singular. Singular matrices are non-invertible. ii. Every non-singular matrix is invertible and possesses a unique inverse.
iii. AadjA
A11 .
iv. IAAAA 11 .
v. AA 11 .
vi. A
A11 .
vii. If A, B, C are square matrices of same order and if A is non-singular then CBACAB and CBCABA .
viii. 111 ABAB .
ix. 1111 ABCABC .
x. If A is invertible then TA is also invertible and TT AA 11 .
xi. II 1 . 14. Orthogonal matrix
A square matrix is called an orthogonal matrix if IAAAA TT or TAA 1 . 15. Indempotent matrix
i. A square matrix is called an indempotent matrix if AA 2 . ii. If A is an indempotent matrix then 2 nAAn .
16. Involutary matrix i. A square matrix is called an involutary matrix if IA 2 . ii. If A is an involutary matrix then AA 1 . iii. If A is an involutary matrix then AAn if n is odd and IAn if n is even.
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.11
17. Periodic matrix i. A square matrix is called a periodic matrix if AAk 1 for some natural number k. The least value of k is
said to be its period. ii. An indempotent matrix is a periodic matrix of period 1.
18. Nilpotent matrix i. A square matrix is called a nilpotent matrix if OAk for some natural number k. The least value of k is
said to be its index. ii. If A is a nilpotent matrix of index k then knOAn . iii. A null square matrix is a nilpotent matrix of index 1.
19. Sub-matrix A matrix obtained by leaving some rows or columns or both of matrix A is called a submatrix of A. The matrix A is itself a sub-matrix of A.
20. Rank of a matrix Rank of a matrix, denoted by r(A), is the order of the highest order non-singular square submatrix. Rank of a null matrix is not defined. Properties:- i. nIr n .
ii. ArAr T .
iii. nmAr nm ,min .
iv. BrArBAr .
v. BrArABr ,min . SECTION-10.5. SOLVING SYSTEM OF LINEAR EQUATIONS BY MATRIX
1. Consider the following system of n non-homogeneous linear equations in n unknowns 1x , 2x , …….., nx
11212111 ....... bxaxaxa nn
22222121 ....... bxaxaxa nn
………………………………… …………………………………
nnnnnn bxaxaxa .......2211
This system of equation can be written in matrix form as
nnnnnn
n
n
b
b
b
x
x
x
aaa
aaa
aaa
......
...
............
...
...
2
1
2
1
21
22221
11211
or AX = B,
where
nnnn
n
n
aaa
aaa
aaa
A
...
............
...
...
21
22221
11211
,
nx
x
x
X...
2
1
and
nb
b
b
B...
2
1
The matrix A is called the coefficient matrix, matrix X is called the variable matrix and matrix B is called the
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.12
constant matrix. Solution of system of non-homogeneous linear equations BAX Case-1: If 0A The system has unique solution, BAX 1
Case-2: If OBAadjA &0 The system has no solution
Case-3: If OBAadjA &0 The system has infinite solutions
2. Solution of system of homogeneous linear equations OAX Case-1: If 0A The system has unique solution, OX (Trivial solution)
Case-2: If 0A The system has infinite solutions (Trivial solution & non-trivial solutions)
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.13
EXERCISE-10
CATEGORY-10.1. EVALUATING DETERMINANTS
1.
1 2 3
3 5 7
8 14 20
{Ans. 0}
2.
43 1 6
35 7 4
17 3 2
{Ans. 0}
3.
111
111
111
{Ans. 4}
4.
151413
141312
131211
{Ans. 0}
5.
38 7 63
16 3 29
27 5 46
{Ans. 0}
6.
265 240 219
240 225 198
219 198 181
{Ans. 0}
7.
18 40 89
40 89 198
89 198 440
{Ans. -1}
8.
13 3 2 5 5
15 26 5 10
3 65 15 5
{Ans. 5 3 6 5 }
9.
21 17 7 10
24 22 6 10
6 8 2 3
5 7 1 2
{Ans. 0}
10.
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7
2 2 2 2
2 2 2 2
2 2 2 2
2 2 2 2
{Ans. 0}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.14
11.
1 1 1 1 1
1 2 3 4 5
1 3 6 10 15
1 4 10 20 35
1 5 15 35 69
{Ans. 0}
12.
a b b c c a
b c c a a b
c a a b b c
{Ans. 0}
13.
a b b c c a
x y y z z x
p q q r r p
{Ans. 0}
14.
1
1
1
a b c
b c a
c a b
{Ans. 0}
15.
abcc
cabb
bcaa
2
2
2
1
1
1
{Ans. 0}
16.
21210
1sincos
1cossin22
22
xx
xx
. {Ans. 0}
17.
1
1
1
log log
log log
log log
x x
y y
z z
y z
x z
x y
{Ans. 0}
18.
b ab b c bc ac
ab a a b b ab
bc ac c a ab a
2
2 2
2
{Ans. 0}
19.
1
1
1
2
2
2
a a bc
b b ca
c c ab
{Ans. 0}
20.
0
0
0
b c
b a
c a
{Ans. 0}
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.15
21.
0
0
0
a b a c
b a b c
c a c b
{Ans. 0}
22.
1
1
1
bc bc b c
ca ca c a
ab ab a b
{Ans. 0}
23.
1
1
1
1
2 3
2 3
2 3
2 3
a a a bcd
b b b cda
c c c abd
d d d abc
{Ans. 0}
24. If Dr =
r xn n
r y n
r zn n
1
22 1
3 23 1
2
2 then prove that Drr
n
01
.
25. If Dr = 2 2 3 4 5
2 1 3 1 5 1
1 1 1r r r
n n n
x y z
then prove that Drr
n
01
.
26. Let nnna
nna
na
a
3331
2421
61
223
22
show that aa
n
c
1
, a constant.
27. Evaluate Unn
N
1
if Un =
n
n N N
n N N
1 5
2 1 2 1
3 3
2
3 2
. {Ans. 0}
28. Express
2
132
112
321
in determinant form and find its value also. {Ans.
21010
486
451
, 196}
CATEGORY-10.2. PROVING IDENTITIES BY DETERMINANTS
29. xy
y
x
111
111
111
.
30.
1
1
1
2
2
2
a a
b b
c c
=
1 1 1
2 2 2
a b c
a b c
.accbba
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.16
31. accbbak
ckkc
bkkb
akka
1
1
1
22
22
22
.
32.
1
1
1
2 4
2 4
2 4
m n l p m n l p
n l m p n l m p
l m n p l m n p
64 l m l n l p m n m p n p .
33.
a b c a
b c a b
c a b c
2
2
2
a b c a b b c c a .
34.
1
1
1
1
1
1
2
2
2
a a
b b
c c
bc b c
ca c a
ab a b
35. 222222
111
baaccb
baaccb
.baaccb
36.
1 1 1
3 3 3
a b c
a b c
= a b b c c a a b c .
37.
2
2
2
a a b c a
b a b b c
c a c b c
4 b c c a a b .
38.
xyzxyz
zyx
zyx222
333
222
111
zyx
zyx x y y z z x xy yz zx .
39.
1
1
1
1
1
1
2
2
2
a bc
b ca
c ab
a a
b b
c c
=
40.
x y z
x y z
x y z
xyz
4
41.
b c c a a b
a b b c c a
c a a b b c
2
a b c
c a b
b c a
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.17
42. 233
22
22
22
2
2
2
ba
aabb
abba
baab
43.
y z z y
z z x x
y x x y
xyz
4
44. xyz
yxzz
yxzy
xxzy
4
45.
a b
cc c
ab c
aa
b bc a
b
abc
2 2
2 2
2 2
4
46.
a b ax by
b c bx cy
ax by bx cy
0 b ac ax bxy cy2 2 22 .
47.
a bc ac c
a ab b ac
ab b bc c
a b c
2 2
2 2
2 2
2 2 24
.
48.
b c ab ac
ab c a bc
ca cb a b
2 2
2 2
2 2
.4 222
2222
2222
2222
cba
bacc
bacb
aacb
49.
a b c
a b b c c a
b c c a a b
a b c abc
3 3 3 3 .
50.
a b c a a
b b c a b
c c c a b
a b c
2 2
2 2
2 2
3 .
51.
bacac
bacbc
bacba
2
2
2
.2 3cba
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.18
52.
4321
4321
4321
4321
1
1
1
1
aaaa
aaaa
aaaa
aaaa
43211 aaaa .
53.
d
c
b
a
1111
1111
1111
1111
dcbaabcd
11111 .
54.
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
103
x
x
x
x
x x .
55.
x a a a
a x a a
a a x a
a a a x
x a x a 3 3.
56.
a ab ac ad
ab b bc bd
ac bc c cd
ad bd cd d
2
2
2
2
1
1
1
1
1
1
1
1
2222
2222
2222
2222
dcba
dcba
dcba
dcba
.1 2222 dcba
57.
ax by cz
x y z2 2 2
1 1 1
a b c
x y z
yz zx xy
.
58.
x x x
x x x x
x x xx
3 2
2 26
3 3 1
2 2 1 1
2 1 2 1
1 3 3 1
1
59.
x x x
x x x
2
3
2 2 1 1
2 1 2 1
3 3 1
1
60.
x x x
x x x
x x x
2 1
1 1
1 2
8
2 2 2
2 2 2
2 2 2
.
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.19
61.
0
0
0
0
2
x y z
x c b
y c a
z b a
ax by cz
.
62. a b c c b
a c b c a
a b b a c
a b c a b c
2 2 2 .
63.
222
222
222
bacc
bacb
aacb
=
2
2
2
acabbc
abcbca
cbcaba
= 32 cbaabc
64. 22
22
22
122
212
221
baab
abaab
babba
3221 ba .
65.
a b c
a b c
a b c
2 2 2
2 2 2
2 2 2
1 1 1
1 1 1
4
1 1 1
2 2 2a b c
a b c .
66.
bc b bc c bc
a ac ac c ac
a ab b ab ab
bc ca ab
2 2
2 2
2 2
3 .
67. .
byaxczcybzazcx
cybzaxczbybxay
azcxbxayczbyax
x y z a b c ax by cz2 2 2 2 2 2 .
68. 2222
2222
2222
1
1
1
dcbacdab
dbacbdca
dacbadbc
.dcdbcbdacaba
69.
x x y z yz
y y z x zx
z z x y xy
2 2 2
2 2 2
2 2 2
x y y z z x x y z x y z2 2 2 .
70. 0
333323231313
323222221212
313121211111
bababa
bababa
bababa
.
71.
cos cos cos
cos cos cos
cos cos cos
A P A Q A R
B P B Q B R
C P C Q C R
0
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.20
72. 222
222
222
222
222
222
ruu
uru
uur
yzxxyzzxy
xyzzxyyzx
zxyyzxxyz
, where r2 = x2 + y2 +z2 and u2 = yz + zx +xy.
73. 1
0cossin
cossinsincos
sinsincoscos2
2
xx
xxxx
xxxx
74. If 0111 cba , prove that abc
c
b
a
111
111
111
.
75. If p + q + r = 0, prove that
pa qb rc
qc ra pb
rb pc qa
pqr
a b c
c a b
b c a
.
76. If 2s = a + b + c, prove that
222
222
222
ccscs
bsbbs
asasa
csbsass 32 .
77. Without expanding at any stage show that x x x x
x x x x
x x x x
xA B
2
2
2
1 2
2 3 1 3 3 3
2 3 2 1 2 1
, where A and B are
determinants of order 3 not involving x.
78. If y =sinpx and yr means rth derivative of y then prove that 0
876
543
21
yyy
yyy
yyy
.
79. If a, b, c (all +ive) are the pth, qth, rth, terms respectively of a geometric progression, then prove that log
log
log
a p
b q
c r
1
1
1
0 .
80. If a, b, c are given to be in A.P., prove that
x x x a
x x x b
x x x c
1 2
2 3
3 4
0 .
81. Given that CBA , prove that ACCBBA
CCCC
BBBB
AAAA
sinsinsin
coscossinsin
coscossinsin
coscossinsin
22
22
22
.
CATEGORY-10.3. EQUATIONS CONTAINING DETERMINANTS
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.21
82. Find the roots of the equation 0
111
111
111
x
x
x
. {Ans. –1. 2}
83.
a a x
x x x
b x b
0 . {Ans. x = 0, a, b}
84. 0
cxba
cbxa
cbax
. {Ans. cba ,0 }
85.
x a a a
x b b b
x c c c
2 3
2 3
2 3
0 . {Ans. acbcab
abcx
}
86.
15 2 11 10
11 3 17 16
7 14 13
0
x
x
x
. {Ans. x = 4}
87.
x x x
x x x
x x x
2 2 3 3 4
4 2 9 3 16
8 2 27 3 64
0 {Ans. x = 4}
88.
4 6 2 8 1
6 2 9 3 12
8 1 12 16 2
0
x x x
x x x
x x x
{Ans. x 1197 }
89.
3 8 3 3
3 3 8 3
3 3 3 8
0
x
x
x
{Ans. x 23
113, }
90.
x x x
x x x
x x x
2 2 3 3 4
2 3 3 4 4 5
3 5 5 8 10 17
0 {Ans. x = -2, -1}
91.
x x x
x x x
x x x
2 6 1
6 1 2
1 2 6
0 {Ans. 3
7x }
92. Show that x = 2 is a root of
x
x x
x x
6 1
2 3 3
3 2 2
0 and solve it completely. {Ans. x = -3, 1, 2}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.22
93. Solve
1 3 9
1
4 6 9
02x x . {Ans. x 3 32, }
94. Show that x= -9 is a root of
x
x
x
3 7
2 2
7 6
0 and find the other two roots. {Ans. x = 2, 7}
95. Given a + b + c = 0, solve
a x c b
c b x a
b a c x
0 {Ans. x a b c 0 32
2 2 2, }
96. If b c a b c a 2 , solve for x
a x b x c x
b x c x a x
c x a x b x
0 {Ans. x a b c 3 }
97. If cba , solve for x
0
0
0
0
x a x b
x a x c
x b x c
. {Ans. x = 0}
CATEGORY-10.4. MISCELLANEOUS QUESTIONS ON DETERMINANTS
98. If rcqbpa ,, and 0
rba
cqa
cbp
, then find the value of p
p a
q
q b
r
r c
. {Ans. 2}
99. Suppose three digit numbers A28, 3B9 and 62C where A, B and C are integers between 0 and 9, are
divisible by a fixed integer k. Prove that the determinant
A
C
B
3 6
8 9
2 2
is also divisible by k.
100. For a fixed positive integer n, if D =
n n n
n n n
n n n
! ! !
! ! !
! ! !
1 2
1 2 3
2 3 4
then show that
D
n! 3 4
is divisible
by n.
101. If x,y and z are all different and given that
x x x
y y y
z z z
2 3
2 3
2 3
1
1
1
0
, prove that 1+xyz = 0
102. If x, y and z are all different and given that
x x x
y y y
z z z
3 4
3 4
3 4
1
1
1
0
, prove that xyz xy yz zx x y z .
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.23
103. Let a, b, c be positive and not all equal. Show that the value of the determinant
a b c
b c a
c a b
is negative.
104. Let α be a repeated root of quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degree 3, 4
and 5 respectively, then show that
A x B x C x
A B C
A B C
is divisible by f(x), where ' denotes the
derivative. CATEGORY-10.5. SOLVING SYSTEM OF LINEAR EQUATIONS BY DETERMINANTS
105. Solve x y z
x y z
x y z
2 3 6
2 4 7
3 2 9 14.
{Ans. (1, 1, 1)} 106. Solve
.2293
1742
0632
yzx
zyx
zyx
{Ans. (1, 4, -1)} 107. Solve
.0243
062
11
zyx
zyx
zyx
{Ans. (-8, -7, 26)} 108. Solve
x y z
x y z
x y z
6
2
2 1.
{Ans. (1, 2, 3)} 109. Solve
5 6 4 15
7 4 3 19
2 6 46
x y z
x y z
x y z
.
{Ans. (3, 4, 6)} 110. Solve
x y z
x y z
x y z
1
3 5 6 4
9 2 36 17.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.24
{Ans.
3
1,1,
3
1}
111. Find a, b, c when f(x) = ax2 + bx + c and f(0) = 6, f(2) = 11, f(-3) = 6. Determine f(x) and find the value of f(1). {Ans. a b c 1
232 6, , , f x x x 1
22 3
2 6 , f 1 8 } 112. Solve
baczyxba
acbyxzac
cbaxzycb
where 0 cba . {Ans.
cba
ab
cba
ca
cba
bc,, }
113. For what value of k, the system of linear equations 423,32,2 kzyxzyxzyx has a unique solution? {Ans. 0k }
114. For what value of k, the system of simultaneous equations 221,12 zykzykx and
32 zk have a unique solution? {Ans. 1,0,2k } 115. Solve
x y z
x y z
x y z
4 2 3
3 5 7
2 3 5.
{Ans. no solution} 116. Solve
1595
42
3232
zyx
zyx
zyx
{Ans.
7
6,
7
411,
ttt }
117. Find k for which the set of equations
kzyx
zyx
zyx
45
032
02
are consistent and find the solution for all such values of k. {Ans. k = 0, ttt ,, } 118. Show that the system of equations
356
232
343
zyx
zyx
zyx
has at least one solution for any real number λ. Find the set of solutions if λ = -5. {Ans.
t
tt,
7
913,
7
54}
119. Solve
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.25
0225
05615
03
zyx
zyx
zyx
{Ans. (0, 0, 0)} 120. Solve
01411
042
023
zyx
zyx
zyx
{Ans.
k
kk,
7
8,
7
10}
121. Find all values of k for which the following system possesses a non-trivial solution x ky z
kx y z
x y z
3 0
2 2 0
2 3 4 0
{Ans. 2, 54 }
122. For what value of k do the following system of equations possess a non-trivial solution over the set of rationals
.0432
023
03
zyx
zkyx
zkyx
For that value of k, find all the solutions of the system. {Ans. 2
33k ,
tt
t3,,
2
15}
123. Given x cy bz
y az cx
z bx ay
where x, y, z are not all zero prove that a b c abc2 2 2 2 1 .
124. If ax
y zb
y
z xc
z
x y
, and , where x, y, z are not all zero, prove that 1 0 ab bc ca .
125. Let α1, α2, and β1, β2 be the roots of 02 cbxax and 02 rqxpx respectively. If the system of equations
0
0
21
21
zy
zy
has non-trivial solution, then prove that b
q
ac
pr
2
2 .
126. If a, b, c are in G.P. with common ratio r1 and α, β, γ also form in G.P.with common ratio r2, then find the conditions that r1 and r2 must satisfy so that the equations ax + αy + z = 0 bx + βy + z = 0
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.26
cx + γy + z = 0 have only zero solution. {Ans. r r r r1 2 1 21 1 , , }
127. Solve
.22
1
232
yx
yx
yx
{Ans. no solution} 128. Solve
.12
12
32
yx
yx
yx
{Ans. (1, 1)} 129. Find the value of λ if the following equations are consistent
x y
x y
x y
3 0
1 2 8 0
1 2 0
{Ans. 1, 53 }
130. If the equations ax hy g
hx by f
gx fy c
0
0
are consistent, show that
abc fgh af bg ch
ab h
2 2 2 2
2 .
131. If the equations b c x c a y a b
cx ay b
ax by c
0
0
0
are consistent then show that either a b c a b c 0 or . 132. If a, b, c are all different and the equations
ax a y a
bx b y b
cx c y c
2 3
2 3
2 3
1 0
1 0
1 0
are consistent then prove that abc + 1 = 0. 133. Find the value of a if the three equations are consistent
a x a y a 1 2 33 3 3
a x a y a 1 2 3 x y 1. {Ans. a = -2}
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.27
134. Are the equations x ay b c
x by c a
x cy a b
where a, b and c are real numbers such that 1222 cba , consistent? {Ans. consistent}
CATEGORY-10.6. SYSTEM OF LINEAR EQUATIONS WITH PARAMETERS
135. ax y
x ay a
2
2
tta
tta
a
Ans
2,,1
2,,1
2,0,1
.
136.
x ay
ax ay a
1 0
3 2 3 0
solution no0
3
1,,3
1,2,3,0
.
a
tta
aa
Ans
137. 3 5
3
2
2
x ay a
x ay a
ta
aaa
Ans
,0,0
2,,0
.2
138.
a x a y a
a x a y a
5 2 3 3 2 0
3 10 5 6 2 4 0
solution no,2
3
52,,0
2
227,
2
1411,2,0
.
a
tta
a
a
a
aa
Ans
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.28
139.
a a x a a y a
a x a y a
1 1 2
1 1 1
3
2 3 4
solution no,1,0
3
5,,2
,2
1,1
1,
1
1,2,1,0
.
2
3
2
23
a
tta
ta
a
aa
aa
aaa
Ans
140. ax y b
bx y a
btatba
baba
Ans
,,0
,1,0
.
141.
a b x a b y a
a b x a b y a
2 2 2 2 2
21,,0
,2
1,0
1,,0,0
,,0,0
2
1,
2
1,0
.
ttba
tba
ttba
ttba
ba
Ans
142. For what values of m does the system of equations mmyx 3 2052 yx
has a solution satisfying the conditions 0,0 yx . {Ans. m , ,152 30 }
143. For what values of a does the system
a x a y a
ax a y a
2 2
5
2 4
2 1 2
possess no solution? {Ans. a = -1, 1} 144. For what values of the parameter a does the system of equations
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.29
ax y a
x a y a
4 1
2 6 3
possess no solutions? {Ans. a = -4} 145. For what values of the parameter a does the system
2 2
1 2 2 4
x ay a
a x ay a
possess infinitely many solutions? {Ans. a = 3} 146. Find the values of the parameters m and p such that the system
24132
1138153
ypmxpm
ypmxpm
possesses infinite solutions. {Ans. 64
19,
16
5 mp }
147. Find the parameters a and b such that the system
a x ay a
bx b y a
2 1
3 2 3
possesses a unique solution x = 1, y = 1. {Ans. a = 1, b = -1} 148. For what values of a and b does the system
a x by a b
bx b y b
2 2
2 2 4
possesses an infinite number of solutions? {Ans. (1, -1), (1, -2), (-1, -1), (-1, -2)} 149. Find the values of and so that the system of equation
6 zyx 1032 zyx zyx 2
has i. Unique solution {Ans. R ,3 } ii. Infinite solutions {Ans. 10,3 } iii. No solution {Ans. 10,3 }
150. For what values of a, b and c, the system
ax by a b
c x cy a b
2
1 10 3
has infinitely many solutions and x = 1, y = 3 is one of the solutions? {Ans. 0 0 2 1194, , , , , }
CATEGORY-10.7. MATRIX AND ITS PROPERTIES
151. Find a 43 matrix A given that
2
2jiaij
. {Ans.
2
4918
2
258
182
258
2
92
258
2
92
A }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.30
152. Find a matrix 32A given that
j
iaij , where [ ] denotes greatest integer function. {Ans.
012
001A }
153. If
9811
970
751
A , then find the trace of matrix A . {Ans. 17}
CATEGORY-10.8. EQUALITY OF MATRICES
154. Find the value of x, y, z and a which satisfy the matrix equation
aaz
xyx
23
70
641
23 {Ans. x = -3,
y = -2, z = 4, a = 3}
155. Find x and y if
71
23
1
2
yx
yx {Ans. x = 5, y = -2}
156. Find x, y, z, w if
130
51
32
2
wzyx
zxyx {Ans. x = 1, y = 2, z = 3, w = 4}
157. Find x, y, z if
11
13
2 xyxz
zyyx {Ans. x = 1, y = 2, z = 3}
158. Find a, b, c, d if
ddc
ba
23
60
641
823 {Ans. a = -3, b = 1, c = -4, d = 3}
CATEGORY-10.9. MULTIPLICATION BY SCALAR, ADDITION, SUBTRACTION OF MATRICES
159. If
510
132A and
310
121B evaluate BA 43 . {Ans.
310
712}
160. If
986
750
321
P ,
075
503
302
Q , evaluate QP 32 . {Ans.
1853
1109
344
}
161. If
03
21A ,
32
41B ,
01
10C , then find CBA 235 . {Ans.
97
208}
162. If
31
01
21
A ,
12
01
54
B ,
21
21
21
C , find CBA 32 . {Ans.
116
62
1410
}
163. If
12
42
36
53
43
12
1
0
y
x, then find x, y. {Ans. 2,3 yx }
164. Given
111
205
321
A and
302
524
213
B . Find the matrix C such that BCA 2 . {Ans.
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.31
12
1
2
12
31
2
12
5
2
31
}
165. Find A and B if
43
21BA and
01
23BA . {Ans.
21
02A ,
22
21B }
166. If
11
01BA and
10
112BA , then find A . {Ans.
3
1
3
23
1
3
1
}
167. Find A and B if
233
0332 BA and
441
5142 AB . {Ans.
021
112A ,
211
211B }
168. If
01
10,
10
01JI and
cossin
sincosB , then show that sincos JIB .
CATEGORY-10.10. MULTIPLICATION OF MATRICES, POWER OF A MATRIX
169. What is the order of
z
y
x
cfg
fbh
gha
zyx . {Ans. 11 }
170. Given
121
111A and
021
102
211
B , find AB and BA. {Ans.
414
130AB , BA is not
defined}
171. Given 21A and
1
3B , find AB and BA. {Ans. 5AB ,
21
63BA }
172. If
10
11A ,
01
10B , then find AB and BA and show that BAAB . {Ans.
01
11AB ,
11
10BA }
173. If
213
132
321
A and
021
210
201
B , obtain the product AB and BA and show that BAAB . {Ans.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.32
451
1011
244
AB ,
145
354
705
BA }
174. If
21
13A , then find 2A and 3A . {Ans.
35
582A ,
118
18193A }
175. If baA 21 , abB 21 and
a
aC 12 , then show that BCAC .
176. Given
00
01A and
10
00B , find AB. {Ans. 22OAB }
177. Given 01A and
1
0B , find AB and BA. {Ans. 11OAB ,
01
00BA }
178. If
510
132A and
310
621B , evaluate 22 BA . {Ans. not defined}
179. If
10
12A and
11
01B , show that 22222 2 BABABBAABABA
180. If
32
21A ,
11
03B . Verify that 22222 2 BABABBAABABA .
181. If
11
10A ,
01
10B , show that 22 BABABA .
182. If
43
21A ,
24
12B ,
47
15C . Show that ACABCBA .
183. The matrix R(t) is defined by
tt
tttR
cossin
sincos. Show that R(s) R(t) = R(s + t).
184. If
cossin
sincosA ,
cossin
sincosB . Show that AB = BA.
185. If a, b, c, d are real numbers and
dc
baA , prove that OIbcadAdaA 2 .
186. Show that EEFFE 22 , where
000
100
100
E ,
100
010
001
F .
187. Find x so that O
x
x
1
1
230
150
231
11 . {Ans. 5392
1 }
188. If
8.06.0
6.08.0A , find 3A . {Ans.
352.0936.0
936.0352.0}
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.33
189. If
11
43X , then find 3X .
190. If
01
10A , then find 11A . {Ans.
01
10}
191. If
01
101J and
01
012J , then find 2
22
1 JJ .
192. If
134
312
231
A ,
21
12
41
B and
123
112C . Show that BCACAB .
193. If Ixxxf 652 , find Af if
011
312
102
A . {Ans.
445
1011
311
}
194. If
34
42A ,
1
nX ,
11
8B and BAX , then find n. {Ans. 2}
195. If
ab
baA and
2A , then show that abba 2,22 .
196. If
21
12A and OnIAA 2
2 4 , then find the value of n . {Ans. -3}
197. If
43
31A and OIkAA 2
2 5 , then find the value of k . {Ans. 5}
198. If
00
10A , I is the unit matrix of order 2 and ba, are arbitrary constants, then show that
abAIabAaI 222 .
CATEGORY-10.11. TRANSPOSE OF A MATRIX
199. If
35
21A , then find TAA . {Ans.
63
32}
200. Verify that TTTTT BAABAB , where
942
322
111
A ,
011
423
110
B .
201. If
524
324A and
42
01
31
B , show that TTT ABAB .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.34
202. If
814
312
011
A and
111
132
014
B , then verify that TTT ABAB .
203. If matrix
bac
acb
cba
A , where a, b, c are real positive numbers, abc = 1 and IAAT , then find the
value of 333 cba . {Ans. 4} CATEGORY-10.12. SYMMETRIC AND SKEW-SYMMETRIC MATRIX
204. If
0
5
y
xA is a symmetric matrix, then show that yx .
205. If
132
24
xx
xA is symmetric, then find x. {Ans. 5}
206. If the matrix
02
03
50
c
b
a
is skew-symmetric, then find a, b, c. {Ans. 5,2,3 cba }
207. If A is a skew-symmetric matrix, then find trace of A . {Ans. 0} 208. If for a square matrix 22, jiaaA ijij , then show that A is a skew-symmetric matrix.
209. Let A be a square matrix, then prove that i. TAA is a symmetric matrix ii. TAA is a skew-symmetric matrix iii. TAA and AAT are symmetric matrices.
210. Prove that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
211. If A and B are symmetric matrices, then show that AB is symmetric iff AB = BA. 212. Show that the matrix ABBT is symmetric or skew-symmetric according as A is symmetric or skew-
symmetric. 213. Show that all natural powers of a symmetric matrix are symmetric. 214. Show that odd natural powers of a skew-symmetric matrix are skew-symmetric and even natural powers of
a skew-symmetric matrix are symmetric. 215. If A and B be symmetric matrices of the same order, then show that
i. BA is a symmetric matrix ii. BAAB is a skew-symmetric matrix iii. BAAB is a symmetric matrix
CATEGORY-10.13. DETERMINANT OF A SQUARE MARIX, SINGULAR AND NON-SINGULAR
MATRIX
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.35
216. If
213
132
321
A , find det A. {Ans. 42}
217. For what value of k,
111
05
321
k is a singular matrix. {Ans. 3
5 }
218. If A and B are two square matrices of order 3 such that 1A , 3B , then find AB3 . {Ans. -81}
219. Prove that a skew-symmetric matrix of odd order must be a singular matrix. CATEGORY-10.14. ADJOINT OF A SQUARE MARIX
220. If
31
25A , then find adjA . {Ans.
51
23}
221. Find the adjoint of
312
321
111
. {Ans.
135
419
543
}
222. Find the adjoint of the matrix
53
21A and verify that AadjAIAadjAA 2 . {Ans.
13
25}
223. Find the adjoint of the matrix
760
543
101
A and verify that
i. AadjAIAadjAA 3
ii. 2
AadjA
iii. TT adjAadjA
iv. AAadjAadj .
{Ans.
4618
8721
462
}
224. For the matrix A, prove that OadjAA , where
10218
032
111
A .
225. If
213
132
321
A and
021
210
201
B , then verify that adjAadjBadjAB .
226. If
xx
xxA
cossin
sincos and
10
01kadjAA , then find the value of k . {Ans. 1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.36
CATEGORY-10.15. INVERSE OF A SQUARE MATRIX
227. Find 1
103
31
. {Ans.
13
310}
228. If
13
25A , then find 1A . {Ans.
53
21}
229. Find 1A , if the matrix A is given by
8.06.0
6.08.0A . {Ans.
8.06.0
6.08.0}
230. For what value of k the matrix
53
2 kA has no inverse. {Ans.
3
10}
231. If
xx
xA
02 and
21
011A , then find the value of x. {Ans. 2
1}
232. If
25
32A be such that kAA 1 , then find the value of k . {Ans.
19
1}
233. X is an unknown square matrix satisfying the equation
10
11
10
31X . Determine the matrix X.
{Ans.
10
41}
234. If
10
01
35
23
23
12A , then find the matrix A . {Ans.
01
11}
235. Find the inverse of the matrix
120
031
221
A . Verify that A
A11 . {Ans.
522
211
623
}
236. If
61
52A , find 1A and verify that IAA
7
8
7
11 . {Ans.
21
56
7
1}
237. If
13
02A and
42
10B . Verify that 111
ABAB .
238. Find the inverse of the matrix
273
143
132
A and verify that IAA 1 . {Ans.
2
1
2
5
2
92
1
2
1
2
32
1
2
1
2
1
}
239. Find the inverse of the matrix
211
103
412
A and verify that IAA 1 . {Ans.
313
1485
161
19
1}
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.37
240. Find the inverse of the matrix
111
232
422
A and verify that IAA 1 . {Ans.
1005
420
1625
10
1}
241. If
100
0cossin
0sincos
F then show that yxFyFxF . Hence prove that xFxF 1 .
242. If
310
015
102
A , prove that IAAA 11621 .
243. If
35
12A and
43
54B , verify that 111
ABAB .
244. If
12
54A , then show that 1623 AIIA .
245. Let
122
212
221
A , prove that OIAA 542 . Hence obtain 1A . {Ans.
322
232
223
5
1}
246. If
ab
ba1
1tan
tan1
1tan
tan1
, then show that 2sin,2cos ba .
247. Show that the inverse of a diagonal matrix is a diagonal matrix. CATEGORY-10.16. ORTHOGONAL MATRIX
248. If
481
744
418
9
1A , prove that A is an orthogonal matrix.
249. Show that if A is an orthogonal matrix, then TA is also orthogonal. 250. If A, B be orthogonal matrices, then prove that AB and BA are also orthogonal matrices. CATEGORY-10.17. INDEMPOTENT, INVOLUTARY, PERIODIC, NILPOTENT MATRIX
251. Prove that the matrix
321
431
422
A is idempotent.
252. Find all idempotent diagonal matrices of order 3. {Ans.
000
000
000
,
100
010
001
}
253. If B is idempotent, show that BIA is also idempotent and that OBAAB . 254. If A and B are idempotent and A and B commute, then show that AB is also idempotent. 255. If A and B are idempotent and OBAAB , then show that BA is also idempotent.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.38
256. Show that
67
56 is an involutary matrix.
257. Show that
1
010
001
ba
A is an involutary matrix.
258. Prove that the matrix
121
033
085
A is involutary.
259. Determine the condition that the matrix
ac
baA is involutary. {Ans. 12 bca }
260. Prove that the matrix
302
913
621
A is periodic whose period is 2.
261. If
001
012
111
A , find 2A and show that 12 AA . Is A a periodic matrix? If yes, find its period. {Ans.
111
210
100
, Period is 3}
262. Show that the matrix
431
431
431
A is nilpotent of index 2.
263. Show that the matrix
aba
babA
2
2
is nilpotent of index 2.
264. Show that the matrix
312
625
311
A is nilpotent of index 3.
CATEGORY-10.18. RANK OF A MATRIX
265. Find the rank of the matrix
122
212
221
A . {Ans. 3}
266. Find the rank of the matrix
543
654
321
A . {Ans. 2}
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.39
267. Find the rank of the matrix
1234
4321A . {Ans. 2}
268. Find the rank of the matrix
321
963
321
A . {Ans. 1}
CATEGORY-10.19. SOLVING SYSTEM OF LINEAR EQUATIONS BY MATRIX
269. Solve
232
22
3
zyx
zyx
zyx
{Ans. (1, 1, 1)} 270. Solve
2
032
4
zyx
zyx
zyx
{Ans. (2, -1, 1)} 271. Solve
2542
1932
1635
zyx
zyx
zyx
{Ans. (1, 2, 5)} 272. Solve
22
2
5582
zyx
zyx
zyx
{Ans. (-3, 2, -1)} 273. Solve
934
323
62
zyx
zyx
zyx
{Ans. (1, 2, -1)} 274. Solve
142
1032
6
zyx
zyx
zyx
{Ans. (-7, 22, -9)} 275. Solve
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.40
577
3252
4
zyx
zyx
zyx
{Ans. No solution} 276. Solve
3074
1432
6
zyx
zyx
zyx
{Ans. kkk ,28,2 } 277. Solve
51027
92263
4735
zyx
zyx
zyx
{Ans.
k
kk,
11
3,
11
167}
278. Solve
13652
043
1083
zyx
zyx
zyx
{Ans. kkk ,23,21 } 279. Solve
0225
05615
03
zyx
zyx
zyx
{Ans. (0, 0, 0)} 280. Solve
01411
042
023
zyx
zyx
zyx
{Ans.
k
kk,
7
8,
7
10}
281. Solve
02395
0234
0723
zyx
zyx
zyx
{Ans. kkk ,2, }
282. Using matrix method, find the values of and so that the system of equation
PART-II/CHAPTER-10
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.41
1787
3
12532
zyx
zyx
zyx
has i. Unique solution {Ans. 2 } ii. Infinite solutions {Ans. 7,2 } iii. No solution {Ans. 7,2 }
CATEGORY-10.20. ADDITIONAL QUESTIONS
283. If every element of a third order determinant of value is multiplied by 5, then find the value of new determinant. {Ans. 125 }
284. If M is a 33 matrix, where IMM T and 1det M , then prove that 0det IM . 285. If A and B are two non-zero square matrices of the same order and OAB , then show that both A and B
must be singular.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
10.42
THIS PAGE IS INTENTIONALLY LEFT BLANK
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-III
TRIGONOMETRY
CHAPTER-11
TRIGONOMETRIC IDENTITIES AND EXPRESSIONS
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.2
CHAPTER-11
TRIGONOMETRIC IDENTITIES AND EXPRESSIONS
LIST OF THEORY SECTIONS
11.1. Angles And Their Measurements 11.2. Standard Trigonometric Identities 11.3. Trigonometric Values As Roots Of Polynomials 11.4. Elimination
LIST OF QUESTION CATEGORIES
11.1. Relationship Between Trigonometric Ratios 11.2. Trigonometric Ratios Of Some Important Angles And Related Angles 11.3. Sum And Difference Of Angles 11.4. Sum And Difference Into Product 11.5. Product Into Sum And Difference 11.6. Double Angle Formulae 11.7. Triple Angle Formulae 11.8. Half Angle Formulae 11.9. Trigonometric Ratios Of The Sum Of Three Or More Angles 11.10. Conditional Identities 11.11. Standard Trigonometric Series 11.12. Product Of Cosines Of Double Angles 11.13. Trigonometric Values As Roots Of Polynomials 11.14. Elimination 11.15. Applications Of Trigonometric Identities 11.16. Additional Questions
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.3
CHAPTER-11
TRIGONOMETRIC IDENTITIES AND EXPRESSIONS
SECTION-11.1. ANGLES AND THEIR MEASUREMENTS
1. Definition of angle i. An angle is the amount of rotation of a revolving line with respect to a fixed line.
2. Measurement of angles i. 1 full rotation = 360; 1 degree (1) = 60 minutes (60’); 1 minute (1’) = 60 seconds (60”).
ii. Angle (in radians) = radius
lengtharc; 1 full rotation = c2 .
iii. 1 radian (1c) =
180 degrees = 5717’45” (approx.).
3. Positive and negative angles i. If the revolving line makes anti-clockwise rotation, then the angle is defined as positive and if the
revolving line makes clockwise rotation, then the angle is defined as negative. ii. When rotated by angle or n 2 or n 360 In , the revolving line is in the same
position.
SECTION-11.2. STANDARD TRIGONOMETRIC IDENTITIES
1. Sign of trigonometric ratios in quadrants i. sin , eccos , cos , sec , tan , cot are positive when is in Ist quadrant. ii. sin , eccos are positive and cos , sec , tan , cot are negative when is in IInd quadrant. iii. tan , cot are positive and sin , eccos , cos , sec are negative when is in IIIrd quadrant. iv. cos , sec are positive and sin , eccos , tan , cot are negative when is in IVth quadrant.
2. Relationship between trigonometric ratios
i.
sin
1eccos ;
cos
1sec ;
cos
sintan ;
sin
coscot .
ii. 1cossin 22 ; 22 sectan1 ; 22 coscot1 ec . 3. Trigonometric ratios of some important angles
i. 10cos;00sin .
ii. 23
30cos;21
30sin .
iii. 2
145cos45sin .
iv. 2
160cos;
2
360sin .
v. 090cos;190sin .
vi. 22
1315cos;
22
1315sin
.
vii. 4
521018cos;
4
1518sin
.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.4
viii. 2
22
2
122cos;
2
22
2
122sin
.
ix. 4
1536cos;
4
521036sin
.
4. Trigonometric ratios of related angles i. tantan;cos)cos(;sin)sin( . ii. cot)90tan(;sin)90cos(;cos)90sin( . iii. cot)90tan(;sin)90cos(;cos)90sin( . iv. tan)180tan(;cos)180cos(;sin)180sin( . v. tan)180tan(;cos)180cos(;sin)180sin( . vi. cot)270tan(;sin)270cos(;cos)270sin( . vii. cot)270tan(;sin)270cos(;cos)270sin( .
viii. tan360tan;cos)360cos(;sin)360sin( .
ix. tan360tan;cos)360cos(;sin)360sin( . 5. Sum and difference of angles
i. BABABA sincoscossin)sin( . ii. BABABA sincoscossin)sin( . iii. BABABA sinsincoscos)cos( . iv. BABABA sinsincoscos)cos( .
v. BA
BABA
tantan1tantan
)tan(
.
vi. BA
BABA
tantan1tantan
)tan(
.
vii. BA
BABA
cotcot
1cotcot)cot(
.
viii. AB
BABA
cotcot
1cotcot)cot(
.
ix. ABBABABA 2222 coscossinsinsinsin .
x. ABBABABA 2222 sincossincoscoscos . 6. Sum and difference into product
i. 2
cos2
sin2sinsinDCDC
DC
.
ii. 2
sin2
cos2sinsinDCDC
DC
.
iii. 2
cos2
cos2coscosDCDC
DC
.
iv. 2
sin2
sin2coscosCDDC
DC
.
v.
BA
BABA
coscos
sintantan
.
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.5
vi.
BA
BABA
coscos
sintantan
.
vii.
BA
BABA
sinsin
sincotcot
.
viii.
BA
ABBA
sinsin
sincotcot
.
7. Product into sum and difference i. )sin()sin(cossin2 BABABA . ii. )sin()sin(sincos2 BABABA . iii. )cos()cos(coscos2 BABABA . iv. )cos()cos(sinsin2 BABABA .
8. Double angle formulae
i. A
AAAA
2tan1
tan2cossin22sin
.
ii. A
AAAAAA
2
22222
tan1
tan11cos2sin21sincos2cos
.
iii. A
AA 2tan1
tan22tan
.
9. Triple angle formulae i. AAA 3sin4sin33sin . ii. AAA cos3cos43cos 3 .
iii. A
AAA 2
3
tan31tantan3
3tan
.
10. Half angle formulae
i.
2tan1
2tan2
2cos
2sin2sin
2 A
AAA
A
.
ii.
2tan1
2tan1
2sin211
2cos2
2sin
2coscos
2
2
2222
A
AAAAA
A
.
iii.
2tan1
2tan2
tan2 A
A
A
.
11. Trigonometric ratios of the sum of three or more angles i. CBACBACBACBACBA sinsinsinsincoscoscossincoscoscossinsin .
ii. CBACBACBACBACBA cossinsinsincossinsinsincoscoscoscoscos .
iii. ACCBBA
CBACBACBA
tantantantantantan1
tantantantantantan)tan(
.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.6
iv. ...........1
..........)........tan(
642
7531321
sss
ssssAAAA n
where iAs tan1 , ji AAs tantan2 , kji AAAs tantantan3 , ……..and so on.
v. 333
11 sincossincossin nnnn CCn
vi. 222 sincoscoscos nnn Cn
vii. ...........tantan1
..........tantantantan
44
22
55
331
ACAC
ACACACnA
nn
nnn
; where
1;!!
!0
n
nnr
n CCrrn
nC .
12. Conditional identities i. Identities valid between two or more angles, which are related by a mathematical relationship, are called
conditional identities. ii. If A + B + C = , then
a. CBA sinsin ; ACB sinsin ; BCA sinsin ,
b. CBA coscos ; ACB coscos ; BCA coscos ,
c. 2
cos2
sinCBA
;
2cos
2sin
ACB
;
2cos
2sin
BCA
,
d. 2
sin2
cosCBA
;
2sin
2cos
ACB
;
2sin
2cos
BCA
,
e. CBACBA tantantantantantan , f. 1cotcotcotcotcotcot BAACCB ,
g. 12
tan2
tan2
tan2
tan2
tan2
tan BAACCB
,
h. 2
cot2
cot2
cot2
cot2
cot2
cotCBACBA
,
i. CBACBA sinsinsin42sin2sin2sin , j. CBACBA coscoscos412cos2cos2cos , k. CBACBA coscoscos21coscoscos 222 ,
l. 2
cos2
cos2
cos4sinsinsinCBA
CBA ,
m. 2
sin2
sin2
sin41coscoscosCBA
CBA .
13. Standard Trigonometric series
i.
mn
n
n 2,2
sin
2sin
21sin
1sin...........2sinsinsin
mn 2,sin .
ii.
mn
n
n 2,2
sin
2sin
21cos
1cos...........2coscoscos
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.7
mn 2,cos . 14. Product of cosines of double angles
i.
m
n
nn ,
sin2
2sin2cos2cos2coscos 12
m2,1
12,1 m . SECTION-11.3. TRIGONOMETRIC VALUES AS ROOTS OF POLYNOMIALS
1. Certain trigonometric values are roots of polynomial with integer coefficients. SECTION-11.4. ELIMINATION
1. If (n + 1) equations containing n unknowns are given, then these n unknowns can be eliminated, i.e. an equation, called eliminant, can be obtained which does not contain these n unknowns.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.8
EXERCISE-11
CATEGORY-11.1. RELATIONSHIP BETWEEN TRIGONOMETRIC RATIOS
1. AAA 244 cos21sincos 2. AAAAAA 33 cossincossin1cossin
3. ecAA
A
A
Acos2
sin
cos1
cos1
sin
4. AAAA 2266 cossin31sincos
5. AAA
Atansec
sin1
sin1
6. AecA
ecA
ecA
ecA 2sec21cos
cos
1cos
cos
7. AAA
ecAcos
tancot
cos
8. AAAAAA 22 sintancosseccossec
9. AAAA
cossintancot
1
10. AAAA
tansectansec
1
11. 1cot
1cot
tan1
tan1
A
A
A
A
12. A
A
A
A2
2
2
2
cos
sin
cot1
tan1
13. AAAAA
AA 2tan2tansec21tansec
tansec
14. 1cossectan1
cot
cot1
tan
ecAA
A
A
A
A
15. AAA
A
A
Acossin
cot1
sin
tan1
cos
16. ecAAAAAA cossectancotcossin
17. AAAA 2424 tantansecsec 18. AecAecAA 2424 coscoscotcot
19. ecAAAec coscos1cos 2
20. 2cottancossec 2222 AAAecA 21. AAAA 2422 secsinsintan 22. 2sectan1coscot1 AAecAA
23. AecAAAAecA cotcos
1
sin
1
sin
1
cotcos
1
24. AA
AA
AA
AA
coscot
coscot
coscot
coscot
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.9
25. BAAB
BAtancot
tancot
tancot
26. AA
AAAA
AAecAA 22
2222
2222 sincos2
sincos1sincos
sincos
1
cossec
1
27. AAAAAA 222288 cossin21cossincossin
28. AecAAA
AAecAAseccos
sincos
secsincoscos
29. A
A
AA
AA
cos
sin1
1sectan
1sectan
30. BecABAABecBA seccoscottan2seccotcostan 22
31. AAAecAecAA 444242 tancotcoscos2secsec2
32. 7cottanseccoscossin 2222 AAAAecAA
33. A
ecA
Aec
AAAAA
22 sec
cos
cos
seccossintancot1
34. If the angle is in the third quadrant and 2tan , then find the value of sin . {Ans. 5
2 }
35. If is an acute angle and 7
1tan , then find the value of
22
22
seccos
seccos
ec
ec. {Ans.
4
3}
36. If q
ptan , show that
22
22
cossin
cossin
qp
qp
qp
qp
.
37. If 22 1tan a , prove that 2
323 2costansec aec .
38. If p
px4
1sec , show that
porpxx
2
12tansec .
CATEGORY-11.2. TRIGONOMETRIC RATIOS OF SOME IMPORTANT ANGLES AND RELATED
ANGLES
39. 1cossinsin 2 ec
40.
2sin2
cos2
sintan
41. 2
375cos75sin
42. 45cos105cos105sin 43. 0165sin255cos 44. 15cos105cos15sin75sin
45. 81522 60sin72sin
46. 16
5144sin108sin72sin36sin .
47. 21
1013sin
10sin
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.10
48. 41
1013sin
10sin
49. 2
1990sin.........15sin10sin5sin 2222 .
50. 2
1890cos.........15cos10cos5cos 2222 .
51. Find the value of 360sin30sin20sin10sin . {Ans. 0}
52. If 3
4cos
3
2cos
zyx , then show that 0 zxyzxy .
CATEGORY-11.3. SUM AND DIFFERENCE OF ANGLES
53. BABABA sin45sin45sin45cos45cos
54. BABABA cos45sin45cos45cos45sin
55.
0coscos
sin
coscos
sin
coscos
sin
AC
AC
CB
CB
BA
BA
56. BABAABA cossinsincoscos
57. ACBACBCBA sinsincoscoscoscos
58. AAnAnAnAn 2cos1cos1cos1sin1sin
59. AAnAnAnAn cos2cos1cos2sin1sin
60. AAA sin2
128
sin28
sin 22
61. 22cossinsin2cos2cos 22
62. 8
1512sin48cos 22
63. 8
156sin24sin 22
.
64. 316cot76cot76cot44cot44cot16cot .
65. If ba
ba
yx
yx
sin
sin, then find the value of
y
x
tan
tan in terms of a and b. {Ans.
b
a}
66. If 2
1tan A ,
3
1tan B , show that 45BA .
67. If 1
tan
m
mA ,
12
1tan
mB , show that 45BA .
CATEGORY-11.4. SUM AND DIFFERENCE INTO PRODUCT
68.
tan
5cos7cos
5sin7sin
69. .tan4sin6sin
4cos6cos
70. .2tan3coscos
3sinsinA
AA
AA
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.11
71. .5sec4cos2sin8sin
sin7sinAA
AA
AA
72. .cotcot2cos2cos
2cos2cosBABA
AB
AB
73.
.tan
tan
2sin2sin
2sin2sin
BA
BA
BA
BA
74. .2
cot2coscos
2sinsin A
AA
AA
75. .tan5cos3cos
3sin5sinA
AA
AA
76. .tan2sin2sin
2cos2cosBA
AB
AB
77. .45cos45sin2sincos BABABA
78. .3cos2cos
sin
2sin4sin
4cos2cos
sin3sin
cos3cos
AA
A
AA
AA
AA
AA
79.
.tan24cos24cos
24sin24sinBA
ABBA
ABBA
80. .3tan2sin2cos5cos3cos2cos
7cos5cos23cos
81. .4tan7cos5cos3coscos
7sin5sin3sinsinA
AAAA
AAAA
82.
.tancoscos2cos
sinsin2sin
83. .5sin
3sin
7sin5sin23sin
5sin3sin2sin
A
A
AAA
AAA
84. B
A
CBBCB
CAACA
sin
sin
sinsin2sin
sinsin2sin
85. AAAAA
AAAA4cot
13cos9cos5coscos
13sin9sin5sinsin
86. .2
cot2
tansinsin
sinsin BABA
BA
BA
87. .2
cot2
cotcoscos
coscos BABA
AB
BA
88. .2
tancoscos
sinsin BA
BA
BA
89. .2
cotcoscos
sinsin BA
AB
BA
90.
BCBACBACBACBA
CBACBACBACBAcot
sinsinsinsin
coscoscoscos
91. .6cos5cos4cos415cos7cos5cos3cos AAAAAAA
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.12
92. .3sin2
3cos
2cos45sin4sin2sinsin a
aaaaaa
93. .coscoscos4coscoscoscos CBACBACBACBACBA
94. DCBADCBADCBADCBADCBA coscossin4sinsinsinsin
95. 040sin10cos70cos . 96. .010sin70sin50sin 97. .80sin70sin50sin40sin20sin10sin 98. 0140cos100cos20cos . 99. 48cos24cos84cos60cos12cos 100. 1cos17sin53sin19sin55sin . CATEGORY-11.5. PRODUCT INTO SUM AND DIFFERENCE
101. .5sin2sin2
11sin
2
3sin
2
7sin
2sin
102. .2
5sin5sin
2
9cos3cos
2cos2cos
103. .sinsin2sinsin2sinsin BABAABBBAA
104. .0coscos3cossinsin3sin AAAAAA
105.
.sin
sin
2sincossin2
2sincossin2
B
A
CBCCB
CACCA
106. .9tan13cos4sin6cos3sin2cossin
13sin4sin6sin3sin2sinsinA
AAAAAA
AAAAAA
107. .5cot6cot7sin4sin5sin2sin3sin4sin
10coscos7cos2cos3cos2cosAA
AAAAAA
AAAAAA
108. .2cos54cos54cos36cos36cos AAAAA
109. .0sincossincossincos BACACBCBA
110. .2cos2
145sin45sin AAA
111. 4
3sin240sin240sin120sin120sinsin AAAAAA .
112. .0cossincossincossincossin ADADDCDCCBCBBABA
113. .0cossincossincossin aaa
114. CBABACACBCBCBCABCBA coscoscossincos2sincos2sin
115. .0sinsinsinsinsinsinsinsinsinsinsinsin ACCBBAACACCBCBBABA
116. .013
5cos
13
3cos
13
9cos
13cos2
117. 50tan220tan70tan .
118. 4
3200cos140cos140cos100cos100cos20cos .
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.13
119. 8
154sin48sin12sin .
120. 8
178cos42cos36cos .
121. 16380sin60sin40sin20sin .
122. 16
178sin66sin42sin6sin .
123. 16
178cos66cos42cos6cos .
124. 178tan66tan42tan6tan . CATEGORY-11.6. DOUBLE ANGLE FORMULAE
125. .tan2cos1
2sinA
A
A
126. .cot2cos1
2sinA
A
A
127. .tan2cos1
2cos1 2 AA
A
128. .2cos2cottan AecAA 129. .2cot2cottan AAA 130. .cot2cot2cos AAAec
131. .4cos2cos43tan5tan
3tan5tan
132. .2tan
8tan
14sec
18sec
A
A
A
A
133.
.2cos45tan1
45tan12
2
AecA
A
134. .tancossincossin
sinsin 22
BABBAA
BA
135. .2tan24
tan4
tan
136. .2tan2sincos
sincos
sincos
sincosA
AA
AA
AA
AA
137. .2sin21
2cos415tan15cot
A
AAA
138. .tan2coscos1
2sinsin
139. .sin1sinsin12sin 22 nAAnAAn
140.
.cos22sin2sin
3sin3sinBA
BA
BABA
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.14
141. .2
3cos
2cossin4sin2sin3sin
AAAAAA
142. .1sec12sec2tan 2 AAA
143. .sincos42cos32cos 663
144. .sincos22cos1 442
145. .2sec22sec1sec2 AAA 146. .sin2cos2cot2cos AAAAec
147. aaa 42 cos8cos814cos 148. AAAAA 33 sincos4cossin44sin 149. AAAAAA tan2tan3tantan2tan3tan
150. cos24cos222 .
151. 2sec24
sec4
sec
152. 23120cos120coscos 222 aaa
153. AAAAAA 222 sec3tan2tan3tan12tan4tan
154. 12cos212cos212cos21cos21cos2
12cos2 12
nn
155. 23
87cos
85cos
83cos
8cos 4444
156. 23
87sin
85sin
83sin
8sin 4444
157. 8
1
8
7cos1
8
5cos1
8
3cos1
8cos1
.
158. 535527sin4 .
159. 410cos
3
10sin
1
.
160. 420sec20cos3 ec . 161. 481tan63tan27tan9tan .
162. 350tan70tan10tan .
163. 3516
7tan
16
6tan
16
5tan
16
4tan
16
3tan
16
2tan
16tan 2222222
.
164. 64322
17cot or 2132
2
182tan .
165. If
2cos3
12cos32cos
, then show that tan2tan .
166. If a
btan , prove that aba 2sin2cos .
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.15
167. If ba
ba
tantan , prove that 2cos2cos baba is independent of and .
168. If 1tan2tan 22 , then prove that 0sin2cos 2 .
169. If
tantan1
tantantan
, prove that
2sin2sin1
2sin2sin2sin
.
CATEGORY-11.7. TRIPLE ANGLE FORMULAE
170. aaaa 3sin4
160sin60sinsin
171. aaaa 3cos4
160cos60coscos
172. aaaa 3cot360cot60cotcot
173. 1cos18cos48cos326cos 246 aaaa
174. 3cos4
3240cos120coscos 333 .
175. If
aa
1
2
1cos , then prove that
33 1
2
13cos
aa .
CATEGORY-11.8. HALF ANGLE FORMULAE
176. 2
cos4sinsincoscos 222
177. 2
cos4sinsincoscos 222
178. 2
sin4sinsincoscos 222
179. AAA
AA tansecsin1
sin12
45tan
180. 2
secsin2
sec2
tan12
sec2
tan1 2
181.
.2
cot2
tancoscoscos1
coscoscos1 BA
BABA
BABA
182. .2
45tansin1
cos
A
A
A
183. .2
tancossin1
cossin1
184. .2
tan2
cot2
1cot
AAA
185.
.
2tan
1coscos21cos
1sin1sin A
AnnAAn
AnAn
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.16
186.
.
2cot
1cos1cos
1sinsin21sin A
AnAn
AnnAAn
187. 2
3
15tan1
15tan12
2
.
188. 33sin29cos28cos466cos58cos56cos1 .
189. If
cos
coscos
ba
ba
, prove that
2tan
2tan
ba
ba
.
190. If
coscos
sinsintan
, prove that one of the values of
2tan
is
2tan
2tan
.
191. If 5
3cos A ,
13
5cos B , prove that
65
1
2sin 2
BA,
65
64
2cos2
BA.
192. If
cos2
1cos2cos
, prove that
2tan3
2tan
, and hence show that
cos2
sin3sin
.
193. If aBA sinsin and bBA coscos , then find the value of BAcos in terms of a and b. {Ans.
22
22
ba
ab
}
CATEGORY-11.9. TRIGONOMETRIC RATIOS OF THE SUM OF THREE OR MORE ANGLES
194. Prove that
coscoscos
sintantantantantantan
.
CATEGORY-11.10. CONDITIONAL IDENTITIES
195. If 45BA , prove that 21cot1cot BA .
196. If 225BA , prove that 2
1
cot1
cot
cot1
cot
B
B
A
A.
197. If thatprove ,180 CBA i. .sincoscos42sin2sin2sin CBACBA ii. .cossinsin412cos2cos2cos CBACBA
iii. .2
cos2
sin2
sin4sinsinsinCBA
CBA
iv. .cossinsin2sinsinsin 222 CBACBA
v. .cossinsin21coscoscos 222 CBACBA
vi. .2
sin2
sin2
sin212
sin2
sin2
sin 222 CBACBA
vii. .2
sin2
cos2
cos212
sin2
sin2
sin 222 CBACBA
viii. .2
sin2
sin2
sin42sin2sin2sinBAACCB
BAACCB
ix. .4
sin4
sin4
sin412
sin2
sin2
sinCBACBA
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.17
x. .4
cos4
cos4
cos42
cos2
cos2
cosCBACBA
xi. .2
sin2
sin2
sin8sinsinsin
2sin2sin2sin CBA
CBA
CBA
xii. .sinsinsin4sinsinsin CBACBABACACB
198. If 2
CBA , prove that
i. CBACBA sinsinsin21sinsinsin 222 ii. CBACBA sinsinsin22coscoscos 222 iii. CBACBA cotcotcotcotcotcot iv. 1tantantantantantan CACBBA
199. If 2 CBA , prove that
i. 2
tan2
tan2
tan2
tan2
tan2
tanCBACBA
ii. 0coscoscos2coscoscos1 222 CBACBA
iii. 2
3sin
2
3sin
2
3sin
2sin
2sin
2sin3sinsinsin 333 CBACBA
CBA
200. If SCBA 2 , prove that i. .sinsinsinsinsinsin BACSSBSAS
ii. .coscoscos2coscoscos1sinsinsinsin4 222 CBACBACSBSASS
iii. .2
sin2
sin2
sin4sinsinsinsinCBA
SCSBSAS
iv. .coscoscos22coscoscoscos 2222 CBACSBSASS
v. .coscoscoscos41coscoscos2coscoscos 222 CSBSASSCBACBA
201. If 0 , prove that coscoscos1sinsinsin22sin2sin2sin 202. If CBA , prove that
i. CBACBA coscoscos21coscoscos 222 . ii. ABCCBA tantantantantantan .
203. If BCA 2 , prove that AC
CAB
coscos
sinsincot
.
204. If 2 , prove that
i. 02
cos2
cos2
cos4coscoscoscos
ii. 02
cos2
sin2
cos4sinsinsinsin
iii. .cotcotcotcottantantantantantantantan 205. If the sum of four angles be 1800, prove that the sum of the products of their cosines taken two and two
together is equal to the sum of the products of their sines taken similarly. CATEGORY-11.11. STANDARD TRIGONOMETRIC SERIES
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.18
206. 2
1
7
6cos
7
4cos
7
2cos
.
207. 2
1
11
9cos
11
7cos
11
5cos
11
3cos
11cos
.
208. 17
6cos
7
5cos
7
4cos
7
3cos
7
2cos
7cos0cos
.
209. 07
12sin
7
10sin
7
8sin
7
6sin
7
4sin
7
2sin
.
210. If n is an integer greater than 2, prove that 0.........6
cos4
cos2
cos termsntonnn
.
211. Prove that ecnntermsnto cos12sinsin124
1.......3sin2sinsin 222
212. Prove that termsnto.......3sin2sinsin 333
2
3cos
2
3sin
2
13sin
4
1
2cos
2sin
2
1sin
4
3 ec
nnec
nn
.
213. Prove that the sum of the product of sines of the angles n,.....,3,2, taking two at a time is
2sin2
sin1cos
2sin
2sin
2
1sin
2
1
2
22
nnnnn
.
CATEGORY-11.12. PRODUCT OF COSINES OF DOUBLE ANGLES
214. 16180cos60cos40cos20cos
215. 115
14cos158cos
154cos
152cos16
216. 64
1
65
32cos
65
16cos
65
8cos
65
4cos
65
2cos
65cos
.
217. If 12
n
, prove that
nn
2
12cos............2cos2coscos 12 .
218. If 12
n
, prove that
nn
2
12cos............2cos2coscos 12 .
219. 72
1157cos
156cos
155cos
154cos
153cos
152cos
15cos
220. 015
8cos
15
4cos
15
2cos
15cos
20
9cos
20
7cos
20
3cos
20cos
.
221. cot2tan2sec1.............8sec14sec12sec1 nn .
222. 380tan40tan20tan .
223. 8
1
14
5sin
14
3sin
14sin
.
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.19
224. 64
1
14
13sin
14
11sin
14
9sin
14
7sin
14
5sin
14
3sin
14sin
.
225. If 12
n
A
, 22
n
B
, then prove that
1
12122
2cos
2cos
2
12cos2cos........2cos2cos2cos2coscoscos
nnnnn BABABABA
.
CATEGORY-11.13. TRIGONOMETRIC VALUES AS ROOTS OF POLYNOMIALS
226. Prove that the roots of the equation 01448 23 xxx are 7
cos
, 7
3cos
and
7
5cos
and hence prove
that
i. 2
1
7
5cos
7
3cos
7cos
ii. 2
1
7
5cos
7
3cos
7
5cos
7cos
7
3cos
7cos
iii. 8
1
7
5cos
7
3cos
7cos
iv. 8
1
7
5cos1
7
3cos1
7cos1
v. the equation whose roots are 7
cos2 , 7
3cos2
and 7
5cos2
is 01248064 23 xxx
vi. 4
5
7
5cos
7
3cos
7cos 222
vii. the equation whose roots are 7
sec
, 7
3sec
and
7
5sec
is 0844 23 xxx
viii. 47
5sec
7
3sec
7sec
ix. the equation whose roots are 7
sec2 , 7
3sec2
and 7
5sec2
is 0648024 23 xxx
x. 247
5sec
7
3sec
7sec 222
xi. the equation whose roots are 7
tan 2 , 7
3tan 2
and 7
5tan 2
is 073521 23 xxx
xii. 217
5tan
7
3tan
7tan 222
xiii. 77
5tan
7
3tan
7tan
xiv. the equation whose roots are 7
cot 2 , 7
3cot 2
and 7
5cot 2
is 0121357 23 xxx
xv. 57
5cot
7
3cot
7cot 222
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.20
227. Find the equation whose roots are 7
2cos
,
7
4cos
and
7
6cos
. {Ans. 01448 23 xxx }
228. 77
3tan
7
2tan
7tan
.
229. 1057
3cot
7
2cot
7cot
7
3tan
7
2tan
7tan 222222
.
230. Find the value of 7
3sin
7
2sin
7sin 222
. {Ans. 4
7}
231. 8
7
7
3sin
7
2sin
7sin
.
232. Find the value of 7
3cos
7
2cos
7cos 222
ececec . {Ans. 8}
233. 2
7
7
8sin
7
4sin
7
2sin
.
234. Show that 14
sin
is a root of the equation 01448 23 xxx and find the other roots. {Ans. 14
3sin
,
14
5sin
}
CATEGORY-11.14. ELIMINATION
235. If n
cos
cos, m
sin
sin, show that 2222 1sin nnm .
236. If 22
sincosba
byax
and 0
sin
cos
cos
sin22
byax, show that 3
222
3
2
3
2
babyax .
237. If a cossec and bec sincos , prove that 132222 baba .
238. If 3sincos aec , 3cossec b , prove that 12222 baba .
239. If m sintan and n sintan , prove that mnnm 422 .
240. If pba sincos and qba cossin , prove that 2222 qpba .
241. If m4sin1cot and n4sin1cot , prove that mnnm 222 .
242. If mcc 23 sincos3cos and ncc sincos3sin 23 , prove that 3
2
3
2
3
2
2cnmnm .
243. If a 2sinsin and b 2coscos , prove that bbaba 232222 .
244. If 13
1cos 22 a and
3
22 tan
2tan , prove that
3
2
3
2
3
2 2sincos
a .
245. If
c
z
b
y
a
x
tantantan prove that
0sinsinsin 222
xz
ac
aczy
cb
cbyx
ba
ba.
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.21
246. If 22
222
tan1
tan2cossin
x
xcxbxa
, prove that 222222 4 bacba .
247. If 2
123cot xxx , 2
11 1cot xx and 2
1123cot
xxx , prove that .
248. If
p
3sin
sin
3cos
cos 33
, show that p
p 12cos
2 .
249. If x tancot , y cossec , eliminate . {Ans. 13
2
3
2
3
2
yxxy }
250. If tan1sec ba , 2222 tan5sec ba , eliminate . {Ans. 094 2222 baba } 251. If 0tansec cba , 0tansec rqp , eliminate . {Ans.
222 bpaqarcpcqbr }
252. If cba 2cottan , cba 2tancot , eliminate . {Ans. aabacbab 22 }
253. If
d
yx
c
yx
b
yx
a
x 3cos2coscoscos
, prove that accdbb .
254. If 1cotcot 22 ba , 1coscos 22 ba and sinsin ba , then prove that 0222 abba .
255. Let coscoscoscoscos and 2
sin2
sin2sin
, prove that 2
tan2
tan2
tan 222 .
256. If coscoscos and coscoscos where 0cos and 2
tan2
tan2
tan
, prove that
1sec1secsin2 . CATEGORY-11.15. APPLICATIONS OF TRIGONOMETRIC IDENTITIES
257. Find the value of 15
tan5
2tan3
15tan
5
2tan
. {Ans. 3 }
258. If ),cos(log)( xxf then evaluate
)(
2
1)()( xyf
y
xfyfxf . {Ans. 0}
259. If sinsin0coscos , then show that cos22cos2cos . 260. If p tansec , obtain the values of sec , tan and sin in terms of p. {Ans.
1
1,
2
1,
2
12
222
p
p
p
p
p
p}
261. If 1sinsin 2 xx , show that 1coscos 42 xx . 262. If 1sinsin 2 xx , then find the value of xxx 468 coscos2cos . {Ans. 1}
263. If cos2sincos , show that sin2sincos .
264. If 8
633cos x , show that the three values of xcos are
6sin
2
6 ,
10sin
2
6 ,
10
3sin
2
6 .
265. If ,0 and 2
3coscoscos , prove that
3
.
266. If cba sincos , show that 222cossin cbaba .
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.22
267. If 5cos5sin3 , show that 33cos3sin5 or .
268. If CBACBA sin1sin1sin1sin1sin1sin1 prove that each side is equal to CBA coscoscos .
269. Prove that
2cot2
coscos
sinsin
sinsin
coscos BA
BA
BA
BA
BA nnn
or 0 according as n is even or odd
positive integer. 270. If DCBADCBA sincossincos , prove that DCBA cotcotcotcot .
271. If 120tan30tan nm , show that nm
nm
22cos .
272. If coscos nm , show that tancot nmnm .
273. If cotcotcot2 , show that cotcot2
12cot .
274. If
24tan
24tan 3 xy
, prove that
x
xxy
2
2
sin31
sin3sinsin
.
275. If B
A
B
A
2sin
2sin
sin
sin
, prove that BA tantantan2 .
276. If ,,0 , prove that
i. sinsinsinsin ii. sinsinsin3sinsinsin
277. If yx tancos , zy tancos , xz tancos , prove that 18sin2sinsinsin zyx .
278. If BBA 3coscoscos2 and BBA 3sinsinsin2 , show that 3
1sin BA .
279. If bab
A
a
A
1cossin 44
, prove that 33
8
3
8 1cossin
bab
A
a
A
.
280. If xzy sin , yxz sin , zyx sin be in A.P., prove that xtan , ytan , ztan are also in A.P.
281. If sec , sec , sec be in A.P. prove that 2
cos2cos
.
282. If ayx sinsin , byx coscos , show that
i. 2
2cos
22
bayx
ii. 22
224
2tan
ba
bayx
.
283. If 5
4cos and
13
5sin and , lie between 0 and
4
, find 2tan . {Ans.
33
56}
284. If xBA tantan and yAB cotcot , prove that yx
BA11
cot .
285. Prove that sinsincos2sinsin 22 is independent of .
PART-III/CHAPTER-11
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.23
286. If 1sin
sin
cos
cos2
4
2
4
y
x
y
x, prove that 1
sin
sin
cos
cos2
4
2
4
x
y
x
y.
287. If xyyx coscos3sinsin , prove that 03sin3sin yx .
288. If cos1cossin1sin x and cos1cossin1sin y , prove that
02sin22sin2 22 yyxx .
289. If CBACBA coscoscoscos , show that CBABAACCB 2sin2sin2sinsinsinsin8 . 290. If 0coscoscos CBA , prove that CBACBA coscoscos123cos3cos3cos .
291. If tancostan , then prove that
sin2
tansin 2 .
292. If tancottan2 , then tan2cot .
293. If 2sinsin n , show that tan1
1tan
n
n
.
294. If an angle be divided into two parts such that the tangent of one part is m times the tangent of the other,
then prove that their difference is obtained by the equation sin1
1sin
m
m.
295. If cossinsin2sinsinsin 2222x , show that tan1
1tan
x
x
.
296. If BmA coscos , then prove that
2tan
1
1
2cot
AB
m
mBA.
297. Given that the angles ,, are connected by the relation
1tantantantantantantantantan2 222222222 , find the value of
222 sinsinsin . {Ans. 1}
298. If axx cossin , evaluate xx 66 cossin . {Ans. 22 14
31 a }
299. If xyxyx sec2secsec , then prove that 2
cos2cosy
x .
300. If xn
xxny
2sin1
cossintan
, prove that xnyx tan1tan .
301. If tantantansecsec , prove that 02cos .
302. Suppose
n
m
mm xCxx
0
3 cos3sinsin , is an identity in x, where nCCCC ..........,,, 210 are real constants and
0nC . Find the value of n. Also find nCCCC ..........,,, 210 . {Ans. n = 6}
303. If 7
2 , prove that 7tan4tan4tan2tan2tantan .
304. If n4 , prove that 112tan22tan..........3tan2tantan nn .
305. Prove that 1cos2 BAxx is a factor of
2coscos42cos2cossinsin42 234 BAxBAxBAxx . Also find the other factor. {Ans.
2cos22 2 BAxx }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
11.24
306. Show that
xxxx
5sin
2sin23sin
2
3sin3 6644 is independent of x.
307. If 0sinsinsincoscoscos . Prove that
i.
cot2
cot
ii. 2
1
2cos
.
iii. 2
3sinsinsincoscoscos 222222 .
308. If 2
11
rr
aa
, prove that 0
21
201
cos atoaa
a
.
CATEGORY-11.16. ADDITIONAL QUESTIONS
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-III
TRIGONOMETRY
CHAPTER-12
TRIGONOMETRIC AND INVERSE TRIGONOMETRIC
FUNCTIONS, EQUATIONS AND INEQUALITIES
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.2
CHAPTER-12
TRIGONOMETRIC AND INVERSE TRIGONOMETRIC
FUNCTIONS, EQUATIONS AND INEQUALITIES
LIST OF THEORY SECTIONS
12.1. Maximum And Minimum Values Of Trigonometric Functions 12.2. Trigonometric Equations 12.3. Inverse Trigonometric Identities 12.4. Inverse Trigonometric Equations 12.5. Trigonometric Inequalities 12.6. Inverse Trigonometric Inequalities
LIST OF QUESTION CATEGORIES
12.1. Maximum And Minimum Values Of Trigonometric Functions 12.2. Trigonometric Equations 12.3. Values Of Inverse Trigonometric Functions 12.4. Inverse Trigonometric Equations 12.5. Trigonometric Inequalities 12.6. Inverse Trigonometric Inequalities 12.7. Limit Of Inverse Trigonometric Functions 12.8. Series Containing Trigonometric And Inverse Trigonometric Functions 12.9. Additional Questions
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.3
CHAPTER-12
TRIGONOMETRIC AND INVERSE TRIGONOMETRIC
FUNCTIONS, EQUATIONS AND INEQUALITIES SECTION-12.1. MAXIMUM AND MINIMUM VALUES OF TRIGONOMETRIC FUNCTIONS
1. Standard results i. 1sin x ; 1cos x ; 1cos ecx ; 1sec x .
ii. 2
0,tansin
xxxx .
iii. 2222 cossin baxbxaba . SECTION-12.2. TRIGONOMETRIC EQUATIONS
1. Simplest Trigonometric equations i. 1,sin)1(sin 1 aanxax n
1, ax
221sin
nxx
2
321sin
nxx
ii. 1,cos2cos 1 aanxax
1, ax
iii. anxax 1tantan iv. anxax 1cotcot
2. Equations of type axf sin , etc. 3. Substitution 4. Homogeneous equation in xsin and xcos 5. Equation of type dxcxxbxa 22 coscossinsin 6. Equation of type cxbxa cossin 7. Equation containing Rational functions of xsin and xcos 8. Lowering high even powers of xsin and xcos 9. Change/break into simpler equations using trigonometric identities
SECTION-12.3. INVERSE TRIGONOMETRIC IDENTITIES
1. i. 11 ,)sin(sin 1 xxx
ii. 11,1)cos(sin 21 xxx
iii. 11,1
)tan(sin2
1
xx
xx
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.4
iv. 11 ,)cos(cos 1 xxx
v. 11 ,1)sin(cos 21 xxx
vi. 1,00,1 ,1
)tan(cos2
1
xx
xx
vii. xx )tan(tan 1
viii. 2
1
1)sin(tan
x
xx
ix. 2
1
1
1)cos(tan
xx
2. i. xx 11 sin)(sin , 11 x
ii. xx 11 tan)(tan
iii. xx 11 cos)(cos , 11 x
iv. xx 11 cot)(cot 3.
i. 2
cossin 11 xx , 11 x
ii. 2
cottan 11 xx
iii. 2
eccossec 11 xx , ,11, x
4.
i. 22
,)(sinsin 1 xxx
ii. xxx 0,)(coscos 1
iii. 22
,)(tantan 1 xxx
5.
i. 10,1cossin 211 xxx
01,1cos 21 xx
ii. 11,1
tansin2
11
xx
xx
iii. 10,1sincos 211 xxx
01,1sin 21 xx
iv. 10,1
tancos2
11
xx
xx
01,1
tan2
1
xx
x
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.5
v. 2
11
1sintan
x
xx
vi. 0,1
1costan
2
11
xx
x
0,1
1cos
2
1
xx
6.
i. 1 ,1
tantantan 111
xy
xy
yxyx
0,0,1 ,1
tan 1
yxxy
xy
yx
0,0,1 ,1
tan 1
yxxy
xy
yx
0,0,1 ,2
yxxy
0,0,1 ,2
yxxy
ii. 11- ,1
2tantan2
211
xx
xx
1 ,1
2tan
2
1
xx
x
1 ,1
2tan
21
xx
x
1 ,2
x
1 ,2
x
SECTION-12.4. INVERSE TRIGONOMETRIC EQUATIONS
1. Simplest inverse trigonometric equations
i. 22
,sinsin 1 aaxax
2
;2
,
aax
ii. aaxax 0,coscos 1 aax ;0,
iii. 22
,tantan 1 aaxax
2
;2
,
aax
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.6
2. Equations of type axf 1sin , etc.
3. Equations of type xgxf 11 sinsin , etc.
i.
xgxf
xg
xf
xgxf 11
11
sinsin 11
4. Substitution. 5. Take sin, cos or tan both the sides converting to non-equivalent algebraic equation. Solve the
algebraic equation, back check its each solution in the original equation and discard extraneous solutions, if any, to get the solutions of the original equation.
6. Change/break into simpler equations using inverse trigonometric identities SECTION-12.5. TRIGONOMETRIC INEQUALITIES
1. Simplest trigonometric inequalities i. axaxaxax sin,sin,sin,sin ii. axaxaxax cos,cos,cos,cos iii. axaxaxax tan,tan,tan,tan iv. axaxaxax cot,cot,cot,cot
2. Inequalities of type axf sin , etc. 3. Substitution 4. Homogeneous inequality in xsin and xcos 5. Inequality of type dxcxxbxa 22 coscossinsin 6. Inequality of type cxbxa cossin 7. Inequalities containing Rational functions of xsin and xcos 8. Lowering high even powers of xsin and xcos 9. Change/break into simpler inequalities using trigonometric identities SECTION-12.6. INVERSE TRIGONOMETRIC INEQUALITIES
1. Simplest inverse trigonometric inequalities i. axaxaxax 1111 sin,sin,sin,sin
ii. axaxaxax 1111 cos,cos,cos,cos
iii. axaxaxax 1111 tan,tan,tan,tan
2. Inequalities of type axf 1sin , etc.
3. Inequalities of type xgxf 11 sinsin , etc. 4. Substitution. 5. Take sin, cos or tan both sides converting to equivalent algebraic inequalities. 6. Change/break into simpler inequalities using inverse trigonometric identities.
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.7
EXERCISE-12
CATEGORY-12.1. MAXIMUM AND MINIMUM VALUES OF TRIGONOMETRIC FUNCTIONS
1. Which of the following statements are correct/incorrect?
i. 5
1sin . {Ans. correct}
ii. 1cos . {Ans. correct}
iii. 2
1sec . {Ans. incorrect}
iv. 20tan . {Ans. correct} 2. Find the maximum and minimum values of sin24cos7 . {Ans. –25, 25} 3. Show that the maximum and minimum values of sin15cos8 are 17 and –17 respectively. 4. Find the maximum and minimum values of 5sin4cos3 xx . {Ans. 0, 10}
5. For what value of x in the interval
2,0
, the maximum value of
6cos
6sin
xx is attained?
{Ans. 12
}
6. Show that 2cossin xx .
7. Prove that 33
cos3cos5
lies between –4 and 10.
8. Find a and b such that the inequality bxxa
6sin5cos3
holds good for all x. {Ans.
19,19 ba } 9. Find the maximum and minimum values of xxx 2cos4cossin6 . {Ans. 5,5 }
10. Show that for all values of , the expression 22 coscossinsin cba lies between
22
22 cabca
and
22
22 cabca
.
11. Express cossin8cos6 2 as 2cosBA and hence show that the greatest and the least values of the expression are 8 and –2 respectively.
12. Find the maximum and minimum values of xx sin92cos . {Ans. 8, -10}
13. Prove that 8
17sin32cos4 xx .
14. If bxxa 6sin52cos , find a and b. {Ans. a = 0, b = 10} 15. Show that the value of 22 cossec is never less than 2.
16. Prove that
2sinsin
sinsin
sin
sin
sin
sin 2
. Hence deduce that if ,0 , 2
sin
sin
sin
sin
.
17. Find the greatest and least values of BAcoscos when 90BA . {Ans. 2
1,
2
1 }
18. If A > 0, B > 0 and ,3
BA find the maximum value of BA tantan . {Ans.
3
1}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.8
19. Prove that x
x
cot
3cot never lies between
3
1 and 3.
20. Prove that xx cot6
tan
cannot lie between 3
1 and 3.
21. Show that
sin
cos3 cannot have any value between 22 and 22 . What are the limits of
cos3
sin
.
{Ans.
22
1,
22
1}
22. Prove that the expression 22 sinsinsincos lies between 2sin1 and 2sin1 ,
20
.
23. Prove that 0coscos and 0sincos for all .
24. For all in
2,0
show that cossinsincos .
25. If 0 , prove that 2cot4
cot
and 1cot2
cot
.
CATEGORY-12.2. TRIGONOMETRIC EQUATIONS
26.
1tan2
1cos
. {Ans.
412
n }
27. 13tan x . {Ans.
123
n}
28. 02cos1
cos
x
x {Ans. }
29. 02cos
cossin
x
xx {Ans. }
30. 03tancos xx {Ans.
3
n}
31. 02tancos4sin xxx {Ans.
2
n}
32. 01sin
1cos1
xx {Ans.
22
n }
33. 02
tancos1 x
x {Ans. n2 }
34. 02sin3sin2 2 . {Ans. 6
71
nn }
35. 043sin53sin2 xx {Ans.
63
2 n}
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.9
36. 01sec6sec8 2 . {Ans. }
37. 3tan3tantan 23 xxx {Ans.
43
kn }
38. 01coscos8cos8 24 xxx {Ans.
4
51cos2
3
2 1
kn
}
39. 0sin2cossin2 3 xxx {Ans.
22
42
k
n }
40. 04sin5cos2 2 xx {Ans.
61
nn }
41. 32cos72sin3 2 xx {Ans.
42
n}
42. 2sincos2 2 xx {Ans.
61
kkn }
43. 0cossin2 2 xx {Ans.
4
32
n }
44. xxxx cossin2cos2sin {Ans.
63
22
kn }
45. xxx sincos2cos2 {Ans.
42
123
2
k
n }
46. xx 2cos3sin {Ans.
105
2 n}
47. xx 15sin5cos {Ans.
2054010
kn }
48. 72cos5sin xx {Ans.
612
22 1
kkn }
49. 32sinsin4 22 xx {Ans.
42
n}
50. 7cos82cos4 22 xx {Ans.
6
n }
51. xxx 2cos13
cos6
sin
{Ans.
32
2
kn }
52. 014cos22cos3sin8 6 xxx {Ans.
42
n}
53. xx 2cos1sin13 {Ans.
61
22
kkn }
54. xx cos4
3sin {Ans.
4
3tan 1n }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.10
55. xx cos2sin3 {Ans.
3
2tan 1n }
56. 0cos6cossin3sin3 22 xxxx {Ans. 2tan4
1
kn }
57. 02sin2cos3sin 22 xxx {Ans. 3tan4
1
kn }
58. 2cossin2sin3 2 xxx {Ans. 31tan 1 n }
59. 3sin5cossin3cos2 22 xxxx {Ans.
4
173tan 1n }
60. 01sincos2 xx .
61. tan 1tan2 xx . {Ans.
6
n }
62. 2cos13sin13 xx . {Ans.
1242
n }
63. xxx sin4sin7sin . {Ans.
93
2
4
nn }
64. xx 3cos2sin .
{Ans. 4
15sin2,
2
3,
4
15sin,
4
15sin,
2,
4
15sin,2 1111
n }
65. 03sin32cos2cossin4 xxxx . {Ans.
6
112
6
52
62
nnn }
66. xx 5tan3tan . {Ans. n }
67. xxxx 7cos9sin3cos5sin {Ans.
24124
kn }
68. xxxxx 2sin3cos5sin2cos6sin {Ans.
23
k
n }
69. 16
7cossin 66 xx {Ans.
62
n}
70. 15coscos2 2 xx {Ans.
33
2
77
2 kn }
71. 03sin2sinsin xxx {Ans.
3
22
2
k
n }
72. 03coscos3sinsin xxxx {Ans.
822
kn }
73. 1cossin xx . {Ans.
441
nn }
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.11
74. 2sin3cos xx . {Ans.
32
n }
75. 22cos2sin3 xx {Ans.
86
n }
76. xxx 5sin3cos2
33sin
2
1 {Ans.
1246
kn }
77. 0cossin33cos2 xxx {Ans.
32
n}
78. xxx 13cos25cos5sin {Ans.
729324
kn }
79. xxx 2sin22cossin2 {Ans.
2
3tan
21
kn }
80. 0cos21cossin10cossin 4322 xxxxx {Ans. 3tan7tan2
11
mkn }
81. 04sin32
sin8 2 xx
{Ans.
3
4tan 1n }
82. xxx 4coscossin 44 {Ans.
2
n}
83. 02sin4
32sinsincos 244 xxxx {Ans. }
84. x
xx
sin
5cot
2tan3 {Ans. }
85.
xxx
xxsincot2cot
11cos32cos {Ans. }
86. x
xx
2tan1
tancos
{Ans. }
87. 2cos1
sincot
x
xx {Ans.
61
nn }
88. 3cos3sin2 xx {Ans.
2
3tan222 1 kn }
89. 22cos2sin3 xx {Ans.
3
63tan 1n }
90. 14sin24cos xx {Ans.
2tan2
1
21
k
n }
91. 2tan2sin xx {Ans.
4
n }
92. 2
1
cos1
sin1
x
x {Ans. 2tan222 1 kn }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.12
93. 1cossin 33 xx {Ans.
kn 22
2
}
94. 05cos33sin4 4 xx {Ans. }
95. 06cos6sin6cossin xxxx {Ans.
kn 2
22 }
96. 02sinsincos44 xxx {Ans.
222
kn }
97. 07cossin112sin5 xxx {Ans.
10
2cos
42 1n }
98. 2
2cos
11
2sin2
4
4
xx
{Ans.
3
22
n }
99. 16
18cos4cos2coscos xxxx {Ans.
817,
2
15;
1717
2
15
2mn
lk
kn }
100. 05sin5cos317sin2 xxx {Ans.
966611
kn }
101. 2
cos8sin232
cos4 3 xx
x {Ans.
2122
kkn }
102. 2
sin4
cos4
cos4
7 3 xxx {Ans.
3
21424
kkn }
103. 44
tan2sin4 2
xx {Ans.
4
n }
104.
xxx 2
6cos52cos32sin
2 {Ans.
12
7n }
105. xx 2cos
3
4cos {Ans.
42
33
kn }
106. xxxx 2sin2coscos2sin {Ans.
22
1cos
42
21
kn }
107. 16coscos32 6 xx {Ans.
4
1cos
2
1
21
kn }
108. 34coscottan xxx {Ans.
2
15sin
2
11
241kk
n
}
109. 0cottancossin12 xxxx {Ans.
4
102cos
42
41
kn }
110. xx
xxsin
1
cos
1cossin 55 {Ans.
4
n }
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.13
111. 128
412cos2sin 88 xx {Ans.
124
n}
112. 64
29cossin 1010 xx {Ans.
84
n}
113. xxx 2cos16
29cossin 41010 {Ans.
84
n}
114. xxx sin2coscos {Ans.
4
522
kn }
115. x
xxsin
1cotcot {Ans.
3
22
n }
116. 1sin6sin25 xx {Ans.
61
nn }
117. xx cos32
1cos42 {Ans.
32
n }
118. 2
tan31tantan23 2 x
xx
{Ans.
4
n }
119. 1sin3cos5sin3 2 xxx {Ans. }
120. 11cos
1cot
9
1tan
2
xxx {Ans.
3
1tan
6
1tan 11 kn }
121. xxx
x cos2sin22
tancos1 {Ans.
22
n }
122. 22
1sin
2
1cos 22 xx {Ans.
42
n }
123. 2cot21tan21 xx {Ans.
8
3n }
124. 0cos32sinsin2sin3 22 xxxx {Ans. 3tan124
2 1
kn }
125. 0cos42sin2sincos 22 xxxx {Ans. 3tan124
52 1
kn }
126. xxxx cossin22sin12cos {Ans.
kn 24
}
127. xxx 2tan3cot32cot2 {Ans. }
128. xxx 2tan3cot5tan6 {Ans.
4
1cos
2
1
3
1cos
2
1 11 kn }
129.
2cot
2tan
cos4
4tantan
4tan
2
xxx
xxx
{Ans. }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.14
130. x
xxxx
xxxxx
5cot1
7cossin2
cos2
3sin
2
3cos
2sincos7sin5sin
22
{Ans.
842
kn }
131. 32
sincoscossinsin 4646 x
xxxx {Ans. n4 }
132. xxx 3sin2
13cos2cos1 2 {Ans. n2 }
133. xxx 2cos2sin5sin {Ans.
22
n }
134. 8sin53
sin3 22 xx
{Ans.
2
33
n }
135. 23sincos3sin xxx {Ans.
6
n }
136. 53
2cos363
2sin2
xx {Ans. }
137. 33
2cos2
4sin
xx {Ans. 224 n }
138. xxxx 2cos32sin10sin18sin 2 {Ans.
4
n }
139. 12sin4
312cos 2
xx {Ans. n }
140. xxx 4sin2cossin 4 {Ans.
42
n }
141. xx 46 sin12cos {Ans. n }
142. 32cos3
cot
x
{Ans.
6
1n }
143. xx 2sincos1cos2
sin2 22
{Ans.
2
1tan
21
kn }
144. If xx sincoscossin , prove that .22
1
4cos
x
145. If xx tancoscotsin , prove that either xec2cos or x2cot is equal to 4
1n , where n is a positive
or negative integer.
146. If sincotcostan , prove that 22
1
4cos
.
147. Find the values of 3600 satisfying 02cos ec . {Ans. 330,210 }
148. Find the solution set of 0cos231cos2 xx in the interval 20 x . {Ans.
3
5
3
}
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.15
149. Find the smallest value of satisfying the equation 4tancot3 . {Ans. 6
}
150. If k20cos and 12cos 2 kx , then find the possible values of x between 0 and 360 . {Ans. 40 and 320 }
151. If and are the solutions to the equation cba sectan , then show that 22
2tan
ca
ac
.
152. If , be unequal values of satisfying the equation 1sectan ba , find a and b in terms of
and and prove that
21
12cossincossin
a
ab
. {Ans.
sinsin
coscos
a ,
sinsin
sin
b }
153. If cba 2sin2cos has and as its solutions, then prove that ac
b
2tantan and
ac
ac
tantan .
154. If and are the solutions of cba sincos , then show that
i. 22
2coscos
ba
ac
ii. 22
22
coscosba
bc
iii. 22
2sinsin
ba
bc
iv. 22
22
sinsinba
ac
.
v. 22
22
cosba
ba
.
155. If and are the roots of the equation 0sinsin2 cba , show that
2
22 2coscos
a
acba .
156. If and are distinct roots of the equation cba sincos , between 0 and 2 , and if also satisfies the equation, show that a = c.
157. If 321 ,, are the values of which satisfy the equation tan2tan , and if no two of these
values differ by a multiple of , then show that 321 is a multiple of .
158. If ,,, are the roots of the equation
3tan34
tan
, no two which have equal tangents, show
that 0tantantantan .
159. If 1 and 2 are two distinct values of 2,0, 21 , satisfying the equation 2sin2
1sin
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.16
prove that
cot
coscos
sinsin
21
21
.
160. Prove that the equation sin1
xx is not possible for any real value of x.
161. Find the values of cos for which the equation x
x1
cos2 is possible, x being real. {Ans. 1 }
162. Equation xxxe 55sin2 has how many solutions? {Ans. No solution}
163. Prove that the equation 2
2 4sec
yx
xy
is possible for real values of x and y only if x = y.
164. For what values of c, the equation cec cossec has two real roots between 0 and 2 ? {Ans. 82 c }
165. If 5k and 3600 , then what is the number of different solutions of k sin4cos3 ? {Ans.
Two} 166. If 2cossin ecxx , then find the value of xecx nn cossin . {Ans. 2} 167. If 3sinsinsin 321 , then find 321 coscoscos . {Ans. 3}
168. Equation xx lnsin has how many solutions? {Ans. One} 169. Equation xx logsin has how many solutions? {Ans. Three} 170. Equation 1sin xx has how many solutions? {Ans. Infinite solutions} 171. Equation 3tanhsec xe x has how many solutions? {Ans. No solution}
172. Equation xx 1cotln has how many solutions? {Ans. Four}
CATEGORY-12.3. VALUES OF INVERSE TRIGONOMETRIC FUNCTIONS
173. Evaluate:-
i. 2
1sin2
2
1cos 11 . {Ans.
3
2}
ii. 1cos2
1
2
1cos1cot
2
3sin2 1111
{Ans.
6
5}
iii.
2
3sin
4
1
3
1tan5tan 11 {Ans. 1 }
iv.
2
1cos23tan3sin 11 {Ans.
2
3 }
v.
2
1cos
2
3sin3cos 11 {Ans.
2
1}
vi.
4coscos 1
{Ans. 4
}
vii.
4
5coscos 1
. {Ans. 4
3}
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.17
viii.
6
7coscos 1
. {Ans. 6
5}
ix.
4
3coscos 1
{Ans. 4
}
x. 6coscos 1 . {Ans. 62 }
xi.
3
2sinsin 1
. {Ans. 3
}
xii.
3
7sinsin 1
{Ans. 3
}
xiii. 2sinsin 1 . {Ans. 2 }
xiv. 10sinsin 1 . {Ans. 103 }
xv.
10
3tantan 1
{Ans. 10
3}
xvi.
3
2tantan 1
{Ans. 3
}
xvii. 5tantan 1 {Ans. 25 }
xviii.
7
46coscos
7
33sinsin 11
{Ans. 7
6}
xix.
8
19cotcot
8
13tantan 11
{Ans. }
xx.
7
1tan
5
3sin 11 . {Ans.
4
}
xxi.
5
33cossin 1
. {Ans. 10
}
xxii.
3
22sin
2
1sin 1 {Ans.
3
1 }
xxiii.
13
5sin
2
1tan 1 {Ans.
5
1}
xxiv.
3
5cos
2
1tan 1 . {Ans.
2
53}
xxv.
7
4cos
2
1cot 1 {Ans.
11
3}
xxvi.
3
2tan
5
4costan 11 . {Ans.
6
17}
xxvii.
17
8sin
15
8tansin 11 {Ans. 0}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.18
xxviii. 8.0sin2sin 1 . {Ans. 0.96}
xxix.
5
3cos
4
1tan2cos 11 {Ans.
85
13}
xxx.
45
1tan2tan 1
. {Ans. 17
7 }
xxxi.
3
5cos
3
5sin2sin 11 {Ans.
81
58}
174. If 5
sin 1 x , find the value of x1cos . {Ans.
10
3}
175. If 3
2sinsin 11
yx , then find the value of yx 11 coscos . {Ans. 3
}
176. If 2
3
2
x , then find the value of xsinsin 1 . {Ans. x }
177. If 2 x , then find the value of xcoscos 1 . {Ans. x2 } 178. Prove that:-
i. .85
77sin
17
8sin
5
3sin 111
ii. .85
77sin
17
15sin
5
4sin 111
iii. .325
253cos
25
7sin
13
5sin 111
iv. .11
27tan
5
3tan
5
4cos 111
v. .65
33cos
13
12cos
5
4cos 111
vi. .25
7cos
2
1
63
16cot
13
3cos2 111
vii. .453
1tan
2
1tan 11
viii. .453cot5
1sin 11
ix. 47
1tan
3
1tan2 11
.
x. .9
2tan
13
1tan
7
1tan 111
xi. .5
12tan
2
1
3
2tan 11
xii. .5
3cos
2
1
9
2tan
4
1tan 111
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.19
xiii. 140
171tan
5
1tan2
17
15cos 111
.
xiv. .48
1tan2
7
1tan
5
1tan2 111
xv. .419
8tan
5
3tan
4
3tan 111
xvi. .48
1tan
7
1tan
5
1tan
3
1tan 1111
xvii. .1985
1tan
420
1tan
4
1tan3 111
xviii. .499
1tan
70
1tan
5
1tan4 111
xix. 24
5cos
5
3sin 11
ec .
xx. .13
5sin2
119
120tan 11
xxi. .3
1tan4sin
7
1tan2cos 11
xxii.
,1
3
1,
3
11,,
31
3tan
1
2tantan
2
31
211 t
t
tt
t
tt
1,
3
1,
31
3tan
2
31 t
t
tt
3
1,1,
31
3tan
2
31 t
t
tt
179. Prove that 0,tan21
1cos 1
2
21
xxx
x
0,tan2 1 xx . CATEGORY-12.4. INVERSE TRIGONOMETRIC EQUATIONS
180. .4
33tan 21 xx {Ans. 21 }
181. .4
3cot3tan 11 xx {Ans.
3
21}
182. .02sin5sin2 121 xx {Ans. 2
1sin }
183. .cot6tan4 11 xx {Ans. 5
2tan
}
184. 01cossin2 11 xx {Ans. 0}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.20
185. xxx 111 cos1sinsin . {Ans.
2
10 }
186. xx 2cossin2 11 {Ans. 2
31}
187. .62
3cos
2sin 11
x
x {Ans. 0}
188. .tancos 11 xx {Ans. 2
15 }
189.
2
3coscossin 111 xx . {Ans.
2
3}
190. .3
1sin1sin
3
2sin 111 x
x {Ans.
3
2}
191. .411
11tan
22
221
xx
xx {Ans. 1 }
192. .4
3tan2tan 11 xx {Ans. 6
1 }
193. .42
1tan
2
1tan 11
x
x
x
x {Ans.
21 }
194. .5
3cos
5
4sin1cot1tan 1111 xx {Ans. 7
34 }
195. 2
1tan1tan 11 xx . {Ans. 0}
196. .31
8tan1tan1tan 111 xx {Ans. 4
1 }
197. .cos2tancostan2 11 ecxx {Ans. 4 n }
198. .3
2cot2tan 11 xx {Ans. 3 }
199. .2
1cotsincostan 11 x {Ans. 3
5 }
200. .152cotcot 11 xx {Ans. 323 }
201. .3
2
1
2tan
1
1cos
21
2
21
x
x
x
x {Ans. 323 }
202. .2cot10cotcot 111 xx {Ans. 73 }
203. .3
2sinsin 11 xx {Ans. 7
321 }
204. .2
12sin
5sin 11
xx {Ans. 13}
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.21
205. .4sec3sec3
sec4
sec 1111 xx
{Ans. 12}
206. 1cos5
1sinsin 11
x . {Ans.
5
1}
207. If xx 11 cossin4 , then find x . {Ans. 2
1}
208. If 4coscoscoscos 1111 uzyx , then find the value of 2002200120001999 uzyx . {Ans. 0}
209. If nxxx n
21
21
11 sinsinsin , then find the value of nxxx 221 . {Ans. 2n}
210. If 2
3sinsinsin 111
zyx , find the value of 101101101
100100100 9
zyxzyx
. {Ans. 0}
CATEGORY-12.5. TRIGONOMETRIC INEQUALITIES
211. 2
1sin x {Ans.
6
72,
62
nn }
212. 2
3cos x {Ans.
6
112,
62
nn }
213. 3
1tan x {Ans.
2,
6
nn }
214. 1cot x {Ans.
nn ,
4
3}
215. 5
1sin x {Ans.
5
1sin22,
5
1sin2 11 nn }
216. 7.0cos x {Ans. 7.0cos2,7.0cos2 11 nn }
217. 5tan x {Ans.
5tan,
21
nn }
218. 4
3cot x {Ans.
4
3cot, 1 nn }
219.
2
1cos
2
1sin
x
x {Ans.
3
52,
6
52
nn }
220.
0tan2
3sin
x
x {Ans.
kknn 2,
222,
32 }
221.
3cot
2
1cos
x
x {Ans.
4
72,2
6
52,
42
kknn }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.22
222.
3
1cot
1tan
x
x
{Ans.
3
2,
24,
kknn }
223.
5
1cos
5
1sin
x
x {Ans.
5
1sin2,
5
1cos2 11 nn }
224.
3tan5
3cos
x
x
{Ans.
3tan2,
5
3cos2
5
3cos2,
223tan2,
22 1111
mmkknn }
225.
2cot7
4sin
x
x
{Ans. 22,2cot22,7
4sin2
7
4sin2,2cot2 1111
mmkknn }
226.
3.0cot
23.0tan
x
x {Ans.
2,3.0cot 1 nn }
227.
05
3sin
0cos
x
x
{Ans.
3
2510,
3
2010510,
2
910
2
710,
3
1010
2
310,
210
llmm
kknn
}
228.
2
12cos
2
1
2sin
x
x
{Ans.
24,
3
54
12
54,4 nnnn }
229. 0cos;3
2sin xx {Ans.
2
32,
3
2sin2 1
nn }
230. 5.3tan;2
1cos xx {Ans. , }
231. 7cot;2
3sin xx {Ans. nn ,7cot 1 }
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.23
232. 2cot;3
1tan xx {Ans.
nnnn ,2cot
6, 1 }
233. 2
1cos 2 x {Ans. 1,
32,
32
32,
32
3,
3
nkknn
}
234. 0cossin . {Ans.
42,
4
32
nn }
235. 12cos2sin3 xx {Ans.
nn ,
3}
236. 23sin33cos xx {Ans.
36
19
3
2,
36
13
3
2 nn}
237. 0cos2cos xx {Ans.
32,
32
nn }
238. 02cos1
cos
x
x {Ans.
2
32,
22
nn }
239. xx 3cos3sin {Ans.
12
5
3
2,
123
2 nn}
240. 04cot3tan xx {Ans.
2,3tan
4, 1
kknn }
241. 02sin3cossin 22 xxx {Ans.
6
52,
22
22,
62
kknn }
242. 02cos2
sin2 2 xx
{Ans.
22,
32
32,
22
kknn }
243. xxx 23 tantan33tan } {Ans.
2,
34,
3
kknn }
244. 03cos3sin
3cos3sin
xx
xx {Ans.
123,
123
nn}
245. 0cos36cossin3sin5 22 xxxx {Ans.
12
5cot,
3
1cot 11 nn }
246. 0cos9cossin4sin2 22 xxxx {Ans. , }
247. 1cossin32sin3cos 22 xxxx {Ans.
nn ,3
}
248. 2cos2sinsin3 22 xxx {Ans.
3
1cot,
41
nn }
249. xx tan4cos3 2 {Ans.
3,
6
nn }
250. 12cot4cos4sin xxx {Ans.
82,
2
nn}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.24
251. 02cot2tan2 xx {Ans.
2,
8282,
42
nnnn }
252. 5tan8coscos2 xxx {Ans.
4
52,
22
22,
42
kknn }
253. x
xxcos
1cossin {Ans.
22,
2
32
4
52,2
22,
42 nnmmkk }
254. 16
7cossin 66 xx {Ans.
32,
62
nn}
255. 02cos
sincot
x
xx {Ans.
kknn 2,
322,
32 }
256. 1cos2cos 22 xx {Ans.
6
5,
6
nn }
257. 04sin32
sin8 2 xx
{Ans.
2
1cot22,2cot22 11 nn }
258. xxx 2cos2cossin {Ans.
4
72,
12
172
4
32,
122
kknn }
259. 03tan2tantan xxx {Ans.
2,
34
3,
3
2
4,
3
1cos
2
1
6,
6,
3
1cos
2
1 11
llpp
mmkknn
}
260. xxx 3cos5cos2cos {Ans.
5
82,
2
32
5
72,
5
62
5
42,
5
32
22,
5
22
52,22,
52
rrll
ppmmkknn
}
261. xxxxx 10sin3cos2cos3sin2sin {Ans.
30
7
5
2,
105
2
305
2,
105
2 kknn }
262. 03
cot22
cotcot
xxx {Ans.
9
5,
29
2,
9,
3
mmkknn }
263. 13sinsinsin2 2 xxx {Ans.
4
52,
6
52
4
32,
42
62,
42
mmkknn }
264. xxxx 4sin3sin2sinsin4 {Ans.
8
3,
88
5,
2,
8
mmkknn }
265. xx
xtan3
cos
2cos2
2
{Ans.
12,
22,
12
7
kknn }
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.25
266. 11cos2cos
3coscos2cos2
2
xx
xxx {Ans.
32,
32
nn }
CATEGORY-12.6. INVERSE TRIGONOMETRIC INEQUALITIES
267. 5sin 1 x {Ans. 1,1 }
268. 2sin 1 x {Ans. 1,1 }
269. 4
1coscos 11 x {Ans.
1,4
1}
270. .6
cos 1 x {Ans.
2
3,1 }
271. .3
tan 1 x {Ans. ,3 }
272. 2cot 1 x {Ans. 2cot, }
273. 06cot5cot 121 xx {Ans. ,2cot3cot, }
274. 03tan4tan 121 xx {Ans. 1tan, }
275. 1tanlog 12 x {Ans. }
276. 22211 tantan xx {Ans. , }
277. 01cos421 x {Ans.
2
1cos,1 }
278. xx 11 cossin {Ans.
1,
2
1}
279. xx 11 cossin {Ans.
2
1,1 }
280. 211 coscos xx {Ans. 0,1 }
281. xx 11 cottan {Ans. ,1 }
282. xx 1sinsin 11 {Ans.
2
1,0 }
283. 1sintan 12 x {Ans.
2
1,11,
2
1 }
CATEGORY-12.7. LIMIT OF INVERSE TRIGONOMETRIC FUNCTIONS
284. lim tanx
xx
x
1 1
2 4
{Ans. 1
2 }
285. lim tan tanx
xx
x
x
x
1 11
2 2 {Ans. 1
2 }
286. limsin tan
x
x x
x
0
1 1
3 {Ans. 12 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.26
287. xxxxx
11 tan2tan2lim
{Ans. }
CATEGORY-12.8. SERIES CONTAINING TRIGONOMETRIC AND INVERSE TRIGONOMETRIC
FUNCTIONS
288. Sum to n terms the series and sum of infinite series
.......1
1tan......
7
1tan
3
1tan
2
111
nn {Ans.
2tan 1
n
n,
4
}
289. Prove that tan3tan2
13secsin and hence find the sum to n terms of the series
...............3sec3sin3sec3sin3secsin 322 . {Ans. tan3tan2
1n }
290. Prove that 2cot2cottan . Hence show that the sum to n terms of the series ..........2tan22tan2tan 22 is nn 2cot2cot .
291. If naaaa ,,, 321 are in AP with common difference d , then prove that the sum of the series
nn aaaaaad secsecsecsecsecsecsin 13221 , is 1tantan aan .
292. If naaaa ,,, 321 are in AP with common difference d , then prove that the sum of the series
nnn aaaecaecaecaecaecaecd cotcotcoscoscoscoscoscossin 113221 .
293. If nn nU secsin , n
n nV seccos , n = 0, 1, 2, …….., prove that tan11 nnn UVV . Hence
deduce that 1coscosseccot....... 1121 nUUU nn
n .
CATEGORY-12.9. ADDITIONAL QUESTIONS
294. If 0,, CBACBA and the angle C is obtuse, then show that 1tantan BA .
295. If
1
1sin
1
1sec 11
x
x
x
xy then find
dx
dy. {Ans. 0}
296. For what value of a the function baxxxxf cossin decreases for all real values of x ? {Ans.
2a }
297. For what value of K the function xx
xxKxf
cossin
cos2sin
is increasing for all values of x ? {Ans. 2K }
298. Show that qp cossin attains a maxima when q
p1tan .
299. Find the values of x for which the function 3
20,cos3sin21)( 2
xxxxf has maxima or
minima. Also find the values of the function at these extremum. {Ans. minima at 2
x , ,3
2
f
maxima at 3
1sin 1x ,
3
13
3
1sin 1
f }
300. If xxx tan,cos,sin6
1 are in GP, then find the value of x . {Ans.
32
n }
301. If 0tantan qp , then show that the values of form a series in A.P.
PART-III/CHAPTER-12
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.27
302. For what value of b , will the roots of the equation 11,cos bbx when arranged in ascending order of their magnitudes, form an AP. {Ans. -1}
303. If 1 zxyzxy , then find the value of zyx 111 tantantan . {Ans. 2
}
304. Find the greatest and least values of 3131 cossin xx . {Ans. 8
7,
32
33 }
305. What is the number of all possible triplets 321 ,, aaa such that 0sin2cos 2321 xaxaa for all x ?
{Ans. infinite}
306. If zyx 111 tan4
tantan
and 1 zyx then arithmetic mean of odd powers of x, y, z is 3
1.
(True/False) {Ans. True} 307. Find the intervals of monotonicity of the function xxxxxf 0,3cos6cos10cos3 234 .
{Ans.
2,0
decreasing,
3
2,
2
increasing,
,
3
2 decreasing}
308. Consider the system of linear equations in x, y, z
sin
cos
.
3 0
2 4 3 0
2 7 7 0
x y z
x y z
x y z
Find the value of θ for which this system has non-trivial solutions. {Ans. n m m 1 6 }
309. Let λ and α be real. Find the set of all values of λ for which the system of linear equations
x y z
x y z
x y z
sin cos
cos sin
sin cos .
0
0
0
has a non-trivial solution. For λ = 1, find all values of α. {Ans. sin cos2 2 , n m 4 }
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
12.28
THIS PAGE IS INTENTIONALLY LEFT BLANK
Mathematics for IIT-JEE By Er. Sanjiva Dayal, B.Tech. (I.I.T. Kanpur)
PART-III
TRIGONOMETRY
CHAPTER-13
PROPERTIES OF TRIANGLES
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS HEAD OFFICE: A-602, TWIN TOWERS, LAKHANPUR, KANPUR-208024, INDIA.
PHONE: +91-512-2581426. MOBILE: +91-9415134052. EMAIL: [email protected]. WEBSITE: sanjivadayal-iitjee.blogspot.com.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.2
CHAPTER-13
PROPERTIES OF TRIANGLES
LIST OF THEORY SECTIONS
13.1. Identities Related To Sides And Angles Of A Triangle 13.2. Solving A Triangle 13.3. Properties Of Median, Altitude And Internal Bisector 13.4. Identities Related To Circumradius 13.5. Identities Related To In-Radius 13.6. Identities Related To Ex-Radii 13.7. Properties Of A Quadrilateral 13.8. Properties Of A Polygon
LIST OF QUESTION CATEGORIES
13.1. Proving Identities Related To Sides And Angles Of A Triangle 13.2. Solving A Triangle 13.3. Relationship Between Sides And Angles Of A Triangle 13.4. Area Of A Triangle 13.5. Ambiguous Case 13.6. Median Of A Triangle 13.7. Altitude Of A Triangle 13.8. Internal Bisector Of A Triangle 13.9. Proving Identities Related To Circumradius 13.10. Circumcircle, Circumcenter, Circumradius 13.11. Proving Identities Related To In-Radius 13.12. Incircle, Incenter, In-Radius 13.13. Proving Identities Related To Ex-Radii 13.14. Ex-Circles, Ex-Centers, Ex-Radii 13.15. System Of Circles 13.16. Maximum And Minimum Values In A Triangle 13.17. General Quadrilateral 13.18. Cyclic Quadrilateral 13.19. Quadrilateral With An Inscribed Circle 13.20. Regular Polygon 13.21. Additional Questions
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.3
CHAPTER-13
PROPERTIES OF TRIANGLES SECTION-13.1. IDENTITIES RELATED TO SIDES AND ANGLES OF A TRIANGLE
1. Convention i. Vertices of a triangle are denoted by A, B, C; internal angles of vertices A, B, C are denoted by angles A,
B, C respectively; sides opposite to angles A, B, C are denoted by a, b, c respectively. 2. Neccessary conditions
i. cba , cab , bac . ii. CBA .
3. Semi-perimeter (s)
i. 2
cbas
.
4. Area (Δ)
i. 321 2
1
2
1
2
1cpbpap .
ii. BacAbcCab sin2
1sin
2
1sin
2
1 .
iii. csbsass .
5. Sine rule
i. abcc
C
b
B
a
A
2sinsinsin.
ii. CBAcba sin:sin:sin:: . 6. Cosine rule
i. ab
cbaC
ac
bacB
bc
acbA
2cos,
2cos,
2cos
222222222
.
7. Projection rule i. AbBacAcCabBcCba coscos,coscos,coscos .
8. Half angle formulae
i. ab
bsasC
ac
csasB
bc
csbsA ))((
2sin,
))((
2sin,
))((
2sin
.
ii. ab
cssC
ac
bssB
bc
assA )(
2cos,
)(
2cos,
)(
2cos
.
iii. )(
))((
2tan,
)(
))((
2tan,
)(
))((
2tan
css
bsasC
bss
csasB
ass
csbsA
.
9. Napier’s analogy
i. 2
cot2
tan,2
cot2
tan,2
cot2
tanC
ba
baBAB
ac
acACA
cb
cbCB
.
SECTION-13.2. SOLVING A TRIANGLE
1. Three sides given: Angles obtained by cosine rule.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.4
2. Two sides, a & b, and included angle, C, given: c obtained by cosine rule, BA obtained by Napier’s analogy.
3. Two sides, a & b, and opposite angle, A, given i. A is acute angle
a. Aba sin No triangle, a
AbB
sinsin has no solution
b. Aba sin Unique Right-angled triangle, 90B , 1sin B c. baAbsin Two triangles (Ambiguous case)
I. There are two set of values of c, B, C; 1c , 1B , 1C and 2c , 2B , 2C .
II. 1B and 2B are solutions of a
AbB
sinsin and 18021 BB .
III. Also, 1c and 2c are roots of quadratic 0cos2 222 abcAbc , and so Abcc cos221
and 2221 abcc .
d. ba Unique triangle, B is acute, a
AbB
sinsin
ii. A is right angle or obtuse angle a. ba No triangle
b. ba Unique triangle, B is acute, a
AbB
sinsin
4. One side and two angles given: obtain third angle, obtain other two sides by sine rule. 5. Three angles given: Infinite similar triangles, Ratios of sides is fixed, i.e. CBAcba sin:sin:sin:: . SECTION-13.3. PROPERTIES OF MEDIAN, ALTITUDE AND INTERNAL BISECTOR
1. Property of median and centroid (G) i. Centroid (G) divides median AD in the ratio 2:1, i.e. AG:DG = 2:1.
2. Property of altitude and orthocenter (P)
i. a
BcCbp
2
sinsin1 , b
AcCap
2
sinsin2 ,
cAbBap
2sinsin3 .
ii. AaAP cot .
iii. A
CBaDP
sin
coscos .
3. Property of internal bisector and incenter (I) i. Internal bisector AD divides side BC in the ratio c:b, i.e. BD:CD = c:b.
ii. cb
acBD
and
cb
abDC
.
SECTION-13.4. IDENTITIES RELATED TO CIRCUMRADIUS
1. Circum-radius (R)
i.
4sin2sin2sin2
abc
C
c
B
b
A
aR .
2. Distance of orthocenter (P) from vertex and side
D
G
A
B C
D
I
A
B C
D
P
A
B C
D
P
A
B C
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.5
i. ARAP cos2 . ii. CBRDP coscos2 .
3. Position of circumcenter (O), centroid (G) and orthocenter (P) i. Circumcenter (O), centroid (G), and orthocenter (P) are collinear & G
divides OP in the ratio 1:2, i.e. OG:GP = 1:2. ii. In an isosceles triangle, O, G, P, I are collinear. iii. In an equilateral triangle, O, G, P, I are coincident.
SECTION-13.5. IDENTITIES RELATED TO IN-RADIUS
1. In-radius (r)
i. s
r
.
ii. 2
tan)(2
tan)(2
tan)(C
csB
bsA
asr .
iii. 2
sin2
sin2
sin4
2cos
2sin
2sin
2cos
2sin
2sin
2cos
2sin
2sin
CBAR
C
ABc
B
CAb
A
CBa
r .
SECTION-13.6. IDENTITIES RELATED TO EX-RADII
1. Ex-radii ( 321 ,, rrr )
i. 2
cos2
cos2
sin4
2cos
2cos
2cos
2tan1
CBAR
A
CBa
As
asr
.
ii. 2
cos2
sin2
cos4
2cos
2cos
2cos
2tan2
CBAR
B
ACb
Bs
bsr
.
iii. 2
sin2
cos2
cos4
2cos
2cos
2cos
2tan3
CBAR
C
BAc
Cs
csr
.
SECTION-13.7. PROPERTIES OF A QUADRILATERAL
1. Convention i. Vertices of a quadrilateral are denoted by A, B, C D in anti-clockwise manner; internal angles of vertices
A, B, C, D are denoted by angles A, B, C, D respectively; sides adjacent to vertices A, B, C, D in anti-clockwise manner are denoted by a, b, c, d respectively.
2. Properties of a general quadrilateral i. 2 DCBA .
ii. 2
dcbas
.
P O G
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.6
iii.
2cos2 DB
abcddscsbsas
2cos2 CA
abcddscsbsas .
3. Properties of a cyclic quadrilateral i. CADB ; .
ii. dscsbsas .
iii. cdab
dcbaB
2cos
2222
.
iv. Circumradius: dscsbsas
bcadbdaccdabR
4
1.
v. Ptolemy’s theorem: bdacADBCCDABBDAC . 4. Properties of a quadrilateral with an inscribed circle
i. sdbca .
ii.
2sin
2sin
CAabcd
DBabcd .
iii. s
r
.
SECTION-13.8. PROPERTIES OF A POLYGON
1. Properties of a general polygon of n sides i. Sum of external angle = 2 . ii. Sum of internal angles = )2( n .
2. Properties of a regular polygon of n sides and side length a
i. Internal angle =
n
n 2.
ii. 2
nas .
iii. n
eca
R
cos2
.
iv. n
ar
cot
2 .
v. n
Rn
nnr
n
an
2sin
2tancot
4
22
2
.
vi. s
r
.
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.7
EXERCISE-13
CATEGORY-13.1. PROVING IDENTITIES RELATED TO SIDES AND ANGLES OF A TRIANGLE
1. .2
cos2
sinA
a
cbCB
2. .sin22sin2sin 22 AbcBcCb
3. .coscos 22 cbBcCba
4. .coscoscos cbaCbaBacAcb
5. .2
sin2coscos 2 AcbCBa
6. .2
cos2coscos 2 AcbBCa
7.
.sin
sin2
22
a
cb
CB
CB
8. ,2
cot2
tanBABA
ba
ba
9. .2
cos2
sin2
sin
CBa
AcbB
Aa
10.
.0sinsin
sin
sinsin
sin
sinsin
sin 222
BA
BAc
AC
ACb
CB
CBa
11. .2
cot22
cot2
cotA
aCB
acb
12. .coscoscos2222 CabBcaAbccba
13. .tantantan 222222222 CcbaBcbaAcba
14. .2
sin2
cos 22222 Cba
Cbac
15. .0sinsinsin BAcACbCBa
16.
.sinsinsin
222222 ba
BAc
ac
ACb
cb
CBa
17. .02
sin2
sin2
sin2
sin2
sin2
sin
BAC
cACB
bCBA
a
18. .0coscoscoscoscoscos 222222222 BAcACbCBa
19. .02sin2sin2sin2
22
2
22
2
22
Cc
baB
b
acA
a
cb
20.
.cotcotcot
2cot
2cot
2cot
222
2
CBA
CBA
cba
cba
21. .3coscoscos 333 abcBAcACbCBa
22. .cos22cos2cos 222 cBAabAbBa
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.8
23. .2
cos2
cos2
cos2coscoscos 222
Cc
Bb
AaCBAcba
24. .0cotcotcot 222222 CbaBacAcb
25. .2
sin4 222 Abccba
26.
.coscos1
coscos122
22
ca
ba
BCA
CBA
27. .coscoscoscoscoscoscoscoscos CBAcBACbACBa
28. .
2tan
2tan
2tan
2tan
BA
BA
ba
c
29. .
2tan
2tan1
2tan
2tan1
BA
BA
ba
c
30. .sin2
sinsin2
A
CBa
31. 3
sinsinsin2
2cos
2cos
2cos CBA
abcCBA
s
32.
.1
2sin
1
2cos
12
22
22 a
CB
cb
CB
cb
33. .02
cot2
cot2
cot C
baB
acA
cb
34. .60cos260cos2 22 AbccCaba
35. .0sinsinsin 333 BAcACbCBa
36. .0coscoscoscoscoscos
222222
BA
ba
AC
ac
CB
cb
37.
Acb
cbabacacbcba 222
sin4
.
38. .2
cot22
tan2
tanC
cBA
cba
39. .2
cot2
sin2
sin2
cot2
cot 22 Cc
Ab
Ba
BA
40. .2
2tan
2tan1
cba
cBA
41. .22
cos2
cos2
cos2 sCBA
abc
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.9
42. .2
cos2
cos2
cos 2222 sC
abB
caA
bc
43. .02
cos2
cos2
cos 222
C
c
baB
b
acA
a
cb
44.
.12
tan2
tan2
tan
bcac
C
abcb
B
caba
A
45. .sinsinsin3cossincossincossin 333 CBABACACBCBA
46. .coscoscos21coscoscoscoscoscos 333 CBAabcBAcACbCBa CATEGORY-13.2. SOLVING A TRIANGLE
47. Solve the triangle, given
i. 3a , 2b and 2
26 c . {Ans. A = 60, B = 45, C = 75}
ii. 3b , 1c and A = 30. {Ans. a = 1, B = 120, C = 30} iii. a = 5, b = 7 and A = 60. {Ans. no triangle}
iv. a = 1, c = 2 and A = 30. {Ans. 3b , B = 60, C = 90}
v. 2a , 13 c and A = 45. {Ans. 21 b , 301B , 1051C & 62 b , 602B ,
752C }
vi. a = 3 , b = 2 and A = 60. {Ans. B = 45, C = 75, 2
13 c }
vii. a = 4, b = 5 and A = 120. {Ans. no triangle}
viii. a = 3 , b = 2 and A = 120. {Ans. B = 45, C = 15, 2
13 c }
ix. a = 2, B = 60 and C = 45. {Ans. A = 75, 13
62
b ,
13
4
c }
x. A = 45, B = 60 and C = 75. {Ans. 13:6:2:: cba }
48. In a ABC , if 45A , 6b , 2a , then find B. {Ans. 60 or 120 }
49. In triangle ABC , 30A , 8b , 6a , then find B. {Ans. 3
2sin 1 }
50. If 30A , 7a , 8b in ABC , then how many values of B are possible? {Ans. Two}
51. If the data given to construct a triangle ABC are 5a , 7b , 4
3sin A , then how many triangles can be
constructed? {Ans. none} CATEGORY-13.3. RELATIONSHIP BETWEEN SIDES AND ANGLES OF A TRIANGLE
52. In a triangle whose sides are 3, 4 and 38 meters respectively, prove that the largest angle is greater then 120.
53. If in any triangle the angle be to one another as 1:2:3, prove that the corresponding sides are as 1: 3 :2.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.10
54. In any triangle, if 6
5
2tan
A and
37
20
2tan
B, find
2tan
C and prove that in this triangle a+c = 2b. {Ans.
52 }
55. In a ABC , 13a , 14b , 15c , then find 2
sinA
. {Ans. 5
1}
56. If a = 4, b = 5 and 32
31cos BA , prove that c = 6.
57. If a, b and c be in A.P. prove that
i. 2
cot and 2
cot,2
cotCBA
are in A.P.
ii. 2
cotcos and 2
cotcos,2
cotcosC
CB
BA
A are in A.P.
iii. 2
3
2cos
2cos 22 bA
cC
a .
iv. 2
cot3
2
2tan
2tan
BCA .
v. 32
cot2
cot CA
.
58. If a2, b2 and c2 be in A.P., prove that i. cot A, cot B and cot C are in A.P.
ii. 222
2sin
3sin
ac
ca
B
B.
59. If a, b and c are in H.P., prove that 2
sin and 2
sin,2
sin 222 CBA are also in H.P.
60. The sides of a triangle are in A.P. and the greatest and least angles are and ; prove that .coscoscos1cos14
61. The sides of a triangle are in A.P. and the greatest angle exceeds the least by 90; prove that the sides are
proportional to 17 , 7 and 17 .
62. If C = 600, then prove that .311
cbacbca
63. In any triangle prove that, if be any angle, then .coscoscos CaAcb
64. If A = 45, B = 75, prove that bca 22 . 65. If kbcacbcba , then prove that 4,0k .
66. The sides of a triangle are a, b, 22 baba , prove that the greatest angle is 120.
67. In any triangle ABC, if CBA 222 sinsinsin , then show that the triangle is right angled.
68. In any ABC if c
aB cos2 , then show that the triangle is isosceles.
69. If in a ABC , BbAa sinsin , then show that the triangle is isosceles.
70. If C
BA
sin2
sincos , prove that the triangle is isosceles.
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.11
71. If C
B
BA
CA
sin
sin
cos2cos
cos2cos
, prove that the triangle is either isosceles or right angled.
72. If ca
b
bc
a
c
C
b
B
a
A
cos2coscos2, find the value of A. {Ans. 90}
73. If A, B, C are in A.P., show that 222
cos2caca
caCA
.
74. If the angles of a triangle are in the ratio 1:2:4, then prove that 222222222 acbcabcba . 75. If BbAa coscos , prove that the triangle is either isosceles or right angled.
76. If BA
BA
ba
ba
sin
sin22
22
, prove that the triangle is either isosceles or right angled.
77. If 1coscoscos 222 CBA , prove that the triangle is right angled.
78. If 3cotcotcot CBA , prove that the triangle is equilateral.
79. If 1sinsinsincoscos CBABA , show that 2:1:1:: cba . 80. If 2coscos2cos CBA , prove that the sides of the triangle are in A.P.
81. If CB
BA
C
A
sin
sin
sin
sin, prove that a2, b2, c2 are in A.P.
82. If acb 3 , prove that 22
cot2
cot CB
.
83. In a triangle ABC , 3
B and
4
C . Let D divide BC internally in the ratio 1:3, then find the
value of CAD
BAD
sin
sin. {Ans.
6
1}
84. The sides of a triangle are three consecutive natural numbers and it’s largest angle is twice the smallest one. Determine the sides of the triangle. {Ans. 4, 5, 6}
85. Find the greatest angle of the triangle whose sides are 12 xx , 12 x and 12 x . {Ans. 120}
86. If
2tantantan
BAbaBbAa , prove that A = B.
87. If 2
cos2
cosC
cabB
bac , prove that the triangle is isosceles.
88. If B = 3C, prove that c
cbC
4cos
,
c
bcC
4
3sin
and
c
cbA
22sin
.
89. If 131211
baaccb
, then prove that
25
cos
19
cos
7
cos CBA .
90. If a, b, c be 5, 4, 3 respectively and D and E are the points of trisection of side BC, then prove that
8
3tan CAE .
91. ABCD is a trapezium such that AB is parallel to CD and CB is perpendicular to them. If ADB , BC =
p and CD = q, show that
sincos
sin22
qp
qpAB
.
92. If the tangents of the angles of a triangle are in A.P., prove that the squares of the sides are in the ratio
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.12
22222 19:3:9 xxxx , where x is the tangent of the least or greatest angle. 93. If p and q be perpendiculars from the angular points A and B on any line passing through the vertex C of
the triangle ABC, then prove that .sincos2 2222222 CbaCabpqqbpa 94. In the triangle ABC, lines OA, OB and OC are drawn so that the angles OAB, OBC and OCA are each equal
to ; prove that cot = cot A + cot B + cot C and cosec2 = cosec2 A + cosec2 B + cosec2 C. 95. In any triangle ABC if D be any point of the base BC, such that BD:DC = m:n, and if BAD = , DAC
= , and CDA = and AD = x, prove that CmBnnmnm cotcotcotcotcot and
.22222 mnancmbnmxnm 96. Two straight roads intersect at an angle of 60 . A bus on one road is 2 km. away from the intersection and
a car on the other is 3 km. away from the intersection. Find the direct distance between the two vehicles.
{Ans. 7 km} 97. A ring, 10 cm. in diameter, is suspended from a point 12 cm. above its center by 6 equal strings attached to
its circumference at equal intervals. Find the cosine of the angle between consecutive strings. {Ans. 338313 }
98. The side of a base of a square pyramid is a meters and it’s vertex is at a height of h meters above the center of the base. If & be respectively the inclinations of any face to the base and of any two faces to one
another, prove that a
h2tan and
2
2
21
2cot
h
a
.
CATEGORY-13.4. AREA OF A TRIANGLE
99. Find the area of the triangle having sides 13, 14 and 15 cm. {Ans. 84 sq. cm.}
100. If the angles of a triangle are 30 and 45 and the included side is 13 cm., then prove that the area of
the triangle is 132
1 sq. cm.
101. If B = 45, 132 a units and 326 sq. units. Determine the side b. {Ans. 4}
102. If one angle of a triangle be 60, the area 310 sq. cm. and the perimeter 20 cm., find the length of the sides. {Ans. 5 cm., 7 cm., 8 cm.}
103. If a = 6, b = 3 and 5
4cos BA , then find it’s area. {Ans. 9 sq. units}
104. If 22 cba , find 2
tanA
. {Ans. 4
1}
105. If C = 60, A = 75 and D is a point on AC such that the area of the BAD is 3 times the area of the BCD, find the ABD . {Ans. 30}
106. If the sides of a triangle are in A.P. and it’s area is 5
3th of an equilateral triangle of same perimeter. Prove
that it’s sides are in the ratio 3:5:7. 107. If 1p , 2p , 3p are altitudes of a triangle ABC from the vertices CBA ,, and the area of the triangle,
then prove that 2
2222
32
22
1 4
cbappp .
108. If 222222 111 nm
c
nm
b
nm
a
, then prove that
n
mA
2tan , mn
B
2tan and
22 nm
mnbc
.
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.13
109. The sides a, b, c of a triangle are the roots of the equation 023 rqxpxx . Prove that
rppqp 844
1 3 .
110. Through the angular points of a triangle are drawn straight lines which make the same angle with the opposite sides of the triangle. Prove that the area of the triangle formed by them is to the area of the original triangle as 1:cos4 2 .
111. In the sides BC, CA and AB are taken three points A’, B’, C’ such that BA’:A’C = CB’:B’A = AC’:C’B = m:n. Prove that if AA’, BB’ and CC’ be joined, they will form by their intersections a triangle whose area is
to that of the triangle ABC as 222 : nmnmnm . CATEGORY-13.5. AMBIGUOUS CASE
112. In a triangle ABC , 60,3,4 Aba . Then show that c is the root of the equation 0732 cc .
113. In the ambiguous case, given a, b, A and 1c , 2c are the two value of side c, then show that
i. Abcc cos221 .
ii. 2221 abcc .
iii. Abacc 22221 sin2~ .
iv. AacAccc 222221
21 cos42cos2 .
v. 22221
221 4tan aAcccc .
vi.
2
2
2
2
1
2
1
2
cos1cos1cos1cos1 C
ab
C
ba
C
ab
C
ba
.
vii. 22
21
2121
2cos
cc
ccCBB
if A = 45.
114. In the ambiguous case, given a, c, A and 12 2bb , where 1b , 2b are the two value of side b, then prove that
Aca 2sin813 . 115. In the ambiguous case, given a, b, A, if the remaining angles of the triangles formed be 1B , 1C and 2B ,
2C , then prove that AB
C
B
Ccos2
sin
sin
sin
sin
2
2
1
1 .
116. In the ambiguous case, given a, b, A, prove that the sum of the areas of the two triangles formed is
Ab 2sin2
1 2 .
117. If ab 32 and 5
3tan 2 A , prove that there are two values of c, one of which is double the other.
118. If amb 12 and
m
mmA
31
2
1cos
, where 31 m , then prove that there are two values of
the third side, one of which is m times the other. CATEGORY-13.6. MEDIAN OF A TRIANGLE
119. Determine the lengths of medians in terms of the sides. {Ans. 2
22 222 acbAD
}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.14
120. If D is the mid-point of BC, then prove that 222 22
sinsin
acb
CaCAD
,
222 22
sinsin
acb
BaBAD
,
222 22
sin2sin
acb
CbADB
and
4cot
22 cbADB .
121. In a triangle ABC, the median to the side BC is of length 3611
1
and it divides angle A into angles of
30 and 45. Prove that side BC is of length 2 units. 122. In an isosceles right-angled triangle a straight line is drawn from the middle point of one of the equal sides
to the opposite angle. Show that it divides the angle into parts whose cotangents are 2 and 3. 123. If in a triangle the median through A is perpendicular to the side AB, prove that 0tan2tan BA .
124. If D is the mid-point of BC and AD is perpendicular to AC, then prove that
ac
acCA
3
2coscos
22 .
125. Prove that the median through A divides it into angles whose cotangents are CA cotcot2 and
BA cotcot2 and makes with the base an angle whose cotangent is BC cot~cot2
1.
CATEGORY-13.7. ALTITUDE OF A TRIANGLE
126. The sides of a right angled triangle are 21 and 28 cm.; find the length of the perpendicular drawn to the hypotenuse from the right angle. {Ans. 16.8 cm.}
127. The perpendicular AD to the base of a triangle ABC divides it into segments such that BD, CD and AD are in the ratio of 2, 3 and 6; prove that the vertical angle of the triangle is 45.
128. If sinA, sinB, sinC are in A.P., then prove that the altitudes are in H.P.
129. If AD is the altitude from A, b>c, C = 23 and 22 cb
abcAD
, find B. {Ans. 113}
130. The base angles of a triangle are 2
122 and
2
1112 . Show that the base is equal to twice the height.
131. Prove that the perpendicular from A divides BC into portions which are proportional to the cotangents of the adjacent angles and that it divides the angle A into portions whose cosines are inversely proportional to the adjacent sides.
132. Prove that the distance between the middle point of BC and the foot of the perpendicular from A is
a
cb
2
~ 22
.
133. If p, q, r are the altitudes of a triangle ABC, prove that
CBA
rqp
cotcotcot111222
.
134. If 1p , 2p , 3p are altitudes of a triangle ABC from the vertices CBA ,, and the area of the triangle,
then prove that
csppp 1
31
21
1 .
135. If p, q, r are the altitudes of a triangle from the vertices A, B, C respectively, prove that
2cos
111 2 C
s
ab
rqp .
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.15
136. If x, y, z are respectively the distance of the vertices from orthocentre, prove that xyz
abc
z
c
y
b
x
a .
137. In a triangle of base a, the ratio of the other sides is r (r < 1). Show that the altitude of the triangle is less
than or equal to 21 r
ar
.
CATEGORY-13.8. INTERNAL BISECTOR OF A TRIANGLE
138. Prove that the length of internal bisector AD is 2
cos2 A
cb
bc
.
139. If p, q, r are the lengths of internal bisectors of the angles A, B, C respectively, then prove that
cba
C
r
B
q
A
p
111
2cos
1
2cos
1
2cos
1 .
140. In a right angled triangle ABC, the bisector of the right angle C divides AB into segments p and q and if
tBA
2tan , then show that ttqp 1:1: .
141. If the bisectors of the angles of a triangle ABC meet the opposite sides in A’, B’ and C’, prove that the ratio
of the areas of the triangles A’B’C’ and ABC is 2
cos2
cos2
cos:2
sin2
sin2
sin2ACCBBACBA
.
CATEGORY-13.9. PROVING IDENTITIES RELATED TO CIRCUMRADIUS
142. CBAR sinsinsin2 2 . 143. CcBbAaCBAR coscoscossinsinsin4 .
144. R
sCBA sinsinsin
CATEGORY-13.10. CIRCUMCIRCLE, CIRCUMCENTER, CIRCUMRADIUS
145. The sides of a triangle are 18, 24 and 30 cm. Find R. {Ans. 15 cm.}
146. Find the circumradius of the equilateral triangle of side 32 cm. {Ans. 2 cm} 147. In the ambiguous case of the triangle, prove that the circumradius of the two triangles are equal. 148. Prove that
i. 22222 9 cbaROP .
ii. 22222
9
1cbaROG .
149. If x, y, z are respectively the perpendiculars from the vertices A, B, C to the opposite sides, prove that
i. 3
222
8R
cbaxyz
ii. Rz
C
y
B
x
A 1coscoscos
iii. R
cba
b
az
a
cy
c
bx
2
222
150. If x, y, z are respectively the perpendiculars from the circumcentre to the sides, prove that
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.16
xyz
abc
z
c
y
b
x
a
4 .
151. If 22228 cbaR , prove that the triangle is right angled. 152. Prove that the radii of the circles circumscribing the triangles BPC, CPA, APB and ABC are all equal. 153. D, E and F are the middle points of the sides of the triangle ABC; prove that the centroid of the triangle
DEF is the same as that of ABC and that it’s orthocentre is the circumcentre of ABC. 154. If 1R , 2R and 3R are respectively the radii of the circumcircles of the triangle OBC, OCA and OAB, prove
that 3
321 R
abc
R
c
R
b
R
a .
155. At the points A, B, C, tangents are drawn to the circumcircle. These tangents enclose a triangle PQR. Prove
that its angles and sides are respectively A2180 , B2180 , C2180 and CB
a
coscos2,
AC
b
coscos2,
BA
c
coscos2.
156. The legs of a tripod are each 10 cm. in length and their points of contact with a horizontal table on which the tripod stands form a triangle whose sides are 7, 8 and 9 cm. in length. Find the inclination of the legs to the horizontal and the height of the apex. {Ans. 62, 8.83 cm.}
CATEGORY-13.11. PROVING IDENTITIES RELATED TO IN-RADIUS
157. 2
cos2
cos2
cos4CBA
Rr .
158. Rrabcabc 2
1111 .
159. R
rCBA 1coscoscos
160. rRCcBbAa 2cotcotcot
161. rRC
baB
acA
cb 42
tan2
tan2
tan
162. R
rCBA
22
2cos
2cos
2cos 222
CATEGORY-13.12. INCIRCLE, INCENTER, IN-RADIUS
163. Find the in-radius of the triangle having sides 13, 14, 15. {Ans. 4} 164. The sides of a triangle are 18, 24 and 30 cm. Find r. {Ans. 6 cm.}
165. Given 6:5:4:: cba . Find the ratio of the radius of the circumcircle to that of the incircle. {Ans. 7
16}
166. If C = 90, prove that barR 2
1 and
a
bc
b
ac
r
c
.
167. Prove that
i. 2
cosA
ecrAI .
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.17
ii. 242
tan2
tan2
tan RrCBA
abcICIBIA .
iii. CBARIO cos2cos2cos2322 .
iv. CBARrIP coscoscos42 222 .
v. Area of 2
sin2
sin2
sin2 2 BAACCBRIOP
.
vi. Area of 2
sin2
sin2
sin3
4 2 BAACCBRIPG
.
168. DEF is the triangle formed by joining the points of contact of the incircle with the sides of the triangle ABC ; prove that:-
i. it’s sides are 2
cos2,2
cos2,2
cos2C
rB
rA
r
ii. it’s angles are 22
,22
,22
CBA
iii. it’s area is abcs
32, i.e.
R
r
2
.
169. AD, BE and CF are the perpendiculars from the angular points of a triangle ABC upon the opposite sides. Prove that the diameters of the circumcircles of the triangles AEF, BDF and CDE are respectively Aa cot ,
Bbcot and Cc cot , and that the perimeters of the triangles DEF and ABC are in the ratio Rr : . 170. If x, y, z are respectively the perpendiculars from the vertices A, B, C to the opposite sides, prove that
rzyx
1111 .
171. If x, y, z are respectively the distance of the vertices from orthocentre, prove that rRzyx 2 . 172. If x, y, z are the distances of the vertices of a triangle from the corresponding points of contact with the
incircle, prove that
2rzyx
xyz
.
173. In a ABC , the line joining the circumcentre to the incentre is parallel to BC , then find the value of CB coscos . {Ans. 1}
174. Let ABC be a triangle and let 1BB , 1CC be respectively the bisectors of B , C with 1B on AC and 1C
on AB. Let E, F be the feet of perpendiculars drawn from A on 1BB , 1CC respectively. Suppose D is the point at which the incircle of ABC touches AB. Prove that AD = EF.
175. Let ABC be a triangle having O and I as its circumcentre and incentre respectively. If R and r are the
circumradius and the inradius respectively, then prove that RrRIO 222 . Further show that the
triangle BIO is right-angled triangle if and only if b is the arithmetic mean of a and c. 176. If 1R , 2R and 3R are respectively the radii of the circumcircles of the triangle IBC, ICA and IAB, prove
that rRRRR 2321 2 .
177. If x, y, z be the lengths of the tangents drawn to the incircle parallel to the sides BC, AC, AB respectively
and intercepted between the other two sides, then prove that 1c
z
b
y
a
x.
178. Prove that the area of the incircle is to the area of the triangle itself is 2
cot2
cot2
cot:CBA
.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.18
179. C is right-angled and a perpendicular CD is drawn to AB. The radii of the circles inscribed into the triangles ACD and BCD are equal to x and y respectively. Find the radius of the circle inscribed into the
triangle ABC. {Ans. 22 yx }
180. If a circle be drawn touching the incircle and circumcircle of a triangle and the side BC externally, prove
that it’s radius is 2
tan 2 A
a
.
CATEGORY-13.13. PROVING IDENTITIES RELATED TO EX-RADII
181. 2
tan 2
32
1 A
rr
rr .
182. 2321 rrrr .
183. 2
cot2
cot2
cot 2223321
CBArrrr .
184. 2
cot1
Arr .
185. 2211332 srrrrrr .
186. rrrr
1111
321
.
187. )()()( 213132321 rrrrcrrrrbrrrra .
188. cC
rrC
rr 2
cot)(2
tan)( 321 .
189. 2
222
23
22
21
2
1111
cba
rrrr.
190. Rrrrr 4321 .
191. 2321 4))()(( Rrrrrrrr .
192. Rrab
r
ca
r
bc
r
2
11321 .
193. 222223
22
21
2 16 cbaRrrrr .
194.
3
21
r
c
b
rr
a
rr
195.
c
rrr
b
rrr
a
rrr 213132321
196.
rr
rrca
r
rrbc
r
rrab
2
13
1
32
3
21
197. CRrrrr cos4321
198. 2321 ))(( arrrr
199. 22
321
16111111
cbar
R
rrrrrr
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.19
200. 222
3
133221
64111111
cba
R
rrrrrr
201. 2133221 4Rsrrrrrr
202.
133221
133221
4 rrrrrr
rrrrrrR
203. rbsas
r
ascs
r
csbs
r 3321
204. 02
tan2
tan2
tan 321
BA
rrAC
rrCB
rr
205.
0321
r
ba
r
ac
r
cb
206.
C
rr
B
rr
A
rr
cos1cos1cos1211332
207.
32
321 c
a
a
c
b
c
c
b
a
b
b
aR
r
ab
r
ca
r
bc
CATEGORY-13.14. EX-CIRCLES, EX-CENTERS, EX-RADII
208. The sides of a triangle are 18, 24 and 30 cm. Find 1r , 2r and 3r . {Ans. 12, 18 and 36 cm.}
209. If a = 13, b = 4 and 13
5cos C , find R, r, 1r , 2r and 3r . {Ans.
8
18 ,
2
11 , 8, 2 and 24}
210. If 321 32 rrr then prove that 4:5: ba .
211. If 1r , 2r , 3r are in H.P., area is 24 sq. cm. and perimeter is 24 cm., find a, b, c. {Ans. 6 cm., 8 cm., 10 cm.}
212. Prove that
i. 2
cos11
AecrAI
ii. 2
sec1
AaII
iii. 2
cos42
cos32
AR
AecaII
iv. rRIIIIII 2321 16
v. 322
32 4 rrRII
vi. 222213
ACBIII
vii. 221
23
213
22
232
21 IIIIIIIIIIII
viii. Area of Rsr
abcCBARIII 2
22cos
2cos
2cos8 2
321
ix. C
IIII
B
IIII
A
IIII
sinsinsin213132321
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.20
213. Prove that the cubic equation whose roots are 1r , 2r and 3r is 04 2223 rsxsxRrx .
214. If u, x, y, z are respectively the areas of the incircle and ex-circles, prove that zyxu
1111 .
215. If rrrr 321 , prove that the triangle is right angled.
216. If 2113
1
2
1
r
r
r
r, prove that the triangle is right angled.
217. If ascbcsba , prove that 1r , 2r , 3r are in A.P.
218. If 1r , 2r , 3r are in H.P., show that a, b, c are in A.P.
219. If x, y, z be the lengths of the tangents from the centers of ex-circles to the circumcircle, prove that
abc
s
zyx
2111222 .
220. If 0 be the area of the triangle formed by joining the points of contact of the incircle with the sides of the
given triangle and 1 , 2 and 3 the corresponding areas for the ex-circles, prove that
20321 .
221. The triangle DEF circumscribes the three ex-circles of the triangle ABC, prove that
Cc
DE
Bb
FD
Aa
EF
coscoscos .
CATEGORY-13.15. SYSTEM OF CIRCLES
222. Two circles, of radii a and b, cut each other at an angle . Prove that the length of the common chord is
cos2
sin222 abba
ab
.
223. Three equal circles touch one another. Find the radius of the circle which touches all three. {Ans.
r
1
3
2}
224. Three circles, whose radii are a, b and c, touch one another externally and the tangents at their points of contact meet in a point. Prove that the distance of this point from either of their points of contact is
2
1
cba
abc.
225. Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from any point of contact is 4. Find the ratio of the product of the radii to the sum of the radii of circles. {Ans. 16:1}
CATEGORY-13.16. MAXIMUM AND MINIMUM VALUES IN A TRIANGLE
226. In an acute angled triangle, prove that 33tantantan CBA . If 33tantantan CBA , prove that the triangle is equilateral.
227. Prove that 332
cot2
cot2
cot CBA
.
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.21
228. Prove that 2
3coscoscos1 CBA . In case of equality, triangle will be equilateral.
229. Prove that 8
1
2sin
2sin
2sin
CBA.
230. Prove that 4
2s .
CATEGORY-13.17. GENERAL QUADRILATERAL
231. The sides of a quadrilateral are respectively 3, 4, 5 and 6 cm., and the sum of a pair of opposite angles is
120. Find the area. {Ans. 303 sq. cm.} 232. Prove that the area of any quadrilateral is one-half the product of the two diagonals and the sine of the
angle between them. 233. a, b, c and d are the sides of a quadrilateral taken in order and is the angle between the diagonals
opposite to b or d, prove that the area of the quadrilateral is tan4
1 2222 dcba .
234. If a, b, c and d be the sides and x and y the diagonals of a quadrilateral, prove that it’s area is
222222244
1cadbyx .
235. The sides of a quadrilateral are divided in order in the ratio m:n and a new quadrilateral is formed by joining the points of division. Prove that it’s area is to the area of the original figure as 22 nm to
2nm . 236. If a, b, c and d be the sides of a quadrilateral, taken in order, prove that
cos2cos2cos22222 caCbcBabcbad , where denotes the angle between the sides a and c. CATEGORY-13.18. CYCLIC QUADRILATERAL
237. Sides of a cyclic quadrilateral ABCD are a = 3 cm., b = 5 cm., c = 7 cm. and d = 9 cm. Find , R, A and D.
{Ans. 1053 sq. cm., 210
4433 cm.,
31
4cos 1 ,
13
8cos 1 }
238. In a cyclic quadrilateral, prove that the angle between its diagonals is
bdac
dscsbsas2sin 1 .
239. If ABCD be a cyclic quadrilateral, prove that dscs
bsasB
2tan .
240. If ABCD be a cyclic quadrilateral, prove that the product of the segments into which one diagonal is
divided by the other diagonal is
bcadcdab
bdacabcd
.
CATEGORY-13.19. QUADRILATERAL WITH AN INSCRIBED CIRCLE
241. The sides of a quadrilateral with an inscribed circle are 7, 10, 5 and 2 cm. and the sum of a pair of opposite
angles is 120. Find area and radius of inscribed circle. {Ans. 215 sq. cm., 12
215 cm.}
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.22
242. A quadrilateral ABCD is circumscribed about a circle, prove that 2
sin2
sin2
sin2
sinDC
cBA
a .
243. The sides of a cyclic quadrilateral are 3, 3, 4 and 4 cm.. Find the radii of the incircle and circumcircle and
the area and the angles. {Ans. 7
12 cm., 2.5 cm., 12 sq. cm., 90, 90,
25
7cos 1 ,
25
7cos180 1 }
244. If a cyclic quadrilateral can be circumscribed about another circle, prove that it’s area is abcd and that
the radius of the inscribed circle is dcba
abcd
2.
245. If a cyclic quadrilateral can be circumscribed about a circle, prove that the angle between its diagonals is
bdac
bdac1cos .
246. Let ABC be a triangle with incentre I and inradius r. Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA and AB respectively. If 1r , 2r and 3r are the radii of circles inscribed in the quadrilateral
AFIE, BDIF and CEID respectively, prove that 321
321
3
3
2
2
1
1
rrrrrr
rrr
rr
r
rr
r
rr
r
.
CATEGORY-13.20. REGULAR POLYGON
247. Find the length of the perimeter of a regular decagon which surrounds a circle of radius 12 cm. {Ans. 77.98 cm.}
248. Find the length of the side of a regular polygon of 12 sides which is circumscribed to a circle of unit radius. {Ans. 0.536 units}
249. Find the area of a pentagon, a hexagon, an octagon and a decagon, each being a regular figure of side 1 meter. {Ans. 1.72 sq. m., 2.6 sq. m., 4.8 sq. m., 7.7 sq. m.}
250. Find the difference between the areas of a regular octagon and a regular hexagon if the perimeter of each is 24 cm. {Ans. 1.887 sq. cm.}
251. A square, whose side is 2 cm., has it’s corners cut away so as to form a regular octagon, find it’s area. {Ans. 3.3 sq. cm.}
252. Compare the areas and perimeters of octagons which are respectively inscribed in and circumscribed to a
given circle. {Ans. 4:22 , 2:22 }
253. Show that the areas of the hexagon and octagon inscribed to a given circle are as 32:27 . 254. If an equilateral triangle and a regular hexagon have the same perimeter, prove that their areas are as 2:3.
255. If a regular pentagon and a regular decagon have the same perimeter, prove that their areas are as 5:2 . 256. Given that the area of a polygon of n sides circumscribed about a circle is to the area of the circumscribed
polygon of 2n sides as 3:2, find n. {Ans. 3} 257. The area of a regular polygon of n sides inscribed in a circle is to that of the same number of sides
circumscribing the same circle as 3:4. Find the value of n. {Ans. 6} 258. The interior angles of a polygon are in A.P., the least angle is 120 and the common difference is 5. Find
the number of sides. {Ans. 9} 259. There are two regular polygons, the number of sides in one being double the number in the other, and an
angle of one polygon is to an angle of the other as 9:8, find the number of sides of each polygon. {Ans. 20 & 10}
260. Show that there are eleven pairs of regular polygons such that the number of degrees in the angle of one is
PART-III/CHAPTER-13
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.23
to the number in the angle of the other as 10:9. Find the number of sides in each. {Ans. 6 & 5, 12 & 8, 18 & 10, 22 & 11, 27 & 12, 42 & 14, 54 & 15, 72 & 16, 102 & 17, 162 & 18, 342 & 19}
261. Prove that the area of the circle and the area of a regular polygon of n sides and of perimeter equal to that
of the circle are in the ratio of nn
:tan .
262. Prove that the sum of the radii of the circles, which are respectively in and circumscribed about a regular
polygon of n sides, is n
a
2cot
2
, where a is a side of the polygon.
263. Of two regular polygons of n sides, one circumscribed and the other is inscribed in a given circle, prove that the perimeters of the circumscribing polygon, the circle and the inscribed polygon are in the ratio
1:cos:secn
ecnn
and that the areas of the polygons are in the ratio 1:cos2
n
.
264. Prove that the area of a regular polygon of 2n sides inscribed in a circle is a mean proportional between the areas of the regular inscribed and circumscribed polygons of n sides.
265. A pyramid stands on a regular hexagon as base. The perpendicular from the vertex of the pyramid on the base passes through the center of the hexagon and it’s length is equal to that of a side of the base. Find the tangent of the angle between the base and any face of the pyramid and also of half the angle between any
two side faces. {Ans. 3
2,
6
1}
266. A regular pyramid has for it’s base a polygon of n sides and each slant face consists of an isosceles triangle of vertical angle 2. If the slant faces are each inclined at an angle to the base and at an angle 2 to one
another, show that n
cottancos and
n
cosseccos .
CATEGORY-13.21. ADDITIONAL QUESTIONS
267. If A, B, C are the angles of a triangle, then show that the system of equations
x y C z B
x C y z A
x B y A z
cos cos
cos cos
cos cos .
0
0
0
has non-zero solution. 268. If A, B, C are the angles of a triangle show that system of equations
x A y C z B
x C y B z A
x B y A z C
sin sin sin
sin sin sin
sin sin sin .
2 0
2 0
2 0
possesses non-trivial solution.
MATHEMATICS FOR IIT-JEE/VOLUME-I
SANJIVA DAYAL CLASSES FOR IIT-JEE MATHEMATICS. Head office: A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA. Phone: +91-512-2581426. Mobile: +91-9415134052. Email: [email protected]. Website: sanjivadayal-iitjee.blogspot.com.
13.24
THIS PAGE IS INTENTIONALLY LEFT BLANK