KENDRIYA VIDYALAYA SANGATHAN MATHS.pdfCLASS XII –MATHEMATICS Workshop on ‘Developing resource...

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1 KENDRIYA VIDYALAYA SANGATHAN NEW DELHI RESOURCE MATERIAL FOR TEACHERS CLASS XIIMATHEMATICS Workshop on ‘Developing resource material for teaching of Mathematics for classes IX-XII’ Venue: Kendriya Vidyalaya Sangathan, ZIET Mysore (20 th April to 25 th April 2015)

Transcript of KENDRIYA VIDYALAYA SANGATHAN MATHS.pdfCLASS XII –MATHEMATICS Workshop on ‘Developing resource...

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KENDRIYA VIDYALAYA SANGATHAN NEW DELHI

RESOURCE MATERIAL FOR TEACHERS

CLASS XII–MATHEMATICS

Workshop on

‘Developing resource material for teaching of Mathematics for classes IX-XII’

Venue: Kendriya Vidyalaya Sangathan, ZIET Mysore

(20th

April to 25th

April 2015)

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OUR PATRONS

SHRI SANTOSH KUMAR MALL,IAS

COMMISSIONER

SHRI.G K SRIVASTAVA IAS SHRI.U N KHAWARE

ADDL. COMMISSIONER (Admin) ADDL.COMMISSIONER (Acad)

Dr. SHACHI KANT Dr. V VIJAYALAKSHMI

JOINT COMMISSIONER (training) JOINT COMMISSIONER (Acad)

Dr. E PRABHAKAR SHRI. M ARUMUGAM

JOINT COMMISSIONER (Personnel) JOINT COMMISSIONER (Finance)

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MATERIAL CONSTRUCTION TEAM

PATRON

Dr. E T ARASU,

DEPUTY COMMISSIONER & DIRECTOR OF ZIET MYSORE

COURSE COORDINATOR

Mrs V.MEENAKSHI,

ASSISTANT COMMISSIONER, KVS RO ERNAKULAM

Resource Material Preparation Team

Mr. E. Ananthan

Principal, KV No : 1, AFS, Tambaram, Chennai – 600073

Mr. S. SreenivasagaPerumal

PGT( Maths), KV Mysore

Mrs. P. Ramalakshmi

PGT( Maths), KV No: 2, Pondicherry

Mr. T. Selvam

PGT( Maths), KV No : 1, Trichy

Mr. P. SenthilKumaran

PGT( Maths), KV HVF , Avadi.

Mr. Philip Shajan

PGT ( Maths), KV, Minambakkam,

COORDINATOR

Mr. K ARUMUGAM, PGT (PHYSICS), KVS ZIET MYSORE

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INDEX

1 Foreword 2 Preface 3 Guidelines to teachers 4 Teaching of Mathematics

Dr.E.T.Arasu, Dy. Commissioner & Director ZIET Mysore 5 New Trends in Assessments

Mrs.V Meenakshi,ASST.Commissioner KVS(RO),Ernakulam 6 Why Mathematics is perceived as a difficult subject?

Shri.E.Ananthan,Principal k. V. No.1 AFS‘ Tambaram ficult 7 Qualities of a successful Mathematics teacher

Shri.E.Krishna Murthy, principal K.V.NFC Nagar, Secundarabad. 8 Teaching strategies in mathematics

Shri.Siby Sebastian ,Principal K. V. Bijapur 9 Teaching learning mathematics with joy

Sharada M, Teacher, DMS,RIE Mysore 10 Resources- chapter wise Expected Learning Outcomes.

Concept mapping in VUE portal.

Three levels of graded exercises including non-routine questions.

Value Based Questions.

Error Analysis and Remediation.

Question Bank.

Power Point Presentations

Web Links

11 Sample papers 12 Tips & techniques to score better 13 Tips & techniques in teaching learning process 14 List of video clippings ,embedded files in the annexure folder

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FOREWORD

There is an adage about Mathematics: ―Mathematics is the Queen of all Sciences‖. This adage

exemplifies the significance, scope and importance of mathematics in the realm of sciences. Being a

‗Queen‘, as a subject, Mathematics deserves to be adored and admired by all. But unfortunately, this

subject is perceived by the students as a most difficult subject. Not only in India, but across the globe,

learning of the subject creates trepidation.

The perception about this subject being difficult in India is rather surprising as ours is a land of great

mathematicians like Ramanujan, Bhaskara, Aryabatta et al. The origin and accomplishments of these

great men should be a source of inspiration to both students and teachers alike. Yet, as the truth being

otherwise, making concerted efforts to identify the reason for perceived fears, initiate suitable damage

control and undertake remedial measures assume paramount importance. Kendriya Vidyalaya Sangathan,

as a pace setting educational organization in the field of School Education which always strives to give

best education to its students, thought it fit to take a pioneering step to empower teachers through teacher

support materials. In-service education too strives to do the same. Yet, providing Teacher Resource

Material in a compact format with word, audio and video inputs is indeed a novel one.

In the name of teacher resources the internet is abound with a lot of materials: books, audio and video

presentations. Yet their validity and usability being debatable, a homemade product by in-house experts

could be a solution. Hence, in response to KVS (HQ)‘s letter dated 03.03.2015 on the subject

―Developing Resource Material for Teaching of English and Maths‖, a six-day workshop was organized

at ZIET, Mysore from 20-25 April 2015. The task allotted to ZIET, Mysore by the KVS is to prepare the

resource materials for teachers of mathematics teaching classes IX to XII.

In the workshop, under the able coordinator ship of Mrs. V. Meenakshi, Assistant Commissioner, KVS,

Ernakulam Region, four material production teams were constituted for preparing materials for classes

IX, X, XI and XII separately. Mr. E.Ananathan, Principal, KV, No.1, Tambaram of Chennai Region

headed Class XII Material Production team; Mr.E.Krishnamurthy, Principal, KV,NFC Nagar, Gatakeshar

of Hyderabad region headed Class XI Material Production Team; Mr. Siby Sebastian, Principal, KV,

Bijapur of Bangalore Region headed Class X Material Production Team and Mr.Govindu Maddipatla,

Principal, KV, Ramavermapuram, Trissur of Ernakulam Region headed Class IX Material Production

Team. Each team was aided ably by a group of five teachers of Mathematics. After a thorough discussion

among KVS faculty members and Mrs.Sharada, TGT (Maths), an invited faculty from Demonstration

Multipurpose School, RIE, Mysore on the ‗Reference Material Framework‘ on the first day, the teams

broke up to complete their allotted work. Their tireless efforts which stretched beyond the prescribed

office hours on all the six days helped complete the task of preparing four Teacher Resource Booklets –

one each for classes IX, X, XI and XII in a time-bound manner.

Even a cursory glance of the index shall reveal the opt areas of support that the Resource Booklet strives

to provide to the teachers of mathematics. The entire material production team deserves appreciation for

the commendable work they did in a short period of six days. It is the earnest hope of KVS that the

effective use of the Resource Materials will serve the purpose of real teacher empowerment which will

result in better classroom teaching, enhanced student learning and above all creating in the minds of the

students an abiding love for the subject of mathematics.

- Dr. E. Thirunavukkarasu

Deputy Commissioner & Course Director

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PREFACE

KVS, Zonal Institute of Education and Training, Mysore organized a 6 Day Workshop on

‘Developing Resource Material for teaching of Mathematics for Class XII’ from 20th April to 25

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April 2015. The Sponsored Four Post Graduate Teachers in Mathematics from Chennai Region and one

Post Graduate Teacher from Bangalore Region were allotted two/ three topics from syllabus of Class XII

to prepare Resource Material for teachers under the heads:

1. Expected Learning Outcomes. 2. Concept mapping in VUE portal.

3. Three levels of graded exercises. 4. Value based questions.

5. Error Analysis and remediation 6.Question Bank

7. Power point presentation. 8. Reference Web links

9. Tips and Techniques to score better 10. Sample papers

11.Tips and Techniques in Teaching Learning process.

As per the given templates and instructions, each member elaborately prepared the Resource Material

under Fourteen heads and presented it for review and suggestions and accordingly the package of

resource materials for teachers were closely reviewed, modified and strengthened to give the qualitative

final shape.The participants shared their rich and potential inputs in the forms of varied experiences, skills

and techniques in dealing with different concepts and content areas and contributed greatly to the

collaborative learning and capacity building for teaching Mathematics with quality result in focus.

I would like to place on record my sincere appreciation to the Team Coordinator

Mr.E.Ananthan,Principal, KV No :1 AFS, Tambaram ,the participants Mr. S. SreenivasagaPerumal, PGT(

Maths), KV Mysore, Mrs. P. Ramalakshmi, PGT( Maths), KV No2, Pondicherry, Mr. P. SenthilKumaran,

PGT( Maths), KV HVF , Avadi, Mr. T. Selvam, PGT( Maths), KV No1, Trichy, Mr. Philip Shajan, PGT (

Maths), KV Minambakkam,the Course Coordinator Mr.Arumugam, PGT (Phy) ZIET Mysore and the

members of faculty for their wholehearted participation and contribution to this programme.

I express my sincere thanks to Dr.E.T.Arasu, Deputy Commissioner and Director KVS, ZIET, Mysore for

giving me an opportunity to be a part of this programme and contribute at my best to the noble cause of

strengthening Mathematics Education in particular and the School Education as a whole in general.My

best wishes to all Post Graduate Teachers in Mathematics for very focused classroom transactions using

this Resource Material to bring in quality and quantity results in the Class XII Examinations.

Mrs.VMeenakshi, Assistant Commissioner, Ernakulum Region

"The result of planning should be effective, efficient, and economical...that is, suitable for the

intended purpose, capable of producing the desired results, and involving the least investment of

resources". - Clark Crouch

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Guidelines to Teachers

The Resource Material has been designed to make learning Mathematics a delightful experience catering

to every kind of learner. As the learners are introduced to a fascinating variety of tools, and participate in

meaningful, fun filled activities their Mathematics competence will grow exponentially. Activities that

cater to different learning styles such as problem solving, reasoning and proof, analytical, logical etc. are

thoughtfully placed in the Resource Material.

1. Expected Learning Outcomes: In this section, the expected learning outcomes are enlisted chapter-wise and these are expected to be

realized among the students on completion of particular chapter. The teachers have to design their

teaching programme which includes mathematical activities, variety of tools and other mathematical

tasks. Teachers may prepare their Power point presentations and use in their regular teaching in order to

realise the desired outcomes.

2. Concept mapping in VUE portal:

The concept mapping works under Visual Understanding Environment portal, which can be downloaded

freely from ―Google‖. A concept map is a type of graphic organizer used to help teachers/students

organize and represent knowledge of a subject. Concept Maps begin with a main idea (or concept) and

then branch out to show that the main idea can be broken down into specific topics. The main idea and

branches are usually enclosed in circles or boxes of any Geometrical figure, and relationship between

concepts indicated by a connecting line linking new concepts. Each concept is embedded into the box,

and those concepts in the form of power point presentation, word document, videos web links etc are

uploaded in the same folder.

How to use a concept Mapping?

The teacher can use as a Teaching Aid for explaining the holistic view of the topic. It can be used as

revision tool. Concept maps are a way to develop logical thinking and study skills by revealing

connections and helping students see how individual ideas form a larger whole. These were developed to

enhance meaningful learning in Mathematics. It enhances metacognition (learning to learn, and thinking

about knowledge). It helps in assessing learner understanding of learning objectives, concepts and the

relationship among those concepts.

Download VUE portal from google and click on this icon to view the content embedded.

3. Three levels of graded exercises including non-routine questions:

In this section, selected questions collected from various reference books and are arranged in graded

manner, in order the child attempt and learn mathematics in that order. Questions are given in three levels

of nature easy, average and difficult respectively. These exercises facilitate the teacher to assign home/

practice works to the students as per their capabilities.

4. Value based questions:

In this section, Value based questions are given in each chapter with an objective to make a

student aware of the moral values along with the value of problem solving. It is an endeavour to inculcate

value system among the students and make them aware of social, moral values and cultural heritage of

our great nation. It is expected that the students develop the values like friendliness, Honesty, Initiative,

Compassion, Loyalty, Patience, Responsibility, Stability, Tactfulness and Tolerance along with problem

solving skills and other applications.

5. ErrorAnalysis and remediation:

It has been observed that the students commit a few common errors. In order to overcome this

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issue,teachers have listed,chapterwise, all possible common errors likely to be committed by the students

and suitable measures to overcome those errors.

5. Question Bank: The questions were prepared chapter wise and kept in order for guiding the students suitably in their

process of learning.Two sets of sample papers were also included for better understanding of the pattern

of the Board Question Paper including weightage of marks.

6. Power point presentation and Video clips:

As educators, our aim is to get students get energized and engaged in the hands-on learning process and

video is clearly an instructional medium that is compelling and generates a much greater amount of

interest and enjoyment than the more traditional printed material. Using sight and sound, video is the

perfect medium for students who are auditory or visual learners. Video stimulates and engages students

creating interest and maintaining that interest for longer periods of time, and it provides an innovative and

effective means for educators to address and deliver the required curriculum content.

PowerPoint is regarded as the most useful, accessible way to create and present visual aids. It is easy to

create colourful, attractive designs using the standard templates and themes; easy to modify compared to

other visual aids, such as charts, and easy to drag and drop slides to re-order presentation. It is easy to

present and maintain eye contact with a large audience by simply advancing the slides with a keystroke,

eliminating the need for handouts to follow the message.

The Resource material contains Power Point Presentations of all lessons of Class X and Video clips/links

to Videos of concepts for clarity in understanding.

Please double click on it to view the Power Point Presentation.

7. Reference Web links:

What is EDMODO?Free, privacy, secure, social learning platform for teachers, students, parents, and

schools.

Provides teachers and students with a secure and easy way to post classroom materials, share

links and videos, and access homework and school notices.

Teachers and students can store and share all forms of digital content – blogs, links, pictures,

video, documents, presentations, Assign and explain online, Attach and links, media, files,

Organize content in Edmodo permanent Library, Create polls and quizzes, Grade online with

rubric, Threaded discussions- prepare for online learning!

8. Sample papers:

In this section, blue print and sample papers are included for SA1 and SA2 which helps teachers to

give practice tests in the board pattern.

9.Annexure folder:

This folder is a collection of all soft copies which are embedded in the section ‗Concept Mapping in

VUE portal‘. ‗Power Point Presentations‘ of the lessons are included for classroom teaching.

Further Video clippings of a few problems and concepts are included.

Feedback:

The Post Graduate Teachers and Trained Graduate Teachers are requested to use this material in

Classroom transaction and send feedback to Mrs.V.Meenakshi, Assistant Commissioner, Ernakulam

Region.

[email protected]

BEST WISHES!

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TEACHING OF MATHEMATICS

Introduction:

India is the land of Aryabhatta, Ramanujam and the like – great luminaries in the field of mathematics.

Yet, this is one subject that our students dread the most. ―It is a nightmarish experience learning this

subject, and even the very thought of the subject sends jitters‖ is a common refrain of school-going

students. Not only in India, a developing country, but also in other countries, be those come under the

category of underdeveloped or developed, Mathematics is a subject of fear among our students. Both

parents and students feel that class room teaching of the subject alone is not adequate for learning it

effectively.

The Parent’s worries:

The parents are worried lot. Getting a ‗good‘ Mathematics teacher, whatever it means, is big problem.

The ones they get in the ‗market‘ are not of any big help to their children; yet they are left with no option

but to depend upon either school teachers or coaches from outside. Class room teaching is woefully

inadequate in enabling the children acquire confidence and interest in the subject. The quantum of

individual attention paid to solving the problems of students in this subjects being almost nil, making

them get through the examination is a challenge. ―Something needs to be done to arrest the rut being set

in Mathematics teaching‖ is the common prayer of parents.

Why do children consider mathematics a difficult subject?

When you ask the teachers of Mathematics as to why Mathematics is considered as a difficult subject, the

answers you get are neither logical nor scientific. Here are a few samples:

The subject requires more of students study time than other subjects (why?)

Students fail to practice problems (what is the reason?)

The subject requires long hours of work, involving practice and drill (why is it so?)

This subject is different from other subjects (in what ways?) Though these answers might partially tell us

the reason why the subject is detested by many, they fail to throw any light on the psychological

prerequisites, if any, specially required for learning this subject.

What goes on in Mathematics classes?

A peep into Mathematics classes and a bit of observation of the ways in which Mathematics is taught by

Mathematics teachers reveal a pattern which is as follows:

Introduction of the new topic: the teacher speaking in general terms for a few minutes about the

topic on hand if it happens to be the beginning of the topic.

Working out problems on the black board either by the teacher himself or by calling out a

student to do it:If the teacher does the problem on the board, one can see him doing it silently or lip-

reading the steps involved in it. If the student does the problem on the board, either continuous

interruption or silent observation of the teacher can be seen to be taking place in the class.

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Occasional fielding of questions by the teacher on the problem area:

When students raise doubts on the steps written how the steps have been arrived at etc., the teacher either

clarifies the doubts or tells them to go through the steps again.

Leaving a large number of problems for the students to solve: Often after having solved one or

two problems given under the exercise questions on the black board (at times those worked out problems

happen to be given as model problems in the text book), teaches tend to leave a large number of

remaining problems as home work to students.

What is wrong with the Existing Teaching Practices?

A critical analysis of what is wrong with the existing practices of Mathematics teaching is of prime

importance. The analysis of commonly existing Mathematics teaching practices is given below:

Introduction of the Topic

Introduction of any new topic is done in not more than 5-10 minutes duration. This duration is not

enough. You cannot throw light on the conceptual frame work of a topic integrating the related concepts

learned in classes down below in a span of 5 to 10 minutes. The teacher cannot say much in such a short

duration. What actually transpires in Mathematics classes in the name of introducing topic can be

illustrated with an example: In the illustration, I have taken here on the topic ‗quadratic equation‘ taught

by a teacher which goes like this. ―We are going to learn quadratic equations today. Any equation of the

form ax2+bx +c =0 in which a , a and b are coefficients of x

2 and x respectively and c is a constant

term is called a quadratic equation. Quadratic Equations have one of the three types of solutions –two

different values for the variable x, same value for the variable x or no solution‖. This type of introduction

with a bit of additional information added or otherwise is observed in many classes.

Obviously the introduction given by the teacher is insufficient. There are concepts learned in other classes

which have vertical connection and relevance with the topic quadratic equation, namely algebraic

equations, linear equations, factors, constants, coefficients, monomials, binomials and of course, algebraic

expressions etc. Sparing 10 to 20 minutes to brush up the memory of the students in the topic is highly

essential, if a teacher has to cater to the needs of the students of varying levels of understanding of the

subject. Introduction given in a generalized manner without taking into account the students previous

knowledge and current knowledge in a topic would serve no purpose.

Working out Problems:

Next the teacher picking up one or two problems randomly from the actual exercise for solving on the

black board is a common practice observed in the Mathematics classes. Even selecting the worked out

examples given in the text book for black board work is not uncommon among the teachers. While the

majority of Mathematics teachers prefer to articulate the steps as they work out on the black board, the

rest does not open their mouth while their written work is in progress. In the absence of any instruction,

students copy down the black board work without listening much to what the teacher says is a common

sight in Mathematics classes. The black board works with our teacher‘s explanation cannot be beneficial

to the entire class. Leaving a few motivated students, who have developed interest in Mathematics

through other sources of learning, the rest would tend to lose interest and accumulate doubts/ignorance

over a period of time, if efforts are not made by the teacher to explain the steps as to why and how those

steps occur in the way they are written on the black board. If the sequential relation and coherence among

the steps in solving a problem is lost sight of, the entire subject matter would present a picture of mystery

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to students. This results in aversion to the subject and ultimately mathematics phobia. The brain develops

a conditioned response to learning mathematics which I call, ‗Mathematics Blindness‘ Anything related to

number, order, sequence, logic and Mathematical operations becoming an anathema to the brain.

Handling students’ questions:

The third aspect of teaching Mathematics, namely, how teachers handle questions posed by students,

requires a closer examination. Questions, as a matter of fact, are not welcome in mathematics classes.

They are perceived as speed breakers to smooth progress (!?) in the completion of syllabus. ―After all

how good a teacher is not important, but are you a teacher who completes the syllabus within the

stipulated period of time is! Where is the time to explain each and everything? Even if you do so, there

are not many takers. Parents have more faith in coaching classes than in our ability than to teach their

children well. ―Explanation likes this fly thick and fast the moment you talk about poor teaching of

Mathematics. Even in the best of mathematics classes, there is no guarantee that the skill and the mental

process of learning the subject, and the components of mastery learning are taken care of. Moving back

and forth in elucidating the concepts of Mathematics is rarely done though it is an essential component to

review and refresh the previous knowledge. When teachers proceed without this exercise, students

stumble with many a doubt and asking that in the class for clarification is straddled with many a pitfall.

Right choice of words for raising questions, receptivity of teacher and the possibility of getting answers

from the teacher are all matters of speculation. Hence the students prefer the permissive coaching classes

for seeking clarification for the doubts that arise in various levels of learning Mathematics. Yet, their

hopes are dashed as coaching classes are as crowded as regular school classes and getting conceptual

understanding of the subject becomes a real challenge.

The Home Assignments:

Teachers give large number of exercise problems for the students to solve at home. As seen earlier, doing

one or two problems for the name sake does not help the students to acquire the insight required for

solving problems at home on their own. The teacher solved problems are inadequate in number and

variety, and the explanation, if any, given about problem solving in the classes is either incomprehensive

or inadequate. As a result students get frustrated when they struggle with problems with answers not in

sight. They lose interest when their woes are not taken care of.

Mathematics-the Queen of All sciences

Mathematics is one of the compulsory subjects of study up to class X and an optional one from class XI

onwards. It is an important subject as it is considered as the ‗Queen of all Sciences‘. The abstract nature

of Mathematics, precision and exactness being its hall mark, makes this subject appears as more difficult

than other subjects. Even the simplest of concepts in it like numbers, addition, subtraction, multiplication,

division prescribed for the primary classes warrant in-depth understanding and imagination and creative

thinking on the part of the teacher for effective teaching. But do we have teachers who possess these

qualities in our schools is a moot question.

Teaching Mathematics in the Primary Classes

Teachers are not having the subject specialization in Mathematics too are allotted this subject for teaching

in Primary Classes. Concepts in mathematics up to class V level in schools, consists of basic operations

such as addition, subtraction etc., that are taught in a routine manner. As a result, the student learns to do

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those basic operations following certain repetitive patterns oblivious of the ―Why‖ aspect of those

patterns. When they reach the middle level (Classes VI to VIII) and later on the secondary level (Classes

IX to X), they understand that the ‗patterns ‗that they learn in the primary classes are not of much use and

that they need to know ‗something more‘. The real problem to them is to know what is that ‗something

more‘. It is the understanding of basic concepts which is more essential than knowing certain patterns of

doing certain problems in Mathematics. But unfortunately, most students come to middle classes without

learning anything about Mathematical concepts and how to use their conceptual understanding for solving

problems. Mathematics learning, hence, becomes a big riddle by then, and the slow but steady process of

developing disinterest in the subject sets in.

The Challenge in Middle and Secondary Classes:

The teachers of the middle and secondary classes have a challenge on hand: opening up the cognitive

domain of students and then taking them to higher order mental abilities though sequential learning

process. In other subjects, knowledge and understanding apart, memory play a vital role in scoring marks.

Even without the former, with the latter (memory) alone, students can score marks in other subjects,

where as in Mathematics, you are expected to do ―problem solving‖ which is a higher order cognitive

skill.

The Cognitive Domain:

The cognitive domain of the human brain is said to be responsible for thinking, understanding,

imagination and creativity. The cognitive domain becomes a fully functional component of human brain

after 10 or 11 years of age in children. This does not mean that this domain remains dormant and

nonfunctional before this stage. In fact ‗concept formation‘- one of the difficult functional outputs of the

cognitive domain –does take place even before 10 or 11 years of age. The lower order cognitive skills

such as knowledge and understanding apart, the elementary level of skills of analysis, and simple problem

solving skills are exhibited by primary class children. Systematic development of these skills is called for

when the children reach the middle classes. Therefore, a thorough understanding of the process involved

in problem solving, which has its genesis in concept learning is a must for teachers of Mathematics.

What Is Concept Learning?

A concept is an abstract idea, and mathematics is full of them. Concept learning involves acquiring a

thorough comprehension and grasp of abstract ideas. Each concept in Mathematics has sub-components.

For example, ‗algebraic expressions‘ is a concept whose sub-components are ‗algebra‘ (What?),

‗expression‘ (What?), and ‗algebraic expressions‘ (definition?).Besides these components, questions like

what are numerical expressions, are algebraic expressions different from numeric expressions, what are

algebraic equations, how do algebraic expressions differ from algebraic equations, and so on may needs

to be answered to bring out clarity in learning the concept ‗algebraic expressions‘.

Knowledge Redundancy:

The information age we live in help us see information explosion taking place all around us. The newer

learning taking place with geometric progression keeps replacing the current and past information, and

hence knowledge is in constant state of flux. Processing information in order to add it to the existing

corpus of knowledge is the need of the hour. Teachers, whose main business is transacting knowledge in

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class room, cannot remain isolated from information processing. They need to keep updating themselves;

else they would become knowledge redundant.

Class Room transactions Cognitive Skills:

Knowledge updated by the teachers is to be transacted in an effective manner, in capsules, in class room

to facilitate students comprehending it. Students‘ language abilities and power of comprehension should

be known to the teacher so as to select the best possible way of communicating knowledge with fosters

comprehension. The real task of the teacher wanting to achieve total comprehension exists in analyzing

and synthesizing knowledge. This is also acquired to develop application skills - in known situations to

start with and progressively in unknown situations. Problem solving requires ‗application skills‘, which

are the by-product of analysis and synthesis. The skills of analyzing and synthesizing, and application of

knowledge at known and unknown situations have an important component called ‗thinking‘. Thinking

has two integral parts: divergent and convergent, while divergent thinking results in creativity, convergent

in conversation.

The vertical Connectivity among the Cognitive Skills:

The skills in various levels of the cognitive domain do not function in isolation. There is a vertical

connectivity among them, which can be presented by a flow chart as given under:

ORDERED SKILLS OF COGNITIVE DOMAIN: THE COGNITIVE LADDER

Higher Order Skills

Middle Order Skills

Lower Order Skills

Analysis

Creativity

Problem Solving

Application: known/ unknown

situations

Synthesis

Understanding/ Comprehension

Information

Knowledge

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Information Processing:

Processing information may require special skills such as skimming and scanning. Yet ‗information‘ is

kept as the bottom as a lower order skill in view of the fact that the information processed as know ledge

is readily made available in text books to study. Information processing is defined as Claude E. Shannon

as the conversion of latent information into manifest information. Latent and manifest information are

defined through the terms equivocation (remaining uncertainty, what value the sender has actually

chosen), dissipation (uncertainty of the sender what the receiver has actually received) and transformation

(saved effort of questioning- equivocation minus dissipation).

Knowledge:

Knowledge too has innumerable components yet the bookish knowledge is emphasized in class room

teaching and hence its categorization as a lower order skill. Understanding of what is given in text is

meant in a limited manner of ‗Knowing what it is‘ rather than why and how. The Wikipedia, free

encyclopedia, defines knowledge as information of which a person, organization or other entities aware.

Knowledge is gained either by experience, learning and perception through association and reasoning.

The term knowledge also means the confident understanding of a subject, potential with the ability to use

it for specific purpose.

What is analysis

There are many definitions given to the term ‘analysis’. Some are given below:

An investigation of the component parts whole and their relations in making up the whole.A form of

literary criticism in which the structure of a piece of writing is analyzed.The use of closed- class words

instead of infections: eg:, the father of a bride‘ instead of ‗the bride‘s father‘

In our article I use the term ‗ analysis‘ to refer to understanding the components that go into making

something. For analogy, think of a TV set. The components are picture tube, condensers, resistances,

speakers etc. are put together to make a composite whole called TV. Similarly any concepts in

Mathematics consist of micro- concepts/ sub- concepts, the understanding, defining and elucidating of

each micro- concept fall under the domain of analysis. In class VII, for example, the concept of ‗rational

number‘ is defined as follows:

―Any number that can be put in the form of p/q where p and q are integers and q not equal to zero is a

rational number‖. Analysis of this concept includes the understanding and elucidation of

i) Why it is said ―that can be put in the form of‖?

ii) What does ‗any number‘ mean?

iii) What are integers?

iv) Why ‗q‘ should not be equal to zero?

v) Why and what for is this new set of numbers called rational numbers?

vi) What is ‗rational‘ about these rational numbers?

A teacher attempting to teach the definition of rational numbers without throwing the light on the above

questions and many more questions related to them is doing disservice to students wanting to learn

Mathematics. Given a permissive and receptive atmosphere, students would come out with many

questions as given above, the answers of which would be an appetizer for developing their analytical

skills.

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Synthesis:

The word ‗synthesis‘ can be defined in many different ways. A few popular definitions are as follows:

The art of putting different representations together and of grasping what is manifold in them in

one act of knowledge.

Synthesis is what first gives rise to knowledge, ie. It is not analysis. It is an act of the imagination.

Synthesis suggests the ability to put together separate ideas to form new wholes of a fabric, or

establish new relationships.

Synthesis involves putting ideas and knowledge in a new and unique form. This is where

innovations truly takes place.

The process of bringing pieces of an analysis together to make a whole.

The process of building a new concept, solution, design for a purpose by putting parts together in

a logical way.

This is fifth level of Bloom‘s taxonomy and deals with the task of putting together parts to form a

new whole. This might involve working with parts and putting them together in a creative new

way or using old ideas to come up with new ones.

Synthesis is to be done for the purpose of establishing the Gestalt view that ‗the whole is more than the

sum total of its parts‘. A suitable analogy can be assembling the components of a TV set and making it

work.

Application:

A few definitions of the ‗application‘ are given as follows:

The act of bringing something to bear; using it for a particular purpose: ―he advocated the application of

statistics to the problem‖; ―a novel application of electronics to medical diagnosis‖

A diligent effort; ―it is a job requiring serious application‖

Utilizing knowledge acquired and processed by the mind for solving problems- both simple and

complicated.

The skills associated with information, knowledge, comprehension, analysis and synthesis are to be

acquired by students in order to go to the next level of the cognitive order called ‗application‘. The

application of knowledge, skills, and attitudes has to be done in known situations to start with so that the

students can progressively move on to unknown situations. Examples suitably selected can help them go

through simple to complex situations and would guide them to acquire insight. This insight is a prime

requisite for problem solving.

In school level Mathematics, the skills of analysis and synthesis and the insight learning that takes place

as a result of the application of those skills would pave the way for solving exercise problems, which the

teachers shy away from under the pretext of lack of time, and other priorities. The skills in the cognitive

ladder are vertically connected, and the acquisition of those skills at each level requires the student‘s to

allow their minds to think and assimilate ideas. This repetitive manner in which the sequential cognitive

skills practiced would train the mind in Mathematical thinking which is otherwise called logical thinking.

Logical Thinking:

Logical thinking is defined as that thinking which is coherent and rational. Reasoning and abstract

thought are synonymous with logical thinking. Logical thinking be in Mathematics or any other subject is

17

required to establish the coherence of facts of matter and formation of logical patterns and sequences.

Mind has the special ability to think and assimilate, and retain subject matter when presented in sequential

manner. Mind grasps matter devoid of gaps quickly. Unanalyzed knowledge in its un-synthesized form

poses difficulty in retaining it in long time memory, as concepts and its components do not function

isolation. Hence, mind rejects fragmented information which lacks patterns.

Problem Solving:

The thought process involved in solving a problem is called problem - solving. Problem solving as a skill

is developed crossing various other skills on its way. The skills lying down below ‗problem solving‘ in

the cognitive ladder can be compared to the floors of a building. You cannot reach the sixth floor without

crossing the floors down below. Similarly when problem solving is attempted in classes with making

explicit efforts to pass through the levels of knowledge, understanding, analysis, synthesis, and

application, students fail miserably.

Often it is said that practice and drill in Mathematics would help learn the subject better. Hence again, by

repeatedly working out problems, one has to ‗memorize‘ the steps, but it does not guarantee success when

problems are differently worded or twisted. Following the cognitive order- moving from information to

problem solving steps in the classroom will help the students know the sequential mental processes

involved in solving problems in Mathematics. As they practice these steps regularly it will boost their

confidence in learning the subject. But classroom learning these days mostly concentrate on problem

solving as a direct hit strategy. Working out the problems first without following the cognitive order is

equivalent to putting the horse behind the cart, which will take the students nowhere. Even conscientious

teachers tend to spend10-20% of their class time on concept teaching and 80-90% on working out the

problems. This is totally incorrect. Concept learning and concept formation require knowledge

comprehension, analysis and synthesis. After going through these steps, as the next stage, ‗application‘

should be dealt with. At the end comes problem solving. The process of going through and gaining

thorough grasp of knowledge, understanding, analysis, and synthesis warrants 80-90% of class room time,

and hence just 10-20% class time is enough for problem solving.

The Cognitive Order Learning:

Failure to recognize orderly thinking, the basic quality of cognitive order, is the main reason for the

difficulty faced by students in learning Mathematics. The earlier the teachers and students recognize the

need for approaching Mathematics logically, the better it would be to create and sustain interest in the

subject. The Mathematics classroom practices may, therefore, be fashioned by following the sequential

steps given as under:

i) Teacher utilizing 5-10 minutes in the beginning of the class on asking questions on the

knowledge, comprehension, analysis and synthesis part of the chapter on hand.

ii) After identifying the skill area in which students have problems, discussion to thrash out

those problems should be taken up. Often the problems of students stem from lack of

understanding of the basic concepts. It is essential therefore to keep doing ‗concept recall‘

and ‗concept clarification‘. Comprehending basic concepts is an essential condition for

moving on the further steps in the cognitive order, namely, analysis, synthesis etc.

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iii) Then the points under application of skills down below may be taken up for discussion. Once

the gray areas in application are cleared, the students may be instructed to do problem

solving.

iv) If students falter in steps, the logical sequence of steps followed to solve any problem in the

given chapter falling under the concepts learnt may be discussed again.

v) Effective questioning to draw out the conceptual understanding of the subject matter learnt by

the students should be done at least every tenth minute in every period to ensure that they are

actually with the teacher.

vi) Free-wheeling of ideas related to the subject by the students should be encouraged as it would

help throw new light on the subject matter under study.

vii) Questions by the students, however silly they may seem, should be welcome in the class and

the teacher should listen to them with patience and convincing answers given.

viii) After a full-fledged concept learning session following cognitive order learning, problem

solving should be taken up where the students should be encouraged to work out the

problems under the watchful eyes of the teacher.

ix) Vertical and horizontal connectivity of concepts in mathematics should always form an

integral part of teaching learning, and students being thorough in the sequential conceptual

elements be taken care of.

The Critical aspects of learning Mathematics:

Besides taking care of the above nine aspects of teaching, teachers desirous of making students love and

do well in mathematics should need to pay heed to the following aspects as well:

A thorough comprehension of the domains - psychological, physical and practical - of effective learning

of the subject by the students;

The process they have to follow scrupulously in acquiring skills for the mastery-level learning of the

subject;

The role the teachers and parents have to play in fostering and sustaining students interest and enthusiasm

so that they learn the subject with ease at class room and face the examination with confidence.

Let me sum up some of the benefits of Cognitive-order-Learning: This methodology- ‗Cognitive-order-

Learning‘ enables the students -

To acquire subject learning competencies

To develop problem solving skills

To boost their confidence in the subject

To widen their interest in the areas of mathematics

To have and sustain self-directed and self-motivated activities in mathematical learning.

To achieve mastery level learning of the subject.

To help apply the skills acquired in Mathematics to other subjects.

To utilize the cognitive domain to its full extent

To remove examination phobia.

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Conclusion:

The subject matter of Mathematics teachers is vast. An attempt has been made to give only the most

rudimentary aspects of it. What requires as a clear understanding on the part of the teachers is that the

subject, Mathematics, is neither difficult unconquerable. Yet, it is perceived to be so mostly owing to

ineffective teaching, which jumps from knowledge to problem solving, leaving a vast territory of skills in

between untouched. As said earlier, Mathematics being the queen of all science deserves an approach to

teaching which is based on the sound scientific principles of human learning. In any class room, if

students declare that they like Mathematics they enjoy learning it, and they have no difficulty in solving

the problems, that class is said to have been blessed with a teacher teaching Mathematics the way it is

deserves to be taught. That way surely is Cognitive-order-Learning approach with the thorough

understanding of the critical aspects of learning Mathematics referred to above.

Dr. E.T. Arasu, Dy. Commissioner &

Director K.V. ZIET Mysore.

20

NEW TRENDS IN ASSESSMENTS

Introduction:

One of the main reasons for teachers to assess student learning is to obtain feedback that will guide

teaching and assist in making modifications to lesson planning and delivery to ensure student progress.

Assessment allows teachers to monitor progress, diagnose individual or group difficulties and adjust

teaching practices. Assessment can support student motivation when students are provided with on-going

information about their progress and with opportunities to set further goals for learning. Assessment is an

interactive process between students and faculty that informs faculty how well their students are learning

what they are teaching. The information is used by faculty to make changes in the learning environment,

and is shared with students to assist them in improving their learning and study habits. This information is

learner-centred, course based, frequently anonymous, and not graded.

Current trends in classroom Assessment:

The terms formative assessment and summative assessment are being redefined in education circles.

Many teachers know formative assessment as the informal, daily type of assessment they use with

students while learning is occurring. Summative assessment was the term used to ―sum it all up,‖ to

indicate a final standing at the end of a unit or a course.Current trends in assessment focus on judging

student progress in three ways: Assessment for learning, assessment as learning and assessment of

learning. Each assessment approach serves a different purpose.

Assessment for learning is especially useful for teachers as they develop, modify and differentiate

teaching and learning activities. It is continuous and sustained throughout the learning process and

indicates to students their progress and growth.

In assessment for learning, teachers monitor the progress made by each student in relation to the program

of studies, outcomes and determine upcoming learning needs. Teachers ensure that learning outcomes are

clear, detailed and ensure that they assess according to these outcomes.

They use a range of methods to gather and to provide students with descriptive feedback to further student

learning. These methods may include checklists and written notes based on observations of students as

they learn. The descriptive feedback gathered is used to inform planning for learning and to assist the

teacher in differentiating instruction in order to meet the needs of all students.

The feedback may be shared in oral or written form with individual students or with the class as a whole.

As the information gathered guides the planning process, it leads to the improvement of future student

performance in relation to specific outcomes.

Assessment as learning focuses on fostering and supporting metacognitive development in students as

they learn to monitor and reflect upon their own learning and to use the information gathered to support

and direct new learning.

It focuses on the role student‘s play in their learning. In this approach to assessment, students are viewed

as the bridge between what they know and the unknown that is still to be learned. Their role is to assess

critically both what and how they are learning. They learn to monitor their thinking and learning

processes; to understand how they are acquiring and retaining new information or developing new skills

and awareness; and how to make adjustments, adaptations and even changes when necessary.

21

For some students, being asked to reflect on their learning by using skills and strategies related to

metacognition (to think about thinking) might seem new and uncomfortable. They may need help to come

to the realization that learning is a conscious process in which knowledge is constructed when the known,

or previously acquired, encounters the new or unknown. This process often results in the restructuring or

reintegration of what was previously learned.

Assessment of learning is cumulative in nature. It is used to confirm what students already know and

what they can do in relation to the program of studies outcomes. Student progress is reported by way of a

mark; e.g., a percentage or letter grade, a few times a year or a term. The report card is usually received

by students, their parents/guardians as well as by school administrators.

Assessment of learning takes place at specific times in the instructional sequence, such as at the end of a

series of lessons, at the end of a unit or at the end of the school year. Its purpose is to determine the

degree of success students have had in attaining the program outcomes. Assessment of learning involves

more than just quizzes and tests. It should allow students to move beyond recall to a demonstration of the

complexities of their understanding and their ability to use the language.

Assessment of learning refers to strategies designed to confirm what students know, demonstrate whether

or not they have met curriculum outcomes or the goals of their individualized programs, or to certify

proficiency and make decisions about students‘ future program or placements.

Teacher reflections Assessment procedures:

It is important for a teacher to reflect on why and when students‘ progress is assessed.

The types of reflective questions that teachers can ask themselves when engaged in assessment for

learning include:

► Am I observing in order to find out what my students know or are able to do?

► Does my assessment strategy allow student learning to be apparent? Are there elements I need to

change in order to minimize anxiety or distractions that might get in the way of learning?

► Will I use the results of my observations to modify my instruction, either with a particular student or

with a group of students, or the next time I teach this concept or skill to a new class?

► Will I share the results of my observations with the individual student so that the student and I can

decide how to improve future performance?

► Will I share the results of my observations with the class in general (without identifying particular

students) in order to provide some indicators as to where they can improve future performance?

The types of reflective questions that teachers can ask themselves when planning opportunities in support

of assessment as learning include:

► Are the students familiar with the purpose of reflective tools, such as the one I am thinking of using?

Will they be able to engage with the questions in a meaningful way?

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► Have I provided/will I provide support for students in accordance with the various points mentioned in

the reflective instrument; i.e., do I provide clear instructions, create a model, share a checklist, ensure that

there are reference materials?

Teacher reflections: The types of reflective questions that teachers can ask themselves when

Planning opportunities in support of assessment of learning include:

► Am I using processes and assessment instruments that allow students to demonstrate fully their

competence and skill?

► Do these assessments align with the manner in which students were taught the material?

► Do these assessments allow students to demonstrate their knowledge and skills as per the program of

studies outcomes?

Student reflection assessment(Assessment as learning):

Students record their reflections by completing sentence starters such as

―Things that went well …‖; ―Things that got in my way …‖; ―Next time I will ….‖

Alternatively, they may check off various statements that apply to themselves or their performance on a

checklist.

An overview of the different practices and variety of instruments that can be used and tailored to meet the

needs of a specific assessment purpose.

Assessment for Learning

Informal observation/Formative

assessments/Peer learning

Assessment as Learning

Conferencing/Learning

conversations/

Peer assessment/Quizzes or

Tests/

Self-assessment and Goal

setting.

Assessment of Learning

Performance Tasks/Projects

Summative assessment

Quizzes/Pen-paper tests

Tests or Examinations.

PSA

Formative Assessmentis a process used by teachers and students as part of instruction that provides

feedback to adjust ongoing teaching and learning to improve students‘ achievement of core content. As

assessment for learning, formative assessment practices provide students with clear learning targets,

examples and models of strong and weak work, regular descriptive feedback, and the ability to self-

assess, track learning, and set goal. Formative assessments are most effective when they are done

frequently and the information is used to effect immediate adjustments in the day-to-day operations of the

course.

Assessment is not formative unless something is ―formed‖ as a result of interpreting evidence elicited. It

informs teacher where the need/problem lies to focus on problem area. It helps teacher give specific

feedback, provide relevant support and plan the next step. It helps student identify the problem areas,

provides feedback and support. It helps to improve performance and provides opportunity to improve

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performance. Peer learning can be encouraged at all stages with variety of tools. Formative Assessment

Strategies:

Tools for Formative Assessment Techniques to check for understanding

One minute answer A one-minute answer question is a focussed question with a specific

goal that can be answered within a minute or two.

Analogy prompt A designated concept, principle, or process is like _______

because_____________________.

Think, pair, share/Turn to your

partner

Students think individually, then pair (discuss with partner), then

share with the class.

10-2 theory/35-5 theory 10 minutes instruction and two minutes reflection/35 minutes

instruction and 5 minutes reflection.

Self -assessment A process in which students collect information about their own

learning, analyze what it reveals about their progress toward

intended learning goals or learning activity or at the end of the day.

Conclusion:

Teachers should continuously use a variety of tools understanding different learning styles and abilities

and share the assessment criteria with the students. Allow peer and self-assessment. Share learning

outcomes and assessment expectations with students. Incorporate student self-assessment and keep a

record of their progress and Teachers keep records of student progress.

*********

24

TEACHING OF MATHEMATICS – MOVING FROM MATHPHOBIA TO MATHPHILIA

Shri. E.Ananthan

Principal, KV No.1 Tambaram, Chennai

―Mathematics is for everyone and all can learn Mathematics‖- NCF 2005

The Little Oxford Dictionary define phobia as fear or aversion.Psychology textbooks describe it as an

abnormal fear. We hear of claustrophobia, acrophobia, nyctophobia, and anthropophobia.

The pioneers in the study of Mathematics anxiety, Richardson and Suinn (1972), defined Mathematics

anxiety in terms of the (debilitating) effect of mathematics anxiety on performance: "feelings of tension

and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in

a wide variety of ordinary life and academic situations".

Is there such a thing as math phobia? To know the answer one needs to only teach mathematics

particularly in the secondary and senior secondary classes. And the reality is that most school drop outs in

the Board exam are due to failure in Mathematics. Studies indicate that students' anxiety about

Mathematics increases between the sixth and twelfth grade.

With this reality check, this write-up aims to analyse the problem and by this parsing, redefine the

teaching learning of Mathematics firmly grounded on foundations of success- for the student for the

teacher, the society and the nation.

The suggestion that Mathematics anxiety threatens both performance and participation in Mathematics,

together with the indications that Mathematics anxiety may be a fairly widespread phenomenon (e.g.

Buxton, 1981), makes a discussion like this, concerning Mathematics anxiety in students, particularly the

Board going students, of extreme importance.

Mathematics is termed as the queen of all sciences, having logical thinking as its crown and problem

solving as its sceptre. Two essential elements which are necessary not just to master nuances of the

numeral world but more importantly to have success in life in qualitative ways- these two are also the

core life skills formulated by WHO for a healthy and successful life.

The question is, ―Does the teaching of Mathematics in our classrooms realise any of these objectives?

The huge population of children who balk at the very mention of the subject is an ever growing one as

generation gives way to another.

The NFG Position paper on the teaching of Mathematics under the section ―Problems in Teaching and

Learning of Mathematics‖ states: four problems which we deem to be the core areas of concern:

Other problems are systemic in nature:

Compartmentalisation- Segregation of Primary, Secondary and Senior secondary

Curricular acceleration- The quantum and scope of the syllabus is much larger and wider with passing

days.

The NFG recommends four fold measures to ensure that all children learn Mathematics:

25

Shifting the focus of mathematics education from achieving ‗narrow‘ goals to ‗higher‘ goals,- whole

range of processes here: formal problem solving, use of heuristics, estimation and approximation,

optimization, use of patterns, visualisation, representation, reasoning and proof, making connections,

mathematical communication. Giving importance to these processes constitutes the difference between

doing mathematics and swallowing mathematics, between mathematisation of thinking and memorising

formulas, between trivial mathematics and important mathematics, between working towards the narrow

aims and addressing the higher aims.

The recommended methods are:

cross curricular and integrated approaches within mathematics and across other disciplines,

Simplifying mathematical communication multiplicity of approaches, procedures, solutions using the

common man‘s mathematics or ―folk algorithm‖- basing problems on authentic real/ daily life contexts

use of technology

1. Engaging every student with a sense of success, while at the same time offering conceptual

challenges to the emerging mathematician- striving to reduce social barriers and gender

stereotypes and focussing on active inclusion of all children in the teaching-learning of

mathematics. Children with math phobia usually seem to have little confidence in themselves.

They feel they are not good in math; they refrain from asking questions (little realizing that more

than half the class is puzzled over the same Problem!); they are afraid to answer any question

directed to them for fear of being labelled "dumb" or "stupid." Such fear or anxiety about math

often begins during the Primary years and continues through life.

1. 1. A sense of fear and failure regarding mathematics among a majority of children

(cumulative nature of mathematics, gender and social biases about math ,use of

language and more importantly symbolic language)

2. A curriculum that disappoints both a talented minority as well as the non-

participating majority at the same time(emphasises procedure and knowledge of

formulas over understanding is bound to enhance anxiety)

3. Crude methods of assessment that encourage perception of mathematics as

mechanical computation, and( only one right answer, sacrificing the process for the

right solution, overemphasis on computation and absolute neglect for development of

mathematical concepts)

4. Lack of teacher preparation and support in the teaching of mathematics.( out dated

methodology, depending on commercial guides due to insufficiency in conceptual

clarity and understanding of the fundamentals of mathematics, inability to link formal

mathematics with experiential learning , particularly in the secondary and senior

secondary stages, incapacity to offer connections within mathematics or across subject

areas to applications in the sciences)

26

Recommendations:

Teacher need to model ―problem –solving‖ particularly in the context of word problems. To work

out diverse problems and build personal repertoire of problem solving skills and model them

with enthusiasm and confidence.

Move from simple step problem solving modes to increasingly complex and multi- step problem

solving.

Inculcate positive, persevering problem solving approaches- solve problems with them building

rapport thus building their self-esteem and confidence.

Use a ―problem solving‖ bulletin board to bring problem solving as part of everyday learning

activity

In problem solving, arriving at the "correct answer" is not the most important step. More

important is choosing the correct strategy for solving the problem. Even though there is only one

correct answer, there will be more than a single correct strategy for solving a problem. When

students are reassured of this fact, they will then be more willing to tackle new problems.

2. Changing modes of assessmentto examine students‘ Mathematisation abilities rather than

procedural knowledge-

3. Enriching teachers with a variety of Mathematical resources. - The development of

teacher knowledge is greatly enhanced by efforts within the wider educational community.

Teachers need the support of others—particularly material, systems, and human and emotional

support. While teachers can learn a great deal by working together with a group of supportive

mathematics colleagues, professional development initiatives are often a necessary catalyst for

major change. Activities like collaborative and strategic approaches, Mathematics Lab and

experiments help in this aspect

Reflecting on and applying these thoughts to the KV context, what should the maths teachers

need to do to ensure that all students learn Mathematics in the true sense of the word i.e. love it,

think, learn and apply it.

Mathematics teachers need to move from emphasis on Computation to holistic Mathematical

concept learning which will mathematise their thoughts and perspectives.

They need to be constantly conscious of and strive to promote a sense of achievement and

comfort in learning of mathematics.

CONCLUSION

UNESCO‘s The International Academy of Education in its paper-Effective Educational Practices

Series on the topic‖ Effective Pedagogy in mathematics‖ by Glenda Anthony and Margaret Walshaw

postulates the following :

1. An Ethic of Care-Caring classroom communities that are focused on mathematical goals help

develop students‘ mathematical identities and proficiencies.

2. Arranging For Learning- Effective teachers provide students with opportunities to work both

independently and collaboratively to make sense of ideas.

27

3. Building on Students’ Thinking- Effective teachers plan mathematics learning experiences

that enable students to build on their existing proficiencies, interests, and experiences.

4. Worthwhile Mathematical Tasks- Effective teachers understand that the tasks and examples

they select influence how students come to view, develop, use, and make sense of mathematics

5. Making Connections Effective teachers support students in creating connections between

different ways of solving problems, between mathematical representations and topics, and

between mathematics and everyday experiences-

6. Assessment for Learning- Effective teachers use a range of assessment, practices to make

students‘ thinking visible and to support students‘ learning.

7. Mathematical Communication- Effective teachers are able to facilitate classroom dialogue

that is focused on mathematical argumentation

8. Mathematical Language- Effective teachers shape mathematical language by modelling

appropriate terms and communicating their meaning in ways that students understand

9. Tools And Representations- Effective teachers carefully select tools and

representations(number system itself, algebraic symbolism, graphs, diagrams, models, equations,

notations, images, analogies, metaphors, stories, textbooks and technology) to provide support for

students‘ thinking

10. Teacher Knowledge-Teacher content knowledge, Teacher pedagogical content knowledge

The referred UNESCO paper can be downloaded from the websites of the IEA (http://www.iaoed.org ) or

of the IBE (http://www.ibe.unesco.org/ publications.htm)

**********************

28

Qualities of a Successful Mathematics Teacher Sri. E. Krishna Murthy

Principal, KV NFC Nagar, Ghatkesar

A teacher who is attempting to teach without inspiring the pupil with a desire to learn is

hammering on a cold iron ---Horace Mann

Not all students like mathematics, but a good mathematics teacher has the power to change that.

A good mathematics teacher can help students who have traditionally struggled with mathematics begin

to build confidence in their skills. Successful mathematics teachers have certain qualities that make them

the experts they are. These are the teachers required by the society, because of their knowledge, style and

handle on the subject; they know what really work for students.

A good mathematics teacher can be thought to need some qualities that are connected to his view

of mathematics. This view consists of knowledge, beliefs, conceptions, attitudes and emotions. Beliefs

and attitudes are formed on the basis of knowledge and emotions and they influence students' reactions to

learn future Mathematics

A good mathematics teacher should have sufficient knowledge and love of mathematics. He

needs to have a profound understanding of basic mathematics and to be able to perceive

connections between different concepts and fields.

A teacher should have a sufficient knowledge of mathematics teaching and learning. He needs to

understand children' thinking in order to be able to arrange meaningful learning situations. It is

important that the teacher be aware of children‘ possible misconceptions. In addition, he needs to

be able to use different strategies to promote children‘ conceptual understanding.

A good mathematics teacher also needs additional pedagogical knowledge: the ability to arrange

successful learning situations (for example, the ability to use group work in an effective way),

knowledge of the context of teaching and knowledge of the goals of education.

A good mathematics teacher's beliefs and conceptions should be as many-sided as possible and be

based on a constructivist view of teaching and learning Mathematics.

In the classroom, a talented mathematics teacher serves as a facilitator of learning, providing

students with the knowledge and tools to solve problems and then encouraging students to solve

them on their own. When students answer a problem incorrectly, he does not allow them to quit.

He encourages students to figure out where they went wrong and to keep working at the problem

until they get the correct answer, providing support and guidance where needed.

A Good Mathematics teacher should have the ability to do quick error analysis, and must be able

to concisely articulate what a student is doing wrong, so they can fix it. This is the trickiest part of

being a good Mathematics Teacher. He should have ability to assign the home work that targeted

what the student is learning in the classroom to minimize the mistakes committed and to have

proper practice on the concepts taught.

A successful Mathematics Teacher is seen as a leader in his classroom and in the school. His

students respect him, not only for his knowledge of Mathematics, but for his overall attitude and

actions. Students can tell that he respects them as well. He has control over the classroom, laying

out clear rules and expectations for students to follow.

29

A good mathematics teacher focuses less on the content being taught than the students being

taught. A good mathematics teacher cares about his students and recognizes when a student needs

some encouragement and addresses the problem to help the student refocus on the content.

A Good mathematics teacher, in particular, possesses enormous amount of patience, because

there are many different ways that students actually learn mathematics. And they learn at many

different speeds. Math teachers are not frustrated by this attitude of students. He should have

sufficient understanding Jean Piaget‘s theory on how youngsters create logic and number

concepts.

A Good mathematics teacher never lives in the past. He knows how to unlearn outmoded

algorithms and outdated mathematical terms and re-learns new ones. He appreciates the change

with all enthusiasm and welcomes it.

He is approachable and explains, demonstrates new concepts/ problems in detail and creates fun.

He commands respect and love by his subject knowledge and transaction skills.

A good teacher sets high expectations for all his students. He expects that all students can and

will achieve in his classroom. He doesn‘t give up on underachievers.

A great teacher has clear, written-out objectives. Effective teacher has lesson plans that give

students a clear idea of what they will be learning, what the assignments are and what the

promoting policy is. Assignments have learning goals and give students ample opportunity to

practice new skills. The teacher is consistent in grading and returns work in a timely manner.

Successful teacher is prepared and organized. He is in his classrooms early and ready to teach. He

presents lessons in a clear and structured way. His classes are organized in such a way as to

minimize distractions.

Successful teacher engages students and get them to look at issues in a variety of ways. He uses

facts as a starting point, not an end point; he asks "why" questions, looks at all sides and

encourages students to predict what will happen next. He asks questions frequently to make sure

students are following along. He tries to engage the whole class, and he doesn‘t allow a few

students to dominate the class. He keeps students motivated with varied, lively approaches.

A good Mathematics teacher forms strong relationships with his students and show that he cares

about them. He is warm, accessible, enthusiastic and caring. Teacher with these qualities is

known to stay after school and make himself available to students and parents who need his

services. He is involved in school-wide committees and activities and demonstrates a

commitment to the school.

A good mathematics teacher communicates frequently with parents. He reaches parents through

conferences and frequent written reports home. He doesn't hesitate to pick up the telephone to call

a parent if he is concerned about a student.

There are five essential characteristics of effective mathematics lessons: the introduction,

development of the concept or skill, guided practice, summary, and independent practice. There

are many ways to implement these five characteristics, and specific instructional decisions will

vary depending on the needs of the students. The successful mathematics teacher should have

these characteristics in his regular teaching practice.

30

In addition, every good Mathematics teacher has the positive values like Accuracy, Alertness,

Courtesy, Empathy, Flexibility, Friendliness, Honesty, Initiative, Kindness, Loyalty, Patience,

Responsibility, Stability, Tactfulness and Tolerance. ―The mathematics teacher is expected to have proficiency in the methodology ‗cognitive –order –

learning’ which enables the students to acquire subject learning competencies, to develop

problem solving skills, to boost their confidence in the subject, to widen their interest in the areas

of Mathematics, to have and sustain self-directed and self-motivated activities in mathematics

learning, to achieve mastery level learning of the subject, to help apply the skills acquire d in

learning mathematics to other subjects, to utilize the cognitive domain to its fullest extent and to

remove examination phobia‖.

National Curriculum Framework – 2005 envisages that

The main goal of mathematics teacher in teaching Mathematics should be Mathematisation

(ability to think logically, formulate and handle abstractions) rather than 'knowledge' of

mathematics (formal and mechanical procedures)

The Mathematics teacher should have ability to teach Mathematics in such a way to enhance

children' ability to think and reason, to visualize and handle abstractions, to formulate and solve

problems. Access to quality mathematics education is the right of every child.

A balanced, comprehensive, and rigorous curriculum is a necessary component for student

success in mathematics. A quality mathematics program which includes best mathematical tasks and

models to assist teachers is essential in making sound instructional decisions that advance student

learning.

*********

31

TEACHING STRATEGIES IN MATHEMATICS FOR EFFECTIVE LEARNING

Mr.Siby Sebastian

Principal,K V Bijapur(Karnataka)

Mathematics by virtue of its boundless practical applications and tasteful bid of its methods and results

has long held a prominent place in human life. From the quick arithmetic that we do in our everyday lives

to the onerous calculations of science and technology, Mathematics shapes and effects about every item

around us.

But for many secondary and senior secondary students, Mathematics consists of facts in a vacuum, to be

memorized because the teacher says so, and to be forgotten when the course of study is completed. In this

common scenario, young learners often miss the chance to develop skills—specifically, reasoning skills—

that can serve them for a lifetime.

In my 20+ years of mathematics teaching in schools across our country and in foreign lands, I have seen

some truly remarkable changes in the way secondary school children perceive Mathematics and their

ability to succeed in it depend upon the pedagogy.

Discovering approaches to make Mathematics exciting for students who are in the middle of the pack

could have a profound effect on their futures. It would attract many students who are apprehensive in their

own abilities into advanced careers. But it is going to require a fundamentally different approach to

teaching mathematics from childhood through secondary school. Here are a few of the many possible

ideas to begin that change.

Recreational Mathematics

Recreational inspiration consists of puzzles, games or contradictions. In addition to being selected for

their specific motivational gain, these procedures must be brief and simple. An effective implementation

of this procedure will allow students to complete the "recreation" without much effort. Using games and

puzzles can make Mathematics classes very amusing, exciting and stimulating. Mathematical games

provide opportunities for students to be dynamically involved in learning. Games allow students to

experience success and satisfaction, thereby building their enthusiasm and self-confidence. But

Mathematical games are not simply about fun and confidence building. Games help students to:

understand Mathematical concepts, develop Mathematical skills, know mathematical facts, learn the

language and vocabulary of Mathematics and develop ability in mental Mathematics.

Investigating Mathematics

Many teachers show students how to do some problems and then ask them to practice. Teachers can set

students a challenge which hints them to discover and practice some new problems for themselves. The

job for the teacher is to find the right challenges for students. The challenges need to be matched to the

ability of the learners. The key point about investigations is that students are stimulated to make their own

decisions about; where to start, how to deal with the challenge, what Mathematics they need to use, how

they can communicate this Mathematics and how to describe what they have discovered. We can say that

investigations are open because they leave many choices open to the student.

32

Creativity in Mathematics

Creativity is a word that is perhaps more easily associated with art, design and writing than it is with

Mathematics, but this is wrong. Mathematics requires as much creativity in its teaching and learning as

any other subject in the curriculum. It is important to remember that creative teaching and learning not

only needs teachers to use creativity in planning inventive and thought provoking learning opportunities

but must also encourage creative thinking and response from learners. A lesson in which the teachers‘

delivery and resources are creatively delivered but which fails to elicit creative thinking and response

from students has not been fully successfully creative lesson

Problem solving is a key to Mathematics and this in itself presents an excellent way of encouraging

creativity in your lessons. It is a common belief that a degree of rote learning is necessary before learners

can engage in problem solving, but such an attitude may have the effect of pre-empting genuine creative

thinking.

Group work

Research evidences has consistently shown that, regardless of the subject being studied learners working

together in small groups tend to make greater progress in learning what is taught than when the same

content is taught in other more didactic ways. Learners working collaboratively also appear more satisfied

with their classes and have been shown to have greater recollection of learning. There are numerous ways

in which you can arrange learners into groups in your class room. Informal groups‘ can be created by

asking learners to turn to a neighbour and spend 2-3 minutes discussing a question you have posed. Such

informal group can be arranged at any time in a class of any size to check on learners understanding, to

provide an opportunity to apply new knowledge, or to provide change of pace within the lesson. A more

formal arrangement can be made by the teacher establishing the groups. There are conflicting ideas for

this but my personal preference is always for mixed ability group.

ICT in Mathematics Teaching and Learning

Appropriate use of ICT can enhance the teaching and learning of Mathematics in secondary and senior

secondary level. ICT offers powerful opportunities for learners to explore Mathematical ideas, to

generalize, explain results and analyse situations, and to receive fast and reliable, and non-judgemental,

feedback. Their use needs careful planning – not just showing a power point presentation but also of

activities that allow for off-computer Mathematical thinking as well as on-computer exploration.

Decisions about when and how ICT should be used to help teach mathematical facts, skills or concepts

should be based on whether or not the ICT supports effective teaching of the lesson objectives. The use of

ICT should allow the teacher or learners to do something that would be more difficult without it, or to

learn something more effectively or efficiently.

Theatre in Mathematics

The individuals who had the delight of being in front of an audience or performing in any capacity before

an audience needn‘t be convinced about the magic of theatre. The world of theatre is one of the most

important ways children learn about actions and implications, about customs and dogmas, about others

and themselves. Students in every class room can claim the supremacy and potential of theatre today. We

don‘t have to wait for costly tools and amenities. An occasion to create their own dramas based on what

33

they learned in math and backing them in implementation improve the communication, leadership and

motivation skills which will have a long lasting effect in their memory.

The National Focus Group Position Papers on all segments linking to education are an extraordinary

repository of ideas, theory and procedure for teachers. The position paper devoted to Arts, music, dance

and theatre clearly mentions why and how it may be integrated in the classroom and invokes what is

called ―Sensitivity Pyramid through Drama‖. In cognizance with the NFG position papers theatre can

work extremely well as micro level experimental innovative and creative math pedagogy. Theatre is an

effective learning tool as it deals with action and imagination, understanding the concept being taught

with a view to applying this understanding to real life situations.

Function dance performed by class XI students

***************************

34

CLASS XII MATHEMATICS

CHAPTER WISE

35

CHAPTER 1

Relations and Functions

LEARNING OBJECTIVES/OUTCOMES

1. RELATION

2. TYPES OF RELATIONS

a) Empty Relation

b) Universal Relation

c) Reflexive Relation

d) Symmetric Relation

e) Transitive Relation

3. EQUIVALENCE RELATION

A Relation which is Reflexive, Symmetric and Transitive.

4. FUNCTION

To recall - identity , constant, linear, polynomial, rational, Signum, modulus,

greatest integer functions.

5. TYPES OF FUNCTIONS

a) One – One ( Injective mapping)

b) Onto ( Surjective mapping)

c) One – One and Onto ( Bijective mapping).

6. COMPOSITION OF FUNCTIONS

To recall algebra of functions and to introduce composition of functions.

7. INVERSE OF A FUNCTION

Problems based on linear, quadratic and rational functions.

8. BINARY OPERATIONS

Operation Table.

9. PROPERTIES OF BINARY OPERATIONS

a) Commutative

b) Associative

c) Existence of Identity element

d) Existence of Inverse element

e) To check properties of binary operation using operation table

36

CONCEPT MAPPING

37

THREE LEVELS OF GRADED QUESTIONS

Level I.

1. Check whether the relation R defined in the set { 1, 2, 3, 4} as R = {(1, 2), (2, 2), (1, 1),

(4, 4),( 1, 3), (3, 3), (3, 2)} is Reflexive, Symmetric and Transitive.

2. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but

neither Reflexive nor Transitive.

3. Show that R defined as R = {(a, b): a ≤ b, a, b R} is Reflexive and Transitive but not

Symmetric.

4. Check whether the following functions are one-one and onto.

a) f: NN given by f(x) = x2

b) f: RR given by f(x) = x2

5. If the functions f: RR given by f(x) = x2

+ 3x + 1 and g : RR is given by

g(x) = 2x – 3, find gf and fg

6. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A

to B.Check whether f is one-one or not.

7. If f(x) = x + 7 and g(x) = x – 7, x R, find )7(gf

8. If f:RR is an invertible function, find the inverse of f(x) = 5

23 x

9. Let * be a binary operation defined by a * b = 3a +4b -2. Find 4 * 5

10. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by

R = {(a, b): |a – b| is a multiple of 4 } is an equivalence relation.

11. Check whether the function f: RR defined by f(x) = 4+ 3x is one – one and onto. If

so, find the inverse of f.

12. Let be defined as f(x) = 10x +7. Find the function such that

g o f = f o g =IR.

13. Let * be a binary operation on the set Q defined by a * b = a + ab, check whether * is

commutative and associative.

14. Let * be the binary operation on the set {1, 2, 3, 4, 5} defined by

a * b = HCF of (a and b). Is the operation * a) Commutative b) Associative

Level II

1. Check whether the relation R defined in the set { 1, 2, 3, 4, 5 , 6} as

R = {(a, b): b = a+ 1} is Reflexive, Symmetric and Transitive.

2. If f : RR be given by f(x) = 31

33 x , then find )(xff

3. Let * be a binary operation on the set Q of rational number such that a * b =4

ab.

Find the identity element for the operation * on Q.

38

4. Let f :R –{-3/5)R defined by f(x) = 35

2

x

xis invertible. Find inverse of f.

5. Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is

similar to T2} is an equivalence relation.

6. Let A and B be sets. Show that f : A x BB x A such that f(a, b) =(b, a) is bijective

function.

7. Le A=N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d)

.Show hat * is commutative and associative. Find the identity element for *on A, if any.

8. Let f : N N be defined by f(x)={

for every n . Check whether the

function is bijective.

9. Consider f:R+ [ given by f(x) = f is invertible with the

inverse of f given by (y) = √ where R+ is the set of all non- negative

real numbers.

Level III

1. If f: R R is defined by f(x) = x2 – 3x +2, find f(f(x))

2. Find the number of binary operations on the set {a, b, c}.

3. If R1 and R2 are Equivalence Relation in a set A, Show that is also an

equivalence relation.

4. Given a non-empty set X, let *: P(X) x P(X) P(X) be defined as A * B = ( A – B)

(B – A), for all A, B P(X). Show that empty set is the identity element for the

operation * and all the elements A of P(X) are invertible with = A

5. Show that the function f: R {x R :-1 < x < 1} defined by f(x) =

, x R is one-

one and onto function.

6. Let f: N R be a function defined as f(x)= 4x2 + 12x + 15. Show that f: NRange f is

invertible. Also find the inverse of f.

VALUE BASED QUESTIONS

1. Let A be the set of all students of class XII in a school and R be the relation, having the

same sex on A, and then prove that R is an equivalence relation.

Do you think, co-education may be helpful in child development and why?

2. Show that the function f: R+ R

+ defined by f(x0 = 3x+ 4 is an invertible function.

If x represents the number of systematic hours of study that a student puts in and f(x)

represents the marks scored by him, from the graph of the above function, which value will

be rewarded?

3. Consider a relation R in the set of people in a colony defined asaRbiff a and b are members

of joint family. Is R an equivalence relation?

Staying with Grandparents in a joint family imbibes the moral valuesin us. Can you elicit 2

such values?

4. Let R be a relation defined as R = { (x,y) : x and y study in the same class}. Show that R is

an equivalence relation.

39

If x is a brilliant student and y is a slow learner and x helps y in his studies, what quality

does x posses?

5. Consider the functions f and g, f:{ 1,2,3 } { a,b,c and

g : { a,b,c { punctuality,honesty,sincerity }

defined as f(1)=a,f(2) = b,f(3) = c,g(a) =punctuality, g(b) = honesty, g(c) =sincerity.

Show that f,g,gof are invertible.

If a,b,c are three students who are awarded prizes for the three values given in the function

g, which value would you prefer to be rewarded and why ?

6) Prove that f: R R is a bijection given by f(x)=x3+3 . Find f

-1(x) .

Do the truthfulness and honesty have any relation ?

7) Set 54321 ,,,, aaaaaA and 4321 ,,, bbbbB were ai‘s and bj‘s are school going

students. A relation R is defined from the set A to the Set B by x R y iff y is a true

friend of x. R= 2524331211 ,,,,,,,,, bababababa . Is R is a bijectivefunction ?

Do you think true friendship is important in life ? How?

8) If h denotes the number of honest people and p denotes the number of punctual people

and a relation between honest people and punctual people is given as h = p+5. If P

denotes the number of people who progress in life and a relation between number of

people who progress and honest people is given as P = (h/8)+5. Find the relation

between number of people who progress in life and punctual people .

How is the punctuality important in the progress of life?

Error analysis and remediation

Concept Error Correction

Proving

Equivalence

Relation

Students Prove it by taking

particular elements

Stress on to prove it by

taking arbitrary elements

Finding inverse of a

function

Some times children don‘t prove

the bijectivity and at the end they

don‘t show the inverse in terms of

the variable of the function

Stress on to verify the

bijectivity and to give the

inverse in terms of the

variable in which the

function is defined

Finding the identity

of a binary

operation defined

on a set A= NxN

Students consider single element

instead of ordered pair of elements

to apply the operation and they

unable to proceed

Insist the students to first

identify the sets and practice

accordingly

40

QUESTION BANK

41

42

43

44

POWER POINT PRESENTATION

PPT-Relations and Functions.pptx

https://youtu.be/TJrTlA1hMVw

https://youtu.be/3P9FFg4eKHw

https://youtu.be/FxLGJLXCdmU

45

CHAPTER 2

Inverse Trigonometric Functions

LEARNING OBJECTIVES/OUTCOMES

1. RECALL ALL THE TRIGONOMETRIC IDENTITIES

2. DOMAIN AND PRINCIPAL VALUE BRANCH OF ALL INVERSE TRIG.

FUNCTION

The following table gives the inverse trigonometric function (principal branches)

along with their domains and ranges

sin-1

: [-1, 1]

2,

2

cos-1

: [-1, 1] [0, ]

cosec-1

: R – (-1, 1)

2,

2

- {0}

sec-1

: R – (-1, 1) [0, ] – {2

}

tan-1

: R

2,

2

cot-1

: R (0, )

3. PROPERTIES OF INVERSE TRIG. FUNCTIONS

4. APPLICATION PROBLEMS

a) Evaluate the Inverse Trig. Functions

b) Evaluating the Problems by using Trig. Identities

i) a2 + x

2 Put x = a tan or x = a cot

ii) a2 - x

2 Put x = a sin or x = a cos

iii) x2 – a

2 Put x = a sec or x = a cosec

c) To reduce into simplest form

d) To solve :- for the unknown variable x or

e) Problems involving cos -sin >0 if

4,0

&

sin -cos >0 if

2,

4

&

46

CONCEPT MAPPING

47

THREE LEVELS OF GRADED QUESTIONS

Level I

1. Find the principal value of

2. Find the principal value of √

3. Find the principal value of √

4. Find the value of

5. Find the value of (√ ) √

6. Find the value of

7. If sin( (

) then find the value of x

8. Find the value of

9. Find the value of cot(

10. Find the value of sin(

11. Find the principal value of √

12. Prove that

13. Solve :

14. Solve :2

Level II

1. Find the value of (

)

2. Find the value of tan( (

)

3. Find the value of

4. Find the value of cosec( √

5. If

6. Write in simplest form:

48

7. Find the value of tan

,

8. Show that

9. Prove that (√ √

√ √ )

10. Prove that 2

11. Prove that

Level III

1. If

find the value of x.

2. Find the value of sin(

3. Find the value of sec( √ )

4. Find the value of ( √

√ )

5. Find the value of cos(

6. Show that

7. Prove that (√ √

√ √ )

,

8. If

9. If

10. If

=

11. Solve for x: (

) (

)

12. Prove that

13. Prove that cos[ ] √

14. Show that

15. Solve:cos(

16. Show that

=

17. Simplify: √

49

18. Prove that sin[ ] √

19. If = prove that a + b + c = abc

20. Show that : 2 (

(

)) =

ERROR ANALYSIS AND REMEDIATION

Concept Error Correction

Finding principal value Wrong application of principal

value

Insist the children to be

thorough with the principle

value of each trigonometric

function

Evaluation of a composition of

trigonometric function & its

inverse

Don‘t check whether the value

lies in the principal value or

not

Stress on to check it before

they write the solution

Solving problems using

properties

Wrong application Insist them to thorough with

the properties

QUESTION BANK

1. Evaluate : ( (

))

2. Write the principal value of ( √ )

3. Write the value of ( (

))

4. Find the value of

5. Find the value of (

)

6. Solve : 2 =

7. Write the principal value of (

)

8. Evaluate : (

)

9. Find the value of

+ 2 (

)

50

POWER POINT PRESENTATION

PPT- INVERSE TRIGONO.pptx

WEB LINKS

https://youtu.be/-FGFEfqs7BI

https://youtu.be/v0VB_xqHLYI

https://youtu.be/3d3HEDhJ_wQ

51

CHAPTER3

MATRICES

EXPECTED LEARNING OUTCOMES:

The teacher as a facilitator should make the students to know about the chapter before going into

the details of the concept. The teacher should keep in mind that all varieties of student are there

in a given classroom situation. The following points can be explained by the teacher in the class.

1. Definition

2. Order of a Matrix

3. Construction of a Matrix of given order &identifying entries(example :Find 23,22,11 aaa

etc)

4. (Construct a matrix of order 2

232

jiaasdefinedareesWhoseentri ij

)

5. Types of matrices

i)column matrix, ii)row matrix, iii)square matrix, iv)diagonal matrix, v) scalar matrix

vi)identity matrix, vii)zero or null matrix

6. Equality of matrices-Finding the unknown values x,y if given two matrices are equal

7. Matrix operations

i) addition ii)subtraction,

iii) scalar multiplication-(multiply all the entries of the matrix with the scalar k,

iv) multiplication of matrices-

a)Matrix multiplication need not be commutative b)Matrix multiplication is

associative c)Product of two non zero matrices can be a zero matrix.

8. Transpose of a matrix &its properties:

i) AA //

, ii) TTTABAB

, iii)(kA)T=kA

T ,

TTTBABAiv )

9. Symmetric &skew –symmetric matrices

10. i)AT=A (Symmetric) , ii)A

T=-A( skew –symmetric)

11. Writing a given square matrix as a sum of a symmetric & a skew symmetric matrices.-

symmetricskewisAAsymmetricisAA // ,

12. //

2

1

2

1AAAAA

13. Elementary row and column transformations.

a).If row transformation is used

i)Use A=IA

Ii)Use only the following three transformations

ji RR , ii kRR

,numberrealnonzeroanyiskwherekRRR jii ,

52

b).If column transformation is used

i)Use A=AI

ii)Use only the following three transformations

ji CC

. ii kCC ,

numberrealnonzeroanyiskwherekCCC jii ,

12. Finding the inverse of a square matrix using elementary row or column transformations

IAAAA

ABAB

11

111

13. Invertible matrix( Let A and B be two square matrices, if AB=BA =I then B is called the

inverse of the matrix A and A is called an invertible matrix . Inverse of A is denoted by A-1

)

53

CONCEPT MAPPING

54

THREE LEVELS OF GRADED QUESTIONS

S.No. LEVEL- 1

1. If a matrix has 5 elements, write all possible order it can have?

2. If *

+ * + *

+ , find the value of x.

3. Construct a 3x3 matrix A, where aij = 2i – 3j .

4. If order of matrix A is 2x3 and order of matrix B is 3x4 , find the order of AB.

5. Write the value of x + y + z if [

] 0 1 [

]

6 For what value of x, is the matrix A=[

] a skew-symmetric

matrix?

7 If A=*

+ and B= [

], write the order of AB and BA.

8 If A, B and C are three non-zero square matrices of same order, find the

condition on A such that AB = AC ⇒ B = C.

9 Give an examples of two non-zero 2×2 matrices A and B such that AB=0.

10 Find the value of x and y if [

] *

+ *

+

11 If A

T=[

] *

+

12 If (

) (

)

, then find x and y.

13 If (

) , find A + A ʹ

14 If (

) (

)

, then find x,y and z.

15 If A is a matrix of order 2×3 and B is of matrix of order 3×5, what is the order

of the matrix (AB)ʹ ?

16 Using elementary operation , find the inverse of the matrix A =*

+

17 I (

) (

), find the value of

18 If (

) (

),find i)3A+2B ii)A- 3B

55

S.No LEVEL II

1 If A is square matrix satisfying A2

= I, then what is the inverse of A?

2 If A =(

) .

3 Find the product[

] [ ]

4 If the matrix[

] is skew symmetric, find a + b + c .

5 If A =*

+ and B = *

+ , 2A + B + X = 0 , find X.

6 If 2 A + 3 B =*

+ and 3 A + 2 B = *

+ , find A and B.

7 If A =*

+ and B = *

+ , then verify that (AB)ʹ= BʹA. ʹ

8 Find A such that A *

+= *

+ .

9 Show that A = [

] satisfies A2 – 4A – 5 I = 0, and hence find A

-1.

10 If A =*

+, Prove that A + A ʹ is a symmetric matrix.

11 Using elementary operation , find the inverse of the matrix A =*

+

12 Find the matrix X, for which *

+ X = *

+

13 Let A = [

]. Express A as a sum of symmetric and skew symmetric matrices.

14 If A ʹ= [

] and B = *

+ , then find A ʹ- B ʹ

15 If A =*

+, f(x) = x2 -2x – 3 , show that f(A) = 0.

16 Solve for x and y: *

+ * + *

+ .

17 If A =*

+, find k so that A2 = 8A + k

56

18 Express the following matrix as the sum of a symmetric and skew symmetric matrix and

Verify your result: [

]

19 Using elementary operations, find the inverse of the following matrix, : [

]

20 If A = *

+ and B =*

+, verify that (AB) ʹ= B ʹA ʹ

21 If A = *

+ prove that An = *

+ for all positive integers n

LEVEL III

1 Find the inverse of the following matrix using elementary operations:

(

)

2 If A

-1 = [

] and B = [

] , find (AB)-1

3 Find X, if X +*

+ = *

+ .

4 Find x, if [ ] *

+ * + = 0.

5 If A =*

+,Prove that A+ A ʹis symmetric and A – A ʹ is skew symmetric

matrices

6 Using elementary operation, find the inverse of the matrix

A = [

].

7 If A = [

] and B = [

] , compute A2B

2

8 If f(x) =[

] , then show that f(x) f(y) = f ( x +y).

9 If A = *

+ and kA = *

+ , find a,b and k.

57

10 If AAʹ= I where A = [

] find a,b and c.

11 If A = *

+ and A2 = [

] Then prove that α + β = (a + b )2.

12 If A = *

+, then prove by principle of mathematical induction

that An = *

+ .

13 Let A = *

+, Show that ( a I + b A )n = a

n I + n a

n-1 b A, where I is the

identity matrix of order 2 and n N.

VALUE BASED QUESTIONS

1 For keeping Fit X people believes in morning walk, Y people believe in yoga and Z people join

Gym. Total no of people are 70.further 20% 30% and 40% people are suffering from any

disease who believe in morning walk, yoga and GYM respectively. Total no. of such people

is 21. If morning walk cost Rs.0 Yoga cost Rs.500/month and GYM cost Rs.400/ month and

total expenditure is Rs.23000.

(i) Formulate a matrix problem.

(ii) Calculate the no. of each type of people.

(iii)Why exercise is important for health?

2. An amount of Rs. 600 crores is spent by the government in three schemes. Scheme A is for

saving girl child from the cruel parents who don‘t want girl child and get the abortion before

her birth. Scheme B is for saving of newlywed girls from death due to dowry. Scheme C is

planning for good health for senior citizen. Now twice the amount spent on Scheme C

together with amount spent on Scheme A is Rs 700 crores. And three times the amount spent

on Scheme A together with amount spent on Scheme B and Scheme C is Rs 1200 crores.

Find the amount spent on each Scheme using matrices? What is the importance of saving girl

child from the cruel parents who don‘t want girl child and get the abortion before her birth?

3. There are three families. First family consists of 2 male members, 4 female members and 3

children. Second family consists of 3 male members, 3 female members and 2 children.

Third family consists of 2 male members, 2 female members and 5 children. Male member

earns Rs 500 per day and spends Rs 300 per day. Female member earns Rs 400 per day and

spends Rs 250 per day child member spends Rs 40 per day. Find the money each family

saves per day using matrices? What is the necessity of saving in the family?

58

4 Two schools A and B decided to award prizes to their students for three values honesty (x),

punctuality (y) and obedience (z). School A decided to award a total of Rs. 11000 for the

three values to 5, 4 and 3 students respectively while school B decided to award Rs. 10700

for the three values to 4, 3 and 5 students respectively. If all the three prizes together amount

to Rs. 2700, then

i)Represent the above situation by a matrix equation and form Linear equations

using matrix multiplication.

ii) Is it possible to solve the system of equations so obtained using matrices?

iii) Which value you prefer to be rewarded most and why?

MATRICES( SOLUTIONS )

Ans.1. (i) x+y+z=70, 2x+3y+4z=210, 5y+4z=230 (ii) x=20, y=30, z=20 (iii) Exercise keeps fit and healthy to a person.

Ans.2. Rs300crores, Rs200crores and Rs100 crores

Our In country, male population is more than female population

(i) It is essential for a human being to save the life of all.

Ans.3. Rs880, Rs970, Rs 500. Saving is necessary for each family as in case of emergency our

saving in good time helps us to survive in bad time.

59

ERROR ANALYSIS AND REMEDIATION

Name of Unit Concept Probable errors by

students

Remediation

MATRICES Order Generally Student

take row as

column and

column as row

Make them understand row means horizontal,

column means vertical more such problems

for practice should be given. The position of

the student in the classroom if it is neatly

arranged.

Product of

Matrices

The meaning of

the order of the

matrix the teacher

should specify.

Not multiplying

the first matrix

row elements with

the second matrix

corresponding

column elements

and add.

By giving tips like Run and Jump remember

while multiplying two matrices. More

practice on various order matrices for

multiplication.

The teacher can have some playway method

devised.

Transpose

of a Matrix

Converting both

column into rows

and rows into

columns

Stress to be given only to change row into

columns or vice versa but not both.

Adjoint of a

Matrix

a) For finding co-

factor not taking

proper sign

b)Not taking

transpose of a co-

factor matrix

Make them to find co-factors by using

and insists them to take transpose.

After getting cofactor matrix the teacher can

insist upon converting rows to column and

colums to row.

60

QUESTION BANK CBSE-XII-HOTS-Matrix-Chapter-3.pdf POWER POINT PRESENTATION

Matrices- Mr D Sreenivasulu.pptx

Matrices- Mr. G Mathilagan.ppt

determinants by Srinivas.pptx WEB LINKS 1. Matrices and Determinants https://youtu.be/mYVbYBZZdW0 2. Order of a matrix, number of elements in a matrix NCERT solution Maths

chapter 3 Matrices

https://youtu.be/KLZlJduSoXY?list=PLauXkHsTK5c_oL12O_7tUwc0GkIHZo-uJ 3. Matrix CBSE Class 12 Math Hindi Medium Ganit Video Lecture https://youtu.be/fzVwSbKu0Uo 4. Class 12 Maths Matrices Properties of Adjoint of a Matrix Video Lecture

https://youtu.be/SK8sN-NErCE

61

CHAPTER 4

DETERMINANTS

LEARNING OBJECTIVES/ OUTCOMES

The teacher should explain the following points before explaining the concepts.

1. Explain the difference between a matrix and a determinant. Determinant is a number or a

function we associate with a matrix.

2. Determinant of a matrix of order 1 and order 2.

3. Determinant of a matrix of order 3. It can be done by expanding about any row or any

column.

4. Properties of determinants.

i)The value of the determinant remains unchanged if its rows and columns are

interchanged.

ii)If any two rows (or columns) of a determinant are interchanged, then sign of determinant

changes.

iii) If any two rows(or columns) of a determinant are identical then the value of the

determinant is zero.

iv )If each element of a row(or a column of a determinant is multiplied by constant k,

then its value gets multiplied by k.

v )If some or all elements of a row or column of a determinant are expressed

as sum of two(or more) terms, then the determinant can be expressed a s sum of two

(or more) determinants.

vi)If to each element of any row or column of a determinant, the equimultiples of

corresponding elements of other row(or column) are adds, the value of determinant

remains the same. Ie.RiRi+kRj orCiCi + kCj then value of determinant remains the

same.

5. Solving problems using properties of determinants ie. problems involving proving LHS =

RHS using properties.

6. Finding the area of a triangle using determinants.

7. Finding minors and co-factors of the matrix .

Aij= (-1)i+j

Mij

8. = a11A11+ a12A12 + a13A13. whereAij are cofactors of aij .

9. If elements of a row (or column) are multiplied with cofactors of any other row (or

column) then their sum is zero.

10. Finding the adjoint and inverse of a matrix.

11. A is singular matrix implies = 0.

12. . But if IAI =I kBI where k is a constant then =kn

where n is the order of Matrix A.

13. = n-1 , where n is the order of the square matrix A

62

14. )T = (A

T)-1

15. adj(AT) = (adj A)

T

16. = (n-1)2

17. If *

+ then adj A =*

+ (Note: to find adjA interchange main

leading diagonal and change the sign of other diagonal. )

18. If A is a square matrix of order n then IkAI= knIAI

19. A-1

=

adj A.

20. (A-1

)T=(A

T)-1

21. Working rule for solving of linear equations in 3 variables.

Step 1: Express the system of linear equation as AX = B find Step 2: If 0, then the system of equation is consistent and has a unique solution given

by

X = A-1

B.

Step 3: If = 0 and (adjA)B 0, then the system is inconsistent.

Step 4: If = 0 and (adjA)B = 0, then the system may or may not be consistent. In case

it is consistent the system has infinite number of solutions.

Step 5: In order to find the solutions when system has infinitely many solution, take any

one variable equate it to t or k. Find the values of other variables in terms of t or k.

63

CONCEPT MAPPING

THREE LEVELS OF GRADED QUESTIONS

LEVEL I

1. *

+then show that = 4 .

2 Find the value of x if |

| |

|.

3 Using property prove that |

| = 0.

Using property prove that |

|= 0.

5 Find the area of the triangle with vertices (-2, -3), (3,2) and (-1, -8) .

64

6 Using co-factors of second row, evaluate = |

|.

7

Using co-factors of elements of third column evaluate = |

|.

8 Find the adjointof *

+.

9 Find the inverse of *

+.

10 If A is of order 3x3 and = 5 then find the .

11 If Aijis the co factor of the element aij if the determinant |

|write the value of

a32 .A32

S.No LEVEL II

1 If A =[

], then show that = 27 .

2 Using properties evaluate |

|.

3 Prove that |

|= 4 abc.

4 Using properties show that |

| = (a-b)(b-c)(c-a).

5 Solve 5x+2y =4 ; 7x+3y = 5.

6 Find the equation of the line joining the points (1,2) and (3,6) using determinants.

65

7 Verify A(adjA) = (adj A) A = I if A = [

].

8 If A = *

+ and B = *

+ verify that (AB)-1

= B-1

A -1

.

9 If A = *

+ then show that A2 – 5A + 7I = 0. Hence find A

-1.

10 11. Evaluate the determinant

11 If A is of order 3 anddet A = 4, find det(3A).

12

Evaluate

13 Show that the points A (a, b+c), B (b,c+a), C(c,a+b) are collinear.

14 Find values of k if area of triangle is 4 sq.units and vertices are (-2,0),(0,4), (0,k).

15 Find equation of line joining (1,2) and (3,6) using determinants.

16 If A is of order 3 and det (A) = -2, find det (AdjA)

17 Find k if

is singular

18 Prove that

= a3

66

19

Show that

= abc + bc + ca + ab =abc ( 1 +

+

+

)

20

Using properties of determinants prove that

= 2

21 Using properties of determinants, Prove that

= 1 +a2 +

b2

+ c2

22 Using properties of determinants, solve the following for x :

|

|

23 Using properties of determinants, solve the following for x :

|

|

24 Using properties of determinants, prove the following:

|

| .

25

If x, y, z are the 10th, 13th and 15th terms of a G.P. find the value of

115log

113log

110log

z

y

x

67

LEVEL III

1 If (

) , find A-1

and hence solve the system of equations:

.

2 If (

) (

) , find AB. Use this to solve the

following system of equations. x - y =3 , 2x +3y + 4z =17 and y + 2z = 7 .

3 Examine the consistency of the following system of equations.

3x-y+7y=3, 2x+y+3z, x+4y-2z

4 Examine the consistency of the followi8ng system of equations

x –y + z=3, 2x+y-z=3, -x-2y+2z=1

5 If A is a matrix of order 3 X 3 then verify that 12 A = 21A

6 If A is a square matrix of order 3 such that AfindadjA ,64

7 If A and B are non singular square matrices of the same order, then write the relationship

between adj AB, adj A and adj B.

8 If A is invertible matrix of 3 X 3 and 7A then find 1A

9 If a, b and c are real numbers and

|

| . Show that either a+b+c=0 or a=b=c.

10 Using properties of determinants, prove that if x, y and z are different and

|

| , show that 1+xyz=0.

11 Using properties of determinants, prove the following:

|

| .

12 Using properties of determinants, solve the following for x :

|

|

68

13 Use the product [

] [

] to Solve the system of equations:

x – y +2z = 1,2y – 3z = 1, 3x – 2y + 4z = 2.

14 Using matrices solve the following system of equations:

2x – 3y +5z = 11, 3x + 2y -4z = -5, x + y -2z = -3

15 Use matrix method, solve the following system of equations:

+

+

= 4,

-

+

= 1,

+

-

= 2: where x

16 There are 3 families A, B and C. The number of men, women and children in these families

are given below:

Men Women Children

Family A 2 3 1

Family B 2 1 3

Family C 4 2 6

Daily expense of men, women and children are Rs200, Rs150 and Rs 200 respectively.

Only men and women earn and children do not. Using matrix multiplication, calculate the

daily expense of each family. What impact does more children in the family create on the

society?

17 Two institutions decided to award their employees for the three values of Resourcefulness,

competence and determination in the form of prizes at the rate Rs x, Rs y and Rs z

respectively per person. The first institution decided to award 4, 3 and 2 respectively with

total prize money of Rs37000 and the second institution decided to award respectively 5, 3

and 4 with total prize money of Rs47000. If all the three prizes per person together amount

Rs 12000, then using matrix method find the values of x, y and z.

VALUEBASED QUESTIONS

Q.1. Three shopkeepers A, B, C are using polythene, handmade bags (prepared by prisoners), and

newspaper‘s envelope as carry bags. it is found that the shopkeepers A, B, C are using

(20,30,40) , (30,40,20,) , (40,20,30) polythene , handmade bags and newspapers envelopes

respectively. The shopkeepers A, B, C spent Rs.250, Rs.220 & Rs.200 on these carry bags

respectively .Find the cost of each carry bags using matrices. Keeping in mind the social &

environmental conditions, which shopkeeper is better? & why?

Q.2 In a Legislative assembly election, a political party hired a public relation firm to promote its

69

candidate in three ways; telephone, house calls and letters. The numbers of contacts of each

type in three cities A, B & C are (500, 1000, and 5000), (3000, 1000, 10000) and (2000, 1500,

4000), respectively. The party paid Rs. 3700, Rs.7200, and Rs.4300 in cities A, B & C

respectively. Find the costs per contact using matrix method. Keeping in mind the economic

condition of the country, which way of promotion is better in your view?

Q.3 A trust fund has Rs. 30,000 is to be invested in two different types of bonds. The first bond

pays 5% interest per annum which will be given to orphanage and second bond pays7% interest per annum which will be given to an N.G.O. cancer aid society. Using matrix multiplication, determine how to divide Rs 30,000 among two types of Bonds if the trust fund obtains an annual total interest of Rs. 1800. What are the values reflected in the question.

Q.4 Using matrix method solve the following system of

equations x + 2y + z = 7 x – y + z =4

x + 3y +2z = 10 If X represents the no. of persons who take food at home. Y represents the no. of persons who take junk food in market and z represent the no. of persons who take food at hotel. Which way of taking food you prefer and why?

Q.5 A school has to reward the students participating in co-curricular activities (Category I) and with 100% attendance (Category II) brave students (Category III) in a function. The sum of the numbers of all the three category students is 6. If we multiply the number of category III by 2 and added to the number of category I to the result, we get 7. By adding second and third category would to three times the first category we get 12.Form the matrix equation and solve it.

DETERMINANTS ( SOLUTIONS ) Ans.1 [Polythene=Re.1] [Handmade bag = Rs.5] [Newspaper‘s envelop=Rs.2]

Shopkeeper A is better for environmental conditions. As he is using least no of polythene.

Shopkeeper B is better for social conditions as he is using handmade bags (Prepared by

prisoners). Ans.2 Cost per Contact:

Telephone = Rs0.40 House calls = Re1.00

Letters

=

Rs0.50 Telephone is better as it is cheap.

Ans.3 Rs.15000 each type of bond.

(i) Charity. (ii) Helping orphans or poor

people. (iii)Awareness about

70

diseases. Ans.4 X = 3, Y =1, Z = 2

Food taken at home is always the best way.

Ans.5 x+y+z=6, x+2z=7, 3x+y+z=12 where x,y,z represent the number of students in

categories I,II,III respectively. X=3, y=1, z=2

71

ERROR ANALYSIS AND REMEDIATION

Name of Unit Concept Probable errors by

students

Remedition taken by teachers

DETERMINANTS Applying

Rules

In finding

Inverse

1.Directly try to

expand

2.While changing

a row or column

multiplying by a

scalar or sign

1.While applying

the elementary

operations up to

some steps row

operations

afterwards column

operations they

use.

2. From word

finding difficult to

convert into

equations

By using rules of determinants try to make

maximum number of zeroes in a row or

column and expand according to the

convenience where more number of zeros are

there..

If a row or column to be changed that

particular row or column is not to be

multiplied by a scalar or sign. If it is

unavoidable to compensate determinants itself

to be dived by such scalar. Such Problems are

to be practiced more.

1.It must be insisted that throughout the

process either use row or column operations

but not both.

2. More drill involving various possibilities

are to be practiced.

In linear equations insist for verification of

solution.

72

QUESTION BANK

CBSE-XII-HOTS-DETERMINANTS-Chapter-4.pdf

1. INTRODUCTION - DETERMINANTS https://www.youtube.com/watch?v=R5jASuOagzU 2. DETERMINANT PART CONCEPTS

https://youtu.be/4C_m0NywM_o?list=PL39FDA035AFD825C6 3. Determinant row transformation

https://youtu.be/Ztb9vUEnB_o

TIPS AND TECHNIQUES

MATRICES AND DETERMINANTS

1. In finding inverse of a matrix by elementary row transformation remember the word RIA

(R-for row transformation, I- for unit matrix, A- for given matrix).

2. For finding the adjoint of a 3 x 3 square matrix (

) for finding first row

co-factors write second and third row elements in order starting from second element

gives similarly second row co-factors

gives for third row co-factors

gives

Web Links

Determinant part 1

https://youtu.be/R5jASuOagzU Lecture on properties of Determinant https://youtu.be/I5XbsLClLc4 Maths – Elementary row operation https://youtu.be/Ztb9vUEnB_o Elementary row operation https://youtu.be/DH2JSYx52nk Properties of adjoint of a matrix https://youtu.be/HxkMFNR7XHg

73

Chapter 5

Continuity and Differentiability

LEARNING OBJECTIVES/OUTCOMES

Understanding the concept of Continuity and differentiability and addressing the problems based on continuity

and derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of

implicit functions.

Learning the concept of exponential and logarithmic functions.

Skills to solve derivatives of logarithmic and exponential function. Logarithmic differentiation, derivative of

functions expressed in parametric forms. Second order derivatives, Rolle‘s and Lagrange‘s Mean Vale

Theorems and their geometric interpretation.

CONCEPT MAPPING

74

75

THREE LEVELS OF GRADED QUESTIONS

Level I

1 Examine the continuity of f(x) = x - 5

2 Examine the continuity of f(x) =

,

3 Find the value of k so that f(x)= {

is continuous at x=2

4 If y=(logx)2

find

5 If y= esinx

find

6 Find

if 2x+3y=sinx

7 Find

if (

)

8 Differentiate with respect to x : y=

9 Find

if x=at

2 , y= 2at

10 If y= asinx+bcosx, then prove that y’’+ y = 0

Second Phase Practice(Self Assessment)

1 Find the derivative of (3x2-9x+5)

9 with respect to x

2 Find the derivative of cos-1

(x/2) with respect to x

3 Find the derivativeof cos2x+cos5x-secx, with respect to x

4 Find the second order derivative of y=x3

+ tanx

5 Differentiate log(logx) with respect to x.

LEVEL II

First Phase

1 If √ , √ , Show that

= -

2 Differentiate [

]with respect to x

3 Differentiate with respect to x

4 Differentiate . with respect to x

5 Differentiate ( (√ )) with respect to x

6 If

in terms of y alone.

7 Find all the points of discontinuity of the function f defined

by

20

212

12

)(

xif

xifx

xifx

xf

8 f , Prove that

9 Find

if y= [

√ √

√ √ ], 0 <

76

10 Show that the function f(x)= is continuous at every x , but fails to be

differentiable at x= 2

11 If y=3cos(logx) + 4sin(logx), Show that x2y2 +xy1 + y = 0

12 Use Lagrange‘s mean value theorem to determine a point P on the curve y=√ ,

where the tangent is parallel to the chord joining (2,0) and (3,1).

13 If

LEVEL III

14 For

{

iscontinuous at x=1.

15 If

16 If

17 If (

) then prove that

18 If √ √ , prove that

Find the relationship between ‗a‘ and ‗b‘ so that the function ‗f‘ defined by

,

19

The function f(x) is defined as f(x) {

.

if f(x) is continuous on [0,8], find the values of a and b.

20 If

21 If

find

22

Find the value of k for which {

√ √

is continuous at x = 0

Second Phase Practice(Self Assessment)

1 Find the relationship between ‗a‘ and ‗b‘ so that the function ‗f‘ defined by :

,

2 Find all the point of discontinuity of the function f defined by

77

f(x)={

3 If ( √ )

, show that

.

4 1. Differentiate the following with respect to x : (

)

5 IF x=√ √

6 Differentiate [

]with respect to x.

VALUE BASED QUESTIONS

1. 0,cos3cos

)(2

xx

xxxf show that f(x) is continuous at x =0

Mention any twofactors that will affect the continuity of thought while writing the examination.

2. The path of a moving bike is given by

012

012)(

xifx

xifxxf Find the point of discontinuity .

Whether the rider should pass that point or not? Justify your answers.

Error analysis and remediation

S.No. Common Error Measures

1. Partially writing the Condition for continuity Insist them to follow the definition

of continuity correctly

2. In Implicit function, starting the differentiation with

Insist them to differentiate the

equation directly w.r.t.x

3. In logarithmic differentiation,

if y = u+v form, log is taken in the beginning

More practice to be given in those

type of questions

4. Writing (logx)n as nlogx Give the power rule

logxn = nlogx

5. In Rolle,s Theorem & Mean Value Theorem, not

writing first two conditions

Stress the conditions of continuity

and differentiability in the specified

interval

6. Not checking the value of c in the given interval for

Rolle,s Theorem & Mean Value Theorem

Insist to check the value of c lies in

the given open interval

7. For Second order derivative in parametric form –

using

=

(

)

Insist them to write the formula

correctly

8. while differentiating inverse trigonometric functions,

applying chain rule

More practice with inverse

trigonometric functions and chain

rule problems

78

QUESTION BANK

5.HOTS-CONTINUITY-AND-DIFFENTIABILITY.pdf

PROJECTS AND PRACTICALS,POWER POINT PRESENTATION,REFERENCE WEB

PPT ON LIMITS AND CONTINUITY.ppt

Differentiability.ppt Rolles and Mean value theorem.ppt

LINKS,VIDEO CLIPINS,VISUALS ETC.

https://youtu.be/7Lr-Ktrq0ck

MULTI DISCIPLINARY ACTIVITIES,CONNECTION WITH DAILY LIFE,PEER GROUP

LEARNIG,DIFFICULT AREAS

functions_&_situations_complete.pdf

79

CHAPTER 6

APPLICATION OF DERIVATIVES

LEARNING OBJECTIVES/OUTCOMES

(I)Rate of change

Recalling : 1. Perimeter and area of plane figures.

2. C.S .A, T.S.A and Volume of solids.

3. Differentiation , chain rule.

Observing the Sign of rate of change –Positive for increasing and negative for decreasing

Definition of cost and marginal cost /Revenue and marginal revenue

Addressing the Questions of the type:- Rate of change of Perimeter/radius/length/side ↔ Rate of change Area

Rate of change of Volume ↔ Rate of change of Surface Area

(II) Increasing and decreasing functions

Learning the Definition of

(i) Increasing, strictly increasing ,Decreasingand strictly decreasing functions and their

graphs.

(ii) Conditions for a function to be increasing and decreasing in the given intervals

(iii) Monotonicity of functions

(iv) Finding intervals – disjoint/ open intervals where the function changes its nature and

solving the problems based on it.

i) Finding f ʹ(x) in factor form.

ii) Solving f ʹ(x) = 0 and finding the roots.

iii) If there are ‗n‘ roots ,then divide the real number line R into (n+1 ) disjoint open

intervals .

iv) Finding the sign of f ʹ(x) in each of the above intervals .

v) f(x) is increasing or decreasing in the intervals when f ʹ(x) is positive or negative

respectively

Solving Problems on Polynomial functions, Trigonometric functions with simple ,

multiple and sub-multiple arguments.

(III) TANGENTS AND NORMAL:-

Geometrical approach to learn about the tangent and normal through differentiation

a)Recalling : 1 . Equation of a line – Point – Slope form

2 .Slope of a line when two points are given.

3. Slope of the line parallel to the axes and parallel to a given line.

4. Conditions for Parallel lines & Perpendicular lines.

b) Finding the equation of Tangent and Normal at a given point using differentiation .

c) Slope of the tangent at p(x0,y0)= dx

dy at (x0,y0).

d) Slope of the normal at p(x0,y0)=

dxdy

1 at (x0,y0).

80

e) Slope of x - axis , slope of y – axis.

f) Definition of orthogonal curves.

(IV) APPROXIMATIONS. :-

1. Recall the formulae for area of plane figures, surface areas and volumes of solids.

2. To know dxx ,dy y and dy = xdx

dy

3. To know relative error , percentage error.

MAXIMA AND MINIMA

1. EXPECTED LEARNING OUTCOMES:

Recalling

Learning the concepts of i)Maximum/Minimum value of a function in an open interval

(ii) Points of Local Maxima & Local Minima

(iii) Local Maxima & Local Minima

(iv) Absolute Maximum & Minimum.

(v) Point of Inflexion

(vi) Concept of Monotonic Function

A) First Derivative Test

B) Second Derivative Test

C) Problems related to MAXIMA AND MINMA

Plane figures- perimeters

& Areas

Solids – Curved & Total Surface

Areas & Volumes

Mensuration formulae

81

THREE LEVELS OF GRADED QUESTIONS

Rate of change

LEVEL I

1 An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the

volume of the cube increasing when the edge is 10 cm long?

2 The total cost C(x)in Rupees associated with the production of x units of an item

is given by C(x)= 0.007x3-

0.003x2+15x+4000. Find the marginal cost when 17

units are produced.

3 The total revenue in Rupees received from the sale of x units of a product is

given by R(x)= 13x2+26x+15 . Find the marginal revenue when x = 7.

82

Second Phase self Assessment

1 The total revenue in Rupees received from the sale ofx units of a product is given

by R(x)=3x2+36x+5. Find the marginal revenue, when x=15.

2 Find the rate of change of the area of a circle with respect to its radius r when

r =5 cm.

3 Find the rate of change of the volume of a cube with respect to its edge when the

edge is 5 cm.

Level II

1 A balloon which always remains spherical on inflation, is being inflated by

pumping in 900 cubic centimeters of gas per second. Find the rate at which the

radius of the balloon increasses when the radius is 15 cm.

2 A stone is dropped into a quiet lake and waves move in circles at a speed of 5

cm/s. At the instant when radius of the circular wave is 8cm., how fast is the

enclosed area increasing?

3 A ladder 5 m long is leaning against a wall. The bottom of the ladder is

pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is

its height on the wall decreasing when the foot of the ladder is 4m away from

the wall?.

4 The length x of a rectangle is decreasing at the rate of 5cm/ min. and the width

y is increasing at the rate of 4 cm / min.When x = 8 cm and y= 6 cm, Find the

rates of changes of a) the perimeter, and b) the area of the rectangle.

Second Phase self Assessment

1 The pressure p and volume v of a gas are connected by the relation pv1/4

=a , where

a is constant. What is the percentage increase in the pressure corresponding to a

decrease of ½% in volume.

2 A spherical ball of salt is dissolving in water in such a manner that the rate of

decrease of the volume at any instant is proportional to the surface. Prove that the

radius is decreasing at a constant rate.

3 A man , 2m , walks at the rate of sm /

3

21 towards a street light which is m

3

15

above the ground. At what rate is the tip of his shadow moving? At what rate is the

length of the shadow changing when he is m3

13 from the base of the light?

83

4 Two equal sides of an isosceles triangle with fixed base ‗a‘ are decreasing at the

rate of 9 cm/ sec. How fast is the area of the triangle decreasing when the two sides

are equal to ‗a‘.

5 A water tank has the shape of a right circular cone with its axis vertical and vertex

lower most. Its semi-vertical angle is

. Water is poured into it at a constant

rate of 5 cubic meters per minute. Find the rate at which the level of water is rising

at the instant when the depth of the water is 10 m .

LEVEL III

1 A water tank has the shape of an inverted right circular cone with its axis vertical

and vertex lowermost . Its semi vertical angle tan-1

(0.5). Water is poured into it at

a constant rate of 5 cubic metre per hour. Find the rate at which the level of the

water is rising at the instant when the depth of water in the tank is 4m.

2 A car starts from a point P at time t=0 seconds and stops at point Q . The distance x

in metres, covered by it, in t seconds is given by 3

22 t

tx Find the time taken by

it to reach Q and also find the distance between P and Q

3 A man of height 2 meters walks at a uniform speed of 5 km/h away from a lamp

post which is 6 metreshigh . Find the rate at which the lenghth of his shadow

increases.

Second Phase self Assessment

1 Sand is pouring from a pipe at the rate of 12cm3. The falling sand forms a cone

on the ground in such a way that the height of the cone is always one-sixth of the

radius of the base. How fast is the height of the sand cone increasing when the

height is 4cm?

2 X and y are the sides of two squares such that y = x – x2. Find the rate of change

of the area of second square with respect to the area of first square.

3 A swimming pool is to be drained for cleaning. If L represents the number of

litres of water in the pool in t seconds after the pool has been plugged off to drain

and L = 200(10-t)2 . How fast is the water running out at the end of 5 seconds?

4 The volume of a cube increases at a constant rate. Prove that the increase in its

surface area varies inversely as the length of the side

84

Increasing and decreasing functions

Level I

1

2

3 Prove that function given by f(x)=cosx is

4

5 Which of the following functions are strictly decreasing on (0, π/2)

a) Cosx (b) cos2x) (c) cos3x (d)tanx

Second Phase self Assessment

1 Determine for which values of x , the function y= x

4 -

3

4xis increasing and for

which values, it is decreasing.

2 Prove that the function f(x)=tan x – 4x is strictly decreasing on

3,

3

LEVEL II

1 Find the intervals in which the function f given by

20,cossin)( xxxxf is strictly increasing or strictly decreasing.

2

85

LEVEL III

1 Prove that

)cos2(

sin4y is increasing function of

2,0

in

2 Show that ,1,

2

2)1log(

x

x

xxy is an increasing function of x

throughout its domain.

3 Find the values of x for which y=[x(x-2)]2 is an increasing function.

4 Find intervals in which the following function

f(x) = 115

363

5

4

10

3 234 xxxx is

a) Strictly increasing.

b) Strictly decreasing.

5 On which of the following intervals is the function f given by f(x) = x100

+sin x -1

strictly decreasing ?

(0,1) b) ( ),2

c) (0, )

2

d) None of these.

Second Phase self Assessment

1 Show that xxxxxf 21 1logcot2)( is increasing in R

2 Show that the function ‘f ‘ given by 0,cossintan)( 1 xxxxf is always

increasing function in

3,0

III) TANGENTS AND NORMAL:-

LEVEL I

1

2

3

86

4

5 Find the equation of tangent and normal to the curve

y = x4-6x

3+13x

2-10x +5 at (0,5).

LEVEL II

1

2 Find the points on the curve 1

169

22

yx

at which the tangents are

(i) parallel to x-axis (ii) parallel to – y axis

3 Find the equation of the tangent line to the curve y= x2-2x+7 which is

a) Parallel to the line 2x-y+9=0

b) perpendicular to the line 5y-15x =13

4 Find the equations of the tangent and normal to the curve

at

. (Mar 2010)

5 Prove that the curves x= y2and xy=k cut at right angle if 8k

2=1

6 At what points will the tangent to the curve be parallel

to x- axis? Also, find the equations of tangents to the curve at those

points.(Mar2008)

7 Prove that x/a + y/b = 1 is touching the curve y = be-x/a

where the curve cuts the Y-

axis.

LEVEL III

1 Find the equations of the tangent and normal to the curve )1,1(23

2

3

2

atyx

2 Find the equation of the tangent to the curve y=√(5x -3 ) -2 which is parallel to the

line 4x-2y +3 =0

3 Show that the curve x2 + y

2 -2x = 0 and x

2 + y

2 -2y = 0 cut orthogonally at the

point (0,0)

4 Find the condition for the curves x2/a

2 - y

2/b

2 =1 andxy = c

2 tointersect

orthogonally.

5 Find the equations of the tangents to the curve which passes through

the point (4/3, 0).(Mar 2013)

87

6 Find the equations of tangent and normal to the curve y=

at the point

where it cuts the x-axis .(Mar 2013)

Second Phase self Assessment

1 Find the angle of intersection of the curves y2=4ax and x

2=4by

2 Show that the equation of normal at any point on the curve

4sin3)sincos(4sinsin3,coscos3 3333 xyisyx

3 Show that the normal at any point to the curve x= a cos +a sin and

y= a sin -a cos is at a constant distance from the origin.

(IV) APPROXIMATIONS. :-

LEVEL I

1

2

3

4

5

LEVEL II

1

2 If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find

the approximate error in calculating its surface area.

3 Using differentials, find the approximate value of (17/81)1/4

88

MAXIMA & MINIMA

LEVEL 1.

1 Find the maximum and minimum values, if any, of the following functions

given by i) f(x)=(2x-1)2 ii)f(x)=9x

2+12x+2 (without using derivatives)

2 Find the maximum and minimum values, if any , of the following functions

given by h(x)=sin(2x)+5 ii) f(x)=I sin4x+3I (without using derivative)

3

i)

ii) (By first derivative test)

4 Find the local maxima and local minima , if any in the following functions

1) f(x)=x3-6x

2+9x+15

2) 0,1)( xxxxf by second derivative test

5 Find the absolute maximum value and absolute minimum value of

f(x)=sinx+cosx , x €[0,π]

.Second Phase self-assessment

1 A function ‗ f ‗ attains local maximum at x = a , Write the nature of f ʹ(x) in the

neighborhood of a .

2 Find the maximum profit that a company can make, if the profit function is given

by p(x)=41-21x-18x2

3 Find the local maximum and the local minimum values, if any, for

f(x)=3x4+4x

3-12x

2+12

LEVEL 2.

1 At what points in the interval [0.2π], does the function sin2x attain its maximum

value?

2 Show that the right circular cylinder of given surface and maximum volume is

such that it height is equal to the diameter of the base.

3 Show that of all the rectangles inscribed ina given fixed circle, the square has the

maximum area.

4 Of all the closed cylindrical cans (right circular) , of a given volume of 100 cubic

centimeters, find the dimensions of the can which has the minimum surface area?

5 A rectangular sheet of tin 45cm by 24cm is to be made into a box whithout top,

by cutting off square from each corner and folding up the flaps . What should be

the side of the square to be cut off so that the volume of the box is maximum?

6 Show that semi-vertical angle of the right circular cone of the maximum volume

and given surface is sin-1

(1/3).

Second Phase self-assessment

1 An open box is to be constructed by removing equal squares from each corner of

89

a 3 meter by 8 metre rectangular sheet of aluminum and folding up the sides.

Find the volume of the largest such box.

2 Find the absolute maximum and minimum value of function ‗ f ‗ given by

3

1

3

4

612)( xxxf , x is in [-1,1].Find the maximum and the local minimum

values, if any, for the function f(x)=-x+2cos x, in the interval [0,π]. Also indicate

the points at which local maximum and local minimum exist.

3 A tank with rectangular base and rectangular sides, open at the top is to be

constructed so that its depth is 2m and volume is 8m3.If building of tank costs Rs.

70 per sq. metre for the base and Rs. 45 per sq. metre for sides , what is the cost

of least expensive tank?

4 A window is in the form of a rectangle surmounted by a semi-circular opening.

The total perimeter of the window is 12 m , find the dimensions of the rectangle

that will produce the largest area of the window

5 An isosceles triangle of vertical angle 2 is inscribed in a circle of radius a.

Show that the area of triangle is maximum when 6

LEVEL 3.

1 Manufacturer can sell x items at a price of rupees(5 –(x/100)) each. The cost

price of x items is Rs((x/5)+500). Find the number of items he should sell to

earn maximum profit.

2 A window is in the form of a rectangle surmounted by semicircular opening .The

total perimeter of the window is 10 m . Find the dimensions of the window to

admit maximum light through the whole opening.

3 A point on the hypotenuse of the triangle is at distance a and b from the sides of

the triangle. Show that the maximum length of the hypotenuse is 2

3

3

2

3

2

ba

4 Show that the height of the cylinder of maximum volume that can be inscribed in

a sphere of radius R is 3

2R.Also find the maximum volume.

5 Show that height of the cylinder of greatest volume which can be inscribed in a

right circular cone of height h and semi vertical angle α is one third that of the

cone and the greatest volume of cylinder is 23 tan27

4h

6 Find the area of the greatest isosceles triangle that can be inscribed in a given

ellipse having its vertex coincident with one extremity of major axis.

7 An open box with a square base is to be made out of a given quantity of

cardboard of area c2 square units. Show that the maximum volume of the box is

c3/6√3 cubic units.

8 The cost of fuel for running a bus is proportional to the square of its speed in

km/hr. This has to be paid in addition to a fixed charge of Rs 108/hr. The cost

of fuel is Rs 48/hr, when the bus moves at the speed of 20 km/hr. What is the

90

most economical speed.

9 Find area of the greatest rectangle that can be inscribed in an ellipse

.

10 Prove that the radius of the base of right circular cylinder of greatest curved

surface area which can be inscribed in a given cone is half that of the cone.

11 Show that the right-circular cone of least curved surface and given volume has

an altitude equal to √ times the radius of the base. (Mar 2011)

12 If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is

given, show that the area of the triangle is maximum when the angle between

them is

.

Second Phase self-assessment

1 Prove that the radius of the base of right circular cylinder of greatest curved

surface area which can be inscribed in a given cone is half that of the cone.

2 An open tank with a square base and vertical sides is to be constructed from a

metal sheetso as to hold a given quantity of water. Show that the total surface

area is least when depth of the tank is half its width.

3 Find the point on the curve

21 x

xy

, where the tangent to the curve has the

greatest slope.

4 A given quantity of metal is to be cast into a half cylinder with a rectangular base

and semi circular ends. Show that in order that total surface area is minimum ,

the ratio of length of cylinder to the diameter of its semi-circular ends is π:(π+2)

5 A jet of an enemy is flying along the curve y= x2+2. A soldier is placed at the

point (3,2) . What is the nearest distance between the soldier and the jet?

VALUE BASED QUESTIONS

1. A window is in the form of rectangle surmounted by a semi-circular opening . Total

perimeter of the window is 10m. What will be the dimensions of the whole opening to

admit maximum light and air?

i) How having large windows help us in saving electrictity and conserving

environment?

ii) Why is optimum use of energy required in the Indian context?

2. A tank with rectangular base and rectangular sides, open at the top is to be consturected

so that its depth is 2m and volume is 8 m3. If building of tank cost Rs. 70 per sqmeters

for the base and Rs.45 per sqare metre for sides. What is the cost of least expensive tank?

What kind of value is hidden in this question and what is its use in practical life?

3. In a competition, a brave child tries to inflate a huge spherical balloon bearing slogans

against child labour at the rate of 900 cubic centimeter of gas per secod. Find the rate at

which the radius of the balloon is increasing when its radus is 15cm.

i) Which values have been reflected in this question?

ii) Why is child labour not good for society?

4. If C=0.00ex3+0.02x

2+6x+250 gives the amount of carbon pollution in air in an area on

the entry of x number of vehicles, then find the marginal carbon pollution in the air, when

3 vehicles have entered in the area and write which value does the question indicate.

*********************************************************************

91

ERROR ANALYSIS AND REMEDIATION

RATE OF CHANGE OF QUATNTITIES

Common Errors Measures to overcome the errors.

Incorrect formula Repeated practice of formulae

Errors in finding derivatives- not applying

chain rule

Proper drill on application of chain rule

Errors in identifying the given rate of

change and required rate of change

Repeated reading and understanding of

questions

Units are not mentioned Emphasis has to be given

INCREASING AND DECREASING FUNCTIONS

Common Errors Measures to overcome the errors.

Finding intervals for T – functions

involving multiples and sub multiples of

angles.

ASTC should be followed.

When the co-efficient of highest power of

the variable in f (x) is negative

Negative sign should not be neglected

while deciding the sign of f‘ (x)

TANGENT AND NORMAL

Common Errors Measures to overcome the errors.

Slope of the lines parallel to both the axis Proper practice

Finding the slope of the lines of the form

ax+by+c = 0

Proper practice using formula

From the slope of the tangent , finding the

slope of the normal

Proper practice and clarity on basic concept

of slope

APPROXIMATIONS

Common Errors Measures to overcome the errors.

Splitting the number under radical sign as x

and

More practice

MAXIMA AND MINIMA

Common Errors Measures to overcome the errors.

Expressing one variable in terms of another

& the constant

Repeated practice of more questions

Differentiating the function by keeping the

radicals as it is

We can square both the sides & assign a

new function in the LHS and then

differentiate and proceed.

Checking the sign of second derivative Emphasis

x

92

QUESTION BANK

CBSE-XII-HOTS-APPLICATION-OF-DERIVATIVES-Chapter-6.pdf

PROJECTS AND PRACTICALS,POWER POINT PRESENTATION,REFERENCE WEB

LINKS,VIDEO CLIPINS,VISUALS ETC.

Application of derivatives.ppt

First order deviative test for maxi mini.ppt

https://youtu.be/D18tbaC3DV0?list=PLl1oW2UaFB2-aeGG_dGOFPgLN6kE3diEn

https://youtu.be/wiB0ydO0l_c

https://youtu.be/IoNJrNrkn2E

93

Chapter -7

INTEGRALS

LEARNING OBJECTIVES/OUTCOMES

Learning Objectives for Indefinite Integrals, Definite Integrals

The goal here is developing the student‘s geometric insight into the concepts of integration, and

applying these concepts to problem solving and ―real world application‖.

Apply arithmetic, algebraic, geometric, statistical and logical reasoning to solve problems.

Represent and evaluate basic mathematical and/or logical information numerically,

graphically, and symbolically.

Interpret mathematical and/or logical models such as formulas, graphs, tables and

schematics, and draw inference from them.

Students will become proficient in techniques of the concept of definite and indefinite integral and

their relations to area and rate of change. In particular, the students will Compute definite and

indefinite integrals

1. Express Calculus I differentiation rules as antidifferentiation rules.

2. Use these antidifferentiation rules and appropriate substitutions to calculate indefinite

integrals.

3. Use identities to prepare indefinite integrals for solution by substitution.

4. Evaluate an indefinite integral using integration by parts.

5. Evaluate an indefinite integral using integration by partial fraction.

6. Evaluate an indefinite integral using integration trigonometric identities.

7. Evaluate an indefinite integral using integration by selecting appropriate technique

8. Evaluate an indefinite integral using integration by using a compilation of techniques.

9. Use fundamental theorem to calculate the definite integrals.

94

CONCEPT MAP

95

96

sumofaitaasdxxEvaluate

dxxx

dxeecxxx

dxx

dxx

dxxxe

dxxx

dxxx

x

xx

dx

xx

dx

x

dx

dxxxx

dxx

dxx

x

dxxx

dxx

e

dxx

xxx

dxxxx

LEVEL

x

x

lim2.18

2.17

)cossin9(.16

41.15

4.14

seclogtan.13

sec.12

11

12.11

13129.10

102.9

94.8

3cos2coscos.7

sin.6

)(logsec.5

cossin

14

13

1

1.2

tansecsec.1

1:

2

0

2

0

2

2

2

2

2

2

3

2

2

tan

23

1

dx

xx

xdx

xx

dx

dxx

xdx

dxxx

dxxx

x

dxxx

x

dxx

x

sin2sin1

cos,

1.25

)(cossin.24

sin.23

15.22

183

32.21

cossin

sin.20

cos1

sin.19

1

0

1

2

2

7

1

1

54

2

2

0

44

4

2

0

26.∫

,∫

,∫

,∫

,∫

dx

27.∫

dx

28.∫

dx

29.∫ ( x +

)dx

97

30.∫ x dx

31 ∫

32 ∫

33 ∫

34 ∫

35 ∫

36 ∫

37 ∫

38 ∫

39 ∫(

)

40 ∫

41 ∫ [ ]

Where[ ]denotes greatest integer.

42 ∫

43 If ∫

then find the value of

44

45

98

18. dxxex

1

0

dxxx

x

dxx

x

dxxx

dxxx

xx

dxxx

x

dxxx

x

dxxx

x

dxxx

x

bxax

dx

xx

dx

dxxx

dxx

dxx

x

x

dx

dxax

x

dxx

x

dxe

Evaluate

x

34

5.17

1

2sin.16

sin15

43

21.14

41.13

45

76.12

45

3.11

562

2.10

coscos.9

67.8

1

1.7

32tan.6

sin1

cos.5

tan1.4

sin

sin.3

sin

4sin.2

1

1.1

2

2

2

1

1

22

22

22

2

2

2

2

6

2

99

19.dx

x

x

2

0

2 4

36

2

0

.20 dxx 12

21. xdxxsin

22

dxxx

xx

12

12

2

23.

dxxxx

x

321

13

24. dxx

xx

0

2cos1

sin.

25.

3

6

tan1

x

dx

26.∫

dx as limit of sum

27.∫

dx

28.∫

dx

29.∫

dx

30∫

dx

3 ∫

32 ∫ ( √ )

33 ∫

34 ∫

35 ∫

100

36 ∫

37 ∫

38 ∫

39 ∫

40 ∫

41 ∫

42

43

44

45

101

.

dxx

x

dxxxEvaluate

dxx

xe

dx

x

x

d

dxxxx

dxxxx

dxex

x

dx

dxx

x

dxx

dx

dxxx

dxxx

x

dxxx

x

xx

dx

dxx

dxx

dxxx

x

dxxx

x

LEVEL

x

x

1

1.19

.18

4sin1.17

30

.16

sin4cos5

cos2sin3.15

5.14

11.13

2cos1

sin2.12

sec.11

1

1cos.10

1.9

42

8.8

45

76.7

1

2.6

25.5

cossin.4

cos.3

cossec

tan.2

cossin

tan1

3:

4

2

2

1

3

0

9

4

2

2

3

2

2

2

3

2

21

3

2

42

2

32

5

102

fullpageexerciseousMiscellane

dxxx

dxx

xx

dxxx

dxxbxa

x

dxxx

dxx

dxxxxEvaluate

Level

dxx

dxxx

x

dxx

357.30

.)tancot(.29

2cos9

2cos2sin.28

)(sin.27

sincos26

6sin16cos.25

sinlog.24

321.23

3

tan1log.22

1

1.21

tan.20

4

2

3

1

0

2222

2

0

4

1

4

0

24

2

31.∫

32.∫

33.∫

- xI dx

34.∫

35.∫

dx

36.∫

37.∫

dx

38.∫

dx

103

39.∫

dx

40

41

42

43 ∫

44 Evaluate ∫

45

VALUE BASED QUESTIONS

1 Evaluate ∫

dx. State any one reason to have integrity in real life

2 Evaluate∫

. What can be the possible limits of integrity in real life

according to you?

3 Find∫ . What is the importance of integration of different religions

for our country peace?

4 Find the integration of∫ . Write any famous proverb for unity

5 Integrate ∫ Write any two characteristics of Mahatma

Gandhi which you adore the most?

104

Error Analysis and Remediation.

Definite Integrals

Students find it difficult to

identify and use the properties

of definite integral

Properties should be

made clear where to use.

While using the properties of

definite integrals for solving a

particular question, mention

the property.

Students make mistakes in

computation while evaluating

the definite integral as limit of

sums

Concept of taking limit and

summation formula should be

made clear

Topic Common Errors Committed

By Students

Measures To Overcome Errors

Indefinite Integrals

Students get confused with

differentiation and integration

formulas

Conduct formula test daily -

oral and written

Students fail to identify the

method, which they have to

opt.

Classification of problems

based on different methods

using different formulas

should be stressed.

In substitution method

students find it difficult to

substitute correctly.

Drilling up the same method

and conduct slip test

frequently.

Answer the question in its

original form.

Making perfect square while

doing the problems of the

type

√ etc.,

Giving sufficient number

of problems for practice.

To reduce a quadratic

expression as a perfect square

keep in mind (i) coefficient of

should be positive.

(ii) coefficient of should be

unity.

Students take negative sign

out from the square root

symbol

Concept should be made clear

Students miss the constant of

integration while writing the

answer & forget to use dx

while integrating

Stress to use constant of

integration & write dx while

teaching.

Integration of inverse

trigonometric functions

For inverse trigonometric

functions, we either reduce

these in trigonometric

functions by using substitution

or directly we can integrate

depending upon the situation

105

QUESTION BANK

Evaluate

1 ∫

2 ∫

3 ∫

4 ∫

5 ∫

6 ∫

7 ∫

8 ∫

√ √

9 ∫

10 ∫

11 ∫

12 ∫

13 ∫

14 ∫ (

)

15 ∫

16 ∫

17 ∫

18

19

∫√

20

∫ √

106

21

22

2

24

25

26

27

∫{

28 ∫

29 ∫

30 ∫

PROJECTS AND PRACTICALS,POWER POINT PRESENTATION,REFERENCE WEB

LINKS,VIDEO CLIPINS,VISUALS ETC.

Integration.pptxIntegration _2.pptx Basic Integration.ppt ppt partial

fraction.pptxIntegration _3.ppt Definite Integral

_1.pptx\\\

Definite Integral_2.pptx

1. https://youtu.be/lwdtYvm06hc

2. https://youtu.be/Znb5v2VfCxU

3. https://youtu.be/fTyRb5OiK6I

107

MULTI DISCIPLINARY ACTIVITIES,CONNECTION WITH DAILY LIFE,PEER GROUP

LEARNIG,DIFFICULT AREAS

1) Abbot, P., and M.E. Wardle. Teach Yourself Calculus. Lincolnwood, IL: NTC Publishing,

1992.

Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press,

1998.

2) https://youtu.be/NkjV1SSJZt8

3) https://youtu.be/r-NeXxpIZX8

108

CHAPTER 8

Application of Integrals

LEARNING OBJECTIVES/OUTCOMES

1. Standard equations of straight lines

2. Equation of circles with Centre at the origin, and center at (h,k)

3. Equation of parabolas

4. Equation of ellipse

5. First fundamental theorem of integral calculus

6. Second fundamental theorem of integral calculus

1.Area under the curve y=f(x)and the x-axis and the ordinates at x=a and at x=b is∫

.

2.Area under the curve x=f(y)and the x-axis and the ordinates at y=c and at y= d is∫

3.The area bounded by the curve y=f(x) and x-axis and the ordinates x=a and x=b is given by A1

+A2=A

4.Area under two curves

If y=f(x) , y= g(x) where f(x) g(x) in the [a,b] such that the point of intersection of these two

curves are given by x=a and x=b obtained by taking common values of y from given equation of

109

two curves then the area between the curves is given by A=∫ [ ]

5.If f(x) g(x) [a,c] and f(x) g(x) in [c,b] where a < c<b , then area of the region bounded by

the curves can be given by

6.Area of a triangle when the coordinates of the vertices or equations of sides are given.

STEPS FOR FINDING THE AREA USING INTEGRATION

STEPS

DRAW THE DIAGRAM

MAKE A SHADED REGION

FIND INTERSECTION POINTS

IDENTIFY THE LIMITS

WRITE THE INTEGRAL(S) FOR

THE REGION

EVALUATE THE INTEGRAL

THE VALUE SHOULD BE

POSITIVE

110

CONCEPT MAP

THREE LEVELS OF GRADED QUESTIONS

LEVEL I

1. Find the area of the region bounded by the curve y 2

= x and the lines x = 1 and

x =4 and the x - axis.

2. Find the area of the region bounded by y = x 2 and the lines y = 2 and y= 4 and the y- axis.

111

3. Find the area of the region bounded by the ellipse

4. Find the area of the region bounded by the parabola y=x2 and y=x

5. Find the area of the region bounded by the curve y = x between x = -1 and x=1.

6. Find the area of the region bounded by the curve x2=4y and the line x=4y-2

7.Find the area of the region bounded by the curve y2= 4x and the line x= 3

8 .Find the area of the region lying in the first quadrant and bounded by y=4x2, x=0,y=1 and y=4

9. Find the area of the region enclosed by the parabola x2 = y, and the line y = x+2 and the

x-axis

10. Find the area lying in the first quadrant and bounded by the circle x2+ y

2=4 and the

lines x=0 and x=2.

11. Find the area of the region bounded by the curve

axis and between

12 Find the area of the region bounded by the curve and the lines

13 Find the area of the region bounded by the curve between

14 Find the area bounded by the curve in first quadrant

LEVEL 2

1. Find the area of the region bounded by the parabola y = x2 and y

2 = x

2. Find the area of the region bounded by the parabola y= x2and y =

3 Find the area of the circle 4x2+4y

2=9 which is interior to the parabola x

2=4y

4. Using Integration find the area of the region bounded by the triangle whose vertices

are (1,0), (2,2) and (3,1)

5. Using integration find the area of the triangular region whose sides

have the equations y = 2x + 1, y = 3x + 1 and x = 4

6. Find the area bounded by the curve y = sin x between x = 0 and x =2 π

7. Find the area of the smaller region enclosed by the circle and the line

8. Find the area of the parabola y2=4ax bounded by the latus rectum.

9. Find the area of the region bounded by the line y=3x+2, the x-axis and the ordinates x=-1 and

x=1

10. Find the area of the region included between 4y=3x2,

and the line 3x-2y+12=0

.

112

11. Find area of the region { ( x, y) :y x2 +1, y x+1, 0 x 2}

12. Draw the rough sketch and find the area of the region bounded by two parabolas

4 y 2 =9x and 3 x

2 =16 y by using the method of integration.

13. FindFind the area enclosed by the curve

14 Find Find the smaller of the two areas enclosed between the ellipse

and the line

15. FindFind the smaller of two areas bounded by the curve and

LEVEL 3

1. Sketch the region bounded by the curves y= √ and y= and find its area.

2. Make a rough sketch of the region given below and find its area using

integration { (x, y); 0≤ y≤x2, 0 ≤ y ≤ 2x+ 3, 0≤x≤ 3 }

3. Find the ratio of the area into which the curve y2=6x divides the region bounded by

x2+y

2 =4 and (x+2 )

2+ y

2= 4 using integration

4. Sketch the graph of . Evaluate∫

dx

5. Sketch the graph of

{

Evaluate ∫ f(x)dx

6.Find the area lying above x-axis and included between the circles x2+y

2=8x and the

parabola y2=4x

7. Find the area of the region between the two curves ( x-1)2+y

2=1 and x

2+y

2=1

8. Prove that the curves y2=4x and x

2=4y divide the area of the square bounded by x=0, x = 4,

y=4and y=0 into three equal parts.

9. Find the area bounded by the curve y=x3, the x-axis and the ordinates x=-2 and x=1

10. Find the area of the region enclosed between two circles x2+y

2=9 and +y

2 =9

11 Using integration , find the area of the region bounded by the triangle whose vertices are

12 Find the area bounded by curves and

13 Find the area bounded by the curves and

14 Find the area bounded by curve and -axis and the ordinates and

113

15 Find the area of the region {

16 Calculate the area of the region enclosed between the circles and

(

)

17 Find the area enclosed by the curve

18 Draw a rough sketch of the given curve and find the

area of the region bounded by them, using integration.

VALUE BASED QUESTIONS

1 A field is in the form of parabola . A farmer has planted trees in

the exterior to the region bounded by the parabola and | and the

remaining part for playing games for children. Find the area of the ground

where the students are playing games. What is the importance of games in

student‘s life

2 The area between and is divided into equal parts by the

line , find the value of a. A man constructed a house in one of the

area and planted trees in the otherarea.Find the area where he constructed

house? What is the importance of plantation of trees

3 A farmer has a plot in the shape of a circle x2+ y

2 = 4. He divides his

property among his son and his daughter in such a way that for son he

gives the area interior to the parabola y2= x and for daughter the area

interior to the parabola y2 = -x. How much area the son got? Have both of

them got equal share? What is the value for life behind this?

4 Using the method of integration find the area bounded by the curve

. What is the value of truthfulness in our life?

5 Find the area lying above x-axis and included between the circle

and the parabola .What is the importance of

non-violence in our life

Error Analysis and Remediation.

Topic Common Errors Measures To Overcome Errors

Application

Of Integrals

Incorrect figures Figures of 4 types of parabolas and 2 types of

Ellipse should be thorough for the students.On

graph paper draw and shade the area as far as

possible, which we have to calculate.

114

Incorrect shading Students should be given enough practice for

shading the area

Making mistakes while

applying limits

Thorough practice

calculate area without

using integration

Instruct the students to use integration method

QUESTION BANK

1 Find the area of the figure bounded by the curve and the straight line

2 Compute the area of the figure which lies on the first quadrant inside the circle

and is bounded by the parabola and

3 Compute the area of the figure bounded by the line and axis.

4 Compute the area bounded by the lines and

5 Draw a rough sketch of the curves and as varies from to

and

find the area of the region enclosed by them and axis.

6 Using integration, find the area of the region {

7 Using integration, find the area of the region bounded by the following curves, after

making a rough sketch

8 Find the area enclosed by the curves .

9 Find the larger are bounded by and

10 Find the area bounded by the –axis and the ordinates and

11 Find the area bounded by the curves and

12 Find the are lying above - axis and included between the circle and the

parabola

13 Find the area enclosed by the curve between and

and axis

115

PROJECTS AND PRACTICALS,POWER POINT PRESENTATION,REFERENCE WEB

LINKS,VIDEO CLIPINS,VISUALS ETC.

AREAS USING INTEGRATION.ppt

FUNDMENTAL THEOREM_PPT.ppt

Integration(area).ppt

1) https://youtu.be/umiUmq5e05o

2) https://youtu.be/3iMgAm6yMU4

3) https://youtu.be/kuC7cmXpwIs

4) https://youtu.be/bhyoybrLmZY

5) http://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx

MULTI DISCIPLINARY ACTIVITIES,CONNECTION WITH DAILY LIFE,PEER

GROUP LEARNIG,DIFFICULT AREAS

https://youtu.be/dUDTdb1DD88

https://youtu.be/rjLJIVoQxz4

https://youtu.be/rjLJIVoQxz4

116

CHAPTER 9

DIFFERENTIAL EQUATIONS

LEARNING OBJECTIVES/OUTCOMES

1. The learner will learn the concept of differential equation

2. Identify an ordinary differential equation its order and degree

3. Verify whether a given function is a solution of a given ordinary differential equation

4. Find solutions of separable differential equations

5. Find solutions of homogenous differential equations

6. Solve first order linear differential equations

7. Model of radioactive, compound interest, and mixing problems using first order equations

8. Model of population dynamics using first order differential equations

9. Mathematical model that bring understanding about the universe and the world around us.

10. Additionally, over the course of the class XII , the child will develop an increased ability to

reason abstractly about mathematical concepts related to differential equations.

CONCEPT MAP

117

THREE LEVELS OF GRADED QUESTIONS

LEVEL 1

1. Find the order and degree of the following differential equations:

a.

b.

c. (

)

(

) .

d. - 2 -y‘ =-1.

2. Verify the given functions is the solution of the corresponding differential equation;

a. y = :

b.

3. Form the differential equation representing the family of curves where

―a‖ is arbitrary constant.

4. Form the differential equation representing y = ( a + bx).

5. Solve the differential equation: .

6. Solve the differential equation –

7. Solve the differential equation: √

8. Solve: given that y=1 when x=1.

9. Solve:

10. Solve:

11. Solve: (

)

12. Solve:

13. Solve:

14. Solve:

.

15. Find the general solution of the differential equation

16. Solve:

17 Solve :

18 Solve :

19 Solve :

20 Solve :

118

21 Solve:

22 Find the differential equation of family straight lines passing through origin

23 Find the differential equation of all non- horizontal lines in a plane

24 The integrating factor of the differential equation

25 Find the order and degree of the differential equations

i)( (

)

)

ii)

iii)

26 Verify is the solution of the differential

27 Find the product of the order and degree of the following differential equation:

.

/

(

)

28 Write a differential equation for where A and B are arbitrary

constants.

LEVEL 2

29. Form the differential equation of the family of circles in the second quadrant and touching the

coordinate axes.

30. Find the particular solution of the differential equation:

, given that

y=1 when x=0.

31. Solve:

32. Solve:

33. Find the equation of the curve passing through the point (0,0) whose differential equation is

34. Solve:

35. Solve:

(

) , y=0 when x=1.

36. Solve: , (

) (

)- , (

) (

)- .

37. Solve:

38. Show that the differential equation

(

) is homogeneous. Find the

particular solution given that x=0 when y=1.

39. Solve:

.

40. Solve:

.

41. Solve:

.

119

42. Solve: .

43 Solve :

44 Solve :

45 Solve :

46 Find order of the differential equation of all circles of given radius

47 Find the general solution of the differential equation

48 Solve the differential equation , given that when

:

49 Show that the general solution of the differential equation

is

50 Solve

(

)

LEVEL 3

43. Find the order and degree of the following differential equation:

(

)

. (b)

(

)

44. Write the integrating factor of the following differential equation:

a.

b.

c.

d.

44. Write the order of the differential equation of family of curves:

a. .(hint: differentiate only once)

b. .(Hint: differentiate only once)

c. ( Hint: differentiate twice)

d. .(hint: differentiate twice)

45 Verify that

46. Solve:

.

47. Solve: .

48. Solve:

49. Solve:

.

50. Solve: ( (

)) ( (

) ) .

51. Solve: √ .

120

52 Solve :

53 Solve :

54 Solve:

55 Solve:

56 Find order of the differential equation of all circles of given radius

57 Find the equation of a curve passing through origin if the slope at any point is equal to the

square of the difference of the abscissa and ordinate of the point.

58 Find the equation of a curve passing through the point if the perpendicular distance

of the orgin from the normal at any point of the curve is equal to the distance of

from the axis.

59 Find the general solution of the differential equation

60 Solve the differential equation , given that when

:

61 Show that the differential equation * (

) + is homogenous. Find

the particular solution of this differential equation , given that

when

VALUE BASED QUESTIONS

1. If the interst is compouned contuniously at 6% p.a , how much worth Rs.1000 will be after

10years? How long will it take to double Rs.1000. What is importance of savings among

students?

2. Form the differential equation of family of circles touching y axis at origin . What are the

values conveyed by family of circles formed?

3. Solve the differential equation

, if y is the distance and x is the

time ,

is veloccity . Ravi rides a vechile beyond the limit in high way. What suggestion would you

give him to help him understand the risk of over speeding.

121

ERROR ANALYSIS AND REMEDIATION.

Topic Common Errors Committed

By Students

Measures To Overcome

Errors

Differential Equations

In finding degree& order

forget to remove radical sign

Proper practice regarding

radical sign to be practiced.

In finding the degree

considering the highest power

rather the power of highest

order.

More practice on finding the

degree

Confusion in finding the

arbitrary constants

Recall the family of curves or

go through the question to

find the arbitrary constants

what mentioned in it.

While finding P if y is not

multiplied by any function not

considering P as 1 rather

considering P=y.

Insists on P is a function of

―x‖ alone

While finding integrating

factor like writing the

final result as 2x

Properties of log to be

recalled and practiced.

Constant of integration

generally not writing

By counseling and insisting

the error can be avoided

While finding the value of

forget to consider the

―– ―sign

By recalling the properties of

logarithms and by practice it

can be avoided.

QUESTION BANK

1 Determine the order and degree of the following differential equations :

a)

√ (

)

122

b)

(

)

(

)

c)

(

)

d)

e)

2 Find the differential equation of the family of curves

where and are arbitrary constants

3 Show that the differential equation that represents the family of all

parabolas having their axis of symmetry coincide with the axis of –

axis is =0

4 Show that is solution of differential equation

5 Verify ( √ ) satisfies the differential equation

6 Solve the differential equations

a)

b)

c)

d)

e) (

)

f)

g)

h)

i) √

j)

7. For the differential equation

find the solution

curve passing through the point .

123

POWER POINT PRESENTATION,.

Basic of differential eqns.ppt

Differential_equations_2013.ppt

VIDEO CLIPPINGS,VISUALS ETC

https://www.youtube.com/watch?v=nlvr3UyMiQ4

https://youtu.be/_x_-9lwUjBw?t=165

https://youtu.be/nlvr3UyMiQ4

https://youtu.be/R3MmS4w6zCY

https://youtu.be/lpEgphNUEcM

124

Chapter 10

Vectors

LEARNING OBJECTIVES/OUTCOMES

Definition of Vectors

Definition of Scalars

Position Vector of a point

Dc‘s and Dr‘s

Types of vectors

Zero Vector

Unit Vector

Unit vector in the direction of + or

Collinear Vectors/parallel vectors

Equal Vectors

Negative of a Vector

Addition of Vectors

Multiplication of a Vector by a Scalar

Unit Vectors along the coordinate axes.

Vector joining two points OAOBAB

Section formula

Dot product of vectors

Projection of vectors on a line

Perpendicular vectors

Finding the angle between the two vectors

Finding I + I, I + + I, I - I

Squaring of a vector

Angle between two vectors.(By scalar product)

Cross product of vectors

Cross product of unit vectors.

Unit vector perpendicular to two given vectors.

Angle between two vectors..(By vector product)

Area of parallelogram when adjacent sides are given.

Area of parallelogram when diagonals are given.

Area of a triangle

Area of a rectangle when position vectors of A,B,C,D are given.

Scalar Triple product of vectors.

Expressing the STP in rectangular coordinates.

Geometrical meaning of STP.

Vector Triple Product.

125

Properties of dot, cross and STP

Dot product Cross product Scalar triple product

| | | | where is

perpendicular to both , .

Notation is [ ]

If If parallel If coplanar [ ]

( ) ( )

[ ] if any two

vectors are equal.

If

Then

If

Then |

|

If

Then [ ]

|

|

Geometrical meaning

projection of

| |

Geometrical meaning

Adjacent sides

Geometrical meaning

Volume of a parallelepiped with

as adjacent sides.

126

CONCEPT MAPPING

127

THREE LEVELS OF GRADED QUESTIONS

Level 1

1. Find the angle between the vectors -2 +3 and 3 -2 +

2. Find projection of the vector +7 on the vector7 - +8

3. Evaluate the product (3 - 5 ).( +7 )

4. Find the dr‘s of the vector 5. Find the area of the parallelogram whose adjacent sides are determined by the

vectors +3 and = + .

6. If P and find the direction ratios of .

7. If and , find a unit vector in the direction .

8. If √ ,| | and , find the angle between .

9. What is the angle between with the vectors of magnitude √ and respectively

and .

10. Write the position vector of the point dividing the line segment joining the points A and B

with position vectors externally in the ratio 1:4, where and

.

11. Given = + and = -4 , find µ such that =3. 12. Find a vector in the direction of that has magnitude ―7‖ units. 13. Find the value of for which the vectors

14. If vector = +3 , 5,find µ.

15. Vectors and act along the adjacent sides of a parallelogram. Find

the angle between the diagonals of the parallelogram.

16. Show that the area of the parallelogram having diagonals and is

5√ Sq.units.

17. If the vertices A,B,C, of a triangle ABC are (1,2,3), (-1,0,0) , (0,1,2) respectively , then

find area of triangle ABC .

18. If a unit vector ―a‖ makes an angle

with ,

with and an acute angle with , then find

and hence the components of .

19. If vectors are such that and ,| | and ,

find the angle between .

20. If the vectors are coplanar show that

21. Find the scalar components of the vector with initial point A(2,1) and terminal point

B(-5,7).

22. Show that the points A(-2,3,5) , B(1,2,3) and C(7,0,-1) are collinear.

23.Find the unit vector in the direction of the resultant of

128

Level 2

1. If = 0 and . = 0 then what can be concluded about the vector ?

2. If = + + , = 2 - +3 and c= -2 + , find a unit vector parallel to the vector

2 – +3 .

3. For given vector = 2 - +2 and = - + - , find the unit vector in the direction of the

vector + .

4. Let , , and be three vectors such that = 3 , | | = 4 and = 5 and each one of

them being perpendicular to the sum of the other two , find | | .

5.Show that + | | is perpendicular to - | | , for any two non- zero vectors

and .

6. If a,b,c are lengths of the opposite sides respectively to the angles A,B,C of a triangle ABC,

Prove that

i)

ii) .

7. Find the volume of the parallelepiped whose adjacent sides are represented by

Level 3

1. If = + , then is it true that = | | + ? Justify your answer.

2. Write down a unit vector in XY plane making an angle of 300 in the positive direction

of X- axis.

3.If and , find the unit vector in the direction of

4.Let | | √

.

5. Write the value of ( ) ( ) .

6.For two non-zero vectors write when | | | | holds.

7. If , , , are unit vectors such that + + = 0, find the value of . + . + .

8.Find a unit vector perpendicular to each of the vectors + and - where = 3 +2 +2

and = +2 -2 .

9.Show that the points A,B and C with PVs and

respectively are the vertices of a right triangle. Also find the remaining angles of the

triangle.

129

10. If are the PVs of the vertices of A,B and C of a ΔABC respectively . Find an

expression for the area of the ΔABC and hence deduce the condition that for the point

A,B and C to be collinear.

11. If vector and are two unit vectors and is the angle between then, show that

Sin

=

| |.

12 If vectors a + + . +b + and + +c are coplanar show that

+

+

=1.a,b.c 1.

13. If with reference to the right handed system of mutually perpendicular unit

vectors, ,and , =3 - , =2 + - , then express in the form = 1

+ 2 Where 1 is parallel to and 2 is perpendicular to .

14. Let = + - , =3 -2 +7 and =2 - + .Find a vector which is

perpendicular to both & and . =15.

VALUE BASED QUESTIONS

1. If , , , are position vectors of vertices A,B,C of triangleABC, show that the area of the

triangle is

| |

A student takes honesty, truthfulness and complacency as the three sides of the triangle.

Which side of the triangle do we prefer to take? Give your suggestion.

2. Let A and B be two points whose position vectors are 3 - 4 + and

- 7 + respectively. Find and .

If point A represents a person who isregular and systematic and B represents a lazy person,

which vector or would you choose for your success?

3. Three hoardings are displayed at the points A,B and C diplaying A(Do not litter), B(Keep

your place clean) and C(Go green). If these 3 points form a triangle ABC, find the area of

triangle ABC using method of vectors if points A,B and C are (0,2,1),(4,8,2) and (8,4,3)

respectively.

Give your views in two lines about ― GO GREEN‖

Error analysis and remediation

Concept Error Correction

Finding a vector of given

magnitude in the direction of

another vector

Forget to find the unit vector

in the direction of given vector

Stress on finding the

magnitude of the given vector

Finding an unit vector

perpendicular to the plane

containing the vectors and

Use the formula as

| |

Stress on to use the correct

formula

130

QUESTION BANK

1

2

3

4

5

6 If the angle between two vectors and of equal magnitudes is 60° and . = 4, find their

magnitudes

7 What is the Projection of the vector = 3 - 2 + on the vector =2 - 2 - ?

If two sides of a triangle are represented by the vectors = + 2 +3 and =-3 - 2 + ,

then what is the area of the triangle?

131

POWER POINT

PPT-VECTORS.pptx

WEB LINKS

https://youtu.be/rd4Yhz35Ilshttps://youtu.be/4Y8UlSzPCCghttps://youtu.be/8SjHpsWP0hw

https://youtu.be/kajf87jOrJEhttps://youtu.be/VnSKy35v3iEhttps://youtu.be/XEwUYGqEgRE

https://youtu.be/rHAIzjTEZEYhttps://youtu.be/8_G4_Q47D6ohttps://youtu.be/IMmATmHZEw

Q

https://youtu.be/4wuCQGkbSrghttps://youtu.be/ehP5MRF4l04https://youtu.be/9fVsoDpEk64

https://youtu.be/_pCyb1Gytbwhttps://youtu.be/qFk4oGPuFVchttps://youtu.be/PtfRUECe8HQ

https://youtu.be/WxqmblhxjEwhttps://youtu.be/byu2MlcChVQhttps://youtu.be/jjtZkD7L50Y

https://youtu.be/sV34QPfdvRMhttps://youtu.be/draA_HD8M7chttps://youtu.be/hgbQ7QmX6W

U

https://youtu.be/2UjGNH_8KAwhttps://youtu.be/lYe_ajktCjwhttps://youtu.be/HMcf3jS8XzU

https://youtu.be/iK2uTIaUktEhttps://youtu.be/kaWyvuL2hi0https://youtu.be/4aUTC8IonBs

https://youtu.be/Hb-6--L0fw4https://youtu.be/pHcUdaPZZ9Y

132

CHAPTER – 11

THREE DIMENSIONAL GEOMETRY

LEARNING OBJECTIVES/OUTCOMES:

To understand the concepts of Direction Cosines and Direction Ratios.

To acquire the different forms of equations of Lines in space.

To find the point of intersection of two lines in space if they intersect.

To find the angle between any two lines in space.

To acquire the concept of Skew Lines and to find the distance between them.

To acquire the different forms of equations of Planes in space.

To find the angle between any two planes.

To find the angle between a line and a plane.

To find the point of intersection of a line with a plane.

To find the equation of family of planes passing through the line of intersection of two

planes.

To find the distance between a point and a plane and distance between two planes.

133

CONCEPT MAP:

134

THREE LEVELS OF GRADED QUESTIONS:

S.No. LEVEL 1

1. Find the direction cosines of the line passing through the two points ( -2,4,-5) and

(1,2,3)

2. Find the direction cosines of X,Y,and Z-axes.

3. If and find the direction ratios of and hence the direction

cosines.

4. What is the cosine of the angle which the vector √2 i + j + k makes with y-axis.

5. Find the Vector and Cartesian equations for the line passing through the points ( -1,0,2)

and( 3,4,6)

6. Find the angle between the pair of lines:

=

=

;

=

=

7. Find the vector and the Cartesian equations of the line through the point ( 5,-2,3) and

which is parallel to the vector 3i+2j-8k.

8. Find the distance of the plane from the origin.

9. If the lines

are perpendicular , then find k.

10. Find the equation of the perpendicular drawn from the point (2,4,-1) to the line

11. Find the coordinates of the image of the point P (1 , 3 , 4) in the plane 2x-y+z+3=0.

12. Prove that the image of the point (3, -2, 1) in the plane lies on the

plane

13. Find the length of the perpendicular from the point (2, -1, 5) to the line

.

14. Find the coordinates of the point where the line

meets the plane

15. Find the angle between the line

=

=

and the plane 10z+2y-11z = 3

16. Show that the lines ( ) and

( )are intersecting. Hence find their point of

intersection.

135

17. Find the vector equation of the line parallel to the line

and passing

through (3 , 0 , -4)

18. Find the shortest distance between the lines, whose equations are

.

19. Find the Co-ordinates of the point where the line through the point (5,1,6) and ( 3,4,1)

crosses the YZ-plane.

LEVEL 2

1. Find the distance of the point (-2, 3, -4) from the line

measured

parallel to the plane 4x+12y-3z+1=0.

2. Find the vector equation of the line parallel to the line

and passing

through (3 , 0 , -4). Also find the distance between these two lines. Also obtain the

equation of the plane containing the line and the point (1, 2, 3).

3. Find the equation of the plane containing the lines ( )and

( ). Find the distance of the plane from origin and also from

the point (1,1,1).

4. Find the equation of the perpendicular drawn from the point (2,4,-1) to the line

5. Find the equation of the perpendicular drawn from the point P(2, 4, -1) to the line

. Also, write down the coordinates of the foot of the

perpendicular from P to the line.

6. If the lines

and

are perpendicular, find the value

of k and hence find the equation of plane containing these lines.

7. Find the foot of the perpendicular from the point P(1, 2, 3)on the line

.

Also obtain the equation of the plane containing the line and the point (1, 2, 3).

8. Find the coordinates of foot of perpendicular and the length of the perpendicular drawn

from the point P ( 5,4,2) to the line ( ) ( ). Also find the

image of P in this line.

9. Find the distance of the point (-2, 3, -4) from the line

measured

parallel to the plane

10. Find the foot of the perpendicular from the point P(1, 2, 3)on the line

136

Also obtain the equation of the plane containing the line and the point (1, 2, 3).

11. Find the equation of the plane containing the lines ( ) and

( ). Find the distance of the plane from origin and also from

the point (1,1,1).

12. Find the distance of the point (3, 4, 5) from the plane measured parallel

to the line 13. Find the distance of the point (-1 ,-5, -10) from the point of intersection of the line

( ) ( ) and the plane ( ) .

14. Find the equation of the plane passing through the line of intersection of the planes

( ) ( ) and parallel to x –axis.

15. Find the equation of the plane passing through the intersection of the planes,

, and parallel to the line

.

16. Find the distance between the point P(6, 5, 9) and the plane determined by thepoints

A(3, -1, 2), B(5, 2, 4), and C(-1, -1, 6).

17. Find the distance of the point (2, 3, 4) from the line

measured parallel

to the plane

18. Show that the four points ( 0,-1,-1) ( 4,5,1) ( 3,9,4) and (-4,4 ,4) are coplanar. Also find

the equation of the plane containing them

LEVEL 3

1. Find the equation of the plane through the intersection of the planes 2x +y-3z = 4 and

3x+4y+8z-1=0 and making equal intercepts on coordinate axes . (Ans: 5x +5y+5z=3)

2. Find the equation of the plane containing the line

and perpendicular to

the plane 2x-y+2z = 11. (Ans: 2x-6y-5z = 3.)

3. Find the vector equation of the plane through the line of intersection of the planes

.( -3 +4 ) =1 and .( - )+ 4 =0 and perpendicular to the plane .( -3 +4 ) =1

.(-5 +2 +12 ) =47 )

4. Find the equation of the plane perpendicular to the line

and passing

through the point where the given line meets the plane x +y +4z =6 .

(Ans : 2x+3y+4z = 9.)

5. Find the equation of the plane passing through the line of intersection of the planes

( ) whose perpendicular distance from origin

is unity.

6. Find the equation of the plane passing through the intersection of the planes

137

and perpendicular to the plane alos find inclination of this plane with XY-plane.

7. Find the vector and Cartesian equations of the plane passing through the intersection of

the planes ( ) = 6 and ( ) = -5 and the point (1,1,1)

8. Let P(3,2,6) be a point in the space and Q be a point on the line ( )

( ) find the value of for which the vector is parallel to the

plane x – 4y + 3z = 1.

9. Find the vector and Cartesian equations of the plane which bisects the line joining the

points ( 3,-2,1) and (1,4,-3) at right angles.

VALUE BASED QUESTIONS:

1. Two groups of students representing ‗SAVE MOTHER EARTH‘ and ‗GO GREEN‘ are

standing on two planes represented by the equations ( ) and

( ) . What is the angle between the two planes?. Also suggest two

activities which should be taken up to save mother Earth.

2. The point A(1,8,4) is situated at QutubMinar. Find the coordinates of the foot of the

perpendicular drawn from A to the line joining the points B(0,-1,3) and C(2,-3,1). Do

you think that the conservation of monument is important why?.

3. Find the image of the point (1,2,3) in the plane x + 2y + 4z = 38. Represent the image of

the point with one life skill.

138

ERROR ANALYSIS AND REMEDIATION

S.No. Error Remedy

1. Using the given improper form of a

line to convert into other form

Bring the given equation into proper form

(i)

3=

7; = 0 should be corrected

as

3=

7=

0

(ii)

3=

7; = 5 should be

corrected as

3=

7=

5

0

2 Improper use of formula to find the

distance between the given two

lines.

If the given question does not mention anything

about the skew lines and usage of ―distance

between the Skew Lines‖ gives 0 as the answer

then the lines must be parallel lines. Then the

―Distance between the Parallel Lines‖ formula to

be used.

3 Misunderstanding the Dr‘s of the

Parallel vector when the equation of

a line is given in Cartesian form

The denominators of the components of X , Y

and Z will give the Dr‘s of the parallel vector

only when the coefficients of X,Y and Z must be

1( one only)

4 Improper use of formula to find the

angle between a line and a plane.

Emphasize to use the proper formula

5 To find the distance between a point

& a plane measured parallel to

another plane (or) line , students

may find the perpendicular distance

between the point and the plane as

the answer

Emphasize to draw the diagram (figure) and

proceed accordingly.

6 While writing the answer of

question like‖ plane passing through

the intersection of two planes and

passing through any point given (or)

perpendicular to the third plane‖

they may stop the answer after

finding the value of ― ‖ (or)‖ ‖

Stress to substitute the value of the scalar in the

combined equation to get the required member of

the family.

7 While finding the image of a point

under a line or under plane ,

sometimes the students may stop the

answer after finding the coordinates

of the foot of the perpendicular.

Emphasize to draw the diagram (figure) and

proceed accordingly

139

QUESTION BANK :

11.HOTS-Three-Dimensional.pdf

1 Find the equation of the plane passing through the point (1,1,1,) and perpendicular to the

planes x+2y+3z-7=0 and 2x-3y+4z=0

2 Find the vector equation of a line joining the points with posiition vectors ,ˆ3ˆ2ˆ kji and

parallel to the line joining the points with position vectors ,ˆ4ˆˆ kji and ,ˆ2ˆˆ2 kji

Also find the cartesian equivalent of this equation.

3 Find the foot of the perpendicualr drawn from the point A (1,0,3) to the join of the points

B(4,7,1) and C(3,5,3)

4 Find the shortest distance between the lines

)ˆ2ˆ2ˆ4()ˆˆˆ2()ˆˆˆ2()ˆˆ( kjikjirandkjijir

5 Find the image of the point (1,-2,1) in the line .

2

3

1

1

3

2

zyx

6 Show that the four points (0,-1,-1), (4,5,1), (3,9,4) and (-4,4,4) are coplanar and find the

equation of the common plane.

7 The foot of the dperpendicular from the origin to the plane is (12,-4,3) . Find the equation

of the plane.

8 Show that the lines

4

3

3

2

2

1

zyx and z

yx

2

1

5

4 intersect. Find their point

of intersection.

9 A line makes angles ,,, with thr four diagonals of a cube, prove that

3

4coscoscoscos 2222

PROJECTS AND PRACTICALS,POWER POINT PRESENTATION,REFERENCE WEB

LINKS,VIDEO CLIPINGS,VISUALS ETC.:

3d coordinate geometry.ppt

3d coordinate geometry-1.ppt

3d ppt -2.ppt

140

CHAPTER – 12

LINEAR PROGRAMMING PROBLEM:

LEARNING OBJECTIVES/OUTCOMES:

To understand the nature of Linear Programming Problem.

To know about the Decision Variables.

To acquire the concept (meaning) of Objective Function.

To know about the Non-Negative Constraints.

To construct the Structural Constraints (Inequalities).

To understand the concept of Feasible Solution.

To find the Optimum Solution from the Feasible Solution.

To develop skills to find the Optimum Solution for the Unbounded Feasible Region.

To be able to solve all practical problems of LPP.

141

CONCEPT MAP:

142

THREE LEVELS OF GRADED QUESTIONS:

LEVEL 1

1. Graphically solve the system of inequations x-y ≤ 2 , x+y ≤ 4 , x 0 , y0

2. Minimise Z = 2x +3y subject to constraints : x 0 , y0 , 1≤x+2y≤ 10

3. A producer has 20 and 10 units of labor and capital respectively which he can

use to produce two kinds of goods X and Y. To produce one unit of goods X,

2 units of capital and 1 unit of labor is required. To produce one unit of goods

Y, 3 units of labor and one unit of capital is required. If X and Y are priced at

Rs.80 and Rs.100 per unit respectively, how should the producer use his

resources to maximize the total revenue?. Solve the problem graphically.

4. A small firm manufactures items A and B. The total number of items A and B that it

can manufacture in a day is at the most 24. Item A takes one hour to make while item

B takes only half an hour. The maximum time available per day is 16 hours. If the profit

on one unit of item A be Rs.300 and one unit of item B be Rs.160, how many of each

type of item be produced to maximize the profit? Solve the problem graphically.

5. Kellogg is a new cereal formed of a mixture of bran and rice that contains at least 88gms

of protein and at least 36 milligrams of iron. Knowing that bran contains 80gms of

protein and 40 milligrams of iron per kilogram, and that rice contains 100gms of protein

and 30 milligrams of iron per kilogram, find the minimum cost of producing this new

cereal if bran costs Rs.5 per kilogram and rice costs Rs.4 per kilogram.

6. A firm manufactures two types of products A and B and sells them at a profit of Rs.5

per unit of type A and Rs.3 per unit of type B. Each product is produced on two

machines M1 and M2, whereas one unit of type B requires one minute of processing time

on M1 and one minute on M2machines M1 and M2 are respectively available for at most

5 hours and 6 hours in a day. Find out how many units of each type of product should

the firm produce a day in order to maximize the profit. Solve the problem graphically.

7. A factory owner wants to purchase two types of machines, A and B , for his factory.

The machine A requires an area of 1000 m2

and 12 skilled men for running it and its

daily output is 50 units, whereas the machine B requires 1200m2 and 8 skilled men and

its daily output is 40 units. If an area of 7600m2

and 72 skilled man be available to

operate the machine, how many machines of each type should be bought to maximize

the daily output.

8. A housewife wishes to mix up two kinds of food X and Y in such a way that the mixture

contains at least 10 units of vitamin A and 12 units of vitamin B and 8 units of

vitamin C. The vitamin contents of 1 kg. of food X and 1 kg of food Y are as given in

the following table:

Food Vitamin A Vitamin B Vitamin c

X 1 2 3

143

Y 2 2 1

If one kg of food X costs Rs.6 and 1 kg of food Y costs Rs.10, find the lest cost of the

mixture which will produce the desired diet.

9. A factory owner wants to purchase two types of machines, A and B , for his factory.

The machine A requires an area of 1000 m2

and 12 skilled men for running it and its

daily output is 50 units, whereas the machine B requires 1200m2 and 8 skilled men and

its daily output is 40 units. If an area of 7600m2

and 72 skilled man be available to

operate the machine, how many machines of each type should be bought to maximize

the daily output.

LEVEL - 2

1 A dietician wishes to mix two types of food in such a way that the vitamin content of the

mixture contain at least 8 unit of vitamin A and 10 unit of vitamin C. Food I contains

2unit/kg of vitamin A and 1unit/kg of vitamin C, while food II contains I unit/kg of

vitamin A and 2unit/kg of vitamin C. It cost Rs.5.00 per kg to purchase food I and

Rs.7.00 per kg to produce food II. Determine the minimum cost of the mixture.

Formulate the LPP and solve it. Why a person should take balanced food?

2. A dietician wants to develop diet using two foods X and Y. Each packet (contains 30 g)

of food X contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6

units of vitamin A. Each packet of the same quantity of food Y contains 3 units of

calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet

requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of

cholesterol. Make as LPP to find how many packets of each food should be used to

minimize the amount of vitamin A in the diet, and solve it graphically.

3. A farmer has a supply of chemical fertilizers of type ‗A‘ which contains 10% nitrogen

and 6% phosphoric acid and type ‗B‘ contains 5% of nitrogen and 10% of phosphoric

acid. After soil testing it is found that at least 7kg of nitrogen and same quantity of

phosphoric acid is required for a good crop. The fertilizers of type A and type B cost

Rs.5 and Rs.8 per kilograms respectively. Using L .P.P, find how many kgs of each type

of fertilizers should be bought to meet the requirement and cost be minimum solve the

problem graphically. What are the side effects of using excessive fertilizers?

4. Vikas has been given two lists of problems from his mathematics teacher with the

instructions to submit not more than 100 of them correctly solved for marks. The

problems in the first list are worth 10 marks each and those in the second list are worth 5

marks each. Vikas knows from past experience that he requires on an average of 4

minutes to solve a problem of 10 marks and 2 minutes to solve a problem of 5 marks.

He has other subjects to worry about; he cannot devote more than 4 hours to his

mathematics assignment. With reference to manage his time in best possible way how

many problems from each list shall he do to maximize his marks? What is the

importance of time management for students?

5. An NGO is helping the poor people of earthquake hit village by providing medicines. In

order to do this they set up a plant to prepare two medicines A and B. There is sufficient

raw material available to make 20000 bottles of medicine A and 40000 bottles of

144

medicine B but there are 45000 bottles into which either of the medicine can be put.

Further it takes 3 hours to prepare enough material to fill 1000 bottles of medicine A

and takes 1 hour to prepare enough material to fill 1000 bottles of medicine B and there

are 66 hours available for the operation. If the bottle of medicine A is used for 8 patients

and bottle of medicine B is used for 7 patients. How the NGO should plan his

production to cover maximum patients? How can you help others in case of natural

disaster

6. A dietician wishes to mix two types of foods in such a way that the vitamin content of

the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I

contains 2 units/kg of vitamin A and 1 units/kg of vitamin C while Food II contains

1unit/kg of vitamin A and 2 units/kg of vitamin C. It costs `5 per kg to purchase Food I

and`7 per kg to purchase Food II. Determine the minimum cost of such a mixture.

Formulate the above as a LPP and solve it graphically.

7. Cottage industry manufactures pedestal lamps and wooden shades, each requiring the

use of grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting

machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes one hour on

the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any

day, the sprayer is available for at the most 20 hours and the grinding/cutting machine

for at the most 12 hours. The profit from the sale of a lamp is ` 5 and that from a shade

is ` 3. Assuming that the manufacturer can sell all the lamps and shades that he

produces, how should he schedule his daily production in order to maximize his profit?

Make an L.P.P. and solve it graphically.

LEVEL - 3

1. An oil company has two depots A and B with capacities of 7000 L and 4000 L

respectively. The company is to supply oil to three petrol pumps D, E and F

whose requirements are 4500 L, 3000 L and 3500 L respectively. The distances

between the depots and the petrol pumps is given in the following table

Distance in (Km)

From/To A B

D 7 3

E 6 4

F 3 2

Assuming that the transportation cost of 10 L of oil is Rs1per km, how should the

deliverybescheduled in order that the transportation cost is minimum? What is the

minimum cost?

2. A manufacturing company makes 2 Models A and of a product .Eachpiece of Model A

requires 9 labour hours for Fabricating and 1 labour hour for finishing.Each piece is of

Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For

fabricating and finishing the maximum labour hours available are 180 and 30

respectively. The company makes a profit of Rs 8000 on each piece of model A and Rs

12000 on each piece of Model B . How many pieces of Model A and Model B should be

manufactured per week to realize a maximum profit? What is the maximum profit per

week

3. An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on

each executive class ticket and a profit of Rs 600 is made on each economy class ticket.

The airline reserves at least 20 seats for executive class .However,at least 4 times as

many passengers prefer to travel by economy class than by the executive class.

Determine how many tickets of each type must be sold in order to maximize the profit

145

for the airline. What is the maximum profit?

4. Every gram of wheat provides 0.1 gram of protein and 0.25 gram of carbohydrates .The

corresponding values for rice are 0.05 gm and 0.5 g respectively. Wheat costs Rs 4 per

Kg and rice Rs 6 per Kg. The minimum daily requirement of protein and carbohydrates

for an average child are50gm 200gm respectively. In what quantities should wheat rice

be mixed in the daily diet to provide minimum daily requirement of proteins and

Carbohydrates at minimum cost. Form an L P P and solve graphically

5. If a class XII student aged 17 years, rides his motor cycle at 40km/hr, the petrol cost is

Rs. 2 per km. If he rides at a speed of 70km/hr, the petrol cost increases Rs.7per km. He

has Rs.100 to spend on petrol and wishes to cover the maximum distance within one

hour.

(a). Express this as an L .P.P and solve graphically.

(b )What is benefit of driving at an economical speed?

(c) Should a child below 18years be allowed to drive a motorcycle? Give reasons.

VALUE BASED QUESTIONS:

1. If a class XII student aged 17 years, rides his motor cycle at 40km/hr, the petrol cost is

Rs. 2 per km. If he rides at a speed of 70km/hr, the petrol cost increases Rs.7per km. He

has Rs.100 to spend on petrol and wishes to cover the maximum distance within one

hour.

(a). Express this as an L .P.P and solve graphically.

(b ) What is benefit of driving at an economical speed?

(c) Should a child below 18years be allowed to drive a motorcycle? Give reasons.

2. A manufacturing company makes two types of teaching aids A and B of mathematics

for class XII. Each type of A requires 9 labour hours for fabricating and 1 labour hour

for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour

hours for finishing. For fabricating and finishing, the maximum labour hours available

are 180 and 30 respectively. The company makes a profit of Rs. 80/- on each piece of

type A andRs. 120/- on each type B. How many pieces of type A and B should be

manufactured per week to get a maximum profit? What is the maximum profit per

week?

Is teaching aid necessary for teaching learning process? If yes, justify your answer.

3. A village has 500 hectares of land to grow two types of plants, X and d Y. The

contributions of total amount of oxygen produced by plant X and plant Y are 60 % and

40 % per hectare respectively. To control weeds, a liquid herbicide has to be used for

X and Y at rates of 20 litres and 10 litres per hectare, respectively. Further no more

than 8000 litres of herbicides should be used in order to protect aquatic animals in a pond

which collects drainage from this land. How much land should be allocated to each crop

so as to maximize the total production of oxygen?

(i) How do you think excess use of herbicides affects our environment?

(ii) What are the general implications of this question towards planting trees around

us?

146

4. In order to supplement daily diet, a person wishes to take some X and some Y tables.

The contents of iron, calcium and vitamins in X and Y ( in milligrams per tablet) are

given below.

Tablets Iron Calcium Vitamins

X 6 3 2

Y 2 3 4

The person needs at least 18 milligrams of iron, 21 milligrams of calcium and 16

milligrams of vitamins. The price of each tablet of X and Y is Rs. 2/- and Rs.1/-

respectively. How many tablets of each should the person take in order to satisfy the

above requirement at the minimum cost?

Do you agree with the message that the nutrients are needful for good health?

ERROR ANALYSIS AND REMEDIATION:

S.No. Error Remedy

1.

Difficulty in converting word

problem into formulating LPP

1. Assume/Fix the Decision Variables such

as x,y….

2. Comprehend /Understand the statements

3. For each input[( Calcium,vitamins….)

(machine A, machineB….)…..] there

should be an inequality.

4. If ‗at least‘ or ‗minimum‘ comes in the

given statement then the unequal symbol

to make the constraint is‖ 5. If ‗atmost‘ or ‗maximum‘ comes in the

given statement then the unequal symbol

to make the constraint is ― 2. Wrong formation of Objective

Function

Repeated drilling in different kinds of problems

3. Sometimes the data, given in the

form of a table in word problems

may make the students to use a row

to make a constraint…

It is not necessary that every row can give an

inequality. Sometimes every column may also

give an inequality. Understanding the data of

the table will enable any one to form correct

inequalities.

4. Forget to write Non-Negative

constraints.

Remind the students that every LPP has Non-

Negative Constraints.

5. Unable to identify the Feasible

Region

Give sufficient practice.

Insist to make Mock (sample) shading for every

inequality.

Decide the common region of all the inequalities.

6. If the Feasible Region is Unbounded

then unable to identify whether the

(open)half plane determined by

ax+by>M or ax+by<M has any

point in common with the Feasible

Region or not

Since the half plane of the New Inequality is to be

drawn with dotted line, and the dotted line goes

through the Feasible Region then the system does

not have any solution. Otherwise the LPP has

Optimum solution.

147

7. Forget to write the final answer Reminding them again and again.

8. Forget to answer the Value Based

Question

Reminding them again and again.

QUESTION BANK:

Problems on LPP.docx

,POWER POINT PRESENTATION

ppt on LPP.ppt

WEB LINKS:

https://youtu.be/g8UI0Ea0Rhw

https://youtu.be/2ACJ9ewUC6U

148

PROBABILITY

LEARNING OBJECTIVES/OUTCOMES

To acquire the concept of Conditional Probability.

To distinguish Independence Events.

To acquire the concept of Partition.

To understand the Theorem on Total Probability.

To acquire the concept of Bayes‘ Theorem.

To distinguish Reverse Conditional from Conditional Probability.

To understand the Random Variable.

To find the Probability Distribution for a Random Variable.

To develop the skills to find the Mean, Variance and SD of a Random Variable.

To know the Bernoulli‘s Trials.

To acquire the Binomial Distribution.

To apply all the concepts in practical/ day to day problems.

149

CONCEPT MAP:

150

THREE LEVELS OF GRADED QUESTIONS:

LEVEL - 1

1. If P(A) = 0.3, P(B) = 0.2, find P(B/A) if A and B are mutually exclusive events

2. Find the probability of drawing two white balls in succession from a bag containing 3 red

and 5 white balls respectively, the ball first drawn is not replaced.

3. If P(A) = 6/11 , P(B) = 5/11 and P(A B)= 7 /11,

find (i) P (A∩B) (ii) P (A|B) (iii) P(B|A)

4. Two cards are drawn at random and without replacement from a pack of 52

playing cards. Find the probability that both the cards are black

5. A die is thrown 6 times. If ‗getting an odd number‘ is a success, what is the

probability of (i) 5 successes? (ii) at least 5 successes? (iii) at most 5 successes?

6. A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is

the conditional probability that the number 4 has appeared at least once?

7. Bag I contains 3 red and 4 black balls while another Bag II contains 5 red

and 6 black balls. One ball is drawn at random from one of the bags and it is found tobe

red. Find the probability that it was drawn from Bag II.

8. A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one

without replacement, find what the probability that none is red is.

9. A bag contains 6 red and 5 blue balls and another bag contains 5 red and 8 blue balls. A

ball is drawn from the first bag and without noticing its colour is put in the second bag.

A ball is drawn from the second bag. Find the probability that the ball drawn is blue in

colour.

10. Two cards are drawn successively with replacement from a well-shuffled deck of 52

cards. Find the probability distribution of the number of spades

11. An experiment succeeds twice as often it fails. Find the probability that in the next six

trials, there will be at least 4 successes.

12. The probability that an event happens in one trial of an experiment is 0.4. Three

independent trials of an experiment are performed. Find the probability that the event

happens at least once.

13. The sum of the mean and variance of a binomial distribution for 5 trials be 1.8.

Find the probability distribution.

Level – 2

1. A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one

without replacement, find the probability that none is red.

2. The probability of A hitting a target is 3/7 and that of B hitting is 1/3. They both fire at

the target. Find the probability that (i) at least one of them will hit the target, (ii) Only

one of them will hit the target.

3. An insurance company insured 2000 scooter and 3000 motorcycles. The probability of an

accident involving scooter is 0.01 and that of motorcycle is 0.02. An insured vehicle met

with an accident. Find the probability that the accidental vehicle was a motorcycle.

4. A purse contains 2 silver and 4 copper coins. A second purse contains 4 silver and 3

copper coins. If a coin is pulled at random from one of the two purses, what is the

151

probability that it is a silver coin?

5. A company has two plants to manufacture bicycles. The first plant manufactures 60 % of

the bicycles and the second plant 40 %. Out of that 80 % of the bicycles are rated of

standard quality at the first plant and 90 % of standard quality at the second plant. A

bicycle is picked up at random and found to be standard quality. Find the probability

that it comes from the second plant.

6. A die is thrown 6 times. If ‗gettingan odd number‘ is a success. What is the probability

of

(i) 5 successes, (ii) at least 5 successes? III) at most 5 successes

7. A family has 2 children. Find the probability that both are boys, if it is known that (i)

at least one of the children is a boy. (ii) the elder child is a boy.

8. In a factory which manufactures bolts, machines A, B and C manufacture respectively

25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively

defective bolts. A bolt is drawn at random from the product and is found to be defective.

What is the probability that it is manufactured by the machine B?

9. 4 defective apples are accidentally mixed with 16 good ones. Three apples are drawn at

random from the mixed lot. Find the probability distribution of the number of defective

apples

10. On a multiple choice examination with three possible answers(out of which only one is

correct) for each of the five questions, what is the probability that a candidate would get

four or more correct answers just by guessing ?

LEVEL- 3

1. A letter is known to have come either from LONDON or CLIFTON. On the envelope

just has two consecutive letters ON are visible. What is the probability that the letter has

come from (i) LONDON (ii) CLIFTON?

2. A test detection of a particular disease is not fool proof. The test will correctly detect the

disease 90 % of the time, but will incorrectly detect the disease 1 % of the time. For a

large population of which an estimated 0.2 % has the disease, a person is selected at

random, given the test, and told that he has the disease. What are the chances that the

person actually have the disease

3. Given three identical boxes I, II and III each containing two coins. In box I, both coins

are gold coins, in box II, both are silver coins and in box III, there is one gold and one

silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold,

what is the probability that the other coin in the box is also of gold ?

4. In a school there are 1000 students , out of which 430 are girls. It is known that out of

430 , 10% of the girls study in class XII. What is the probability that a student chosen

randomly studies in class XII, given that the chosen student is a girl?.

Colored balls are distributed in three bags as shown in the following table

152

5.

Bag Color of the ball

Black White Red

I 2 1 3

II 4 2 1

III 5 4 3

A bag is selected at random and then two balls are randomly drawn from the

selected bag. They happen to be white and red. What is the probability that they have

come from bag II?

6. In a bulb factory,machines A,B and C manufacture 60%,30% and 10% bulbsrespectively

1% , 2% and 3% of the bulbs produced respectively by A,B and C are found to be

defective. A bulb is picked up at random from the total production and found to be

defective . Find the probability that the bulb was produced by the machine A.

7. There is a group of 50 people who are patriotic out of which 20 believe in non- violence.

Two persons are selected at random out of them, write the probability distribution for

the selected persons who are non-violent. Also find the mean of the distribution.

8. A man is known to speak the truth 3 times out of 5 times. He throws a die and reports

that it is a number greater than 4. Find the probability that it is actually a number greater

than 4.

9. Two cards are drawn simultaneously (without replacement) from a well shuffled pack

of 52 cards. Find the probability distribution of the number of aces. Also find mean of

the distribution.

10. A man takes a step forward with probability 0.4 and backward with probability 0.6. Find

the probability that at the end of eleven steps he is one step away from the starting point.

11. In a game, a man wins a rupee for a six and looses a rupee for any other number when a

fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a

six. Find the expected value of the amount he wins/looses.

VALUE BASED QUESTIONS:

1. In answering a question on a MCQ test with 4 choices per question, a student knows the

answer, guesses or copies the answer. Let ½ be the probability that he knows the answer,

¼ be the probability that he guesses and ¼ that he copies it. Assuming that a student, who

copies the answer, will be correct with the probability ¾ , what is the probability that the

student knows the answer, given that he answered it correctly?

Arjun does not know the answer to one of the questions in the test. Theevaluation process

has negative marking. Which value would Arjun violateif he resorts to unfair means? How

would an act like the above hamper hischaracter development in the coming years?

2) An insurance company insured 2000 cyclists, 4000 scooter drivers and 6000 motorbike

drivers. The probability of an accident involving a cyclist, scooter driver and a motorbike

driver are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an

153

accident. What is the probability that he is a scooter driver? Which mode of transport

would you suggest to a student and why?

3) The probabilities of two students A and B coming to school in time are 3/7 and 5/7

respectively. Assuming that the events , ‗A come in time ‗ and ‗B come in time ‗ are

independent , find the probability of only one of them coming to the school in time.

Write at least one advantage of coming to the school in time.

4) In a hockey match , both teams A and B scored same number of goals up to the end of

the game, so as to decide the winner , the referee asked both the captains to throw a die

alternately and decided that the team, whose captain gets a six first, will be declared

the winner. If the captain of team A was asked to start, find their respective probabilities

of winning the match and state whether the decision of the referee was fair or not.

5) Of the students in a school , it is known that 30 % has 100 % attendance and 70 %

students are irregular.Previous year results show that 70 % of all students who has 100 %

attendance attain A grade and 10 % of irregular students attain A grade in their annual

examination. At the end of the year, one student is chosen at random from the school

and he has A grade. What is the probability that the student has 100 % attendance.

(i) Write any two values reflected in this question (ii) Is regularity required only in school?

Justify your answer.

154

ERROR ANALYSIS AND REMEDIATION

S.No. Error Remedy

1 Unable to identify the idea to be

employed whether Conditional or

Baye‘s Theorem

Reverse conditional is the Baye‘s Theorem.

1. If the events are in proper sequence

then it the Conditional Prob.

2. If the events are in reverse sequence

then it is the Baye‘s Therorem

2 Mistakes in identifying events that

constitute the Partition

These events must be mutually exclusive and

exhaustive.

That is Pairwise disjoint and their union must be

the Sample Space.

3 If in the problems of Baye‘s

Theorem, the values of the

conditional prob. such as P(A/E1),

P(A/E2) … are not given

They should be either Sure Event with Prob 1 or

Impossible Event with prob 0

4 In problems of Baye‘s Theorem

involving % (percentages )

confusion to find every conditional

prob such as P(A/E1), P(A/E2)

Take E1 or E2 as the sample space that is 100 %

and take the value given in terms of % and do not

multiply /subtract the percentages

5 Unable to take the values of the

Random Variable for a probability

distribution

Practise more questions of such type.

6 Unable to form the probability

distribution table

Identify whether it is with replacement or

without replacement.

Sum of all the probabilities is equal to 1.

Practise more questions of such type

7. Forget to write the final answer Emphasize on writing the final answer

8. Forget to answer the Value Based

Question

Reminding them again and again.

QUESTION BANK :

13.HOTS-PROBABILITY.pdf

PROJECTS AND PRACTICALS,POWER POINT PRESENTATION,REFERENCE WEB

LINKS,VIDEO CLIPINGS,VISUALS ETC.:

https://youtu.be/8_DdIEzmMTs

- Conditional Probability ppt -1.ppt

https://youtu.be/yBn-eGrRHJU

155

SAMPLE PAPER

156

SAMPLE PAPER - 1

157

158

159

160

161

162

TIPS &TECHNIQUES IN TEACHING LEARNIG PROCESS

1: Interest and explanation – ―When our interest is aroused in something, whether it is an

academic or a hobby, we enjoy working hard at it. We come to feel that we can in some way own

it and use it to make sense of the world around us.‖ . Coupled with the need to establish the

relevance of content, instructors need to craft explanations that enable students to understand the

material. This involves knowing what students understand and then forging connections between

what is known and what is new.

2: Concern and respect for students and student learning –

Ramsden (an educationist ) stars with the negative about which he is assertive and unequivocal.

―Truly awful teaching in higher education is most often revealed by a sheer lack of interest in

and compassion for students and student learning. It repeatedly displays the classic symptom of

making a subject seem more demanding than it actually is. Some people may get pleasure from

this kind of masquerade. They are teaching very badly if they do. Good teaching is nothing to do

with making things hard. It is nothing to do with frightening students. It is everything to do with

benevolence and humility; it always tries to help students feel that a subject can be mastered; it

encourages them to try things out for themselves and succeed at something quickly.‖

3: Appropriate assessment and feedback –

This principle involves using a variety of assessment techniques and allowing students to

demonstrate their mastery of the material in different ways. It avoids those assessment methods

that encourage students to memorize and regurgitate. It recognizes the power of feedback to

motivate more effort to learn.

4: Clear goals and intellectual challenge –

Effective teachers set high standards for student. They also articulate clear goals. Students should

know what they will learn and what they will be expected to do with what they know.

5: Independence, control and active engagement –

―Good teaching fosters a sense of student control over learning and interest in

the subject matter.‖ . Good teachers create learning tasks appropriate to the student‘s level of

understanding. They also recognize the uniqueness of individual learners and avoid the

temptation to impose ―mass production‖ standards that treat all learners as if they were exactly

the same. ―It is worth stressing that we know that students who experience teaching of the kind

that permits control by the learner not only learn better, but that they enjoy learning more.‖

6: Learning from students –

―Effective teaching refuses to take its effect on students for granted. It sees the relation between

teaching and learning as problematic, uncertain and relative. Good teaching is open to change: it

involves constantly trying to find out what the effects of instruction are on learning, and

modifying the instruction in the light of the evidence collected.‖

163

TIPS AND TECHNIQUES TO SECURE BETTER MARKS

1. Usage of CAL/TAL/Concept mapping to be extensively and exhaustively done

to captivate all students so that understanding of concepts( which forms basis for

good scoring) will be made as easy as possible.

2. Teaching – Learning process should be paced such a way that it caters to the

needs of all types of students, Late bloomers, Average students then

automatically the Gifted students.

3. Recapitulation, at the end of every period , to be done in the form of Oral

Questions.

4. Identify the Late Bloomers and Gifted Students at the beginning of the year.

For Late Bloomers

Identify the strengths and weaknesses in every topic/chapter

Thorough revision is to be done to make these Late Bloomers feel

confident.

Give time for them to interact with the teacher/peer group student to

dispel any doubt, if they have

Conduct slip test.(The question paper should contain very simple

concepts/questions to boost their self-confidence)

After the distribution of evaluated answer scripts, interact with

every individual to clear all their doubts if exists.

As far as possible give separate home assignments for these late

bloomers so that mere copying from the peer group can be avoided .

And it will make them to work independently

As far as possible inform the students that a surprise test of 1 or 2

questions can be conducted on the following day of every home

assignment . This will enable every late bloomer to work

independently.

These students should always be appreciated in the class room in front

of all students for every good work/answer…. This will enthuse them

in the subject.

Once these late bloomers grow in self-confidence (Or) show the sign

of self confidence (in slip test), they should be graduated to Average

students. This will prompt them to work harder and smarter.

After the completion of syllabus, a Common Minimum Programme to

be devised by selecting a few chapters, which will provide a minimum

of 50 marks in the Board Examination.

Rigourous drilling to be done in the CMP so that their minimum pass

mark can be ensured

164

For Gifted Students

These Gifted Students should be encouraged by providing the concept

wise HOTS questions.

They should be encouraged to solve more challenging questions

which will improve their calculational or computational skill , mostly

required for the Entrance Examinations.

More thought provoking questions are to be collected and a question

bank to be prepared to be given to gifted students to develop their

analyzing and reasoning capabilities.

Class seminars can be allotted the these students so that the late

bloomers will also be motivated automatically.

Guide them to solve previous Entrance Examination question papers

to prepare them for their entrance examination also.

5. While doing revision, Micro learning process can be implemented by dividing

the entire students into different groups of 5 students (comprising late bloomers,

average students), headed by a gifted student. Each group will revise under the

control of the group leader, supervised and guided by the teacher.

6. Previous years Board papers are to be practiced thoroughly.

7. The students have to be advised to practice all problems including the examples

and problems of Misc. Exercises.

8. Advice each student to have LONG TERM GOAL . To achieve the Long Term

Goal, some short term goals can be planned and meticulously executed.

9. Sufficient tips should be given for time management.

10. Parents- Teachers meeting is a must after every Cumulative Examination

including Pre-Board Examinations.

*************

165

Useful web links for Maths

1- Aplus Math

Aplus Math provides Interactive math resources for teachers, parents, and students

featuring free math worksheets, math games, math flashcards, and more.

2- Math TV

Math TV is a platform that features a wide range of math videos covering a plethora of

mathematical concepts. These videos are browsable via topic or by textbook.

3- AAA Math

AAA Math offers thousands of arithmetic lessons from kindergarten through eightgrade.

Unlimited practice is also available on each topic which allows thorough mastery of the

concepts.

4- Math's Fun

166

This philosophy behind this website is to make math learning fun and enjoyable. It

features a myriad of lessons and activities provided by teachers and math community

from all around the world.

5- Math Central

Math Central is an Internet service for mathematics students and teachers. This site is

maintained by faculty and students in Mathematics and Statistics and Mathematics

Education at the University of Regina in Canada.

6- Ten Marks

TenMarks provides students with access to hints and video lessons on every problem, so

if they can't recall something, or didn't quite get the topic when it was covered in class,

they can quickly review the content, and move forward.

7- Maths Frame

Mathsframe has more than 170 free interactive maths games. All resources are designed,

by an experienced KS2 teacher, to help children to visualise numbers, patterns and

numerical relationships and to develop their mathematical thinking. New games are

added most weeks

8- SMILE

The SMILE program was designed to enhance the elementary and high school learning of

Science and Mathematics through the use of the phenomenological approach.

9- The Math Forum

167

This is a community of teachers, mathematicians, researchers, students, and parents using

the power of the Web to learn math and improve math education. The forum offers a

wealth of problems and puzzles; online mentoring; research; team problem solving;

collaborations; and professional development. Students have fun and learn a lot.

Educators share ideas and acquire new skills.

10- Simpsons math

The Simpsonsmath contains over a hundred instances of mathematics ranging from

arithmetic to geometry to calculus, many designed to expose and poke fun at innumeracy.

11- SuperKids

SuperKids lets you create your own Math drills. Simply select the type of problem, the

maximum and minimum numbers to be used in the problems, then click on the button! A

worksheet will be created to your specifications, ready to be printed for use.

12- Math Words

This is an interactive math dictionary with enough math words, math terms, math

formulas, pictures, diagrams, tables, and examples to satisfy your inner math geek.

13- Math Guide

MATHguide offers a variety of mathematics lessons. Numerous lessons in algebra,

geometry, and pre-calculus are available. One can also utilize an assessment resource,

168

called quizmasters.

14- Math League

Mathleague.org offers a number of services focused on enhancing the quality and

quantity of competitive mathematical opportunities available to students everywhere. We

offer a variety of programs for students in grades 3-12.

15- Math Drills

Math-Drills has thousands of Free Math Worksheets for teachers and parents on a variety

of math topics.

16- Math Goodies

Math Goodies is your free math help portal featuring interactive lessons, worksheets, and

homework help.

17- Math Aids

Math-Aids is a free resource for teachers, parents, students, and home schoolers. The

math worksheets are randomly and dynamically generated by our math worksheet

generators. This allows you to make an unlimited number of printable math worksheets to

your specifications instantly

18- Math Basics

169

Math Basics offers help with basic math like subtraction, multiplication, division,

fractions, times tables and percents. It provides math tutorials and learning interactives to

make learning math easier, and allow you to practice basic math skills at your own level

and pace.

19- Get The Math

Get the Math is about algebra in the real world. See how professionals use math in music,

fashion, videogames, restaurants, basketball, and special effects.

20- Absurd Math

Absurd Math is an interactive mathematical problem solving game series. The player

proceeds on missions in a strange world where the ultimate power consists of

mathematical skill and knowledge.

THE END

170

Annexure list

Topic S.No Name of the

documents

Type

VUE

Determinant

1 Consistency or

Inconsistency

Word

document

2 Minor and Cofactor Word

document

3 Properties of adjoint

matrix

Word

document

4 Area of a triangle Word

document

5 Properties of a

Determinant

Word

document

Matrices

1 Addition of two

Matrices using scalar

PPT

2 Entries of a matrix Word

document

3 Addition of two

Matrices

Word

document

4 Equality of two

Matrices

Word

document

5 Sum addition using

Transpose

Word

document

6 Rule for Matrix

multiplication

Word

document

7 Types of Matrices Word

document

Power Point

Presentation

Determinant ppt

Matrices ppt

171

Integrals

Indefinite Integrals

1) VUE Map

2) Definition

3) Fundamental Formula

4) Integration by Substitution

5) Trigonometric Identities

6) Special Integrals

7) Standard form

8) Partial Fraction

9) Integration by parts

10) PPT_ Basic Integration

11) PPT_ Integration

12) PPT_ Integration_2

13) PPT_ Integration_3

14) PPT _Partial Fraction

Definite Integrals

1) VUE Map

2) Definition

3) Fundamental Theorem

4) Limit of sums

5) Properties of Definite Integrals

6) PPT_ Definite Integrals _ 1

7) PPT_ Definite Integrals _2

Application of Integrals

Application of Integrals

1) VUE Mapping

2) Area

3) Area Under Simple Curve

4) Area Enclosed

5) Area of Triangle

6) Circle

7) Circle and line

8) Ellipse

9) Ellipse and line

10) Ellipse and Parabola

11) Modulus Function

12) PPT_ Fundamental Theorem

13) PPT_ Ares using Integration

14) PPT_ Integration Area

Differential Equations

Differential Equations

1) VUE Mapping

2) Definition

3) Order and Degree

4) Solution of differential equation

5) Formation of differential equation

172

6) Variable Separable

7) Homogenous Differential Equations

8) Linear Differential Equations Type I

9) Linear Differential Equations

10) PPT_ Basic of Differential Equations

11) PPT_ Differential Equations

Topic S.No Name of the documents Type

VUE

RELATIONS AND

FUNCTIONS

1 Definition of Relation Word document

2 Types of Relations Word document

3 Domain and Range Word document

4 Equivalence Relation Word document

5 Definition of Function Word document

6 Types of Functions Word document

7 Bijective Function Word document

8 Composition Function Word document

9 Invertible Function Word document

INVERSE

TRIGONOMETRIC

FUNCTIONS

1 Function and its type Word document

2 Bijective Function Word document

3 Invertible Function Word document

4 Trigonometric Function Word document

5 Restricted Domain Word document

6 Principal Value Branch Word document

7 Inverse Trigonometric

Functions

Word document

8 Graph of Trig Functi&Its

Inverse

Word document

9 Properties Word document

VECTORS

1 Definition PPT

2 Types of Vectors PPT

3 dc‘s&dr‘s of a Vector PPT

4 Position Vector of a point PPT

5 Addition of two Vectors PPT

6 Properties of Addition PPT

7 Difference of two Vectors PPT

8 Multiplication of a Vector

by a scalar

PPT

9 Components of a Vector PPT

10 Section Formula PPT

11 Vector joining two points PPT

12 Dot Product of Two

Vectors

(a) Definition

(b) Properties

(c) Angle between two

vectors

PPT

173

(d) Condition for

Perpendicularity

(e) Projection of a

Vector

13 Cross Product of two

Vectors

(a) Definition

(b) Properties

(c) Angle between two

vectors

(d) Condition for

Parallal vectors

(e) Area of Triangle

(f) Area of

Parallelogram

PPT

ANNEXURE; 11- 3 Dimensional Geometry

S.No. Software Linked/Embedded word doc.

1 VUE on 3 –d-

geometry

1. Direction Cosines

2. Direction Ratios

3. Line- Point , Parallel Vector

4. Line- Two Point

5. Skew Lines

6. Parallel Lines

7. Angle between two lines

8. Angle between a Line and a Plane

9. Plane- Normal form

10. Plane- Point, Normal form

11. Plane- Three point form

12. Plane - Intercepts form

13. Angle between two planes

14. Intersection of two planes

15. Coplanarity of two lines

16. Distance between a point and a plane

2 PPT Three Ppt on 3 dimensional geometry

3 Geo-Gebra Geogebra on on 3 dimensional geometry

4 VUE- on LPP 1. Optimization problem

2. Corner Point Methor

3. Objective Function

4. Constraints

5. Non-negative Constraints

6. Feasible Region

7. Feasible Solution

8. Infeasible Solution

9. Manufacture Problem

174

10. Diet Problem

11. Transportation Problem.

5 Geo-Gebra on

LPP

Manufacturing Problem

6 PPT Ppt on LPP

7 VUE (Probability) 1. Probability- Basics

2. Conditional Probability

3. Properties of Conditional Probability

4. Independent Events

5. Partition of Events

6. Theorem on Total Probability

7. Baye‘s Theorem

8. Bernoulli‘s Trials

9. Binomial Distribution

10. Random Variable

11. Probability Distribution of Random Variable.

12. Mean of Random Variable

13. Variance and Standard Deviation of Random

Variable

8 PPT(Probability) Ppt on Conditional Probability

THE END