CBSEmaths-i.pdf

केंद्रीय विद्यालय संगठननई दिल्ऱी KENDRIYA VIDYLAYA SANGTHAN NEW DELHI

Venue: KVS ZIET MYSORE Date: 15th to 17th July, 2014 RESOURCE MATERIALS CLASS XII(2014-15)(Mathematics)

आंचलऱकलिऺाएवंप्रलिऺणसंस्थान जीआईटीबीप्रेसकैम्पस, लसद्धाथथनगर, 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

केंद्रीय विद्याऱय संगठन नईदिल्ऱी KENDRIYA VIDYLAYA SANGTHAN

NEW DELHI

आंचलऱक लिक्षा एिं प्रलिक्षण संस्थान मैसूर ZONAL MINSTITUTE OF EDUCATION AND TRAININGMYSOREिप्र

3-Day Strategic Action plan workshop आं15th-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)

ऺाएवंप्रलिऺ णसंस्थान

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 “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 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 6. Fine-tuning of study material supplied in Learning process 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

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

KVS-ZIET-MYSORE 03-day Workshop on Strategic Action Plan 15-17.07.14 - Details for Contact Sl No 1 2 3

Name in English Mrs. V. Meenakshi Mr. E. Krishna Murthy Mr. T. Prabhudas

Design. Asstt.Commr. Principal Principal

K.V in English Regional Office NFC Nagar,Ghatkesar Malleswaram

Region Ernakulam Hyderabad Bangalore

Phone No. 9496146333 9989063749 8762665990

E-mail Address [email protected] [email protected] [email protected]

4 5 6 7 8 9

Dr(Mrs.) S. Nalayini Mr. M. Krishna Mohan Mr. R. Sankar Mr. E.N. Kannan Mr. D.B. Patnaik Mr. K.S.V. Someswara Rao

Principal Principal Principal PGT(Phy) PGT(Bio) PGT(Phy)

Kanjikode CRPF Hyderabad No.2 Uppal BEML Nagar Railway Colony MEG & Centre

Ernakulam Hyderabad Hyderabad Bangalore Bangalore Bangalore

9446361186 9440865761 9491073600 8762208431 8971240593 9448708790

[email protected] [email protected] [email protected] [email protected] [email protected] [email protected]/[email protected]

10 11 12 13 14 15 16

Mr. K.R. Krishna Das Mr. G.N. Hegde Dr. Vivek Kumar Mr. RangaNayakulu .A Mrs. G.K. Vinayagam Mr. D. Rami Reddy Mrs. T.M. Sushma

PGT(Maths) PGT(Maths) PGT(Chem) PGT(Chem) PGT(Bio) PGT(Eco) PGT(Eco)

No.1 AFS Sambra Dharwad CRPF Yelahanka Hebbal No.2 Belgaum Cantt. Railway Colony Hebbal

Bangalore Bangalore Bangalore Bangalore Bangalore Bangalore Bangalore

8951648275 9448626331 8970720895 7899287264 9448120612 9740398644 8762691800

[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

17 18 19 20 21 22

Mrs. Asha Rani Sahu Mrs. Joan Santhi Joseph Mrs. C.V. Varalakshmi Mr. Siby Sebastian Mr. S. Vasudhevan Mrs. Sathya Vijaya Raghavan

PGT(Maths) PGT(Chem) PGT(Phy) PGT(Maths) PGT(Chem) PGT(Eco)

Mysore IS Grounds, Chennai AFS Avadi, Chennai Gill Nagar DGQA Complex Minambakkam

Bangalore Chennai Chennai Chennai Chennai Chennai

9902663226 9940945578 9003080057 8056179311 9444209820 9445390058

[email protected] [email protected] [email protected] [email protected] [email protected]

23 24 25 26

Mr. S. Kumar Mrs. A. Daisy Mrs. C.K. Vedapathi Mrs. J. Uma

PGT(Phy) PGT(Bio) PGT(Bio) PGT(Eco)

No.1 Kalpakkam Minambakkam IIT Chennai Annanagar

Chennai Chennai Chennai Chennai

8015374237 9840764240 9841583882 9840988755

[email protected] [email protected] [email protected] [email protected]

[email protected]

27 28

Mrs. Sulekha Rani .R Mrs. Mary V. Cherian

PGT(Chem) PGT(Bio)

NTPC Kayamkulam SAP Peroorkada

Ernakulam Ernakulam

9745814475 9447107895

[email protected] [email protected]

29 30 31 32 33 34

Mrs. Susmitha Mary Robbins Mr. Joseph K.A Mrs. Jyolsna K.P Mrs. UshaMalayappan Mrs. Sujatha M. Poduval Mr. Prashanth Kumar .M

PGT(Phy) PGT(Eco) PGT(Maths) PGT(Eco) PGT(Bio) PGT(Phy)

Kalpetta R.B Kottayam No.1 Calicut Kanjikode Keltron Nagar Keltron Nagar

Ernakulam Ernakulam Ernakulam Ernakulam Ernakulam Ernakulam

9495528585 9446369351 9447365433 9496519079 9446494503 9400566365

[email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

35 36 37 38 39 40 41

Mr. Sibu John Mr. N.S. Subramanian Mrs. Josephine Balraj Mr. B. Sesha Sai Mr. V.V..S.Kesava Rao Kum. SanuRajappan Mr. M.T. Raju

PGT(Chem) PGT(Maths) PGT(Maths) PGT(Phy) PGT(Phy) PGT(Eco) PGT(Bio)

Ernakulam Gooty No.1 AFA Dundigal AFS Hakimpet Gachibowli Gachibowli AFS Begumpet

Ernakulam Hyderabad Hyderabad Hyderabad Hyderabad Hyderabad Hyderabad

9544594068 9490039741 9440066208 9912384681 9490221144 9640646189 9652680800

[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

42 43 44 45

Mr. D. Ashok Mr. D. Saidulu Mrs. Surya KumariBarma Dr. K.V. Rajendra Prasad

PGT(Chem) PGT(Chem) PGT(Eco) PGT(Bio)

CRPF Hyderabad AFS Begumpet AFS Begumpet NTPC Ramagundam

Hyderabad Hyderabad Hyderabad Hyderabad

9618012035 9908609099 9441779166

[email protected] [email protected] [email protected]

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) 15/07/14 TUESDAY

Inauguration

16//07/14 WEDNESDAY

17/07/14 THURSDAY

Insight into VUE& Concept Mapping

SESSION 2 (11:15-01:00 PM) Presentation of Concept Mapping

Strategic action plan to achieve quality result.

SESSION III ( 02:0003:30PM)

SESSION IV (03:45- 05:30 )

Review of Study Material

Presentation of fine-tuned study material

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

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

0

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. 01 Top sheet

INDEX

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

1

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: 2

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

3

STUDY MATERIAL

SUBJECT : MATHEMATICS

CLASS : XII

4

सहायकसामग्री २०१४ - २०१५

SUPPORT MATERIAL 2014-2015 कऺा१२

Class : XII

MATHS

5

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

17. Bibliography

141 170 6

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

7

8

9

10

11

CHAPTER I RELATIONS&FUNCTIONSSCHEMA TIC DIAGRAM Topic Relations& Functions

Concepts (i).Domain,Codomain& Rangeofarelation (ii).Typesofrelations (iii).One-on e,onto&inverse ofafunction (iv).Compositionoffunction (v).BinaryOperations

Degreeof impo1tance

* *** *** * ***

References NCERTTextBookXII Ed.2007 (PreviousKnowledge) ExI.IQ.No-5,9,12,14 Ex1.2Q.No-7,9 Example12 Ex1.3QNo-3,7,8,9,13 Example25,26 MiscExample45,42,Misc.Ex2,8,12,14

Ex1.4QNo-5,9,II

SOMEI MPORTANTRESULTS/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. 12

**Function:Arelation f:A Bissaidtobeafunctionifeveryclementof Aiscorrelated 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.

13

14

15

16

13

3. ShowthattherelationRdefinedinthesetAofalltrianglesasR={(T1,T2):T1issimilarto T2},isequiv alencerelation.ConsiderthreerightangledtrianglesT1withsides3, 4,5, T2withsides5,12,13andT3withsides6,8,I0.WhichtrianglesamongT1,T2andT3arerelated? 4. IfR1 andR2areequivalencerelationsinasetA,showthatR1 R2isalsoan equivalencerelation. 5. LetA=R-{3}andB=R-{l}.Considerthefunctionf:A→Bdefinedbyf(x)= Isfone-oneandonto?Justifyyouranswer. 2

6. Considerf: R+→ [-5,∞)givenbyf(x)=9x +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. 14

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] 15

(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 16

CHAPTER II

17

18

19

9. Prove that √ 10. Simplify

11. Prove that ( )

( )

( )

12. Simplify

.

/ 20

ANSWERS

10.

π/4 + x

11.

-

12. - 2

21

CHAPTER III & IV MATRICES&DETERMINANTS SCHEMATIC DIAGRAM

Topic

Matrices& Determinants

Concepts

Degreeofi mportance

(i)Order, Addition, Multiplication and transpose of matrices (ii)Cofactors&Adjointofamat rix (iii)lnverseof a matrix& applications

***

(iv)To find difference between ·AI, adjA, kAI,A.adjA (v)Properties of Determinants

*

..

References NCERTTextBookXIEd.2007 Ex3.1-Q.No4,6 Ex3.2-Q.No7,9,13, 17,18 Ex3.3-0.NoIO Ex4.4-Q.No5

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

** *

**

Ex4.6-Q.No15,16 Example-29,30,32,33 MiscEx4-Q.No4,5,8,12,15 Ex4.1-Q.No3,4,7,8

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

amI

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

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

23

elements are unity is called identity or unit matrix

24

25

26

27

28

29

30

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) (ii) (iii)

Represent the above situation by a matrix equation and form linear equations using matrix multiplication. Is it possible to solve the system of equations so obtained using matrices? 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]

31

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[ ] + and

(2) IF A=*

=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 [ + , find 0

(4)If A =*

] =* , when A+

+ =I

(ii)LEVEL I | write the minor of the element

(1) If A=|

(2) If

is the cofactor of

of|

| find

(iii)LEVEL 1 32

(1) If A is a square matrix such that (2) If A =*

+ and B =*

=A then write the value of +, then verify that

-3A

=

.

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 33

Prove that |

|=

Answers

34

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

35

=

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 36

CHAPTER V

37

38

39

40

41

42

ANSWERS TO

43

45

46

47

48

49

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

12. Verify Role’s theorem for the function f given by f(x) =

sq.cm (sinx – cosx) on [ ,

].

13. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h 50

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

51

52

53

54

CHAPTER VII

54

55

56

57

58

Log sinx dx

59

60

ADDITIONAL QUESTIONS (Indefinite & Definite Integrals) 1. Evaluate ∫

dx

2. Evaluate ∫

ans : (

dx

ans :

3. Evaluate∫

6.Evaluate∫

7.Evaluate∫

11

]

+C ans: tanx + C

[secx + log(secx+tanx)]dx ans: log(secx+tanx) + C

dx

ans: - log

dx

ans:

8.Evaluate∫ 9.

sin2x+b2cos2 x I +C)

[(a+bx) -2alogIa+bxI -

ans :

4. Evaluate∫ 5. Evaluate∫

logI

dx

+

[

ans: ta

log

+C

] +C

[tanx+ ]+C

10

12

61

62

63

64

2

Log 2

65

CHAPTER VIII

66

67

68

67

69 68

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

70

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= π and the x-axis. Ans : π/2 Sq.units 69

ANSWERS

70

71

CHAPTER IX 9

72

(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) Obtainthedifferentialequationofthefamilyofellipseshavingfocionyaxisandcentreattheorigin. 4) Findthedifferentialequationofthefamilyofcurvesy=

74

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.

75

Additional Questions (for self practice)

1. Write the order and degree of the following differential equation 4

d2y  dy   2   cos   0  dx   dx 

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 7.Solve

dy = ( 1 + x2)( 1 + y2) dx

8.Solve

dy =e-ycos x Given that y(0) =0 dx

(1,2)

9.Solvecos ( ) = a (a Ɛ R) ; y=2 when x=0 10. (x3+x2+x+1) 11. Solve 12.Solve

=2x2 +x ; y=1 when x =0

dy x  2 y  dx x y

y 3  2x 2 y dy = 3 dx x  2y2x

13.Solve y dx + x log ( ) dy – 2x dy = 0 76

x

x

y y 14.Solve y e dx = ( x e +y) dy

15. Solve cos 2 x 16.Solve x 2  1

dy  y  tan x dx







dy  2 xy  x 2  2 x 2  1 dx

17.Solve 1  x dy  e 3 x x  12  y dx 18.Solve ( 1 + y + x2y) dx + ( x + x3) dy = 0  e 2

19.Solve  

x

x



y  dx  1 , x≠0 ; when x=0 , y=1 x  dy

Answers

2.Ans:

3:

=0

4:

77

4.

2.,:

3.

=Sinx+1

5.Siny-

4.:

logx=c6: (x-1)

=C

5.

CHAPTER X

Answers

CHAPTER XI

ADDITIONAL QUESTIONS FOR SELF EVALUATION

1. Write the direction cosines of the line parallel to Z-axis. (Ans 0,0,1)

2.Find the distance between the parallel planes. r.(2i-j+3k)=4 and r.(6i-3j+9k)+13=0 (Ans 25/3√14)

3.The Cartesian equation of the line is 3x+1= 6y-2=1-z. Find the direction ratios of the line (Ans (2,1,-6))

4.Find the length and foot of the perpendicular from the point (2,-1,5) to the line (x11)/10 = (y+2)/-4 = (z+8)/-11. (ans (1,2,3) √14 ) 5.Write the intercept cut off by the plane 2x+y-z=5 on x axis (Ans x = 5/2) 6.Find the equation of a line passing through the point (1,2,3) and parallel to the planes xy+2z=5 and 3x+ y+z=6. 7. Show that the lines r = -i-3j-5k+α(3i+5j+7k) and r = (2i+4j+6k) + β(i+3j+5k) intersect each other.

ANSWERS

LEVEL II

CHAPTER XII

LINEAR PROGRAMMING LINEAR PROGRAMMING SCHEMATIC DIAGRAM Topic Linear Programming

Concepts (i)LPP and its Formulation Mathematical (ii)Graphical method of Solving LPP (bounded unbounded solutions) and (iii)Diet Problem

Degree of Importance ** **

***

References NCERT Book Vol. II Articles 12.2 and 12.2.1 Article12.2.2 Solved Ex. 1 to 5 Q. Nos 5 to 8 EX.12.1 Q. Nos 1,2 and 9 Ex. 12.2 Solved Ex. 9 Q. Nos 2and3 Misc. Ex.

(iv)Manufacturing

***

Problem (v)Allocation Problem

Solved Ex. 8 Q. Nos 3,4,5,6,7 of Ex.12.2 Solved EX.10 Q. Nos4 &10 Misc. Ex.

**

(vi)Transportation

*

Solved Example 7Q. No 10 Ex.12.2, Q. No 5 &8 Misc. Ex. Solved EX.11 Q. Nos 6 &7 Misc. Ex.

Problem (vii)Miscellaneous

**

Q. No 8 Ex. 12.2

Problems SOME IMPORTANT RESULTS /CONCEPTS **Solving linear programming problem using Corner Point Method. The method comprises of the following steps: I.Find the feasible region of the linear programming problem and determine its corner points (vertices) either by inspection or by solving the two equations of the lines intersecting at that point. 2.Evaluate the objective function Z= ax + by at each corner point. Let M and m, respectively denote the largest and smallest values of these points. 3.(i)When the feasible region is bounded, M and m are the maximum and minimum values of Z. (ii) in case, the feasible region is unbounded, we have: 4.(a) M is the maximum value of Z, if the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, Z has no maximum value. (b)Similarly, m is the minimum value of Z, if the open half plane determined by ax + by < m has no point in common with the feasible region. Otherwise, Z has no minimum value.

(i)LPP and its Mathematical Formulation

LEVEL I I). Avinash 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 getting assignment marks. The problems in the first list carry 10 marks each and those in the second list carry 5 marks each. He 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. Formulate this problem as a linear programming problem to maximize his marks? What is the importance of time management for students?

(ii)Graphical method of solving LPP (bounded and unbounded solutions) LEVEL I Solve

the

following

Linear

Programming Problems graphically:

1) Minimize Z= - 3x+4y subject to x+2y≤8, 3x+2y≤12, x ≥0,y ≥0. 2) Maximize Z=5x+3y subject to 3x+5y≤I5, 5x+2y≤10, x ≥0,y ≥0. 3) Minimize Z=3x+5y such that x+3y≥3, x+y≥2, x,y≥0.

(iii)Diet Problem LEVEL ll 1) A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2units/kg of vitamin A and 1 unit/kg of vitamin C, while food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs.5.00 per kg to purchase food I and Rs.7.00 per kg to purchase food II. Formulate this problem as a linear programming problem to minimize the cost of such mixture. Why should a person take balanced food?

2. Every gram of wheat provides 0 .1 g of proteins and 0.25 g of carbohydrates. The corresponding values for rice are 0.05 g and 0.5 g respectively. Wheat costs Rs. 20 per kg and rice costs Rs.20 per kg. The minimum daily requirements of protein and carbohydrates for an average child are 50 gm and 200 gm respectively. In what quantities, should wheat and rice be mixed in the daily diet to provide the minimum daily requirements of protein and carbohydrates at minimum cost? Which type of food an average child should consume? (iv)

Manufacturing Problem LEVEL ll

ILLUSTRATIVE EXAMPLE

A company manufactures two types of sweaters, type A and type B. It costs Rs. 360 to make one unit of type A and Rs. 120 to make a unit of type B. The company can make atmost 300 sweaters and can spend atmost Rs. 72000 a day. The number of sweaters of type A cannot exceed the number of type B by more than 100. The company makes a profit of Rs. 200 on each unit of type A but considering the difficulties of a common man the company charges a nominal profit of Rs. 20 on a unit of type B. Using LPP, solve the problem for maximum profit.(CBSE Sample Paper 2014). Ans: let the company manufactures sweaters of type A = x, and that of type B = y daily LPP is to maximise P = 200x + 20y subject to the constraints: x+y ≤ 300 360 x + 120y ≤ 72000 x – y ≤ 100 x ≥ 0, y ≥ 0

Vertices of the feasible region are A (100, 0), B (175, 75), C (150, 150) and D (0, 300) Maximum profit is at B So Maximum Profit = 200 (175) + 20 (75) = 35000 + 1500 = Rs. 36500

1.

A company manufactures two articles A and B. There are two departments through which these

articles are processed: (i ) assembling and (ii) finishing departments. The maximum capacity of the assembling department is 60 hours a week and that of the finishing department is 48 hours a week. The

production of each article of A requires 4 hours in assembling and 2 hours in finishing and that of each unit of B requires 2 hours in assembling and 4 hours in finishing. If the profit is Rs. 6 for each unit of A and Rs. 8 for each unit of B, find the number of units of A and B to be produced per week in order to have maximum profit. 2. A company sells two different products A and B. The two products are produced in a common production process which has a total capacity of 500 man hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The demand in the market shows that the maximum number of units of A that can be sold is 70 and that for B is 125. Profit on each unit of A is Rs. 20 and that on B is Rs. 15. How many units of A and B should be produced to maximize the profit? Solve it graphically.Which are the factors affecting the demand of a product in the market ? LEVELIII 1. 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 medicine B but there are 45000 bottles into which either of the medicines 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. 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 its production to cover maximum patients? How can you help others in case of natural disasters?

(v)AllocationProblem LEVELII 1. Ramesh wants to invest at most Rs.70,000 in Bonds A and B .According to the rules, he has to invest at least Rs.10,000 in Bond A and at least Rs.30,000 in Bond B. lf the rate of interest on bond A is 8% per annum and the rate of interest on bond B is 10% per annum, how much money should he invest to earn maximum yearly income? Find also his maximum yearly income. Why investment is important for future life?

2. lf 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 to Rs.7per km. He has Rs.100 to spend on petrol and wishes to cover the maximum distance within one hour. (i) Express this as an L .P.P and solve it graphically. (ii) What is the benefit of driving at an economical speed? (iii) Should a child below 18 years be allowed to drive a motorcycle? Give reasons.

LEVELIII 1. An aero plane can carry a maximum of 250 passengers. A profit of Rs 500 is made on each executive class ticket and a profit of Rs 350 is made on each economy class ticket. The airline reserves at least 25 seats for executive class. However, at least 3 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 for the airline. What is the maximum profit? Suggest necessary preparations to be made before going on a trip? 2. 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 costs Rs 5 and Rs 8 per kilograms respectively. Using L.P.P, find out what quantity of each type of fertilizers should be bought to meet the requirement so that the cost is minimum. Solve the problem graphically. What are the side-effects of using excessive fertilizers?

(vi) Transportation Problem LEVEL III ILLUSTRATIVE EXAMPLE Q-1Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table: How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

From/To

A

B

D E F

6 3 2.50

4 2 3

X ≥ 0, Y ≥ 0, and 100 – X – Y ≥ 0 60 – X ≥ 0, 50 – Y ≥ 0, and X + Y – 60 ≥ 0 X ≤ 60, Y ≤ 50, and X + Y ≥ 60 Total transportation cost Z is given by, Z= 6x + 3y +2.5(100 – x – y) + 4(60 – x) + 2(50 – y) + 3(x + y – 60) = 6x + 3y + 250 -2.5x – 2.5y + 240 – 4x + 100 – 2y +3x + 3y – 180 = 2.5x + 1.5y +410 The given problem can be formulated as Minimize Z= 2.5x + 1.5y + 410 … (1) subject to the constraints, X + Y ≤ 100 ……(2) X ≤ 60 …….(3) Y ≤ 50 …….(4) X + Y ≥ 60 …….(5) X, Y ≥ 0 …….(6)

Z=2.5x + 1.5y + 410 1) In point A (60, 0) Z= 2.5 x 60 + 1.5 x 0 + 410 Z= 560 2) In point B (60, 40) ( Checking by solving the two lines x + y = 100 and x=60 we get x = 60, y = 40). Z= 2.5 x 60 + 1.5 x 40 + 410 Z= 620 3) In point C (50, 50) (Checking by solving the two lines x + y = 100 and y = 50 we get x = 50, y = 50.) Z= 2.5 x 50 + 1.5 x 50 + 410 Z= 610

4) In point D(10 ,50) (Checking by solving the two lines x + y = 60 and y = 50 we get x = 10, y = 50).Z=2.5 x 10 + 1.5 x 50 + 410 = 510

The minimum value of Z is 510 at (10, 50). RESULT :

Thus, the amount of grain transported from A to D = 10 quintals A to E = 50 quintals A to F =40 quintals B to D = 50 quintals B to E = 0 quintals B to F = 0 quintals respectively. The minimum cost is Rs 510 1. A medicine company has factories at two places A and B. From these places, suppIy is to be made to each of its three agencies P, Q and R. The monthly requirement of these agencies are respectively 40, 40 and 50 packets of the medicines, While the production capacity of the factories at A and B are 60 and 70 packets are respectively. The transportation cost per packet from these factories to the agencies are given: Transportation cost per packet (in Rs.) A

B

P

5

4

Q

4

2

R

3

5

From

To

How many packets from each factory be transported to each agency so that the cost of transportation is minimum? Also find the minimum cost. What should be the features of best location for a factory? CBSE PREVIOUS YEAR QUESTIONS

LEVEL-II 1.A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs.5760.00 to invest and has space for at most 20 items. An electronic sewing machine costs him Rs.360.00 and a manually operated sewing machine Rs.240.00. He can sell an electronic sewing machine at a profit of Rs.22.00 and a manually operated sewing machine at a profit of Rs.18.00. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise his profit? Make it as a linear programming problem and then, solve it graphically. Keeping the rural background in mind justify the

'values' to be promoted for the selection of the manually operated machine (CBSE sample paper 2013). 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 per week are 180 and 30 respectively. The company makes a profit of Rs 80 on each piece of type A and Rs 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week? (CBSE 2014) LEVEL III If a young man drives his scooter at 25 kmph, he has to spend Rs 2 per kilometer on petrol. If he drives the scooter at a speed of 40 kmph, it produces more pollution and increases his expenditure on petrol to Rs 5 per km. He has a maximum of Rs 100 to spend on petrol and wishes to travel a maximum distance in 1 hour time with less pollution. Express this problem as an LPP and solve it graphically. What value do you find here? [CBSE 2013 C (DB)]

LEVEL-II I A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs.5760.00 to invest and has space for at most 20 items. An electronic sewing machine costs him Rs.360.00 and a manually operated sewing machine Rs.240.00. He can sell an Electronic Sewing Machine at a profit of Rs.22.00 and a manually operated sewing machine at a profit of Rs.18.00. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a linear programming problem and then, solve it graphically. Keeping the rural background in mind justify the 'values' to be promoted for the selection of the manually operated machine Questions for self evaluation l. Solve the following linear programming problem graphically: maximize Z =x - 7y+ 190 subject to the constraints x + y 8, x 5, y 5, x+y 4, x 0, y 0. 2. Solve the following linear programming problem graphically: Maximize z=3x+5y subject to the constraints x+ y 2, x+3y 3, x 0, y 0. 3. Kelloggis a new cereal formed of a mixture of bran and rice that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains, 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams 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. 4. A shopkeeper deals only in two items- tables and chairs. He has Rs. 6,000 to invest and a space to store at most 20 pieces. A table costs him Rs. 400 and a chair Rs. 250. He can sell a table at a profit of Rs. 25 and a chair at a profit of Rs. 40. Assume that he can sell all items that he buys. Using linear programming formulate the problem for maximum profit and solve it graphically. What would be your criteria to select a good piece of furniture? 5. A small firm manufactures items A and B. The total number of items A and B it can manufacture a day is at most 24 items. 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. A firm has 2 types of machines. Machine A operates on electricity, Machine B operates on coal. Which machine would you prefer? 6. A chemist requires 10, 12 and 12 units of chemicals A, Band C respectively for his analysis. A liquid product contains 5, 2 and 1 units of A, Band C respectively and it costs Rs. 3 per jar. A dry product contains 1, 2 and 4 units of A. Band C per carton and costs Rs. 2 per carton. How many of each should he purchase in order to minimize the cost and meet the requirement? 7. A person wants to invest at most Rs. 18,000 in Bonds A and E. According to the rules, he has to invest at least Rs. 4,000 in Bond A and at least Rs. 5,000 in Bond B. If the rate of interest on bond A is 9% per annum and the rate of interest on bond B is 11 %per annum, how much money should he invest to earn maximum yearly income? Explain the importance of investment for future life?

8. Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively by stitching uniform. A can stitch 6 shirts and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to stitch at least 60 shirts and 32 pants at a minimum labour cost. What should be the features of uniform of a student?

ANSWERS LINEAR PROGRAMMING (i)LPP and its Mathematical Formulation LEVEL l 1. X+y:

100

4x+2y 240 Z=10 x+5y Students who divide the time for each subject per day according to their need don't feel burden of any subject before the examination (ii) Graphical method of solving LPP (bounded and unbounded solutions) I.

Minimum z= - 12 at (4.0).

2. Maximum Z= 235 at 20, 45

..

19

19 19

3. Minimum Z=7 at (3/2, 1/2) (iii)Diet Problem LEVELII I. Minimum cost = Rs.38.00 at x = 2, Y = 4. Balanced diet keeps fit, healthy and disease free life for a person 2. Minimum cost = Rs.6 at x = 400 and y = 200 Qualities of food are a) It should not contain more fats b) It should not contain more carbohydrates c) It should contain enough fiber, vitamin etc (iv)Manufacturing Problem LEVELII 1). Maximum profit is Rs.120 when 12 units of A and 6 units of B are produced 2). For maximum profit, 25 units of product A and125 units of product B are produced and sold. The factors affecting the demand of a product in the market are a) Quality of the product b) Timely supply of the product c) Customer's satisfaction LEVEL III 1.10500 bottles of medicine A and 34500 bottles of medicine B and they can cover 325500 patients. We should not get panic and should not create panic in case of natural disaster. We must have the helpline numbers of government agencies and NGO working in case of natural disaster. (v)Allocation Problem

LEVEL-II Maximum annual income =Rs. 6,200 on investment of Rs. 40,000 on Bond A and Rs.30, 000 on Bond B. We save money with a purpose of making use of it when we face any kind of financial crisis in our life. We will also be to able to achieve our goals of life if we have enough investment. Max. Z = x + y. Subject to constraints: x/40 + y/70 1, 2x + 7y 100; x, y O. Here x & y represents the distance travelled by the boy at speed of 40km/hr&70km/h respectively. (i) x= 1560/41km, y = 140/41km. (ii) It saves petrol. It saves money. (iii) No, because according to the law driving license is issued when a person is above the 18 years of age. LEVEL-III 1) For maximum profit, 62executive class tickets and 188 economy class ticket should be sold. 1) Plan the trip 2) Check the journey tickets 3) Check the weather forecast 4) Do not take too much of cash 3.Type A fertilizers = 50 kg, Type B = 40 kg. Minimum cost = RS.570. Side effects: Excessive use of fertilizers can spoil the quality of crop also it may cause infertility of land. (vi)Transportation Problem LEVEL-III I. Minimum transportation cost is Rs. 400 when 10, 0 and 50 packets are transported from factory at A and 30, 40 and 0 packets are transported from factory at B to the agencies at P, Q and R respectively. The location for a factory should have the following features 1) enough transport facility 2) enough natural resources 3) enough water 4) availability of electricity 5) availability of labours

CBSE PREVIOUS YEAR QUESTIONS LEVEL-II 1. Max. Z = Rs.392. No. of electronic machines = 8 and no. of manually operated machines = 12. Keeping the 'save environment' factor in mind the manually operated machine should be promoted so that – maximum use of man power and thereby leading to minimum use of energy resources – providing more opportunities for employment in the rural areas (CBSE sample paper 2013) 2. Max profit = Rs 1680 when 12 pieces of type A and 6 pieces of type B are manufactured per week (CBSE 2014) 3. Max distance = 30 Km. at (50/3, 40/3) value save natural resources / our earth [CBSE 2013 C(DB)]

Questions for self evaluation 1) Minimum 155 at (0 , 5) 2) Minimum value is 5 at(3/2, I /2) 3) Maximum is Rs 4.60 at (0.6 , 0.4) 4) Maximum is Rs.800 at(0, 20) The criteria which we have to take into consideration for selecting a good piece of furniture are a) durability b) cost effectiveness c) attractive d) occupy minimum area 5). 8 items of type A and 16 items of type B I would prefer machine A because machine B is not eco-friendly 6. 1 jar of liquid and 5 cartons of dry product. 7.

8.

Rs.4,000 in Bond A and Rs.14,000 in Bond B. We save money with a purpose of making use of it when we face any kind of financial crisis in our life. We will also be to able to achieve our goals of life if we have enough investment. Minimum cost Rs.1350 at (5, 3) The uniform of a student should be a) b) c) d)

well pressed neat and tidy properly stitched shoe must be polished

ADDITIONAL IMPORTANT QUESTIONS: 1. A manufacturer makes two types of cups A and B. Three machines are required to manufacture

the cups and time in minutes required by each is as given below : Types of Cup A B

Machines I 12 6

II 18 0

III 6 9

Each machine is available for a maximum period of 6 hours per day. If the profit on each cup A is 75 paise and on B is 50 paise. Find how many cups of each type should be manufactured to maximize the profit per day. [ Ans : Cup A: 15, Cup B: 30 ] 2. A catering agency has two kitchens to prepare food at two places A and B. From these places, mid-day meal is to be supplied to three different schools situated at P, Q, R. The monthly requirement of these schools are respectively 40, 40 and 50 food packets. A packet contains lunch for 1000 students. Preparing capacity of kitchens A and B are 60 and 70 packets per month respectively. The transportation cost per packet from the kitchens to the school is given below:

Transportation Cost per packet (in Rs.) To P Q R

FROM A 5 4 3

B 4 2 5

How many packets from each kitchen should be transported to schools so that the the cost of transportation is minimum? Also find the minimum cost. [ Ans : Min cost = Rs. 400] 3. Every gram of wheat provides 0.1 gm of proteins and 0.25 gram of carbohydrates. The corresponding values for rice are 0.05 gram and 0.5 gram respectively. Wheat costs Rs 4 per kg. and rice Rs 6 per kg. The minimum daily requirements of protein and carbohydrates for an average child are 50 grams and 200 grams respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of protein and carbohydrates at minimum cost. Frame an L.P.P and solve it graphically. [ Ans : wheat = 400 gm and rice = 200 gm ]

CHAPTER XIII

ADDITIONAL IMPORTANT QUESTIONS: 1. There are three coins .One is a two-headed coin (having head on both faces),another is a biased coin that comes up heads75% of the times and third is also a biased coin that comes up tails 40% of the times. One of the three coins is chosen at random &tossed, and it shows heads What is the probability that it was the two-headed coin? [Ans :4/9] 2. In a bolt factory, three machines A, B, and C manufactures 25,35 and 40 percent of the total bolts manufactured. Of their outputs, 5, 4 and 2 percent are defective respectively. A bolt is drawn at random and is found defective. Find the probability that it was manufactured by either machine A or C. [ Ans : 41/69] 3. Coloured balls are distributed in three bags as shown in the following table: Colour of the ball Bag

Black

White

Red

I

1

2

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 black and red. What is the probability that they came from bag I ?[ Ans : 231/551] 4. A bag contains 4 balls. Two balls are drawn at random, and are found to be white. What is the probability that all balls are white ?[ Ans : 3/5]

5. Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable, and hence find the mean of the distribution.

[Ans: X

2

3

4

5

P(X) 1/15 2/15 3/15 Mean=14/3

6

4/15 5/15

6.In a game ,a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as when he gets a six .Find the expected value of the amount he wins/loses. [ Ans: 11/216 ]

7. Two balls are drawn one by one with replacement from a bag containing 4 red and 6 black balls. Find the probability distribution of ‘ number of red balls ‘. [Ans:

X: 0

1

P(X) : 9/25

2 12/25

4/25

]

8. Find the probability distribution of the number of doublets in three throws of a pair of dice. [ Ans :

X :

0

1

2

3 P(X)

: 125/216

75/216 15/216 1/216 ]

VALUE BASED QUESTIONS 1. In a school, 30% of the student has 100% attendance. Previous year result report tells that 70% of all students having 100% attendance attain A grade and 10% of remaining students attain A grade in their annual examination. At the end of the year, One student is chosen at random and he has an A grade. What is the probability that the student has 100% attendance? Also state the factors which affect the result of a student in the examination. [Ans.45 3/4 Factors :-(i) Regular study management (v) Writing skills]

(ii) Hard work

(iii) Good memory

(iv) Well time

2. A company has two plants of scooter manufacturing. Plant I manufacture 70% Scooter and plant II manufactures 30%. At plant I 80% of the scooter’s are maintaining pollution norms and in plant II 90% of the scooter maintaining Pollution norms. A Scooter is chosen at random and is found to be fit on pollution norms. What is the probability that it has come from plant II. What is importance of pollution norms for a vehicle? [ Ans: 27/53, Pollution free environment minimize the health problems in the human being.] 3. In a group of students, 200 attend coaching classes, 400 students attend school regularly and 600 students study themselves with help of peers. The probability that a student will succeed in life who attend coaching

classes, attend school regularly and study themselves with help of peers are 0.1, 0.2 and 0.5 respectively. One student is selected who succeeded in life, what is the probability that he study himself with help of peers. What type of study can be considered for the success in life and why? [Ans:0.75self studies with the help of peers is best as through it students can get the knowledge in depth of each concept. But students should be regular in school and if they feel need they could join different classes]. 4. A clever student used a biased coin so that the head is 3 times as likely to occur as tail. If the coin tossed twice find the probability distribution and mean of numbers of tails. Is this a good tendency? Justify your answer. [ Ans: X : 0 P(X) : 9/16 Mean = ½.

1

2

6/16

1/16

No, it may be good once or twice but not forever.

Honesty pays in a long run. ]

SYLLABUS

SAMPLEPAPER BLUE PRINT

S.No.

Topics

1.(a)

Relations&Functions

(b) 2.(a) (b). 3(a) (c) (e)

VSA(1) SA(1)

LA (6) Total

4(1)

4(1)

InverseTrigonometricFunctions

6(1)

Matrices

1(2)

Determinants

1(1)

6(1) 4(1)

Continuity&Differentiability

4(2)

ApplicationsofDerivatives

4(1)

Integrals

4(3)

6(1)

13(5)

11(3) 8(2)

6(1)

10(2) 12(3)

6(1)

DifferentialEquation

(b)

2(2)

44(10)

Applicationof Integrals

4.(a)

10(2)

8(2)

Vectors

1(2)

4(2) 4(1)

Three DimensionalGeometry

1(1)

4(1)

5.

LinearProgramming

6.

Probability Total

6(1)

6(6)

6(3) 6(1)

11(3)

6(1)

6(1)

4(1)

6(1)

10(2)

52(13)

42(7)

100(26)

17(6)

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

IITJEE 2014

159

160

161

162

163

164

165

166

167

168

VALUE BASED PROBLEMS MATHEMATICS CLASS-XII RELATIONS AND FUNCTIONS

Q.4

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, coeducation may be helpful in child development and why?

Q.5

Consider a relation R in the set A of people in a colony, defined as aRb, if and only if a and b are members of a joint family. Is R an equivalence relation? Staying with Grandparents in a joint family imbibes the moral values in us. Can you elicit two such values?

Q.6

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. If x is a brilliant student and y is a slow learner and x helps y in his studies. What quality does x possess?

Q.7

Let L be the set of all lines in a plane and R be the relation in L defined by R = { (L1, L2): L1 is parallel to L2 } Show that R is an Equivalence Relation. L1 represents the ideologies of Gandhi, L2 represents ideologies of NetajiSubhash Chandra Bose. Even though their ideologies ran on parallel tracks both had the common goal to achieve independence for India. Which common value did they both exhibit?

MATRICES & DETERMINANTS 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 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 parsons 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 way?

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. Q.6

F 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? Q.7.

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?

Q.8.

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? CONTINUITY AND DIFFERENTIABILITY

Q.1.

A car driver is driving a car on the dangerous path given by

m ∈N

Find the dangerous point (point of discontinuity) on the path. Whether the driver should pass that point or not? Justify your answers. APPLICATION OF DERIVATIVES Q.1

Q.2.

Q.3.

Q.4.

A car parking company has 500 subscribers and collects fixed charges of Rs.300 per subscriber per month. The company proposes to increase the monthly subscription and it is believed that for every increase of Re.1, one subscriber will discontinue the service. What increase will bring maximum income of the company? What values are driven by this problem? Check whether the function + is strictly increasing or strictly decreasing or none of both on . Should the nature of a man be like this function? Justify your answers. If , when denotes the number of hours worked and denotes the amount (in Rupees) earned. Then find the value of (in interval) for which the income remains increasing? Explain the importance of earning in life? If performance of the students ‘y’ depends on the number of hours ‘x’ of hard work done per day is given by the relation. Find the number of hours, the students work to have the best performance.

‘Hours of hard work are necessary for success’ Justify. Q.5.

A farmer wants to construct a circular well and a square garden in his field. He wants to keep sum of their perimeters fixed. Then prove that the sum of their areas is least when the side of square garden is double the radius of the circular well. Do you think good planning can save energy, time and money?

Q.6.

Profit function of a company is given as where x is the number of units produced. What is the maximum profit of the company? Company feels its social responsibility and decided to contribute 10% of his profit for the orphanage. What is the amount contributed by the company for the charity? Justify that every company should do it. In a competition a brave child tries to inflate a huge spherical balloon bearing slogans against child labour at the rate of 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon is increasing when its radius is 15cm. Also write any three values/life skill reflected in this question.

Q.7.

Q.8.

In a kite festival, a kite is at a height of 120m and 130m string is out. If the kite is moving horizontally at the rate of 5.2m/sec, find the rate at which the string is being pulled out at that instant. How a festival enhance national integration. Q.9. An expensive square piece of golden color board of side 24 centimeters. is to be made into a box without top by cutting a square from each corner and folding the flaps to form a box. What should be the side of the square piece to be cut from each corner of the board to hold maximum volume and minimize the wastage? What is the importance of minimizing the wastage in utilizing the resources? Q.10. A student is given card board of area 27 square centimeters. He wishes to form a box with square base to have maximum capacity and no wastage of the board. What are the dimensions of the box so formed? Do you agree that students don’t utilize the resources properly? Justify. INTEGRATION Q.1

Evaluate,

, Discuss the importance of integration (unity) in life. APPLICATIONS OF INTEGRALS

Q.1.

A farmer has a piece of land. He wishes to divide equally in his two sons to maintain peace and harmony in the family. If his land is denoted by area bounded by curve and and to divide the area equally he draws a line what is the value of a? What is the importance of equality among the people?

Q.2. A circular Olympic gold medal has a radius 2cm and taking the centre at the origin, Find its area by method of integration. What is the importance of Olympic Games for a sportsman and why? Olympic game is a supreme platform for a sportsman. In Olympic Games all countries of the world participate and try their best and make their country proud. Q.3.

A poor deceased farmer has agriculture land bounded by the curve y= , between x = 0 and x=2 π. He has two sons. Now they want to distribute this land in three parts (As

already partitioned).Find the area of each part. Which parts should be given to the farmer & why? Justify your answer. Q.4

Q.5

Q.6

Ans.

If a triangular field is bounded by the lines x+2y = 2, y-x = 1 and 2x+y = 7Using integration compute the area of the field (i) If in each square unit area 4 trees may be planted. Find the number of trees can be planted In the field. (ii) Why plantation of trees is necessary? A parking lot in an IT company has an area bounded by the curve y= 4-x2 and the lines y=0 and y = 3 divides the area in to two equal parts out of which the greater area is allotted for car owners who practice carpooling. Find this area using integration. Write any two benefits of carpooling. Ans. Fuel saving, Less pollution Find the area of the region enclosed by the curve y= x 2 and the lines x=0, y=1 and y=4. A farmer plans to construct an electrical fence around this bounded region to protect his crop. But his son rejects this idea and wants wooden fence to be erected. Who would you favour? Mention two values demonstrated by the son . Concern for animals, kind hearted, not being cruel, bold DIFFERENTIAL EQUATIONS

1.

Solve the differential equation (x+ 2y2 )y’=y. Given that when x= 2, y=1. If x denotes the % of people who are polite and y denote the % of people who are intelligent. Find x when y=2%. A polite child is always liked by all in society. Do you agree? Justify.

2.

y’ +

= 0 where x denotes the percentage of population living in a city and y denotes the

area for living a healthy life of population .Find the particular solution when x=100 , y=1. Is higher density of population is harmful? Justify your VECTORS & 3-DIMENSIONAL GEOMETRY Q.1.

considering the earth as a plane having equation , A monument is standing vertically such that its peak is at the point (1, 2, -3). Find the height of the monument. How can we save our monument?

Q.2.

Let the point p (5, 9, 3) lies on the top of QutubMinar, Delhi. Find the image of the point on the line = = Do you think that the conservation of monuments is important and why?

Q.3

Two bikers are running at the Speed more than allowed speed on the road along the Lines =

and

= Using Shortest distance formula check whether they meet to an accident or not? While driving should driver maintain the speed limit as allowed. Justify?

LINEAR PROGRAMMING PROBLEMS Q.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? Q.2. 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? Q.3 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. 1. Express this as an L .P.P and solve graphically. 2. What is benefit of driving at an economical speed? 3. Should a child below 18years be allowed to drive a motorcycle? Give reasons. Q.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? Q.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 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? Q.6 A retired person has Rs. 70,000 to invest in two types of bonds. First type of bond yields 10% per annum. As per norms he has to invest minimum to Rs. 10,000 in first type and not more than Rs. 30,000 in second type. How should he plan his investment so as to get maximum return after one year of investment? What values have to be inculcated by a person for a peaceful retired life.

Q.7 A company manufactures two types of stickers A: ‘SAVE ENVIRONMENT’ and B: ‘BE COURTEOUS’. Type A requires five minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There 3 hours and 20 minutes available for cutting and 4 hours available for assembling in a day. He earns a profit of Rs. 50 on each type A and Rs. 60 on each type B. How stickers of each type should company manufacture in a day of each type should company manufacture in a day to maximize profit? Give your views about ‘SAVE ENVIRONMENT’ and ‘BE COURTEOUS’ Q.8 Suppose every gram of wheat produces 0.1 g of protein and 0.25 g of carbohydrates and corresponding values for rice are 0.05 g and 0.5 g respectively. Wheat cost Rupees 25 and rice Rs.100 per kilogram. The minimum daily requirements of proteins and carbohydrates for an average man are 50 g and 200 g respectively. In what quantities should wheat and rice be mixed in daily diet to provide minimum daily requirements or proteins of carbohydrates at minimum cost, assuming that wheat and rice are to be taken in a diet? What is your opinion about healthy diet? PROBABILITY Q.1

Probability of winning when batting coach A and bowling coach B working independently are ½ and ⅓ respectively. If both try for the win independently find the probability that there is a win. Will the independently working may be effective? And why?

Q.2.

A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike and 0.32 that the construction job will be completed on time if there is strike. Determine the probability that the construction job will be completed on time. What values are driven by this question?

Q.3.

A clever student used a biased coin so that the head is 3 times as likely to occur as tail. If the coin tossed twice find the probability distribution and mean of numbers of tails. Is this a good tendency? Justify your answer.

Q.4

A man is known to speak truth 5 out of 6 times. He draws a ball from the bag containing 4 white and 6 black balls and reports that it is white. Find the probability that it is actually white? Do you think that speaking truth is always good?

Q.5

A drunkard man takes a step forward with probability 0.6 and takes a step backward with probability 0.4. He takes 9 steps in all. Find the probability that he is just one step away from the initial point. Do you think drinking habit can ruin one’s family life?

Q.6.

If group A contains the students who try to solve the problem by knowledge, Group B contains the students who guess to solve the problem Group C contains the students who give answer by cheating. If n (A) = 20, n (B) = 15, n(C) = 10, 2 Students are selected at random. Find the probability that they are from group c. Do you think that cheating habit spoils the career?

Q.7

In a school, 30% of the student has 100% attendance. Previous year result report tells that 70% of all students having 100% attendance attain A grade and 10% of remaining students attain A grade in their annual examination. At the end of the year, One student is chosen at random and he has an A grade. What is the probability that the student has 100% attendance? Also state the factors which affect the result of a student in the examination.

Q.8

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is six. Find the probability that it is actually a six. Write any three benefits of speaking the truth.

Q.9.

There are 20 People in a group. Out of them 7 people are non –vegetarian, 2 people are selected randomly. Write the probability distribution of non–vegetarian people. Explain whether you would like to be vegetarian or non- vegetarian and why? Also keeping life of animals in mind how would you promote a person to be vegetarian?

Q.10

Two third of the students in a class are sincere about their study and rest are careless Probability of passing in examination are 0.7 and 0.2 for sincere and careless studentsrespectively, A Student is chosen and is found to be passed what is the probability that he/she was sincere. Explain the importance of sincerity for a student.

Q.11. A company has two plants of scooter manufacturing. Plant I manufacture 70% Scooter and plant II manufactures 30%. At plant I 80% of the scooter’s are maintaining pollution norms and in plant II 90% of the scooter maintaining Pollution norms. A Scooter is chosen at random and is found to be fit on pollution norms. What is the probability that it has come from plant II. What is importance of pollution norms for a vehicle? Q. 12 A chairman is biased so that he selects his relatives for a job 3 times as likely as others. If there are 3 posts for a job. Find the probability distribution for selection of persons other than their relatives. If the chairman is biased than which value of life will be demolished? Q.13 A manufacturer has three machine operators A (skilled) B (Semi- skilled) and C (nonskilled).The first operator A Produces 1% defective items where as the other two operators B and C produces 5% and 7 % defective items respectively. A is on the job for 50% of time B in the job for 30% of the time and C is on the job for 20 % of the time. A defective item is produced what is the probability that it was produced by B? What is the value of skill? Q.14

In a group of 100 families, 30 families like male child, 25 families like female child and 45 families feel both children are equal. If two families are selected at random out of 100 families, find the probability distribution of the number of families feel both children are equal. What is the importance in the society to develop the feeling that both children are equal?

Q.15

In a group of 200 people, 50% believe in that anger and violence will ruin the country, 30% do not believe in that anger and violence will ruin the country and 20% are not sure about anything. If 3 people are selected at random find the probability that 2 people believe and 1 does not believe that anger and violence will ruin the country. How do you consider that anger

Q.16

Q.17

Q.18

Q.19

Q.20

Q.21

and violence will ruin the country? In a group of students, 200 attend coaching classes, 400 students attend school regularly and 600 students study themselves with help of peers. The probability that a student will succeed in life who attend coaching classes, attend school regularly and study themselves with help of peers are 0.1, 0.2 and 0.5 respectively. One student is selected who succeeded in life, what is the probability that he study himself with help of peers. What type of study can be considered for the success in life and why? Ramesh appears for an interview for two posts A and B for which selection is independent .The probability of his selection for post A is 1/6 and for post B is 1/7. He prepared well for the two posts by getting all the possible information. What is the probability that he is selected for at least one of the post? Which values in life he is representing? Past experience shows that 80% of operations performed by a doctor are successful. If he performs 4 operations in a day, what is the probability that at least three operations will be successful? Which values are reflected by the doctor? A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise it is rejected. Find the probability that the box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale. In answering a multiple choice question test with four choices per question, a student knows the answer, guesses or copies the answer. If ½ be the probability that he knows the answer, 1/4 be the probability he guesses and ¼ that he copies it .Assuming that a student who copies the answer will be correct with the probability 3/4, what is the probability that the student knows the answer given that he answered it correctly? Mehul does not know the answer to one of the question in the test. The evaluation process has negative marking. Which value would Mehul violate if he restores to unfair means? In a class, having 60% boys, 5% of the boys and 10% of the girls have an IQ of more than 150. A student is selected at random and found to have an IQ of more than 150. Find the probability that the selected student is a boy. It has been seen that students with not high IQ have also performed well. What values have been inculcated by the student? **********************

RELATIONS AND FUNCTION ( answers) Ans.1 f-1(x) = , Truthfulness and honesty among people may have the bijective (one-one onto) relation as people who are honest usually truthful and vice versa. Ans.2 Neither one-one nor onto hence not bijective Yes, true friendship makes life easier. Ans.3 P= , Punctuality develops discipline in life and hence progressive in life. Ans.4 The relation R is reflexive, symmetric and transitive .Co-education is very helpful because it leads to the balanced development of the children and in future they become good citizens. Expected Answers 5. Love and concerned for grandparents. Respect for Grand Parents. Tolerance 6. Concern about fellow student, helping others, sharing of knowledge. 7. Patriotism, sacrifice, Leadership and Pride in our country. MATRICES & DETERMINANTS 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 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 Participating in co-curricular activities is very important. It is very essential for all round development.

Ans.6. (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.7. Rs300crores, Rs200crores and Rs100 crores (i) Our In country, male population is more than female population. (ii) It is essential for a human being to save the life of all. Ans.8. 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. CONTINUITY AND DIFFERENTIABILITY Ans.1

[Point No, because Life is precious. Or Drive carefully.

]

APPLICATION OF DERIVATIVES

Ans.1 Increase of Rs.100 monthly subscription for Max. Income of the company. 1. The sharing (2-3 persons on the same route) will be promoted. 2. Decrease pollution 3. Decrease vehicle density on road. 4. Saving of energy. Ans.2 [Neither strictly increasing nor strictly decreasing]. Yes, because strictness in not always good in life. Ans.3 To support the family, regular increasing income is must. Ans.4. 4 hours per day. By hard work, we can create skill in using the things Learnt by us. So we Don’t make mistake in the competition when the things are asked. Ans.5. Yes, every work done in a planned way proves to be more fruitful. If a student makes a planning for his studies he can do wonders. Ans.6. Maximum profit = Rs76 when x=240. Yes it is good for society Ans.7 15/2π Cm. /Sec. (i) Bravery (ii) Awareness about child labour (iii)Right of a child Ans.8 4.8m/sec. In a festival many people participated with full happiness and share their lives and enjoy it.

Ans.9. 4 centimeters. As our country is still developing and most of the Indian people are from the middle class, so we should utilize our resources in proper way. Students should buy only those books which they feel really important. Instead of buying books for only one or two chapters. They should borrow it from the library. Ans.10. length of square base is 3 centimeters and height of the box is 1.5 centimeters. Yes, I agree that students don’t utilize the resources properly. They get various notes photocopies and waste one side of the paper. Whereas other side of paper can be utilized for making comments on those notes. INTEGRATION

Ans.1.

- log|

+ tan -1 + C

1. United we stand, divided we fall. 2. Union is strength. APPLICATIONS OF INTEGRALS 1/3

Ans.1

. Equality helps to maintain peace and harmony in all aspect of society

Ans.2 4π Ans.3.

1, 2, 1 1. Respect the parents 2. Help the elders (parents)

Ans.4

Area of the field= 6 Sq. unit (i) 24 trees (ii) Plants provide us oxygen and play major role in rain, so plantation is essential for all human beings. DIFFERENTIAL EQUATIONS

Ans.1

, 8. Yes polite child has a peaceful mind and peaceful mind grasps the ideas easily and understand the complicated concept

Ans.2 Yes, as the population increases area for living decreases, that is very harmful for us.

VECTORS & 3-DIMENSIONAL GEOMETRY

Ans.1 (i) Units (ii) We should not harm any monument. (iii)We should not write anything on it. (iv) We should respect our national heritage. Ans.2 The point of image is (3, 5, 7) Conservation of monuments is very important because it is a part of our history and their contribution. Ans.3 S.D =0, this means they meet to an accident. If a driver follow speed limit there will be minimum chance of accident. LINEAR PROGRAMMING PROBLEMS Ans.1 Minimum cost = Rs. 8.00 x=2, y=4 Balanced diet keeps fit, healthy and disease free life to a person. Ans.2 Type A fertilizers = 50 kg, Type B = 40 kg. Minimum cost =Rs. 570/-infertility of land. Excessive use of fertilizers can spoil the quality of crop also it may cause. Ans.3 . Max. Z= x + y, + 2x+7y X

,y

Where x & y represents the distance travelled by the speed of 40km/hr& 70 km/h respectively. 1. X=1560/41Km., y= 140/41Km. 2. It Saves petrol. It saves money. 3. No because according to the law driving license is issued when a person is above the 18 years of age. Ans.4. 20 problems from first list and 80 problems from second list. Students who divide the time for each subject per day according to their need don’t feel burden of any subject before the examination. Ans.5. 10500 bottles of medicine A and 34500 bottles of medicine B and they can cover 325500 patients. We should not get panic and should not create panic in case of natural disaster. Must have the helpline numbers of government agencies and NGO working in case of Natural Disaster. 6

Rs.40000 must be invested in 8 % bonds and Rs.30000 in 10% bonds for a maximum return of Rs.6200. One should start saving at early age of retirement.

7

8

8 stickers of type A and 20 stickers of type B should be manufactured for a maximum profit of Rs. 1600. Saving environment is a big challenge which is very important and necessary to survive . Be courteous is life skill which everyone must acquire to be compassionate . Cost Rs. 30 is minimum for 400 g of wheat and 200 g of rice to provide minimum daily requirements. We must take balanced healthy diet for good health. PROBABILITY

Ans.1 1. Chances of success increase when ideas flow independently. 2. Hard work pays the fruits. Ans.2 [0.488] Peace is better than strike. As the probability of completion of job on time if there is strike is less then ½. Ans.3 x P(x)

0

1

2

Mean = 1. No, it may be good once or twice but not forever. 2. Honesty pays in a long run. Ans.4 , speaking truth pays in the long run. Sometimes lie told for a good cause is not bad. Ans5 Yes, addiction of wine or smoking is definitely harmful for a person and its family. Ans.6 (i) (ii) Yes, because a cheater finds it to do any work independently. But it is harmful in long run. Ans.7

3/4

Factors :-(i) Regular study (ii) Hard work (iii) Good memory (iv) Well time management (v) Writing skills

Ans.8

3/8 (i) (ii) (iii)

It gives positive thinking &satisfaction Everyone loves it. It is good life skill

Ans.9

I would like to be a vegetarian because vegetarian food is much easier to digest than nonvegetarian (may be given other reason) Or For non- vegetarian food we have to kill animals this is not good thing because everybody has right to survive, etc. Ans.10 A Student is future of a country. If a student is sincere then he/she can serve the country in a better way. Ans.11 Pollution free environment minimize the health problems in the human being. Ans12 X

0

1

2

3

P(x)

Values lost by chairman – Honesty, Integrity Ans.13 skilled person can complete a work in better way than other person Ans.14 X P(x)

0

1

2

2.

To maintain the ratio of male and female equally. This is important to consider both children are equal. Ans15. 0.225, People in anger cannot use their presence of mind and become violent and destroy public property in riots which is indirectly their own property.

Ans.16. 0.75 self-studies with the help of peers is best as through it students can get the knowledge in depth of each concept. But students should be regular in school and if they feel need they could join different classes. 17. P(at least one post) = 1 – P(none posts) 5 6 2 =1- x = 6 7 7 He represents hard work,honesty, zest to excel. 18. P(at least 3) = P(3) + P(4) 3

8 4 x  5 5 The values reflected are responsibility, love for life, dedicated to work. 12 19. c .3 c 44 Required probability = 153 0 = 91 c3 20. 1 . 1 2 2 Required probability = = 1 1 1 1 3 3 1   2 4 4 4 4 If he restores to unfair means he violates honesty. 21 3 1 3 5 20 Required probability = = 3 1 2 1 7  5 20 5 10 A student can perform well if he is hard working, sincere and well-focused.

=

GRADED EXERCISE QUESTIONS (LEVEL I, II, III) Relations and Functions ( Level 1)Easy (1M) 1. Prove that f: R R is defined by f(x)= x3 is one- one function. 2. * be a binary operation defined on Q given by a*b = a+ab , a, b ∈ Q . Is * is commutative? 3. 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. Show that ‘f’ is one- one. Relations and Functions ( Level 2)Average (1M) 1) If functions f and g are given by f= { (1,2),(3,5),(4,1),(2,6) } g= { (2,6),(5,4), (1,3),(6,1) } find fog and gof . 2) Let f: A B where set A= {1,2,3} B={a , c} defined as f(1)=a f(2) =c and f(3) =a find f-1 if exist. 3) Prove that the Greatest Integer function f: R R by f(x) = is neither one- one nor onto.

Relations and Functions ( Level 3)Difficult(1M) 1) f : R R defined by f(x) = | | . Is function f onto ? give Reasons . 2) Let f: R R defined by f(x) = x2 +1 , Find the pre-image of i) 17 ii) 5 3) Let f and g be two real valued functions defined as f(x) = 2x-3 and g(x) =

, find fog

=================================================================== Relations and Functions ( Level 1)Easy (4M) 1) Let A=R – {3} and B= R-{1} . Consider the function f : A that f is one –one and onto and hence find f 2) Find the Inverse of f(x) =

,x

B by f(x) =

, Show

-1

-1 , and verify that fof-1 is an identity function.

3) Let R be the set of real numbers and * be a binary operation defined on R as a*b = a+b-ab , for every a, b ∈ R , Find the identity element with respect to the binary operation *. Relations and Functions ( Level 2)Average (4M) 1) Let A be the a set of all 46 students of class XII in a school. Let f: A N be a function defined by f(x) Roll Number of the student ‘x’. Show that ‘f’ is one-one but not onto. 2) Let R be the relation on N defined as R = { (x,y) : x,y∈ N , 2x+y =41. } Find the domain and range of R . Also verify whether R is reflexive, symmetric and transitive. 3) A relation R :N N defined as (a,b) R (c, d) a+d= b+c , show that R is an Equivalence Relation. Relations and Functions ( Level 3)difficult (4M) 1) Show that the relation R in the set of real numbers defined as R = { (a,b) : a b3} is neither reflexive nor symmetric nor transitive. 2) Let A= {1,2,3,4,…………………9} and R be the relation in AxA defined by (a,b) R )c,d) if a+d= b+c , for (a,b) ,(c,d) ∈ AXA . Prove that R is an Equivalenc e Relation. 3) Let f: N

N defined by f(n) = {

} for all n∈ N. Find whether the

function ‘f’ is bijective Inverse Trigonometric Function Level 1 –Easy (1M) 1) Evaluate Sin-1 (-

+ Cos-1(-

2) Find the principal value of Sec-1(-2) 3) Prove that

=

)

Inverse Trigonometric Function Level 2 –Average (1M) 1) If Sin {

=1 , then find the value of ‘x’.

+

2) Evaluate Cos { -

}

3) Evaluate Inverse Trigonometric Function Level 3 – Difficult (1M) 1) Evaluate

+

2) Find the value of Sec(

) in terms of y

3) Write the simplest form :

)

Inverse Trigonometric Function Level 1 –Easy (4M) 1) Prove that 2) Write in its simplest form: 3) Solve for x :

=

Inverse Trigonometric Function Level 2 – Average (4M) 1) Prove that 2) Prove that

*

+

*

,

+ -

3)

*

+=0

, x∈ (0 ,

=

Inverse Trigonometric Function Level 3 – Difficult (4M) }=√

1) Prove that Cos { 2)

Prove that

3)

Solve for x :

)+

) =

===============================================================

Answers Relations and Functions ( Level 1)Easy (1M) 1. Proof 2. No 3. Different elements have different images

Relations and Functions ( Level 2)Average (1M) 1. G of = {(1,6),(3,4),(4,3),(2,1)} fog is not defined 2. Not 1-1 and hence f-1 doesn’t exist 3. Not 1-1

Relations and Functions ( Level 3)Difficult(1M) 1. No, the Negative Real numbers have no pre-images, 2. i) 4,-4 ii) 3,-3 3. proof Relations and Functions ( Level 1)Easy (4M) 1. f-1(x) = 2. Show that fof-1(x) = x 3. e=0 , b = Relations and Functions ( Level 2)Average (4M) 1. 1-1 because each student related to unique Roll numbers. But not 1-1 because the remaining Natural numbers in the co-domain are having no pre-images. 2. R= { (1,39), (2,37) ,(3,35),………………………(20,1) } Domain = {1,2,3,………20} Range = { 39,37,35, …………………..1} Not reflexive and symmetric but transitive. 3. Proof. Relations and Functions ( Level 3)difficult (4M) 1. Proof 2. Proof 3. F is not 1-1 but Onto.

Inverse Trigonometric Function Level 1 –Easy (1M) 1. 2. 3. Proof Inverse Trigonometric Function Level 2 –Average (1M) 1. 2. 3. Inverse Trigonometric Function Level 3 – Difficult (1M) 1. 2.

√

3. Inverse Trigonometric Function Level 1 –Easy (4M) 1. Proof 2. 3. Inverse Trigonometric Function Level 2 – Average (4M) 1. Proof 2, Proof

Inverse Trigonometric Function Level 3 – Difficult (4M) 1.Proof 2.Proof 3.

MATRICES LEVEL -I QUESTIONS 1. If the matrix

(

)

then find p.

2. Find the values of x, y if (

)

(

)

(

3. Construct a 2 x 3 matrix whose elements in the

). row and

column are given by

. LEVEL – 2 QUESTIONS 1. Construct a 2 x 2 matrix 2. For the matrix

(

[

] whose elements

) verify that

3. If the matrix (

is symmetric.

) is skew symmetric find the value of a, b, and c.

4. If A, B are two given matrices such that the order of A is 3 x 4 , if defined then find the order of . 5. If

(

{

are given by

) find x , satisfying

when

and

are both

.

LEVEL – 3 QUESTIONS 1. Construct a 2 x 2 matrix

[

] whose elements

are given by

*+

represent greatest integer function. 2. If

(

) find A(adjA) without computing adjA.

3. If A is a square matrix such that

, then find the value of

.

DETERMINANTS LEVEL -I QUESTIONS 1. For what value of x is the matrix (

) singular?

| | | 2. If A is the square matrix of order 3 such that | 3. If A , B and C are n x n matrices and det(A) = 2, det (B) = 3 and det (c) = 5. Find the value of det ( LEVEL – 2 QUESTIONS | | |. 1. If A is a square matrix of order 3 such that | 2. If A= ( 3. Let

) find Adj(AdjA). (

)

(

). If B is the inverse of A then find

. By using properties of determinants prove that: 4. |

|

.

5. |

|

6. |

|

[

1. If matrix Prove that 2. Let and

. |

|.

LEVEL – 3 QUESTIONS [ ]

] is singular.

|

| where a, b are real constants . |

3. If a,b,c are positive and unequal, show that vale of determinant negative. 4. Prove that |

5. Prove that |

|

|

.

.

| is

ANSWERS : MATRICES LEVEL -I QUESTIONS 2. X=2, y=-8. 3. (

1. P=4

)

LEVEL – 2 QUESTIONS (

1.

) 3. A=-2, b=0, c=-3. 4. Order of

= 4 x 4. 5. x= .

LEVEL – 3 QUESTIONS (

1.

) 2. (

) 3. I

ANSWERS : DETERMINANTS LEVEL -I QUESTIONS 1. X=4. 2. | |

3.

LEVEL – 2 QUESTIONS 1. |

|

2. A= (

) 3. .=5.

LEVEL – 3 QUESTIONS 1. Since A is skew symmetric det(A)=0. Therefore det(

)=0 . 2.

.

Level I Continuity and differentiability Section A ( 1 Mark ) 1)Discuss the continuity of the function f given by f(x) =x3 +x2-1 2) Is the function defined by f (x) = | x |, a continuous function? 3) Check the points where the constant function f (x) = k is continuous Section B ( 4 Mark ) 4) Differentiate xsinx , x> 0 w.r.t. x. 5) Find dy/dx , if x = a

y = a (1 –

6) Verify Mean Value Theorem for the function f (x) = x2 in the interval [2, 4]. Section C ( 6 mark ) 7) Differentiate the following w.r.t. x. 8)

9) Differentiate w.r.t. x, the following function Answers 1. f is contfn 2. f is contfn 3. f is contfn 4. xsix-1.sinx + xsinx.cosxlogx 5. tanθ/2 6. verified 7. i) -1 ii) ½ 8. proving 9. i) ii) 2sec2xtanx

+3 (

)

LEVEL –III ( Continuity and Derivatives ) 1 MARK 1. Check the continuity of the function f ( x)  sin x  x 2 at x =  2. Give an example of a function which is continuous but not differentiable. 3. . Discuss the continuity of the function f given by f (x) = | x | at x = 0. 4 Marks

4. Differentiate

5)

6)

6 MARKS 7)

8)

9)

Answers 1. Continuous 2. Example 3. Continuous 4.

[

log (x+1/x)} +

[

]

5. 6. Proving 7. Proving 8.

t<

9. APPLICATIONS OF DERIVATIVES

1. 2.

3. 4. 5.

6. 7. 8. 9.

RATE OF CHANGE (4 MARKS) 3 A particle moves along the curve 6y = x + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as x-coordinate. A man 160 cm tall walks away from a source of light situated at the top of the pole 6 m high at the rate of 1.1 m/sec. How fast is the length of the shadow increasing when he is 1 m away from the pole? The surface area of a spherical bubble is increasing at the rate of 2 cm2/sec. Find the rate of which the volume of the bubble is increasing at the instant if its radius is 6 cm. Water is passed into an inverted cone of base radius 5 cm and depth 10 cm at the rate of 3/2 c.c/sec. Find the rate at which level of water is rising when depth is 4 cm. Find the total revenue received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5. Find the marginal revenue when x = 5. INCREASING AND DECREASING FUNCTIONS (4 MARKS) Find the intervals in which function f(x) = 6 + 12x + 3x2 – 2x3 is increasing or decreasing. Find the intervals in which function f(x) = 4x2 + 1 is increasing or decreasing. X Show that y = log (1+x) – 2x , x> -1 is an increasing function of x, throughout its domain 2+x Find the intervals in which function f(x) = sin x + cos x in [0,2П] is increasing or decreasing.

10. 11. 12.

Find the intervals in which function f(x) = (x+1)3 (x-3)3is increasing or decreasing. Find the intervals in which function f(x) = sin4x + cos4x in [ 0, П/2] is increasing or decreasing. Find the intervals in which function f(x) = sin 3x , x Є [ 0,П/2] is increasing or decreasing. TANGENTS AND NORMALS

13. 14. 15. 16.

Find a point on the parabola f(x) = (x-3)2, where the tangent is parallel to the chord joining the points, (3,0) and (4,1) Prove that the curves y2 = 4ax and xy = c2 cut at right angles, if c4 = 32a4. At what points will the tangent to the curve y = 2x3 – 15x2 + 36x -21 be parallel to x-axis ? Also find the equations of the tangents to the curve at these points. Find the equations of the normals to the curve3x2 – y2 = 8 parallel to the line x + 3y = 4. APPROXIMATIONS

17. 18. 19.

(4 MARKS)

Using differentials, find the approximates value of √0.037. Find the approximates value of f(5.001), where f(x) = x3 – 7x2 + 15. A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. Find the rate at which its area is increasing when radius is 3.2 cm. MAXIMA AND MINIMA

20.

21.

22. 23.

24. 25. 26.

27. 28. 29.

(4 MARKS )

( 6 MARKS )

A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semi-circle. Find the dimensions of the rectangle , so that its area is maximum. Also find maximum area. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be the least when the depth of the tank is half of its width. An open box with a square base is to be made out of a given quantity of sheet of area a2 s.u.. Show that the maximum volume of the box is a3 /6√3 c.u. A window is in the form of a rectangle above which there is a semicircle. If the perimeter of the window is p cm. Show that the window will allow the maximum possible light only when the radius of the semicircle is p/(П+4) Find the absolute maximum value and the absolute minimum value for the function f(x) = 4x – x2/2 Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 times the radius of the base. A wire of length 36 cm is cut into two pieces. One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two be minimum. Show that the semi-vertical angle of a right circular cone of maximum volume and given slant height is tan‫־‬1√2 Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere. Show that volume of greatest cylinder which can be inscribed in a cone of height h and semi vertical angle α is (4/27) Пh3 tan2α.

1.

(4,11),(-4,-31/3)

2.

0.4 cm/sec

3.

6 cm3/sec

4.

3/(8П) cm/sec

5.

66.

6.

↑ in [-1,2) and ↓ (-∞,-1)U[2,∞)

7.

↑(-∞,-1/2)U(1/2,∞)and ↓ (-

9. ↑ in (0, П/4)U(5П/4,2П) and ↓ (

1/2,1/2)-{0} 10.

↑ in (1,∞) and ↓ (-∞,1)

5 4 , 4

11. ↑ ( 4 , 2 ) and ↓ (0, 4 )

12.

13. ↑ (0, 6 ) and ↓ ( 6 , 2 )

(

7 2

,

1 4)

15.

(2,7); y-7 = 0 (3,6);y-6 = 0

16. x + 3y = 8 ; x + 3y = -8

17.

0.1924

18. -34.995

19.

0.32  cm2/sec

20

r 2 units, 2 r units, r2sq.units

24.

ab.max is 8 at x=4, ab

26. 24.96cm for equilateral triangle.

min.is -10 at x=-2

INTEGRATION LEVELWISE GRADED QUESTIONS LEVEL 1. 1. Evaluate:∫ 2. Evaluate:∫

ANS. .Ans.

3. Evaluate:∫ 4. Write a value of ∫ 5. .Evaluate:∫

.

Ans. . Ans.

)

6. Evaluate:∫

7. Evaluate: ∫ √

8. . Evaluate :∫

LEVEL 2. 1.Find f(x) satisfying the following :∫ Ans. F(x) = tan x 2.Evaluate : ∫

Ans.

3.Write the value of ∫ 4.

∫

5.Evaluate: ∫

Ans.

Ans. zero

.

6. Evaluate:∫

7. .Evaluate:∫

(

)

8. Evaluate.∫

LEVEL 3. 1. Evaluate :∫

Ans.

2. Evaluate:∫

Ans

3. Given ∫ (tan x + 1) sec x dx = exf(x) + c. then f(x) =? 4. Evaluate :∫ Ans.

Ans. Secx

5.Evaluate: ∫

5. Evaluate: ∫

7.. Evaluate ∫

using limit as sum.

6. Evaluate: ∫

7. Evaluate: ∫

8. Evaluate:∫

.

9. Evaluate: ∫

APPLICATIONS OF INTEGRALS 1. Find the area enclosed by circle the 2. Find the area enclosed by ellipse the 3. Find the area of the region in the first quadrant enclosed by x-axis, the line

and the

circle 4. Find the area of the region bounded by 5. Find the area bounded by the curves 6. Sketch the graph

and evaluate the integral

7. Find the area between the curves

and

.

.

8. Find the area lying above x-axis and included between the circle the parabola

and inside of

.

9. Find the area of the region enclosed between the two circles

10. Find the region by the curves

and

and

11. Find the area bounded by the curves

and

.

12. Using the method of the integration find the area of the region bounded by the triangle whose vertices are (-1,0), (1,3) and (3,2). 13. Using the method of the integration find the area of the region bounded by the lines and

.

14. Find area the of region 15. Find the area of the region 16. Sketch the curves and identify the region bounded by

and

1.Ans.π 2. Ans. πab 3. Ans. 4π 4. Ans. 5. Ans. 6. Ans. 9 7. Ans. 9.Ans. 15. Ans.

10. Ans. 16. Ans.

11. Ans.

12. Ans. 4 13. Ans.

14. Ans.

and

8. Ans.

Differential Equations 4:Marks Q1 . Form the differential equation of the family of Parabolas having vertex at origin and axis along positive y axis ? Q2. Find the particular solution satisfying the given condition of differential equation(X3+X2+X+1)dy/dx = 2X2+X :

y=1 when X=0

Q3 Solve : log(dy/dx) = ax+by Q4. Solve :dy/dx =( y-x)/(x+y) Q5. Solve :-xdy/dx + 2y = xcosx Q6 : Solve - dy/dx+secy = tanx (0≤x M or ax+by< M has a point in common with feasible region or not . Forget to write the final answer Reminding again and again Forget to attempt the value based question Reminding again and again

ERROR ANALYSIS INPROBABILITY

Sl.No. COMMON ERRORS COMMITTED 1 Unable to identify the question (whether independent events or Bayes’ theorem or Binomial distribution) 2 Difficulty in converting word problem into mathematical terms 3 Mistakes in identifying different ‘EVENTS’ in Bayes’ theorem 4 Mistakes in identifying the probability of different events in Bayes’ theorem 5 Computational mistakes Inability to find out the correct random variable Unable to form the probability distribution table Unable to identifY the values of n , p , q in binomial distribution Forget to write the final answer Forget to attempt the value based question

SUGGETED REMEDIES More practice of questions of various types . Drilling in conversion of different kinds of problems More practice of such questions Drilling in such problems More concentration and attention. More practice of such questions More practice of such questions More practice to be given Emphasis on writing the final answer. Reminding again and again

TIPS AND TECHNIQUES 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 i.e

gives

similarly second

row co-factors

gives

row co-factors

gives

for third

Tips and techniques Continuity and differentiability 1. 2. 3. 4.

Learn the basics of Limit , LHL ,RHL. Learn the concept of continuity A function is continuous when LHL=RHL =f(a) , at a Derivative of implicit function of the type xy+y2= tanx + y, stress the fact that xy is product of two functions. 5. For differentiation of Parametric form Start with y=f(x) dy/dx = f’(x) dy/dt= ? dy/dz= ? dy/dθ = ? again start with y=f(x) , where x=g(θ ) dy/dx= f’(x) d2y/dx2=f’’(x) dθ/dx 6. Rolle’s theorem … checking 3 conditions mandatory.

TIPS AND TECHNIQUES IN APPLICATIONS OF DERIVATIVES RATE OF CHANGE OF QUANTITIES 1. From given units like cm/sec.,cm2/sec , cm3/ sec. etc, identifying the given quantities and assess the possible answer to be found. 2. Taking the independent variable as t (time) when the independent variable is not mentioned in the question. 3. Work out recently asked Board Questions 4. Prepare a work sheet of Important Questions. 5. Conduct slip test from the questions of work sheet.

INCREASING AND DECREASING FUNCTIONS 1. Use of the number line for finding the various intervals and putting + and – sign over the intervals to show increasing and decreasing part of the function for polynomial and trigonometric functions. 2. Draw the table to describe the nature of f’(x) and f(x) in various intervals. 3. If f’(x) is a square function it is always positive and hence increasing. 4. Work out recently asked Board Questions 5. Prepare a work sheet of Important Questions. 6. Conduct slip test from the questions of work sheet. TANGENTS AND NORMALS 1. Taking f’ (xo) = slope of tangent at x = xo and

= slope of the normal at x= xo

2. Remembering the equation of a straight line as y = m x + c 3. Finding the point of contact of the tangent or normal with the curve when the function is in Parametric form with the given initial conditions. 4. Work out recently asked Board Questions 5. Prepare a work sheet of Important Questions 6. Conduct slip test from the questions of work sheet.

ERRORS AND APPROXIMATIONS 1. Remember the rules

(Appx.) and

2. Work out recently asked Board Questions 3. Prepare a work sheet of Important Questions. 4. Conduct slip test from the questions of work sheet.

= f(x) +

(Appx.)

MAXIMA AND MINIMA 1. If the function is of the form f(x) =√ , square the function and find maximum or minimum of the function g(x) and hence give the conclusion regarding the given function f(x). 2. Enough to check whether

< 0 for maximum and

> 0 for minimum instead of going for

actual substitution and long calculation at critical points. 3. Work out recently asked Board Questions 4. Prepare a work sheet of Important Questions. 5. Conduct slip test from the questions of work sheet.

INTEGRATION TIPS AND TECHNIQUES 1. Drilling of formulae (direct formulae, trig. Formulas,∑ ∑ etc.) has to be done. 2. Insist to write the relevant formulae as it carry 1 mark 3. To teach integration start with simple questions before starting Text book questions 1. Differentiation of fns ,and integration of same fns to be repeated in the initial teaching of the topic. 2. Direct substitutions related qns, more drilling can be given for slow learners 3. Integration by parts, ILATE can be used for choosing first and second fns 4. Slips test based on one or two concepts with variation in questions can be given periodically. 5. During remedial classes questions of level -1 must be worked out by the students. 6. HOW TO REMEMBER THE FORMULAE:

Make them to learn the dr. formulas, and for Nr. Formulas make them to repeat as below given. log |

1) LHS =

|

2) LHS = 3)

log |

|

7. Selected patterns from board qn. Papers can be solved at the end of the topic. 8. For recapitulation make the students to draw concept mapping .

 Make them to write LPP in the form  Objective function  Subject to the constraints  Non negativity restrictions  Insist to write non negativity restrictions  Train to draw the lines and mark the inequality region  Train to shade the feasible region  Train to deal with the feasible region with bounded as well as unbounded solution  Insist to write the answer and conclusion al last  Insist to attempt the value based question and to write the answer in a sentence , rather than writing in a single word.  Maximum practice to be given to different types of LPP as it is a sure question of 6 marks for the board examination. Make the student to practice thoroughly all the problems of NCERT text book first.  To work out maximum number of extra problems from various reference books and sample papers  To work out previous year board question papers and sample papers

 Train to identify the question (whether independent events or Bayes’ theorem or Binomial distribution)  Train to convert word problem to mathematical terms  Maximum practice to be given to Bayes’ theorem as it is a question of 6 marks  Train to identify the values of n , p and q for questions on binomial distribution  Insist to attempt the value based questions for both 4 marks and 6 marks and to write the answer in a sentence , rather than writing in a single word  Make the student to practice thoroughly all the problems of NCERT text book first.  To work out maximum number of extra problems from various reference books and sample papers  To work out previous year board question papers and sample papers.