J{UV ©KTBS textbooks/class7/7th-marathi-maths-1.pdf · making education relevant to the present and future needs. softening the subject boundaries-integrated knowledge and the joy

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J{UVMATHEMATICS

(Revised)

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Seventh Standard‘amR>r ‘mÜ¶‘

Marathi Medium

^mJ - 1PART - 1

KARNATAKA TEXT BOOK SOCIETY (R).,

100 Feet Ring Road, Banashankari 3rd Stage,Bengaluru - 560 085

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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 Letter

Dear 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 signifi cance 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, fi gures, 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 far as possible. Our Committee is 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 scrutinize 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 personel 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 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 Committee

Chair 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, 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, Hospet Ballari

District- 583218.Sri Pramod G. Kulkarni V. B. Darbar P. B. 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 Hospet. Ballary 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 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, Education RIE

Mysuru.Dr Prasad S. V. Lecturer RIE Mysuru.Dr Sharad Sure Assistant Proffessor Azim Premji university Bengalure -10 PES

School of Engineering Campus Komappa Agra Hasa.Translation Committee :Sri P. C. Halgekar Assistant Teacher, Rangubai Bhosale High School, Belagavi

Sri R. G. Kulkarni Assistant Teacher, Govt. M.H.P.S. Gunji, Tq: Khanapur, Dist: Belagavi.

Sri B. G. Vaghamare Assistant Teacher, Govt. M.H.P.S. Nilawade, Tq: Khanapur, Dist: Belagavi.

Chief - Coordinators :Sri G. S. Mudambaditaya Co-ordinator, Curriculum revision and Text Book Preparation.

Guidance :Sri Nagendra Kumar Managing Director, Karanataka, Text Book Society.Sri Panduranga DDPI (incharge), Karnataka, Text Book Society.Programme Coordinators :

Smt Uma Devi Assistant Director, Karanataka Text Book Society.Smt Vijaya Kulkarni Assistant Director, Karanataka Text Book Society.Smt Prema B. R. Technical Assistant, Karanataka 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 Revision Committee Bengaluru.

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Text Books Revision CommitteeChairman-in-chief.Prof. Barguru Ramchandrappa, State Review Committee, Karnataka

textbooks Society®, Bengaluru. Revision Committee

ChairpersonDr. Narasimhamurthy S.K. Professor and Chairman, Department

of Mathematics , Kuvempu University, Shankaraghatta-577 451. Shivamogga

MembersDr. B . Chaluvaraju, Professor, Department of Mathematics,

Bengaluru University, Bengaluru.Sri. B. K. VishwanathRao, Rtd., Principal, No.94, ''Prashanthi'', 30th

Cross, BSK 2nd Stage, Bengaluru.Sri Narasimha murthy 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. Thara, Headmistress, Govt. High School,

Mavattur, K.R. Nagar taluk, Mysuru Dist,Smt Sushma NagarajRao, High School Teacher, Govt. Model Higher

Primary School, RamnagarSri Shrinath Shastri, Kannada Ganak Parishat, Chamrajpete,

Bengaluru.High Power CommitteeDr.Kashinath Biradar, Plot No.7, Gangasiri, Jayanagar,

Kalburgi - 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 TranslatorsSri. R.G. Kulkarni, Rtd. Teacher, Khanapur, Belagavi.Sri. P.C. Halgekar, A.M. Rangubai Bhosale, H.S. Belagavi.Chief Advisors Sri Narasimaiah, Managing Director, Karnataka Textbooks

Society®, Banashankari 3rd stage, Bengaluru-85.

Smt Nagamani C. Karnataka Textbooks Society®, Banashankari 3rd stage, Bengaluru-85.

Programme co-ordinator:Smt. Vijaya Kulkarni, Asst.Director, Karnataka Textbooks

Society®, Banashankari 3rd stage, Bengaluru-85.

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nyUmªH$ [INTEGERS]

`m àH$aUmÀ`m AÜ``ZmZ§Va Vwåhr :

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G$U nyUmªH$mbm G$U nyUmªH$mZo JwUmb.

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EH$m nyUmªH$mbm Xþgè`m nyUmªH$mZo ^mJmb.

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amonmda {e„H$ am{hbobo Jwbm~ = ?

{e„H$ Jwbm~ 5-3 = 2

∴{e„H am{hbobo Jwbm~ = 2

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Amåhmbm g§»¶m aofm YZ nyUmªH$, G$U nyUmªH$ d eyÝ` _mhrV AmhoV. YZ nyUmªH$, G$U nyUmªH$ d eyÝ` ho gd© nyUmªH$ AmhoV. nyUmªH$m§Mr ~oarO Am{U dOm~mH$s H$er H$aVmV ho Amåhmbm _mhrV Amho.CXmhaU 1 : gwgmZ Ioi Ioiy BpÀN>Vo. {VZo Amnë`m COdrH$S>o 3 g_mZ CS>çm _maë`m Am{U VoWyZ S>mdrH$S>o 5 g_mZ CS>çm _maÊ`mMo R>a{dbo. {VZ ^maboë¶m CS>çm g§»¶m aofmda H$em Xe©dmb ? g_Om n{hë`m§Xm gwgmZ eyÝ` ñWmZr Amho; Va {VZo _maboë`m CS>çm g§»`maofoda Xe©dy eH$Vmo. Vr H$moR>o nmohMbr Amho?

0 1 2 3

-2 2-1 13 30

0

3 + (-5) = -2

bjmV R>odm :

_moO g§»`m / Z¡g{J©H g§»`m `m YZ nyUmªH$ AmhoV.

eyÝ` hr Z¡g{J©H$ g§»`m Zmhr.

Z¡g{J©H$ g§»¶m Am{U "0' {_iyZ "nyU© g§»`m" ~ZVmV. nyUmªH$m_Üò nyU©g§»`m VgoM G$U nyUmªH$m§Mm g_mdoe hmoVmo.

ñdmÜ¶m¶ 1.1

I. Imbrb CXmhaUo g§»`maofoda Xe©dm.

1) 8 + 5 2) 2 + (-7)

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3) (- 3) + (4) 4) (– 7) + (- 2)

nyUmªH$m§Mm JwUmH$ma :

YZ nyUmª©H$mbm YZ nyUmªH$mZo JwUUo :

Á`m g§»òg_moa {MÝh {Xbobo ZgVo. Vr g§»`m YZg§»`m g_Omdr.

JwUmH$ma åhUOo H$m`?

JwUmH$ma åhUOo nwZamdVu ~oarO ho Amåhmg _mhrV Amho.

AmVm AmnU Imbrb CXmhaUo nmhÿ.

CXmhaU 1 :

âVZo EH$m XþH$mZmVyZ nopÝgbMo 3 ~m°Šg {dH$V AmUbo. àËòH$ ~m°Šg_Üò 6 nopÝgbr AmhoV. Va âVH$S>o EHy$U {H$Vr nopÝgbr AmhoV?

6 + 6 + 6 = 18 (~oarO)

ho g§»`maofoda H$go Xe©dmb?

0 6 12 18

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nyUmªH$.

Va Ë`m§Zr EHy$U {H$Vr Am§~o {dH$V KoVbo?

CH$b : × =

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YZ nyUmªH$mbm G$U nyUmªH$mZo JwUUo :

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AmVm, AmnU àË¶oH$ {Xder H$‘r Pmbobo ~Xm‘ -2 G$U nyUmªH$mZo Xe©dy.

-2, -2, -2. `m§Mr ~oarO Ho$ë`mg

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åhUyZ, S>ã`mV 6 ~Xm_ hmoVo.

ho AmnU g§»`maofoda Xe©dy.

-6 -5 -4 -3 -2 -1 0-7

-2 × 3 = -6

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-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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CXmhaU 2 : à`ËZ H$am.

4 × -5 = ?

-20 -15 -10 -5 0 5 104 × (-5) = (-20)

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CXmhaU 3 : (+5) × (-6) =?+ 5 × (-6) = -30 5 doim -6

5 ×(-6) = (-6) + (-6) + (-6) + (-6) + (-6) = -30

Z E R _bm H$moUVo {MÝh Amho?

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{MÝh Zmhr

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G$U nyUmªH$mbm YZ nyUmªH$mZo JwUUo :

G$U nyUmªH$mbm YZ nyUmªH$mZo JwUë`mg H$m` hmoVo?Imbrb CXmhaUo nhm. 5 × 4 = 20 4 × 4 = 16 = 20 - 4 3 × 4 = 12 = 16 - 42 × 4 = 8 = 12 – 41 × 4 = 4 = 8 – 40 × 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

åhUyZ (- 5) × 4 = - 20

{_Ìm, Vwbm _mhrV Amho H$m gdm©V _moR>m G$U

nyUmªH$ H$moUVm?

hmo` _mÂ¶m {_Ìm, Vmo Amho - 1

OmUyZ ¿`m :

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Ooìhm Amåhr G$U nyUmªH$ Am{U YZ nyUmªH$mMm JwUmH$ma H$aVmo, Voìhm nyU© g§»`m g_OyZ JwUmH$ma H$amdm Am{U JwUmH$mambm dOm {MÝh R>odmdo. åhUOo Amåhmg G$U nyUmªH$ {_iob.

CXmhaU 1 : (- 6) × 7 =?

Ho$di qH$_VrMm JwUmH$ma H$am6 × 7 = 42

OmUyZ ¿`m : Ag_mZ {MÝhm§À`m XmoZ nyUmªH$m§Mm JwUmH$ma Zoh_r G$U nyUmªH$ òVmo.

5 × 7 = 35 = 42 – 7

4 × 7 = 28 = 35 – 73 × 7 = 21 = 28 – 72 × 7 = 14 = 21 – 71 × 7 = 7 = 14 – 70 × 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)

XmoZ nyUmªH$m§Mr {MÝho Ag‘mZ AgVm§Zm Ambobm JwUmH$ma G$U AgVmo.

CXmhaU 2 : -8 × +2 = -16darb à_mUo Z_wZm {bhm.

CXmhaU 3 :

EH$ nmU~wS>m g_wÐmV EH$m {_{ZQ>mbm 9 _rQ>a doJmZo ~wS>Vmo. Ë`mMm ~wS>Ê`mMm doJ EH$ g_mZ Agob Va 6 {_{ZQ>mV Vmo {H$Vr A§Va nmÊ`mV Jobm Agob?

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g‘wÐmVrb nmÊ¶mV Imobda OmD$Z emoY KoUmè¶m ì¶{º$bm nmU~wS>m (SCUBA DIVER) åhUVmV.

Oa AmnU nmÊ`mMr nmVir eyÝ` (0) _mZbr Va, nmU~wS>çmMo nmÊ`mVrb ñWmZ G$Ug§»`m Yabr nm{hOo.

VoWo 9 hr G$U g§»`m Amho.

åhUyZ 6 {_{ZQ>mV nmU~wS>m Jobobo A§Va = -9 × 6 = -54

nmU~wS>m nmÊ`mÀ`m nmVirnmgyZ 54 _rQ>a Imob Amho qH$dm Vmo -54 _rQ>a A§Vamda Amho.

nyUmªH$mbm 0 (eyÝ¶) Zo JwUUo :

6 × 2 = 12 ;12 = 12 - 0

5 × 2 = 10 ; 10 = 12 - 2

4 × 2 = 8 ; 8 = 10 - 2

3 × 2 = 6 = 8 - 2

2 × 2 = 4 = 6 - 2

1 × 2 = 2 = 4 - 2

0 × 2 = ?

¶m Z‘wÝ¶mZwgma

0 × 2 ho 2 - 2 Ago hr {b{hVmV.

∴ 0 × 2 = 0

EH$m nyUmªH$mbm "0' Zo JwUë¶mg JwUmH$ma "0' ¶oVmo.

CXmhaU :

1) 5 × 0 = 0

2) - 10 × 0 = 0

3) 0 × 0 = 0

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G$U nyUmª©H$mbm G$U nyUmªH$mZo JwUUo :

ho dmMm :

dmMboë`m n{hë`m d eodQ>À`m dmŠ`m§Mm AW© EH$M Amho.

Am{U Xþgè`m d {Vgè`m dmŠ`m§Mm AW© EH$M Amho.

_r åhUmbmo "_bm Vo A_mÝ` Zmhr' `mMm AW© "Vo _bm _mÝ` Amho'.

`mdê$Z AmnU Ago åhUy eH$Vmo H$s H$moUË`mhr dmŠ`mMm ZH$mamMm ZH$ma åhUOo "hmoH$ma' hmo`.

x = 3 × 5 ¶m‘Ü¶o y = 3 × (-5) Mm ~oarOì¶ñV -3 × (-5) x + 0 = 3 × 5 + 0

x + y - y = 3 × 5 + [3 × (-5)] + [(-3) × (-5)]

= 3 [5 + (-5)] - [(3) × (-5)] (x + y bm gai ê$n XoD$Z)

= [3 × 0] - [(3) × (-5)]

= 0- [(3) × (-5)]

= - 3 × -5

∴ 3 × 5 = - 3 × -5

15 = - 3 × -5

‘r Imëb Amho.

‘r Imëb Zmhr.

‘r Cnmer Amho.

‘r Cnmer Zmhr.

H$moUË¶mhr G$U g§»¶onojm gdm©V ‘moR>r g§»¶m H$moUVr? Am{U H$moUË¶mhr YZg§»¶onojm gdm©V

bhmZ g§»¶m H$moUVr?

ho ‘mÂ¶m {‘Ìm.... Vr Amho eyÝ¶ !!

OmUyZ ¿`m :

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gm‘mÝ¶nUo 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

∴ XmoZ G$U nyUmªH$m§Mm JwUmH$ma Zoh‘r YZ nyUmªH$ ¶oVmo.

`m g§»`mMm JwUmH$ma Z_yÝ`mMm Cn`moJ H$ê$Z H$am

-3 × - 6 = ?

XmoZ F$U² nyUmªH$m§Mm JwUmH$ma hm YZnyUmªH$ Wobo àW‘ Amåhr Ho$db g§»¶m§Mm JwUmH$ma

H$aVmo Am{U JwUmH$mambm YZ {MÝh XoVmo.ho Z_yÝ`mà_mUo {b{hë`mg -3 × 6 = -18 (6 ì`m B`ÎmoV {eH$boë`m ~oarO ì`ñVMr

AmR>dU H$am.)-3 × 5 =-15 =-18 –(-3) -3 × 4 = - 12 = -15 – (-3) -3 × -1 = 0-(- 3) = 0 +3 = 3 -3 × 3 = -9 = - 12 – (- 3) -3 × -2 = 3 - (-3) = 3 + 3 = 6

-3 × 2 = - 6 = - 9 - (- 3) -3 × -4 = 9 - (-3) = 9 + 9 = 12

-3 × 1 = - 3 = -6 - (- 3) -3 × -5 = 12 - (-3)= 12 + 3 =15

-3 × 0 = 0 = - 3 - (- 3) -3 × -6 = 15 - (-3)= 15 + 3 =18

-3 × 1= ?

∴ H$moUË¶mhr g§»¶obm 0 Zo JwUë¶mg Ë¶m§Mm JwUmH$ma Zo‘r 0 ¶oVmo. -3 ×0 = 0

XmoZ g_mZ {MÝhmÀ`m nyUmªH$m§Mm JwUmH$ma Zoh_r YZ òVmo.

CXmhaU 1 : -4 × -3 = ?nyU© qH$_VrMm JwUmH$ma H$am. 4 ×3 = 12

na§Vy XmoZ G$U nyUmªH$m§Mm JwUmH$ma YZ AgVmo. åhUyZ -4 × -3 =12

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XmoZ nojm OmñV nyUmªH$m§Mm JwUmH$ma :

CXmhaU 1 : +3 × +5 × +6 = ? +3 × +5 × +6 = +15 × +6 = +90

CXmhaU 2 : +3 × +5 × -6 =? +3 × +5 × -6 =+15 × -6 = -90

CXmhaU 3 : +2 × +3 × +5 × +6 = ? +2 × +3 × +5 × +6 = +6 × +30 = +180

CXmhaU 4 : +2 × +3 × -5 × -6 =? +2 × +3 × -5 × -6 =+6 × +30 = +180

CXmhaU 5 : -2 × -3 × +5 × -6 = ? -2 × -3 × +5 × -6 = +6 × -30 =-180

CXmhaU 6 : -2 × -3 × -5 × -6 = ? -2 × -3 × -5 × -6 = +6 × +30 =+180

OmUyZ ¿¶m.

nmàr 1 : {Xboë`m g§»`m§À`m g§»`mË_H$ qH$_VrMm JwUmH$ma emoYm.

nmàr 2 : {Xboë`m g§»`m§À`m JwUmH$mamVrb G$U nyUmªH$ _moOm.

nmàr 3 : nmàr 2 _Yrb _moObobr g§»`m g_ Agob Va Ambobm JwUmH$ma YZ ¶oVmo

d _moObobr g§»`m {df_ Agob Va JwUmH$ma G$U òVmo.

eãX g_ñ`m :

CXmhaU 1 :

Z_ZZo EH$m \$imÀ`m XþH$mZmVrb EH$ S>PZ Ho$ir IaoXr

Ho$br. Oa EH$m Ho$ù`mMr {H$_V ` 3 Agob Va Ë`mZo {H$Vr

énò {Xbo nm{hOoV?

‘mhrV Amho H$m?

EH$ S>PZ = ~mam

1 J«mog = ~mam S>PZ

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CH$b : EH$ S>PZ = 12EH$m Ho$ù`mMr qH$_V = ` 3 EHy$U Úmd`mMr a¸$_ H$mT>Ê`mg 12 bm 3 Zo Jw{Ubo nm{hOo. Z_ZZo Úmd`mMr EHy$U a¸$_ = 12 × 3 = ` 36.åhUyZ, Z_ZZo XþH$mZXmambm ` 36 {Xbo nm{hOoV.

CXmhaU 2 : EH$ nmU~yS>r g_wÐmVrb nmÊ`mÀ`m nmVirnmgyZ 15 _r/{_{ZQ> doJmZo Imbr OmVo. Va

5 {_{ZQ>mV nmU~wS>r Jobobr Imobr {H$Vr?CH$b : g_wÐ gnmQ>rnmgyZ Imbr OmUo åhUOo G$U. EH$m {_{ZQ>mV nmU~wS>rMo ñWmZ = -15 _rQ>a 5 {_{ZQ>mZ§Va nmU~wS>rMo ñWmZ = -15 × 5 = -75 _rQ>a

åhUyZ 5 {_{ZQ>mV nmU~wS>r 75 _rQ>a A§Va g_wÐ nmVirnmgyZ Imob Jobobr Amho.

ñdmÜ`m` 1.2

I. g§»`m§er Ioim Am{U _‚mm H$am.

1) n§º$sVrb àËòH$ g§»òbm ñV§^mVrb g§»òZo JwUm Am{U Ambobm JwUmH$ma H$moï>H$mV Xe©{dë`mà_mUo {bhm.

× -3 6 11 -5

7 -21

4 44

-8

-2

12

-9

-14

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2) dV©wimVrb g§» oZo Ë`mÀ`m ~mhoarb dV©wimVrb g§»`m§Zm JwUm Am{U Ambobm JwUmH$ma dV©wimÀ`m ~mhoarb Mm¡H$moZm§V {bhm.

AmVrb dV©wimV doJdoJir g§»`m KoD$Z JwUmH$ma H$am d _‚mm H$am.

3. hm Ioi Ioim.

i) {MÌmV Xe©{dë`mà_mUo (– 50) Vo 50 g§»`m {b{hbobm EH$ ~moS>© ¿`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

50 49 47 46 45 44 43 42 41 40

85

-5-67

-2

1

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ii) XmoZ {Zio Am{U XmoZ bmb a§JmMo Ago EHy$U Mma \$mgo EH$m {nedrV ¿`m. {Zù`m \$memdarb {R>nHo$ YZ nyUmªH$ d bmb \$memdarb {R>nHo$ G$U nyUmªH$ g_Om.

IoimMo {Z`_ :

àË oH$ IoimSy>Zo Amnbo ß`mXo eyÝ` {R>H$mUr R>odmdo.

àËòH$ IoimSy>Zo EH$m doir Z ~KVm XmoZ \$mgo KoD$Z WmoS>o da \o$H$mdoV.

\$mgo Imbr nS>ë`mZ§Va {R>nHo$ nmhÿZ Ë`m§Mm JwUmH$ma H$amdm.

JwUmH$mamZwgma Amnbo ß`mXo ~moS>©darb Mm¡H$moZmV R>odV Omdo.

15 {_{ZQ>mÀ`m H$mbmdYrV Á`m IoimSy>Mo ß`mXo _moR>çm g§»òÀ`m Mm¡H$moZmV nmohMbo Agob

Vmo IoimSy> {dO`r Pmbm Ago g_Omdo.

ñdmÜ`m` 1.3

I. Imbrb nyUmªH$m§Mm JwUmH$ma H$am :

1) 5, (- 3) 2) (– 3), 8

3) 7, (- 34) 4) (- 23), (- 12)

5) (- 21), 6 6) (-2), (3), (-4)

7) (-3), 4, (-5), (-1)

II. _moH$ù`m OmJm âm :

1) 4 × + 7 = + 28

2) - 3 × 5 = - ….

3) + 9 × - 7 = ….

4) 6 × - 8 = + ….

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III) Imbrb boIr J{UVmdê$Z `mo½` CÎmambm ~amo~a {MÝhmZo IyU H$am :

1) EH$m B_maVrda EH$ nmÊ`mZo abobr Q>mH$s Amho. ~mînr^dZmZo Xa {Xder nmÊ`mMr nmVir 2 cm Zo H$_r hmoVo. Oa n{hë`m {Xder nmÊ`mMr nmVir A cm Yaë`mg, 6 {XdgmZ§Va nmÊ`mMr nmVir

1) A cm + (2 × 6) cm 2) A cm – (2 × 6) cm3) A cm – (2 × -6) cm4) 6 ×(A cm - 2) cm

2) {Jare Amnë`m ~±H$ ImË`m_Üò Xa AmR>dS>¶mbm ` 7500 à_mUo ‘mJrb 8 AmR>dS>o a¸$_ O_m H$aV Amho. AmVm Ë`mÀ`m ImË`mVrb {e„H$ ` 62,000 Amho. `m H$mbmdYrV ImË`mda H$moUVrhr ì`mOmMr a¸$_ O_m Pmbobr Zmhr. Va 8 AmR>dmS>¶m Ë`mÀ`m ImË`mV Agbobr a¸$_ -

1) ` (62000 – 8 × 7 × 7500) 2) ` (62000 – 8 × 7500) 3) ` (62000 + 8 × 7500) 4) ` (8 × 7500 – 62000)

{dMma H$am : i) gdm©V _moR>m YZ nyUmªH$ Am{U gdm©V bhmZ G$U nyUmªH$ H$moUVm?

ii) gdm©V _moR>m G$U nyUmªH$ Am{U gdm©V bhmZ YZ nyUmªH$ H$moUVm?

nyUmªH$m§Mm ^mJmH$ma

bVm Am{U {VÀ`m _¡qÌUr : C_m, Vwbm dmT>{XdgmÀ`m hmXuH$ ewôÀN>m.

C_m : Am^mar Amho. ho _¡{ÌUrZm|, òWo noT>çmMm EH$ ~m°Šg Amho. Amnë`mV Vmo g_mZ dmQy>Z KoD$`m.

a{~`m : R>rH$ Amho, _r _XV H$aVo. (_moOVo Am{U gm§JVo) `m ~m°Šg_Üò 20 noT>o AmhoV Am{U C_m Yê$Z nmM OUr AmhmoV.

lr{ZYr : N>mZ! H$mhr haH$V Zmhr. 20 ^m{Jbo 5 åhUOo 20 ÷ 5 = 4

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bVm : àËòH$sZo 4 noT>o ¿`m.

Om°Wgr : bVm Vwbm _mhrV Amho H$m? 5 × 4 = 20 gwÜXm.

lr{ZYr : JwUmH$ma Am{U ^mJmH$ma `m§Mm g§~§Y AgVmo. bVm, Om°Wgr, lr{ZYr, a{~`m : Agy Xo C_m, VwPo Amåhr Am^mar AmhmoV.

C_m : _o¡{ÌUrZm|§, Oa Vw_À`m ~amo~a _r noT>o KoVbo ZgVo Va, Vwåhmbm OmñV noT>o {_imbo AgVo.

bVm : hmo`, 20 ÷ 4 = 5.

AmVm AmnU EH$ Ioi Ioiy. O°H$ Am{U {O„ ho XmoKo _¡XmZmda IoiV hmoVo. H$mhr doimZ§Va Ë`m§Zm VhmZ bmJbr. Ë`m§Zm {nÊ`mg nmUr hdo hmoVo åhUyZ Vo ~mOyÀ`m {d{harda Jobo. VoWo Ë`m§Zm EH$ ~mXbr d Xmoa nU {_imbm. Ë`mZr {d{harVyZ nmUr H$mT>Ê`mMo R>a{dbo. nmUr H$mTy>Z {nD$Z Pmë`mZ§Va Ë`m§À`mVrb MMm© Aer -

O°H$ : {Ob², `m {d{harMr Imobr {H$Vr Agob?

{Ob² : AmnU Ë`m ~mOyÀ`m H$mH$m§Zm {dMmê$. H$mH$m `m {d{harMr Imobr {H$Vr Amho?

am_y H$mH$m : _wbm§Zmo, O{_ZrnmgyZ {d{ha 35 \y$Q> Imob Amho.

O°H$ : AmVm nmÊ`mMr nmVir {H$Vr Amho?

am_y H$mH$m : 28 \y$Q>

{Ob² : H$mH$m Amåhr Vw_Mo Am^mar AmhmoV ho O°H$, AmnU O_rZ nmVir eyÝ` g_Oy.

O°H$ : hmo`, {d{harV 28 \y$Q> nmUr Amho åhUOo O_rZ nmVirnmgyZ 35-28= 7 \y$Q>mda nmUr Amho.

{Ob² : Vo -7 Zo Xe©{dVmV.

O°H$ : nmUr gwÜXm O_rZ nmVirÀ`m Imbr Amho åhUyZ Amåhr nmÊ`mMr Imobr -28 \y$Q> Ago Xe©{dVmo.

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{Ob² : Oa ~mXbr {d{harV nS>br Am{U 4 {_{ZQ>mV nmÊ`mÀ`m nmVirnmgyZ Vimer Jobr Va ~mXbrMm nmÊ`mVrb doJ {H$Vr?

O°H$ : {Ob² Vy EH$ Mm§Jbm àý {dMmabmg. Amåhr {dkmZm_Üò doJ ~Ôb Aä`mg Ho$bobm Amho.

{Ob² : hmo`, doJ = AmH«${_bobo A§Va ^m{Jbo doi.

O°H$ : doJ = 28 ÷ 4 = 7 \y$Q> / {_{ZQ>.

{Ob² : na§Vy Amåhr nmÊ`mMr Imobr - 28 \y$Q> g_Obr Amho.

O°H$ : _J 4 × ? = -28

{Ob² : -7. åhUOo -28 ÷ 4 = -7 ~mXbrMm nmÊ`mVrb doJ -7 \y$Q> / {_{ZQ.> dOm {MÝh ho ~mXbrMr Imbrb {XeoZo OmÊ`mMr {Xem Xe©{dVo.

O°H$ : {Ob² AmVm nwao. AmVm AmnU nwÝhm Ioiy`m Mb.

CXmhaU 2 : i) 6 × 2 = 12 = 2 × 6

åhUyZ 12 ÷ 6 = 2

Am{U 12 ÷ 2 = 6

ii) (- 4) × 5 = - 20 darb (i) _Yrb nÜXV dmnê$Z

– 20 ÷ 5 = - 4

Am{U – 20 ÷ - 4 = 5.

m nÜXVrZo CXmhaUo gmoS>{dÊ`mMm gamd H$am. àË oH$ JwUmH$mamÀ`m {dYmZmbm mJmH$mamMr

XmoZ {dYmZo AgVmV. Imbrb Vº$m nyU© H$am.

JwUmH$mamMo {dYmZ g§~§YrV ^mJmH$mamMo {dYmZ

( - 8) × 2 = - 16 -16 ÷ 2 = - 8, -16 ÷ - 8 = 2

(- 5 ) × (- 4 ) = 20 20 ÷ - 4 = - 5, ----------------

9 × (- 4) = - 36 -36 ÷ - 4 = 9, - 36 ÷ 9 = - 4

( - 7)× (- 6) = 42 ---------------, 42 ÷ - 7= - 6

3 × (- 10) = - 30 - 30 ÷ - 10 = 3, ---------------

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`m VŠË`mdê$Z Ago AZw_mZ H$mT>Vm òVo H$s,

• XmoZ g_mZ {MÝhm§À`m g§»`m§Mm ^mJmH$ma Zoh_r YZg§»`m òVo.

• XmoZ {^Þ qH$dm Ag_mZ {MÝhm§À`m g§»`m§Mm ^mJmH$ma G$U g§»`m òVo.

nyUmªH$mbm eyÝ`Zo ^mJë`mg H$m` hmoB©b?

à{gÜX ^maVr` J{UVk lr{Zdmg am_mZwOZ `m§Zr darb àý Vo {dÚmWuXeoV AgVm§Zm dJm©V {dMmabm hmoVm.

ho AmnU H¥$VrÛmao nmhÿ : gma»`m AmH$mamMo 8 H$mJX ¿`m

8 Ë`mMo Ago 8, 1 doim = 8gd© H$mJXmMo g_mZ 2 JQ> H$am. 4 + 4 Ë`mMo Ago 4, 2 doim = 8`m àËòH$ JQ>mMo AmUIr g_mZ 2 JQ> 2 + 2 + 2 + 2 Ë`mMo Ago 2, 4 doim = 8nwÝhm `m àËòH$ JQ>mMo AmUIr g_mZ 2 JQ>1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 Ë`mMo Ago 1, 8 doim = 8

8 H$mJXmMo àËòH$s 2 g_mZ VwH$S>o H$am12 + 1

2 + 12 + 1

2 + 12 + 1

2 + 12 + 1

2 + 12 + 1

2 + 12 + 1

2 + 12 + 1

2 + 12 + 1

2 Ë mMo Ago 12 , 16 doim = 8

`m gmoim VwH$S>çm§Mo àË¶oH$s naV 2 g_mZ VwH$S>o H$am

14 + 1

4 + 14 + 1

4 + 14 + 1

4 + 14 + 1

4 +.........+ 14 + 1

4 + 14 + 1

4 + 14 Ë`mMo Ago 1

4 , 32 doim = 8

qH$dm 80.25=32 0.25+0.25+…………+0.25 (32 doim) = 8

Ë¶mM à‘mUo 80.1 =80 0.1+0.1+………….+0.1 (80 doim) = 8

80.01 =800 0.01+0.01+…………+0.01 (800 doim) = 8

=800 åhUOo 0.0001+0.0001+………… 8000 doim

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80 = ? 8 òÊ`mg {H$Vr eyÝ` {_idmdoV?

åhUOo 8 òÊ`mg eyÝ¶ Mm JQ> {_iV Zmhr. åhUyZ AmnU H$moUË`mhr nyUmªH$mbm eyÝ`Zo ^mJy eH$V Zmhr.

[Q>rn : H$moUË`mhr g§Iobm eyÝ`Zo ^mJUoo `mMo {ddaU H$aVm òV Zmhr.]

ñdmÜ`m` 1.4

I. gaiê$n Úm :

1) (- 40) ÷ (- 10) 2) 34 ÷ (- 2) 3) (- 44) ÷ 4

4) (- 28) ÷ (- 7) 5) 0 ÷ (- 8)

II. _moH$ù`m OmJm âm :

1) (- 45) ÷ ----- = - 45 2) (- 27) ÷ (- 27) = ----- 3) 30 ÷ ---- = - 15 4) ---- ÷ 4 = - 3 5) ---- ÷ ( - 3) = 10

III. Vº$m âm :

nyUmªH$m§Mm ^mJmH$ma ^mJbãY

+24 ÷ +12 =

+24 ÷ -12 =

-24 ÷ +12 =

-24 ÷ -12 =

IV. lo`gZo YmadmS> noT>çm§Mo 8 ~m°Šg IaoXr Ho$bo. àËòH$ ~m°Šg_Üò g_mZ noT>o AmhoV. EHy$U noT>o 320 AgVrb Va àËWoH$ ~m°³g‘Yrb noT>¶m§Mr {H$Vr?

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nyUmªH$m§Mo JwUY‘©

a) H«$_{Zanoj JwUY_©

~oarO : 2 + - 5 = -3 Am{U -5 + 2 = -3.Oa a Am{U b ho H$moUVohr XmoZ nyUmªH$ AgVrb Va, a+ b = b +a.`mbm ~oaOoMm H«$_{Zanoj JwUY_© Ago åhUVmV. ~oaOoV g§»`m§À`m H«$_mbm _hËd ZgVo.

nyUmªH$m§Mm gQ> ~oaOoÀ`m ~m~VrV H«$_m{Zanoj JwUY_© Xe©{dVmo.

JwUmH$ma : 2 × −5 = −5 × 2 = -10

a Am{U b `m H$moUË`mhr XmoZ nyUmªH$mgmR>r a × b = b × aJwUmH$mam_Üò g§»`mÀ¶m H«$_mbm _hËd ZgVo.

nyUmªH$m§Mm gQ> JwUmH$mamÀ`m ~m~VrV H«$_{Zanoj JwUY_© Xe©{dVmo.

dOm~mH$s : 18 – 15 = 3 Am{U 15 – 18 = -3 18 – 15 ≠ 15 – 18

dOm~mH$s_Üò g§»`§mMm H«$_ ~Xbë¶mg CÎmao doJdoJir òVmV.

åhUyZ H«$_{Zanoj JwUY_© dOm~mH$s_Üò bmJy nS>V Zmhr.

^mJmH$ma : 12 ÷3 = 4 Am{U 3 ÷12 = 312 = 1

412 ÷3 ≠ 3 ÷12

^mÁ` Am{U ^mOH$ `m§Mo ñWmZ~Xb H$ê$Z ^mJmH$ma AXbm~Xb Ho$ë`mg ^mJmH$ma doJdoJim òVmo.

åhUyZ ^mJmH$mamÀ`m ~m~VrV H«$_{Zanoj JwUY_© bmJy nS>V Zmhr.

b) gmhM`© JwUY_©

~oarO : (4 + −2) + −5 = 2 + -5 = -3. 4 + (−2 + −5) = 4 + -7 = -3.a, b Am{U c `m H$moUË`mhr VrZ nyUmªH$mgmR>r (a+b)+c = a+(b+c) VrZ nyUmªH$m§Mr ~oarO H$aVm§Zm n{hbm XmoZ nyUmªH$m§Mr ~oarO H$ê$Z Ë`mV {Vgam nyUmªH$

{_i{dë`mZo ¶oUmao CÎma qH$dm Xþgè`m XmoZ nyUmªH$m§Mr ~oarO H$ê$Z n{hbm nyUmªH$ {_idyZ òUmao CÎma hr XmoÝhr CÎmao g_mZ oVmV. nyUmªH$mMm gQ> hm ~oaOoÀ`m ~m~VrV gmhM`© JwUY_© Xe©{dVmo.

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JwUmH$ma : (4 × −2) × −5 = -8 × -5 = 40

4 × (−2 × −5) = 4 × 10 = 40

a, b Am{U c VrZ nyUmªH$ AgVrb Va, (a × b) × c = a × (b × c)

VrZ nyUmªH$m§Mm JwUmH$ma H$aVm§Zm n{hë`m XmoZ nyUmªH$mMm JwUmH$ma H$ê$Z {Vgè`m nyUmªH$mbm

JwUë`mg qH$dm eodQ>À`m XmoZ nyUmªH$m§Mm JwUmH$ma H$ê$Z n{hë`m nyUmªH$mbm JwUë`mg CÎmao

g_mZ òVmV.

nyUmªH$mMm gQ> JwUmH$mamÀ`m ~m~VrV gmhM`© JwUY_© Xe©{dVmo.

dOm~mH$s : 6 – (3 – 7) = 6 – (-4) = 10

(6 - 3) – 7 = 3 – 7 = -4 6 – (3 – 7)≠ (6 - 3) – 7

`mdê$Z nyUmªH$m§À¶m gQm>bm dOm~mH$sÀ`m ~m~VrV gmhM¶© JwUY‘© bmJy nS>V Zmhr.

^mJmH$ma : 8 ÷ (4 ÷ 2) = 4 (8 ÷ 4) ÷ 2 = 1 8 ÷ (4 ÷ 2) ≠(8 ÷ 4) ÷ 2

`mdê$Z nyUmªH$m§À¶m gQmbm> ^mJmH$mamÀ`m ~m~VrV gmhM¶© JwUY‘© bmJy nS>V Zmhr.

c) ~oarO A{dH$maH$ KQ>H$−5 + 0 = −50 + −5 = −5

a hm nyUmªH$ Agob Va, a + 0 = a = 0 + a

eyÝ` hm ~oaOoMm A{dH$maH$ KQ>H$ Amho. H$moUË`mhr g§»òV eyÝ` {_i{dë`mg CÎma VrM

g§»`m òVo.

d) JwUmH$ma ì`ñV−3 × 1 = 1 × −3 = −3

a hm nyUmªH$ Agob Va, a × 1 = a = 1 × a

1 hm JwUmH$mamMm A{dH$maH$ KQ>H$ Amho. H$moUË`mhr g§»òbm 1 Zo JwUë`mg CÎma VrM g§»`m òVo.

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e) {dVaU JwUY_©

Imbrb CXmhaUm‘Ü¶o H§$gmÀ`m ~mhoarb g§» oZo H§$gmVrb g§»`mZm ñdV§ÌnUo JwUbo Amho nhm.

CXmhaU 1 : 3 × [(2 )+ −4)] = (3 × 2) + [(3) × (−4)]CXmhaU 2 : [(-2) × (5 − 7)] = [(-2) × 5) – (-2) × (−7)] a, b, c ¶m H$moUË¶mhr VrZ nyUmªH$mgmR>r

a × (b + c) = (a × b) + (a × c)

a × (b - c) = (a × b) - (a × c)

¶m JwUY‘m©bm {dVaU JwUY‘© åhUVmV.

ñdmÜ`m` 1.5

I. `mo½` JwUY_© dmnê$Z Imbrb CXmhaUo gmoS>dm.

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)]

boIr CXmhaUo :CXmhaU 1 :

aå`m Amnë`m VrZ mdm§Zm H$nS>o KoÊ`mg EH$m H$mnS> XþH$mZmV Jobr. {VZo àË oH$s 450 Mo

3 n±Q>{ng d 320 Mo 3 eQ>©nrg {dH$V KoVbo. Va {VZo XþH$mZXmambm {H$Vr a¸$_ {Xbr nm{hOo?

CH$b :

EH$m n±Q>{ngMr qH$_V = ` 450

åhUyZ 3 n±Q>{ngMr qH$_V = 3×450 = ` 1350

EH$m eQ>©{ngMr qH$_V = ` 320

åhUyZ 3 eQ>©{ngMr qH$_V = 3×320 = ` 960

åhUOo aå`mZo XþH$mZXmambm Úmd`mMr EHy$U a¸$_ = 1350+960 = ` 2310

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CXmhaU 2 : gmoZmÀ`m ~±H$ImË`mV ` 1020 hmoVo. {VZo gmo_dmar ` 200 ImË`mV O_m Ho$bo. Am{U

Ë`mM {Xder {VZo ` 500 H$mT>bo. naV ~wYdmar ` 580 O_m Ho$bo. e{Zdmar ` 1000 naV

ImË`mV O_m Ho$bo. Va e{Zdmar {VÀ`m ImË`mVrb EHy$U {e„H$ a¸$_ {H$Vr?

CH$b :

òWo ImË`mda O_m Ho$bobr a¸$_ YZ Am{U ImË`mVyZ H$mT>bobr a¸$_ G$U g_Oy.

gmoZmÀ`m ImË`mVrb _mJrb {e„H$ a¸$_ = ` 1020

gmo_dma =1020 + 200 – 500 =1220 – 500 = ` 720

~wYdma = 720 + 580 = ` 1300

e{Zdma = 1300 + 1000 = ` 2300

e{Zdmar {VÀ`m ImË`mVrb {e„H$ a¸$_ = ` 2300.

CXmhaU 3 :

EH$m H§$nZrbm EH$ noZ {dH$ë`mZo 12 Z\$m hmoVmo Am{U EH$ nopÝgb ~m°Šg {dH$ë¶mZo

` 8 VmoQ>m hmoVmo. Oa H§$nZrZo 2500 noZ Am{U 4000 nopÝgëg ~m°Šg EH$m _{hÝ`mV V`ma

H$ê$Z {dH$bo Va H§$nZrbm Pmbobm EHy$U Z\$m qH$dm VmoQ>m {H$Vr?

CH$b :EH$ noZ {dH$ë`mZo hmoUmam Z\$m = `12

{dH$bobr EHy$U noZ = 2500 2500 noZ {dHy$Z Pmbobm Z\$m = 12 × 2500

= `30000

EH$ nopÝgb ~m°Šg {dH$ë`mZo hmoUmam VmoQ>m = ` 8

nopÝgb ~m°Šg {dH$bobr g§»`m = 4000

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Pmbobm EHy$U VmoQ>m = 8 × 4000

= ` 32000

åhUyZ Pmbobm VmoQ>m = 32000 – 30000

= ` 2000

CXmhaU 4 :

EH$m àý_§Owfm àýn{ÌHo$V 12 àý AmhoV. àËòH$ àýmbm 5 JwU AmhoV. CÎmao

VnmgUrdoir AMyH$ CÎmambm +5 JwU d MyH$sÀ`m CÎmambm -2 JwU Úmd`mMo AmhoV.

i) ^maVrZo gd© àý gmoS>{dbo. Ë`mVrb 9 ~amo~a d 3 MyH$sMr CÎmao Ambr.

ii) _§OyZo \$º$ 10 àý gmoS>{dbo. Ë`mVrb 6 ~amo~a d Cabobo MyH$ Pmbo.

Va àý_§Owfm ñnY}Mr {dOoVr H$moU?

CH$b :

i) EH$m ~amo~a CÎmambm {_iUmao JwU = 5 9 ~amo~a CÎmamZm {_imbobo JwU = 5 × 9 = 45

EH$m MwH$sÀ`m CÎmambm {_iUmao JwU = -2 VrZ MwH$sÀ`m CÎmambm {_iUmao JwU = (-2) × 3 = (-6) åhUyZ ^maVrbm {_imbobo EHy$U JwU = 45 + (-6) = 39

ii) EH$m ~amo~a CÎmambm {_iUmao JwU = 5 6 ~amo~a CÎmamZm {_imbobo JwU = 5 × 6 = 30

EH$m MwH$sÀ`m CÎmambm {_iUmao JwU = -2 (10 – 6) = 4 MwH$sÀ` CÎmam§Zm {_imbobo JwU = (-2) × 4 = (-8) åhUyZ _§Owbm {_imbobo EHy$U JwU = 30 + (-8) = 22

39 >22, ^maVrMo JwU OmñV AmhoV.

∴ åhUyZ ñnY}Vrb {dOoVr ^maVr.

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ñdmÜ`m` 1.6

I. Imbrb Vm|S>r CXmhaUo gmoSdm.

1) _mo{ZH$mH$S>o Mm°H$boQ>g²Mo 5 ~m°Šg AmhoV. àËòH$ ~m°Šg_Üò 25 Mm°H$boQ>g² Agë¶mg {VÀ`mH$S>o EHy$U {H$Vr Mm°H$boQ>g² AmhoV?

2) A\$Vm~Zo EH$m _{hÝ`mV ~|Jiyê$ ehamMo gm_mÝ` Vmn_mZmMo {ZarjU Ho$bo AgVm Vo 210C Agbobo AmT>ibo. EH$m R>am{dH$ {Xder gm_mÝ` Vmn_mZmÀ`m VwbZoV {XgyZ Ambobo Vmn_mZmVrb ~Xb nwT>rb à_mUo :

gH$mir : 5 dmOVm - 50C, 10 dm. +50C,

Xþnmar : 12 dmOVm +70C,

g§Ü¶mH$mir : 3 dmOVm +20C.

Va gH$mir 5 dmOVm, gH$ir10 dm, Xÿnmar 12 dm, Am{U g§Ü¶mH$mir 3dm. AgUmao.

AMyH$ Vmn_mZ H$mT>Ê¶mg A\$Vm~bm _XV H$am.

3) aí_rÀ`m ~±H$ ImË`mV ` 12,350 hmoVo. {VZo 3 doim àËòH$s ` 200 H$mT>bo. Va {VÀ`m ~±H$ ImË`mVrb {e„H$ {H$Vr?

4) EH$m noZMr qH$_V ` 8 Am{U EH$m nopÝgbMr qH$_V ` 5 Amho. Va EH$ S>PZ noZ Am{U AYm© S>PZ nopÝgëg²Mr EHy$U qH$‘Vr {H$Vr?

5) EH$m B`ÎmoÀ`m EH$m nwñVH$ g§MMr qH$_V ` 1235 Amho. Va Aem 38 g§MMr qH$_V {H$Vr hmoB©b?

6) EH$m nmoË`mV45 {H$.J«°. gmIa Amho. VrZ df} d`mÀ`m {dO`Zo Ë`mVrb 750 J«°‘ gmIa O{_Zrda gm§S>br. Va Caboë`m gmIaoMr 250 J«°‘ dOZmMr {H$Vr nm{H$Q>o ~ZVrb?

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àH$aU - 2

AnyUmªH$ [FRACTIONS]

`m àH$aUmÀ`m AÜ``ZmZ§Va Vwåhr :

nyU© g§»¶oZo {Xboë`m AnyUmªH$mbm JwUmb.

EH$m AnyUmªH$mbm Xþgè`m AnyUmªH$mZo JwUmb.

"Mo' Mm `mo½` H«$_m_Üò Cn`moJ g_OyZ ¿`mb.

{Xboë¶m AnyUmªH$mMm ì¶ñV AnyUmªH$ {bhmb Am{U Cn¶moJ H$amb.

EH$m AnyUmªH$mbm EH$m nyUmªH$mZo ^mJmb Am{U nyUmªH$mbm EH$m AnyUmªH$mZo ^mJmb.

EH$m AnyUmªH$mbm Xþgè`m AnyUmªH$mZo ^mJmb.

AnyUmªH$m§À¶m JwUmH$ma Am{U ^mJmH$ma ¶m g§~§Yr boIr CXmhaUo gmoS>dmb.

Vwåhr AnyUmªH$mMm AW©, AnyUmªH$mMo àH$ma, g__yë` AnyUmªH$ Am{U AnyUmªH$m§Mr ~oarO Am{U dOm~mH$s {eH$bm AmhmV. Ë`mMo ñ_aU H$am.

AnyUmªH$m§Mo dJuH$aU

1) nwT>rb AnyUmªH$m§Mo g_, {df_ Am{U {_l AnyUmªH$mV dJuH$aU H$am.

, , , , ,2 , , ,43152 2 5

347911

65

811361315

61

àËòH$ AnyUmªH$mMo {ZarjU H$am Am{U `mo½` n§{º$V {bhm.

A) g_ AnyUmªH$ : , , ,315265136

~) {df_ AnyUmªH$ : , , ,47

911

1315

33

H$) {_l AnyUmªH$ : , ,2 53 2 8

1 461

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2) A `mXr_Üò {df_ AnyUmªH$m§Mo {_l AnyUmªH$mV ê$nm§Va Ho$bobo Amho Am{U B `mXr_Üò {_l AnyUmªH$m§Mo {df_ AnyUmªH$mV ê$nm§Va Ho$bobo Amho. H$go Vo nmhm.

, , ,29

512

615

1112

A`mXr

, , ,2 43 5 6

1 4 72 6 5

3B`mXr

4

2

2

29

21

512

52

615

63

1112 111

1

=

=

=

=

2 43

411

561

631

4 72

730

6 53

533

=

=

=

=

CXmhaU 1 :

_h_XZo EH$m Mm°H$boQ>À¶m 8 g_mZ mJmn¡H$s 6 mJ KoVbo AmhoV. Am{U _ÝZZo Mm°H$boQ>À¶m 4 g_mZ mJmn¡H$s 3 mJ KoVbo AmhoV Va H$moÊmmbm OmñV Mm°H$boQ> {‘imbo? ({MÌm§Mo {ZarjU H$am Am{U CÎmam§Mr VwbZm H$am.)

A ~

KoVboë¶m XmoÝhr Mm°H$boQ>g²Mm AmH$ma g_mZ Amhoo.åhUyZ, ,8

643

86

43= = ho g__wë` AnyUmªH$ AmhoV

{Q>n : ÷÷

××

)

)

a

b

86

8 26 2

43

43

4 23 2

86

''

##

= =

= =

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CXmhaU 2 :

e¡bmZo EH$m g^mJ¥hmMm 125 mJ gO{dbm Am{U aoUyZo Ë`mM g^mJ¥hmMm 6

3 mJ gO{dbm. Va XmoKtZr {_iyZ Ë`m g^mJ¥hmMm {H$Vr ^mJ gO{dbm?

e¡bmZo gO{dbobm g^mJ¥hmMm ^mJ = 125

aoUyZo gO{dbobm g^mJ¥hmMm ^mJ = 63

e¡bm Am{U aoUy XmoKtZr gO{dbobm ^mJ125

63

125

6 23 2

125

126

125 6

1211

##

= +

= +

= +

= + =

CXmhaU 3 :

EH$m ^m§S>çmV 128

kg gmIa Amho. Ë`mVrb 83 kg gmIa dmnabr. Va am{hboë`m

gmIaoMo à_mU H$mT>m. ^m§S>çmVrb gmIa dmnabobr gmIa

e¡bm Am{U aoUy XmoKtZr gO{dbobm ^mJ =

125

63

125

6 23 2

125

126

125 6

1211

##

= +

= +

= +

= + =

N>oXm§Mr b. gm. {d.

^m§S>çmVrb gmIa = 128 kg

dmnabobr gmIa = 83 kg

am{hbobr gmIa = kg128

83- kg

= kg128

83- -

2 ,2 ,3.2

2 2 3 2

12 86 4

24# # #

=

128

83

12 28 2

8 33 3

2416

249

2416 9

247

##

##

-

= -

= -

= - -

∴am{hbobr gmIa = 247

kg

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ñdmÜ`m` 2.1

I. Imbrb AnyUmªH$m§Mo g_, {df_ Am{U {_l AnyUmªH$mV dJuH$aU H$am.

2 , , ,6 , ,611271618

9245203

II. Imbrb àËòH$mgmR>r XmoZ g__wë` AnyUmªH$ {bhm.

1) 63 2) 5

4 3) 107

III. Imbrb AnyUmªH$m§Mo g§{já ê$n {bhm

1) 128 2) 30

18 3) 4228

IV. Imbrb AnyUmªH$m§Mo {_l AnyUmªH$mV ê$nm§Va H$am.

1) 815 2)12

25 3) 497

V. Imbrb AnyUmªH$m§Mo {df_ AnyUmªH$mV ê$nm§Va H$am.

1) 3 43 2) 8 2

1 3) 465

VI. gmoS>dm.

1) a) 43

53+ 2) 3 2

143

52+ +

3) a) 159

52- 4) 4 5

3 265-

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AnyUmª©H$m§Mm JwUmH$ma :

dJ©[ejH$mZr Imbrb {MÌê$n CXmhaUo gmoS>{dÊ`mg {Xbr. gmZdr Am{U amohZZo Vr Imbrb à_mUo gmoS>{dbr.

1) 41 Mo 5 ^mJ òWo AmhoV Vo OmoS>m Am{U

{bhm.

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) àËòH$ 43 Mo 3 ^mJ òWo AmhoV. Vo OmoS>m

Am{U {bhm.

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

49 2 4

1

+ +

= + +

= = =

Ë¶mM dJm©Vrb lo`mZo darb ~oaOm§Mo {ZarjU Ho$bo Am{U {ejH$mbm {dMmabo.

darb XmoZ ~oaOm `m EH$m AnyUmªH$m§Mr nwZamd¥Vr ~oarO Amho. Aem AnyUmªH$mbm nyU© g§»òZo

Amåhr JwUy eH$Vmo H$m?

hmo`, H$m Zmhr, VwPo ~amo~a Amho. darb CXmhaUo Vy JwUmH$ma {H«$òZo

gmoS>dy eH$Vog.

AnyUmªH$m§Mm JwUmH$ma H$gm H$aVmV? Vo {eHy$¶m.

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AnyUmªH$m§Zm nyU© g§»òZo JwUUo :CXmhaU 1 : Imbrb {MÌo nhm. àËòH$ dVw©imMm {H$Vdm ^mJ a§J{dbobm Amho?

àËòH$ dVw©imMm 41 ^mJ a§J{dbobm Amho.

∴EHy$U a§J{dbobo ^mJ = 41

41

41

41 3#= + + =

AmVm a§J{dbobo {VÝhr ^mJ OmoSy>Z AmnU Ago hr XmIdy eH$Vmo.

hm 43 ^mJ Amho.

åhUyZ 41 3 4

1 343

# #= =

¶oWo A§embm nyU© g§» oZo JwUwZ JwUmH$ma A§eñWm§Zr {b{hbm Amho. Am{U N>oX

4 VgmM N>oXñWmZr R>odbm Amho.

OmUyZ ¿`m : H$moUË`mhr nyU© g§»òbm 1 hm N>oX AgVmo.åhUyZ, 3 ,5 ,81

315

18

41 3 4

113

4 11 3

43

# ###

= = =

= = =

CXmhaU 2 : 2 ?52# = darb JwUmH$mamà‘mUo 5

2 À¶m XmoZ nQ²>Q>çm EH$Ì OmoSy>Z nmhÿ.

52

+ =

52

54

ho JwUmH$ma énmV ¶m [aVrZo gm§JVm ¶oVo.

252

52

12

5 12 2

54

# ###= = =

nyU© g§»`m Am{U A§e `mMm JwUmH$ma →

N>oXmMm JwUmH$ma →

×

×

× ××

×

×

× ××

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× ×

××

××

CXmhaU 3 : ?92 3# =

+ + =

392

92

13

9 12 3

96

# ###= = =

←A§emMm JwUmH$ma←N>oXm§Mm JwUmH$ma

←A§emMm JwUmH$ma

g§{já ê$n nhm

33

96

32

32

#

#= =

n¶m©¶r nÜXV

CXmhaU 4 :

22

125 2

125

12

12 15 2

1210

2 62 5

65

65

#

###

##

#

#

= =

= =

= =

← [gm_mB©H$ Ad`d emoYyZ]

← [g§{já ê$n]

à¶ËZ H$am : a) 73 2# b) 10

3 3# c) 2 94

# d) 5 153

#

{df_ AnyUmªH$mZm nyU© g§»òZo JwUUo :nwT>rb CXmhaU nhm.

CXmhaU 1 : òWo EHy$U {H$Vr ^mJ AmhoV?

41

41

41

41

41

41

41

41

41

41

41

41

41

41

41

←N>oXm§Mm JwUmH$ma

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àËòH$ {MÌm‘Ü¶o 45 Mo 5 ^mJ AmhoV. [ 5

414

×5 = ]EHy$U ^mJ

45 3

45

13

4 15 3

415

415 3 4

3

#

###

=

= = =

= =

¶mMo {‘l AnyUmªH$mV énm§Va H$aVm ¶oVo.

åhUOo 415 3 4

3= ∴ EHy$U ^mJ = 3 43

CXmhaU 1 : 3 ?79

79 3 7

913

7 19 3

727 3 7

6

#

# ###

=

= = = =

CXmhaU 2 : ?10 56

10 56

110

56

1 510 6

560

5 15 12 12

#

# ###

#

#

=

= = = = =

Vwåhr ñdV… H$am : a) 58 3# b) 6

13 2# c) 6 35

# d) 12 78

#

{_l AnyUmªH$m§Zm nyU© g§»òZo JwUUo :CXmhaUmMo {ZarjU H$am.CXmhaU 1 : òWo EHy$U {H$Vr ^mJ {Xbobo AmhoV?

1 1 41 + 1 1 4

1 + 1 1 41

?2 41 2 4

1 2 41+ + =

åhUOo 2 41 Mo 3.nQ>

2 341

49

13

4 19 3

427 6 4

3

#

###

=

= =

= =

nmàr 1 : {_l AnyUmªH$mMo {df_ AnyUmªH$mV ê$nm§Va H$am.

nmàr 2 : Z§Va JwUmH$ma H$am.

∴ EHy$U ^mJ = 3 43

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CXmhaU 2 : 2 7 ?54

514

17

5 114 7

598

19 53

#

###

=

= = =

=

à¶ËZ H$am : a) 5 21 7# b) 3 6

2 3# c) 8 2 41

# d) 12 3 21

#

"Mo' Agboë`m AnyUmªH$m g§~§Yr {H«$`m :CXmhaU 1 : amo{hUrZo 3 dVw©im§Mm àËòH$s AYm© ^mJ Jwbm~r a§JmZo a§J{dbm.

amo{hUrZo EHy$U {H$Vr ^mJ a§J{dbo AmhoV?

ho AmnU Ago åhUVmo 3 Mo 21 AmnU "Mm' qH$dm "Mo' Mm g_mdoe AgUmao AnyUmªH$ gmoS>{dÊ¶mMo

{eHy$.

3 Mo 21 ¶mMr qH$‘V {H$Vr?

3 21

#= ¶oWo Mo åhUOo JwUmH$ma

1 23 1

23 1 2

1##= = = Z§Va JwUmH$ma {H«$¶m H$amdr.

∴ EHy$U a§J{dbobo ^mJ = 1 21

CXmhaU 2 : a{d Kam‘Ü¶o XþY {dH«$sMm ì¶dgm¶ H$aVmo. EHo$ {Xder a{dZo àË oH$s 54 {bQ>a XÿY

8 ~mQ>ë`m_Üò âbo. Va 8 ~mQë`mVrb EHy$U XÿY {H$Vr?

19598

548453

n`m©`r nÜXV

3 Mo 21

321

2 11 3

23 1 2

1

###= =

= =

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34

54 l

54 l 5

4 l 54 l 5

4 l 54 l 5

4 l 54 l

8 ~mQ>ë`mVrb EHy$U XÿY

n`m©`r nÜXV

8 Mo 54

8

l54

1 58 4

532 6 5

2

#

##

=

= = =

gÀ l

l l

l

l

8 54

8 54

18

54

1 58 4

532

6 52

# #

##

=

= =

= =

=

a{dZo 8 ~mQ>ë`mVyZ âbobo EHy$U XÿY = 652 {bQ>a.

CXmhaU 3 : EH$m dJm©Vrb 42 {dÚmÏ`m©n¡H$s, 72 {dÚmWu ìhm°br~m°b IoiVmV. 7

3 {dÚmWu H$amQ>o Am{U am{hbobo {dÚmWu H$ã~S>r IoiVmV. Va -

a) ìhm°br~m°b IoiUmè`m {dÚmÏ`mªMr g§»`m {H$Vr? b) H$amQ>o IoiUmè`m {dÚmÏ`mªMr g§»`m {H$Vr? c) H$ã~S>r IoiUmè`m {dÚmÏ`mªMr g§»`m {H$Vr?

a) ìhm°br~m°b IoiUmè`m {dÚmÏ`mªMr g§»`m

of42 72

142

72

1 742 2 12

121

6

##

#= = =

=

∴ ìhm°br ~m°b IoiUmè`m {dÚmÏ¶mªMr g§»¶m = 12 b) H$amQ>o IoiUmè`m {dÚmÏ`mªMr g§»`m

42 73

142

73 18

18

6

1# #= = =

=

∴ H$amQ>o IoiUmè`m {dÚmÏ¶mªMr g§»¶m = 18

Mo

Mo

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35

c) H$ã~S>r IoiUmè`m {dÚmÏ`mªMr g§»`m = dJm©Vrb EHy$U {dÚmWu - (ìhm°br~m°b IoiUmao {dÚmWu + H$amQ>o IoiUmao

{dÚmWu) = 42 - (12 + 18)

= 42 - 30 = 12

∴ H$ã~S>r IoiUmao {dÚmWu = 12

à¶ËZ H$am : I . 1) a) 9 Mo 34

b) 15 Mo 26

c) 57 Mo 4 d) 20 Mo 5

62) EH$m dJm©V 45 {dÚmWu AmhoV. Ë`mn¡H$s 5

3 ^mJ _wbr AmhoV. Va dJm©Vrb _wbtMr g§»`m {H$Vr? Am{U _wbm§Mr g§»`m {H$Vr?

ñdmÜ`m` 2.2

I. ¶m {MÌmVrb a§Jdbobm EHy$Z ^mJ JwUmH$ma {H«$¶oZo H$mT>m.

Z‘wZm : 3

4

6

4× 2 =+ =

a) 41

41

41

41

41+ + + + =?

b) =?+21 +2

121

c) + =?

d) 31

31 + 3

1

31 + 3

1

31 + 3

1

31 =?

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36

II. JwUmH$ma H$am Am{U CÎma g§{já ê$nmV {bhm.

A. 1) 72 2# 2) 5

4 3# 3) 92 5# 4) 8

7 6#

5) 9 42

# 6) 10 32

# 7) 15 53

# 8) 13 41

#

B. 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. gai ê$n Úm Am{U CÎma g§{já ê$nmV {bhm.

A. 1) 6 of 32 Mo 6 of 32 2) 10 of 5

3 Mo 10 of 53 3) 16 of 9

1 Mo 16 of 91 4) 28 of 7

2 Mo 28 of 72

5) 20of43 Mo 20of43 6) 16of8

2 Mo 16of82 7) 6of10

7 Mo 6of107 8) 4of9

5 Mo 4of95

B. 1)_mYdrZo J{UV {df`mÀ`m MmMUr narjoV 50 Mo 53 JwU {_i{dbo. Va {VZo

{_i{dbobo JwU {H$Vr?

2) EH$m emioZo 60 {dÚmÏ`mªMr EH$ ghb Am`mo{OV Ho$br hmoVr. {dÚmÏ`mªZm Ë`m§À`m

KamH$Sy>Z OodU AmUÊ`mg gm§{JVbo hmoVo. EHy$U {dÚmÏ`m©n¡H$s 123 {dÚmÏ`mªZr MnmVr,

52 {dÚmÏ`mªZr nwbmdm Am{U 20

7 {dÚmÏ`mªZr Xhr^mV AmUbm hmoVm. Va emoYm.

i) MnmVr AmUboë`m {dÚmÏ`mªMr g§»`m {H$Vr?

ii) nwbmdm AmUboë`m {dÚmÏ`mªMr g§»`m {H$Vr?

iii) Xhr^mV AmUboë`m {dÚmÏ`mªMr g§»`m {H$Vr?

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37

AnyUmªH$mbm AnyUmªH$mZo JwUUo

aoí_mZo XþH$mZmVyZ 43 kg gmIa {dH$V AmUbr. Ë`mVrb 2

1 ^mJ gmIa {VÀ`m eoOmaUrZo

Zobr. Va eoOmaUrbm {H$Vr gmIa {_imbr?

aoí‘mZo {dH$V AmUbobr gmIa = 43 kg

eoOmaUrbm {_imbobr gmIa = gÀ43

21= Mo gÀ4

321= kg

= kg43

21

83

# =

`m CXmhaUmV XmoZ AnyUmªH$ AmhoV. ¶oWo AnyUmªH$mbm AnyUmªH$mZo JwUmH$ma H$aÊ`mMo

g‘OmdyZ KoD$.

CXmhaU 1 :

{MÌmVrb a§J{dboë`m ^mJmMm 41 ^mJ {H$Vr?

a§J{dbobm ^mJ 21=

`m a§J{dboë`m ^mJmMm åhUOo gÀ21

41 Mm gÀ2

141 ^mJ

`mMm AW© a§J{dboë`m mJmÀ`m 4 g_mZ mJmn¡H$s 1 mJ. of41

21 Mm of4

121 ^mJ

hm nyU© dñVyMm 81 ^mJ hmo`. ho Imbrb {MÌmV XmI{dbo Amho.

åhUOo

XmoÝhr AnyUmªH$m§À`m A§emMm JwUmH$ma H$ê$Z

A§eñWmZr {bhm. `mMà_mUo XmoÝhr N>oXm§Mm

JwUmH$ma H$ê$Z N>oXñWmZr {bhm.

of

of

21

41

21

41

2 41 1

81

21

41

81

#

##

`

=

= =

=

AmVm AmnU aoí_mZo gmoS>{dboë`m CXmhaUmMr MMm© H$ê$.

aoí_mZo {dH$V AmUbobr gmIa = 43 {H$.J°«.

Mo

Mo

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38

eoOmaUrbm {_imbobr gmIa = of21

43 Mm of2

143^mJ

Aem§Mm JwUmH$ma H$am.N>oXm§Mm JwUmH$ma H$am.kg

43

21

4 23 1

83

#

##

=

= =

[nS>VmiyZ nhm : of43

21Mm of4

321 Am{U of2

143 Mm of2

143 ho g_mZ hmoVmV H$m?]

à¶ËZ H$am : a) 31

41

# b) 53

31

#

c) 65

72

# d)83

52

#

X¡Z§{XZ OrdZmV AmnU g_ AnyUmª©H$ Am{U {_l AnyUmªH$ m§Mm JwUmH$ma H$aV AgVmo. H$gm Vmo nmhÿ.

CXmhaU 1 : {dUmZo ~mhþbr V`ma H$aÊ`mgmR>r 4 21 m H$mnS> {dH$V AmUbo. na§Vy Ë`mn¡H$s

31 ^mJ H$mnS> EH$ ~mhþbr V`ma H$aÊ`mg bmJbo. Va AmUboë`m H$mnS>mMm {H$Vr ^mJ H$mnS>

~mhþbrbm bmJbo?

{dUmZo {dH$V AmUbobo H$mnS> = 4 21= m

EH$m ~mhþbrbm Ë`mn¡H$s bmJbobo H$mnS> = of4 21

31= Mo of4 2

131= ^mJ

4 21

31

29

31

2 39 1

69

2 33 3

23 1 2

1

#

#

##

#

#

=

=

= =

= = =

[421 Mo {df__Üò ê$nm§Va]

∴ eoOmaUrbm {_imbobr gmIa = 83= kg

~mhþbr V¶ma H$aÊ¶mg bmJbobo H$mnS>=

4 21

31

29

31

2 39 1

69

2 33 3

23 1 2

1

#

#

##

#

#

=

=

= =

= = = m

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39

CXmhaU 2 : EH$ _rQ>a [a~ZMr qH$_V ` 4 53 Amho. Jm¡V_rZo 2 2

1 _rQ>a [a~Z IaoXr Ho$br.

Va [a~ZMr qH$_V {H$Vr hmoB©b?

EH$ _rQ>a [a~ZMr qH$_V = `4 53

2 21 m [a~ZMr qH$‘V = ` 4 5

3 2 21

#

= 4 53 2 2

1# (¶oWo XmoZ {‘l AnyUmªH$ AmhoV)

5

5232

1

1

#= ← [{‘l AnwUmªH$mMo {df‘ AnyUmªH$mV ê$nm§Va]

223 11 2

1= =

ñdmÜ`m` 2.3

I. JwUmH$ma H$am Am{U Ambobm JwUmH$ma g§{já ê$nmV {bhm.

1) 65

32

# 2) 87

43

# 3) 54

97

# 4) 159

35

#

5) 31

97

# 6) 311

103

# 7) 74

59

# 8) 21

1213

#

II. JwUmH$ma H$ê$Z CÎmamMo g§{já ê$n {bhm.

1) 43 25

1# 2) 5

4 365

# 3) 35 2 2

1# 4) 7

3 4 31

#

5) 4 53 2 2

1# 6) 5 8

223

# 7) 3 95 5# 8) 6 4

1 351

#

III. ñdV… H$am :

1) 3 52 4# 2) 4 3

253

# 3) 5 43 2 2

1# 4) 25 3 2

1#

∴ 2 21 _r. [a~ZMr qH$_V = ` 11 2

1

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40

IV. Imbrb CXmhaUo gmoS>dm.

1) EH$ emim dmhZ EH$ {bQ>a {S>Pobbm 8 {H$._r. A§Va OmVo. Oa dmhZm_Üò 8 43 {bQ>a

{S>Pob Agob Va, Vo dmhZ {H$Vr {H$._r. A§Va OmB©b?

2) EH$ {bQ>a XÿYmMr qH$_V ` 34 21 . Amho. Va 12 {bQ>a XÿYmMr qH$_V {H$Vr?

3) amoeZrZo EH$ Eo{Vhm{gH$ H$mX§~ar EH$m {Xder 1 43 Vmg BVŠ`m doioÀ`m à_mUmV dmMÊ`mg

gwédmV Ho$br. {VZo Vr H$mX§~ar 6 {XdgmV dmMyZ nyU© Ho$br. Va {VZo EHy$U {H$Vr Vmg dmMZ Ho$bo?

4) EH$ _rQ>a MoZMr qH$_V ` 5 41 Amho. Va 8 5

4 _rQ>a MoZMr qH$_V {H$Vr?

5) EH$m Am`VmH¥$Vr nmVi nÍ`mMr bm§~r 3 21 m Am{U ê§$Xr 2 2

1 m Amho. Va nÍ`mMo joÌ\$i {H$Vr?

[Am¶VmMo joÌ’$i = bm§~r ×é§Xr]

AnyUmªH$m§Mm ^mJmH$ma

amYmZo 4 {~ñH$sQ>o Amnë`m _¡{ÌUtZm g_mZ dmQ>br. àËòH$ _¡{ÌUrbm AYm© {~ñH$sQ>

{_imbm. Va amYmÀ`m {H$Vr _¡{ÌUrZm§ 21 {~ñH$sQ> {_imbo ho Imbr {MÌmV XmI{dbo Amho.

EHy$U {~ñH$sQ>o Ooìhm Vo AY} AY} Ho$bo Ooìhm dmQ>bo

4 {~ñH$sQ>o àËòH$sbm 21 à_mUo 8 _¡{ÌUtZm dmQ>bo Jobo.

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41

ho Ago {bhVmV 4 8 ,21

21 8 4' #= =

åhUyZ 21 8 4# =

AmVm dmnaboë`m {H«$òMm Aä`mg H$ê$.

4 Mo àËòH$s 21 ^mJ åhUOo 4 1

2# [ 4 bm 1

2 Zo JwUyZ]åhUyZ 4 bm 2 Zo JwUyZ

JwUmH$ma ì¶ñVm§H$ ê$n

Ooìhm AnyUmªH$m§Mo A§e d N>oX `m§À¶m ñWmZm§Mr AXbm-~Xb hmoVo Voìhm AmnUmbm _yi

AnyUmªH$m§Mo JwUmH$ma ì`ñV ê$n {_iVo. {Xboë`m CXmhaUm§Mo {ZarjU H$am.

53 Mm ì`ñVm§H$ 3

587 Mm ì`ñVm§H$ 7

8

49 Mm ì`ñVm§H$ 9

432 Mm ì`ñVm§H$ 2

3

• 2 43 Mm JwUmH$ma ì`ñV {H$Vr?

àW_ {_l AnyUmªH$mMo {df_ AnyUmªH$mV ê$nm§Va H$ê$ Z§Va ì`ñV H$mTy>.

2 43 Mm {df_ AnyUmªH$ = 4

11

411 Mm ì`ñV 11

4

• 8 Mm ì`ñV {H$Vr?

8 Mm ì¶ñVm§H$ 81 hmoVmo.

4

421

12

18

8

'

#= =

=

21 À`m A§e d N>oXmMo ñWmZ AXbm-~Xb

Ho$ë`mg AmnUmg 12 {_iVmV. 1

2 hm 21 Mm

JwUmH$ma ì`ñV hmo`.

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42

EH$m nyUmªH$mbm Ë`mÀ`m ì`ñVm§H$mZo JwUë`mg òUmè`m JwUmH$mamMo {ZarjU H$am.

1

1

1

41

14

44

11

52

25

1010

11

32

23

66

11

5 51

55

11 1

#

#

#

#

= = =

= = =

= = =

= = =

hm JwUmH$ma nmhÿZ Vw_Mo åhUUo H$m`? ñnï> H$am

Q>rn : H$moUË`mhr g§»òbm eyÝ¶m{edm` {VÀ`m JwUmH$ma ì`ñV g§»òZo JwUë`mg JwUmH$ma 1 òVmo.

à¶ËZ H$am : Imbrb g§»`m§Mo JwUmH$ma ì`ñV {bhm a) 4

3 b) 95 c) 13 d) 3 4

1

nyU© g§»òbm AnyUmªH$mZo ^mJUoCXmhaU 1 : gai ê$n Úm

nmàr 1 : ^mJmH$ma {MÝhmMo JwUmH$ma {MÝhmV ê$nm§Va H$am d ^mOH$mMm JwUmH$ma ì`ñV {bhm. nmàr 2 : JwUmH$mamMr {H«$`m Mmby R>odm.

CXmhaU 2 : VrZ dVw©imH$ma V~H$S>çmnmgyZ 41 g_mZ ^mJ Agbobo EHy$U {H$Vr ^mJ V`ma

hmoVrb?

8

843

34

18

34

232 10 3

2

'

#

#

=

=

= =

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43

VrZ V~H$S>çmnmgyZ àËòH$s 41 ^mJ Agbobo 12 g_mZ ^mJ V`ma hmoVrb.

ho Ago {b{hVm òVo.

3 41

' nmàr 1 : ^mOH$mMm ì`ñVm§H$ {bhÿZ ^mJmH$ma {MÝhmMo JwUmH$ma {MÝhmV ê$nm§Va H$am

3 14

#= nmàr 2 : JwUmH$ma {H«$òZo gmoS>dm

1 13 4

112 12

##= = =

∴ 3 dVw©imH$ma V~H$S>tMo àËòH$s à_mUo hmoUmao ^mJ = 12

CXmhaU 3 : gai ê$n Úm : 12 265

' Oa ^mOH$ {_l AnyUmªH$ Agob Va Ë`mMo {df_ AnyUmªH$mV H$am Am{U gmoS>dm.

12 2

12

12

65

617

176

1772 417

4

'

'

#

=

=

= =

à¶ËZ H$am : a) 6 53

' b) 8 41

' c) 2 53

'

d) 9 73

' e) 15 2 21

' f) 12 3 41

'

AnyUmªH$mbm nyU© g§»òZo ^mJUo

CXmhaU 1 : 53 2'

òWo ^mOH$ 2 Amho. ^mÁ`mbm 2 À`m ì`ñVm§H$mZo JwUm.

53 2 5

321

103' #= =

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Not to

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44

Aem àH$mao

1) 587

87

51' #= =

2) 1065' = = =

[_l AnyUmªH$mbm nyU© g§»òZo ^mJUo. {‘l AnyUmªH$mbm nyU© g§»¶oZo ^mJVmZm àW‘ H$m¶ Ho$bo nm{hOo? {_l AnyUmªH$mMo {df_ AnyUmªH$mV ê$nm§Va H$edo. Am{U Z§Va ^mJmH$ma H$amdm.

CXmhaU 2 : ?5 41 9

421 9 4

2191

3621

'

' #

=

= =

Aem àH$mao

1) 3 5 521

27

27

51' ' #= = =

2) 4 732' = = =

AnyUmªH$mbm AnyUmªH$mZo ^mJUoCXmhaU 1 : 4

153

'

41 ^m{Jbo 5

3 òWo 41 bm 5

3 À`m ì`ñVm§H$mZo JwUm.

Z§Va 41

53

41

35

125' #= =

CXmhaU 2 : 32

83

'

132

83

32

38

916

97' #= = =

Aem àH$mao1) 10

731

107

13' #= = =

2) 65

73

' = =

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Not to

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shed

45

ñdmÜ`m` 2.4

I. 1. gai ê$n Úm

1) 9 21

' 2) 13 43

' 3) 15 61

'

4) 20 73

' 5) 15 2 31

' 6) 10 3 72

'

2. {Xboë`m AnyUmªH$m§Mm ì`ñVm§H$ {bhm.

1) 52 2) 9

7 3) 121

4) 261 5) 9 6) 4 3

2

3. gai ê$n Úm

1) 32 5' 2) 8

7 3' 3) 76 13'

4) 59 4' 5) 2 4

3 7' 6) 3 21 14'

4. gai ê$n Úm

1) 85

31

' 2) 65

72

' 3) 94

58

'

4) 54 1 2

1' 5) 3 4

3 1 32

' 6) 3 72 15

1'

II. gmoS>dm

1) 6 Mm°H$boQ>g²Mr qH$_V `10 21 Amho. Va EH$m Mm°H$boQ>Mr qH$_V {H$Vr?

2) EH$m emiobm {dÚmÏ`mªZm XoÊ`mg EHo$ {Xder 12 43 l XÿY XoÊ`mV Ambo hmoVo. àËòH$

{dÚmÏ`mªZm 203 l XÿY {_imbo. Va EHy$U {H$Vr {dÚmÏ`mªZm XÿY {_imbo.

3) EH$m {dÚmÏ`mªZo 15 døm ` 142 21 bm IaoXr Ho$ë`m Va EH$m dhrMr qH$_V {H$Vr?

4) EH$m m§S>çmV 10 54 l Xhr Amho. ho Xhr 5

2 l Xhr Agboë`m bhmZ nm{H$Q>mVyZ aë`mg EHy$U {H$Vr nm{H$Q>o hmoVrb?

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46

{_l {H«$`m

Amnë`m X¡Z§{XZ OrdZmV AZoH$ {H«$`m Aem KS>V AgVmV H$s OoWo AmnU AnyUmªH$mdarb {d{dY {H«$`m§Mm dmna H$aVmo.

H$mhr CXmhaUo nmhÿ :

CXmhaU 1 : aoí_mH$S>o `3 21 AmhoV. {VZo Ë`mVyZ `2 4

3 Mr EH$ nopÝgb Am{U ` 21 Mm EH$

a~a {dH$V KoVbm. Va {VÀ`mH$S>o {H$Vr a¸$_ {e„H$ am{hb?

CH$b : oWo Vwåhr 3 21 _YyZ nopÝgb d a~a IaoXr Ho$bobr EHy$_ a¸$_ dOm Ho$br nm{hOo.

nwT>rb nÜXVrMo {ZarjU H$am.

← {_l AnyUmªH$m§Mo {df_ AnyUmªH$mV ê$nm§Va H$ê$Z N>oXm§Mm b.gm.{d. H$mTy>Z

← N>oXm§Mm b.gm.{d. KoD$Z g_mZ AnyUmªH$mV H$ê$Z

3 12 - (2 3

4 + 12 )

= 72 - (11

4 +1 ×22 ×2)

= 72 - (11

4 + 24 )

= 7 ×22 ×2 - 13

4

= 144 - 13

4 =14 -134 = 1

4

bjmV R>odm : {_l {H«$`m H$aVmZm BODMAS {Z¶‘m§Mm H«$_mZo dmna H$am.

CXmhaU 2 : EH$m qeß`mZo àË oH$s 52 _rQ>a bm§~rMo 12 H$mnS>mMo VwH$S>o IaoXr Ho$bo. Ë`mnmgyZ Ë`mZo

2 81 _rQ>a bm§~rMo XmoZ nS>Xo {edbo. Va {H$Vr bm§~rMo H$mnS> {e„H$ am{hbo?

CH$b : H$nS>çmMr EHy$U bm§~r = 52 12#=

XmoZ nS>Xo {edÊ`mg dmnaboë`m H$mnS>mMr bm§~r = 2 81 2#=

{e„H$ am{hboë`m H$nS>çmMr bm§~r ( 25 ×12) - (2 1

8 ×2)

∴ aoí_mH$S>o amhUmar {e„H$ ` 41=

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47

bjmV R>odm : ¶oWo JwUmH$ma Am{U ì¶dH$bZ (dOm~mH$s) {H«$¶m AmhoV. åhUyZ BODMAS {Z¶‘m§Mm Cn¶moJ H$ê$Z JwUmH$ma H$ê$Z dOm~mH$s H$am.

= ( 25 ×12) - (2 1

8 ×2)= ( 2

5 ×121 ) - (17

8 × 21 )

= 245 -17

4

= 24 ×45 ×4 -17 ×5

4 ×5

= 9620 -85

20

= 96 -8520 = 11

20

∴{e„H$ am{hbobo H$mnS> = 1120 m

← N>oXm§Mm b.gm.{d. ¿¶m. 4 d 5 Mm b.gm.{d. 20

CXmhaU 3 : gai ê$n Úm. 2 21

21

41

52

21# '+ -

`m CXmhaUmV ~oarO, JwUmH$ma, mJmH$ma d dOm~mH$s m gd© {H«$`m EH$Ì AmhoV. BODMAS À`m {Z`_mZwgma AmnU àW_ ^mJmH$ma Z§Va JwUmH$ma Z§Va ~oarO d eodQ>r dOm~mH$s H$amdr.

2 12 × 1

2 + ( 14 ÷ 2

5 ) - 12

nm¶ar 141

52

41

25

85' #= =2 1

2 × 12 + ( 5

8 ) - 12

= (2 12 × 1

2 ) + 58 - 1

2

nm¶ar 2 54 + 5

8 - 12

= ( 54 + 5

8 ) - 12

2 21

21

25

21

45

# #= =

nm¶ar 3815

21-

815

2 41 4

815

84

815 4

811

##-

= -

= - =

45

85

4 25 2

85

810 5

815

##+ = +

= + =

nm¶ar 4 1811

83=

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48

ñdmÜ`m` 2.5

I. gai ê$n Úm

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

'- +

II. Imbrb CXmhaUo gmoS>dm

1) emioVrb EH$m H$m ©H«$_mgmR>r 15 m [a~Z AmUbr. Ë`mn¡H$s m441 m [a~Z VmoaUgmR>r

dmnabr, 4107 m ~°OoggmR>r dmnabr d Cabobr [a~Z hñV{b{IV _m{gH$mgmR>r

dmnabr. Va _m{gH$mgmR>r dmnaboë`m [a~ZMr bm§~r {H$Vr?

2) EH$m H$m`©H«$_mÀ`m Am`moOH$mZo 3 21 kg gmIa 4 doim {dH$V AmUbr. Ë`mn¡H$s 9 4

3

kg gmIa dmnabr. Va {e„H$ am{hbobr gmIa {H$Vr?

3) am‘ßnmbm EH$ eQ>© {edÊ¶mg 2 21 m bm§~rMo H$mnS> bmJVo. XþH$mZH$mamH$S>o 2

1 m

bm§~Mr ‘moOnÅ>r Amho. Va 2 21 m H$mnS> ‘moOÊ¶mg {H$Vr doim ‘moOnÅ>rZo ‘moObo

nm{hOo?

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àH$aU - 3

n[a‘o¶ g§»¶m [ RATIONAL NUMBERS ]

`m àH$aUmÀ¶m AÜ``ZmZ§Va Vwåhr :

n[a_o` g§»`mMm AW©, à_m{UV ê$n Am{U g_ê$nVm g_OmdyZ ¿`mb.

g§»`maofoda n[a_o` g§»`m XmI{dÊ`mMo H$m¡eë` àmá H$amb.

n[a_o` g§»`m§Mr VwbZm, XmoZ n[a_o` g§»ò_Yrb n[a_o` g§»`m emoYÊ`mÀ`m nÜXVr {eH$mb.

n[a_o g§»`m_Yrb {H«$`m (~oarO, dOm~mH$s, JwUmH$ma, mJmH$ma) g_OmdyZ ¿`mb.

n[a_o` g§»`m Xem§e AnyUmªH$mV {b{hÊ`mÀ`m nÜXVr g‘OyZ ¿¶mb.

Xem§e AnyUmªH$mVrb JwUmH$ma, mJmH$ma {H«$`m g_OmdyZ ¿`mb. VgoM H$mhr _hËdmMr CXmhaUo gmoS{dÊ`mÀ`m nÜXVr g_OyZ ¿`mb.

n[a_mU _moOÊ`mÀ`m nÜXVr g_OmdyZ ¿`mb (bm§~r Am{U ametMo ‘mnZ B.)

n[a_o` g§»`mMm AW© :

Vwåhmbm Z¡g{J©H$ g§»`m (N), nyU© g§»`m (w), nyUmªH$ (Z) ~Ôb ‘m{hVr Amho.

N = { 1, 2, 3, 4....................}

W = { 0, 1, 2, 3....................}

Z = {....................-4, -3, -2, -1, 0, 1, 2, 3....................}

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gwédmVrbm \$º$ Z¡g{J©H$ g§»`m hmoË`m. dñVy Zmhr ho Xe©{dÊ`mgmR>r "0' (eyÝ`) Mm emoY bmdÊ`mV Ambm. Aem àH$mao nyU© g§»`m§Mm gQ> CX`mg Ambm. Z§Va YZ nyUmªH$ d G$U nyUmªH$À`m gQ>m§Mm {dH$mg Pmbm. Ë`m_wio XmoZ Z¡g{J©H$ g§»`mMr dOm~mH$s H$aUo gmono Pmbo. CXm: 3-5=-2, 10-15=-5 Aem dOm~mH$sVyZ Z{dZ gQ> V¶ma Pmbm -2, -5 .... ¶m§Zm

G$U nyUmªH$ åhUVmV. Aem [aVrZo {d{dY JaOmVyZ "g§»`mgQ>' {dH${gV Pmbm;.

OmUyZ ¿`m : gQ> hm eãX {Z{üV AmoiIVm òUmè`m dñVy qH$dm g§»`m `m§Mm JQ> Xe©{dVmo.

`mMà_mUo, Ooìhm EH$m nyUmªH$mZo Xþgè`m nyUmªH$mbm ^mJbo AgVm H$mhr doim nyUmªH$

Zgboë¶m g§»¶m {‘iVmV. CXm : ,53

710- . `m g§»`m nyUmªH$ gQ>mV òV ZmhrV. åhUyZ n[a_o`

g§»`m gQ> (Q) ApñVËdmV Ambm.

Amåhr _mJrb dJm©V AnyUmªH$m {df`r {eH$bmo AmhmoV. AnyUmªH$ ho pq `m ñdê$nmV

AgVmV. ¶oWo q ≠ o Am{U p Am{U q `m Z¡g{J©H$ g§»`m. nU Ooìhm pq `m ñdê$nmVrb

AnyUmªH$m§Mr dOm~mH$s H$aVmo Voìhm H$mhr doim G$U g§»`m {_iVmV. Aem p Am{U q d

q > 0 nyUmªH$ Agboë¶m pq ñdê$nmVrb g§»¶m§Zm n[a‘o¶ g§»¶m åhUVmV.

pq ñdê$nmV Agboë`m g§»`m n[a_o` g§»`m AgVmV. p

q ‘Ü¶o q ≠ 0 Am{U p, q ho

nyUmªH$ AmhoV.

n[a_o` g§»`m§Mm gQ> hm Zoh_r 'Q' AjamZo Xe©{dVmV.

Q = { pq `m ñdê$nmVrb gd© g§»`m, q > 0, Am{U p, p ho nyUmªH$ }.

n[a‘o¶ g§»¶oMo CXmhaU : , , , , , , ...etc32453074

715

94103- - -

- BË`mXr

gd© Z¡g{J©H$ g§»`m, nyUmªH$ Am{U nyU© g§»`m m n[a_o` g§»`m åhUyZ AmnU {bhÿ eH$Vmo.

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CXmhaU : 3 ,12 ,8

0 ,0 ,0

, ,

13

112

18

10

20

100

5 15 7 1

7 10 110

= = =

= = =

- = - - = - - = -

åhUyZ n[a‘o¶ g§»¶mgQ>m‘Ü¶o N, W, Z,Q ¶m gd© gQ>m§Mm

g‘mdoe hmoVmo.

åhUyZM n[a_o` g§»`m§Mm gQ> hm _moR>m gQ> Amho.

NW

ZQ

{dMma H$am! gd© AnyUmªH$ n[a_o` g§»`m AgVmV. na§Vy gd© n[a_o` g§»`m AnyUm§H$ ZgVmV?

OmUyZ ¿`m : n[a‘o¶ g§»¶m (Rational Numbers) hm eãX Ratio (JwUmoÎma) ¶m eãXmdéZ Ambobm Amho.

pq = p:q ¶mdê$Z n[a‘o¶ g§»¶m§Zm JwUmoÎmar¶ g§»¶m gwÜXm åhUVmV.

n[a_o` g§»`m§Mo à_m{UV ê$n (g§{já ê$n) :

¶m n[a_o` g§»`m§Mo {ZarjU H$am

, , , ,5341107

73114-

Aem gd© g§»ò_Yrb A§e d N>oX `m§Mm g_mB©H$ Ad`d 1 Agob Va Ë`m n[a_o` g§»`m

à_m{UV ê$nmV (g§{já ê$nmV) AmhoV Ago åhUVmV.

104 nhm (A§embm d N>oXmbm 2 Zo ^mJm)

104

52` = ( is10

452 Mo à‘m{UV ê$n is10

452 Amho)

2515 = 25

1553

5

3

= (A§e Am{U N>oX ¶m§Zm 5 Zo mJ XoD$Z) 2515 Mo à_m{UV$ ê$n 5

3 Amho.

3627

43- = - (à_m{UV ê$n) 36

2743- = - (A§embm d N>oXmbm 9 Zo ^mJyZ)

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n[a_o` g§»ò_Yrb A§e d NoXm§Ë¶m Ë¶m§À¶m> ‘hÎm‘ gmYmaU Ad¶d Zo ^mJë`mg Ë`m

n[a_o` g§»òMo à_m{UV ê$n (g§{já ê$n) {_iVo.

CXmhaUo :

a) 5522

52- = - , b) 30

16158

-- = , c) 39

2632- = - , d) 63

392113=

g_mZ n[a_o` g§»`m :g_mZ AnyUmªH$ {_i{dÊ`mÀ¶m nÜXVrMo ñ‘aU H$am. n[a_o` g§»òVrb A§embm Am{U

N>oXmbm eyÝà{hV nyUmªH$mZo JwUyZ qH$dm ^mJyZ AmnU {Xboë`m n[a_o` g§»òMr g_mZ n[a_o` g§»`m {_idy eH$Vmo.

CXmhaU 1 : 32 n[a_o` g§»òÀ`m 4 g_mZ n[a_o` g§»`m {bhm.

a) 3 22 2

64

## = b) 3 3

2 396

## = c) 3 5

2 51510

## = d) 3 4

2 4128

## =

CXmhaU 2 : 75- n[a_o` g§»òÀ`m 4 g_mZ n[a_o` g§»`m {bhm.

a) 7 25 2

1410

##- = - b) 7 3

5 32115

##- = -

c) 7 45 4

2820

##- = - d) 7 5

5 53525

##- = -

CXmhaU 3 : 8040 À`m 4 g_mZ n[a_o` g§»`m {bhm.

a) 80 240 2

4020

'' = b) 80 4

40 42010

'' =

c) 80 540 5

168

'' = d) 80 10

40 1084

'' =

OmUyZ ¿`m : Oa p1

q1 Am{U

p2

q2 `m g_mZ n[a_o` g§»`m AgVrb Va p1 × q2 = p2 × q1.

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g§»`maofoda n[a_o` g§»`m Xe©{dUo :

`m AJmoXa nyU© g§»`m Am{U nyUmªH$ g§»`maofoda H$go Xe©{dVmV mMr _mhrVr Vwåhmbm Amho.

g§»`maofoda eyÝ` (0) ho ‘Ü¶mda AgyZ YZ nyUmªH$ g§»`maofoÀ`m COì`m ~mOybm Am{U

S>mì`m ~mOybm G$U nyUmªH$ eyÝ`mnmgyZ g_mZ A§Vamda AgVmV.

CXmhaU 1 : 53 g§»`maofoda XmIdm.

g§»`maofoda n[a_o` g§»`m XmI{dÊ`mgmR>r n[a_o` g§»ò_Yrb N>oXmBVHo$ g§»`maofoMo g_mZ

^mJ H$am. Z§Va n[a_o` g§»ò_Yrb A§emBVHo$ g_mZ ^mJ H$am.

0

0 +1-1

53+

0 Vo 1 n¶ªVÀ¶m A§VamMo 5 g_mZ ^mJ H$am.

CXmhaU 2 : 25 g§»`maofoda XmIdm.

0

-3 -2 -1 0 +1 +2 +3

25

òWo àËòH$ EH$H$mMo 2 g_mZ ^mJ Ho$bo AmhoV.

CXmhaU 3 : 47- g§»`maofoda XmIdm. ¶oWo àË¶oH$ EH$H$ 4 g‘mZ ^mJmV {d^mJbo Amho.

-2 -1 0 +1 +2

047- 4

744

44-

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ñdmÜ`m` 3.1

I. Imbrb MyH$ H$s ~amo~a Vo gm§JyZ, MyH$ Agë`mg ~amo~a H$ê$Z {bhm.

1) - 12 hr n[a_o` g§»`m Amho. 2) 03 hr n[a_o` g§»`m Amho.

3) 43- hr n[a_o` g§»`m Amho. 4) 7

5- hr n[a_o` g§»`m Amho.

5) 40 hr n[a_o` g§»`m Amho. 6) 8

5-- hr YZ n[a_o` g§»`m Amho.

7) 2812 hr g§{já ê$n Amho.

II. n[a_o` g§»`m à_m{UV ê$nmV {bhm.

1) 2028 2) 150

120 3) 4015- 4) 56

32--

III. Imbrb àËòH$ g§»`m§À¶m Mma g_mZ n[a_o` g§»`m {bhm.

1) 92- 2) 10

3 3) 54- 4) 96

80

IV. n[a_o` g§»`m g§»`maofoda XmIdm (àËòH$ JQ>mgmR>r doJir g§»`maofm dmnam)

1) , ,73

7475- 2) , ,4

74549-

3) , ,35

3238- 4) , ,5

65758-

V. Imbrb n[a_o` g§»ò§Mo YZ n[a_o` g§»`m d G$U n[a_o` g§»`m Ago JQ> V`ma H$am.

, , , , , ,5467

35

710

97152

611-

- -- -

--

VI. [aH$må`m OmJm âm.

1) 43 12= 2) 6

524

- = 3) 2812 3- = -

4) 2015

4= 5) 18 14

9- = 6) 12 2

5- = -

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n[a_o` g§»òdarb {H«$`m

AnyUmªH$ d nyUmªH$ `m§Mr ~oarO, dOm~mH$s, JwUmH$ma d ^mJmH$ma `m {H«$`m Vwåhr `mnyduM {eH$bm AmhmV. AmVm AmnU n[a_o` g§»òdarb ¶m {H«$`m nmhÿ.

~oarO : AnyUmªH$mMr ~oarO H$aVmZm Oo {Z`_ AmnU dmnaVmo VoM {Z`_ oWo gwÜXm dmnaVmV. Oa N>oX g_mZ Agob Va VmoM N>oX {bhÿZ A§emMr ~oarO H$ê$Z Vmo {bhmdm.

CXmhaU 1 :

Oa N>oX doJdoJim Agob Va N>oXm§Mm b.gm.{d. KoD$Z Ë`m§Mo N>oX g_mZ H$ê$Z Z§Va Ë`mMr ~oarO H$amdr.

a) 32

31

32 1

33 1+ = + = =

b) 52

51

52 1

53+ = + =

c) 87

82

87 2

85- + = - + = -

d) ( )107

103

107 3

104

52+ - = + - = =

CXmhaU 2 :

a)32

41

128 3

1211+ = + = 3 d 4 Mm b.gm.{d. = 12

b) (-35)+ 7

10

-6 + 77 =

110

5 Am{U 10 Mm b.gm.{d. = 10

c) (-56)+ (-4

5)

-25 - 2430 = -

4930

6 d 5 Mm b.gm.{d. = 30

d)35

21

43

1220 6 9

1235

+ +

= + + =

2, 3 d 4 Mm b.gm.{d. = 12 3,2,42

32

3,1,21,1,21,1,12 × 3 × 2 = 12

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CXmhaU 3 : em§Vå_m mOrÀ`m XþH$mZmVyZ 21 kg {~Zrg, 4

1 kg {hadr {_aMr, 23 kg ~Q>mQ>o

d 101 kg Ambo IaoXr H$aVo. Va {VZo IaoXr Ho$boë¶m ^mOrMo EHy$U dOZ {H$Vr?

CH$b :

{~ZrgMo dOZ = 21 kg

{haì¶m {_aMrMo dOZ = 41 kg

~Q>mQ>çmMo>o dOZ = 23 kg

Amë¶mMo dOZ = 101 kg

∴ EHy$U dOZ 21

41

23

101

2010 5 30 2

2047

2 207

= + + +

= + + +

=

- kg

2, 4, 2, 10 Mm b.gm.{d. = 20

2,4,2,101,2,1,51,1,1,51,1,1,1

225

2 × 2 × 5 = 20

CXmhaU 4 : EH$m _wbmbm eQ>©gmR>r 45 m H$mnS>, n°ÝQ>gmR>r 3

2 m H$mnS> Am{U Q>monrgmR>r

41 m H$mnS> bmJVo. Va Ë`mbm EHy$U {H$Vr ‘rQ>a H$mnS> bmJob?

CH$b : eQ>©gmR>r bmJUmao H$mnS> = 45 m

n°ÝQ>gmR>r bmJUmao H$mnS> = 32 m

Q>monrgmR>r bmJUmao H$mnS> = 41 m

∴ EHy$U bmJUmao H$mnS> 45

32

41

1215 8 3

1226

2122

= + +

= + +

=

=

4, 3 d 4 Mm b.gm.{d.

4,3,41,3,11,1,1

43

b.gm.{d. 4 × 3 = 12

= 2 16

m

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dOm~mH$s

nyUmªH$m_Üò 8 ‘YyZ 5 dOm H$am. åhUOoM 8 + (-5). Vo åhUOoM dOm hmoUmè`m g§»òMm ~oarO ì¶ñV KoUo hmo`.

∴ 8 + (-5) = 3

Ë`mMà_mUo, 7 _YyZ -4 dOm H$am.∴ 7+4 = 11 (-4 Mm ~oarO ì`ñV $+4)n[a_o` g§»`m§À`m dOm~mH$s_Üò gwÜXm hmM {Z`_ dmnaVmV.

CXmhaU 1 :

a) 75 ‘YyZ 7

1 dOm H$am.

= 57 +(- 1

7 ) = 57 - 1

7 = 5 -17 = 4

7b) 87 _YyZ 8

1- dOm H$am.

= 7

8 -(- 18 ) = 7

8 + 18 = 7 +1

8 = 81

81 = 1

doJdoJù`m N>oXmÀ`m n[a_o` g§»`m§Mr dOm~mH$s H$aVmZm Ë`m§Mo N>oX àW_ g_mZ H$ê$Z nwT>o gmoS>dmdo.

CXmhaU 2 : from32

65 _YyZ from3

265 dOm H$am.

65

32

65 4

61

-

= - =

6 d 3 Mm b.gm.{d = 66,32,11,1

32

3 × 2 = 6

CXmhaU 3 : from43

711 _YyZ from4

3711 dOm H$am.

711

43

2844 21

2823

-

= -

=

7 d 4 Mm b.gm.{d = 28

7,41,41,1

74

7 × 4 = 28

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CXmhaU 4 : ( -49 ) _YyZ ( 5

18) dOm

H$am.94

185

188 5

1813

= -

= - -

= -

9 d 18 Mm b.gm.{d = 18

9,181,21,1

92

9 × 2 = 18

CXmhaU 5 : Jm§Yr O`§VrÀ`m {Xder {ejH$m§Zr {dÚmÏ`mªZm dmQ>Ê`mgmR>r 13 43 kg {‘R>mB©

AmUbr. Oa Ë`m§Zr 12 87 kg {‘R>mB© {dÚmÏ`mªZm dmQ>brVa {e„H$ am{hboë`m {‘R>mB©ModOZ

{H$Vr?

AmUboë`m {‘R>mB© dOZ = 13 43 kg

{dÚmÏ`mªZm Mm°H$boQ>Mo dOZ = 12 87 kg

∴ {e„H$ am{hboë`m {‘R>mB©Mo dOZ = 13 12

1043

87

455

83

8110 103

87

= -

= -

= -

= kg

4 ,2 ,

,

4 81 21 1

Mm b.gm.{d =4 × 2 = 8

∴ {e„H$ am{hboë`m Mm°H$boQ>Mo dOZ = 78 kg

CXmhaU 6 : ahr_Zo Amnë`m nJmamVrb 21 mJ AmhmamgmR>r, 5

1 mJ _wbm§À`m {ejUmgmR>r,

41 ^mJ BVa Jmoï>rgmR>r IM© Ho$bm d am{hbobm ^mJ ~MVgmR>r R>odbm. Va nJmamVrb {H$Vr

^mJ ~MVrgmR>r R>odbm?

ahr‘Zo nJmamVrb AmhmamgmR>r IM© Ho$bobm ^mJ 21=

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nJmamVrb _wbm§À`m {ejUmVrb IM© Ho$bobm ^mJ 51=

nJmamVrb BVa Jmoï>rgmR>r IM© Ho$bobm ^mJ 41=

∴ nJmamVrb EHy$U IM© Ho$bobo ^mJ = 21

51

41

2010 4 5

2019

= + +

= + +

= ^mJ

2,5,41,5,21,1,21,1,1

252

2 × 5 × 2 = 20

b.gm.{d. 2, 5, 2 = 20Oa EHy$U nJma 1 _mZbm Va

∴ {e„H$ am{hbobm ^mJ = 1 2019

2020 19

201

= -

= -

=

∴ ahr_Zo ~MVgmR>r 201 ^mJ {e„H$ R>odbm.

ñdmÜ`m` 3.2

1. ~oarO H$am.

1) 52

107+ 2) 3

292+ 3) 5

365- + 4) 7

531

23+ +

2. dOm~mH$s H$am.

1) 85 _YyZ 4

1 2) 65 _YyZ 6

1-

3) 53 _YyZ 15

4 4) 87 _YyZ 4

3

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3. CfmZo ~mOmamVyZ 5 43 kg H$S>YmÝò Am{U 2 3

2 kg ^mOr IaoXr Ho$br. Va IaoXr Ho$boë`m H$S>YmÝò d ^mOrMo EHy$U dOZ {H$Vr?

4. emiobm "jra ^m½`' `moOZo A§VJ©V 15 34 kg XÿY nmdS>a EH$m AmR>dS>çmV

{_imbr. Ë`mVrb 14 15 kg XÿYnmdS>a dmnabr. Va {H$Vr XÿY nmdS>a {e„H$

am[hbr?

5. adrZo H$mnS> XþH$mZmVyZ 8 75 m H$mnS> {dH$V KoVbo. Ë`mVrb 4 3

1 m H$mnS>

_wbmgmR>r d 3 35

m H$mnS> _wbrgmR>r H$nS>o {ebmB©gmR>r dmnabo. Va {H$Vr H$mnS> {e„H$ am{hbo?

6. {bbmÀ¶m KamVrb 4 JmBªZr EHo$ {Xder 20 35 l XÿY {Xbo. Ë¶mVrb

18 23 l XÿY {bbmZo {dH$bo. Va {H$Vr XþY {e„H$ am{hbo?

n[a_o` g§»`m§Mm JwUmH$ma :

AnyUmªH$m§À¶m JwUmH$mamà_mUoM n[a_o` g§»`m§Mm JwUmH$ma H$aVmV.

CXmhaU 1 :

1) 43

13

49

# = gyMZm : n[a_o` g§»`m§Mm JwUmH$ma H$aVmZm A§emZo A§embm d N>oXmZo N>oXmbm JwUmdo.

2) 32

34

98

# =

3) 43 5 4

15#- = -

4) -21

5 ×(- 342

) = 310

5) 43

21

65

4 2 21 5

165

1

2= =# #

# ##

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61

CXmhaU 2 : EH$ {dÚmWu 100 nmZr nwñVH$mVrb 52 ^mJ EH$ {XdgmV dmMVmo. Va

Ë`m {dÚmÏ`mªZo EHy$U {H$Vr nmZo dmMbr?

CH$b : EHy$U nmZo = 100

{dÚmÏ`m©Zo dmMbobm EHy$U ^mJ = 52 Mo 100

10052 4020= = nmZo

CXmhaU 3 : EH$ _rQ>a H$mnS>mMr qH$_V 30 43 Amho, Va Ë`mM qH$_VrZo 14 2

1 _rQ>a H$mnS>mMr qH$_V {H$Vr?

CH$b : 1 _rQ>a H$mnS>mMr qH$_V = 30` 32=

14 21` _rQ>a H$mnS>mMr qH$_V =

30 43 14 2

1

4123

229

3567

14 445§mÉÖAiÀÄ¨É¯É `m8 445 8

1

21

81

=

=

#

#=

=

=

30 43 14 2

1

4123

229

3567

14 445§mÉÖAiÀÄ¨É¯É `m8 445 8

1

21

81

=

=

#

#=

=

=

n[a_o` g§»`mMm JwUmH$ma ì`ñVn[a_o` g§»òÀ`m A§emMr d N>oXmMr AXbm~Xb H$ê$Z À`m n[a_o` g§»òMm JwUmH$ma

ì`ñV {_i{dVm òVmo.

CXmhaU : 1) 43 Mm JwUmH$ma ì`ñV 3

4

2) 25 Mm JwUmH$ma ì`ñV 5

2

3) 103- Mm JwUmH$ma ì`ñV 3

10-

4) 4 Mm JwUmH$ma ì`ñV 41

5) a Mm JwUmH$ma ì`ñV a1 .

OmUyZ ¿`m : JwUmH$ma ì`ñV {b{hVmZm {MÝh ~XbV Zmhr. g§»`m d {VMm ì`ñVm§H$ `m§Mm JwUmH$ma 1 AgVmo. (JwUmH$mamMm A{dH$maH$ KQ>H$ 1 Amho) a a

1 1# =

.

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62

{dMma H$am ! : eyÝ` (0) bm JwUmH$ma ì`ñV KQ>H$ ZgVmo H$m?

n[a_o` g§»`m§Mm ^mJmH$ma :

Vwåhmbm _mhrV Amho H$s, EH$m AnyUmªH$mbm Xþgè`m AnyUmªH$mZo ^mJUo åhUOoM ^mÁ` VgmM R>odyZ ^mOH$mMm ì`ñV KoD$Z JwUmH$ma H$aUo. `mMà_mUo n[a_o` g§»`m§Mm ^mJmH$ma Ho$bm OmVmo.

CXmhaU 1 : a) 43

21

' åhUOoM 43

12

23

2

1=#

b) 52

103

'- åhUOoM

52

310

34

1

2- = -#

c) 65

34

65

43

85

2- - = - - =+' #

d) 83

76

83

67

167

1

2= =' #

e) 54

32

54

23

56

2- =

-= -' #

f) 1512 10

101

756

1512

252

5

6

25

2- = =

-' #

-= -

CXmhaU 2 : 45 {dÚmÏ`mªZm 11 41 l XÿY 45 {dÚmÏ¶mªZm g_mZ dmQ>bo AgVm, àË oH$ {dÚmÏ`m©bm

{H$Vr XÿY {_imbo?

CH$b :

EHy$U XÿYmMo dOZ =11 41 l

EHy$U {dÚmWu = 45

∴ àËòH$ {dÚmÏ`m©bm {_iUmao XÿY = 111 45

45 45

41 l

4

4

445

451

=

=

=

=

'

'

#

∴

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63

∴ àËòH$ {dÚmÏ`m©bm 41 l XÿY {_iob.

CXmhaU 3 : 3 {_Ìm§Zm EHy$U 3 53 kg g\$aM§X g_mZ dmQ>br, Va àËòH$mbm {H$Vr dOZmMr

g\$aM§X {_imbr?

CH$b :

g\$aM§XmMo EHy$U dOZ 3 53 kg

EHy$U {_Ì = 3

∴ àËòH$mbm {_iboë`m g\$aM§XmMo dOZ = 3 53 3

518 3

518

31

56

151

6

=

=

= =

=

'

'

#

kg

àËòH$mbm 151 kg dOZmMr g\$aM§Xo {_imbr.

ñdmÜ`m` 3.3

1. JwUmH$ma H$am :

1) 45

103

# 2) 87

425

#- 3) 15

7143

#-

4) 65

152

#- - 5) 9

786

72

# #- - 6) 10

935

32

# #

2. ^mJmH$ma H$am :

1) 91

73

' 2) 87

67

' 3) 32

95

' 4) 54

107

'- -

3. Imbrb g§»¶m§Mo ~oarO ì`ñV {bhm :

1) 53- 2) 4

9 3) -10

4) 27 5) 5

1- 6) 0.

4. Imbrb g§»¶m§Mo JwUmH$ma ì`ñV {bhm.

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64

1) 109 2) 6

5- 3) 1813- 4) 7

5-

5. EH$m S>ã¶m_Üò 15 43 kg gmIa Amho. àËòH$ n°Ho$Q>_Üò 1 4

3 kg gmIa aë`mg EHy$U

n°Ho$Q> V`ma hmoVrb?

6. EH$m {dÚmÏ`m©bm emioMm S´>og {edÊ`mgmR>r 2 43 m H$mnS> bmJVo. Va 44 {dÚmÏ`mªZm

S´>og {edÊ`mgmR>r EHy$U {H$Vr ‘rQ>a H$mnS> bmJob?

7. EH$m Am`VmH¥$Vr ~mJoMr bm§~r 351 m Am{U é§Xr 2 4

3 m Agob Va Ë`m ~mJoMo joÌ\$i

{H$Vr?

8. EH$m emioV 20 54 l XÿY XoÊ`mgmR>r AmUÊ`mV Ambo. àËòH$ {dÚmÏ`m©bm 5

1 l XÿY dmQ>bo

Va emioV EHy$U {H$Vr {dÚmWu AmhoV?

n[a_o` g§»`m Xem§e AnyUmªH$mV {b{hUo

AnyUmªH$ Xem§e énmV H$go {bhrVmV ho AJmoXaM Vwåhmbm _mhrV Amho. Ë`mMà_mUo n[a_o` g§»`m gwÜXm Xem§e AnyUmªH$mV {b{hVmV.

Xem§e g§»òV XmoZ ^mJ AgVmV.

61.35 _Üò 61 hm nyUmªH$ ^mJ d 35 hm Xem§e AnyUmªH$ ^mJ Amho.

61.35 dmMÊ`mMr nÜXV → EH$ï> q~Xÿ VrZ nmM

Ë`mMà_mUo 27.834 dmMÊ`mMr nÜXV → gÎmmdrg q~Xÿ AmR> VrZ Mma.

H¥$Vr :

EH$mM AmH$mamMr âb°e H$mS>© KoD$Z Ë`mda doJdoJù`m Xem§e g§»`m {bhÿZ EH$m ~m°Šg_Üò

R>odm, Z§Va àËòH$ {dÚmÏ`m©bm EH$ âb°e H$mS>© KoÊ¶mg gm§JyZ Ë`mdarb nyUmªH$ mJ d Xem§e

^mJ ì`dpñWV dmMm`bm gm§Jm. àËòH$ A§H$mMo ñWmZ_yë` gm§JÊ¶mg gm§Jm.

nyUmªH$m§Mo Xem§emV ê$nm§Va H$aUo.ho Vwåhmg ‘mhrZ Amho.

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65

CXmhaU 1 :

a) .107 0 7= b) .10

27 2 7=

c) .53 0 6= d) .4

3 0 75=

e) 0.002001 5= f) .3

1 0 3333= ...

g) .65 0 8333= h) .11

1 0 090909= ...

Aem àH$mao H$moUË¶mhr n[a‘o¶ g§»¶obm Xem§e g§»¶oV {b{hVm ¶oVo.

darb CXmhaUmdê$Z EH$m n[a‘o¶ g§»¶oMo Xem§e ê$n {b{hVmZm Ë¶m Xem§emV EH$ A§H$ qH$dm EH$ nojm OmñV A§H$ ¶oVmV. H$mhr R>am{dH$ A§H$m§Mo nwZamdV©Z hmoVo.

CXmhaU : a , b , c , d Am{U e ‘Ü¶o Xem§e ñWm§Zm‘Ü¶o ‘¶m©{XV A§H$ ¶oVmV. åhUyZ Aem AnyUmªH$m§Zm gm§V Xem§e AnyUmªH$ åhUVmV.

f , g , h ¶m CXmhaUm‘Ü¶o Xem§eñWmZm‘Ü¶o EH$ A§H$ qH$dm EH$nojm OmñV A§H$m§Mo nwZamdV©Z Pmbobo {XgVo. nwT>o {H$Vr hr mJmH$ma Ho$bm Var Ë¶mM A§H$m§Mo nwZamdV©Z hmoVo. åhUyZ ¶m§Zm AmdVu qH$dm nwZamdVu Xem§e AnyUmªH$ åhUVmV.

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66

1) gm§V Xem§e AnyUmªH$ :

a) 3.57 b) 15.035 c) 0.0004321 d) 10.23576

2) AmdV© qH$dm nyZamdVu Xem§e AnyUmªH$ :

a) 2.544 ................ b) 10.272727 ................

c) 15.145614561456 ................ d) 18.321321321.............

{dMma H$am : gd© n[a_o` g§»`m A§V Xem§e AnyUmªH$mV qH$dm nwZamdVuV Xem§e AnyUmªH$mV {bhrVm oVmV. na§Vy, Xem§e ñWimMm eodQ> Pmbm Zmhr qH$dm ñWimMr nwZamd¥Îmr Pmbr Zmhr Va Vr H$moUË`m àH$maMr g§»`m Amho? Vr b

a `m ñdê$nmV {bhrVmV.

CXmhaU : ¶m n[a‘o¶ g§»¶m§Mo Xem§emV ê$nm§Va H$am.

A) ... .72 0 285714285714= b) 1 1.9

5 555= ...

.914 1 555= ...

c) .4152

1562 4 133= = ...

d) 0.1253 024=

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ñdmÜ`m` 3.4

1. Imbrb n[a_o` g§»`m Xem§e AnyUmªH$mV {bhm.

1) 54 2) 8

3 3) 165 4) 14

3

2. Imbrb n[a_o` g§»`mn¡H$s A§V Xem§e AnyUmªH$ d A§V Zgbobo nwZamdVuV Xem§e AnyUmªH$ AmoiIm.

1) 85 2) 12

1 3) 92 4) 16

1

3. Imbrb Xem§e An©UmªH$m§Vrb Xem§e AnyUmªH$, gm§V Zgbobo nwZamdVuV Xem§e AnyUmªH$ Am{U g§V Zgbobo nwZamdVuV Zgbobo Xem§e AnyUmªH$ emoYm.

1) 8.751 2) 2.5444 .......... 3) 0.030.03

4) 9.2874 5) 2.456731456731456731 ..........

6) 10.56173824931685 .......... 7) 3.147521896397 ..........

Xem§e AnyUmªH$ n[a_o` g§»òV {b{hUo

Xem§e AnyUmªH$mMo n[a‘o¶ g§»¶oV ê$nm§Va H$aVmZm q~XÿZ§VaMr g§»¶m A§emñWmZr {bhmdr.

N>oXñWmZr 1 {bhÿZ Xem§emV q~XÿZ§Va {H$Vr A§H$ AmhoV ho nmhÿZ {VVHo$ eyÝ¶ 1 Z§Va {bhmdoV.

CXmhaU : 1) .0 3 103= 2) 0.12 100

12= 3) .1 4 1104

1014= =

Xem§e AnyUmªH$m§Mm JwUmH$ma

Xem§e g§»òMm JwUmH$ma H$aVmZm Xem§e {MÝh Z KoVm g§»ò§Mm JwUmH$ma H$amdm d Z§Va JwÊ¶

d ^mJH$ g§»`m `m§Mo Xem§e ñWimZ§Va AgUmao A§H$ _moOyZ Amboë`m JwUmH$mam_Üò COì`m

~mOyH$Sy>Z A§H$ _moOyZ Xem§e ñWi Úmdo. EImXo doir JwUmH$mamV {VVHo$ A§H$ AgVrb Va

A§H$m§À¶m g‘moa {VVHo$ eyÝ¶ {bhmdoV.

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CXmhaU 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

CXmhaU 2 : EH$m Am`VmMr bm§~r 4.7 cm d é§Xr 3.5 cm Agob Va Am`VmMo joÌ\$i {H$Vr?

Am`VmMr bm§~r = 4.7 cm

Am`VmMr é§Xr = 3.5 cm

Am`VmMo joÌ\$i = bm§~r × é§Xr

= 4.7 × 3.5

= 16.45 sq.cm

CXmhaU 3 : 1m H$mnS>mMr qH$_V ` 98.75 Amho. Va 3.20m H$mnS>mMr qH$_V H$mT>m.

1m H$mnS>mMr qH$_V = ` 98.75

∴ 3.20m H$mnS>mMr qH$_V = 98.75 × 3.20 = ` 316

CXmhaU 4 : _Ü`mÝh AmhmamZwgma EH$m {dÚmÏ`m©bm EH$m {Xdgmbm 0.035 kg S>mi {Xbr

OmVo. Va 38 {dÚmÏ`mªZm EHy$U {H$Vr S>mi Úmdr bmJob?

1 {dÚmÏ`m©bm EH$m {Xdgmbm {Xbobr S>mi = 0.035 kg

∴ 38 {dÚmÏ`mªZm bmJUmar EHy$U S>mi = 0.035×38

= 1.330 kg

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Xem§e AnyUmªH$m§Mm ^mJmH$ma

nyUmªH$mZo Xem§e AnyUmªH$ g§»òbm ^mJUo

CXmhaU 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=

Xem§e AnyUmªH$mZo Xem§e AnyUmªH$mbm ^mJUo :

CXmhaU 1 :a)

.

.

.

.

8 48 4

0 4 108 4 10

484 2121

##= = =b)

..

..

.

1 83 24

1 8 1003 24 100

180324

1018 1 8

10

18##=

= = =

Q>rn :

Xem§e {MÝh H$mT>Ê¶mg àW‘ A§e Am{U N>oX ¶m§Zm g‘mZ nyUmªH$mZo JwUmdo. OmñV Xem§e ñWmZo AgVrb Va g§»¶oVrb Xem§e ñWmZ H$mT>Ê¶mg H$moUË¶m g§»¶oZo A§embm JwUbo OmVo Ë¶m g§»¶oZo N>oXmbm JwUmdo. Aem àH$mao nyUmªH$ {‘iVmV. Z§Va ^mJmH$ma H$amdm.

c)..

..

12 56 25

12 5 1006 25 100

1250625

21

502

125##= = =

d) . .

..

..

0 729 1 80

1 800 729

1 80 10000 729 1000

1800729

20081

600200

24381

'

##

=

=

=

=

OmUyZ ¿¶m :

{Xboë`m g§»òMo nyUmªH$m_Üò ê$nm§Va H$aVmZm

Oa Xem§e {MÝhmZ§Va EH$ ñWi Agob Va 10

Zo, 2 ñWio AgVrb Va 100 Zo JwUmdo.......

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CXmhaU 2 : EH$ ~g 1.1 VmgmV 48.4 km A§Va OmVo Va Ë`m ~gMm 1 Vmgmbm AgUmam doJ H$mT>m.

~gZo AmH«${_bobo A§Va = 48.4 km KoVbobm doi = 1.1 Vmg

..

1 1 1048 4 10

1148444

##

=doJ

A§Vadoi

doJ ..

/km hr1 148 4

44

=

=

CXmhaU 3 : H${dVmZo Amnë`m _wbm§Zm S´>og {edÊ`mgmR>r ` 427.50 bm 12.5 _rQ>a H$mnS> IaoXr Ho$bo. Va 1 _rQ>a H$mnS>mMr qH$_V H$mT>m.

H${dVmZo IaoXr Ho$bobo H$mnS> $ = 12.5 _rQ>a

EHy$U qH$_V = ` 427.50

∴ 1 _rQ>a H$mnS>mMr qH$_V = 427.50 ÷ 12.5

12.50 100427.50 100

05171

34.20

12505

4275 171= =

##

∴ 1 _rQ>a H$mnS>mMr qH$_V = ` 34.20

OmUyZ ¿¶m :

ba `m ñdê$nmV XmoZ n[a_o` g§»`m AgVrb Ooìhm a < b AgyZ Ë`m§Mm JwUmH$ma hm Ë`m XmoZ n[a_o` g§»`mnojm H$_r òVmo. VgoM EH$m g§»òbm Xþgè`m g§»òZo ^mJbo Va {_iUmam ^mJmH$ma hm Ë`m XmoZ n[a_o` g§»`m§À`mnojm _moR>m AgVmo.

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ñdmÜ`m` 3.5

1. JwUmH$ma H$am

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

2. ^mJmH$ma H$am

1) 0.42÷6 2) 0.144÷12 3) 4.97÷10

4) 6.75÷0.25 5) 2.86÷1.3 6) 68.8÷0.16

3. Oa 1 kg gmIaoMr qH$_V ` 35.45 Agob Va 20.25 kg gmIaoMr qH$_V H$mT>m.

4. EH$ ~g 28.25 km/Vmg doJmZo OmVo. Va Vr 4.25 VmgmV {H$Vr A§Va OmB©b?

5. EH$m H$mabm 21.5 km A§Va OmÊ¶mg 1 {bQ>a noQ´>mob bmJVo. Va 219.30 km A§Va OmÊ¶mgmR>r {H$Vr {bQ>a noQ´>mob bmJob?

6. Oa {ebm 12.5 m a§JrV [a~Z ` 257.5 bm IaoXr H$aVo. Va EH$ ‘rQ>a [a~ZMr qH$_V {H$Vr?

‘moOÊ`mÀ`m n[a_mUm§Mo ê$nm§Va :

‘moOÊ`mMo n[a_mU ho g§»`mË_H$ qH$_V qH$dm ‘m¡ë¶ AgVo. n[a_mUmMr VwbZm EH$H$m ~amo~a

H$aVmV. Amåhr H$mhr Am§Vaamï´>r` n[a_mUo pñdH$mabr AmhoV. Ë`m _moO_mnmZm à_m{UV EH$Ho$

(Standard Unit) åhUVmV. _w»` EH$H$mnmgyZ bhmZ Am{U _moR>r _moO_mno {Z_m©U hmoVmV.

bm§~rMo Am§Vaamï´>r` (‘yb^yV) EH$H$ Amho _rQ>a (m)

dñVw_mZmMo Am§Vaamï´>r` (‘yb^yV) EH$H$ Amho {H$bmoJ«°_ (Kg)

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OmUyZ Ü¶m:

1 B§M = 2.54 cm1 ’y$Q> = 30.48 cm1 `mS>© = 0.914 m12 B§M> = 1 ’y$Q3 ’y$Q = 1 `mS>©220 ’y$Q = 1 \$bmªJ8 \$bmªJ = 1 _¡b1 _¡b = 1.61 km1 km = 0.62 _¡b

1 Am¡g = 28.35 gm1 nm¡§S> = 0.45 kg1 kg = 2.204623 nm¡§S>

1 km = 1000 m1 m = 100 cm1 cm = 10 mm1 kg = 1000 gm1 gm = 1000 mg

Vwåhmbm ho ‘mhrV Amho H$m?

{H$bmo‘rQ>a

ho³Q>mo‘rQ>a

S>oH$m

‘rQ>a

S>ogr‘rQ>a

g|Q>r‘rQ>a

{‘{b‘rQ>a

÷ 10× 10

¶mMmM AW©

1 {H$bmo‘rQ>a = 10 ho³Q>mo‘rQ>aË¶mMà‘mUo

1 cm = 10 mm

CXmhaU 1 : g|Q>r‘rQ>a‘Ü¶o énm§Va

a) 45 m = 45 ×100 = 4500 cm

b) 2.7 m = 2.7 ×100 = 270 cm

c) 4 43 m = 4

19 100 19 25 475# #= = cm

d) 3.02 m = 3.02×100 = 302.00 = 302 cm

gyMZm : m Mo cm ‘Ü¶o énm§Va H$aVmZm g§»¶obm 100 Zo JwUmdo

a 1m = 100 cm 1 cm = 1001 m

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73

CXmhaU 2 : _rQ>a (m) _Üò ê$nm§Va

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

gyMZm : m Mo cm ‘Ü¶o énm§Va H$aVmZm g§»¶obm 100 Zo ^mJmdo

∴ 100 cm = 1 m 1 cm = 101 m

CXmhaU 3 : g| ‘r cm _Üò {bhm.

a) 6mm

= 6 101

53

# = cm

b) 23 mm

= .1023 2 3= cm

c) 276 mm

= .10276 27 6= cm

CXmhaU 4 : {H$._r. (km) _Üò {bhm.

a) 105 m

1051000

12002121

200#= = km

b) 725 m

7251000

1402529

40#= = km

CXmhaU 5 : {H$bmoJ«°_ (kg) _Üò {bhm.

a) 720 g

1000720

10072

2518

25

18= = = kg

b) 3150 g

3.1510003150

100315= = = kg

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CXmhaU 6 : J«°__Üò (g) {bhm.

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

CXmhaU 7 : J«°__Üò (g) {bhm.

a) 740 mg

00

10074

10074

5037= = g

b) 210 mg

00

10021

10021= g

CXmhaU 8 : 125 mg Mo kg _Üò énm§Va H$am.

gm kgmg kg

mg kg

kg

125 10000001

400005

1000 11000000 1

1 10000001

80001

540000

1

8000# =

==

=

=

CXmhaU 9 : EH$ ~g EH$m Vmgm_Üò 16.2 _¡b A§Va OmVo. Ë`mMo {H$bmo_rQ>a_Üò énm§Va H$am.

1 _¡b = 1.61 km∴ 16.2 _¡b = 16.2 × 1.61

= 26.082 km

CXmhaU 10 : EH$m ImobrMr bm§~r 45 \y$Q> Agob Va, Vr bm§~r cm _Üò H$mT>m.CH$b : 1 \w$Q> = 30.48 cm∴ ImobrMr bm§~r g|Q>r_rQ>a_Üò = 30.48 × 4.5

=137.160 cm

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ñdmÜ`m` 3.6

1. Imbrb gm _Üò ê$nm§Va H$am

1) 2.51 kg 2) 72.5 mg

2. Imbrb kg _Üò ê$nm§Va H$am

1) 625 gm 2) 10825 mg

3. Imbrb km _Üò {bhm.

1) 1450 m 2) 17.3 m 3) 21350 cm

4. Imbrb m _Üò {bhm.

1) 2.7 km 2) 7525 cm 3) 1.58 km

5. Imbrb cm _Üò {bhm.

1) 12.5 m 2) 4.7 km

6. 7.2 _¡bMo ê$nm§Va km ‘Ü¶o H$am.

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àH$aU - 4

~¡{OH$ amer [ALGEBRIC EXPRESSIONS]

¶m àH$aUmÀ¶m AÜ``ZmZ§Va Vwåhr :

~¡{OH$ amer g‘OmdyZ ¿¶mb.

~¡{OH$ amerMm ì`m»`m gm§Jy eH$mb.

MbnXm§Mm Cn¶moJ H$ê$Z ~¡{OH$ amer aMmb.

Vm|S>r {dYmZmMo ê$nm§Va ~¡{OH$ amerV H$ê$ eH$mb.

~¡{OH$ ametÀ`m KmVm§H$m§Mm AW© g_OmdyZ ¿¶mb.

J{UVmÀ¶m ‘yi g§H$ënZm ~¡{OH$ amer_Üò dmnê$ eH$mb.

àñVmdZm : ~¡{OH$ amer hr ~rOJ{UVmVrb AË`§V _hËdmMr g§H$ënZm Amho. X¡Z§{XZ OrdZmVrb

AZoH$ g‘ñ¶m gmoS{dÊ`mgmR>r ¶m§Mr ‘XV hmoVo. g_rH$aUmVrb Aì¶º$ (MbnX) nXmMr qH$‘V H$mT>Ê¶mg qH$dm CH$b H$aÊ`mg _XV H$aVmV.

~rOJ{UVmVrb _yi nXm§Mr g§H$ënZm

MbnX qH$dm Aì¶º$nX :

gw_m d _Yy _mMrgÀ`m H$mS>çm KoD$Z IoiV AmhoV. Ë`mZr _mMrgÀ`m H$mS>çmnmgyZ B§J«Or_Yrb _wimjao ~Z{dÊ`mMo R>a{dbo.

gw_mZo VrZ _mMrgÀ`m H$mS>çm KoD$Z 'C' Aja ~Z{dbo.

Z§Va _YyZo VrZ _mMrgÀ`m H$mS>çm KoD$Z AmUIr EH$ 'C' Aja ~Z{dbo d Vo gw_mZo ~Z{dboë`m 'C' AjamÀ`m nwT>o R>odbo.

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nwÝhm _mMrgÀ`m H$mS>çm dmnê$Z ‘YyZo AmUIr EH$ 'C' Aja ~Z{dbo Am{U Vo Aja EH$mg_moa EH$ R>odbo.

Aem [aVrZo ‘Yy Am{U gw‘m ¶m§Zr 'C' Ajao ~ZdyZ aMV

Joë¶m.

eodQ>r Ë`m§Zr EHy$U 'C' Aja ~Z{dÊ`mgmR>r bmJUmè`m d EHy$U _mMrg H$mS>çmMr g§»`m `m§Mm Vº$m V`ma Ho$bm.

V`ma Ho$boë`m 'C' AjamMr g§»`m

1 2 3 4 5 ... 20 ... 25 ... n

dmna Ho$boë`m

_mMrg H$mS>çm§Mr

g§»`m

3 6 9 12 15 ... 60 ... 75 ... 3n

1×3 2×3 3×3 4×3 5×3 ... 20×3 ... 25×3 ... n×3

Vº$m {b{hVmZm Ë`m§Zm Ago AmT>iyZ Ambo H$s 'C' Ajam§À`m EHy$U g§» oÀ`m VrZ nQ> _mMrg H$mS>çm bmJVmV. åhUOo :

'C' Aja aMÊ¶mg bmJUmè`m _mMrg H$mS>çmMr g§»`m = 3 × 'C' Ajam§Mr EHy$U g§»`m. Oa 'n' åhUOoM 'C' Ajam§Mr EHy$U g§»`m Agob Va EHy$U _mMrg H$mS>çmMr g§»`m

= 3 × n = 3n

¶oWo 'n' ho MbnX Amho ho g‘OVo.

Oa,n=1, nmhrOo Agboë`m EHy$U _mMrg H$mS>çm = 3 ×1 = 3n=2, nmhrOo Agboë`m EHy$U _mMrg H$mS>çm = 3 ×2 = 6n=3, nmhrOo Agboë`m EHy$U _mMrg H$mS>çm = 3 ×3 = 9

òWo n Mr qH$_V ~XbV Joë`mg, bmJUmè`m EHy$U _mMrg H$mS>çmMr g§»`m gwÜXm ~XbV OmVo.

∴ n ho EH$ MbnXmMo CXmhaU Amho. MbnX g§»`m hr MbnX åhUyZ dmnaVmV.

'n' Mr qH$_V 1, 2, 3, 4............ KoD$ eH$Vmo.

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MbnX åhUOoM pñWa Zgbobo qH$dm ~XbUmao nX Agm AW© hmoVmo. MbnXmMr qH$_V hr pñWa ZgVo. Ë`mMr qH$_V doJdoJir Agy eH$Vo.

MbnXmMr AmUIr CXmhaUo ;

Am`VmMo joÌ\$i = bm§~r × é§Xr hmo!, a,b,c,....x,y,z

hr MbnXo VgoM Aì`º$ nXo åhUyZ dmnê$ eH$Vmo.A = l × b

¶oWo A = joÌ’$i

l = bm§~r, b = é§Xr

Imbrb V³Ë¶m‘Ü¶o doJdoJù¶m ‘mnmÀ¶m Am¶Vm§Mo joÌ’$i {Xbobo Amho.

Am`V‘moO‘mn

bm§~r (l) é§Xr (b) joÌ\$i 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

darb VŠË`mdê$Z Ago {XgVo H$s doJdoJù`m Am`VmgmR>r 'l' Mr qH$_V d 'b' Mr qH$_V doJdoJir Amho. åhUOoM Ajar g§»`m 'l ' d 'b' `m§Zm MbnXo åhUVmV.

MbnX åhUOoM MbnX g§»`m qH$dm H$moUVohr g§km Á`mMr H$moUË`mhr àg§Jr doJdoJir qH$_V AgVo.

Aì¶º$ nX gm§JÊ¶mg dmnaë¶m OmUmè¶m H$moUË¶mhr g§kmZm MbnXo åhUVmV.

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OmUyZ ¿`m :Eo{Vhm{gH$ nmíd©^y_r : {I.nyd© 300 À`m gw_mamg EImXo _mhrV Zgbobo nX dU©_mboVrb EImÚm AjamZo

Xe©dyZ {dYmZ ì`º$ H$aÊ`mMr nÜXV ^maVmV hmoVr.

AZoH$ à{gÜX J{UVk : Am`©^Q> (OÝ_ B.g. 476), ~«÷Jwá (OÝ_ B.g. 598), _hmdra

(OÝ_ B.g. 850) Am{U mñH$a II (OÝ_ B.g. 1114) m§Zr ~rOJ{UVmÀ`m Aä`mgm_Üò

OmñV `moJXmZ {Xbo Amho. Ë`m§Zr ~rO, dJ© B. eãX Aì`º$ nXm§gmR>r dmnabo. "Ab{O~«m'

`m B§J«Or eãXmbm ^maVmV "~rOJ{UV' åhUVmV. \«$mÝgH$mB©g pìhQ>o ho 16 ì`m eVH$mVrb EH$ à{gÜX J{UVk hmoVo. hr n{hbr ì`º$s

hmoVr. Á`m§Zr gm_mÝ` Ajao A§H$J{UVm_Üò dmnabr.

pñWam§H$ :

EH$m {ÌH$moUmÀ`m EHy$U ~mOy 3 nojm OmñV Agy eH$VmV? {Z[üVM Zmhr. åhUyZM {ÌH$moUmÀ`m EHy$U ~mOy§Mr {ZpíMV g§»`m AgVo. Vr g§»`m åhUOoM pñWam§H$ hmo`.

Mm¡H$moZmMo EHy$U {eamoq~Xÿ 4 nojm OmñV Agy eH$VmV? Z¸$sM Zmhr, åhUyZM Mm¡H$moZmÀ`m EHy$U {eamoq~XÿMr g§»`m {ZpíMV AgVo.

Ooìhm AmnU 4 A§H$ {bhVmo, Vmo 4 A§H$ Xe©{dVmo.

Ooìhm Amåhr "5' ho {MÝh {b{hVmo Vo ‘amR>r ^mfoV A§H$mVrb nmM A§H$ Xe©{dVmo.

pñWam§H$ åhUOoM EH$ g§»`m qH$dm H$moUVohr {MÝh

Á`mMr EH$M qH$_V AgVo.

5, 9, 10, 3/4, -8, 0.01,............ pñWam§H$ AmhoV.

n[aK ì¶mg = π hm pñWam§H$ Amho

c = 3 ×108 _rQ>a/goH§$X pñWam§H$ Amho

({Zdm©V nmoH$irVrb àH$memMm doJ)

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~¡{OH$ nX : Vwåhr `m nydu {eH$bm AmhmV H$s 6 d 8 Mm JwUmH$ma 6 × 8 = 48 ¶oVmo. Ë¶mMà_mUo 18 bm 2 Zo ^mJë`mZ§Va 9 hm mJmH$ma {_iVmo. 18 ÷ 2 = 9

¶mà‘mUo,

x Am{U 3 Mm JwUmH$ma AmnU

H$ê$ eH$Vmo?

hmo`, `m§Mm JwUmH$ma 3x ¶oVmo.

y bm 4 Zo ^mJyZ ^mJmH$ma H$mTy> eH$Vmo?

hmo`, ^mJmH$ma y4 ¶oVmo.

3x, y4 BË`mXr ~¡{OH$ nXo AmhoV. ~¡{OH$ nXm§Mr AmUIr H$mhr CXmhaUo

3x , -6x y, 2x2y, y2 , 2x3-

d a r b ~ ¡ { O H $ n X m _ Ü ` o p ñ W a m § H $ d M b n X o AgVmV. 3x, `m ~¡{OH$ nXmV 3 ho pñWam§H$ d

x ho MbnX Amho.

-6x y m ~¡{OH$ nXmV - 6 ho pñWam§H$ d x, y MbnXo AmhoV.

2x2y `m ~¡{OH$ nXmV 2 ho pñWam§H$ x VgoM y hr MbnXo AmhoV.

EH$ qH$dm AZoH$ MbnXo AgUmao, nyU© g§»¶m§Mm KmVm§H$ Agboë¶m pñWam§H$mZo JwUbobo qH$dm ^mJbobo nX åhUOo ~¡{OH$ nX åhUVmV.

2xy, 4pq, x y43 2- , 10m

hr ~¡{OH$ nXo AmhoV

~¡{OH$ nXo hr g§»¶m, Ajar g§»¶m Am{U g§»¶m Am{U Ajar

g§»¶m XmoÝhr nU AgVmV

{dMma H$am x-2, x/y,

1/x hr ~¡OrH$ nXo AmhoV

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ñdmÜ`m` 4.1

I. Imbrb àËòH$ ~¡{OH$ nXm§_Yrb MbnX qH$dm MbnXm§Mr `mXr H$am. 1) 5x 2) -3a 3) 7x y 4) 4

3 x 2y5) 0.8a2b 6) m2n2 7) xyz 8) -5m2np

2. Imbrb àËòH$ ~¡{OH$ nXm§_Yrb g§»`mdmMH$ pñWam§H$mMr `mXr H$am. 1) 2x 2) -5x 2y 3) m2n 4) -8p

5) 9p2qr 6) 43 x 2y 7) 0.5pqr2 8) 0.008mn

~¡{OH$ amer (Algebraic Expression)

J{UVmV amer H$er V`ma hmoVo? Imbrb J{UVr {dYmZ nhm.

EH$ g§»`m d 5 `m§Mr ~oarO 59 Amho.

+ 5 = 59

åhUOo, 59 = 54 + 5

nm{hOo Agbobr g§»`m 54 Amho.

Ë`mMà_mUo,

∴ EH$ g§»`m d 8 `m§Mm JwUmH$ma 88 Amho.

8 × = 88

8 × 11 = 88

∴ nm{hOo Agbobr g§»`m 11 Amho.

Oa AZoH$ Aì`º$ nXo AgVrb Va J{UVr {dYmZ ~Zdm`bm H$R>rU OmVo. Aì`º$ nXo

gmoS>{dÊ`mgmR>r Ë`m Aì`º$ nXm§Zm AmnU x, y, z B. Mbmjao XoD$ eH$Vmo.

~rOJ{UV åhUOo _Om!"‘mhrV Zgbobo emoYÊ¶mMm

AZw^d'

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Imbrb CXmhaU nhm :

EH$ _wbJr AmVm 10 dfm©Mr Amho. Va Imbrb à‘mUo Vr {H$Vr dfm©Mr hmoVr qH$dm hmoB©b?

1) 5 dfm©Z§Va ?

2) 3 dfm©nydu ?

3) x dfm©Z§Va ?

4) y dfm©nydu ?

CH$b : _wbJrMo d` Xa dfm©bm 1 dfm©Zo dmT>Uma Amho.

1) 5 dfm©Z§Va Vr (10 + 5) = 15 dfm©Mr hmoB©b. 2) 3 dfm©nydu Vr (10 - 3) = 7 dfm©Mr hmoVr. 3) 'x' dfm©Z§Va Vr (10 + x) dfmªMr hmoB©b.

4) 'y' dfm©nydu Vr (10 - y) dfm©Mr hmoVr.

(10 + x) d (10 - y) `m amerV MbnXo AmhoV åhUyZ Ë`m§Zm ~¡{OH$ amer åhUVmV.

Imbrb amer nhm :

i) x2 ii) 2y2

OmUyZ ¿`m : Imbrb {MÝho Aer dmMVmV :ly - Ë¶mMà‘mUo

i.e.- åhUyZ≅ g‘mZVm

Ë¶m AWu (H$maU)

iii)3x + 2 iv) 2pq + 7

x2 amer åhUOoM x Mm x Mm JwUmH$ma. åhUyZ x × x = x2 Ë`mMà_mUo 2 × y × y = 2y2

3x + 2 ho H$go {_iVo?

x bm 3 Zo JwUyZ d Ë`m§V 2 {_idyZ 3x + 2 hr amer {_iVo.

åhUyZ (x × 3) + 2 = 3x + 2

2pq + 7 H$go {_iVo?

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2pq + 7 hr amer Imbrb à_mUo {_iob. àW_ pq {_i{dÊ`mgmR>r p d q Mm JwUmH$ma H$éZ Z§Va Ë`m§Mm 2 er JwUmH$ma Ho$bm

Va 2pq {_iVmV Am{U Ë`mV 7 {_i{dë`mg 2pq + 7 hr amer {_iVo.

darb x2, 2y2, 3x + 2, 2pq + 7 `m gd© ametZm ~¡{OH$ amer åhUVmV.

Imbrb Vº$m nhm;

{dYmZ ~¡{OH$ amer

x _Üò 8 {_i{dbo x + 8

y _YyZ 5 dOm Ho$bo y - 5

6 d p Mm JwUmH$ma 6 × p

'm' bm n Zo ^mJbo m ÷ n

òWo darb {dYmZm§Mo {ZarjU Ho$bo AgVm ‘yb^yV {H«$`m ~oarO, dOm~mH$s, JwUmH$ma qH$dm ^mJmH$ma `m MbnXmda Ho$ë`m AgVm Ë`m§Mo ~¡{OH$ amer_Üò énm§Va hoVo.åhUyZM AmnU åhUy eH$Vmo -

"" ~¡OrH$ nXm§~amo~a _wb^yV {H«$`m Ho$ë`m AgVm ~¡{OH$ amer {_iVmV''

~¡OrH$ amer d Ë¶mVrb nXo

~¡{OH$ amer amerVrb nXo EHy$U nXo

2x + 3y 2x, + 3y 2

x2 - 4x + 3 x2, - 4x, + 3 3

3ab - 4bc - 6cd 3ab, - 4bc, 6cd 3

-2p2q + 3q2r - rp + 4pqr - 2p2r, + 3q2r, - rp, 4pqr 4

AmVm 2x + 3y hr amer nhm -

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`m amerV x d y hr MbnXo Am{U 2 d 3 `m g§»`m AmhoV. amerVrb 2x ho EH$ nX d

3y ho Xwgao nX Amho. Ë`mMà_mUo BVa amer_Üò ‘mJo nm{hë`mà_mUo nXo òVmV.

JwUmH$ma, Ad`d Am{U ghJwUH$ :

Amnë`mbm _mhrV Amho 4 × 5 = 20

òWo 20 hm JwUmH$ma Am{U 4 d 5 ho 20 Mo Ad`d Amho.

Ooìhm XmoZ MbnXm§Mm JwUmH$ma hmoVmo Voìhm Ë`m§Mm JwUmH$ma H$m¶ AgVmo?

g_Om, 'x' d 'y' Mm JwUmH$ma Ho$ë`mg Ë`m§Mm JwUmH$ma {H$Vr Amho?

'x' d 'y' Mm JwUmH$ma Amåhr Agm {bhVmo {H$ (x × y) qH$dm xy`mMà_mUo,

2 d p Mm JwUmH$ma = 2p {Q>n :Oa MbnXo EH$_oH$m§À`m ~mOybm {b{hbr Va Ë`m§À`mV JwUmH$ma AgVmo.

2p Mo 2 d p ho Ad`d AmhoV

5 × m × n = 5mn

5, m d n Mm JwUmH$ma = 5mn

5mn Mo 5, m d n ho Ad`d AmhoV.

7 x × x × y Mm JwUmH$ma = 7x2y

7x2y Mo 7, x, y ho Ad`d AmhoV.

~¡OrH$ amer d Ë`m§Mr nXo `m§Mr AmH¥$Vr

2

8x2y

8 5x x xy y

+5xy - 8

amer :

nXo :

Ad`d :

8x2 + 5 xy - 8

4

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ghJwUH$ :

x d 8 Mm JwUmH$ma = 8x

8x ‘Ü¶oo 8 hr g§»`m d x ho MbnX Amho. XmoÝhr {_iyZ 8x åhUOoM + 8x 8 hm x Mm g§»`mË_H$ Ad`d qH$dm A§H$J{UVr Ad`d qH$dm g§»`mË_H$ ghJwUH$ Amho.

x bm 8 Mm Ajar Ad`d qH$dm Ajar ghJwUH$ åhUVmV.

x = x

y = 1 y

z = 1z

Ooìhm MbnX g§»`mË_H$ ghJwUH$m{edm` AgVo Ë`mdoir g§»`mË_H$ ghJwUH$ 1

AgVmo.

H$mhr OmñVrMr CXmhaUo nhm

JwUmH$ma ghJwUH$ g§»`mË_H$ghJwUH$

AjarghJwUH$

8xy y Mm ghJwUH$ 8xx Mm ghJwUH$ 8y8 Mm ghJwUH$ xy

881

xyxy

-21x2y=-21x × x × y

y Mm ghJwUH$ -21x2 xy Mm ghJwUH$ -21x x2 Mm ghJwUH$ -21y 21 Mm ghJwUH$ -x2y

-21-21-211

x2

xy

x2y

ñdmÜ`m` 4.2

I. Imbrb amer g§»`mË_H$ amer Am{U MbnX ~¡OrH$ amer_Üò dJuH$aU H$am.

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. a) Imbrb ~¡{OH$ amer_Yrb nXo {bhm.

1) 3x + 4y 2) 2pq - 8qr 3) 3 x 2 - 3x + z

4) ab + bc - ca 5) 9m + 6n 6) - 3xy + x - y

b) Imbrb ~¡{OH$ amerVrb nXo Am{U Ad`d AmoiIm.

1) 2xy + 5 2) x2 + xy + 4 3) 3p - 5y2 4) 2ab + bc + ca

III. Imbrb {dYmZo ~¡{OH$ amer_Üò {bhm.

1) x _Üò 8 {_i{dbo.

2) y _YyZ 7 dOm Ho$bo.

3) p d 12 Mm JwUmH$ma

4) q bm 5 Zo ^mJbo

5) x Mr MmanQ> d y Mr VrZnQ> {_i{dbr.

6) 10 _YyZ y Mr 5 nQ> dOm Ho$br.

7) p d q Mm JwUmH$ma hm 3x _Üò {_i{dbm.

8) m À¶m nmMnQ> _YyZ l Mr VrZ nQ> ho dOm H$am.

9) 10 d y À¶m JwUmH$ma_Üò 15 {_idm.

10) x d y À¶m JwUmH$mambm z Zo ^mJm.

IV. Imbrb amerVrb x Mm g§»`mË_H$ ghJwUH$ {bhm. 1) 3x 2) -4x 3) x4

3 4) 10.5x 5) 168x 6) x9

8-

V. Imbrb amerVrb Ajar ghJwUH$ {bhm. 1) 3m 2) -9xy 3) 16pqr 4) 10c 5) mn4

5- 6) 0.8x2y

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VI. 27xyz Mo Imbr {Xë`mà_mUo ghJwUH$ {bhm. 1) xyz Mm g§»`mË_H$ ghJwUH$ ______ Amho.

2) xy Mm Ajar ghJw_H$ ______ Amho.

3) 27z Mm Ajar ghJwUH$ ______ Amho.

4) 27 Mm Ajar ghJwUH$ ______ Amho.

gOmVr` Am{U {dOmVr` nXo

dJm©_Üò am_m, gw_m, gb_m Am{U am°~Q>© ho Mma {dÚmWu MMm© H$aV AmhoV :

am_m åhUmbm : 20 noZ + 5 noZ = 25 noZ

gw_m åhUmbr : 50 døm - 5 døm = 45 døm

gb_m åhUmbr : 20 noZ + 15 dø = 35

am°~Q>©Zo àý Ho$bm H$s 35 døm qH$dm 35 noZ? Vy `m à_mUo ~oarO H$ê$ eH$V Zmhrg? gw_m åhUmbr Vr `mà_mUo ~oarO H$m H$ê$ eH$V Zmhr?

am°~Q>© : Ë`m doJdoJù`m dñVy AmhoV. doJdoJù`m dñVy§Mr AmnU ~oarO qH$dm dOm~mH$s H$ê$ eH$V Zmhr.

gb_m : hmo`, Amåhr 3 {H$.J«°. gmIa d 2 {H$.J«°. _rR> EH$Ì H$ê$ eH$V Zmhr.

Ooìhm darb Mma {_Ì MMm© H$aV hmoVo Voìhm {ejH$m§Zr dJm©V àdoe Ho$bm d MmQ>© \$ù`mda bmdbm d Ë`mMo {ZarjU H$am`bm gm§{JVbo.

Imbrb VŠË`mVrb ^mJ-1 d ^mJ-2 Mo {ZarjU H$am :

^mJ-11) x, 3x, -6x2) 2ab, -3ba, 2

1 ab3) y2, 3y2, -52 y2

^mJ-21) x, 3y, -6z2) 2ab, -3bc, 2

1 ca

3) x, x2, -2x3

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{ejH$m§Zr {dMmabo : ^mJ-1 H$m` Xe©{dVmo?

^mJ -1 _Üò ~¡{OH$ nXmMo JQ> AgyZ Ë`m§Mo MbnX Ad`d g_mZ AgyZ KmVm§H$ gwÜXm g_mZ AmhoV.

{ejH$m§Zr {dMmabo : ^mJ-2 H$m` Xe©{dVmo?

^mJ -2 _Üò ~¡{OH$ nXmMo JQ> AgyZ Ë`m§Mo MbnX Ad`d doJdoJio qH$dm MbnX Ad`d g_mZ AgyZ VrMo KmVm§H$ doJdoJio AmhoV.

åhUyZM -^mJ-1 _Yrb 1, 2 d 3 JQ> hr gOmVr` nXo AmhoV

^mJ-2 _Yrb 1, 2 d 3 JQ> hr {dOmVr` nXo AmhoV.

Á`m ~¡{OH$ nXm§_Üò g_mZ Ajar Ad`d AgyZ KmVm§H$ gwÜXm g_mZ AgVmo Ë`m§Zm gOmVr` nXo åhUVmV. Á`m ~¡{OH$nXm§_Üò doJdoJir Ajar Ad`d nU KmVm§H$ g_mZ qH$dm g_mZ Ajar Ad`d nU KmVm§H$ doJdoJio AgVrb Va Ë`m§Zm {dOmVr` nXo åhUVmV.

~¡{OH$ ametMo dJuH$aU :Imbrb ^mJm§Mo dJuH$aU H$am;

^mJ-1 ^mJ-2 ^mJ-3 ^mJ-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

^mJ - 1 _Üò EH$ nX Agboë`m amer AmhoV. EH$ åhUOo EH${Û åhUOo XmoZ{Ì åhUOo VrZ~hþ åhUOo AZoH$ ^mJ - 2 _Üò XmoZ nXo Agboë`m amer AmhoV.

^mJ - 3 _Üò VrZ nXo Agboë`m amer AmhoV.

^mJ - 4 _Üò AZoH$ nXo Agboë`m amer AmhoV.

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¶m§Mo Imbrbà_mUo dJuH$aU H$aVmV :

i) EH$nXr : EH$M nX AgUmar amer.

ii) pìXnXr : XmoZ {dOmVr` nXo AgUmar amer.

iii) {ÌnXr : VrZ {dOmVr` nXo AgUmar amer.

iv) ~hþnXr : ~rOJ{UVmV AZoH$ {dOmVr¶ nXo AgUmar amer. na§Vw \$º$

YZKmVm§H$ AgUmè`m ~¡{OH$ amerbm ~hþnXr Ago åhUVmV.

Q>rn : ~rOJ{UVmV gOmVr` nXm§Mr ~oarO qH$dm dOm~mH$s H$aVm òVo. a + 3a = 4a, 3ab + 5ab + 6ab = 14ab ∴ a + 3a hr {ÛnXr Zmhr, hr EH$nXr Amho. Ë`mMà_mUo 3ab + 5ab + 6ab ho {ÌnXr Zmhr, EH$nXr Amho.

~¡{OH$ amer

pñWam§H$ EH$nXr {ÛnXr {ÌnXr

~hþnXr

OmUyZ ¿¶m :

+ Am{U - {MÝh ~¡{OH$nXo doJdoJir H$aVmV.

× Am{U ÷ {MÝh ~¡{OH$nXo doJdoJir H$aV ZmhrV.

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~hþnXtMo H$mhr {deof JwUY_©

1) H$moUË`mhr amer_Üò EH$ qH$dm OmñV$ nXo AgUmè¶m gd© ~¡{OH$ ametZm ~hþnXr åhUVmV. åhUyZM EH$nXr, {ÛnXr Am{U {ÌnXr `m ~hþnXr AgVmV.

2) àËòH$ nXmVrb MbnXmMm KmVm§H$ hr nyU©g§»¶m Agbr nm{hOo.3) x½, 5x - 2, 1/x, p1/3 `m gma»¶m ~¡{OH$ nXo AgUmè¶m amer ~hþnXr ZgVmV.

H¥$Vr : EH$nXr, {ÛnXr, {ÌnXr amer AmoiIm.

nmàr-1 : A, B, C Ago 3 ~m°Šg KoD$Z Ë¶mda AZwH«$_o EH$nXr, {ÛnXr, {ÌnXr Aer Zmdo

{bhm.

nmàr-2 : ~¡{OH$ amer {b{hbobo âb°e H$mS>© KoD$Z doJdoJù`m àH$maÀ`m ñQy>bda ~¡{OH$

amer {b{hbobr ~mOy Imbr H$ê$Z R>odm.

nmàr-3 : {dÚmÏ`mªZm EH$m nmR>monmR> EH$ Ago ~mobmdyZ Ë`m_Yrb EH$ H$mS>© ¿`m`bm bmdyZ

amer dmMm¶bm gm§Jm d `mo½` Ë`m ~m°Šg_Üò Q>mH$m`bm gm§Jm.

nmàr-4 : àËòH$ ~m°Šg_Yrb H$mS>} _moOyZ Ë`mVrb MyH$sMr H$mS>} ~mOybm H$am. Imbrb

VŠË`m à_mUo `mXr H$ê$Z {ZarjU H$am.

JQ> EHy$U H$mS>} nXm§Mr g§»`m amerMm àH$ma

A ___________ EH$ EH$nXr

B ___________ XmoZ ÛrnXr

C ___________ VrZ {ÌnXr

Vwåhmbm g_Obo AgobM H$s, ~¡OrH$ ametMo H$moUË`m AmYmamda dJuH$aU H$aVmV?

H$moUVo H$maU Amho H$s `mà_mUo Ë`m§Zm Zmdo XoVmV?

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KmVm§H$ g§»`m :

Vwåhr ¶m AJmoXa {eH$bm AmhmV H$s, Mm¡agmMo joÌ\$i = ~mOy × ~mOy

Oa Mm¡agmMr àËòH$ ~mOy 2 cm Agob Va Mm¡agmMo joÌ\$i = 2 × 2 = 22 Mm¡.cm

= 4 Mm¡.cm

Oa Mm¡agmÀ`m àËòH$ ~mOyMr bm§~r 'x' cm Agob Va Mm¡agmMo joÌ\$i = x cm × x cm

= x2 Mm¡.cm

Ë`mMà_mUo,

KZmMo KZ\$i = bm§~r × é§Xr × C§Mr Oa bm§~r, é§Xr d C§Mr 2 àË¶oH$s cm Agob Va,

KZmMo KZ\$i = (2 × 2 × 2) KZ cm = 23 KZ cm

= 8 KZ cm

Oa bm§~r, é§Xr d C§Mr àË¶oH$s 'y' cm Agob Va, KZmMo KZ\$i = (y × y × y) KZ cm = y3 KZ cms

åhUyZM, x2, y3, p4, q6, d7 Am{U m9 BË`mXtZm KmVm§{H$V g§»`m åhUVmV.

KmVm§{H$V g§»`m H$er dmMVmV?

• a2 ho a dJ© Ago dmMVmV qH$dm 'a Mm KmVm§H$ 2'

• x3 ho x KZ Ago dmMVmV qH$dm 'x Mm KmVm§H$ 3'

• p4 ho p Mm KmVm§H 4 Amho Ago dmMVmV

• d7 ho d Mm KmVm§H 7 Amho Ago dmMVmV

•m9 ho m Mm KmVm§H 9 Amho Ago dmMVmV

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KmVm§H$ g§»òer g§~§YrV nXo :

x2, y3, n7, r4 B. _Yrb KmVm§H$ g§»`m.

2, 3, 7, 4 `m g§»`m MbnX g§»òÀ`m

KmVm§H$/H$moQ>r

AmYmam§H$

x2da COì¶m ~mOyÀ`m H$monè`mbm {b{hVmV. Ë`m§Zm KmVm§H$ åhUVmV.

x, y, m, k m g§»`m§Zm AmYmam§H$ åhUVmV.

a2 = a × a (AmYmam§H$ a Mm 2 doim JwUmH$ma)

x3 = x × x × x (AmYmam§H$ x Mm 3 doim JwUmH$ma)

(ab)4 = ab × ab × ab × ab (AmYmam§H$ ab Mm Mma doim JwUmH$ma)

{Q>n : KmVm§{H$V g§»¶m {b{hVmZm KmVm§H$ g§»¶m, AmYmam§H$ g§»¶onojm bhmZ AmH$mamV {bhmdr.

~¡{OH$ ametMr H$moQ>r :Amåhr `mnydu ~¡{OH$ ametMr nXo Am{U KmVm§H$ g§»`m `m~Ôb {eH$bmo AmhmoV.hr amer nhm x4 + x2 - 3x + 4amerVrb nXo x4, x2, -3x, 4 AmhoV.amerVrb H$moUË`m nXmMr H$moQ>r _moR>r Amho?x4 Mm nXmMm KmVm§H$ _moR>m Amho Vmo 4 Amho.åhUyZM, x4 + x2 - 3x + 4 ¶m ametMr 4 hr H$moQ>r Amho.åhUyZ `m amerbm 4 H$moQ>rMr ~¡{OH$ amer åhUVmV.a2 + 5a3 - 4a - 8 `m amerV 5a3 `m nXm§Mm 3 hm _moR>m KmVm§H$ Amho. åhUyZM

ametMr H$moQ>r 3 Amho. åhUyZ `m amerbm 3 H$moQ>r amer åhUVmV.

EH$ MbnX Agboë`m ~hþnXr ~¡{OH$ amerVrb nXmMm _moR>çmV _moR>m KmVm§H$ hm Ë`m amerÀ`m MbnXmMr H$moQ>r AgVo.

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~hþnXrÀ`m H$moQ>r~Ôb A{YH$ _m{hVr :

Imbrb amer nhm :

3x4 - 4x3y2 + 8xy + 7

àËòH$ nX KoD$Z Ë`m§Mm KmVm§H$ Vnmgm.

3x4, _Üò KmVm§H$ 4 Amho.

x3 y2 ‘Ü¶o x d y À`m KmVm§H$m§Mr ~oarO H$am.

3+2 = 5

- 4x3 y2 Mm KmVm§H$ _moR>m Amho. `mMr H$moQ>r 5 Amho.

∴ ~hþnXrMr H$moQ>r 5 Amho.

XmoZ qH$dm XmoZ nojm OmñV MbnXo Agboë`m ~hþnXr amerÀ`m MbnXm§À`m KmVm§H$mMr _moR>çmV

_moR>r ~oarO åhUOoM Ë`m ~hþnXrMr H$moQ>r hmo`.

ñdmÜ`m` 4.3

I. Imbrb ametMo EH$nXr, ÛrnXr d {ÌnXr amer_Üò dJuH$aU H$am.

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

II. Imbrb amerVrb ~hþnXr AmoiIm. 1) x + 2yq 2) x3 + x - 2 + 1

3) y2 + xy½ - 6 4) m3 + 2m2 + 3m - 4

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III. gOmVr` nXm§Mo 'L' Zo Am{U {dOmVr` nXm§Mo 'U' Zo JQ> ~Zdm.

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. gOmVr` nXm§Mo JQ> H$ê$Z {bhm.

1) x, y3, 2x, -7y3, -8x, 23 y3, 6x, -y3

2) 7ab, 6bc, -8ba, 2ca, -3ab, 2abc, 4ab, 2a2b

3) 7p, 8pq, -5pq, 2p and 3p

V. Imbrb _Yrb g§»`mË_H$ ghJwUH$ {bhm.

1) 3xy 2) ab32- 3) 0.3p

4) 24xyz 5) -18p2q 6) m np119 2-

VI. Imbr gwM{dë¶mà‘mUo ghJwUH$ {bhm.

1) -3axy _Yrb xy Mm ghJwUH$

2) 4a2b _Yrb ab Mm ghJwUH$

3) p2yz2 _Yrb z2 Mm ghJwUH$

4) 10xy _Yrb xy Mm ghJwUH$

5) -15p2 _Yrb 15 Mm ghJwUH$

6) -mn _Yrb mn Mm ghJwUH$

VII. Imbrb _Yrb H$moQ>r {bhm.

1) 3x2 2) 3 - p2 + p3 3) 7m2n

4) z - p 5) a2 + 2ab + abc 6) m2 + 2mn2 + n2

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VIII. Imbrb Vº$m nyU© H$am.

KmVm§{H$V g§»`m AmYmam§H$ KmVm§H$

x5

(ab)6

(3 p)9

(-y)10

(xyz)7

(0.59)20

IX. KmVm§H$ê$nmV {bhm. 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 doim

~¡{OH$ amerVrb _wb^yV {H«$`m :A§H$J{UVmà_mUoM, ~¡{OH$ amer_Yrb gd© ‘yb^yV {H«$`m H$aVmZm ~oarO, dOm~mH$s,

JwUmH$ma Am{U ^mJmH$ma `m§Mo _yb^yV JwUY_© nmibo OmVmV. VgoM {MÝho gwÜXm dmnabr

OmVmV.

~¡{OH$ ametMr ~oarO Am{U dOm~mH$s :

VrZ Q>monë¶m> KoD$Z Ë`mV H$mhr g\$aM§Xo, H$mhr Am§~o d H$mhr g§Ìr Imbrbà_mUo EH$Ì

R>odbr AmhoV.

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1 ë`m Q>monbrV 8 g\$aM§X Am{U 8 Am§~o R>odbo AmhoV.

2 è`m Q>monbrV 6 g\$aM§X Am{U 4 g§Ìr R>odbo AmhoV.

3 è`m Q>monbrV 5 g\$aM§X Am{U 6 g§Ìr R>odbo AmhoV.

Oa gd© \$io EH$m Q>monbr‘Ü¶o R>odbr Va Ë`m_Üò

8 + 6 =14 g\$aM§X

8 + 5 = 13 Am§~o

4 + 6 = 10 g§Ìr

VerM {H«$`m ~¡{OH$ ametMr ~oarO Am{U dOm~mH$s H$aVmZm dmnaVmV. Amåhr gOmVr`

nXm§Mr ~oarO qH$dm dOm~mH$s H$aVmo.

~¡{OH$ ametMr ~oarO Am{U dOm~mH$s H$aVmZm nyUmªH$mMr ~oarO Am{U dOm~mH$s H$aVmZmMo

{Z`_ dmnaVmo.

AmR>dm :

1) YZ nyUmªH$m§Mr ~oarO YZ nyUmªH$ AgVo.

2) G$U nyUmªH$m§Mr ~oarO G$U nyUmªH$ AgVo.

3) YZ nyUmªH$ _moR>m AgVmZm YZnyUmªH$ d G$U nyUmªH$ `m§Mr ~oarO Zoh_r YZ òVo. G$U

nyUmªH$ _moR>m AgVmZm YZ nyUmªH$ d G$U nyUmªH$ `m§Mr ~oarO Zoh_r G$U òVo.

~¡{OH$ ametMr ~oarO -

a) EH$nXtMr ~oarO i) \$º$ gOmVr` nXm§Mr ~oarO hmoVo.

ii) Ë`m§Mr ~oarO H$aÊ`mgmR>r Ë`m§À`m g§»`mË_H$ ghJwUH$m§Mr ~oarO H$amdr.

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CXmhaU 1 : 4x, 8x Am{U 5x {_idm.

CH$b 4x + 8x + 5x gd© gOmVr` nXo AmhoV.

= (4 + 8 + 5)x MbnX x À`m g§»`mË_H$ ghJwUH$m§Mr ~oarO H$am.

= 17x g§»`mË_H$ ghJwUH$m§~amo~a ‘x’ {bhm.

CXmhaU 2 : 3a, 5b, 8a Am{U 2b {_idm

CH$b 3a + 5b + 8a + 2b= 3a + 8a + 5b + 2b= (3+8)a + (5+2)b= 11a + 7b

oWo a d b hr XmoZ MbnXo AmhoV. gOmVr¶ MbnXm§Mo JQ> H$am. g§»¶mË‘H$ ghJwUH$m§Mr ~oarO H$am Am{U MbnXo {bhm.

CXmhaU 3 : 3x2y, 2xy2 Am{U 7yx2 {_idm.

CH$b 3x2y + 2xy2 + 7yx2

= 3x2y + 7yx2 + 2xy2 = (3+7)x2y + 2xy2

= 10x2y + 2xy2

gyMZm :

i) x2y = yx2

ii) x2y ≠ xy2

b) {ÛnXr Am{U {ÌnXtMr ~oarO

i) AmS>dr nÜXV : gd© ~¡{OH$ amerVrb nXo EH$m AmS>ì`m aofoV {bhÿZ Ë`mVrb gOmVr` nXo EH$Ì {bhÿZ Ë`m§Mr ~oarO H$amdr.

CXmhaU 1 : (3a + 2b) Am{U (5a + 7b) Mr ~oarO H$am.

CH$b : AmS>dr nÜXV

(3a + 2b) + (5a + 7b)

= 3a + 2b + 5a + 7b

= 3a + 5a + 2b + 7b

= 8a + 9b

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ii) C^r nÜXV : amerVrb gOmVr` nXo Aer ñV§^m_Üò _m§S>m H$s Vr EH$mImbr EH$

òVrb. Z§Va nXm§Mr ñV§^mZwgma ~oarO H$am.

CXmhaU 2 : (3a + 2b) Am{U (5a + 7b) ¶m§Mr ~oarO H$am.

3a + 2b ... g§H$ë¶5a + 7b ... g§H$bH$8a + 9b ... ~oarO/g§H$bZ

CXmhaU 3 : (14m2n-10n2) Am{U (10m2n+6n2) Mr ~oarO H$am.

CH$b AmS>dr nÜXV C^m nÜXV

(14m2n -10n2) + (10m2n + 6n2) 14m2n-10n2

= 14m2n - 10n2 + 10m2n + 6n2 +10m2n+6n2

= 14m2n + 10m2n -10n2 + 6n2 + 24m2n-4n2

= 24m2n - 4n2

CXmhaU -4 : (4x+3y+5z), (2x+5y-2z) Am{U (3x-4y-z) Mr ~oarO H$am.

CH$b AmS>dr nÜXV C^r nÜXV

(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

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ñdmÜ`m` 4.4

I. ~oarO H$am :1) 4a, 7a, 9a, 21a 2) 6p2, 4p2, -10p2, 12p2

3) 6y, 7x, -3y, -4y, -4x 4) 2a2, a, 3b2, 4a2, 11a, -4b2

II. gaiê$n Úm :1) 5x + 7x - 3x + 2x - 6x 2) 5a2 + 2a2-a2 + 7a2

3) 5xy + 6xy - 9xy - xy + 10xy - 2xy4) 2x 2 + 4y - 3z + x2 - y + z + 4x2 + 3y + z

III. Imbrb {ÛnXtMr ~oarO H$am :

1) (3x + 4y) Am{U (5x + 7y)

2) (13a - 4b) Am{U (4a - 6b)

3) (xy - 7z) Am{U (6z -10xy)

4) (20a2bc - 3ab2c) Am{U (10ab2c - 3a2bc)

IV. Imbrb {ÌnXtMr ~oarO H$am :

1) (4x + 3y + 5z) Am{U (2x - 6y - 2z)

2) (p2 + 3q2 + 5r2) Am{U (6p2 - 2q2 + 6r2)

3) (4a + b - 2c) Am{U (3b + 2a + 5c)

4) (4x2y - 5y2z + 6z2x) Am{U (7z2x - 2x2y + 3y2z)

V. Imbrb ametMr ~oarO H$am :

1) (5mn - 3p), (mn + p), (6mn - 2p) 2) (x + y - z), (2x + 3y + 2z) Am{U (5x - 6y + 3z) 3) (5a2 + 2a-3), (2a2 + 3a + 1) Am{U (3a2 - 4a + 5)

VI. gd© nXm§Mr ~oarO H$am :

5x2, 2x2 + y, 3 x2 + 3y + 7, y2+ 5z + 6y + 4

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~¡{OH$ ametMr dOm~mH$s :

a) EH$nXtMr dOm~mH$s : i) \$º$ gOmVr` nXm§Mr dOm~mH$s hmoVo. ii) g§»`mË_H$ ghJwUH$m§Mr dOm~mH$s Ho$br AgVm Ë`m§À`m _Yrb \$aH$ g_OVmo.

CXmhaU 1 : 12x _YyZ 5x dOm H$am.

CH$b 12x - 5x

= (12 - 5) x

= 7x

CXmhaU 2 : 7xy d 3xy Mr ~oarO H$ê$Z Ë`m_YyZ 5xy dOm H$am.

CH$b (7xy + 3xy) - 5xy

= 10 xy - 5 xy

= 5xy

b) {ÛnXr d {ÌnXtMr dOm~mH$s :

i) AmS>dr nÜXV : gd© amer_Yrb nXo EH$m AmS>ì`m aofoV _m§Sy>Z gOmVr` nXm§Mo

JQ> EH$Ì H$am. amerVrb dOm hmoUmè`m nXm§Mr {MÝho ~XbyZ (+ {MÝhmMo - qH$dm -

{MÝhmMo +) Z§Va Ë`m§Mr ~oarO H$am.

CXmhaU 1 : (9x+7y) _YyZ (5x+3y) dOm H$am.

CH$b (9x + 7y) - (5x + 3y) = 9x + 7y - 5x - 3y = 9x - 5x + 7y - 3y = (9 - 5)x + (7 - 3)y

= 4x + 4y

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ii) C^r nÜXV : `m nÜXVrV amerVrb gOmVr` nXo doJdoJù`m ñV§^m_Üò EH$mImbr

EH$ òVrb. Aer ‘m§Sy>Z ¿¶m. Z§Va dOm hmoUmè`m àËòH$ nXm§Mo {MÝh ~XbyZ gOmVr`

nXm§Mr ~oarO H$am.

CXmhaU 1 : (9x+7y) _YyZ (5x+3y) dOm H$am.

CH$b 9x + 7y ì¶dH$ë¶

5x + 3y ì¶dH$bH$

(-) (-)

4x + 4y \$aH$

CXmhaU 2 : (2ax + 5by) _YyZ (ax + by) dOm H$am.

(2ax + 5by) - (ax + by) 2ax + 5by ì¶dH$ë¶

= 2ax + 5by - ax - by (-)ax + (-)by ì¶dH$bH$

= (2 - 1)ax + (5 - 1) by ax + 4by \$aH$/ì¶dH$bH$

= ax + 4by

CXmhaU 3 : (10a - 5b + 12c) _YyZ (2a + b + 6c) dOm H$am.

CH$b AmS>dr nÜXV C^r nÜXV

(10a - 5b + 12c) - (2a + b + 6c) 10a - 5b + 12c ì¶dH$ë¶

= 10a - 5b + 12c - 2a - b - 6c 2a + b + 6c ì¶dH$bH$

= (10a - 2a) + ( - 5b - b) + (12c - 6c) ( - ) ( - ) ( - ) = 8a - 6b + 6c 8a - 6b + 6c \$aH$

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ñdmÜ`m` 4.5

I. Imbr Xþgè`m nXm_YyZ n{hbo nX dOm H$am.

1) 8x, 15x 2) 6a2b, 14a2b

3) -6ab, 10ab 4) 8x2y, -14x2y

II. dOm~mH$s H$am.

1) (7x + 4y) ho (12x + 6y) ‘YyZ dOm H$am.

2) (13a - 6b) ho (17a - 4b) ‘YyZ dOm H$am.

3) (- 6m2n - mn2) ho ( - 10m2n + 4mn2) ‘YyZ dOm H$am.

4) (7pqr - 8q + 6) ho (10 - 4pqr + 3q) ‘YyZ dOm H$am.

III. 1) (6x + 10y) hr amer (2x + 6y) nojm {H$VrZo ‘moR>r Amho.

2) (5c - 2d) nojm (2d - c) hr {H$VrZo ‘moR>r Amho.

3) (3x + 4y + 6) ‘Ü¶o H$m¶ {‘i{dë¶mZo (4x - 2y - 8) amer {‘iob.

4) (9m2n2 - 12xy + 10) ‘Ü¶o H$m¶ {‘i{dë¶mZo (15m2n - 10xy) {‘iob.

~¡{OH$ ametMm JwUmH$ma

Imbrb ½bmgMr ‘m§S>Ur nhm.

5 ½bmg

5 ½bmg

5 ½bmg

5 ½bmg ½bmgMr EH$ n§³Vr H$‘o H$adr.

4 n§º$s_Üò ½bmgMr aMZm Ho$br Amho. àËòH$ n§º$sV 5 ½bmg AmhoV. 4 n§º$s_Üò EHy$U

{H$Vr ½bmg AmhoV?

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Amåhr `mnyduM {eH$bmo AmhoV H$s JwUmH$ma åhUOoM ~oaOoMr nwZamd¥Îmr hmo`.

5 + 5 + 5 + 5 = 20 ½bmg

5 × 4 = 20 ½bmg

g_Om Oa EH$m n§º$sV '2x' ½bmg AmhoV.

Va 4 eoë\$ _Üò Agboë`m ½bmgMr EHy$U g§»`m

: 2x + 2x + 2x + 2x = 8x 2x × 4 = 8x ½bmg

nyUmªH$m§Mm JwUmH$ma H$aVmZm dmnabobr {MÝho AmR>dm :

YZ nyUmªH$ (+a) × YZ nyUmªH$ (+b) = YZ nyUmªH$ (+ab)

YZ nyUmªH$ (+a) × G$U nyUmªH$ (-b) = G$U nyUmªH$ (-ab)

G$U nyUmªH$ (-a) × YZ nyUmªH$ (+b) = G$U nyUmªH$ (-ab)

G$U nyUmªH$ (-a) × G$U nyUmªH$ (-b) = YZ nyUmªH$ (+ab)

EH$nXtMm JwUmH$maCXmhaU 1 : 3x d 5y Mm JwUmH$ma H$am

CH$b : àW_ g§»`mË_H$ ghJwUH$m§Mm JwUmH$ma H$ê$Z Z§Va MbnX ghJwUH$m§Mm JwUmH$ma H$am.(3x) (5y) = (3 × x) × (5 × y) Q>rn :

3×5≠35na§Vy a × b = abOa MbnXo ~mOy-~mOybm {b{hbr Va Ë`m_Üò JwUmH$ma AgVmo.

= (3 × 5) × ( x × y) = 15xy

CXmhaU 2 : (-20x) d (+3y) Mm JwUmH$ma H$am.

CH$b : (-20x) (+3y) = (-20 × x) (3x y)

= (-20 × 3) (x × y)

= -60xy

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CXmhaU 3 : (2x), (-3x), (4x) `m§Mm JwUmH$ma H$am.

CH$b : (-2x) (-3x) (4x) = (-2 ×-3 × 4) (x × x × x)

= +24x 3

CXmhaU 4 : 3x 2 Am{U 5x 3 ¶m§Mm JwUmH$ma H$am.

CH$b : (3x 2) (5x 3)

= (3 × 5) (x2 × x3)

= 15x 5

{ÛnXr d {ÌnXtMm JwUmH$ma

5 Am{U 108 ¶m§Mm JwUmH$ma nhm.

5 × 108. = 540

hm 5 (100+8) Agm {b{hVm ¶oVmo H$m?

CH$b : (5 × 100) + (5 × 8)

= 500 + 40

= 540

nyUmªH$m‘Ü¶o {dVaU JwUY‘m©Mr AmR>dU H$am.

a (b + c) = (a × b) + (a × c)

CXmhaU 1 : (3x + 2y) bm 5y Zo JwUm

CH$b : 5y (3x + 2y)

= (5y × 3x) + (5y × 2y)

= 15xy + 10y2

CXmhaU 2 : (8y - 3) Am{U 4x Mm JwUmH$ma H$am.

CH$b : 4x (8y - 3)

= (4x × 8y) + (4x × - 3)

= 32xy - 12x

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CXmhaU 3 : (8x -17y) Am{U (- 2x) Mm JwUmH$ma H$am. CH$b : (8x - 17y) ( - 2x)

= (8x × -2x) + (-17y - 2x)

= -16x2 + 34xyÛrnXrZo ÛrnXrbm JwUUo :

AmnU {dVaU JwUY_© dmnê$Z {ÛnXrZo {ÛnXrbm JwUy eH$Vmo.{ÛnXtMm JwUmH$ma nmhÿ .

CXmhaU 1 : (x + 1) (x+ 2) = x (x + 2) + 1(x + 2)

= x2 + 2x + x +2

= x2 + 3x + 2

{ÛnXrbm {ÛnXrZo JwUVmZm Amåhr FOIL nÜXV gwÜXm dmnê$ eH$Vmo.

(x + 1) (x + 2)

F - (First) n{hë`m XmoZ nXm§Mm JwUmH$ma x × x = x2

O - (Outer) ~mhoarb XmoZ nXm§Mm JwUmH$ma x × 2 = 2xI - (Inner) AmVrb XmoZ nXm§Mm JwUmH$ma 1 × x = xL - (Last) eodQ>À`m XmoZ nXm§Mm JwUmH$ma 1 × 2 = 2 åhUyZ, (x + 1) (x +2) = x2 + 2x + x + 2 = x2 + 3x +2

CXmhaU 2 :

(2a + 5) (a - 7) `m§Mm JwUmH$ma H$am.

CH$b : (2a + 5) (a - 7)

= 2a (a-7) + 5 (a-7)

= 2a2 - 14a + 5a - 35

= 2a2 - 9a - 35

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CXmhaU 3 : (n - 11) Am{U (n - 5) `m§Mm JwUmH$ma H$am.

CH$b : (n - 11) (n - 5)

= n (n - 5) - 11 ( n - 5)

= [(n × n) - (n × 5)]- [(11 × n) - (11 × 5)]

= n2 - 5n - (11n - 55)

= n2 - 5n - 11n + 55

= n2 - 16n + 55

ñdmÜ`m` 4.6

I. JwUmH$ma H$am. 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. JwUmH$ma H$am.

1) 6xy, 2yz Am{U 7xz 2) -7ab, 5bc Am{U -6ca

3) 4x2y, -3y2z Am{U x2 4) -3a2, -2b3 Am{U -4e4

III. JwUmH$ma H$am. 1) (6x + 7y) bm 2z Zo JwUm 2) (3p - 5q) bm -5p Zo JwUm

3) (-18x - y) bm 2z Zo JwUm 4) (2ab + 3bc) bm 6abc Zo JwUm

IV. JwUmH$ma H$am. 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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àH$aU- 5

H$moZm§Mr OmoS>r (PAIR OF ANGLES)

¶m àH$aUmÀ¶m AÜ¶¶ZmZ§Va Vwåhr :

gb¾ H$moZ, H$moQ>rH$moZ, nyaH$H$moZ, aofr¶ OmoS>r Am{U {eamo{déÜX H$moZm§Mm AW© g‘OyZ ¿¶mb.

gb¾ H$moZ, H$moQ>rH$moZ, nyaH$ H$moZ, {eamo{déÜX H$moZ Am{U aofr¶ OmoS>çm AmoiImb.

gb¾ H$moZ, H$moQ>rH$moZ, nwaH$ H$moZ, aofr¶ OmoS>r Am{U {eamo{déÜX H$moZ ¶mda AmYmarV CXmhaUo gmoS>dmb.

X¡Z§{XZ OrdZmV AmnU doJdoJù¶m ì¶dgm¶m‘Ü¶o ^m¡{‘VrH$ kmZmMm Cn¶moJ H$aVmo. OrdZmon¶moJr ì¶dgm¶ Agbobo gwVmaH$m‘, {edUH$m‘, A{^¶m§{ÌH$s BË¶mXr‘Ü¶o ^m¡{‘VrH$ H$ënZm Agboë¶m aofm, H$moZ BË¶mXrMo kmZ Iyn Cn¶moJr AgVo. ¶m n[aH$ënZm§Mm g‘mdoe AgUmao g§X^© Amnë¶m OrdZmV nwîH$i AmhoV.

CXmhaUmW© ImobrMo H$monao, q^Vrbm bmdbobr {eS>r, S>m|JamMm CVma ¶m gdmª‘Ü¶o H$moZ V¶ma hmoVmV. ¶m§Zr {Z‘m©U Ho$bobo H$moZ AmnU emoYy eH$Vmo.

‘mJrb B¶ÎmoV Vwåhr doJdoJù¶m àH$maÀ¶m aofm Am{U H$moZm~Ôb {eH$bm AmhmV.

AmVm Imbrb aofm H$moUË¶m àH$maÀ¶m AmhoV AmoiIm.

(i)

(ii) (iii)

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CXmhaUmW© : {MÌ (i) {jVrO g‘m§Va aofm

{MÌ (ii) -------------- {MÌ (iii) --------------

Imbr {Xboë¶m {MÌmVrb H$moZm§Mm àH$ma AmoiIy eH$mb?

O

A

B

{MÌ i

OP

Q

{MÌ iiO

K

L{MÌ iii

O

X

Y

{MÌ iv

CXmhaUmW© : {MÌ (i) ‘Ü¶o H$moZ AOB hm bKwH$moZ Amho.

Imbr B§J«Or dU©‘mboVrb H$mhr Ajao {Xbr AmhoV. Ë¶mVrb H$moZ AmoiIm. Ë¶m H$moZm§Zm bKwH$moZ, H$mQ>H$moZ Am{U {dembH$moZ Aer Zmdo Úm.

CXmhaUmW© : ''T'' Ajam‘Ü¶o Xe©{dbobm H$moZ H$mQ>H$moZ Amho.

ñ‘aU H$am :

• q~Xÿ hr y{‘Vr‘Yrb gdm©V gmonr AmH¥$Vr Amho. q~Xÿbm AmH$ma Zmhr. na§Vw ""ñWmZ'' Amho. EH$m H$mJXmda Q>moH$Xma noÝgrbZo Ho$bobr IyU, m¡{VH$arË¶m q~XÿMo à[V{ZYrËd H$aVo.

• aofobm àma§^ Am{U A§Ë¶ q~Xÿ ZgVmo. hr XmoÝhr ~mOybm A‘¶m©XnUo dmTy> eH$Vo Aer H$ënZm H$am. ho XmoÝhr Q>moH$mbm ~mU Xe©dyZ gyM{dVmV.

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• aofmI§S>m‘Ü¶o àma§^ Am{U A§Ë¶ q~Xÿ AgVmV. hm aofoÀ¶m XmoZ q~Xÿ‘Yrb EH$ ^mJ AgVmo.

X

M

K

L O

N

PQY

KLO {ÌH$moUmÀ¶m gd© ~mOy aofmI§S> AmhoV.

• {H$aUmbm EH$ Ama§^ q~Xÿ AgyZ, A§{V‘ q~Xÿ ZgVmo. {dMma H$am. gy¶© {H$aUm§Mm {dMma H$am. Vo gy¶m©nmgyZ gd© {Xeobm ngabobo AgVmV.

gy¶©

• H$moZ EH$mM Ama§^q~XÿVyZ XmoZ {H$aU {ZKmë¶mZo V¶ma hmoUmar AmH¥$Vr åhUOo H$moZ hmo¶. AWdm g‘mZ q~XÿVyZ {ZKmboë¶m XmoZ aofmI§S>mZo ¶wº$ AmH¥$Vrg H$moZ Ago åhUVmV. H$moZ Xe©{dUmè¶m {H$aUm§Zm ~mOy Am{U Ë¶m gm§JUmè¶m q~Xÿbm {eamoq~Xÿ åhUVmV.

H$moZmMo ‘mn ho XmoZ ~mOy ‘Yrb Ho$di CVaË¶m qH$dm {ZaH$g ê$nm‘Ü¶o

O

A

B

AgVo.

{Xboë¶m {MÌm‘Ü¶o AO Am{U OB ho XmoZ O q~XÿVyZ {ZKmbobo {H$aU AmhoV. {jVrO aofm OB Am{U b§~ aofm AO À¶m ‘Ü¶o V¶ma Pmbobm H$moZ H$mQ>H$moZ hmo¶. åhUOoM AOB 900=

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OmUyZ ¿¶m :

H$moZmMo ‘mn : EH$m {H$aUmÀ¶m Ama§^ñWmZmnmgyZ A§{V‘ ñWmZmn¶ªVÀ¶m dVw©imH$ma H$joMoo n[a‘mU åhUOoM H$moZ hmo¶. ho n[a‘mU KS>çmimÀ¶m H$mQ>çmMm {XeoZ qH$dm {déÜX {XeoZo Agy eH$Vo. H$mQ>çmÀ¶m {XeoZo AgUmao n[a‘mU YZ Am{U {déÜX {XeoZo Agbobo F$U AgVo.

Ama§^ ñWmZ

A§{V‘ ñWmZ

Ama§^ ñWmZ

A§{V‘ ñWmZKS>çmimÀ¶m H$mQ>çmÀ¶m

{déÜX {XeoZoKS>çmimÀ¶m H$mQ>çmÀ¶m

{XeoZo

gyMZm : KS>çmimÀ¶m H$mQ>mÀ¶m {’$aÊ¶mÀ¶m {XeoZo qH$dm KS>çmimÀ¶m {déÜX {XeoZo H$moZ V¶ma Pmbm Var Ë¶mMo ‘yë¶ (value) EH$M AgVo.

g‘H$moZ :

Imbrb AmH¥$Ë¶m nhm. àË¶oH$ H$moZmMo ‘mn {H$Vr Amho?

àVb - 1

400400

400

àVb - 2

300 300

àVb - 3

500

500

àVb - 1 ‘Yrb àË¶oH$ H$moZmMo ‘mn 400 Amho. àVb - 2 ‘Ü¶o Vo 300 Am{U àVb - 3 ‘Ü¶o 500 Amho.

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EH$m àVbmVrb g‘mZ ‘mno AgUmè¶m H$moZm§Zm g‘H$moZ åhUVmV.

H$m±½¶«w¶§Q> ¶m eãXmMm AW© ‘wbV… "g‘' Agm hmoVmo. hm b°Q>rZ eãXm‘YyZ Ambm Amho. H$m±J«mo åhUOo "‘mÝ¶ H$aUo' Ë¶m‘wio g‘mZ AgUo Agm AW© hmoVmo.

OmoS>rH$moZ :

^y{‘Vr‘Ü¶o OmoS>r H$moZm§Mo H$mhr {deof JwUY‘© AmhoV. Amnë¶mbm ‘m{hV Agbobo bKwH$moZ, H$mQ>H$moZ Am{U {dembH$moZ ¶m§À¶m ghmæ¶mZo OmoS>r H$moZ ‘Yrb g§~§Y g‘OyZ KoD$.

gb¾ H$moZ :

{Xbobr AmH¥$Vr nmhÿZ EH$mÀ¶m ~mOybm EH$ AgUmao H$moZ AmoiIm.

• AmH¥$Vr‘Ü¶o V¶ma Pmboë¶m H$moZm§Zm Zmdo Úm.

• AmH¥$Vr‘Ü¶o EH$mÀ¶m ~mOybm EH$ AgUmao H$moZ AmoiIm.

• EH$mÀ¶m ~mOybm EH$ AgUmè¶m H$moZmÀ¶m ~mOy Am{U Ë¶m§À¶m {eamoq~XÿMo Zmd {bhm.

OB Am{U OC ¶m BOC H$moZmÀ¶m ~mOy AmhoV. Am{U 'O' hm {eamoq~Xÿ Amho.

OA Am{U OB ¶m AOB H$moZmÀ¶m ~mOy AmhoV.

åhUOo ;

i) ¶oWo EH$ gm‘mB©H$ {eamoq~Xÿ Amho.

ii) ¶oWo EH$ gm‘mB©H$ ~mOy Amho.

EH$ gm‘mB©H$ {eamoq~Xÿ Am{U EH$ ~mOy gm‘mB©H$ AgUmè¶m OmoS>r H$moZm§Zm gb¾H$moZ Ago åhUVmV.

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{Xboë¶m {MÌmV AOB Am{U BOC ho gb¾ H$moZ AmhoV.

H¥$Vr - 1 :

BC

O

A

{Xboë¶m {MÌm‘Yrb gb¾ H$moZm§À¶m VrZ OmoS>çm AmoiIm.

{dMma H$am :

• XmoZ {demb H$moZ gb¾ H$moZ hmoVrb H$m? VH©$ H$am.

• EH$ bKwH$moZ EH$m {dembH$moZmÀ¶m gb¾ H$moZ hmoBb H$m?

{dMma H$am :

AOB Am{U AOC ho gb¾ H$moZ AmhoV.

ñdmÜ¶m¶ 5.1

1. Imbrb AmH¥$Ë¶m§À¶m AmYmao {Xbobm Vº$m nyU© H$am :

(i)(ii)

(iii)

(iv) (v)(vi)

(vii)

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AmH¥$Vr H$moZ ~mOy gm‘mB©H$ {eamoq~Xÿ

gm‘mB©H$ ~mOy

H$moZ, gb¾ H$moZ AmhoV qH$dm

ZmhrV

1 2 3 4 5

iAOB ,

BOCOA, OB, OC O OB hmo¶

AOBc Am{U

BOCc

ii

iii

iv

v

vi

vii

2. {Xboë¶m {MÌm‘Yrb gb¾ H$moZm§À¶m OmoS>çm AmoiIÊ¶mgmR>r Ë¶m§Mo gm‘mB©H$ {eamoq~Xÿ Am{U gm‘mB©H$ ~mOy nwT>rb V³Ë¶mV Z‘yX H$am.

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114

H$moZ

{eamoq

~XÿMo

Zmd

gm‘m

B©H$

~mOy|Mo Z

md

gm‘m

B©H$

{eamoq

~Xÿ

gm‘m

B©H$ ~

mOy

gb¾

H$moZ

m§Mr OmoS>r

(hmo¶

/Zmhr)

AOB and BOD Am{U AOB and BOD

AOD and BOC Am{U AOD and BOC

AOC and BOC Am{U AOC and BOC

BOC and AOB Am{U BOC and AOB

AOD and AOB Am{U AOD and AOB

AmUIr H$mhr OmoS>r H$moZ {bhm. Vo gb¾ H$moZ AmhoV H$m? narjU H$am.

3. ¶m AmH¥$Vr‘Ü¶o EH$m H§$gmZo AWdm XmoZ H§$gmZo Xe©{dboë¶m H$moZm§Mo {ZarjU H$am. Vo gb¾ H$moZ AmhoV H$m?

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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115

H$moQ>r H$moZ :

Imbrb AmH¥$Vr‘Yrb EH$mM àVbmV AgUmè¶m OmoS>r H$moZm§Mo {ZarjU H$am.

(i) (ii)

(iii)

{Xboë¶m AmH¥$Vr : i), ii) Am{U iii) ‘Yrb OmoS>r H$moZm§À¶m ‘mnZmMr qH$‘V emoYm.

EH$mM àVbm‘Yrb XmoZ H$moZm§Mr qH$‘V 900 hmoV Agob Va Ë¶m H$moZm§Zm H$moQ>r H$moZ Ago åhUVmV.

åhUOo ho XmoÝhr H$moZ EH$Ì {‘i{dë¶mg V¶ma hmoUmam ZdrZ H$moZ 900 Mm AgVmo.

darb AmH¥$Vr‘Yrb 90ABC PQR 0+ = ¶oWo B hm Q Mm Am{U Q hmB Mm H$moQ>rH$moZ Amho.

""EH$m OmoS>r H$moZmMo ‘mn {‘i{dë¶mg Ë¶m§Mr qH$‘V EH$m H$mQ>H$moZm BVH$s hmoV Agë¶mg Ë¶m§Zm H$moQ>rH$moZ åhUVmV.''

bjmV R>odm : XmoZ H$moZ H$moQ>rH$moZ Agë¶mg EH$ H$moZ Xþgè¶mMm H$moQ>rH$moZ AgVmo.

{dMma H$am : EH$ H$moZ x0 Agë¶mg Ë¶mÀ¶m H$moQ>rH$moZ {H$Vr?

CXmhaU 1 : 350 À¶m H$moZmMm H$moQ>rH$moZ emoYm. 350 H$moZmMm H$moQ>rH$moZ x0 g‘Oy

350 + x0 = 900 (H$moQ>rH$moZmMr qH$‘V 900)

x0 = 900 -350

x0 = 550

350 H$moZmMm H$moQ>rH$moZ 550 Amho.

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CXmhaU 2 : EH$m H$moZmMm H$moQ>rH$moZ Ë¶mÀ¶m nmMnQ> Amho. Va OmoS>r H$moZmMr ‘mno H$mT>m.

EH$m H$moZmMo ‘mn x0 g‘Oy

¶mÀ¶m 5 nQ> H$moZ x0 + 5 = 5x0

x0 + 5 x0 = 900

6x0 = 900

x0 = 900

60

x0 = 150

150 À¶m H$moZmMm H$moQ>rH$moZ 750 hmoVmo.

{dMma H$am :

* H$mQ>H$moZmÀ¶m H$moQ>rH$moZmMo ‘mn {H$Vr?

* 600 À¶m H$moZmÀ¶m H$moQ>rH$moZmMo ‘mn {H$Vr?

{dMma H$am :

XmoZ bKwH$moZ nañna H$moQ>rH$moZ AgVrb H$m?

XmoZ {demb H$moZ nañna H$moQ>rH$moZ hmoVrb H$m?

XmoZ H$mQ>H$moZ nañna H$moQ>rH$moZ hmoVrb H$m?

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117

H¥$Vr : EH$ nm§T>am H$mJX ¿¶m. {MÌmV

Xe©{dë¶mà‘mUo Ë¶mMr KS>r Kmbm. Z§Va H$mJX CbJSy>Z "0' Xe©dm Am{U KS>r Ho$boë¶m ^mJm‘Ü¶o V¶ma Pmbobm H$moZ ‘moOm. AmH¥$Vr (1) AmH¥$Vr (2) AmH¥$Vr (3)

o o o

¶mdê$Z Vwåhr H$moUVm {ZîH$f© H$mT>mb?H¥$Vr :

EH$m H$mS>©erQ>da EH$mM ‘mnmÀ¶m {ÌOoMr XmoZ dVw©io H$mT>m.

dVw©im§Zm A Am{U B Aer Zmdo Úm.

dVw©i A À¶m n[aKmn¶ªV {ÌÁ¶m H$mT>m.

{MÌmV Xe©{dë¶mà‘mUo H$moZ‘mnH$ Am{U ‘moOnÅ>rÀ¶m ghmæ¶mZo AZwH«$‘o 100, 200, 300, 400, 500, 600, 700 Am{U 800 Mo H$moZ dVw©imV Xe©dm.

àË¶oH$ H$moZmMm dVw©iI§S> mJ H$mnyZ doJim H$am. Vwåhmbm 8 dVw©iI§S> mJ {‘iVmV.

dVw©i B ‘Ü¶o 900 Xe©dm.

40 302010

80706050

A

{MÌmV Xe©{dë¶mà‘mUo 8 dVw©iI§S>mn¡H$s EH$ dVw©imÀ¶m ~mOy

OX ~amo~a ¶mo½¶arË¶m Owiob Aem àH$mao R>odm. Vmo dVw©iI§S> mJ LOX hmoD$ Úm. AmVm AmUIr EH$ dVw©iI§S> ^mJ {ZdSy>Z dVw©imÀ¶m ^mJmbm Owi{dÊ¶mMm à¶ËZ H$am.

doJdoJù¶m dVw©iI§S>mMo mJ KoD$Z H¥$Vr nwÝhm H$am. H$moUVo XmoZ dVw©iI§S> ^mJ XOY er OyiVmV Ë¶m§Mr ¶mXr H$am. Ë¶m§Zr XOY mJ PmH$bm

OmVmo.

Vw‘À¶m H¥$Vr‘Ü¶o Cn¶moJ Ho$boë¶m H$moQ>rH$moZm§Mr ¶mXr H$am.

y

O

Lx

B

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ñdmÜ¶m¶ 5.2

1. AmH¥$Vr‘Yrb àË¶oH$ OmoS>r H$moZmMr ‘mno EH$‘oH$mMo H$moQ>rH$moZ AmhoV. Ë¶mn¡H$s EH$mMo ‘mn {Xbo Amho. Xþgè¶mMo ‘mn H$mT>m.

(i) (ii) (iii) (iv)

2. ¶oWrb àË¶oH$ H$moZmÀ¶m H$moQ>rH$moZmMo ‘mn emoYm

a) 220 b) 840 c) 20.50 d) 40.50

3. EH$ H$moZ Ë¶mÀ¶m H$moQ>rH$moZmÀ¶m 3 nQ> Agë¶mg Ë¶m OmoS>rH$moZmMo ‘mn H$mT>m.

nyaH$ H$moZ (Supplementary angles) :

EH$m àVbmda AgUmao OmoS>rH$moZ nhm.O

500 1300

S P Q R

M

AmH¥$Vr - i AmH¥$Vr - ii

O

7201080

S P Q R

M

AmH¥$Vr - iii

O 800

1000

S

P Q R

M

{Xboë¶m AmH¥$Ë¶m$ i), ii) Am{U iii) ‘Ü¶o ∠SPO d ∠RQM OmoS>r H$moZm§Mr ~oarO H$mT>m.

EH$m àVbmVrb OmoS>r H$moZm§Mr qH$‘V gaiH$moZmBVH$s Agob Va Ë¶m H$moZm§Zm nyaH$H$moZ Ago åhUVmV.

EH$m àVbm‘Yrb OmoS>r H$moZm§Mr qH$‘V gaiH$moZ qH$dm 1800 Agë¶mg Ë¶m H$moZm§Zm nyaH$H$moZ Ago åhUVmV.

darb AmH¥$Vr‘Ü¶o SPO MQR 1800+ =

¶oWo ,P Q Mm nyaH$ H$moZ Amho Am{U ,Q hm P Mm nyaH$ H$moZ Amho.

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gyMZm : XmoZ H$moZ nyaH$ AgVrb Va EH$ H$moZ Xþgè¶mMm nyaH$ AgVmo.

{dMma H$am : EH$ H$moZ x0 Agë¶mg Ë¶mÀ¶m nyaH$ H$moZ {H$Vr Agob?

CXmhaUo :

CXmhaU 1 : 130.50 H$moZmÀ¶m nyaH$ H$moZmMo ‘mn H$mT>m.

1300 H$moZmMm nyaH$ H$moZ x0 g‘Oy

x0 + 130.50 = 1800 [nyaH$ H$moZmMr qH$‘V]

åhUOo x0 = 1800-130.50

x0 = 49.50.

CXmhaU 2 : EH$ H$moZ Xþgè¶m H$moZmMm nyaH$ H$moZ Amho. ‘moR>çm H$moZmMo ‘mn bhmZ H$moZmÀ¶m ‘mnmnojm 900 OmñV Agë¶mg àË¶oH$ H$moZmMo ‘mn H$mT>m.

bhmZ H$moZ x 0 g‘Oy.

‘moR>m H$moZ (x 0 + 900) hmoVmo.

AmVm x 0 + (x 0 + 900) = 1800

x 0 + x 0 + 900 = 1800

2x 0 + 900 = 1800

2x 0 = 1800 - 900

x

x290

45

0=

=2x0 = 900

∴ bhmZ H$moZmMo ‘mn 450

‘moR>çm H$moZmMo ‘mn

= x0 + 900 = 450 + 900 = 1350

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120

{dMma H$am :ho H$moZ nyaH$ H$moZ AmhoV H$m?

RO

1450450

P

Q

{dMma H$am :

XmoZ bKwH$moZ nañna nwaH$ H$moZ hmoVrb H$m? ñnï>rH$aU Úm.

XmoZ {demb H$moZ nañna nyaH$ H$moZ hmoVrb H$m? ñnï>rH$aU Úm.

XmoZ H$mQ>H$moZ nañna nyaH$ H$moZ hmoVrb H$m? ñnï>rH$aU Úm.

gyMZm : nyaH$ H$moZ ho gb¾ H$moZ Agbo nm{hOoV Ago Zmhr.

ñdmÜ¶m¶ : 5.3

I. nwT>rb AmH¥$VrV H$moUVr OmoS>r H$moZ {‘i{dë¶mg nyaH$ H$moZ hmoVmV?

A O B

C

D

E

600700

200

300

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121

II. nwT>rb OmoS>r H$moZm§Mo {ZarjU H$am. Vo nyaH$ H$moZ AmhoV?

O

700 1100

P

QRAmH¥$Vr (i)

O

1350

450

P

M

NAmH¥$Vr (ii)

O

750 1050

A C

B

AmH¥$Vr (iii)

A

200 1600

EBC F

D

AmH¥$Vr (iv)O

8001100

Q

P

S

R

T

AmH¥$Vr (v)

III. àË¶oH$ H$moZmMm nyaH$ H$moZ H$mT>m.

1) 950 2) 1100 3) 120.50 4) 125.50

IV. XmoZ nyaH$ H$moZm‘Yrb ‘moR>çm ‘mnmMm H$moZ bhmZ ‘mnmÀ¶m H$moZmnojm 500 OmñV

Amho. Va XmoÝhr H$moZmMr ‘mno H$mT>m.

V. EH$ H$moZ Ë¶mÀ¶m nyaH$ H$moZmÀ¶m 5 nQ> Agë¶mg Vo H$moZ emoYm.

VI. {Xboë¶m AmH¥$Vr‘Ü¶o ABC 350= Am{U DBE 900= Agë¶mg Imbrb H$moZm§Mo ‘mn H$mT>m.

i) CBDMm nyaH$ H$moZ

B350900

A

C

D

E

ii) EBC Mm nyaH$ H$moZ

iii) ABDMm nyaH$ H$moZ

iv) EBDMm nyaH$ H$moZ

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aofr¶ H$moZm§Mr OmoS>r:

{Xboë¶m AmH¥$VrVrb gb¾ H$moZmMr ‘mno nhm.

O3001500

B

C

A

AmH¥$Vr - i

O900 900

B

C

A

AmH¥$Vr - ii

O600 1200

B

C

A

AmH¥$Vr - iii

¶m§Mr qH$‘V H$mT>m. Z§Va V³Ë¶mV âm.

AmH¥$Vr AOC BOC AOC BOC+

i. 1500 300 1500 + 300 = 1800

ii. 900

iii. 600

AOC Am{U BOC gb¾ H$moZ AmhoV?AmH¥$Vr i, ii, Am{U iii ‘Ü¶o gb¾ H$moZm§Mr ~oarO 1800 Amho. OA Am{U OB Am{U

AB Am{U {H$aZm‘wio gai aofm V¶ma Pmbr Amho.

Aem OmoS>r H$moZm§Zm H$m¶ åhUVmV?Ë¶m§Zm aofr¶ OmoS>r H$moZ Ago åhUVmV.

XmoZ H$moZ aofr¶ OmoS>r H$moZ hmoÊ¶mgmR>r Vo gb¾ H$moZ Agbo nm{hOoV.

H$moZmÀ¶m ‘mnZmMr qH$‘V 1800 Agbr nm{hOo.

bjmV R>odm : OmoS>r aofr¶ H$moZ EH$‘oH$mg gb¾ Agë¶mg Vo nyaH$ AgVmV.

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ñdmÜ¶m¶ : 5.4

1. {MÌm‘Yrb H$moZ AOB Am{U BOC ¶m§Mo {ZarjU H$am. AOB Am{U BOC ho OmS>r aofr¶ H$moZ AmhoV H$m? MMm© H$am Am{U {ZîH$f© H$mT>m.

O

650

1350

B

C

A

O600

1300B

C

A

2. XmoÝhr H$moZ gb¾ Agë¶mg, H$moUVr OmoSr H$moZ OmoS>r aofr¶ hmoVmo ho emoYm.

Q

B

O

A

1250

350

R

P

AmH¥$Vr - i

Q700

R

1100

O B

AP

AmH¥$Vr - ii

P

A

O BQ

R

400

1500

AmH¥$Vr - iii

A

O B

R

P Q900

900

AmH¥$Vr - iv

A

O BR

PQ

900

900

AmH¥$Vr - v

P

QR

1400

A

B

O400

AmH¥$Vr - vi

3. Imbrb H$moZ, OmoS>r aofr¶ H$moZ hmoÊ¶mgmR>r Xþgè¶m àH$maMm H$moZ emoYm.

a. bKwH$moZ b. EH$ {demb H$moZ c. EH$ H$mQ>H$moZ

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Ioi

XmoZ S´>°mBªJ H$mJX ¿¶m. àË¶oH$ H$mJXmMo 40 VwH$S>o H$am. AmVm Imbr {Xboë¶m ‘mnmMm H$moZ

àË¶oH$ VwH$S>çmda H$mT>m. AemàH$mao XmoZ JQ> H$am. H$moZ 10, 20, 30, 40, 50, 60, 70,

80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180 Am{U 25, 35, 45, 55,

65, 75, 85, 95, 105, 115, 125, 135, 145, 155, 165, 175 (Vwåhr H$moUVohr ‘mn

{ZdS>m. na§Vw Vo H$moQ>r qH$dm nyaH$ H$moZ hmoVrb Ago nhm) àË¶oH$ H$mS>m©À¶m ‘mJrb ~mOyg 1,

2, 3, 4, 5 Am{U 6 ¶m n¡H$s H$moUVrhr EH$ g§»¶m {bhm. EH$m Z§Va EH$ gd© H$mS>© Ë¶mdarb

g§»¶m {XgVrb Ago R>odm.

1 Vo 6 g§»¶m AgUmao qH$dm {Q>n³¶m AgUmao ’$mgo ¿¶m. (XmoZ qH$dm Ë¶mnojm A{YH$

{‘Ì hm Ioi Ioiy eH$VmV)

{Z¶‘ : àË¶oH$ IoimSy>Zo ’$mgo Q>mH$bo nm{hOoV Am{U Ë¶m g§»¶oMo H$mS>© KoVbo nm{hOo.

eodQ>r gd© H$mS>© g§në¶mZ§Va IoimSy>§Zr Amnbo H$mS>© EH$‘oH$mV ~XbyZ, Ë¶m§Mm H$moQ>r AWdm nyaH$

H$moZ hmoB©b AemàH$mao Owidm. ""Á¶m IoimSy>H$S>o OmñV OmoS>r H$mS>© AgVrb Vmo {dOoVm R>aob''.

{eamo{déÜX H$moZ :

XmoZ H$mS>çm KoD$Z Ë¶m {MÌmV Xe©{dë¶mà‘mUo Xmoè¶mZo ~m§Ym. AB Am{U CD H$mS>çmZm

aofm g‘Om Am{U Ë¶m ~m§Yboë¶m {R>H$mUmbm Ò' Ago g‘Om.

AB Am{U CD aofmI§S> Agë¶mg Ò' {eamoq~Xÿ Amho., ,AOC BOC AOD Am{U BOD ho H$moZ

AmhoV.

BOC bm {H$Vr gb¾ H$moZ AmhoV? Ë¶m§Mr Zmdo {bhm.

AOC bm {H$Vr gb¾ H$moZ AmhoV? Ë¶m§Mr Zmdo {bhm.

A

B

D

C

O

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AmVm BOC Am{U AOD Mo {ZarjU H$am.

AOC Am{U BOD ¶m§Mo {ZarjU H$am.

Vo gb¾ H$moZ AmhoV?

Ë¶m§Mo g§~§Y H$m¶?

BOC Am{U AOD ho {eamoq~XÿÀ¶m AmOy~mOyMo H$moZ AmhoV. Ë¶mMà‘mUo AOC

Am{U BOD XoIrb {eamoq~XÿÀ¶m AmOy~mOyMo H$moZ AmhoV. Aem OmoS>r H$moZm§Zm H$m¶ åhUVmV?

XmoZ aofm EH$‘oH$sbm N>oXë¶mZo V¶ma hmoUmao H$moZ nañna {déÜX Agë¶mg Ë¶m§Zm {eamo{déÜX H$moZ Ago åhUVmV.

{eamo{déÜX H$moZm§Zm EH$M gm‘mB©H$ {eamoq~Xÿ AgVmo.

EH$m OmoS>r {eamo{déÜX H$moZm‘Yrb g§~§Y g‘OyZ KoÊ¶mgmR>r nwT>rb H¥$Vr H$é¶m.

H¥$Vr - 1

A

B

D

C

O

A

B

D

C

O

EH$m H$mJXmda XmoZ aofm H$mT>m. Á¶m EH$‘oqH$Zm O' q~XÿV N>oXVrb, Ë¶m§Zm AB Am{U CD Aer Zmdo Úm. Z§Va {MÌmV Xe©{dë¶mà‘mUo MmaH$moZ , ,COB BOD AOD

Am{U AOC Xe©dm.

AmVm Ë¶mda EH$ Q´>oqgJ H$mJX AWdm nmaXe©H$ H$mJX R>odm. Q´>ooqgJ H$mJXmda AB

Am{U CD aofm Q´>og H$am.

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AmVm Q´>og Pmbobr àV ‘yi à{Vda Amho. Vr ‘yi H$mJXmdarb aofm Q´>oqgJ H$mJXmdarb

AB aofoer EH$ê$n Amho.

AmVm aofm gm§Yboë¶m q~Xÿda EH$ {nZ Q>moMm. Q´>og H$mJX {’$adm. ‘yi H$mJXmda

Agbobr AB aofm Q´>og à{VÀ¶m AB aofobm {déÜX AgVo.

Vwåhr H$m¶ {ZarjU Ho$bm?

darb H¥$Vrdê$Z AmnU H$m` {ZarjU Ho$bo ?

1) ‘yi H$mJXmda H$mT>bobm COB hm Q>og H$mJXmda H$mT>boë¶m AOD bm g§nyU©nUo

PmH$Vmo. åhUOo COB hm AOD er EH$ê$n hmoVmo.

2) ‘yi H$mJXmda H$mT>bobm AOD hm Q>og H$mJXmda H$mT>boë¶m COB bm g§nyU©nUo

PmH$Vmo. åhUOo AOD hm COB er EH$ê$n hmoVmo.

3) Q>og H$mJXmda H$mT>bobm BOD hm ‘yi H$mJXmda H$mT>boë¶m AOC bm g§nyU©nUo

PmH$Vmo. åhUOo BOD hm AOC er EH$ê$n hmoVmo.

4) Q>og H$mJXmda H$mT>bobm AOC hm ‘yi H$mJXmda H$mT>boë¶m BOD bm g§nyU©nUo

PmH$Vmo. åhUOo AOC hm BOD er EH$ê$n hmoVmo.

COB AOD= Am{U BOD AOC= Agm {ZîH$f© H$mTy> eH$Vmo.

Ë¶m‘wio, XmoZ aofm EH$‘oqH$Zm N>oXë¶mg V¶ma hmoUmao OmoS>r {eamo{déÜX H$moZ g‘mZ AgVmV.

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H¥$Vr - 2 EH$ nm§T>am H$mJX ¿¶m. Ë¶mÀ¶m Cä¶m KS>çm KmbyZ CKS>m. Z§Va Ë¶m‘Ü¶o H$U© V¶ma

hmoVrb Aem naV KS>çm KmbyZ nwÝhm CbJS>m. Ë¶m‘wio Vwåhmbm nwaogo N>oXZ q~Xÿ {‘iVmV. àË¶oH$s A, B, C........ Aer Zmdo Úm. {eamoq~XÿVrb H$moZ ‘moOyZ Ë¶m§Zm A1, A2, A3.............. Aer Zmdo Úm. H$moZ ‘moOm Am{U Ë¶m‘Yrb {eamo{déÜX H$moZm§À¶m OmoS>çm ‘moOyZ Imbrb V³Ë¶mV âm.

H«$.g§. {eamoq~Xÿ{eamo{déÜX H$moZ n{hbr

OmoS>r{eamo{déÜX H$moZ Xþgar OmoS>r

i. AA1 = A3 =

A2 =A4 =

ii.

X¡Z§{XZ OrdZmV AmT>iyZ ¶oUmè¶m H$m§hr {eamo{déÜX H$moZm§À¶m g§X^mªMr Zmdo {bhm.

ñdmÜ¶m¶ : 5.5

1. {Xboë¶m AmH¥$Vr‘Ü¶o AOC 600=

Agë¶mg BVa H$moZ H$mT>m.

A

B

D

C

600O

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2. {Xboë¶m AmH¥$Vr‘Ü¶o OmoS>r {eamo{déÜX H$moZ AmoiIm Zmdo Úm. BOE 1400= .

OC=EO Agë¶mg gd© H$moZm§Mr ‘mno H$mT>m.

E

A B1400

C

D

O

3. {Xboë¶m AmH¥$Vr‘Ü¶o AOC 300=

Agë¶mg BVa H$moZm§Mr ‘mno H$mT>m.A

B

D

C

300 O

{dMma H$am :

XmoZ aofm N>oXë¶mg V¶ma hmoUmè¶m XmoZ OmoS>r {eamo{déÜX H$moZm‘Ü¶o EH$ OmoS>r {eamo{déÜX H$moZ Imbrb à‘mUo AmhoV. Xþgè¶m OmoS>r {eamo{déÜX H$moZmMo ñdê$n H$go Agob?

bKwH$moZm§Mr EH$ OmoS>r

H$mQ>H$moZmMr EH$ OmoS>r

{demb H$moZmMr EH$ OmoS>r

ñdmÜ¶m¶ : 5.6

I. Imbrb n¡H$s H$moUË¶m àH$maMo H$moU aofr¶ OmoS>r H$moZ V¶ma H$aVmV.

1) bKwH$moZ

2) {demb H$moZ

3) H$mQH$moZ

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II. ¶m‘Ü¶o `x' Mr qH$‘V H$mT>m Am{U àË¶oH$ H$moZmMo ‘mn emoYm.

x 750

AmH¥$Vr - i

3x x+20

AmH¥$Vr - ii

x2x x+50

AmH¥$Vr - iii

x+15 2x+15x+10

AmH¥$Vr - iv

III. {Xboë¶m AmH¥$Vr‘Ü¶o AOC 900= BOD bm PQ g‘mZ ^mJmV {d^mJVmo. Am{U AOD bm g‘mZ ^mJmV RS {d^mJVmo. Va BVa H$moU emoYm.

Q

D

R

A

900P

C

S

B

O

IV. ¶m‘Ü¶o x, y Am{U z Mr qH$‘V H$mT>m.

B

D

2z 3z

450

A

C E

Ox

y

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V. {Xboë¶m AmH¥$VrV OE ⊥AB [OE, AB bm b§~ Amho] Agë¶mg Imbrb OmoS>r H$moZ emoYm.

1) OmoS>r aofr¶ H$moZ

B

D

A

CE

O2) nyaH$ H$moZ

3) {eamo{déÜX H$moZ

4) H$moQ>r H$moZ

Vi. [aH$må¶m OmJm âm :

1) XmoZ H$moZ H$moQ>rH$moZ Agë¶mg Ë¶m§Mr qH$‘V ....................... AgVo.

2) XmoZ H$moZ nyaH$ H$moZ Agë¶mg, Ë¶m§Mr qH$‘V ..................... AgVo.

3) XmoZ H$moZ gai aofr¶ OmoS>r Agë¶mg Vo ..................... AgVmV.

4) XmoZ gb¾ H$moZ nyaH$ H$moZ Agë¶mg, Vo .................... AgVmV.

5) XmoZ aofm EH$m q~XÿV N>oXë¶mg VoWo V¶ma hmoUmar EH$ OmoS>r {eamo{déÜX H$moZ bKwH$moZ Agë¶mg Xÿgar OmoS>r {eamo{déÜX H$moZ .................... AgVo.

Vii. {Xboë¶m AmH¥$VrV KL, PQ bm b§~ Amho. AmVm ¶m§À¶m OmoS>r H$moZm§Zm Zmdo Úm.

1) g‘mZ nyaH$ H$moZ

Q

S

K

450

P

R

L

O

2) {eamo{déÜX H$moZ

3) g‘mZ Zgbobo nyaH$ H$moZ

4) nyaH$ gb¾ H$moZ

5) {demb H$moZ

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àH$aU - 6

aofr` OmoS>çm [PAIR OF LINES]

¶m àH$aUmÀ¶m Aä¶mgmZ§Va Vwåhr :

aofm§À¶m OmoS>rMr ì¶m»¶m gm§Jmb.

N>oXZ aofoÀ¶m AW© g‘OyZ ì¶m»¶m H$amb.

g‘m§Va aofm d N>oXZ aofm ¶m§Mm AW© g‘OyZ ¿¶mb.

X¡Z§{XZ OrdZmV g‘m§Va aofm Am{U N>oXZ aofm ¶mV ¶oUmao g§X^© AmoiImb.

N>oXZ aofo‘wio V¶ma Pmbobo g§JV H$moZ, ì¶wËH«$‘ H$moZ, A§Va H$moZ N>oXZ aofoÀ¶m

EH$mM ~mOybm AgUmao H$moZ AmoiImb.

aofr¶ OmoS>çm§er g§~§YrV H$moZm§Mr aMZm H$amb.

{IS>H$sMo JO qH$dm Vw‘À¶m dhr‘Yrb aofm§Mo {ZarjU H$am.

AmnU OmoS>r H$moZm~Ôb ¶m AmYr {eH$bmo AmhmoV.

hr Imbrb AmH¥$Ë¶m nhm. àË¶oH$ OmoS>r‘Ü¶o AgUmè¶m gaiaofm EH$mM àVbmV AmhoV.

AmH¥$Vr (a)

AmH¥$Vr (b) AmH¥$Vr (c)

AmH¥$Vr (d)

àË¶oH$ AmH¥$Vr‘Yrb aofm ‘moOm. àË¶oH$ {MÌmV {H$Vr aofm AmhoV? àË¶oH$ XmoZ aofm AmhoV. AmH¥$Vr (a) Am{U AmH¥$Vr (c) Mo {ZarjU H$am. aofm EH$‘o{H$dê$Z AmS>ì¶m Joboë¶m Vwåhmbm {XgVrb.

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H$moUË¶mhr XmoZ aofm (Ë¶m§À¶m ñWmZm§Mo ~§YZ ZgVmZm) EH$mM àVbmV Agë¶mg Ë¶m§Zm aofr¶ OmoS>r åhUVmV.

gyMZm : H$moUË¶mhr XmoZ qH$dm A{YH$ aofm qH$dm aofmI§S> EH$mM àVbmV Agë¶mg Ë¶m§Zm EH$ àVbr¶ aofm qH$dm aofmI§S> åhUVmV.

N>oXZ aofm

i) EH$m àVbmda EH$ q~Xÿ ¿¶m Am{U Ë¶mVyZ {ZKUmao XmoZ {H$aU H$mT>m. Ë¶m§Zm Zmdo Úm.

ii) PQ aofmI§S> H$mT>m Am{U Ë¶mda `R' q~Xÿ Xe©dm.

P QR

AmUIr EH$ aofmI§S> RT H$mT>m.

T

QP R

PQ aofmI§S>mda RT C^m Agy Úm.

TR aofmI§S> nwT>o dmT>dm. Voìhm TS {‘iVo.P R

S

Q

T

ho darb AmH¥$Vr nhm. PQ aofmI§S>mdarb R' q~Xÿ‘Ü¶o SR Am{U TR aofmI§S> gm§Ybobo

AmhoV. PQ bm ST aofmI§S> `R' q~XÿV N>oXbm Amho. ST aofmI§S>mbm PQ aofmI§S>mMr N>oXZ

aofm Am{U R bm N>oXZ q~Xÿ åhUVmV.

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133

{H$aU Am{U aofmI§S>mZo ~Zboë¶m Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am.


AmH¥$Vr - iii

AmH¥$Vr - (i) ‘Ü¶o BA Am{U BC {H$aU B nmgyZ {ZKmbo.

AmH¥$Vr -(ii) Am{U (iii) ‘Ü¶ AB Am{U CD aofm nañna O' q~Xÿ‘YyZ Joë¶m AmhoV.

¶oWo Ò' bm N>oXZ q~Xÿ, AB Am{U CD aofm§Zm N>oXZ aofm åhUVmV.

EH$m g‘mZ q~Xÿ‘YyZ H$moUË¶mhr XmoZ aofm N>oXÿZ Joë¶mg Ë¶m§Zm N>oXZ aofm åhUVmV. ¶m aofm N>oXUmè¶m q~Xÿbm N>oXZ q~Xÿ åhUVmV.

{Xboë¶m AmH¥$Vr‘Ü¶o PQ Am{U RS aofmI§S> nañna

N>oXbo AmhoV. PQ Am{U RS H$moR>o N>oXbo AmhoV.

PQ Am{U RS aofmI§S>m§À¶m gm‘mB©H$ q~Xÿ H$moUVm?

¶m gm‘mB©H$ q~Xÿbm H$m¶ åhUVmV?

H¥$Vr 1 : {Xboë¶m AmH¥$Vr‘Ü¶o N>oXZ aofmI§S> AmoiIm. CXmhaUmW©

O

AmH¥$Vr‘Ü¶o AC Am{U BD ho O q~Xÿ‘Ü¶o N>oXbo AmhoV. BVa

N>oXZ aofmI§S>mÀ¶m OmoS>çm> emoYÊ¶mMm à¶ËZ H$ê$Z Ë¶m§Mr ¶mXr

H$am.

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{dMma H$am : H$moUË¶mhr XmoZ aofm qH$dm aofmI§S> Zoh‘r N>oXbobo AgVmV H$m?

H¥$Vr 2 : AB aofmI§S> H$mT>m. Ë¶mda K, L, M, N, q~Xÿ Xe©dm. M q~Xÿ‘Ü¶o 900 Mm H$moZ Agy Úm. H$moZ oXe©{dUmar aofm H$mT>m. Z§Va Ë¶m aofoÀ¶m da `P' q~Xÿ Xe©dm. AmVm PK, PL, PM, PN H$mT>m. ‘moOnÅ>rÀ¶m ghmæ¶mZo PK, PL, PM Am{U PN ‘moOm. VgoM

, ,PMB PMA PNMAm{U PKL H$moZ‘mnH$mÀ¶m ghmæ¶mZo ‘moOm.

gdm©V H$‘r ‘mnmMm aofmI§S> H$moUVm?

H$‘r ‘mn Agboë¶m H$moZmMo ‘mn {H$Vr?

~mhoarb P q~Xÿ‘YyZ AB aofmI§S>mda H$mT>boë¶m aofmI§S>m‘Ü¶o PM bhmZ aofmI§S>

Amho. M = 900 AgyZ ,PM AB hm ,PM AB bm b§~ Amho. Ë¶m‘wio P nmgyZ AB aofmI§S>mda

H$mT>boë¶m gd© aofmI§S>m‘Ü¶o b§~ AgUmam aofmI§S> gdm©V H$‘r ‘mnZmMm Amho.

gyMZm : PM aofm AB bm b§~ Amho.ho {MÝhmÀ¶m ê$nmV PM ⊥ AB Ago Xe©{dbo OmVo.

g‘m§Va aofm

‘moOnÅ>r

nmoñQ>H$mS>©

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‘moOnÅ>rÀ¶m Am{U nmoñQ>H$mS>m©À¶m ~mOy nhm. ‘moO nÅ>rÀ¶m bm§~rÀ¶m aofm H$mT>m. AmH¥$VrV Xe©{dë¶mà‘mUo Ë¶m§Zm PQ Am{U RS Aer Zmdo Úm. Z§Va PQ Am{U RS XmoÝhr ~mOybm dmTdm.

P

R

Q

S

XmoÝhr aofm nañnamZm {‘iVmV H$m?

doJdoJù¶m q~XÿVyZ aofm ‘Yrb b§~m§Va ‘moOm.

¶mVyZ Vwåhmbm H$m¶ AmT>iVo?

Aem àH$maÀ¶m aofm§Zm H$moUVo Zmd Úmb?

¶oWo PQ Am{U RS aofm§Mo gd© {R>H$mUMo q~Xÿ g‘mZ A§Vamda AmhoV. Ë¶m‘wio Aem àH$maÀ¶m aofm§Zm ""g‘m§Va aofm'' åhUVmV.

aofr¶ OmoS>çm g‘m§Va AgÊ¶mg Ë¶m§À¶m darb gd© q~Xÿ‘Yrb A§Va g‘mZ AgVo.

åhUyZ EH$m àVbmVrb XmoZ g‘m§Va aofmdarb q~XÿÀ¶m OmoS>r‘Yrb A§Va g‘mZ Agë¶mg Ë¶m§Zm g‘m§Va aofm åhUVmV.

gyMZm :

1) XmoZ ""g‘m§Va aofm'' XmoÝhr aofoda EH$mM {Xeobm AgUmè¶m ~mUmÀ¶m {MÝhmZo Xe©{dë¶m

OmVmV.

2) EH$mM àVbmVrb XmoZ g‘m§Va aofm XmoÝhr ~mOyg {H$Vrhr dmT>{dë¶mg N>oXV ZmhrV.

{Xboë¶m AmH¥$Vr‘Ü¶o AB Am{U CD aofm doJdoJù¶m {R>H$mUr H$mT>ë¶m AmhoV. Am{U

Ë¶m‘Yrb A§Va EH$M Amho.

¶m aofm H$m gm§Yë¶m AWdm N>oXë¶m OmV ZmhrV?

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XmoZ g‘m§Va aofm‘Yrb A§Va H$‘r Pmbo qH$dm OmñV Pmbo Var H$moUVmhr ~Xb hmoV Zmhr. Ë¶m‘wio g‘m§Va aofm H$moR>ohr gm§Yë¶m OmV ZmhrV.

H¥$Vr 3 : AmboI dhrV H$mT>boë¶m aofm§Mo {Z[ajU H$am Am{U AmoiIm.

i) g‘m§Va aofm VgoM

ii) b§~ aofm

¶m‘Ü¶o Vwåhmbm H$moUVm ’$aH$ AmT>iVmo?

H¥$Vr 4 : Vw‘À¶m g^modVmbr Vwåhr nm{hboë¶m 5 CXmhaUm§Mr ¶mXr H$am.

(i) g‘m§Va aofm

(ii) b§~aofm

gyMZm :

(i) AB Am{U CD XmoZ aofm g‘m§Va Agë¶mg Ë¶m AB || CD Ago {bhrVmV.

(ii) Ë¶m b§~ Agë¶mg AB ⊥ CD Ago {bhrVmV.

N>oXZ aofm

H¥$Vr 5 : AmH¥$VrV Xe©{dë¶mà‘mUo AB Am{U PQ aofmI§S> H$mT>m. Vo EH$‘oH$m§Zm H$moR>o OmoS>ë¶m AmhoV?

AmH¥$VrV Xe©{dë¶mà‘mUo AmVm AB hm AmUIr EH$ aofmI§S>mda CD aofmI§S> H$mT>m. ho PQ

aofobm L ‘Ü¶o N>oXÿ Úm.

A

A B

DQQ

L

KK

PP

C

AB B

¶mVyZ Vwåhmbm H$m¶ {XgyZ ¶oVo? PQ hr AB bm K ‘Ü¶o Am{U CD bm L ‘Ü¶o N>oXVo. åhUOoM PQ aofm AB Am{U CD bm N>oXVo.

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¶m AmH¥$Ë¶m§Mo {ZarjU H$am :


C

AB

D

P

KLM

QAmH¥$Vr - iii

A C

B D

LKQP

AmH¥$Vr - iv

darb AmH¥$Vr‘Ü¶o àË¶oH$ AB Am{U CD aofm§Zm N>oXUmè¶m aofoMo Zmd {bhm.

AmH¥$Vr (i) ‘Ü¶o Xe©{dë¶mà‘mUo N>oXbobo q~Xÿ K Am{U L Zo Xe©dm.

darb `PQ' aofm H$‘rVH$‘r XmoZ aofm§Zm N>oXbr Amho. Ë¶m N>oXboë¶m aofobm N>oXZ aofm åhUVmV.

EH$mM àVbmVrb XmoZ gaiaofm§Zm NoXUmè¶m aofobm N>oXZ aofm åhUVmV.

H¥$Vr 6 :

EH$ nm§T>am H$mJX ¿¶m.

Ë¶mÀ¶m Cä¶m XmoZ qH$dm VrZ KS>çm Kmbm Am{U Z§Va CbJS>m.

AmVm Ë¶mÀ¶m AmS>ì¶m KS>çm Kmbm Am{U CbJS>m.

noÝgrbÀ¶m ghmæ¶mZo KS>r KmVboë¶m {R>H$mUr V¶ma Pmboë¶m aofm H$mT>m.

aofm Am{U q~XÿMr Zmdo {bhÿZ N>oXZ aofm Am{U N>oXH$ aofm§Mr ¶mXr H$am.

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H¥$Vr 7 : Imbr {Xboë¶m AmH¥$Vr‘Ü¶o N>oXZ aofm§Zm Zmdo Úm Am{U {ZîH$f© Úm.

A

B

C

D

P

Q

L

K

AB

CD

P

Q

L

K

A

B

C

D

P QLK

{dMma H$am :

{Xboë¶m {MÌm‘Ü¶o EF hr AB Am{U CD aofm§Zm N>oXVo.

na§Vy Vr N>oXZ aofm Zmhr. Ago H$m?

AmnU aofobm N>oXZ aofm Ho$ìhm åhUy eH$Vmo?

N>oXZ aofo‘wio V¶ma hmoUmao H$moZ :

AmH¥$VrMo {ZarjU H$am.

aofm, N>oXZ aofm Am{U N>oXZ aofo‘wio V¶ma Pmboë¶m N>oXZ q~XÿZm

Zmdo Úm.

AB, CD Am{U PQ øm aofm AmhoV. Á¶mV PQ EH$ N>oXZ aofm. K Am{U L N>oXZ q~Xÿ.

¶oWo ,PKB AKQ ho {eamo{déÜX H$moZ AmhoV.

K Am{U L q~Xÿ‘Ü¶o V¶ma Pmbobo BVa {eamo{déÜX H$moZ Xe©dmb? PKB hm AKP

Mm nyaH$H$moZ AmhoV. K Am{U L q~Xÿ‘Ü¶o V¶ma Pmboë¶m BVa nyaH$ H$moZm§À¶m OmoS>çm AmoiIm.

N>oXZ q~Xÿ K Am{U L ‘Ü¶o AmR> doJdoJio H$moZ V¶ma Pmbo AmhoV. Ë¶m§Mo dJuH$aU H$go

H$ê$ eH$Vmo.

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Am§VaH$moZ :

C

A

D

BP

2K

143

LQ

~mOyÀ¶m AmH¥$Vr‘Ü¶o AB, CD Am{U PQ VrZ aofm AmhoV. ¶m‘Yrb N>oXZ aofm H$moUVr?

PQ À¶m K q~Xÿ‘Ü¶o V¶ma Pmbobo H$moZ H$moUH$moUVo?

¶m Mma H$moZm§Mo {ZarjU H$am.

K N>oXZ q~Xÿ‘Ü¶o V¶ma Pmbobo AKQ Am{U BKQ

L N>oXZ q~Xÿ‘Ü¶o V¶ma Pmbobo CLP Am{U DLP

¶m Mma H$moZm‘Yrb gmå¶ H$m¶?

AmnU n{hë¶mà‘mUo ho gd© H$moZ AB Am{U CD aofo‘Ü¶o AmhoV Am{U ,KL PQ N>oXZ

aofoMo ^mJ AmhoV. ¶m H$moZm§Zm Zmd XoD$ eH$mb?

hmo¶, ¶m H$moZm§Zm Am§VaH$moZ åhUVmV.

N>oXZ aofoÀ¶m aofmI§S>m‘wio V¶ma Pmbobo Am{U EH$m aofr¶ OmoS>rÀ¶m ‘Ü¶o V¶ma Pmboë¶m

H$moZm§Zm Am§VaH$moZ åhUVmV.

Ë¶m‘wio darb AmH¥$Vr‘Ü¶o ,AKL KLC Am{U ,BKL KLD Am§VaH$moZ AmhoV.

~møH$moZ :

A

CL

Q

D

BK

P ~mOyÀ¶m AmH¥$Vr‘Ü¶o AB, CD Am{U PQ VrZ aofm AmhoV. ¶m‘Yrb N>oXZ aofm H$moUVr?

AmVm PQ N>oXZ aofoÀ¶m EH$m ~mOybm AgUmè¶m Am§VaH$moZm§Zm Zmdo Úm.

¶m Imbrb OmoS>r H$moZm§Mo {ZarjU H$am.

1) AKP Am{U CLQ

2) PKB Am{U DLQ

¶m OmoS>r H$moZm‘Yrb gmå¶ H$m¶?

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ho OmoS>r H$moZ AB,CD g‘m§Va aofm§À¶m ~mhoarb ~mOyg Agë¶mMo AmnU nmhÿ eH$Vmo.

¶m OmoS>r H$moZm§Zm H$m¶ åhUy eH$Vmo?

hmo¶, ho H$moZ AB,CD aofoÀ¶m ~mhoarb ~mOyg AgyZ N>oXZ aofoÀ¶m EH$mM ~mOybm AgUmao

H$moZ åhUOoM ~møH$moZ hmo¶.

aofr¶ OmoS>rÀ¶m ~mhoarb ~mOyg AgUmao, åhUOo N>oXZ aofoÀ¶m EH$mM ~mOybm AgUmè¶m H$moZm§Zm ~mø H$moZ åhUVmV.

H¥$Vr 8 :

¶m AmH¥$Vr‘Yrb OmoS>r Am§VaH$moZ Am{U OmoS>r ~møH$moZ AmoiIm

A

C

ED

B

F G

Am{U Ë¶m§Mr ¶mXr H$am.

H«$‘mZo ¶oUmao Am§VaH$moZ :

{Xboë¶m AmH¥$Vr‘Ü¶o AKL Am{U KLC OmoS>rH$moZ Am{U Xþgao OmoS>rH$moZ BKL

Am{U KLD ¶m§Mr PQ ¶m N>oXH$aofoer {ZarjU H$am.

A

BC

D

P

Q

L

K

OmoS>r Am§VaH$moZ Agbobo AKL Am{U KLC ho PQ N>oXZ

aofoÀ¶m EH$mM ~mOybm AmhoV. BKL Am{U KLD OmoS>r

Am§VaH$moZ N>oXZ aofoÀ¶m COì¶m ~mOybm AmhoV. Aem OmoS>r H$moZm§Zm

H«$‘mZo ¶oUmao OmoS>r H$moZ åhUVmV.

darb AmH¥$Vr‘Ü¶o EH$ H«$‘mZo ¶oUmar OmoS>r Am§VaH$moZmZo

AKL Am{U KLC Zo V¶ma Pmbr Amho. Ë¶mM {MÌm‘Ü¶o

AmUIr EH$ H«$‘mZo ¶oUmam OmoS>r Am§VaH$moZm§Mr Zmdo {bhÿ eH$mb?

N>oXZ aofoÀ¶m EH$mM ~mOybm AgUmè¶m Am§VaH$moZm§Zm H«$‘mZo ¶oUmao Am§VaH$moZ åhUVmV.

{dMma H$am : Vwåhmbm H«$‘mZo ¶oUmao Am§VaH$moZ Xe©{dÊ¶mgmR>r B§J«Or Ajao N Am{U Z ghmæ¶H$ R>aVmV H$go?

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g§JV H$moZ

àË¶oH$ AmH¥$Vr‘Ü¶o Xe©{dbobo H$moZ nhm.

AB

D

K

LCQ

P

AmH¥$Vr iAmH¥$Vr ii

AmH¥$Vr iii

A B

D

K

LC

Q

P

AmH¥$Vr iv

darb AmH¥$Vr‘Ü¶o CD bm AB g‘m§Va Amho. PQ N>oXZ aofm Amho. hr ,AB CD

g‘m§Va aofm§Zm AZwH«$‘o K Am{U L ‘Ü¶o N>oXVo.

{Xboë¶m AmH¥$Vr (i) ‘Ü¶o AKP Am{U CLK ho PQ N>oXZ aofoÀ¶m S>mì¶m ~mOybm Xe©{dbo AmhoV. Am{U AKP hm AB À¶m da CLK hm CD À¶m darb ^mJmV Amho.

{Xboë¶m AmH¥$Vr (ii) ‘Ü¶o AKL Am{U CLQ OmoS>r H$moZ N>oXZ aofm PQ À¶m S>mì¶m ~mOybm AgyZ AB Am{U CD À¶m Imbrb ^mJmV Amho.

{Xboë¶m AmH¥$Vr (iii) ‘Ü¶o PKB Am{U KLD OmoS>r H$moZ N>oXZ aofm PQ À¶m COì¶m ~mOybm AgyZ AB Am{U CD À¶m darb ^mJmV AmhoV.

{Xboë¶m AmH¥$Vr (iv) ‘Ü¶o BKL Am{U DLQ OmoS>r H$moZ N>oXZ aofm PQ

À¶m COì¶m ~mOybm AgyZ AB Am{U CD À¶m Imbrb ^mJmV AmhoV.

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¶oWo nm{hë¶mà‘mUo AmnU H$moUVm {ZîH$f© H$mTy> eH$Vmo?

àË¶oH$ AmH¥$Vr‘Yrb OmoS>rH$moZ.

a) àË¶oH$ {eamoq~Xÿ AmhoV.

b) N>oXZ aofoÀ¶m EH$mM ~mOybm AmhoV.

c) AmH¥$Vr (i) AmH¥$Vr (iii) ‘Ü¶o g‘m§Va aofoÀ¶m darb ^mJmV AmhoV Am{U AmH¥$Vr

ii) Am{U iv) ‘Ü¶o g‘m§Va aofoÀ¶m Imbrb mJmV AmhoV. Aem H$moZm§Zm g§JV H$moZ åhUVmV.

OmoS>r H$moZ

àË¶oH$s {eamoq~Xÿ AgVmV.

N>oXZ aofoÀ¶m EH$mM ~mOyg AmhoV Am{U

XmoZ OmoS>r aofoÀ¶m ‘Ü¶o V¶ma Pmboë¶m H$moZm§Zm g§JV H$moZ åhUVmV.

darb {MÌm‘Ü¶o OmoS>r H$moZ i) ( AKP Am{U CLK ), ii) ( AKL Am{U CLQ )

iii) ( PKB Am{U KLD ) VgoM iv) ( BKL Am{U DLQ ) ho

H¥$Vr 9 : AmH¥$Vr‘Ü¶o Xe©{dë¶mà‘mUo g‘m§Va aofm Am{U N>oXZ aofm V¶ma hmoVrb Aer EH$m H$mJXmMr KS>r Kmbm. aofm Am{U N>oXZ q~XÿZm Zmdo Úm. àË¶oH$ {MÌm‘Yrb OmoS>r g§JV H$moZ {bhm.

{dMma H$am! Vwåhmbm g§JV H$moZ Xe©{dÊ¶mgmR>r B§J«Or Aja "F' ghmæ¶ H$aVo H$go?

ì¶wËH«$‘ H$moZ

{Xboë¶m AmH¥$Vr‘Ü¶o Xe©{dbobo H$moZ nhm.

AmH¥$Vr (i) AmH¥$Vr (ii)

A

B

D

K

LCQ

P

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AmH¥$Vr (i) ‘Ü¶o PQ N>oXZ aofoÀ¶m S>mdrH$S>o Am{U AB aofoÀ¶m Imbrb mJmV AKL

H$moZ Amho. VgoM KLD hm PQ N>oXZ aofoÀ¶m S>mdrH$S>o Am{U CD aofoÀ¶m darb ^mJmV AmhoV.

AmH¥$Vr (ii) ‘Ü¶o BKL hm PQ N>oXZ aofoÀ¶m COdrH$S>o Am{U AB aofoÀ¶m Imbr Amho. VgoM KLC hm N>oXZ aofoÀ¶m S>mdrH$S>o Am{U CD aofoÀ¶m da Amho.

darb ‘m{hVrdê$Z H$moUVm {ZîH$f© H$mT>mb ¶mMr ¶mXr H$am.

a) {^Þ {eamoq~Xÿ AmhoV.

b) N>oXZ aofoÀ¶m AmVrb ^mJmV AmhoV.

c) g‘m§Va aofoÀ¶m AmVrb ^mJmV AmhoV.

d) g‘m§Va aofoÀ¶m doJdoJù¶m ñWmZr AmhoV.

EH$ H$moZ aofoÀ¶m darb ^mJmV Va Xþgam Imbrb ^mJmV AgVmo.

darb AmH¥$Vr‘Ü¶o OmoS>r H$moZ Agbobo ,AKL KLD Am{U ,BKL KLC ho ì¶wËH«$‘ H$moZ AmhoV.

OmoS>r H$moZ

i) {^Þ {eamoq~Xÿ AgVmV.

ii) N>oXZ aofoÀ¶m {déÜX ~mOybm doJdoJio AgVmV.

iii) g‘m§Va aofoÀ¶m Am{Vb ^mJmV AgVmV.

iv) g‘m§Va aofoÀ¶m doJdoJù¶m ñWmZr AgVmV. Aem H$moZm§Zm ì¶wËH«$‘ H$moZ åhUVmV.

H¥$Vr 10 : ‘mJrb H¥$Vr‘Ü¶o gyM{dë¶mà‘mUo H$mJXmMr KS>r KmbyZ aofm H$mT>m. Am{U ì¶wËH«$‘ H$moZ Xe©dm.

{dMma H$am ! Vwåhmbm ì¶wËH«$‘ H$moZ AmoiIÊ¶mgmR>r N Am{U Z Ajam§Mr ‘XV H$er hmoVo?

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g‘m§Va aofm§Mr N>oXH$ aofm :Vw‘À¶m X¡Z§{XZ OrdZmV AmT>iUmar hr {MÌo nhm.

AmH¥$Vr i AmH¥$Vr ii AmH¥$Vr iii

darb AmH¥$Vr (i) {eS>rMo AmH¥$Vr, {MÌ (ii) {IS>H$sMo AmH¥$Vr (iii) aoëdo éimMo

AmH¥$Vr ¶m‘Yrb H$moUVo (i) g‘m§Va aofm Am{U (ii) N>oXZ aofm Xe©{dVmV.

(N>oXZ aofo‘wio V¶ma Pmboë¶m OmoS>r g§JV H$moZm‘Yrb g§~§Y) hm {df¶ g§JV H$moZm~amo~a KoD$ eH$Vmo.

H¥$Vr 11 :

EH$ H$mJX ¿¶m.

Ë¶mda AB Am{U CD g‘m§Va aofm H$mT>m.

Z§Va ¶m aofm§Zm PQ N>oXZ aofm H$mT>m.

¶m {MÌm‘Yrb g§JV H$moZ AmoiIm

AmH¥$Vr i AmH¥$Vr ii

AmH¥$Vr iii AmH¥$Vr iv

AmH¥$Vr-1 da Q´>oqgJ nona R>odm. AmH¥$Vr-1 bm ì¶dpñWV Q´>og H$am. L q~Xÿda Q´>oqgJ nona ¿¶m. Vmo AB aofm d CD aofm EH$ê$n hmoVrb Am{U K Am{U L q~Xÿ ‘Ü¶o EH$ê$nVm ¶oB©b Ago R>odm.

darb AmH¥$Vrdê$Z H$m¶ g‘Obo?

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Amåhmbm g‘Obo H$s àË¶oH$ g§X^m©‘Ü¶o Q´>og Ho$bobo AKP hm KLC H$moZmbm {eamoq~XÿV PmH$Vmo.

AKP Am{U KLC ho H$moUË¶m àH$maMo H$moZ AmhoV?

AKP d KLC ho g§JV H$moZ AmhoV. ¶m H¥$Vrdê$Z AmnU

AKP = KLC Ago g‘Oy eH$Vmo.

AmVm AmH¥$Vr iii) ‘Ü¶o Xe©{dë¶mà‘mUo PKB Xe©dm. Xe©{dboë¶m Q´>oqgJ H$mJX Ë¶mda R>odm. Ë¶mda AB aofm CD bm EH$ê$n hmoB©b Am{U K q~Xÿ L q~Xÿ bm EH$ê$n hmoB©b Ago R>odm.

PKB Am{U KLD Mo {ZarjU H$am. Vw‘Mm ..... Amho?

hmo¶, PKB KLD=

,LKB QLD AKL CLQ= = AmhoV ho {XgyZ ¶oVo.

Ë¶mdê$Z Am‘À¶m {ZîH$fm©à‘mUo

XmoZ qH$dm A{YH$ g‘m§Va aofm N>oXZ aofo‘wio N>oXë¶mg àË¶oH$ OmoS>rVrb g§JV H$moZ g‘mZ AgVmV.

N>oXZ aofo‘wio V¶ma Pmboë¶m OmoS>r g§JV H$moZm‘Yrb g§~§Y

H¥$Vr 12 : EH$m H$mJXmda AB Am{U CD XmoZ g‘m§Va aofm H$mT>m. Am{U Imbrb AmH¥$Vr‘Ü¶o Xe©{dë¶mà‘mUo PQ N>oXZ aofm H$mTy>Z OmoS>r ì¶wËH¥$‘ H$moZ Xe©dm.

AmH¥$Vr (i) AmH¥$Vr (ii) AmH¥$Vr (iii) AmH¥$Vr (iv)

AmH¥$Vr (i) nmaXe©H$ H$mJXmda Q´>og H$am. Q´>og Ho$boë¶m H$mJXmMr KS>r Kmbm. AmVm CD

aofoer AB aofm EH$ê$n hmoVo Am{U L q~Xÿgmo~V K q~Xÿ EH$ê$n hmoVrb.

Z§Va KS>r KmVbobm nmaXe©H$ H$mJX CbKS>m Am{U PQ aofoda N>oXbobm q~Xÿ Xe©dm. Vmo

"X' Zo Xe©dm.

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Z§Va Ë¶mM AmH¥$Vrda EH$ nmaXe©H$ H$mJX R>odm. Ë¶mda AmH¥$Vr H$mT>m. åhUOo AmH¥$Vr

(i) à‘mUo nmaXe©H$ H$mJXmdarb aofm AB hr H$mJXmdarb aofm AB er EH$ê$n hmoVo.

Z§Va nmaXe©H$ H$mJXmda Am{U H$mJXmda "X' q~Xÿ‘Ü¶o EH$ {nZ Q>moMm. Am{U AKL

Xe©dm. AmVm nmaXe©H$ H$mJX {’$adm.Voìhm H$mJXmdarb CD er nmaXe©H$ H$mJXmdarb AB

aofm EH$ê$n hmoB©b. KLD er AKL nyU©nUo EH$ê$n Pmbobo AmT>iyZ ¶oVrb.

¶m H¥$Vrdê$Z H$moUVm {ZîH$f© H$mT>mb?

hmo¶, AKL KLD= ho g‘OyZ ¶oVo.

AmH¥$VrV Xe©{dë¶mà‘mUo BKL Xe©dm. Z§Va "X' Ajamda {’$adm. OmoS>r H$moZ

BKL KLC= Ago gm§Jy eH$Vmo?

Ë¶m‘wio darb H¥$Vrdê$Z AmnU Agm {ZîH$f© H$mTy> eH$Vmo H$s,

N>oXZ aofoÀ¶m EH$mM ~mOyg AgUmao Am§Va H$moZ

H¥$Vr 13 : EH$m H$mJXmda AB Am{U CD g‘m§Va aofm H$mT>m. Am{U N>oXZ aofm PQ H$mT>m.


AmVm, Q´>oqgJ H$mJXmda AmH¥$Vr (i) Q´>og H$am. AmVm Q´>og Ho$bobm Q´>oqgJ H$mJX ‘yi {MÌmda

R>odm. hr Q´>oqgJ H$mJXmdarb AB aofm ‘yi H$mJXmdarb CD aofoda EH$ê$n hmoD$ Úm. Ë¶m‘wio

"A' hm "D' gmo~V EH$ê$n hmoVmo.

(gyMZm : Q´>o[g§J H$mJXmdarb AB aofoÀ¶m ~mUmMr {Xem Am{U ‘yi H$mJXmÀ¶m CD

aofoÀ¶m ~mUmMr {Xem {déÜX AgVo)

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ho AdbmoH$Z Ho$ë¶mg AKL KLD= AmVm BKL Am{U KLD ¶m§Mo ‘mnZ

H$ê$¶m Am{U Ë¶m§À¶m H$moZm§Mr qH$‘V H$mTy>¶m.

AmnUmg ‘m{hV Agë¶mà‘mUo AKL BKL 180+ = ∴gb¾ H$moZ.

AmVm AKL EdOr KLD {Xbobm Amho.

Voìhm KLD BKL 1800+ =

∴ BKL CLK= Ë¶m‘wio AKL CLK 1800+ =

¶m H¥$Vrgmo~V OmoS>r H$moZ Am{U aofmI§S>m‘Yrb g§~§Y nmhÿ¶m.

nm¶ar 1 : EH$ H$mJX ¿¶m. AB aofm H$mT>m. Ë¶mda K

q~Xþ Xe©dm.

A BK

nm¶ar 2 : K q~Xÿ‘Ü¶o 1000 Mm H$moZ aMm. H$moZmMr

aofm H$mTy>Z Vr dmT>dm. A BK1000

nm¶ar 3 : EH$ Q´>oqgJ nona ¿¶m. ‘mH©$a noZÀ¶m ghmæ¶mZo

CD aofm Ë¶mda L q~Xÿ Xe©dm.

C DL

nm¶ar 4 : "L' ‘Ü¶o1000 Mm H$moZ aMm. ‘mJo gm§{JVë¶m

à‘mUo H$moZmMr aofm H$mTy>Z dmT>dm. C DL1000

nm¶ar 5 : Q´>oqgJ nona AB aofoÀ¶m da qH$dm Imbr R>odm.

AB Am{U CD Mo ñWmZ H$mJX gaH$dyZ da qH$dm Imbr

OwS>dm. K Am{U L ‘YyZ H$mT>bobr aofm nyU©nUo EH$ê$n

AgVo. Ë¶m‘wio Vr aofm N>oXZ aofm AgVo.

A

C D

B

P

Q

K

L

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ªÀÄvÀÄÛAKP CLK Am{U ªÀÄvÀÄÛAKP CLK ¶m§Mo {ZarjU H$am. ho H$moUË¶m àH$maMo H$moZ AmhoV?

Vo g§JV H$moZ AmhoV Am{U g‘mZ AmhoV. ªÀÄvÀÄÛAB CD Am{U ªÀÄvÀÄÛAB CD ‘Yrb A§Va doJdoiJù¶m ñWmZr ‘moOm. Vo g‘mZ Amho? Vwåhr nm{hë¶mà‘mUo AB Am{U CD aofr¶ OmoS>tMm g§~§Y H$m¶ Amho?

N>oXZ aofm XmoZ aofm§Zm N>oXë¶mg V¶ma hmoUmao g§JV H$moZ g‘mZ Agë¶mg Ë¶m aofm g‘m§Va AgVmV.

{dMma H$am :

XmoZ aofm EH$m N>oXZ aofoZo N>oXë¶mg V¶ma hmoUmao OmoS>r g§JV H$moZ g‘mZ AgVmV.

Voìhm Ë¶m aofm g‘m§Va AgVmV.

XmoZ aofm§Zm EH$m N>oXZ aofoZo N>oXë¶mg V¶ma hmo‘mao ì¶wËH«$‘ H$moZ g‘mZ AgVmV.

Voìhm Ë¶m aofm g‘m§Va AgVmV.

XmoZ aofm§Zm EH$m N>oXZ aofoZo N>oXë¶mg V¶ma hmoUmè¶m N>o{XHo$À¶m EH$mM ~mOyg

AgUmè¶m H$moZm§Mr qH$‘V nyaH$ H$moZm§BVH$s AgVo. Voìhm Ë¶m aofm g‘m§Va AgVmV.

{dMma H$am!

1) ¶m Imbr {Xboë¶m AmH$mam§Mo {ZarjU H$am. Vwåhmbm N>oXZ aofobm EH$mM ~mOyg AgUmao Am§VaH$moZ Xe©{dÊ¶mg ghmæ¶H$ N>aVmV H$go?

2) XmoZ aofmI§S>mZm EH$ AmS>dr aofm N>oXë¶mg Am§Va H$moZm§Mr qH$‘V nyaH$ H$moZmBVH$s AgVo. Voìhm Vo aofmI§S> H$go AgVrb?

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H¥$Vr 14 : XmoZ g‘m§Va ZgUmè¶m AB Am{U CD aofm H$mT>m. ¶m aofm§Zm AZwH«$‘o K Am{U

L q~Xÿ‘Ü¶o N>oXVrb Agr PQ N>oXZ aofm H$mT>m. Z§Va OmoS>r g§JV H$moZ, OmoS>r ì¶wËH«$‘ H$moZ

Am{U N>oXZ aofoÀ¶m EH$mM ~mOybm AgUmao Am§Va H$moZ H$moZ‘mnH$mÀ¶m ghmæ¶mZo ‘moOm. Vwåhr

H$mT>boë¶m {ZîH$fm©Mo ‘wÔo {bhm.

aoer¶ OmoS>r g‘m§Va Amho H$s Zmhr ho gmoß¶m nÜXVrZo {ZpíMV H$é eH$Vmo.{MÌ - (i) Am{U (ii) ‘Ü¶o Xe©{dë¶mà‘mUo XmoZ OmoS>r aofm EH$mM àVbmV H$mT>m.

AmH¥$Vr iAmH¥$Vr ii

A

C

B

D

A

C

B

D

XmoZ aofm§Zm AB Am{U CD Aer Zmdo Úm. AB Am{U CD aofm§Zm N>oXoV Aer

EH$ ‘moOnÅ>r R>odm. ‘moOnÅ>rÀ¶m A§H$mÀ¶m IyUm Xe©dm. ‘moOnÅ>r H$mTy>Z ¿¶m Am{U AB da

Xe©{dboë¶m nÅ>rdarb A§H$mÀ¶m IyUm N>oXboë¶m q~Xÿ§Zm PQ Aer Zmdo Úm. VgoM CD da

N>oXboë¶m q~XÿZm RS Aer Zmdo Úm. PQ, RS aofmI§S>mMr ‘mno Úm.


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AmH¥$Vr - i ‘Ü¶o PR Mr bm§~r = QS Mr bm§~r Agbobr AmT>iyZ ¶oVo.

Ë¶m‘wio AB Am{U CD aofm nañna g‘mZ AmhoV Agm {ZîH$f© H$mTy> eH$Vmo.

AmH¥$Vr - ii ‘Ü¶o PQ Mr bm§~r ≠ RS Mr bm§~r. Ë¶m‘wio AB Am{U CD aofm g‘m§Va ZmhrV.

Ago Agë¶mg, AmnU ¶mdê$Z PQ Am{U RS ho AB Am{U CD aofobm b§~ Agë¶mg Vo g‘m§Va AgVmV Agm {ZîH$f© H$mTy> eH$Vmo.

XmoZ aofm‘Yrb b§~m§Va g‘mZ Agë¶mg, Ë¶m aofm g‘m§Va AgVmV.

CXmhaU 1 : {Xboë¶m AmH¥$Vr‘Ü¶o ABIICDIIEF.

GF=EF Am{U BGD = 500. Agë¶mg BVa

H$moZ emoYm.

A B

D500

FE

C G

[ ]EFG CGF CD EFCGFCGF

180180 9090

0

0 0

0

a <+ == -=

ªÀÄvÀÄÛABG BGD Am{U ªÀÄvÀÄÛABG BGD ho ì¶ËH«$‘ H$moZ AmhoV.

ABG BGDABG

5050

0

0

`

`

= ==

ªÀÄvÀÄÛCGB BGD Am{U ªÀÄvÀÄÛCGB BGD ho aofr¶ H$moZ

CGB BGDCGBCGB

180180 50130

0

0 0

0

` + == -=

CXmhaU 2 : {Xboë¶m AmH¥$Vr‘Ü¶o {MÝhmZo Xe©{dboë¶m H$moZm§Mr ‘mno H$mT>m.

18040 180180 401 0

CFE EFGCFECFECFE 4

0

0 0

0 0

0

+ =+ == -=

A

C DF

E

400

G

B

{MÝhmZo Xe©{dbobm H$moZ FEB

ªÀÄvÀÄÛCFE FEB Am{U ªÀÄvÀÄÛCFE FEB ho ì¶wËH«$‘ H$moZ AmhoV [a ABIICD]∴ CFE FEB=FEB 1400=

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CXmhaU 3 : {Xboë¶m AmH¥$Vr‘Ü¶o gyM{dbobo A

C

D

r

700

EF

600

B

pq?

s

p,

H$moZ q, r Am{U s ¶m§Mr ‘mno H$mT>m.

(i) 60ABC DE BC0 z=

ADF ABCp ABC

ABCp

6060

0

0

`

`

====

[g§JV H$moZ]

(ii) ªÀÄvÀÄÛAFD EFC Am{U ªÀÄvÀÄÛAFD EFC [{eamo{déÜX H$moZ]

AFD EFC=

∴ 700 = q

∴ q = 700

(ii) ªÀÄvÀÄÛEFC ACBACB EFC

r qr 70

0

0

`

`

===

[ho ì¶wËH«$‘ H$moZ]

110110

ªÀÄvÀÄÛ

AFD CFDCFDCFDCFD

CFD AFECFD AFE

ss

18070 180

180 70110

0

0 0

0 0

0

0

0

`

`

`

+ =+ =

= -=

===

[g§JZ H$moZ] [aDEIIBC Am{U EC N>oXZ aofm]

P = 600, q = 700, r = 700, Am{U S = 1100

Am{U

Am{U

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ñdmÜ¶m¶ 6.1

1. Imbr Xe©{dboë¶m H$mù¶m ^mJmÀ¶m H$moZmMm g§JV H$moZ {bhm.

a) b) c)

d) e) f)

2. AmH¥$Vr‘Ü¶o Xe©{dboë¶m àË¶oH$ H$moZmMo ‘mn H$mT>m.

a)

A

C

F H

K DL

I J B

GE700

b) c)

3. {Xboë¶m AmH¥$Vr‘Ü¶o PQRS g‘m§Va^wO Mm¡H$moZ Agë¶mg gd© Am§Va H$moZm§Mr ‘mno H$mT>m.

P

Q R

S

T1300

4. {Xboë¶m AmH¥$Vr‘Ü¶o MN||BCAgë¶mg ABC {ÌH$moUmÀ¶m gd© Am§Va H$moZm§Mr ‘mno H$mT>m.

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5. {Xboë¶m AmH¥$Vr‘Ü¶o AB g‘m§Va CD ho Xe©dm.

AC

F

D

BG 1260

440H

E

{MÌ i

750750

AC

F

DB

E

{MÌ ii

6. {Xboë¶m AmH¥$Vr‘Ü¶o XmoZ H$moZm§À¶m ~mOw g‘m§Va AmhoV. ABC 700= Agë¶mg,DGC DEF H$mT>m.

7. {Xboë¶m AmH¥$Vr‘Ü¶o a, b, c Am{U d Zo Xe©{dboë¶m H$moZm§Mr ‘mno H$mT>m.

i) ii) iii)

iv) v)

D

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àH$aU - 7

{ÌH$moU [TRIANGLES]


{ÌH$moUmÀ¶m Am§VaH$moZm§Mr ~oarO H¥$Vr H$éZ H$mT>mb.

H$mJXmbm KS>çm KmbÊ¶mÀ¶m {H«$¶oVyZ {ÌH$moUmÀ¶m Am§Va H$moZm§Mr ~oarO 1800 AgVo ho {ZYm©arV H$amb.

g‘m§Va aofoÀ¶m JwUY‘mªÀ¶m ghmæ¶mZo {ÌH$moUmÀ¶m {VÝhr Am§Va H$moZm§Mr ~oarO 1800 AgVo ho {gÜX H$amb.

Vm{H©$H$ {dYmZo Am{U H¥$Vr AmYm[aV {ZîH$f© ¶m§À¶mVrb ’$aH$ AmoiImb,

{ÌH$moUmMm ~møH$moZ hm Am§Va{déÜX H$moZm§À¶m ~oaOoer g‘mZ AgVmo ho {gÜX H$amb,

{ÌH$moUmÀ¶m XmoZ ~mOy§À¶m bm§~rMr ~oarO hr {Vgè¶m ~mOyÀ¶m bm§~rnojm OmñV AgVo ho g‘OyZ ¿¶mb.

nm¶Wm°JmoagÀ¶m à‘o¶mMm AW© H¥$VrÛmao g‘OyZ ¿¶mb.

àñVmdZm :

{ÌH$moU ^y{‘Vr‘Yrb n[aMrV AmH¥$Vrn¡H$s EH$ Amho. darb {MÌm‘Ü¶o AmnU AZoH$ {ÌH$moU nmhÿ eH$Vmo. {ÌH$moUmH$mam‘Ü¶o OmoS>bobo Im§~ OmñV pñWa amhVmV. Ë¶m‘wio Amnë¶m g^modVmbÀ¶m AZoH$ aMZo‘Ü¶o g§aMZo‘Ü¶o {ÌH$moUmH$ma nmhÿ eH$Vmo. Aem {ÌH$moUmMo JwUY‘© nwT>o g‘OyZ KoD$¶m.

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{ÌH$moUmÀ¶m H$moZm§À¶m ~oaOoMo JwUY‘© :

H¥$Vr 1 :

A

B CAmH¥$Vr (i)

A

BC AmH¥$Vr (ii)

AmH¥$Vr (i) ‘Ü¶o Xe©{dë¶mà‘mUo EH$ {ÌH$moU aMm. Ë¶mÀ¶m {eamoq~XÿZm A, B, C Aer Zmdo Úm.

H$moZ‘mnH$mÀ¶m ghmæ¶mZo Ë¶mMoo {VÝhr Am§VaH$moZ H$moZ ‘moOm (AmH¥$Vr (ii) ‘Ü¶o Xe©{dë¶mà‘mUo)

H$moZm§Mr ‘mno Imbrb V³Ë¶mV {bhm.

Ë¶m VrZ H$moZm§Mr ~oarO H$mT>m.

{ÌH$moUmMo Am§VaH$moZ {ÌH$moU 1(darb {MÌ)

{ÌH$moU 2

{ÌH$moU 3

{ÌH$moU 4

{ÌH$moU 5

H$moZ 1 640

H$moZ 2 560

H$moZ 3 600

{VÝhr H$moZm§Mr ~oarO 1800

AemàH$mao, doJdoJù¶m àH$maMo {ÌH$moU (CXm : g‘^wO, {df‘^wO, bKwH$moZ, {demb H$moZ BË¶mXr) aMm.

àË¶oH$ {ÌH$moUmÀ¶m H$moZm§Mr ‘mno V³Ë¶mV {bhm. Ë¶m H$moZm§Mr ~oarO H$mT>m.

Z‘wÝ¶mgmR>r {ÌH$moU 1 Mo H$moZ {Xbo AmhoV.

darb H¥$Vrdê$Z {ÌH$moUmÀ¶m {VÝhr Am§VaH$moZm§À¶m ~oaOo~m~V Vwåhmbm H$m¶ g‘Obo?

AmUIr EH$ H¥$Vr H$ê$Z nmhÿ¶m.

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H¥$Vr 2 : AmH¥$Vr (i) ‘Ü¶o Xe©{dë¶mà‘mUo,

EH$m OmS> H$mJXmda H$moUË¶mhr ‘mnmMm EH$ {ÌH$moU aMm. 1

2 3

AmH¥$Vr (i)

Ë¶mÀ¶m {VÝhr H$moZm§Zm 1, 2 Am{U 3 Ago H«$‘ Úm.

H$moZ 1, H$moZ 2 Am{U H$moZ 3 AmH¥$VrV Xe©{dë¶mà‘mUo H$mnyZ àË¶oH$s doJio H$am.

AmH¥$Vr (ii) ‘Ü¶o Xe©{dë¶mà‘mUo, Vo {VÝhr H$moZ EH$mÀ¶m ~mOybm EH$ ¶mà‘mUo OmoS>m.

{VÝhr H$moZ EH$mM {eamoq~XÿV {‘iV ZmhrV Zm?

AmH¥$Vr (ii)

¶mdê$Z {ÌH$moUmÀ¶m {VÝhr H$moZmÀ¶m ~oaOo~Ôb Vw‘Mo {ZarjU H$m¶?

AmUIr EH$ H¥$Vr H$é¶m (H$mJX Z H$mnVm)


OmS> H$mJXmda EH$ {ÌH$moU ABC aMm. Ë¶mÀ¶m {VÝhr H$moZm§Zm 1, 2, Am{U 3 Aer Zmdo Úm.AB Am{U AC ~mOwMo ‘Ü¶q~Xÿ P Am{U Q Xe©dm.

AmH¥$Vr (ii) ‘Ü¶o Xe©{dë¶mà‘mUo,

{ÌH$moU APQ bm PQ da KS>r Kmbm. Voìhm {eamoq~Xÿ 1 BC da M ¶oWo N>oXVmo.

BP hm MP er Owiob AemàH$mao Xþ‘S>m H$s {eamoq~XÿÀ¶m ~mOyg H$moZ 2 ¶oB©b.

AemàH$mao Xþ‘S>m QC hm MQ er Owiob. H$moZ 1 À¶m ~mOyg H$moZ 3 ¶oB©b.

132

MB

A

P Q

C

AmH¥$Vr (ii)

AmH¥$Vr (i)

AmVm H$moZ 1, H$moZ 2 Am{U H$moZ 3 EH$mÀ¶m ~mOyg EH$ ¶oD$Z gaiH$moZ V¶ma Pmbobm {Xgob

darb {VÝhr H¥$Vrdê$Z Vwåhmbm g‘Obobo A§e H$moUVo?∴ {ÌH$moUmÀ¶m {VÝhr Am§VaH$moZm§Mr ~oarO 1800 AgVo.

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ñdmÜ¶m¶ - 7.1

1) EH$ Am¶VmH$ma H$mJX KoD$Z Ë¶m‘YyZ EH$ H$mQ>H$moZ {ÌH$moZ H$mnyZ ¿¶m. Ë¶m‘Ü¶o H$mQ>H$moZ

Zgbobo XmoZ H$moZ H$mnyZ H$mQ>H$moZ AgUmè¶m {eamoq~Xÿ‘Ü¶o nañna {‘iVrb Ago OmoS>m.

Ë¶m {ÌH$moUmÀ¶m {VÝhr H$moZmÀ¶m ~oaOo~Ôb Vw‘Mm {ZîH$f© gm§Jm.

2) H$mJXmda {MÌmV Xe©{dë¶mà‘mUo EH$ {dembH$moZ {ÌH$moU

B

A

C

aMyZ H$mnyZ ¿¶m. {ÌH$moUmMo H$moZ Z H$mnVm Ë¶mÀ¶m {VÝhr

H$moZm§À¶m ~oaOoMo JwUY‘© nS>Vmim.

{dMma H$am :

AmnU Ho$boë¶m gd© H¥$Vr‘Yrb {ÌH$moUmÀ¶m H$moZm§Mr ~oarO 1800 ¶oÊ¶mMo H$maU

H$m¶? Ë¶m§Mr ~oarO 1500 qH$dm 2000 H$m Pmbr Zmhr?

AmH$ma, ñdê$n doJdoJio AgUmao AZoH$ {ÌH$moU AgVmV, Ë¶m‘Yrb gd© H$moZm§Mr

~oarO 1800 Agy eHo$b H$m?

gd© H$moZm§Mr ~oarO 1800 ZgUmao {ÌH$moU AgVrb H$m?

{ÌH$moUmÀ¶m H$moZm§Mr ~oarO VH$m©Zo H$mTUo :

AmVmn¶ªV Ho$boë¶m AZoH$ H¥$Vr‘YyZ {ÌH$moUmÀ¶m {VÝhr H$moZm§Mr ~oarO 1800 AgVo ho

gyM{dVmV. na§Vw hm JwUY‘© gd© {ÌH$moUmZm bmJy hmoB©b Ë¶mMr ImÌr Zmhr. hOmamo {ÌH$moUm§À¶m

H$moZm§Mr N>oXboë¶m ‘mno KoD$Z XoIrb hm JwUY‘© gd© {ÌH$moUm§Zm bmJy hmoB©b Ago gm§JVm ¶oV Zmhr.

KoVboë¶m CXmhaUmdê$Z Agboë¶m ‘¶m©Xm AmoiIyZ {Z¶‘ ‘m§Sy> eH$Vmo.

J{UVm‘Ü¶o {ZU©¶mMo gmd©{ÌH$aU H$aÊ¶mg dmnabo OmUmao {dYmZ Amho. Vo åhUOo "Vm{H©$H$

{dYmZ' Vm{H©$H$ {dYmZ J{UVr {dYmZm§Zm {gÜX H$aÊ¶mg dmnabo OmVo.

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AmVm AmnU {ÌH$moUm§À¶m {VÝhr H$moZm§Mr ~oarO 1800 AgVo ho Vm{H©$H$ {dYmZmZo gmÜ¶ H$ê$¶m.

AmH¥$Vr‘Ü¶o ∆ABC Mo ∠A , ∠B Am{U ∠C Mr ~oarO 1800

B C

A

AgVo ho Vm{H©$H$ {dYmZmZo {gÜX H$amd¶mMo Amho.

∆ABC Mo VrZ H$moZ EH$m gaiaofoÀ¶m EH$m q~Xÿ‘Ü¶o ¶oVrb Aer H$mhrVar aMZm AmH¥$Vr‘Ü¶o H$ê$ eH$Vmo? hmo¶, A q~XÿVyZ EH$ gaiaofm RS H$mTy>¶m.

AmVm AmH¥$Vr‘Ü¶o,

i) ∠ABC Am{U ∠BAR (∗ Zo Xe©{dbo Amho)

B C

A

R

S

∗

∗

ii) ∠ACB Am{U ∠SAC ( Zo Xe©{dbo Amho)

na§Vw ho H$moZ g‘mZ ZmhrV

darb H$moZ ∠ACB ‘Ü¶o, Oa VoWo EH$ g‘m§Va aofm§Mr OmoS>r Amho Va VoWo ì¶wËH«$‘ H$moZ AmhoV. Oo g‘mZ AgVmV.

Amnë¶mbm ‘mhrV Amho H$s, Oa N>o{XHo$Zo g‘m§Va aofm§À¶m OmoS>rbm N>oXbo Va Aem [aVrZo V¶ma hmoUmao ì¶ËH«$‘ H$moZ g‘mZ AgVmV.

Ë¶m‘wio, A q~XÿVyZ BC bm g‘m§Va PQ aofm H$mTy>¶m.

¶oWo PQ&&BC Am{U AB hr N>o{XH$m Amho.

B C

AP Q∗

∗

AmVm AmH¥$Vr‘Ü¶o nmhm.

Vwåhmbm H$moUVr g‘mZ H$moZm§Mr OmoS>r {‘iVo H$m?

∴∠PAB = ∠ABC ---(i) (∗ {MÝhmZo Xe©{dbo ì¶wËH«$‘ H$moZ AmhoV)

PQ&&BC Am{U AC hr N>o{XH$m Amho

Vwåhmbm H$moUVr g‘mZ H$moZm§Mr OmoS>r {‘iVo H$m?

∴ ∠QAC = ∠ACB ------(ii) ( ho ì¶wËH«$‘ H$moZ AmhoV.)

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PA , AB Am{U AC aofm nhm.

∠PAB + ∠BAC + ∠QAC = 1800 = gai H$moZ (iii)

g‘rH$aU (iii) ‘Ü¶o ∠RAB Am{U ∠SAC À¶m {R>H$mUr AZwH«$‘o ∠ABC Am{U ∠ACB KoVbo.

∠ABC + ∠BAC + ∠ACB = 1800 Amåhmbm {‘iVo.

ho {ÌH$moUmMo A§mVaH$moZ AmhoV.

Ë¶m‘wio {ÌH$moUmÀ¶m Am§VaH$moZm§Mr ~oarO 1800 AgVo.

MMm© Ho$bobr Vm{H©$H$ {gÜXVm {ÌH$moUmÀ¶m gd© AmH$ma Am{U ‘mnmgmR>r gË¶ Amho. hm {ZîH$f© gd© {ÌH$moUm§Zm bmJy nS>Vmo.

Vm{H©$H$ nÜXVrZo {ÌH$moUmMo Am§VaH$moZ H$mT>Uo.

gmoS>{dbobr CXmhaUo 1 :

~mOyÀ¶m AmH¥$Vr‘Ü¶o PQ || RS

A hm PQ darb EH$ q~Xÿ Amho.

A q~XÿVyZ AB Am{U AC aofm H$mT>ë¶m AmhoV.

∠PAB = 500 Am{U ∠QAC = 700 Agë¶mg, ABC {ÌH$moUmÀ¶m gd© H$moZm§Mr ‘mno VH$m©Zo H$mT>m.

PQ Am{U RS ¶m g‘m§Vaaofm Am{U AB aofm nm{hë¶mg ∠PAB Am{U ∠ABC ho

H$moUË¶m àH$maMo H$moZ AmhoV?

Vo ì¶wËH¥$‘ H$moZ AmhoV.

∴ ∠ABC = ∠PAB = 500

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AmVm PQ, RS Am{U AC aofm nhm.

∠QAC Am{U ∠ACB H$moUË¶m àH$mao H$moZ AmhoV?

Vo ì¶wËH«$‘ H$moZ AmhoV. ∠ACB = ∠QAC = 700

PQ, AB, AC ¶m aofm nhm.

∠PAB, ∠QAC Am{U ∠BAC ho H$moUË¶m àH$maMo H$moZ AmhoV?

Vo nyaH$ H$moZ AmhoV H$m?

∠PAB + ∠BAC + ∠QAC = 1800

500 + ∠BAC + 700 = 1800

= 1800 - 500- 700

∴ ∠BAC = 600

gmoS>{dbobo CXmhaU 2 :

AmH¥$Vr‘Ü¶o {ÌH$moUmÀ¶m XmoZ H$moZm§Mr ‘mno

{Xbr AmhoV. (400 Am{U 300) {ÌH$moUmÀ¶m

{Vgam H$moZ VH$m©Zo H$mT>m.

P

400 300

?

Q R

P q~XÿVyZ QR bm g‘m§Va aofm H$mT>m. Q

AmH¥$Vr‘Ü¶o P q~XÿV 400 Am{U 300 Mo ì¶wËH«$‘

H$moZ Xe©dm.

P

400

400 300

300

?

Q R

~mOyÀ¶m {ÌH$moU PQR ‘Ü¶o, ∠Q = 400, ∠R = 300

'P' ¶m q~XÿVrb VrZ H$moZm§Mr ~moarO 1800 Amho

åhUyZ H$moZ ∠PQR = 1800 (-400-300) = 1100.

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CXmhaU 1 : AmH¥$Vr‘Ü¶o, ∠A = 600, ∠B = 500 Va VH$m©Zo ∠C ‘mn H$mT>m.

∠A + ∠B + ∠C = 1800

{ÌH$moUmÀ¶m {VÝhr H$moZm§Mr ~oarO 1800

600 + 500 + ∠C = 1800

1100 + ∠C = 1800

∠C = 1800 - 1100

∠C = 700

CXmhaU 2 : ~mOyÀ¶m AmH¥$Vr‘Yrb x Mr qH$‘V H$mT>m.

∠A + ∠B + ∠C = 1800 ({ÌH$moUmÀ¶m {VÝhr H$moZm§Mr ~oarO 1800)

600 + x 0 + 800 = 1800 (∠A Am{U ∠C Mr ‘mno dmnabr) 1400 + x 0 = 1800

x 0 = 1800 - 1400

x 0 = 400

ñdmÜ¶m¶ 7.2

1. Imbrb AmH¥$VrVrb x Mr qH$‘V VH$m©Zo H$mT>m.

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2. {ÌH$moU ABC Mo XmoZ H$moZ {Xbobo AmhoV. {Vgam H$moZ H$mTy>Z V³Ë¶mV âm.

∠A ∠B ∠C

1 600 700

2 1000 350

3 650 650

4 1100 360

3. Imbrb V³Ë¶mV {VÝhr H$moZ {Xbobo AmhoV. Ë¶m {VÝhr H$moZm§Mr ~oarO H$mT>m. Vo {ÌH$moUmMo

VrZ H$moZ AmhoV H$m?

A.Z§. ∠A ∠B ∠C ~oarO{ÌH$moU

hmoB©b/Zmhr

1 500 600 700

2 600 700 800

3 650 750 550

4 560 640 600

5 570 640 790

4. {ÌH$moU ABC ‘Ü¶o Oa ∠A = 450, ∠B = 650 Agë¶mg ∠C Mo ‘mn H$mT>m.

5. Imbrb ‘mno Agbobm {ÌH$moU ABC aMm.

AB = 5 cm, ∠PAB = 500 Am{U ∠QBA = 600 Amho.

Ë¶m‘Ü¶o ∠ACB ‘moOm.

Am{U ∠ACB Mo ‘mn J{UVmÀ¶m ghmæ¶mZo nS>VmiyZ nmhm.

6. Oa H$mQ>H$moZ {ÌH$moUmV EH$ H$moZ 350 Agë¶mg BVa XmoZ

H$moZm§Mr ‘mno H$mT>m.

7. 4 cm ~mOy AgUmam g‘^wO {ÌH$moU aMm. Am{U Ë¶mÀ¶m àË¶oH$ H$moZmMo ‘mn H$mT>m.

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8. {ÌH$moU ABC ‘Ü¶o, ∠A = 350 Am{U ∠B hm ∠A nojm 150 Vo A{YH$ Agë¶mg,

∠C H$mT>m.

9. H$mQ>H$moZ {ÌH$moUm‘Ü¶o, H$mQ>H$moZ Zgbobm EH$ H$moZ Xþgè¶m H$mQ>H$moZ Zgboë¶m H$moZmÀ¶m

XþßnQ> Amho. Va H$mQ>H$moZ Zgboë¶m XmoZ H$moZm§Mr ‘mno H$mT>m.

{ÌH$moUmMo ~møH$moZ

{ÌH$moUmbm VrZ ~mOy AgVmV. Oa {ÌH$moUmMrH$m oUVrhr EH$ ~mO y dmT > {dbr Va

{ÌH$moUmÀ¶m ~mhoa {eamoq~XÿV V¶ma hmoUmam H$moZ ¶mbm ""{ÌH$moUmMm ~møH$moZ'' åhUVmV. Am§Va gb¾ H$moZ ~møH$moZ

¶m ~møH$moZmÀ¶m ~mOybm {ÌH$moUmÀ¶m AmV AgUmè¶m H$moZmg "Am§Va gb¾ H$moZ'

åhUVmV.

Am§Va gb¾ H$moZ ~møH$moZ

Am§Va gb¾ H$moZ

~møH$moZ

Am§Va gb¾ H$moZ~møH$moZ

Am§Va gb¾ H$moZ~møH$moZ

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Oa {ÌH$moUmMr H$moUVrhr EH$ ~mOy H$moUË¶mhr

Am§Va gb¾ H$moZ ~møH$moZ

~mOybm dmT>{dë¶mg ~møH$moZ ~ZVmV. {ÌH$moUmMm ~møH$moZ Am{U A§Va gb¾ H$moZm§Mr ~oarO {H$Vr?

ho XmoZ H$moZ nyaH$ H$moZ AgVmV qH$dm aofr¶ H$moZ AgVmV.

Ë¶m‘wio gb¾ Am§VaH$moZ Am{U ~møH$moZm§Mr ~oarO 1800 AgVo.

A§Vam{déÜX H$moZ :

{ÌH$moUm‘Ü¶o Am§Va gb¾ H$moZmg‘moarb H$moZm§Zm Am§Va{déÜX H$moZ åhUVmV.

Am§Va{déÜX H$moZ - 2

Am§Va{déÜX H$moZ - 1 Am§Va gb¾ H$moZ~møH$moZ

EH$m ~mø H$moZmg EH$ Am§Va gb¾ H$moZ Am{U XmoZ Am§Va{déÜX H$moZ AgVmV.

{ÌH$moUmMo ~møH$moZ, Am§Va gb¾ H$moZ, A§mVa{déÜX H$moZ ¶m‘Ü¶o H$mhr g§~§Y Amho H$m?


OmS> H$mJXmda H$moUË¶mhr AmH$mamMm EH$ {ÌH$moU aMm.

Ë¶m {ÌH$moUmMm nm¶m dmT>dm.

{ÌH$moUmÀ¶m VrZ H$moZm§Zm 1, 2 Am{U 3 AerZmdo Úm.

~møH$moZmg 4 ho Zmd Úm. AmH¥$Vr (i)

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AmH¥$Vr (ii) ‘Ü¶o Xe©{dë¶mà‘mUo,H$moZ 1 Am{U H$moZ 2 H$mnyZ ¿¶m.Vo XmoÝhr H$moZ ~møH$moZ 4 da OmoS>m

Vw‘Mm {ZîH$f© H$moUVm?

H$moZ 1+ H$moZ 2 = ~møH$moZ 4

AmH¥$Vr (ii)∴ {ÌH$moUmMm ~møH$moZ Am§Va{déÜX H$moZmÀ¶m ~oarOoer g‘mZ AgVmo ho ¶m H¥$Vrdê$Z

g‘OVo.darb JwUY‘m©da AmYmarV H$mhr CXmhaUo VnmgyZ nmhÿ.

CXmhaU 1 : Imbr {Xboë¶m AmH¥$Vr‘Ü¶o

700

500

C D

A

B

~møH$moZ = ?

{ÌH$moU ABC Mo XmoZ H$moZ {Xbobo AmhoV. Ë¶mMm ~møH$moZ J{UVmÀ¶m ghmæ¶mZo H$mT>m.

∠A + ∠B + ∠C = 1800 ({ÌH$moUmÀ¶m {VÝhr H$moUm§Mr ~oarO 1800) 700 + 500 + ∠C = 1800 (∠A Am{U ∠B Mr qH$‘V {Xbobr Amho.) 1200 + ∠C = 1800

∠C = 1800 - 1200 ∴ Am§VaH$moZ ∠ACB = 600

Am§VaH$moZ ∠ACB Mm ~møH$moZ ∠ACD = 1800 (EH$ gb¾ Am§Va H$moZ Am{U ~møH$moZmMr ~oarO 1800)

∴ ∠ACB + ∠ACD = 1800

600 + ∠ACD = 1800

∠ACD = 1800 - 600 ∠ACD = 1200

~møH$moZ ∠ACD = 1200

Am§Va {déÜX H$moZm§Mr ~oarO = 700 + 500 = 1200

~møH$moZ hm Am§Va {déÜX H$moZm§À¶m ~oaOo BVH$m AgVmo.

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CXmhaU 2 :

AmH¥$Vr‘Ü¶o {ÌH$moU PQR Mo XmoZ H$moZ {Xbobo 400

550RS

P

Q~møH$moZ=?

AmhoV. Ë¶mÀ¶m ~møH$moZmMo ‘mn H$mT>m.

∠P + ∠Q + ∠R = 1800 ({ÌH$moUmÀ¶m {VÝhr H$moZm§Mr ~oarO 1800)

400 + 550 + ∠R = 1800 (∠P Am{U ∠Q À¶m qH$‘V dmnê$Z)

950 + ∠R = 1800

∠R = 1800 - 950 = 850

∴ Am§VaH$moZ ∠PRQ = 850

Am§VaH$moZ ∠PRQ Am{U ~møH$moZ ∠PRS ho nyaH$ H$moZ AmhoV.Ë¶m‘wio, ∠PRQ + ∠PRS = 1800

850 + ∠PRS = 1800

∠PRS = 1800 - 850 = 950

darb CXmhaUmV, ~møH$moZ = 950 Am§Va{déÜX H$moZm§Mr qH$‘V = 400 + 550 = 950

∴ ~møH$moZ Am§Va{déÜX H$moZm§À¶m ~oaOoBVH$m AgVmo.

CXmhaU 3 : AmH¥$Vr‘Ü¶o ∠BAC = 500 Am{U ∠ABC = 850 A

500

850

DCB

Agë¶mg ∠ACB Am{U ∠ACD Mr ‘mno H$mT>m.

∠BAC + ∠ABC + ∠ACB = 1800 ({ÌH$moUmÀ¶m {VÝhr H$moUm§Mr ~oarO 1800)

500 + 850 + ∠ACB = 1800 (∠A Am{U ∠B Mr qH$‘Vr dmnê$Z)

1350 + ∠ACB = 1800

∠ACB = 1800 - 1350 = 450

∠ACB + ∠ACD = 1800 (nyaH$ H$moZ)

450 + ∠ACD = 1800

∠ACD = 1800 - 450 = 1350

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ñdmÜ¶m¶ 7.3

1. Imbrb AmH¥$Vr‘Yrb x Am{U y H$moZm§Mr qH$‘V H$mT>m.

a)600

400 x y

c)

x y

550

550

b) 1450 700

x

y d)

500

650x y

2. {ÌH$moU ABC ‘Ü¶o Oa

∠BAC = 450 Am{U ∠ABC = 750 Agë¶mg ∠ACB Am{U ∠ACD Mr ‘mno H$mT>m.

A

450

750

DCB

3. AmH¥$Vr‘Ü¶o ∠PQR = 500, ∠PRS = 1300 Agë¶mg ∠QPR Am{U ∠PRQ Mr ‘mno H$mT>m. 500 1300

R S

P

Q

4. AmH¥$Vr‘Ü¶o ∠NKL = 350, ∠KNL = 300 Am{U ∠LNM = 750 Agë¶mg ∠NLM Am{U ∠NML H$mT>m.

300 750

MLK

N

350

5. {ÌH$moU ABC ‘Ü¶o BC ~mOy D n¶ªV dmT>{dbr Am{U A d D gm§Ybo. Oa ∠BAC = 600, ∠ABC = 450 Agë¶mg ∠ACD H$mT>m.

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{ÌH$moUmÀ¶m H$moUË¶mhr XmoZ ~mOy§Mr ~oarO

H¥$Vr 1 :

àVbmda EH$m gai aofoV Zgbobo (Z¡H$aofr¶) VrZ q~Xÿ

A

B C

A, B Am{U C Xe©dm. MyÝ¶mÀ¶m nmdS>aZo AB, BC Am{U AC aofm H$mT>m.

¶m VrZ aofm ∆ABC À¶m VrZ ~mOy Xe©{dVmV.

Vw‘À¶m {‘Ìmbm A nmgyZ gwédmV H$ê$Z H$moUË¶mhr ‘mJm©Zo C n¶ªV OmÊ¶mg gm§Jm.

A nmgyZ C n¶ªV OmÊ¶mg XmoZ ‘mJ© AmhoV. EH$ ‘mJ© A nmgyZ gai C n¶ªV OmVmo. Xþgam

A nmgyZ B bm OmD$Z, B nmgyZ C bm OmVmo.

n{hbm ‘mJ© H$‘r A§VamMm Amho Am{U àË¶oH$OU Ë¶mM ‘mJm©À¶m Cn¶moJ H$aVmV.

A nmgyZ B bm OmD$Z Am{U B nmgyZ C bm OmÊ¶mg OmñV A§Va Omdo bmJVo.

VoM ∆ABC À¶m ~mOy~m~V Ago gm§Jy eH$Vmo.

AB + BC > AC ---- (i)

Ë¶mMà‘mUo, B nmgyZ A bm OmÊ¶mg XmoZ ‘mJ© AmhoV. B nmgyZ gai A bm OmÊ¶mMm

‘mJ©, B nmgyZ C bm OmD$Z C nmgyZ A bm OmÊ¶mÀ¶m Xþgam ‘mJ© Amho.

Vo Ago gm§Jy eH$Vmo -----

BC + CA > BA ---- (ii)

VgoM B nmgyZ C bm OmÊ¶mMo XmoZ ‘mJ© AmhoV. EH$ gai B nmgyZ C H$S>o OmVmo.

AmUIr EH$ B nmgyZ A bm OmVmo Z§Va nmgyZ H$S>o

∴ BA + AC > BC ---- (iii)

¶mbm g§~§YrV AmUIr EH$ H¥$Vr H$é¶m.

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H¥$Vr 2 :

3cm, 5cm, 7cm, 9cm, 11cm, 13cm, 15cm, 17cm, 19cm, 21cm

‘mnmÀ¶m Xhm H$mS>çm ¿¶m. Ë¶mn¡H$s H$moUË¶mhr VrZ H$mS>çm KoD$Z {ÌH$moU aMÊ¶mMm à¶ËZ

H$am. (VgoM doJdoJù¶m H$mS>çm KoD$Z doJù¶m bm§~rMm {ÌH$moU aMm.)

CXmhaUmW©, Vwåhr 5cm. Am{U 7cm À¶m XmoZ H$mS>çm ¿¶m. {ÌH$moU aMÊ¶mgmR>r

AmUIr EH$ H$mS>r {ZdS>mdr bmJob. Vr H$mS>r 7 - 5 = 2 cm nojm ‘moR>r Am{U

7 + 5 = 12 cm nojm bhmZ Agë¶mg ‘mÌ {ÌH$moU aMUo e³¶ Amho H$m Vo nhm.

{ÌH$moU aMÊ¶mg bmJUmè¶m VrZ H$mS>çmn¡H$s H$moUË¶mhr XmoZ H$mS>çm§À¶m bm§~rMr ~oarO

{Vgè¶m H$mS>rÀ¶m bm§~rnojm OmñV Agbr nm{hOo.

H¥$Vr 1 Am{U 2 dê$Z Amåhr Agm {ZîH$f© ¶oVmo H$s {ÌH$moUmÀ¶m H$moUË¶mhr XmoZ ~mOy§Mr

~oarO {Vgè¶m ~mOynojm A{YH$ AgVo.

CXmhaU 1 :

{ÌH$moUmÀ¶m XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOynojm ‘moR>r Amho H$m? Vo Vnmgy¶m.

a) 3cm + 4cm > 6cm {ÌH$moUmÀ¶m XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOynojm ‘moR>r Amho.

b) 4cm + 6cm > 3cm {ÌH$moUmÀ¶m XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOynojm ‘moR>r Amho.

c) 6cm + 3cm > 4cm {ÌH$moUmÀ¶m XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOynojm ‘moR>r Amho.

∴ ¶m ‘mnmMm {ÌH$moU aMUo e³¶ Amho.

CXmhaU 2 : 3cm, 4cm, Am{U 7cm, ~mOy AgUmam {ÌH$moU aMUo e³¶ Amho H$m? Vnmgm.

{ÌH$moUmÀ¶m XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOynojm ‘moR>r Amho H$m ho Vnmgy¶m.

a) 3cm + 4cm = 7cm {ÌH$moUmÀ¶m XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOynojm ‘moR>r Zmhr.

b) 3cm + 7cm > 4cm {ÌH$moUmÀ¶m XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOynojm ‘moR>r Amho.

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c) 4cm + 7cm > 3cm {ÌH$moUmÀ¶m XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOynojm ‘moR>r Amho.

∴ EH$m g§X^m©V XmoZ ~mOy§Mr ~oarO {Vgè¶m ~mOy nojm ‘moR>r Zmhr. Ë¶m‘wio ¶m ‘mnmMm {ÌH$moU

H$mT>Uo e³¶ Zmhr.

ñdmÜ¶m¶ 7.4

1) Imbrb ‘mnmMm {ÌH$moU H$mT>Uo e³¶ Amho? gm§Jm.

1) 3cm, 4cm, 5cm.

2) 3cm, 2cm, 6cm.

3) 2cm, 6cm, 6cm.

4) 5cm, 3cm, 8cm.

2) {ÌH$moU ABC ‘Ü¶o M hm H$moUVmhr {ÌH$moUmÀ¶m AmVrb EH$ q~Xÿ Agë¶mg

Imbrb ~amo~a H$s MyH$ gm§Jm. A

MB C

1) MA + MB > AB ?

2) MB + MC < BC ?

3) MC + MA > CA ?

nm¶W°JmoagMm à‘o¶ (~moYm¶Z gyÌ)

nm¶W°Jmoag ho {J«H$ VËdkmZr hmoVo. Ë¶m§Zr {¼. ny. 6 ì¶m eVH$m‘Ü¶o H$mQ>H$moZ {ÌH$moUmg§~§Yr EH$ ‘hËdmÀ¶m Am{U ~hþ Cn¶moJr JwUY‘m©Mo g§emoYZ Ho$bo. Ë¶m‘wio H$mQ>H$moZ {ÌH$moUmer g§~§YrV ¶m à‘o¶mbm Ë¶m§À¶m ZmdmZo AmoiIbo OmVo.

Oa {ÌH$moUmMm EH$ H$moZ H$mQ>H$moZ Agë¶mg Ë¶m {ÌH$moUmbm H$mQ>H$moZ {ÌH$moU åhUVmV. Ë¶m‘Ü¶o H$mQ>H$moZm g‘moarb ~mOyg "H$U©' Ago åhUVmV. H$mQ>H$moZ {ÌH$moUmV H$U© hr gdm©V ‘moR>r ~mOy AgVo.

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H¥$Vr 1 : EH$ Am¶VmH$ma nm§T>am H$mJX ¿¶m. Ë¶mÀ¶m Mma H$monè¶mV Imbrb {MÌmV {Xboë¶m ‘mnmMo Mma H$mQ>H$moZ {ÌH$moU aMm. Vo ∆ABC, ∆KLM, ∆PQR Am{U ∆XYZ Agy Úm. Ë¶m {ÌH$moUmMo H$U© Agbobo AC, KM, PR Am{U XZ aofmI§S> H$mTy>Z H$Um©À¶m ~mOwZo H$mJX Xþ‘S>m.

A X 12cm

12cm8cm

3cm

5cm

4cm

9cm6c

m

Y

ZR

QPML

KC

B

Ë¶m {ÌH$moUmÀ¶m bm§~rMr ‘mno Imbrb V³Ë¶mV am. V³Ë¶m‘Yrb BVa aH$mÝ¶mV ∆ABC

À¶m g‘moa {bhboë¶m gyMZoZwgma {bhm.

{ÌH$moU H$Um©Mo‘mn

H$Um©MmdJ©

EH$m~mOwMr bm§~r

{VMm dJ©

Xÿgè¶m ~mOyMr bm§~r

{VMm dJ©

XmoZ ~mOy§À¶mdJmªMr qH$‘V

∆ABC 5 25 3 9 4 16 9 + 16 = 25

∆KLM

∆PQR

∆XYZ

∆ABC ‘Ü¶o AB = 3cm. BC = 4cm. Amho. H$U© AC ‘moOm.

AC = 5cm. Amho H$s Zmhr?

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AmVm H$mQ>H$moZ {ÌH$moUmÀ¶m XmoZ ~mOyÀ¶m dJm©Mr ~oarO J{UVmÀ¶m ghmæ¶mZo H$mT>m. AB2 + BC2 = 32 + 42

= 9 + 16 = 25

AC Mm dJ© H$am. AC2 = 52 = 25 ∴ AC2 = AB2 + BC2

52 = 32 + 42

25 = 9 + 16 = 25

darb J{UVmdê$Z Ago g‘OVo H$s,

H$mQH$moZ {ÌH$moUm‘Ü¶o H$Um©Mm dJ© hm BVa XmoZ ~mOy§À¶m dJm©À¶m ~oaOoBVH$m AgVmo.

¶mbmM nm¶W°JmoagMm à‘o¶ åhUVmV.

nm¶W°JmoagMm à‘o¶ (~moYm¶Z gyÌ)

""H$mQ>H$moZ {ÌH$moUm‘Ü¶o H$Um©Mm dJ© hm BVa XmoZ ~mOy§À¶m dJm©À¶m ~oaOoBVH$m AgVmo''.

AB ~mOyMm

dJ© AB2

BC ~mOyMm

dJ© BC2

AC ~mOyMm dJ©

AC 2 (H$Um©Mm dJ© )

{ÌH$moU ABC ‘Ü¶o ∠B = 900

Agë¶mg AB2 + BC2 = AC2

A

B C

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nm¶W°Jmoag ({¼.ny. 475-569) À¶m gw‘mao e§â dfm©nydu ""~moYm¶Z'' ({¼.ny. 800-600) ¶m§Zr aMboë¶m ""gwb~m gyÌ'' ‘Ü¶o ¶mM {Z¶‘mMm C„oI Amho.

XrK©MaVwaòñ¶múU¶m a‚mw : nmíd©‘mZr {V¶©J ‘mZrM

¶V² n¥W½^yVo Hw$éVñVXþ^¶§ H$amoVr &

¶mMm AW©,

Am¶VmMr bm§~r d é§Xr ¶m§À¶m dJmªMr ~oarO hr Ë¶m Am¶VmÀ¶m H$Um©À¶m dJm©BVH$s AgVo.

AmnU nm¶W°JmoagMm à‘o¶ àm¶mo{JH$arË¶m Vnmgbm. ¶mMo Vm{H©$H$ {díbofU nwT>rb B¶ÎmoV {eH$mb.

CXmhaU 1 :

∆ABC ‘Ü¶o AB = 6cm, BC = 8cm. Am{U ∠B = 900 Agë¶mg AC H$mT>m.

CH$b : nm¶W°JmoagÀ¶m {Z¶‘mZwgma.

AC2 = AB2 + BC2

A

900

6cm

8cmB C

= 62 + 82

= 36 + 64

= 100

= 102

∴ AC = 10 cm

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CXmhaU 2 : 13 m. bm§~rMr {eS>r EH$m q^Vrbm {VaH$g bmdbr Amho. Ë¶m {eS>rMo Q>moH$$ O{‘ZrnmgyZ 12 m. C§Mrda Amho. Va Ë¶m {eS>rMo q^VrnmgyZ O{‘Zrdarb A§Va {H$Vr?

CH$b : nm¶W°JmoagÀ¶m {Z¶‘mZwgma,

AB2 + BC2 = AC2

A

12 ‘r13 ‘r

?B C

q^V

O‘rZ

{eS>r

122 + BC2 = 132

144 + BC2 = 169

BC2 = 169 - 144

BC2 = 25

∴ BC = 5

∴ {eS>rMm nm¶m q^VrnmgyZ 5 m. A§Vamda Amho.

ñdmÜ¶m¶ 7.5

1. EH$ Im§~ O{‘Zrda gai C^m H$amd¶mMm Amho. O{‘ZrnmgyZ 8 m.> C§Mrda>

86?

Ë¶m Im§~mbm Ë¶mÀ¶m VimnmgyZ 6 m.> bm§~ AgUmè¶m Iw§Q>rbm ~m§YÊ¶mgmR>r {H$Vr bm§~rMr Vma bmJob?

2. ∆ABC ‘Ü¶o AB = 5 cm, BC = 12 cm, ∠ABC = 900 Agë¶mg AC Mr bm§~r H$mT>m.

3. 15 m. C§MrÀ¶m {eS>rMo Q>moH$ O{‘ZrnmgyZ 12 m. C§Mrda q^Vrbm Q>oH$bo Amho. Va {eS>rMm nm¶m q^VrnmgyZ {H$Vr A§Vamda Amho?

4. Imbrb ‘mnmÀ¶m ~mOy AgUmè¶m {ÌH$moUm‘Ü¶o H$moUVm {ÌH$moU H$mQ>H$moZ {ÌH$moU Agob?

a) 2 cm, 2 cm, 5 cm.

b) 1.5 cm, 2 cm, 2.5 cm.

c) 6 cm, 8 cm, 10 cm.

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àH$aU - 8

g‘{‘Vr (SYMMETRY)


g‘{‘Vr Am{U à{Vq~~rV g‘{‘Vr AmoiImb,

2D ({Û{‘Vr¶) AmH¥$Ë¶m§Mr à{Vq~~rV g‘{‘VrMr ì¶m»¶m gm§Jmb,

AmH¥$Ë¶m§Mo 900, 1200, 1800 H$moZm‘Ü¶o n[a^«‘U hmoÊ¶mMo AmoiImb,

900, 1800 H$moZm‘Ü¶o {’$a{dÊ¶mMo H$m¡eë¶ dmnê$Z AmH¥$Ë¶m§Mr g‘{‘Vr AmoiImb,

n[a^«‘U Am{U à{Vq~~rV g‘{‘Vr {H«$¶m Agboë¶m AmH¥$Ë¶m§À¶m ghmæ¶mZo H¥$Vr

H$ê$ eH$mb,

n[a^«‘U Am{U à{Vq~{~V g‘{‘Vr¶ {H«$¶m AmH¥$Vr‘Ü¶o Xe©dmb.

8.1 g‘{‘Vr

‘mJrb B¶ÎmoV Vwåhr g‘{‘Vr¶ AmH¥$Ë¶m Am{U g‘{‘Vr {df¶r {eH$bm AmhmV.

g‘{‘Vr AmH¥$Vr åhUOo H$m¶?

EH$ AmH¥$Vr EH$m H$mën{ZH$ aofoZo XmoZ g‘mZ ^mJmV {d^mJë¶mda Vo ^mJ EH$‘oH$mda R>odë¶mg Vo ^mJ nañnamg ¶mo½¶ Owië¶mg Ë¶m§Zm "g‘{‘Vr AmH¥$Vr' åhUVmV. Eo{Vhm{gH$ B‘maVr, gmoÝ¶mMo Xm{JZo BË¶mXr gw§Xa {XgÊ¶mMo H$maU åhUOo Ë¶m‘Yrb g‘{‘Vr. Amnë¶m X¡Z§{XZ OrdZmV dmnaUmè¶m dñVy‘Ü¶o XoIrb AmnU g‘{‘Vr nmhÿ eH$Vmo.

a{d Am{U ‘mohZ XmoKo {‘Ì AmhoV.

a{d : Vwbm {ZgJ© gm¢X¶© AmdS>Vo H$m?

‘mohZ : hmo¶ a{d, Iyn AmdS>Vo.

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a{d : ‘r EH$ ’$moQ>mo H$mT>bm Amho ~K.

‘mohZ : dmd! IynM gw§Xa Amho a{d. AmH$memMo

à{Vq~~ nmÊ¶mV {XgV Amho. gw §Xa

Amho!!!

¶m {MÌm‘Ü¶o Vwbm {deof H$m¶ {XgVo?

nmÊ¶mÀ¶m XmoÝhr ~mOyg EH$mM àH$maMo {MÌ {XgVo H$m?

¶mbm EH$ {deof Zmd Amho. Vo åhUOo g‘{‘Vr.

{ZgJm©Vrb AZoH$ dñVy g‘{‘VrV Agë¶mà‘mUo {XgVmV. g‘{‘Vr J{UVm‘Yrb EH$ {d{eð> n[aH$ënZm Amho.

~mOyÀ¶m {MÌmMo {ZarjU H$am.

ho gw§Xa {XgVo H$m?

ho {MÌ gw§Xa {XgÊ¶mgmR>r ^y{‘VrMm H$moUVm KQ>H$ ghmæ¶ H$aVmo.

hr AmH¥$Vr g‘{‘Vr‘Ü¶o Amho.

¶m‘Yrb g‘{‘Vr¶ ^mJ AmoiIÊ¶mMm à¶ËZ H$am.

H¥$Vr 1 : EH$m H$mJXmda WmoS>r emB© Q>mH$m. H$mJXmMo Cô XmoZ g‘mZ ^mJ hmoVrb Agm Xþ‘S>m.

H$mJXmda Xm~ Úm. H$mJX CbKS>m. V¶ma Pmbobr aMZm nhm. Ë¶mbm g‘{‘Vr Aj H$mT>m.

¶oWo g‘{‘Vr V¶ma hmoÊ¶mg H$moUVo H$maU Am{U H$m¶ ¶mMm {dMma H$am.

EH$m gai aofoZo AmH¥$VrMo XmoZ g‘mZ ^mJ hmoV Agë¶mg Aem AmH¥$Vrbm aofr¶ g‘{‘Vr

AmH¥$Vr Ago åhUVmV. ¶m gaiaofobm g‘{‘Vr Aj Ago åhUVmV.

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¶m Imbrb AmH¥$Ë¶m nhm.

{R>n³¶m§Mr aofm g‘{‘Vr Aj gyM{dVo.

g‘{‘Vr‘wio dñVy§Zm narnyU©Vm Am{U g‘Vmob {‘iVmo.

à{V{~§~rV g‘{‘Vr Am{U AmaemVrb à{Vq~~ g‘{‘Vr ¶m§Mo EH$‘oH$m‘Ü¶o g§~§Y AmhoV. à{Vq~~rV g‘{‘Vr AmH¥$Vr‘Ü¶o, AmaemÀ¶m à{V‘oMm AYm©^mJ AgVmo.

H¥$Vr 2 : Imbrb AmH¥$Ë¶m§Zm g‘{‘Vr Aj H$mT>m.

ñ_aU H$am

{MÌmbm XmoZ g‘{‘Vr¶ ^mJmV {d^mJUmè¶m aofobm g‘{‘Vr Aj Ago åhUVmV.

Vwåhmbm ‘mhrV Amho?

Amnbm amï´>ÜdO g‘{‘VrMo EH$ CÎm‘ CXmhaU Amho.

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H¥$Vr 3 :

{ZË¶mbm AmoarJm‘r AmdS>o. {VZo nonaMo bm§~rÀ¶m ghmæ¶mZo XmoZ

i) ii)

g‘mZ mJmV KS>r H$ê$Z, Ë¶mda gwaHw$Ë¶m nmS>ë¶m. Z§Va {MÌmV

Xe©{dë¶mà‘mUo H$mJX H$mnbm.

Z§Va Vr nonamMr KS>r CbKS>br. VrZo H$m¶ {ZarjU Ho$bo ho

Vwåhr gm§Jy eH$mb? AmH¥$Vr (ii) nmhÿZ CÎma XoÊ¶mMm à¶ËZ H$am.

hmo¶, VrZo H$mJXmMr KS>r KmVboë¶m aofmÀ¶m XmoÝhr mJmV AmH¥$Vr EH$mM à‘mUmV Am{U

EH$mM AmH$mamV {d^mJbr Jobobr Agë¶mMo {ZarjU Ho$bo.

H$mJX KS>r KmVboë¶m aofoMo Zmd Vwåhr AmR>dyZ gm§Jm.

H$mJX KS>r KmVboë¶m aofobm g‘{‘VrMm g‘{‘Vr Aj åhUVmV.

H¥$Vr 4 :

EH$ {ÌH$moUr AmH$mamMm H$mJX ¿¶m Am{U Ë¶mÀ¶m ~mOy AmH¥$VrV Xe©{dë¶mà‘mUo PQ =

PR Agy Úm.

Q S R

PAmH¥$VrMm AYm©^mJ hm Caboë¶m

AÜ¶m©^mJmÀ¶m AmaemVrb à{Vq~~m gmaIm {XgV Agë¶mg, Aem AmH¥$Ë¶m§Zm à{Vq~~rV g‘{‘Vr åhUVmV.

hm {ÌH$moU PS AjmÀ¶m ghmæ¶mZo PR Am{U PQ ~mOy EH$mV EH$ {‘giVrb Agm Xþ‘S>m. Ë¶mbm gwaHw$Ë¶m nmS>m Am{U CbKS>m. ∆PQS Am{U ∆ PRS EH$‘oH$mÀ¶m AmaemVrb à{Vq~~mgmaIo {XgVmV. Zmhr H$m? AmnU ¶m {ÌH$moUmbm AmUIr EH$ g‘{‘Vr Aj H$mTy> eH$Vmo? Zmhr.

g‘{Û^wO {ÌH$moUm‘Ü¶o g‘{‘Vr AjmÀ¶m ghmæ¶mZo à{Vq~~rV g‘{‘Vr V¶ma hmoVo. ¶oWo EH$M g‘{‘Vr Aj Amho.

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H¥$Vr 5 : EH$ Am¶VmH$ma H$mJX PQRS {MÌmV Xe©{dë¶mà‘mUo H$mnyZ ¿¶m.

S R

P Q

QR Am{U PS ~mOy EH$‘oH$mV {‘giVrb AemàH$mao Am¶VmH$ma H$mJXmMr KS>r Kmbm. H$mJXmda ~moQ>mZo gwaHw$Ë¶m nmSy>Z H$mJX CbKS>m.

AmVm PQ Am{U SR ~mOy EH$‘oH$mV {‘giVrb AemàH$mao Am¶VmH$ma H$mJXmMr KS>r Kmbm. H$mJXmda ~moQ>mZo gwaHw$Ë¶m nmSy>Z H$mJX CbKS>m.AmVm Vwåhmbm KS>r KmVboë¶m {H$Vr aofm {XgVmV?

XmoZ aofm {XgVmV.Am¶Vmbm XmoZ g‘{‘Vr Aj AgVmV.

H¥$Vr 6 : EH$m H$mJXmda g‘^wO {ÌH$moU H$mT>m. àË¶oH$ ~mOyda ‘Ü¶q~Xÿ Xe©dm. ‘Ü¶q~Xÿ Ë¶mÀ¶m g‘moarb {eamoq~Xÿer OmoS>m.

B

F

A

E

CD

∆ABC hm g‘^wO {ÌH$moU Amho. ∆ADB Mo AmaemVrb à{Vq~~mMr ∆ADC hr AmH¥$Vr Amho. AD, BE Am{U CF ~mOy ¶m g‘{‘Vr¶ Aj AmhoV. g‘ wO {ÌH$moUmV VrZ g‘{‘Vr Aj AgVmV.Caboë¶m AmaemVrb à{Vq~~mÀ¶m AmH¥$Ë¶m emoYm.

H¥$Vr 7 : PQRS Mm¡agm‘Ü¶o gd© ~mOy g‘mZ AmhoV. ¶mbm g‘{‘Vr Aj H$mTy>¶m.

SN

K M

R

P QL

PR Am{U NL ho g‘{‘Vr Aj AmhoV. Caboë¶m XmoZ g‘{‘Vr Ajm§Mr Zmdo {bhm.

Mm¡agm‘Ü¶o {H$Vr g‘{‘Vr Aj AmhoV?

Mm¡agm‘Ü¶o Mma g‘{‘Vr Aj AmhoV.

¶m‘Ü¶o XmoZ H$U© Am{U ~mOyZm nañna b§~m‘Ü¶o N>oXUmè¶m XmoZ aofm AmhoV.

{d{dY AmH¥$Ë¶m§Zm AmnU doJdoJù¶m g§»¶oV g‘{‘Vr Aj H$mTy> eH$Vmo.

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bjmV R>odm : • g‘^wO {ÌH$moUmV VrZ g‘{‘Vr Aj AgVmV.

• Mm¡H$moZm‘Ü¶o Mma g‘{‘Vr Aj AgVmV.

• {Z¶{‘V n§MH$moZmH¥$Vr‘Ü¶o nmM g‘{‘Vr Aj AgVmV.

• {Z¶{‘V fQ>H$moZmH¥$Vr‘Ü¶o ghm g‘{‘Vr Aj AgVmV.

Ë¶m‘wio {Z¶{‘V ~hþ wOmH¥$Vr‘Ü¶o, AmH¥$VrÀ¶m ~mOyÀ¶m g§»¶oBVHo$ g‘{‘Vr Aj AgVmV.

dVw©im‘Ü¶o {H$Vr g‘{‘Vr Aj H$mTy> eH$Vmo? Imbrb AmH¥$Ë¶m§Mo {ZarjU H$am.

{dMma H$am : EH$m dVw©imbm AZ§V g‘{‘Vr Aj AgVmV H$m?

gyMZm : AmnU EH$m ì¶{º$Mo à{Vq~~ AmaemV nmhÿ eH$Vmo. AmaemVrb à{Vq~~ {ZpíMVnUo à{Vq~~rV g‘{‘Vr ZgVo. dñVyMo à{Vq~~ Am{U à{Vq~~rV g‘{‘Vr XmoÝhr EH$M Am{U g‘mZ ZgVmV.

ñdmÜ¶m¶ 8. 1

I. Imbrb n¡H$s H$moUË¶m AmH¥$Vr g‘{‘Vr¶ Amho?

a) b) c) d)

II. Imbrb Ajam§Zm e³¶ AgUmao gd© g‘{‘Vr Aj H$mT>m.

a) A b) H c) X

©

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III. Imbrb àË¶oH$ AmH¥$VrVrb VwQ>H$ aofm g‘{‘V Aj Amho H$m? hmo¶ qH$dm Zmhr gm§JyZ Vw‘À¶m CÎmambm g‘W©Z Úm.

a) b) c) d)

e) E f) g) h)

IV. Imbrb AmH¥$Vr‘Yrb g‘{‘V Ajm§Mr g§»¶m {bhm.

a) g‘{Û^wO {ÌH$moU b) g‘^wO {ÌH$moU

c) Mmoag d) g‘^wO MmoH$moZ

e) dVw©i f) g‘m§Va^wO Mm¡H$moZ

g) {Z¶{‘V n§MH$moZ h) {Z¶{‘V fQ>H$moZ

V. B§J«Or Ajam§Mo Ë¶mÀ¶m AmaemVrb à{Vq~~mer OmoS>çm Oyidm.

Ajao AmaemVrb à{Vq~~

B E

C B

D E

E C

R D

R

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n[a^«‘U :

Amnë¶m X¡Z§{XZ OrdZmV, {’$aUmè¶m AWdm ^«‘U hmoUmè¶m AZoH$ dñVy AmnU nmhVmo.

CXm : i) n§»¶mÀ¶m nmË¶m§Mo {’$aUo.

ii) dmhZm§À¶m MmH$mMo {’$aUo.

iii) KS>çmimÀ¶m H$mQ>çm§Mo {’$aUo.

KS>çmù¶mVrb H$mQ>çm§À¶m MbZmMo {ZarjU H$am. ¶m MbZmg AmnU KS>çmimÀ¶m {XeoZo

MbZ åhUVmo. Oa {déÜX {Xeobm {’$aë¶mg KS>çmimÀ¶m {déÜX {XeoZo MbZ åhUVmo.

^«‘Um‘Ü¶o XmoZ àH$maMo MbZ AmT>iyZ ¶oVo. Vo åhUOo :

i) KS>çmimÀ¶m H$mQ>çmÀ¶m {’$aÊ¶mÀ¶m {XeoZo

ii) KS>çmimÀ¶m H$mQ>çmQ>çm {’$aÊ¶mÀ¶m {déÜX {XeoZo

KS>çmimÀ¶m {MÌmMo {ZarjU H$am. KS>çmimMo XmoZ H$mQ>o

KS>çmimÀ¶m ‘Ü¶mda EH$m pñWa q~Xÿ^modVr {’$aVmV. ¶m pñWa

q~Xÿg n[a^«‘U H|$Ð Ago åhUVmV.

{MÌm‘Yrb MmH$m§Mo {ZarjU H$am. {MÌmdarb {R>n³¶mMo {ZarjU H$am.

XmoÝhr {MÌmVrb MmH$m§Mm AmH$ma EH$ gmaImM Amho Am{U MmHo$ EH$m {ZpíMV q~Xÿ modVr

{’$aVmV. MmH$mdarb {R>n³¶mÀ¶m ñWmZ~Xbm‘wio n[a^«‘U {XgyZ ¶oVo.

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AmUIr EH$ CXmhaU nmhÿ. Om°Z EH$ ImoS>H$a ‘wbJm Amho. Ë¶mMr AmB© Ë¶mbm {¼g‘g

gmR>r EH$ Mm§XUr AmUyZ XoVo. Vmo Vr {’$a{dV {MÌmV XmI{dë¶mà‘mUo IoiVmo.

A

B

C

D

D

A

B

CD

A

B

C

D

A

B

C

Vmo Vr Mm§XUr gmÜ¶ AgUmè¶m gd© {Xeobm {’$a{dVmo.

Z§Va Vmo Vr Mm§XUr S>mdrH$Sy>Z COdrH$S>o Am{U COdrH$Sy>Z S>mdrH$S>o {’$a{dVmo.

Ë¶mZo Mm§XUr KS>çmimÀ¶m {XeoZo qH$dm {déÜX {XeoZo {’$a{dbr.

EH$mM àVbm‘Ü¶o EH$m pñWa q~XþÀ¶m ghmæ¶mZo dñVyÀ¶m H$moUË¶mhr {Xeobm {’$aÊ¶mÀ¶m

à{H«$¶obm n[a «‘U Ago åhUVmV Am{U pñWa q~Xÿbm n[a «‘U H|$Ð qH$dm n[a «‘U q~Xÿ åhUVmV.

H¥$Vr 8 :

EH$m H$mS>©~moS>©À¶m ‘Ü¶mda EH$ bhmZ {Iim ~gdm. AmH¥$VrV

90 90

9090

Xe©{dë¶mà‘mUo H$mS>©erQ>nmgyZ ~Z{dbobm EH$ ~mU Ë¶mV

AS>H$dm. ¶oWo {Iim n[a^«‘U H|$Ð Amho. AmVm ~mUmMo Q>moH$

900, 1800, 2700 Am{U EH$ nyU© ’o$ar‘Ü¶o {’$adm. Vwåhr

H$m¶ {ZarjU H$amb?

3600 À¶m H$moZmV qH$dm EH$ nyU© ’o$ar {’$a{dë¶mg ~mUmMo Q>moH$ Ë¶mÀ¶m ‘yi ñWmZr ¶oVo.

åhUyZ gd© dñVy 3600 À¶m H$moZmV qH$dm EH$ nyU© ’o$ar {’$a{dë¶mg Ë¶m§À¶m ‘yi ñWmZr

¶oVmV.

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¶m H¥$Vr‘YyZ Vwåhr {ZarjU Ho$bobo AmUIr H$mhr A§e H$moUVo?

nmd ’o$ar {’$a{dë¶mg ~mU 900 À¶m H$moZmV {’$aVmo. AYu ’o$ar {’$a{dë¶mg Vmo 1800

H$moZmV {’$aVmo. nmD$Z ’o$ar {’$a{dë¶mg Vmo 2700 H$moZmV {’$aVmo Am{U EH$ ’o$ar nyU© {’$a{dë¶mg

Vmo 3600 H$moZmV {’$aVmo. ho A§e Amnë¶m bjmV ¶oVmV.

Imbrb {MÌm§Mo n[a^«‘U nhm.

1)

2)

3)

B

CD

A C

DA

B D

AB

C A

BC

D B

CD

A

4)

5)

B

CD

A

DA

B CD

AB

C

BC

D AB

CD

A

H¥$Vr 9 : OABC Am¶V Ò' q~XÿÀ¶m ghmæ¶mZo Imbr {Xboë¶m H$moZm‘Ü¶o {’$a{dë¶mg

V¶ma hmoUmè¶m AmH¥$Ë¶m H$mT>m.

(i) 900 ii) 1800 iii) 2700 iv) 3600

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C''

C'

C

BA

O A'

B'

B''

B'''

A''

C'''

A'''

nm¶ar 1 : OABC Am¶V H$mT>m Am{U Ò' ho

Ë¶mÀ¶m n[a^«‘UmMo H|$Ð Agy Úm.

nm¶ar 2 : 900 À¶m H$moZmV Am¶V {’$adm. OA ~mOy

OA ~mOy AmH«${‘Vmo.

nm¶ar 3 : {MÌ OABC ho 900 H$moZmV {’$a{dë¶mg

{MÌ OA'B'C' Am¶VmMo ñWmZ AmH«${‘Vmo.

nm¶ar 4 : nwT>rb 900 À¶m H$moZmV Am¶V {’$a{dë¶mg OABC Am¶V OA"B"C"

Am¶VmMo ñWmZ AmH«${‘Vmo.

nm¶ar 5 : OABC Am¶V ‘ yiñWmZmnmg yZ 2700 {’$a{dë¶mg Am¶V

OA"'B"'C"' Am¶VmMo ñWmZ AmH«${‘Vmo.

nm¶ar 6 : 3600 H$moZm‘Ü¶o nyU© EH$ ’o$ar {’$a{dë¶mg Am¶V ‘yi ñWmZr ¶oVmo.

8.3 n[a^«‘Ur¶ g‘{‘Vr

CXmhaU 1 : Vwåhr ho {’$ady eH$mb? XmoZ doim {’$a{dë¶mg AmH¥$Vr

n{hë¶m gmaIr {XgVo hmo¶ H$s Zmhr?

gyMZm : n[a^«‘U g‘{‘Vr Zoh‘r EH$m gmnojmÀ¶m AmYmao JUbr OmVo.

CXmhaU 2 : ho H$m¶ Amho Vwåhmbm ‘mhrV Amho H$m?

ho ndZMH«$ ({â{âo) Amho.

ho gwÜXm Ë¶mÀ¶m H|$Ðq~Xÿ^modVr {’$aVo.

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CXmhaU 3 : gdmªZm ’y$bo AmdS>VmV. {MÌm‘Yrb ’y$b gw§Xa Zmhr H$m? lmdUrbm ho 900

À¶m H$moZm‘Ü¶o ho {’$adm¶Mo Amho.

’y$bmbm 900, 1800, 2700 Am{U 3600 À¶m H$moZm‘Ü¶o Vr {’$a{dVo.

CXmhaU 4 : EH$m H$mJXmÀ¶m VwH$S>çmMm {ÌH$moU ¿¶m Am{U KS>çmimÀ¶m {déÜX {XeoZo Vmo

900 À¶m H$moZm‘Ü¶o {’$adm.

900

1800

2700

3600

AmnU Vmo 900, 1800, 2700 Am{U 3600 ‘Ü¶o KS>çmimÀ¶m H$mQ>çmÀ¶m {déÜX {XeoZo

{’$a{dë¶mg darb AmH¥$Vr à‘mUo {XgVmo. EH$m nyU© ’o$arZ§Va ‘yi ñWmZ àmá hmoVo.CXmhaU 5 :

1200 2400 3600

AmH¥$Vr 1200, 2400 Am{U 3600 À¶m H$moZm‘Ü¶o KS>çmimÀ¶m H$mQ>çmÀ¶m {XeoZo {’$a{dbr Amho. AmH¥$VrÀ¶m gd© ñWmZm§Mo {ZarjU H$am. AmH¥$Vr ‘yi AmH¥$Vr gmaIr {XgVo?

hr AmH¥$Vr g‘{‘V Amho. Ë¶m‘wio ¶mbm n[a^«‘U g‘{‘Vr åhUVmV. AmH¥$Vr H$mhr H$moZmV n[a^«‘U H$|ÐmÀ¶m ghmæ¶mZo {’$a{dë¶mg Ë¶m‘Ü¶o H$moUVmhr ~Xb AmT>iyZ ¶oV Zmhr. Aem AmH¥$Ë¶m§Zm n[a^«‘U g‘{‘Vr åhUVmV. AmH¥$Vr {’$aÊ¶mÀ¶m H$moZmbm n[a^«‘U H$moZ åhUVmV.

EH$ AmH¥$Vr {’$a{dVmZm n[a^«‘U H|$Ð, {’$aÊ¶mMm H$moZ VgoM {’$aÊ¶mMr {Xem Xe©{dUo

Amdí¶H$ AgVo.

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CXmhaU 5 ‘Ü¶o O q~Xÿ ^«‘U H|$Ð Amho. Am{U n[a^«‘U H$moZ 1200 Amho.

EH$ AmH¥$Vr n[a «‘U g‘{‘Vr¶ Amho åhUÊ¶mg Vr {d{eð> n[a «‘U q~Xÿ‘Ü¶o VgoM {d{eð>

‘mnmÀ¶m H$moZmV {’$aV ‘yi AmH¥$Vrer {‘iVr OyiVr Agbr nm{hOo.

Imbrb {MÌ nyU© EH$ ’o$ar {’$adm.

OQ

RS

P

OR

SP

Q

OS

PQ

R

OP

QR

S

OQ

RS

P

nyU© EH$ ’o$ar {’$a{dÊ¶mgmR>r AmH¥$Vr 3600 H$moZmV {’$a{dVmo. ¶oWrb 4 hr ñWmZm‘Ü¶o

AmH¥$Vr ‘yi AmH¥$VrgmaIr {XgVo.

¶mbm AmnU n[a^«‘U g‘{‘VrMm H«$‘ 4 Ago åhUVmo.

nyU© EH$ ’o$ar {’$a{dVmZm V¶ma hmoUmar AmH¥$Vr ‘yi AmH¥$VrgmaIr {XgUmè¶m {’$aÊ¶mÀ¶m

g§»¶obm n[a^«‘U g‘{‘Vr H«$‘ Ago åhUVmV.

CXmhaU 1 ‘Ü¶o, n[a^«‘U g‘{‘VrMm H«$‘ 2 Amho. AmnU AmH¥$Vr 900 ‘Ü¶o {’$a{dbr

Amho.

CXmhaU 2 ‘Ü¶o, n[a^«‘U g‘{‘VrMm H«$‘ 3 Amho. AmnU AmH¥$Vr 1200 ‘Ü¶o VrZ doim

{’$a{dë¶mZ§Va ‘yi AmH¥$Vr {‘iVo.

CXmhaU 3 ‘Ü¶o, n[a^«‘U g‘{‘VrMm H«$‘ 4 Amho. ¶oWo Mma doim {’$a{dë¶m Z§Va ‘yi

AmH¥$Vr {‘iVo.

CXmhaU 4 ‘Ü¶o, n[a^«‘U g‘{‘VrMm H«$‘ 4 Amho.

CXmhaU 5 ‘Ü¶o, n[a^«‘U g‘{‘VrMm H«$‘ 3 Amho.

AmVm AmnU Imbrb AmH¥$Ë¶m§Mo {ZarjU H$ê$¶m.

P

O

RQ

S O

S

QP

R O

R

PS

Q O

Q

SR

P O

P

RQ

S

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188

O q~XÿÀ¶m ghmæ¶mZo AmH¥$Vr nyU©nUo {’$a{dë¶mg Vr 4 doim ‘yi AmH¥$Vrer nyU©nUo

OwiVo. hr 900, 1800, 2700 Am{U 3600 H$moZm‘Ü¶o {’$a{dë¶mg ‘yi AmH¥$Vrer OwiVo.

Ë¶m‘wio n[a^«‘U g‘{‘VrMm H«$‘ 4 Amho.

PQRS g‘m§Va^wO Mm¡H$moZmMo {ZarjU H$am. hm àË¶oH$doir 1800 À¶m H$moZm‘Ü¶o

{’$a{dë¶mg AmnUmg Imbrb {MÌo {‘iVmV.

S R

QP

S

R Q

P S R

QP

• g‘m§Va ^yO Mm¡H$moZmMm n[a^«‘U g‘{‘Vr H«$‘ 2 Aho.

{dMma H$am : dVw©imMm n[a^«‘U g‘{‘VrMm H«$‘ {H$Vr?

ABCDE EH$ {Z¶{‘V n§M^wOmH¥$Vr Amho. Vr 720 À¶m H$moZmV {’$a{dë¶mg Imbrb AmH¥$Ë¶m AmT>iyZ ¶oVmV.

EH$ n§MH$moZmH¥$Vr ABCDE AmH$mamMm H$mJX ¿¶m. EH$ {Iim Ë¶mÀ¶m ‘Ü¶^mJr pñWa H$am Am{U Vmo {’$adm. EH$ ’o$ar nyU© hmoD$ n¶ªV Vmo ‘yi AmH¥$Vrer {H$Vr doim OwiVmo ¶mH$S>o bj Úm.

A B

CE

D

B C

DA

E

C D

EB

A

D E

AC

B

E A

BD

C

A B

CE

D

• {Z¶{‘V n§MH$moZmH¥$Vr‘Ü¶o n[a^«U g‘{‘Vr¶ H«$‘ 5 Amho.

∴ n[a^«‘U g‘{‘Vr H«$‘ =3600

n[a^«‘U H$moZ = 3600

A0

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H¥$Vr 10 : EH$m H$mS>© ~moS>©da EH$ H$mJX {MH$dm. Ë¶mda ~mOy§Mr bm§~r 6 cm AgUmam EH$

g‘ yO {ÌH$moU aMm. gd© ~mOyda ‘Ü¶q~Xÿ Xe©dm Am{U ‘Ü¶q~Xÿ d g‘moarb

{eamoq~Xÿ gm§Ym. gd© ~mOy gm§YUmè¶m q~Xÿbm "O' Ago Zmd Úm. AmVm

nmaXe©H$ H$mJX KoD$Z ‘yi {ÌH$moUmda R>oD$Z AmUIr EH$ Ë¶mM àH$maMm

g‘^wO {ÌH$moU aMm. Aem àH$mao aMboë¶m {ÌH$moUmÀ¶m ~mOyMo ‘Ü¶q~Xÿ

Xe©dm d g‘moarb {eamoq~Xÿ gm§Ym. gd© ~mOyZm gm§YUmè¶m q~Xÿbm O Ago

Zmd Úm.

AmVm XmoÝhr {ÌH$moUmMo ‘Ü¶q~Xÿ O q~Xÿda O1 q~Xÿ ¶oB©b Am{U gd© ~mOy nañna

{‘iVrb Ago n{hë¶m {ÌH$moUmda Xÿgam {ÌH$moU R>odm. O q~Xÿ Am{U O1 q~Xÿ‘YyZ OmB©b

Agm EH$ {Iim Q>moMm. da R>odbobm {ÌH$moU gwb^ nUo {’$ê$ Úm. nmaXe©H$ H$mJXmdarb

g‘^wO {ÌH$moU KS>çmimÀ¶m {XeoZo EH$ ’o$ar {’$adm Am{U hm ‘yi g‘^wO {ÌH$moUmer {H$Vr

doim OyiVmo ho nhm. hm VrZ doim nañna OwiVmo. Ë¶m‘wio ¶mMm n[a^«‘U H«$‘ 3 Amho Am{U

n[a^«‘U H$moZ 1200 Amho.

hr H¥$Vr g‘m§Va^yO Mm¡H$moZ Am{U g‘^yO Mm¡H$moZmgmo~V H$aÊ¶mMm à¶ËZ H$am.

EImXo doir AmnUmg n[a «‘U H«$‘ ‘mhrV Pmë¶mg Imbrb gwÌmMm Cn¶moJ H$ê$Z n[a «‘U H$moZ H$mTy> eH$Vmo.

darb H¥${Vdê$Z,

n[a^«‘U H$moZ = 3600

n[a^«‘U H«$‘ =

3360c

= 1200

n[a^«‘U g‘{‘Vr H«$‘ = 3600

n[a^«‘U H$moZ

120360

30

0

=

=

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190

gd© AmH¥$Ë¶m 3600 À¶m H$moZm‘Ü¶o {’$a{dë¶mg Ë¶m ‘yi AmH¥$VrÀ¶m àW‘ ñWmZr ¶oVmV. Aem AmH¥$Ë¶m§Mm n[a^«‘U g‘{‘Vr H«$‘ EH$ AgVmo.

ho H$m¶ Amho Vwåhmbm ‘m{hV Amho? hr EH$ dVw©imH$ma V~H$S>r Amho. V~H$S>r ghOnUo {’$aÊ¶mgmR>r ¶mÀ¶m H|$Ð^mJr EH$ {Iim ~g{dbm Amho.ho XoIrb EH$ ^«‘U g‘{‘VrMo CXmhaU Amho. Vwåhr ¶mMm H«$‘ emoYy eH$mb?hmo¶, nm§T>è¶m qH$dm H$mù¶m a§JmÀ¶m {ÌH$moUm§À¶m g§»¶oer g‘mZ Amho. Ë¶m‘wio ¶m V~H$S>rMm g‘{‘Vr H«$‘ 10 Amho.

AmH¥$VrMr EH$ ’o$ar nyU© Ho$ë¶mg EH$mnojm OmñV doim Vr ‘yi AmH¥$Vrer OwiV Agë¶mg Ë¶m AmH¥$VrMo doJdoJio ^«‘U g‘{‘Vr¶ H«$‘ AgVmV.

• EH$m AmH¥$VrMm n[a^«‘U H$moZ 1800 nojm H$‘r qH$dm g‘mZ Agë¶mg Vr AmH¥$Vr n[a^«‘U g‘{‘Vr¶ AmhoV.

• gd© AmH¥$Ë¶m 3600 H$moZm‘Ü¶o {’$a{dë¶mg Ë¶m ‘yi AmH¥$VrÀ¶m ñWmZr ¶oVmV.

• gd© AmH¥$Ë¶m§Mm ^«‘U g‘{‘Vr H$moZ H${‘V H$‘r EH$ AgVmo.

H¥$Vr 11 :

• EH$m OmS> H$mJXmMo XmoZ Mm¡H$moZ H$mnyZ ¿¶m.

• Ë¶m§Zm H$U© H$mTy>Z ‘Ü¶q~Xÿ Xe©dm.

• Mm¡H$moZmMo XmoÝhr H$U© N>oXUmè¶m q~Xÿg ‘Ü¶q~Xÿ åhUVmV.

• XmoÝhr Mm¡H$moZ EH$mda EH$ ¶oB©b Ago OmoSy>Z ¿¶m.

• XmoÝhr H$mJXmÀ¶m ‘Ü¶q~Xÿda EH$ {nZ Q>moMm.

• darb H$mJX {’$adm Vmo Imbrb H$mJXmer {H$Vr doim OwiVmo nhm.

n[a^«‘U H$moZ {H$Vr? Am{U Ë¶mMm n[a^«‘U H«$‘ {H$Vr? n[a^«‘UmMr {Xem ~Xbë¶mg Ë¶m‘Ü¶o H$m¶ ~Xb hmoB©b?

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CXmhaU 1 : AmH¥$Vr O q~XÿÀ¶m ghmæ¶mZo 900, 1200, 1800 H$moZm‘Ü¶o {’$a{dë¶mg

H$moUË¶m H$moZm‘Ü¶o AmH¥$Vr ‘yi AmH¥$Vrer OyiVo? ¶mMm n[a^«‘U H$moZ {bhm.

CH$b : AmnU AmH¥$Vr O q~XÿÀ¶m ghmæ¶mZo 900 {’$a{dë¶mg hr AmH¥$Vr {‘iVo.

1200 H$moZm‘Ü¶o {’$a{dë¶mg AmH¥$Vr {‘iVo (g‘{‘Vr‘Ü¶o)

1800 H$moZm‘Ü¶o {’$a{dë¶mg AmH¥$Vr {‘iVo.

AmH¥$Vr 1800 H$moZm‘Ü¶o {’$a{dë¶mg AmH¥$Vr n[a^«‘U g‘{‘VrV AgVo.

∴ n[a^«‘U g‘{‘Vr H«$‘ = 3600 = 2

1800

CXmhaU 2 : ABC g‘^yO {ÌH$moUmMm n[a^«‘U H«$‘ emoYm.

CH$b : g‘^yO {ÌH$moU n[a^«‘U H|$Ð O, À¶m ghmæ¶mZo O

A

CB

1200 H$moZm‘Ü¶o {’$a{dë¶mg, Vmo ‘yi AmH¥$VrMo ñWmZ {‘i{dVmo.

∴ n[a^«‘U g‘{‘Vr H«$‘ = 3600 = 3

1200

H$mhr AmH¥$Ë¶m‘Ü¶o à{Vq~~ Am{U n[a^«‘U ¶m XmoÝhrhr g‘{‘Vr nmhÿ eH$Vmo.

{MÌ A

{MÌ B

{MÌ B ho {MÌ A Mr à{Vq~~rV g‘{‘Vr Amho.

hr AmH¥$Vr n[a «‘UH|$ÐmÀ¶m ghmæ¶mZo 900 H$moZm‘Ü¶o {’$a{dë¶mg V¶ma hmoUmè¶m AmH¥$Ë¶m

Imbr Xe©{dë¶m AmhoV.

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Mm¡agmbm n[a^«‘U Am{U à{Vq~~ ¶m XmoÝhr g‘{‘Vr AgVmV.

AmVm Vwåhr Imbrb AmH¥$Vr‘Ü¶o {MÌ 1 Mr à{Vq~~ g‘{‘Vr AmoiIm.

{MÌ 1 (a) (b) (c)

g‘^yO {ÌH$moUm‘Ü¶o à{Vq~~ g‘{‘Vr Am{U n[a^«‘U g‘{‘Vr ¶m XmoÝhrhr nmhÿ eH$Vmo.

{Z¶{‘V n§MH$moZmH¥$VrMo {ZarjU H$am.

{MÌ A {MÌ B

{MÌ B {MÌ A Mo à{Vq~~ g‘{‘Vr Amho. AmVm AmH¥$Vr

n[a «‘U H|o$ÐmÀ¶m ghmæ¶mZo 720 À¶m H$moZm‘Ü¶o {’$adm. AmnUmg

Imbr Xe©{dboë¶m AmH¥$Ë¶m {‘iVmV. 4 n[a^«UUmZ§Va nmMdr

AmH¥$Vr ‘yi n§MH$moZmH¥$Vrer {‘iVo.

{MÌ (1) {MÌ (2) {MÌ (3) {MÌ (4) {MÌ (5)

{Z¶‘rV n§MH$moZmH¥$Vr‘Ü¶o à{Vq~~ g‘{‘Vr Am{U n[a «‘U g‘{‘Vr XmoÝhrhr nmhÿ eH$Vmo.

à¶ËZ H$am : à{Vq~~rV g‘{‘Vr Am{U n[a^«‘U g‘{‘Vr XmoÝhr AgUmar B§J«Or ^mfoVrb Ajao

AmoiIm.

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193

H, N, O, S, X Am{U Z Ajao H$mnyZ KoD$Z {ZarjU H$am.

Imbrb {ddaUmË‘H$ V³Ë¶mMo {ZarjU H$am.

AmH¥$Vr ~hþ^wOmH¥$VrMo Zmd

~mOy§Mr g§»¶m

g‘{‘Vr¶ Ajm§Mr g§»¶m

n[a^«‘U H$moZ

n[a^«‘U g‘{‘VrMm

H«$‘

g‘^wO {ÌH$moU

3 3 1200 3

g‘m§Va^wO Mm¡H$moZ

4 4 900 4

{Z¶‘rV n§MH$moZ

5 5 720 5

{Z¶‘rV fQ>H$moZ

6 6 600 6

V³Ë¶mdéZ Vwåhmbm H$m¶ g‘OyZ Ambo?{Z¶‘rV ~hþ^wOmH¥$Vr‘Ü¶o ~mOy§Mr g§»¶m, g‘{‘Vr Ajm§Mr g§»¶m, ^«‘U g‘{‘VrMm H«$‘

EH$M AgVmo ho g‘OyZ ¶oVo.

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194

à¶ËZ H$am : ‘moR>çm dVw©imH$ma nmiÊ¶mMo {MÌ nhm. ¶mMm n[a^«‘U g‘{‘Vr H«$‘ emoYÊ¶mMm à¶ËZ H$am. Aaoao! ‘r {H$Vr ‘moObo? ‘r {dgabmo..... Am{U..... ‘bm ‘moOVmM Ambo Zmhr.

H¥$Vr 12 : Vwåhr EH$m ImobrV g‘{‘Vr‘Ü¶o AgUmar EH$ dñVy R>odm. Vr dñVy H$moUVr ho AmoiIÊ¶mMo H$m‘ Vw‘À¶m {‘ÌmMo. Ë¶mbm AmoiIÊ¶mg ‘XV H$aÊ¶mgmR>r Vwåhr 10 g§Ho$V EH$m Z§Va EH$ gm§Jy eH$Vm. Ë¶mZo g‘{‘V {MÌ emoYë¶mZ§Va Ë¶mZo R>odbobr g‘{‘Vr Agbobr dñVw Vwåhr AmoiIm.

bjmV R>odÊ¶mMo A§e :

• n[a^«‘U KS>çmimÀ¶m {XeoZo AWdm {déÜX {XeoZo Agy eH$Vo.

• gd© AmH¥$Ë¶m§Mm n[a^«‘U g‘{‘Vr H«$‘ H${‘VH$‘r EH$ AgVmo.

• AmH¥$Vr Á¶m q~XÿÀ¶m ghmæ¶mZo {’$aVo Ë¶mbm n[a^«‘U q~Xÿ åhUVmV.

• AmH¥$Vr Á¶m H$moZmV {’$aVo Ë¶mbm n[a^«‘U H$moZ åhUVmV.

• nyU© EH$ ’o$ar {’$aÊ¶m‘Ü¶o V¶ma hmoUmè¶m AmH¥$Vr ‘yi AmH¥$Vr gmaIr {XgÊ¶mÀ¶m g§»¶obm n[a^«‘U g‘{‘VrMm H«$‘ åhUVmV.

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ñdmÜ¶m¶ 8.2

I 1) S Am{U I Ajam‘Ü¶o g‘{‘Vr¶ n[a^«‘U q~Xÿ AmoiIm Am{U g‘{‘Vr H«$‘ emoYm.

2) g‘{‘Vr‘Ü¶o AgUmar na§Vw n[a^«‘U g‘{‘Vr Zgbobr B§J«Or‘Yrb VrZ Ajao {bhm.

3) g‘{‘Vr‘Ü¶o AgUmar Am{U n[a^«‘U g‘{‘Vr AgUmar B§J«Or‘Yrb VrZ Ajao {bhm.

II 1) Imbrb n§»¶mÀ¶m nmË¶m§Mo n[a^«‘U g‘{‘Vr¶ H«$‘ {bhm. (i) 3 nmVr (ii) 4 nmVr

2) n[a^«‘U g‘{‘Vr AgUmè¶m H$mhr g§»¶m {bhm.

3) AmH¥$Vr‘Ü¶o {Xboë¶m Vmè¶mMo nm§T>ao Am{U H$mio ^mJ g‘^yO

{ÌH$moUm‘Ü¶o AmhoV. Va Vmè¶mÀ¶m n[a^«‘U g‘{‘VrMm H«$‘ {bhm.

III 1) 2cm bm§~r AgUmam g‘^wO {ÌH$moU ∆ ABC aMm. B q~Xÿ n[a^«‘U H|$Ð R>oD$Z

{ÌH$moU 900 Am{U 1800 H$moZm‘Ü¶o {’$a{dÊ¶mg V¶ma hmoUmao {MÌ H$mT>m.

2) g‘{Û^wO {ÌH$moU OAB ‘Ü¶o OA = AB. O hm EH$ pñWa q~Xÿ Amho. ¶mg 900,

1800 Am{U 2700 H$moUm‘Ü¶o {’$a{dë¶mg V¶ma hmoUmao {MÌ H$mT>m.

IV 1) Imbrb n¡H$s H$moUVr AmH¥$Vr à{Vq~~ g‘{‘Vr Am{U H$moUVr n[a^«‘U g‘{‘Vr‘Ü¶o Amho? gm§Jm.

E E E EE E(a) (b) (c)

(a) (b) (c)

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2) n[a^«‘U g‘{‘VrMm H«$‘ emoYm.

(i) Am¶V (ii) g‘m§Va^wO Mm¡H$moZ

V Imbrb Vº$m nyU© H$am.

B§J«Or Ajao

à{Vq~~rV g‘{‘Vr hmo¶/Zmhr

g‘{‘Vr Ajm§Mr g§»¶m

n[a^«‘U g‘{‘Vr hmo¶/Zmhr

n[a^«‘U g‘{‘VrMm H«$‘

I

B

O

H

E

G

Z

S

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CÎmaoàH$aU - 1 nyUmªH$

ñdmÜ¶m¶ 1.1 I. 1) 13, 2) -5, 3) 1 4) - 9 ñdmÜ¶m¶ 1.21. x -3 6 11 -5

7 -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

ñdmÜ¶m¶ 1.3 I. 1) -15, 2) -24, 3) -238, 4) 36, 5) -126, 6) 24, 7) -60

II. 1) 4 7 28#+ + = +

2) 3 5 15#- + = -

3) 9 7 63#+ - = -

4) 6 8 48#- - = +

III. 1) b) A cm - 2 × 6 cm 2) ` 62000 - 8 × 7500

ñdmÜ¶m¶ 1.4 I. 1) + 4, 2) -17, 3) -11, 4) 4 , 5) 0 IV. 40

II. 1) 1, 2) 1, 3) -2, 4) -12, 5) -30 III. nyUmªH$ ^mJbãY+ 24 ÷ + 12 + 2

+ 24 ÷ -12 - 2

- 24 ÷ +12 - 2

- 24 ÷ -12 + 2

ñdmÜ¶m¶ 1.5 I. 1) -10, 2) -9, 3) -18 4) 0,

5) -2, 6) -19

ñdmÜ¶m¶ 1.6 I. 1) 125, 2) 5am 16oC, 10am 26oC, 12noon 28oC, 3pm 23oC

3) ` 11,750, 4) ` 126, 5) ` 46930, 6) 177

××

××

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àH$aU - 2 AnyUmªH$

ñdmÜ¶m¶ 2.1: I. g‘ AnyUmªH$ : ,127

203 {df‘ AnyUmªH$ : ,16

1845

{‘l AnyUmªH$ : ,2 61 6 9

2

II 1) ,126189 , 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 3) 51 4) 1 30

23

ñdmÜ¶m¶ 2.2: I a) 41 5# , b) 2

1 3# , c) 83 2# , d) 3

2 4#

II A. 1) 74 , 2) 5

12 , c) 910 , 3) 4

21 , 4) 29 , 5) 3

20 , 6) 9, 7) 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) i)15, ii) 24, iii) 21

ñdmÜ¶m¶ 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) 461 , 4) 1 7

6 , 5) 11 21 , 6) 7 8

7 , 7) 17 97 , 8) 20

III 1) 685 2) 15

72 3) 8115 4) 2

175

IV 1) 70 km 2) ` 414, 3) 10 21 Vmg, 4) `465

1 , 5) 8 43 km

× × × ×

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ñdmÜ¶m¶ 2.4: I 1. 1) 18, 2) 352 , 3) 90, 4) 3

140 , 5) 6 73 , 6) 3 23

1

2. 1) 25 , 2) 7

9 , 3) 12, 4) 136 , 5) 9

1 , 6) 43 3. 1) 15

2 2) 247 3) 91

6 4) 209

5) 2811 6) 4

1 4. 1) 815 2) 12

35 3) 185 4) 15

8 5) 2 41 6) 2 42

31

II 1) ` 1 43 2) 85 3) ` 9 2

1 4) 27

ñdmÜ¶m¶ 2.5 : I 1) 43 2) 310

2 3)47

801 4) 2 20

13

II 1) m6201 2) kg4 4

1 3) 5m

àH$aU - 3 n[a‘o¶ g§»¶m

ñdmÜ¶m¶ 3.1: I 1) ~amo~a 2) MyH$ 3) ~amo~a 4) MyH$ 5) ~amo~a 6) ~amo~a 7) MyH$

II 1) 57 2) 5

4 3) 83- 4) 7

4 III 1) , , ,184276368

4510- - - - 2) , , ,20

6309

4012

5015

3) , , ,108

1512

2016

2520

- - - - 4) , , ,192160

288240

384320

480400 V , , ,6

7710

97

611

--

--

, ,54

35152-

-- VI 1) 16 2) -20 3) 7 4) 3 5) -28 6) 30

ñdmÜ¶m¶ 3.2:1)1) 1011 2) 9

8 3) 307 4) 42

107 2) 1) 83- 2) -1 3) 3

1- 4) 81-

3. 8 kg125 4. 1 kg20

11 5. m10582 6. 1 l15

14

ñdmÜ¶m¶ 3.3: 1) 1) 83 2) 48

5- 3) 101- 4) 9

1 5) 61 6) 1 2) 1) 27

7 2) 43 3) 5

6

4) 78 3. 1) 5

3 2) 49- 3) 10. 4) 2

7- 5) 51 6) 0 4) 1) 9

10 2) 56- 3) 13

18-

4) 57- 5. 9 6. 121 m 7. 8 5

4 sq.m 8 104

ñdmÜ¶m¶ 3.4:1)1) 0.8 2) 0.375 3) 0.3125 4) 0.2142857142...

2) AZ§V g§»¶m - ,8561 gm§V g§»¶m ,12

192-

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ñdmÜ¶m¶ 3.5:1) 1) 34.4 2) 7.5 3) 36.945 4) 3.328 5) 0.04368 6) 1.682) 1) 0.07 2) 0.012 3) 0.497 4) 27 5) 2.2 6) 430 3. ` 717.86 4. 120.06 km 5. 10.2l 6) ` 20.6ñdmÜ¶m¶ 3.6:

1. 1) 2510 g 2) 0.0725 g

2. 1) 0.625 kg 2) 0.0108 kg

3. 1) 1.45 km 2) 0.173 km 3) 0.2135 km

4. 1) 2700 m 2) 75.25 m 3) 1580 m

5. 1) 1250 cm 2) 470000 cm

6. 11.59 km

àH$aU - 4 ~¡{OH$ amerñdmÜ¶m¶ 4.1. 1. x 2) a 3) xy 4) x2y 5) a2b 6) m2n2 7) xyz 8) m2np

2. 1) 2 2) -5 3) 1 4) -8 5) 9 6) 43 7) 0.5 8) 0.008

ñdmÜ¶m¶ 4.2: I Ajar g§»¶m :1) 8 + 5 - 3, 2) 3x - 8, 5) (20 × 7) - 5 × 10) - 45

~¡{OH$ nX :3) (7 × 6) -4m 4) 3p + 4q 6) 2y + 6 - 4z

II 1) 3x, + 4y 2) 2pq, - 8qr 3) 3x2, -3x, +z 4) ab, + bc, -ca

5) 9m, + 6n 6) 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) q5 5) 4x + 3y 6) 10 - 5y

7) pq + 3x 8) 5m - 3l 9) 10y + 15 10) zyx

IV 1) 3 2) -4 c) 43 4) 10.5

5) 168 6) q8-

V 1) m 2) -xy 3) pqr 4) c 5) mn 6) x2y

VI 1) 27 2) 27z 3) xy 4) xyz

ñdmÜ¶m¶ 4.3: I EH$nXr: 2xy, xyz, 3xp ÷ q {ÛnXr: 2x + y, 3x2 + 5x, ab - bc

{ÌnXr: 5 + 6a + 4b, 2y z x37- + , a2 - 3ab + c, x2 + 4 - 3x

II x3 + x - 2 + 1 m3 + 2m2 + 3m - 4, x + 2yz

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III gOmVr¶ nXo (L) {dOmVr¶ nXo (U)1. 3x, 5x, 8x

3. -8p2, 6p2, 10p2

4. 2ab, 6ba, 8ab

2. x3, - 3x2, 8x

5.m 3a2b, -2ab2, 7a2b2

6. -a3, 2a2, -8a

IV 1) y3, -7y3, 23y3, -y3 2) 7ab, -8ba, -3ab, 4ab 3) 7p, 2p, 3p

V 1) 3 2) 32- 3) 0.3 4) 24 5) -18 6) 11

9- 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 AmYmam§H$ KmVm§H$x 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

ñdmÜ¶m¶ 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

ñdmÜ¶m¶ 4.5: I 1) 7x 2) 8a2b 3) 16ab 4) -22x2y II 1) 5x + 2y 2) 4a + 2b

3) -4m2n + 5mn2 4) -11pqr + 11q + 4 III 1) 4x + 4y 2) -6c + 4d

3) x-6y - 14 4) 6m2n2 + 2xy - 10

ñdmÜ¶m¶ 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

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àH$aU - 5 H$moZm§Mr OmoS>r

ñdmÜ¶m¶ 5.1: 1. Imbrb AmH¥$Ë¶m nmhm Am{U Vº$m nyU© H$am.

AmH¥$Vr H$moZ ~mOy gm‘mB©H$ {eamoq~Xÿ

gm‘mB©H$ ~mOy

gb¾ H$moZAmhoV/Zmhr

i ,AOB BOC OA, OB, OC,

O OB AmhoV ,AOB BOC

ii , ,PQR RQS SQJ QP,QR,QS,QJ

Q QRQS

Zmhr PQR qH$dm RQS AmhoV

RQS qH$dm SQJiii ,LMN KLM LK, LM,

MN- LM Zmhr

iv ,DEG FEG ED, EF, EG

E DE Zmhr

v ,EOG GFH OE, OG, FG, GH

- - Zmhr

vi ,OAB DOC AB, OA,OC, OD

O OC AmhoV

&COD AOC

vii ,HOG GOJ OH, OG, OJ

O OG AmhoV

2. {Xboë¶m AmH¥$Vr‘Ü¶o Imbrb OmoS>çm gb¾ H$m oZ AmhoV. Ë¶mVrb gm‘mB©H$ {eamoq~Xÿ Am{U gm‘mB©H$ ~mOy.

H$moZ {eamoq~XÿMo Zmd

~mOy§Mm Zmd gm‘mB©H$ {eamoq~Xÿ

gm‘mB©H$ ~mOy

gb¾ H$moZm§Mr OmoS>r Amho/Zmhr

,AOBBOD

0 , ,OA OB OD Amho Amho

OBAmho

,AODBOC

0 , , ,OA OD OB OC Amho Zmhr Zmhr

,AOCBOC

0 , ,OA OB OC Amho Amho

OCAmho

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,BOCAOB

0 , ,OA OB OC Amho Amho

OBAmho

,AODAOB

0 , ,OA OB OD Amho Amho

OAAmho

3. gb¾ H$moZ : i) ªÀÄvÀÄÛDOC COE Am{U ªÀÄvÀÄÛDOC COE ii) ªÀÄvÀÄÛPOR POQ Am{U ªÀÄvÀÄÛPOR POQ

iv) ªÀÄvÀÄÛJOK KOL Am{U ªÀÄvÀÄÛJOK KOL vii) ªÀÄvÀÄÛQOM MON Am{U ªÀÄvÀÄÛQOM MON , O, MWñdmÜ¶m¶ 5.2: I. i) 500, ii) 700, iii) 350 iv) 520 2. a) 680, b) 60 c) 69.50 d) 49.50 3. 22.50 Am{U 67.50

ñdmÜ¶m¶ 5.3: 1.a) AOD BOD AOB+ = b) AOE EOB AOB+ = c) AOC COB AOB+ = II. (i), (ii), (iii), (iv) nyaH$ H$moZIII. 1) 850 2) 700 3) 59.50 4) 54.50 IV. 650 V. 360 Am{U 1440 VI.i) 550, ii) 1450 iii) 900 iv) 900

ñdmÜ¶m¶ 5.4: 1) nyaH$ H$moZ Zmhr 2) (ii), (iv), (v) Am{U (vi) H$moZm§Mr OmoS>r

3) a) {demb H$moZ b) bKwH$moZ c) H$mQ>H$moZ

ñdmÜ¶m¶ 5.5:1)

,,

AODBODBOC

12060120

0

0

0

===

2) ,,

AOEBOCEOD

405090

0

0

0

===

3) BOCBOD

15030

0

0

==

ñdmÜ¶m¶ 5.6: I a) {demb H$moZ b) bKwH$moZ c) H$mQ>H$moZII i) 1050 ii) 400 iii) 32.50 iv) 350

III

90454545

AOC BODAOD COBAOR SOBDOR COSPOC DOQAOP BOQ

90

45

0

0

0

0

0

0

`

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

IV x = 450, z = 67.5, y = 1350

V 1) &AOD AOC

2) a) &AOE BOE b) &COA AOD

3) &AOC BOD 4) &AOC COE

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VI) 1) 900 2)1800 3) ì¶wËH«$‘ H$moZ 4) H$moZm§Mr OmoS>r 5) {dembH$moZ

àH$aU - 6 aofr¶ OmoS>rñdmÜ¶m¶ 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

== == == == == =

{eamo{déÜX H$moZ

gb¾ H$moZ

{eamo{déÜX H$moZgb¾ H$moZ{eamo{déÜX H$moZ

b) {déÜX H$moZ

gb¾ H$moZ

{déÜX H$moZ

GIB AIJAIJ CJKDJKDJK FKH

6060

180 60 120120

0

0

0 0 0

0

= == == - == =

c) gb¾ H$moZ

{déÜX H$moZEHB GHFGHF ACD

5050

0

0

= == =

3. ì¶wËH«$‘ H$moZ

gb¾ H$moZ

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. AmH¥$Vr a b c d

(i) 800 500 1000 1300

(ii) 700 800 700 -

(iii) 500 1300 1300 -

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(iv) 400 2600 600 -

(v) 1300 700 1300 1100

àH$aU - 7 {ÌH$moUñdmÜ¶m¶ 7.1: 1. 1800

ñdmÜ¶m¶ 7.2: 1. a) x = 500 2) x = 800 3) x = 200

2. 1) C = 500 2) B = 450 3) C = 500 4) C = 340

3. 1) 500 + 600 + 700 = 1800 Amho 2) 600 + 700 + 800 = 2100 Zmhr

3) 650 + 750 + 550 = 19500 Zmhr 4) 560 + 640 + 600 = 1800 Amho

5) 570 + 640 + 790 = 2000 Zmhr

4. C = 700 5. C = 700 6. x = 900 7. x = 600 8. C = 950

ñdmÜ¶m¶ 7.3:1. 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, ACB 600= , 3. ∠QRP = 500, ∠RPQ = 800

4. ∠KLN = 1150, ∠NLM = 650, ∠LMN = 400 5. ∠ACD = 1050

ñdmÜ¶m¶ 7.5: 1. AC = 10, 2. AC = 13 cms, 3. AB = 9 m 4. b), c)

àH$aU - 8 g‘{‘Vr

ñdmÜ¶m¶ 8.1: I. i. iii. iv.

II. a) 1 b) 2 c) 2III a) b) c) d) e) f) g)

Zmhr Amho Amho Amho Amho Zmhr Amhoh) i) j) k) l)

Zmhr Zmhr Amho Amho Amho

IV a)1 b) 3 c) 4 d) 4 e) AZ§V f) 4 g) 4 h) 5 i) 6

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V C

D

E

R

1 - 2i) C -

ii) D -

iii) E -

iv) R -

ñdmÜ¶m¶ 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)

B§J«Or ‘yimjao

à{Vq~~rV g‘{‘Vr

Amho/Zmhr

g‘{‘VrÀ¶m aofm§Mr g§»¶m

n[a^«‘U g‘{‘Vr

Amho/Zmhr

n[a^«‘U g‘{‘VrMm H«$‘

I Amho 1 Amho 2

B Amho 1 Amho 1

O Amho AZoH$ Amho AZoH$

H Amho 2 Amho 2

E Amho 1 Amho 1

G Zmhr - Amho 1

Z Amho - Amho 2

S Amho - Amho 2

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