KENDRIYA VIDYLAYA SANGTHAN NEW DELHI

क द र य व द य लय स गठननई द ल ऱ KENDRIYA VIDYLAYA SANGTHAN NEW DELHI Venue: KVS ZIET MYSORE Date: 15 th to 17 th July, 2014 RESOURCE MATERIALS CLASS XII(2014-15)(Mathematics) आ चलऱकल एव प रल णस स थ न ज आईट ब प र सक म पस, लसद ध थथनगर, म स र-570011 KVS Zonal Institute of Education and Training GITB Press Campus, Siddartha Nagar, Mysore Website: www.zietmysore.org, Email: kvszietmysore@gmail.com/zietmysore@rediffmail.com Phone: 0821 2470345 Fax: 0821 24785

क द र य व द य ऱय स गठन नईद ल ऱ KENDRIYA VIDYLAYA SANGTHAN NEW DELHI आ चलऱक ल क ष ए प रल क षण स स थ न म स र ZONAL MINSTITUTE OF EDUCATION AND TRAININGMYSORE प र 3-Day Strategic Action plan workshop आ 15th-1555((15 th to 17 th 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) एव प रल णस स थ न

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 15 th to 17 th 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 15 th July to 17 th 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 Name in English Design. K.V in English Region Phone No. E-mail Address 1 Mrs. V. Meenakshi Asstt.Commr. Regional Office Ernakulam 9496146333 minakviswa@gmil.com 2 Mr. E. Krishna Murthy Principal NFC Nagar,Ghatkesar Hyderabad 9989063749 krishnamurthykvs@gmail.com 3 Mr. T. Prabhudas Principal Malleswaram Bangalore 8762665990 t.prabhudas@rediffmail.com 4 Dr(Mrs.) S. Nalayini Principal Kanjikode Ernakulam 9446361186 dr.s.nalayini@gmail.com 5 Mr. M. Krishna Mohan Principal CRPF Hyderabad Hyderabad 9440865761 mohanmk58@yahoo.co.in 6 Mr. R. Sankar Principal No.2 Uppal Hyderabad 9491073600 ramisettysankar@gmail.com 7 Mr. E.N. Kannan PGT(Phy) BEML Nagar Bangalore 8762208431 kannaen2007@yahoo.com 8 Mr. D.B. Patnaik PGT(Bio) Railway Colony Bangalore 8971240593 dabbirubharath@gmail.com 9 Mr. K.S.V. Someswara Rao PGT(Phy) MEG & Centre Bangalore 9448708790 ksvsrao64@yahoo.com/ksvsrao@gmail.com 10 Mr. K.R. Krishna Das PGT(Maths) No.1 AFS Sambra Bangalore 8951648275 krishnadasdwd@gmail.com 11 Mr. G.N. Hegde PGT(Maths) Dharwad Bangalore 9448626331 gnhegde130@gmail.com 12 Dr. Vivek Kumar PGT(Chem) CRPF Yelahanka Bangalore 8970720895 kumar.v.vivek@gmail.com 13 Mr. RangaNayakulu.A PGT(Chem) Hebbal Bangalore 7899287264 androjula@gmail.com 14 Mrs. G.K. Vinayagam PGT(Bio) No.2 Belgaum Cantt. Bangalore 9448120612 vgayatri3@gmail.com 15 Mr. D. Rami Reddy PGT(Eco) Railway Colony Bangalore 9740398644 ramireddy_dondeti@yahoo.co.in 16 Mrs. T.M. Sushma PGT(Eco) Hebbal Bangalore 8762691800 t.msushma@gmail.com 17 Mrs. Asha Rani Sahu PGT(Maths) Mysore Bangalore 9902663226 asharanisahu@gmail.com 18 Mrs. Joan Santhi Joseph PGT(Chem) IS Grounds, Chennai Chennai 9940945578 joansanthi@gmail.com 19 Mrs. C.V. Varalakshmi PGT(Phy) AFS Avadi, Chennai Chennai 9003080057 c.venkatavaralakshmi@gmail.com 20 Mr. Siby Sebastian PGT(Maths) Gill Nagar Chennai 8056179311 siby_kavil@yahoo.co.in 21 Mr. S. Vasudhevan PGT(Chem) DGQA Complex Chennai 9444209820 vasu_sbm@yahoo.com 22 Mrs. Sathya Vijaya Raghavan PGT(Eco) Minambakkam Chennai 9445390058 sathyavijayaraghavan@gmail.com 23 Mr. S. Kumar PGT(Phy) No.1 Kalpakkam Chennai 8015374237 soorareddyk@gmail.com 24 Mrs. A. Daisy PGT(Bio) Minambakkam Chennai 9840764240 daisy.devakumar@gmail.com 25 Mrs. C.K. Vedapathi PGT(Bio) IIT Chennai Chennai 9841583882 ved.kviitc@gmail.com 26 Mrs. J. Uma PGT(Eco) Annanagar Chennai 9840988755 umakrkumar@gmail.com

27 Mrs. Sulekha Rani.R PGT(Chem) NTPC Kayamkulam Ernakulam 9745814475 sulekharanir@gmail.com 28 Mrs. Mary V. Cherian PGT(Bio) SAP Peroorkada Ernakulam 9447107895 maryvcherian@yahoo.co.in 29 Mrs. Susmitha Mary Robbins PGT(Phy) Kalpetta Ernakulam 9495528585 ssmth.robbins@gmail.com 30 Mr. Joseph K.A PGT(Eco) R.B Kottayam Ernakulam 9446369351 kallikattu31@rediffmail.com 31 Mrs. Jyolsna K.P PGT(Maths) No.1 Calicut Ernakulam 9447365433 jyolsnakp@gmail.com 32 Mrs. UshaMalayappan PGT(Eco) Kanjikode Ernakulam 9496519079 ushamalayappan@gmail.com 33 Mrs. Sujatha M. Poduval PGT(Bio) Keltron Nagar Ernakulam 9446494503 sujathampoduval@gmail.com 34 Mr. Prashanth Kumar.M PGT(Phy) Keltron Nagar Ernakulam 9400566365 mpknambiar@gmail.com 35 Mr. Sibu John PGT(Chem) Ernakulam Ernakulam 9544594068 sibu_jhn@yahoo.co.in 36 Mr. N.S. Subramanian PGT(Maths) Gooty Hyderabad 9490039741 nssss_math@yahoo.co.in 37 Mrs. Josephine Balraj PGT(Maths) No.1 AFA Dundigal Hyderabad 9440066208 balraj_jb@yahoo.com 38 Mr. B. Sesha Sai PGT(Phy) AFS Hakimpet Hyderabad 9912384681 seshasai2005@yahoo.com 39 Mr. V.V..S.Kesava Rao PGT(Phy) Gachibowli Hyderabad 9490221144 vvskr2007@gmail.com 40 Kum. SanuRajappan PGT(Eco) Gachibowli Hyderabad 9640646189 sanu_rajappan@yahoo.com 41 Mr. M.T. Raju PGT(Bio) AFS Begumpet Hyderabad 9652680800 mtrajuhyderabad@gmail.com 42 Mr. D. Ashok PGT(Chem) CRPF Hyderabad Hyderabad 9618012035 saiashokdomakonda@gmail.com 43 Mr. D. Saidulu PGT(Chem) AFS Begumpet Hyderabad 9908609099 saidulu_dongari@yahoo.com 44 Mrs. Surya KumariBarma PGT(Eco) AFS Begumpet Hyderabad 9441779166 tell_rama@rediffmail.com 45 Dr. K.V. Rajendra Prasad PGT(Bio) NTPC Ramagundam Hyderabad

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 15/07/14 TUESDAY 16//07/14 WEDNESDAY 17/07/14 THURSDAY SESSION 1 (09:00-11:00 AM) Inauguration Preparation of Value based questions. Tips and techniques (Chapter wise) in teaching learning process Insight into VUE& Concept Mapping Presentation of Value Based questions. Presentation of tips and techniques. SESSION 2 (11:15-01:00 PM) Presentation of Concept Mapping Preparation of 3 levels of question papers. Subject wise specific issues 11.00-11.15 Tea break Strategic action plan to achieve quality result. Preparation of 3 levels of questions Consolidation of material 1.00-2.00 Lunch break SESSION III ( 02:00-03:30PM) Review of Study Material Presentation of 3 levels of questions. Consolidation of material 3.30-3.45 Tea Break SESSION IV (03:45-05:30 ) Presentation of fine-tuned study material Error analysis and remediation. Un attempted questions in tests and examinations Valedictory Function 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

सह यकस मग र २०१४-२०१५ SUPPORT MATERIAL 2014-2015 क १२ 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 141 17. Bibliography 170 6

Level I, Level II & Level III indicate the difficulty level of questions 7

CHAPTER I RELATIONS&FUNCTIONSSCHEMA TIC DIAGRAM Topic Concepts Degreeof impo1tance Relations& Functions (i).domain,codomain& Rangeofarelation References NCERTTextBookXII Ed.2007 * (PreviousKnowledge) (ii).typesofrelations *** ExI.IQ.No-5,9,12,14 (iii).one-one,onto&inverse ofafunction *** Ex1.2Q.No-7,9 Example12 (iv).compositionoffunction * Ex1.3QNo-3,7,8,9,13 Example25,26 (v).binaryoperations *** MiscExample45,42,Misc.Ex2,8,12,14 Ex1.4QNo-5,9,II SOMEIMPORTANTRESULTS/CONCEPTS TYPES OF RELATIONS A relation R in a set A is called reflexive if (a, a) R for every a A. A relation R in a set A is called symmetric if (a 1, a 2 ) R implies that (a 2, a 1 ) R, for all a 1, a 2 A. A relation R in a set A is called transitive if (a 1, a 2 ) R, and (a 2, a 3 ) R together imply that (a 1,a 3 ) R, for all a 1, a 2, a 3 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 (A i ) called partitions or subdivisions of X satisfying the following conditions: All elements of A i are related to each other for all i No element of A i is related to any element of A j whenever i j A i A j = X and A i A j = Φ, i j These subsets (A i ) 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 BissaidtobeafunctionifeveryclementofAiscorrelated to a uniqueelementinb. *Aisdomain * Biscodomain * Forany xelement of A,function f correlatesittoanelementinb,whichisdenotedbyf(x)andiscalledimageofxunder/.againify=f( x),thenxiscalledaspre-imageofy. * Range={f(x)Ix A}. Range Co domain ** Composite function ** Let f: A B and g: B C be two functions. Accordingly, the composition of f and g is denoted bygof and is defined as the function gof: A C given by gof(x) = g(f(x)), for all x A. 13

3. ShowthattherelationRdefinedinthesetAofalltrianglesasR={(T 1,T 2 ):T1issimilartoT 2 },isequiv alencerelation.considerthreerightangledtrianglest1withsides3, 4,5, T 2 withsides5,12,13andt 3 withsides6,8,i0.whichtrianglesamongt 1,T 2 andt 3 arerelated? 4. IfR 1 andr 2 areequivalencerelationsinaseta,showthatr 1 R 2 isalsoan equivalencerelation. 5. LetA=R-{3}andB=R-{l}.Considerthefunctionf:A Bdefinedbyf(x)= Isfone-oneandonto?Justifyyouranswer. 6. Considerf: R + [-5, )givenbyf(x)=9x 2 +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) =x 2 +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

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 Concepts Degreeofi mportance Matrices& Determinants (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 OR A=[a.ij], where i=1,2,...,m;j=1,2,...,n. ami am2. amnmxn * Row Matrix:A matrix which has one row is called row matrix. *Column Matrix: A matrix which has one column is called column matrix *SquareMatrix:A matrix in which number of rows are equal to number of columns, is called a square matrix * Diagonal Matrix:Asquare matrix is called!adiagonal Matrix if all the elements, except the diagonal elements are zero * Scalar Matrix: A square matrix is called scalarmatrix if all the elements, except diagonal elements are zero and diagonal elements are same non-zero quantity. * Identity or UnitMatrix: A square matrix in which all the non diagonalelements are zero and diagonal 23

elements are unity is called identity or unit matrix 24

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[ ] (2) IF A=* + and =ka find k (ii)level II (1)If[ ] =* + find p (2) Give examples of a square matrix of order 2 which is both symmetric and skew symmetric (3)Find the value of x and y if [ ] =* + (4)If A =* +, find 0, when A+ =I (ii)level I (1) If A= write the minor of the element (2) If is the cofactor of of find (iii)level 1 32

(1) If A is a square matrix such that =A then write the value of -3A (2) If A =* + and B =* +, then verify that =. LEVELIII (1) If =[ ] and B = [ ] Find (2) Using elementary transformations, find the inverse of the matrix [ ] (3) The management committee of a residential colony decided to award some of its members (say x) For honesty,some(say y)for helping others and some others(say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33 If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others,using matrix method find the number of awardees for each category. apart from these values,namely, honesty,cooperation and super vision, suggest one more value which the management of the colony must include for awards (iv)levelii (1) If A is asquare matrix of order 3 such that =225 Find (V) LEVELI (1) Evaluate (2)Find the value of QUESTIONS FOR SELF EVALUATION 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- 1 exists, so equations have a unique solution. X= A- 1 B 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

ANSWERS TO 43

11. If the length of three sides of a trapezium, other than the base is equal to 10cm each, then find the area of trapezium when it is maximum. Ans.75 sq.cm 12. Verify Role s theorem for the function f given by f(x) = (sinx cosx) on [, ]. 13. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h 50

and semi-vertical angle is tan 2. 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

CHAPTER VII 54

Log sinx dx 59

ADDITIONAL QUESTIONS (Indefinite & Definite Integrals) 1. Evaluate dx ans : ( logi sin 2 x+b 2 cos 2 x I +C) 2. Evaluate dx ans : [(a+bx) -2alogIa+bxI - ] 3. Evaluate ans : + C 4. Evaluate ans: tanx + C 5. Evaluate [secx + log(secx+tanx)]dx ans: log(secx+tanx) + C 6.Evaluate dx ans: - log + log +C 7.Evaluate dx ans: [ ] +C 8.Evaluate dx ans: ta [tanx+ ]+C 9. 10 11 12 61

2 Log 2 65

CHAPTER VIII 66 67

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 = cos 2 x in [0, π ] and find the area enclosed by the curve, the line x=0, x= 69

ANSWERS 70

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 2 d y 2 dx 4 dy cos 0 dx 2. Show that y=3e 2x + 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 x 2 y + xy + y =0 4. Obtain the differential equation of family of parabola having vertex at the origin and axis along the positive direction of x-axis LEVEL III 5.Obtain the differential equation of family of ellipses having foci on y-axis and centre at the origin. 6.Find the differential equation of system of concentric circles with centre at (1,2) dy 7.Solve = ( 1 + x 2 )( 1 + y 2 ) dx dy 8.Solve =e -y cos x Given that y(0) =0 dx 9.Solvecos ( ) = a (a Ɛ R) ; y=2 when x=0 10. (x 3 +x 2 +x+1) =2x 2 +x ; y=1 when x =0 11. Solve dy dx x 2y x y dy 12.Solve = dx y x 3 3 2 2x y 2 2y x 13.Solve y dx + x log ( ) dy 2x dy = 0 76

14.Solve y x y e dx = ( x e y +y) dy x dy dx 15. Solve cos 2 x y tan x 2 dy 2 2 16.Solve x 1 2xy x 2 x 1 dx 3x 17.Solve 1 x dy e x 1 2 y dx 18.Solve ( 1 + y + x 2 y) dx + ( x + x 3 ) dy = 0 e 2 x y dx x dy 19.Solve 1 x, x 0 ; when x=0, y=1 Answers 2.Ans: 3: =0 4: 77

4. =Sinx+1 5.Siny- logx=c6: (x-1) =C 2.,: 3. 4.: 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 (x- 11)/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 x- y+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 Concepts Degree of References Importance NCERT Book Vol. II Linear (i)lpp and its ** Articles 12.2 and 12.2.1 Programming Formulation (ii)graphical method of ** Article12.2.2 Solving LPP (bounded Solved Ex. 1 to 5 unbounded solutions) Q. Nos 5 to 8 EX.12.1 (iii)diet Problem *** Q. Nos 1,2 and 9 Ex. 12.2 Solved Ex. 9 Q. Nos 2and3 Misc. Ex. (iv)manufacturing *** Solved Ex. 8 Q. Nos 3,4,5,6,7 of Ex.12.2 Solved EX.10 Q. Nos4 &10 Misc. Ex. (v)allocation Problem ** Solved Example 7Q. No 10 Ex.12.2, Q. No 5 &8 Misc. Ex. (vi)transportation * Solved EX.11 Q. Nos 6 &7 Misc. Ex. (vii)miscellaneous Problems ** Q. No 8 Ex. 12.2 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.) From To A B P 5 4 Q 4 2 R 3 5 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)]

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) Questions for self evaluation 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. 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. 8. Minimum cost Rs.1350 at (5, 3) The uniform of a student should be a) well pressed b) neat and tidy c) properly stitched d) 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 Machines I II III A 12 18 6 B 6 0 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 FROM A B P 5 4 Q 4 2 R 3 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 6 P(X) 1/15 2/15 3/15 4/15 5/15 Mean=14/3 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 2 P(X) : 9/25 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 (ii) Hard work (iii) Good memory (iv) Well time management (v) Writing skills] 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 1 2 P(X) : 9/16 6/16 1/16 Mean = ½. No, it may be good once or twice but not forever. Honesty pays in a long run. ]

SYLLABUS