Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry Sanjiva Dayal IIT Kanpur

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


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


1.3

CHAPTER-1


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



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


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



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


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 .



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


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



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


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



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


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



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


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



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


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



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


axxf )( , 10 a , q


axxf )( , 10 a , q


axxf )( , 0a , q


axxf )( , 0a , q


0 x ab

b

PART-I/CHAPTER-1


1.19

axxf )( , 0a , q


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 .



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


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



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


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 .



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


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}



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


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 }



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


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}



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


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}



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


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



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


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.



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


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 }


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



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


PART-I


CHAPTER-2

LIMIT





2.2

CHAPTER-2

LIMIT


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


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


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



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


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 ....................



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


2.7

2. Finding values of the parameters given the value of limit 3. Finding the limit as function of parameter



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


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}



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}


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


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}



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


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}



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


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}



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


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}



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


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}



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


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}



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


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 }



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


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



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


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 }



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


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 }



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


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 }



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


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}



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


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}



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 }


PART-I


CHAPTER-3

CONTINUITY





3.2

CHAPTER-3

CONTINUITY


3.1. Continuity At A Point 3.2. Continuity In A Set/ Domain


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


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 ,

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 ,

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



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


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.



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


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



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


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}



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


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



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}


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


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.



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 .


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}


PART-I


CHAPTER-4

DERIVATIVES





4.2

CHAPTER-4

DERIVATIVES


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


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


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.



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


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 ;



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


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



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}


0,0 x Find ).0(f {Ans. 0 }


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


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



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}


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


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 }



4.12

xxiii. xxxf1

. {Ans.

2

1

ln1

x

xx x }

39. If 20&, 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


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

}



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


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}



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


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( }



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


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

}



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


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}


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}


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}



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


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 }



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


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

}



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


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

}



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


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 }




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


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 }


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}



4.32

THIS PAGE IS INTENTIONALLY LEFT BLANK


PART-I


CHAPTER-5

APPLICATIONS OF DERIVATIVES





5.2

CHAPTER-5



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


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


5.3

CHAPTER-5


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


iii. A function xf is said to be non-decreasing in an interval ba, if


iv. A function xf is said to be non-increasing in an interval ba, if


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



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


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



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


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:-



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


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}



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


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 }



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


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:-



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


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 .



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 .


PART-I/CHAPTER-5


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.



5.18



PART-I


CHAPTER-6

INVESTIGATION OF FUNCTIONS AND THEIR GRAPHS





6.2

CHAPTER-6



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)


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


6.3

CHAPTER-6


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)(



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


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



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


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



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


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 .



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


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 }



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


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}



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


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



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


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 }



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


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}



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


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



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


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 }



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


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 .



6.26



PART-II

ALGEBRA

CHAPTER-7

EQUATIONS AND INEQUALITIES





7.2

CHAPTER-7



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


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


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



7.4

CHAPTER-7


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


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.



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


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.



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


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



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


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.



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


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:



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


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.



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


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



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


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



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


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



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


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.



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


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 }



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


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 }



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


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 }



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


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 }



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


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, }



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


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 }



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


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 .



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


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 }



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


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}



7.42



PART-II

ALGEBRA

CHAPTER-8

QUADRATIC EXPRESSIONS





8.2

CHAPTER-8



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


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


8.3

CHAPTER-8


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 ;



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


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



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


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.



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


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 }



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


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}



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


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 }



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 }


12

xx

xxf . {Ans. 1,3

1 }


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


8.15


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.



8.16



PART-II

ALGEBRA

CHAPTER-9

PROGRESSIONS





9.2

CHAPTER-9

PROGRESSIONS


9.1. Sequence And Series 9.2. Arithmetic Progressions 9.3. Geometric Progressions 9.4. Harmonic Progressions 9.5. Sum Of Standard Series


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


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 .



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


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 .



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


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.



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


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}



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


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 .



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


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



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


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}


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



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


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

.



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


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.



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


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



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


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}


1

75

1

53

1

{Ans.

323 n

n,

6

1}


1

21

11

{Ans.

1

2

n

n, 2}



9.24


1

753

1

531

1

{Ans.

32123

2

nn

nn,

12

1}


7

32

5

21

3222222

{Ans.

21

2

n

nn, 1}


3

531

2

31

1

{Ans.

12............531

11

2

1

n,

2

1}


1

23

1

12

1

{Ans.

11n , diverging}


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 }


4

!3

3

!2

21

222

. {Ans. e2 }

PART-II/CHAPTER-9


9.25


4

!3

3

!2

21

333

. {Ans. e5 }


8

!7

6

!5

4

!3

2. {Ans.

e

1}


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 }


!4

4321

!3

321

!2

211 . {Ans.

2

3e}


!4

321

!3

21

!2

1. {Ans.

2

e}


!3

43

!2

32

!1

21 222

. {Ans. e7 }


10

!3

6

!2

31 . {Ans.

2

3e}


78

!4

50

!3

28

!2

12. {Ans. e5 }


753 37

1

35

1

33

1

3

1. {Ans. 2ln

2

1}


642 27

1

25

1

23

11 . {Ans. 3ln }


65

1

43

1

21

1. {Ans. 2ln }


76

1

54

1

32

1. {Ans. 2ln1 }


94

1

73

1

52

1

31

1. {Ans. 2ln2 }


765

9

543

7

321

5. {Ans. 12ln3 }


765

1

543

1

321

1. {Ans.

2

12ln }


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 }



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 }


PART-II

ALGEBRA

CHAPTER-10

DETERMINANTS AND MATRICES





10.2

CHAPTER-10



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


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


10.3

CHAPTER-10


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



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


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.



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


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



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


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 .



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


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



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


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}



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


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



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


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



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


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



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


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}



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


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.



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


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



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


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



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


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 }



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


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.



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


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 .



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


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}



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 .


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

}


211

103

412


313

1485

161

19

1}

PART-II/CHAPTER-10


10.37


111

232

422


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.



10.38

256. Show that

67

56 is an involutary matrix.

257. Show that

1

010

001

ba

A is an involutary matrix.


121

033

085

A is involutary.

259. Determine the condition that the matrix

ac

baA is involutary. {Ans. 12 bca }


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.


aba

babA

2

2

is nilpotent of index 2.


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}


543

654

321

A . {Ans. 2}

PART-II/CHAPTER-10


10.39


1234

4321A . {Ans. 2}


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



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


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 }


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.



10.42



PART-III

TRIGONOMETRY

CHAPTER-11

TRIGONOMETRIC IDENTITIES AND EXPRESSIONS





11.2

CHAPTER-11



11.1. Angles And Their Measurements 11.2. Standard Trigonometric Identities 11.3. Trigonometric Values As Roots Of Polynomials 11.4. Elimination


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


11.3

CHAPTER-11


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

.



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


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(

.



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


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.



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


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



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


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



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


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



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


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



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


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



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


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



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

.


3sin

7

2sin

7sin 222

. {Ans. 4

7}

231. 8

7

7

3sin

7

2sin

7sin

.


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


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


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 .



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


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 }



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

.



PART-III

TRIGONOMETRY

CHAPTER-12

TRIGONOMETRIC AND INVERSE TRIGONOMETRIC

FUNCTIONS, EQUATIONS AND INEQUALITIES





12.2

CHAPTER-12


FUNCTIONS, EQUATIONS AND INEQUALITIES


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


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


12.3

CHAPTER-12


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



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


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



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


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}



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


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 }



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


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 }


2

3tan222 1 kn }


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 }



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


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. }



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


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



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


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}



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


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}



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


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 }



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


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 }


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}



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


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 }



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 .


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


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 }



12.28



PART-III

TRIGONOMETRY

CHAPTER-13

PROPERTIES OF TRIANGLES





13.2

CHAPTER-13

PROPERTIES OF TRIANGLES


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


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


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.



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


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



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


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



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


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.



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


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



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


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

}



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


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



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


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

.



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


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



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


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.}



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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


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 .


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.



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Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry Sanjiva Dayal IIT Kanpur