आचं लक श ाएवं श णसं थान

जीआईट बी ेसकै पस, स ाथनगर, मैसूर-570011 KVS Zonal Institute of Education and Training GITB Press Campus, Siddartha Nagar, Mysore

Website: www.zietmysore.org, Email: kvszietmysore@gmail.com/zietmysore@rediffmail.com

Phone: 0821 2470345 Fax: 0821 24785

Venue: KVS ZIET MYSORE

Date: 15th to 17th July, 2014

RESOURCE MATERIALS

CLASS XII(2014-15)(Mathematics)

क य िव ालय सगंठननई द ल

KENDRIYA VIDYLAYA SANGTHAN NEW DELHI

क य व यालय संगठन नई द ल KENDRIYA VIDYLAYA SANGTHAN NEW DELHI

आंच लक श ा एवं श ण सं थान मैसूर

ZONAL MINSTITUTE OF EDUCATION AND TRAININGMYSOREश

3-Day Strategic Action plan workshop

आ1ं5th-1555((15th to 17th July, 2014

DIRECTOR

Mr.S Selvaraj DEPUTY COMMISSIONER

KVS ZIET Mysore

COURSE DIRECTOR

Mrs.V. Meenakshi ASSISTANT COMMISSIONER

KVS ERNAKULUM REGION

ASSOCIATE COURSE DIRECTOR

Mr. E. Krishna Murthy

PRINCIPAL, KV NFC Nagar

OUR PATRONS

Shri AvinashDikshit , IDAS

Commissioner

Sh. G.K. Srivastava, IAS

Addl. Commissioner (Admn)

Dr. Dinesh Kumar

Addl. Commissioner (Acad.)

Dr. Shachi Kant

Joint Commissioner (Training)

ाएवं श

णसं थान

FOREWORD Excellence and perfection has always been the hallmark of KendriyaVidyalayaSangathan in all its activities. In academics, year after year, KVS has been showing improved performance in CBSE Examinations, thanks to the consistent and committed efforts of the loyal KVS employees, the teachers, Principals and officials collectively. Every year begins with a new strategic academic planning, carefully calibrated to achieve the targeted results. In line with the holistic plan of KVS, ZIET Mysore took the initiative to organize a 3-day Strategic Action Plan Workshop from 15th to 17th July, 2014, in the subjects of Physics, Chemistry, Mathematics, Biology and Economics to produce Support Materials for students as well as teachers so that the teaching and learning process is significantly strengthened and made effective and efficient. For the purpose of the Workshop, each of the four Regions namely Bengaluru, Chennai, Ernakulam and Hyderabad was requested to sponsor two highly competent and resourceful Postgraduate Teachers in each of the above mentioned subjects. Further, in order to guide and monitor their work, five Principals with the respective subject background were invited to function as Associate Course Directors:

1. Mr. E. Krishna Murthy, K.V. NFC Nagar, (Mathematics) 2. Mr. M. Krishna Mohan, KV CRPF Hyderabad(Economics) 3. Mr. R. Sankar, KV No.2 Uppal, Hyderabad (Biology) 4. Dr. (Mrs.) S. Nalayini, K.V. Kanjikode (Physics) 5. Mr. T. Prabhudas, K.V. Malleswaram (Chemistry)

In addition to the above, Mrs. V. Meenakshi, Assistant Commissioner, KVS, Regional Office, Ernakulam willingly agreed to support our endeavor in the capacity of the Course Director to oversee the workshop activities. The Workshop was aimed at creating such support materials that both the teachers and the students could rely upon them for complementing the efforts of each other to come out with flying colours in the CBSE Examinations. Accordingly, it was decided that the components of the package for each subject would be:

(1) Chapter-wise concept Map. (2) Three levels of topic-wise questions. (3) Tips and Techniques for teaching/learning each chapter. (4) Students’ common errors, un-attempted questions and their remediation. (5) Reviewed Support Materials of the previous year.

In order to ensure that the participants come well-prepared for the Workshop, the topics/chapters were distributed among them well in advance. During the Workshop the materials prepared by each participant were thoroughly reviewed by their co-participantS and necessary rectification of deficiencies was carried out then and there, followed by consolidation of all the materials into comprehensive study package. Since, so many brilliant minds have worked together in the making of this study package, it is hoped that every user- be it a teacher or a student – will find it extremely useful and get greatly benefitted by it. I am deeply indebted to the Course Director, Smt. V. Meenakshi, the Associate Course Directors viz., Mr. E. Krishna Murthy, Mr.M. Krishna Mohan, Mr. R. Sankar, Dr.(Mrs.) S. Nalayani and Mr. T. Prabhudas and also all participants for their significant contribution for making the workshop highly successful, achieving the desired goal. I am also greatly thankful to Mr. M. Reddenna, PGT [Geog](Course Coordinator) and Mr. V.L. Vernekar, Librarian and other staff members of ZIET Mysore for extending their valuable support for the success of the Workshop. Mysore ( S. SELVARAJ ) 17.07.2014 DIRECTOR

Three Day workshop on Strategic planning for achieving quality results in Mathematics

KVS, Zonal Institute of Education and Training, Mysore organized a 3 Day Workshop on ‘Strategic Planning for Achieving Quality Results in Mathematics’ for Bangalore, Chennai, Hyderabad, & Ernakulum Regions from 15th July to 17th July 2014.

The Sponsored Seven Post Graduate Teachers in Mathematics from four regions were allotted one/ two topics from syllabus of Class XII to prepare concrete and objective Action Plan under the heads:

1. Concept mapping in VUE portal 2. Three levels of graded exercises3 3.Value based questions 4. Error Analysis and remediation 5.Tips and Techniques in Teaching Learning process

6. Fine-tuning of study material supplied in 2013-14.

As per the given templates and instructions, each member elaborately prepared the action plan under six heads and presented it for review and suggestions and accordingly the package of study materials 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 wish to place on record my sincere appreciation to the Associate Course Director Mr.E Krishnamurthy, Principal, K.V.NFC Nagar, Hyderabad, the Resource Persons, the Course Coordinator Mr.M.Reddenna, PGT (Geo) ZIET Mysore and the members of faculty for their wholehearted participation and contribution to this programme.

I thank Mr. S.Selvaraj, 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 of Bangalore, Chennai, Ernakulum and Hyderabad Regions for very focused classroom transactions using this Resource Material (available at www.zietmysore.org) to bring in quality and quantity results in the Class XII Board Examinations 2015.

Mrs.V Meenakshi Assistant Commissioner

Ernakulum Region

“With a clever strategy, each action is self-reinforcing. Each action creates more options that are mutually beneficial. Each victory is not just for today but for tomorrow.”

― Max McKeon

From Associate Director’s Desk: In-service Courses, Orientation Programmes and workshops on various issues are integral part of Kendriya Vidyalaya Sangathan. These courses provide the teachers opportunities to learn not only the latest in the field of Mathematics teaching, latest technologies in teaching learning process to update themselves to become professional teachers but also help the teachers to face the emerging challenges of present day world. The 03 day workshop for preparation of Practice papers and strategic plan for achieving quality result in CBSE Examinations for class XII in Mathematics organized at ZIET, Mysore, is designed with time table which gives sufficient room for Concept mapping on various Chapters, Strategic plan to improve results of Class XII, Preparation of Value based and graded questions, common errors committed by students and methods of remediation, methods to make the students to attempt questions from difficult areas of Mathematics and Chapter- wise tips and techniques to maximize the scores in the CBSE Examinations. This time table has been carried out with utmost care and lot of material has been prepared by the team of well experienced teachers selected for this purpose from KVS Hyderabad Bangalore, Chennai and Ernakulam Regions. The material prepared is so useful to the teachers to produce better and quality results and make the teaching – learning is easier and effective. I record my sincere appreciations to all the Resource persons for their sincere efforts, dedication, commitment and contribution in preparing the material and Strategic plan to improve the performance of students in CBSE Examinations. I too have learned and enjoyed working with the Resource persons during three day workshop in preparing the strategic plan. I express my sincere gratitude to KVS authorities particularly Shri. S Selvaraj, Director, ZIET Mysore and Mrs. V. Meenakshi, Asst. Commissioner, Ernakulam Region and Course Director for providing me the opportunity to participate in 03 day workshop as Associate Director.

Also I express my sincere thanks to the faculty and staff of ZIET Mysore for their kind support in successful organization of 03 day workshop.

My best wishes to all the students and teachers.

E KRISHNA MURTHY Associate Director and Principal

Kendriya Vidyalaya, NFC Nagar, Hyderabad Region

KV

S-ZI

ET

-MY

SOR

E

03-d

ay W

orks

hop

on S

trat

egic

Act

ion

Plan

15-

17.0

7.14

- D

etai

ls fo

r C

onta

ct

Sl N

o N

ame

in E

nglis

h D

esig

n.

K.V

in E

nglis

h R

egio

n Ph

one

No.

E

-mai

l Add

ress

1

Mrs

. V. M

eena

kshi

A

sstt.

Com

mr.

R

egio

nal O

ffic

e Er

naku

lam

94

9614

6333

m

inak

visw

a@gm

il.co

m

2 M

r. E.

Kris

hna

Mur

thy

Prin

cipa

l N

FC N

agar

,Gha

tkes

ar

Hyd

erab

ad

9989

0637

49

kris

hnam

urth

ykvs

@gm

ail.c

om

3 M

r. T.

Pra

bhud

as

Prin

cipa

l M

alle

swar

am

Ban

galo

re

8762

6659

90

t.pra

bhud

as@

redi

ffmai

l.com

4 D

r(M

rs.)

S. N

alay

ini

Prin

cipa

l K

anjik

ode

Erna

kula

m

9446

3611

86

dr.s.

nala

yini

@gm

ail.c

om

5 M

r. M

. Kris

hna

Moh

an

Prin

cipa

l C

RPF

Hyd

erab

ad

Hyd

erab

ad

9440

8657

61

moh

anm

k58@

yaho

o.co

.in

6 M

r. R

. San

kar

Prin

cipa

l N

o.2

Upp

al

Hyd

erab

ad

9491

0736

00

ram

iset

tysa

nkar

@gm

ail.c

om

7 M

r. E.

N. K

anna

n PG

T(Ph

y)

BEM

L N

agar

B

anga

lore

87

6220

8431

ka

nnae

n200

7@ya

hoo.

com

8 M

r. D

.B. P

atna

ik

PGT(

Bio

) R

ailw

ay C

olon

y B

anga

lore

89

7124

0593

da

bbiru

bhar

ath@

gmai

l.com

9 M

r. K

.S.V

. Som

esw

ara

Rao

PG

T(Ph

y)

MEG

& C

entre

B

anga

lore

94

4870

8790

ks

vsra

o64@

yaho

o.co

m/k

svsr

ao@

gmai

l.com

10

Mr.

K.R

. Kris

hna

Das

PG

T(M

aths

) N

o.1

AFS

Sam

bra

Ban

galo

re

8951

6482

75

kris

hnad

asdw

d@gm

ail.c

om

11

Mr.

G.N

. Heg

de

PGT(

Mat

hs)

Dha

rwad

B

anga

lore

94

4862

6331

gn

hegd

e130

@gm

ail.c

om

12

Dr.

Viv

ek K

umar

PG

T(C

hem

) C

RPF

Yel

ahan

ka

Ban

galo

re

8970

7208

95

kum

ar.v

.viv

ek@

gmai

l.com

13

Mr.

Ran

gaN

ayak

ulu

.A

PGT(

Che

m)

Heb

bal

Ban

galo

re

7899

2872

64

andr

ojul

a@gm

ail.c

om

14

Mrs

. G.K

. Vin

ayag

am

PGT(

Bio

) N

o.2

Bel

gaum

Can

tt.

Ban

galo

re

9448

1206

12

vgay

atri3

@gm

ail.c

om

15

Mr.

D. R

ami R

eddy

PG

T(Ec

o)

Rai

lway

Col

ony

Ban

galo

re

9740

3986

44

ram

iredd

y_do

ndet

i@ya

hoo.

co.in

16

Mrs

. T.M

. Sus

hma

PGT(

Eco)

H

ebba

l B

anga

lore

87

6269

1800

t.m

sush

ma@

gmai

l.com

17

Mrs

. Ash

a R

ani S

ahu

PGT(

Mat

hs)

Mys

ore

Ban

galo

re

9902

6632

26

asha

rani

sahu

@gm

ail.c

om

18

Mrs

. Joa

n Sa

nthi

Jose

ph

PGT(

Che

m)

IS G

roun

ds, C

henn

ai

Che

nnai

99

4094

5578

jo

ansa

nthi

@gm

ail.c

om

19

Mrs

. C.V

. Var

alak

shm

i PG

T(Ph

y)

AFS

Ava

di, C

henn

ai

Che

nnai

90

0308

0057

c.

venk

atav

aral

aksh

mi@

gmai

l.com

20

Mr.

Siby

Seb

astia

n PG

T(M

aths

) G

ill N

agar

C

henn

ai

8056

1793

11

siby

_kav

il@ya

hoo.

co.in

21

Mr.

S. V

asud

heva

n PG

T(C

hem

) D

GQ

A C

ompl

ex

Che

nnai

94

4420

9820

va

su_s

bm@

yaho

o.co

m

22

Mrs

. Sat

hya

Vija

ya R

agha

van

PGT(

Eco)

M

inam

bakk

am

Che

nnai

94

4539

0058

sa

thya

vija

yara

ghav

an@

gmai

l.com

23

Mr.

S. K

umar

PG

T(Ph

y)

No.

1 K

alpa

kkam

C

henn

ai

8015

3742

37

soor

ared

dyk@

gmai

l.com

24

Mrs

. A. D

aisy

PG

T(B

io)

Min

amba

kkam

C

henn

ai

9840

7642

40

dais

y.de

vaku

mar

@gm

ail.c

om

25

Mrs

. C.K

. Ved

apat

hi

PGT(

Bio

) II

T C

henn

ai

Che

nnai

98

4158

3882

ve

d.kv

iitc@

gmai

l.com

26

Mrs

. J. U

ma

PGT(

Eco)

A

nnan

agar

C

henn

ai

9840

9887

55

umak

rkum

ar@

gmai

l.com

27

Mrs

. Sul

ekha

Ran

i .R

PG

T(C

hem

) N

TPC

Kay

amku

lam

Er

naku

lam

97

4581

4475

su

lekh

aran

ir@gm

ail.c

om

28

Mrs

. Mar

y V

. Che

rian

PGT(

Bio

) SA

P Pe

roor

kada

Er

naku

lam

94

4710

7895

m

aryv

cher

ian@

yaho

o.co

.in

29

Mrs

. Sus

mith

a M

ary

Rob

bins

PG

T(Ph

y)

Kal

petta

Er

naku

lam

94

9552

8585

ss

mth

.robb

ins@

gmai

l.com

30

Mr.

Jose

ph K

.A

PGT(

Eco)

R

.B K

otta

yam

Er

naku

lam

94

4636

9351

ka

llika

ttu31

@re

diff

mai

l.com

31

Mrs

. Jyo

lsna

K.P

PG

T(M

aths

) N

o.1

Cal

icut

Er

naku

lam

94

4736

5433

jy

olsn

akp@

gmai

l.com

32

Mrs

. Ush

aMal

ayap

pan

PGT(

Eco)

K

anjik

ode

Erna

kula

m

9496

5190

79

usha

mal

ayap

pan@

gmai

l.com

33

Mrs

. Suj

atha

M. P

oduv

al

PGT(

Bio

) K

eltro

n N

agar

Er

naku

lam

94

4649

4503

su

jath

ampo

duva

l@gm

ail.c

om

34

Mr.

Pras

hant

h K

umar

.M

PGT(

Phy)

K

eltro

n N

agar

Er

naku

lam

94

0056

6365

m

pkna

mbi

ar@

gmai

l.com

35

Mr.

Sibu

John

PG

T(C

hem

) Er

naku

lam

Er

naku

lam

95

4459

4068

si

bu_j

hn@

yaho

o.co

.in

36

Mr.

N.S

. Sub

ram

ania

n PG

T(M

aths

) G

ooty

H

yder

abad

94

9003

9741

ns

sss_

mat

h@ya

hoo.

co.in

37

Mrs

. Jos

ephi

ne B

alra

j PG

T(M

aths

) N

o.1

AFA

Dun

diga

l H

yder

abad

94

4006

6208

ba

lraj_

jb@

yaho

o.co

m

38

Mr.

B. S

esha

Sai

PG

T(Ph

y)

AFS

Hak

impe

t H

yder

abad

99

1238

4681

se

shas

ai20

05@

yaho

o.co

m

39

Mr.

V.V

..S.K

esav

a R

ao

PGT(

Phy)

G

achi

bow

li H

yder

abad

94

9022

1144

vv

skr2

007@

gmai

l.com

40

Kum

. San

uRaj

appa

n PG

T(Ec

o)

Gac

hibo

wli

Hyd

erab

ad

9640

6461

89

sanu

_raj

appa

n@ya

hoo.

com

41

Mr.

M.T

. Raj

u PG

T(B

io)

AFS

Beg

umpe

t H

yder

abad

96

5268

0800

m

traju

hyde

raba

d@gm

ail.c

om

42

Mr.

D. A

shok

PG

T(C

hem

) C

RPF

Hyd

erab

ad

Hyd

erab

ad

9618

0120

35

saia

shok

dom

akon

da@

gmai

l.com

43

Mr.

D. S

aidu

lu

PGT(

Che

m)

AFS

Beg

umpe

t H

yder

abad

99

0860

9099

sa

idul

u_do

ngar

i@ya

hoo.

com

44

Mrs

. Sur

ya K

umar

iBar

ma

PGT(

Eco)

A

FS B

egum

pet

Hyd

erab

ad

9441

7791

66

tell_

ram

a@re

diff

mai

l.com

45

Dr.

K.V

. Raj

endr

a Pr

asad

PG

T(B

io)

NTP

C R

amag

unda

m

Hyd

erab

ad

0

3 DAY WORK SHOP ON STRATEGIC PLANNING FOR ACHIEVING QUALITY RESULT IN MATHEMATICS,PHYSICS, CHEMISTRY,BIOLOGY, & ECONOMICS

15/07/14 TO 17/07/14 TIME TABLE

DATE/DAY SESSION 1

(09:00-11:00 AM)

SESSION 2

(11:15-01:00 PM)

SESSION III

( 02:00- 03:30PM)

SESSION IV (03:45- 05:30

) 15/07/14

TUESDAY Inauguration

Insight into VUE&

Concept Mapping

Presentation of Concept Mapping

Strategic action plan to

achieve quality result.

Review of Study Material

Presentation of fine-tuned study material

16//07/14 WEDNESDAY

Preparation of Value based questions.

Presentation of Value

Based questions.

Preparation of 3 levels of

question papers.

Preparation of 3 levels of questions

Presentation of 3 levels of questions.

Error analysis and

remediation. Un attempted questions in

tests and examinations

17/07/14 THURSDAY

Tips and techniques

(Chapter wise) in teaching

learning process

Presentation of tips and techniques.

Subject wise specific issues

Consolidation of material

Consolidation of material

Valedictory Function

11.00 -11.15

Tea break 1.00 - 2.00

Lunch break 3.30-3.45 Tea Break

1

Workshop on Preparation of Strategic Action plan and Resource material in Maths/Physics/Chemistry/Biology/Economics

Venue: ZIET, MYSORE15.07.14 to 17.07.14

S.No. INDEX 01 Top sheet

02 Opening page

03 Our patrons

04 FOREWORD

05 MESSAGE BY COURSE DIRECTOR

06 MESSAGE BY ASSOCIATE COURSE DIRECTOR

07 LIST OF RESOURCE PERSONS (address,e-mail id,phone no.)

08 Time table

09 Strategic action plan to achieve quality result

10 Fine-tuned Study material

11 Value based question bank

12 Graded exercise questions (Level I,II,III)

13 Error analysis, remediation, unattended questions in exams.

14 Tips and Techniques

15 Strategic action plan to achieve quality result

16 Concept mapping

2

STRATEGIES TO ACHIEVE QUALITATIVE AND QUANTITAIVE RESULTS IN MATHEMATICSCLASS XII

Strategies for Slow learners: 1. Identify the slow learners at the beginning of the year. Set achievable targets and motivate them

throughout the year so that they will not be depressed and discouraged.

2. Question papers of last five years (both main and supplementary examinations) are to be collected and the list out all repeated, important concepts/problems. The slow learners are to be given sufficient practice in these areas/concepts.

3. The Latest Blue Print prepared by the CBSE to be given to each child especially to the slow learners in the beginning of the session.(From 2014-2015 onwards , pattern is changed)

4. The strengths and weaknesses are to be diagnosed in these areas. Thorough revision in these concepts is to be given by conducting frequent slip tests and re-teaching.

5. Preparation of Question-wise analysis of each examination including slip tests to be done to locate the weak areas and thorough revision is to be conducted.

6. Collect the drilling problems of a particular concept, and solve two or three problems in the class. Then allow the slow learners to solve the remaining problems as per their capacity to attain a good command and confidence over that particular method/type (Drilling Exercises).

7. Three model papers based on the Sample Papers issued by CBSE (SET I, II, III) along with marking scheme should be prepared by the teacher. Copies of these papers are to be issued to all the slow learners. This will help the child to know the type of questions/methods important for board exams. They will get more confidence to face the board exam.

8. Concept wise, specially designed home assignments are to be given to students daily. The assignments are to be corrected by giving proper suggestions in front of students.

9. After the completion of each concept/topic allow the low achiever to solve the problem pertaining to that method. If possible every day at least one low achiever should come on to the board to solve a problem.

10. Whenever possible, teach Mathematics by using PP Presentations in an effective way. 11. Weekly test pertaining to these formulae has to be conducted regularly. 12. The students have to be asked to read the entire text book thoroughly. 13. The students are to be made aware about the chapter wise distribution of marks or marking scheme.

14. Sufficient tips should be given for time management.

15. Few easy topics are to be identified from examination point of view and are to be assigned to the

slow learners. The slow learners are to be prepared for reduced, identified syllabus. Strategies for bright and Gifted Student:

3

16. Bright Children are the back bones to improve the overall Performance Index of the Vidyalaya. So

they should be encouraged by providing concepts wise HOTS questions. They should be encouraged to solve more challenging questions which have more concepts and challenging tasks. More thought provoking questions are tobe collected and a question bank is to be given to gifted students to develop their analyzing and reasoning capabilities.

17. Instead of preparing the PP presentation by the teacher, better to handover all the necessary content to the students and ask the bright students, to prepare one PPT each. After submission of completed PP Presentation, check the PPT and the same can be used effectively in the teaching learning process.

18. On completion of syllabus topic wise revision plan is to be framed for both slow learners and gifted

students.

19. The students have to be asked to read the entire text book thoroughly. 20. The students are to be made aware about the chapter wise distribution of marks or marking scheme. 21. Sufficient tips should be given for time management.

Revision Plan:

After completion of coverage of syllabus, proper revision plan is to be prepared Concept-wise (questions for slow learners/gifted students), HOTS questions/optional exercises (for

gifted students) is to be prepared and given to the students. Minimum learning programme for slow learners is to be prepared and identified/reduced syllabus is

to assigned to slow learners. CBSE Board pattern question papers (at least 10 papers should be solved) CBSE Board papers 2014 (3 sets) CBSE Board Compartment Paper 2014 (1 set) CBSE Board papers 2011. 2012, 2013 (3 sets) CBSE Board Compartment Paper 2013 (1 set) Common Pre-board Board Examination 2013, 2014 (2 sets) CBSE sample papers

4

STUDY MATERIAL

SUBJECT : MATHEMATICS

CLASS : XII

5

सहायकसाम ी

२०१४ - २०१५

SUPPORT MATERIAL 2014-2015

क ा१२

Class : XII

MATHS

6

INDEX

SlNO. Topics PageNo.

1. Detail of the concepts 3

2. Relations &Functions 8

3. Inverse Trigonometric Functions 17

4. Matrices &Determinants 22

5. Continuity &Differentiability 36

6. Application of derivative 44

7. Indefinite Integrals 54

8. Application of Integration 66

9. Differential Equations 72

10. Vector Algebra 80

11. Three Dimensional Geometry 92

12. Linear Programming 105

13. Probability 119

14. Syllabus 2014-15 128

15. Sample paper 2014-15 133

16. IIT JEE question paper with solutions 141

17. Bibliography 170

7

Level I, Level II & Level III indicate the difficulty level of questions

8

9

10

11

12

CHAPTER I RELATIONS&FUNCTIONSSCHEMA

TIC DIAGRAM

Topic Concepts Degreeof impo1tance

References NCERTTextBookXII Ed.2007

Relations& Functions

(i).Domain,Codomain& Rangeofarelation

* (PreviousKnowledge)

(ii).Typesofrelations *** ExI.IQ.No-5,9,12,14 (iii).One-one,onto&inverse ofafunction

*** Ex1.2Q.No-7,9 Example12

(iv).Compositionoffunction * Ex1.3QNo-3,7,8,9,13 Example25,26

(v).BinaryOperations *** MiscExample45,42,Misc.Ex2,8,12,14 Ex1.4QNo-5,9,II

SOMEIMPORTANTRESULTS/CONCEPTS

TYPES OF RELATIONS

A relation R in a set A is called reflexive if (a, a) R for every a A. A relation R in a set A is called symmetric if (a1, a2) R implies that (a2, a1) R, for all a1, a2 A. A relation R in a set A is called transitive if (a1, a2) R, and (a2, a3) R together imply that (a1,a3) R, for

all a1, a2, a3 A. ** EQUIVALENCE RELATION

A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

Equivalence Classes Every arbitrary equivalence relation R in a set X divides X into mutually disjoint subsets (Ai) called partitions

or subdivisions of X satisfying the following conditions: All elements of Ai are related to each other for all i No element of Ai is related to any element of Aj whenever i ≠ j Ai Aj = X and Ai ∩ Aj = Φ, i ≠ j

These subsets (Ai) are called equivalence classes. For an equivalence relation in a set X, the equivalence class containing a X, denoted by [a], is the subset

of X containing all elements b related to a.

13

**Function:Arelation f:A BissaidtobeafunctionifeveryclementofAiscorrelated to a uniqueelementinB.

*Aisdomain * Biscodomain

* Forany xelement of A,function f correlatesittoanelementinB,whichisdenotedbyf(x)andiscalledimageofxunder/.Againify=f(x),thenxiscalledaspre-imageofy.

* Range={f(x)Ix A}. Range Co domain ** Composite function ** Let f: A → B and g: B → C be two functions. Accordingly, the composition of f and g is denoted bygof and

is defined as the function gof: A → C given by gof(x) = g(f(x)), for all x A.

14

15

16

13

14

•

3. ShowthattherelationRdefinedinthesetAofalltrianglesasR={(T1,T2):T1issimilartoT2},isequivalencerelation.ConsiderthreerightangledtrianglesT1withsides3, 4,5, T2withsides5,12,13andT3withsides6,8,I0.WhichtrianglesamongT1,T2andT3arerelated?

4. IfR1andR2areequivalencerelationsinasetA,showthatR1R2isalsoan equivalencerelation.

5. LetA=R-{3}andB=R-{l}.Considerthefunctionf:A→Bdefinedbyf(x)= Isfone-oneandonto?Justifyyouranswer.

6. Considerf: R+→ [-5,∞)givenbyf(x)=9x2+6x-5.Showthatfisinvertibleandfind f-1

7. OnR-{l}abinaryoperation*isdefinedasa* b=a+b-ab.Provethat *iscommutativeandassociative. Findtheidentityelementfor*.AlsoprovethateveryelementofR-{1)isinvertible.

8. If A=Q xQand*beabinaryoperationdefinedby(a,b)*(c,d)=(ac,b+ad),for (a,b),(c,d)€A.Thenwithrespectto* onA

(i) examinewhether*iscommutative&associative (i) findtheidentityelementinA, (ii) )findtheinvertibleelementsofA.

9. Considerf: R→ [4,∞)givenbyf(x) =x2+4.Showthatfisinvertiblewith

theinversef'offbyf'(y) = whereRisthesetofallnonnegativerealnumbers.

EXTRA ADDED QUESTIONS (FOR SELF EVALUATION):

1. If f : R→ R and g : R→ R defined by f(x)=2x + 3 and g(x) = x+ 7, then

find the value of x for which f(g(x))=25 .

2. Find the Total number of equivalence relations defined in the set

S = {a, b, c}

3. Find whether the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2),

(3, 3), (1, 2), (2, 3)} is reflexive, symmetric or transitive.

15

4. Show that the function f: N X N , given by f (x) = 2x, is one-one but not

onto.

5. Find gof and fog, if f: R→R and g: R to R are given by f (x) = cos x and g (x) =

6. Find the number of all one-one functions from set A = {1, 2, 3} to itself.

7.Check the injectivity and surjectivity of the following:

i) f from N→N given by f(x)= and ii) f from R→R given by g(x)=

8.If f: R→ R and g: R→ R defined by f(x) =2x + 3 and g(x) = x+ 7, then

find the value of x for which f(g(x))=25 .

9. Find the Total number of equivalence relations defined in the set S = {a, b, c} 10. Show that f: [–1, 1] R, given by f (x) = x/(x+2) is one-one. Find the

inverse of the function f : [–1, 1] & Range f.

11) Prove that the inverse of an equivalence relation is an equivalence relation.

12) Let f: A →B be a given function. A relation R in the set A is given by

R = {(a ,b) ε A x A :f(a) = f(b)} . Check, if R is an equivalence relation. Ans: Yes

13. Determine which of the following functions

f: R → R are (a) One - One (b) Onto

(i) f(x) = |x| + x

(ii) f(x) = x - [x]

16

(Ans: (i) and (ii) → Neither One-One nor Onto)

14). On the set N of natural numbers, define the operation * on N by m*n = gcd (m, n)

for all m, n ε N. Show that * is commutative as well as associative.

HOTQUESTIONS:

http://www.kv1alwar.org/admin/downloads/19.pdf

17

CHAPTER II

18

19

20

9. Prove that

10. Simplify

11. Prove that

12. Simplify

21

ANSWERS

10. π/4 + x 11. - 12. - 2

23

CHAPTER III & IV

MATRICES&DETERMINANTS

SCHEMATIC DIAGRAM

Topic Concepts Degreeofimportance

References NCERTTextBookXIEd.2007

Matrices& Determinants

(i)Order, Addition, Multiplication and transpose of matrices

*** .. Ex3.1-Q.No4,6 Ex3.2-Q.No7,9,13,17,18 Ex3.3-0.NoIO

(ii)Cofactors&Adjointofamatrix

Ex4.4-Q.No5 Ex4.5-Q.No12,13,17,18

(iii)lnverseof a matrix& applications

***

Ex4.6-Q.No15,16 Example-29,30,32,33 MiscEx4-Q.No4,5,8,12,15

(iv)To find difference between AI, adjA,

kAI,A.adjA * Ex4.1-Q.No3,4,7,8

(v)Properties of Determinants

** Ex4.2-Q.No11,12,13 Example-16,I8

SOME IMPORTANT RESULTS/CONCEPTS

A matrix is a rectangular array of mxnnumbers arranged in m rows and n columns.

a11 a12………….a1n

a22………….a2n OR A=[a..ij] , where i=1,2,....,m;j=1,2,....,n.

amI am2·……….amnmxn

* Row Matrix:A matrix which has one row is called row matrix.

*Column Matrix: A matrix which has one column is called column matrix

*SquareMatrix:A matrix in which number of rows are equal to number of columns, is called a square matrix

* Diagonal Matrix:Asquare matrix is called!aDiagonal Matrix if all the elements, except the diagonal elements are zero

* Scalar Matrix: A square matrix is called scalarmatrix if all the elements, except diagonal elements are zero and diagonal elements are same non-zero quantity.

* Identity or UnitMatrix: A square matrix in which all the non diagonalelements are zero and diagonal

24

elements are unity is called identity or unit matrix

25

26

27

28

29

30

31

VALUE BASED QUESTIONS.

1. 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 totalof Rs 11,000 for the three values to 5,4 and3 students respectively while school B decided to award Rs 10,700 for the three values to 4,3 and5 students respectively .I fall the three prizes together amount to Rs2,700then

(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?

[CBSE sample paper, 4 marks]

2. Using matrix method , solve the following system of equations.

x-y+2z = 7

3x+4y-5z=-5

2x-y+3z=12

If x represents the number of who take food at home represents the number of persons who take junk food in market and z represents the number of persons who take food at hotel. Which way of taking food you prefer and why?

3. The management committee of a residential colony decided to award some of its member(say x)

for honesty ,some(say y) for helping others and some other(say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is33.If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others , using matrix method , find the number of awardees of each category. Apart from these values, namely ,honesty, cooperation and supervision ,suggest one more value which the management of the colony must include for awards.-

[CBSE2013 6marks]

32

4. 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 sand second bond pays 7% interest per annum which will be given to an NGO cancer aid society. Using matrix multiplication method determine how to divide Rs.30000 among two types of bonds if the trust fund obtains an annual total interest of Rs.1800.Whatarethevaluesreflected in the question.

5.Three shopkeepers A B C are using polythene, hand made bags, and newspaper envelopes as carry bags. Itis found that the shop keepers A B C are using (40,30,20),(20,40,60) (60,20,30), polythene, hand made bags and newspapers envelopes respectively. The shopkeepers A B C spend Rs.600, Rs.900, Rs.700 on these carry bags respectively. Find the cost of each carry bags using matrices keeping in mind the social and environmental conditions which shopkeeper is better? And why?

Additional Questions

(I) LEVEL I

(1) Write the order of the product matrix

(2) IF A= and =kA find k

(ii)LEVEL II

(1)If = find p

(2) Give examples of a square matrix of order 2 which is both symmetric and skew symmetric

(3)Find the value of x and y if =

(4)If A = , find 0 , when A+ =I

(ii)LEVEL I

(1) If A= write the minor of the element

(2) If is the cofactor of of find

(iii)LEVEL 1

33

(1) If A is a square matrix such that =A then write the value of -3A

(2) If A = and B = , then verify that = .

LEVELIII

(1) If = and B = Find

(2) Using elementary transformations, find the inverse of the matrix

(3) The management committee of a residential colony decided to award some of its members (say x)

For honesty ,some(say y)for helping others and some others(say z) for supervising the workers to

keep the colony neat and clean . The sum of all the awardees is 12. Three times the sum of awardees

for cooperation and supervision added to two times the number of awardees for honesty is 33 If

the sum of the number of awardees for honesty and supervision is twice the number of awardees

for helping others ,using matrix method find the number of awardees for each category . apart from

these values ,namely , honesty ,cooperation and super vision , suggest one more value which the

management of the colony must include for awards

(iv)LEVELII

(1) If A is asquare matrix of order 3 such that =225 Find

(V) LEVELI

(1) Evaluate

(2)Find the value of

QUESTIONS FOR SELF EVALUATION

34

Prove that =

Answers

35

Value based question answers Answer: 1The given situation can be written as a system of linear equations as

5x + 4y + 3z = 11000,

4x + 3y + 5z = 10700

X + y +z =2700

35

(i) This system of equations can be written in the matrix form as

=

This equation is of the form AX=B, where A = =, X =

and B=

(ii) =5(-2) -4(-1) +3 (1)=-3≠0

Therefore exists, so equations have a unique solution.

(iii)Any answer of the three values with proper reasoning will be considered correct.

Answer 2 : X=2, Y=1, Z=3

Answer3: The given situation can be written as a system of linear equations as

x +y+ z=12

3(y + z)+2x=33 or 2x+3y+3z=33

x+ z=2y or x- 2y+z=0

this system of equations can be written in the matrix form as

36

=

This equation is of the form AX=B, where A=

X= and B

=1(9)-1(-1)+1(-7)=3≠0

Therefore A-1exists, so equations have a unique solution. X= A-1B

x =3,y=4,z=5

Those who keep their surroundings clean.

Answer4:Rs.1500,Rs.1500

Answer5:50,80,80

Additional Questions (Answer)

(i) LEVELI (1) order3x3, (2) 2

LEVELII (1)12 (2)any example (3) X=1, Y=-2 (4)

(ii)LEVELI (1) 7 (2) 110

(iii) LEVELIII (1) = = (2) (3) = ,X=3 Y=4,

Z=5

(IV)LEVELII (1) 15 (V)LEVEL I (1) 1, (2) 0

37

CHAPTER V

38

39

40

41

42

43

ANSWERS TO

45

46

47

48

49

50

11. If the length of three sides of a trapezium, other than the base is equal to 10cm each, then find

the area of trapezium when it is maximum. Ans.75 sq.cm

12. Verify Role’s theorem for the function f given by f(x) = (sinx – cosx) on [ , ].

13. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h

51

and semi-vertical angle is tan2 .

14. A window is in the form of a rectangle surrounded by a semi –circular opening. The total

perimeter of the window is 10 metres. Find the dimensions of the window so as to

admit maximum light through the whole opening. Ans . , ,

15. A window is in the form of a rectangle surmounted by a semi –circular opening. The total

perimeter of the window is p metres. Show that the window will allow the maximum possible light only when the radius of the semi circle is p/ π+ 4 m

16. A window is in the form of a rectangle surmounted by an equilatral triangle. The total

perimeter of the window is 12 metres, find the dimensions of the rectangle that will produce the largest area of the window. Ans : 12/ 6- m

52

53

54

54

CHAPTER VII

55

56

57

58

59

Log sinx dx

60

61

ADDITIONAL QUESTIONS (Indefinite & Definite Integrals)

1. Evaluate ∫ dx ans : ( logI sin2x+b2cos2 x I +C)

2. Evaluate ∫ dx ans : [(a+bx) -2alogIa+bxI - ]

3. Evaluate ans : + C

4. Evaluate ans: tanx + C

5. Evaluate∫ [secx + log(secx+tanx)]dx ans: log(secx+tanx) + C

6.Evaluate dx ans: - log + log +C

7.Evaluate dx ans: [ ] +C

8.Evaluate dx ans: ta [tanx+ ]+C

9. 10

11 12

62

63

64

65

2 Log 2

67

CHAPTER VIII

66

68

69

67

68

70

HOTS QUESTIONS

1. Using integration, find the area of the following region

{ (x,y): + 1 + }Ans :( - 3)Sq.units

2. Find the area of the region bounded by the curve y= , line y=x and the positive x- axis Ans : π/8Sq.units

3. Draw a rough sketch of the curve y = cos2x in [0, π ] and find the area enclosed by the curve, the line x=0 , x=

69

70

ANSWERS

71

72

9

CHAPTER IX

74

(2) Showthaty=3 isthesolutionofthedifferentialequation -4y=12x.

(3) Verifythatthefunctiony=3Cos(logx)+4Sin(logx),isasolutionofthedifferentialequ

ation

2) ObtainthedifferentialequationbyeliminatingAandBfromtheequation

y=ACos2x+BSin2x,where‘A’and‘B’areconstants.

3) Obtainthedifferentialequationofthefamilyofellipseshavingfociony-axisandcentreattheorigin.

4) Findthedifferentialequationofthefamilyofcurvesy=

75

2) Solve thedifferentialequation :

3) Solvethed.e. ,

4) Findtheparticularsolutionofthedifferentialequation: ,giventhaty=πandx=3

5) Solve:

75

6) Solvethed.e. ,

7) Solve: ,

8) Solve: ,

9) Therateofgrowthofapopulation is proportional to the numberpresent.Ifthepopulation of acitydoubled in the past 25years , andthepresentpopulation is 100000, when will the cityhaveapopulation of 500000?(log5=1.609and log2=0.6931). Writeyourcomments about adverse effectsofpopulation explosion.

76

Additional Questions (for self practice)

1. Write the order and degree of the following differential equation

0cos4

2

2

dxdy

dxyd

2. Show that y=3e2x + e-2x – 3x is the solution of the differential equation

y”- 4y = 12x

3. Verify that y = 3 cos(log x) + 4 sin(log x) is a solution of the differential equation x2 y” + xy’ + y =0

4. Obtain the differential equation of family of parabola having vertex at the origin and axis along the positive direction of x-axis LEVEL III

5.Obtain the differential equation of family of ellipses having foci on y-axis and centre at the origin .

6.Find the differential equation of system of concentric circles with centre at (1,2)

7.Solvedxdy = ( 1 + x2)( 1 + y2)

8.Solvedxdy =e-ycos x Given that y(0) =0

9.Solvecos ( ) = a (a Ɛ R) ; y=2 when x=0

10. (x3+x2+x+1) =2x2 +x ; y=1 when x =0

11. Solve yxyx

dxdy 2

12.Solvedxdy =

xyxyxy

23

23

22

13.Solve y dx + x log dy – 2x dy = 0

77

14.Solve y yx

e dx = ( x yx

e +y) dy

15. Solve xydxdyx tancos2

16.Solve 1221 222 xxxydxdyx

17.Solve dxyxedyx x 23 11

18.Solve ( 1 + y + x2y) dx + ( x + x3) dy = 0

19.Solve 12

dydx

xy

xe x

, x≠0 ; when x=0 , y=1

Answers

2.Ans: 3: =0

4:

4. =Sinx+1 5.Siny- logx=c6: (x-1) =C

2.,: 3. 4.: 5.

CHAPTER X

Answers

CHAPTER XI