Upload
dangkhuong
View
214
Download
0
Embed Size (px)
Citation preview
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: [email protected]/[email protected]
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
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
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
•
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
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
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
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
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
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
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
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: