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J{UVMATHEMATICS
(Revised)
H$Zm©Q>H$ gaH$ma
{hÝXr ‘mܶ‘Hindi Medium
gmVdr H$jmSeventh Standard
^mJ - 1Part - I
KARNATAKA TEXT BOOK SOCIETY (R).,
100 Feet Ring Road, Banashankari 3rd Stage,Bangalore - 560 085
7
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PrefaceThe Textbook Society, Karnataka, has been engaged in producing
new textbooks according to the new syllabi which in turn are designed on NCF - 2005 since June 2010. Textbooks are prepared in 12 languages; seven of them serve as the media of instruction. From Standard 1 to 4 there is the EVS, mathematics and 5th to 10th there are three core subjects, namely, mathematics, science and social science.
NCF - 2005 has a number of special features and they are: connecting knowledge to life activities. learning to shift from rote methods. enriching the curriculum beyond textbooks. learning experiences for the construction of knowledge. making examinations flexible and integrating them with
classroom experiences. caring concerns within the democratic policy of the country. making education relevant to the present and future needs. softening the subject boundaries-integrated knowledge and the
joy of learning. the child is the constructor of knowledge.The new books are produced based on three fundamental approaches
namely, Constructive approach, Spiral approach and Intergrated approach.
The learner is encouraged to think, engage in activities, master skills and competencies. The materials presented in these books are integrated with values. The new books are not examination oriented in their nature. On the other hand they help the learner in the all round development of his/her personality, thus help him/her become a healthy member of a healthy society and a productive citizen of this great country, India.
Mathematics is essential in the study of various subjects and in real life. NCF 2005 proposes moving away from complete calculations, construction of a framework of concepts, relate mathematics to real life experiences and cooperative learning. Many students have a maths phobia and in order to help them overcome this phobia, jokes, puzzles,
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riddles, stories and games have been included in textbooks. Each concept is intoduced through an activity or an interesting story at the primary level. The contributions of great Indian mathematicians are mentioned at appropriate places.
We live in an age of science and technology. During the past five decades man has achieved great things and realized his dreams and reached pinnacle of glory. He has produced everything to make life comfortable. In the same way he has given himself to pleasures and reached the stage in which he seems to have forgotten basic sciences. We hope that at least a good number of young learners take to science in higher studies and become leading scientists and contribute their share to the existing stock of knowledge in order to make life prosperous. Ample opportunity has been given to learners to think, read, discuss and learn on their own with very little help from teachers. Learning is expected to be activity centered with the learners doing experiments, assignments and projects.
In Social Science especially in Standard 5 the first chapter deals with the historical, geographical, cultural and local study of the division in which learners live. Chapters on sociology, business studies and commerce are introduced in Standard VIII as per the guidelines of NCF-2005. A lot of additional information is given through box items. Learners are encouraged to work towards construction of knowledge through assignments and projects. Learning load of memorizing dates has been reduced to the minimum. Life values have been integrated with content of each chapter. The Textbook Society expresses grateful thanks to the Chairpersons, writers, Scrutinisers, artists, staff of DIETs and CTEs and the Members of the Editorial Board and printers in helping the Textbook Society in producing these textbooks.
Prof. G.S. MudambadithayaCoordinator
Curriculum Revision and TextbookPreparation
Karnataka Textbook Society®Bengaluru, Karnataka
NagendrakumarManaging Director
Karnataka Textbook Society®Bengaluru, Karnataka
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Chairperson's LetterDear All,NCF 2005 intends that the aim of learning Mathematics to be of higher value rather than mere learning
algorithms and this is stated as learning of Mathematics is 'Mathemetisation. To achieve this objective our educational system, that is administration, class, school, society and persons in these agencies has to provide opportunities to the child to gain excellence experiences and fecilitate to construct knowledge himself.
The logical thinking adopted to solve a problem is to be of higher significance rather than knowing the solution to the problem. This develops rational and logical thinking among learners and enables them to explore their own method or approach of finding the solution and keeps them to be active participants. To make it possible, teachers have to become friendly facilitators to provide learning opportunities to the learners; as well as encourage pupils to learn with co - operation, in peer groups.
'Mathematics' is a challenge at 7th standard because it has to be nearer to the experiences and environment of learner and the learner has to comprehend the logical process and the abstract idea/concept.
Scope for exploration and creation is required in Mathematics instead of the rote problems and complicated calculations. Students have to be encouraged to solve the problems in different ways. To achieve this, previous learning is linked with present topics of learning Model activities, activities that can be done by the students and activities that have to be performed by the students are given to the possible extent Illustration, figures, laws/ principles and worked (model) problems are given to facilitate better comprehension of the subject. Additional information / issues are included here and there to enrich learning. To make learning easier and meaningful simple language along with wider scope and social situations are used. For reinforcement of each unit exercises, based on knowledge, understanding, skills and application- are given as for as possible. Our Committee is are grateful to the chief co-ordinator, the managing director, the joint director and programme co-ordinator of karnataka textbooks society for providing an opportunity and responsibility of producing this textbook through which we have the pleasure to reach and serve larger educational community. We thank the editorial board and the scrutinizes for the valuable guidance and suggestions the members and DIETS involved in 'Try - out' for the feedback. All of this is attended to improve the text.
I am thankful to the members, translators, artists of this committee, technical assistants and other departmental, non departmental who have assisted me in this service .
It is tried to make this text to be free from doubts, confusions, ambiguity and printing mistakes. In spite of it, I request you to kindly to communicate with me for any such things. Any suggestions for the improvement will be acknowledged whole heartedly. Please, share your opinion regarding the quality of the textbook with us and the department.
Yours truly,N. Kaleshwara Rao
Chair Person
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Text Book CommitteeChair Person :
Sri Kaleshwar Rao Bagoor 3 Santhasa, 1st Cross, Aravinda Marga, J.P. Nagar, IInd Phase, Bengaluru.
Members :
Smt B. Keerthi Principal, Siliconcity, P. U. College, Konanakunte cross, Bengaluru -62.
Sri Subramanya Bhatt Assistant Teacher, K.V.S.M High school, Kanchana, Puttur (T)
Sri Sadananda Kumar Government Girls P. U. College, Hampi Road, Hosapete Ballari District- 583218.
Sri Pramod G. Kulkarni V. B. Darbar P. U. College, Vijayapura.
Sri C. L. Bhaskar Assistant Proffessor Vijaya Teachers College, Jayanagar 4th Block Bengaluru - 56001.
Smt Renuka Assistant Teacher, Goverment P. U. College composite Kamalapur Taluk Hosapete Ballari District.
Sri Jangi Art Teacher. DSERT Bengaluru.Scrutinizers :
Prof. Prabhakar R. V. Deen Vijaya College Jayanagar and Assistant Secretary B. H. S. H. E. S. Jayanagar 4th Block. Bengaluru
Dr R. Latha Kumari Principal, Sanjaya Gandhi College of Education, cholanagar, Hebbal Bengaluru.
Editorial Committee :Dr Ravindra Former NCERT Director Arehalli BSK 3rd stage Bengaluru - 61.
Dr Upadya B. S. Lecturer and Head of Department Maths, Regional Institute of Mysuru.
Dr Prasad S. V. Lecturer RIE Mysuru.
Dr Sharad Sure Assistant Proffessor Aziam Premji university PES School of Engi-neering Campus Komappana Agra Hara Bengaluru -10
Translation Committee :R. Shanwaz Begam A.M, Sri DPH High School Bengaluru -47
Vilas Pudale Sri Shanthinath Hindi High School Ganthi Keri Hubballi 580020
Chief - Coordinators :Sri G. S. Mudambaditaya Co-ordinator, Curriculum revision and Text Book Preparation.
Guidance :Sri Nagendra Kumar Managing Director, Karnataka, Text Book Society.Sri Panduranga Deputy Director(incharge), Karnataka, Text Book Society.Programme Coordinators :Smt Vijaya Kulkarni Assistant Director, Karnataka Text Book Society.Smt Prema B. R. Technical Assistant ,Karnataka Text Book Society.
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About the Review of Textbooks
Honourable Chief Minister Sri Siddaramaiah who is also the Finance Minister of Karnataka, in his response to the public opinion about the New Textbooks from standard one to ten, announced, in his 2014-15 budget speech of constituting an expert-committee, to look into the matter. He also spoke of the basic expectations there in, which the Textbook experts should follow: “ The Textbooks should aim at inculcating social equality, moral values, development of personality, scientific temper, critical acumen, secularism and the sense of national commitment”, he said.
Later, for the production of the Textbooks from class one to tenth, the Department of Education constituted twenty seven committees and passed an order on 24-11-2014. The committees so constituted were subject and class-wise and were in accordance with the standards prescribed. Teachers who are experts in matters of subjects and syllabi were in the committees.
There were already many complaints, and analyses about the Textbooks. So, a freehand was given in the order dated 24-11-2014 to the responsible committees to examine and review text and even to prepare new text and review if necessary. Afterwards, a new order was passed on 19-9-2015 which also give freedom even to re-write the Textbooks if necessary. In the same order, it was said that the completely reviewed Textbooks could be put to force from 2017-18 instead of 2016-17.
Many self inspired individuals and institutions, listing out the wrong information and mistakes there in the text, had send them to the Education Minister and to the ‘ Textbook Society’. They were rectified. Before rectification we had exchanged ideas by arranging debates.
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Discussions had taken place with Primary and Secondary Education ‘Teacher Associations’. Questionnaires were administered among teachers to pool up opinions. Separate meeting were held with teachers, subject inspectors and DIET Principals. Analytical opinions had been collected. To the subject experts of science, social science, mathematics and languages, textbooks were sent in advance and later meeting were held for discussions. Women associations and science related organistation were also invited for discussions. Thus, on the basis of all inputs received from various sources, the textbooks have been reviewed where ever necessary.
Another very important thing has to be shared here. We constituted three expert committees. They were constituted to make suggestions after making a critical study of the text of science, mathematics and social science subjects of central schools (N.C.E.R.T), along with state textbooks. Thus, the state text books have been enriched basing on the critical analysis and suggestions made by the experts. The state textbooks have been guarded not to go lower in standards than the textbooks of central school. Besides, these textbooks have been examined along side with the textbooks of Andhra, Kerala, Tamil Nadu and Maharashtra states.
Another clarification has to be given here. Whatever we have done in the committees is only review, it is not the total preparation of the textbooks. Therefore, the structure of already prepared textbooks have in no way been affected or distorted. They have only been reviewed in the background of gender equality, regional representation, national integrity, equality and social harmony. While doing so, the curriculum frames of both central and state have not been transgressed. Besides, the aspirations of the constitution are incorporated carefully. Further, the reviews of the committees were once given to higher expert committees for examination and their opinions have been effected into the textbooks.
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Finally, we express our grateful thanks to those who strived in all those 27 committees with complete dedication and also to those who served in higher committees. At the same time, we thank all the supervising officers of the ‘Textbook Society’ who sincerely worked hard in forming the committees and managed to see the task reach it’s logical completion. We thank all the members of the staff who co-operated in this venture. Our thanks are also due to the subject experts and to the associations who gave valuable suggestions.
Narasimhaiah
Mangaging Director Karnataka Textbook Society
Bengaluru.
Prof. Baraguru Ramachandrappa Chairman-in-Chief
Textbook Review Committee Bengaluru.
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Text Books Review CommitteeChairman-in-chief.Prof. Barguru Ramchandrappa, State Review Committee, Karnataka
textbooks Society®, Bengaluru. Review Committee
ChairpersonDr. Narasimhamurthy S.K. Professor and Chairman, Department
of Mathematics, Kuvempu University, Shankaraghatta-577 451. Shivamogga
MembersDr. B . Chaluvraju, Professor, Department of Mathematics,
Bengaluru University, Bengaluru.Sri. B. K. VishwanathRao, Rtd., Principal, No.94, ''Prashanthi'', 30th
Cross, BSK 2nd Stage, Bengaluru.Sri Narasimha murty G. N., ‘Beladingalu’ No.23/1,5th cross, Hosalli,
Bengaluru.Sri Shankarmurthy M.V. Rtd Headmaster, Sarvodaya High-school,
BengaluruSri H.N.Subbarao, Headmaster, Sadvidya Highschool,
N.S.Road, Mysuru.Smt S.S. Tara, Headmistress, Govt. High School,
Mavattur, K.R. Nagar taluk, Mysuru Dist,Smt Sushma NagarajRao, High School Teacher, Govt. Higher Primary
School, RamnagarSri Shrinath Shastri, Kannada Ganak Parishat, Chamrajpete,
Bengaluru.High Power CommitteeDr.Kashinath Biradar, Plot No.7, Gangasiri, Jayanagar,
Kalaburgi - 585 105.Smt. L. Padmavati, Vice-principal, Empress Girls
High-school, Tumkur.Sri T Gangadharaiah, Associate Professor, Department of
Mathematics, Govt. women’s college, Kolar Translators Sri. Mahadev K. Jadhav A.M. Dayanand Hindi Vidyalaya H.S. Shaha
Bazar, Kalaburgi.Smt. Shanwaz Begum, A.M. Sri VSK H.S. Srinivasnagar, Bengaluru.Chief Advisors Sri Narasimaiah, Managing Director, Karnataka Textbooks
Society®, Banashankari 3rd stage, Bengaluru-85.
Smt Nagamani C. Deputy Director Karnataka Textbooks Society®, Banashankari 3rd stage, Bengaluru-85.
Programme co-ordinator:Smt. Vijaya Kulkarni, Asst.Director, KTBS, Banashankari
3rd stage, Bengaluru-85.
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^mJ - 1
efJe<e³e metef®e
¬eÀ. mebK³ee IeìkeÀ He=<þ
1 HetCee¥keÀ 1 - 24
2 efYeVe 25 - 49
3 Heefjces³e mebK³eeSB 50 - 77
4 yeerpeer³e J³ebpekeÀ 78 - 109
5 keÀesCe 110 - 134
6 jsKeeSB 135 - 158
7 ef$eYegpe 159 - 181
8 meceefceefle 182 - 203
GÊej ceeefuekeÀe 204 - 214
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IeìkeÀ - 1
HetCee¥keÀ
Fme IeìkeÀ kesÀ DeO³e³eve kesÀ yeeo DeeHe:
IeveelcekeÀ HetCee¥keÀ keÀes IeveelcekeÀ HetCee¥keÀ mes, Deewj $eÝCeelcekeÀ HetCee¥keÀ mes iegCee keÀj mekeWÀies
$eÝCeelcekeÀ HetCee¥keÀ keÀes $eÝCeelcekeÀ HetCee¥keÀ mes iegCee keÀj mekeWÀieW,
mebK³eejsKee Hej iegC³e, iegCekeÀ Deewj iegCeveHeÀue DebefkeÀle keÀj mekeWÀieW,
HetCee¥keÀeW keÀer leguevee keÀj mekeWÀies, efo³es HetCee¥keÀeW ceW mes keÀewvemee íesìe keÀewvemee ye[e Hen®eeve mekeWÀies,
iegCee keÀjves ceW mener efJeOeeve DevegmejCe keÀj mekeWÀieW leLee mebK³eeDeeW kesÀ iegCeveHeÀue Deewj Yeeie ueieeves ceW mebyebOe %eele keÀj mekeWÀies,
SkeÀ HetCee¥keÀ keÀes otmejs HetCee¥keÀ mes Yeeie ueiee mekeWÀieW,
SkeÀ HetCee¥keÀ keÀes Metv³e mes Yeeie ke̳eeW Yeeie veneR ueiee mekeÀles mecePe mekeWÀies,
DeeJejCe, ¬eÀceefJeefve³ece, meen®e³e& Deewj efJelejCe iegCeeW mes Heefjef®ele nes pee³eWieW~
megnemeerveer ves SkeÀ iegueeye kesÀ HeewOes Hej HeeB®e iegueeye ueies osKeleer nw~ Gmeves 3 iegueeye efvekeÀeue
efue³es~ yeleeFS, HeewOes Hej efkeÀleves iegueeye yeekeÀer nQ?
nue :
HeewOes Hej ueies iegueeyeeW keÀer kegÀue mebK³ee = 5
efvekeÀeues ngS iegueeyeeW keÀer mebK³ee = 3
HeewOes Hej yeekeÀer iegueeyeeW keÀer mebK³ee = 5 - 3 = 2
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IeveelcekeÀ HetCee¥keÀ, $eÝCeelcekeÀ HetCee¥keÀ Deewj Metv³e keÀes efceueekeÀj HetCee¥keÀ keÀnles nwb~ nyUmªH$ H$m ì`dH$bZ Ed§ g§H$bZ H¡$go H$aVo h¡§ &
GoenjCe 1 : [aVm SkeÀ Kesue Kesuevee ®eenleer nw~ Jen oeefnveer Deesj 3 keÀoce ketÀoleer nw~ Deewj yeeo ceW 5 keÀoce yee³eeR Deesj ketÀoves keÀe efveM®e³e keÀjlee nw~ ketÀoves keÀer jerefle keÀes mebK³ee jKee Hej yeleeFS.
³eefo nce [aVm keÀer Henueer efmLeefle keÀes Metv³e ceeveles nQ, lees GmekesÀ íueebie keÀes mebK³ee jsKee Hej oMee& mekeÀles nwb~ Jen keÀneB HengB®eleer nw?
0 1 2 3
-2 2-1 13 30
0
3 + (-5) = -2
³eeo jefKeS :
♦ efieveleer/ mJeeYeeefJekeÀ mebK³eeSW IeveelcekeÀ HetCee¥keÀ nesleer nw~
♦ Metv³e ³en mJeeYeeefJekeÀ mebK³ee veneR nw~
♦ mJeeYeeefJekeÀ mebK³ee Deewj `0' keÀes efceueekeÀj ``HetCe& mebK³ee'' keÀnles nQ~
♦ HetCe¥keÀeW ceW HetCe& mebK³ee Deewj $eÝ=CeelcekeÀ HetCee¥keÀ meceeefJe<ì nesles nw~
DeY³eeme : 1.1
I. efvecveeW keÀesmejue keÀerefpeS:
a) 8 + 5 b) 2 + (-7)
c) (- 3) + (4) d) (– 7) + (- 2)
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HetCee¥keÀesb keÀe iegCeveHeÀue :
IeveelcekeÀ HetCee¥keÀ keÀes OeveelcekeÀ HetCee¥keÀ mes iegCee keÀjvee~
efkeÀmeer mebK³ee mes ceW keÀesF& mebkesÀle peg[e veneR nw lees Gmes OeveelcekeÀ ceeveles nQ~
iegCeveHeÀueve ke̳ee nw?
iegCeveHeÀueve oesnjerle pees[vee nw~ efvecve GoenjCe keÀe efvejer#eCe keÀerefpeS~
GoenjCe 1 : Yejle ves SkeÀ otkeÀeve mes 3 HeWefmeue kesÀ ef[yyes IarXo~ Òel³eskeÀ {S>ã~o ceW 6 Hebefmeue
nQ~ Yejle kesÀ Heeme kegÀue efkeÀleves Hesbefmeue nwb?
: 6 + 6 + 6 = 18 (³eesieHeÀue)
mebK³ee jsKee Hej Fmes kewÀmes oMee&³esbies?
0 6 12 18
mebK³ee jsKee Hej efoKee³es ieS leerj ef®evn O³eeve mes osefKeS~ mebK³ee jsKee Hej 3 leerj nw Òel³eskeÀ
leerj 6 Deevele keÀjlee nQ~ leermeje leerj 18 HengB®elee nw~
FmeefueS 6 × 3 = 18 ³en mebK³ee jsKee Hej efve©efHele efkeÀ³ee ie³ee nw~
GoenjCe 2 : 5 efce$e SkeÀ otkeÀeve Hej ieS~ Òel³eskeÀ ves 12 Deece KejeroW~ kegÀue efkeÀleves Deece GvneWves KejeroW h¡?
× =
Fmes peeve ueerefpeS:
HetCee¥keÀeW keÀes metef®ele keÀjves Z De#ej GHe³eesie keÀjles nQ~ peneuesve (Zahlen) pece&veer Meyo HetCee&keÀeW keÀes metef®ele keÀjlee nw~
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SkeÀ IeveelcekeÀ Deewj $eÝCeelcekeÀ HetCee¥keÀ keÀe iegCeveHeÀue:
efveOeer keÀes metKes HeÀue (Dry Fruits) yengle efÒe³e Les~ GmekeÀer ceeB ves kegÀí yeeoece SkeÀ ef[yyes ceW jKekeÀj 3 efove kesÀ efueS yeenj ieeBJe ®eueer ieF&~ efveOeer ves Òeefleefove 2 yeeoece Kee³eW~ Òeefleefove yeeoeceeW keÀer mebK³ee 2 mes Ieìles ie³es~ peye GvekeÀer ceeB Iej ueewìer, ef[yyee Keeueer Lee~ ef[yyes ceW efkeÀleves yeeoece Les?
DeeF&S, nce Ieìleer mebK³ee keÀes $eÝCeelcekeÀ mebK³ee mes metef®ele keÀjW~
-2, -2, -2 Deye Fve HetCee¥keÀesb keÀes peesef[S~
(-2) + (-2) + (-2) = -6
∴ 3 × (-2) = -6
3 efoveeW ceW efveOeer ves 6 yeeoece Kee³eW~ FmeefueS ef[yyes ceW 6 yeeoece Les~
Fmes nce mebK³ee jsKee Hej efve©efHele keÀjW~
-6 -5 -4 -3 -2 -1 0-7
(-2) × 3 = -6
GoenjCe 1 : 5 × (-4) %eele keÀerefpeSnce (-4) kesÀ 5 mecetn Fmelejn ues mekeÀles nQ
-20 -16 -12 -8 -4 0 4 8
1 × - 4 = - 4 2 × - 4 = - 8 3 × - 4 = - 12 4 × - 4 = - 16 5 × - 4 = - 20
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GoenjCe 2 : nce Fmes Òe³elve keÀjW! 4 × (-5) = ?
-20 -15 -10 -5 0 5 10
4 × (-5) = (-20)
³eefo nce, SkeÀ IeveelcekeÀ HetCee¥keÀ keÀes $eÝCeelcekeÀ HetCee¥keÀ mes iegCee keÀjles nQ lees SkeÀ $eÝCeelcekeÀ mebK³ee ÒeeHle nesleer nw~
GoenjCe 3 : (+ 5) × (-6) = ?
DeLee&le +5 × (-6) = -30
DeLee&le 5 iegCee (-6)
5 × (-6) = (-6) + (-6) + (-6) + (-6) + (-6) = - 30
cegPes ve IeveelcekeÀ ef®evn nw, vee $eÝCeelcekeÀ~ ceQ ef®evn mes efyeuekegÀue cegkeÌle ntB~
ke̳ee cegPes keÀesF& ef®evn
nw?
ne! ne ne keÀesF& ef®evn veneR
$eÝCeelcekeÀ HetCee¥keÀ keÀes IeveelcekeÀ HetCee¥keÀ mes iegCee keÀjvee :$eÝCeelcekeÀ HetCee¥keÀ keÀes IeveelcekeÀ HetCee¥keÀ mes iegCee keÀjves mes ke̳ee neslee nw?
Fme vecetves keÀes osefKeS : 5 × 4 = 20 4 × 4 = 16 = 20 - 4 3 × 4 = 12 = 16 - 4 2 × 4 = 8 = 12 - 4 1 × 4 = 4 = 8 - 4 0 × 4 = 0 = 4 - 4-1 × 4 = 0 - 4 = - 4- 2 × 4 = - 4 - 4 = -8- 3 × 4 = - 8 – 4 = -12 - 4 × 4 = - 12 – 4 = -16- 5 × 4 = - 16 - 4 = -20 (- 5) × 4 = - 20
Fmelejn nceW, (- 5) × 4 = -20 ÒeeHle neslee nw~
ke̳ee legce peeveles nes meye mes ye[er $eÝCeelcekeÀ mebK³ee keÀewvemeer nw?
neB, -1 nesleer nw,efce$e
Bgo Om{ZE
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peye nce $eÝCeelcekeÀ HetCee¥keÀ keÀes IeveelcekeÀ HetCee¥keÀ mes iegCee keÀjles nQ, lees nce Henues HetCe& mebK³eeDeeW keÀes iegCee keÀjles nwb Deewj iegCeveHeÀue keÀes $eÝCeelcekeÀ HetCee¥keÀ ÒeeHle neslee nw~
GoenjCe 1: (- 6) × (+ 7) = ?
Hejce cetu³eeW keÀes iegCee keÀerefpeS6 × 7 = 42
5 × 7 = 35 = 42 – 7
4 × 7 = 28 = 35 – 7
3 × 7 = 21 = 28 – 7
2 × 7 = 14 = 21 – 7
1 × 7 = 7 = 14 – 7
0 × 7 = 0 = 7 – 7
-1 × 7 = 0 -7 = -7
-2 × 7 = -7 – 7 = -14
-3 × 7 = -14 – 7 = -21
-4 × 7 = -21 – 7 = -28
-5 × 7 = -28 – 7 = -35
-6 × 7 = -35 -7 = -42
∴ (-6) × 7 = - 42
peye oes efJe<ece ef®evn ³egkeÌle HetCee¥keÀeW keÀes iegCee keÀjles nwb, iegCeveHeÀue keÀe ef®evn $eÝCeelcekeÀ neslee nw~
GoenjCe 2 : (- 8) × 2 = -16
Deye Fmes, GHejeskeÌle vecetves keÀer lejn efueefKeS~
GoenjCe 3 : SkeÀ ieesleeKeesj mecegê ceW ieeslee ueieelee nw~ Jen 9m (9 ceerìj) Òeefle efceveì kesÀ ®eeue mes ieeslee ueieelee nw~ ³eefo Jen DeHeveer ®eeue keÀes yevee³es jKelee nw lees 6 efceveì kesÀ yeeo Jen keÀneB nesiee %eele keÀerefpeS~
mecegê ceW ieeslee ueieevesJeeues keÀes mketÀyee ieesleeKeesj SCUBA DIVER keÀnles nw,
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nue : ³eefo Heeveer ceW ieesleeKeesj kesÀ mLeeve (0) ceeveles nQ, mecegê ceW ieerleeKeesj kesÀ mLeeve keÀes $eÝCeelcekeÀ mebK³ee ceeve mekeÀles nQ~
³eneB $eÝCeelcekeÀ mebK³ee nw 9
FmeefueS 6 efceveì kesÀ yeeo GmekeÀe mLeeve = -9 × 6 = -54
ieesleeKeesj 54 ceerìj Heeveer kesÀ veer®es nesiee DeLeJee Jen - 54 ceerìj keÀer otjer Hej nesiee~
`0' keÀes HetCee¥keÀ iegCee keÀjves mes :
efvecve vecedves Hej O³eeve oerefpeS.
6 × 2 = 12 = 12 - 0
5 × 2 = 10 = 12 - 2
4 × 2 = 8 = 10 - 2
3 × 2 = 6 = 8 - 2
2 × 2 = 4 = 6 - 2
1 × 2 = 2 = 4 - 2
0 × 2 = 0 = 2 - 2
0 × 2 keÀes 2 - 2 efueKe mekeÀles nQ ∴ 0 × 2 = 0
SkeÀ HetCee¥keÀ keÀer `0' mes iegCee keÀjves mes `0' ÒeeHle neslee nw~
GoenjCe :
1) 5 × 0 = 0
2) - 10 × 0 = 0
3) 0 × 0 = 0
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$eÝCeelcekeÀ mebK³ee keÀes $eÝCeelcekeÀ mebK³ee mes iegCee keÀjvee :Fmes Heef{S :
ceQ Keelee ntB~ceQ veneR Keelee ~ceQ GHeJeeme keÀjlee ntB ~ceQ GHeJeeme vener keÀjlee~
efkeÀmeer Yeer $eÝCeelcekeÀ mebK³ee mes ye[er, Deewj IeveelcekeÀ mebK³ee mes íesìer mebK³ee keÀewvemeer nw?
ns! Jen Metv³e nw cesjs efce$e!!...
Henues Deewj Deefvlece Jeeke̳e keÀe SkeÀ ner DeLe& neslee nw~Deewj otmejs Deewj leermejs keÀe SkeÀ ner DeLe& neslee nw~³eefo ceQ keÀnlee ntB ``ceQ Demencele veneR ntB~ DeLee&le ceQ mencele ntB~nce ³en efveCe&³e ues mekeÀles nQ efkeÀ keÀesF& Jeeke̳e ceW oesnje³ee ngDee vekeÀjelcekeÀ DeLe& nw lees
mekeÀejelcekeÀ Jeeke̳e neslee nw~1) x = 3 × 5 ceeve ueerefpeS y = 3 × (-5) Deewj
GmekeÀe ³eesieHeÀueer³e efJeueesce -3 × (-5) x + 0 = 3 × 5 + 0
x + y- y = 3 × 5 + [3 × (-5)] + [(-3) × (-5)]
= 3 [5 + (-5)] + [(-3) × (-5)] (x + y mejue efkeÀ³ee ie³ee nw~) = 3 × 0 + [(-3) × (-5)]
= 0- [(-3) × (-5)]
= - 3 × (-5)
∴ 3 × 5 = - 3 × -5
15 = - 3 × -5
Bgo Om{ZE
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Fmes meeceev³e ©He cesb efueKeves mes a × b = a × b + 0 = a × b + a (-b) - a (-b) = a ( b + -b) - a (-b) = a (0) - a (-b) = 0 - a (-b) ∴ ab = - a (- b) = - a × - b = a × b
∴ $eÝCeelcekeÀ HetCee¥keÀesb keÀe iegCeveHeÀue IeveelcekeÀ HetCee¥keÀ neslee nw~
GvneR mebK³eeDeeW keÀes SkeÀ ¬eÀce kesÀ Devegmeej iegCee keÀjles lees -3 × - 1 = ? Xmo $eÝUmË_H$ nwUmªHo$ H$m JwUZ\$b KZmË_H$& hmoVm h¡ & Ed§
Cgo KZmË_H$ {MÝh go {XImVo h¡ &Fmes efvecve vecetves ceW efueKeles nQ~
(íþer keÀ#ee ceW DeeHeves ³eesieHeÀueer³e efJeueesce kesÀ yeejs ceW DeO³e³eve efkeÀ³ee nw~
– 3 × 6 = – 18 – 3 × 5 = – 15 = - 18 – (–3) – 3 × 4 = – 12 = – 15 – (–3) – 3 × 3 = – 9 = – 12 – (– 3) – 3 × 2 = – 6 = – 9 – (– 3) – 3 × 1 = – 3 = – 6 – (– 3) – 3 × 0 = 0 = - 3 - (- 3) – 3 × –1 = 0 – (–3) = 0 + 3 = 3 – 3 × –2 = 3 – (–3) = 3 + 3 = 6 – 3 × –3 = 6 – (–3) = 6 + 3 = 9 – 3 × –4 = 9 – (–3) = 9 + 3 = 12 – 3 × –5 = 12 – (–3) = 12 + 3 = 15 – 3 × –6 = 15 – (–3) = 15 + 3 = 18
nce GvnW HetCe& mebK³eeDeeW keÀer lejn iegCee keÀjles nwb Deewj iegCeveHeÀue kesÀ meeceves IeveelcekeÀ ef®evn ueieeles nwb~
GoenjCe 1 : (-4) × (-3) = ?
Hejce cetu³eeW keÀes iegCee keÀerefpeS 4 × 3 = 12
Hejvleg oes $eÝCeelcekeÀ HetCee¥keÀes keÀe iegCeveHeÀue ncesMee IeveelcekeÀ neslee nw~
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Fmelejn (-4) × (-3) =12
oes mes DeefOekeÀ HetCee¥keÀeW keÀe iegCeveHeÀue :GoenjCe 1 : 3 × 5 × 6 = ?
3 × 5 × 6 = 15 × 6 = 90
GoenjCe 2 : +3 × +5 × (-6) =?
+3 × +5 × (-6) =+15 × (-6) = -90
GoenjCe 3 : +2 × +3 × +5 × +6 = ?
+2 × +3 × +5 × +6 = +6 × +30 = +180
GoenjCe 4 : +2 × +3 × (-5) × (-6) =?
+2 × +3 × (-5) × (-6) =+6 × +30 = +180
GoenjCe 5 : (-2) × (-3) × +5 × (-6) = ?
(-2) × (-3) × +5 × (-6) = +6 × (-30) =-180
GoenjCe 6 : (-2) × (-3) × (-5) × (-6) = ?
(-2) × (-3) × (-5) × (-6) = +6 × +30 =+180
³eeo jefKeS
®ejCe 1 : oÊe mebK³eeDeeW kesÀ mebK³eelcekeÀ cetu³eeW keÀe iegCeveHeÀue %eele keÀerefpeS~
®ejCe 2 : iegCeveHeÀue kesÀ $eÝCeelcekeÀ HetCee¥keÀeW keÀes efieve uMeeerefpeS~
®ejCe 3 : ³eefo $eÝCeelcekeÀ HetCee¥keÀes keÀer mece mebK³ee nw (®ejCe 2 kesÀ Devegmeej) lees iegCeveHeÀue ®ejCe 1 ceW ÒeeHle iegCeveHeÀue ner nesiee, ³eefo $eÝCeelcekeÀ HetCee¥keÀeW keÀer mebK³ee efJe<ece nw, lees iegCeveHeÀue ®ejCe 1 ceW ÒeeHle iegCeveHeÀue ner nesiee Hejvleg meeLe ceW $eÝCeelcekeÀ ef®evn nesiee~
MeeqyokeÀ ieefCele:GoenjCe 1 :
vebove SkeÀ HeÀue efJe¬esÀlee kesÀ Heeme ie³ee~ Gmeves SkeÀ ope&ve ke̳ee DeeHe peeveles nw?
1 ope&ve = 12
kesÀues KejeroW~ ³eefo SkeÀ kesÀues keÀe oece ` 3 nw lees Gmes efkeÀlevee Hewmee osvee nesiee ?
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1 ope&ve = 12 SkeÀ kesÀues keÀe oece = ` 3 Gmes efkeÀlevee Hewmee osvee nw ³eefo peevevee nw lees, 12 keÀes 3 mes iegCee keÀerefpeSvebove keÀes kegÀue = 12 × 3 = ` 36
osves neWies FmeefueS, vebove ves ` 36 HeÀue efJe¬esÀlee keÀe efo³es~
GoenjCe 2 : SkeÀ Heve[tyeer melen mes 15 ceer / efceveì keÀer mes Heeveer ceW [tyeves ueieleer nw~ 5 efceveì kesÀ yeeo Heve[tyeer efkeÀleveer ienjeF& Hej nesieer?
O³eeve oW, veer®es peevee $eÝCeelcekeÀ nw.SkeÀ efceveì ceW Heve[gyyeer keÀe mLeeveeblejie = -15 ceer∴ 5 efceveì kesÀ yeeo keÀe mLeeve = -15 ceer × 5
= -75 ceerFmeefueS, 5 efceveì kesÀ yeeo Heve[tyyeer melen mes 75 ceer kesÀ ienjeF& Hej nesiee~
DeY³eeme 1.2
I. DeeDees mebK³eeDeeW kesÀ meeLe KesuekeÀj Deevebo ueW
1) ef®e$e ceW efoKee³es pewmes HebeqkeÌle kesÀ Òel³eskeÀ mebK³ee keÀes mlebYe kesÀ mebK³eeDeeW kesÀ meeLe iegCee keÀerefpeS Deewj iegCeveHeÀue efueefKeS.
× -3 6 11 -5
7 -21
4 44
-8
-2
12
-9
-4
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2) yeenjer Je=Êe ceW oer mebK³ee keÀes Devoj kesÀ Je=Êe ceW oer mebK³ee kesÀ meeLe iegCee keÀj, iegCeveHeÀue keÀes Je=Êe kesÀ yeenj oer Jeieex ceW efueefKeS~
®e¬eÀ ceW mes keÀesF& mebK³ee ®egvekeÀj otmejer mebK³ee 85
-5-67
-2
1 DeLeJee meK³eeDeeW mes iegCee keÀj Deevevo ueerefpeS.
3) SkeÀ Kesue KesueW :
i) ef®e$e ceW efoKee³es pewmes (- 50) mes 50 DebeqkeÀle SkeÀ HeÀuekeÀ ueerefpeS. KZmË_H$ nwUm©Ho$ H$mo ........... $eÝUmË_H$ nwUmªHo$ H$m ......................
-50 -49 -48 -47 -46 -45 -44 -43 -42 -41
-31 -32 -33 -34 -35 -36 -37 -38 -39 -40
-30 -29 -28 -27 -26 -25 -24 -23 -22 -21
-11 -12 -13 -14 -15 -16 -17 -18 -19 -20
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
9 8 7 6 5 4 3 2 1 0
10 11 12 13 14 15 16 17 18 19
29 28 27 26 25 24 23 22 21 20
30 31 32 33 34 35 36 37 38 39
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50 49 47 46 45 44 43 42 41 40
ii) oes vee rues Deewj oes ueeue Heemes ve Lewuee r ce W ueerefpeS~ veerues Heemes keÀer efyevog HetCee¥keÀ keÀe mLeeve metef®ele keÀjles nQ Deewj ueeue Heemes keÀer efyevog $eÝCeelcekeÀ HetCee¥keÀ metef®ele keÀjles nQ~
Kesue kesÀ efve³ece :-
Òel³eskeÀ efKeuee[er keÀes H³eeos keÀes Metv³e Hej jKevee nw~
Òel³eskeÀ efKeuee[er keÀes SkeÀ mece³e Hej oes Heemes Lewueer mes ueskeÀj HeWÀkeÀvee nw~
Òel³eskeÀ Gíeue ceW Heemes Hej ÒeeHle mebK³eeDeeW keÀes iegCee keÀjvee nw~
efKeuee[er iegCeveHeÀue kesÀ Devegmeej DeHeves H³eeos keÀes Deeies ye{e³esiee~ efKeuee[er pees 15 efceveì ceW meyemes ye[er mebK³ee keÀes HengB®esiee Jen efJepeslee nesiee~
DeY³eeme 1.3
I. efvecve HetCee¥keÀeW kesÀ iegCeveHeÀue %eele keÀerefpeS a) 5, (- 3) b) (- 3), 8
c) 7, (- 34) d) (- 23), (- 12)
e) (- 21), 6 f) (-2), (3), -(4)
g) (-3), 4, (-5), (-1)
II. efjkeÌle mLeeveeW keÀer Hetefle& keÀerefpeS:
a) 4 × + 7 = + 28
b) - 3 × + 5 = - .......
c) + 9 × - 7 = .......
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d) - 6 × - 7 = + .......
III. efvecveefueefKele meboYe& ceW mener GÊej keÀes efìkeÀ ceeke&À ueieeFS :
1) íle Hej jKeer Kegueer ìbkeÀer Heeveer mes Yejer nw~ yee<HeerkeÀjCe kesÀ keÀejCe Heeveer keÀe mlej Òeefleefove 2 cm mes yeouelee nw~ ³eefo Henues efove Heeveer keÀe mlej mes.ceer. nw lees 6 efoveeW yeeo Heeveer keÀe mlej nesiee.
ÒeLece efove ìbkeÀer keÀer mlej x cm, 6 efove kesÀ yeeo Heeveer keÀe mlej
a) x cm + (2 × 6) cm
b) x cm – (2 × 6) cm
c) x cm – (2 × -6) cm
d) 6 × x cm – 2
2) efHeíues 8 meHleen ceW efiejerMe ves ` 7500 Òeefle meHleen DeHeves yeQkeÀ Keeles ceW pecee keÀjlee Dee³ee nw~ Deye GmekeÀe yeQkeÀ yekeÀe³ee ` 62000 nw~ FmeefueS 8 meHleen kesÀ yeeo, Fme DeJeefOe ceW GmekeÀe yeQkeÀ yekeÀe³ee, ³eefo keÀesF& y³eepe ve pees[e ngDee nes lees, pecee
a) ` (62000 – 8 × 7 × 7500) b) ` (62000 – 8 × 7500)
c) ` (62000 + 8 × 7500) d) ` (8 × 7500 – 62000)
meesef®eS : i) ke̳ee DeeHe meyemes ye[er IeveelcekeÀ HetCee¥keÀ Deewj meye mes íesìe $eÝCeelcekeÀ HetCee¥keÀ efueKe mekeÀles nQ?ii) meyemes ye[er IeveelcekeÀ HetCee¥keÀ Deewj meyemes íesìer $eÝCeelcekeÀ HetCee¥keÀ veneR nw~
HetCee¥keÀeW keÀe efJeYeepeve
meboYe& 1 :
mensefue³eeB : pevceefove keÀer neefo&keÀ MegYekeÀecevee³eW, Gcee!
Gcee : Oev³eJeeo, ³en efceþeF& keÀe keÀmee nw~ DeeDees, nce meYeer Fmes meceeve ©He mes yeeBì uesves nQ~
jeefye³ee : þerkeÀ nw, ceQ legcnW ceoo keÀjleer ntB~ (efievekeÀj keÀnleer nw) Fme yekeÌmes ceW
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20 efceþeF& Deewj Gcee keÀes efceueekeÀj nce ueesie 5 nQ~
ÞeerefveefOe : þerkeÀ nw, 20 keÀes 5 mes Yeeie ueieeles nw DeLee&le 20 ÷ 5 = 4
uelee : DeeDees nce Òel³eskeÀ 4 efceþeF&³eeB uesles nQ~
peesmed : uelee, ke̳ee legce peeveleer nes, 5 × 4 = 20 Yeer neslee nw~
ÞeerefveefOe : iegCeveHeÀue Deewj efJeYeepeve mebyebefOele nw~
uelee, peesmed, ÞeerefveefOe, jeefye³ee : keÀesF& yeele vener, Oev³eJeeo, Gcee~
Gcee : ³eefo ceQ legcnejs meeLe yeeBì veneR uesleer, Mee³eo legcnW DeefOekeÀ efceþeF&³eeB efceueleer~
uelee : neB 20 ÷ 4 = 5.meboYe& 2 :
DeeDees, nce Lees[e cepee keÀjles nQ~ peskeÀ Deewj efpeue SkeÀ ieeBJe ceW Kesue jns Les~ kegÀí mece³e kesÀ yeeo GvekeÀes H³eeme ueieer~ Jes Heeveer Heervee ®eenles Les~ GvneWves SkeÀ kegBÀDee Deewj SkeÀ yekesÀì osKee~ GvneWves kegBÀS mes Heeveer uesves keÀe mees®ee~ Heeveer Heerves kesÀ yeeo GvekeÀe mebJeeo Fme ÒekeÀej Lee~
peskeÀ : efpeue, kegBÀS keÀer ienjeF& ke̳ee nesieer?peerue : ce... nce Gme DebkeÀue mes Hetíles nwb~ Fme kegBÀS keÀer ienjeF& ke̳ee nw?jepet : keÀjeryeve, peceerve mes 35 HeÀerì nesieer, ye®®eeW~peskeÀ : Heeveer keÀe mlej ke̳ee nesiee?jepet : 28 HeÀerìpeerue : Oev³eJeeo DebkeÀue, peskeÀ, DeeDees nce ceeve uesles nQ. efkeÀ peceerve keÀe mlej
Metv³e nw~peskeÀ : neB, lees 28 HeÀerì Heeveer keÀe DeLe& nw, Heeveer keÀe mlej nw 35-28 = 7 HeÀerì peceerve
kesÀ veer®es.peerue : nce Gmes -7 mes metef®ele keÀjles nwb~ peskeÀ : Heeveer, peceerve kesÀ veer®es nesves mes, nce Heeveer keÀer ienjeF& keÀes -28 HeÀerì metef®ele keÀj
mekeÀles nQ~
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peerue : ³ee fo keÀ e sF & yeke s Àì ke g B ÀS ce W e f i eje Dee wj 4 e f c ev eì ce W H e ev e e r k e s À mlej mes veer®es pee HengB®ee lees yekesÀì keÀer jHeÀleej ke̳ee nesieer?
peskeÀ : efpeue, yengle De®íe ÒeMve nw~ DeeDees nce Fmes HeefjkeÀueve keÀjles nw b~ nceves efJe%eeve ceW ®eeue kesÀ yeejs ces DeO³e³eve efkeÀ³ee nw~
efpeue : neB, ieefle Mmb = otjer yeìe mece³e
peskeÀ : lees ieefle Mmb = 28 ÷ 4 = 7 HeÀerì Òeefle efceveì
efpeue : uesefkeÀve nceves Heeveer keÀer ienjeF& keÀes -28 ceeveer nw~
peskeÀ : Deewj 4 × ? = -28
efpeue : ³en -7, nw~ FmeefueS -28 ÷ 4 = -7
yekesÀì keÀer ieefle -7 HeÀerì Òeefle Òeefle efceveì nw $eÝCeelcekeÀ ef®evn metef®ele keÀjlee nw~ yekesÀì veer®es keÀer Deesj ieefleMeerue nw~
peskeÀ : yeme efpeue~ Deye nce Kesueles nwb~ peskeÀ Deewj efpeue oesveeW Kesueves peeles nQ~ nce iegCeveHeÀue peeveles nQ~
GoenjCe 2:
i) 6 × 2 = 12 = 2 × 6
FmeefueS 12 ÷ 6 = 2 Deewj 12 ÷ 2 = 6
ii) (- 4) × 5 = - 20 (i) ceW GHe³eesie efkeÀ³es efJeOeeve mes nce.
(– 20) ÷ 5 = - 4 efueKe mekeÀles nwb Deewj – 20 ÷ - 4 = 5.
DeeDees; Fme Òeef¬eÀ³ee Hej Lees[e Deewj DeY³eeme keÀjles nQ~ Òel³eskeÀ iegCeveHeÀue keÀLeve kesÀ oes efJeYeepeve kesÀ keÀLeve nw~ DeHeves GÊejeW mes Fme leeefuekeÀe keÀes HetCe& keÀerefpeS~
iegCeveHeÀue keÀLeve Deveg©He efJeYeepeve keÀLeve( - 8) × 2 = - 16 -16 ÷ 2 = - 8, -16 ÷- 8 = 2
(- 5 ) × (- 4 ) = 20 20 ÷ - 4 = - 5, ----------------
9 × (- 4) = - 36 -36 ÷ - 4 = 9, - 36 ÷ 9 = - 4
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( - 7) × (- 6) = 42 ---------------, 42 ÷- 7= - 6
3 × (- 10) = - 30 - 30 ÷- 10 = 3, ---------------
Fme leeefuekeÀe kesÀ efve<keÀ<e& nQ :
• meceeve ef®evn kesÀ oes mebK³eeDeeW keÀe efJeYeepeve ncesMee IeveelcekeÀ neslee nw~
• Deueie ef®evn kesÀ, oes mebK³eeDeeW keÀe efJeYeepeve ncesMee $eÝCeelcekeÀ neslee nw~
SkeÀ HetCee¥keÀ keÀes Metv³e mes Yeeie ueieeves Hej keÀ³ee neslee nw?
Yeejle kesÀ ceneve ieefCele%e Sme jecevegpeve ves ÒeMve keÀes DeHeves keÀ#ee ceW Gþe³ee Lee~
DeeDees, Fmes SkeÀ keÀe³e&keÀueeHe Üeje keÀjkesÀ osKeles nwb
meceeve ceeHe kesÀ 8 keÀeiepe kesÀ Hevves ueerefpeS,
8 8, 1 yeej = 8
GvekesÀ oes meceeve mecetn yeveeFS
4 + 4 4, 2 yeej = 8
Fve mecetn kesÀ Deewj 2 mecetn yeveeFS
2 + 2 + 2 + 2 2, 4 yeej = 8
Hegve: Fve Òel³eskeÀ mecetn kesÀ 2 mecetn yeveeFS
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 1, 8 yeej = 8
Fve HevveeW kesÀ Òel³eskeÀ kesÀ oes Yeeie keÀeì ueerefpeS :
21 + 2
1 + 21 + 2
1 + 21 + 2
1 + 21 + 2
1 + 21 + 2
1 + 21 + 2
1 + 21 + 2
1 + 21 + 2
1 21 , 16 yeej = 8
Deye 16 ìgkeÀ[eW kesÀ Òel³eskeÀ Yeeie yeveeFS :
41 + 4
1 + 41 + 4
1 + 41 + 4
1 + 41 + 4
1 +..........+ 41 + 4
1 + 41 + 4
1 + 41 4
1 , 32 yeej = 8
Deye efkeÀleves yeej Metv³e pees[ves Hej 8 DeeSiee? nce %eele veneR keÀj mekeÀles~
DeeDees, Fmes efvecve efoS efJeOeeve mes peeve uesles nwb :
.0 258 = 32 0.25 + 0.25+…………+0.25, 32 yeej = 8
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Fmeer ÒekeÀej
.0 18 = 80 0.1 + 0.1+………….+0.1,80 yeej = 8
.0 018 = 800 0.01 + 0.01+…………+0.0,800 yeej = 8
08 = ? efkeÀleves Metv³e 8 keÀes pees[vee nw?
Fme ceW Metv³e keÀe mecetn venerb nw~ 8 yeveeves kesÀ efueS 0 mes Yeeie venerb keÀj mekeÀles
(met®evee : Metv³e keÀe efJeYeepeve HeefjYeef<ele veneR nesles nwb)DeY³eeme 1.4
I. mejue keÀerefpeS :
a) (- 40) ÷ (- 10) b) 34 ÷ (- 2) c) (- 44) ÷ 4
d) (- 28) ÷ (- 7) e) 0 ÷ (- 8)
II. efjkeÌle mLeeveeW keÀer Hetefle& keÀerefpeS :
a) (- 45) ÷ ----- = - 45 b) (- 27) ÷ (- 27) = -----
c) 30 ÷ ---- = - 15 d) ---- ÷ 4 = - 3
e) ---- ÷ ( - 3) = 10
III. Fme leeefuekeÀe keÀes HetCe& keÀerefpeS :
HetCee¥keÀ keÀe efJeYeepeve YeeieHeÀue
24 ÷ +12
24 ÷ (-12)
(-24) ÷ +12
(-24) ÷ (-12)
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VI. Heeìerue ves OeejKee[ Hes[s kesÀ 8 n§H$oQ> KejeroW~ Òel³eskeÀ n§H$oQ> ceW meceeve Hes[s nwb~ kegÀue Gve
ceW 320 Hes[s Les~ Òel³eskeÀ n§H$oQ> ceW efkeÀleves Hes{s Les %eele keÀerefpeS~
HetCee¥keÀeW keÀe iegCeOece&
a) ¬eÀce efJeefvece³e efve³ece:pees[ : 2 + - 5 = -3 Deewj -5 + 2 = -3.
keÀesF& oes HetCee¥keÀ a Deewj b kesÀ efueS, ³eefo a + b = b + a nes lees Fme iegCe keÀes ¬eÀceefJe efvece³e iegCe keÀnles nQ~peye nce oes HetCee¥keÀ pees[les nQ, ¬eÀce keÀe cenlJe veneR neslee nceW Jener GÊej ÒeeHle neslee nw~b) HetCee¥keÀeW keÀe meceg®®e³e, pees[ Hej ¬eÀceefJeefvece³e efve³ece keÀe Heeueve keÀjlee nQ~JwUm: 2 × −5 = −5 × 2 = -10
keÀesF& oes HetCee¥keÀ a Deewj b kesÀ efueS, a × b = b × a
peye nce oes HetCee¥keÀ iegCee keÀjles nQ, ¬eÀce keÀe cenlJe veneR neslee nceW Jener GÊej ÒeeHle neslee nw~
c) H etCee¥keÀes keÀe meceg®®e³e, iegCee Hej ¬eÀceefJeefvece³e efve³ece keÀe Heeueve keÀjvee nw~KQ>m : 18 – 15 = 3 Deewj 15 – 18 = -3 ∴18 – 15 ≠ 15 – 18
peye ¬eÀce yeouelee nw leye Ieìeves Hej GÊej Yeer yeouelee nw~ FmeefueS,
d) HetCee¥keÀes keÀe meceg®®e³e ceW HetCee¥keÀesb kesÀ Ieìeves Hej ¬eÀce efJeefvece³e keÀe Heeueve veneR neslee~efJeYeepeve : 12 ÷ 3 = 3
12 4= Deewj 3 ÷ 12 = 123
41=
∴ 12 ÷ 3 ≠ 3 ÷ 12
Yeep³e Deewj YeepekeÀ kesÀ mLeeveeW keÀes yeoueves Hej, YeeieHeÀue Yeer yeouelee nw~ FmeefueS, HetCee¥keÀes kesÀ meceg®®e³e ceW efJeYeepeve kesÀ meboYe& ceW ¬eÀce efJeefvece³e efve³ece keÀe Heeueve veneR neslee~b) meen®e³e& iegCe :pees[: (4 + −2) + −5 = 2 + -5 = -3.
4 + (−2 + −5) = 4 + -7 = -3.
efo³es ieS leerve HetCee¥keÀ a,b Deewj c kesÀ efueS (a + b) + c = a + (b + c)
leerve HetCee¥keÀesb keÀes pees[les mece³e, Yeues nce Henues pees[er keÀes pees[ uesles nQ ³ee Deefvlece, keÀesF&
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Devlej veneR neslee, GÊej Jener jnlee nw~HetCee¥keÀes keÀe meceg®®e³e, pees[ kesÀ meboYe& ceW meen®e³e& iegCe mev³e nw~
iegCeveHeÀue: (4 × −2) × −5 = -8 × -5 = 40
4 × (−2 × −5) = 4 × 10 = 40
a, b Deewj c leerve HetCee¥keÀ, (a × b) × c = a × (b × c)
peye leerve HetCee¥keÀeW keÀe iegCee keÀjles nQ, Yeues nce Henues pees[er keÀes iegCee keÀjles nQ ³ee Debeflece, keÀesF& Deblej veneR neslee, GÊej Jener Deelee nw~HetCee¥keÀeW keÀe meceg®®e³e ceW iegCeveHeÀue kesÀ meboYe& ceW, meen®e³e& iegCe keÀe Heeueve neslee nw~ J³eJekeÀueve (Ieìevee): 6 – (3 – 7) = 6 – (-4) = 10
(6 - 3) – 7 = 3 – 7 = -4
∴6 – (3 – 7) ≠ (6 - 3) – 7
³en mHe<ì nw efkeÀ, HetCee¥keÀeW kesÀ meceg®®e³e ceW, J³eJekeÀueve kesÀ meboYe& ceW, meen®e³e& iegCe keÀe Heeueve veneR neslee~ efJeYeepeve: 8 ÷ (4 ÷ 2) = 4
(8 ÷ 4) ÷ 2 = 1
8 ÷ (4 ÷ 2) ≠ (8 ÷ 4) ÷ 2
³en mHe<ì nw efkeÀ, HetCee¥keÀeW kesÀ meceg®®e³e ceW, efJeYeepeve kesÀ meboYe& Hej meen®e³e& efve³ece keÀe Heeueve veneR neslee~c) pees[ keÀe lelmecekeÀ DeJeHeJe (Additive Identity): (−5) + 0 = −5, 0 + (−5) = −5
a SkeÀ HetCee¥keÀ, a + 0 = a = 0 + a
Metv³e, pees[ keÀe lelmecekeÀ DeJe³eJe nw~Metv³e keÀes oesveeW He#e ceW pees[ves Hej, mebK³ee ceW keÀesF& Devlej veneR neslee~
d) iegCee keÀe lelmecekeÀ DeJe³eJe (Multiplicative Identity) : −3 × 1 = -3 = 1 × -3
a SkeÀ HetCee¥keÀ a × 1 = a = 1 × a
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iegCeveHeÀue kesÀ efueS 1 ³en lelmecekeÀ DeJe³eJe nw~ oesveeW He#eeW ceW 1 mes iegCee keÀjves Hej, mebK³ee
ceW keÀesF& Devlej veneR neslee~
e) efJelejCe iegCe :
nce keÀes<ìkeÀ kesÀ Devoj keÀer Òel³eskeÀ mebK³ee keÀes yeenj keÀer mebK³ee mes iegCee keÀjles nw Deewj yeer®e ceW Ieìeves keÀe ef®evn Jewmes ner jKeles nQ~
GoenjCe 1 : 3 × [2 + (−4)]
GoenjCe 2 : - 2 × (5-7) =[(-2) × 5)] - (-2 × (+7))
a, b, c leerve HetCee¥keÀ nes lees
a × (b + c) = (a × b) + (a × c)
a × (b - c) = (a × b) - (a × c)
Fmes efJelejCe iegCe keÀnles nQ~
DeY³eeme 1.5
I. mecegef®ele iegCeeW keÀe GHe³eesie keÀjles ngS efvecveeW keÀes mejue keÀerefpeS.
1) 5 × [(-4)] - 2)= - 2) -3 ×( 8 – 5) 3) [4 × -2] + [5 ×-2]
4) [(5+-3)]-2 5) -7+(8-3) 6) [(7×(-2)]+(4-7)-6(2 ×(-3)
Deye nce _m¡[IH$ ÌíZ nue keÀjles nQ~
GoenjCe 1 : jc³ee DeHeves 3 YeeF³eesb kesÀ efueS HeQì keÀe keÀHe[e Deewj Meì& keÀe keÀHe[e Kejeroves kesÀ efueS SkeÀ keÀHe[s kesÀ otkeÀeve Hej peeleer nQ~ SkeÀ HeQì Heerme kesÀ keÀHe[s keÀe oece ` 450 Deewj Meì& Heerme kesÀ keÀHe[s keÀe oece ` 320 nw~ keÀHe[s kesÀ otkeÀeve ceW Gmes kegÀue efkeÀlevee Hewmee osvee nesiee? nue:
SkeÀ HeQì Heerme keÀHe[s keÀe oece = ` 450
FmeefueS 3 HeQì Heerme keÀe oece = 3 × 450 = ` 1350
SkeÀ Meì& Heerme keÀe oece = ` 320
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FmeefueS 3 Meì& Heerme keÀe oece = 3 × 320
= ` 960
FmeefueS, keÀHe[s Hej jc³ee ves = 1350 + 960
= ` 2310 Ke®e& efkeÀ³es~
GoenjCe 2 : meesvee kesÀ yeQkeÀ Keeles ceW ` 1020 Les~ meesceJeej keÀes Gmeves ` 200 pecee keÀjJee³es, Deewj Gmeer efove ` 500 efvekeÀeue efue³es, Deewj yegOeJeej keÀes ` 580 pecee keÀjJee³eW~ MeefveJeej keÀes ³eefo Jen ` 1000 pecee keÀjleer nw lees MeefveJeej kesÀ efove GmekesÀ Keeles ceW efkeÀlevee yekeÀe³ee Lee?
³eneB Oeve pecee keÀjevee IeveelcekeÀ Deewj efvekeÀeuevee $eÝCeelcekeÀ nw~
meesvee kesÀ Keeles ceW Henues Oeve Lee = ` 1020
meesceJeej kesÀ efove = 1020 + 200 -500 = ` 720
yegOeJeej = 720 + 580 = ` 1300
MeefveJeej = 1300 + 1000 = ` 2300
MeefveJeej kesÀ efove GmekesÀ Keeles ceW yekeÀe³ee ` 2300
GoenjCe 3 : SkeÀ kebÀHeveer keÀes SkeÀ Hesve yes®eves Hej Òel³eskeÀ Hesbefmeue yekeÌmes kesÀ Heerís ` 12 keÀe ueeYe neslee nw Òel³eskeÀ HeWefmeue yekeÌmes kesÀ Heerís ` 8 keÀer neefve nesleer nw~ Jen kebÀHeveer 2500 Hesve Deewj 4000 HeWefmeue yes®eleer nw~ ueeYe DeLeJee neefve efkeÀleveer nw?
1 Hesve yes®eves Hej ueeYe = `12
yes®es HesveeW keÀer mebK³ee = 2500
Hesve yes®eves Hej ÒeeHle ueeYe = 2500×12 = `30000
SkeÀ HeWefmeue yekeÀmee yes®eves Hej ÒeeHle neefve = ` 8
yes®es ieS HeseqvmeueeW keÀer yekeÌmes = 4000
HeWefmeue yes®eves Hej ngF& neefve = 8 × 4000
= ` 32000
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FmeefueS neefve = 32000 - 30000
= ` 2000
GoenjCe 4 :
SkeÀ eqkeÌJepe ÒeMveeJeueer ceW 12 ÒeMve nQ Òev³eskeÀ kesÀ 5 DebkeÀ nw~ cetu³eebkeÀve ÒeCeeueer ceW mener GÊej kesÀ efueS +5 DebkeÀ Deewj ieuele GÊej kesÀ efueS (-2) DebkeÀ nw~
i) Yeejleer meYeer ÒeMveeW kesÀ GÊej efueKeleer nw, FveceW 9 mener Deewj 3 ieuele nw~
ii) cebpeguee kesÀJeue 10 ÒeMveeW keÀes nue keÀjleer nw efpeveceW 6 mener Deewj yeekeÀer ieuele nw~ yeleeFS eqkeÌJepe ceW keÀewve efJepeslee yevelee nw?
i) Òel³eskeÀ mener GÊej kesÀ efueS DebkeÀ = 5
9 mener GÊej kesÀ DebkeÀ = 5 × 9 = 45
Òel³eskeÀ ieuele GÊej kesÀ efueS DebkeÀ = -2
3 ieuele GÊej kesÀ DebkeÀ = (-2) × 3 = (-6)
Dele: Yeejleer kesÀ ÒeeHle DebkeÀ = 45 + (-6) = 39
ii) Òel³eskeÀ mener GÊej kesÀ DebkeÀ 6 mener GÊej kesÀ DebkeÀ = 5 × 6 = 30
SkeÀ ieuele GÊej keÀe DebkeÀ = -2 × 4 = - 8
4(10-6) ieuele GÊej kesÀ DebkeÀ
Dele: cebpeguee kesÀ ÒeeHle DebkeÀ = 30 + (-8) = 22
39 > 22, cebpeg kesÀ kegÀue DebkeÀ
Yeejleer ner efJepeslee jner~
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DeY³eeme 1.6
I. efvecveefueefKele MeeqyokeÀ ÒeMveeW keÀes nue keÀerefpeS.
1) ceesefvekeÀe kesÀ Heeme 5 ®ee@keÀuesì kesÀ ef[yyes nQ~ ³eefo Òel³eskeÀ yekeÌmes ceW 25 ®ee@keÀuesì kesÀ yekeÌmes nQ~ GmekesÀ Heeme kegÀue efkeÀleves ®ee@keÀuesì nQ?
2) SkeÀ efove DeHeÀleeye ves osKee efkeÀ yesiebuet© keÀe meeceev³e leeHeceeve 210 C nw~ SkeÀ efJeefMe<ì efove Hej meeceev³e leeHeceeve keÀer leguevee ceW leeHeceeve ceW ngS HeefjJele&ve Fme lejn nw~
megyen : 5am: - 50 C; 10am : + 50 C,
oesHenj : 12pm: + 70 C
em_ : 5pm: + 20 C,
DeHeÀleeye keÀes megyen kesÀ 5 yepes, 10 yepes, 12 yepes Deewj 3 yepes mener leeHeceeve ceeuetce keÀjves ceW mene³elee keÀerefpeS~
3) jMceer kesÀ yeQkeÀ Keeles ̀ 12350 nw~ Gmeves DeHeves Keeles ceW mes ̀ 200, 3 yeej efvekeÀeue efue³es nQ~ GmekesÀ yeQkeÀ Keeles ceW efkeÀlevee yekeÀe³ee nw?
4) SkeÀ Hesve keÀe oece ` 8 nw Deewj HeWefmeue keÀe oece ` 5 nw~ SkeÀ ope&ve Hesve Deewj DeeOes ope&ve HeWefmeue keÀe cetu³e %eele keÀerefpeS~
5) SkeÀ keÀ#ee ceW 38 efJeOeeLeea nQ~ HegmlekeÀeW kesÀ SkeÀ mesì keÀe oece ` 1235 nw~ 38 HegmlekeÀeW kesÀ mesì keÀe cetu³e ke̳ee nw?
6) SkeÀ Hee$e ceW 45kg MekeÌkeÀj nw~ ceerje 750 MekeÌkeÀj veer®es efiejeleer nw~ yeekeÀer jns MekeÌkeÀj ceW 250 gm kesÀ efkeÀleves HeskeÀì yeve mekeÀles nQ?
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IeìkeÀ - 2
efYevve
Fme IeìkeÀ kesÀ DeO³e³eve keÀjves kesÀ yeeo DeeHe:
SkeÀ efYevve keÀes HetCe& mebK³ee mes iegCee keÀjvee meerKeWies,
SkeÀ efYevve keÀes otmejs efYevve mes iegCee keÀjvee meerKeWieW,
mener ¬eÀce ceW FmekeÀe DeLe& mecePeWieW,
SkeÀ efYevve keÀe Òeefleuees_ efueKevee Deewj GHe³eesie keÀj mekeWÀieW,
SkeÀ efYevve keÀes SkeÀ HetCe& mebK³ee mes Deewj HetCe& mebK³ee keÀes efYevve mes Yeeie ueieevee meerKeWies,
SkeÀ efYevve keÀes otmejs efYevve mes Yeeie ueieevee meerKeWieW,
iegCee keÀjves Deewj Yeeie ueieeves kesÀ keÀLeveelcekeÀ ieefCele nue keÀjWieW~.
DeeHeves efYevve keÀe DeLe&, efYevveeW kesÀ ÒekeÀej, mecelegu³e efYevve Deewj efYevveeW keÀes pees[vee Deewj Ieìevee meerKe efue³ee nw~ GvnW mcejCe keÀerefpeS :
efYevveeW keÀe JeieeakeÀjCe (Classification of Fractions) :
1) efvecvee efueefKele efYevveeW keÀes mece, efJe<ece Deewj efceÞe efYevve ceW Jeieeake=Àle keÀerefpeS :
, , 2 , , , , 2 , , , 431
52
53
47
911
65
81
136
1315
61
Òel³eskeÀ efYevve keÀes O³eeve mes osefKeS Deewj GvekeÀes HebeqkeÌle³eeW ceW efueefKeS :
a) mece efYevve : , , ,31
52
65
136
b) efJe<ece efYevve : , ,47
911
1315
c) efceÞe efYevve : 2 ,2 , 453
81
61
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2) A met®eer ceW efo³es ieS efJe<ece efYevveeW keÀes efceÞe efYevveeW ceW B met®eer kesÀ efceÞe efYevveeW keÀes efJe<ece efYevveeW ceW HeefjJeefle&le keÀj mekeÀles nQ~ O³eeve mes osefKeS~
, , ,29
512
615
1112
met®eer `A' met®eer `B'
, , ,2 43 5 6
1 4 72 6 5
3
29 4 2
1
512 2 5
2
615 2 6
3
1112 111
1
=
=
=
=
2 43
411
5 61
631
4 72
730
6 53
533
=
=
=
=
GoenjCe 1:ceesnceo kesÀ Heeme 8
6 ®eekeÀuesì Deewj ceev³ee kesÀ Heeme nQ~ 43 ®eekeÀuesì efkeÀmekesÀ Heeme p³eeoe
®eekeÀuesì nw? DeeHekeÀe efveCe&³e ke̳ee nw?
i) ii)
oesveeW kesÀ Heeme meceeve iee$e kesÀ ®ee@keÀuesì nwb~
86
43= ³es mece efYeVejeefMe nQ~
met®evee : )a 86
8 26 2
43
''= = )b 4
34 23 2
86
##= =
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GonjCe 2:
Mewuee ves mketÀue ne@ue keÀe 125 Yeeie mepee³ee Deewj jsCeg ves Gmeer ne@ue keÀe 6
3 Yeeie mepee³ee~ ne@ue keÀe mepee³ee ie³ee kegÀue Yeeie efkeÀlevee nw? ceeuetce keÀerefpeS~
Mewuee mes mepee³ee ie³ee ne@ue keÀe Yeeie = 125
jsCeg mes mepee³ee ie³ee ne@ue keÀe Yeeie = 63
Mewuee Deewj jsCeg mes mepee³ee ie³ee neue keÀe Yeeie 125
63
125
6 23 2
125
126
125 6
1211
##
= +
= +
= +
= + =
Mewuee Deewj jCeg mes mepee³ee ie³ee ne@ue keÀe Yeeie = 1211
GonjCe 3:
SkeÀ Hee$e ceW 128 kg MekeÌkeÀj Lee~ 8
3 kg Yeeie GHe³eesie efkeÀ³ee~ Hee$e ceW yeekeÀer MekeÌkeÀj keÀer cee$ee efkeÀleveer nw?
Hee$e ceW jne MekeÌkeÀj 128 kg
GHe³eesie keÀer ieF& MekeÌkeÀj 83 kg
yeekeÀer MekeÌkeÀj =
128
83
12 28 2
8 33 3
2416
249
2416 9
247
##
##
= -
= -
= -
= - =
ue. mee. De uesves Hej12,8 6,4 3,2
22
2×2×3×2=24
∴ yeekeÀer MekeÌkeÀj = 247 efkeÀueesûeece =
247 kg
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DeY³eeme 2.1
I. efvecveeW keÀes mece efYevve, efJe<ece efYevve Deewj efceÞe efYevve ceW Jeieeake=Àle keÀerefpeS :
2 , , , 6 , ,61
127
1618
92
45
203
II. efvecve Òel³eskeÀ kesÀ efueS oes mecelegu³e efYevve efueefKeS :
a) 63 b) 5
4 c) 107
III. efvecveeW keÀes mejue keÀerefpeS (efvecveeW keÀes meyemes efvecve ©He ceW efueefKeS).
a) 128 b) 30
18 c) 5628
IV. efvecveeW keÀes efceÞe efYevve ceW efueefKeS.
a) 815 b) 12
25 c) 497
V. efvecveeW keÀes efJe<ece efYevve ceW efueefKeS.
a) 3 43 b) 8 2
1 c) 4 65
VI. mejue keÀerefpeS.
a) 43
35+ b) 3 2
143
52+ +
c) 159
52- d) 4 5
3 2 65-
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efYevveeW keÀe iegCeveHeÀue:
SkeÀ keÀ#eeO³eeHekeÀ ves ye®®eeW keÀes SkeÀ ef®e$eelcekeÀ ieefCele nue keÀjves efo³ee~ Hegveerle Deewj meveJeer ves Fve ieefCeleeW keÀes efvecve ©He mes nue efkeÀ³ee:
1) 41 kesÀ 5 Yeeie nw~ GvekeÀes pees[keÀj efueefKeS~
14 + 1
4 + 14 + 1
4 + 14
=(1+1+1+1+14
)= 54
= 54 = 1 1
4
41
41
41
41
41
41 1 1 1 1
45
45 1 4
1
+ + + +
= + + + + =
= =
` j41
41
41
41
41
41 1 1 1 1
45
45 1 4
1
+ + + +
= + + + + =
= =
` j41
41
41
41
41
41 1 1 1 1
45
45 1 4
1
+ + + +
= + + + + =
= =
` j41
41
41
41
41
41 1 1 1 1
45
45 1 4
1
+ + + +
= + + + + =
= =
` j41
41
41
41
41
41 1 1 1 1
45
45 1 4
1
+ + + +
= + + + + =
= =
` j
2) Òel³eskeÀ 43 kesÀ 3 Yeeie nw~ pees[keÀj
efueefKeS~
43
43
43
43 3 3
49
49 2 4
1
+ +
= + +
=
= =
43
43
43
43 3 3
49
49 2 4
1
+ +
= + +
=
= =
43
43
43
43 3 3
49
49 2 4
1
+ +
= + +
=
= =
43
43
43
43 3 3
49 2 4
1
+ +
= - + +
= =
Þes³ee pees Fve ieefCeleeW keÀes osKe jner Leer DeHeves efMe#ekeÀ mes ÒeMve Hetíe~
Fve oes ieefCeleeW ceW yeejb yeej efYevveeW keÀes pees[e ie³ee nw~ keÀ³ee nce Fve
efYevveeW keÀes iegCee keÀj mekeÀles? pewmes nce HetCe& mebK³eeDees ceW keÀjles nQ~
efve: mebosn, legce mener nes, Fve ieefCeleeW keÀes iegCeveHeÀue Òeef¬eÀ³ee Üeje Yeer nue keÀj mekeÀles nwb~
Deye nce efYevveeW keÀes iegCee keÀjvee meerKeWieW~
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efYevveeW keÀes HetCe& mebK³eeDeeW mes iegCee keÀjvee:GoenjCe 1 : veer®es kesÀ ef®e$e osefKeS~ Òel³eskeÀ Je=Êe keÀe efkeÀlevee Yeeie íe³eebefkeÀle ngDee nw?
Òel³eskeÀ Je=le ceW 41 Yeeie íe³eebefkeÀle ngDee nw~
∴ kegÀue efceueekeÀj íe³eebeqkeÀle Yeeie
41
41
41
41 3#= + + =
Deye nce íe³eebeqkeÀle Yeeie keÀes Fmelejn efueKe mekeÀles nQ,
43 ,
Dele: 41 3 4
1 343
# #= =
DebMe keÀes HetCe& mebK³ee mes iegCee efkeÀ³ee ie³ee nw Deewj nj 4 Jewmes jKee ie³ee nw~
Fmes peeve ueerefpeS : HetCe& mebK³ee keÀe nj 1 neslee nw~ FmeefueS, ,3 1
3 5 15 8 1
8
41 3 4
113
4 11 3
43
# ###
= = =
= = =
GoenjCe 2 : 2 ?52# = iegCeveHeÀue %eele keÀjves kesÀ efueS 5
2 pees[vee nQ~
52
+ =
52
54
HetCe¥ mebK³ee Deewj DebMees keÀe iegCeveHeÀue nj keÀe iegCeveHeÀue
252
52
12
5 12 2
54
# ###= = = DebMeeW keÀe iegCeveHeÀue
nj keÀe iegCeeHeÀue=
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31
GoenjCe 3 : ?92 3# =
+ + =
92 3 9
213
9 12 3
96
# ###= = = ← DebMeeW keÀe iegCeveHeÀue
← nj keÀe iegCeveHeÀue
mejue efkeÀ³ee ngDee ©He osefKeS
33
96
32
32
#
#= =
He³ee&³e efJeOeeve3 9
2
96
32
3
2
#
= - -
GoenjCe 4 :
?125 2
125
12
12 15 2
1210
2 62 5
2 62 5
65
#
###
##
#
#
=
= =
= =
= =
← meeceev³e iegCeveKeC[ %eele keÀerefpeS
← mejue ©He
mJe³eb keÀerefpeS: a) 73 2# b) 10
3 3# c) 2 94
# d) 5 153
#
efJe<ece efYevve keÀes HetCe& mebK³ee mes iegCee keÀjvee~
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32
GoenjCe 1 :
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
E` g»`m _o [ 54
14
× 5 = ] Hw$b g§»`m 45
345
45
13
4 15 3
415
#
###
=
= = = E` g»`m _o
E` 415 3 4
3=
Goe 2 : 3 ?79
79 3 7
913
7 19 3
727 3 7
6
#
# ###
=
= = = =
Goe 3 : ?10 56
10 56
110
56
1 510 6
560
5 15 12 12
#
# ###
#
#
=
= = = = =
mJeb³e efkeÀefpeS : a) 58 3# b) 6
13 2# c) 6 35
# d) 12 78
#
HetCe& mebK³eeDeeW keÀe efceÞe efYevve mes iegCee keÀjveeFve GoenjCeeW keÀes efveefj#eCe keÀerefpeS~
GoenjCe 1 kegÀue efkeÀleves Yeeie efo³es ie³es nw?
1 1 41 + 1 1 4
1 + 1 1 41
©KTBS
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33
?2 41 2 4
1 2 41+ + =
FmeefueS 3 keÀe 2 41
2 341
49
13
4 19 3
427 6 4
3
#
###
=
= =
= =
2 41 keÀes leerve yeej ueer³es peeles nw~
®ejCe 1 : efce$e efYeVe mes efJe<ece efYeVe ceW HeefjJele&ve keÀerefpeS
®ejCe 2 : iegCee keÀerefpeS kegÀue Yeeie = 6 43
GoenjCe 2 : 2 7 ?54
514
17
5 114 7
598
19 53
#
###
=
= = =
=
19
5 98
485
345
h
mJe³eb nue keÀerefpeS: a) 5 21 7# b) 3 6
2 3# c) 8 2 4
1# d) 12 3 2
1#
efYevve mes mecyebefOele Òeef¬eÀ³eeBS :
GoenjCe 1 : jsKee ves Òesl³eskeÀ Je=Êe keÀe DeeIee Yeeie í³eebefkeÀle efkeÀefpeS
jsKee ves kegÀue efkeÀleves Yeeie íe³eebeqkeÀle efkeÀ³es?
Fmes nce 3 keÀe 21 Yeeie~ Deye nce efYevveeW ceW keÀe GHe³eesie keÀjvee meerKeWieW~
3 keÀe 21
13
#= (³eneB `keÀe' keÀe celeueye iegCee keÀjvee)
©KTBS
Not to
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shed
34
3 21
#= ie gCeveHeÀue ce s De³ee sie keÀjke sÀ ie gCee keÀe re fpeS iegCee keÀe ef®evn keÀe GHe³eesie
21 332
12 11 3
23 1 2
1
###= =
= =
1 2
3 123 1 2
1##= = = iegCee keÀerefpeS
∴íe³eebefkeÀle kegÀue Yeeie 1 21
GoenjCe 2 : Jesoeble kesÀ Iej Hej otOe keÀe J³eeHeej keÀjles nQ~ SkeÀ efove Jesoeble ves Òel³eskeÀ yeesleue ceW
54 ueerìj HeHkeÀj mes 8 yeesleue Yejs~ yeleeFS 8 yeesleue ceW kegÀue efkeÀleves ueerìj otOe Yeje ie³ee?
54 l 5
4 l 54 l 5
4 l 54 l 5
4 l 54 l 5
4 l
yeesleueeW ceW kegÀue otOe keÀer cee$e gÀ l
l l
l
l
8 54
8 54
18
54
1 58 4
532
6 52
# #
##
=
= =
= =
=
He³ee&³e efJeOeeve 5
4 kesÀ 8
6 l54 8
5 14 8
532
52
#
##
=
= = =
Jesoevle ves kegÀue 6 l52= ueerìj otOe yeerleueeW ceW Yeje~
GoenjCe 3 : 42 efJeOeeefLe&³eeW kesÀ SkeÀ keÀ#ee ceW, GveceW 72 Jeeueeryee@ue ceW Yeeie efue³ee, 7
3 keÀjeìs ceW Yeeie efue³ee Deewj byeekeÀer efJeOeeLeea keÀye·er ceW Yeeie efue³es~
a) efkeÀleves efJeOeeefLe&³eeW ves Jee@ueeryee@ue ceW Yeeie efue³ee?
b) efkeÀleves efJeOeeefLe&³eeW ves keÀjeìs ceW Yeeie efue³ee?
c) efkeÀleves efJeÐeeefLe&³eeW ves keÀye·er ceW Yeeie efue³ee?
©KTBS
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35
a) Jee@ueeryeeue ceW Yeeie uesvesJeeues efJeOeeLeea 42 kesÀ 72
7
4242 72
142
72
12 121
6
# ##
#= = =
∴ 12 efJeÐeeefLe&³eeW ves yee@ueeryeeue= ceW Yeeie efue³ee~ b) keÀjeìs ceW Yeeie uesvesJeeues efJeOeeLeea 42 kesÀ 7
3
42 73
142
73
# #= =
181 742 3
1
6
#
#= = ye®®es
∴ keÀjeìs Òe³eesie keÀjvesJeeue, efJeÐeeLeea keÀer mebK³ee = 18
c) keÀye·er ceW Yeeie uesves Jeeues efJeOeeLeea = kegÀue efJeÐeeefLe&³eeW keÀer mebK³ee - (Jee@ueeryeeue Deewj keÀjeìs kesÀ efJeÐeeLeea) = 42 - (12 + 18) = 42 - 30 = 12 ∴ 12 efJeÐeeefLe&³eeW ves keÀyye[er ceW Yeeie efue³ee~
mJe³eb nue keÀerefpeS : 1) a) keÀe9 43 b) keÀe15 6
2 c) keÀe 475 d) keÀe20 6
5
2) SkeÀ keÀ#ee ceW 45 efJeÐeeLeea nQ~ GveceW 53 ue[efkeÀ³eeB nQ~ keÀe
keÀ#ee ceW ue[efkeÀ³eeW keÀer mebK³ee efkeÀleveer nw? keÀ#ee ceW efkeÀleves ue[kesÀ nQ %eele keÀerefpeS~
DeY³eeme 2.2
I. GÊej keÀes iegCeveHeÀue kesÀ ©He ceW efueefKeS.
GoenjCe : 34
64× 2 =+ =
©KTBS
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36
a) 41
41
41
41
41+ + + + = ___× ___ =
b) = ___× ___ =+21 +2
121
c) + = ___× ___ =
d) 31
31 + 3
1
31 + 3
1
31 + 3
1
31 = _× _ =
II. iegCee keÀjkesÀ iegCeveHeÀue keÀes mejue ©He ceW efueefKeS
1) 1) 72 2# 2) 5
4 3# 3) 92 5# 4) 8
7 6#
5) 9 21
# 6) 10 32
# 7) 15 53
# 8) 13 41
#
2) 1) 2 41 5# 2) 3 2
1 8# 3) 6 53 4# 4) 9 2 2
1#
5) 3 4 53
# 6) 2 1103
# 7) 14 2 71
# 8) 30 3107
#
III. mejue keÀerefpeS Deewj GÊej keÀes efvecve ©He ceW efueefKeS.
A) 1) keÀe6 32 2) keÀe10 5
3 3) keÀe16 81 4) keÀe28 7
2
5) keÀe 2043 6) keÀe 568
2 7) keÀe 6107 8) keÀe 49
5
©KTBS
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37
B) 1) ceOeg ves ieefCele ceW 50 kesÀ 53 DebkeÀ ÒeeHle efkeÀ³es~ Gmemes ÒeeHle DebkeÀ %eele keÀerefpeS~
2) SkeÀ mketÀue ves 60 efJeÐeeefLe&³eeW kesÀ efueS SkeÀ Ye´ceCe keÀer J³eJemLee keÀer~ ye®®eeW keÀes DeHevee Yeespeve ueeves keÀes keÀne ie³ee~ efJeÐeeefLe&³eeW kesÀ 12
3 ves ®eHeeleer ueskeÀj DeeS, 52 ves HeueeJe
Deewj 207 ves ef®e$evvee uee³es~ %eele keÀerefpeS
i) efkeÀleves efJeÐeeLeea ®eHeeleer ueskeÀj Dee³es?
ii) efkeÀleves efJeÐeeLeea HeueeJe uee³es?
iii) efkeÀleves efJeÐeeLeea ef®e$evvee uee³es?
efYevve keÀes efYevve mes iegCee keÀjvee
jsMcee ves SkeÀ otkeÀeve mes 43 kg MekeÌkeÀj Kejeroe~ FmekeÀe 2
1 Yeeie GmekesÀ He[esmeer ves
efue³ee~ %eele keÀerefpeS He[emeer ves efkeÀleveer MekeÌkeÀj ueer?
jsMcee mes Kejeroer ngF& MekeÌkeÀj keÀer cee$ee = 43 kg
He[esmeer mes efue³ee ngDee Yeeie keÀe43
21= kg
kg43
21
83
# =
³eneB oes efYevve nQ, Fme ieefCele ceW nce iegCeveHeÀue %eele keÀjvee meerKeWies~ Deye nce SkeÀ efYevve
keÀes otmejs efYevve mes iegCe keÀjvee meerKeWieW ~
GoenjCe 1 : Deeke=Àefle ceW íe³eebeqkeÀle keÀe 41 Yeeie efkeÀlevee nw?
íe³eebeqkeÀle Yeeie 21=
íe³eebeqkeÀle Yeeie keÀe 41 keÀe keÀe2
141 .
FmekeÀe DeLe& nw íe³eebeqkeÌle Yeeie ceW 4 meceeve Yeeie keÀjvee nw~ GmeceW mes 1 Yeeie ueerefpeS~
21 keÀe 4
1 nw mebHetCe& Jemleg keÀe 81 Jee@ Yeeie~
©KTBS
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38
oesveeW efYevveeW keÀes iegCee keÀjkesÀ Gmes DebMe ceW efueefKeS~
oesveeW efYevveeW kesÀ nj iegCee keÀjkesÀ Gmes nj ceW efueefKeS~
keÀe
keÀe
21
41
21
41
2 41 1
81
21
41
81
#
##
`
=
= =
=
Deye nce jsMcee kesÀ GHejeskeÌle ieefCele nue keÀjles nwb~
jsMcee mes Kejeroer ngF& MekeÌkeÀj keÀer cee$ee = 43 kg
He[esmeer mes ÒeeHle MekeÌkeÀj keÀer cee$ee 21= keÀe 4
3
kg43
21
4 23 1
83
#
##
=
= =
DebMe keÀe iegCeveHeÀue %eele keÀerefpeSnj keÀe iegCeveHeÀue %eele keÀerefpeS
∴He[esmeer mes ÒeeHle MekeÌkeÀj keÀer cee$ee 83= kg
mel³eeHeve keÀerefpeS: 43 keÀe 2
1 Deewj ke̳ee 21 keÀe 4
3 meceeve nw
mJe³eb nue keÀerefpeS :a) 3
141
# b) 53
31
#
c) 65
72
# d) 83
52
#
oewefvekeÀ peerJeve ceW nce mece, efJe<ece efYevve Deewj efceÞe efYevve iegCee keÀjles nQ~ Deye nce iegCee
keÀjvee meerKeWies~
GoenjCe 1 : SkeÀ iegef[³ee yeveeves kesÀ efue³es JeerCee ves 4 21 m keÀHe[e Kejeroe~ Gmeves keÀHe[s keÀe
31 m Yeeie iegef[³ee yeveeves ceW GHe³eesie efkeÀ³ee~ iegef[³ee yeveeves kesÀ efueS JeerCee ves efkeÀlevee keÀHe[s~ GHe³eesie efkeÀ³ee?
©KTBS
Not to
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publi
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39
JeerCee mes Kejeroer ngF& keÀHe[s keÀer cee$ee = 4 21= m
iegef[³ee yeveeves GHe³eesie efkeÀ³ee ie³ee keÀHe[e = 4 21= keÀe 3
1
4
33
21
31
29
31
2 39 1
69
23
23 1 2
1
#
#
##
#
#
=
=
= =
= = =
4 21 keÀes efJe<ece efYevve ceW yeoefueS]
ce. mee. De. ueskeÀj mejue keÀerefpeS
∴ 1 JeerCee ves iegef[³ee yeveeves kesÀ efueS 1 21 m keÀHe[e GHe³eesie efkeÀ³ee~
GoenjCe 2 :
SkeÀ ceerìj efjyyeve keÀe oece ̀ 4 53 nw~ ieewleceer ves 2 2
1 ceerìj efjyyeve Kejeroe~ Gmemes Kejeroe
ie³ee efjyyeve keÀe cetu³e %eele keÀerefpeS~
1 m efjyyeve keÀe oece = 4 53
2 21 m efjyyeve keÀe oece = 4 5
3 2 21
# ³eneB oesveeW efceÞe efYevve
= (oesveeW efceÞe efYevveeW keÀes efJe<ece efYevve ceW efueefKeS)
223 11 2
1= =
∴ ieewleceer mes Kejeros ieS 2 21 ceer efjyyeve keÀe cetu³e ` 11 2
1
©KTBS
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40
DeY³eeme 2.3
I. iegCee keÀjkesÀ iegCeveHeÀue keÀes mejue ©He ceW J³ekeÌle keÀerefpeS :
1) 65
32
# 2) 87
43
# 3) 54
97
# 4) 59
35
#
5) 31
97
# 6) 311
103
# 7) 74
59
# 8) 21
1213
#
II. iegCee keÀjkesÀ iegCeveHeÀue keÀes meyemes efvecve ©He ceW J³ekeÌle keÀerefpeS :
1) 43 2 5
1# 2) 5
4 3 65
# 3) 35 2 2
1# 4) 7
3 4 31
#
5) 4 53 2 2
1# 6) 5 4
123
# 7) 3 95 5# 8) 6 4
1 3 51
#
III. mJe³eb keÀerefpeS :1) 3 5
2 4# 2) 4 32
53
# 3) 5 43 2 2
1# 4) 25 3 2
1#
IV. nue keÀerefpeS :
1) SkeÀ mketÀue yevoiee[er(van) SkeÀ ueerìj ef[peue ceW 8 km otjer le³e keÀjlee nw~ ³eefo iee[er ceW
8 43 l ueerìj ef[peue nes lees, Jen efkeÀleveer otjer le³e keÀj mekeÀlee nw?
2) SkeÀ ueerìj otOe keÀe oece `34 21 nw~ 12 l ueerìj otOe keÀe oece ke̳ee nesiee?
3) jesMeveer SkeÀ SsefleneefmekeÀ GHev³eeme 1 43 Iebìs Òeefle efove He{vee Meg© efkeÀ³ee~ Gmeves Gme GHev³eeme
keÀes 6 efoveeW ceW He{keÀj HetCee& efkeÀ³ee~ Gme GHev³eeme keÀes He{ves efkeÀleves IeCìs efue³ee? %eele
keÀerefpeS~
4) SkeÀ ceerìj efpeHe keÀe cetu³e ` 5 41 nw~ 8 5
4 m efpeHe keÀe cetu³e %eele keÀerefpeS~
5) SkeÀ Heleuee Dee³eleekeÀej OeeeqlJekeÀ He$ekeÀ keÀer uebyeeF& 3 21 m. Deewj ®eew[eF& 2 2
1 m. nw~
GmekeÀe He=<þer³e #es$eHeÀue %eele keÀerefpeS~
©KTBS
Not to
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41
efYevveeW keÀe efJeYeepeve
jeOee ves 4 efyemeefkeÀì DeHeves mensefue³eeW kesÀ yeer®e Fme lejn efJeleefjle efkeÀ³ee leeefkeÀ Òel³eskeÀ 21
efyemeefkeÀì efceueW~ jeOee keÀer mensefue³eeW 21 efyemeefkeÀì ³eneB efoKee³ee ie³ee nw~
kegÀue efyemeefkeÀì
→
peye GvekeÀe DeeOee efkeÀ³ee ie³ee
→
peye efyeleefjle efkeÀ³ee ie³ee
4 efyemeefkeÀìeW keÀes 8 mensefue³eeW kesÀ yeer®e ceW efJeleefjle efkeÀ³ee ie³ee, efpevnW Òel³eskeÀ keÀes 21
efyemeefkeÀì efceuee nw~
Fmes Fmelejn efueKe mekeÀles nQ 4 8 ,21
21 8 4' #= =
Deye nce Fme Òeef¬eÀ³ee keÀes meerKeles nw~ peye 4 keÀes 21 Yeeie ueieeves keÀe DeLe& nw
4 × 12 [4 keÀes 1
2 mes iegCee keÀjvee ]
4
421
12
18
8
'
#= =
=
DebMe Deewj nj HejmHej mLeeve yeoueves Hej
21 mes 1
2 ÒeeHle keÀjles nQ~
Fmes 21 keÀe Òeefleueesce nw~
©KTBS
Not to
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publi
shed
42
Òeefleueesce ©He (Inverse Form)
peye efYevveeW kesÀ DebMe Deewj nj kesÀ mLeeve HejmHej yeoueles nQ, lees cetue efYevveeW kesÀ Òeefleueesce
nesles nQ~
53 keÀe Òeefleueesce = 3
587 keÀe Òeefleueesce = 7
8
49 keÀe Òeefleueesce = 9
432 keÀe Òeefleueesce = 2
3
• 2 43 keÀe Òeefleueesce ke̳ee nw?
Henues Fmes efJe<ece efYevve ceW yeoefueS, yeeo ceW Òeefleueesce ceeuetce keÀerefpeS :
2 43 keÀe efJe<ece efYevve = 4
11
411 keÀe Òeefleueesce = 11
4
• 8 keÀe Òeefleueesce ke̳ee nw?
∴ 8 keÀes 18 kesÀ ©He ceW efueKe mekeÀles nQ~ 8 1
8=` j
∴ 8 keÀe Òeefleueesce = 81
SkeÀ efYevve keÀes GmekesÀ Òeefleueesce mes iegCee keÀjves Hej ÒeeHle iegCeveHeÀue Hej O³eeve oerefpeS.
1
1
1
41
14
44
11
52
25
1010
11
32
23
66
11
5 51
55
11 1
#
#
#
#
= = =
= = =
= = =
= = =
DeeHekeÀe efveCe&³e ke̳ee nQ?JeCe&ve keÀerefpeS
met®evee : Metv³e kesÀ DeueeJee, SkeÀ mebK³ee keÀes GmekesÀ Òeefleueesce mes iegCee keÀjves Hej iegCeveHeÀue `1' neslee nw~
©KTBS
Not to
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43
mJe³eb nue keÀerefpeS: efvecveeW kesÀ Òeefleueesce efueefKeS :a) 4
3 b) 95 c) 1
13 d) 3 41
SkeÀ HetCe& mebK³ee keÀes efYevve mes Yeeie ueieevee
GoenjCe 1 : mejue keÀerefpeS 8 43
'
8 43
8 34
18
34
332 10 3
2
'
#
#
=
=
= =
®ejCe 1 : Yeeie kesÀ ef®evn keÀes iegCee ceW yeoefueS Deewj YeepekeÀ keÀe Òeefleueesce ueerefpeS~®ejCe 2 : iegCeveHeÀue Òeef¬eÀ³ee HetCe& keÀerefpeS.
GoenjCe 2 : leerve Je=Êeer³e ef[mkeÀ mes efkeÀleves 41 Yeeie ÒeeHle keÀj mekeÀles nQ ?
3 ef[mkeÀ mes nceW 12 meceeve Yeeie ÒeeHle nesles nQ pees Je=Êe kesÀ 41 kesÀ nQ~
Fmes Fmelejn efueKe mekeÀles nQ.
3 41
' Yeeie ef®evn keÀes iegCee ceW yeoefueS Deewj YeepekeÀ keÀe Òeefleueesce ueerefpeS~ 41 .
3 14
#= yeeo ceW iegCeveHeÀue Òeef¬eÀ³ee Deeies ye{eFS~
1 13 4
112 12
##= = =
∴Je=leer³e ef[mkeÀ mes 41 kesÀ 12 Yeeie ÒeeHle nesles nw~
©KTBS
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44
GoenjCe 3 : mejue keÀerefpeS 12 2 65
'
peye YeepekeÀ efceÞe efYevve ceW nes lees Henues Gmes efJe<ece efYevve ceW efueefKeS Deewj yeeo SW nue keÀerefpeS
12 2
12
12
65
617
176
1772 417
4
'
'
#
=
=
= =
mJe³eb nue keÀerefpeS:1) 6 5
3' 2) 8 4
1' 3) 2 5
3'
4) 9 73
' 5) 15 2 21
' 6) 12 3 41
'
SkeÀ efYevve keÀes HetCe& mebK³ee mes Yeeie ueieevee :
GoenjCe 1 : 53 2'
³eneB YeepekeÀ 2 nw, Yeep³e keÀes 2 kesÀ Òeefleueesce mes iegCee keÀerefpeS~
53 2 5
321
103' #= =
Fmeerlejn,
1) 587
87
51' #= _________
2) 1065' = _________ = _________
GoenjCe 2 : SkeÀ efceÞe efYevve keÀes SkeÀ HetCe& mebK³ee mes iegCee keÀjvee,
?5 41 9
421 9 4
2191
3621
'
' #
=
= = (efceÞe efYevve keÀes efJe<ece efYevve ceW yeoefueS)
©KTBS
Not to
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shed
45
Fmeerlejn,
1) 3 5 521
27
27
51' ' #= = = _________
2) 4 732' = _________ = _________ = _________
SkeÀ efYevve keÀes otmejs efYevve mes Yeeie keÀjvee
GoenjCe 1 : 41
53
' 41 keÀes 5
3 mes Yeeie ueiee³ee ie³ee nw~
³eneB 41 keÀe YeepekeÀ 5
3 kesÀ Òeefleueesce mes iegCee keÀjvee nw~
lees 41
53
41
35
125' #= =
GoenjCe 2 : ?32
83
' =
132
83
32
38
916
97' #= = =
Fmeer ÒekeÀej 1) 107
31
107
13' #= = _________ = _________
2) 65
73
' = _________ = _________
DeY³eeme 2.4
I. mejue keÀerefpeS~
1) 9 21
' 2) 13 43
' 3) 15 61
'
4) 20 73
' 5) 15 2 31
' 6) 10 3 72
'
II. FvekesÀ Òeefleueesce efueefKeS~
1) 52 2) 9
7 3) 121 4) 2 6
1 5) 9 6) 4 32
©KTBS
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46
III. mejue keÀerefpeS~
1) 32 5' 2) 8
7 3' 3) 76 13'
4) 59 4' 5) 2 4
3 7' 6) 3 21 14'
IV. mejue keÀerefpeS~
1) 85
31
' 2) 65
72
' 3) 94
58
'
4) 54 1 2
1' 5) 3 4
3 1 32
' 6) 3 72 1 5
1'
V. nue keÀerefpeS~
1) 6 ®ee@keÀuesì keÀe oece ` 10 21 nw~ SkeÀ ®ee@keÀuesì keÀe oece ke̳ee nw?
2) SkeÀ mketÀue keÀes 12 43 ueer. otOe SkeÀ efove kesÀ efueS efo³ee ie³ee~ Òel³eskeÀ ye®®es keÀes 20
3
ueer. otOe osves Hej efkeÀleves ye®®eeW keÀes otOe Gme efove efceuelee nw?
3) oes ye®®es ves 15 veesì yegkeÀ Kejeroles nQ~ ³eefo 15 veesì yegkeÀ keÀe cetu³e ` 142 21 , nw lees SkeÀ
veesì yegkeÀ keÀe cetu³e %eele keÀerefpeS~
4) 10 54 ueer otOe ceW mes 5
2 ueer otOe kesÀ efkeÀleves Hee@kesÀì yevee mekeÀles nQ?
efceÞe Òeef¬eÀ³eeSB (Mixed Operations)
owefvekeÀ peerJeve ceW DeveskeÀ Òemebie Deeles nQ efpeveceW nce efYevveeW kesÀ efJeefYevve Òeef¬eÀ³eeDeeW keÀe GHe³eesie keÀjles nQ~
DeYeer nce kegÀí GoenjCeeW Hej efJe®eej keÀjles nQ~ GoenjCe Demecee kesÀ Heeme ̀ 3 21 Les~ Gmeves
` 2 43 mes 1 HeWefmeue Deewj ` 2
1 mes jye[ Kejeroe~ GmekesÀ Heeme efkeÀlevee Hewmee yeekeÀer jnlee nw?nue : nceW HeWefmeue Deewj jye[ Hej Ke®e& efkeÀ³es HewmeeW keÀes Ieìevee He[siee~
©KTBS
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47
3 21 2 4
321
27
411
2 21 2
27
411
42
2 27 2
413
414
413
414 13
41
##
##
- +
= - +
= - +
= -
= - = - =
`
c
`
j
m
j
← efpeve efYevveeW keÀes pees[vee nw GvekesÀ nj kesÀ ue. mee. De efvekeÀeefueS~
← nj keÀes meceeve keÀjves Hej
∴ Demecee kesÀ Heeme ` 41 yeekeÀer jnles nQ~
GoenjCe 2 :
SkeÀ opeea ves keÀHe[eW kesÀ 12 ìgkeÀ[s Kejeroe pees Òel³eskeÀ 52 m kesÀ nw~ Gmeves 2 8
1 m kesÀ oes
HejoeW keÀer efmeueeF& keÀer~ yeekeÀer keÀHe[s keÀer uebyeeF& %eele keÀerefpeS~
keÀHe[s keÀer kegÀue uebyeeF& 52 12#=
oes HejoeW keÀer efmeueeF& kesÀ efueS 2 81 2#=
GHe³eesie efkeÀ³ee ngDee keÀHe[e
52 12 2 8
1 2# #= -` `j j
G½zÀ§mÉÖ m
52 12 2 8
1 2
52
112
817
12
524
417
5 424 4
4 517 5
2096
2085
2096 85
2011
2011
4
1
# #
# #
##
##
`
-
= -
= -
= -
= -
= - =
=
` `
` e
j j
j o
← nj keÀe ue. mee. De ueerefpeS5 Deewj 4 keÀe uee. mee. De 20 nw~
©KTBS
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48
GoenjCe 3 : mejue keÀerefpeS : 2 21
21
41
52
21# '+ -
GHejeskeÌle GoenjCe ceW iegCee, pees[, Yeeie Deewj Ieìeves mes peg[e nw~ (BODMAS efve³ece kesÀ Devegmeej nceW Henues Yeeie ueieevee nw yeeo ceW iegCee, yeeo ceW pees[vee Deewj Devle ceW Ieìevee nw~)
2 21
21
41
52
21# '+ -` j
®ejCe 1852 2
121
21
2 21
21
85
21
#
#
+ -
= + -
`
`
j
j
a41
52
41
25
85' #= =
®ejCe 245
85
21
45
85
21
+ -
= + -` j
a 2 21
21
25
21
45
# #= =
®ejCe 38
1521-
815
2 41 4
815
84
815 4
811
##-
= -
= - =
a45
85
4 25 2
85
810 5
815
##+ = +
= + =
®ejCe 4 1811
83=
DeY³eeme 2.5
I. mejue keÀerefpeS.
1) 87
41
21
#- 2) 2 53
53 1 2
1103
#+ -
3) 2 41
43
41 2 2
1# '- 4) 4 21
41
51
53
'- +
©KTBS
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49
5) 2 53
53
27
#+ 6 153
54
31
41' #+
II. efvecve àíZmo keÀe nue keÀerefpeS :
1) SkeÀ keÀe³e&¬eÀce kesÀ efueS mketÀue ves 15 m efjyyeve Kejeroe~ efpemeceW 441 m ceer leesjCe kesÀ
efueS GHe³eesie. efkeÀ³ee ie³ee, 4107 m efyeuuee yeveeves, yeekeÀer efj<yeve nmleefueefHe Heef$ekeÀe keÀes
mepeeves kesÀ GHe³eesie keÀjles nQ~ nmleHeef$ekeÀe mepeeves kesÀ efueS efkeÀleveer efjyyeve keÀe GHe³eesie
%eele keÀerefpeS ngDee~
2) SkeÀ keÀe³e&¬eÀce kesÀ Dee³eespekeÀ ves 3 21 kg MekeÌkeÀj kesÀ 4 iegvee Kejeroe~ GmeceW 9 4
3 kg
MekeÌkeÀj GHe³eesie efkeÀ³ee ie³ee~ efkeÀleveer MekeÌkeÀj yeekeÀer nw %eele keÀerefpeS~
3) SkeÀ Meì& efmekeÀeves kesÀ efueS jepeg keÀes 2 21 m keÀHe[s keÀer DeJej³ekeÀlee nw~ otkeÀeveoej kesÀ
Heeme 21 m mkesÀue nw~ 2 2
1 m keÀe ceeHe keÀves kesÀ efueS Gmes efkeÀleveer yeej GHe³eesie keÀjvee
®eefnS?
©KTBS
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50
DeO³ee³e - 3
Heefjces³e mebK³eeSB
Fme IeìkeÀ kesÀ DeO³eve kesÀ yeeo DeeHe :
Heefjces³e mebKe³eeDeeW keÀe DeLe&, ceevekeÀ ©He Deewj GvekesÀ mecelegu³e peeveWies~
mebK³ee jsKee Hej Heefjces³e mebK³eeDeeW keÀes efve©efHele keÀjves keÀe keÀewMeewu³e ye{e³esbies,
Heefjces³e mebK³eeDeeW keÀer leguevee oes Heefjcesce mebK³eeDeeW kesÀ yeer®e Heefjces³e mebK³ee %eele keÀjWies,
Heefjces³e mebK³eeDeeW keÀer Òeef¬eÀ³eeDeeW keÀes peeveWies, (pees[vee, Ieìevee, iegCee keÀjvee, Yeeie ueieevee) Deewj Gvemes mebyebefOele keÀLeveeW ceW oled ieefCele nue keÀjves kesÀ efJeOeeve peevesieW,
Heefjces³e mebK³eeDeeW keÀes oMeceueJe ©He ceW J³ekeÌle keÀjves keÀe efJeOeeve peeveWies,
oMeceueJeeW keÀes iegCee Deewj Yeeie ueieeves keÀer Òeef¬eÀ³ee peeveWiesb Deewj Gvemes mebyebefOele ieefCele nue keÀjvee peeveWies,
ceeHeve FkeÀeF&³eeW keÀes HeefjJele&ve keÀjves keÀe efJeOeeve mecePeWies~
(uebyeeF& Deewj êJ³eceeve kesÀ ceeHeve kesÀ yeejs ceW)
Heefjces³e mebK³eeDeeW keÀe DeLe&:
DeeHe mJeYeeefJekeÀ mebK³ee (N), HetCe& mebK³ee (W), HetCee¥keÀeW (Z) kesÀ yeejs ceW peeveles nQ~
N = { 1, 2, 3, 4....................}
W = { 0, 1, 2, 3....................}
Z = {....................-4, -3, -2, -1, 0, 1, 2, 3....................}
©KTBS
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51
Meg©Jeele ceW kesÀJeue mJeYeeefJekeÀ mebK³eeSB cee$e Leer~ SkeÀ Jemleg vee nesves keÀes metef®ele keÀjves `0'
(Metv³e) keÀe DeeefJe<³ekeÀej efkeÀ³ee ie³ee~ Fme lejn, HetCe& mebK³eeSB GHe³eesie ceW Dee ieF&~ yeeo ceW,
HetCee¥keÀesb keÀe meceg®®e³e efJekeÀefmele ngDee keÀ³eeWefkeÀ $eÝCeelcekeÀ mebK³eeSB efJekeÀefmele ngS ke̳ee ngS~
oes mJeYeeefJekeÀ mebK³eeDeeW keÀe WDeblej %eele keÀjles mece³e (GoenjCe: 3-5=-2, 10-15=-5) SkeÀ
mebK³eeDeeW keÀe ve³ee meceg®®e³e ÒeeHle neslee nw FvekeÀes $eÝCeelcekeÀ HetCee¥keÀ keÀnles nQ~
FmeefueS, mebK³eeDeeW kesÀ meceg®®e³e keÀe efJekeÀeme DeeJeM³ekeÀe kesÀ keÀejCe ngDee nw~
Fmes peeve ueerefpeS : meceg®®e³e Meyo, megHeefjYeeef<ele mebyeebefOele Jemleg efJe®eej DeLeJee mebK³eebkeÀeW keÀes metef®ele keÀjlee nw~
Fmeer lejn, SkeÀ HetCee¥keÀ keÀes otmejs HetCee¥keÀ mes Yeeie ueieeves Hej SkeÀ ve³ee meceg®®e³e ÒeeHle
neslee nw~
GoenjCe : ,53
710- Ssmeer mebK³eeSB HetCee¥keÀesb kesÀ meceg®®e³e ceW veneR nw FmeefueS Heefjces³e
mebK³eeDeeW keÀe meceg®®e³e (Q) DeefmlelJe ceW Dee ie³ee~
efHeíueer keÀ#eeDeeW ceW nce ves efYevveeW kesÀ yeejs ceW meerKee nw~ efYevve pq kesÀ ©He ceW nesvee ®eeefnS,
peneB p Deewj q oesveeW mJeYeeefJekeÀ mebK³ee nesleer nwb~ Hejvleg pq ©He kesÀ efYevveeW keÀes Ieìeles mece³e
GoenjCe ceW efo³es pewmes $eÝCeelcekeÀ mebK³ee ÒeeHle keÀjves keÀer mebYeeJevee nw~ FmeefueS, veF& mebK³eeSB
efpeve ceW GHeueyOe nw pees velees efYevve nw ve HetCee¥keÀ nQ~pq ©He kesÀ meYeer mebK³eeSB Heefjces³e mebK³eeSB nQ~
pq b>0, p Deewj b HetCee¥keÀ nw~
Heefjces³e mebK³eeDeeW kesÀ meceg®®e³e keÀes Q mes metef®ele keÀjles nwb~
Q = pq ©He kesÀ meYeer mebK³eeSB meceeefJe<ì nQ, peneB b > 0, Deewj a Deewj b HetCee¥keÀ nQ ~
GoenjCe: 32 , 4
5 , 30 , 7
47
15 , 94 , 10
3- - -- ........
nce meYeer mJeYeeefJekeÀ mebK³eeDeeW keÀes, HetCee¥keÀeW Deewj HetCe& mebK³eeDeeW keÀes Heefjces³e mebK³ee kesÀ
©KTBS
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52
©He ceW efueKe mekeÀles nQ~
GoenjCe : 3 ,12 ,8
0 ,0 ,0
, ,
13
112
18
10
20
100
5 15 7 1
7 10 110
= = =
= = =
- = - - = - - = -
FmeefueS Heefjces³e mebK³eeDeeW keÀe meceg®®e³e ye[e meceg®®e³e nw~
NW
ZQ
meesef®eS ! meYeer efYevve Heefjces³e mebK³eeSB nQ~ Hejvleg meYeer Heefjces³e mebK³eeSB efYevve veneR nw? keÀejCe oerefpeS~
Fmes peeve ueerefpeS : Rational Numbers Meyo Ratio :ba a b= mes yevee nw~ :2
1 1 2- = - DevegHeele nw
Heefjces³e mebK³eeDeeW keÀe ceevekeÀ ©He (mejuelece ©He)
efvecve Heefjces³e mebK³eeDeeW keÀes O³eeve mes osefKeS
, , , ,53
41
107
73
114-
Fve mebK³eeDeeW ceW DebMe Deewj nj keÀe meceev³e iegCeveKeC[ 1 (DeLeJee ce. mee. De. 1 nw) Fve
Heefjces³e mebK³eeDeeW keÀes ceevekeÀ ©He (mejue ©He) ceW ceevee peelee nw~
104 Hej efJe®eej keÀerefpeS 10
452= (DebMe Deewj nj keÀes 2 mes Yeeie oerefpeS)
104
52` = ( 10
4 keÀe ceevekeÀ ©He 52 nw)
2515
53= (DeeoMe& ©He) 25
1553
5
3
= ceevekeÀ ©He (DeMe Deewj nj keÀes 5 mes Yeeie keÀerefpeS)
3627
43- = - (ceevekeÀ ©He) 36
2743- = - (DebMe Deewj nj keÀes9 mes Yeeie keÀerefpeS)
©KTBS
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53
³eefo keÀesF& Heefjces³e mebK³ee ceevekeÀ ©He ceW veneR nw lees GmekesÀ DebMe Deewj nj keÀes SkeÀ mebK³ee
mes Yeeie oskeÀj ceevekeÀ ©He DeLeJee mejuelece ©He ÒeeHle keÀj mekeÀles nQ~
GoenjCe: 1) 5522
52- = - 2) 30
16158
-- = 3) 39
2632- = - 4) 63
392113=
mecelegu³e Heefjces³e mebK³eeSB :
mecelegu³e efYevve ÒeeHle keÀjves keÀe efJeOeeve ³eeo keÀerpefS~ Fmelejn oÊe Heefjces³e mebK³ee keÀer
mecelegu³e Heefjces³e mebK³ee, GmekesÀ DebMe Deewj nj keÀes SkeÀ Metv³e jefnle mebK³ee mes iegCee keÀjves
DeLeJee Yeeie osves mes %eele keÀj mekeÀles nQ~
GoenjCe 1 : 32 kesÀ 4 mecelegu³e Heefjces³e mebK³eeSB efueefKeS~
1) 3 22 2
64
## = 2) 3 3
2 396
## = 3) 3 4
2 4128
## = 4) 3 5
2 51510
## =
GoenjCe 2 : 75- kesÀ 4 mecelegu³e Heefjces³e mebK³eeSB efueefKeS~
1) 7 25 2
1410
##- = - 2) 7 3
5 32115
##- = -
3) 7 45 4
2820
##- = - 4) 7 5
5 53525
##- = -
GoenjCe 3 : 8040 kesÀ 4 Heefjces³e mebK³eeSB efueefKeS~
1) 80 240 2
4020
'' = 2) 80 4
40 42010
'' =
3) 80 540 5
168
'' = 4) 80 10
40 1084
'' =
©KTBS
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54
Fmes peeve ueerefpeS: ³eefo ba
dc= mecelegu³e Heefjces³e mebK³eeSB nQ lees a × d = b × c
mebK³eejsKee Hej Heefjces³e mebK³eeDeeW keÀes efve©efHele (DebefkeÀle keÀjvee) :
HetCe& mebK³ee Deewj HetCee¥keÀes keÀes mebK³ee jsKee Hej efve©efHele keÀjvee nce peeveles nw~
Metv³e kesÀ oeefnveer Deesj IeveelcekeÀ mebK³ee Deewj yee³eeR Deesj $eÝCeelcekeÀ mebK³ee meceeve otjer
Hej nesles nwb~
GoenjCe 1 : 53 keÀes mebK³eejsKee Hej efve©efHele keÀerefpeS~
mebK³eejsKee Hej Heefjces³e mebK³ee efve©efHele keÀjves, mebK³ee jsKee kesÀ FkeÀeF& ceW, Heefjces³e mebK³ee
kesÀ nj kesÀ Devegmeej meceeve Yeeie yeveeFS~ yeeo ceW, Heefjces³e mebK³ee kesÀ DebMe kesÀ Devegmeej YeeieeW
keÀes DebeqkeÌle keÀerefpeS~
0
0 +1-1
53+
O³eeve oerefpeS, 0 Deewj 1 kesÀ yeer®e keÀer otjer keÀes 5 meceeve YeeieeW ceW efJeYeeefpele efkeÀ³ee ie³ee nw~
GoenjCe 2 : SkeÀ mebK³ee jsKee Hej 25 efve©efHele keÀerefpeS~
0
-3 -2 -1 0 +1 +2 +3
25
O³eeve oerefpeS Òel³eskeÀ FkeÀeF& ceW oes Yeeie yevee³es ieS nQ~
GoenjCe 3: mebK³ee jsKee Hej 47- efve©efHele keÀerefpeS~
©KTBS
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55
-2 -1 0 +1 +2
047-
DeY³eeme 3.1
I. yeleeFS efvecve keÀLeve mener nw ³ee ieuele~
1) - 12 SkeÀ Heefjces³e mebK³ee nw~
2) 03 SkeÀ Heefjces³e mebK³ee nw~
3) 43- SkeÀ Heefjces³e mebK³ee nw~
4) 75- SkeÀ Heefjces³e mebK³ee nw~
5) 40 SkeÀ Heefjces³e mebK³ee nw~
6) 85- SkeÀ IeveelcekeÀ Heefjces³e mebK³ee nw~
7) 2812 DeHeves mejuelece ©He ceW nw~
II. Fve Heefjces³e mebK³eeDeeW keÀes ceevekeÀ ©He ceW efueefKeS~
1) 2028 2) 150
120 3) 4015- 4) 56
32--
III. efvecve Òel³eskeÀ kesÀ 4 mecelegu³e Heefjces³e mebK³eeSB efueefKeS~
1) 92- 2) 10
3 3) 54- 4) 96
80
IV. Fve Heefjces³e mebK³eeDeeW keÀes mebK³eejsKee Hej efve©efHele keÀerefpeS~(Òel³eskeÀ mecetn kesÀ efueS
Deueie jsKeeDeeW keÀes GHe³eesie keÀerefpeS)
1) , ,73
74
75- 2) , ,4
745
49- 3) , ,3
532
38- 4) , ,5
657
58-
©KTBS
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56
V. efvecveefueefKele Heefjces³e mebK³eeDeeW keÀes IeveelcekeÀ Deewj $eÝCeelcekeÀ Heefjces³e mebK³eeDeeW
ceW Jeieeake=Àle keÀerefpeS~
, , , , , ,54
67
35
710
97
152
611-
- -- -
--
VI. efjkeÌle mLeeveeW keÀer Hetefle& keÀerefpeS.
1) 43 12= 2) 6
524
- = 3) 2812 3- = -
4) 2015
4= 5) 18149- = 6) 12 2
5- = -
Heefjces³e mebK³eeDeeW keÀer leguevee :
oes efYevveeW keÀer leguevee keÀjvee nce peeveles nQ; Deewj Gmeer lejn Heefjces³e mebK³eeDeeW keÀer leguevee keÀjles nQ~
43 Deewj 4
1 ceW 43 ye[e nw~ ³eefo oes efYevveeW kesÀ nj meceeve nw lees efpeme
efYevve keÀe DebMe ye[e nw Jen ye[e nw~107 Deewj 10
3 ceW 107 ye[e nw~
52- Deewj 5
4- ceW 52- ye[e nw~
Heefjces³e mebK³eeDeeW Hej Òeef¬eÀ³eeSB
HetCee¥keÀeW Deewj efYevveeW Hej DeeHe ves pees[vee, Ieìevee, iegCee keÀjvee Deewj Yeeie ueieevee meerKee nw~ DeeDees FvneR Òeef¬eÀ³eeDeeW keÀes Heefjces³e mebK³eeDeeW Hej keÀj kesÀ osKeles nQ~
pees[ (³eesieHeÀue): efYevveeW keÀe efve³ece ner ³eneB Hej GHe³eesie keÀjvee nw~ ³eefo nj meceeve nw, lees nj Jenerb jKekeÀj, DebMeeW keÀes pees[keÀj, Gmes DebMe ceW efueefKeS~
©KTBS
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57
GoenjCe 1 :
³eefo nj Deueie - Deueie nes lees, nj keÀe ue. mee. De ueskeÀj GvnW SkeÀ ner nj ceW yeoue oerefpeS Deewj yeeo ceW peesef[S~
1) 32
31
32 1
33 1+ = + = =
2) 52
51
52 1
53+ = + =
3) 87
82
87 2
85- + = - + = -
4) ( )107
103
107 3
104
52+ - = + -
= =
GoenjCe 2 :
1)32
41
128 3
1211+ = + = ue. mee. De Deewj 4 = 12
2)53
107
106 7
101
- +
= - + =
` j
ue. mee. De 5 Deewj 10 = 10
3)65
54
3025 24
3049
- + -
= - - =-
` `j j ue. mee. De 6 Deewj 5 = 30
4)35
21
43
1220 6 9
1235
+ +
= + + =
ue. mee. De 3,4,2 = 12 3,2,42
32
3,1,21,1,21,1,1
2 × 3 × 2 = 12
GoenjCe 3 : Meebleccee ves 21 kg mesce, 4
1 kg njer efce®e&, 23 kg Deeuet
101 kg DeojkeÀ SkeÀ
meypeer kesÀ otkeÀeve mes Kejeroleer nw~
Gmeves kegÀue efkeÀleves efkeÀueesûeece meypeer Kejeroer nw?
meceeOeeve : mesce keÀe Jepeve = 21 kg
©KTBS
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njer efce®e& keÀe Jepeve = 41 kg
Deeuet keÀe Jepeve = 23 kg
DeojkeÀ keÀe Jepeve = 101 kg
∴ kegÀue Jepeve
kg
21
41
23
101
2010 5 30 2
2047
2 207
= + + +
= + + +
=
=
2, 4, 2, 10
2,4,2,101,2,1,51,1,1,51,1,1,1
225
2 × 2 × 5 = 20
GoenjCe 4 : SkeÀ yeeuekeÀ DeHeves keÀHe[s efmeueJeeves kesÀ efueS keÀHe[e Kejerovee ®eenle nw~ Meì&, efmeueJeeves 4
5 m 32 m HeQì efmeueJeeves Deewj 4
1 m keÀHe[e ìesHeer efmeueJeeves ke sÀefueS Kejerolee nw~ kegÀue efceueekeÀj Gmes efkeÀlevee ceerìj keÀHe[e ®eeefnS?
meceeOeeve : Meì& kesÀ efueS DeeJeM³ekeÀ keÀHe[e = 45 m
HeQì kesÀ efueS DeeJeM³ekeÀ keÀHe[e = 32 m
ìesHeer kesÀ efueS DeeJeM³ekeÀ keÀHe[e = 41 m
∴kegÀue DeeJeM³ekeÀ keÀHe[e
m
45
32
41
1215 8 3
1226
613
2 61
= + +
= + +
=
=
=
4, 3, 4 keÀe ue. mee. De
4,3,41,3,11,1,1
43
ue. mee. De = 4 × 3 = 12
J³eJekeÀueve (Ieìevee) :
HetCee¥keÀeW ceW, 8 ceW mes 5 Ieìeves keÀe DeLe& nw 8 + (-5) DeLee&le SkeÀ mebK³ee keÀer pees[ keÀer Òeefleueesce keÀes pees Ieìevee nw Gmes pees[vee nw~ 8 + (-5) = 3
Fmeerlejn 7 keÀes (- 4) ceW mes Ieìevee nw~
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7 + 4 = 11 (a -4 keÀer pees[ keÀer Òeefleueesce mebK³ee + 4 nw)
Heefjces³e mebK³ee keÀes Ieìeles mece³e ceW Yeer ³ener efve³ece DevegmejCe keÀjles nQ~
GoenjCe 1 : a) 75 ceW mes 7
1 keÀes IeìeFS~
= 57 +(- 1
7 ) = 57 - 1
7 = 5 -17 = 4
7
b) 87 keÀes 8
1-
= 7
8 -(- 18 ) = 7
8 + 18 = 7 +1
8 = 81
81 = 1
efYevve nj kesÀ Heefjces³e mebK³eeDeeW keÀes Ieìeles mece³e SkeÀ ner nj ceW HeefjJeefle&le keÀjkesÀ IeìeFBS~
GoenjCe 2 :65
32
65 4
61
-
= - =
6 Deewj 3 keÀe ue. mee. De = 66,32,11,1
32
3 × 2 = 6
GoenjCe 3 :7
1143
2844 21
2823
-
= -
=
7 Deewj 4 keÀe ue. mee. De = 28 7,41,41,1
74
7 × 4 = 28
GoenjCe 4 : ( 49 ) ceW mes (15
18) keÀes IeìeFS
94
185
188 5
1813
= - -
= - -
= -
9 Deewj 18 keÀe ue. mee. De = 18
9,181,21,1
92
9 × 2 = 18
GoenjCe 5 :
ieebOeer pe³ebefle kesÀ DeJemej Hej SkeÀ efMe#ekeÀ ves ye®®eeW cesb efJeleefjle keÀjves 13 43 kg efceþeF&
Kejeroe~ ³eefo 12 87 kg efceþeF& efJeleefjle yeekeÀer efkeÀ³ee lees efkeÀleves ®ee@keÀuesì jnles nw?
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60
Kejeros ngS kegÀue efceþeF& keÀe Jepeve = 13 43 kg
efJeleefjle efceþeF& keÀe Jepeve = 12 87 kg 4,8
1,21,1
42
∴ yeekeÀer efceþeF& keÀe Jepeve 13 43 12 8
7
455
8103
8110 103
= -
= -
= -
ue. mee. De = 4 × 2 = 8
∴ yeekeÀer efceþeF& keÀe Jepeve 8
7= kg
GoenjCe 6 :
jefnce ves DeHeves Jesleve ceW mes 21 Yeeie Deenej kesÀ efueS, Yeeie ye®®eeW kesÀ efMe#ee kesÀ efueS,
41 Yeeie Dev³e Ke®ee& Hej Deewj yeekeÀer Yeeie ye®ele keÀjlee nw~ lees Gmeves DeHeves Jesleve keÀe efkeÀlevee
Yeeie ye®ele keÀjlee nw?
meceeOeeve:
jefnce mes Deenej Hej Ke®e& efkeÀ³ee ie³ee Jesleve keÀe Yeeie = 21
ye®®eeW kesÀ efMe#ee Hej Ke®e& Jesleve keÀe Yeeie = 51=
Dev³e Ke®eeX Hej GHe³eesie efkeÀ³ee Jesleve keÀe Yeeie = 41=
∴ Ke®e& efkeÀ³ee ie³ee Jesleve keÀe Yeeie 21
51
41
2010 4 5
2019
= + +
= + +
= Yeeie
2,5,41,5,21,1,21,1,1
252
2 × 5 × 2 = 20
ceevee ueerefpeS Jesleve 1 nw
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∴ Ke®e& keÀjves kesÀ yeekeÀer Jesleve keÀe Yeeie 1 2019
2020 19
201
= -
= -
= Yeeie
∴ jefnce DeHeves Jesleve keÀe 201 Yeeie Jesleve keÀjlee nw~
DeY³eeme 3.2
I. efvecveeW keÀe pees[ %eele keÀerefpeS~
1) 52
107+ 2) 3
292+ 3) 5
365- + 4) 7
531
23+ +
II. IeìeFS~
1) 85 keÀes 4
1 ceW mes 2) 65 keÀes 6
1- ceW mes
3) 53 keÀes 15
4 ceW mes 4) 87 keÀes 4
3 ceW mes
III. mejue keÀerefpeS~
1) G<ee ves 5 43 kg efkeÀueesûeece oeue Deewj 2 3
8 kg meypeer Kejeroe~ kegÀue meeqype³eeW keÀe Jepeve ke̳ee nw?
2) #eerj Yeei³e ³eespevee kesÀ Debleie&le SkeÀ mketÀue keÀes 5 431 kg otOe keÀe HeeT[j Òeefle meHleen
ÒeeHle neslee nw~ GmeceW mes 14 51 kg otOe keÀe HeeT[j GHe³eesie neslee nw, yeleeFS otOe kesÀ
HeeT[j keÀe efkeÀlevee Yeeie yeekeÀer jnlee nw?
3) jefJe ves 8 75 m keÀHe[e SkeÀ otkeÀeve mes Kejeroe~ GmeceW mes 4 3
1 m ceerìj Yeeie Heg$e kesÀ efueS Deewj 3 5
3 m ceerìj GmekesÀ Heg$eer kesÀ efueS GHe³eesie ngDee~ efkeÀlevee keÀHe[e yeekeÀer jnlee nw %eele keÀerefpeS~
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4) ueeruee kesÀ Iej 4 iee³e SkeÀ efove 20 53 l otOe osleer nQ~ ueeruee 18 3
2 l otOe yes®eleer nw~ Mes<e ye®ee ngDee otOe efkeÀlevee nw?
Heefjces³e mebK³eeDeeW keÀe iegCeveHeÀue:
efYevveeW ceW iegCee efkeÀ³es pewmes Heefjces³e mebK³eeDeeW keÀes Yeer iegCee keÀj mekeÀles nwb~
GoenjCe 1 :
1) 43 3 4
9# = met®evee : Heefjces³e mebK³eeDeeW keÀes iegCee keÀjles mece³e,
DebMe keÀes DebMe mes Deewj nj keÀes nj mes iegCee keÀjles nwb~2) 3
234
98
# =
3) 43 5 4
15#- = -
4) -21
5 ×(- 342
) = 310
5) 43
21
65
4 2 21 5
165
1
2= =# #
# ##
GoenjCe 2 :
SkeÀ efJeOeeLeea ves100 HevveeW kesÀ SkeÀ HegmlekeÀ keÀe 52 Yeeie SkeÀ efove ceW He{lee nw; lees
Gmeves efkeÀleves Hevves Gme efove He{e?
meceeOeeve: kegÀue Hevves = 100
efJeOeeLeea mes He{e ngDee HegmlekeÀ keÀe Yeeie = 100 keÀe 52 Yeeie
10052 4020= =# Hevves.
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GoenjCe 3 :
SkeÀ ceerìj keÀHe[s keÀe cetu³e ` 30 43 nw Gmeer oj Hej 14 2
1 m keÀHe[s keÀe cetu³e ke̳ee
nesiee? 1 ceerìj keÀHe[s keÀe cetu³e = ` 30 43 nw~
meceeOeeve : 1 m keÀHe[s keÀe cetu³e 30` 43=
14 21` m keÀHe[s keÀe cetu³e
kegÀue cetu³e
30 43 14 2
1
4123
229
3567
14 445 `m8 445 8
1
21
81
=
=
#
#=
=
=
Heefjces³e mebK³ee keÀe efJeYeepeve:
DeeHe peeveles nQ efkeÀ, SkeÀ efYevve keÀes, otmejs efYevve mes Yeeie ueieeves keÀe DeLe& nw, Yeep³e keÀes YeepekeÀ kesÀ Òeefleueesce mes iegCee keÀjvee nw~ Fmeer lejn, Heefjces³e mebK³eeDeeW keÀe efJeYeepeve keÀj mekeÀles nQ~
GoenjCe 1 : 1) 43
21
' DeLee&le 43
12
23
2
1=#
2) 52
103
'- DeLee&le
52
310
34
1
2- = -
#
3) 65
34
65
43
85
1
2- - = - -
=+' #
4) 8
376
83
67
167
1
2= =' #
5) 5
432
52
23
56- = - = -' #
6) 1512 10
101
756
1512
252
5
6
25
2- = =
-' #
-= -
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2 : peye 11 82 ueerìj otOe 45 efJeOeeefLe&³eeW kesÀ yeer®e efJeleefjle keÀjles nQ lees Òel³eskeÀ efJeOeeLeea
keÀes ÒeeHle otOe efkeÀlevee nesiee?
kegÀue otOe keÀer cee$ee 11 41= l.
kegÀue efJeOeeefLe&³eeW keÀer mebK³ee =45
∴Òel³eskeÀ efJeOeeLeea keÀes ueerìj otOe ÒeeHle neslee nw 111 45
45 45
41 l
4
4
445
451
=
=
=
=
'
'
#
3) ³eefo 3 53 kg mesye 3 efce$e DeeHeme ceW yeeBì uesles nQ lees Òel³eskeÀ keÀes efkeÀleves efkeÀueesûeece mesye
ÒeeHle neslee nw?meceeOeeve:
mesye keÀe kegÀue Jepeve = 3 53 kg
efce$eeW keÀer mebK³ee = 3
∴Òel³eskeÀ keÀes ÒeeHle mesye keÀe Jepeve 3 53 3
518 3
518 1
56
1 51 kg
3
6
1
=
=
=
=
=
'
'
#
Òel³eskeÀ efce$e keÀes 1 51 efkeÀueesûeece mesye ÒeeHle neslee nw~
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DeY³eeme 3.3
1) iegCee keÀerefpeS :
1) 45
103
# 2) 87
425
#- 3) 15
7143
#-
4) 65
152
#- - 5) 9
786
72
# #- - 6) 10
935
32
# #
2) Yeeie ueieeF&S :
1) 91
73
' 2) 87
67
'
3) 32
95
' 4) 54
107
'- -
3) efvecveeW kesÀ ³eesieHeÀueer³e (pees[ keÀer Òeefleueesie mebK³ee efueefKeS) :
1) 53- 2) 4
9 3) -10
4) 27 5) 5
1- 6) 0.
4) efvecveeW iegCeveHeÀueer³e Òeefleueesce mebK³ee efueefKeS.
1) 109 2) 6
5- 3) 1813- 4) 7
5-
5) SkeÀ Hesìer ceW 15 43 kg MekeÌkeÀj nw~ 1 4
3 kg kesÀ efkeÀleves HewkesÀì yeve mekeÀles nQ?
6) SkeÀ efJeOeeLeea keÀes meceJem$e kesÀ efueS 2 43 m keÀHe[e ®eeefnS~ Ssmes 44 efJeO³eeefLe&³eeW kesÀ
meceJem$e kesÀ efueS efkeÀlevee keÀHe[e ®eeefnS?
7) SkeÀ Dee³eleekeÀej yeeie keÀer uebyeeF& 3 51 m Deewj ®eew[eF& 2 4
3 m nw~ GmekeÀe #es$eHeÀue ke̳ee
nw?
8) SkeÀ mketÀue ceW 20 54 l otOe ceBie³ee ie³ee~ ³eefo Òel³eskeÀ efJeOeeLeea keÀes 5
1 l otOe efo³ee ie³ee
lees mketÀue kesÀ ye®®eeW keÀer mebK³ee %eele keÀerefpeS~
©KTBS
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Heefjces³e mebK³eeDeeW keÀes oMeceueJe ©He ceW efueKevee
DeeHe efYevveeW keÀes oMeceueJe ©He ceW efueKevee peeveles nQ~ Fmeerlejn Heefjces³e mebK³eeDeeW keÀes Yeer
oMeceueJe ceW efueKe mekeÀles nw~
oMeceueJe mebK³ee ceW oes Yeeie nesles nQ~ 61.35 ceW 62 HetCee¥keÀ Yeeie Deewj 35 oMeceueJe Yeeie nw~
GoenjCe: 61.35 He{ves keÀe efJeOeeve → FkeÀmeþ oMeceueJe leerve HeeB®e
Fmeerlejn, 27.834 keÀes He{ves keÀe efJeOeeve → meleeF&me oMeceueJe Deeþ leerve ®eej
keÀe³e&keÀueeHe:
SkeÀ Hesìer ceW kegÀí HeÀuesMe keÀe[ex keÀes jefKeS efpeve Hej Deueie - Deueie oMeceueJe mebK³eeSB efueKeer ieF& nQ~ Òel³eskeÀ efJeOeeLeea keÀes SkeÀ keÀe[& ueskeÀj GmeHej efueefKele oMeceueJe mebK³ee He{ves oerefpeS Deewj GmekeÀe HetCee¥keÀ Yeeie Deewj oMeceueJe Yeeie He{ves oerefpeS~ Òel³eskeÀ DebkeÀ keÀe mLeeve cetu³e yeleeves oerefpeS~
GoenjCe: 1 :
a) .107 0 7= b) .10
27 2 7=
c) .53 0 6=
0.6530
30 0
d) .43 0 75=
0.75430
28 202000
e) 0.002001 5=
0.0052001000
1000 0
f) .31 0 3333= ... 0.333
3109 10
910
91
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g) .65 0 8333= 0.833
650 48
20182018
2
h) .111 0 090909= ... 0.090909
11100 99
1009910099
1
³eefo DeeHe GHejeskeÌle GoenjCeeW keÀes O³eeve mes osKeles nes, Heefjces³e mebK³eeDeeW keÀes oMeceueJe
ceW efueKeles mece³e, GmekesÀ DebkeÀ meeble (terminating) nes mekeÀles nQ DeLeJee GmekesÀ SkeÀ Deewj
DeefOekeÀ DebkeÀ HegvejeJeefle&le nesles nQ~
GoenjCe: a , b , c , d , e Fve ceW oMeceueJe kesÀ mLeeveeW kesÀ DebkeÀeW keÀe Deble neslee
nw~ FmeefueS FvnW meele oMeceueJe (terminating) keÀnles nw~
Hejvleg f , g , h GoenjCeeW ceW, oMeceueJe mLeeve kesÀ SkeÀ DeLeJee DeefOekeÀ DebkeÀ
HegvejeJeefle&le nesles nQ~ efJeYeepeve peejer jns lees Yeer Jener DebkeÀ oesnjeles nQ~ oMeceueJe Yeeie ceW efyeefo³eeW
keÀes ueieekeÀj nce DebkeÀeW keÀe HegvejeJele&ve oMee&les nw~
1) meeble oMeceueJe :
a) 3.57 b) 15.035 c) 0.0004321 d) 10.23576
2) HegvejeJeefle&le DeLeJee DeeJele& oMeceueJe:
a) 2.544 ... b) 10.272727 ...
c) 15.145614561456 ... d) 18.321321321 ...
meesef®eS ! meYeer Heefjces³e mebK³eeDeeW keÀes meeble DeLeJee HegvejeJeefle&le oMeceueJe kesÀ ©He ceW J³ekeÌle keÀj mekeÀles nQ~ Hejvleg oMeceueJe mLeeve kesÀ DebkeÀ ³eefo ve Deble nesles nw vee oesnjeles nwb lees Jes kewÀmes oMeceueJe nw? ke̳ee GvekeÀes
pq ©He ceW J³ekeÌle keÀj mekeÀles nQ?
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GoenjCe: oMeceueJe mes Heefjces³e meK³eeDeeW keÀe HeefjJele&ve~
a) .72 0 285714285714=
0.285714285714720
146056
4035
5049
1073028
2014
6056
4035
5049
1073
b) 1 1.95 555= ... 1.555...
9149 50455045
5045
5
.914 1 555= ...
c) .4152
1562 4 133= = ... 4.133...
156260 2015
5045
5045
5
d) 0.1253 024= 0.024
125300 250 500 500
0
DeY³eeme 3.4
1) Fve Heefjces³e mebK³eeDeeW keÀes oMeceueJe ©He ceW efueefKeS~
1) 54 2) 8
3 3) 165 4) 14
3
2) efvecveeW ceW meeble Deewj HegvejeJeefle&le Heefjces³e mebK³eeDeeW keÀes Hen®eevekeÀj efueefKeS~
©KTBS
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1) 85 2) 12
1 3) 92 4) 16
1
3) efvecve oMeceueJeeW ceW meeble, Deveble HegvejeJeefle&le oMeceueJe Hen®eeefveS~
1) 8.751 2) 2.5444 ...
3) 0.030303 4) 9.2874
5) 2.456731456731456731 ... 6) 10.56173824931685 ...
7) 3.147521896397 ...
oMeueJeeW keÀes Heefjces³e mebK³eeDeeW ceW J³ekeÌle keÀjvee:
oMeceueJe mes Heefjces³e meKb³ee HeefjJele&ve keÀjves kesÀ efueS 1 meKb³ee keÀes Gmeer ÒekeÀej efueefKeS
DeMe keÀes oMeceueJe Deewj nj~ kesÀ meeLe Metv³e ueieeves nw~
GoenjCe : a) .0 3 103= b) 0.12 100
12= c) .1 4 1104
1014= =
oMeceueJeeW keÀe iegCeveHeÀue (Multiplication of decimal numbers) :
oMeceueJeeW keÀes iegCee keÀjles mece³e, oMeceueJeeW keÀes íes[keÀj mebK³eeDeeW keÀe iegCeveHeÀue %eele
keÀerefpeS~ yeeo ceW iegCeebkeÀ Deewj iegC³e kesÀ Devegmeej, iegCeveHeÀue kesÀ FkeÀeF& mLeeve mes DebkeÀeW keÀes
íes[keÀj oMeceueJe efyevoer ueieeFS~ ³eefo mebK³ee ceW DebkeÀ veneR nQ lees GmekesÀ meceeve Metv³e ueieekeÀj
eEyeefo ueieeFS~
GoenjCe 1 : a) 0.2 × 0.3 = 0.06
b) 0.12×0.6 = 0.072
c) 1.2×1.5 = 1.80
d) 0.003×0.12 = 0.00036
e) 0.13×0.0005 = 0.000065
f) 3.41×2.678 = 9.13198
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GoenjCe 2 : SkeÀ Dee³ele keÀer uebyeeF& 4.7 cm Deewj ®eew[eF& 3.5 cm nw lees GmekeÀe #es$eHeÀue
ke̳ee nw?
meceeOeeve :
Dee³ele keÀer uebyeeF& = 4.7 cm
Dee³ele keÀer ®eew[eF& = 3.5 cm
∴Dee³ele keÀe #es$eHeÀue = uebyeeF& × ®eew[eF&
= 4.7 × 3.5 = 16.45 sq cm
GoenjCe 3 : ³eefo 1 m keÀHe[e keÀe oece 98.75 nes lees keÀHe[e 3.20 m keÀe oece %eele
keÀerefpeS~
meceeOeeve: 1 m keÀHe[s keÀe oece = ` 98.75
∴ 3.20 m keÀHe[s keÀe oece = 98.75 × 3.20 = 316.0000 = 316
3.20 m ce´er keÀHe[s keÀe oece = ` 316
GoenjCe 4 : Òeefleefove SkeÀ efJeÐeeLeea keÀes oer peeveer Jeeueer oeue keÀer cee$ee 0.035 kg 38 efJeÐeeLeea
keÀes oer ngF& oeue keÀer cee$ee ke̳ee nw?
= 0.035 kg
∴ 38 efJeÐeeefLe&³eeW keÀes ÒeeHle oeue keÀer cee$ee = 0.035×38
= 1.330 kg
oMeceueJees keÀe efJeYeepeveoMeceueJe mebK³eeDeeW keÀes HetCee¥keÀes mes Yeeie ueieevee:
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GoenjCe 1 : a) . . .3 5 7 73 5 0 5' = = b) . .4
6 4 1 6=
c) . .88 24 1 03= d) . .25
15 625 0 625=
SkeÀ oMeceueJe keÀes otmejs oMeceueJe mes Yeeie ueieevee:
GoenjCe 1 :
) ..
.
.
a 0 48 4
0 4 108 4 10
484 21
21
##= = =
) ..
..
.
b 1 83 24
1 8 1003 24 100
180324
1018 1 8
10
18##=
= = =
met®evee :
oMeceueJe mLeeveeW kesÀ Devegmeej DebMe Deewj nj keÀes SkeÀ ner mebK³ee mes iegCee keÀerefpeS Deewj yeeo ceW Yeeie ueieeFS.
) ..
..
c 12 56 25
12 5 1006 25 100
1250625
21
1
502
25
##= = =
) . .
..
..
d 0 729 1 80
1 800 729
1 80 10000 729 1000
1800729
20081
600
243
200
81
'
##
=
= = =
O³eeve oerefpeS: oMeceueJe keÀes HetCee¥keÀ ceW yeoueves kesÀ efueS SkeÀ oMeceueJe mLeeve jns lees 10, 2 mes 100............ Fl³eeefo mes iegCee keÀjles nQ~
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GoenjCe 2 :
³eefo SkeÀ yeme 1.1 Iebìs ceW 48.4 km le³e keÀjleer nw lees GmekeÀer Òeefle Iebìs keÀe jHeÀleej ke̳ee nw?
kegÀue le³e keÀer ieF& otjer = 48.4 km
efue³ee ngDee mece³e = 1.1 km jHeÀleej = otjer / mece³e ∴®eeue = 44 efkeÀ. ceer / Iebìe .
.
/km hr1 148 4
44=
GoenjCe 3 :
keÀefJelee ves 12.5 ceerìj keÀHe[e ` 127.50 keÀer oj mes, DeHeves ye®®eeW kesÀ [dsme kesÀ efueS Kejeroe~ Òel³eskeÀ ceerìj keÀHe[s keÀe oece ke̳ee nw?
keÀefJelee mes Kejeroe ngDee keÀHe[e = 12.5 m
kegÀue cetu³e = `127.50
∴Òeefle ceerìj keÀHe[s keÀe cetu³e = 127.50 ÷12.5 m
keÀHe[s keÀe oece = 12.5 m
keÀHe[s keÀe kegÀue oece = ` 427.50
∴Òeefle ceerìj keÀHe[s keÀe oece = 427.50 ÷ 12.5
12.50 100427.50 100
05
171
34.20
12505
4275 171= =
##
Òeefle ceerìj keÀHe[s keÀe oece = `
ke̳ee legce peeveles nes?pq ©He kesÀ oes Heefjces³e mebK³ee, peneB p < q nw, iegCee keÀjves Hej ÒeeHle iegCeveHeÀue oesveeW Heefjces³e
mebK³ee mes keÀce neslee nw~ ³eefo SkeÀ mebK³ee keÀes otmejs mes Yeeie ueieeles nwb lees ÒeeHle YeeieHeÀue oesveeW Heefjces³e mes ye[e neslee nw~
..
1 1 1048 4 10
11484 44
44##
= =
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DeY³eeme 3.5
I. iegCee keÀerefpeS~
1) 8.6×4 2) 3.75×2
3) 4.105×9 4) 2.56×1.3
5) 0.03×1.456 6) 11.2×0.15
II. Yeeie ueieeFS~
1) 0.42÷6 2) 0.144÷12
3) 4.97÷10 3) 6.75÷0.25
5) 2.86÷1.3 6) 68.8÷0.16
III. efvecveefueefKele ÒeMveesb keÀe nue keÀerefpeS~
1) ³eefo MekeÌkeÀj keÀe oece ` 35.45 Òeefle efkeÀuees ûeece nes lees 20.25 kg MekeÌkeÀj keÀe oece %eele keÀerefpeS~
4) ³eefo SkeÀ yeme `28.25 km Òeefle IeCìs keÀer jHeÀleej mes 4.25 IeCìs lekeÀ ®eueleer nw lees le³e keÀer ieF& otjer %eele keÀerefpeS~
5) ³eefo SkeÀ keÀej 1 ueerìj ceW 21.5 km keÀer otjer le³e keÀjlee nw lees 219.30 km otjer le³e keÀjves kesÀ efueS efkeÀlevee Hewì^esue DeeJeM³ekeÀ nw?
6) ³eefo Meeruee `257.50 ceW 12.5 m ceerìj efjyyeve Kejeroleer nw lees Òeefle ceerìj efjyyeve keÀe oece keÀ³ee nw?
Heefjces³e FkeÀeF&³eeW keÀe ©HeevlejCe :cee$ee keÀe ceeHeve SkeÀ mebK³eelcekeÀ cetu³e oslee nw~ HeefjceeCeeW keÀes FkeÀeF&³eeW Üeje leguevee
keÀj mekeÀles nQ~ nceves Debleje&<ì^er³e mlej Hej kegÀí ceeHeeW keÀes mJeerkeÀeje nw~ Fve ceeHeeW keÀes ceevekeÀ FkeÀeF&³eeB keÀnles nQ~ ceewefuekeÀ FkeÀeF&³eeW (Fundamental Units) mes íesìs Deewj ye[s ceeHeeW keÀes ÒeeHle keÀj mekeÀles nQ~
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uebyeeF& keÀer Debleje&<ì^er³e FkeÀeF& ceerìj nw~ êJ³eceeve keÀer Debleje&<ì^er³e FkeÀeF& efkeÀuees ûeece nw~
keÀ³ee DeeHe peeveles nw?
1 Fb®e = 2.54 cm
1 HeÀerì = 30.48 cm
1 ³ee[& = 0.914 m12 Fb®e = 1 HeÀeRì3 HeÀerì = 1 m220 ³ee[& = 1 HeÀjueebie8 HeÀjueebie = 1 ceerue1 ceerue = 1.61 km
1 km = 0.62 ceerue
1 DeeTvme = 28.35 gm
1 HeewC[ = 0.45 kg
1 kg = 2.204623 Heewb[
1 km = 1000 m 1 m = 100 cm
1 cm = 10 mm
1 kg = 1000 gm
1 gm = 1000 mg
ke̳ee DeeHe peeveles nQ?
efkeÀuees ceerìjnskeÌìeW ceerìj[skeÀe ceerìjceerìj[smeer ceerìjmesCìer ceerìjefceueer ceerìj
÷ 10× 10
DeLee&le 1 efkeÀueesceerìj = 10 nskeÌì
Fmeerlejn 1 cm = 10 mm
GoenjCe 1 : FvnW mesCìer ceerìj ceW yeoefueS~
a) 45 m = 45 × 100 = 4500 cm
b) 2.7 m = 2.7 × 100 = 270 = 270 cm
c) 4 43 m = 4
19 100 19 25 475# #= = cm
d) 3.02 m = 3.02 × 100 = 302.00 = 302 cm
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met®evee: m keÀes cm ceW yeoueves kesÀ efueS Gmes 100 mes iegvee keÀjvee ®eeefnS~
a 1m = 100 cm 1 cm = 1001 m
GoenjCe 2 : FvnW mes. ceer. ceW efueefKeS~
a) 20 cm 10020
51= = m
b) 2 cm 1002
501= = m
c) 415 cm 4.15100415= = m
d) 0.003 cm 0.003 1001
10003
1001
1000003
# #= = = m
e) 4.5 km = 4.5 × 1000 = 4500 m
f) 25 km = 25 × 1000 = 25000 m
g) 2.1 km = 2.1 × 1000 = 2100 m
GoenjCe 3 : FvnW mes. ceer. ceW efueefKeS~a) 6mm
= 6 101
53
# = cm
b) 23 mm
= .1023 2 3= cm
c) 276 mm
= .10276 27 6= cm
GoenjCe 4 : FvekeÀes efkeÀuees ceerìj cesb efueefKeS~a) 105 m
1051000
12002121
200#= = km
b) 725 m
7251000
1402529
40#= = 29
40 km
GoenjCe 5 : FvekeÀes efkeÀueesûeece cesb efueefKeS~a) 720 g
1000720
10072
2518
25
18= = = kg
b) 3150 g
3.1510003150
100315= = = kg
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GoenjCe 6 : FvekeÀes ûeece cesb efueefKeS~
a) 1.4 kg
1.4×1000
= 1400 g
b) 10 21 kg
10 21 kg =
221 1000 10500
1
500# = g
c) 3.42 kg
3.42 kg
= 3.42 × 1000 = 3420 g
GoenjCe 7 : FvekeÀes ûeece cesb efueefKeS~
a) 740 mg
00
10074
10074
5037= = g
b) 210 mg
00
10021
10021= g
GoenjCe 8 : 125 mg efJeueesûeece ces efueefKeS
gm kgmg kg
mg kg
kg
125 10000001
400005
1000 11000000 1
1 10000001
80001
540000
1
8000# =
==
=
=
GoenjCe 9 : SkeÀ yeme SkeÀ Iebìs cesb 16.2 ceerue otjer le³e keÀjleer nw Fmes km ceW efueefKeS~
meceeOeeve : 1 ceerue = 1.61 km
∴ 16.2 ceerue = 16.2 × 1.61
= 26.082 km
GoenjCe 10 : SkeÀ keÀcejs keÀer 4.5 HeÀerì uebyeeF& nw~ mesCìerceerìj cesb efueefKeS~
meceeOeeve : 1 HeÀerì = 30.48 cm
∴ 4.5 HeÀerì = 137.160 cm = 30.48 ×4.5 = 137.160 meWceer
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DeY³eeme 3.6
1) FvnW ûeeceeW ceW J³ekeÌle keÀerefpeS~
1) 2.51 kg 2) 72.5 mg
2) FvnW efkeÀueesûeece ceW J³ekeÌle keÀerefpeS~
1) 625 gm 2) 10825 mg
3) FvnW efkeÀuees ceerìj ceW J³ekeÌle keÀerefpeS~
1) 1450 m 2) 17.3 m 3) 21350 cm
4) FvnW ceerìjeW ceW yeoefueS~
1) 2.7 km 2) 7525 cm 3) 1.58 km
5) FvnW mes. ceer ceW efueefKeS~
1) 12.5 m 2) 4.7 km
6) 7.2 ceerueeW keÀes efkeÀ.ceer. ceW yeoefueS~
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IeìkeÀ - 4
yeerpeer³e J³ebpekeÀ
Fme IeìkeÀ kesÀ DeO³e³eve kesÀ yeeo DeeHe :
yeerpeer³e J³ebpekeÀeW mes mebyebefOele ceewefuekeÀ HeoeW keÀes J³eke̳e keÀj mekeWÀieW, yeerpeer³e J³ebpekeÀ kesÀ DeLe& keÀer HeefjYee<ee os mekeWÀie, ®ejebkeÀeW keÀes mJe³eb kesÀ meeLe DeLeJee Dev³e ®ejebkeÀeW kesÀ meeLe meb³eesie keÀjkesÀ yeerpeer³e J³ebpekeÀ
ÒeeHle keÀj mekeWÀieW, MeeqyokeÀ keÀLeveeW keÀes yeerpeer³e J³ebpekeÀeW ceW HeefjJele&ve keÀj mekeWÀieW, yeerpeer³e J³ebpekeÀ kesÀ IeeleebkeÀ keÀe DeLe& mecePe mekeWÀieW, ceewefuekeÀ Òeef¬eÀ³eeDeeW keÀes yeerpeer³e J³ebpekeÀeW kesÀ meeLe GHe³eesie keÀj mekeWÀies~
ÒemleeJevee:
yeerpeer³e J³ebpekeÀ, yeerpe ieefCele kesÀ ÒecegKe HeefjkeÀuHeveeSB nwb~ ³es nceejs owefvekeÀ peerJeve keÀer mecem³eeDeeW keÀes nue keÀjves kesÀ MeeqkeÌleMeeueer meeOeve nw~
meceerkeÀjCeeW keÀes DeO³e³eve keÀjles mece³e De%eele cetu³eeW keÀes %eele keÀjves kesÀ efueS yeerpeer³e J³ebpeve mene³ekeÀ nesles nQ~yeerpeieefCele kesÀ ceewefuekeÀ HeefjkeÀuHeveeSB:
megcee Deewj ceOeg efo³eemeueeF& keÀer leerefue³eeW kesÀ meeLe Kesue jns nQ~ GvneWves leerefue³eeW keÀe GHe³eesie keÀjles ngS Debûespeer De#ejceeuee yeveeves keÀe efve½e³e efkeÀ³ee~
megcee ves leerve leerefue³eeW kesÀ GHe³eesie mes `C' De#ej yevee³ee~
ceOeg leerve leerefue³eeW keÀes Gþeleer nw Deewj SkeÀ ̀ C' De#ej yeveekeÀj, megcee mes yevee³es ̀ C' De#ej kesÀ Heeme jKeleer nw~
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megcee Deewj ceOeg leerefue³eeW keÀe GHe³eesie keÀjles ngS Fmelejn `C' De#ej yeveekeÀj, SkeÀ kesÀ yeeo SkeÀ jKeleer peeleer nQ~
Devle ceW, GvneWves leerefue³eeW keÀer mebK³ee Deewj yevee³es ̀ C' De#ej oMeeves kesÀ efueS leeefuekeÀe yeveeF&~
yevee³es ieS`C' De#ej 1 2 3 4 5 ... 20 ... 25 ... n
GHe³eesie efkeÀ³es ieS leeueef³eeW keÀer mebK³ee
3 6 9 12 15 ... 60 ... 75 ... 3n
1×3 2×3 3×3 4×3 5×3 ... 20×3 ... 25×3 ... n×3
`n' SkeÀ De#ejer³e mebK³ee nw~
leeefuekeÀe ceW efueKeles mece³e, GvneWves DevegYeJe efkeÀ³ee efkeÀ leeueef³eeW keÀer mebK³ee, yevee³es ie³es `C' De#ej keÀer mebK³ee keÀer efleiegveer nw~
DeeJeM³ekeÀ leerefue³eeW keÀer mebK³ee = 3 × yevee³es ie³es `C' kesÀ De#ej ³eefo `n' yevee³es `C' De#ej metef®ele keÀjlee nw lees DeeJeM³ekeÀ leerefue³eeW keÀer mebK³ee = 3 × n = 3n
Fmelejn,
n = 1 kesÀ efueS, DeeJeM³ekeÀ leerefue³eeW keÀer mebK³ee = 3 × 1 = 3
n = 2 kesÀ efueS, DeeJeM³ekeÀ leerefue³eeW keÀer mebK³ee = 3 × 2 = 6
n = 3 kesÀ efueS, DeeJeM³ekeÀ leerefue³eeW keÀer mebK³ee = 3 × 3 = 9
³eneB, `n' yevee³es `C' vecetves keÀer mebK³ee nw Deewj leeefuekeÀe ceW 1, 2, 3, 4 ... kesÀ cetu³e ueslee nw~ `n' keÀe cetu³e yeouelee peelee nw~ HeefjCeecele : leerefue³eeW keÀer mebK³ee Yeer yeoueleer peeleer nw~
`n' ³en ®ejebkeÀ keÀe GoenjCe nw~ De#ejer³e mebK³eeDeeW keÀes ®ejebkeÀ kesÀ ©He ceW GHe³eesie keÀjles nw~
³en 1, 2, 3, 4,....... Deeefo keÀesF& Yeer cetu³e ues mekeÀlee nw~
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®ejebkeÀ keÀe DeLe& nw pees yeoue mekeÀlee nw DeLeJee HeefjJeefle&le neslee nw~ ®ejebkeÀ keÀe cetu³e efmLej veneR neslee~ ³en Deueie - Deueie cetu³e ues mekeÀlee nw~
®ejebkeÀ kesÀ kegÀí Deewj GoenjCe : neB,! a,b,c,....x,y,z ®ejebkeÀ Yeer metef®ele keÀjles nQ Deewj meeLe De%eele Heo Yeer
nce peeveles nw b e fkeÀ; Dee³ele keÀe #es$eHeÀue = uebyeeF& × ®eew[eF&
DeLee&le, A = l × b
efvecve leeefuekeÀe keÀes O³eeve mes osefKeS :
Dee³ele keÀe cetu³e
uebyeeF& (l) cm ®eew[eF&(b) cm #es$eHeÀue A = l × b
ABCD 20 cm 3 cm 60 sq cm
PQRS 15 cm 4 cm 60 sq cm
KLMN 12 cm 5 cm 60 sq cm
WXYZ 10 cm 6 cm 60 sq cm
GHejeskeÌle leeefuekeÀe oMee&lee nw efkeÀ efJeefYevve Dee³eleeW ceW `l ' Deueie - Deueie cetu³e ûenCe
keÀjlee nw Deewj efJefYevve Dee³eleeW ceW `b' Deueie - Deueie cetu³e ÒeeHle keÀjlee nw~ Fmelejn `l '
Deewj `b ' pewmes De#ejer³e mebK³ee ®ejebkeÀ keÀnueeles nQ~
SkeÀ ®ejebkeÀ, SkeÀ De#ejer³e mebK³ee DeLeJee SkeÀ mebkesÀle nw pees meboYe& kesÀ Devegmeej Deueie -
Deueie cetu³e ÒeeHle keÀjlee nw~
Dev³e MeyoeW ceW, De%eele mebK³eeDeeW keÀes metef®ele keÀjves GHe³egkeÌle mebkesÀleeW keÀes ®ejebkeÀ keÀnles nQ~
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Fmes peeve ueerefpeS: SsefleneefmekeÀ YetefcekeÀe
ef¬eÀ. Het 300 kesÀ DeemeHeeme, De#ejeW Üeje De%eeleeW keÀes metef®ele keÀjvee Deewj J³ebpekeÀ lew³eej keÀjvee Yeejle ceW meJe&meeceev³e Lee~ Dee³e& Yeìdì (F. meved 476 ceW pevceW), yeïeiegHle (Fmeved 598), ceneJeerje (pees meve 850 kesÀ Deeme Heeme Les) Deewj YeemkeÀj II (meved 1114 ceW pevceW) yeerpe ieefCele kesÀ efueS GuuesKeveer³e ³eesieoeve efo³es nQ~ GvneWves ®ejebkeÀeW kesÀ efueS yeerpe, JeCe& Deefo veece metef®ele efkeÀ³ee~ Alegbra kesÀ efueS Yeejleer³e veece Lee yeerpe ieefCele~
ÖeÀebkeÀesFme Jesìs,16 Jes Meleeyoer kesÀ ceneve ieefCele%e meJe&ÒeLece meeceev³e DebkeÀ kesÀ vecetveeW kesÀ efueS De#ejeW keÀe GHe³eesie efkeÀ³ee~
efmLejebkeÀ (Constant) :
ke̳ee ef$eYegpe kesÀ YegpeeDeeW keÀer mebK³ee 3 kesÀ DeueeJee otmejer mebK³ee nes mekeÀleer nw? efveef½ele ©He mes veneR, FmeefueS ef$eYegpe kesÀ YegpeeDeeW keÀer mebK³ee SkeÀ efveef½ele mebK³ee nw, ³en efmLejebkeÀ nw~
ke̳ee ®elegYe&pe kesÀ Meer<eeX keÀer mebK³ee 4 kesÀ DeueeJee otmejer keÀesF& nes mekeÀleer nw?
efveef½ele ©He mes veneR, FmeefueS ®elegYe&gpe kesÀ Meer<eeX keÀer mebK³ee efmLej nesleer nw Deewj Fmelejn ³en efmLejebkeÀ nw~
peye nce mebK³eebkeÀ 4 efueKeles nQ, Jen mebK³ee 4 metef®ele keÀjlee nw~
peye nce 5 mebkesÀle GHe³eesie keÀjles nwb Jen keÀvve[ mebK³ee ÒeCeeueer ceW 5 (HeeB®e) efve©efHele keÀjlee nw~
SkeÀ efmLejebkeÀ, SkeÀ mebK³ee DeLeJee Dev³e mebkesÀle nw pees SkeÀ ner cetu³e keÀes metef®ele keÀjlee nw~
5, 9, 10, 3/4, -8, 0.01.......... efmLejebkeÀ nwb~
= π pees SkeÀ efmLejebkeÀ nw~HeefjefOeJ³eeme
= C = 3 × 108 m/s SkeÀ efmLejebkeÀ nw~
DeLee&le Jee³eg DeLeJee efveJee&le ceW ÒekeÀeMe keÀe jHeÀleej efmLejebkeÀ neslee nw~
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yeerpeer³e Heo (Algebraic term)
nce peeveles nQ. efkeÀ 6 Deewj 8
2xy, 4pq, x y4
3 2- , 10, m yeerpeer³e Heo nQ~
yeerpeer³e HeoeW ceW mebK³ee, De#ejer³e mebK³ee Deewj
oesveeW mebK³ee Deewj De#ejer³e mebK³ee nes mekeÀles nQ~
keÀe iegCeveHeÀue~
6 × 8 = 48 Fmeerlejn,18 keÀes 2 mes Yeeie osves Hej YeeieHeÀue 9 nw DeLee&le 18 ÷ 2 = 9
ke̳ee legce x Deewj 3 keÀe iegCeveHeÀue %eele keÀj mekeÀles nes?
neB, iegCeveHeÀue 3 x nw~
ke̳ee legce y keÀes 4 mes Yeeie ueieeves Hej ÒeeHle YeeieHeÀue %eele keÀj mekeÀles nes?
neB, YeeieHeÀue y4 nw~
3x, y4 yeerpeer³e Heo nw~
3x , -6x y, 2x2y, y2 , 2x
3- Deeefo kegÀí Deewj yeerpeer³e HeoeW kesÀ GoenjCe nQ~
Fve yeerpeer³e HeoeW kesÀ efmLejebkeÀ Deewj ®ejebkeÀ Hej O³eeve oerefpeS ~
yeerpeer³e Heo 3xceW, 3efmLejebkeÀ nw Deewj
x SkeÀ ®ejebkeÀ nw~
yeerpeer³e Heo -6xy,ceW, -6 efmLejebkeÀ
nw Deewj x Deewj y ®ejebkeÀ nwb~
yeerpeer³e Heo 2x2y, ceW 2 efmLejebkeÀ
Deewj x leLee y ®ejebkeÀ nwb~
met®evee:x2, x/y, 1/x nQ yeerpeieefCeleer³e MeleX?
SkeÀ yeerpeer³e Heo SkeÀ efmLejebkeÀ nw DeLeJee HetCe& mebK³ee kesÀ Ieele kesÀ SkeÀ DeLeJee DeefOekeÀ ®ejebkeÀ ³egkeÌle efmLejebkeÀ keÀe iegCeveHeÀue DeLeJee YeeieHeÀue nw~
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DeY³eeme 4.1
I. efvecveefueefKele yeerpeer³e HeoeW ceW ®ejebkeÀ DeLeJee ®ejebkeÀeW keÀer met®eer yeveeFS~
1) 5x 2) -3a 3) 7x y 4) 43 x 2y
5) 0.8a2b 6) m2n2 7) xyz 8) -5m2np
II. mebK³elcekeÀ efmLejebkeÀ %eele keÀerefpeS~
1) 2x 2) -5x2y 3) m2n 4) -8p
5) 9p2qr 6) 43 x 2y 7) 0.5pqr2 8) 0.008mn
yeerpeer³e J³ebpekeÀ
J³ebpekeÀ kewÀmes yeveles nQ? efvecve ieefCeleer³e keÀLeveeW Hej efJe®eej keÀerefpeS SkeÀ mebK³ee Deewj Heeb®e keÀe pees[ 59 nw + 5 = 59
DeLee&le 59 = 54 + 5
∴ Dele : DeHesef#ele mebK³ee 54 nw ~
Fmeerlejn,
SkeÀ mebK³ee Deewj 8 keÀe iegCeveHeÀue 88 nw~
8 × = 88
∴8 × 11 = 88
∴ FmeefueS, DeHesef#ele mebK³ee 11 nw~
³eefo DeefOekeÀ De%eele HeoeW nes lees ieefCeleer³e keÀLeve yeveevee keÀefþve nw~ mejue yeveeves kesÀ efueS nce ®ejebkeÀeW keÀes x, y, z Deeefo mes metef®ele keÀjles nQ~
yeerpeieefCele mes De%eele keÀes nele keÀjves ueecepee
ueerefpeS~
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efvecve GoenjCe Hej efJe®eej keÀerefpeS SkeÀ yeeefuekeÀe Deye 10 Je<e& keÀer nw~ GmekeÀer Dee³eg
1) 5 Je<eeX kesÀ yeeo ke̳ee nesieer ?
2) 3 Je<eeX kesÀ Henues ke̳ee Leer ?
3) x Je<eeX kesÀ yeeo ke̳ee nesieer ?
4) y Je<eeX kesÀ Henues ke̳ee Leer?
nue: Jen nj Je<e&1 Je<e& ye[er nesleer peeleer nw~
1) 5 Je<e& yeeo yen = (10 + 5) = 15 Je<e& keÀer nesieer~
2) 3 Je<e& kesÀ Henues Jen = (10 - 3) = 7 Je<e& keÀer Leer~
3) `x' Je<e& yeeo Jen = (10 + x) Je<e& keÀer nesieer~
4) `y' Je<e& Henues Jen = (10 - y) Je<e& keÀer Leer~
(10 + x), (10 - y) pewmes J³ebpekeÀ efpemeceW ®ejebkeÀ meceeefJe<ì nesles nw yeerpeer³e J³ebpekeÀ keÀnueeles nw~
efvecve efueefKele J³ebpekeÀeW Hej efJe®eej keÀerefpeS:
i) x2 ii) 2y2
iii)3x + 2 iv) 2pq + 7
x2 J³ebpekeÀ keÀe DeLe& nw x keÀes mJe³eb mes iegCee keÀjvee DeLee&le ∴ x × x = x2
Fwmes peeve ueerefpeS: Fves Ssmes He{les nw
ly Fmeerlejn
i.e. DeLee&le
∴ FmeefueS
≅ meJee¥iemece
ke̳eeW efkeÀ Fmeer ÒekeÀej 2 × y × y = 2y2
3x + 2 kewÀmes ÒeeHle efkeÀ³ee peelee nw?
J³ebpekeÀ 3x + 2 ÒeeHle keÀjves kesÀ efueS 3 iegCee keÀjkesÀ iegCeveHeÀue keÀes 2 pees[vee nw~
∴ (x) × 3 + 2 = 3x + 2
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Fmeer ÒekeÀej 2pq + 7 kewÀmes ÒeeHle keÀjles nQ?
2pq keÀes efvecve ÒekeÀej mes ÒeeHle keÀjles nQ Henues pq keÀes p Deewj q iegCee keÀjkesÀ ÒeeHle keÀjles nQ, yeeo ceW Gmes 2 mes iegCee keÀjkesÀ 2pq ÒeeHle keÀjles nQ Deewj iegCeveHeÀue ceW 7 pees[keÀj efo³ee J³ebpekeÀ ÒeeHle keÀjles nQ~
GHejeskeÌle x2, 2y2, 3x + 2, 2pq + 7 J³ebpekeÀeW keÀes yeerpeer³e J³ebpekeÀ keÀnles nQ~
efvecve leeefuekeÀe Hej O³eeve oerefpeS :
keÀLeve yeerpeer³e J³ebpekeÀ
x keÀes 8 pees[e nw x + 8
y ceW mes 5 Ieìe³ee nw y - 5
p keÀes 6 mes iegCee efkeÀ³ee nw 6 × p
`m' keÀes n mes Yeeie ueiee³ee m ÷ n
³eneB, nce osKeles nQ efkeÀ keÀLeve kesÀ Devegmeej, pees[ves, Ieìeves, iegCee keÀjves DeLeJee Yeeie ueieeves keÀer Òeef¬eÀ³ee ®ejebkeÀes Hej GHe³eesie keÀjkesÀ yeerpeer³e J³ebpekeÀeW keÀes ®ejebkeÀeW Hej GHe³eesie keÀjkesÀ yeerpeer³e J³ebpekeÀeW keÀes ÒeeHle efkeÀ³ee ie³ee nQ~
``yeerpeer³e HeoeW kesÀ meeLe ceewefuekeÀ Òeef¬eÀ³eeDeeW keÀes DeHeveekeÀj yeerpeer³e J³ebpekeÀ ÒeeHle keÀjle nQ~''
yeerpeer³e J³ebpekeÀ kesÀ Heoefvecve efueefKele J³ebpekeÀeW Hej O³eeve oerefpeS:
yeerpeer³e J³ebpekeÀ yeerpeer³e J³ebpekeÀ kesÀ Heo HeoeW keÀer mebK³ee2x + 3y 2x, + 3y 2
x2 - 4x + 3 x2, - 4x, + 3 3
3ab - 4bc - 6cd 3ab, - 4bc, 6cd 3
-2p2q + 3q2r - rp + 4pqr - 2p2q, + 3q2r, - rp, 4pqr 4
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Deye 2x + 3y J³ebpekeÀ Hej efJe®eej keÀerefpeS Fme J³ebpekeÀ ceW oes ®ejebkeÀ x Deewj y Deewj 2 mebK³ee 2 Deewj 3 nw, 2x SkeÀ Heo ³egkeÌle J³ebpekeÀ nw Deewj 3y Deewj SkeÀ Heo, Fmelejn J³ebpekeÀ kesÀ Heo DeeHekesÀ O³eeve Deeles nw~
iegCeueyOe iegCeveKeC[ Deewj men iegCeebkeÀ :nce peeveles nQ efkeÀ 4 × 5 = 20 nw~ met®evee:
³eefo ®ejebkeÀ SkeÀ otmejs kesÀ Heeme Heeme efueKeW nw lees Jen GvekeÀe iegCeueyOe nw~
³eneB 20 iegCeueyOe nw, Deewj 4 Deewj 5 oesveeW 20kesÀ iegCeveKeC[ nQ~peye oes ®ejebkeÀ iegCee keÀjles nQ lees iegCeveueyOe ke̳ee nw?
ceeve ueerefpeS, nce `x' Deewj `y', keÀes iegCee keÀjles nQ iegCeueyOe (iegCeveHeÀue) ke̳ee nw ?
`x' Deewj `y' kesÀ iegCeueyOe keÀes (x × y) DeLeJee xy efueKeles nQ~
Fmeerlejn,
2 Deewj p keÀe iegCeueyOe = 2p
2p kesÀ iegCeveKeC[ 2 Deewj p nwb~
5 × m × n keÀe iegCeveHeÀue = 5mn nw~
ly 7 × x × x × y keÀe iegCeveHeÀue = 7x2y
7x2y kesÀ iegCeveKeC[ 7, x, y nQ~
yeerpeer³e J³ebpekeÀ Deewj GmekesÀ HeoeW keÀe ef®e$e Üeje efve©HeCe
8x2y + 5xy - 8
8x2y
8 5x x xy y
5xy -8
J³ebpekeÀ:
Heo:
iegCeveKeC[:42
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meniegCeebkeÀ :
x Deewj 8 iegCeveHeue Hej efJe®eej keÀerefpeS, DeLee&le 8x ceW 8 SkeÀ mebK³ee Deewj x SkeÀ meniegCeebkeÀ nw~ oesveeW keÀes efceueekeÀj 8x DeLee&le +8x, 8 ³en `x' keÀe mebK³eelcekeÀ DeLeJee DebkeÀer³e iegCeveKeC[ nw~
FmeefueS, x keÀes 8 keÀe De#ejer³e iegCeveKeC[ DeLeJee De#ejer³e meniegCeebkeÀ keÀnles nQ~
x = 1 x
y = + y
z = +1z
mebK³eelcekeÀ meniegCeebkeÀ kesÀ efyevee efueKes ieS De#ejer³e mebK³ee ceW GmekeÀe mebK³eelcekeÀ meniegCeebkeÀ 1 nw~
kegÀí Deewj GoenjCeeW Hej efJe®eej keÀjles nQ:
iegCeveHeÀue men iegCeebkeÀ mebK³eelcekeÀ men iegCeebkeÀ
De#ejer³e meniegCeebkeÀ
8xy = 8 × x × y 8x keÀe meniegCeebkeÀ y nw~
8y keÀe meniegCeebkeÀ x nw~
xy keÀe meniegCeebkeÀ 8 nw~
881
xyxy
-21x2y=-21x × x × y
-21x2 keÀe meniegCeebkeÀ y nw~
-21x keÀe meniegCeebkeÀ xy nw~
-21y keÀe meniegCeebkeÀ x2 nw~
-21-21-21
-1
x2
xy
x2y keÀe meniegCeebkeÀ -21 nw~ _ x2 y
DeY³eeme 4.2
I. efvecve J³ebpekeÀeW ceW mebK³eelcekeÀ J³ebpekeÀ Deewj De#ejer³e mebK³eelcekeÀ J³ebpekeÀ (yeerpeer³e J³ebpekeÀeW ceW Jeieeake=Àle keÀerefpeS).
1) 8 + 5 - 3 2) 3×-8 3) (7 × 6) - 4m
4) 3p + 4q 5) (20 × 7) - (5 × 10) - 45 6) 2y + 6 - 4z
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II. 1) efvecveefueefKele Jeerpeer³e J³ebpekeÀ kesÀ Heo efueefKeS~
1) 3x + 4y 2) 2pq - 8qr 3) 3 x 2 - 3x + z
4) ab + bc - ca 5) 9m + 6n 6) - 3xy + x - y
2) efvecve yeerpeer³e J³ebpekeÀeW kesÀ Heo Deewj iegCeveKeC[ Hen®eeefveS~
1) 2xy + 5 2) x2 + xy + 4 3) 3p - 5y2 4) 2ab + bc + ca
III. efvecve keÀLeveeW keÀes yeerpeer³e J³ebpekeÀeW ceW J³ekeÌle keÀerefpeS~
1) `x'keÀes 8 pees[e nw~
2) `y' ceW mes 7 Ieìe³ee ie³ee nw~
3) 12 kesÀ meeLe P iegCee efkeÀ³ee ie³ee nw~
4) q keÀes 5 mes Yeeie ueiee³ee ie³ee nw~
5) x kesÀ 4 iegvee keÀes y kesÀ 3 iegvee ceW pees[e ie³ee nw~
6) y kesÀ 5 iegCee keÀes 10 cesb mes Ieìe³ee ie³ee nw~
7) p Deewj q kesÀ iegCeveHeÀue keÀes 3x pees[e ie³ee nw~
8) `l'kesÀ 3 iegCee ceW mes kesÀ iegCee keÀes Ieìe³ee ie³ee nw~
9) y keÀes 10 iegCee keÀjkesÀ Deewj iegCeveHeÀue ceW 15 pees[e ie³ee nw~
10) x Deewj y kesÀ iegCeveHeÀue keÀes z mes Yeeie ueiee³ee nw~
IV. efvecveeW ceW `x' kesÀ mebK³eelcekeÀ meniegCeebkeÀ efueefKeS~
1) 3x 2) -4x 3) x43
4) 10.5x 5) 168x 6) x98-
V. efvecveeW ceW De#ejer³e meniegCeebkeÀ efueefKeS~
1) 3m 2) -9xy 3) 16pqr
4) 10c 5) mn45- 6) 0.8x2y
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VI. DeeosMeevegmeej 27xyz kesÀ meniegCeebkeÀ efueefKeS
a) xyz keÀe mebK³eelcekeÀ meniegCeebkeÀ __________ nw~
b) xy keÀe De#ejer³e meniegCeebkeÀ __________ nw~
c) 27z keÀe De#ejer³e meniegCeebkeÀ __________ nw~
d) 27 keÀe De#ejer³e meniegCeebkeÀ __________ nw~
mepeeefle Deewj efJepeeefle Heo (Like terms and unlike terms)
jece, megcee, meuecee Deewj je@yeì& SkeÀ keÀ#ee ceW ®e®ee& keÀj jnW nQ~
jece keÀnlee nw : 20 Hesve + 5 Hesve = 25 Hesve
megcee keÀnleer nw : 50 veesì yegkeÀ - 5 veesìyegkeÀ = 45 veesìyegkeÀ
meuecee keÀnleer nw : 20 Hesve + 15 veesìyegkeÀ = 35.
jeyeì& ves Hetíe 35 veesìyegkeÀ DeLeJee 35 Hesve? legce Ssmes pees[ veneR mekeÀles~
meuecee ves Hetíe : ke̳eeW ceQ Ssmes pees[ veneR mekeÀleer?
jeyeì& : Jes efJeefYevve JemlegSB nQ; nce efJeefYevve JemlegDeeW keÀes pees[ Yeer veneR, vee Ieìe mekeÀles nQ~
meuecee : neB, nce 3 efkeÀ. ûeeb MekeÌkeÀj keÀes 2 efkeÀuees ûeece vecekeÀ kesÀ meeLe veneR pees[ mekeÀles~
peye efJeOeeLeea Fme lejn ®e®ee& keÀj jns Les GvekeÀer efMeef#ekeÀe DeekeÀj SkeÀ yees[& SkeÀ ®eeì& ueìkeÀe³ee Deewj GvnW Gmes O³eeve mes osKeves keÀne~
®eeì& kesÀ Hesìer-1 Deewj Hesìer -2 keÀes O³eeve mes osefKeS
Hesìer - 11) x, 3x, -6x2) 2ab, -3ba, 2
1 ab
3) y2, 3y2, -52 y2
Hesìer - 21) x, 3y, -6z2) 2ab, -3bc, 2
1 ca
3) x, x2, -2x3
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efMeef#ekeÀe ves Hetíe : Hesìer 1- ceW kewÀmeW Heo nQ?
Hesìer : 1 ceW SkeÀ De#ejer³e iegCeveKeC[ Deewj SkeÀ ner IeeleebkeÀ kesÀ yeerpeer³e J³ebpekeÀ nwb~
efMeef#ekeÀe : 2 otmejs Hesìer ceW ke̳ee nw?
Hesìer : 2-ceW oesveeW efJeefYevve De#ejer³e iegCeveKeC[ DeLeJee SkeÀ ner De#ejer³e iegCeveKeC[ Hejvleg efYevve Ieele kesÀ nw~
FmeefueS,
Hesìer -1 kesÀ 1, 2, 3 mecetn mepeeefle Heo kesÀ mecetn efve©efHele keÀjles nw~
Hesìer -2 kesÀ 1, 2, 3 mecetn, efJepeeefle HeoeW kesÀ mecetn efve©efHele keÀjles nw~
SkeÀ ner IeeleebkeÀ ³egkeÌle leLee SkeÀ De#ejer³e iegCeveKeC[ kesÀ yeerpeer³e HeoeW keÀes mepeeefle Heo keÀnles nw~ efYevve De#ejer³e iegCeveKeC[ DeLeJee SkeÀ De#ejer³e iegCeveKeC[ efYevve IeeleebkeÀ kesÀ yeerpeer³e HeoeW keÀes efJepeeefle Heo keÀnles nQ~
yeerpeer³e J³ebpekeÀeW keÀe JeieeakeÀjCeefvecve Hesefì³eeW keÀes O³eeve mes osefKeS :
Hesìer - 1 Hesìer - 2 Hesìer - 3 Hesìer - 4-3x, 2xy, 7x2y,
4pqr, -10/7mn
x+y, 2ab+c,
4p2+3y, m-7
x+y+z,
2x2-3x+4,
ab+bc+ca
5x4+4x3-3x2+2x+8,
x5-3x2+7x+8,
9p2+q2-rq+pq-8
Hesìer 1: ceW SkeÀ Heo mes ³egkeÌle J³ebpekeÀ meceeefJe<ì nQ~
Hesìer 2: ceW oes HeoeW mes ³egkeÌle J³ebpekeÀ meceeefJe<ì nQ~
Hesìer 3: ceW leerve HeoeW mes ³egkeÌle J³ebpekeÀ meceeefJe<ì nQ~
Hesìer 4: ceW DeveskeÀ HeoeW mes ³egkeÌle J³ebpekeÀ nQ~
ceesvees keÀe DeLe& SkeÀSkeÀ → SkeÀ
yeeF → oes
$eÝ → leerve
Heewueer → DeefOekeÀ
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JeieeakeÀjCe efvecve ÒekeÀej mes efkeÀ³ee peelee nw~
i) SkeÀ Heoer³e : efpeme J³ebpekeÀ ceW kesÀJeue SkeÀ Heo nw~
ii) efÜ Heoer³e : efpeme J³ebpekeÀ ceW oes efJepeeefle Heo nesles nQ ~
iii) ef$e Heefo³e : efpeme J³ebpekeÀ ceW leerve efJepeeefle Heo nesles nQ ~
iv) yengHeoer³e : HeoeW keÀer mebK³ee keÀes O³eeve ceW ve ueskeÀj J³ebpekeÀ keÀes yengHeoer keÀnles nQ ~
meeceev³ele : IeveelcekeÀ IeeleebkeÀ kesÀ HeoeW kesÀ J³ebpekeÀ keÀes yengHeoer keÀnles nw~
met®evee : yeerpe ieefCele ceW, mepeeefle HeoeW keÀes pees[ mekeÀles nw DeLeJee Ieìe mekeÀles nw~ a + 3a = 4a, 3ab + 5ab + 6ab = 14ab ∴a + 3a SkeÀ Heoer³e veneR nw Hejvleg Jen SkeÀ Fmeerlejn 3ab + 5ab + 6ab ef$eHeoer veneR nQ Hejvleg SkeÀ Heoer³e nw~
peeefveS
+ Deewj - ef®evn J³ebpekeÀ keÀes Deueie keÀjles nQ~
× Deewj ÷ yeerpeer³e J³ebpekeÀ keÀes Deueie vener keÀjles nQ~
yeerpeer³e J³ebpekeÀ
efmLejebkeÀ SkeÀ Heoer efÜHeoer ef$eHeoer
yengHeoer
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yengHeoer³eeW kesÀ kegÀí efJeMes<e iegCeOece&
1) SkeÀ DeLeJee DeefOekeÀ HeoeW mes ³egkeÌle J³ebpekeÀ keÀes yengHeoer³e keÀnles nw~ Dele: SkeÀ Heoer, efÜHeoer Deewj ef$eHeoer yengHeoer nQ~
2) Òel³eskeÀ Heo kesÀ ®ejebkeÀ keÀe IeeleebkeÀ HetCe& mebK³ee nesvee ®eeefnS~
3) `{X Ajar` g§»` H$m EŠgmo_oQ> G$UmË_H$ Wm {^Ýh h¡ & `h ~hþnXr Z§hr h¡ &
x½, 5x - 2, 1/x, p1/3, SkeÀ yengHeoer kesÀ Heo veneR yeve mekeÀles~
keÀe³e&keÀueeHe: SkeÀ Heoer, efÜHeoer, ef$eHeoer Deewj yengHeoer Hen®eevevee.
®ejCe-1 : ®eej Kegues ef[yyeeW Hej SkeÀ Heoer, efÜHeoer, ef$eHeoer, Deewj yengHeoer efueKekeÀj jefKeS~®ejCe-2 : efJeefYevve ÒekeÀej kesÀ J³ebpekeÀeW kesÀ HeÀueeMe keÀe[X keÀes mìtue Hej Guìs jefKeS~®ejCe-3 : SkeÀ SkeÀ efJeOeeLeea keÀes yegueekeÀj, SkeÀ keÀe[& ueskeÀj, Gmes peesj mes He{keÀj, ³eesi³e
ef[yyeeW ceW jKeves keÀefnS~®ejCe-4 : Òel³eskeÀ ef[yyes ceW jKes ieS keÀe[& efievekeÀj Deewj ieuele ef[yyeeW ceW jKes keÀe[eX keÀes
efvekeÀeueles ngS, DeHeves efvejer#eCeeW keÀes SkeÀ leeefuekeÀe ceW efueKekeÀj jefKeS~
mecetn keÀe[eX keÀer mebK³ee HeoeW keÀer mebK³ee J³ebpekeÀ keÀe ÒekeÀejA SkeÀ SkeÀ Heoer --------------B oes efoHeoer --------------C leerve ef$eHeoer --------------
ke̳ee DeeHeves yeerpeer³e J³ebpekeÀeW kesÀ JeieeakeÀjCe keÀe DeeOeej mecePe efue³ee nw? Fme lejn veeceebkeÀve keÀjves keÀe keÀejCe ke̳ee nw?
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IeeleebkeÀ ³egkeÌle mebK³ee (Powered numbers) :nceves meerKee nw efkeÀ Jeie& keÀe #es$eHeÀue = Yegpee × Yepee³eefo Jeie& keÀer Òel³eskeÀ Yegpee keÀer uebyeeF& 2 cm nes lees
Jeie& keÀe #es$eHeÀue = 2 × 2 = 22 Jeie& cm = 4 Jeie& cm
³eefo Òel³eskeÀ Yegpee keÀer uebyeeF& `x' cm, nes lees,Jeie& keÀe #es$eHeÀue = x cm × x cm
= x2 sq cm
Fmeerlejn, Ieve keÀe Dee³eleve = uebyeeF& × ®eew[eF& × TB®eeF&³eefo uebyeeF& ®eew[eF& Deewj TB®eeF& `2'cm kesÀ nes leesIeve keÀe Dee³eleve = 2cm × 2cm × 2cm
= 8 Ieve cm
= 8 Ieve cm
³eefo Ieve keÀer uebyeeF&, ®eew[eF& Deewj ®eew[eF& kesÀ 'y' cm nes leesIeve keÀe Deeceleve = y cm × y cm × y cm
= y3 cm
= y3 cubic cm
x 2, y3, p4, q6, d7 Deewj m9 Fl³eeefo IeeleebkeÀ ³egkeÌle mebK³eeSB nw~IeeleebkeÀ ³egkeÌle mebK³eeDeeW keÀes kewÀmes He{s?• a2 keÀes a IeeleebkeÀ a DeLeJee 2 keÀe Jeie& He{les nw~ • a3 keÀes a IeeleebkeÀ a DeLeJee a keÀe Ieve He{les nw~ • p4 keÀes p Ieele 4 He{les nw~• d7 keÀes `d' Ieele 7 He{les nw~• m9 keÀes `m' Ieele 9 He{les nw~
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IeeleebkeÀ ³egkeÌle mebK³eeDeeW mes peg[s Heo:
x2, y3, n7, r4 Fl³eeefo IeeleebkeÀ ³egke Ìle mebK³eeDeeW ceW, IeeleebkeÀ
DeeOeej
x2
De#ejer³e mebK³eeDeesb kesÀ THej oeefnves Yeeie ceW efueKeer mebK³ee keÀes Ieele
DeLeJee IeeleebkeÀ keÀnles nw~
x, y, n, m mebK³eeDeeW keÀes DeeOeej keÀnles nQ~
IeeleebkeÀ ³en metef®ele keÀjlee nw efkeÀ DeeOeej keÀes efkeÀleveer yeej
iegCeveKeC[ kesÀ ©He ceW GHe³eesie keÀjvee nw~.
a2 = a × a (DeeOeej `a' oes yeej GHe³eesie ngDee nw~)
x3 = x × x × x (DeeOeej `x' leerve yeej iegCeveKeC[ kesÀ ©He ceW GHe³eesie ngDee nw~)
(ab)4 = ab × ab × ab × ab (DeeOeej ab 4 yeej iegCeveKeC[ kesÀ ©He ceW GHe³eesie ngDee nw~)
met®evee : DeeOeej keÀer leguevee ceW IeeleebkeÀ keÀer mebK³ee keÀe iee$e íesìe nesvee ®eeefnS~
SkeÀ yeerpeer³e J³ebpekeÀ keÀe Ieele (Degree of Algebraic Expression) :
Deye lekeÀ nceves yeerpeer³e J³ebpekeÀ kesÀ Heo Deewj OeeleebkeÀ kesÀ yeejW ceW DeO³e³eve efkeÀ³ee nw~
x 4 + x2 - 3x + 4 Hej efJe®eej keÀerefpeS~
J³epekeÀ kesÀ Heo nw x 4, x2, -3x Deewj 4
Fme J³ebpekeÀ kesÀ keÀewve mes Heo keÀe IeeleebkeÀ Dev³eeefOekeÀ nw? neB, x 4 Heo IeeleebkeÀ Dev³eeefOekeÀ nw DeLee&le 4 nw~
Fmelejn x 4 + x2 - 3x + 4 J³ebpekeÀ keÀe Ieele 4 keÀne peelee nw~
a2 + 5a3 - 4a - 8 Heo 5a3 keÀe Ieele Deel³eeefOekeÀ nw~ FmeefueS J³ebpekeÀ keÀe Ieele 3 nw~
FmeefueS ³en Ieele 3 keÀe J³ebpekeÀ nw~
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SkeÀ ®ejebkeÀ ³egkeÌle yengHeoer keÀe Ieele J³ebpekeÀ ceW GHeefmLele Del³eeefOekeÀ Ieele kesÀ ®ejebkeÀ keÀe Ieele neslee nw~.
yengHeoer kesÀ Ieele kesÀ yeejs ceW DeefOekeÀ peevekeÀejer:
3x4 - 4x3y2 + 8xy + 7 J³ebpekeÀ Hej efJe®eej keÀerefpeS~ Òel³eskeÀ Heo keÀe Ieele osefKeS~
3x4 → ceW, IeeleebkeÀ 4 nw Deewj Ieele 4 keÀe nw~
- 4x3 y2 → x kesÀ Ieele keÀes y kesÀ Ieele ceW peesef[S DeLee&le 3 + 2 = 5.³en Ieele keÀe nw~
8xy Heo keÀe Ieele (1 + 1) = 2
FmeefueS, Del³eeefOekeÀ IeeleebkeÀ keÀe Heo nw (- 4x3y2) Deewj GmekeÀe Ieele 5 nw~
SkeÀ mes DeefOekeÀ ®ejebkeÀ kesÀ yengHeoer keÀe Ieele Gme J³ebpekeÀ kesÀ HeoeW kesÀ IeeleebkeÀeW keÀe ³eesieHeÀue nw~
DeY³eeme 4.3
I. Fve J³ebpekeÀeW keÀes SkeÀ Heoer, efÜHeoer DeLeJee ef$eHeoer ceW Jeieeake=Àle keÀerefpeS~
1) 2x + y 2) 2xy
3) 5 + 6a + 4b 4) 3x2 + 5x
5) xyz 6) ab - bc
7) 2y z x37- + 8) 3xp ÷ q
9) a2 - 3ab + c 10) x2 + 4 - 3x
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II. efvecve J³ebpekeÀeW ceW keÀewve mes yengHeoer nQ?
1) x + 2yq 2) x3 + x - 2 + 1
3) y2 + xy½ - 6 4) m3 + 2m2 + 3m - 4
III. mepeeefle HeoeW kesÀ mecetneW keÀes `L' Deewj efJepeeefle HeoeW keÀes `U' metef®ele keÀerefpeS~
1) 3x, 5x, 8x 2) x3, -3x2, 8x 3) -8p2, 6p2, 10p2
4) 2ab, 6ba, 8ab 5) 3a2b, - 2ab2, 7a2b2 6) - a3, 2a2, - 8a
IV. mepeeefle HeoeW kesÀ mecetn yeveeFS~
1) x, y3, 2x, -7y3, -8x, 23 y3, 6x, -y3
2) 7ab, 6bc, -8ba, 2ca, -3ab, 2abc, 4ab, 2a2b
3) 7p, 8pq, -5pq, 2p Deewj 3p
V. efvecveeW kesÀ mebK³eelcekeÀ meniegCeebkeÀ efueefKeS~
1) 3xy 2) ab32- 3) 0.3p
4) 24xyz 5) -18p2q 6) m np119 2-
VI. met®eveevegmeej meniegCeebkeÀ efueefKeS~
1) 3axy ceW xy keÀe
2) 4a2b ceW ab keÀe
3) p2yz2 ceW z2 keÀe
4) 10xy ceW xy keÀe
5) -15p2 ceW 15 keÀe
6) -mn ceW mn keÀe
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VII. efvecveefueefKele Òel³eskeÀ yengHeoer keÀe Ieele efueefKeS~
1) 3x2 2) 3 - p2 + p3 3) 7m2n
4) z - p 5) a2 + 2ab + abc 6) m2 + 2mn2 + n2
VIII. efvecve leeefuekeÀe HetCe& keÀerefpeS :
IeeleebkeÀ ³egkeÌle mebK³ee DeeOeej IeeleebkeÀ
x 5
(ab)6
(13 p)9
(-y)10
(xyz)7
(0.59)20
IX. efvecveeW keÀes IeeleebkeÀ ©He ceW efueefKeS:
1) p × p × p × p × p 2) m × m × m × m
3) ab × ab × ab × ab × ab × ab 4) z × z × z × z × z × z × z × z
5) abc × abc × abc 6) b × b × b × ...... 10 yeej
yeerpeer³e J³ebpekeÀes keÀer ceewefuekeÀ Òeef¬eÀ³eeSB
DebkeÀ ieefCele keÀer lejn, yeerpeer³e J³ebpekeÀeW Hej meYeer ceewefuekeÀ Òeef¬eÀ³eeSB HetCe& keÀj mekeÀles nwb Deewj Jes mebK³eeDeeW keÀer lejn pees[ves, Ieìeves, iegCee keÀjves Deewj Yeeie ueieeves kesÀ meYeer ceewefuekeÀ efve³eceeW keÀe Heeueve keÀjles nQ~ HetCee¥keÀeW keÀer lejn ner mebkesÀleeW keÀe GHe³eesie keÀjles nQ~
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yeerpeer³e J³ebpekeÀeW keÀe pees[vee Deewj Ieìevee
ceeve ueerefpeS 3 ìeskeÀefj³eeW ceW kegÀí mesye, kegÀí Deece Deewj veejbieer nQ Deewj nce GvnW SkeÀ meeLe efceueekeÀj jKeles nQ~
ceeve ueerefpeS 1 ìeskeÀjer ceW 8 mesye Deewj 8 Deece nQ,
2 js ìeskeÀjer ceW 6 mesye Deewj 4 veejbieer
Deewj 3 js ìeskeÀjer ceW 5 Deece Deewj 6 veejbieer nQ~
peye nce Fve HeÀueeW keÀes efceueekeÀj SkeÀ ìeskeÀjer ceW jKeles nQ~ lees.
8 + 6 =14 mesye
8 + 5 = 13 Deece
4 + 6 = 10 veejbieer nesles nQ~
³ener efJeOeeve, yeerpeer³e J³ebpekeÀeW kesÀ pees[ves Deewj Ieìeves ceW DeHeveeles nw~ nce mepeeefle HeoeW keÀes pees[les DeLeJee Ieìeles nQ~
yeerpeer³e J³ebpekeÀeW kesÀ pees[les Deewj Ieìeles mece³e HetCee¥keÀeW kesÀ pees[ves Deewj Ieìeves kesÀ efve³eceeW keÀe DevegmejCe keÀerefpeS~
³eeo keÀerefpeS:
1) IeveelcekeÀ HetCee¥keÀeW keÀe pees[ IeveelcekeÀ neslee nw~
2) $eÝCeelcekeÀ HetCee¥keÀeW keÀe pees[ $eÝCeelcekeÀ neslee nw~
3) IeveelcekeÀ HetCee¥keÀ Deewj $eÝCeelcekeÀ HetCee¥keÀ keÀe pees[ IeveelcekeÀ neslee nw ³eefo ye[e HetCee¥keÀ IeveelcekeÀ nes~
4) IeveelcekeÀ HetCee¥keÀ Deewj $eÝCeelcekeÀ HetCee¥keÀ keÀe pees[ $eÝCeelcekeÀ neslee nw ³eefo ye[e HetCee¥keÀ $eÝCeelcekeÀ nes~
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yeerpeer³e J³ebpekeÀeW keÀe pees[
a) SkeÀ Heefo³eeW keÀe pees[
i) kesÀJeue mepeeefle HeoeW keÀes pees[ mekeÀles nQ~
ii)GvekeÀe pees[ %eele keÀjves mebK³eelcekeÀ meniegCeebkeÀes keÀe pees[ uesles nQ~
GoenjCe 1 : 4x, 8x Deewj 5x keÀes peesef[S.
nue 4x + 8x + 5x meYeer mepeeefle Heo nQ~ = (4 + 8 + 5)x meYeer x keÀes pees[ ueerefpeS
= 17x Fme mebK³eelcekeÀ meniegCeebkeÀ kesÀ meeLe x efueefKeS
GoenjCe 2 : 3a, 5b, 8a Deewj 2b keÀe ³eesieHeÀue %eele keÀerefpeS
nue 3a + 5b + 8a + 2b
= 3a + 8a + 5b + 2b
= (3+8)a + (5+2)b
= 11a + 7b
³eneB a Deewj b ®ejebkeÀ nw mepeefle HeoeW keÀes Deueie mecetn yeveefnS.
mebK³eelcekeÀ meniegCeebkeÀ peesef[S Deewj mebK³eelcekeÀ meniegCeebkeÀ kesÀ meeLe x efueefKeS
GoenjCe 3 : 3x2y, 2xy2 Deewj 7yx2 keÀes peesef[S~nue 3x2y + 2xy2 + 7yx2
= 3x2y + 7yx2 + 2xy2
= (3+7)x2y + 2xy2
= 10x2y + 2xy2
met®evee:
i) x2y = yx2
ii) x2y ≠ xy2
b) efÜHeoer Deewj ef$eHeefo³eeW keÀe pees[:
GoenjCe 1 : (3a + 2b) Deewj (5a + 7b) keÀes peesef[S~
i) #eweflepe efJeOeeve: (HebefkeÀle) meYeer J³ebpekeÀeW keÀes SkeÀ #eweflepe jsKee efueKekeÀj, mepeeefle HeoeW
keÀes SkeÀef$ele keÀjkesÀ pees[les nQ~
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nue: #eweflepe efJeOeeve
(3a + 2b) + (5a + 7b)
= 3a + 2b + 5a + 7b
= 3a + 5a + 2b + 7b
= 8a + 9b
ii) TOJee&Oej (mlebYe) efJeOeeve : J³ebpekeÀeW keÀes Fmelejn J³eJeefmLele keÀjles nQ leeefkeÀ mepeeefle Heo SkeÀ otmejs kesÀ veer®es mlebYe kesÀ ©He ceW Deeles nQ~ yeeo ceW, mlebYeeW ceW HeoeW keÀes pees[les nQ~
TOJee&Oe&j (mlebYe) efJeOeeve:
3a + 2b ... ³eesp³e
5a + 7b ... ³eesp³e8a + 9b ... pees[ (³eesieHeÀue)
GoenjCe 2 : (14m2n-10n2) Deewj (10m2n+6n2) keÀes peesef[S ~
nue: #eweflepe efJeOeeve TOJee&Oe&j efJeOeeve
(14m2n -10n2) + (10m2n + 6n2) 14m2n - 10n2
= 14m2n - 10n2 + 10m2n + 6n2 +10m2n + 6n2
= 14m2n + 10m2n -10n2 + 6n2 24m2n - 4n2
= 24m2n - 4n2
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GoenjCe 3 : (4x+3y+5z), (2x+5y-2z) Deewj (3x-4y-z) keÀes ³eesieHeÀue %eele keÀerefpeS
nue: #eweflepe efJeOeeve TOJee&Oe&j efJeOeeve
(4x + 3y + 5z) + (2x + 5y -2z) + (3x - 4y - z) 4x + 3y + 5z
= 4x + 3y + 5z + 2x + 5y - 2z + 3x - 4y - z + 2x + 5y - 2z
= (4x + 2x + 3x) + (3y + 5y - 4y) + (5z - 2z - z) + 3x - 4y -z
= 9x + 4y + 2z 9x + 4y + 2z
DeY³eeme 4.4
I efvecveeW keÀe pees[ %eele keÀerefpeS~ 1) 4a, 7a, 9a, 21a 2) 6p2, 4p2, -10p2, 12p2
3) 6y, 7x, -3y, -4y, -4x 4) 2a2, a, 3b2, 4a2, 11a, -4b2
II. mejue keÀerefpeS~ 1) 5x + 7x - 3x + 2x - 6x 2) 5a2 + 2a2-a2 + 7a2
3) 5xy + 6xy - 9xy - xy + 10xy - 2xy
4) 2x2 + 4y - 3z + x2 - y + z + 4x2 + 3y + z
III. efvecve efÜHeefo³eeW keÀe pees[ %eele keÀerefpeS~
1) (3x+ 4y) Deewj (5x+ 7y)
2) (13a - 4b) Deewj (4a - 6b)
3) (xy - 7z) Deewj (6z -10xy)
4) (20a2bc - 3ab2c) Deewj (10ab2c - 3a2bc)
IV. efvecve ef$eHeefo³eeW keÀe pees[ %eele keÀerefpeS~
1) (4x + 3y + 5z) Deewj (2x- 6y - 2z)
2) (p2 + 3q2 + 5r2) Deewj (6p2 - 2q2 + 6r2)
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3) (4a + b - 2c) Deewj (3b + 2a + 5c)
4) (4x2y - 5y2z + 6z2x) Deewj (7z2x- 2x2y + 3y2z)
V. efvecve J³ebpekeÀeW keÀes peesef[S~
1) (5mn - 3p), (mn + p), (6mn - 2p)
2) (x+ y - z), (2x+ 3y + 2z) Deewj (5x- 6y + 3z)
3) (5a2 + 2a-3), (2a2 + 3a + 1) Deewj (3a2 - 4a + 5)
VI. peesef[S :
5x2, 2x2 + y, 3 x2 + 3y + 7, y2+ 5Z + 6y + 4
yeerpeer³e J³ebpekeÀeW keÀe Ieìevee (J³eJekeÀueve)
a) SkeÀ Heefo³eeW keÀe J³eJekeÀueve :
i) kesÀJeue mepeeleer³e HeoeW keÀes Ieìe mekeÀles nwb~
ii) GvekeÀe Devlej %eele keÀjves GvekesÀ mebK³eelcekeÀ menlegCeebkeÀ Ieìe mekeÀles nQ~
GoenjCe 1 : 12x ceW mes 5x IeìeF&S~ (12x) - (5x)
= (12 - 5) x
= 7x
GoenjCe 2 : 7xy Deewj 3xy kesÀ pees[ ceW mes 5xy keÀes IeìeF&S~ (7xy + 3xy) - (5xy)
= 10 xy -5 xy
= 5xy
b) efÜHeefo³eeW leLee ef$eHeefo³eeW keÀe J³eJekeÀueve
GoenjCe 1 : (9x+7y) ceW mes (5x+3y)
ii) TOJee&Oej DeLeJee mlebYe efJeOeeve:
Fme efJeOeeve ceW J³ebpekeÀeW keÀes Òel³eskeÀ HebeqkeÌle ceW efueKeles nQ leeefkeÀ mepeeefle Heo SkeÀ otmejs kesÀ
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veer®es mlebYeekeÀej ceW Deeles nw~ efpeme J³ebpekeÀes keÀes Ieìevee nw GmekesÀ meYeer HeoeW kesÀ cetue ef®evn yeoueles nQ~ yeeo ceW mepeeefle HeoeW keÀes pees[les nQ~
#eweflepe efJeOeeve :
(9x + 7y) - (5x + 3y)
= 9x + 7y - 5x - 3y
= 9x - 5x + 7y - 3y
= (9 - 5)x + (7 - 3)y
= 4x + 4y
ii) TOJee&Oej mlebYe efJeOeeve:
J³ebpekeÀeW keÀes oes Òel³eskeÀ mlebYe kesÀ ©He ceW Ssmee efueKevee ®eeefnS efkeÀ mepeeefle Heo SkeÀ otmejs
kesÀ veer®es Deevee nw (pees J³ebpekeÀes keÀe J³eJekeÀue£À keÀjlee nw, GvekeÀe ef®evn yeouekeÀj GmekesÀ Devegmeej
pees[vee ³ee Ieìevee ®eeefnS~
9x + 7y → J³eJekeÀuekeÀ
(-)5x + (-)3y → J³eJekeÀu³e
4x + 4y → Deblej
GoenjCe 2 : (ax + by) ceW mes (2ax + 5by) keÀes IeìeF&S
#eweflepe efJeOeeve TOJee&Oej efJeOeeve
(2ax + 5by) - (ax + by) 2ax + 5by J³eJekeÀuekeÀ
= 2ax + 5by - ax - by (-)ax + (-)by J³eJekeÀu³e
= (2 - 1)ax + (5 - 1) by ax + 4by Deblej
= ax + 4by
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GoenjCe 3 : (10a - 5b + 12c) ceW mes (2a + b + 6c) keÀes IeìeFS~
#eweflepe efJeOeeve
(10a - 5b + 12c) - (2a + b + 6c) 10a - 5b + 12c J³eJekeÀuekeÀ
= 10a - 5b + 12c - 2a - b - 6c (-)2a + b + (-)6c J³eJekeÀu³e
= (10a - 2a) + ( - 5b - b) + (12c - 6c) 8a - 6b + 6c Deblej
= 8a - 6b + 6c
DeY³eeme 4.5
I. efvecveeW ceW, ÒeLece Heo keÀes otmejs Heo ceW mes IeìeFS~
1) 8x; 15x 2) 6a2b; 14a2b
3) -6ab, 10ab 4) 8x2y; -14x2y
II. 1) (12x + 6y) ceW mes (7x + 4y)
2) (17a - 4b) ceW mes (13a - 6b)
3) ( - 10m2n + 4mn2) ceW mes (- 6m2n - mn2)
4) (10 - 4pqr + 3q) ceW mes (7pqr - 8q + 6)
III. 1) (2x + 6y) mes (6x + 10y) efkeÀlevee DeefOekeÀ nw?
2) (5c - 2d) mes (2d - c) efkeÀlevee DeefOekeÀ nw?
3) (3x + 4y + 6) keÀes efkeÀlevee pees[vee ®eeefnS leeefkeÀ (4x - 2y - 8) ÒeeHle nes~
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4) (9m2n2 - 12xy + 10) keÀes ke̳ee pees[vee ®eeefnS~(15m2n2 - 10xy)
yeerpeer³e J³ebpekeÀeW keÀe iegCeveHeÀue
Fme leekeÀ (Shelf) keÀes osefKeS
5 efieueeme
5 efieueeme
5 efieueeme
5 efieueeme
iueemeeW keÀes 4 HebeqkeÌle³eeW ceW J³eJeefmLele efkeÀ³ee ie³ee nw~ Òel³eskeÀ HebeqkeÌle ceW 5 iueeme nQ~ ®eejeW
HebeqkeÌle³eeW ceW kegÀue efkeÀleves iueeme nQ? nce Henues meerKeW nQ efkeÀ iegCeveHeÀue HegvejeJeefle&le pees[ nw~
DeLee&le :
5 + 5 + 5 + 5 = 20 iueeme
5 × 4 = 20 iueeme
ceeve efueefpeS Òel³eskeÀ HebeqkeÌle ceW `2x' iueeme nw leekeÀ ceW kegÀue iueeme Fmelejn %eele keÀj
mekeÀles nQ~
: 2x + 2x + 2x + 2x = 8x iueeme
2x × 4 = 8x iueeme nwb~
HetCee¥keÀeW keÀes iegCee keÀjles mece³e ef®evneW kesÀ GHe³eesie ³eeo keÀerefpeS:
IeveelcekeÀ HetCee¥keÀ × IeveelcekeÀ HetCee¥keÀ = IeveelcekeÀ HetCee¥keÀ
IeveelcekeÀ HetCee¥keÀ × $eÝCeelcekeÀ HetCee¥keÀ = $eÝCeelcekeÀ HetCee¥keÀ
$eÝCeelcekeÀ HetCee¥keÀ × IeveelcekeÀ HetCee¥keÀ = $eÝCeelcekeÀ HetCee¥keÀ
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$eÝCeelcekeÀ HetCee¥keÀ × $eÝCeelcekeÀ HetCee¥keÀ = IeveelcekeÀ HetCee¥keÀ
1) SkeÀ Heefo³eeW keÀe iegCeveHeÀue (Multiplication of Monomials)
GoenjCe 1 : 3x keÀes 5y mes iegCee keÀerefpeS~
nue : Henues mebK³eelcekeÀ meniegCeebkeÀ iegCee keÀerefpeS Deewj yeeo ceW De#ejer³e men iegCeebkeÀ. (3x) × (5y)
= (3 × x) × (5 × y) met®evee:
3 × 5 ≠ 35uesefkeÀve a × b = ab³eefo ®ejebkeÀ keÀes Heeme Heeme efueKeles nQ lees Jen iegCeveHeÀue metef®ele keÀjlee nw~
= (3 × 5) × ( x × y) = 15xy
GoenjCe 2 : (-20x) keÀes (+3y) mes iegCee keÀerefpeS~
nue : (-20x) (+3y) = (-20 × x) (3 × y)
= (-20 × 3) (x × y)
= -60xy
GoenjCe 3 : (-2x), (-3x), (4x) keÀe iegCeveHeÀue %eele keÀerefpeS~
(-2x) (-3x) (4x)
= (-2 ×-3 ×4) (x × x × x)
= +24x 3
GoenjCe 4 : 3x 2 Deewj 5x3 keÀe iegCeveHeÀue %eele keÀerefpeS~
(3x 2) (5x 3)
= (3 × 5) (x 2 × x 3)
= 15x 5
2) SkeÀ efÜHeoer keÀes SkeÀ Heoer mes iegCee keÀjvee.
5 iegCeve HeÀue keÀer ueerefpeS :
5 × 108 = 540
efueKeves kesÀ efueS 5 × (100+8)
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= (5 × 100) + (5 × 8)
= 500 + 40
= 540
HetCee¥keÀes ceW nceves pees efJelejCe efve³ece meerKee Gmes mcejCe keÀerefpeS~
ie a (b + c) = (a × b) + (a × c)
GoenjCe 1 : (3x + 2y) keÀes 5y mes iegCee keÀerefpeS~
5y (3x + 2y)
= (5y × 3x ) + (5y × 2y)
= 15x y + 10y 2
GoenjCe 2 : (8y - 3) Deewj 4x keÀe iegCeveHeÀue %eele keÀerefpeS~
4x × (8y - 3)
= (4x × 8y) + (4x × - 3)
= 32x y - 12x
GoenjCe 3 : (8x -17y) Deewj ( - 2x ) keÀe iegCeveHeÀue %eele keÀerefpeS.
(8x - 17y) × ( - 2x )
= (8x × -2x ) + (-17y × - 2x )
= -16x 2 + 34x y
SkeÀ efÜHeoer keÀes SkeÀ efÜHeoer mes iegCee keÀjvee.
pewmes nceves efJelejCe efve³ece keÀjles ngS SkeÀ SkeÀ Heoer keÀes efÜHeoer mes iegCee efkeÀ³ee nw Jewmes ner nce SkeÀ efÜHeoer keÀes otmejs efÜHeoer mes iegCee keÀj mekeÀles nw~.
GoenjCe 1 : (x + 1) Deewj (x + 2) keÀe iegCeveHeÀue
(x + 1) (x + 2) = x (x + 2) + 1(x + 2)
= x2 + 2x + x +2
= x2 + 3x + 2
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GoenjCe 2 : (2a + 5) (a - 7)
nue: (2a + 5) (a - 7)
= 2a (a-7) + 5 (a-7)
= 2a2 - 14a + 5a - 35
= 2a2 - 9a - 35
GoenjCe 3 : (n - 11) Deewj (n - 5) keÀe iegCeveHeÀue %eele keÀerefpeS~ nue: (n - 11) (n - 5)
= n (n - 5) - 11 ( n - 5)
= [(n × n) - (n × 5)] - [(11 × n) - (11 × 5)]
= [(n × n) - (n × 5)] - [(11 × n) - (11 × 5)]
= n2 - 5n - (11n - 55)
= n2 - 5n - 11n + 55
= n2 - 16n + 55
DeY³eeme 4.6
I. iegCee keÀerefpeS~ 1) 7x , 8y, 5z 2) -2p, 59, -4r 3) (6m) , (-8n) , (+3p)
4) (-3a) , (-b) , (-6c) 5) -3x , 4x , 5x 6) (-2p) (+3p2) (-3p3)
II. efvecveeW keÀe iegCeveHeÀue %eele keÀerefpeS~
1) 6x y, 2yz Deewj 7x z 2) -7ab, 5bc Deewj -6ca
3) 4x 2y, -3y2z Deewj x 2 4) -3a2, -2b3 Deewj -4e4
III. iegCee keÀerefpeS~
1) (6x + 7y) mes 2z 2) (3p - 5q) mes -5p
3) (-18x - y) mes 2z 4) (2ab + 3bc) mes 6abc
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IV. iegCee keÀerefpeS~ 1) (x + 4) (x + 8) 2) (5n + 2) (n -3)
3) (3a -b) (2a + b) 4) (5x - 2p) (5x + 2p)
5) (2x - 7) (x - 3)
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DeO³ee³e - 5
keÀesCeFme IeìkeÀ keÀes DeO³e³ee keÀjves kesÀ yeeo:
mebueive keÀesCe, HetjkeÀ keÀesCe, mebHetjkeÀ keÀesCe, jwefKekeÀ pees[er, Meer<e& DeefYecegKe keÀesCe keÀe
DeLe& mecePeles nQ,
meuebile keÀesCe, HetjkeÀ Deewj mebHetjkeÀ, keÀesCe nw jefKekeÀ pees[er, Meer<ee& efYecegKe keÀesCe keÀes Hen®eeveles nwb,
mebueive keÀesCe, HetjkeÀ mebHetjkeÀ keÀesCe, jwefKekeÀ pees[er Deewj Meer<ee&efYecegKe keÀesCe kesÀ DeeOeej
Hej ÒeMve nue keÀjles nQ~
uekeÀ[er kesÀ keÀe³e& Fbefpeveer³eeEjie Deeefo ceW jsKeeieCeerle keÀer %eele GHe³eesefiekeÀ nw~ jsKeeieefCele ceW
jsKee Deewj keÀesCe keÀer %eeve DeJeM³ekeÀ nw~ kewÀcejs kesÀ keÀesCeW, meer{er oerJeej Hej þnjves keÀer lejerkesÀ,
Hene[ keÀe peceerve Hej PegkeÀve, ³en meYeer keÀesCe keÀe ceeHeve nw~
HetJe& keÀ#ee cesb DeeHe efJeefYevve keÀesCe Deewj jsKeeDeeW kesÀ yeejW ceW He{s nQ~
efvecveefueefKeleeW mes efJeefYevve ÒekeÀej kesÀ jsKeeDeeW keÀes Hen®eeefveS~
(i) (ii) (iii)
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GoenjCe : Deeke=Àefleæ (i) SkeÀ #eweflepe jsKee
Deeke=Àefle (ii) ----------- Deeke=Àefle (iii) -----------
efvecveefueefKele Deeke=Àefle³eeW ceW yeves ngS efJeefYevve keÀesCeeW keÀes Hen®eeefveS?
O
A
B
ef®e$e iOP
Q
ef®e$e ii O
K
L
ef®e$e iii
O
M
N
ef®e$e iv
GoenjCe: GHejesÊeg Deeke=Àefle AOB v³etve keÀesCe veer®es kegÀí Debûespeer kesÀ De#ej efo³es ie³es nw, yeves keÀesCeesb keÀes DebefkeÀle keÀerefpeS Deewj v³etve, DeefOekeÀ Deewj uebye keÀesCeeW keÀes DebefkeÀle keÀerefpeS~
GoenjCe: De#ej T cesb DebefkeÀle keÀesCe uebye keÀesCe nw~
³eeo jefKeS
• jsKeeieefCele ceW mejuelece efJe<e³e efyevog nw~ efyevog DeekeÀej venerb kesÀJeue mLeeve nw~ Hesveefmeue keÀer veeskeÀ mes [@eì HesHej Hej yeveeSB peeles nw pees efyevog keÀe YeeweflekeÀ GoenjCe nw~
• jsKee ceW keÀesF& DeejbefYekeÀ ³ee Deebeflece efyevog vener nw~ keÀuHevee keÀerefpeS efkeÀ oesveeW efoMee Demeerefcele lejerkesÀ oesveeW Debeflece efyevog leerjs mes efveefo&<ì efkeÀ³ee ie³ee nw~
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• jsKeeKeC[ ceW efyevog Deewj SkeÀ Debeflece efyevog nwb~ ³en SkeÀ mejue jsKee keÀe Yeeie nQ~
X
M
K
L O
N
PQY
KLO ef$eYegpe kesÀ meYeer Yegpee jsKee KeC[ nQ~
• SkeÀ efkeÀjCe ceW ÒeejbefYekeÀ efyevog nw Hejvleg Debeflece efyevog veneR nQ~ met³e& kesÀ efkeÀjCeeW kesÀ yeejW ceW meesef®eS met³e& mes Meg© nes keÀj meYeer efoMee ceW ®eueles nQ~
megjped
peye oes efkeÀjCe meeceev³e efyevog Hej efceueves mes keÀesCe yeveles nwb~ keÀesCe kesÀ oes efkeÀjCe yeeng keÀnles nwb Deewj efyevog peneB efceueles nwb GvekeÀes Meer<e& keÀnles nQ~
keÀesCe keÀe DeekeÀej GmekesÀ PegkeÀeJe Hej efveYe&j keÀjles nQ ³ee keÀesCe kesÀ
O
A
B
yeeng otmejs yeeng kesÀ Ke[ves Hej~
oÊe Deeke=Àefle OA Deewj OB oes efkeÀjCe pees O mes Meg© nesles nw~ #eweflepe efkeÀjCe OB Deewj TOJee&Oej efkeÀjCe OA keÀe yeer®e keÀe keÀesCe uebye keÀesCe nw~ AOB 900=
leye Fve efkeÀjCeesb keÀes ke̳ee keÀnles nwb?
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keÀesCeeW keÀe ceeHeve:
efkeÀjCe keÀe IegceeJe ÒeejbefYekeÀ mLeeve mes mLeeveer³e mLeeve lekeÀ~ ³en IegceeJe Ie[er keÀer IegceeJe keÀer efoMee ceW nesleer ³ee efJe©× efoMee ceW.
ÒeejbefYekeÀ mLeeve
mLeeveer³e
ÒeejbeefYekeÀ mLeeve
mLeeveer³eefJe©× Ie[er keÀer IegceeJeIe[er keÀer IegceeJe
met®evee : yeves ngS Ie[er IegceeJe ³ee efJe©× Ieb[er OegceeJe efoMee ces keÀesCe keÀe cetu³e meceeve nw~
meceleue - 1
400400
400
meceleue - 2
300 300
meceleue - 3
500
500
meceleue - 1 ces keÀesCe keÀe ceeHe 400 meceleue- 2 ces 300 nQ~ Deewj meceleue -3 ceW 500 nw~.
meJee¥iemece (Congruent) ueefìve Meyo mes efue³ee ie³es efpemekeÀe DeLe& meceeve nw~
keÀesCe keÀer pees[er :jsKeeieefCele ceW kegÀí keÀesCeeW keÀer pees[er ceW efJeMes<e DebkeÀ nw~ v³etve, uebye Deewj DeefOekeÀ keÀesCeeW
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keÀer %eeve mes keÀesCeeW keÀer pees[er keÀe mecyebOe peeves ceW mene³ekeÀ nw~meuebive keÀesCe :
oÊe Deeke=Àefle ceW osefKe³es, keÀesCeeW keÀes Hen®eeves, Òel³eskeÀ Deeke=Àefle Yegpee mes pees[s ie³es nwb~
• Deeke=Àefle ces efkeÀleves keÀesCe yeveles nwb? DebefkeÀle keÀerefpeS~
• keÀesCeeW keÀes Hen®eeves pees Deeke=Àefle ceW yeves nw~
• Meer<e& keÀe veece efueefKeS Deewj keÀesCe kesÀ yeengDeeW kesÀ veece efueefKeS~.
³eneB, Meer<e& OB ceW `0' MeerJe& Deewj SkeÀ keÀesCe yeng OC nw~ Gmeer ÒekeÀej otmejs keÀesCe kesÀ
yeeng OA Deewj OB nw~
FmekeÀe DeLe& nw
i) FmeceW SkeÀ meceeve Meer<e& nw~
ii) FmeceW SkeÀ meceeve Yegpee nwb~
keÀesCeeW keÀer pees[er ceW SkeÀ meceeve Meer<e& Deewj meceeve Yegpee nesves mes mebueive keÀesCe keÀnles nwb~
oÊe Deeke=Àefle ceW AOB Deewj BOC meuebive keÀesCe nw~
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keÀe³e& keÀueeHe - 1
BC
O
A
oÊe Deeke=Àefle ceW leerve meueiCe keÀesCe keÀes DebefkeÀle keÀerefpeS.
meesef®eS
• ke̳ee oes DeefOekeÀ keÀesCe meueiee keÀesCe nes mekeÀlee nw? peeefveS~
• ke̳ee oes v³etve keÀesCe DeefOekeÀ nes mekeÀlee nw? peeefveS~
meesef®eS
AOB Deewj AOC meueiCe keÀesCe neslee nw?
DeY³eeme 5.1
1. efvecve Deeke=Àefle³eeW keÀes osefKeS Deewj mejCeer keÀes YeefjS~
(i)(ii)
(iii)
(iv) (v)(vi)
(vii)
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Deeke=Àefle keÀesCe Yegpee meeceev³e Meer<e&
meeceev³e Yegpee
keÀesCe mebueive nw ³ee veneR
iAOB ,
BOCOA, OB, OC O OB
nQ
AOB Deewj
BOC
ii
iii
iv
v
vi
vii
2. oÊe Deeke=Àefle ceW efvecveefueefKele mebueive keÀesCeeW keÀer pees[er keÀewve meer nw~ meeceev³e Meer<e& Deewj meeceev³e Yegpee yeleeFS~
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keÀesCe
Meer<e
& keÀe
veece
Yegpee
keÀe v
eece
meecee
v³e M
eer<e&
meecee
v³e Y
egpee
mebueiv
e keÀe
sCeeW ke
sÀ peer[
er
DeewjAOB BOD
DeewjAOD BOC
DeewjAOC BOC
DeewjBOC AOB
DeewjAOD AOB
kegÀí keÀesCeeW keÀer pees[er keÀe mecetn yeveeFS Deewj yeleeF³es efkeÀ Jen mebueile keÀesCe nw?3. oÊe Deeke=Àefle ces keÀesCeeW ceW SkeÀ ®eeHe Deewj oes ®eeHe keÀes efvejer#eCe keÀerefpeS~ ke̳ee Jen mebueive
keÀesCeeW keÀer pees[er nw? ke̳eeW?
A
DO 1
2 C
F
E
B
R
Q
PO 1
2
X
X
Z
O
1
2
W
O L
J
21
K
R S
P
21
Q
B1 2
C
A D M
O
P
NQ
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HetjkeÀ keÀesCe:meceeve meceleue veer®es efo³es ie³es Deeke=Àefle³eeW ceW peesef[³eeW keÀes efvejer#eCe keÀerefpeS~
meceleue 1
AC
Q
RPB
700
200
meceleue 2
A B650
250
Q
PRC
meceleue 3
A
A
C
800
Q
P
100
oÊe Deeke=Àefle i), ii) Deewj iii) keÀesCeeW kesÀ pees[er keÀe ceeHeve keÀe ³eesie ke̳ee nQ?
meceeve meceleue ceW ³eefo oes keÀesCeeW keÀe ³eesie 900 nes lees Gmes HetjkeÀ keÀesCe keÀnles nw~
oes keÀesCeeW keÀes efceuelves mes SkeÀ ve³es keÀesCe yeveles nw efpemekeÀe ceeHe 900 nQ~
GHejeskeÌle Deeke=Àefle ceW 90ABC PQR 0+ = ³en ABC , PQR mes mebHetjkeÀ nw~
Deewj PQR , ABC mes mebHetjkeÀ nQ~
HetjkeÀ keÀesCe : meceeve meceleue veer®es efo³es ie³es Deeke=Àefle³eeW ceW peesef[³eeW keÀes efvejer#eCe keÀerefpeS~
meesef®eS : ³eefo x 0 nes lees HetjkeÀ keÀesCe %eele
GoenjCe 1 : 350 keÀesCe keÀe HetjkeÀ keÀesCe %eele keÀerefpeS 350 %eele keÀerefpeS350 + x0 = 900 [∴ HetjkeÀ keÀesCe 900]
x0 = 900 = 350
x0 = 550
350 keÀesCe keÀe HetjkeÀ keÀesCe 550 nw~
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GoenjCe 2 :SkeÀ keÀesCe keÀe 5 iegCee HetjkeÀ keÀesCe nw~ceeefveS keÀesF& keÀesCe x0 keÀesCe keÀe 5 iegCee = 5x0
x0 + 5x0 = 900
6x0 = 900
x0 = 906
0
x0 = 150
5x0 = 5 × 15 = 750
150 keÀe HetjkeÀ keÀesCe 750 nw~
meesef®eS :uebye keÀesCe keÀe mebHetjkeÀ keÀesCe keÀe ceeHeve ke̳ee nw?600 keÀe meHetjbkeÀ keÀesCe keÀe ceeHeve ke̳ee nw?
meesef®eS : ke̳ee oes v³etve keÀesCe SkeÀ otmejs mes mebHetjkeÀ nes mekeÀlee nw? ke̳ee oes DeefOekeÀ keÀesCe SkeÀ otmejs mes mebHetjkeÀ nes mekeÀlee nw? ke̳ee oes uebye keÀesCe SkeÀ otmejs mes mebHetjkeÀ nes mekeÀlee nw?
ef¬eÀ³ee:
Dee³eelekeÀej HesHej Meerì keÀes ceesef[S Hej efmekegÀ[ves mes Deeke=Àefle keÀes pewmes Keesefue³es Deeke=Àefle kesÀ pewmes Debeflece keÀesCeeW keÀes ceeefHeS~
Fmemes DeeHe ke̳ee efveM®e³e uesles nwb?
ef®e$eæ-(1) ef®e$eæ-(2) ef®e$e-(3)
o o o
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ef¬eÀ³ee
keÀþesj keÀe[& yees[& Meerì Hej oes Je=ÊeeW keÀer j®evee keÀerefpeS 2 DeHeveer megefJeOee Devegmeej ef$ep³ee ueerefpeS~
Je=Êe keÀer HeefjefOe A ef$ep³ee mes j®evee keÀerefpeS ~
Je=Êe kesÀ keÀesCe 100, 200, 300, 400, 500, 600, 700 Deewj 800 keÀes DebefkeÀle keÀerefpeS~ ®eeBoe Deewj mkesÀue keÀe GHe³eesie keÀerefpeS~
Òel³eskeÀ keÀesCe pees KeC[ Deueie keÀeefìS efpememes 8 KeC[ nes mekeÀles nQ~
Je=Êe B Hej Meerì Hej 900 yeveeFS~
Je=Êe B Hej Meerì Hej 900 yeveeFS~
403020
8070
60
50
10
A
y
O
Lx
B
8 KeC[eW ceW SkeÀ KeæC[ Ssmes jefKeS pewmes OX Òeefle®ís Deeke=Àefle ceW pewmes keÀjlee nw~ Fme KeC[ LOX jns~ otmejs KeC[ ueerefpeS LOY
Òeefleíefole neslee nw~
Fme ef¬eÀ³ee keÀes ogyeeje keÀerefpeS~ 12 KeC[eW keÀes ueerefpeS XOY Deewj LOX .
jsKeeKeC[es Hej mebHetjkeÀ keÀesCe keÀer met®eer yeveFS pees Fme ef¬eÀ³ee ceW GHe³eesie keÀjles nw~
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DeY³eeme 5.2
1. efvecveefueefKele Deeke=Àefle cesb Òel³eskeÀ keÀesCe SkeÀ otmejs mes HetjkeÀ~ SkeÀ keÀesCe efo³es otmeje keÀesCe %eele keÀerefpeS~
(i) (ii) (iii) (iv)
2. efvecveefueefKele keÀesCe keÀe HetjkeÀ keÀesCe %eele keÀerefpeS~
a) 220 b) 840 c) 20.50 d) 40.50
3. ³eefo SkeÀ keÀesCe otmejs keÀesCe mes HetjkeÀ keÀesCe
mebHetjkeÀ keÀesCe:
efvecveefueefKele meceeve meceleue ceW keÀesCeeW keÀer pees[er keÀes osefKe³es.O
500 1300
S P Q R
M
ef®e$e- i ef®e$e- ii
O
7201080
S P Q R
M
ef®e$e- iii
O 800
1000
S
P Q R
M
oÊe Deeke=Àefle ceW ef®e$e i), ii) Deewj iii) ,SPO RQM/ /
keÀe keÀesCeeW keÀe ³eesie %eele keÀerefpeS~
³eefo meceeve meceleue ces yeves keÀesCeeW keÀe ³eesie mejue keÀesCe nesves mes Gmes HetjkeÀ keÀesCe keÀnles nwb~
³eefo keÀesCeeW keÀe ³eesie mejue keÀesCe 1800 nes lees Gmes HetjkeÀ keÀesCe keÀnles nwb~
meHetjkeÀ keÀesCe SkeÀ meceleue ces mejue keÀesCe keÀe ³eesie 1800
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GHejeskeÌle Deeke=Àefle ceW: SPO MQR 1800+ =
FmeefueS ,P Q keÀe HetjkeÀ keÀesCe Deewj Q keÀe P HetjkeÀ keÀesCe Q nw~
met®evee :1) ³eefo oes keÀesCe HetjkeÀ nes lees Òel³eskeÀ keÀesCe otmejs keÀesCe mes HetjkeÀ neslee nw~2) ³eefo SkeÀ keÀesCe nes lees otmeje HetjkeÀ keÀesCe (180-x)0.
meesef®eS : SkeÀ keÀesCe x 0 FmekeÀe mebHetjkeÀ keÀesCe ke̳ee nw?
GoenjCe 1 : 130.50 meesef®eS SkeÀ keÀesCe x 0 FmekeÀe mebHetjkeÀ keÀesCe ke̳ee nw?
130.500 keÀe mebHetjkeÀ keÀesCe ceeefveS mebHetjkeÀ keÀesCe x 0
x 0 + 130.50 = 1800 mebHetjkeÀ keÀesCe]
x 0 = 1800-130.50
x 0 = 49.50.
GoenjCe 2 : SkeÀ keÀesCe otmejs keÀesCe mes mebHetjkeÀ nw ye[e keÀesCe 900 íesìs keÀesCe mes p³eeoe nw Òel³eskeÀ keÀesCe %eele keÀerefpeS~
ceeefveS íesì keÀesCe x0 nw
Deewj ye[e keÀesCe (x 0 + 900)
x 0 + (x 0 + 900) = 1800
x 0 + x 0 + 900 = 1800
2x 0 + 90 0 = 180 0
2x 0 = 180 0 - 90 0
x
x2
90
45
0=
=2x0 = 900
∴ íesìer keÀesCe 450 ye[e keÀesCe = x0 + 900
= 450 + 900 = 1350
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meesef®eS:
ke̳ee efvecveefueefKele keÀesCe mebHetjkeÀ nw?
RO
1450450
P
Q
meesef®eS :
meesef®eS ke̳ee oes v³etve keÀesCe HejmHej mebHetjkeÀ nw peeefveS~
ke̳ee oes DeefOekeÀ keÀesCe HejmHej mebHetjkeÀ nw?
ke̳ee oes uecye keÀesCe HejmHej mebHetjkeÀ nw?
met®evee: mebHetjkeÀ keÀesCe mebueiCe keÀesCe vener nes mekeÀlee nw~
mebHetjkeÀ ueefìve Meyo mes efue³ee ie³ee nw efpemekeÀe DeLe& mebHetCe& nw~
DeY³eeme : 5.3
I. efvecveefueefKele keÀesCeeW keÀer pees[er HetjkeÀ nw - mel³eeHeve keÀerefpeS~
A O B
C
D
E
600700
200
300
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II. efvecveefueefKele keÀewve mee keÀesCe HetjkeÀ nesles nwb? mel³eeHeve keÀerefpeS~
O
700 1100
P
QRef®e$e- (i)
O
1350
450
P
M
Nef®e$e- (ii)
O
750 1050
A C
B
ef®e$e- (iii)
A
200 1600
EBC F
D
ef®e$e- (iv)O
8001100
Q
P
S
R
T
ef®e$e- (v)
III. efvecveefueefKele keÀesCeeW keÀe HetjkeÀ keÀesCe efueefKeS~
a) 950 b) 1100 c) 120.50 d) 125.50
IV. oes HetjkeÀ keÀesCe keÀe ceeHe 500 otmejs mes DeefOekeÀ nw~ Òel³eskeÀ keÀesCe %eele keÀerefpeS~
V. ³eefo SkeÀ keÀesCe HetjkeÀ keÀesCe ®eej iegCee nw lees keÀesCe %eele keÀerefpeS~
VI. oÊe Deeke=Àefle ceW ABC 350= Deewj DBE 900= nes lees ceeHe %eele keÀerefpeS :
i) CBD keÀe HetjkeÀ keÀesCe
B350900
A
C
D
E
ii) EBC keÀe HetjkeÀ keÀesCe
iii) ABD keÀe HetjkeÀ keÀesCe
iv) EBD keÀe HetjkeÀ keÀesCe
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keÀesCeeW keÀer jwefKekeÀ pees[er :oÊe Deeke=Àefle ces mebueive keÀesCeeW keÀe ceeHe osefKeS~
O3001500
B
C
A
ef®e$e- i
O900 900
B
C
A
ef®e$e- ii
O600 1200
B
C
A
ef®e$e- iii
³eesie %eele keÀerefpeS~ leeefuekeÀe ceW efueefKeS;
Deeke=Àefle AOC BOC AOC BOC+
i. 1500 3001500 + 300 = 1800
ii. 900 900900 + 900 = 1800
iii. 600 1200600 + 1200 = 1800
ke̳ee AOC Deewj BOC mebueive keÀesCe nQ?Deeke=Àefleæ i, ii, Deewj iii ceW mebueive keÀesCe keÀe ³eesie 1800 nw~ Deewj mejue jsKee kesÀ efkeÀjCe
OA Deewj OB nw~.keÀesCe keÀer pees[er keÀes ke̳ee keÀnles nQ?FvekeÀes keÀesCeeW keÀer jwefKekeÀ pees[er keÀnles nQ~oes keÀesCees keÀes jwefKekeÀ pees[er keÀnles nQ ³eefo
Gveces meeceev³e Meer<e& Deewj meeceev³e Yegpee nesleer nw~
mejue jsKee yeveeves kesÀ efueS oes Demeceeve Yegpee nesveer ®eefnS~
keÀesCeeW keÀe ceeHe keÀe ³eesie 1800 nw~
met®evee : jwefKekeÀ pees[er HetjkeÀ keÀesCeeW keÀer pees[er nesleer nw peye SkeÀ otmejs kesÀ HetjkeÀ nesles nQ~
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DeY³eeme: 5.4
I. Deeke=Àefle ceW AOB Deewj BOC keÀes efvejer#eCe keÀerefpeS~ ke̳ee AOB Deewj BOC jwefKekeÀ pees[er nw? ®e®ee& keÀjkesÀ keÀejCe oerefpeS~
O
650
1350
B
C
A
O600
1300B
C
A
II. efvecveefueefKele jwefKekeÀ pees[er ces peye oesvees keÀesCeeW keÀes yeepet cesb jKelee nw keÀewve meer jwefKekeÀ pees[e nw~
Q
B
O
A
1250
350
R
P
ef®e$e- i
Q700
R
1100
O B
AP
ef®e$e- ii
P
A
O BQ
R
400
1500
ef®e$e- iii
A
O B
R
P Q900
900
ef®e$e- iv
A
O BR
PQ
900
900
ef®e$e- v
P
QR
1400
A
B
O400
ef®e$e- vi
III. efvecveefueefKele keÀesCe otmejs ÒekeÀej keÀe keÀesCe %eele keÀerefpeS ³eefo keÀesCe jwefKekeÀ pees[er yeveles nQ~
a. v³etve keÀesCe b. DeefOekeÀ keÀesCe c. uebye keÀesCe
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Kesue
oes [^eF¥ie Meerì HesHej, 40 {gkeÀ[eW ces keÀeefìS Deewj efvecveefueefKele Òel³eskeÀ ogkeÀ[e keÀe keÀesCe
yeveeF&S~ FmekesÀ oes mecetn yeveeFS~ keÀesCe 10, 20, 30, 40, 50, 60, 70, 80, 90, 100,
110, 120, 130, 140, 150, 160, 170, 180 Deewj 25, 35, 45, 55, 65, 75, 85,
95, 105, 115, 125, 135, 145, 155, 165, 175 (keÀesF& ceeHeve keÀe GHe³eesie keÀj me-
keÀles nQ Hejvleg mebHetjkeÀ Deewj HetbjkeÀ keÀesCe meesef®eS) Òel³eskeÀ keÀe[& kesÀ Heerís efvecveefueefKele mebK³eeSB
1, 2, 3, 4, 5 Deewj 6 DebefkeÀle keÀerefpeS~ meYeer keÀe[& keÀes Ssmes jefKeS efpememes mebK³ee osKe mekesÀ~
SkeÀ Heemee ueerefpeS efpemeceW 1 mes 6 lekeÀ DebefkeÀle nw ³ee [eìme nw (oes ³ee leerve oesmle Fme
Kesue keÀes Kesueles nw~
efve³ece: Òel³eskeÀ meom³e keÀes Heemee HeWÀkeÀkeÀj Deewj mebK³ee mecyevOe keÀe[& keÀes uesves nw~ keÀe[&
nesves Hej efKeueeef[³eeW keÀes keÀe[& efJeefve³ece keÀjvee nw~ efpememes mebHetjkeÀ Deewj HetjkeÀ keÀesCe nes
mekeÀlee nw~
J³eeqkeÌle efpemekesÀ Heeme DeefOekeÀ keÀe[& keÀer pees[er jnleer nw Jener peerlelee nw~
Meer<ee&efYecegKe keÀesCe :
ef®e$e kesÀ Devegmeej oes uekeÀef[³eeW keÀes yeebOee nw~ ³eefo AB Deewj CD oes uekeÀ[er Deewj `O'
efyevog Hej yeebOee nw~
³eefo AB Deewj CD jsKeeKeb[ nes lees `O' Meer<e& nw~
, ,AOC BOC AOD Deewj BOD keÀesCe nQ~
BOC kesÀ efkeÀleves mebueive keÀesCe nw? DebefkeÀle keÀerefpeS~
AOC kesÀ efkeÀleves mebueive keÀesCe nQ? DebefkeÀle keÀerefpeS~
keÀesCeeW keÀes DebefkeÀle keÀerefpeS~
A
B
D
C
O
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BOC Deewj AOD keÀes efvejer#eCe keÀerefpeS~
Gmeer ÒekeÀej |||1y AOC Deewj BOD keÀes efvejer#eCe keÀerefpeS~ GvekeÀe mecyevOe ke̳ee nw?
Meer<e& kesÀ Heeme BOC Deewj AOD nQ~ Gmeer ÒekeÀej AOC Deewj BOD Meer<e& kesÀ
Heeme nQ~
Fve keÀesCeeW keÀer pees[er keÀes ke̳ee keÀnles nQ? oes jsKeeDeeW kesÀ Òeefle®ísove efyevog Hej yeves ngS
Òel³eskeÀ efJe©× keÀesCe keÀes Meer<ee&efYecegKe keÀesCe keÀnles nw~
Meer<ee&efYecegKe keÀesCe ceW meeceev³e Meer<e& neslee nQ~
efvecveeefueefKele keÀe³e& keÀueeHe mes Meer<e&efYecegKe keÀesCe keÀe mecyevOe %eele keÀj mekeÀles nQ~
ZÀlĪÀnPÉ -1
A
B
D
C
O
A
B
D
C
O
oes jsKeeDeeW AB Deewj CD keÀer j®evee SkeÀ Meerì ceW keÀerefpeS pees ̀ O' efyevog Hej Òeefle®ísefole nesleer nw~ ef®e$e kesÀ pewmes ®eej keÀesCe , ,COB BOD AOD Deewj AOC yeveeFS~
GmekesÀ THej HeejoMe&keÀ Meerì jKekeÀj AB Deewj CD jsKee yeveeFS~
HeejoMe&keÀ Meerì cetue Meerì Hej jKekeÀj AB jsKee yeveeFS~
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jsKee keÀer Òeefle®ísove Hej efHeve ueieeFS~ HeejoMe&keÀ Meerì keÀes IegceeF³es~ efpememes jsKee AB cetue Meerì mes efJe©× nw DeeHe ke̳ee efvejer#eCe keÀjles nw?
nce efvejer#eCe keÀjles nwb efkeÀ
COB cetue Meerì mes AOD keÀes Iesj ueslee nw HeejoMe&keÀ Meerì Hej~
COB HeejoMe&keÀ Meerì mes AOD keÀes uesles nwb~
AOC cetue Meerì mes BOD HejoMe&keÀ Meerì keÀes Iesj ueslee nw~
BOD meHesÀo Meerì mes AOC HejoMe&keÀ Meerì keÀes Iesj uesles nw~
FmeefueS
COB AOD= Deewj BOD AOC= Iesj ³eefo oes mejue jsKeeSB DeeHeme ceW
Òeefle®ísove keÀjles nwb lees Meer<ee&efYecegKe keÀesCe meceeve nw (Jen meJee¥iemece ³ee meceeve keÀesCe)
keÀe³e& keÀueeHe - 2 : meHesÀo Meerì HesHej ueerefpeS~ uebyeeF& kesÀ Devegmeej ceesef[³es Deewj efmekegÀef[³es~ HesHej keÀes Deewj efJekeÀeCe& kesÀ
Devegmeej ceesef[³es Deewj Keesefue³es A, B, C........ Òeefle®ísove efyevog A, B Deewj C........ nw~ Meer<e& Hej
yeves keÀesCe A1, A2, A3.............. ³eefo nw~ meejCeer ceW efueKekeÀj Meer<ee&efYecegKe keÀesCe yeepeg ceW efueefKeS~
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¬eÀceebkeÀ Meer<e& Meer<ee&efYecegKe keÀesCe keÀer Henueer pees[er Meer<ee&efYecegKe keÀesCe keÀer otmejer pees[er
i. AA1 = A3 =
A2 =A4 =
ii.
owefvekeÀ peerJeve ceW mes GoenjCe oerefpeS peneB Meer<ee&efYecegKe keÀesCe efoKeeF& osles nwb~
DeY³eeme : 5.5
I. oÊe Deeke=Àefle ceW AOC 600= Mes<e keÀesCees keÀe ceeHe yeleeF³es
A
B
D
C
600O
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II. oÊe Deeke=Àefle ceW, Meer<ee&efYecegKe keÀesCe Hen®eeefveS~ DebefkeÀle keÀerefpeS ³eefo, BOE 1400= . OC⊥EO Mes<e keÀesCe %eele keÀerefpeS~
E
A B1400
C
D
O
III. oÊe Deeke=Àefle ceW AOC 300= Mes<e keÀesCeeW keÀe ceeHe %eele keÀerefpeS~ A
B
D
C
300 O
meesef®eS :
³eefo oes jsKeeSB SkeÀ efyevog Hej Òeefle®ísefole keÀjles nw lees Meer<ee&efYecegKe keÀesCe keÀe mJeYeeJe ke̳ee
nw?
v³etve keÀesCe keÀer SkeÀ pees[er~
uebye keÀesCe keÀer SkeÀ pees[er~
DeefOekeÀ keÀesCe keÀer SkeÀ pees[er~
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DeY³eeme 5.6
I. jsefKekeÀ pees[er yeveves kesÀ efueS keÀewvemee ÒekeÀej keÀe keÀesCe yevelee nQ?
1) v³etve keÀesCe
2) DeefOekeÀ keÀesCe
3) uebye keÀesCe
II. efvecveefueefKeleeW ceW mes keÀesCe keÀe ceeHe Deewj `x' keÀe cetu³e %eele keÀerefpeS~
x 750
ef®e$e- i
3x x+20
ef®e$e- ii
x2x x+50
ef®e$e- iii
x+15 2x+15x+10
ef®e$e- iv
III. oÊe Deeke=Àefle ceW AOC 900= keÀesCe BOD keÀe uecyeefÜYeepekeÀ PQ nw Deewj AOD keÀe uecyeef×YeepekeÀ RS nw~ otmejs keÀesCe %eele keÀerefpeS~
Q
D
R
A
900P
C
S
B
O
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IV. efvecveefueefKeleeW ceW x, y Deewj z keÀe cetu³e %eele keÀerefpeS
B
D
2z 3z
450
A
C E
Ox
y
V. oÊe Deeke=Àefle ceW OE ⊥AB nes lees efvecveefueefKele pees[er keÀes %eele keÀerefpeS~
1) jsefKekeÀ pees[er~
B
D
A
CE
O 2) HetjbkeÀ keÀesCe~
3) Meer<ee&efYecegKe keÀesCe
4) HetbjkeÀ keÀesCe
VI. Keeueer mLeeve YeeefjS :
1) ³ee fo oe s keÀe sCe me b He tjkeÀ ne s lee s Gveke s À keÀe sCee W keÀe ceeHee W keÀe ³ee s ie
____________ nQ~
2) ³eefo oes keÀesCe HetjkeÀ nes lees GvekesÀ keÀesCeeW keÀe ³eesie ____________ nQ~
3) jwefKekeÀ pees[er yeveves Jeeues oes keÀesCe ____________ nQ~
4) ³eefo oes mebueive keÀesCe mebHetjkeÀ nes lees Jen ____________ yeveeles nQ~
5) ³eefo SkeÀ efyevog Hej oes jsKeeSB Òeefle®ísefole nes lees Meer<ee&efYecegKe keÀesCe keÀer
pees[er otmejs Meer<ee&efYecegKe keÀesCe v³etve keÀesCe ____________ nQ~
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VII. efvecveefueefKele keÀesCeeW keÀer pees[er keÀes DebefkeÀle keÀerefpeS~
1) mecelegu³e mebHetjkeÀ keÀesCe~
Q
S
K
450
P
R
L
O
2) Meer<ee&efYecegKe keÀesCe~
3) Demeceeve mebHetjkeÀ keÀesCe~
4) mebueive HetjkeÀ keÀesCe~
5) DeefOekeÀ Meer<ee& keÀesCe~
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DeO³ee³e- 6
pees[er jsKeeSB
Fme IeìkeÀ kesÀ DeO³e³eve keÀjves kesÀ yeeo DeeHe:
jsKeeDeeW keÀer pees[er kesÀ DeLe& keÀer HeefjYee<ee oWieW,
Òeefleísefole jsKeeDeeW kesÀ DeLe& keÀer HeefjYee<ee oWies,
meceeblej jsKee Deewj efle³e&keÀ jsKeeDeeW keÀe DeLe& mecePesies,
meceeblej jsKee Deewj efle³e&keÀ jsKeeDeeW keÀes owefvekeÀ peerJeve ceW Hen®eeveWieW,
efle³e&keÀ jsKee mes yeves keÀesCe, Deveg©He keÀesCe, SkeÀeblej keÀesCe, efle³e&keÀ kesÀ He#e ceW yeveW efle³e&keÀ kesÀ He#e ceW yeveW Devle: mLe keÀesCe Hen®eeveWies,
keÀesCeeW kesÀ mebcyeefOele ÒeMveeW keÀes nue keÀjWieW ~
efKe[keÀer kesÀ meueeKeeW keÀe HeÀesìes veesì yegkeÀ Hej yeveer jsKee Fl³eefo, keÀe O³eeve mes efvejer#eCe keÀerefpeS ~
FmekesÀ HetJe& kesÀ DeO³ee³e ceW keÀesCeeW keÀer pees[er kesÀ yeejs ceW DeeHe He{s ®egkesÀ nQ~
veer®es efo³es ie³es Deeke=Àefle ceW osefKeS; DeeHekeÀes %eele nesiee efkeÀ jsKeeDeeW~
ef®e$e (a)
ef®e$e (b) ef®e$e (c)
ef®e$e (d)
Òel³eskeÀ Deeke=Àefle ceW jsKeeDeeW keÀer mebK³ee efieefve³es~ Òel³eskeÀ Deeke=Àefle ceW efkeÀleves jsKeeSB nwb? neB Òel³eskeÀ Deeke=Àefle ceW oes jsKeeSB nwb~ Deeke=Àefle³eeW keÀes efvejer#eCe keÀerefpeS~ Deeke=Àefle (a) Deewj Deeke=Àefle (C) ceW Òeefleíso keÀjves Jeeues jsKeeDeeW keÀes keÀ³ee DeeHe Hen®eeveles kesÀ pees[er keÀer Òel³eskeÀ jsKee SkeÀ meceleue ceW nw~
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SkeÀ : ner meceleue ceW efmLeve, efkeÀmeer Yeer mLeeve ³ee ÒekeÀej keÀer oes jsKeeDeeW keÀes pees[ jsKeeSB keÀnles nwB~
met®evee : keÀesF& oes DeefOekeÀ jsKeeSB DeLeJee jsKeeKeC[ SkeÀ ner ceW meceleue ceW nes lees GvnW meceleueer³e jsKeeSB DeLeJee jsKeeKeC[ keÀnles nQ~
Òeefle®ísefole jsKeeSB
i) meceleue ces keÀesF& efyevog ueerefpeS Deewj oes efkeÀjCe KeeRef®eS~ GvekeÀes DebefkeÀle keÀerefpeS~
ii) PQ jsKeeKeC[ keÀer j®evee keÀerefpeS Deewj GmeHej `R' efyevog DebefkeÀle keÀerefpeS~
p QR
SkeÀ Deewj jsKeeKeC[ RT keÀer j®evee keÀerefpeS~
T
QP R
ceeefveS RT , PQ Hej Ke[e nw~TR keÀes TS lekeÀ ye{eFS~
P R
S
Q
T
GHejesÊeÀ Deeke=Àefle osefKeS, SR Deewj TR jsKeeKeC[ PQ keÀes `R' Hej efceueles nQ~ jsKee ST, jsKeeKeC[ keÀes PQ keÀes Òeefle®ísove keÀjvesJeeuee jsKeeKeC[ keÀnles nQ, P Deewj R Òeefle®ísefole efyevog nQ~
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efkeÀjCe Deewj jsKeeDees mes yevesb Fve Deeke=Àefle³eeW keÀes osefKeS~
ef®e$e - i ef®e$e - ii
ef®e$e- iii
Deeke=Àefle - (i) Deewj , B efyevog mes Meg© nesles nQ~
Deeke=Àefle - (ii) Deewj (iii) Deewj , `O' Hej efceueles nQ~
³eneB ̀ O' keÀes Òeefle®ísefole efyevog keÀnles nwb~ Deewj keÀes Òeefle®ísefole jsKeeSB keÀnles nQ~
oes jsKeeSB pees meeceev³e efyevog Hej efceuelee nQ GvnW Òeefle®ísefole jsKeeBS keÀnles nQ~ Deewj meeceev³e efyevog keÀes Òeefle®ísefole efyevog keÀnles nwb~
ole Deeke=ef]le ceW PQ Deewj RS Hej HejmHej Òeefle®ísefole p S
QR
O
keÀjles nwb~ PQ Deewj RS keÀneB Òeefle®ísefole nesleer nQ?
oesveeW jsKeeDeeW kesÀ efueS meeceev³e efyevog keÀewve meer nw? Fme meeceev³e efyevog keÀes ke̳ee keÀnles nQ?
keÀe³e& keÀueeHe 1 :
oÊe Deeke=Àefle Òeefle®ísefole jsKeeKeC[ Hen®eefveS~ Deeke=Àefle ceW AC
oDeewj BD, O Hej Òeefle®ísefole nw~ otmejs Òeefle®ísefove jsKeeKeC[eW keÀes Hen®eeveves keÀer keÀesefMeMe keÀerefpeS Deewj GvekeÀer met®eer yeveeF&S~
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meesef®eS! oes jsKeeSB ³ee jsKeeKeC[ ncesMee Òeefle®ísefole nesles nQ ?meceeblej jsKeeSB
keÀe³e& keÀueeHe : AB jsKeeKeC[ keÀer j®evee keÀerefpeS~ jsKeeKeC[ AB Hej KL, M Deewj N keÀes DebefkeÀle keÀerefpeS~ AB
kesÀ GHej P efyevog keÀes DebefkeÀle keÀerefpeS~ ef®e$e kesÀ Devegmeej PK, PL, PM Deewj PN keÀes peesef[S~ PK, PL, PM Deewj PN keÀes ceeefHeS~ PMB , PMA , PNM
Deewj PKL keÀes ceeefHe³es~ PK, PL, PM Deewj PN uebyeeF& keÀer leguevee keÀerefpeS~.
keÀewve meer uebyeeF& meyemes keÀce nw? ke̳ee AB keÀer meyemes v³etvelece uebyeeF& nw?
jsKeeKeC[ Deewj yee¿e efyevog keÀer v³etvelece otjer PM nQ~ M Hej keÀesCe 900 nQ~ PM jsKee AB keÀes uebye nQ~ SkeÀ mejue jsKee keÀes efkeÀmeer yee¿e efyevog mes KeeR®eer jsKeeDeeW ceW uebye jsKee ner íesìer nesleer nw~
met®evee: uebye keÀe mebkesÀle ┴ nQ~ PM , AB keÀes uebye nw metef®ele keÀjlee nw~
g_m§Va aoImE±ceeHeveer
Heesmì keÀe[&
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mkesÀue Deewj Heesmì keÀe[& kesÀ uebyeeF& kesÀ Devegmeej jsKeeDeeW keÀer j®evee keÀerefpeS~ PQ Deewj RS
keÀes DebefkeÀle keÀerefpeS~ Deewj keÀes ye{eFS~
P
R
Q
S
ke̳ee oesveeW jsKeeSB DeeHeme ceW efceueleer nQ?
efJeefYevve efyevogDeeW Hej jsKeeDeeW kesÀ yeer®e keÀer uebye ceeefHe³es~
Fmemes DeeHe ke̳ee efveCe&³e ueWieW?
Fve jsKeeDeeW keÀes ke̳ee veece osles nwb?
Deewj Deveg©He efyevogDeeW Hej meceeve otjer nw~ Fve jsKeeDeeW keÀes meceeblej jsKee SW keÀnles nwb~
mejue jsKeeDeeW keÀer pees[er meceeblej nw, ³eefo Deveg©He efyevogDeeW Hej jsKeeDeeW kesÀ yeer®e keÀer otjer meceeve jnleer nQ~
met®evee:
1) oes meceeblej jsKee efve©efHele keÀjves oesveeW jsKeeDeeW keÀes SkeÀ ner efoMee ceW leerj ueie³ee peelee nQ~
2) ³eefo oes meceeblej jsKeeDeeW keÀes oesvees efoMee ceW ye{eves Hej Jes meceeve meceleue ceW keÀYeer veneR efceueleer~
oÊe Deeke=Àefle ceW, efJeefYevve mLeeveeW Hej KeeR®eer AB Deewj CD jsKeeDeeW kesÀ yeer®e keÀer otjer meceeve nwb~
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Ssmeer jsKeeSB ke̳eeW veneR efceueleer DeLeJee Òeefle®ísefole keÀjleer?
oes jsKeeDeeW keÀer yeer®e keÀes otjer veneR yeoueleer nw~ FmeefueS meceeblej jsKeeSW veneR efceueleer nQ~
keÀe³e&keÀueeHe 3 : ûeeHeÀ yegkeÀ ceW jsKeeDeeW keÀes efvejer#eCe keÀerefpeS Deewj Hen®eefveS;
i) meceeblej jsKeeSW
ii) uecye jsKeeSW
oesvees jsKeeDeeW kesÀ yeer®e ceW Deblej ke̳ee nw?
keÀe³e& keÀueeHe 4 : HeeB®e GoenjCe keÀer met®eer yeveeFS efpemeces legce osKe mekeÀles nes;
i) meceeblej jsKeeSW
ii) uecye jsKeeSW
met®evee : (i) ³eefo oes jsKeeSW AB Deewj CD meceeblej nQ lees AB║CD efueKeles nQ~
(ii) ³eefo uecye nes lees AB ┴CD efueKeles nQ~
efle³e&keÀ jsKeeSB
keÀe³e& keÀueeHe 5 : AB jsKeeKeC[ keÀer j®evee keÀerefpeS Deeke=Àefle kesÀ pewmes otmejer jsKeeKeC[ PQ yeveeFS~ oesveesb jsKeeSB keÀneB Hej efceuelee nQ~
otmejer jsKeeKeC[ CD yeveeFS jsKeeKeC[ AB, kesÀ THej Deewj otmeje jsKeeKeC[ KeeRef®eS pees Deeke=Àefle SW efoKee³es pewmes PQ keÀes Òeefle®ísefole keÀjlee nQ~
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AB Deewj CD keÀer Òeefle®ísove jsKee keÀes efueefKeS~ Deeke=Àefle (i) K Deewj L Òeefle®ísove efyevog nw~keÀe³e& keÀueeHe: Fve Deeke=Àefle³eeW keÀes O³eeve mes osefKeS
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ef®e$e - i ef®e$e - ii
c
ab
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KLM
qef®e$e - iii
a c
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LKqp
ef®e$e - iv
GHejesÊeÀ Deeke=Àefle³eeW ceW
AB Deewj CD keÀer Òeefle®ísove jsKee keÀes efueefKeS~ Deeke=Àefle (i) ceW Òeefle®ísove efyevog DeeW keÀes K Deewj L mes metef®ele keÀerefpeS~
GHejeskeÌle Deeke=Àefle³eeW ceW, ` ' keÀce mes keÀce oes jsKeeDeeW keÀes Òeefle®ísefole keÀjlee nw; Ssmeer Òeefle®ísefove jsKee keÀes efle³e&keÀ jsKee keÀnles nQ~ pees jsKee SkeÀ ner meceleue ceW keÀce mes keÀce oes
efyevogDeeW keÀes Òeefle®ísefole keÀjleer nw, Gmes efle³e&keÀ jsKee keÀnles nQ~
keÀe³e& keÀueeHe 6 :
meHesÀo Meerì HesHej ueerefpeS~
oes ³ee leerve yeej ceesef[S Deewj KeesefueS~
ueyeebF& kesÀ Devegmeej Deye otmejer Deesj cees[keÀj KeesefueS~
cees[ves ÒeeHle jsKeeDeeW keÀes HeWefmeue mes DebefkeÀle keÀerefpeS~
jsKeeDeeW Deewj Òeefle®ísove efyevog keÀes DebefkeÀle keÀerefpeS~ Òesefle®ísove jsKee Deewj efle³e&keÀ jsKeeDeeW keÀer met®eer yeveeFS~
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keÀe³e& keÀueeHe 7 : veer®es efo³es Deeke=Àefle ceW efle³e&keÀ jsKeDeeW keÀes DebefkeÀle keÀerefpeS~
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p qlk
meesef®eS!
oÊe Deeke=Àefle ceW EF efle³e&keÀ veneR nw ³eIeefHe Jen AB Deewj CD
keÀes Òeefle®ísefole keÀjleer nw?
GHejesÊe efvejer#eCe mes SkeÀ jsKee efle³e&keÀ keÀye nesleer nw?
efle³e&keÀ jsKee mes yeveW keÀesCe
Deeke=Àefle keÀes osefKe³es~ jsKeeKeC[, DevegÒemLe jsKee Deewj Òeefle®ísove efyevog keÀe veece efueefKeS,
pees efle³e&keÀ mes yeves nQ~ neB, AB, CD Deewj PQ jsKeeSB nQ efpemeces PQ efle³e&keÀ nw~ K Deewj L
Òeefle®ísove kesÀ efJeefYevve efyevog nw~
³eneB PKB , AKQ Meer<ee&efYecegKe keÀesCe nQ~
K Deewj L Hej otmejs Meer<ee&efYecegKe keÀesCeeW keÀer pees[er keÀes Hen®eeveles nwb? PKB , AKP
mebHetjkeÀ keÀesCe nQ~ K Deewj L Hej yeveW Dev³e keÀesCeeW keÀer pees[er Hen®eeefveS Òeefle®ísove efyevog K Deewj
L mes yeveW Deeþ efJeefYevve keÀesCe nQ~ Fve keÀesCeeW keÀes kewÀmes Jeieeake=Àle keÀj mekeÀles nQ?
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Deble : keÀesCe
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2K
143
Lq
Deeke=Àefle ceW AB, CD Deewj PQ leerve mejue jsKeeBS nQ~ efle³e&keÀ jsKee keÀewve meer nQ?
K Deewj L efyevogDeeW Hej PQ mes yeveW keÀesCe keÀewve mes nQ?
efvecve ®eej keÀesCeeW keÀes O³eeve mes osefKeS :
efyevog K Hej keÀesCe AKQ Deewj BKQ Deewj
Gmeer ÒekeÀej L efyevog Hej CLP Deewj DLP
Fve ®eej keÀesCeeW ceW ke̳ee meeceev³e nw !
nceW %eele neslee nw ³es keÀesCe pees AB Deewj CD kesÀ yeer®e yeves nQ, ³en KL Yeeie efle³e&keÀ PQ kesÀ DevlemLe Yeeie nw~ jsKeeDeeW kesÀ pees[er kesÀ DevlemLe Yeeie ceW yeveves kesÀ keÀejCe Fve keÀesCeesb keÀes Devele: keÀesCe keÀnles nQ~
efle³e¥keÀ jsKee kesÀ jsKeeKeC[ Deewj jsKeeDeeW kesÀ pees[er kesÀ yeer®e yeves keÀesCeeW keÀes Devle: keÀesCe keÀnles nw~
GHej kesÀ ef®e$e ceW AKL , KLC Deewj BKL , KLD keÀes Deble: keÀesCe keÀnles nQ~
yee¿e keÀesCe:
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cl
q
d
bk
p Deeke=Àefle cesb , Deewj leerve mejue jsKee SW nQ~ efle³e&keÀ
jsKee keÀewvemeer nw~
efle³e&keÀ Hej Deble: keÀesCeW kesÀ veece efueefKeS~
³eneB K Hej AKP Deewj L Hej CLQ keÀesCeeW keÀer pees[er nw~ Gmeer
ÒekeÀej K Hej PKB Deewj L Hej QLD keÀesCeeW keÀer pees[er nw~
Fve keÀesCeeW keÀer pees[er ceW ke̳ee meceevelee nw? efle³e&keÀ kesÀ SkeÀ ner He#e ceW jsKeeDeeW kesÀ pees[er kesÀ
yeenj yeveW keÀesCeeW keÀer pees[er Hej O³eeve oerefpeS~
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³en keÀesCeeW keÀer pees[er, meceevlej jsKee AB Deewj CD kesÀ yeenj efle³e&keÀ jsKee kesÀ SkeÀ ner He#e ceW yeveeR nw~
Fve keÀesCeeW keÀer pees[er keÀes ke̳ee veece os mekeÀles nQ?
efle³e&keÀ kesÀ DeueeJee ³en keÀesCe jsKeeDeeW kesÀ yeenj nesles nQ Deewj FmeefueS Fvns yee¿e keÀesCe keÀnles nQ~
jsKeeDeeW kesÀ pees[er kesÀ yeenj Hejvle efle³e&keÀ jsKee kesÀ SkeÀ ner He#e ceW yeveW keÀesCeeW keÀes yee¿e keÀesCe keÀnles nQ~
keÀe³e& keÀueeHe :
efvecveefueefKele Deeke=Àefle ceW mes Deble: keÀesCe Deewj yee¿e keÀesCe A
C
ED
B
F G
Hen®eefveS Deewj met®eer yeveeFS~
¬eÀceeiele Deble : keÀesCe
oÊe Deeke=Àefle ceW, efle³e&keÀ jsKee PQ kesÀ meboYe& ceW keÀesCeeW keÀer pees[er AKL , KLC keÀes Deewj otmejer pees[er BKL Deewj KLD efvejer#eCe keÀerefpeS~ AKL Deewj KLC , PQ efle³e&keÀ kesÀ SkeÀ ner He#e kesÀ Hej Deble: keÀesCe nQ~ Gmeer ÒekeÀej BKL Deewj KLD kesÀ keÀesCeeW keÀer pees[er nQ~ Ssmes keÀesCeeW keÀer pees[er keÀes ¬eÀceeiele Deble: keÀesCe keÀnles nQ~
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GHejesÊeÀ Deeke=Àefle ceW AKL Deewj KLC ¬eÀceeiele Deble: keÀesCe keÀer pees[er nw~ Gmeer Deeke=Àefle ceW ¬eÀceeiele otmejer Deble: keÀesCeeW keÀer pees[er keÀe ke̳ee veece nw?
efle³e&keÀ jsKee kesÀ SkeÀ ner He#e ceW yeveW Deble: keÀesCeeW keÀes ¬eÀceeiele Deble: keÀesCe keÀnles nw~
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Deveg©He keÀesCe (Corresponding Angles)
efvecveefueefKele ef®e$eeW ceW DebefkeÀle keÀesCeeW keÀe efvejer#eCe keÀerefpeS~
AB
D
K
LCQ
P
ef®e$e i ef®e$e ii
ef®e$e iii
A B
D
K
LC
Q
P
ef®e$e iv
PQ efle³e&keÀ jsKee meceeblej jsKee AB Deewj CD keÀes K Deewj L Hej Òeefle®ísefove keÀjlee nw~
Deeke=Àefle (i) ceW AKP Deewj CLK , PQ efle³e&keÀ jsKee kesÀ yee³eeR Deesj nwb Deewj AKP , AB
kesÀ THej efmLele nQ leLee CLK , CD kesÀ THej efmLele nQ~
DevegÒemLe PQ kesÀ yee³esb cesb AKP Deewj CLK keÀes DebefkeÀle keÀerefpeS Deewj AB kesÀ THej
AKP Deewj CD kesÀ THej CLK nQ~
Deeke=Àefle (ii) ceW AKP Deewj CLQ keÀesCeeW keÀer pees[er PQ efle³e&keÀ jsKee kesÀ yee³eeR Deesj nQ AB Deewj CD jsKeeDeeW kesÀ veer®es nQ~
Deeke=Àefle (iii) ceW PKB Deewj KLD, keÀesCeeW efle³e&keÀ jsKee PQ kesÀ oenerveer Deesj nQ Deewj AB Deewj CD kesÀ THej yeveW nwb~
Deeke=Àefle (iv) ceW BKL Deewj DLQ, keÀesCeeW keÀer pees[er efle³e&keÀ jsKee PQ keÀer {enerveer Deesj
nQ leLee AB Deewj CD kesÀ veer®es efmLele nQ~
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GHejesÊeÀ efvejer#eCe mes efveefM®ele keÀjles nQ efkeÀ Òel³eskeÀ Deeke=Àefle ceW keÀesCeeW keÀer pees[er ceW
a) efJeefYevve Meer<e& nwb~
b) efle³e&keÀ jsKee kesÀ SkeÀ ner He#e ceW yeveW nQ~
c) Deeke=Àefle~ (i) Deewj (ii) ceW meceeblej Yegpee kesÀ THej nw; Deeke=Àefle (iii) Deewj (iv) ceW meceeblej Yegpee kesÀ veer®es nwb~ FmeefueS Ssmes keÀesCeeW keÀer pees[er keÀes Deveg©He keÀesCe keÀnles nQ~
keÀesCeeW keÀer pees[er efpemekesÀ,
efJeefYevve Meer<e& nw~
efle³e&keÀ jsKee kesÀ SkeÀ ner He#e ceW yeveW nw~
oes meceeblej jsKeeSB efJeefYevve mLeeveeW Hej nw, Deveg©He keÀesCe keÀnueeles nwb~
Deeke=Àefle (i, ii, iii Deewj iv) ceW keÀesCeeW keÀer pees[er
(i) ( AKP Deewj CLK), (ii) ( AKL Deewj CLQ) (iii) ( PKB Deewj KLD) leLee ( BKL Deewj DLQ) Deveg©He keÀesCe nwb~
keÀe³e& keÀueeHe ! SkeÀ Meerì HesHej keÀes ceesef[S Deewj meceeblej Deewj efle³e&keÀ jsKeeSB Òeefle®ísove
efyevog DebefkeÀle keÀerefpeS~ Òel³eskeÀ ces Deveg©He keÀesCeeW keÀer pees[er efueefKeS~ meesef®eS De#ej F
Deveg©He keÀesCe Hen®eeveves ceW mene³elee keÀjlee nw...~ ? kewÀmes?
SkeÀebvlej Deble: keÀesCeoÊe Deeke=Àefle ces keÀesCeeW keÀes efvejer#eCe keÀerefpeS~
ef®e$e (i) ef®e$e (ii)
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Deeke=Àefle (i) ceW AKL efle³e&keÀ jsKee PQ kesÀ yee³eW ceW nQ~ AB Deewj kesÀ veer®es nw~ peyeefkeÀ
KLD efle³e&keÀ PQ kesÀ oe³eW Deewj CD kesÀ THej nw~
Deeke=Àefle (ii) ceW BKL efle³e&keÀ DevegÒemLe PQ kesÀ oe³es ceW Deewj pees AB kesÀ veer®es nw, KLC
efle³e&keÀ PQ kesÀ yee³esb Deewj CD kesÀ GHej nw~
Fmemes ke̳ee efveCe&³e ues mekeÀles nQ? neB Òel³eskeÀ keÀesCees keÀer pees[er kesÀ
a) efJeefYevve Meer<e& nw~
b) efle³e&keÀ kesÀ efJe©× He#e ceW yeveW nQ~
c) meceeblej jsKee kesÀ Deble: Yeeie ceW nw~
d) meceeblej jsKee kesÀ efJeefYevve mLeeve nQ~ SkeÀ keÀesCe jsKee kesÀ THej nw, Deewj otmeje jsKee kesÀ
veer®es nQ~
Fme ÒekeÀej kesÀ keÀesve keÀes SkeÀebvlej Deble: keÀesCe keÀesCeeW keÀer pees[er keÀnles nQ~ keÀesCeeW keÀer pees[er
GHejeskeÌle Deeke=Àefle ceW AKL , KLD Deewj BKL , KLC SkeÀevlebj Deble: keÀesCe nw~
keÀesCeeW keÀer pees[er efpeme kesÀ
i) efJeefYevve Meer<e& nw~
ii) efle³e&keÀ kesÀ efJe©× He#e ceW nesles nwb~
iii) meceeblej jsKee kesÀ Deble: Yeeie ceW nesles nwb~
iv) meceeblej jsKee kesÀ efJeefYevve mLeeveeW Hej yeveW SkeÀevlej Deble: keÀesCe keÀnueeles nwb~
keÀe³e& keÀueeHe 10 : HesHej cees[keÀj Henues keÀe³e& keÀueeHe kesÀ pewmes keÀerefpeS, Deewj Deble: keÀesCeeW keÀes Hen®eeefveS~
meesef®eS De#ej N Deewj Z SkeÀeblej Deble: keÀesCeW keÀes Hen®eeveves ceW mene³ekeÀ nQ, kewÀmes?
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meceeblej jsKeeDeeW kesÀ efle³e&keÀ
keÀe³e& keÀueeHe 11 : owefvekeÀ peerJeve ces ef®e$eeW keÀe efvejer#eCe keÀerefpeS~
ef®e$e i ef®e$e ii ef®e$eæ iii
THej kesÀ ef®e$eeW ceW
(i) meer[er keÀe ef®e$e (ii) efKe[keÀer keÀe HeÀesìes (iii) jsueJes keÀer Heìjer nw~ Fve ceW
(i) meceelej jsKeeSB (ii) efle³e&keÀ jsKee Hen®eeefveS~
keÀe³e& keÀueeHe 12 :
Meerì HesHej keÀes ueerefpeS~
AB Deewj CD oes meceeblej HesHej Hej yeveeFS~ PQ mes efle³e&keÀ jsKee yeveeFS~
veer®es efoKeeS pewmes Deveg©He keÀesCeeW keÀer pees[er DebefkeÀle keÀerefpeS
ef®e$e i ef®e$e ii
ef®e$e iii ef®e$e iv
ì^wme HesHej keÀes Deeke=Àefle (i) Hej jKekeÀj ì^wme keÀerefpeS~ Meer<e& L Hej ì^wme HesHej Ssmee jefKeS efpememes AB, CD kesÀ meeLe Deewj Meer<e& K, Meer<e& L Hej mecceefuele nes~
efvecve Deeke=Àefle ceW, ì^sme efkeÀ³ee ngDee AKP mebMCe& ©He mes {keÀ ie³ee nw?
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ì^sme keÀesCe AKP mebHetCe& ©He mes íe keÀesCe Hej nesles nwb~
KLC keÀe Meer<e& L nw~
AKP Deewj KLC efkeÀme ÒekeÀej kesÀ keÀesCe nw?
AKP Deewj KLC Deveg©He keÀesCe nQ~ AKP = KLC
PKB DebefkeÀle keÀerefpeS Deewj ì^Wme HesHej jefKeS~ leeefkeÀ AB, CD mes efceues~ Deewj K Deewj
L mes efceues~
PKB Deewj KLD keÀes efvejer#eCe keÀerefpeS~
PKB - KLD
LKB - QLD - AKL - CLQ ³en efveCe&³e ues mekeÀles nQ~
³eefo oes ³ee DeefOekeÀ meceeblej jsKeeDeeW keÀes efle³e&keÀ jsKee Òeefle®ísefole keÀjleer lees Deveg©He keÀesCe meceeve nesles nQ~
efle³e&keÀ jsKee Deewj meceeblej jsKeeDeesb kesÀ yeer®e yeves SkeÀevlej keÀesCees keÀe mecyevOe.keÀe³e& keÀueeHe 13 : AB Deewj CD meceeblej jsKee keÀer j®evee keÀerefpeS~ Deewj HesHej Meerì PQ
DevegÒemLe jsKee yeveeFS Deewj SkeÀebvlej keÀesCe keÀes DebefkeÀle keÀerefpeS.
ef®e$e (i) ef®e$eæ (ii) ef®e$eæ (iii) ef®e$eÀæ (iv)
Deeke=Àefle (i) keÀes ì^wme HesHej mes ì^wme keÀerefpeS~ ì^wme HesHej keÀes ceesef[S~ efpememes AB jsKee CD mes efceueleer nw~ Meer<e& K, L mes efceueleer nw~
HesHej keÀes KeesefueS jsKeeKeC[ PQ keÀer Òeefle®ísove DebefkeÀle keÀerefpeS~ `X' efyevog DebefkeÀle keÀerefpeS~
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ì^Wme HesHej keÀes ef®e$e Hej jKekeÀj AB jsKee yeveeFS efpememes AB jsKee Hej efceueleer nw~ (Deeke=Àefle (iii).
ì^Wme HesHej keÀes efHeve mes ueieeFS Deewj AKL keÀes DebefkeÀle keÀerefpeS~ ì^Wme HesHej keÀes IegceekeÀj
AB jsKee keÀes CD Hej efceues~ Hejvleg CD keÀe efJe©× efoMee nw~ AKL Deewj KLD efceueves keÀe
efvejer#eCe keÀerefpeS~
Fmemes AKL = KLD
Deeke=Àefle (iii) ceW BKL keÀes DebefkeÀle keÀerefpeS~ (iii) `X' De#e mes IegceeFS efpememes
BKL - KLC
³eefo meceeblej jsKee mes DevegÒemLe jsKee mes yeveW SkeÀbevlej keÀesCe meceeve nesles nw~
DevegÒemLe keÀer meceeve Yegpee Hej Deble: keÀesCe:
keÀe³e& keÀueeHe 14 : SkeÀ Meerì HesHej Hej AB Deewj CD oes meceeblej jsKee Deewj PQ DevegÒemLe jsKee keÀer j®evee keÀerefpeS~
ef®e$e i ef®e$e ii
ì^Wme HesHej Hej Deeke=Àefle (i) keÀes ì^Wme keÀerefpeS peye AB jsKee CD Hej efceueleer nw~ efpememes `A', `D' mes efceueleer nw~
met®evee : ì^Wme HesHej AB keÀer efoMee CD mes efJe©× nw~
AKL - KLD Deewj BKL Deewj KLD kesÀ keÀesCe keÀes ceeefHeS Deewj GvekesÀ keÀesCeeW keÀe ³eesie %eele keÀerefpeS~ AKL + BKL = 180 ∴meuebive keÀesCe jsefKekeÀ pees[er yeveleer nw~
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AKL kesÀ yeoues ceW KLD efueefKeS~
KLD + BKL = 180
Fmeer ÒekeÀej, ||| ly BKL = CLK Deewj AKL + CLK = 180
Fmemes
³eefo oes meceeblej jsKeDeeW keÀes efle³e&keÀ jsKee Òeefle®ísefole keÀjleer nw lees efle³e&keÀ jsKee kesÀ meceeve Yegpee kesÀ Deble: keÀesCe mebHetjkeÀ nesles nw~
®ejCe 1 : SkeÀ Meerì HesHej ueerefpeS~ mejue jsKee AB keÀer j®evee keÀerefpeS~
a bk
®ejCe 2 : K Hej 1000 keÀer j®evee keÀerefpeS~ jsKee keÀer j®evee keÀjkesÀ ye{eF&S~ a bk
1000
®ejCe 3: ì^wme HesHej CD jsKee yeveeF&S Deewj `L' keÀes DebefkeÀle keÀerefpeS~
c dl
®ejCe 4: `L' Hej 1000 keÀer j®evee keÀerefpeS ~ keÀesCe keÀer Yegpee keÀes ye{eF&S~ c dl
1000
®ejCe 5: jsKee AB Hej ì^wme HesHej jsefKeS~ AB Deewj CD keÀer otjer yeveeF&S efpememes K Deewj L efceues leeefkeÀ ³en jsKee efle³e&keÀ jsKee nw~
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AKP Deewj CLK Hej O³eeve oerefpeS~ ³es efkeÀme ÒekeÀej kesÀ keÀesCe nw~
³es Deveg©He keÀesCe nw Deewj meceeve nw~ efYevve efYevve mLeeveeW Hej AB Deewj CD kesÀ yeer®e TB®eeF& ceefHeS~ ke̳ee Jen meceeve nw? AB Deewj CD jsKeeDeeW ceW ke̳ee mebyebOe nw~
peye SkeÀ efle³e&keÀ jsKee oes jsKeeDeeW keÀes Fme lejn Òeefle®ísefole keÀjleer nQ leeefkeÀ yeveW Deveg©He keÀesCe meceeve nQ lees Jes meceevlej nesles nQ~
Hegve : ³eeo jefKeS
peye SkeÀ efle³e&keÀ oes jsKeeDeeW keÀes Òeefle®ísefole keÀjleer nQ efpemeceW SkeÀeblej keÀesCe meceeve nesles nw lees Jes kewÀmeer jsKeeSW nw? dh g_m§Va hmoVr h¡.
peye SkeÀ efle³e&keÀ oes jsKeeDeeW keÀes Òeefle®ísefole keÀjleer nw efpemeces Deveg©He keÀesCe meceeve nesles nw lees Jes kewÀmeer jsKeeSW nQ? dh g_m§Va jsKeeSW hmoVr h¡.
peye SkeÀ efle³e&keÀ oes jsKeeDeeW keÀes Òeefle®ísefole keÀjleer nw efpemeceW efle³e&keÀ jsKee kesÀ meceeve Yegpee kesÀ Devle: keÀesCe mebHetjkeÀ nesles lees Jes kewÀmeer jsKeeSW nQ~ dh g_m§Va jsKeeSW hmoVr h¡.
meesef®eS!
1) efvecve efJeefYevve DeekeÀej O³eeve mes owefKeS efle³e&keÀ jsKee kesÀ Yegpee Hej yeveW Devle: keÀesCe kewÀmes Hen®eeve mekeÀles nQ?
2) oes jsKeeKeC[eW keÀes efle³e&keÀ jsKee keÀes keÀeìleer nw lees Deble: keÀesCe keÀe ³eesie mebHetjkeÀ nes lees Fve jsKee KeC[eW keÀes ke̳ee keÀnles nw?
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keÀe³e& keÀueeHe 14 : AB Deewj CD oes Demeceeblej jsKeeSW KeeRef®eS Deewj efle³e&keÀ jsKee PQ Yeer KeeRef®eS~ pees jsKeeDeeW keÀes ¬eÀceMe : K Deewj L Hej Òeefle®ísefole keÀjW~ lees Deveg©He keÀesCe, SkeÀeblej keÀesCe Deewj efle³e&keÀ jsKee SkeÀ Yegpee Hej yeveW keÀesCeeW keÀes ceeefHeS~ DeeHekesÀ efveCe&³e efueefKeS~
oÊe jsKeeDeeW keÀer pees[er meceeblej nw J³ee veneR peeb®eves keÀe mejue efJeOeeve~SkeÀ keÀeiepe Hej Deeke=Àefle- (i) Deewj Deeke=Àefle (ii) ceW efoKee³eW pewmes oes jsKeeDeeW keÀer peesef[³eeB
KeeRef®eS~
ef®e$e ief®e$e ii
a
c
b
d
a
c
b
d
oes jsKeeDeeW keÀes AB Deewj CD DebefkeÀle keÀerefpeS~ mkewÀue Ssmes jefKeS leeefkeÀ AB Deewj CD
keÀes efceues~ mkesÀue keÀer jsKeeDeeW keÀes DebefkeÀle keÀerefpeS~
mkesÀue keÀes efvekeÀeue keÀj P Deewj Q efyevogDeeW mes AB jsKee yeveeFS~ Fmeer ÒekeÀej R Deewj
S mes CD jsKee yeveeFS~ jsKeeKeC[ PQ Deewj RS ceeefHeS~
ef®e$e i ef®e$e ii
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oÊe ef®e$e - i) ceW nce efvejer#eCe keÀjles nw efkeÀ AB keÀer uebyeeF& = RS keÀer uebyeeF&~ nce efveef½ele
keÀjles nw efkeÀ AB Deewj CD HejmHej meceelej nw~
ef®e$e - ii) ceW PR = QS AB Deewj CD meceeblej veneR nw~
FmeefueS PQ Deewj RS, AB Deewj CD kesÀ efueS uecye nes lees Jes meceeblej nw~
oes jsKeeDeeW kesÀ yeer®e keÀer uebye otjer meceeve nes lees, Jes jsKeeSsb meceevlej nesleer nw~
GoenjCe1: oÊe Deeke=Àefle ces ABIICDIIEF.
GF T
EF Deewj BGD = 500. Mes<e keÀesCe %eele keÀerefpeS~
A B
D500
FE
C G
+ = 180O [ CD║EF]
= 180O
- 90O
= 90O
ABG Deewj BGD SkeÀeblej keÀesCe keÀnles nw~
= = 50O
= 50O
CGB Deewj BGD jsefKekeÀ pees[er
∴ CGB + BGD = 180O
CGB = 180O - 50O
CGB = 130O
CGB Deewj BGD jsefKekeÀ pees[er h¢
GoenjCe 2 : oÊe Deeke=Àefle ces DebefkeÀle keÀesCe keÀes ceeefHeS~
CFE + EFG = 180O
CFE + 40O =180O
CFE +180O + 40O
CFE +140O
A
c df
E
400
G
B
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DeefkeÀle keÀesCe FEB
CFE Deewj FEB SkeÀeblej keÀesCe nw ∴ ABIICD]
∴ FEB = FEBFEB = 1400
GoenjCe 3 : obÊe Deeke=Àefle ces keÀesCe p, q, r Deewj s DebefkeÀle keÀesCe %eele keÀerefpeS~
(i) ABC = 60O [ DE║BC]
ADF = ABC Deveg©He keÀesCe meceeve nw p = ABC ABC = 600
p = 600
A
C
D
R
700
EF
600
B
PQ?
S
(ii) AFD Deewj EFC Meer<eee&efYecegKe keÀesCe
AFD = EFC
∴ 700 = q
∴ q = 700
(ii) EFC Deewj ACB SkeÀeblej keÀesCe nw~
∴ ACB = EFC
r = q0
∴ r = 700
700 + CFD = 1800 jsKeer³e keÀesCe
∴ CFD = 1800 - 700
∴ CFD = 1100
CFE Deewj AFE Deveg©He keÀesCe
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∴ CFD = AFE
s = 1100
∴ s = 1100
∴ s = 600, q = 700, r = 700 Deewj s = 1100
DeY³eeme 6.1
1. íe³eeke=Àle keÀesCe keÀe Deveg©He keÀesCe efueefKeS~a) b) c)
d) e) f)
2) Òel³eskeÀ Deeke=Àefle ceW DebefkeÀle De#ej kesÀ keÀesCeeW keÀe cetu³e %eele keÀerefpeS~a)
A
C
F H
K DL
I J B
GE700
b) c)
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3) oÊe Deeke=Àefle ceW PQRS nes lees Deble: keÀesCe %eele keÀerefpeS~ P
Q R
S
T1300
4) Deeke=Àefle ceW MN||BC nes lees ABC keÀe Deble: keÀesCe %eele keÀerefpeS~
5) oÊe Deeke=Àefle ceW efveef½ele keÀerefpeS AB Ed§ CD mes meceeblej nQ~
AC
F
D
BG 1260
440H
E
ef®e$e i
750750
AC
F
DB
E
ef®e$e ii
6) oÊe Deeke=Àefle oes keÀesCe kesÀ Yegpee meceeblej nw~ ³eefo ABC = 700 Deewj DGC, DEF%eele keÀerefpeS~
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7) oÊe Deeke=Àefle ceW mes a, b, c Deewj d DebefkeÀle keÀesCe %eele keÀerefpeS~ meceeblej jsKeeS yeleeF&S
p Q
i) ii) iii)
iv) iv)
a
C
F
BD 80● 50●
Ea
a
bdc
a
P
B D
Q
C
50●
a
b
c
Q
SR
P
50●
110●a
c d
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DeO³ee³e - 7
ef$eYegpe
Fme DeO³ee³e keÀes He{ves kesÀ yeeo :
ef$eYegpe keÀe Deble: keÀesCeeW keÀe ³eesie keÀe³e& keÀueeHe mes %eele keÀjWieW,
ef$eYegpe keÀe Deble: keÀesCe keÀe ³eesie 1800 ~ HesHej cees[ves mes keÀjvee,
meceeblej jsKee kesÀ iegCeeW keÀes GHe³eesie efmeOo keÀjvee efkeÀ ef$eYegpe kesÀ Deble: keÀesCe keÀe ³eesie 1800,
ue@peerkeÀue GHeHeefle Deewj keÀe³e& DeeOeeefjle GHeHeefle kesÀ Deblej Hen®eeveWies,
yee¿e keÀesCe Deewj Deble: efJe©× keÀesCe kesÀ yeer®e mecyevOe mLeeefHele keÀjWies,
ef$eYegpe keÀer keÀesF& oes YegpeeDeeW keÀe pees[ ncesMee leermejer Yegpee mes p³eeoe nw,
HewLeeieewefj³eve Òeces³e keÀes keÀe³e& keÀueeHe mes mecePevee~
ÒemleeJeve:
ef$eYegpe SkeÀ Dev³eble Heefjef®ele jsKeeieefCeleer³e DeekeÀej nw~ GHejeskeÀle Deeke=Àefle³eeW ceW DeeHe DeveskeÀ ef$eYegpe osKeles nQ~ ef$eYegpeer³e DeekeÀej cesb mlebYe yeebOeves mes DeefOekeÀ efmLej nesles nwb~ owefvekeÀ peerJeve ceW ef$eYegpeer³e JemlegDee WkeÀes efvejer#eCe keÀjles nQ~ FmeefueS ef$eYegpeeW kesÀ iegCeOece& kesÀ yeejW ces meerKevee DeeJeM³ekeÀ nw~
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ef$eYegpe kesÀ keÀesCees keÀe ³eesie iegCeOece& :
keÀe³e& keÀueeHe 1
A
B Cef®e$e -(i)
A
BC ef®e$e-(ii)
SkeÀ ef$eYegpe keÀer j®evee keÀerefpeS (Deeke=Àefle (i) pewmes) Meer<eex keÀes A, B, C veece oerefpeS
ef$eYegpe kesÀ Deble : keÀesCe keÀes ®eeBoe kesÀ GHe³eesie mes ceeefHeS~
veer®es meejCeer ceW ceeHe efueefKeS~ GoenjCe kesÀ efueS Deeke=Àefle (i) kesÀ keÀesCe efueKeW nwb~
leerve keÀesCeeW keÀe ³eesie %eele keÀerefpeS~
ef$eYegpe kesÀ Deble: keÀesCe ef$eYegpe (1) ef$eYegpe (2) ef$eYegpe (3) ef$eYegpe (4) ef$eYegpe (5)
keÀesCe 1 640
keÀesCe 2 560
keÀesCe 3 600
leerve keÀesCe keÀe ³eesie 1800
Gmeer ÒekeÀej ef$eYegpe efJeefYevve ÒekeÀej keÀer j®evee keÀerefpeS~ (Goe: meceyeeng, efye<ece v³etve keÀesCe, efJe<ece DeefOekeÀ keÀesCe) Òel³eskeÀ ef$eYegpe kesÀ Deble: keÀesCe keÀes ceeefHeS~ oÊe meejCeer ceW ceeHe efueefKeS~ ef$eYegpe kesÀ leerve keÀesCe keÀe ³eesie %eele keÀerefpeS~
GHejesÊeÀ keÀe³e& keÀueeHe mes ef$eYegpe keÀs leerve Deble: keÀesCeeW keÀe kesÀ yeejs ceW DeeHe ke̳ee efvejer#eCe keÀjles nQ~
DeeF&S, SkeÀ, Deewj keÀe³e&keÀueeHe keÀjles nQ~
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keÀe³e& keÀueeHe (2) :
SkeÀ HesHej ueerefpeS~ Gmeces ef$eYegpe keÀeefì³es~ 1
2 3
ef®e$e (i)
leerve keÀesCe 1,2 Deewj 3 keÀes DebefkeÀle keÀerefpeS~
keÀesCe ef®e$e kesÀ Devegmeej keÀesCe keÀeì ueerefpeS~
keÀeìs ngS ìgkeÀ[s keÀes Ssmes peesef[³es efpememes Meer<e& SkeÀ efyevog Hej efceues Deeke=Àefle (ii) pewmes leerveeW keÀesCeeW kesÀ ³eesie kesÀ yeejs ceW DeeHekeÀe ke̳ee efvejer#eCe nw?
∴ef$eYegpe kesÀ Deble: keÀesCeeW keÀe ³eesie ke̳ee nw?
ef®e$e (ii)
otmeje keÀe³e& keÀueeHe (HesHej efyevee keÀeìs)
keÀe³e& keÀueeHe (3) :
Deeke=Àefle kesÀ pewmes ABC keÀer j®evee Meerì HesHej Hej yeveeF&S~
keÀesCe 1, 2 Deewj 3 keÀes DebefkeÀle keÀerefpeS~AB Deewj AC kesÀ ceO³e efyevog P Deewj Q Deeke=Àefle
(i) ceW pewmes DebefkeÀle keÀerefpeS
jsKee PQ Hej keÀesæCe APQ keÀes ceesef[S~ Meer<e& A,BC keÀes M Hej efceueleer nw~
Meer<e& B keÀes ceesef[S leeefkeÀ BP, PM mes efceueW~ Meer<e& C Hej ceesef[S leeefkeÀ QC, QM mes efceueW~ 1 , 2 Dee wj 3 me s mejue keÀe sCe yeveele s nw~ leerveeW keÀe³e&keÀueHeesb mes mHe<ì neslee efkeÀ ef$eYegpe kesÀ Deble: keÀesCeeW keÀe ³eesie 1800.
132
mb
a
p q
cef®e$e (ii)
ef®e$e (i)
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DeY³eeme 7.1
1) SkeÀ Meerì Hej ef$eYegpe keÀer j®evee keÀerefpeS~ ef$eYegpe keÀer Yegpee kesÀ Devegmeej keÀeefì³es~ leerve
keÀesCeeW keÀes keÀeì keÀj Deueie keÀerefpeS~ ef$eYegpe kesÀ leerve Meer<e& keÀes pees[ keÀj SkeÀ efyevog Hej
efceueles nw~ keÀesCe keÀe ueieYeie ³eesie %eele keÀerefpeS~
2) mebueive Deeke=Àefle kesÀ pewmes SkeÀ DeefOekeÀ keÀesCe keÀer
b
a
c
j®evee keÀerefpeS~ Fme ef$eYegpe kesÀ keÀesCe efyevee keÀeìW, ef$eYegpe
keÀeì ueerefpeS~ ef$eYegpe kesÀ keÀesCeeW keÀe ³eesie iegCe keÀe
mel³eeHeve keÀerefpeS~
meesef®eS :
Fve keÀe³e& keÀueeHeeW ceW ef$eYegpe kesÀ Dele: keÀesCeeW keÀe ³eesie1800 nw~ ke̳eeW1500 ³ee 2000
veneR nw?
ef$eYegpe kesÀ efJeefYevve ÒekeÀej nesves mes Yeer ke̳ee ef$eYegpe kesÀ Deble: keÀesCeeW keÀe ³eesie 1800 nw~
ke̳ee keÀesF& ef$eYegpe keÀe Deble: keÀesCeeW keÀe ³eesie 1800 veneR nw?
ef$eYegpe kesÀ Deble: keÀesCeeW keÀe ³eesie leke&À efJeOeeve mes %eele keÀjvee~
meYeer keÀe³e&keÀuHe metef®ele keÀjles nQ efkeÀ ef$eYegpe kesÀ Deble: keÀesCeeW keÀe ³eesieHeÀue 1800 neslee nw~
Hejvleg Ssmee keÀesF& efveef½ele iegCeOece& veneR nw pees meYeer ef$eYegpeeW kesÀ efueS GHe³egkeÌle nw~ npeejeW ef$eYegpeeW
kesÀ keÀesCeeW keÀes ceeHeves kesÀ yeeo Yeer keÀesF& J³eeqkeÌle keÀesCeeW kesÀ ³eesie iegCe keÀes HetCe&le : mel³e efme× keÀjW~
efJeOeeve ceW kegÀí keÀefce³eeB nesleer nw efpemeceW GoenjCe efue³es ieS nw~ Fme keÀceer kesÀ otmejs efJeOeeve mes
otj keÀj mekeÀles nQ~
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leke&À efJeOeeve mes efve<keÀ<e& keÀes meeceev³e ©He os mekeÀles nQ~ ieefCeleer³e keÀLeveeW keÀes efme× keÀjves
leke&À efJeOeeve GHe³eesie keÀjles nQ~
DeeF&S, ``ef$eYegpe kesÀ leerveeW keÀesCeeW keÀe ³eesie 1800 neslee nw'' efme× keÀjves leke&À efJeOeeve
GHe³eesie keÀjW~
DeeF&S, Deeke=Àefle kesÀ ∆ABC kesÀ keÀesCe ∠A, ∠B Deewj ∠C keÀes ³eesie
B c
A
1800 nw efme× keÀjves leke&À efJeOeeve keÀjW~
ke̳ee j®evee Üeje ∆ABC kesÀ leerveeW keÀesCeeW keÀes efkeÀmeer mejue jsKee kesÀ
SkeÀ efyevog (keÀnes A,B DeLeJee C) ueevee mebYeJe nw? neB, DeeF&S, A Üeje
mejue jsKee RS KeeR®es~
Deeke=Àefle ceW oes SkeÀeblej keÀesCeeW keÀer peesef[³eeB nQ~ Jes nQ~
B c
A
R
s
∗
∗
i) ∠ABC Deewj ∠BAR (∗ mes metef®ele )
ii) ∠ACB Deewj ∠SAC ( mes metef®ele venerb nw~)
Hejvleg ³es SkeÀeblej keÀesCe meceeve veneR nw~
GHejeskeÌle ef$eYegpe ABC ceW SkeÀ meceevlej jsKeeDeeW keÀer pees[er nesleer lees SkeÀeblej keÀesCe meceeve
nesles~
nce peeveles nw efkeÀ, ³eefo meceeblej jsKeeDeeW keÀes SkeÀ efle³e&keÀ jsKee Òeefle®ísefole keÀjleer nQ lees
yeveW SkeÀeblej keÀesCe meceeve nesles nw~
nce peeveles nw efkeÀ DevegÒemLe jsKee ³eefo meceeblej jsKee keÀes keÀeìleer nw lees yeves SkebÀevlej keÀesCe meceeve nesles nw~
FmeefueS, A Üeje PQ jsKee BC kesÀ efueS meceeblej KeeRef®eS~ ³eneB PQ||BC Deewj AB
SkeÀ efve³e&keÀ jsKee nw?
Deye Deeke=Àefle O³eeve mes osefKeS ke̳ee keÀesF& keÀesCeeW keÀer pees[er meceeve nw?
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PAB = ABC SkeÀevlej keÀesCe mes metef®ele
PQ Deewj BC meceeblej jsKeeDeeW keÀes keÀeìvesJeeueer AB jsKee
B c
Ap q∗
∗
osefKeS~
ke̳ee JeneB HejmHej mece keÀesCe nQ?
Deeke=Àefle ceW efueKee keÀesCe SkeÀeblej keÀesCe nQ~ PAB = ABC
Fmeer ÒekeÀej PQ||BC Deewj AC DevegÒemLe jsKee nw~
ke̳ee keÀesF& keÀesCeeW keÀer pees[er meceeve?
CQAC = ACB ii) (SkeÀevlegj keÀesCe)
PA, AB Deewj AC jsKee keÀe efvejer#eCe keÀerefpeS~
PAB + BAC + CAQ = mejue keÀesCe - 1800
PAB Deewj QAC kesÀ yeoues ceW ABC Deewj ACB ceW keÀerefpeS
nceW ÒeeHle nw efkeÀ ABC + BAC + ACB + BAC =1800
³en ef$eYegpe kesÀ Deble: keÀesCe nw~
ef$eYegpe kesÀ Deleb: keÀesCe keÀe ³eesie 1800 nw~
leke&À mebiele efJeOeeve mes efve½e³e~
leke&À mebiele drYmZ mes efve½e³e~
GoenjCe 1: hb
Deeke=Àefle ceW PQ||RS. PQ Hej A efyevog nw~ AB Deewj AC A efyevog Hej nw~
∠PAB = 500 ∠QAC = 700
ef$eYegpe kesÀ meYeer keÀesCe kesÀ ceeHe leke&À mebiele nw~
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meceeOeeve
meceelebj jsKee PQ Deewj RS Deewj otmejer jsKee AB nw~
∠PAB Deewj ∠ABC ceW keÀewvemes ÒekeÀej kesÀ keÀesCe nw?
SkeÀevlej keÀesCe nw~ FmeefueS
∠ABC = ∠PAB = 500
PQ, RS Deewj AC jsKeeDees keÀe efveefj#eCe keÀerefpeS~
∠QAC Deewj ∠ACB keÀewvemes ÒekeÀej kesÀ keÀesCe nw?
³es SkeÀevlej keÀesCe nQ~
FmeefueS ∠ACB = ∠QAC = 700
PQ, AB, AC jsKeeSB efvejer#eCe keÀerefpeS~
∠PAB, ∠QAC Deewj ∠BAC keÀewvemes ÒekeÀej kesÀ keÀesCe nw?
ke̳ee Jes mebHetjkeÀ keÀesCe nQ?
∠PAB + ∠BAC + ∠QAC = 1800
500 + ∠BAC + 700 = 1800
= 1800 - 500- 700
∴ ∠BAC = 600
GoenjCe 2 : hb
meuebive ef$eYegpe PQR ceW ∠Q = 400 ∠R =
300 leke&À mebiele mes leermeje keÀesCe %eele keÀerefpeS~
P kesÀ oteje QR meceeblej jsKee yeveeFS~
p
400 300
?
q r
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∴ ef$eYegpe kesÀ leerve keÀesCeeW keÀe ³eesie 1800 nw ~
p
400
400 300
300
?
q r
HeeMJe& ef$eYegpe PQR cesb ∠Q = 400 ∠R = 300 efyevog Hej keÀesCe keÀe ³eesie leermeje keÀesCe = 1800-400-300 = 1100
Goenj£Ce 1 : Deeke=Àefle ceW ∠A = 600, ∠B = 500 lees leke&À mebiele efJeOeeve mes ∠C %eele keÀerefpeS~
meceeOeeve
∠A + ∠B + ∠C = 1800
(leerve keÀesCeeW keÀe ³eesie 1800)
600 + 500 + ∠C = 1800
1100 + ∠C = 1800
∠C = 1800 - 1100
∠C = 700
Goenj£Ce 2 : HeeMJe& Deeke=Àefle ceW x keÀe cetu³e %eele keÀerefpeS~ meceeOeeve
∠A + ∠B + ∠C = 1800 (leerve keÀesCeeW keÀe ³eesie 1800 nw)
600 + x 0 + 800 = 1800 (∠A Deewj ∠C keÀe cetu³e efo³ee nw~)
1400 + x 0 = 1800
x 0 = 1800 - 1400
x 0 = 400
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DeY³eeme 7.2
1) efvecveefueefKele Deeke=Àefle ces x keÀe cetu³e leke&À mebiele efJeOeeve mes %eele keÀerefpeS~
2) ef$eYegpe ABC kesÀ oes keÀesCe efo³es ie³es nw~ meejCeer ces Keeueer mLeeve YeefjS Deewj leermeje keÀesCe %eele keÀerefpeS~
∠A ∠B ∠C
1 600 700
2 1000 350
3 650 650
4 1100 360
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3) meejCeer ceW leerve keÀesCe efo³es ie³es nwb~ leerve keÀesCeeW keÀe ³eesie %eele keÀerefpeS~ ke̳ee ³en ef$eYegpe kesÀ keÀesCe yeveles nwb?
∠A ∠B ∠C ³eesie ef$eYegpe yevelee nw ³ee veneR~
1 500 600 700
2 600 700 800
3 650 750 550
4 560 640 600
5 570 640 790
4) ef$eYegpe ABC ceW, ³eefo ∠A = 450, ∠B = 650 nes lees ∠C keÀe ceeHe %eele keÀerefpeS~
5) efvecveefueefKele ceeHe mes ∠ABC keÀer j®evee keÀerefpeS~
∠PAB = 500 Deewj ∠QBA = 600 ∠ACB keÀe ceeHe ke̳ee
nw? Deewj ∠ACB leke&À mebiele efJeOeeve mes %eele keÀerefpeS~
6) uebye keÀesCe ef$eYegpe ceW ³eefo SkeÀ keÀesCe keÀe ceeHe 350
otmejs oes keÀesCe keÀe ceeHe %eele keÀerefpeS~
7) meceyeeng ef$eYegpe keÀer Yegpee 4 cm nw~ Òel³eskeÀ keÀesCe keÀe ceeHe %eele keÀerefpeS~
8) ef$eYegpe ABC ceW ∠A = 350 Deewj ∠B, ∠A mes 150 p³eeoe nw lees, ∠C %eele keÀerefpeS~
9) uebye keÀesCe ef$eYegpe ceW, oes keÀesCeeW ceW SkeÀ keÀesCe otmejs mes ogiegvee nw~ otmejs oes keÀesCe %eele
keÀerefpeS Jen ke̳eeW uebye keÀesCe ef$eYegpe vener nw?
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keÀesCe kesÀ yee¿e keÀesCe
SkeÀ ef$eYegpe kesÀ leerve Yegpee nw~ ³eefo keÀesF&
SkeÀ Yegpee keÀes ye{eles nQ, lees Meer<e& Hej ef$eYegpe
kesÀ yeenj yeves keÀesCe pees yee¿e keÀesCe mes mebueive
nw lees Gmes mebueile Deble: keÀesCe keÀnles nw~
Deble: meueive keÀesCeye¿e
yee¿e keÀesCe
Deble: mebueive keÀesCe
Deble: metueive keÀesCeyee¿e keÀesCe meueive keÀesCeyee¿e keÀesCe
ef$eYegpe kesÀ keÀesF& Yegpee ye{ves mes yee¿e keÀesCe Deewj Deble keÀesCe yeveles nQ~ yee¿e keÀesCe Dele: keÀesCe mebueive keÀesCe keÀe ³eesie keÀ³ee nw?
³es oesveeW keÀesCe mebHetjkeÀ keÀesCe nQ DeLeJee SkeÀ jsKeer³e
Deble: meueive yee¿e
pees[ keÀesCe nw~
Deble : keÀesCe Deewj yee¿e keÀesCe keÀe ³eesie 1800 neslee nw~
ef$eYegpe kesÀ keÀesF& Yegpee ye{ves mes yee¿e keÀesCe Deewj Deble keÀesCe yeveles nQ~ yee¿e keÀesCe Deble: keÀesCe mebueive keÀesCe keÀe ³eesie ke̳ee nw?
meueiCe Deble: keÀesCe yee¿e
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Deble: efJe©× keÀesCe~
ef$eYegpe kesÀ Deble : meueive keÀesCe kesÀ efJe©× kesÀ keÀesCe keÀes
Deble : efJe©× keÀesCe keÀnles nQ~
ef$eYegpe kesÀ Òel³eskeÀ yee¿e keÀesCe kesÀ oes Dóble efJe©× : keÀesCe Deewj SkeÀ meuebive Deble: keÀesCe neslee nwb~
Deble: efJe©× keÀesCe - 2
Deble: efJe©× keÀesCe - 1meueie efJe©× keÀesCe
yee¿e
ke̳ee : yee¿e keÀesCe, ef$eYegpe kesÀ Deble : efJe©× keÀesCe kesÀ yeer®e keÀe mecyevOe ceW keÀesF& mebyebOe neslee nw?
keÀe³e&keÀueeHe 2 :
Deeke=Àefle kesÀ Devegmeej SkeÀ Meerì HesHej ceW ef$eYegpe keÀer j®evee keÀerefpeS
ef$eYegpe kesÀ DeeOeej keÀes ye{eFSef$eYegpe kesÀ leerve keÀesCe 1, 2 DebefkeÀle
keÀerefpeS~yee¿e keÀesCe 4 keÀes DebefkeÀle keÀerefpeS~ Deeke=Àefle (i)
Deeke=Àefle 2 : ceW efoKee³eW pewmes
keÀesCe (1) Deewj keÀesCe (2) keÀes keÀeìkeÀj keÀesCe yeen³e keÀesCe 4 Hej peesef[S nw~ keÀesCe~ DeeHekeÀ efve<keÀ<e& ke̳ee nw? keÀesCe 1 + keÀesCe 2 = yeen³ekeÀesCe 4 ef®e$e (ii)
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∴ef$eYegpe kesÀ yee¿e keÀesCe Deble: efJe©× keÀesCeW kesÀ ³eesie meceeve neslee nw~
kegÀí GoenjCe keÀe efvejer#eCe keÀjWies~GoenjCe 1 :
ef®e$e ces ef$eYegpe kesÀ oes keÀesCe efo³es nQ~ 700
500
c d
a
byee¿e keÀesCe ?
yee¿e keÀesCe keÀe ceeHe %eele keÀerefpeS~
meceeOeeve : ∠A + ∠B + ∠C = 1800 (ef$eYegpe kesÀ leerve keÀesCeeW keÀe ³eesie = 1800)
700 + 500 + ∠C = 1800 (∠A Deewj ∠B kesÀ yeoues ceW)
1200 + ∠C = 1800
∠C = 1800 - 1200
∠C = 600
∴ Deble : keÀesCe ∠ACB = 600
∠ACB + yee¿e keÀesCe∠ACD = 1800
(∴ SkeÀ mebueive Deble : keÀesCe Deewj yee¿e keÀesCe keÀe ³eesie = 1800)
∴ 600 + ∠ACD = 1800
∠ACD = 1800 - 600
= 1200
yee¿e keÀesCe ∠ACD = 1200
efJe©× keÀesCe keÀe ³eesie = 700 + 500 = 1200
FmeefueS yee¿e keÀesCe keÀes meceeve Deble : efJe©× keÀesCe keÀe ³eesie nw~
nue efkeÀ³ee ngDee GoenjCe 2 :
Deeke=Àefle ceW ³eefo ABC ef®e$e ceW PQR ceW oes keÀesCe 400
550RS
P
Qyee¿e keÀesCe ?
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efo³es nwb yee¿e keÀesCe %eele keÀerefpeS~∠P + ∠Q + ∠R = 1800 (leerve keÀesCe keÀe ³eesie 1800)
400 + 550 + ∠R = 1800 (∠P Deewj ∠Q keÀe cetu³e [eueves mes.)
950 + ∠R = 1800
∠R = 1800 - 950 = 850
∴ Deble : keÀesCe ∠PRQ = 850
Deble keÀesCe ∠PRQ Deewj ∠PRS HetjkeÀ nQ~
∴ ∠PRQ + ∠PRS = 1800
850 + ∠PRS = 1800
∠PRS = 1800 - 850 = 950
yee¿e keÀesCe = 950 Deewj Deble: keÀesCeeW keÀe ³eesie = 400 + 550 = 950
∴ FmeefueS ef$eYegpe keÀe yee¿e keÀesCe = efJe©× Deble: keÀesCeeW keÀe ³eesie
GoenjCe 3 : ef®e$e ces ∠BAC = 500 Deewj ∠ABC = 850 ∠ACB a
500
850
dcb
Deewj ∠ACD %eele keÀerefpeS~
∠BAC + ∠ABC + ∠ACB = 1800 (ef$eYegpe keÀe ³eesie 1800)
500 + 850 + ∠ACB = 1800 (∠A Deewj ∠B keÀe cetu³e [eueves ces)
1350 + ∠ACB = 1800
∠ACB = 1800 - 1350 = 450
∠ACB + ∠ACD = 1800 (HetjkeÀ keÀes)
450 + ∠ACD = 1800
∠ACD = 1800 - 450 = 1350
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DeY³eeme 7.3
1. efvecveefueefKele Deeke=Àefle ceW x Deewj y keÀe cetu³e %eele keÀerefpeS ~
a)600
400 x y
c)
x y
550
550
b)
1450 700
x
y
d)500
650x y
2) Deeke=Àefle ceW, ³eefo ∠BAC = 450 Deewj ∠ABC = 750. ∠ACB Deewj ∠ACD keÀe ceeHe %eele keÀerefpeS~
a
450
750
dcb
3) Deeke=Àefle ceW ³eefo ∠PQR = 500, ∠PRS = 1300 ∠QPR = ∠PRQ keÀe ceeHe %eele keÀerefpeS~ 500 1300
R S
P
Q
4) Deeke=Àefle ceW ³eefo ∠NKL = 350, ∠KNL = 300 Deewj ∠LNM = 750 nes lees ∠NLM Deewj ∠NML %eele keÀerefpeS~
300 750
MLK
N
350
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5) ef$eYegpe ABC ceW BC keÀes D lekeÀ yeæ{e³es ie³es nQ~ A Deewj D keÀes peesef[S~ ³eefo ∠BAC = 600,
∠ABC = 450 nes lees ∠ACD %eele keÀerefpeS~
ef$eYegpe kesÀ keÀesF& oes Yegpee keÀe ³eesie
keÀe³e& keÀueeHe - 1
Deeieve ceW leerve DemecejsKe efyevog A, B Deewj C DebeqkeÌle keÀerefpeS~ ®ee@keÀ mes AB, BC Deewj
keÀes C A peesef[S~ ³es jsKeeSB ef$eYegpe keÀer leerve YegpeeDeeW keÀes metef®ele keÀjles nQ~
A mes C lekeÀ ®eueves oerefpeS A mes C lekeÀ oes jemles nQ~
A mes C lekeÀ meerOee nw otmeje A mes B yeeo ceW B mes C lekeÀ Henuee jemlee yengle íesìe nw pees
keÀesF& Yeer ®euelee nw~ otmeje jemlee ye[e nw ke̳eeWefkeÀ GmekeÀes A mes B HeÀerj B mes C lekeÀ ®euevee nw,
FmekeÀe DeLe& AB + BC > AC ---- (i)
Fmeer ÒekeÀej, B mes A lekeÀ HengB®eves ceW oes lejerkesÀ nw SkeÀ B mes A lekeÀ otmeje B mes C
Deewj C mes A lekeÀ~
peye ----
BC + CA > BA ---- (ii)
Fmeer ÒekeÀej, B mes C lekeÀ HengB®eves kesÀ efueS oes jemles nw~
∴ BA + AC > BC ---- (iii)
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keÀe³e& keÀueeHe 2 :
3cm, 5cm, 7cm, 9cm, 11cm, 13cm, 15cm, 17cm, 19cm, 21cm ueyeer
uekeÀ[er ueerefpeS~ 3 uekeÀ[er keÀes ®egefve³es efpememes ef$eYegpe yevelee nw~ Fmeer ÒekeÀej efJeefYevve uebyeeF&
keÀer uekeÀef[³eeB ueerefpeS efpememes ef$eYegpe yeveleer nw~
GoenjCe 5cm Deewj 7cm oes uekeÀef[³eeB ueerefpeS leermejer uekeÀ[er ef$eYegpe yeveves kesÀ efueS
®eeefnS~ leermejer uekeÀ[er keÀer uebyeeF& keÀ³ee nw?
Jen uekeÀ[er
7 - 5 = 2 cm mes p³eeoe nesvee ®eeefnS Deewj
7 + 5 = 12 cm mes keÀce jnvee ®eeefnS~
Fme efueS ef$eYegpe yeveves kesÀ efueS oes uekeÀæ[er keÀe uebyeeF& keÀe ³eesie ®eeefn³es ³ee leermejer uekeÀ[er
mes p³eeoe~ keÀe³e& keÀueeHe mes, ef$eYegpe kesÀ keÀesF& oes Yegpee keÀe ³eesie leermejer Yegpee mes p³eeoe nesvee nw,
ef$eYegpe kesÀ oes YegpeeDeeW keÀe ³eesie leermejer Yegpee mes p³eeoe nw~
GoenjCe 1 : 3cm, 4cm, Deewj 6cm, YegpeeDeeW keÀe ef$eYegpe jnlee nw ke̳ee?
ef$eYegpe kesÀ oes Yegpee keÀe ³eesie leermejer Yegpee mes p³eeoe.
GoenjCe mel³eeHeve keÀerefpeS efkeÀ ef$eYegpe kesÀ keÀesF& oes Yegpee keÀe ³eesie leermejer mes p³eeoe nw~
a) 3cm + 4cm > 6 cm ef$eYegpe yevelee nw~
b) 4cm + 6cm > 3 cm ef$eYegpe yevelee nw~
c) 6cm + 3cm > 4 cm ef$eYegpe yevelee nw~
∴ Fve kesÀ ceeHe mes ef$eYegpe yevelee nw~
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GoenjCe 2 : 3cm, 4cm, Deewj 7cm, ef$eYegpe yeveeves keÀer meYeeJevee nw~ peeBef®eS mev³eeHeve
ef$eYegpe kesÀ oes Yegpee keÀe ³eesie.
a) 3cm, + 4cm, = 7cm, ef$eYegpe kesÀ oes YegpeeDeeW keÀe ³eesie leermejer Yegpee mes p³eeoe vener~
b) 3cm, + 7cm, > 4cm, ef$eYegpe kesÀ YegpeeDeeW keÀe ³eesie leermejer Yegpee mes p³eeoe nw~
c) 4cm, + 7cm, > 3cm, ef$eYegpe kesÀ YegpeeDeeW keÀe ³eesie leermejer Yegpee mes p³eeoe nw~
∴ SkeÀ meoYe& (a) oes YegpeeDees keÀe ³eesie leermejer Yegpee mes ye[es vener nw~ ef$eYegpe yeveeves keÀer mebYeJevee
venerb nw~ FmeefueS Fme ceeHe keÀe ef$eYegpe keÀer j®evee veneR keÀj mekeÀles~
DeY³eeme 7.4
1) efvecveefueefKele ceeHeeW mes mel³eeHeve ef$eYegpe yevelee nw~
i) 3cm, 4cm, 5cm. ii) 3cm, 2cm, 6cm.
iii) 2cm, 6cm, 6cm. iv) 5cm, 3cm, 8cm.
2) ef$eYegpe ABC ceW M keÀesF& efyevog nw~ efvecveefueefKele keÀLeve mener nw ³ee ieuele nw~
yeleeFS~
a) MA + MB > AB ? a
mb c
b) MB + MC < BC ?
c) MC + MA > CA ?
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HewLeeieesjme, Òeces³e (yeesOee³eve met$e)
HewLeeieesjme ûeerkeÀ oeMe&efvekeÀ nw~ uebye keÀesCe ef$eYegpe keÀe cegK³e iegCeOece& keÀe ÒemleeJe 600 BC
ces efkeÀ³es nwb~ Fme efueS uebye keÀesCe ef$eYegpe FvekeÀe veece efo³es ie³es nwb~
³eefo ef$eYegpe keÀe SkeÀ keÀesCe 900 nw lees ef$eYegpe keÀes uebye keÀesCe ef$eYegpe keÀnles nwb~ uebye keÀesCe
keÀer efJe©× Yegpee keÀes efJekeÀCe& keÀnles nwb pees uebye keÀesCe ef$eYegpe keÀer meyemes yeæ[er Yegpee nw~
keÀe³e& keÀueeHe 1 : SkeÀ Dee³eelekeÀej Meerì HesHej ueerefpeS~ Deeke=Àefle ces oMee&³ee pewmes, uebye keÀesCe
ef$eYegpe Meerì kesÀ keÀesves Hej ∆ABC, ∆KLM, ∆PQR Deewj ∆XYZ ef$eYegpe nw~ Fve ef$eYegpe kesÀ
efJekeÀCe& keÀes ceesef[S~ AC, KM, PR Deewj XZ nw~A x 12cm
12cm8cm
3cm
5cm
4cm
9cm6c
m
y
zr
qpml
kc
b
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efvecveefueefKele meejCeer keÀesCe keÀer Yegpee keÀer uebyeeF& nw~ ∆ABC keÀer met®evee efueefKeS~
ef$eYegpe efJekeÀCe& keÀe ceeHe meWceer
efJekeÀeCe& keÀe Jeie&
SkeÀ Yegpee keÀer
uebyeeF&
GmekeÀe Jeie&
otmejer Yegpee keÀer uebyeeF&
FmekeÀe Jeie&
oes Yegpee keÀe Jeie& keÀe ³eesie nw~
∆ABC 5 25 3 9 4 16 9 + 16 = 25
∆KLM
∆PQR
∆XYZ
∆ABC ceW AB = 3cm. BC = 4cm. AC %eele keÀerefpeS ke̳ee AC = 5cm.
oes Yegpee keÀe JeieeX keÀe ³eesie %eele keÀjWies~
AB2 + BC2 = 32 + 42
= 9 + 16
= 25
AC keÀe Jeie& %eele keÀerefpeS, AC2 = 52 = 25
∴ AC2 = AB2 + BC2
52 = 32 + 42
25 = 9 + 16
= 25
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GHejesÊeÀ mes nce mecePeles nw efkeÀ uebye keÀesCe ef$eYegpe ceW, efJekeÀeCe& keÀe Jeie& kesÀ meceeve Dev³e oes
YegpeeDeeW kesÀ Jeie& nw~ FmekeÀes efJeMJe ceev³elee mes HewLeeieesjme Òeces³e keÀnles nw~
HewLeeieesjme Òeces³e (yeesOee³eve met$e)
SkeÀ uebye keÀesCe ef$eYegpe ces efJekeÀCe& keÀe Jeie& Dev³e oes YegpeeDeeW kesÀ JeieeX kesÀ ³eesie kesÀ meceeve nw~
AB SkeÀ Yegpee keÀe Jeie& AB2
BC otmejer Yegpee keÀe Jeie&
BC2
BC efJekeÀeCe& keÀe Jeie&
AC 2
∆ ABC ceW ∠B = 900 AB2 + BC2 = AC2
SkeÀ Yegpee keÀe Jeie& AB2
a
b c
Megueyee met$e 800-BC ces yeves ie³es nw~ yeesOee³eve 600 kesÀ Devegmeej Dee³eele kesÀ efJekeÀCe&
pees oesveeW #es$eeW efpemeceW uebyeeF& Deewj ®eew[eF& Deueie GlHevve keÀjles nwb~
mebmke=Àle ñbmoH$ efpemekeÀ DeLe& Dee³ele keÀe efJekeÀCe& oesveesb #es$eHeÀue efpemeceW uebyeeF& Deewj
®eew[eF& Deueie GlHevve keÀjlee.
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GoenjCe 1 : HeeMJe& Deeke=Àefle ∆ABC, AB = 6cm, BC = 8cm Deewj ∠B = 900. AC
%eele keÀerefpeS~
meceeOeeve
HewLeeieesjme efve³ece Devegmeej
AC2 = AB2 + BC2
a
900
6cm
8cmb c
= 62 + 82
= 36 + 64
= 100
= 102
∴ AC = 10 g|_r
GoenjCe 2 : SkeÀ 13 ceerìj meer{er oerJeej Hej jKee peelee nw~ ³eefo meeræ{er 12 ceerìj otjer Hej
nw lees meer{er Deewj oerJeej keÀer otjer %eele keÀerefpeS~
HewLeeieesjme efve³ece mes
AB2 + BC2 = AC2
a
12 13
?B c
efoJeej
peceerve
efme[er 122 + BC2 = 132
144 + BC2 = 169
BC2 = 169 - 144
BC2 = 25
∴ BC = 5 _r
∴ meeræ{er Deewj oerJeej keÀer otjer 5 ceerìj.
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DeY³eeme 7.5
1) peceerve Hej PeC[e ueiee³ee ie³ee nw~ mlecYe keÀer T®eeBF&. 8 ceerìj Deewj 6 ceerìj
otjer Hej nw~ leej uebyeeF& %eele keÀerefpeS~
2) oerJeej keÀer TB®eeF& 12 ceerìj Deewj otjer 15 ceerìj efkeÀleveer otjer Hej efme{er nw?
3) ∆ABC ceWè AB = 5 cm. BC = 12 cm. ∠ABC = 900 AC keÀer uebyeeF& %eele
keÀerefpeS~
4) ef$eYegpe kesÀ Yegpee veer®es efo³es ie³es nQ~ keÀewvemee meboYe& ces uebye keÀesCe ef$eYegpe nw?
a) 3 cm, 4 cm, 5 cm.
b) 1.5 cm, 2 cm, 2.5 cm.
c) 6 cm, 8 cm, 10 cm.
86?
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DeO³ee³e - 8
meceefceefle
Fme IeìkeÀ keÀes meerKeves kesÀ yeeo DeeHe :
Òeke=Àefle ceW meceefceefle Deewj HejeJele&ve meceefceefle keÀes Hen®eeve mekeÀles,
2D Jemleg kesÀ ®e¬eÀer³e meceefceefle keÀer HeefjYee<ee oWies,
900, 1200, 1800 kesÀ IegceeJe kesÀ ®e¬eÀer³e meceefceefle keÀes Hen®eeveWieW,
900, 1800 IegceeJe mes meceefcele oMee&³eWieW~,
HejeJele&ve Deewj ®e¬eÀer³e IegceeJe oesveeW meceefceefle keÀes Heefj®eeefuele keÀjWieW
HejeJele&ve Deewj ®e¬eÀer³e meceefceefle JemlegDeeW keÀes Hen®eeveWies~
meceefceefle
meceefceefle Deeke=Àefle Deewj HejeJele&ve meceefceefle kesÀ yeejs ceW He{s nQ~
meceefceefle Deeke=Àefle ke̳ee nw?
SkeÀ jsKee kesÀ GHeue#e ceW efpeme Deeke=Àefle ce WDeekeÀej Deewj iee$e kesÀ meblegefuele meceevegHeele nesles nw Jes meceefceefle Deeke=Àefleb keÀnueeleer nQ ~ keÀF& mceejkeÀ, FceejleW meesves kesÀ DeeYet<eCe neles nQ Jes meceefceefle efveceeCe& kesÀ keÀejCe megvoj nesleer nQ~ mepeerJe Deewj efvepeeaJe kesÀ ef®e$e Yeer meceefceefle oMee&les nQ~
jefJe Deewj ceesnve oesmle nQ~
jefJe : ke̳ee Òeke=Àefle keÀes Hemebo keÀjles nwb? ceesnve
ceesnve : neB jefJe, yengle kegÀí~
jefJe : ceQ ves SkeÀ HeÀesìes KeeR®ee nw~ osKee ~
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ceesnve : Jeen ! jefJe Ketyemetjle~ Heeveer ceW DeekeÀeMe efoKeeF& oslee nw~ megvoj~
Deeke=Àefle ces ke̳ee efJeMes<e efvejer#eCe keÀjles nes?
ke̳ee oesvees lejHeÀ meceeve ÒeefleeEyeye efoKeeF& oslee nw?
Fme efJeMes<elee keÀe veece meceeefceefle nw~
meceefceefle jsKeeieeCeerleer³e iegCeOece& pees Òeke=Àefle ces efoKeeF& oslee nw~
Òeke=Àefle ces keÀF& JemlegDeeW ceW meceefceefle nesleer nwb~
Fme ef®e$e keÀes osKee~
ke̳ee osKeves kesÀ efueS yengle megvoj nw?
megvoj ke̳eeW yevelee nw?
efmeceWìerjer keÀe DeLe& meye keÀe ceeHe nw~
meceefceefle kesÀ keÀejCe~
meceefceefle kesÀ YeeieeW keÀes Hen®eefveS~
³eefo SkeÀ jsKee Deeke=Àefle keÀes meceeve YeeieeW cesb efJeYeeefpele keÀjles nwb lees Gme Deeke=Àefle keÀes meceefceefle Deeke=Àefle keÀnles nw~ Deewj FmeceW HejeJele&ve meceefceefle nesleer nwb~ Fme jsKee keÀes meceefceefle De#e keÀnles nwb~
keÀe³e& keÀueeHe 1 : SkeÀ Meerì HesHej ceW m³eener [eefueS~ HesHej keÀes ceesef[S~ HesHej keÀes KeesefueS~ meefceefcele ke̳eeW Deewj kewÀmee yevee meesef®eS~
³eefo SkeÀ jsKee ef®e$e keÀes oes YeeieeW ceW efJeYeeefpele efkeÀ³ee lees Deeke=Àefle keÀes meceeefceefle keÀnles nwb~
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efvecveefueefKele Deeke=Àefle keÀe efvejer#eCe keÀerefpeS:
DebefkeÀle jsKee keÀes meceeefceefle jsKee keÀnles nwb~ Fmes meceefceefle De#e keÀnles nwb~
meceefceefle JemlegDeeW keÀes meblegueve Deewj mebHetCe&le: osleer nw~HejeJele&ve meceefceefle Deewj oHe&Ce meceefceefle mes mebyebefOele nw~
keÀe³e& keÀueeHe 2: efvecveefueefKele Deeke=Àefle³eeW ceW jsKee/ meceefceefle jsKeeDeeW keÀer j®evee keÀerefpeS~
ke̳ee DeeHe peeveles nw?
nceeje je<ì^er³e PeC[e meceefceefle keÀe megvoj GoenjCe nw~
peeefve³es~
jsKee pees Deeke=Àefle keÀes oes Yeeie ces efJeYeeefpele keÀjleer nw lees meceefceefle De#e keÀnles nwb~
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keÀe³e& keÀueeHe: efvel³ee keÀes Deesjerieceer Hemevo nw~ HesHej Meerì keÀes oes meceeve
i) ii)
YeeieeW ceW ceesef[S Deewj ef®e$e kesÀ pewmes keÀeefì³es~
Meerì Keesueves mes efvejer#eCe keÀerefpeS ~ ef®e$e keÀes osKekeÀj GÊej osves keÀer keÀesefMeMe keÀerefpeS~ neB efvejer#eCe mes Deeke=Àefle kesÀ oes meceeve YeeieeW ceW DeekeÀej Deewj ceeHe mes ceesæ[e ie³ee nw~
oyeer jsKee keÀe veece Hegve: ³eeo keÀerefpeS~oyeer jsKee keÀes meceefceefle keÀer jsKee ³ee meceefceefle keÀe De#e keÀnles nwb~
keÀe³e& keÀueeHe :ef®e$e kesÀ pewmes ef$eYegpeekeÀej HesHej Hej yeveeF&S efpemeceW PQ = PR
Q S R
P
peye Deeke=Àefle keÀe DeeOee Yeeie oHe&Ce keÀe ÒeefleeEyeye neslee nw lees Gmes HejeJele&ve meceefceefle keÀnles nw~
Meerì keÀes Ssmes ceesef[S efpememes PR, PQ mes efceueleer nw~ HesHej keÀes KeesefueS~ oHe&Ce ces ∆ PQS keÀer Òeefleefyecye ∆ PRS efoKeeF& oslee nw~
meceefÜyeeng ef$eYegpe ceW meceefceefle kesÀ De#e Hej, HejeJele&ve meceefceefle neslee nw~ Fme ef$eYegpe ceW meceefceefle SkeÀ De#e neslee nw~
S R
P Q
keÀe³e& keÀueeHe : Deeke=Àefle kesÀ pewmes PQRS Dee³eelekeÀej DeekeÀej kesÀ
HesHej ueerefpeS~ Meerì keÀes cees[keÀj PQ, SR Hej efceues~ KeesefueS~
FmeceW efkeÀleveer jsKeeSB osKe mekeÀles nQ~ oes jsKeeSB osKe mekeÀles nw~
Dee³ele ceW oes meceefceefle jsKeeSW nesleer nw~
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B
F
A
E
CD
keÀe³e& keÀueeHe : Meerì cesb HesHej meceyeeng ef$eYegpe keÀes ueerefpeS~ Òel³eskeÀ Yegpee keÀe ceO³e efyevog DeebefkeÀle keÀerefpeS~ Òel³eskeÀ efyevog efJe©× Meer<e& Hej peesef[S~
∆ABC meceyeeng ef$eYegpe nw~ ∆ADB keÀe oHe&Ce Òeefleefyecye ∆ADC nw~ AD, BE Deewj CF leerve De#e nQ~
SN
K M
R
P QL
keÀe³e& keÀueeHe : PQRS Jeie& ces meYeer Yegpee meceeve nw~ meceef-
ceefle keÀer jsKeeDeeW keÀer j®evee keÀerefpeS~
PR, NL meceefceefle kesÀ oes De#e nwb~ otmejs oes DebefkeÀle keÀerefpeS~
meceefceefle kesÀ efkeÀleves De#e nwb~ meceefceefle kesÀ ®eej De#e nwb~
FmeceW oes efJekeÀCe& Deewj efJe©× Yegpee kesÀ oes uecye efÜYeepekeÀ
nwb~ efJeefYevve Deeke=Àefle³eeW ceW efJeefYevve meceefceefle De#e KeeR®e
mekeÀles nQ~
met®evee : • meceyeeng ef$eYegpe ceW 3 meceefceefle keÀe jsKeeSB nQ~ • Jeie& ceW ®eej meceefceefle keÀe jsKeeSB nQ~ • SkeÀ efve³eefcele Heb®eceeYegpeeke=Àleer ceW HeeB®e meceefceefle keÀe jsKeeSB nw~ • SkeÀ efve³eefcele <eìYegpeeke=Àleer ces í: meceeefceefle keÀe jsKeeSB nw~efve³eefcele yengYegpeeke=Àefle meceefceefle keÀer jsKeeDeeW keÀer mebK³ee YegpeeDeeW keÀer mebK³ee kesÀ meceeve
nesleer nw~Je=Êe cesb efkeÀleves meceefceefle keÀer jsKeeBS nesleer nQ? osefKeS~
Je=Êe mes meceefceefle jsKeeDeeW keÀer mebK³ee %eele keÀj mekeÀles nwb? keÀ³eesb~
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meesef®eS!!! Je=Êe cesb Demeerefcele meceefceefle jsKeeSB nwb~ ke̳eeW?
DeY³eeme 8. 1
I. efvecveefueefKele ceW keÀewvemeer Deeke=Àefle cesb meceefceefle jsKeeSB nwb?
a) b) c) d)
I. efvecveefueefKele De#ejeW mes mebYeeefJele meceefceefle De#e keÀer j®evee keÀerefpeS~
a) A b) H c) XIII. Òel³eskeÀ Deeke=Àefle ces [eìdì jsKee ke̳ee meceefceefle keÀer jsKee nw? JeekeÌme ces neB ³ee vee efueefKeS~
GÊej keÀes peebef®eS~
a) b) c) d)
e) E f) g) h)
IV. efvecveefueefKele cesb meceefceefle keÀer mebK³ee efueefKeS~
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a) meceefÜyeeng ef$eYegpe b) meceyeeng ef$eYegpe c) SkeÀ Jeie& d) SkeÀ mece ®elegYe&gpe
e) Je=Êe f) meceeblej ®elegYe&gpe
g) efve³eefcele Heb®eceeYegpeeke=Àefle h) efve³eefcele <eìdYegpeeke=Àefle
V. efvecveefueefKele Debûespeer mlecYe mes oHeCe& mlecYe B ÒeefleeEyeye mes peesef[S
De#ej oHe&Ce& ÒeefleefyebyeB E
C B
D E
E C
R D
R
DeeJele&ve (Rotation) :
owefvekeÀ peerJeve ceW keÀF& JemlegSB nwb pees DeeJele&ve ®eueve ceW nesleer nQ~
GoenjCe : i) HebKeeW kesÀ HebKeeW keÀe DeeJele&ve
ii) Jeenve kesÀ Heefn³eeW keÀe DeeJele&ve
iii) Ieæ[er kesÀ metF&³eeW keÀe DeeJele&ve
DeeHeves Ieæ[er keÀer metF&³eeW keÀe ®eueve osKee nw~ Fme ÒekeÀej kesÀ ®eueve keÀes Ieæ[er IegceeJe ®eueve keÀnles nwb~ ®eueve peye Ieæ[er keÀer efJe©× efoMee cesb nes lees Gmes Ie[er keÀer efJe©× efoMee ceW IegceeJe keÀnles nwb~
IegceeJe ®eueve oes ÒekeÀej kesÀ nesles nwb~:
i) Ie[er IegceeJe ®eueve ii) Ie[er keÀer efJe©× efoMee ®eueve
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Ieæ[er keÀes osefKeS~ Ieæ[er kesÀ oes meF& efveef½ele efyevog ceO³e cesb Oetceleer nQ~ Fme efveef½ele efyevog keÀes DeJele&ve keÀe kesÀvê keÀnles nwb~
Deeke=Àefle³eeW keÀes efvejer#eCe keÀerefpeS~ ®e¬eÀ Deewj eEyeoer keÀes efvejer#eCe keÀerefpeS~
oesveeW Deeke=Àefle³eeW ceW DeekeÀej Deewj ®e¬eÀ keÀe ceeHe veneR yeoueles Deewj ®e¬eÀ efveef½ele efyevog ceW yeoueles nwb~ eEyeoer DeJele&ve metef®ele keÀjleer nQ ~
pee@ve veìjJeì ue[keÀe nw~ ef¬eÀmeceme ígÆer ceW GmekeÀer ceeB ves GmekesÀ efueS SkeÀ leeje Kejeroe~ Gmes Iegcee keÀj Jen Kesuee leejs keÀes Ssmes Gmeves Iegcee³ee~
A
B
C
D
D
A
B
CD
A
B
C
D
A
B
C
Gmeves mebYeJe veer³e efoMeeDeeW ceW Iegcee³ee ~ yeeo ces yee³esb mes oe³esb Deewj oe³es mes yee³esb~
Gmeves oesveesb lejerkeÀeW mes Ie[er keÀer IegceeJe Deewj Ie[er kesÀ efJe©× efoMee ceW Iegcee³ee~ DeJele&ve efJeOeeve ceW Jemleg meceleue kesÀ efveef½ele efyevog Hej keÀesF& efoMee cesb Iegceleer nw~
Fve efveef½ele efyeog keÀe DeeJele&ve keÀe kesÀvê ³ee DeJele&ve efyevog keÀnles nwb~
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keÀe³e& keÀueeHe :
keÀeF& yee s[& Hej e fHeve ueieeFS Deewj keÀe[& yee s[& keÀe s 90 90
9090
leerj mes DebefkeÀle keÀerefpeS~ efHeve DeJele&ve keÀe kesÀvê nw~ Deye leerj keÀes 900, 1800, 2700 ceW IegceeF&S Deewj yeeo cesb SkeÀ HetCe& Heefj¬eÀcee HetCe& keÀerefpeS~ Fmemes ke̳ee efveef½eCe keÀjles nQ?
3600 Hetje keÀjves Hej ³ee SkeÀ Hetje DeeJele&ve kesÀ yeeo eqmì^He DeHeves cetue mLeeve Hej Deelee nw~
meYeer Deeke=Àefle ³eeB 3600 Hetje keÀjves Hej ³ee SkeÀ Hetje DeJele&ve kesÀ yeeo ceW DeHeves cetue mLeeve Hej Deeles nw~
GHejeskeÌle keÀe³e&keÀueeHe mes Deewj ke̳ee mecePeles nQ?
nceW %eele neslee nw efkeÀ 41 (SkeÀ ®eewLeeF&) kesÀ IegceeJe ceW 900 DeeOees IegceeJe ceW 1800 leerve
IegceeJe ceW 2700 Deewj mebHetCe& IegceeJe ceW 3600 keÀesCe HetCe& keÀjlee nw~
efvecveefueefKele G$eeke=Àefle³eeW kesÀ DeeJe&leve keÀe efvejer#eCe keÀerefpeS~
1)
2)
3)
B
CD
A C
DA
B D
AB
C A
BC
D B
CD
A
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4)
5)
B
CD
A
DA
B CD
AB
C
BC
D AB
CD
A
keÀe³e& keÀueeHe : Dee³ele OABC keÀe ÒeefleeEyeye %eele keÀerefpeS peye GmekeÀe DeJele&ve keÀe keÀesCe nesiee : (i) 900 ii) 1800 iii) 2700 iv) 3600
C''
C'
C
BA
O A'
B'
B''
B'''
A''
C'''
A'''
®ejCe 1: Dee³ele OABC keÀer j®evee keÀerefpeS Deewj DeJele&ve efyevog `O' ceeve ueerefpeS~
®ejCe 2: Dee³ele keÀes 900 IegceeF³es~ OA keÀes ve³ee mLeeve OA' nQ~
®ejCe 3: OABC keÀer Òeefleefyecye OA1B1C nw 900 Hej Iegceelee nQ~
®ejCe 4 : Deewj 900 kesÀ IegceeJe ceW Dee³ele keÀe mLeeve OA"B"C" nesiee~
®ejCe 5 : efHeÀj 900 IegceeF³es (æcetue mLeeve mes 2700) Deewj Dee³ele keÀe mLeeve OA"B"C" nesiee~
®ejCe 6 : SkeÀ HetCe& IegceeJe 3600 ceW Dee³eele DeHeves cetue mLeeve Hej Deelee nw~
8.3 DeJele&ve meceefceefle
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GoenjCe 1 :S ke̳ee DeeHe FmekeÀe DeeJele&ve keÀj mekeÀles nQ?
Soes yeej DeJele&ve keÀjves mes Jen
cetue DeekeÀej ueslee nQ~Smet®evee : DeeJele&ve meceefceefle keÀes ncesMee SkeÀ meeHes#e DeeOeej Hej efJe®eej keÀjles nQ~
GoenjCe 2 : ³en ke̳ee nw?
³en Jee³eg ®ekeÌkeÀer (HeJeve ®ekeÌkeÀer)
³en Yeer DeHeves kesÀvê Hej Iegcelee nw~
GoenjCe 3 : meeceev³ele: HetÀue meyekeÀes Hemebo Deeles nQ~ ke̳ee ³en megboj veneR nw?
DeHes#ee HetÀue keÀes Ie[er keÀer efoMee ceW 900 kesÀ keÀesCe ceW Iegceevee ®eenleer nw~
HetÀue keÀes Ie[er keÀer efoMee ceW, 900, 1800, 2700 Deewj 3600 ceW Iegcee³ee~
GoenjCe 4 : Fme ef$eYegpeekeÀej Heìue Hej efJe®eej keÀerefpeS~ Ie[er keÀer efJe©× efoMee ceW 900 Hej
IegceeF³es~
900
1800
2700
3600
peye Ie[er keÀer efJe©× efoMee 900, 1800, 2700 Deewj 3600 Iegceeves Hej Deeke=Àefle Fve mLeeveeW
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keÀes ÒeeHle keÀjleer nw~ SkeÀ HetCe& IegceeJe mes cetue mLeeve ÒeeHle neslee nw~
GoenjCe 5 :
1200 2400 3600
³eneB Deeke=Àefle keÀes Ie[er keÀes IegceekeÀj 1200, 2400 Deewj 3600 Fve mLeeveeW keÀe efvejer#eCe keÀerefpeS~ Deeke=Àefle cetue Deeke=Àefle pewmeer nw~
Deeke=Àefle meceefceefle cesb nw~ IegceeJe kesÀvê efyevog mes neslee nw~ lees keÀesF& HeefjJele&ve veneR neslee Ssmeer
Deeke=Àefle³eeW keÀes DeeJele&ve meceefceefle ceW nw keÀnles nQ~
Jemleg keÀes Iegceeves mes cetue DeekeÀej ÒeeHle keÀjves kesÀ keÀesCe keÀes DeJele&ve keÀesCe keÀnles nQ~ GoenjCe
5 ceW, O DeJele&ve efyevog Deewj DeJele&ve keÀesCe 1200 nw~
SkeÀ Deeke=Àefle keÀes DeJele&ve meceefceefle ceW keÀnles nw peye Jen cetue Deeke=Àefle kesÀ pewmes efoKeeF&
osleer nw~ Deeke=Àefle 3600 Ietceves Hej cetue mLeeve ÒeeHle keÀjlee nw~
HesHej Jee³eg ®ekeÌkeÀer yeveeFS Deewj Deeke=Àefle ces DeJele&ve keÀer leguevee keÀerefpeS~ efvecveefueefKele
Deeke=Àefle keÀes mebHetCe& DeeJele&ve HetCe& keÀerefpeS~
OQ
RS
P
OR
SP
Q
OS
PQ
R
OP
QR
S
OQ
RS
P
peye nce Jemleg keÀes 3600 keÀesCe ceW Iegceeles nQ SkeÀ Heefj¬eÀcee MCe& nesleer nw~ Deeke=Àefle keÀes DeHeves
cetue mLeeve ÒeeHle keÀjves kesÀ 4 mLeeve nes DeJele&ve mececeefle keÀe ¬eÀce ®eej nwb~ SkeÀ mebHetCe& DeeJele&ve
ceW Jemleg meceeve nes lees Gmes DeJele&ve meceefceefle keÀe ®e¬eÀ keÀnles nwb~
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GHejeskeÌle GoenjCe 1 ceW DeeJele&ve keÀe ¬eÀce 2 nQ~ nceves Deeke=Àefle keÀes 900 mes Iegcee³ee Lee~
GoenjCe 2 ceW DeeJele&ve keÀe ¬eÀce 3 nw~ 1200 kesÀ 3 DeeJele&ve mes cetue mLeeve ÒeeHle keÀjles nQ~
GoenjCe 3 ces DeeJele&ve keÀe ¬eÀce 4 nw~ 4 mLeeve kesÀ yeeo cetue mLeeve ÒeeHle keÀjlee nw~
GoenjCe 4 ces DeeJele&ve keÀe ¬eÀce 4 nw~
GoenjCe 5 ces DeeJele&ve keÀe ¬eÀce 3 nw~
efvecveefueefKele Deeke=Àefle keÀes ueerefpeS~
P
O
RQ
S O
S
QP
R O
R
PS
Q O
Q
SR
P O
P
RQ
S
³en Deeke=Àefle DeHeves cetue DeekeÀej ceW Deeleer nw peye Jen efyevog 0 Hej SkeÀ Heefj¬eÀcee HetCe& keÀjlee
nw~ peye Gmes 900, 1800, 2700 Deewj 3600 Hej Iegceeves Hej DeHeves cetue DeekeÀej ceW ®eej yeej
efoKeeF& oslee nw~ 4 nw~
meceeblej ®elegYe&gpe PQRS ueerefpeS~1800 DeJele&ve ceW Iegceeves Hej efvecveefueefKele Deeke=Àefle ÒeeHle
nesleer nw~S R
QP
S
R Q
P S R
QP
• meceeblej ®elegYe&gpe keÀe DeJele&ve meceefceefle ¬eÀceebkeÀ 2 nw~
meesef®eS! Je=Êe keÀer DeJele&ve meceefceefle ¬eÀceebkeÀ ke̳ee nw?
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efve³eefcele Heb®eceeYegpeeke=Àefle ABCDE keÀe Meerì ueerefpeS~ kesÀvê Hej efHeve ueieekeÀj IegceefFS~ SkeÀ DeeJevele& HetCe& keÀjveW ceW efkeÀleveer yeej cetue DeekeÀej ceW efoKeeF& osleer O³eeve oerSefpe nw~
A B
CE
D
B C
DA
E
C D
EB
A
D E
AC
B
E A
BD
C
A B
CE
D
• efve³eefcele Heb®eceeYegpeeke=Àefle keÀe DeJele&ve meceefceefle keÀe ¬eÀceebkeÀ 5 nw~
∴ DeeJele&ve meceefceefle keÀe ¬eÀce 3600
DeeJele&ve keÀesCe3600
A0=
SkeÀ DeeJele&ve HetCe& keÀjveW ceW efkeÀleveer yeej cetue DeekeÀej ceW efoKeeF& osleer O³eeve oerefpeS~keÀe³e& keÀueeHe : HesHej keÀes keÀe[& yees[& Hej ueieeFS~ Òel³eskeÀ Yegpee 6 cm keÀer
meceyeeng ef$eYegpe keÀer j®evee keÀerefpeS~ YegpeeDeeW kesÀ ceO³e efyevogDeeW keÀes DebefkeÀle
keÀjkesÀ Meer<e& mes peesef[S O ÒeeHle nesles nw~ HeejoMe&keÀ HesHej ueskeÀj ef$eYegpe
yeveeFS Dee wj ceO³e e fyevogDee W keÀe s Mee r<e X me s pee s[keÀj O e fyevog
ÒeeHle keÀerefpeS~ veS ef$eYegpe keÀes jefKeS leeefkeÀ O, O1 mes meceefceefuele nw~
Deewj Yegpee³eW HejmHej cesue Keeles nQ~ HeejoMe&keÀ ef$eYegpe keÀes Ie[er keÀer efoMee ceW IegceekeÀj HeÊee
ueieeF&S SkeÀ Heefj¬eÀcee ceW efkeÀleveer yeej Jen DeHeves cetue DeekeÀej keÀes meceefceefuele neslee nw Jen
3 yeej efceuelee nw~ FmeefueS GmekeÀer DeeJele&ve meceefceefle ¬eÀceebkeÀ 3 nw Deewj DeeJe&leve meceefceefle
keÀesCe 1200 nw~
meceeblej ®elegYe&gpe Deewj mece®elegYe&gpe kesÀ efueS keÀe³e& keÀueeHe keÀer keÀesefMeMe keÀerefpeS~
DeJele&ve meceefceefle ¬eÀceebkeÀ ceeuetce nesves mes DeJele&ve keÀesCe efvecve met$e mes %eele keÀj mekeÀles nw~
DeJele&ve keÀesCe = 3600
DeJele&ve ¬eÀceebkeÀ
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196
= 3
360c
= 1200
DeJele&ve ¬eÀce = 3600
DeJele&ve keÀesCe
120360
30
0
=
=
Òel³eskeÀ DeeJele&ve Deeke=Àefle keÀe cetue mLeeve 3600 kesÀ yeeo ÒeeHle neslee nw~ meceefceefle keÀe
¬eÀce~ nw~
ke̳ee DeeHe peeveles nwb? ³en ke̳ee nw?³en Je=Êeer³e ef[mkeÀ nw~ kesÀvê keÀes keÀerue mes ueieeF& ef[mkeÀ keÀes Deemeeveer mes Iegcee mekeÀles nQ~ ³en DeJele&ve meceefceefle keÀe GoenjCe nw~FmekeÀe ¬eÀce ke̳ee nw?neB yees[& keÀer keÀeueer ³ee meHesÀo ef$eYegpe keÀer mebK³ee nw~
ef[mkeÀ keÀe ¬eÀce 10 nw ~
Deeke=Àefle pees meceeve efoKeeF& osleer nw cetue Deeke=Àefle kesÀ pewmes nesleer nw SkeÀ mes DeefOekeÀ mecHetCe& DeJele&ve keÀes DeJele&ve meceefceefle keÀe efJeefYevve ¬eÀce keÀnles nwb~
• Deeke=Àefle DeeJele&ve meceefceefle nesleer nw peye DeeJele&ve keÀesCe 1800 kesÀ meceeve ³ee keÀce nw~
• meYeer Deeke=Àefle 3600 DeeJele&ve keÀjkesÀ cetue mLeeve ÒeeHle keÀjles nwb~
• meYeer Deeke=Àefle keÀer keÀce mes keÀce 1 DeeJele&ve meceefceefle ¬eÀceebkeÀ neslee nw~
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keÀe³e& keÀueeHe :
• Meerì HesHej kesÀ 2 Jeie& meceeve DeekeÀej kesÀ keÀeefìS~
• efJekeÀCe& yeveekeÀj, kesÀvê keÀes DebefkeÀle keÀerefpeS~
• efJekeÀCe& peneB Òeefle®ísefole nesles nwb JeneB kesÀvê nw~
• Meerì keÀes SkeÀ kesÀ THej jefKeS peneB oes Deeke=Àefle³eeB meceefceefuele nes~
• kesÀvê efyevog Hej efHeve ueieekeÀj oesveeW Meerì efceueeF³es~
• THej kesÀ Meerì keÀes Deewj osefKeS efkeÀ veer®es keÀer Deeke=Àefle efceueleer nw~
• DeeJele&ve keÀe keÀesCe Deewj ¬eÀce ke̳ee nw? ke̳ee DeeJele&ve keÀesCe efoMee yeoueves Hej yeouelee
nw?
GoenjCe 1 : Deeke=Àefle keÀes 900, 1200, 1800, O Hej Iegceeves Hej~ keÀewve mes keÀesCe
Hej Jen DeHeves cetue DeekeÀej pewmes efoKesiee? DeJele&ve meceefceefle keÀe ¬eÀce Yeer %eele keÀerefpeS~
meceeOeeve: keÀes O Hej 900 Hej Oegceeves Hej ÒeeCle neslee nw~
1200 DeJele&ve mes (meceefceefle veneR nw), ÒeeHle neslee nw
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1800 DeJele&ve mes ÒeeHle neslee nw
DeJele&ve meceefceefle peye1800 Iegceeves Hej ÒeeHle neslee nw~
∴ DeJele&ve meceefceefle keÀe ¬eÀce 180360 2
cc= =
GoenjCe 2 : meceyeeng ef$eYegpe keÀe DeJele&ve meceefceefle ¬eÀce %eele keÀerefpeS~
meceeOeeve : meceyeeng ef$eYegpe kesÀvêg efyevogO, Hej Iegceeves mes cetue ©He
O
A
CB
1200 Hej ÒeeHle keÀjles nQ~
∴ DeJele&ve meceefceefle keÀe ¬eÀce 120360 3
cc= =
kegÀí Deeke=Àefle³eeW ceW HejeJele&ve Deewj DeJele&ve meceefceefle nesles nwb~
ef®e$e A
ef®e$e B
Deeke=Àefle B Deeke=Àefle A keÀe HejeJele&ve nw~Deeke=Àefle keÀe DeJele&ve, DeJele&ve efyevog mes 900 keÀesCe efvecveefueefKele efoKee³es pewmes~
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Jeie& ces oesvees HejeJele&ve Deewj DeJele&ve meceefceefle nQ~
efvecveefueefKele Deeke=Àefle³eeW ceW Deeke=Àefle 1 HejeJele&ve Hen®eeefveS~
ef®e$e 1 (a) (b) (c)
meceyeeng ef$eYegpe oesvees HejeJele&ve Deewj DeJele&ve meceefceefle neslee nw~
efve³eefcele Heb®eceeYegpeeke=Àefle keÀes ueerefpeS :
ef®e$e A ef®e$e B
Deeke=Àefle B Deeke=Àefle A keÀe HejeJele&ve nw~ FmekeÀes 720 DeJele&ve
keÀerefpeS~ efvecveeefueefKele Deeke=Àefle³eeB ÒeeHle nesles nQ~
Heeb®eJeeb Deeke=Àefle 4 DeJele&ve keÀjkesÀ cetue Deeke=Àefle ÒeeHle keÀjles nQ~
ef®e$e (1) ef®e$e (2) ef®e$e (3) ef®e$e (4) ef®e$e (5)
efve³eefcele Heb®eceeYegpeeke=Àefle ceW oesveeW HejeJele&ve Deewj DeJele&ve meceefceefle nesles nQ~
keÀesefMeMe keÀerefpeS: HejeJele&ve Deewj DeJele&ve meceefceefle ceW nesvesJeeues Debûespeer De#ejes keÀes Hen®eefveS~
H, N, O, S, X Deewj Z Deeke=Àefle³eeW keÀes keÀeBì keÀj osefKe³es~
mejCeer keÀes oesefKe³es pees yengle peevekeÀejer oslee nw~
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DeekeÀej yengYegpeeke=Àefle Yegpee meceefceefle keÀer jsKeeSB
DeeJele&ve keÀesCe
DeJele&ve meceefceefle
meceyeeng ef$eYegpe 3 3 1200 3
Dee³ele 4 4 900 4
efve³eefcele Heb®eceeYegpee
ke=Àefle5 5 720 5
efve³eefcele <eìdYegpee ke=Àefle
6 6 600 6
mejCeer mes efvecveefueefKele met®evee ÒeeHle nesleer nw~YegpeeDeeW keÀer mebK³ee, meceefceefle jsKeeDeeW keÀer mebK³ee Deewj DeJele&ve meceefceefle keÀe ¬eÀce Òel³eskeÀ
yengYegpeeke=Àefle ceW meceeve nw~
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keÀesefMeMe keÀerefpeS :
peeFvì& ®e¬eÀ keÀe ¬eÀce %eele keÀerefpeS~ Deesn! cesjer efieveleer keÀes ítì ie³eer.....
keÀe³e& keÀueeHe 12 : keÀcejs keÀer Jemleg keÀes DebefkeÀle keÀerefpeS ~ pees mececeefle nw Deewj oesmle keÀes DeeHemes ®egveer Jemleg keÀes Devegceeve ueieekeÀj peeveves keÀefnS~ ome efJekeÀuHeeW ceW Òel³eskeÀ efJekeÀuHe kesÀ yeeo Jemleg yeesueves oerefpeS~ ³eefo oesmle mener Jemleg yeesueW lees þerkeÀ veneR lees otmejesb keÀer yeejer nw~
³eeo jefKeS:
• DeJele&ve Ie[er kesÀ IegceeJe ³ee Ieæ[er keÀer efJe©× efoMee kesÀ IegceeJe ceW nes mekeÀleer nQ~
• Òel³eskeÀ Deeke=Àefle cesb DeJele&ve meceefceefle keÀe ¬eÀce keÀce mes SkeÀ 1 nes~
• efyevog efpemeHej Deeke=Àefle keÀe DeJele&ve neslee nw Gmes DeJele&ve efyevog keÀnles nwb~
• DeJele&ve Hej Iegceves kesÀ keÀesCe keÀes DeJele&ve keÀesCe keÀnles nwb~
• SkeÀ mecHetCe& DeJele&ve ceW Deeke=Àefle efpeleveer yeej cetue Deeke=Àefle mes efceueleer nw Gmes DeJele&ve keÀe
¬eÀce keÀnles nQ~
DeY³eeme 8.2
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I 1) S Deewj I De#ej keÀes ueerefpeS~ meceefceefle efyevog DebefkeÀle keÀerefpeS~ DeJele&ve meceefceefle ¬eÀce Yeer %eele keÀerefpeS~
2) leerve Debûespeer De#ej efueefKeS efpemeceW HejeJele&ve meceefceefle nesleer nw Hejvleg DeeJele&ve meceefceefle veneR nw~
3) leerve Debûespeer De#ej efueefKeS~ efpemeceW oesveeW HejeJele&ve meceefceefle Deewj DeJele&ve meceefceefle nw~II 1) meereEueie HeBKes keÀe DeJele&ve meceeefceefle keÀe ¬eÀce ke̳ee nw? efpemeceW
(i) 3 yues[ (ii) 4 ®eej yues[ nesles nQ~
2) kegÀí mebK³ee efueefKeS efpemeceW DeeJele&ve meceefceefle nesleer nw~
3) Fme leeje ceW meHesÀo Deewj keÀeues meceyeeng ef$eYegpe yeves nw~
GvekeÀe DeeJele&ve meceefceefle ¬eÀS efueefKeS~III 1) 2 meWceer Yegpee keÀe ABC meceyeeng ef$eYegpe keÀer j®evee keÀerefpeS~ B eEyeog Hej ef$eYegpe keÀes 900
Deewj 1800 kesÀ keÀesCe mes IegceeF&S~ Deewj ÒeefleeEyeye %eele keÀerefpeS~
2) meceefÜyeeng ef$eYegpe OAB ceW OA=OB nw GmekeÀe efveef½ele efyevog 0 nw~ 900,1800 Deewj 270
kesÀ keÀesCe mes IegceeF&S~
Deeke=Àefle³eeW Üeje DeeJele&ve efoKeeF&S~
IV 9) efvecveefueefKeleeW ceW HejeJele&ve meceefceefle Deewj DeJele&ve meceefceefle Hen®eeefveS~
E E E EE E(a) (b) (c)
(a) (b) (c)
10) DeJele&ve meceefceefle keÀe ¬eÀce %eele keÀerefpeS :
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(i) Dee³ele (ii) mece®elegYe&gpe kesÀ
11) Keeueer mLeeve Yeefj³es~-
De#ej HejeJele&ve meceefceefle neB~ venerb
meceefceefle keÀer jsKeeDeeW keÀer
mebK³ee
DeJele&ve meceefceefle neB~ vener
DeJele&ve meceeefceefle keÀe ¬eÀce
I
B
O
H
E
G
Z
S
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IeìkeÀ - 1 HetCee¥keÀDeY³eeme 1.1 I. I. 1) 13, 2) -5, 3) 1 4) - 9
DeY³eeme 1.2
1. x -3 6 11 -57 -21 42 77 -354 -12 24 44 -20-8 +24 -48 -88 +40-2 +6 -12 -22 +1012 -36 72 132 -60-9 +27 -54 -99 +45-14 +42 -84 -154 +70
2.
85
-5-6
7-2
1-25
+30+10
-5
-40
-35
DeY³eeme 1.3 I.1) -15, 2) -24, 3) -238, 4) 36, 5) -126, 6) 24, 7) -60
II. 1)
2) 3)
4)
III. 1) b) A cm - (2 × 6) cm IV. b) (` 62000 - 8 × 7500)
DeY³eeme 1.4 I.
1) = + 4, 2) -17, 3) -11, 4) 4, 5) 0
II. 1) 1, 2) 1, 3) -2, 4) -12, 5) -30
DeY³eeme 1.5 I. 1) -10, 2) -9, 3) -18
4) 0, 5) -2, 6) 19
DeY³eeme 1.6 1) 125, 2) 16 0C, 260C, 280C,
3) ` 11,750, 4) ` 126,
5) ` 10880, 6) 177
III.
HetCee¥keÀ YeeieHeÀue
+ 24 ÷ + 12 + 2
+ 24 ÷ -12 - 2
- 24 ÷ +12 - 2
- 24 ÷ -12 + 2
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IeìkeÀ -2 efYeVe
DeY³eeme 2.1 : I. mece efYeVe : ,127
203 efJe<ece efYeVe : ,16
1845 efce$e efYeVe : ,2 6
1 6 92
II 1) ,126
189 , 2) ,10
81512 , 3) ,20
143021 III 1) 3
2 , 2) 53 , 3) 2
1
IV 1) 1 87 , 2) 212
1 , 3) 24 41 , V 1) 4
15 , 2) 217 , 3) 6
29
VI A. 1) 1229 2) 20
93 B. 1) 51 2) 1 30
23
DeY³eeme 2.2 : I 1) 41 5# , 2) 2
1 3# , 3) 83 2# , 4) 3
2 4#
II A. 1) 74 , 2) 5
12 , 3) 910 , 4) 4
21 , 5) 29 , 6) 3
20 , 7) 9, 8) 143 ,
B. 1) 11 41 , 2) 28,3) 26 5
2 , 4) 22 21 , 5) 13 5
4 , 6) 2 53 , 7) 30, 8) 111
III A. 1) 4, 2) 6, 3) 2, 4) 8, 5) 15, 6) 14, 7) 521 , 8) 9
20
B. 1) 30, 2) a) 15, b) 24, c) 1
DeY³eeme 2.3 : I 1) 95 , 2) 32
21 , 3) 4528 4) 3 5) 27
7 , 6) 1011 7) 35
36 8) 2413
II 1) 1 2013 , 2) 315
1 , 3) 4 61 , 4) 1 7
6 , 5) 11 21 , 6) 7 8
7 , 7) 17 97 8) 20
III 1) 2) 3) 4)
IV 1) 70 km 2) ` 414, 3) h 4) ` , 5) km
DeY³eeme 2.4 : I. 1) 18, 2) 352 , 3) 90, 4) 3
140 , 5) 6 73 , 6) 3 23
1
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II. 1) 25 , 2) 7
9 , 3) 12, 4) 136 , 5) 9
1 , 6) 43 III. 1) 15
2 2) 247 3) 91
6 4) 209
5) 2811 6) 4
1 IV. 1) 815 2) 12
35 3) 185 4) 15
8 5) 2 41 6) 2 42
31
V. 1) ` 1 43 2) 85 3) ` 9 2
1 4) 27
DeY³eeme 2.5 : I 1) 43 2) 310
2 3) 1 4780 4) 2 20
13
II 1) m6 201 2) kg4 4
1 3) 5
IeìkeÀ -3 Heefjces³e mebK³eeSB
DeY³eeme 3.1 : I. 1) mener 2) ieuele 3) mener 4) ieuele 5) mener 6) mener 7) ieuele
II 1) 57 2) 5
4 3) 83- 4) 7
4 III 1) , , ,184
276
368
4510- - - - 2) , , ,20
6309
4012
5015
3) , , ,108
1512
2016
2520
- - - - 4) , , ,192160
288240
384320
480400 IV , , ,6
7710
97
611
--
--
, ,54
35
152-
-- V 1) 16 2) -20 3) 7 4) 3 5) -28 6) 30
DeY³eeme 3.2 : I 1) < 2) < 3) > 4) < II 1) 43 2) 5
6 3) 38
DeY³eeme 3.3 :I. 1) 1011 2) 9
8 3) 307 4) 42
107 II. 1) 83- 2) -1 3) 3
1- 4) 81-
III. 1) 8 kg125 2) 1 kg20
11 3) m10582 4) 1 l15
14
DeY³eeme 3.4 : I. 1) 83 2) 48
5- 3) 101- 4) 9
1 5) 61 6) 1 II. 1) 27
7 2) 43 3) 5
6 4)
78 III. 1) 5
3 2) 49- 3) 10 4) 2
7- 5) 51 6) 0 IV. 1) 9
10 2) 56- 3) 13
18- 4) 57-
V. 9 VI. 121 m VII. 8 54 sq.m VIII. 104
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DeY³eeme 3.5 :I. 1) 0.8 2) 0.375 3) 0.3125 4) 0.2142857142...
II. Deleble - ,85
61 meele ,12
192- 3)
DeY³eeme 3.6 :I. 1) 34.4 2) 7.5 3) 36.945 4) 3.328 5) 0.04368 6) 1.68
II. 1) 0.07 2) 0.012 3) 0.497 4) 27 5) 2.2 6) 430 III. 1)` 717.86
2) 120.06 km 3) 10.2l 4) ` 20.6
DeY³eeme 3.7 :
1) a) 2510 g b) 0.0725 g
2) a) 0.625 kg b) 0.0108 kg
3) a) 1.45 km b) 0.173 km c) 0.2135 km
4) a) 2700 m b) 75.25 m c) 1580 m
5) a) 1250 cm b) 470000 cm
6) 11.59 km
IeìkeÀ - 4 yeerpeer³e J³ebpekeÀDeY³eeme 4.1 1) x 2) a 3) xy 4) x2y 5) a2b 6) m2n2 7) xyz 8) m2np
II 1) 2 2) -5 3) 1 4) -8 5) 9 6) 7) 0.5 8) 0.008
DeY³eeme 4.2 I De#ejer³e mebK³eeSB :: 1) 8 + 5 - 3, 2) 3x - 8, 5) (20 × 7) - (5 × 10) - 45
3) (7 × 6) -4m 4) 3p + 4q 6) 2y + 6 - 4z
II a) 3x, + 4y b) 2pq, - 8qr c) 3x2, -3x, +z
d) ab, + bc, -ca e) 9m, + 6n f) 6, -3xy, +x, -y
b)1) xy, 2, 5 2) x2, 1, xy, 4 3) p, 3, y2, -5 4) ab, 2, bc, 1, ca, 1
III 1) x + 8 2) y - 7 3) p 12 = 12p 4) 5) 4x + 3y 6) 10 - 5y
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7) pq + 3x 8) 5m - 3l 9) 10y + 15 10)
IV 1) 3 2) -4 3) 4) 10.5
5) 168 6)
V 1) m 2) -xy 3) pqr 4) c 5) mn 6) x2y
VI 1) 27 2) 27z 3) xy 4) xyz
DeY³eeme 4.3 I meHeÀHeo: 2xy, xyz, 3xp ÷ q efÜHeo: 2x + y, 3x2 + 5x, ab - bc
ef$eHeo: 5 + 6a + 4b, , a2 - 3ab + c, x2 + 4 - 3x
II x3 + x - 2 + 1 m3 + 2m2 + 3m - 4, x+2y3
III mepeeefle Heo (L) efJepeeefle Heo (U)1. 3x, 5x, 8x
3. -8p2, 6p2, 10p2
5. 2ab, 6ba, 8ab
2. x3, - 3x2, 8x
4.m 3a2b, -2ab2,
6. 7a2b2-a3, 2a2, -8a
IV 1) y3, -7y3, 23y3, -y3 2) 7ab, -8ba, -3ab, 4ab 3) 7p, 2p, 3p
V 1) 3 2) 3) 0.3 4) 24 5) -18 6) VI 1) 3a 2) 4a3) p2y 4) 10
5) -p2 6) -1 VII 1) 2, 2) 3, 3) 3, 4) 1, 5) 3,6) 3
VIII IeeleebkeÀ ³egkeÌle mebK³eex 5
ab 613p 9-y 10
xyz 70.59 20
IX a) p5 b) m4 c) (ab)6
d) z8 e) (abc)3 f) b10
DeY³eeme 4.4 I 1) 41 a 2) 12 p2 3) -1y + 3x 4) 6a2 - b2 + 12a
II 1) 5x 2) 13-a2 3) 9xy 4) 7x2 + 6y - z III 1) 8x + 11y 2) 17a - 10b
3) -9xy-z 4) 17a2bc + 7ab2c IV 1) 6x - 3y + 3z 2) 7p2 + q2 + 11r2
3) 6a + 4b + 3c 4) 2x2y - 2y2z + 13z2x V 1) 12mn - 4p 2) 8x - 2y - 4z
3) 10a2 + a + 3 VI 10x2 + 10y + y2 + 5z + 11
VII 1) 27 2) Z 3) xy 4) xyz
DeY³eeme 4.5 I 1) 7x 2) 8a2b 3) 16ab 4) -22x2y II 1) 5x + 2y 2) 4a + 2b
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3) -4m2n + 5mn2 4) -11pqr + 11q + 4 III 1) 4x + 4y 2) -6c + 4d
3) x-6y - 14 5) 6m2n2 + 2xy - 10
DeY³eeme 4.6 I 1) 280 xyz 2) 40 pqr 3) -144 mnp 4) -18abc 5) -60x3
6) +18 p6 II 1) 84 x2y2z2 2) + 210 a2b2c2 3) -12x4y3z 4) -24a2b3c4
III1) 12xz + 14yz 2) -15p2 + 25pq 3) -36xz - 2yz 4) 12a2b2c + 18ab2c2
IV 1) x2 + 12x + 32 2) 5n2 - 13n - 6 3) 6a2 + ab - b2 4) 25x2 -
4p2 5) 2x2 - 13x + 21
IeìkeÀ - 5 mebueive keÀesCeDeY³eeme 5.11. efvecve Deeke=Àefle osKekeÀj mejCeer YeefjS~ :
ef®e$e keÀesCe Yegpe (yeeng) mebueive keÀesCe nw ³ee veneR
i OA, OB, OC,
O OB neB
ii QP,QR,QS,QJ
Q QRQS
veneR PQR or RQS neBRQS or SQJ
iii LK, LM, MN - LM veneRiv ED, EF, EG E DE veneR
v OE, OG, FG, GH
- - veneR
vi AB, OA,OC, OD
O OC neB
vii OH, OG, OJ O m e e c e e v ³ e Meer<e&
OG m e e c e e v ³ e Meer<e
neB
2. oÊe Deeke=Àefle ceW efvecveefueefKele mebueive keÀesCeeW keÀer pees[er keÀewve meer nw meeceev³e Meer<e& Deewj meeceev³e Yegpee yeleeFS :
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keÀesCe Meer<e& keÀe veece
meeceev³e Yegpe keÀe veece meeceev³e Meer<e& neB ³ee vener
meeceev³e Yegpee neB ³ee veneR
mebueive keÀesCeeW keÀer pees[er
,AOBBOD
0 , ,OA OB OD neB neBOB
neB
,AODBOC
0 , , ,OA OD OB OC neB veneR veneR
,AOCBOC
0 , ,OA OB OC neB neBOC
neB
,BOCAOB
0 , ,OA OB OC neB neBOB
neB
,AODAOB
0 , ,OA OB OD neB neBOB
neB
3. Deveg©He keÀesCe : i) ªÀÄvÀÄÛDOC COE Deewj ªÀÄvÀÄÛDOC COE ii) ªÀÄvÀÄÛPOR POQ Deewj ªÀÄvÀÄÛPOR POQ
iv) ªÀÄvÀÄÛJOK KOL Deewj ªÀÄvÀÄÛJOK KOL vii) ªÀÄvÀÄÛQOM MON Deewj ªÀÄvÀÄÛQOM MON
DeY³eeme 5.2 I. 1) 500, 2) 700, 3) 350 4) 520 II. a) 680, b) 60 c) 69.50 d)
49.50 III. 22.50 Deewj 67.50
DeY³eeme 5.3 1.a) AOD BOD AOB+ = b) AOE EOB AOB+ =
c) AOC COB AOB+ = 2. (i), (ii), (iii), (iv) mebHetCe& keÀesCe
3. a) 850 b) 700 c) 59.50 d) 54.50 4. 650 5. 360 Deewj 1440 6. i) 1250,
ii) 350 iii) 900 iv) 900
DeY³eeme 5.4 1) mebHetjkeÀ keÀesCe vener 2) (ii), (iv), (v) Deewj (vi) jsKeer³e pees[er keÀesCe
3) a) DeefOekeÀ keÀesCe b) ueIeg keÀesCe c) uebye keÀesCe
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DeY³eeme 5.51)
,,
AODBODBOC
12060120
0
0
0
===
2) ,,
AOEBOCEOD
405090
0
0
0
===
3) BOCBOD
15030
0
0
==
DeY³eeme 5.6 I a) DeefOekeÀ keÀesCe b) ueIeg keÀesCe c) uebye keÀesCe
II i) 1050, ii) 400 iii) 32.50 iv) 350
III
90454545
AOC BODAOD COB
AOR SOBDOR COSPOC DOQAOP BOQ
90
45
0
0
0
0
0
0
`
= == =
= == == == =
IV x = 450, z = 67.5, y = 1350
V i) &AOD AOC
ii) a) &AOE BOE b) &COA AOD
iii) &AOC BOD iv) &AOC COE
iv) 1) 900 2) 1800 3) mebHetjkeÀ keÀesCe
4) jsKeer³e pees[er keÀesCe
5) DeefOekeÀ keÀesCe
IeìkeÀ - 6 pees[er jsKeeSBDeY³eeme 6.1 1.a) RNK b) QMN c) MNR d) SNK e) MNR f) QMN
2. a) EIJEIJ AIKEIJ IKLIKL CKFJIK BJLAIE JIK
70707070110110
0
0
0
0
0
0
== == == == == =
Meer<ee&efYecegKe keÀesCeDeveg©He keÀesCeMeer<ee&efYecegKe keÀesCeDeveg©He keÀesCeMeer<ee&efYecegKe keÀesCe
b) Meer<ee&efYecegKe keÀesCePeveg©He keÀesCe
Deveg©He keÀesCe
GIB AIJAIJ CJKDJKDJK FKH
6060
180 60 120120
0
0
0 0 0
0
= == == - == =
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c) Meer<ee&efYecegKe keÀesCeDeveg©He keÀesCe
EHB GHFGHF ACD
5050
0
0
= == =
3.
He³ee&³e keÀesCeDeveg©He keÀesCe
PARPST QPSPST QRS
50130130
0
0
0
== == =
4. , ,BAC ABC ACB80 45 550 0 0= = =
5. a) AB # CD b) AB CD 6. 70DGCGEF 70
0
0
==
7. ƒçºð® a b c d(i) 800 500 1000 1300
(ii) 700 800 700 -(iii) 500 1300 1300 -(iv) 400 2600 600 -(v) 1300 700 1300 1100
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IeìkeÀ - 7 ef$eYegpeDeY³eeme 7.2 1. a) x = 500 b) x = 800 c) x = 200
2. 1) C = 500 b) B = 450 c) C = 500 d) C = 340
3. 1) 500 + 600 + 700 = 1800 neB 2) 600 + 700 + 800 = 2100 veneR
3) 650 + 750 + 550 = 19500 veneR 4) 560 + 640 + 600 = 1800 neB
5) 570 + 640 + 790 = 2000 veneR
4. C = 700 5. C = 700 6. x = 550 7. x = 600 8. C = 950
DeY³eeme 7.3 I. i) ii) iii) Çò»Îðæ® iv) Çò»Îðæ®
II. a) mener b) mener c) mener
DeY³eeme 7.31. a) x = 800, y = 1000 b) y = 700, x = 1100
c) y = 350, x = 750 d) x = 650, y = 1150
2. ∠ACB = 600 ∠ACD = 1200 3. ∠QRP = 500, ∠RPQ = 800
4. ∠KLN = 1150, ∠NLM = 650, ∠LMN = 400 5. ∠ACD = 1050
DeY³eeme 7.5 1. AC = 10, 2. AC = 13 cms, 3. AB = 9 m 4. b) c)
DeO³ee³e - 8 meceefceefleDeY³eeme 8.1 I. i iii. iv. iv.
III a) b) c) d) e) f) g)
veneR neB neB neB neB veneR neB
h) i) j) k) l)
veneR veneR neB neB neB
IV a)1 b) 3 c) 4 d) 4 e) neB f) 4 g) 4 h) 5 i) 6
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C
D
E
R
1 - 2i) C -
ii) D -
iii) E -
iv) R -
DeY³eeme 8.2 I. 1) SI 2.2
2) M W E C U 3) O N 2
II. 1) 1200, 3 2) 900, 4
III 1) 0.8169 2) 3
IV)
De#ej HejeJele&ve meceefceefle neB ³ee veneR
meceefceefle jsKeeDeeW keÀes
mebK³ee
DeeJele&ve meceefceefle neB ³ee veneR
DeeJele&ve meceefceefle keÀe ¬eÀce
I neB 1 neB 2
B neB 1 neB 1
O neB yengle neB yengle
H neB 2 neB 2
E neB 1 neB 1
G veneR - neB 1
Z neB - neB 2
S neB - neB 2
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