H$Zm©Q>H$ gaH$ma J{UV 8 - Kar textbooks/class8/8th...KZ Am{U KZ_yi 3 VŠVm - 2 Amåhr g_ AmhmoV. åhUyZ Am_Mo KZ g_ AmhoV. Amåhr {df_ AmhmoV. åhUyZ Am_Mo KZ {df_ AmhoV. g »`m KZ

$Page 1: H$Zm©Q>H$ gaH$ma J{UV 8 - Kar textbooks/class8/8th...KZ Am{U KZ_yi 3 VŠVm - 2 Amåhr g_ AmhmoV. åhUyZ Am_Mo KZ g_ AmhoV. Amåhr {df_ AmhmoV. åhUyZ Am_Mo KZ {df_ AmhoV. g »`m KZ$
H$Zm©Q>H$ gaH$ma

J{UV8

B`Îmm AmR>drEIGHT STANDARD

_amR>r _mÜ`_MARATHI MEDIUM

^mJ - 2PART – 2

amï´>r` e¡j{UH g§emoYZ Am{U à{ejU n[afX²

H$Zm©Q>H$ nmR>²çnwñVH$ g§K (a.)100 \y$Q> [a¨J amoS>, ~Ze§H$ar 3 am Q>ßnm

~|Jiyê$ - 85

^mJ - 2

A. Z§. àH$aU n¥ð> H«$_m§H$

1. KZ Am{U KZ_yi 1-11

2. KmVm§H$ Am{U KmV 12-24

3. _m{hVr hmVmiUo 25-48

4. g_à_mU Am{U ì`ñVà_mU 49-66

5. AmboIm§Mm n[aM` 67-87

6 Ad`drH$aU 88-105

7. KZmH¥$VtMo Ñí`rH$aU 106-122

8. à_mUm§Mr VwbZm 123-145

9. joÌ_mnZ 146-176

CÎmao 177-190

1.1.n[aM`

^maVr` _hmZ J{UVk Eg². am_mZwOZ `m§À`m{df`tMr EH$ KQ>Zm Amho. EH$Xm àmo\o$ga Or. EM². hmS>u Ë`m§Zm ôQ>m`bm Ambo hmoVo. Ë`m§À`m Q>°ŠgrMm Z§~a 1729 hmoVm. am_mZwOZ ~amo~a ~mobVmZm hmS>uZr Ë`m Z§~aMo dU©Z "{ZñVoO g§»`m' Ago Ho$bo. `mda am_mZwOZm§Zr MQ²>H$Z 1729 hr IamoIa _Zmoa§OH$ g§»`m Amho Ago Xe©{dbo. hr EH$ bhmZmV bhmZ g§»`m AgyZ Vr XmoZ doJdoJù`m àH$mao XmoZ KZ g§»`m§À`m ~oaOoZo XmI{dVm òVo. 1729 = 1728 + 1 = 123 + 13

1729 = 1000 + 729 = 103 + 93

1729 `m g§»òbm Oar hmS>u- am_mZwOZ g§»`m `m ZmdmZo AmoiIV Agbo Var 1729 `m g§»òMo bjU am_mZwOZ `m§À`m nydu 300 df©o _mhrV hmoVo.

am_mZwOZ `m§Zm ho H$go g_Obo? Vo g§»`m§da ào_ H$aV hmoVo, Amnë`m Am`wî`â Vo g§»`m§da à`moJ H$aV hmoVo. Ë`m§Zr XmoZ dJ©g§»`m§Mr ~oarO Am{U XmoZ KZ g§»`m§À`m ~oaOoÀ`m ñdê$nmV gwÜXm Á`m g§»`m ì`ŠV H$aVm oVmV Aem Ag§»` g§»`m emoYë`m hmoË`m.

`m {R>H$mUr KZg§»ò§Mo AZoH$ _Zmoa§OH$ Z_wZo AmhoV. AmnU KZ Am{U KZ_yim{df`r Aä`mg H$ê Am{U Ë`m§À`m {df`r g§~§{YV _Zmoa§OH$ KQ>Zm OmUyZ KoD$.

Á`m {Ì{_Vr` AmH¥$Ë`m AmhoV Ë`m§Zm KZmH¥$Vr åhUVmV.

1.2.KZ (cube)

"KZ' hm eãX ^y{_Vr_Üò gwÜXm dmnabm OmVmo ho AmnUmg _mhrV Amho. KZ hr EH$ ârd AmH¥$Vr Amho {OÀ`m gd© ~mOy g_mZ AgVmV. 1 cm ~mOyÀ`m {H$Vr KZmZr 2 cm ~mOwMm KZ ~ZVmo?

hmS>u-am_mZwOZ g§»`m

1729 hr bhmZmV bhmZ hmS>u-am_mZwOZ g§»`m Amho. `m àH$maÀ`m AZ§V g§»`m AmhoV. Ë`mVrb H$mhr 4104 (2, 16 ; 9, 15), 13832 (18, 20; 2, 24) H§$gmV {Xboë`m g§»`m§À`m OmoS>rer ho VnmgyZ nhm.

àH$aU

1 KZ Am{U KZ_yi

2 J{UV

1 cm ~mOyMr bm§~r AgUmè`m {H$Vr KZmZr 3 cm ~mOy§Mm KZ ~ZVmo? 1, 8, 27, .....

`m g§»`m {dMmamV ¿`m. `m g§»`m§Zm nyU© KZ qH$dm KZg§»`m åhUVmV. Ë`m§Zm ho Zmd H$m XoÊ`mV

Ambo Amho? hr àËòH$ g§»`m EH$ g§»`m VrZ doim JwUyZ Ambobr Amho.

AmnU bjmV ¿`m H$s, 1 = 1 × 1 × 1 = 13, 8 = 2 × 2 × 2 = 23, 27 = 3 × 3 × 3 = 33. Á`mAWu 53= 5 × 5 × 5 = 125 åhUyZ, 125 hr KZg§»`m Amho. 9 hr

KZg§»`m Amho H$m? Zmhr. 9 = 3 × 3 Am{U òWo Z¡g{J©H$ g§»`m Zmhr {OMm VrZ doim JwUmH$ma Ho$bm

AgVm 9 {_iVmV. AmnU Ago gwÜXm nmhÿ eH$Vmo H$s, 2 × 2 × 2 = 8 Am{U 3 × 3 × 3 = 27 mdê$Z

9 hr nyU© KZg§»`m Zmhr. Imbr 1 Vo 10 n`ªVÀ`m KZg§»`m {Xboë`m AmhoV.

VŠVm - 1

729, 1000, 1728 `m g§»`m gwÜXm nyU© KZg§»`m

AmhoV.

g§»`m KZ

ho nyU© H$am

1 13 = 1

2 23 = 8

3 33 = 27

4 43 = 64

5 53 = ––––

6 63 = ––––

7 73 = ––––

8 83 = ––––

9 93 = ––––

10 103 = ––––

1 Vo 1000 _Üò \$ŠV Xhm nyU© KZg§»`m AmhoV (Vnmgm). 1 Vo 100 _Üò {H$Vr nyU©

KZg§»`m AmhoV?

g_g§»`|Mo KZ nhm. Vo gd© g_ AmhoV H$m? {df_ g§»`m§À`m KZm{df`r Vwåhr H$m` gm§Jmb?

nwT>o 11 Vo 20 n`ªVÀ`m g§»`m§Mo KZ AmhoV.

KZ Am{U KZ_yi 3

VŠVm - 2

Amåhr g_ AmhmoV. åhUyZ

Am_Mo KZ g_ AmhoV.

Amåhr {df_ AmhmoV. åhUyZ

Am_Mo KZ {df_ AmhoV.

g§»`m KZ

11 1331

12 1728

13 2197

14 2744

15 3375

16 4096

17 4913

18 5832

19 6859

20 8000

EH$H$ñWmZr 1 AgUmè`m H$mhr g§»`m {dMmamV ¿`m. Ë`m§Mo àËòH$mMo KZ H$am. Ë`m~X²Xb Vwåhr H$m` gm§Jmb? `m KZm§À`m EH$H$ñWmZr H$moUVm A§H$ òVmo. Ë`mM à_mUo EH$H$ñWmZr 2,3,4.... BË`mXr A§H$ AgUmè`m g§»`m§Mo KZ H$mT>m d `m KZm§À`m EH$H$ñWmZr (gwQ>oñWmZr) H$moUVm A§H$ òVmo Vo nhm.

.

Imbrb g§»ò§À`m KZg§»òV EH$H$ñWmZr H$moUVm A§H$ òVmo Vo gm§Jm.

(i) 3331 (ii) 8888 (iii) 149 (iv) 1005

(v) 1024 (vi) 77 (vii) 5022 (viii) 53

à`ËZ H$am :

1.2.1 H$mhr _Ooera Z_wZo (Patterns)

1. H«$_dma {df_ g§»`m§Mr ~oarO H$éZ

Imbrb {df_ g§»`m§Mr ~oarO Ho$boë`m KQ>Zm nhm.

1 = 1 = 13

3 + 5 = 8 = 23

7 + 9 + 11 = 27 = 33

13 + 15 + 17 + 19 = 64 = 43

21 + 23 + 25 + 27 + 29 = 125 = 53

ho _Ooera Amho H$s Zmhr. 103 {_i{dÊ`mgmR>r {H$Vr H«$_dma {df_ g§»`m§Mr ~oarO òVo?

4 J{UV

.

_mJrb nmZmdarb Z_wÝ`mà_mUo Imbrb g§»`m {df_ g§»`m§À`m ~oaOoZo XmIdm. (i) 63 (ii) 83 (iii) 73

Imbrb Z_wZo {dMmamV ¿`m.

23 – 13 = 1 + 2 × 1 × 3

33 – 23 = 1 + 3 × 2 × 3

43 – 33 = 1 + 4 × 3 × 3

darb Z_wÝ`m§Mm Cn`moJ H$ê$Z Imbrb qH$_Vr H$mT>m.

(i) 73 – 63 (ii) 123 – 113 (iii) 203 – 193 (iv) 513 – 503

à`ËZ H$am :

2. KZg§»`m Am{U Ë`m§Mo _yi Ad`d.

Imbr g§»`m§Mo d KZg§»`m§Mo _yi Ad`d {Xbo AmhoV Ë`m§Mm {dMma H$am.

g§»`mMo _yi Ad`drH$aU

KZm§Mo _yi Ad`drH$aU

àËòH$ _yi Ad`d {VÀ`m KZmV VrZ

doim òVmo.

4 = 2 × 2 43 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 23 × 23

6 = 2 × 3 63 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 × 33

15 = 3 × 5 153 = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 33 × 53

12 = 2 × 2 × 3 123 = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 23 × 23 × 33

KZ g§»`m§À`m Ad`drH$aUmV àËòH$ _yi Ad`d VrZ doim {XgyZ òVmV.

H$moUË`mhr g§»`m§À`m _yi Ad`drH$aUmV Ooìhm àËòH$ Ad`d 3 doim {XgyZ òVmV, Voìhm Vr g§»`m KZ g§»`m AgVo H$m? `mMm {dMma H$am.

216 nyU© KZ Amho H$m?

KZ Am{U KZ_yi 5

222333

2161085427931

_yi Ad`drH$aUmZo 216 = 2 × 2 × 2 × 3 × 3 × 3

àËòH$ Ad`d 3 doim {XgVmo.216 = 23 × 33

= (2 × 3)3 = 63 Omo nyU© KZ Amho.

729 g§»`m nyU© KZ Amho H$m? 729 = 3 × 3 × 3 × 3 × 3 × 3

hmo`. 729 nyU© KZg§»`m Amho.

AmVm, 500 gmR>r Vnmgm.

500 Mo _yi Ad`d 2×2×5 × 5 × 5

åhUyZ 500 hr nyU© KZg§»`m Zmhr.

Vwåhmbm _mhrV Amho H$s,am × bm = (a × b)m

Ad`d 3 À`m JQ>mV AmhoV.

`m {R>H$mUr JwUmH$mamV 5 ho VrZXm d 2 ho XmoZXm AmhoV.

CXmhaU 1 : 243 hr nyU© KZg§»`m Amho H$m?

CH$b : 243 = 3 × 3 × 3 × 3 × 3

darb Ad`dmV VrZÀ`m g_mZ Ad`dm§À`m JQ>mZ§Va 3 × 3 ho VrZ doim ZmhrV. åhUyZ 243 nyU© KZg§»`m Zmhr.

à`ËZ H$amImbrb n¡H$s H$moUË`m g§»`m nyU© KZg§»`m AmhoV? 1. 400 2. 3375 3. 8000 4. 15625

5. 9000 6. 6859 7. 2025 8. 10648

1.2.2. bhmZmV bhmZ JwUH$ Omo nyU© KZ Amho.

amOZo {MH$U_mVrMm EH$ KZm`V ~Z{dbm. KZm`VmMr bm§~r, é§Xr Am{U C§Mr AZwH«$_o 15 g|._r, 30 g|._r. Am{U 15 g|._r Amho. AZwZo {dMmabo nyU© KZ ~Z{dÊ`mg Aem {H$Vr KZm`Vm§Mr JaO Amho?

Vwåhr gm§Jy eH$mb H$m?

amO åhUbm, KZm`VmMo KZ\$i = 15 × 30 × 15

= 3 × 5 × 2 × 3 × 5 × 3 × 5

= 2 × 3 × 3 × 3 × 5 × 5 × 5

6 J{UV

m {R>H$mUr 2 hm _yi Ad`d EH$XmM òVmo. åhUyZ AmnUmg AmUIr 2 × 2 Mr JaO Amho. åhUOoM 4 nyU© KZ. åhUyZ AmnUmg nyU© KZ ~Z{dÊ`mgmR>r Aem 4 KZm`Vm§Mr JaO Amho.

CXmhaU 2 : 392 hr nyU© KZ g§»`m Amho H$m? Oa Zmhr Va bhmZmV bhmZ Aer g§»`m emoYm H$s {VZo 392 bm JwUë`mg nyU© KZg§»`m {_iob?

CH$b : 392 = 2 × 2 × 2 × 7 × 7

7 À`m _yi Ad`dm§À`m VrZÀ`m JQ>mV AmUIr EH$ 7 Zmhr. åhUyZ 392 hr nyU© KZg§»`m Zmhr, hr nyU© KZg§»`m ~ZÊ`mgmR>r Amåhmbm AmUIr EH$ 7 nm{hOo.

392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744, Or nyU© KZg§»`m Amho.

åhUyZ 392 bm 7 `m bhmZmV bhmZ g§»òZo Jw{Ubo AgVm Vr nyU© KZg§»`m ~ZVo.

CXmhaU 3 : 53240 hr nyU© KZg§»`m Amho H$m? Oa Zmhr, Va {Vbm H$moUË`m bhmZmV bhmZ g§»òZo ^mJbo AgVm ^mJmH$ma hm nyU© KZg§»`m hmoB©b?

CH$b : 53240 = 2 × 2 × 2 × 11 × 11 × 11× 5

_yi Ad`d 5 hm JQ>mV EH$XmM Amho. åhUyZ 53240 hr nyU© KZg§»`m Zmhr. Oa Amåhr {Xboë`m g§»òbm 5 Zo ^mJbo Va g§»òÀ`m _yi Ad`dm_Üò 5 hm Ad`d Zgob.

åhUyZ 53240 ÷ 5 = 2 × 2 × 2 × 11 × 11 × 11

åhUyZ 53240 hr nyU© KZg§»`m ~ZÊ`mgmR>r {Vbm ^mJmd`mMr bhmZm§V bhmZ g§»`m 5 hr hmo`.

`m pñWVrV nyU© KZg§»`m = 10648 hr Amho.

CXmhaU 4 : 1188 hr nyU© KZg§»`m Amho H$m? Oa Zmhr, Va 1188 bm H$moUË`m bhmZm§V bhmZ g§»òZo ^mJbo AgVm Vr nyU© KZg§»`m ~Zob?

CH$b : 1188 = 2 × 2 × 3 × 3 × 3 × 11

_yi Ad`d 2 Am{U 11 `m§Mo VrZMo JQ> {XgyZ òV ZmhrV. åhUyZ 1188 hr nyU© KZg§»`m Zmhr. 1188 À`m _yi Ad`dmV 2 hm Ad`d \$ŠV XmoZXm Am{U 11 hm Ad`d \$ŠV EH$Xm AmT>iVmo åhUyZ 1188 bm 2 × 2 × 11 = 44 Zo ^mJ {Xbm Va ^mJmH$mamÀ`m _yi Ad`dmV 2 Am{U11 ho Ad`d AgUma ZmhrV.

åhUyZ 1188 bm nyU© KZg§»`m ~ZÊ`mgmR>r bhmZmV bhmZ 44 Zo ^mJmdo Am{U ^mJmH$ma åhUyZ Ambobr nyU© KZg§»`m 1188 ÷ 44 = 27 (=33) nyU© KZg§»`m Amho.

KZ Am{U KZ_yi 7

CXmhaU 5 : 68600 hr nyU© KZg§»`m Amho H$m? Zgë`mg bhmZm§V bhmZ g§»`m emoYm, {OÀ`m JwUmH$mamZo 68600 nyU© KZg§»`m ~Zob.

CH$b : 68600 = 2 × 2 × 2 × 5 × 5 × 7 × 7 × 7

`m _yi Ad`d _m§S>UrV, Amnë`mbm AmT>iVo H$s, 5 hm VrZ doim òV Zmhr, nU XmoZ doim Amho. åhUyZ 68600 hr nyU© KZg§»`m Zmhr. {hbm nyU© KZg§»`m ~Z{dÊ`mgmR>r 5 Zo AmnUmg JwUmdo bmJob.

68600 × 5 = 2 × 2 × 2 × 5 × 5 × 5 × 7 × 7 × 7

= 343000 Or nyU© KZg§»`m Amho. bjnyd©H$ nhm H$s 343 hr nyU© KZg§»`m Amho. CXmhaU 5 déZ AmnUmg 343000 hr gwÜXm nyU© KZg§»`m {_iVo ho _mhrV hmoVo.

{dMma H$am, MMm© H$am Am{U {bhm

Imbrb n¡H$s H$moUË`m g§»`m nyU© KZg§»`m AmhoV ho Vnmgm.

(i) 2700 (ii) 16000 (iii) 64000 (iv) 900 (v) 125000

(vi) 36000 (vii) 21600 (viii) 10000 (ix) 27000000 (x) 1000

`m KZg§»`m_Üò H$moUVm Z_wZm Vwåhmbm AmT>iVmo?

ñdmÜ`m` 1.1

1. Imbrb n¡H$s H$moUË`m g§»`m nyU© KZg§»`m ZmhrV?

(i) 216 (ii) 128 (iii) 1000 (iv) 100 (v) 46656

2. Imbrb àËòH$ g§»`m nyU© KZg§»`m hmoÊ`mgmR>r H$moUË`m bhmZmV bhmZ g§»òZo JwUmdo bmJob?

(i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100

3. Imbrb àËòH$ g§»`m nyU© KZg§»`m hmoÊ`mgmR>r H$moUË`m bhmZmV bhmZ g§»òZo ^mJmdo bmJob?

(i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704

4. n[a{jVZo {MZr_mVrMm EH$ KZm`V ~Z{dbm Amho. Ë`mÀ`m ~mOy 5 g|._t., 2 g|._t., 5 g|._t., AmhoV. EH$ KZ {_i{dÊ`mH$[aVm Aem {H$Vr KZm§Mr Amdí`H$Vm Amho?

8 J{UV

1.3 KZ_yi (cube roots)

Oa EH$m KZmMo KZ\$i 125 cm3 Amho. Va Ë`mÀ`m EH$m ~mOwMr bm§~r {H$Vr? KZmÀ`m

~mOwMr bm§~r H$mT>Ê`mgmR>r 125 hr g§»`m H$moUË`m g§»òMm KZ Amho ho Amåhmbm _mhrV H$ê$Z KoUo

JaOoMo Amho.

dJ©_yi H$mT>Uo hr dJ© H$aÊ`mÀ`m {H«$òÀ`m {déÜX {H«$`m Amho. Ë`mMà_mUo KZ_yi H$mT>Uo

hr KZ H$aÊ`mÀ`m {H«$òÀ`m {déÜX {H«$`m Amho.

AmnUmg _mhrV Amho 23 = 8. åhUyZ AmnU gm§Jy eH$Vmo H$s 8 Mo KZ_yi 2 Amho. ho AmnU

83 = 2 Ago {b{hVmo. KZ_yi Xe©{dÊ`mgmR>r 3 ho {MÝh dmnVamV.

{dYmZ AZw_mZ {dYmZ AZw_mZ

13 = 1 1 13 = 63 = 216 216 63 =

23 = 88 2 23 33= =

73 = 343 343 73 =

33 = 2727 3 33 33= =

83 = 512 512 83 =

43 = 64 64 43 = 93 = 729 729 93 =

53 = 125 125 53 = 103 = 1000 1000 103 =

1.3.1 _yi Ad`drH$aU nÜXVrZo KZ_yi H$mT>Uo.

g§»`m 3375 {dMmamV ¿`m. Amåhr _yi Ad`d nÜXVrZo `m g§»òMo KZ_yi H$mTy>`m. 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 33 × 53 = (3 × 5)3

åhUyZ 3375 Mo KZ_yi = 33753 = 3 × 5 = 15

Ë`mMà_mUo, 740883 KZ_yi H$mT>Ê`mgmR>r, Amåhmb {_iVo,

74088 = 2 × 2 × 2 × 3 × 3 × 3 × 7 × 7 × 7

= 23 × 33 × 73 = (2 × 3 × 7)3

åhUyZ, 740883 = 2 × 3 × 7 = 42

KZ Am{U KZ_yi 9

CXmhaU 6 : 8000 Mo KZ_yi H$mT>m.

CH$b : 8000 Mo _yi Ad`drH$aU 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5

80003 = 2 × 2 × 5 = 20

CXmhaU 7 : 13824 Mo KZ_yi _yi Ad`drH$aU nÜXVrZo H$mT>m.

CH$b : 13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 23 × 23 × 23 × 33

åhUyZ, 138243 = 2 × 2 × 2 × 3 = 24


H$moUË`mhr nyUmªH$ m gmR>r m2 < m3 ho ~amo~a H$s MyH$ gm§Jm. H$m?

1.3.2 KZ g§»òMo KZ_yi

{Xbobr g§»`m KZg§»`m Amho ho Oa Vwåhmbm _mhrV Agob Va Imbrb nÜXVrMm Cn`moJ H$ê$ eH$Vmo.

nmàr 1 : 857375 hr g§»`m ¿`m Am{U COdrH$Sy>Z S>mdrH$S>o VrZ-VrZ A§H$m§Mo JQ> H$am.

857

↓Xþgam JQ>

375

↓n{hbm JQ>

{Xboë`m g§»òMo KZ_yi nmàr nmàrZo H$mTy>`m.

AmnUmg 375 Am{U 857 Ago XmoZ JQ> àËòH$s VrZ A§H$m§Mo {_imbo AmhoV.

nmàr 2 : n{hbm JQ> 375 AmnUmg Amdí`H$ KZ_yiMo gwQ>o EH$H$ ñWmZ XoB©b.

375 hr g§»`m 5 Zo eodQ> Pmbobr Amho. AmnUm§g _mhrV Amho H$s, EImÚm KZ g§»òÀ`m EH§$ñWmZr Ooìhm 5 hm A§H$ òVmo, Voìhm Ë`m g§»òÀ`m KZ_yimMm eodQ>Mm A§H$ 5 AgVmo.>

åhUyZ AmnUmg KZ_yimV EH$§ñWmZr 5 hm A§H$ {_iVmo.

10 J{UV

nmàr 3 : AmVm Xþgam JQ> ¿`m, 857

AmnUmg _mhrV Amho H$s, 93 = 729 Am{U 103 = 1000, VgoM 729 < 857 < 1000 `mVrb gdm©V bhmZ g§»`m 729 Amho. {VÀ`m EH§$ñWmZmMm A§H$ Amdí`H$ KZ_yimV

XeH$ñWmZr KoVbm nm{hOo. åhUyZ AmnUmg 857375 953 = {_iVmV.

CXmhaU 8 : 17576 Mo KZ_yi A§XmOmZo H$mT>m.

CH$b : {Xbobr g§»`m 17576 Amho.

nmàr 1 : 17576 Mo VrZ A§H$m§Mo JQ> COdrH$Sy>Z H$am.

17576 _Üò EH$ JQ> 576 VrZ A§H$mMm hmoVmo Am{U 17 _Üò \$ŠV XmoZ A§H$ AmhoV.

nmàr 2 : 576 ¿`m

EH§$ñWmZr 6 hm A§H$ Amho.

åhUyZ AmnUmg hì`m AgUmè`m KZ_yimÀ`m EH§$ñWmZr 6 KoD$.

nmàr 3 : Xþgam JQ> ¿`m. Vmo åhUOo 17

2 Mm KZ 8 Am{U 3 Mm KZ 27. 17 hr g§»`m 8 Am{U 27 À`m _Üò Amho.

2 Am{U 3 _Yrb bhmZ g§»`m 2 Amho. åhUyZ 2 A§H$ XeH$ñWmZr ¿`m.

åhUyZ, Ogo 175763 = 26 (VnmgyZ nhm)

ñdmÜ`m` 1.2

1. Imbrb g§»`m§Mo KZ_yi _yi Ad`drH$aU nÜXVrZo H$mT>m. (i) 64 (ii) 512 (iii) 10648 (iv) 27000 (v) 15625 (vi) 13824 (vii) 110592 (viii) 46656 (ix) 175616 (x) 91125

2. ~amo~a H$s MyH$ gm§Jm.

(i) {df_ g§»òMr KZg§»`m g_g§»`m AgVo.

(ii) nyU© KZg§»òMm eodQ> XmoZ eyÝ`m§Zr hmoV Zmhr.

(iii) Oa g§»òMm dJm©Mm eodQ> 5 Zo hmoV Agob, Va {VÀ`m KZmMm eodQ> 25 Zo hmoVmo.

(iv) g§»òMm eodQ>Mm (EH$H$ñWmZMm) A§H$ 8 Agob, Va Vr g§»`m nyU© KZg§»`m ZgVo.

KZ Am{U KZ_yi 11

(v) XmoZ A§H$s g§»òÀ`m KZm_Üò VrZ A§H$ Agy eH$VmV.

(vi) XmoZ A§H$s g§»òÀ`m KZm_Üò gmV qH$dm A{YH$ A§H$ Agy eH$VmV.

(vii) EH$ A§H$s g§»òMm KZ EH$ A§H$s Agy eH$Vmo.

3. Vwåhmbm gm§{JVbo H$s 1331 hr nyU© KZg§»`m Amho. Ad`drH$aU Z H$aVm {VMo KZ_yi H$m` Agob ho VH©$ H$éZ gm§Jy eH$mb H$m? Ë`mMà_mUo 4913, 12167, 32768 `m g§»`m§À`m KZ_yim~Ôb VH©$ H$am.

AmnU H$m` MMm© Ho$br?

1. 1729, 4104, 13832 `m g§»`m§Zm hmS>u-am_mZwOZ g§»`m åhUVmV. `m g§»`m XmoZ g§»`m§À`m KZm§À`m ~oaOoZo Xe©{dVm òVmV.

2. EH$m g§»òMm Ë`mM g§»òer VrZ doim JwUmH$ma H$ê$Z Amboë`m g§»òbm Ë`m g§»òMr KZ g§»`m åhUVmV. CXmhaUmW©, 1, 8, 27, ........... BË`mXr.

3. g§»òÀ`m _yi Ad`drH$aUmV Oa àËòH$ Ad`d 3 doim òV Agob, Va Vr g§»`m nyU© KZg§»`m AgVo.

4. 3 ho {MÝh KZ_yi Xe©{dVo. CXmhaUmW©, 27 3 33 33= = = 3.

2.1. n[aM`

AmnUmg _mhrV Amho H$m?

n¥ÏdrMo dñVw_mZ 5,970,000,000,000,000,000,000,000 kg BVHo$ Amho. AemàH$maÀ`m _moR>çmV _moR>çm g§»`m ghO[aË`m g_OÊ`mgmR>r KmVmH$m§Mm Cn`moJ H$gm Ho$bm OmVmo ho Vwåhr _mJrb B`ÎmoV {eH$bm AmhmV, åhUOo `mà_mUo 5.97 × 1024 kg.

Amåhr 1024 ho XhmMm KmVm§H$ Mmodrg Ago dmMVmo.AmnUmg _mhrV Amho,25 = 2 × 2 × 2 × 2 × 2 Am{U 2m = 2 × 2 × 2 × 2 × ......... × 2 × 2 .... (m doim)

AmVm AmnU 2 – 2 H$m` òVo? Vo emoYy.

KmVm§H$ gyMH$ G$UnyUmªH$ Amho.

KmVm§H$

AmYmam§H$

ho Amåhr 10 Mm KmVm§H$ 24 Ago dmMVmo.

1024

2.2.F$U KmVm§H$ Agbobo AmYmam§H$

AmnUmg _mhrV Amho H$s, 102 = 10 × 10 = 100

101 = 10 = 10010

100 = 1 = 1010

10–1 = ?

KmVm§H$ 1 Zo H$_r Ho$ë`mg g§»òMr qH$_V

nyduÀ`m qH$_Vrnojm 110

nQ> hmoVo.

darb Z_yZm AgoM Mmby R>odyZ Amåhmg {_iVo 10–1 = 110 .

VgoM, 10–2 = 110

10110

110

1100

1102

÷ = × = =

àH$aU

2 KmVm§H$ Am{U KmV

KmVm§H$ Am{U KmV$ 13

10–3 = 1100

101100

110

11000

1103

÷ = × = =

Va 10–10 {H$Vr hmoB©b?

AmVm Imbrb _wÔm nhm.

33 = 3 × 3 ×3 = 27

{_imboë`m qH$_Vrbm AmYmam§H$ 3 Zo ^mJbo.

32 = 3 × 3 = 9 = 273

31 = 3 = 93

30 = 1 = 3

3

darb Z_wZm nm{hbm AgVm, Amåhr Ago gm§Jy eH$Vmo,

3–1 = 1 ÷ 3 = 13

3–2 = 13

÷ 3 = 1

3 3132×

=

3–3 = 1

3 3132×

= ÷ 3 = 13

13

132 3× =

AmVm, darb à_mUo AmnU 2–2 Mr qH$_V H$mTy>.

Amåhmg _mhrV Amho, 10–2 = 2

110 qH$dm 102 = 2

110−

10–3 = 3

110 qH$dm 103 = 3

110−

3–2 = 2

13

qH$dm 32 = 2

13− ...... BË`mXr

gm_mÝ`nUo, 'a' hr eyÝà{hV nyUmªH$ g§»`m KoD$Z,

am = 1am

{bhÿ eH$Vmo. òWo 'm' hm KZ nyUmªH$ Amho.

am hm am Mm JwUmH$ma ì`ñV Amho.

14 J{UV

à`ËZ H$am 1. Imbrb KQ>H$m§Mo JwUmH$ma ì`ñV KQ>H$ emoYm. (i) 2–4 (ii) 10–5

(iii) 7–2 (iv) 5–3 (v) 10–100

1425 `mgma»`m g§»ò§Mo {dñVmarV ñdê$n KmVm§H$mMm Cn`moJ H$éZ H$go {bhmdo ho AmnU {eH$bmo AmhmoV.

1 × 103 + 4 × 102 + 2 × 101 + 5 × 100

AmVm, 1425.36 `m g§»òMo {dñVm[aV ñdê$n Ë`mÀ`m KmVm§H$m_Üò H$go hmoVo Vo nmhÿ.

1425.36 = 1 × 1000 + 4 × 100 + 2 × 10 + 5 × 1 + 310

6100

+

= 1 × 103 + 4 × 102 + 2 × 10 + 5 × 1 + 3 × 10–1 + 6 × 10–2

10 110

10 110

1100

1 22

− −= = =,

.

KmVmH$m§Mm dmna H$ê$Z Imbrb g§»`m§Mm {dñVma H$am.

(i) 1025 . 63 (ii) 1256.249

darb à`ËZ H$am

2.3.KmVm§H$m§Mo {Z`_

AmnU {eH$bm AmhmV, Oa a hr H$moUVmhr eyÝòVa nyUmªH$ Agob Va am × an = am+n, òWo m d n Z¡g{J©H$ g§»`m AmhoV. Oa KmVm§H$ G$U Agob Va hm {Z`_ VoWo bmJy hmoVmo H$m? `mg§~§Yr nmhÿ.


AmnUmg _mhrV Amho H$s,

(i). 2–3 = 123 Am{U 2–2 = 1

22 am = 1am . òWo a hm

H$moUVmhr eyÝòÎma nyUmªH$ Amho.

åhUyZ, 2–3 × 2–2 = 123 × 1

22

= 12 2

12

123 2 3 2 5×

= =+

= 2–5

–3 Am{U –2 `m XmoÝhr KmVm§H$m§Mr ~oarO –5 hmoVo.

(ii). (–3)–4 × (–3)–3 ¿`m.

(–3)–4 × (–3)–3 = 13

134 3−( )

×−( )

= 1

3 34 3−( ) × −( )

= 13

134 3 7−( )

=−( )+

= (–3)–7

(–4) + (–3) = –7

(iii). AmVm 5–2 × 54

5–2 × 54 = 15

524×

= 55

4

2

= 5 4 – 2

= 5(2) (–2) + 4 = 2

B`Îmm 7 dr _Üò Vwåhr {eH$bm AmhmV

H$s a hm H$moUVmhr eyÝòÎma nyUmªH$

Agob Va aa

am

nm n= − . òWo m d n

Z¡g{J©H$ g§»`m Am{U m > n.

(iv). AmVm (–5)–4 × (–5)2

(–5)–4 × (–5)2 = 15

542

−( )× −( )

16 J{UV

= −( )−( )

55

2

4 = 1

5 54 2−( ) × −( )−

= 15

54 22

−( )= −( )−

−( )

(–4) + 2 = –2

gm_mÝ`nUo AmnU gm§Jy eH$Vmo H$s a hm H$moUVmhr eyÝòÎma nyUmªH$ Agob Va am × an = am+n òWo m d n ho nyUmªH$ AmhoV.

à`ËZ H$am KmVm§H$ ê$nmV {bhm Am{U gmoS>dm (i) (–2)–3 × (–2)–4 (ii) P3 × P–10

(iii) 32 × 3–5 × 36

Oa a Am{U b ho eyÝòÎma nyUmªH$ Am{U m, n ho H$moUVohr nyUmªH$ AgVrb Va AmnU KmVm§H$m§Mo {Z`_ Imbrb à_mUo {bhÿ eH$Vmo.

(i). aa

am

nm n= − (ii). (am)n = amn (iii). am × bm = (ab)m

AgoM {Z`_ AmnU B`Îmm 7 dr_Üò YZ nyUmªH$m§gmR>r nm{hbm AmhmV.

(iv). ab

ab

m

m

m

=

(v). a0 = 1

AmVm, AmnU darb KmVm§H$m§À`m {Z`_mZwgma H$mhr CXmhaUo gmoS>dy.

CXmhaU 1 : qH$_V H$mT>m.

(i) 2–3 (ii) 13 2−

CH$b : (i) 2–3 = 12

183 =

(ii) 13 2−

= 32 = 3 × 3 = 9

CXmhaU 2 : gai ê$n Úm.

(i) (– 4)5 × (– 4)–10

(ii) 25 ÷ 2–6


CH$b : (i) (– 4)5 × (– 4)–10 = (– 4)(5 – 10) = (– 4)–5

= 14

15−( )

× = =

+ −a a a aa

m n m n mm,

(ii) 25 ÷ 2–6 = 25–(–6) = 211 (am ÷ an =am – n)

CXmhaU 3 : 4–3 hr g§»`m AmYmam§H$ 2 KoD$Z KmVm§H$mÀ`m ñdê$nmV {bhm.

CH$b : AmnUmg _mhrV Amho, 4 = 2 × 2 = 22

åhUyZ (4)–3 = (2 × 2)–3

= (22)–3 = 22 × ( – 3)

= 2–6 [(am)n = amn]

CXmhaU 4 : Imbrb g§»`m§Zm gai ê$n XoD$Z KmVm§H$m_Üò {bhm.

(i) (25 ÷ 28)5 × 2–5 (ii) (–4)–3 × (5)–3 × (–5)–3

(iii) 18

3 3× ( )− (iv) −( ) ×

3 53

44

CH$b :

(i) (25 ÷ 28)5 × 2–5 = (25 – 8)5 × 2–5

= (2– 3)5 × 2–5 = 2–15 – 5

= 2–15 – 5 = 2–20

= 1220

(ii) (–4)–3 × (5)–3 × (–5)–3 = [(–4) × (5) × (–5)]–3 = (100)–3

= 1

1001

3 a b ab aa

m m m mm× = ( ) =

−,

(iii) 18

3 3× ( )− = 12

333× ( )−

= 2–3 × 3–3 = (2 × 3)–3 = (6)–3

= 163

18 J{UV

(iv) −( ) ×

3 53

44 = (–1 × 3)4 × 5

3

4

4

= (–1)4 × 34 × 53

4

4

= (–1)4 × 54 = 54 [(–1)4 = 1]

CXmhaU 5 : (–3)m + 1 ×(–3)5 = (–3)7 _Yrb m Mr qH$_V H$mT>m.

CH$b : (–3)m + 1 ×(–3)5 = (–3)7

(–3)m + 1 + 5 = (–3)7

(–3)m + 6 = (–3)7

òWo XmoÝhr ~mOybm AmYmam§H$ g_mZ Agë`mZo `m g_rH$aUmV Ë`m§Mo KmVm§H$ gwÜXm nañnamer g_mZ AgVmV.

åhUOoM m + 6 = 7

qH$dm m = 7 – 6

\ m = 1

an = 1 \$ŠV n = 0 AgVmZm. H$moUË`mhr a gmR>r ho gË` hmoVo. Oa a = 1 Va, 11 = 12 = 13 = 1–2 = .... = 1. n À`m AZ§V qH$_Vrg (1)n = 1.

Oa a = –1 Va, (–1)0 = (–1)2 = (–1)4 = (–1)–2 = ........ = 1 qH$dm (–1)p =1 ho H$moUË`mhr g_ nyUmªH$ p gmR>r.

CXmhaU 6 : 23

2

−

Mr qH$_V H$mT>m.

CH$bÀ: 23

23

2 2

2

=− −

−

= 32

94

2

2 =

CXmhaU 7: (i) 13

12

14

2 3 2

−

÷

− − −

(ii) 58

85

7 5

×

− −


CH$bÀ :

(i) 13

12

14

2 3 2

−

÷

− − −

= 13

12

14

2

2

3

3

2

2

−

−

−

−

−

−−

÷

= 31

21

41

2

2

3

3

2

2−

÷

= {9 – 8} ÷ 16 = 1

16

23

23

32

32

2 2

2

2

2

2

= = =

− −

−

gm_mÝ`nUo, ab

ba

m m

=

−

(ii) 58

85

7 5

×

− −

= 58

85

7

7

5

5

−

−

−

−×

= 55

88

7

5

5

7

−

−

−

−×

= 5–7–(–5) × 8–5–(–7)

= 5–2 × 82

= 85

6425

2

2 =

ñdmÜ`m` 2.1

1. gmoS>dm :

(i) 3–2 (ii) (–4)–2 (iii) 12

5

−

2. Imbrb g§»`m§Zm gaiê$n XoD$Z CÎma YZ KmVm§H$ ê$nmV {bhm :

(i) (–4)5 ÷ (–4)8 (ii) 2

3

12

(iii) −( ) ×

3 53

44

(iv) (3–7 ÷ 3–10) × 3–5 (v) 2–3 × (–7)–3

3. qH$_Vr H$mT>m :

(i) (30 + 4–1) × 22 (ii) (2–1 × 4–1) ÷ 2–2

20 J{UV

(iii) 12

13

14

2 2 2

+

+

− − −

(iv) (3–1 + 4–1 + 5–1)0 (v) −

−23

2 2

4. gmoS>dm :

(i) 8 52

1 3

4

−

−

× (ii) (5–1 × 2–1) × 6–1

5. 5m ÷ 5–3 = 55 _Yrb m Mr qH$_Vr H$mT>m.

6. gmoS>dm :

(i) 13

14

1 1 1

−

− − −

(ii) 58

85

7 4

×

− −

7. gaiê$n Úm :

(i) 255 10

04

3 8

×× ×

≠( )−

− −

tt

t (ii) 3 10 1255 6

5 5

7 5

− −

− −

× ××

2.4. bhmZ g§»`m à_m{UV ñdénmV ì`ŠV H$aÊ`mg KmVm§H$m§Mm Cn`moJ H$aUo.

Imbrb {dYmZm§Mo {ZarjU H$am.

1. gy`m©nmgyZ n¥Ïdrn`ªVMo A§Va 149,600,000,000 m Amho.

2. àH$memMm AdH$memVrb doJ 300,000,000 m/sec Amho.

3. B`Îmm 7 dr À`m J{UV nwñVH$mMr gamgar OmS>r 20 mm Amho.

4. Vm§~S>çm aŠVnoerMm gamgar ì`mg 0.000007 mm AgVmo.

5. _mZdmÀ`m EH$m Ho$gmMr OmS>r 0.005 cm Vo 0.01 cm `m Xaå`mZ AgVo.

6. n¥ÏdrnmgyZ M§Ðmn`ªVMo A§XmOo A§Va 384,467,000m Amho.

7. dZñnVr noerMm AmH$ma 0.00001275 m AgVmo.

8. gy`m©Mr gamgar {ÌÁ`m 695000 km Amho.

9. AdH$memV OmUmè`m Ap½Z~mUmÀ`m B§YZmÀ`m OmJoMo dñVy_mZ 503600 kg Amho.

10. EH$m H$mJXmÀ`m VwH$S>çmMr OmS>r 0.0016 cm Amho.


11. g§JUH$ {MnÀ`m (computer chip) VmaoMm ì`mg 0.000003m Amho.

12. _mD§$Q> EìhaoñQ> {eIamMr C§Mr 8848m Amho.

{ZarjU H$am H$s 2 cm , 8848m , 6,95,000 km Aem H$mhr bhmZ g§»`m AmnU ghO dmMy eH$Vmo. òWo 150,000,000,000m gma»`m AVr _moR>çm Am{U 0.000007 m gma»`m A{V bhmZ g§»`m AmhoV.

darb {Xboë`m {dYmZmVrb gdm©V _moR>çm Am{U gdm©V bhmZ g§»`m emoYyZ Ë`m Imbrb VŠË`mV {bhm.

AVr _moR>çm g§»`m AVr bhmZ g§»`m

150,000,000,000 m------------------------------------------------------------

0.000007 m------------------------------------------------------------

_mJrb B`ÎmoV AmnU {eH$bm AmhmV H$s, _moR>çm g§»`m§Zm à_mUrV ñdê$nmV H$go ê$nm§Va H$amdo.

CXmhaUmW© : 150,000,000,000 = 1.5 × 1011

Xem§e q~Xÿ S>mdrH$S>o 11 ñWmZo gaH$Vmo.

AmVm 0.000007 `m g§» obm à_mUrV ê$nmV {b{hÊ`mMm à`ËZ H$am.

0 000007 7

10000007

106. = = = 7 × 10–6

0.000007 = 7 × 10–6m

Xem§e q~Xÿ COdrH$S>o 6 ñWmZo gaH$Vmo.

Ë`mMà_mUo, 0.0016 cm AgUmè`m H$mJXmÀ`m VwH$S>çmMr OmS>r,

0.0016 = 16

100001 6 10

104= ×.

= 1.6 × 10 × 10–4

= 1.6 × 10–3

Xem§e q~Xÿ COdrH$S>o 3 ñWmZo gaH$Vmo.

nwÝhm nhm

åhUyZ, H$mJXmÀ`m VwH$S>çmMr OmS>r 1.6 × 10–3 cm Amho.

22 J{UV

à`ËZ H$am 1. Imbrb g§»`m gm_mÝ` KmVm§H$ ñdê$nmV {bhm. (i) 0.000000564 (ii) 0.0000021

(iii) 21600000 (iv) 15240000

2. (2.4) _Üò {Xbobr gd© {dYmZo gm_mÝ` ñdê$nmV {bhm.

2.4.1 AVr _moR>çm Am{U AVr bhmZ g§»`m§Mr VwbZm

n¥ÏdrMm ì`mg 1.2756 × 107 m Amho Am{U gy`m©Mm ì`mg 1.4 × 109 m Amho. g_Om, Vwåhmbm gy`m©À`m ì`mgm~amo~a n¥ÏdrÀ`m ì`mgmMr VwbZm H$amd`mMr Agob Va,

gy`m©Mm ì`mg = 1.4 × 107 m

n¥ÏdrMm ì`mg = 1.2756 × 107 m

åhUOoM, 1 4 10

1 2756 101 4 10

1 27561 4 1001 2756

9

7

9 7..

..

..

××

= × = ×−

= 110, òWo A§XmOo 100 hmoB©b.

åhUyZ, gy`m©Mm ì`mg hm n¥ÏdrMm ì`mgmÀ`m 100 nQ> Amho.

AmVm, Vm§~S>çm aŠVnoerÀ`m AmH$ma 0.000007m Am{U dZñnVrnoerMm AmH$ma 0.00001275m `m§Mr VwbZm H$ê$.

Vm§~S>`m aŠVnoerMm AmH$ma = 0.000007m = 7 × 10–6 m

dZñnVrnoerMm AmH$ma = 0.00001275 m =1.275 × 10–5 m

åhUOoM, 7 10

1 275 10

6

5

××

−

−. = × − − −( )7 101 275

6 5

. = × −7 101 275

1

.

= 0 71 275

..

= 0 71 3

.

. = 12 (A§XmOo)

åhUyZ, Vm§~S>çm aŠVnoerMm AmH$ma dZñnVrnoerÀ`m AmH$mamÀ`m {Zå_nQ> (AYm©) Amho.

n¥ÏdrMo dñVy_mZ 5.97 × 1024 kg Amho Am{U M§ÐmMo dñVy_mZ 7.35 × 1022 kg Amho. Va EHy$U dñVy_mZ {H$Vr hmoB©b?


EHy$U dñVy_mZ = 5.97 × 1024 kg + 7.35 × 1022 kg

= 5.97 × 100 × 1022 + 7.35 × 1022

= 597 × 1022 + 7.35 × 1022

= (597 + 7.35) × 1022

= 604.35 × 1022 kg

Oa gm_mÝ` ñdê$nmÀ`m g§»ò§Mr ~oarO H$am`Mr Agob Va Ë`m g§»ò§Mo KmVm§H$

g_mZ H$ê$Z ¿`mdo.

gy`© Am{U n¥Ïdr `m_Yrb A§Va 1.496 × 1011 m Amho. n¥Ïdr Am{U M§Ð `m_Yrb A§Va 3.84 × 108 m Amho.

gy`©J«hUmdoir M§Ð hm gy`© Am{U n¥ÏdrÀ`m _Üò òVmo. Ë`mdoir gy`© Am{U M§Ð `m_Yrb A§Va {H$Vr AgVo?

gy`© Am{U n¥Ïdr `m_Yrb A§Va = 1.496 × 1011 m

n¥Ïdr Am{U M§Ð `m_Yrb A§Va = 3.84 × 108 m

gy`© Am{U M§Ð `m_Yrb A§Va = 1.496 × 1011 – 3.84 × 108

= 1.496 × 1000 × 108 – 3.84 × 108

= (1496 – 3.84) × 108 m

= 1492 . 16 × 108 m.

CXmhaU 8 : Imbrb g§»`m à_m{UV ê$nmV ê$nm§Va H$am.

(i) 0.000035 (ii) 4050000

CH$b : (i) 0.000035 = 3.5 × 10–5 (ii) 4050000 = 4.05 × 106

CXmhaU 9 : Imbrb g§»`mMo ê$nm§Va gm_mÝ` énmV H$am.

(i) 3.52 × 105 (ii) 7.54 × 10–4 (iii) 3 × 10–5

CH$b : (i) 3.52 × 105 = 3.52 × 100000 = 352000

(ii) 7.54 × 10–4 = 7 54104

.

= 7 5410000

. = 0.000754

nwÝhm AmnUmbm gm_mÝ` ñdénmVrb g§»`m g_mZ KmVm§H$ g§»òV ê$nm§Va H$amì`m bmJVmV.

(iii) 3 × 10–5 = 3

103

1000000 000035 = = .

24 J{UV

ñdmÜ`m` 2.2

1. Imbrb g§»`m à_m{UV ñdê$nmV {bhm.

(i) 0.0000000000085 (ii) 0.00000000000942

(iii) 6020000000000000 (iv) 0.00000000837 (v) 31860000000

2. Imbrb g§»`m gm_mÝ` ñdénmV {bhm.

(i) 3.02 × 10–6 (ii) 4.5 × 104 (iii) 3 × 10–8

(iv) 1.0001 × 109 (v) 5.8 × 1012 (vi) 3.61492 × 106

3. Imbr {Xboë`m {dYmZm§_Yrb g§»`m à_m{UV ñdê$nmV {bhm.

(i) 1 _m`H«$moZ åhUOo 1

1000000 m hmo`.

(ii) EH$m BboŠQ´>m°ZMm {dÚwV^ma 0.000,000,000,000,000,000,16 Hw$b_ Amho.

(iii) ~°ŠQ>o[a`mMm AmH$ma 0.0000005 m Amho.

(iv) dZñnVr noerMm AmH$ma 0.00001275 m Amho.

(v) EH$m H$mJXmMr OmS>r 0.07 mm Amho.

4. 5 nwñVHo$ d 5 H$mS>©erQ²>g EH$_oH$mda R>odbr AmhoV (Wßnr à_mUo). àËòH$ nwñVH$mMr OmS>r 20 mm Am{U àËòH$ H$mS>©erQ>Mr OmS>r 0.016 mm Amho. Va Ë`m§Mr (WßnrMr) EHy$U OmS>r (C§Mr) {H$Vr hmoB©b?


1. G$U KmVmqH$V g§»`m§Zm KmVm§H$m§Mo Imbrb {Z`_ bmJy hmoVmV.

(a) am × an = am + n (b) am ÷ an = am – n (c) (am)n = amn

(d) am × bm = (ab)m (e) a0 = 1 (f) aa

ab

m

m

m

=

2. G$U KmVm§H$m§Mm dmna H$ê$Z AVr bhmZ g§»`m à_m{UV ñdê$nmV ì`ŠV H$aVm òVmV.

3.1.n[aM`

Vw_À`m X¡Z§{XZ OrdZmV Vwåhmbm Aem H$mhr _m{hVr {_iVmV Ogo

(a) _mJrb 10 H$gmoQ>r gm_Ý`m§V EH$m \$bXm§OmZo H$mT>boë`m EHy$U Ymdm.

(b) EH$m Jmob§XmOmZo _mJrb 10 Am§Vaamï´>r` EH${Xder` {H«$Ho$Q> gm_Ý`m§V KoVbobo EHy$U ~ir.

(c) Vw_À`m dJm©Vrb {dÚm§Ï`mªZr J{UVmÀ`m MmMUr n[ajoV KoVbobo JwU.

(d) Vw_À`m àËòH$ {_ÌmZo dmMbobr H$Wm nwñVH$m§Mr g§»`m BË`mXr.

Aem KQ>Zm_Üò {_imboë`m qH$dm EH$Ì Ho$boë`m kmZmbm _m{hVr (data) åhUVmV. hr _m{hVr AmnUmg hì`m AgUmè`m KQ>ZmnmgyZ g§J«hrV Ho$br OmVo. CXm : {ejH$m§Zm Ë`m§À`m dJm©Vrb _wbm§Mr gamgar C§Mr _mhrV AgUo Amdí`H$ Amho. ho {_i{dÊ`mgmR>r {ejH$ àËòH$ _wbmMr C§Mr _moOVmV d `m C§MrÀ`m g§»`m§Mr _m{hVr nÜXVeranUo _m§Sy>Z Ë`mVrb {ZîH$f© H$mT>VmV.

Aer _m{hVr H$mhr doim AmboImZo Xe©{dVmV Á`m_wio Ë`mVrb _m{hVr ñnï> hmoVo. Vwåhr _mJrb B`Îmo_Üò doJdoJù`m àH$maMo AmboI {eH$bm AmhmV. ho Vw_À`m Ü`mZmV Amho Zm?

1. {MÌmboI (A pictograph) : {Xbobr _m{hVr H$mhr {MÝhm§Mm dmna H$ê$Z {MÌ_` ê$nmV Xe©{dVmV.

= 100 H$ma ← 100 H$magmR>r EH$ H$maMo {MÝh.

Owb¡ = 250 ho {MÝh 100 Mo 12 Xe©{dVo.

Am°JñQ> = 300

gßQ>|~a = ?

àH$aU

3 _m{hVr hmVmiUo

26 J{UV

(i) Owb¡ _{hÝ`mV {H$Vr H$maMo CËnmXZ Pmbo?

(ii) H$moUË`m _{hÝ`mV OmñVrV OmñV H$maMo CËnmXZ hmoVo?

2. ñV§^mboI (Bar graph) : {Xboë`m _m{hVrdéZ EH$g_mZ ê§$Xr d Ë`m§Mr C§Mr qH$_VrÀ`m à_mUmV KoD$Z Am`VmH$ma ñV§^mÀ`m ê$nmV Xe©dyZ hm AmboI H$mT>Vm òVmo.

àËòH$ ñV§^ hm Ë`m KQ>H$mMr qH$_V Xe©{dVmo.

XmoZ ñV§^mVrb A§Va ho Ë`mÀ`m é§Xrer g_mZ AgVo.

e¡j{UH$ df©2003-04 2004-05 2005-06 2006-07 2007-08

B`Îmm

8 d

rÀ`m

{dÚm

Ï`mªM

r nQ>g

§»`m

(i) ñV§^mboImZo H$moUVr _m{hVr {Xbr Amho?

(ii) H$moUË`m dfm©nmgyZ {dÚmÏ`mªMr g§»`m dmT>V Jobr Amho?

(iii) H$moUË`m dfu {dÚmÏ`mªMr g§»`m gdm©V OmñV Amho?

(iv) ~amo~a H$s MyH$ gm§Jm.

2003-04 gmbmnojm 2005-06 gmbr {dÚmÏ`mªMr g§»`m XþßnQ> hmoVr.

3. Xþhoar ñV§^mboI ({ÛñV§^mboI) (Double Bar Graph) : `m ñV§^mboImV EH$m nmR>monmR> EH$ {Xbobo XmoZ _m{hVrMo JQ> Xe©{dbo OmVmV. `mMm Cn`moJ {Xboë`m _m{hVrMr VwbZm H$aÊ`mgmR>r> hmoVo.

J{UV g_mO {dkmZ B§J«Or qhXr{df`

{dÚm

Ï`mªZ

r {_i

{dbob

o JwU

(i) Xþhoar ñV§^mboImZo H$moUVr _m{hVr {Xbr Amho?

_m{hVr hmVmiUo 27

(ii) H$moUË`m {df`mV A{YH$ àJVr {XgyZ òVo?

(iii) H$moUË`m {df`mMr àJVr gdm©V H$_r hmoV Amho?

(iv) H$moUË`m {df`mMr àJVr gmaIrM {XgyZ òVo?


Oa AmnU ñV§^m§boImVrb H$moUË`mhr ñV§^mMr pñW{V ~Xbbr, Va Ë`mÀ`m _m{hVrV ~Xb hmoVmo H$m? hmoVmo Va H$m?

.à`ËZ H$am

Imbrb _m{hVr Xe©{dUmam `mo½` AmboI H$mT>m.

1. _{hZm Owb¡ Am°JñQ> gßQ>|~a Am°ŠQ>mo~a Zmoìh|~a {S>g|~a

K‹S>`mi {d{H«$Mr g§»`m 1000 1500 1500 2000 2500 1500

2. {dÚmÏ`mªMr AmdS> emim A

emim B

emim C

MmbUo 40 55 15

gm`H$b MmbdUo 45 25 35

3. AmR> Aìdb {H«$Ho$Q> g§KmZr EH${Xder` Am§Vaamï´>r` gm_Zo qOH$Ê`mMr Q>¸o$dmar.

g§K M±nr`Z MfH$ Vo{dídH$n 2006

2007 _Yrb eodQ>Mo 10 EH${Xder` Am§Vaamï´>r` gm_Zo

X{jU Am{\«$H$m 75% 78%

Am°ñQ´>o{b`m 61% 40%

lrb§H$m 54% 38%

Ý`w{Pb§S> 47% 50%

B§½b§S> 46% 50%

nm{H$ñVmZ 45% 44%

doñQ>B§S>rO 44% 30%

^maV 43% 56%

28 J{UV

3.2._m{hVrMr aMZm H$aUo

Amnë`mbm _mhrV Agbobr gm_wJ«r hr dJuH¥$V Zgob Va {Vbm H$ÀMr _m{hVr åhUVmV. Ë`mVrb AW©nyU© {ZîH$f© H$mT>Ê`mgmR>r Amåhmbm gm_wJ«rMr `mo½` OwiUr qH$dm dJuH$aU H$aUo Amdí`H$ Amho. CXmhaUmW©, {dÚmÏ`mªÀ`m JQ>m_Üò Ë`m§À`m AmdS>rÀ`m {df`m{df`r {dMmabo Ë`mM {ZîH$fm©Mr `mXr Imbrb à_mUo.

H$bm, J{UV, {dkmZ, B§p½be, J{UV, H$bm, B§p½be, J{UV, B§p½be, H$bm, {dkmZ, H$bm, {dkmZ, {dkmZ, J{UV, H$bm, B§p½be, H$bm, {dkmZ, J{UV, {dkmZ, H$bm,

H$moUVm {df` OmñV AmdS>rMm d H$moUVm {df` H$_r AmdS>rMm Amho?

{dÚmÏ`mªÀ`m ng§VrMo A§XmO H$aUo BVHo$ gmono Zmhr. AmnU hr gm_wJ«r VŠVm 3.1 _Üò {b{hboë`m à_mUo _m§Sy>.

VŠVm 3.1

{df` JUVr IwUm {dÚmÏ`mªMr g§»`m

H$bm |||| || 7

J{UV ||||| 5

{dkmZ |||| || 6

B§p½be |||| 4

àËòH$ {df`mÀ`m g_moa Agboë`m JUVr IwUm§Mr g§»`m hr Ë`m {df`mMr AmdS> AgUmè`m {dÚmÏ`mªMr g§»`m Xe©{dVo.

`mbm Ë`m {df`mMr dma§dmaVm åhUVmV.

dma§dmaVm hr EH$ R>am{dH$ Zm|XUr {H$Vr doim `mMr g§»`m Xe©{dVo.

3.1 `m VŠË`mdéZ B§p½be {df` AmdS>V AgUmè`m {dÚmÏ`mªMr g§»`m 4. J{UV {df` AmdS>V AgUmè`m {dÚmÏ`mªMr g§»`m 5. Aem V`ma Ho$boë`m VŠË`mbm dma§dmaVm {dVaU VŠVm (Frequency distribution table) åhUVmV.

_m{hVr hmVmiUo 29

à`ËZ H$am. 1. EH$m {dÚmÏ`mªÀ`m JQ>m_Üò Ë`m§Zm OmñV AmdS>rMm nmird àmUr

H$moUVm Vo gm§JÊ`mg gm§{JVbo. Ë`mMm {ZîH$f© Imbr {Xbobm Amho.

Hw$Ìm, _m§Oa, _m§Oa, _mgm, _m§Oa,ggm, Hw$Ìm, _m§Oa, Hw$Ìm, Hw$Ìm, Hw$Ìm, _m§Oa, Jm`, _mgm, ggm, Hw$Ìm, _m§Oa, Hw$Ìm, _m§Oa, _m§Oa, Hw$Ìm, ggm, _m§Oa, _mgm Hw$Ìm.

`mdéZ dma§dmaVm {dVaU VŠVm V`ma H$am.

3.3. _m{hVrMo dJuH$aU (JQ> H$aUo)

EH${ÌV Ho$boë`m gm_wJ«r_Üò Amnë`m AmdS>rÀ`m {df`m§À`m {ZdS>rMr Zm|X Ho$bobr AmT>iVo. CXm. H$bm {df` 7 {dÚmÏ`mªZm AmdS>Vmo, 5 {dÚmÏ`mªZm J{UV AmdS>Vmo Am{U (VŠVm 3.1) `mà_mUo hr gm_wJ«r ñV§^mboI qH$dm {MÌmboImZo XmI{dbr OmVo. H$mhr doim _moR>r gm_wJ«r hmVmimdr bmJVo. CXm. B`Îmm VIII À`m 60 {dÚmÏ`mªZr J{UV {df`mV {_i{dbobo JwU (50 n¡H$s) Imbrbà_mUo AmhoV.

21, 10, 30, 22, 33, 5, 37, 12, 25, 42, 15, 39, 26, 32, 18, 27, 28, 19, 29, 35, 31,

24, 36, 18, 20, 38, 22, 44, 16, 24, 10, 27, 39, 28, 49, 29, 32, 23, 31, 21, 34, 22,

23, 36, 24, 36, 33, 47, 48, 50, 39, 20, 7, 16, 36, 45, 47, 30, 22, 17.

`mMm AmnU Oa dma§dmaVm {dVaU VŠVm V`ma Ho$bm Va Vmo \$maM {dñV[aV hmoB©b. åhUyZ gmo`rgmR>r AmnU {ZarjUm§Mo JQ> V`ma H$ê$ Ogo 0-10, 10-20 `m à_mUo {dVaU Ho$boë`m Am{U àËòH$ JQ>mVrb dm§admaVoÀ`m g§»`m§Mr _m§S>Ur H$é. `mMm dma§dmaVm {dVaU VŠVm Imbrb à_mUo Agob.

VŠVm 3.2

JQ> JUVr IwUm dma§dmaVm

0-10 || 2

10-20 ||||| ||||| 10

20-30 ||||| ||||| ||||| ||||| | 21

30-40 ||||| ||||| ||||| |||| 19

40-50 ||||| || 7

50-60 | 1

EHy$U 60

30 J{UV

Aem àH$mao _m§S>boë`m gm_J«rbm dJuH¥$V H$aUo åhUVmV Am{U {_imboë`m {dVaUmg dJuH¥$V> dma§dmaVm {dVaU åhUVmV. ho AmnUm§g _hËdmMo {ZîH$f© H$mT>Ê`mg _XV H$aVo.

(1) OmñVrV OmñV {dÚmÏ`mªZr 20 Am{U 40 À`m _Üò JwU {_i{dbo AmhoV.

(2) 50 n¡H$s 40 nojm OmñV JwU AmR> {dÚmÏ`mªZr {_i{dbo AmhoV Am{U `m à_mUo.

> 0-10, 10-20, 20-30 BË`mXr àËòH$ JQ>mbm g§^mJ loUr (Class Interval) åhUVmV.

Ago {XgyZ òVo H$s, 10 A§H$ XmoÝhr g§^mJloUr _Üò Amho. 0-10 d 10-20 _Üò. Ë`mMà_mUo 20 A§H$ 10-20 Am{U 20-30 _Üò AmT>iVmo. na§Vw 10 qH$dm 20 ho XmoÝhr g§^mJloUrV Z_yX H$aVm òV ZmhrV. Ago g_mB©H$ A§H$ ho Zoh_r _moR>çm g§^mJloUrV KoVbo OmVmV.

CXm. 10 hm A§H$ 0-10 EodOr 10-20 _Üò KoVmV, 20 A§H$ 10-20 EodOr 20-30 _Üò KoVmV. 10-20 `m loUr_Üò 10 bm Ë`mMr {ZƒV_² _`m©Xm (lower class limit) åhUVmV Am{U 20 bm CÀMV_² _`m©Xm (upper class limit) åhUVmV. Ë`mMà_mUo loUr 20-30 _Üò 20 bm {ZƒV_² d 30 bm CƒV_² _`m©Xm åhUVmV. CÀMV_² _`m©Xm Am{U {ZƒV_² _`m©Xm `mVrb-\$aH$ g_mZ {XgyZ òVmo. `m CÀMV_ d {ZƒV_² _`m©XoVrb \$aH$mg g§^mJ loUrMr ì`mßVr (n„m) Ago åhUVmV.

à`ËZ H$am. 1. Imbrb dma§dmaVm {dVaU VŠË`mMm Aä`mg H$am Am{U Ë`mImbr

{Xboë`m àûZm§Mr CÎmao Úm. EH$m H$maImÝ`mVrb 550 H$m_Jmam§À`m XaamoOÀ`m CËnÝZmMm

dma§dmaVm {dVaU VŠVm.

VŠVm 3.3

g§^mJ loUr XaamoOMo CËnÞ (é.)

dma§dmaVm H$m_Jmam§Mr g§»`m

100-125 45125-150 25150-175 55175-200 125200-225 140225-250 55250-275 35275-300 50300-325 20

EHy$U 550

_m{hVr hmVmiUo 31

(i) g§^mJloUrMr ì`mßVr {H$Vr Amho?

(ii) H$moUË`m g§^mJloUrMr dma§dmaVm gdm©V OmñV Amho?

(iii) H$moUË`m g§^mJloUrMr dma§dmaVm gdm©V H$_r Amho?

(iv) 250-275 `m g§^mJloUrMr CÀMV_ _`m©Xm {H$Vr?

(v) H$moUË`m XmoZ loUr_Üò g_mZ dma§dmaVm Amho?

2. EH$m dJm©Vrb 20 {dÚmÏ`mªMo dOZ (kg _Üò) {Xbo Amho. m _m{hVrdê$Z 30-35, 35-40 `m à_mUo g§^mJ loUr dmnéZ dma§dmaVm {dVaU VŠVm V`ma H$am.

40, 38, 33, 48, 60, 53, 31, 46, 34, 36, 49, 41, 55, 49, 65, 42, 44, 47, 38, 39.

3.3.1. \$aH$mg{hV ñV§^

AmVm AmnU 60 {dÚmÏ`mªZr J{UV {df`mÀ`m MmMUr_Üò {_i{dboë`m JwUm§Mm dJuH¥$V dma§dmaVm VŠË`mMm {dMma H$é. (VŠVm 3.4)

VŠVm 3.4

g§^mJloUr dma§dmaVm

0-10 2

10-20 10

20-30 21

30-40 19

40-50 7

50-60 1

EHy$U 60

{dÚmÏ`mªZr {_i{dbobo JwU

{dÚm

Ï`mªM

r g§»`

m

AmH¥$Vr 3.1

ho AmH¥$Vr 3.1 _Üò AmboImÀ`m ñdénmV Xe©{dbo Amho. B`Îmm 7 dr _Üò Vwåhr H$mT>boë`m `màH$maÀ`m ñV§^mboImhÿZ hm doJim Amho H$m? {ZarjU H$am H$s, òWo AmnU g§^mJloUr hr AmS>ì`m (x - Aj) Ajmda Xe©{dbr Amho. òWo ñV§^mMr C§Mr g§^mJ loUrMr dma§dmaVm Xe©{dVo. VgoM àËòH$ g§^mJ loUr_Üò Ogo A§Va Zmhr Vgo ñV§^m_Üò gwÜXm A§Va AgV Zmhr.

Aem àH$mao _m{hVr AmboImÛmao Xe©{dÊ`mbm {hñQ>moJ«m_ (Histogram) Ago åhUVmV.

32 J{UV

Imbrb AmboI EH$ Xþgam {hñQ>moJ«m_ Amho. (AmH¥$Vr 3.2)

ñV§^mMr C§Mr hr gm_wJ«r_Yrb JQ>m§Mr g§»`m Am{U dm§admaVm Xe©{dVo.

_Yo A§Va Zgbobo g_mZ é§XrMo ñV§^

{ejH

$m§Mr g

§»`m

d` (dfm©_Üò)g§^mJloUr 0-20 _Üò g§»`m Zgë`mZo Vo AmS>ì`maofoda Xe©{dÊ`mgmR>r ZmJ_moS>r aofm dmnaVmV.

g§^mJjoUrMr g_mZ ì`már(òWo 5 df©).

EH$m emioVrb 25 {ejH$m§Mr d`mo_`m©Xm

AmH¥$Vr 3.2`m {hñQ>moJ«m_ ñV§^m§Mm dmna H$éZ AmnU Imbrb àíZm§Mr CÎmao XoD$ eH$Vmo.

1. 50 dfm©nojm H$_r Am{U 45 df© dm A{YH$ d` AgUmè`m {ejH$m§Mr g§»`m {H$Vr? 2. 35 dfm©nojm H$_r d` AgUmè`m {ejH$m§Mr g§»`m {H$Vr?

à`ËZ H$am.

1. {Xboë`m {hñQ>moJ«m_ (AmH$¥Vr 3.3) Mo {ZarjU H$éZ nwT>rb àíZm§Mr CÎmao {bhm.

7

6

5

4

3

2

11 1

2 2

3

4

7

1300 125 135 140 145 150 155 160

B`Îmm

7 d

r _Yr

b {d

Úm{W

©ZtMr

g§»`

m

C§Mr (cm _Üò)

AmH¥$Vr 3.3

_m{hVr hmVmiUo 33

(i) hm {hñQ>moJ«m_ H$moUË`m àH$maMr _m{hVr XoVmo?

(ii) H$moUVm JQ> {dÚmWuqZMr g§»`m A{YH$ Xe©{dVmo?

(iii) {H$Vr {dÚmWuqZMr C§Mr 145 cm Am{U OmñV Amho?

(iv) Oa AmnU {dÚmWuqZZm Imbrb VrZ àH$mamV {d^mJbo Va àËòH$ JQ>mV {dÚmWuqZMr g§»`m {H$Vr Agob?

150 cm Am{U OmñV - JQ> A

140 cm Vo 150 cm nojm H$_r - JQ> B

140 cm nojm H$_r - JQ> C

ñdmÜ`m` 3.1

1. `mn¡H$s H$moUVr _m{hVr XmI{dÊ`mgmR>r Vwåhr {hñQ>moJ«°_Mm Cn`moJ H$amb (a) nmoñQ>_ZÀ`m {nedrVrb doJdoJù`m ^mJmVrb nÌm§Mr g§»`m (b) niÊ`mÀ`m e`©VrVrb ñnY©H$m§Mr C§Mr. (c) nmM H§$nÝ`mZr V`ma Ho$boë`m H°$goQ>Mr g§»`m. (d) 7 a.m. Vo 7 p.m. `m doioV aoëdogmR>r aoëdoñQ>oeZda Amboë`m

àdmí`m§Mr g§»`m. àËòH$mgmR>r H$maU Úm.

2. EH$m {S>nmQ>©_|Q>b ñQ>moAa (Departmental Store) _Üò Amboë`m {JèhmB©H$m§Zm `mà_mUo IwUm (g§Ho$V) Ho$ë`m AmhoV. nwéf (M), _{hbm(W), _wbJm (B) qH$dm _wbJr (G). gH$miÀ`m n{hë`m Vmgm_Üò Amboë`m {JèhmB©H$m§Mr `mXr nwT>rb à_mUo {Xbr Amho.

W W W G B W W M G G M M W W W W G B M W B G G M W W

M M W W W M W B W G M W W WW G W M M W W M W G W

M G W M M B G G W

JUVr IwUm H$ê$Z dma§dmaVm {dVaU VŠVm V`ma H$am. Ë`mMo ñnîQ>rH$aU XoÊ`mgmR>r ñV§^mboI H$mT>m.

3. EH$m H$maImÝ`mVrb 30 H$m_Jmam§Mr EH$m AmR>dS>çmMr _Owar (` _Üò) {Xbr Amho.

830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845, 804, 808, 812, 840, 885, 835, 835, 836, 878, 840, 868, 890, 806, 840.

34 J{UV

JUVr IwUm§Mm dmna H$ê$Z d g§^mJloUr 800-810, 810-820 Am{U nwT>o `mà_mUo KoD$Z dma§dmaVm VŠVm V`ma H$am.

4. àý H«$ 3 _Yrb dma§dmaVm VŠË`mMm {hñQ>moJ«m_ H$mTy>Z Imbr {Xboë`m àûZm§Mr CÎmao {bhm.

(i) H$moUË`m JQ>m_Üò gdm©V OmñV H$m_Jma AmhoV?

(ii) {H$Vr H$m_Jmam§Zm ` 850 qH$dm A{YH$ _Oyar {_iVo?

(iii) {H$Vr H$m_Jmam§Zm ` 850 nojr H$_r _Oyar {_iVo?

5. gwÅ>rÀ`m doir EH$m R>am{dH$ dJm©Vrb {dÚmÏ`mªZr {Q>.ìhr. nm{hboë`m Vmgm§Mr g§»`m AmboImV {Xbr Amho. `mdéZ Imbrb àûZm§Mr CÎmao {bhm.

(i) {Q>.ìhr. nmhUmao OmñVrV OmñV {dÚmWu {H$Vr Vmg Q>r.ìhr. nmhVmV?

(ii) 4 Vmgmnojm H$_r doi {Q>.ìhr. nm{hboë`m {dÚmÏ`mªMr g§»`m {H$Vr?

(iii) 5 Vmgmnojm A{YH$ doi {Q>.ìhr nm{hboë`m {dÚmÏ`mªMr g§»`m {H$Vr?

32

28

24

20

16

12

8

4

1 2 3 4 5 6 7

4

8 86

22

32

àËòH$ {Xder {Q>.ìhr. nm{hbobo Vmg

{dÚm

Ï`mªM

r g§»`

m

3.4. dVw©imboI qH$dm {ÌÁ`m§Va I§S>mboI

AmH¥$Vr 3.4 _Üò Xe©{dë`mà_mUo EImXr _m{hVr dVw©imH$ma ñdénmV {Xbobr H$Yr nm{hbm AmhmV H$m?

_m{hVr hmVmiUo 35

EH$m _wbmZo {XdgâmV Kmb{dbobm doi. EH$m ehamVrb bmoH$m§Mo d`moJQ>.

3 Vmg

4 Vmg

3 Vmg

6 Vmg

8 Vmg

J¥hnmR>

BVaIoi

emim

Pmon 50 hOma30 hOma

20 hOma

0-14 df©o

15-60 df©o

60 df©o Am{UA{YH$

(i) AmH¥$Vr 3.4 (ii)

`m§Zm dVw©imboI åhUVmV. `m AmboImV EH$ nyU© dVw©i Am{U Ë`mMo ^mJ `m_Yrb g§~§Y XmI{dVmV. òWo nyU© dVw©i Ë`mÀ`m I§S>m_Üò {d^mJbo OmVo. àËòH$ I§S>mMm AmH$ma hm _m{hVrÀ`m à_mUmV XmI{dbm OmVmo. CXmhaUmW©, darb AmboImV PmonogmR>r IM© Ho$bobm doi.

=PmonoMo Vmg

= 824

13

VmgVmg

=nyU© {Xdg

åhUyZ dVw©imMm 13

14 mJ (I§S>) hm PmonoÀ`m doiogmR>r XmI{dbm Amho. Ë`mMà_mUo emiogmR>r

IM© Ho$bobm doi

=emioVrb Vmg

=6 Vmg

= 13

14nyU© {Xdg 24 Vmg

åhUyZ nyU©> dVw©imMm 13

14 dVw©iI§S> H$mT>bm Amho. `mM à_mUo BVagmR>r gwÜXm I§S> H$mT>Vm

òVrb. gd© AnyUmªH$m§Mr ~oarO H$am. Vwåhmbm ~oarO 1 {_iVo H$m? `m dVw©imboImbm {ÌÁ`m§Va

I§S>mboI Ago gwÜXm åhUVmV.

à`ËZ H$am.

1. nwT>rb àËòH$ {ÌÁ`m§Va I§S>mboIm_Üò (AmH¥$Vr 3.5) Vwåhmbm Vw_À`m dJm©À`m doJdoJù`m _m{hVrMo I§S> (^mJ) {Xbo AmhoV. Va àËòH$gmR>r dVw©imMm ^mJ Xe©{dUmè`m _m{hVrMo AnyUmªH$ gm§Jm.

36 J{UV

_wbr 50%

_wbo 50%

_wbr qH$dm _wbo

20%gm`H$b

MmbUo40%

40%~g qH$dm H$ma

emiobm òÊ`mMr dmhVyH$

AmdS>

Ûof 15%

J{UV {df`r AmdS>/Ûof

(i) (ii) (iii)

AmH¥$Vr 3.5

2. {Xboë`m {ÌÁ`m§Va I§S>mboImdê$Z (AmH¥$Vr 3.6) `m àûZm§Mr CÎmao Úm.

(i) H$moUË`m àH$maMm H$m`©H«$_ OmñV àojH$$ nmhVmV?

(ii) H$moUË`m XmoZ àH$maÀ`m àojH$m§Mr g§»`m EH$m Ioi M°ZobÀ`m àojH$m§À`m g§»ò BVH$s Amho?

_m{hVr

10%~mbå`m

15%

Ioi25%

_Zmoa§OZ50%

{Q>.ìhr H$m`©H«$_ nmhUmao àojH$

AmH¥$Vr 3.6

3.4.1 {ÌÁ`m§Va I§S>mboI aMUo

EH$m emioVrb {dÚmÏ`mªMr AmB©gH«$s_À`m MdtÀ`m {df`rMr eoH$S>odmar Imbr {Xbr Amho.

Md{dÚmÏ`mªZm AmdS>Ê`mè`m

MdrMr eoH$S>odmar

Mm°H$boQ> 50%

ìh°{Zbm 25%

BVa Mdr 25%

hr _m{hVr {ÌÁ`m§Va I§S>mboIZo Xe©dm.

dVw©i_Ü`mer gd© H$moZm§Mr ~oarO 360° AgVo. _Ü`mer AgUmè`m àËòH$ I§S>mMm H$moZ hm 360° Mm AnyUmªH$s ^mJ AgVmo.

dVw©iI§S>mMm dVw©iH|$Ðmer hmoUmam H$moZ emoYÊ`mgmR>r EH$ VŠVm V`ma H$ê$.

_m{hVr hmVmiUo 37

VŠVm 3.5

Md{dÚmÏ`mªZm AmdS>Umè`m

MdrMr eoH$S>odmarAnyUmªH$mV 360° Mm ^mJ

Mm°H$boQ> 50%50

10012

= 50100

12

= × 360° = 180°

ìh°{Zbm 25%25

10014

= 25100

14

= × 360° = 90°

BVa Mdr 25%25

10014

= 25100

14

= × 360° = 90°

1. gmoB©ñH$a {ÌÁòMo EH$ dVw©i H$mT>m. dVw©iH|$Ð (O) Am{U {ÌÁ`m (OA) Xe©dm.

A

O

2. Mm°H$boQ> MdrgmR>r H$moZ_mnH$m§À`m ghmæ`mZo ∠AOB = 180° H$éZ dVw©iI§S> H$mT>m.

AO

B180°

3. Aem àH$mao BVagmR>r dVw©iI§S> H$mT>m. A B90°

Mm°H$boQ>

BVa ìh°{Zbm

CXmhaU 1 : EH$m Hw$Qw>§~mMr EH$m _{hÝ`mVrb ~MV Am{U {d{dY KQ>H$mgmR>r hmoUmè`m IMm©Mm Vnerb (Q>¸o$dmarV) {ÌÁ`m§Va I§S>m_Üò (AmH¥$Vr 3.7) Xe©{dbm Amho. Ë`mdéZ Imbrb àíZm§Mr CÎmao {bhm.

(i) H$moUË`m KQ>H$mda OmñV IM© hmoVmo? (ii) Hw$Qw>§~mÀ`m ~MVr EdT>m IM© H$moUË`m

KQ>H$mda Ho$bm OmVmo?

_wbm§Mo {ejU15%^mS>o 10%

dmhVyH$ 5%

BVa20%

~MV 15% H$nS>o 10%

Amhma 25%

AmH¥$Vr 3.7

38 J{UV

(iii) Oa _m{gH$ ~MV ` 3000 Agob, Va H$nS>çmgmR>r H$amdm bmJUmam _m{gH$ IM© {H$Vr?

CH$b : (i) AmhmamgmR>r gdm©V OmñV IM© hmoVmo.

(ii) _wbm§À`m {ejUmdarb IM© (15%) hm Hw$Qw>§~mÀ`m ~MVr EdT>m Amho.

(iii) 15% ho ` 3000 Xe©{dVo. åhUyZ 10% ho 300015

10× = ` 2000

CXmhaU 2: EH$m R>am{dH$ {Xder ~oH$arVrb doJdoJù`m nXmWmªMr {dH«$s (` _Üò) Imbr {Xbr Amho.

gmYm ~«oS> 320

\$im§Mm ~«oS> 80

Ho$H$ d n°Q>rg 160

{~pñH$Q>o 120

BVa 40

EHy$U 720

`m _m{hVrMm {ÌÁ`m§Va I§S>mboI H$mT>m.

CH$b : dVw©i H|$Ðmer hmoUmao àËòH$ dVw©iI§S>mMo H$moZ H$mTy>. òWo EHy$U {dH«$s = ` 720.

`mdéZ Agm VŠVm Amåhmbm {_iVmo.

nXmW© {dH«$s (` _Üò) AnyUmªH$mV H|$Ðmer H$moZ

gmYm ~«oS> 320320720

49

= 320720

49

= × 360° = 160°

{~ñH$sQ>o 120120720

16

= 120720

16

= × 360° = 60°

Ho$H$ d n°Q>rg 160160720

29

= 160720

29

= × 360° = 80°

_m{hVr hmVmiUo 39

\$im§Mm ~«oS> 80 80720

19

= 80720

19

= × 360° = 40°

BVa 40 40720

118

= 40720

118

= × 360° = 20°

AmVm {ÌÁ`m§Va I§S>mboI V`ma H$am (AmH¥$Vr 3.8).

{~ñH$sQ>o

Ho$H$ Am{U n°Q>rg

\$im§Mm ~o«S>BVa

gmYm ~«oS>

AmH¥$Vr 3.8

à`ËZ H$am.

EH$ _wbJm Amnbm {XdgmMm doi Agm Kmb{dVmo. `m _m{hVrdê$Z {ÌÁ`m§Va I§S>mboI H$mT>m.

Pmon 8 Vmgemim 6 VmgJ¥hnmR> 4 VmgIoi 4 VmgBVa 2 Vmg

{dMma H$am, MMm© H$am Am{U {bhmImbrb _m{hVrgmR>r H$moUVm AmboI OmñV `mo½` hmoB©b.

1. amÁ`mVrb AÝZYmÝ`m§Mo CËnmXZ

df© 2001 2002 2003 2004 2005 2006CËnmXZ

(bmI Q>ZmV)60 50 70 55 80 85

40 J{UV

2. EH$m JQ>mVrb bmoH$m§À`m OodUmMr ng§Vr.

ng§VrMo OodU bmoH$m§Mr g§»`m

X{jU ^maVr` 30

CÎma ^maVr` 40

Mm`ZrµO 25

BVa 25

EHy$U 120

3. EH$m H$maImÝ`mVrb H$m_Jmam§Mo XaamoOMo CËnÝZ

XaamoOMo CËnÝZ (ê$n`m§V)

H$maImÝ`m_Yrb H$m_Jmam§Mr g§»`m

75-100 45

100-125 35

125-150 55

150-175 30

175-200 50

200-225 125

225-250 140

EHy$U 480

ñdmÜ`m` 3.2

1. EH$m ehamVrb H$mhr VéUm§À`m g§JrVmÀ`m ng§VrMo gd}jU H$aÊ`mV Ambo. ~mOwbm {Xboë`m {ÌÁ`m§Va I§S>mboImV Ë`m gd}jUmMr _m{hVr {Xbr Amho. `m {ÌÁ`m§Va I§S>mboImdê$Z Imbrb àûZm§Mr CÎmao Úm.

(i) Oa emór` g§JrV 20 VéUm§Zm AmdS>Vo Va {H$Vr VéUm§Mo gd}jU Ho$bo hmoVo.

20%

10% 40%

30%

bmoH$JrV

_Zmoa§OH$

Cnemór`

emór`

(ii) OmñVrV OmñV bmoH$m§Zm AmdS>Umao g§JrV H$moUË`m àH$maMo Amho?

(iii) Oa H°$goQ> H§$nZrZo doJdoJù`m g§JrVm§À`m EHy$U 1000 grS>r V`ma Ho$ë`m Va, àËòH$ àH$maÀ`m {H$Vr grS>r V`ma Ho$ë`m?

_m{hVr hmVmiUo 41

2. 360 bmoH$m§À`m JQ>mbm Ë`m§À`m ng§VrÀ`m G$VwgmR>r _VXmZ H$aÊ`mg gm§{JVbo. Vo VrZ G$Vw CÝhmim, nmdgmim Am{U {hdmim Ago AmhoV. àmá _m{hVr ~mOybm {Xbr Amho. ImbrbgmR>r CÎmao Úm.

(i) H$moUË`m G$Vwbm OmñV _Vo {_imbr Amhob?

(ii) àËòH$mgmR>r dVw©iI§S>mMm H|$Ðmer hmoUmam H$moZ H$mT>m.

(iii) hr _m{hVr XmI{dUmam {ÌÁ`m§Va I§S>mboI H$mT>m.

G$Vy _Vm§Mr g§»`m

CÝhmim

nmdgmim

qhdmim

90

120

150

3. Imbrb VŠË`m_Üò H$mhr bmoH$m§Zr ng§X Ho$bobo a§J Xe©{dbo AmhoV. `m _m{hVrdê$Z {ÌÁ`m§Va I§S>mboI H$mT>m.

a§J bmoH$m§Mr g§»`m

àËòH$ I§S>mMo à_mU emoYm. CXm. {Zim 1836

12

936

14=

1836

12

936

14

; {hadm 1836

12

936

14=

1836

12

936

14 , Am{U `mà_mUo `mMm

Cn`moJ g§JV H$moZ H$mT>Ê`mgmR>r H$am.

{Zim 18

{hadm 9

bmb 6

{ndim 3

EHy$U 36

4. ~mOybm {Xboë`m {ÌÁ`m§Va I§S>mboImV EH$m {dÚmÏ`mªZo EH$m n[ajoV qhXr, B§½bre, J{UV, g_mO emñÌ Am{U {dkmZ {df`m§V {_i{dbobo JwU {Xbo AmhoV. Ë`m {dÚmÏ`m©Zo EHy$U 540 JwU {_i{dbo AmhoV. Va Imbrb àûZm§Mr CÎmao Úm.

(i) {dÚmÏ`m©Zo H$moUË`m {df`mV 105 JwU {_i{dbo AmhoV? (gyMZm : 540 JwUm§gmR>r EHy$U H|$ÐñW H$moZ = 360° åhUyZ 105 JwUmgmR>r H|$ÐñW H$moZ {H$Vr?)

(ii) qhXrnojm J{UVm_Üò {H$Vr OmñV JwU {_i{dbo AmhoV?

90°

55°70°

80°

g_mO emó

{dkmZB§½bre

J{UV

qhXr

65°

(iii) g_mO emñÌ Am{U J{UV_Üò {_i{dboë`m JwUm§Mr ~oarO hr {dkmZ Am{U qhXr _Üò {_i{dboë`m JwUmnojm OmñV Amho H$m ho Vnmgm. (gyMZm : H|$ÐñW H$moZ Aä`mgm)

42 J{UV

5. EH$m dgVrJ¥hmVrb {d{dY ^mfm ~mobUmè`m {dÚmÏ`mªMr g§»`m Imbr {Xbr Amho. `m _m{hVrdê$Z {ÌÁ`m§Va I§S>mboI V`ma H$am.

^mfm qhXr B§½bre _amR>r V{_i H$ÞS> EHy$U

{dÚmÏ`mªMr g§»`m 40 12 9 7 4 72

3.5. g§Yr Am{U g§^mì`Vm

H$mhr doim Ago KS>Vo H$s, nmdgmù`mV Vwåhr XaamoO aoZH$moQ> ~amo~a KoVm nU H$mhr {Xdg nmD$g òV Zmhr. na§Vy EH$ {Xdg MwHy$Z Vwåhr aoZH$moQ> KoÊ`mg {dgaVm Am{U Ë`mM {Xder ânya nmD$g nS>Vmo.

H$mhr doiobm EH$ {dÚmWu n[ajogmR>r 5 n¡H$s 4 àH$aUmMr CÎm_ V`mar H$aVmo. na§Vy AZoH$ àûZ V`mar Z Ho$boë`m àH$aUmda {dMmabo OmVmV.

gdmªZm _mhrV Amho H$s aoëdo R>oaboë`m doir YmdV AgVmV. na§Vy H$mhr doim Vwåhr doioV ñQ>oeZda nmohMVm nU Vw_Mr aoëdo Ceram òVo.

Aao! _mPm aoZH$moQ>

Aem àH$maÀ`m AZoH$ KQ>Zm§Zm AmnUmg gm_moao Omdo bmJVo. Vwåhmbm H$mhr doim g§Yr {_iVo Am{U Or Vwåhmbm hdr Agbr Var H$mhr doim {_iV Zmhr. Vwåhr Aer H$mhr OmXm CXmhaUo Úmb H$m? òWo H$mhr JmoîQ>r KS>Ê`mMr g§Yr qH$dm Z KS>Ê`mMr g§Yr `m g_mZ ZmhrV. aoëdo doioV òVo qH$dm òV Zmhr hr g§Yr g_mZ ZgVo. Ooìhm Vwåhr AmajUmMo {Q>H$sQ> IaoXr H$aVm Voìhm Vo à{Vjm `mXrV AgVo. Ooìhm Vwåhmbm òWo g§Yr {_iVo, Voìhm Vwåhr Amem H$ê$ eH$Vm H$s Vo Vw_À`m àdmemÀ`m doir {ZpíMV hmoB©b.

AmnU Ago H$mhr à`moJ H$ê$`m H$s Á`m§Mo {ZîH$f© g_mZ g§Yr {_iÊ`mMr AgVo.

3.5.1 H$mhr {ZînÎmr {_i{dUo (Getting a Result)AmnUmbm _mhrV Amho H$s, {H«$Ho$Q> gm_Zm gwê$ hmoÊ`mnydu H$moU n{hë`m§Xm \$b§XmOr H$aUma ho R>a{dÊ`gmR>r XmoÝhr g§KmMo H$U©Yma ZmUo\o$H$sgmR>r OmVmV.

Ooìhm ZmUo da CS>{dbo OmVo Voìhm H$moUVr {ZînÎmr da òÊ`mMr eŠ`Vm AgVo Ago Vwåhmbm dmQ>Vo? AWm©VM N>mn qH$dm H$mQ>m.

H$ënZm H$am H$s, Vwåhr EH$m g§KmMo H$U©Yma AmhmV Am{U Vw_Mm {_Ì Xþgè`m g§KmMm H$U©Yma Amho. Vwåhr ZmUo CS>{dbm Am{U Vw_À`m {_Ìmbm {dMmaVm. Vwåhr {ZînÎmrda {Z`§ÌU R>ody eH$Vm H$m? Vwåhmbm hdo Agbobo N>mn qH$dm H$mQ>m {_idy eH$Vm H$m? Oa Vwåhmbm H$mQ>m hdo Agob?

_m{hVr hmVmiUo 43

hr eŠ`Vm Zmhr. Aem à`moJmbm `mÑpÀN>H$ à`moJ åhUVmV. `m à`moJmV N>mn qH$dm H$mQ>m `m XmoZ {ZînÎmr {_iVmV.

à`ËZ H$am.

1. Oa Vwåhr ñHy$Q>a Mmby H$aÊ`mMm à`ËZ H$aV Agmb, Va g§^mì` {ZînÎmr H$moUË`m?

2. Ooìhm EH$ \$mgm da \o$H$bm OmVmo, Voìhm H$moUË`m ghm {ZînÎmtMr eŠ`Vm AgVo?

3. AmH¥$Vr (3.9) _Üò XmI{dë`mà_mUo Ooìhm AmnU MmH$ {\$a{dVmo Voìhm H$moUË`m {ZînÎmtMr eŠ`Vm Amho? `mXr H$am.

(`mMr {ZînÎmr åhUOo Xe©H$ H$moUË`m I§S>mV Wm§~Vmo.)

A

B

C

avÀæ 3.9

W R B

G Y

avÀæ 3.10AmH¥$Vr 3.9 AmH¥$Vr 3.10

4. Vw_À`mH$S>o Agboë`m EH$m {nedrV g_mZ AmH$mamMo nU doJdoJù`m a§JmMo nmM M|Sy> AmhoV. Am{U Vwåhr {nedrV Z nmhVm Ë`mVrb M|Sy> CMbmd`mMm Amho. Ë`mVyZ Vwåhmbm {_imboë`m {ZînÎmtMr `mXr ~Zdm. (AmH¥$Vr 3.10)


EH$m \$mgm da \o$H$Ê`mÀ`m KQ>ZoV : * n{hë`m IoimSy>bm ghm {_iÊ`mMr gdm©V _moR>r g§Yr Amho H$m? * Ë`mÀ`mZ§Va IoiUmè`m Xþgè`m IoimSy>bm ghm {_iÊ`mMr H$_r g§Yr Amho H$m? * g_Om Xþgè`m IoimSy>bm ghm {_imbo. Va `mMm AW© Agm hmoVmo H$m H$s,

{Vgè`m IoimSy>bm ghm {_iÊ`mMr g§Yr A{O~mV Zmhr?

3.5.2 g_ g§^mì` {ZînÎmr

EH$ ZmUo AZoH$ doim da CS>{dbo Am{U àËòH$ doir N>mn qH$dm H$mQ>m da Amboë`m§À`m Zm|Xr Zm|X Ho$ë`m. Imbrb {ZînÎmtMr VŠVm nhm Á`mV ZmUo da CS>{dÊ`mÀ`m KQ>Zm dmT>V Joë`m AmhoV.

44 J{UV

ZmUo CS>{dÊ`mMr

g§»`mN>mngmR>r JUVrIwUm (H)

N>mn da òÊ`mMr g§»`m

H$mQ>mgmR>r JUVrIwUm (T)

H$mQ>m da òÊ`mMr g§»`m

50 ||||| ||||| ||||| ||||| ||||| ||| 27 ||||| ||||| ||||| ||||| ||| 23

60 ||||| ||||| ||||| ||||| ||||| |||

28||||| ||||| ||||| |||||

||||| ||||| |||32

70 ... 33 ... 3780 ... 38 ... 4290 ... 44 ... 46100 ... 48 ... 52

Vwåhr OgOer ZmUo CS>{dÊ`mMr g§»`m dmT>{dV Omb VgVer N>mn da òÊ`mMr g§»`m d H$mQ>m da òÊ`mMr g§»`m Odinmg gmaIr òVo. hr {H«$`m \$mgm \o$Hy$Zhr H$aVm òVo. Ooìhm Amåhr \$mgm AZoH$ doim \o$H$Vmo Voìhm 6 n¡H$s àËòH$ {ZînÎmtÀ`m g§»`m hr EH$_oH$mbm Odinmg g_mZ {_iVrb. `m àH$maÀ`m KQ>Zm_Üò, AmnU Ago åhUy eH$Vmo H$s, à`moJmÀ`m doJdoJù`m {ZînÎmr g_ g§^mì` AgVmV. `mMm AW© Agm H$s àËòH$ {ZînÎmr hmoÊ`mMr g§Yr g_mZ Amho.

AemàH$mao _r nS>oZ qH$dm Wm§~oZ.

AemàH$mao _rq^Vrbm bmJyZ nS>oZ.

qMVm H$é Z\$m, `màH$maMr KQ>Zm KS>Uo H$_r Agë`mZo _mPr Vr {ZînÎmr hmoV Zmhr.

3.5.3 g§Yr g§^mì`Vobm OmoS>Uo (Linking Chances to probability)

EH$ ZmUo EH$Xm da CS>{dÊ`mÀ`m à`moJmMm {dMma H$am. n[aUm_ H$m` {_iVmo? oWo \$ŠV XmoZ n[aUm_ ({ZînÎmr) AmhoV. N>mn qH$dm H$mQ>m. XmoÝhr {ZînÎmr g_g§^mì` AmhoV. N>mn da òÊ`mMr Ano{jV

g§^mì`Vm XmoZ {ZînÎmrn¡H$s EH$ AgVo AWm©V 50

10012

= . Xþgè`m eãXmV Amåhr Ago gm§Jy eH$Vmo H$s, N>mn

òÊ`mMr g§^mì`Vm = 50100

12

= . H$mQ>m da òÊ`mMr g§^mì`Vm {H$Vr?

_m{hVr hmVmiUo 45

AmVm \$mgm da \o$H$Ê`mMo CXmhaU ¿`m. \$memda 1,2,3,4,5,6 IwUm (EH$m ~mOwda EH$ g§»`m) AgVmV. Oa \$mgm EH$Xm \o$H$bm, Va H$moUVr {ZînÎmr {_iVo?

{ZînÎmr 1, 2, 3, 4, 5, qH$dm 6 Aer AgVo. Va òWo ghm g_g§^mì` {ZînÎmr.

2 A§H$mMr {ZînÎmr òÊ`mMr g§^mì`Vm {H$Vr?

Vr Amho1 → {ZîH$f© òÊ`mMr g§»`m

6 → g_g§^mì` {ZînÎmr§ òÊ`mMr g§»`m

5 hr g§»`m da òÊ`mMr g§^mì`Vm {H$Vr? 7 hr g§»`m da òÊ`mMr g§^mì`Vm {H$Vr? 1 Vo 6 n`ªV g§»`m da òÊ`mMr g§^mì`Vm {H$Vr?

3.5.4 KQ>Zm§À`m ñdénmV {ZînÎmr (Outcomes as Event)

à`moJmVrb àËòH$ {ZînÎmr qH$dm {ZînÎmrMm JQ> `mZr KQ>Zm (event) V`ma hmoVo.

CXm. ZmUo \o$H$Ê`mÀ`m à`moJmV da N>mn {_iUo hr EH$ KQ>Zm Amho Am{U H$mQ>m {_iUo hr gwÜXm KQ>Zm Amho.

\$mgm \o$H$Ê`mÀ`m {H«$òV {_iUmar àËòH$ {ZînÎmr 1, 2, 3, 4, 5 qH$dm 6 `m KQ>Zm (event) AgVmV.

g_ g§»`m {_iUo hr KQ>Zm Amho H$m? Á`m AWr°© g_g§»`m 2,4 AWdm 6 AgVrb. g_g§»`m {_iUo hr gwÜXm KQ>Zm Amho.g_g§»`m {_iÊ`mMr g§^mì`Vm {H$Vr?

Vr Amho 36 . òWo

3 → {ZîH$f© òÊ`mMr g§»`m

6 → g_g§^mì` {ZînÎmr òÊ`mMr g§»`m

CXmhaU 3 : EH$m {nedrV 4 bmb Am{U 2 {ndio M|Sy> AmhoV (M|Sy> doJdoJù`m a§JmMo AgyZ BVa ~m~VrV gmaIoM AmhoV) {nedrVrb EH$ M|Sy> {nedrV Z nmhVm H$mT>bm Va bmb M|Sy> {_iÊ`mMr g§^mì`Vm H$m`? Vo {ndim M|Sy> òÊ`mÀ`m g§^mì`Vonojm OmXm qH$dm H$_r Amho H$m?

CH$b : òWo gd© {_iyZ (4 + 2) = 6 g§^d{Z` KQ>Zm AmhoV. bmb M|Sy> òÊ`mÀ`m g§^dZr`

KQ>Zm 4 AmhoV (H$m?). åhUyZ bmb M|Sy> òÊ`mMr g§^mì`Vm 46

23

= Amho. `mM nÜXVrZo {ndim

M|Sy> òÊ`mMr g§^mì`Vm 26

13

= . (H$m)? åhUyZ bmb M|Sy> òÊ`mMr g§^mì`Vm hr {ndim M|Sy>

òÊ`mÀ`m g§^mì`Vonojm OmñV Amho.

46 J{UV

à`ËZ H$am.

g_Om Vwåhr MmH$ {\$a{dbo Va, Xe©H$ -

(i) {haì`m {ÌÁ`m§VaI§S>mda òÊ`mMr {ZînÎmr g§»`m d {haì`m {ÌÁ`m§VaI§S>mda> Z òÊ`mMr {ZînÎmr g§»`m `m§Mr `mXr H$am (AmH¥$Vr 3.11).

(ii) {haì`m {ÌÁ`m§VaI§S>mda> òÊ`mMr g§^mì`Vm H$mT>m.

(iii) {haì`m {ÌÁ`m§VaI§S>mda Z òÊ`mMr g§^mì`Vm H$mT>m.

RR

R

G

GGG

G

AmH¥$Vr 3.11>

3.5.5 g§Yr Am{U g§^mì`Vm `m§Mm dmñVd OrdZmer g§~§Y

Amåhr g§Yr{df`r ~mobVmo H$s, Ooìhm AmnU Á`m{Xder aoZH$moQ> KoD$Z OmV Zmhr Ë`m{Xder nmD$g nS>Vmo. òWo g§Yr hr g§^mì`Vm_Üò H$m` gm§Jy eH$mb? Oa 10 {XdgmVyZ EH$Xm nmD$g nS>V Agob

Va nmD$g òÊ`mMr g§^mì`Vm 110

910 Am{U nmD$g Z òÊ`mMr g§^mì`Vm

110

910 . (EHo$ {Xder nmD$g nS>Uo

qH$dm Z nS>Uo ho g_g§^mì` g_Omdo.)

1. _moR>çm JQ>mMr {deof bjUo Ë`mVrb EH$ N>moQ>m JQ> KoD$Z emoYUo. CXmhaUmW©, {ZdS>UyH$sÀ`m doir EŠPrQ> nmob KoVbm OmVmo. g§nyU© _VXmZ joÌmV _VXmZ H$éZ Amë`mZ§Va H$mhr bmoH$m§Zm {dMmabo OmVo H$s Ë`m§Zr H$moUmbm _VXmZ Ho$bo. mdéZ C_oXdmamMo {dO`mMo ^mH$sV dV©{dbo OmVo.

2. hdm_mZ ImVo _mJrb AZoH$ dfmªÀ`m _m{hVrÀ`m {ZarjUmdéZ Mmby hdm_mZmMm A§XmO ì`ŠV H$aV AgVo.

ñdmÜ`m` 3.3 1. `m à`moJm_Üò {XgyZ òUmè`m {ZîH$fmªMr `mXr H$am.

(a) MmH$ {\$a{dUo

(b) XmoZ ZmUr EH$mMdoir da CS>{dUo.

A B

C

D

A

2. Ooìhm EH$ \$mgm da \o$H$bm OmVmo, Voìhm {_iUmè`m g§^mì` KQ>Zm§À`m {ZîH$fmªMr `mXr H$am.

(i) (a) _yi g§»`m Amho (b) _yi g§»`m Zmhr.

(ii) (a) 5 nojm _moR>r EH$ g§»`m (b) 5 nojm _moR>r g§»`m Zmhr.

_m{hVr hmVmiUo 47

3. emoYm.

(a) Xe©H$ D da Wm§~Ê`mMr g§^mì`Vm? (àûZ 1-(a)) _Üò

(b) {ngboë`m 52 nË`m§À`m OmoS>mVyZ EH$ E¸$m òÊ`mMr g§^mì`Vm?

(c) bmb g\$aM§X òÊ`mMr g§^mì`Vm (Imbr {Xbobr AmH¥$Vr nhm).

G R R

RR G

G

4. Xhm doJdoJù`m {MÇ>`mda 1 Vo 10 g§»`m {b{hboë`m AmhoV (EH$m {MÇ>rda EH$ g§»`m) d Ë`m {MÇ>`m EH$m noQ>r_Üò CÎm_ [aVrZo {_giyZ R>odë`m AmhoV. noQ>rV Z nmhVm EH$ {MÇ>r CMbbr Va Imbrb ~m~VrV g§^mì`Vm gm§Jm.

(i) 6 hr g§»`m {_iVo.

(ii) 6 nojm bhmZ EH$ g§»`m {_iVo.

(iii) 6 nojm _moR>r EH$ g§»`m {_iVo.

(iv) 1 A§H$s EH$ g§»`m {_iVo.

5. Vw_À`mH$S>o 3 {hado {ÌÁ`m§VaI§S>, 1 {Zim {ÌÁ`m§VaI§S> Am{U 1 bmb {ÌÁ`m§VaI§S> Ag-bobo EH$ {\$aUmao MmH$ Amho. Va EH$ {ÌÁ`m§VaI§S> {_iÊ`mMr g§^mì`Vm {H$Vr? {Zim {ÌÁ`m§VaI§S> Zgbobo {ÌÁ`m§VaI§S> {_iÊ`mMr g§^mì`Vm {H$Vr?

6. àíZ 2 _Yrb KQ>Zm§Mr g§^mì`Vm H$mT>m.

AmnU H$moUVr MMm© Ho$br?

1. {deofVm Amåhmbm {_imbobr _m{hVr EH$ÌrV ZgVo. {Vbm H$ÀMr _m{hVr åhUVmV.

2. _m{hVrnmgyZ AW©nyU© {ZîH$f© H$mT>Ê`mgmR>r _m{hVr ì`dpñWV aMbobr Agbr nm{hOo.

3. EImXr R>am{dH$ Zm|X AZoH$ doim oV Agob Va {Vbm dma§dmaVm åhUVmV.

4. H$ÀÀ`m _m{hVrMo EH$Ì JQ> H$ê$Z dJuH¥$V> dma§dmaVm {dVaUmÀ`m ñdê$nmV ì`dpñWV _m§S>Vm òVo.

5. dJuH¥$V _m{hVr {hñQ>moJ«m_Zo XmI{dbr OmVo. {hñQ>moJ«m_ hm EH$$ ñV§^mboI AgyZ Ë`mÀ`m AmS>ì`m aofoda g§^mJloUr Am{U ñV§^m§Mr C§Mr hr Ë`m§Mr dmadm§aVm Xe©{dVo. VgoM XmoZ ñV§^m_Üò _moH$ir OmJm ZgVo H$maU XmoZ g§^mJ loUr_Üòhr _moH$ir OmJm ZgVo.

48 J{UV

6. _m{hVr hr dVw©imboImZo qH$dm {ÌÁ`m§VaI§S>mboImZo Xe©{dVm òVo. dVw©imboI nyU© Am{U Ë`mMo ^mJ `mÀ`mVrb g§~§Y XmI{dVmo.

7. H$mhr à`moJmVrb KQ>Zm_Ü o àËòH$mbm {ZîH$fm©À`m g_mZ g§Yr AgVmV.

8. `mÑpÀN>H$ à`moJm_Üò {ZîH$f© AJmoXa {ZpíMV H$mTy> eH$V Zmhr.

9. Ooìhm àËòH$mbm g_mZ g§Yr CnbãY AgVo, Voìhm à`moJmVrb {ZînÎmr g_g§^mì` AgVo.

10. Ooìhm {ZînÎmr g_g§^mì` AgVo, Voìhm,

EH$m KQ>ZoMr g§^mì`Vm = KQ>Zoer AZwHy$b KQ>Zm§Mr g§»`m

à`moJmVrb EHy$U {ZînÎmtMr g§»`m

11. à`moJmVrb EH$ qH$dm AZoH$ {ZînÎmr KQ>Zm ~Z{dVmV.

12. g§Yr Am{U g§^mì`Vm dmñVd OrdZm§er g§~§{YV AgVmV.

4.1 n[aM`

_mohZZo ñdV…gmR>r Am{U Ë`mÀ`m ~{hUrgmR>r Mhm ~Z{dbm. Ë`mgmR>r Ë`mZo 300 {_.br. nmUr, 2 M_Mo gmIa, 1 M_Mm MhmnmdS>a Am{U 50 {_.br. XÿY dmnabo. Va 5 _mUgm§À`mgmR>r AgmM Mhm ~Zdm`Mm Agob Va Ë`mbm darb àËòH$ KQ>H$ {H$Vr bmJob?

Oa EH$m gôV 2 {dÚmÏ`mªZm IwÀ`m© bmdÊ`mgmR>r 20 {_{ZQ>o bmJV AgVrb, Va VodT>çmM IwÀ`m© bmdÊ`mg 5 {dÚmÏ`mªZm {H$Vr doi bmJob?

Amåhmbm X¡Z§{XZ OrdZmV Aem AZoH$ KQ>Zm§Zm gm_moao Omdo bmJVo, OoWo EH$m KQ>H$mÀ`m amer_Üò ~Xb Pmë`mg Xþgè`m KQ>H$mÀ`m amer_Üò gwÜXm ~Xb hmoVmo.

CXmhaUmW© :

(i) Oa IaoXr Ho$boë`m dñVw§Mr g§»`m dmT>br H$s Ë`m§Mr EHy$U IaoXr qH$_V gwÜXm dmT>Vo.

(ii) Oa ~±Ho$V OmñV a¸$_ R>odbr Va ì`mO OmXm {_iVo.

(iii) Oa dmhZmMr JVr dmT>dbr Va R>am{dH$ A§Va OmÊ`mg doi H$_r bmJVmo.

(iv) EImXo H$m_ nyU© H$aÊ`mgmR>r OmXm H$m_Jma bmdë`mg Vo H$m_ H$_r doioV nyU© hmoVo.

darb CXmhaUmdê$Z EH$m KQ>H$m_Yrb amerVrb ~Xb Xþgè`m KQ>H$m_Yrb amerbm gwÜXm ~XbVmo.

Aem 5 KQ>Zm {bhm Á`m§_Üò EH$m KQ>H$mÀ`m g§»òV hmoUmè`m ~XbmZo Xþgè`m KQ>H$mÀ`m g§»òV ~Xb hmoVmo.

_mohZbm 5 _mUgm§Zm Mhm H$aÊ`mgmR>r àËòH$ KQ>H$ {H$Vr bmJob? qH$dm nmM {dÚmÏ`mªZm IwÀ`m© bmdÊ`mg {H$Vr doi bmJVmo? ho H$go emoYy eH$mb?

Aem àíZm§Mo CÎma emoYÊ`mgmR>r Amåhr H$mhr ~Xbm§À`m KQ>H$m§Mm Aä`mg H$é`m.

àH$aU

4 g_à_mU Am{U ì`ñVà_mU

50 J{UV

4.2 g_à_mU (DIRECT PROPORTION)

1 kg gmIaoMr Mr qH$_V ` 36 Amho, Va 3 kg gmIaoMr qH$_V {H$Vr? Vr ` 108 hmoB©b. `mà_mUo 5 kg d 8 kg gmIaoMr qH$_V Amåhr H$mTy> eH$Vmo. Imbrb VŠË`mMm Aä`mg H$am.

gmIaMo dOZ (kg _Üò)

qH$_V (` _Üò)

{ZarjU H$am, gmIaoMo dOZ dmT>bo AgVm Ë`mMr qH$_Vhr Aer dmT>Vo H$s Ë`m§À`mVrb JwUmoÎma pñWa amhVo. Xþgao EH$ CXmhaU KoD$. g_Om EH$m H$mabm 60 km A§Va OmÊ`mg 4 {bQ>a noQ´>mob bmJVo, Va 12 {bQ>a noQ´>mob_Üò Vr H$ma {H$Vr A§Va OmB©b? `mMo CÎma Amho 180 km. ho AmnU H$go H$mT>Vmo? Ogo H$s Xþgè`m KQ>ZoV 12 {bQ>a noQ´>mob dmnaVmo åhUOoM 4 {bQ>aÀ`m {VßnQ>. åhUyZ H$maZo AmH«${_bobo A§Va gwÜXm 60 km À`m {VßnQ> Agob. Xþgè`m eãXmV gm§Jm`Mo Pmë`mg Ooìhm noQ´>mob {VßnQ>rZo dmT>{dbo Voìhm AmH«${_bobo A§Va gwÜXm {VßnQ>> hmoVo. g_Om dmnabobo noQ´>mob x {bQ>a Amho d Ë`mZwgma AmH«${_bobo A§Va y km Amho. Imbrb VŠVm nyU© H$am.

noQ´>mob ({bQ>a_Üò) (x) 4 8 12 15 20 25

A§Va (km _Üò) (y) 60 ..... 180 ..... ..... .....

òWo AmnUmg Ago {XgyZ òVo H$s, Oer x Mr qH$_V dmT>V OmVo Ver y Mr qH$_V Aer dmT>Vo H$s x

y ho JwUmoVa ~XbV Zmhr. Vo H$m`_ pñWa amhVo. (g_Om k). `m KQ>ZoV JwUmoÎma 1

15

Amho (Vnmgm!). Aem Jmoï>r_Üò Amåhr gm§Jy eH$Vmo H$s, x Am{U y ho g_ à_mUmV AmhoV, Oa xy= k qH$dm x = k y.

`m CXmhaUmV, 460

12180

= , OoWo 4 Am{U 12 `m dmnaboë`m noQ´>mobogmR>r g§»`m {bQ>a (x)

_Üò Am{U 60 Am{U 180 ho AZwH«$_o AmH«${_bobo A§Va (y) Agob (km) _Üò.

g_à_mU Am{U ì`ñVà_mU 51

åhUyZ Ooìhm x Am{U y ho g_à_mUmV AgVrb Voìhm Amåhr Ago {bhÿ eH$Vmo xy

xy

1

1

2

2

= . (y À`m y1, y2 `m qH$_Vr x À`m x1, x2 À`m qH$_Vrer AZwH«$_o g§JV AgVrb).

H$ma_Üò Agbobo noQ´>mob Am{U H$maZo AmH«${_bobo A§Va hr g_à_mUmMr KQ>Zm Amho. `mMà_mUo, EHy$U IM© Ho$bobr a¸$_ Am{U {dH$V KoVboë`m dñVy ho gwÜXm g_à_mUMo CXmhaU Amho.

AmUIr H$mhr g_à_mUm§À`m CXmhaUm§Mm {dMma H$ê$.

_mJrb CXmhaUmV, _mohZZo 750 ml nmUr, 5 M_Mo gmIa, 2 12

M_Mm MhmnmdS>a Am{U

125 ml XÿY dmnê$Z 5 OUm§À`mgmR>r Mhm ~Z{dbm. darb CXmhaUmVrb KQ>H$ g_à_mUmV AmhoV H$m? `mMm {dMma H$am. Imbrb CXmhaUm§dê$Z g_à_mUmMr g§H$ënZm g_OyZ KoÊ`mMm à`ËZ H$am.

ho H$am : (i) * EH$ KS>çmi ¿`m d Ë`mMm {_{ZQ>H$mQ>m 12 da R>odm.

* {_{ZQ> H$mQ>m _yi ñWmZmnmgyZ nwT>o-nwT>o OmV Agob Vgo VoWo Pmbobo H$moZ _moOm d `m§Mr Zm|X Imbrb VŠË`mV H$am.

Pmbobm doi (T) {_{ZQm>_Üò(T1) 15

(T2) 30

(T3) 45

(T4) 60

Pmbobm H$moZ (A) A§em_Üò(A1) 90

(A2)....

(A3) .....

(A4) .....

TA

T Am{U A {df`r Vwåhmbm H$m` AmT>ibo? Vo XmoÝhrhr dmT>VmV> H$m? TA

ho JwUmoÎma àËòH$ doir g_mZ Amho H$m?

Oa {_{ZQ> H$mQ>çmZo Pmbobo H$moZ nwT>o-nwT>o Pmboë`m doio§À`m g_à_mUmV AmhoV H$m? hmo`!

darb VŠË`mdê$Z Vwåhr Ago nmhÿ eH$mb.

T1 : T2 = A1: A2, H$maU T1 : T2 = 15:30 = 1:2 A1 : A2 = 90:180 = 1:2 Vnmgm T2 : T3 = A2 : A3 d T3 : T4 = A3 : A4. darb H¥$Vr Vw_À`m ñd…VmÀ`m doioZwgma nwÝhm nwÝhm VnmgyZ nhm.

ii. nwT>rb VŠVm Vw_À`m {_Ìmbm amd`mg gm§Jm Am{U Ë`mMo d` d Ë`mÀ`m AmB©Mo g§JV d` `m§Mo JwUmoÎma H$mT>m.

52 J{UV

5 dfm©nyduMo d` AmVmMo d` 5 dfmªZ§VaMo d`

{_ÌmMo d` (F)

AmB©Mo d` (M)

FM

Vwåhmbm H$m` AmT>iVo?

F Am{U M `m qH$_Vr EH${ÌVnUo dmT>ë`m qH$dm H$_r Pmë`m H$m?FM

JwUmoÎma àËòH$ doir g_mZ Amho H$m? Zmhr!

Vw_À`m BVa {_Ìm§gmo~V Vwåhr hr H¥$Vr nwÝhm nwÝhm H$éZ Vw_Mo {ZîH$f© {bhÿ eH$Vm.

`mdê$Z ho Amdí`H$ Zmhr H$s EH${ÌV dmT>Umar qH$dm H$_r hmoUmar MbnXo g_à_mUmV AgmdrV.CXmhaUmW© :

(i) _mZdmVrb emar[aH$ ~Xb Ë`mÀ`m d`mZwgma hmoVmV. na§Vw Ë`mÀ`m nyduÀ`m JwUmoÎmamZwgma hmoUo JaOoMo Zmhr.

(ii) àËòH$ ì`ŠVrMr C§Mr Am{U Ë`mÀ`m dOZmVrb ~Xb ho _mhrV Agboë`m à_mUmV ZgVmV.

(iii) EImÚm PmS>mMr C§Mr Am{U Ë`mÀ`m \$m§Úmda dmT>Umè`m nmZm§Mr g§»`m m§Mm_Üò àË`j g§~§Y qH$dm JwUmoÎma ZgVo.

à`ËZ H$am 1. Imbrb VŠË`mMo {ZarjU H$éZ x d y ho g_à_mUmV AmhoV H$m Vo

emoYm.

(i) x 20 17 14 11 8 5 2y 40 34 28 22 16 10 4

(ii) x 6 10 14 18 22 26 30y 4 8 12 16 20 24 28

(iii) x 5 8 12 15 18 20y 15 24 36 60 72 100


p r tp1

100+ −

p t r× ×100

2. _wÔb = ` 1000, ì`mO Xa = 8% Xagmb. `mdéZ Imbrb VŠVm âm Am{U H$moUË`m àH$maMo ì`mO (gai qH$dm MH«$dmT>ì`mO) _wXVrZwgma g_à_mUmV ~XbVo Vo emoYm.

_wXV 1 df© 2 df© 3 df©

gaiì`mO

MH«$dmT>ì`mO


AmnU _wXV Am{U ì`mOmMm Xa {ZpíMV Ho$bm Va gaiì`mO ho _wÔbm~amo~a à_mUmV ~XbVo. AemM àH$maMm g§~§Y MH«$dmT>ì`mOmgmR>r Amho H$m`? H$m?

Mbm AmnU H$mhr gmoS>{dbobr CXmhaUo nmhÿ Á`m_Üò g_à_mUmMr g§H$ënZm dmnabr Amho.

CXmhaU 1 : 5m H$mnS>mMr qH$_V ` 210 Amho. Va 2m, 4m, 10m Am{U 13m H$mnS>m§Mr qH$_V Xe©{dUmam VŠVm V`ma H$am.

CH$b : g_Om H$mnS>mMr bm§~r x _rQ>a Amho Am{U Ë`mMr qH$_V ` y Amho.

x 2 4 5 10 13y y2 y3 210 y4 y5

H$mnS>mMr bm§~r Oer dmT>ob, Ë`m à_mUmV Ë`mMr qH$_V gwÜXm dmT>Vo. hr g_à_mUmMr KQ>Zm Amho.

Amåhr xy

xy

1

1

2

2

= àH$maMm g§~§Y Cn`moJmV AmUy.

(i) òWo x1 = 5, y1 = 210 Am{U x2 = 2.

åhUyZ xy

xy

1

1

2

2

= ho XoVo 5210

2

2

=y

qH$dm 5y2 = 2 × 210

qH$dm y22 210

584=

×=

(ii) Oa x3 = 4, Va, 5

2104

3

=y qH$dm 5 y3= 4×210

åhUOo, y34 210

5168= × =

xy

xy

2

2

3

3

=

ho AmnU dmnê$ eH$Vmo H$m? à`ËZ H$am

54 J{UV

(iii) Oa x4= 10, Va 5

21010

4

=y

qH$dm y410 210

5420=

×=

(iv) Oa x5= 13, Va 5210

13

5

=y

qH$dm y513 210

5546=

×=

[gyMZm - oWo 2

84 qH$dm

4168

qH$dm 10420

m§Mm dmna 5

210 À`m EodOr H$ê$ eH$Vmo.]

CXmhaU 2 : 14m C§MrÀ`m {dÚwV Im§~mMr gmdbr 10m nS>Vo. Va Ë`mM pñWVrV (doiobm) 15m gmdbr Agboë`m PmS>mMr C§Mr H$mT>m.

CH$b : PmS>mMr C§Mr x m. _mZy. AmnU Imbrbà_mUo VŠVm V`ma H$ê$.

dñVyMr C§Mr (_rQ>a_Üò) 14 x

gmdbrMr bm§~r (_rQ>a_Üò) 10 15

dñVyMr C§Mr OmñV Agob Va Ë`mÀ`m gmdbrMr bm§~r gwÜXm OmñV AgVo. hr g_à_mUmMr KQ>Zm

Amho. åhUyZ xy

xy

1

1

2

2

=

1410 15

=x

(H$m?)

qH$dm 14 1510×

= x

qH$dm 14 32×

= x

åhUyZ 21 = x

åhUyZ PmS>mMr C§Mr 21 _rQ>a (m).

`m CbQ> Amåhr Ago hr {bhÿ eH$Vmo. xx

yy

1

2

1

2

= ho xy

xy

1

1

2

2

=

åhUyZ x1: x2 = y1 : y2 qH$dm 14 : x = 10:15

åhUyZ, 10 × x = 15 × 14

qH$dm x = ×=

15 1410

21


CXmhaU 3 : Oa 12 OmS> H$mJXm§Mo dOZ 40 gms Amho, Va 2 12

kg dOZm_Üò Ago {H$Vr H$mJX AgVrb?

CH$b : 2 12

kg dOZm_Üò x H$mJX AmhoV Ago _mZy. BVa _m{hVr Imbrb VŠË`mV ^ê$.

H$mJXm§Mr g§»`m 12 x

H$mJXm§Mo dOZ (gms) 40 2500

H$mJXm§Mr g§»`m dmT>br H$s Ë`m§Mo dOZhr Ë`mM à_mUmV dmT>Vo. ho g_à_mU Amho.

åhUyZ, 1240 2500

=x

1 kg = 1000 gms2½ kg = 2500 gms

qH$dm 12 250040×

= x

qH$dm 750 = x

åhUyZ BpÀN>V H$mJXm§Mr g§»`m = 750.

n`m©`r nÜXV :

XmoZ KQ>H$ x Am{U y ho g_à_mUmV ~XbVmV Voìhm Ë`m§À`mVrb g§~§Y x = ky qH$dm xy

k= AgVmo.

òWo K = H$mJXm§Mr g§»`mË`m§Mo dOZ _Üògm ( ) = =

1240

310

AmVm x hr H$mJXm§Mm g§»`m AgyZ Ë`m§Mo dOZ 2 12

kg (2500 gm)

x = ky g§~§Y dmnéZ x = × =3

102500 750

åhUyZ 750 H$mJXm§Mo dOZ 2 12

kg.

CXmhaU 4 : EH$ aoëdo 75 km/Vmg(h) `m g_mZ JVrZo OmV Amho.

(i) Va Vr aoëdo 20 {_{ZQ>mV {H$Vr A§Va OmB©b?

(ii) Ë`mM JVrZo 250 km A§Va OmÊ`mg {Vbm {H$Vr doi bmJob?

CH$b : g_Om 20 {_{ZQ>mV aoëdoZo AmH«${_bobo A§Va x _mZy Am{U 250 km A§Va OmÊ`mg y {_{ZQ>o bmJVrb Ago _mZy.

56 J{UV

1 Vmg =60 {_{ZQ>oAmH«${_bobo A§Va 75 x 250

bmJbobm doi 60 20 y

JVr EH$ g_mZ Agë`mZo AmH«${_bobo A§Va {Vbm bmJUmè`m doioÀ`m g_à_mUmV AgVo.

(i) 7560 20

=x qH$dm

7560

20× = x qH$dm x = 25

åhUyZ aoëdoZo 20 {_{ZQ>mV AmH«${_bobo A§Va 25 km Agob.

(ii) 7560

250=

y qH$dm y = ×

=250 60

75200 {_{ZQ>o

aoëdobm 250 km A§Va OmÊ`mg bmJbobm doi 3 Vmg 20 {_{ZQ>o Agob. x _mhrV

Agë`mg y Mr H$mT>Ê`mgmR>r xy20

250= hm g§~§Y dmnamdm.

Vwåhmbm _mhrV Amho H$s ZH$mem hm \$ma _moR>çm joÌmMo à{V{Z{YËd H$aVmo. ZH$memÀ`m Imbrb mJmV à_mU {Xbobo AgVo. à_mU ho àË`j bm§~r Am{U ZH$memVrb bm§~r `m§Mo à_mU Xe©{dVo. ZH$memMo à_mU ho XmoZ q~Xÿ_Yrb àË`j Iao A§Va Am{U ZH$memVrb XmoZ q~Xÿ_Yrb A§Va m§Mo JwUmoÎma AgVo. CXm. Oa ZH$memV 1 cm ho àË`j 8 km A§Va. Va (Ë`mgmR>r ñHo$b 1cm : 8km qH$dm 1 : 800, 000) Va 2 cm ZH$memVrb A§Va ho àË`j 16 km Agob. åhUyZ ZH$memMo à_mU (ñHo$b) ho g_à_mUmÀ`m g§H$ënZoda AmYmarV AgVo.

CXmhaU 5 : EH$m ZH$memMo à_mU Ago {Xbo Amho 1 : 30000000. ZH$memV XmoZ ehao EH$_oH$m§nmgyZ 4 cm A§Vamda XmI{dbobr AmhoV. Va Ë`m§À`mVrb àË`j A§Va H$mT>m.

CH$b : ZH$memVrb A§Va x cm Am{U àË`j A§Va y cm _mZy. Va

1 : 30000000 = x : y qH$dm 1

3 107×=xy

x = 4 åhUyZ, 1

3 104

7×=

y

y = 4 × 3 × 107 = 12 × 107 cm = 1200 km.

qH$dm XmoZ ehao Or ZH$memV 4cm A§Vamda AmhoV Vr àË`jmV 1200km EH$_oH$mnmgyZ Xÿa AmhoV.

amO{H$` ZH$mem

à_mU 1 cm = 300 km


ho H$am :

Vw_À`m amÁ`mMm EH$ ZH$mem ¿`m. Ë`mda dmnabobo à_mU nhm. _moOnÅ>r KoD$Z XmoZ eham_Yrb ZH$mem A§Va _moOm. Ë`m§Mo àË`j A§Va gm§Jm.

ñdmÜ`m` 4.1

1. EH$m aoëdoñQ>oeZ Odi H$ma nmH$sªJMo Xa Imbrb à_mUo AmhoV.

4 VmgmgmR>r ` 60

8 VmgmgmR>r ` 100

12 VmgmgmR>r ` 140

24 VmgmgmR>r ` 180

nmH$sªJMo ^mS>o ho nmH$sªJ doioÀ`m g_à_mUmV Amho H$m? Vnmgm.

2. EH$m a§JmÀ`m _yi {_lUmÀ`m 8 ^mJm_Üò bmb a§JmMm 1 ^mJ {_giyZ EH$ a§J V`ma Ho$bm

OmVmo. Va VŠË`mV {Xë`mà_mUo bmb a§JmÀ`m {d{dY ^mJmV _yi {_lUmMm ^mJ {H$Vr

à_mUmV {_gimdm, ho AmoiIyZ VŠVm nyU© H$am.

bmb a§JmMo ^mJ 1 4 7 12 20

_yi {_lUmMo ^mJ 8 .... .... .... ....

3. àíZ H«$_m§H$ 2 _Üò Oa 1 ^mJ bmb a§JmV 25 ml _yi {_lU bmJVo. Va 180 ml _yi

{_lUm_Üò {H$Vr ^mJ bmb a§J bmJob?

4. EH$m {eVno` V`ma H$aÊ`mÀ`m H$maImÝ`mV EH _erZ ghm VmgmV 840 ~mQ>ë`m no` âVo.

Va `mM à_mUmV nmM VmgmV Vr _erZ {H$Vr ~mQ>ë`m$ âob.

5. EH$m OrdmUyMm _yi AmH$mamMm \$moQ>mo ({MÌ)

50,000 nQ> _moR>m Ho$bm AgVm \$moQ>moMr

bm§~r 5cm hmoVo (AmH¥$Vr nhm). Va Ë`m

OrdmUyMr _yi bm§~r {H$Vr? Oa OrdmUyÀ`m

_yi \$moQ>moMm AmH$ma \$ŠV 20,000 nQ> _moR>m

Ho$bm. Va Ë`m \$moQ>moMr bm§~r {H$Vr hmoB©b?

58 J{UV

6. EH$m OhmOmÀ`m _m°S>ob (Z_wZm) _Üò

àH$meXrn 9 cm C§Mrda Amho, na§Vy àË`j

OhmOmMm _yi àH$me {Xn 12 _r. C§Mrda

Amho. Oa _yi OhmOmMr bm§~r 28 _r. Agob

Va Ë`m Z_wZm OhmOmMr bm§~r {H$Vr Agob?

7. g_Om 2 kg gmIa_Üò 9×106 ñ\${Q>H$ H$U AmhoV. Va (i) 5 kg gmIao_Üò (ii) 1.2 kg gmIao_Üò {H$Vr ñ\$Q>{H$ H$U AgVrb?

8. aí_rH$S>o EH$ añË`m§Mm ZH$mem Amho H$s `m_Üò 1cm ho 18km à_mU Xe©{dVo. Oa {VZo añË`mdéZ JmS>rZo 72km àdmg Ho$bm Va ho A§Va ZH$memV {H$Vr hmoB©b?

9. EH$m 5m. 60cm bm§~rÀ`m Cä`m Im§~mMr gmdbr 3m. 20cm nS>Vo. Ë`mM doiobm (i) 10 _r. 50 cm C§MrÀ`m Im§~mMr gmdbr {H$Vr bm§~rMr Agob? (ii) 5m. 50cm bm§~rMr gmdbr AgUmè`m Im§~mMr C§Mr {H$Vr Agob?

10. _mb âbobm EH$ Q´>H$ 25 {_{ZQ>mV 14 km A§Va OmVmo. Oa JVr gmaIrM Agob Va 5 VmgmV Vmo {H$Vr A§Va OmB©b?

ho H$am

1. EH$m Mm¡agmH¥$Vr H$mJXmda doJdoJù`m ~mOw§Mo 5 Mm¡ag H$mT>m d Imbrb

VŠË`mV _m{hVr âm.

Mm¡ag 1 Mm¡ag 2 Mm¡ag 3 Mm¡ag 4 Mm¡ag 5

~mOwMr bm§~r (L)

n[a{_Vr (P)LP

joÌ\$i (A)

LA

àËòH$ Mm¡agmÀ`m ~mOwMr bm§~r hr

(a) Ë`m Mm¡agmMr n[a_rVr

(b) Ë`m Mm¡agmMo joÌ\$i `m§À`mer g_à_mUmV Amho H$m ho emoYm.


2. 5 _mUgm§À`mgmR>r hbdm ~Z{dÊ`mg bmJUmao gm{hË` nwT>rb à_mUo bmJVo. adm 250 gm, gmIa 300 gm, Vwn 200 gm, nmUr 500 ml `m à_mUmÀ`m g§H$ënZoMm Cn`moJ H$ê$Z Vw_À`m dJm©Vrb _wbm§À`mgmR>r bmJUmè`m hbì`mgmR>r {H$Vr à_mUmV darb gm{hË` bmJob `mMm {hemo~ H$am.

3. `mo½` à_mU {ZdS>m d Vw_À`m dJm©Mm ZH$mem V`ma H$am. Ë`m_Üò {ISŠ`m, XadmOo d \$im B. XmIdm.

(~mOybm EH$ CXmhaU {Xbo Amho.)

cm

Scale


AmVmn`ªV MMm© Ho$boë`m g_à_mU g§~§Yr H$mhr CXmhaUo ¿`m. Aer CXmhaUo EH$_mZ nÜXVrZo (Unitary Methods) gmoS>{dVm òVrb, `m~Ôb Vwåhr {dMma H$é eH$mb H$m?

4.3 ì`ñVà_mU (INVERSE PROPOTION)

XmoZ amer_Üò H$mhr doim Agm ~Xb hmoVmo H$s EH$m amerMr qH$_V dmT>ë`mg Xþgè`m amerMr qH$_V H$_r hmoVo Am{U `m CbQ> gwÜXm>. CXmhaUmW©, Oa H$m_Jmam§Mr g§»`m dmT>{dbr Va Vo H$m_ nyU© H$aÊ`mg bmJUmam H$mbmdYr H$_r bmJVmo. Ë`mMà_mUo Oa AmnU JmS>rMr JVr dmT>{dbr Va {Xbobo R>am{dH$ A§Va OmÊ`mg bmJUmam doi H$_r bmJVmo.

ho g_OyZ KoÊ`mgmR>r AmnU Imbrb H$mhr KQ>ZmH$S>o nmhÿ.

Pm{hXm Amnë`m emiobm Mma doJdoJù`m àH$mao OmD$ eH$Vo. Vr MmbV, niV, gm`H$bdéZ. qH$dm H$maZo OmD$ eH$Vo. Imbrb VŠVm Aä`mgm.

MmbUo niUo gm`H$b _maUo H$maZo

bmJbobm doi ({_{ZQ>_Üò)JVr (km/h _Üò)

òWo Ago {XgyZ òVo H$s JVr Oer dmT>ob Ë`mà_mUmV g_mZ A§Va OmÊ`mg bmJUmam doi H$_r H$_r hmoV OmVmo.

60 J{UV

Oa Pm{hXmZo niÊ`mMr JVr XþßnÅ>rZo dmT>{dbr Va doi {Zå`mZo H$_r hmoVmo.

{VZo gm`H$bMr JVr VrZ nQ> dmT>{dbr

Va {Vbm bmJUmam doi 13

Zo H$_r hmoVmo.

Ë`mMà_mUo H$maMr JVr 15 nQ>rZo dmT>{dbr Va

{Vbm bmJUmam doi 1

15 Zo H$_r hmoVmo. [qH$dm Xþgè`m eãXmV gm§Jmd`mMo Va doi H$_r hmoÊ`mMo

JwUmoÎma ho JVr dmT>{dÊ`mÀ`m JwUmoÎmamÀ`m ì`ñV

AgVo.]

EH$m g§»òMm JwUmH$ma ì`ñV hm nañnam§Mm

ì`wËH«$_ [reciprocal] AgVmo. Ogo 12 hm

2 Mm ì`ñV d 2 hm 12 Mm JwUmH$ma ì`ñV.

bjmV R>odm : 2 12

12

2 1× = × =

AmnU Ago åhUy eH$Vmo H$m doi Am{U JVr ho ì`ñV JwUmoÎmamV ~XbV AgVmV?

Xþgao EH$ CXmhaU {dMmamV ¿`m. EH$ emim J{UV nwñVHo$ IaoXrgmR>r `.6000 IM© H$aÊ`mMo R>a{dVo. àËòH$s ` 40 XamZo `m a¸$_oV {H$Vr nwñVHo$ {_iVrb? ñnï>nUo 150 nwñVHo$. Oa EH$m nwñVH$mMr qH$_V ` 40 nojm OmñV Agob Va Ë`mM qH$_VrV nwñVHo$ 150 nojm H$_r$ {_iVrb. Imbrb VŠVm nhm.

EH$m nwñVH$mMr qH$_V (` _Üò) 40 50 60 75 80 100

IaoXr Ho$boë`m nwñVH$m§Mr g§»`m 150 120 100 80 75 60

òWo Vwåhmbm H$m` AmT>iVo? Vwåhmbm Ago {XgyZ oVo H$s Ooìhm nwñVH$mMr qH$_V dmT>Vo Voìhm Ë`mM qH$_VrV nwñVH$m§Mr g§»`m H$_r hmoVo. EH$m nwñVH$mMr qH$_V Ooìhm ` 40 déZ ` 50 hmoVo Voìhm qH$_VtMo JwUmoÎma 4:5 hmoVo. na§Vy `mdoir nwñVH$m§Mr g§»`m 150 déZ 120 Aer H$_r hmoVo d `m§Mo JwUmoÎma 5:4 hmoVo. `mMm AW© Agm H$s XmoÝhr JwUmoÎmao hr EH$_oH$m§er ì`ñV AmhoV.

bjmV ¿`m H$s bJVÀ`m (g§JV) XmoZ ametMm JwUmH$ma pñWa AgVmo, åhUOoM 40 × 150 = 50 × 120 = 6000.

Oa EH$m nwñVH$mMr qH$_V x d `m XamZo IaoXr Ho$boë`m nwñVH$m§Mr g§»`m y _mZë`mg, Ogo x dmT>Vo Vgo y H$_r hmoVo Am{U CbQ> gwbQ>.

_hËdmMr Jmoï> bjmV ¿`m H$s JwUmH$ma xy pñWa amhVmo. AmnU Ago åhUy eH$Vmo H$s òWo x ho y À`m ì`ñVà_mUmV ~XbVo Am{U y ho x À`m ì`ñVà_mUmV ~XbVo. åhUyZ XmoZ qH$_Vr x Am{U

y `m ì`ñVà_mUmV ~XbVmV. Ë`m§À`m_Üò xy = k hm g§~§Y AgVmo. k ho pñWanX. Oa y1, y2 `m y

À`m g§JV qH$_Vr AZwH«$_o x À`m x1, x2 er ì`ñV AgVrb Va, x1 y1 = x2 y2 (=k) qH$dm xx

yy

1

2

2

1

= . AmnU x Am{U y ho ì`ñVà_mUmV AmhoV Ago åhUVmo.


åhUyZ darb CXmhaUmV EH$m R>am{dH$ qH$_Vr_Üò nwñVH$m§Mr qH$_V Am{U IaoXr Ho$boë`m nwñVH$m§Mr g§»`m EH$_oH$mg ì`ñVà_mUmV AgVmV. mà_mUo dmhZmMr JVr Am{U EH$ R>am{dH$ A§Va OmÊ`mgmR>r bmJUmam doi> ì`ñVà_mUmV AgVmV.

màH$maÀ`m H$mhr OmXm CXmhaUm§Mm {dMma H$é.

oWo H$mhr H¥$Vr AmhoV Ë`m_wio ì`ñVà_mU CÎm_[aË`m g_OyZ òB©b.

ho H$am

EH$ Mm¡agmH¥$Vr H$mJX ¿`m Am{U Ë`mda 48 MH$Ë`m doJdoJù`m n§ŠVrV Imbr XmI{dë`mà_mUo aMm.

4 n§ŠVr, 12 ñV§^

8 n§ŠVr, 6 ñV§^

n§ŠVtMr g§»`m (R)(R1) 2

(R2) 3

(R3) 4

(R4) 6

(R5) 8

ñV^m§Mr g§»`m (C)(C1)....

(C2)....

(C3) 12

(C4) 8

(C5) ....

Vwåhmbm H$m` {XgyZ Ambo? Ogo R Mr g§»`m dmT>Vo, Vgo C Mr g§»`m H$_r hmoVo.(i) R1 : R2 = C2 : C1 Ago Amho H$m`? (ii)R3 : R4 = C4 : C3 Ago Amho H$m`? (iii) R Am{U C ho nañnamg ì`ñV à_mUmV ~XbVmV H$m?hrM H¥$Vr 36 MH$Ë`m dmnéZ H$ê$Z nhm.

à`ËZ H$amImbrb VŠËò nhm Am{U Ë`mVrb ~XbUmar g§JV MbnXm§Mr OmoS>r (òWo x Am{U y) hr ì`ñVà_mUmV Amho H$m Vo nhm.

(i)x 50 40 30 20

(ii)x 100 200 300 400

y 5 6 7 8 y 60 30 20 15

62 J{UV

(iii)x 90 60 45 30 20 5y 10 15 20 25 30 35

Amåhr ì`ñVà_mUmMr g§H$ënZm dmnaboë`m H$mhr CXmhaUm§Mm {dMma H$ê$.

Oa x Am{U y `m XmoZ amer g_à_mUmV AgVrb Va Vo x a y Ago hr {b{hVmV.

Oa x Am{U y `m XmoZ amer ì`ñVà_mUmV AgVrb Vo x yα

1 Ago {b{hVmV.

CXmhaU 7 : EH$ nmÊ`mMr Q>mH$s nyU© âÊ`mgmR>r 6 nmB©n OmoS>ë`mg 1 Vmg 20 {_{ZQ>o bmJVmV. Va Ë`mM àH$maMo 5 nmB©n OmoS>ë`mg Vr Q>mH$s nyU© aÊ`mg {H$Vr doi bmJob?

CH$b : g_Om Q>mH$s aÊ`mg bmJUmam H$mbmdYr x {_{ZQ>o _mZy. mdéZ AmnU Imbrb VŠVm V`ma H$ê$.

nmB©nMr g§»`m 6 5doi ({_{ZQ>_Üò) 80 x

nmB©nMr g§»`m H$_rVH$_r Agob Va Q>mH$s aÊ`mg bmJUmam H$bmdYr A{YH$m{YH$ Agob. åhUyZ ho ì`ñV à_mUmMo CXmhaU Amho.

åhUyZ 80 × 6 = x × 5 [x1y1 = x2y2]

qH$dm 80 65× = x

qH$dm x = 96

Vr Q>mH$s 5 nmB©nZo nyU© aÊ`mg 96 {_{ZQ>o doi bmJob qH$dm 1 Vmg 36 {_{ZQ>o doi bmJob.

CXmhaU 8: EH$m hmoñQ>ob_Üò 100 {dÚmWu AmhoV. Ë`m§À`mgmR>r 20 {Xdg nwaob BVH$m AÞgmR>m Amho. Oa 25 {dÚmWu Ë`mV dmT>bo Va {H$Vr {Xdg Vmo AÞgmR>m nwaob?

CH$b : {dÚmÏ`mªMr g§»`m 125 AgVm§Zm Vmo AÞgmR>m y {Xdg nwaob Ago g_Oy.

{dÚmWu g§»`m 100 125

{Xdgm§Mr g§»`m 20 y

{dÚmÏ`mªMr g§»`m A{YH$ Pmë`mg AÞgmR>m nwaÊ`mMo {Xdg H$_r hmoVrb. åhUyZ ho ì`ñV à_mU Amho.


åhUyZ 100 × 20 = 125× y

qH$dm 100 20125

× = y

qH$dm 16 = y

`mdê$Z, Oa 25 {dÚmWu hm°ñQ>ob_Üò A{YH$ Pmë`mg Vmo AÞgmR>m 16 {Xdg nwaob.

`mCbQ> Amåhr hoM Ago {bhÿ eH$Vmo.

x1y1 = x2y2 Ogo xx

yy

1

2

2

1

= .

x1 : x2 = y2 : y1

100:125 = y : 20

y = 100 20125

× = 16

CXmhaU 9 : EH$ q^V 15 H$m_Jma 48 VmgmV ~m§YVmV. Va 30 VmgmV VoM ~m§YH$m_ nyU© H$aÊ`mgmR>r {H$Vr H$m_Jma bmJVrb?

CH$b : 30 VmgmV ~m§YH$m_ nyU© H$aÊ`mg y H$m_Jma bmJVrb Ago _mZy AmnUmg Imbrb VŠVm {_iVmo.

Vmgm§Mr g§»`m 48 30

H$m_Jmam§Mr g§»`m 15 y

Oa OmXm H$m_Jma AgVrb Va ~m§YH$m_ ObX hmoUma åhUyZ H$m_Jmam§Mr g§»`m d ~m§YH$m_mgmR>r bmJUmam doi ì`ñV à_mUmV AmhoV.

åhUyZ 48 × 15 = 30 × y

åhUyZ 48 15

30×

= y qH$dm y =24

30 VmgmV ~m§YH$m_ nyU© H$aÊ`mg 24 H$m_Jma bmJVrb.

ñdmÜ`m` 4.2

1. Imbrb n¡H$s H$moUVo ì`ñV à_mUmV AmhoV? (i) EH$ H$m_ nyU© H$aÊ`mg bmJUmè`m H$m_Jmam§Mr g§»`m d bmJUmam doi. (ii) EH$ R>am{dH$ A§Va OmÊ`mgmR>r bmJUmam doi Am{U EH$ g_mZ doJ. (iii) _emJV Ho$boë`m O{_ZrMo joÌ\$i d Ë`mVyZ KoVbobo nrH$. (iv) EH$m R>am{dH$ àdmgmgmR>r bmJUmam doi Am{U dmhZmMr JVr. (v) EH$m XoemMr bmoH$g§»`m Am{U Xa _mUgr O{_ZrMo joÌ\$i.

64 J{UV

2. XÿaXe©Zdarb (T.V) EH$m IoimÀ`m H$m`©H«$_mV `1,00,000 hr ~{jgmMr a¸$_ {dO`r ñnY©H$m§Zm g_mZ ^mJmV {d^mJbr nm{hOo. Imbrb VŠVm nyU© H$am Am{U d¡`pŠVH$ {Xbobo ~{jg ho {dOoË`m§À`m g§»òÀ`m g_à_mUmV qH$dm ì`ñVà_mUmV Amho H$m` Vo gm§Jm.

{dOoË`m§Mr g§»`m 1 2 4 5 8 10 20

d¡`pŠVH$ ~{jg a¸$_ (`._Üò) 1,00,000 50,000 ... ... ... ... ...

3. aoh_mZ Amao (Xm§S>o) dmnéZ EH$ MmH$ ~Z{dV Amho. Vmo g_mZ Amao Ago bmdy BpÀN>Vmo H$s H$moUË`mhr XmoZ H«$_mJV Amè`m§À`m OmoS>rVrb H$moZ g_mZ Agbo nm{hOo. Imbrb VŠVm nyU© H$aÊ`mg Ë`mbm _XV H$am.

Amè`m§Mr g§»`m 4 6 8 10 12

XmoZ H«$_mJV Amè`m_Yrb H$moZ 90° 60° .... .... ....

(i) Amè`m§Mr g§»`m Am{U H«$_mJV Amè`m§À`m OmoS>rVrb$ H$moZ ho EH$_oH$mg ì`ñV JwUmoÎmamV AmhoV H$m?

(ii) 15 Amao AgUmè`m MmH$mÀ`m XmoZ H«$_mJV Amè`m§À`m OmoS>r_Yrb H$moZ {H$Vr Agob?

(iii) XmoZ H«$_mJV Amè`m§À`m OmoS>r_Yrb H$moZ 40° Mm Agob Va Ë`mbm {H$Vr Amao bmJVrb?

4. noT>o ({_R>mB©) Agbobm EH$ ~m°Šg 24 _wbmZm g_mZ dmQ>bm Va àËòH$mbm 5 noT>o {_iVrb. Oa _wbm§Mr g§»`m 4 Zo H$_r Pmë`mg àËòH$mbm {_iUmao noT>o {H$Vr?

5. EH$m eoVH$è`mH$S>o 20 OZmdam§Zm 6 {Xdg nwaob BVH$m Mmam Amho. Oa Ë`mÀ`mH$S>o AmUIr OmXm 10 OZmdao Agë`mg Vmo Mmam {H$Vr {Xdg nwaob?

6. EH$m H§$ÌmQ>XmamZo 3 ì`pŠV§Zm Ogq_XaÀ`m Kar 4 {Xdgm§V dmàtJ H$aÊ`mgmR>r R>odbo. Oa Ë`mZo 4 ì`pŠV§Zm dmàtJ H$aÊ`mgmR>r R>odbo, Va Vo H$m_ {XdgmV {H$Vr nyU© hmoB©b?

7. àËòH$ bhmZ noQ>r_Üò 12 ~mQ>ë`m ^éZ Aem 25 _moR>çm noQ>çm âë`m AmhoV. Oa VemM d VodT>çmM ~mQ>ë`m àËòH$ bhmZ noQ>rV 20 âë`mg EHy$U _moR>çm noQ>çm {H$Vr âVrb?


8. EH$m H$maImÝ`mV 63 {XdgmV H$mhr dñVy V`ma H$aÊ`mgmR>r 42 `§ÌmMr JaO Amho. VodT>çmM dñVy 54 {XdgmV V`ma H$aÊ`mgmR>r {H$Vr `§Ìo bmJVrb?

9. EH$m H$mabm 60 km/h `m JVrZo EH$ R>am{dH$ A§Va OmÊ`mgmR>r 2 Vmg bmJVmV. Ë`mM H$mabm 80 km/h `m JVrZo VodT>oM A§Va OmÊ`mgmR>r {H$Vr doi bmJob?

10. EH$m Kambm XmoZ ì`ŠVr H$mhr ZdrZ {IS>Š`m 3 {XdgmV ~g{dVmV.

(i) Ë`mVrb EH$ ì`ŠVr H$m_ gwê$ H$aÊ`mAmYr AmOmar nS>Vo. Voìhm Vo H$m_ nyU© H$aÊ`mg Xþgè`m ì`ŠVrbm {H$Vr {Xdg bmJVrb?

(ii) EH$m {XdgmV Ë`m {IS>Š`m ~g{dÊ`mg {H$Vr ì`ŠVr bmJVrb?

11. EH$m emioVm EH$m {Xdgmbm àËòH$s 45 {_{ZQ>mÀ`m 8 Vm{gH$m (Periods) AmhoV. Oa emio_Üò {Xdgmbm 9 Vm{gH$m (Periods) Agë`mg àËòH$ Vm{gH$m {H$Vr {_{ZQ>m§Mr Agob? (emioÀ`m H$m_mMm AdYr gmaImM YéZ).

ho H$am

1. EH$ H$mJX ¿`m. AmH¥$VrV Xe©{dë`mà_mUo Vmo KS>r Kmbm. àËòH$ KQ>ZoVrb ^mJmMr g§»`m _moOm Am{U Ë`m ^mJm§Mo joÌ\$i H$mT>m.

Imbrb VŠË`m_Üò Vw_Mo {ZarjU Zm|Xdm d {_Ìmgmo~V MMm© H$am. hr ì`ñVà_mUmMr KQ>Zm Amho H$m? Oa Agob Va H$m?

^mJm§Mr g§»`m 1 2 4 8 16àËòH$ ^mJmMo joÌ\$i nyU© H$mJXmMo joÌ\$i 1/2 H$mJXmMo joÌ\$i

2. doJdoJù`m AmH$mamMr dVw©imH$ma VimMr H$mhr ^m§S>r ¿`m. àËòH$mV g_mZ _mnZmMo nmUr âm. àËòH$ m§S>çmMm ì`mg Am{U Ë`m§Mr AZwH«$_o C§Mr OoWo nmÊ`mMr nmVir Amho Vr _moOm.Vw_Mr {ZarjUo VŠË`mV âm. hr ì`ñVà_mUmMr KQ>Zm Amho H$m`?

^m§S>çmMm ì`mg (go§._r. _Üò)nmÊ`mÀ`m nmVirMr C§Mr (g§o._r._Üò)

66 J{UV

Amåhr H$m` MMm© Ho$br?

1. XmoZ amer x Am{U y g_à_mUmV AmhoV Ago åhUVmV, Ooìhm g_mZ[aË`m Ë`m§À`m qH$_Vr

dmT>VmV qH$dm H$_r hmoVmV Aem H$s, Ë`m§Mo JwUmoÎma Zoh_r pñWa amhVo. åhUOoM Oa xy

k=

(k YZ g§»`m Amho), Aem KQ>Zm_Üò Oa y1, y2 `m y À`m qH$_Vr x1,x2 `m x À`m AZwH«$_o

g§JV qH$_Vrer Aem AgVrb, Va xy

xy

1

1

2

2

= AgVmV.

2. XmoZ amer> x Am{U y `m ì`ñVà_mUmV AgVmV, Ooìhm x À`m qH$_VrV dmT> Pmë`mg

Ë`mÀ`m à_mUmV y À`m qH$_VrV KQ> hmoVo (d CbQ> gwbQ)> Am{U Ë`m§Mm qH$_VtMm JwUmH$ma

pñWa AgVmo.

Oa x y = k Voìhm x Am{U y ho ì`ñVà_mUmV ~XbVmV. `m KQ>ZoV Oa y1, y2 `m y À`m qH$_Vr x1, x2 `m x À`m g§JV qH$_Vr, Va x1y1 = x2y2 qH$dm

xx

yy

1

2

2

1

= .

5.1.n[aM`

Vwåhr dV©_mZnÌo, XÿaXe©Z, _m{gHo$, nwñVHo$ BË`mXr_YyZ à{gÕ hmoUmao AmboI nm{hbo AmhoV H$m? g§»`mË_H$ g§J«{hV _m{hVr {MÌê$nmV Xe©{dë`mZo _m{hVrMo {ddaU A{VObX g_OVo, VgoM _m{hVrMo gmoß`m nÜXVrZo Am{U ñnîQ>nUo AmH$bZ hmoVo. g§J«{hV _m{hVr {MÌê$nmV Xe©{dUo åhUOo AmboI. g§J«m{hV _m{hVrgmaUr ñdê$nmV gwÜXm Xe©{dVm òVo. AmboImÛmao Xe©{dboë`m g§J«{hV _m{hVrMo ObX AmH$bZ hmoVo. g§»`m§Mo {ddaU qH$dm Ë`m§Mr VwbZm XmI{dÊ`mgmR>r AmboI AË`§V Cn`wŠV AgVmV. AmVmn`ªV Amåhr AZoH$ àH$maMo AmboI g_OyZ KoVbo AmhoV. Ë`m§Mr AmnU AmR>dU H$ê$`m.

5.1.1. ñV§^mboI (Bar graph)

ñV§^mboI {d{dY loUtÀ`m_Yrb VwbZm H$aÊ`mg Cn`moJr nS>Vmo. `m AmboImV XmoZ qH$dm XmoZ nojm A{YH$ Cô (qH$dm AmS>do) ñV§^ (Am`V) EH$_oH$mer g_m§Va AgVmV. AmH¥$Vr 5.1 _Yrb ñV§^mboImV AZwZo J{UV {df`mV VrZ gÌ n[ajo_Üò {_i{dbobo JwU Xe©{dbobo AmhoV. `m_wio {VZo {_i{dboë`m JwUm§Mr VwbZm H$aUo ghmæ`H$ hmoVo Am{U `mdê$Z AmnU åhUy eH$Vmo H$s {VMr J{UV {df`mÀ`m Aä`mgmVrb àJVr Mm§Jbr Amho.

gÌ2 ao 3 ao1 bo

J{UV

{df`

mVrb

JwU

AmH¥$Vr 5.1

ñV§^mboIm_Üò XmoZ ñV§^ OmoSy>Z gwÜXm AgVmV (AmH¥$Vr 5.2 _Üò Xe©{dë`mà_mUo). hm AmboI, XmoZ {XdgmV {d{dY àH$maÀ`m \$im§Mr Pmbobr {dH«$s (` _Üò) `m§Mo VwbZmË_H$ {ddaU XoVmo. AmH¥$Vr 5.2 hr AmH¥$Vr 5.1 nojm doJir H$er Amho? Vw_À`m {_Ìm§er MMm© H$am.

àH$aU

5 AmboIm§Mm n[aM`

68 J{UV

Ho$irnnB©g§Ìrg\$aM§XAm§~m\$io

{dH«$s

(kg

_Ü`

o)

gmo_dma _§Jidma

AmH¥$Vr 5.2

5.1.2. {ÌÁ`m§Va I§S>mboI (pie-graph) (qH$dm dVw©imboI) (circle graph){ÌÁ`m§Va I§S>mboI hm EH$m nyU© ^mJmer BVa ^mJm§Mr VwbZm H$aÊ`mg dmnaVmV. òWo dVw©i ho nyU© ^mJ AgVo. AmH¥$Vr 5.3 hm EH$ {ÌÁ`m§Va I§S>mboI Amho. `m I§S>mboImÛmao EH$ ì`ŠVr Q>r. ìhr. darb {d{dY dm{hÝ`m (M°Zoëg) nmhÊ`mMo eoH$S>m à_mU g_OVo.

5.1.3. {hñQ>moJ«m_ (Am`VmboI) (Histogram)

{hñQ>moJ«m_ (Am`VmboI) hm ñV§^mboIà_mUo g§~§YrV

_m{hVr dma§dmaVoÀ`m à_mUmV Xe©{dVmo. òWo x - Ajmda

Cô ñV§^ gbJ XmI{dbobo AgVmV. Imbr AmH¥$Vr

5.4 _Üò EH$m n[agamVrb 40 ì`pŠV¨À`m dOZmMr

(kg _Üò) dma§dmaVm {hñQ>moJ«m_Zo Xe©{dbr Amho.

_m{hVr`wŠV

_Zmoa§OZ

H«$sS>m

~mVå`m

AmH¥$Vr 5.3

AmboIm§Mm n[aM` 69

dOZ (kg) 40-45 45-50 50-55 55-60 60-65

ì`pŠV¨Mr g§»`m 4 12 13 6 5

dOZ (kg _Üò)

ì`pŠ

V¨Mr g

§»`m

AmH¥$Vr 5.4 _Üò EH$ ZmJ_moS>r aofm (x - Ajmda) dmnabobr Amho Or {jVrOAjmda 0 Vo 40 n`ªV g§»`m ZmhrV Ago Xe©{dVo.

AmH¥$Vr 5.4

bjmV ¿`m H$s, ñV§^m§À`m_Üò _moH$ir OmJm ZgVo. ñV§^ gbJ OmoS>bobo AgVmV. `m {hñQ>moJ«m_dê$Z Vwåhmbm H$m` g_OVo? Ë`m§Mr `mXr H$am.

5.1.4. aofm AmboI (line graph)

EH$ aofm AmboI, Aer _m{hVr àñVyV H$aVmo Omo doio ~amo~a gVV ~XbV amhVmo.

Ooìhm aoZy AmOma nS>br, Voìhm S>m°ŠQ>amZr àËòH$ Mma VmgmZ§Va {VÀ`m earamÀ`m Vmn_mZmMr Zm|X Ho$br. hr Zm|X AmboI ñdê$nmV hmoVr (AmH¥$Vr 5.5 d 5.6 nhm).

AmnU `mbm "doi-Vmn_mZMm AmboI' Ago åhUVmo. Imbr {Xboë`m gmaUr ñdê$nmVrb _m{hVrMo ho {MÌê$n gmXarH$aU Amho.

doi 6 a.m 10 a.m 2 p.m 6 p.m

Vmn_mZ (°C) 37 40 38 35

{jVrO g_m§Va aofm ( {hbm x- Aj åhUVmV.) hr OoWo Vmn_mZmMr Zm|X Ho$br OmVo Vr doi Xe©{dVo. Cä`m aofodarb Zm|Xr H$moUË`m? (`mbm AmnU y-Aj åhUVmo).

70 J{UV

Vmn_

mZ (

°C)

Vmn_

mZ (

°C _

Üò)

q~Xÿ EH$_oH$m§Zm OmoS>bo Joë`mZo aofm AmboI V`ma Pmbm Amho.

_m{hVrVrb àËòH$ g§»`m Mm¡agmH¥$Vr Omirda ñWm{nV Ho$br Amho.

doi doi

AmH¥$Vr 5.5 AmH¥$Vr 5.6

`m AmboImÛmao Vwåhmbm H$m` g_OVo? CXmhaUmW©, AmnU `m_Üò Vmn_mZmMo Z_wZo nmhÿ eH$Vm ; gH$mir 10 dmOVm Vmn_mZ OmñV hmoVo. Z§Va gm`§H$mi 6 p.m. n`ªV H$_r hmoV Jobo Amho (AmH¥$Vr 5.5 nhm). bjmV ¿`m H$s gH$mir 6 nmgyZ gH$mir 10 n`ªV Vmn_mZmV 3°C (= 40°C – 37°C) dmT> Pmbr.

gH$mir 8 dmOVm Vmn_mZmMr Zm|X Zmhr Var gwÜXm AmboI nmhÿZ Ago {XgVo H$s `m doioMo Vmn_mZ 37°C nojm OmñV hmoVo (H$go Vo gm§Jm?).

CXmhaU 1 : AmH¥$Vr 5.7 _Yrb AmboI 2007 gmbr {H«$Ho$Q>À`m 10 gm_mÝ`m_Üò A Am{U B m \$bXm§OmZr H$mT>boë`m Ymdm Xe©{dVmo. `m AmboImMm Aä`mg H$am Am{U Imbrb àûZm§Mr CÎmao Úm.

(i) XmoÝhr Ajmda H$moUVr _m{hVr {Xbobr Amho?

(ii) AmboImVrb H$moUVr aofm A \$bXm§OmZo H$mT>boë`m Ymdm XmI{dVo.

(iii) gZ 2007 _Üò H$moUË`m gm_Ý`mV XmoÝhr \$b§XmOmZr H$mT>boë`m Ymdm g_mZ hmoË`m?

(iv) XmoÝhr \$b§XmOm_Üò H$moUmMm Ioi A{YH$ pñWa Amho? hm {ZU©` Vwåhr H$gm KoVbmV?

CH$b : (i) {jVrO g_m§Va Aj (x -Aj) gZ 2007 _Üò Ioië`m Joboë`m {H«$Ho$Q> gm_mÝ`m§Mr g§»`m Xe©{dVmo.

(ii) {R>nHo$`wŠV aofm A \$b§XmOmZo H$mT>boë`m Ymdm Xe©{dVo. Ogo AmboImÀ`m daVr CëboI gwÜXm Ho$bobm Amho.

AmboIm§Mm n[aM` 71

(iii) 4 Ï`m gm_Ý`mV XmoÝhr \$b§XmOmZr g_mZ 60 Ymdm H$mT>boë`m AmhoV.

(iv) \$b§XmO A Mo AmboIm_Üò EH$ C§M "{eIa' Amho. na§Vy AZoH$ Imob "Xè`m' AmhoV. Vmo Ymdm H$mT>Ê`mV pñWa Zmhr Ago {XgVo. Va Xþgè`m ~mOwbm B \$b§XmOmZo Ho$ìhmhr 40 nojm H$_r Ymdm H$mT>boë`m ZmhrV.

A \$b§XmOmÀ`m OmñVrV OmñV 115 Ymdm§Mr VwbZm Ho$ë`mg B \$bXm§OmÀ`m gdm©V OmñV Ymdm 100 AmhoV. A \$b§XmOmZo XmoZ gm_Ý`mV eyÝ` Ymdm d EHy$U nmM gm_mmÝ`mV 40 nojm H$_r Ymdm H$mT>boë`m AmhoV. \$b§XmO A Zo H$mT>boë`m Ymdm_Üò AZoH$ MT>-CVma AmhoV. åhUyZ B hm EH$ {dídgZr` d pñWa nb§XmO Amho.

\$b§XmO A

gm_Zo

H$mT>b

oë`m Y

mdm

\$b§XmO B

AmH¥$Vr 5.7

CXmhaU 2 : AmH¥$Vr 5.8 _Yrb AmboI EH$m H$maZo P ehamnmgyZ 350 km Xÿa Agboë`m Q ehamn`ªV doJdoJù`m H$mbmV àdmg Ho$boë`m A§VamMo {ddaU H$aVmo. AmboImMo {ZarjU H$ê$Z Imbrb àûZm§Mr CÎmao Úm.

(i) XmoZ Ajmda H$moUVr _m{hVr {Xbobr Amho?

(ii) H$maMm àdmg H$moRy>Z Am{U Ho$ìhm gwê$ hmoVmo?

(iii) n{hë`m Vmgm_Üò H$ma {H$Vr A§Va àdmg H$aVo?

72 J{UV

(iv) H$ma : (a) Xþgè`m VmgmV (b) {Vgè`m VmgmV {H$Vr A§Va OmVo?

(v) n{hë`m VrZ Vmgm_Üò H$maMm doJ EH$ g_mZ hmoVm H$m? ho Vwåhr H$go OmUyZ KoVbm?

(vi) H$moUË`mhr EH$m {R>H$mUr H$mhr H$mbmgmR>r H$ma Wm§~br hmoVr H$m? Vw_Mo CÎma VnmgyZ nhm.

(vii) H$ma Q ehambm Ho$ìhm OmD$Z nmohMbr?A

mH«${_

bobo A

§Va (p

ehamn

mgyZ)

H$mb (doi)AmH¥$Vr 5.8

CH$b :

(i) AmboIm_Üò {j{VO g_m§Va x - Aj H$mb (doi) Xe©{dVmo Am{U C^m y - Aj H$maMo P ehamnmgyZMo A§Va Xe©{dVmo.

(ii) P ehamnmgyZ H$maMm àdmg 8 a.m. bm gwê$ hmoVmo.

(iii) H$ma n{hë`m Vmgm_Üò 50 km àdmg H$aVo. [H$ma P ehamnmgyZ 8.00 a.m. gwê$ hmoVo Am{U 9.00 a.m. bm 50 km A§Va OmVo (AmboImH$ê$Z)].

(iv) H$maZo AmH«${_bobo A§Va

(a) Xþgè`m VmgmV (åhUOo 9.00 a.m Vo 10.00 a.m) 100 km (150-50).

(b) {Vgè`m VmgmV (åhUOo 10.00 a.m to 11.00 a.m) 50 km (200-150).

(v) àíZ (iii) Am{U (iv) À`m CÎmamdê$Z Amåhmbm AmT>iyZ oVo H$s àËòH$ doiobm H$maMm doJ g_mZ Zmhr.

AmboIm§Mm n[aM` 73

(vi) 11 a.m Am{U Xþnmar 12 dmOVm H$ma P ehamnmgyZ 200 km. A§Va Xÿa Amho. `mdê$Z Ago {XgyZ òVo Vo H$s 11 a.m Vo Xþnmar 2 n`ªV H$maZo àdmg Ho$bm Zmhr. {j{VO g_m§Va aofmI§S> `m H$mbmdYrV ho Xe©{dVmo.

(vii) H$ma Q ehambm 2 a.m. bm nmohMbr.

ñdmÜ`m` 5.1 1. EH$m hm°pñnQ>b_Yrb EH$m é½UmMo Xa Vmgmbm AmT>iyZ Ambobo

Vmn_mZ Imbrb AmboImV Xe©{dbo Amho.

(a) 1 pm bm é½UmMo Vmn_mZ {H$Vr hmoVo?

(b) {H$Vr dmOVm é½UmMo Vmn_mZ 38.5°C hmoVo?

H$mb (doi)

Vmn_

mZ (

°C _

Üò)

(c) é½UmMo Vmn_mZ H$moUË`m XmoZ doir g_mZ AmT>iyZ Ambobo Amho?

(d) 1.30 pm bm Vmn_mZ {H$Vr hmoVo? Vwåhr `mMo CÎma H$go H$mT>mb?

(e) H$moUË`m H$mbmdYrV é½UmMo Vmn_mZ dmT>bobo AmT>ibo?

2. nwT>rb AmboI EH$m H§$nZrÀ`m CËnmXZm§À`m {d{H«$Mr dm{f©H$ CbmT>mb Xe©{dVmo.

(a) (i) 2002 d (ii) 2006 gmbmVrb àËòH$s {dH«$s {H$Vr?

(b) (i) 2003 d (ii) 2005 gmbmVrb àËòH$s {dH«$s {H$Vr?

74 J{UV

(c) 2002 d 2006 gmbmVrb {dH«$sMm \$aH$ H$mT>m.

(d) _mJrb df© d Ë`m nwT>rb df© `m§À`mV {d{H«$Vrb _moR>m \$aH$ AgUmao df© H$moUVo?

df}

{dH«$s

(`

H$moQ>r_

Üò)

3. dZñnVremñÌmÀ`m EH$m à`moJemioV amon A d amon B `m doJdoJù`m amonm§da Ë`m§À`m dmT>rMo gma»`mM pñWVrV à`moJ H$aÊ`mV Ambo. àËòH$ AmR>dS>çmÀ`m eodQ>r Ë`m§À`m C§MrMr _moO_mn Ho$bo d Ago 3 AmR>dS>o Ho$bo. Ë`mV {XgyZ Amboë`m n[aUm_m§Mm AmboI Imbr {Xbobm Amho.

AmR>dS>o

C§Mr (

cm _

Üò)

Ama§^

amonQ>o

amonQ>o

(a) (i) 2 AmR>dS>çmZ§Va (ii) 3 AmR>dS>çmZ§Va A amonmMr C§Mr {H$Vr hmoVr?

(b) (i) 2 AmR>dS>çmZ§Va d (ii) 3 AmR>dS>çmZ§Va B amonmMr C§Mr {H$Vr hmoVr?

(c) 3 è`m AmR>dS>çmV Pmbobr A amonmMr dmT> {H$Vr?

(d) 2 am AmR>dS>m Vo {Vgam AmR>dS>m `m H$mimV B amonmMr Pmbobr dmT> {H$Vr?

(e) H$moUË`m AmR>dS>çmV A amonmMr dmT> OmñV Pmbobr {XgVo?

AmboIm§Mm n[aM` 75

(f) H$moUË`m AmR>dS>çmV B amonmMr dmT> gdm©V H$_r Pmbobr {XgVo?

(g) XmoÝhr amonm§Mr g_mZ C§Mr H$moUË`m AmR>>dS>çmV {XgyZ òVo.

4. EH$m AmR>dS>çmVrb àËòH$ {XdgmMo dV©{dbobo Vmn_mZ d dmñVd Vmn_mZ Imbrb AmboI Xe©{dVmo.

gmo_ _§Jb ~wY Jwê$ ewH«$ e{Z a{d{Xdg

dV©{dbobo Vmn_mZ dmñVd Vmn_mZ

Vmn_

mZ (

°C _

Üò)

(a) dV©{dbobo Vmn_mZ d dmñVd (gË`) Vmn_mZ H$moUË`m {Xder hmoVo?

(b) H$moUË`m {Xder dV©{dbobo Vmn_mZ OmñVrV OmñV Agob Ago dV©{dbo hmoVo?

(c) gdm©V H$_r dmñVd Vmn_mZ AmR>dS>`mVrb H$moUË`m {Xder hmoVo?

(d) H$moUË`m {Xder dV©{dboë`m Vmn_mZmnojm dmñVd Vmn_mZ gdm©V OmñV hmoVo.

5. aofr` AmboI H$mT>Ê`mg Imbrb gmaUtMm Cn`moJ H$am

(a) EH$m W§S> hdoÀ`m {R>H$mUr doJdoJù`m dfu ~\©$ nS>boë`m {Xdgm§Mr g§»`m.

df© 2003 2004 2005 2006

{Xdg 8 10 5 12

(b) EH$m JmdmVrb doJdoJù`m dfu nwéf Am{U pñÌ`m `m§Mr bmoH$g§»`m (hOmam_Üò)

df© 2003 2004 2005 2006 2007nwéfm§Mr g§»`m 12 12.5 13 13.2 13.5pñÌ`m§Mr g§»`m 11.3 11.9 13 13.6 12.8

6. Hw$[aAa dmQ>n H$aUmar EH$ ì`ŠVr EH$m ehamnmgyZ Ë`mÀ`m CnZJam_Üò Agboë`m ì`mnmè`mbm Ë`m§Mo Ambobo nmg©b XoÊ`mg gm`H$brdê$Z {\$aVo. ehamÀ`m _w»` ^mJmnmgyZ Vmo CnZJamV {\$aV Agbobo A§Va d Ë`mgmR>r bmJUmam doJdoJim doi (H$mb) `m§Mm AmboI nwT>o {Xbobm Amho.

76 J{UV

doi (H$mb)

A§Va

(km

_Ü`

o)

(a) doi Xe©{dUmè`m AmboImÀ`m Ajmda H$moUVo à_mU (scale) KoVbo Amho?

(b) EHy$U Ho$boë`m àdmgmgmR>r KoVbobr EHy$U doi {H$Vr?

(c) ehamnmgyZ ì`mnmè`mMo {R>H$mU {H$Vr Xÿa Amho?

(d) nmg©b XoUmar ì`ŠVr Ë`mÀ`m _mJm©V H$moR>o Wm§~br hmoVr H$m? ñnîQ>rH$aU H$am.

(e) H$moUË`m doio_Üò Ë`mZo gm`H$b OmoamV Mmb{dbr Amho.

7. doi (H$mb) - Vmn_mZMm AmboI Imbrbà_mUo Agy eH$Vmo H$m? CÎma VnmgyZ nhm.

H$mb

Vmn_

mZVm

n_mZ

Vmn_

mZVm

n_mZ

H$mb H$mb

H$mb

AmboIm§Mm n[aM` 77

5.2.aofr` AmboI (linear graph)

aofm AmboI hm bhmZ bhmZ aofmI§S> EH$_oH$m§Zm OmoSy>Z V`ma hmoVmo. H$mhr doim AmboI AI§S> nyU© aofmI§S>mZo ~Zbobm AgVmo. Aem AmboImbm aofr` AmboI Ago åhUVmV Aer AmboImMr aofm H$mT>Ê`mg Amåhmbm AmboIH$mJXmda H$mhr q~Xÿ KoUo ({ZX}erV H$aUo) Amdí`H$ AgVo. Ago q~Xÿ KoÊ`mg§~§Yr AmnU AmVm _m{hVr H$ê$Z KoD$.

5.2.1. q~XÿMo ñWmZ (q~Xÿ {ZX}eZ) (location of a point){ejH$m§Zr \$bH$mda (~moS>©da) EH$ {R>nH$m H$mT>bm. Ë`mZr {dÚmÏ`mªZm {dMmabo H$s `m {R>nŠ`mÀ`m ñWmZmMo Vwåhr H$go dU©Z H$ê$ eH$mb. `mbm AZoH$ CÎmao hmoVr (AmH¥$Vr 5.9).

{R>nH$m ~moS>©À`m darb A¿`m© ^mJmV Amho.

{R>nH$m ~moS>©À`m S>mì`m~mOyÀ`m H$S>m Odi

Amho.

{R>nH$m ~moS>©À`m darb S>mì`m H$monè`mÀ`m AJXr Odi Amho.

AmH¥$Vr 5.9

`mVrb H$moUVo {dYmZ {R>nŠ`mMo ñWmZ {ZpûMV H$aÊ`mg _XV H$aVo?

Om°ZZo Amnbm gëbm {Xbm, Ë`mZo ~moS>©À`m S>mdrH$S>rb H$S>mnmgyZ {R>nŠ`mMo A§Va _moObo Am{U gm§{JVbo H$s ""{R>nH$m ~moS>©À`m S>mdrH$S>rb H$S>mnmgyZ 90cm A§Vamda Amho.'' Om°ZMo ho {dYmZ IamoIa _XV H$aob H$m? (AmH¥$Vr 5.10)


A, A1, A2, A3.... ho gd© {R>nHo$ S>mì`m H$S>mnmgyZ 90cm Xÿa AmhoV.

A {R>nH$m S>mì`m H$S>onmgyZ 90 cm Am{U VimÀ`m H$S>mnmgyZ 160 cm A§Vamda Amho.

78 J{UV

aoImZo Amnbo gwYm[aV {dYmZ _m§S>bo. Vr åhUmbr ""A {R>nH$m S>mì`m ~mOyÀ`m H$S>mnmgyZ 90cm Am{U VimÀ`m H$S>mnmgyZ 160 cm A§Vamda Amho''. ho {dYmZ mo½` Amho. (AmH¥$Vr 5.11). `mda {ejH$ åhUmbo `m {R>nŠ`m§À`m pñWVrMo dU©Z Amåhr (90,160) Ago {bhÿZ H$aVmo. q~Xÿ (160,90) hm q~Xÿ (90, 160) nojm doJim AgVmo H$m? `mda MMm© H$am.

Ago gm§{JVbo OmVo H$s, 17 ì`m eVH$m_Üò J{UVk aoZo {S>ñH$mQ>© `m§Zr EH$ H$sQ>H$ ImobrÀ`m N>VmÀ`m H$monè`mOdi OmVmZm nm{hbm Am{U Vo nmVir_Üò EH$m q~XÿMr pñWVr {ZYm©[aV H$aÊ`m~Ôb {dMma H$ê$ bmJbo. {j{VO g_m§Va d C^r aofm Aem XmoZ aofm§_wio {Xboë`m q~XÿMo A§Va _moOyZ q~XÿMo ñWmZ gm§JÊ`mÀ`m nÜXVrbm Ë`m§À`m gÝ_mZmW© mbm ""H$mV}{e`Z nÜXV (Cartesian System) Ago åhQ>bo OmVo.

aoZo {S>ñH$mQ>©> (1596-1650)

5.2.2. gh{ZX}eH$ (coordinates)g_Om H$s Vwåhr EH$m {WEQ>a_Ü o OmD$Z Vw_Mo Ama{jV AmgZ (Seat) emoYV AmhmV. `mgmR>r Vwåhmbm XmoZ g§»`m nm{hOoV. EH$ n§ŠVrMr g§»`m Am{U Xþgar AmgZ H«$_m§H$. H$moUË`mhr nmVirV q~XÿMo ñWmZ {ZpûMV H$aÊ`mgmR>r hm EH$ AmYma Amho.

AmH¥$Vr 5.12 Mo {ZarjU H$am. q~Xÿ (3,4) hm AmboI H$mJXmda S>mì`m H$S>onmgyZ 3 EH$H$ COdrH$S>o Am{U VimÀ`m H$S>onmgyZ 4 EH$H$ da ñWm{nV Ho$bm Amho. AmboI H$mJX hm EH$ Mm¡agmH¥$Vr Omir AgVo. Ë`mda Amåhr x-Aj d y- Aj H$mTy>Z Amdí`H$ Vo q~Xÿ {ZpíMV H$aVmo. òWo 3 bm q~XÿMm x-gh{ZX}eH$ d 4 bm y- gh{ZX}eH$ åhUVmV. Aem àH$mao (3,4) ho q~XÿMo gh{ZX}eH$ AmhoV Ago AmnU åhUVmo.

Ama§^ q~Xÿ(0, 0) {jVrO Aj

(AmS>dm Aj) (x - Aj)

C^m A

j(y

- A

j)

COdrH$S>o {H$Vr EH$Ho$ gaH$bo nm{hOo.

3 EH$Ho$

4 EH

$Ho$

da {H$Vr EH$Ho$ gaH$bo nm{hOo.

AmH¥$Vr 5.12

AmboIm§Mm n[aM` 79

CXmhaU 3 : AmboIH$mJXmda q~Xÿ (4,3) ñWmnZ H$am. hm q~Xÿ, q~Xÿ (3,4) er g_mZ Amho H$m?

CH$b : EH$m AmboIH$mJXmda x-Aj d y-Aj H$mT>m. (ho XmoÝhr Aj Ia åhUOo g§»`maofm AgVmV). Ama§^ q~Xÿ O (0, 0) nmgyZ gwadmV H$am. VoWyZ 4 EH$Ho$ COdrH$S>o Mbm, Z§Va 3 EH$Ho$ daVr Mbm. Vwåhr q~Xÿ (4,3) er nmohMmb. AmH¥$Vr 5.13 dê$Z Vwåhr nmhÿ eH$mb H$s q~Xÿ (3,4) Am{U q~Xÿ (4,3) ho doJdoJio q~Xÿ AmhoV.

3 EH

$Ho$

4 EH$Ho$

AmH¥$Vr 5.13

AmH¥$Vr 5.14

CXmhaU 4 : AmH¥$Vr 5.14 Mo {ZarjU H$am. Imbr {Xboë`m q~Xÿ§Mo ñWmZ Xe©{dUmar Ajao AmoiIm.

(i) (2, 1) (ii) (0, 5) (iii) (2,0)

ho {bhm - (iv) A q~XÿMo gh{ZX}eH$

(v) F q~XÿMo gh{ZX}eH$

CH$b : (i) (2, 1) hm q~Xÿ E Amho (Vmo D Zmhr. H$m?)

(ii) (0, 5) hm q~Xÿ B Amho (H$m? Vw_À`m {_Ìm§er MMm© H$am!)

(iii) (2, 0) hm q~Xÿ G Amho.

(iv) A q~XÿMo gh{ZX}eH$ (4,5) ho AmhoV.

(v) F q~Xÿ q~XÿMo gh{ZX}eH$ (5.5, 0) AmhoV.

CXmhaU 5 : Imbrb q~Xÿ AmboInQ>mda ñWmnZ H$am Am{U Vo EH$m aofoda òVmV H$m Vo Vnmgm. Oa

aofoda òV AgVrb Va aofoMo Zmd gm§Jm.

(i) (0, 2), (0, 5), (0, 6), (0, 3.5)

(ii) A(1, 1), B(1, 2), C(1, 3), D(1, 4)

(iii) K(1, 3), L(2, 3), M(3, 3), N(4, 3)

(iii) W(2, 6), X(3, 5), Y(5, 3), Z(6, 2)

80 J{UV

CH$b :

(i) ho gd© q~Xÿ EH$m aofoda òVmV. hr aofm y - Aj Amho.

(ii) ho gd© q~Xÿ EH$m aofoda òVmV. Vo gd© AD aofoda AmhoV. (Vwåhr Xþgè`m àH$mao dmnê$Z Zmd XoD$ eH$Vm). hr aofm y-Ajmbm g_m§Va Amho.

(iii) ho gd© q~Xÿ EH$m aofoda òVmV. Amåhr `m aoofobm KL qH$dm KM qH$dm MN Zmd XoD$ eH$Vmo. hr aofm x-Ajmbm g_m§Va Amho.

(iv) ho gd© q~Xÿ EH$m aofoda òVmV. Amåhr {Vbm XY qH$dm WY qH$dm YZ Zmd XoD$ eH$Vmo.)

AmH¥$Vr 5.15

bjmV ¿`m H$s, darb gd© AmboIm_Üò ñWmnZ Ho$bobo q~Xÿ OmoSy>Z EH$ gai aofm {_iVo. Aem AmboIm§Zm aofr` AmboI (linear graphs) åhUVmV.

ñdmÜ`m` 5.2

1. AmboIH$mJXmda Imbrb q~Xÿ Xe©dm. Vo EH$mM aofoV òVmV H$m Vo Vnmgm.

(a) A (4, 0), B (4, 2), C(4, 6), D(4, 2.5)

(b) P (1, 1), Q (2, 2), R(3, 3), S(4, 4)

(c) K (2, 3), L (5, 3), M(5, 5), N(2, 5)

AmboIm§Mm n[aM` 81

2. AmboI H$mJXmda (2,3) Am{U (3,2) `m q~XÿVyZ OmUmar gai aofm H$mT>m. hr aofm x- Aj Am{U y-Aj `m§Zm OoWo {_iVo VoWrb gh{ZX}eH$ emoYm.

3. Imbrb AmboImVrb AmH¥$Ë`m§Mo {eamoq~Xÿ nhm Am{U Ë`m§Mo gh{ZX}eH$ {bhm.

4. MyH$ H$s ~amo~a Vo gm§Jm. MwH$sMo Agbobo ~amo~a H$am.

(i) EH$ q~Xÿ Á`mMm x - gh{ZX}eH$ eyÝ` Amho Am{U y -gh{ZX}eH$ eyÝòVa g§»`m hm y -

Ajmda AgVmo.

(ii) EH$ q~Xÿ Á`mMm y - gh{ZX}eH$ eyÝ` Amho Am{U x - gh{ZX}eH$ 5 AgyZ Vmo y - Ajmda òVmo.

(iii) Ama§^ q~XÿMo gh{ZX}eH$ (0, 0) AgVmV.

5.3 H$mhr Cn`moJ

X¡Z§{XZ OrdZmV Vwåhmbm Ago AmT>iyZ òVo H$s Vwåhr Oem gw{dYm dmnamb Vgo Vwåhmbm n¡go hr OmñV IM© H$amdo bmJVmV.Oa Vwåhr Kam_Üò {dÚwVMm OmñV Cn`moJ H$[aV Agmb, Va Ë`mMo {~b OmñV âmdo bmJVo. `m CbQ> Oa H$_r {dÚwV dmnabr Va {dÚwV {~b H$_r òVo. òWo EH$m KQ>H$mMo à_mU Xþgè`m KQ>H$mÀ`m à_mUmda n[aUm_ H$aVo. {dÚwV {~bmMr aŠH$_ hr dmnaboë`m {dÚwVÀ`m à_mUmda Adb§~yZ AgVo. òWo Amåhr {dÚwVMo à_mU ho ñdV§Ì MbnX (independent variable) (qH$dm H$mhr doim {Z`§{ÌV MbnX (control variable) Ago åhUVmo. Am{U {dÚwV {~bmMr a¸$_ ho Adb§{~V MbnX (dependent variable) Ago åhUVmo. Aem MbnXm§Vrb g§~§Y AmboImZo Xe©{dVm òVmo.

82 J{UV


H$maJmS>rÀ`m noQ´>mobÀ`m Q>mH$sV noQ´>mob âÊ`mg noQ´>mobÀ`m {bQ>am§Mr g§»`m hr Vwåhr {H$Vr a¸$_ XoUma `m Vw_À`m {ZU©`mda Adb§~yZ AgVo. `mVrb Adb§{~V MbnX H$moUVo? {dMma H$am.

CXmhaU 6 : (à_mU Am{U qH$_V)

Imbrb VŠË`m_Üò noQ´>mobMo à_mU Am{U Ë`mMr qH$_V {Xbobr Amho.

noQ´>mob ({bQ>a_Üò) 10 15 20 25

noQ´>mobMr qH$_V (` _Üò) 500 750 1000 1250

`m _m{hVrMm AmboI H$mT>m.

CH$b :

qH$_V

(` _

Üò)

{bQ>a

AmH¥$Vr 5.16

AmboIm§Mm n[aM` 83

(i) XmoÝhr Ajmda `mo½` à_mU ¿`m (AmH¥$Vr 5.16).

(ii) {jVrO g_m§Va (AmS>ì`m) x - Ajmda {bQ>aÀ`m IwUm H$am.

(iii) Cä`m y - Ajmda qH$_VtÀ`m IwUm H$am.

(iv) (10, 500), (15, 750), (20, 1000), (25, 1250) ho q~Xÿ AmboInQ>mda ñWmnZ

H$am.

(v) gd© q~Xÿ OmoS>Umar aofm H$mT>m.

òWo AmnUmbm AmboI EH$ aofm {_iVo. (hm aofr` AmboI hmo`.) `m AmboImMr aofm

Ama§^ q~XÿVyZ OmVo. H$m? {dMma H$am.

hm AmboI H$mhr JmoîQ>r{df`r A§XmO H$aÊ`mg _XV H$aVmo. g_Om AmnUmbm 12 {bQ>a

noQ´>mobMr qH$_V emoYm`Mr Amho. Va AmS>ì`m Ajmda 12 g§»òdê$Z aofm AmboIn`ªV

da Mbm g_Om Vwåhr P q~Xÿbm nmohMbm.

AmVm P nmgyZ gaiaofoV AmS>do Cä`m AjmH$S>o `m. Cä`m Ajmdarb Á`m qH$_Vrbm

Vwåhr nmohMmb Vr 12 {bQ>a noQ´>mobMr qH$_V hmoB©b.

hm AmboI XmoZ KQ>H$m§Mr pñW{V g_à_mUmV Xe©{dVmo. (H$go?). `m pñWVrV AmboI

hm Zoh_r aofr` AmboI AgVmo.

à`ËZ H$am 1. darb CXmhaUmVrb AmboImMm Cn`moJ H$ê$Z `800 _Üò {H$Vr

{bQ>a noQ´>mob {dH$V KoVm òB©b Vo emoYm.

CXmhaU 7 : (_wÔb Am{U gaiì`mO)

EH$ ~±H$, OoîR> ZmJ[aH$m§Zr ~±Ho$V R>odboë`m _wXV R>odrVrb a¸$_oda 10% gaiì`mO XoVo. EHy$U

R>od R>odbobr a¸$_ Am{U {VMo {_iUmao gaiì`mO `m§Mm g§~§Y AmboImZo Xe©dm. `m AmboImdê$Z

Imbrb KQ>H$m§Mr _m{hVr emoYm.

(a) ` 250 À`m Jw§VdUyH$sda {_iUmao dm{f©H$ ì`mO.

(b) ` 70 dm{f©H$ gaiì`mO {_iÊ`mg H$amdr bmJUmar Jw§VdUyH$.

84 J{UV

CH$b :

R>od EH$m dfm©Mo gaiì`mO bjmV KoÊ`mÀ`m nm`è`m

` 100 ` 100 1 10

100× ×

= ` 10 1. R>od Am{U gaiì`mO `m§À`m

à_mUm§Mm emoY ¿`m.

` 200 `. 200 1 10

100× ×

= ` 20 2. x - Ajmda Am{U y - Ajmda

KoÊ`mMo à_mU R>adm.

` 300 ` 300 1 10

100× ×

= ` 30 3. `mo½` à_mUmMr {ZdS> H$am.

` 500 ` 500 1 10

100× ×

= ` 50 4. q~Xÿ ñWmnZ H$am.

` 1000 ` 1000 1 10

100× ×

= ` 100 5. gd© q~Xÿ aofoZo OmoS>m.

`mdê$Z Amåhmbm qH$_VtMr gmaUr {_iob.

R>od (` _Üò) 100 200 300 500 1000

dm{f©H$ gaiì`mO (` _Üò) 10 20 30 50 100

(i) à_mU : 1 EH$H$ = ` 100

AmS>ì`m Ajmda, 1 EH$H$ = `

10 Cä`m Ajmda.

(ii) AmS>ì`m g_m§Va Ajmda R>odr§À`m

IwUm H$am.

(iii) Cä`m Ajmda gaiì`mOÀ`m

IwUm H$am.

(iv) (100, 10), (200, 20), (300, 30), (500, 50), (1000, 100)

B. q~Xÿ ñWmnZ H$am.

dm{f

©H$ ga

iì`m

O (`

_Ü`

o)

_wXV R>od (` _Üò)

AmH¥$Vr 5.17

(v) q~Xÿ Omo‹S>m. Amåhmbm EH$ gai aofm {_iVo (AmH¥$Vr 5.17)

(a) AmS>ì`m x - Ajmdê$Z ` 250 bm, Cä`m y - Ajmda ` 25 gaiì`mO {_iob.

(b) Cä`m y - Ajmdê$Z ` 70 bm, g_m§Va x - Ajmda ` 700 R>od qH$_V {_iob.

AmboIm§Mm n[aM` 85

à`ËZ H$am

CXmhaU 7 ho g_à_mUmMo CXmhaU Amho H$m?

CXmhaU 8 : (doi Am{U A§Va)

AOrV Amnbr ñHw$Q>a 30 km/hr `m g_mZ doJmZo Mmb{dVmo. `m pñWVrMm doi-A§Va Xe©{dUmam

AmboI H$mT>m. AmboImdê$Z emoYm

(i) 75 km A§Va OmÊ`mg AOrVbm bmJUmam doi.

(ii) 3 12 VmgmV AOrVZo AmH«${_bobo A§Va.

CH$b :

ñHy$Q>a Mmb{dbobo Vmg AmH«${_bobo A§Va

1 Vmg 30 km2 Vmg 2 × 30 km = 60 km3 Vmg 3 × 30 km = 90 km4 Vmg 4 × 30 km = 120 km

Am{U `m àH$mao

`mdê$Z Amåhmbm qH$_VrMr gmaUr Aer {_iob.

doi (Vmgm_Üò) 1 2 3 4

AmH«${_bobo A§Va (km _Üò) 30 60 90 120

(i) à_mU : (AmH¥$Vr 5.18 gmR>r)

AmS>dm x - Aj : 2 EH$Ho$ = 1 Vmg

C^m y - Aj : 1 EH$H$ = 10 km.

(ii) AmS>ì`m x - Ajmda doioÀ`m IwUm H$am.

(iii) Cä`m y - Ajmda A§VamÀ`m IwUm H$am.

(iv) (1, 30), (2, 60), (3, 90), (4, 120) ho q~Xÿ AmboInQ>mda Xe©dm.

(v) gd© q~Xÿ OmoS>m. Vwåhmbm aofr` AmboI

{_iVmo.

A§Va

(km

_Ü`

o)

doi (Vmgm_Üò)

AmH¥$Vr 5.18

86 J{UV

(a) Cä`m y - Ajmda 75 km A§Va Yê$Z Amåhmbm AmS>ì`m x - Ajmda 2.5 Vmg hr doi {_iVo. åhUOo 75 km A§Va OmÊ`mg 2.5 Vmg doi Amdí`H$ Amho.

(b) AmS>ì`m x - Ajmda 3 12 Vmg doi Yê$Z Amåhmb Cä`m y - Ajmda AmH«${_bobo

A§Va 105 km {_iVo.

ñdmÜ`m` 5.3

1. `mo½` à_mU KoD$Z Imbrb VŠË`mVrb qH$_VtMm AmboI H$mT>m.

(a) g\$aM§Xm§Mr qH$_V

g\$aM§Xm§Mr g§»`m 1 2 3 4 5

qH$_V (` _Üò) 5 10 15 20 25

(b) H$maZo AmH«${_bobo A§Va

doi (Vmgm_Üò) 6 am 7 am 8 am 9 am

A§Va (km _Üò) 40 50 120 160

(i) 7.30 am Vo 8 am `m H$mbmdYrV H$maZo AmH«${_bobo A§Va {H$Vr?

(ii) H$ma gwê$ Pmë`mnmgyZ 100 km A§Va AmH«${_Ê`mg {H$Vr dmOobobo AgVrb?

(c) EH$m dfm©gmR>r R>odrdarb ì`mO.

R>od (` _Üò) 1000 2000 3000 4000 5000

gaiì`mO (` _Üò) 80 160 240 320 400

(i) AmboImMr aofm Ama§^q~XÿVyZ OmVo H$m?

(ii) AmboImMm Cn`moJ H$ê$Z ` 2500 Mo EH$m dfm©Mo gaiì`mO H$mT>m.

(iii) ` 280 à{V df© ì`mO {_iÊ`mg {H$Vr a¸$_ R>od åhUyZ Jw§Vdmdr bmJob?

2. ImbrbgmR>r AmboI H$mT>m.

(i) Mm¡agmMr ~mOy (cm _Üò) 2 3 3.5 5 6

Mm¡agmMr n[a{_Vr (cm _Üò) 8 12 14 20 24

hm aofr` AmboI Amho H$m?

(ii) Mm¡agmMr ~mOy (cm _Üò ) 2 3 4 5 6

Mm¡agmMo joÌ\$i (cm2 _Üò) 4 9 16 25 36

hm aofr` AmboI Amho H$m?

AmboIm§Mm n[aM` 87


1. _m{hVrMo AmboImÛmao gmXarH$aU g_OÊ`mg gmono AgVo.

2. (i) VwbZm Xe©{dÊ`mg ñV§^mboI Mm Cn`moJ hmoVmo.

(ii) EH$m nyU© ^mJmer BVa ^mJm§Mr VwbZm H$aÊ`mg {ÌÁ`m§Va I§S>mboImMm Cn`moJ hmoVmo.

(iii) {hñQ>moJ«m_ (Am`VmboI) hm ñV§^mboIM AgVmo. `m_Üò _m{hVr hr {d{dY loUr_Üò Xe©{dbr OmVo.

3. doioÀ`m H$mbmdYrà_mUo gVV ~XbUmar _m{hVr aofm AmboI Xe©{dVmo.

4. AmboI EH$ nyU© AI§S> aofm Agob Va Ë`mbm aofr` AmboI åhUVmV.

5. AmboInQ>mda q~Xÿ {ZpíMV H$aÊ`mg x - gh{ZX}eH$ d y - gh{ZX}eH$ `m§Mr Amdí`H$Vm AgVo.

6. Adb§{~V MbnX Am{U ñdV§Ì MbnX `m§À`mVrb g§~§Y AmboImZo Xe©{dVm òVmo.

6.1 n[aM`

6.1.1 Z¡g{J©H$ g§»`m§Mo Ad`dB`Îmm 6 dr _Üò AmnU Ad`d {df`r {eH$bm AmhmV Ë`mMo ñ_aU H$am. AmVm 30 hr EH$ Z¡g{J©H$ g§»`m ¿`m Am{U Vr BVa Z¡g{J©H$ g§»`m§À`m JwUH$ma ñdê$nmV {bhm.

30 = 2 × 15 = 3 × 10 = 5 × 6

òWo 1,2,3,5,6,10,15 Am{U 30 ho 30 Mo Ad`d AmhoV. `mVrb 2,3 Am{U 5 ho 30 Mo _yi Ad`d AmhoV. (H$m?)

EImÚm g§»òMo _yi Ad`d JwUmH$ma ñdén åhUOoM, Ë`m g§»òMo _yi Ad`dmV Ho$bobo ^mJ hmo`. CXmhaUmW© : 30 Mo _yi Ad`d ñdén 2 × 3 × 5 Ago {bhÿ eH$Vmo. 70 Mo _yi Ad`d ñdén 2 × 5 × 7 Amho. 90 Mo _yi Ad`d ñdê$n 2 × 3 × 3 × 5 Amho.

`mMà_mUo, AmnU ~¡{OH$ ametMo gwÜXm \$moS> H$éZ Ad`d nmSy> eH$Vmo Am{U Ë`mMm Aä`mg AmnUmg `m àH$aUmV H$am`Mm Amho.

AmnUmg _m{hVM Amho H$s, AmnU 30 Mo Ad`d 30 = 1 × 30 Ago {b{hVmo. åhUyZ 1 Am{U 30 ho 30 Mo Ad`d AmhoV. Vwåhr {ZarjU Ho$bmV H$m, 1 hm àËòH$ g§»òMm Ad`d AgVmo. CXmhaUmW©, 101=1×101. _mÌ, Ooìhm AmnU EImÚm g§»òMo Ad`d nmS>Vmo Voìhm Amdí`H$Vm Zgob. VoWo 1 {b{hV Zmhr.

6.1.2 ~¡{OH$ ametMo Ad`d

B`Îmm 7 dr _Üò AmnU ~¡{OH$ amerMo \$moS> H$éZ Ad`d (JwUmH$ma nXo) H$mT>>Ê`mMo ñdén nm{hbo Amho. CXmhaUmW©, 5xy + 3x `m ~¡{OH$ amer_Üò 5xy Mo Ad`d 5, x Am{U y `m ñdénmV {_iVmV.

5xy = 5 × x × y

{ZarjU H$am H$s 5xy Mo Ad`d 5, x Am{U y. `m§Mo nwÝhm Ad`d H$mT>Vm òV ZmhrV. AmVm AmnU Ago gm§Jy eH$Vmo H$s 5, x Am{U y ho 5xy Mo "_yi' Ad`d AmhoV. ~¡{OH$ amerÀ`m ~m~VrV AmnU "_yi' bm "A{ZdV©Zr`' (AmUIr H$_r qH$dm g§{já H$aVm Z oUmao) (Irreducible) hm eãX dmnê$ eH$Vmo. Va AmnU 5 × x × y ho 5xy Mo A{ZdV©Zr` ñdén Amho Ago gm§Jy eH$Vmo.

Qrn : 1 hm 5xy Mm Ad`d Amho. åhUOoM 5xy =1×5×x×y. 1 hm àËòH$mMm Ad`d Amho. Z¡g{J©H$ g§»`m§À`m ~m~VrV JaO ZgVmZm 1 hm ñdV§Ì Amdí`H$ Ad`d Amho Ago XmI{dÊ`mMr Amdí`H$Vm Zmhr.

àH$aU

6 Ad`drH$aU

Ad`drH$aU 89

Qrn : 5 × (xy) ho 5xy Mo A{ZdV©Zr` ñdén Zmhr. xy ho x Am{U y À`m Ad`dm§Mr \$moS> Amho. åhUOoM xy = x × y.

AmVm, 3x(x + 2) `m amerMm {dñVma H$é. AmnU 3, x Am{U (x + 2) ho Ë`m amerMo Ad`dr KQ>H$ AmhoV Ago {bhÿ eH$Vmo.

3x (x + 2) = 3 × x × (x + 2)

òWo 3, x Am{U (x + 2) ho 3x (x + 2) Mo A{ZdV©Zr` Ad`d AmhoV.

Ë`mMà_mUo, 10x (x + 2) (y + 3) Mo A{ZdV©Zr` Ad`dmÀ`m {dñVmamMo ñdén, 10x (x + 2) (y + 3) = 2 × 5 × x × (x + 2) × (y + 3) Amho.

6.2 Ad`drH$aU åhUOo H$m`?

Ooìhm AmnU ~¡{OH$ amer§Mo Ad`drH$aU H$aVmo Voìhm Ë`m§À`m Ad`dm§Mm JwUmH$ma {b{hbm nm{hOo. Am{U ho Ad`d g§»`m, ~¡{OH$ nXo qH$dm ~¡{OH$ amer Agy eH$VmV.

amer 3xy, 5x2y, 2x (y + 2), 5(y + 1) (y + 2) `m§Mo Var AJmoXaM Ad`d ñdén Amho. AmnUmg _mhrV Amho H$s, `màH$maMo Ad`d AmnU \$ŠV dmMyZ H$mTy> eH$Vmo.

Xþgè`m ~mOwZo {dMma Ho$ë`mg, 2x + 4, 3x + 3y, x2+ 5x, x2 + 5x + 6 `màH$maÀ`m amer Ooìhm AgVmV Voìhm Ë`m§Mo Ad`d ñnï> H$é eH$V Zmhr. `màH$maÀ`m amerMo Ad`d H$mT>>Ê`mgmR>r AmnUmg EH$ gmoB©ñH$a nÜXV Amdí`H$ Amho.

6.2.1 gm_mB©H$ Ad`d nÜXV

* `mgmR>r EH$ gmYo CXmhaU KoD$ : (2x + 4) Mo Ad`d H$mT>>m. AmVm àËòH$ nXmMo JwUmH$mamÀ`m A{ZdV©Zr` ñdénmV Ad`d H$mT>y.

2x = 2 × x

4 = 2 × 2

∴ 2x + 4 = (2 × x) + (2 × 2).

{ZarjU H$am H$s, XmoÝhr nXm_Üò 2 hm gm_mB©H$ Ad`d Amho.

{dVaU JwUY_m©déZ {ZarjU H$am, 2 × (x +2) = (2 × x) + (2 × 2)

∴ 2x + 4 = 2 × (x + 2) = 2 (x + 2) Ago {b{hVmo.

2x + 4 Mm {dñVma hm 2(x + 2) er g_mZ Amho. AmVm AmnU ho Ad`d dmMVmZm 2 Am{U (x+2) Ago dmMVmo. Va ho Ad`d A{ZdV©Zr` Ad`d AmhoV.

AmVm, 5xy + 10x Mo Ad`d H$mT>m.

90 J{UV

5xy Am{U 10x Mo A{ZdV©Zr` Ad`d Imbrb à_mUo 5xy = 5 × x × y

10x = 2 × 5 × x

{ZarjU H$am H$s, òWo 5 Am{U x ho gm_mB©H$ Ad`d AmhoV. 5xy + 10x = (5 × x × y) + (5 × x × 2)

= (5x × y) + (5x × 2)

`m§Zm {dVaU JwUY_m©déZ EH${ÌV H$é.

(5x × y) + (5x × 2) = 5x × (y + 2)

åhUyZ, 5xy + 10x = 5x (y + 2) [ho BpÀN>V Ad`d ñdén Amho]CXmhaU 1 : 12a2b + 15ab2 Mo Ad`d H$mT>m.

CH$b : 12a2b = 2 × 2 × 3 × a × a × b 15ab2 = 3 × 5 × a × b × b

òWo 3, a Am{U b ho gm_mB©H$ Ad`d AmhoV.

åhUyZ, 12a2b + 15ab2 = (3 × a × b × 2 × 2 × a) + (3 × a × b × 5 × b) = 3 × a × b × [(2 × 2 × a) + (5 × b)] = 3ab × (4a + 5b)

= 3ab (4a + 5b) [Amdí`H$ Ad`d ñdén]CXmhaU 2 : 10x2 – 18x3 + 14x4 Mo Ad`d H$mT>m.

CH$b : 10x2 = 2 × 5 × x × x 18x3 = 2 × 3 × 3 × x × x × x 14x4 = 2 × 7 × x × x × x × x

òWo 2, x Am{U x ho gm_mB©H$ Ad`d AmhoV.

åhUyZ, 10x2 – 18x3 + 14x4 = (2×x×x×5) – (2×x×x×3×3×x) + (2×x×x×7×x×x) = 2 × x × x ×[(5 – (3 × 3 × x) + (7 × x × x)] = 2x2 × (5 – 9x + 7x2)= 2x2 (7x2 – 9x + 5)

{dñVm[aV Ho$bobo Ad`d ñdén ho \$ŠV EH$M nX Agy eH$Vo `mMo Vwåhr {ZarjU

Ho$bmV H$m?

Ad`drH$aU 91

à`ËZ H$am :

Ad`d H$mT>m : (i) 12x + 36 (ii) 22y – 33z (iii) 14pq + 35pqr

6.2.2 JQ> H$ê$Z Ad`drH$aU

2xy + 2y + 3x + 3 hr amer nhm.AmnU {ZarjU Ho$bmV H$m, n{hë`m XmoZ nXm_YyZ 2 hm gm_mB©H$ Ad`d Amho Am{U eodQ>À`m XmoZ nXm_YyZ 3 hm gm_mB©H$ Ad`d Amho. nU `m gd© nXm_YyZ H$moUVmhr EH$ Agm gm_mB©H$ Ad`d Zmhr. ho H$go gmoS>dy eH$Vmo?

AmVm (2xy + 2y) ho Ad`d ñdénmV {bhÿ, 2xy + 2y = (2 × x × y) + (2 × y) = (2 × y × x) + (2 × y × 1)

= (2 y × x) + (2y × 1) = 2y (x + 1)3x + 3 = (3 × x) + (3 × 1) = 3 × ( x + 1) = 3 ( x + 1)

åhUyZ 2xy + 2y + 3x + 3 = 2y (x + 1) + 3 (x +1)

Q>rn → òWo AmnU 1 hm Ad`d KoUo

Amdí`H$ Amho. H$m?

{ZarjU H$am H$s, COì`m ~mOybm XmoÝhr nXm_Üò (x + 1) hm gm_mB©H$ Ad`d Amho. Vmo gm_mB©H$ KoD$.

∴2xy + 2y + 3x + 3 = 2y (x + 1) + 3 (x + 1) = (x + 1) (2y + 3)

{Xbobr amer 2xy + 2y + 3x + 3 hr AmVm JwUmH$mamÀ`m Ad`dmÀ`m ñdénmV Amho Am{U `mMo Ad`d (x + 1) Am{U (2y + 3) AmhoV. Va ho Ad`d A{ZdV©Zr` (Irreducible) AmhoV.

nwZaJQ>rH$aU åhUOo H$m`? (What is regrouping)

g_Om Oa darb amer 2xy + 3 + 2y + 3x Aer {Xbr Agob Va Ë`mMo Ad`drH$aU H$aUo gmono Pmbo ZgVo. Oa `m amerbm 2xy + 2y + 3x + 3 `m ñdénmV _m§Sy>Z {VMo (2xy + 2y) Am{U (3x + 3) `m_Üò JQ> H$ê$Z Ad`drH$aU H$aVm òVo. `mbm nwZaJQ>rH$aU åhUVmV.

nwZaJQ>rH$aU EH$ nojm OmñV nÜXVtZr H$aVm oVo. g_Om, 2xy + 3x + 2y + 3 Mo nwZaJQ>rH$aU H$aVmZm {hMo Xþgao gm_mB©H$ Ad`d KoD$ eH$Vmo. Mbm à`ËZ H$é.

2xy + 3x + 2y + 3 = 2 × x × y + 3 × x + 2 × y + 3 = x × (2y + 3) + 1 × (2y + 3)

= (2y + 3) (x + 1)

_mJrb à_mUoM Ad`d òVmV, nU Ë`m§Mo H«$_ _mÌ ~XbVmV.

CXmhaU 3 : 6xy – 4y + 6 – 9x Mo Ad`d H$mT>m.

CH$b :

nmàr 1 : àW_ gd© nXm_YyZ gm_mB©H$ Ad`d KoVm òD$ eH$Vmo H$m? Vo Vnmgm. Zmhr.

92 J{UV

nmàr 2 : nwZaJQ>rH$aUmMm {dMma H$am, n{hë`m XmoZ nXm_YyZ 2y hm gm_mB©H$ Ad`d KoVm òVmo.

6xy – 4y =2y (3x – 2) (a)

{ZarjU H$am H$s, eodQ>À`m XmoZ nXm_Yrb gm_mB©H$ Ad`d H$m` òB©b? Oa Vwåhr `mMm H«$_ – 9x + 6 `m à_mUo ~XbmV Va (3x – 2) Ad`d {_iVmo.

–9x + 6 = –3 (3x) + 3 (2) = – 3 (3x – 2) (b)

nmàr 3 : a Am{U b EH$Ì H$éZ, 6xy – 4y + 6 – 9x = 6xy – 4y – 9x + 6 =

= 2y (3x – 2) – 3 (3x – 2) = (3x – 2) (2y – 3)

åhUyZ, (6xy – 4y + 6 – 9x) Mo (3x – 2) (2y – 3) ho Ad`d AmhoV.

ñdmÜ`m` 6.1

1. {Xboë`m nXm_Yrb gm_mB©H$ Ad`d emoYm.

(i) 12x, 36 (ii) 2y, 22xy (iii) 14 pq, 28p2q2

(iv) 2x, 3x2, 4 (v) 6 abc, 24ab2, 12 a2b (vi) 16 x3, – 4x2, 32x (vii) 10 pq, 20qr, 30rp (viii) 3x2 y3, 10x3 y2,6 x2 y2z

2. Ad`d H$mT>m.

(i) 7x – 42 (ii) 6p – 12q (iii) 7a2 + 14a

(iv) – 16z + 20z3 (v) 20l2 m + 30 alm

(vi) 5 x2y – 15xy2 (vii) 10a2 – 15b2 + 20c2

(viii) – 4a2 + 4ab – 4ca (ix) x2yz + xy2z + xyz2

(x) ax2y + bxy2 + cxyz 3. Ad`d H$mT>m.

(i) x2 + xy + 8x + 8y (ii) 15xy – 6x + 5y – 2

(iii) ax + bx – ay – by (iv) 15pq + 15 + 9q + 25p

(v) z – 7 + 7 xy – xyz

6.2.3 {ZË`g_rH$aUm§Mm Cn`moJ H$éZ Ad`d H$mT>Uo (Factorisation using identities)

AmnUmg _mhrV Amho H$s, (a + b)2 = a2 + 2ab + b2 .... .... ..... (I)

(a – b)2 = a2 – 2ab + b2 .... .... ..... (II)

(a + b) (a – b) = a2 – b2 .... .... ..... (III)

Ad`drH$aU 93

`m {ZË`g_rH$aUm§Mm Cn`moJ H$éZ Ad`drH$aU H$aUo ho Imbrb gmoS>{dboë`m CXmhaUmZo ñnï> H$aVm òVo. `m {dñVmam§Mo AmnU {ZarjU H$é. {ZË`g_rH$aUm_Yrb S>mì`m ~mOyMo {dñVmarV ê$n COdr ~mOy XoVo.

CXmhaU 4 : x2 + 8x + 16 Mo Ad`d H$mT>m.

CH$b : {Xbobm amer nhm, òWo VrZ nXo AmhoV. Ë`m_wio òWo {ZË`g_rH$aU (III) bmJy nS>V Zmhr. VgoM _Ü` nXmÀ`m AJmoXa n{hbo nX Am{U {Vgao nX ho YZ nX AgUmao nyU© dJ© AmhoV. åhUyZ ho a2 + 2ab + b2 `m ñdénmV Amho. OoWo a = x, b = 4

∴ a2 + 2ab + b2 = x2 + 2(x) (4) + (4)2

= x2 + 8x + 16

∴a2 + 2ab + b2 = (a + b)2

Va x2 + 8x + 16 = (x + 4)2

CXmhaU 5 : 4y2 – 12y + 9 Mo Ad`d H$mT>m.

CH$b : {ZarjU H$am 4y2 = (2y)2, 9 = (3)2 Am{U 12y = 2 × 3 × (2y)

4y2 – 12y + 9 = (2y)2 – 2 × 3 × (2y) + (3)2

{ZarjU H$am H$s, {Xbobr amer a2 – 2ab + b2 `m ñdénmV

Agob d a=2y Am{U b=3, Va 2ab = 2 × 2y × 3 = 12y hmoVo.

= ( 2y – 3)2

CXmhaU 6 : 49p2 – 36 Mo Ad`d H$mT>m.

CH$b : òWo XmoZ nXo AgyZ XmoÝhr nXo dJ© AmhoV Am{U Xþgao nX G$U Amho. {Xbobr amer (a2 – b2) `m ñdénmV Amho. òWo {ZË`g_rH$aU (III) Mm Cn`moJ H$é.

49p2 – 36 = (7p)2 – (6)2

= (7p – 6 ) ( 7p + 6)

CXmhaU 7 : a2 – 2ab + b2 – c2 Mo Ad`d H$mT>m.

CH$b : n{hbr VrZ nXo (a–b)2 À`m énmV AmhoV. Va Mm¡Wo nX dJ© nX Amho. åhUyZ {Xbobr nXmdbr XmoZ nXm§À`m dJm©_Üò énm§VarV H$é.

∴ a2 – 2ab + b2 – c2 = (a – b)2– c2 ({ZË`g_rH$aU II dmnéZ)

= [(a – b) – c) ((a – b) + c)] ({ZË`g_rH$aU III dmnéZ)

= (a – b – c) (a – b + c) (Amdí`H$ Ad`d)

Q>rn : AmnUmg Amdí`H$ Ad`drH$aU {_iÊ`mgmR>r AmnU EH$m nmR>monmR> EH$ {ZË`g_rH$aUm§Mm Cn`moJ H$gm H$é eH$Vmo.

94 J{UV

CXmhaU 8 : m4 – 256 Mo Ad`d H$mT>m.

CH$b : AmnU nmhÿ m4 = (m2)2 Am{U 256 = (16)2

`mgmR>r {ZË`g_rH$aU III `mo½` R>aVo

åhUyZ, m2 – 256 = (m2)2 – (16)2

= (m2 –16) (m2+16)

AmVm, (m2 + 16) Mo naV Ad`drH$aU hmoV Zmhr, nU (m2 – 16) Mo _mÌ {ZË`g_rH$aU III Zo naV Ad`drH$aU H$aVm òVo.

∴ m2–16 = (m2) – (42) = (m – 4) (m + 4)

åhUyZ, m4 – 256 = (m – 4) (m + 4) (m2 +16) ({ZË`g_rH$aU III dmnéZ)

6.2.4 (x + a) ( x + b) `m ñdénmVrb Ad`d

AmVm, x2+5x+6, y2–7y+12, z2–4z–12, 3m2+9m+6 B. `mgma»`m EH$ MbnX AgUmè`m ~¡{OH$ amer§Mo ({ÌnXtMo) Ad`d H$emàH$mao H$mT>Vm òVrb mMr MMm© H$é. m amer (a+b)2 qH$dm (a–b)2 `m àH$maÀ`m ZmhrV. H$maU Ë`m nyU© dJ© ZmhrV. {Xboë`m x2+5x+6 `m CXmhaUm_Üò 6 hm nyU© dJ© Zmhr. hr amer (a2–b2) `m àH$maÀ`m gwÌmgmR>r gwÜXm `mo½` Zmhr.

`mgmR>r x2+(a+b)x+ab ho g_rH$aU `mo½` Amho. eodQ>r `mgmR>r AmnU _mJrb àH$aUm_Üò Ad`drH$aUmgmR>r dmnaboë`m {ZË`g_rH$aU IV Mm Cn`moJ H$éZ `m amerMo Ad`d H$mTy>.

(x + a) (x + b) = x2 + (a + b) x + ab (IV)

òWo x `m MbnXmMo ghJwUH$ Am{U pñWa nX `m§Mo AmnU {ZarjU H$é. AmVm ho CXmhaU H$go gmoS>{dVm òVo Vo nmhÿ.

CXmhaU 9 : x2 + 5x + 6 Mo Ad`d H$mT>m.

CH$b : x2 + 5x + 6 Mr {ZË`g_rH$aU (IV) _Yrb AgUmè`m COì`m ~mOyÀ`m KQ>mH$m§er VwbZm H$am. AmnU ab = 6 Am{U a + b = 5 H$mTy> eH$Vmo. `mdéZ AmnU a Am{U b H$mTy> eH$Vmo. Ë`m_wio `m§Mo Ad`d (x + a) (x + b) KoD$ eH$Vmo.

Oa ab = 6 Agob Va a Am{U b ho 6 Mo Ad`d AmhoV Agm Ë`mMm AW© hmoVmo. Va à`ËZ H$é a = 6, b = 1 AgVmZm a + b = 7 hmoVo. nU 5 Zmhr. åhUyZ hr {ZdS> `mo½` Zmhr.

AmVm AmnU a = 2, b = 3 KoD$Z à`ËZ H$é.

òWo a + b = 2 + 3 = 5, Oo `mo½` Amho.

hr Oa {ZdS> `mo½` Amho, Va {Xboë`m amerMo Ad`d (x + 2) (x + 3).

Ad`drH$aU 95

gm_mÝ`nUo, x2 + px + q `màH$maÀ`m ~¡{OH$ amerMo Ad`drH$aU H$aVmZm AmnU q Mo a Am{U b ho XmoZ Ad`d H$mTy> eH$Vmo. (åhUOoM, pñWa nX).

åhUyZ, ab = q Am{U a + b = PVa Vr amer, x2 + (a + b) x + abqH$dm x2 + ax + bx + abqH$dm x(x + a) + b(x + a)qH$dm (x + a) (x + b) ho Amdí`H$ Ad`d AmhoV.

CXmhaU 10 : y2 – 7y + 12 Mo Ad`d H$mT>m.

CH$b : AmnU nmhÿ eH$Vmo H$s, 12 = 3 × 4 Am{U 3 + 4 = 7

∴ y2 – 7y+ 12 = y2 – 3y – 4y + 12

= y (y –3) – 4 (y –3) = (y –3) (y – 4)

AmVm, òWo AmnU a Am{U b emoYÊ`mgmR>r {Xboë`m amerMr {ZË`g_rH$aU IV er VwbZm H$é eH$V Zmhr. nwaoem gamdmZ§Va {Xboë`m amerMo {ZË`g_rH$aUmer VwbZm Z H$aVm WoQ> Ë`m amerMo Ad`d nmSy> eH$Vmo.

CXmhaU 11 : z2 – 4z – 12 Mo Ad`d H$mT>m.

CH$b : òWo ab = –12 Amho `mMmM AW© a qH$dm b ho G$U Amho. Ë`mMà_mUo a+b= – 4 AgVmZm òWo _moR>çm g§»òMo {MÝh G$U Amho. a = – 4, b = 3 `mdê$Z à`ËZ H$é. nU a + b = –1 ho `mo½` Zmhr. nwT>rb qH$_Vr a = –6, b = 2 AgVmZm a + b = –4 Amdí`H$ KQ>H$ AmhoV.

åhUyZ, z2 – 4z –12 = z2 – 6z + 2z –12 = z(z – 6) + 2(z – 6 ) = ( z – 6) (z + 2)

CXmhaU 12 : 3m2 + 9m + 6 Mo Ad`d H$mT>m.

CH$b : AmnU nmhÿ eH$Vmo H$s gd© nXm_YyZ 3 hm gm_mB©H$ Amho.

åhUyZ, 3m2 + 9m + 6 = 3(m2 + 3m + 2)

AmVm m2 + 3m + 2 = m2 + m + 2m + 2 (Ogo 2 = 1 × 2) = m(m + 1)+ 2( m + 1) = (m + 1) (m + 2)

åhUyZ, 3m2 + 9m + 6 = 3(m + 1) (m + 2)

96 J{UV

ñdmÜ`m` 6.2

1. Imbrb ametMo Ad`d H$mT>m.

(i) a2 + 8a + 16 (ii) p2 – 10p + 25 (iii) 25m2 + 30m + 9 (iv) 49y2 + 84yz + 36z2 (v) 4x2 – 8x + 4: (vi) 121b2 – 88bc + 16c2

(vii) (l + m)2 – 4lm (gyMZm : òWo àW_ (l + m)2 Mm {dñVma H$é) (viii) a4 + 2a2b2 + b4

2. Ad`d H$mT>m.

(i) 4p2 – 9q2 (ii) 63a2 – 112b2

(iii) 49x2 – 36 (iv) 16x5 – 144x3 (v) (l + m)2 – (l – m)2 (vi) 9x2 y2 – 16 (vii) (x2 – 2xy + y2) – z2 (viii) 25a2 – 4b2 + 28bc – 49c2


(i) ax2 + bx (ii) 7p2 + 21q2 (iii) 2x3 + 2xy2 + 2xz2 (iv) am2 + bm2 + bn2 + an2

(v) (lm + l) + m + 1 (vi) y (y + z) + 9 (y + z) (vii) 5y2 – 20y – 8z + 2yz (viii) 10ab + 4a + 5b + 2 (ix) 6xy – 4y + 6 – 9x 4. Ad`d H$mT>m.

(i) a4 – b4 (ii) p4 – 81 (iii) x4 – (y + z)4 (iv) x4 – (x – z)4

(v) a4 – 2a2 b2 + b4


(i) p2 + 6p + 8 (ii) q2 – 10q + 21 (iii) p2 + 6p – 16

6.3 ~¡{OH$ ametMm ^mJmH$ma (Division of Algebraic Expressions)~¡{OH$ amerMr ~oarO Am{U dOmdmH$s Vwåhr {eH$bm AmhmV. VgoM XmoZ amer§Mm JwUmH$ma H$aUo gwÜXm Vwåhr {eH$bm AmhmV. Vwåhr EH$m ~¡{OH$ amerbm Xþgè`m amerZo ^mJUo ho AmVmn`ªV {eH$bm Zmhr. Va ho AmnU `m ^mJmV nmhÿ.

^mJmH$ma hr JwUmH$mamMr ì`ñV {H«$`m Amho, ho AmnU AmR>dy. åhUOoM 7 × 8 = 56 AgVmZm 56 ÷ 8 = 7 qH$dm 56 ÷ 7 = 8 hmoVo.

Ad`drH$aU 97

AmVm, AmnU `mgma»`mM H$mhr ~¡{OH$ amerdarb {H«$`m nmhÿ., CXmhaUmW© -

(i) 2x × 3x2 =6x3

åhUyZ, 6x3 ÷ 2x =3x2

Am{U 6x3 ÷ 3x2 =2x. (ii) 5x (x + 4) = 5x2 + 20x

åhUyZ, (5x2 + 20x) ÷ 5x = x + 4

Am{U (5x2 + 20x) ÷ (x + 4) = 5x.

EH$m ~¡{OH$ amerZo Xþgè`m ~¡{OH$ amerbm H$go ^mJbo OmVo ho AmnU AmVm nmhÿ. AmVm AmnU EH$nXrbm EH$nXrZo ^mJ XoD$.

6.3.1 EH$nXrbm EH$nXrZo ^mJUo

6x3 ÷ 2x AgVmZm

2x Am{U 6x3 Mo A{ZdV©Zr` Ad`d AmnU {bhÿ. 2x = 2 × x 6x3 = 2 × 3 × x × x × x

AmVm, AmnU 6x3 À`m Ad`dmÀ`m JQ>mVyZ 2x doJio H$é. 6x3 = 2 × x (3 × x × x)

= (2x) × (3x2)

åhUyZ, 6x3 ÷ 2x = 3x2.

VgoM, EH$m gmoß`m nÜXVrZwgma g§»`m§Mm ^mJmH$ma H$aV AgVmZm Ë`m_Yrb gm_mB©H$ Ad`d H$mTy>Z Vo aÔ H$ê$Z {e„H$ nX| ~mOybm H$mT>bm OmVmV.

77 ÷ 7 = 777

7 117

11= × =

Ë`mMà_mUo, 6x3 ÷ 2x = 62

3xx =

2 32

× × × ××x x x

x = 3 × x × x = 3x2

CXmhaU 13 : Imbrb ^mJmH$ma H$am.

(i) –20x4 ÷ 10x2 (ii) 7x2y2z2 ÷ 14xyz.

CH$b :

(i) –20x4 = –2 × 2 × 5 × x × x × x × x 10x2 = 2 × 5 × x × x

åhUyZ, –20x4 ÷ 10x2 = − × × × × × ×

× × ×2 2 5

2 5x x x xx x

= –2 × x × x = –2x2

98 J{UV

(ii) 714

72 7 2

12

2 2 2x y zxyz

x x y y z zx y z

xyz xyz= × × × × × ×× × × ×

= =

à`ËZ H$am

^mJmH$ma H$am :

(i) 24xy2z3 bm 6yz2 Zo (ii) 63a2b4c6 bm 7a2b2c3 Zo

6.3.2 ~hþnXrbm EH$nXrZo ^mJUo

AmVm 2y `m EH$nXrZo 4y3 + 5y2 + 6y `m {ÌnXrbm ^mJ XoUo nmhÿ`m.

4y3 + 5y2 + 6y = (2 × 2 × y × y × y) + (5 × y × y) + (2 × 3 × y) (òWo, àËòH$ nXmMm {dñVma Ad`d nÜXVrZo H$amdm).

gd© {R>H$mUr 2y hm gm_mB©H$ KQ>H$ Amho. Ë`m_wio àËòH$ nXm_YyZ 2y doJio H$am.

åhUOoM, 4y3 + 5y2 + 6y = 2 × y × (2 × y × y) + 2 × y × 52

×

y + 2 × y × 3

= 2y (2y2) + 2y 52

×

y + 2y(3)

= + +

2 2 5

232y y y (2y gm_mB©H$ Ad`d KoD$Z)

åhUyZ, (4y3 + 5y2 + 6y) ÷ 2y = 4 5 62

2 2 52

3

22 5

23

3 22

2y y yy

y y y

yy y+ + =

+ +

= + +

n`m©`rnUo aÔH$aU nÜXVr (Cancellation method) Mm dmna H$éZ {ÌnXrÀ`m àËòH$ nXmbm EH$nXrZo ^mJy.

(4y3 + 5y2 + 6y) ÷ 2y = 4 5 6

2

3 2y y yy

+ +

= 42

52

62

2 52

33 2

3yy

yy

yy

y y+ + + +

= 42

52

62

2 52

33 2

3yy

yy

yy

y y+ + + +

òWo AmnU A§eñWmZr Agboë`m {ÌnXrÀ`m àËòH$ nXmbm N>oXñWmZr Agboë`m EH$nXrZo ^mJ XoVmo.

Ad`drH$aU 99

CXmhaU 14 : Vwåhr {eH$boë`m XmoÝhr nÜXVr§Mm Cn`moJ H$éZ 24 (x2yz + xy2z + xyz2) bm 8xyz Zo ^mJm.

CH$b : 24 (x2yz + xy2z + xyz2) = 2 × 2 × 2 × 3 × [(x × x × y × z) + (x × y × y × z) + (x × y × z × z)]

= 2 × 2 × 2 × 3 × x × y × z × (x + y + z) = 8 × 3 × xyz × (x + y + z) (òWo g_mB©H$ KQ>H$ ~mhoa KoD$Z.)

åhUyZ, 24 (x2yz + xy2z + xyz2) ÷ 8xyz

= 8 38

× × × + +×

=xyz x y zxyz( ) 3 × (x + y + z) = 3 (x + y + z)

24 (x2yz + xy2z + xyz2) ÷ 8xyz = 248

248

248

2 2 2x yzxyz

xy zxyz

xyzxyz

+ +

= 3x + 3y + 3z = 3 (x + y + z)

6.4 ~hþnXrbm ~hþnXrZo ^mJUo

òWo A§eñWmZr Agboë`m ~hþnXrÀ`m àËòH$ nXmbm N>oXñWmZÀ`m {ÛnXrZo ^mJÊ`mg _XV hmoB©b H$m?

(7x2 + 14x) ÷ (x + 2)àW_ AmnU (7x2 + 14x) Mo Ad`d nmSy>Z N>oXñWmZr Agboë`m Ad`dmbm g_mZ H$é.7x2 + 14x = (7 × x × x) + (2 × 7 × x) = 7 × x × (x + 2) = 7x(x + 2)

AmVm (7x2 + 14x) ÷ (x + 2) = 7 14

27 2

2

2x xx

x xx

++

=+( )

+( )

= 7x [(x + 2) Ad`d aÔ H$éZ]CXmhaU 15 : 44 (x4 – 5x3 – 24 x2) bm 11x(x – 8) Zo ^mJmH$ma H$am.CH$b : 44 (x4 – 5x3 – 24 x2) Mo Ad`d H$mTy>. 44(x4 – 5x3 – 24x2) = 2 × 2 × 11 × x2(x2 – 5x – 24) (H§$gm_YyZ x2 hm gm_mB©H$ Ad`d KoVbm) = 2 × 2 × 11 × x2(x2 – 8x + 3x – 24) = 2 × 2 × 11 × x2 [x (x – 8) + 3(x – 8)] = 2 × 2 × 11 × x2 (x + 3) (x – 8)

åhUyZ,44 (x4 – 5x3+ 24x2) ÷ 11x (x – 8)

= 44 5 24

11 8

4 3 2x x xx x

− −( )−( )

= 2 2 11 3 8

11 8× × × × × +( )× −( )

× × −( )x x x x

x x

= 2 × 2 × x (x + 3) = 4x(x + 3)

A§eñWmZr Am{U N>oXñWmZr Agbobo gm_mB©H$ KQ>H$ 11, x Am{U (x – 8) aÔ Ho$bo.

100 J{UV

CXmhaU 16 : z(5x2 – 80) bm 5z (z + 4) Zo ^mJm.

CH$b : {d^mÁ` = z(5z2 – 80) = z [(5 × z2)] – (5 × 16)] = z × 5 × (z2 – 16)

= 5z × (z + 4) (z – 4) [a2 – b2 = (a + b) (a – b) Mm Cn`moJ H$éZ]

5 5 80 5 45 4 4

5 42z z z

z z zz z

−( ) ÷ +( ) = +( ) −( )+( )

= (z – 4)

ñdmÜ`m` 6.3

1. ^mJmH$ma H$am.

(i) 28x4 ÷ 56x (ii) –36y3 ÷ 9y2 (iii) 66pq2r3 ÷ 11qr2 (iv) 34x3y3z3 ÷ 51xy2z3

(v) 12a8b8 ÷ (– 6a6b4)

2. {Xboë`m ~hþnXrbm EH$nXrZo ^mJmH$ma H$am.

(i) (5x2 – 6x) ÷ 3x (ii) (3y8 – 4y6 + 5y4) ÷ y4 (iii) 8(x3y2z2 + x2y3z2 + x2y2z3) ÷ 4x2y2z2 (iv) (x3 + 2x2 + 3x) ÷ 2x (v) (p3q6 – p6q3) ÷ p3q3

3. ^mJmH$ma H$am.

(i) (10x – 25) ÷ 5 (ii) (10x – 25) ÷ (2x – 5) (iii) 10y(6y + 21) ÷ 5(2y + 7) (iv) 9x2y2(3z – 24) ÷ 27xy(z – 8) (v) 96abc(3a – 12) (5b – 30) ÷ 144(a – 4) (b – 6)

4. ^mJmH$ma H$am.

(i) 5(2x + 1) (3x + 5) ÷ (2x + 1) (ii) 26xy(x + 5) (y – 4) ÷ 13x(y – 4) (iii) 52pqr (p + q) (q + r) (r + p) ÷ 104pq(q + r) (r + p) (iv) 20(y + 4) (y2 + 5y + 3) ÷ 5(y + 4) (v) x(x + 1) (x + 2) (x + 3) ÷ x(x + 1)

Ad`drH$aU 101

5. Imbrb ametMo Ad`d H$mT>m Am{U ^mJmH$ma H$am. (i) (y2 + 7y + 10) ÷ (y + 5) (ii) (m2 – 14m – 32) ÷ (m + 2) (iii) (5p2 – 25p + 20) ÷ (p – 1) (iv) 4yz(z2 + 6z – 16) ÷ 2y(z + 8) (v) 5pq(p2 – q2) ÷ 2p(p + q) (vi) 12xy(9x2 – 16y2) ÷ 4xy(3x + 4y) (vii) 39y3(50y2 – 98) ÷ 26y2(5y + 7)

6.5 Vwåhr MwH$m (ÌwQ>r) emoYy eH$mb H$m?

H$m`© 1 ; g_rH$aU gmoS>{dVmZm g[aVmZo Imbrb {H«$`m Ho$br.

3x + x + 5x = 72 8x = 72

x Mm 1 hm ghJwUH$ gm_mÝ`nUo Xe©{dbm OmV

Zmhr. nU Ooìhm AmnU ~oarO H$aVmo Voìhm Vmo

KQ>H$ VoWo {_i{dbm nm{hOo.

x = 728

x = 9

{VZo ho MwH$sMo Ho$bobo Amho H$m? Agob Va ~amo~a CÎma emoYm

H$m`© 2 : Aßny nwT>rb à_mUo H$aVmo,

x = –3, 5x = 5 – 3 = 2 hr {H«$`m ~amo~a Amho H$m? Zgob Va ~amo~a H$am.

H$m`© 3 : Z_«Vm Am{U gb_m `m§Zr ~¡{OH$ amerÀ`m JwUmH$mamMr {H«$`m Imbrb nÕVrZo Ho$br.

bjmV R>odm H$s, dOm~mH$s H$aVmZm G$U KQ>H$ Zoh_r H§$gmMm dmna H$éZ gmoS>dm.

Z_«Vm gb_mbjmV R>odm H$s, H§$gmÀ`m ~mhoa AgUmè`m nXmZo H$§gm§Vrb àËòH$ nXmbm JwUmH$ma H$aUo Amdí`H$ Amho.

(a) 3(x – 4) = 3x – 4 3(x – 4) = 3x – 12

(b) (2x)2 = 2x2 (2x)2 = 4x2

bjmV R>odm, Ooìhm Vwåhr EH$nXrMm dJ© H$aVm Voìhm Ë`m§À`m ghJwUH$m§Mm gwÜXm dJ© H$aUo.

(c) (2a – 3) (a + 2) = 2a2 – 6

(2a – 3) (a + 2) = 2a2 + a – 6

(d) (x + 8)2 = x2 + 64 (x + 8)2

= x2 + 16x + 64

102 J{UV

(e) (x – 5)2 = x2 – 25 (x – 5)2

= x2 – 10x + 25

H$moUË`mhr gwÌmMm Cn`moJ

H$aÊ`mnydu Vo gwÌ Cn`moJr

nS>Vo H$m nhm.

Z_«Vm Am{U g_bmZo Ho$bobm JwUmH$ma ~amo~a Amho H$m? `m Vw_À`m CÎmambm H$maUo Úm.

H$m`© 4 : Omogo\$Zo nwT>rb ^mJmH$ma Agm Ho$bm, a a+ = +55

1

Ooìhm AmnU ~hþnXrbm EH$nXrZo

^mJVmo Voìhm A§eñWmZr Agboë`m

~hþnXrÀ`m àËòH$ nXmg N>oXñWmZr

Agboë`m EH$nXrZo ^mJ Úm.

Ë`mMm {_Ì {earf `mZo hmM ^mJmH$ma Agm Ho$bm, a a+ = +5

51 Am{U Ë`mMr Xþgar _¡ÌrU gw_ZZo hr {H«$`m

Aer Ho$br, a a+ = +5

5 51.

^mJmH$ma H$moUr ~amo~a Ho$bm Amho? H$moUr MwH$sMm Ho$bm Amho? H$m?

H$mhr _‚mm

AVwb Zoh_r EH$ doJimM {dMma H$aVmo. Ë`mZo gw_Vr {Q>MaZm§ {dMmabo H$s, Vwåhr gm§{JVbobo

~amo~a H$m` Amho? 6416

41

4= = ho _mPo CÎma ~amo~a H$go Ambo? {Q>MaZr ñnï> Ho$bo >`mMo H$maU

åhUOo, 64 ho 16 × 4 Ago hmoVo, åhUyZ 6416

16 416 1

41

= ××

= . åhUOoM òWo AmnU 6 gm_mB©H$ Z

H$mT>Vm 16 gm_mB©H$ H$mT>bo. H$maU, 16 Am{U 64 Mm 6 hm {d^mOH$ hmoD$ eH$V Zmhr. ""VgoM

{Q>MaZr {Xbo 664166

41

66641666

41

= =, , Am{U `mMàH$mao.'' ho _Ooera Zmhr H$m? Va Aem àH$mao

Vwåhr AVwbbm 6416 Aem àH$maMr BVa CXmhaUo emoYÊ`mg _XV H$amb H$m?

ñdmÜ`m` 6.4

Imbrb J{UVr {dYmZmVrb MwH$m emoYm Am{U ~amo~a H$am.

1. 4(x – 5) = 4x – 5

2. x(3x + 2) = 3x2 + 2

Ad`drH$aU 103

3. 2x + 3y = 5xy

4. x + 2x + 3x = 5x

5. 5y + 2y + y – 7y = 0

6. 3x + 2x = 5x2

7. (2x)2 + 4(2x) + 7 = 2x2 + 8x + 7

8. (2x)2 + 5x = 4x + 5x = 9x

9. (3x + 2)2 = 3x2 + 6x + 4

10. Imbrb amer_Üò x = – 3 KoD$Z.

(a) x2 + 5x + 4 {Xbo AgVm(– 3)2 + 5 (– 3) + 4 = 9+2+4 = 15

(b) x2 – 5x + 4 {Xbo AgVm (– 3)2 –5 ( – 3) + 4 = 9–15+4 = – 2

(c) x2 + 5x {Xbo AgVm (– 3)2 + 5 (–3) = – 9 – 15 = – 24

11. (y – 3)2 = y2 – 9

12. (z + 5)2 = z2 + 25

13. (2a + 3b) (a – b) = 2a2 – 3b2

14. (a + 4) (a + 2) = a2 + 8

15. (a – 4) (a – 2) = a2 – 8

16. 33

02

2

xx

= 17. 3 13

1 1 22

2

xx+ = + =

18. 3

3 212

xx +

= 19. 3

4 314x x+

=

20. 4 54

5xx+ = 21.

7 55

7x x+ =

104 J{UV


1. Ooìhm AmnU amerMo Ad`drH$aU H$aVmo, Voìhm Vo Ad`d JwUmH$ma ñdénmV {b{hVmo. ho Ad`d g§»`m, ~¡{OH$ MbnXo qH$dm ~¡{OH$ amer_Üò Agy eH$VmV.

2. A{ZdV©Zr` Ad`d åhUOo, Oo Ad`d JwUmH$ma Ad`dm§À`m ñdénmV nwÝhm {dñVma H$é eH$V Zmhr.

3. EImÚm amerMm Ad`drH$aU H$aÊ`mMm nÜXVera _mJ© åhUOo gm_mÝ` Ad`d nÜXV. `m nÜXVrV VrZ nm`è`m AgVmV.

(i) amer_Yrb àËòH$ nX A{ZdV©Zr` Ad`dm§À`m JwUmH$mamV {b{hUo.

(ii) nhm Am{U Ë`m_YyZ gm_mB©H$ Ad`d doJio H$am.

(iii) {dVaU {Z`_mZwgma àËòH$ nXm_Yrb BVa Ad`d EH$Ì H$am.

4. H$mhr doim, {Xboë`m amer_Yrb àËòH$ nXmMo gm_mÝ` Ad`d Agy eH$V ZmhrV. nU nXm§Mo Ago JQ> H$am H$s gd© JQ>mV gm_mB©H$ Ad`d Agob. Ooìhm Ë`m JQ>mMo Ad`drH$aU ho Ë`m JQ>mVrb gm_mB©H$ Ad`dmda Adb§~yZ AgVo. Voìhm `m nÕVrbm nwZa²JQ>rH$aU nÕV åhUVmV.

5. bjmV R>odm H$s, nwZa²JQ>rH$aU Ad`drH$aUm_Üò amerÀ`m nwZa²aMZoV KQ>H$ ho amerÀ`m Ad`drH$aUmer AmKmS>r KoD$ eH$V ZmhrV. `mMo AmnU Ago {ZarjU H$é eH$Vmo H$s, amer Am{U g§nyU© nwZa²JQ>rH$aU> ho AmnU MmMUr Am{U Ì¥Q>r nÕVrZo (Trial and Error) H$mTy> eH$Vmo.

6. a2 + 2ab + b2, a2 – 2ab + b2, a2 – b2 Am{U x2 + (a+b) + ab `m ñdénmV AgUmè`m qH$dm Ago H$aVm òUmè`m ametMo Ad`drH$aU ho àH$aU 9 _Yrb {ZË`g_rH$aUo I, II, III Am{U IV `mMm Cn`moJ H$éZ H$mTy> eH$Vmo.

a2 + 2ab + b2 =(a + b)2 a2 – 2ab + b2 =(a – b)2 a2 – b2 =(a + b) (a – b)

x2 + (a + b) x + ab =(x + a) (x + b)

7. bjmV R>odm H$s, (x + a) (x + b) `m àH$maMo Ad`d AgUmè`m amer_Üò ab ho g§»`m nX AgVo. òWo a Am{U b À`m {MÝhm§Mr H$miOr KoD$Z x Mm ghJwUH$ {_i{dbm nm{hOo. d a Am{U b Mm JwUmH$ma ab À`m {MÝhmer Ambm nm{hOo.

8. AmnUmg _mhrV Amho H$s, g§»`mÀ`m ~m~VrV g§»`m ^mJmH$ma hm JwUmH$mamMm ì`ñV AgVmo. `mMm Cn`moJ ~¡{OH$ amerÀ`m ^mJmH$mamgmR>r gwÜXm Ho$bm OmVmo.

Ad`drH$aU 105

9. Ooìhm AmnU EH$nXrZo ~hþnXrbm ^mJ XoVmo. Voìhm AmnU ~hþnXrÀ`m àËòH$ nXmbm Ë`m EH$nXrZo ^mJ XoVmo qH$dm gm_mB©H$ Ad`d ~mhoa H$mTy>Z EH$nXrZo ^mJ XoVmo.

10. Ooìhm ~hþnXrZo ~hþnXrbm ^mJ XoVmo Voìhm {d^mÁ` àËòH$ nXmbm ^mOH$Zo ^mJ Z XoVm, XmoÝhr {d^mÁ` Am{U {d^mOH$Mo Ad`d H$mTy>Z Ë`mVrb gm_mB©H$ Ad`d H$mTy>Z Vo aÔ Ho$bo nm{hOoV.

11. `m àH$maUm_Üò AmnU ~¡{OH$$ amerÀ`m ^mJmH$mam~Ôb {eH$bmo AmhmoV, Vo åhUOo,

{d^mÁ` = {d^mOH$ × ^mJmH$ma

gm_mÝ`nUo, hm g~§Y Imbrbà_mUo Xe©{dVmV

{d^mÁ` = {d^mOH$ × ^mJmH$ma + ~mH$s

AmnU m àH$aUm_Üò mJmH$ma Ho$ë`mZ§Va ~mH$s eyÝ` oB©b Aer CXmhaUoM \$ŠV nm{hbr.

12. ~¡{OH$ amerMr CXmhaUo gmoS>{dVm§Zm {dÚmWu gm_mÝ`nUo AZoH$ MwH$m H$aVmZm AmT>iVmV. Aem MwH$m Vwåhr H$aÊ`mMo Q>mim.

7.1.n[aM`

B`Îmm gmVdr_Üò Vwåhr g_Vb AmH¥$Ë`m Am{U KZmH¥$Vr m~Ôb {eH$bm AmhmV. g_Vb AmH¥$Ë`m§Zm XmoZ _mnZo AgVmV Oer bm§~r Am{U é§Xr. Ë`m_wio Ë`m§Zm {Û{_Vr` AmH¥$Ë`m åhUVmV. na§Vy KZmH¥$VtZm VrZ _mnZo AgVmV Oer bm§~r, é§Xr, C§Mr AWdm Imobr. Ë`m_wio Ë`m§Zm {Ì{_Vr` AmH¥$Ë`m åhUVmV. Ë`mMà_mUo KZmH¥$Vr H$mhr OmJm ì`mnVmV. {Û{_Vr` Am{U {Ì{_Vr` AmH¥$Ë`m§Zm g§{jßV ê$nmV 2-D Am{U 3-D AmH¥$Ë`m Ago g§~moYVmV. {ÌH$moU, Am`V, dVw©i BË`mXr 2-D AmH¥$Ë`m Am{U KZ, d¥Îm{MVr, e§Hy$, Jmob `m 3-D AmH¥$Ë`m AmhoV.

. ho H$am

OmoS>çm Owidm : (n{hbo CXmhaU Vw_À`mgmR>r Ho$bo Amho)

AmH$ma AmH$mamMm àH$ma AmH$mamMo Zmd

{Ì{_Vr` Jmob

{Û{_Vr` d¥Îmm{MVr

{Ì{_Vr` Mm¡ag

{Û{_Vr` dVw©i

àH$aU

7 KZmH¥$VtMo Ñí`rH$aU

KZmH¥$VtMo Ñí`rH$aU 107

{Ì{_Vr` KZm`V

{Ì{_Vr` KZ

{Û{_Vr` e§Hy$

{Ì{_Vr` {ÌH$moU

bjmV ¿`m H$s, darb gd© AmH$ma \$ŠV EH$mH$ma AmhoV. na§Vy X¡Z§{XZ OrdZmV AmnU AZoH$Xm g§`wŠV AmH$mam§À`m g§nH$m©V òVmo. CXmhaUmW©, Imbrb dñVy nhm.

V§~y d¥Îm{MVrda e§Hy$

H$WrbmMm S>~m d¥Îm{MVrMo H$dM

AmB©g²H«$s_ e§Hy$da AY©JmobmMo AmdaU

\$moQ>mo\«o$_ Am`VmH¥$Vr _mJ©

dmQ>r (~mD$b) AY©JmobmH¥$Vr H$dM Im§~mdarb Kw_Q>

d¥Îm{MVrdarb AY©JmobmMo AmdaU

108 J{UV

. ho H$am

Imbrb {MÌm§Zm (dñVw§Zm) Ë`m§À`m AmH$mamer OmoS>m.

dñVy {MÌ AmH$ma

(i) eoV O_rZ Am`VmH¥$Vr ~mJo_Üò EH$_oH$m§Zm N>oXUmao Am`VmH¥$Vr añVo.

(ii) Imo~U dVw©imH$ma ~mJobm dVw©imH$ma añVm.

(iii) ^modam {ÌH$moUmH¥$Vr eoVr Mm¡agmH¥$Vr eoVrbm Owibr Amho.

(iv) dVw©imH$ma ~mJ d¥Îm{MVrVyZ e§Hy$ ~mhoa H$mT>bm Amho.

(v) N>oXUmao añVo e§Hy$da AY©Jmob ~g{dbm Amho.

7.2 3D AmH$mam§Mo Ñî`Vwåhr {eH$bm AmhmV H$s {Ì{_Vr` dñVy doJdoJù`m ñWmZmdê$Z ({ñWVrdê$Z) doJdoJù`m ñdê$nmV {XgVmV. Ë`mH$[aVm Ë`m§Mr AmH¥$Vr doJdoJù`m ÑîQ>rVyZ H$mT>br OmVo. CXmhaUmW©, Imbr {Xbobr PmonS>rMr doJdoJù`m ~mOyZo {XgUmar Ñîò nhm.

~mOyg_moa

darb

PmonS>r g_moarb Ñî`~mOyMo Ñî` dê$Z {XgUmao Ñî`

`mMà_mUo EH$m ½bmgMr Ñîò nhm.

½bmg ~mOyMo Ñî` dê$Z {XgUmao Ñî`


EH$m ½bmgMo dê$Z Ñî` nm{hbo AgVm Vo EH$H|$Ðr` dVw©im§À`m OmoS>r gmaIo H$m {XgVo? Ë`mbm ~mOyZo nm{hbobo Ñî` doJio {Xgob H$m? `m~m~V {dMma H$am. AmVm AmnU EH$m {dQ>oÀ`m doJdoJù`m Ñî`m§H$S>o nmhÿ.

g_moarb Ñî`g_moa

darb

~mOy~mOyMo Ñî` dê$Z {XgUmao Ñî`

H$mhr KZ OmoSy>Z V`ma hmoUmè`m aMZm§Mr doJdoJir Ñîò {XgVmV, CXmhaUmW©,

g_moa~mOy

da

g_moarb Ñî`~mOyMo Ñî` déZ {XgUmao Ñî`

VrZ KZm§Mr aMZm

g_moa~mOy

da

g_moarb Ñî`déZ {XgUmao Ñî` ~mOyMo Ñî`

Mma KZm§Mr aMZm

g_moa~mOy

da

g_moarb Ñî`~mOyMo Ñî` déZ {XgUmao Ñî`

Mma KZm§Mr aMZm

. ho H$am

Vw_À`m g^modVmbr Agboë`m AZoH$ dñVw§Mo {d{dY ~mOyZo {ZarjU H$am. Ë`m~Ôb Vw_À`m {_Ì _¡{ÌUr§er MMm© H$am.

110 J{UV

ñdmÜ`m` 7.1

1. Imbrb àËòH$ KZmH¥$VrMr XmoZ Ñî` {Xbr AmhoV. Ë`mÀ`m darb ~mOy Am{U g_moarb ~mOyÀ`m OmoSçm Owidm. EH$ CXmhaU Vw_À`mgmR>r {Xbo Amho.

dñVy g_moarb Ñî` déZ {XgUmao Ñî`

(a)

~mQ>br

(i) (i)

(b)

dOZ

(ii) (ii)

(c)

W_m©g

(iii) (iii)

(d)

H$n~er

(iv) (iv)

(e)

S>~m

(v) (v)


2. Imbr àËòH$ KZmH¥$VrMr VrZ Ñî` {Xbr AmhoV. àËòH$mMo dê$Z, g_moê$Z Am{U ~mOyZo {XgUmao Ñî` AmoiIm.

(a)

g_moa

~mOy

da

H$nmQ>

(i) (ii) (iii)

(b)

g_moa~mOy

da

_mMrg

(c)

g_moa

~mOy

da

{Q>ìhr

(d)

g_moa~mOy

da

H$ma

112 J{UV

3. {Xboë`m KZmH¥$VtMo darb Ñî`, ~mOyMo Ñî`, g_moarb Ñî` AmoiIm.

(a)

g_moa

~mOy

da

(b)

g_moa

~mOy

da

(c)

g_moa

~mOy

da

(d)

g_moa

~mOy

da

(e)

g_moa~mOy

da


4. Imbr Xe©{dboë`m dñVw§Mo g_moarb Ñî`, ~mOyZo {XgUmao Ñî` Am{U dê$Z {XgUmao Ñî` H$mT>m.

(a) {_brQ>ar V§~y

g_moa

~mOy

da

(b) Q>o~b

g_moa

~mOy

da

(c) ZQ>

g_moa

~mOy

da

(d) fQ>H$moZr R>moH$im

g_moa

~mOy

da

(e) \$mgm

g_moa

~mOy

da(f) KZ

g_moa

~mOy

da

7.3 g^modVmbÀ`m ^mJmMo ZH$mem {MÌU

àmW{_H$ dJm©V Agë`mnmgyZ Vw_Mm ZH$mem§er g§nH©$ òVmo Amho. ^yJmob {eH$VmZm ZH$memda Vwåhmbm EImXo {d{eîQ> amÁ`, {d{eîQ> ZXr, {d{eîQ> nd©V XmI{dÊ`mg gm§{JVbo AgUma. B{Vhmg {eH$VmZm àmMrZ H$mir EImXr KQ>Zm KS>bobo {R>H$mU Xe©{dÊ`mg gm§{JVbo AgUma. ZÚm§Mo _mJ©, añVo, aoëdo_mJ©, ì`mnmar_mJ© ho XoIrb Vwåhr emoYbo AgUma.

114 J{UV

ZH$mem H$gm dmMm`Mm? ZH$mem dmMVmZm H$m` g_OyZ ¿`mdo Am{U AmnU H$moUVm {ZîH$f© H$mTy> eH$Vmo? ZH$mem_Üò H$moUH$moUVr _m{hVr AgVo Am{U H$moUH$moUVr ZgVo? ZH$mem {MÌm§nojm doJim AgVmo H$m? `m ^mJmV AmnU `m n¡H$s H$mhr àým§Mr CÎmao emoYÊ`mMm à`ËZ H$ê$. KamÀ`m {MÌmÀ`m ~mOybm ZH$mem {Xbm Amho Vmo nhm (AmH¥$Vr 7.1).

AmH¥$Vr 7.1darb CXmhaUmdê$Z Vwåhr H$m` {ZîH$f© H$mT>mb? Amåhr Á`m doir {MÌ H$mT>Vmo Ë`mdoir

AmnU Ë`m_Üò g{dñVanUo H$m` nm{hbo Amho Vo Xe©{dÊ`mMm à`ËZ H$aVmo na§Vy ZH$mem Ë`m dñVyMo ñWmZ Am{U BVa dñVyer Agbobm g§~§Y Xe©{dVmo. Xþgam _wÔm åhUOo EH$mÚm {MÌmMo {ddaU àËòH$ OU doJdoJù`m [aVrZo H$aVmo. H$maU darb {MÌmVrb Ka nhmVmZm Ë`mMo C^maÊ`mMo ñWmZ doJdoJio AgVo na§Vy ZH$memÀ`m ~m~VrV Vgo KS>V Zmhr. KamMm ZH$mem ~XbV Zmhr. Oar nhmUmè`mZo Amnbo ñWmZ ~Xbbo Var \$aH$ nS>V Zmhr. {MÌ H$mT>VmZm à_mU~ÜXVobm \$ma _hËd AgVo na§Vy ZH$mem_Üò Ë`mMm g§~§Y òV Zmhr.

AmVm gmV df} d` Agboë`m amKdZo H$mT>boë`m ZH$memH$S>o nhm (AmH¥$Vr 7.2). `mV Ë`mZo Amnë`m KamnmgyZ Amnë`m emion`ªVMm _mJ© Xe©[dbm Amho. `m ZH$memÀ`m ghmæ`mZo Vwåhr gm§Jy eH$mb H$m -

(i) amKdMr emim Ë`mÀ`m KamnmgyZ {H$Vr A§Vamda Amho?

(ii) ZH$memV H$mT>bobo àËòH$ dVw©i Mma añVo EH$Ì òD$Z V`ma Pmbobo _Yrb dVw©i Xe©{dVo H$m?

(iii) H$moUmMr emim KamnmgyZ Odi Amho. amKdMr H$m Ë`mÀ`m ~{hUrMr?

darb àým§Mr CÎmao ZH$memÀ`m ghmæ`mZo XoUo \$ma H$R>rU Amho. H$m? ho Vwåhr gm§Jy eH$mb?

_mÂ`m ~{hUrMr emim

_mPr emim

_mPo Ka

AmH¥$Vr 7.2

`mbm _w»` H$maU åhUOo Xe©{dbobo A§Va `mo½` à_mUmV Amho H$s Zmhr ho _mhrV Zmhr AWdm Ë`mV Xe©{dbobr dVw©io hr Mma añVo {_imë`mZ§Va ~ZUmao dVw©i (circle) Amho H$s AmUIr H$m`.


Ë`mMr Xhm dfm©Mr ~{hU _rZm {hZo H$mT>bobm Xþgam ZH$mem nmhÿ. Á`mV {VZo Amnë`m KamnmgyZ emion`ªV OmÊ`mMm _mJ© Xe©{dbm Amho (AmH¥$Vr 7.3).

hm ZH$mem AmYrÀ`m ZH$memnojm doJim Amho. òWo _rZmZo _mJ©Xe©H$ IwUm Xe©{dÊ`mgmR>r doJdoJù`m {MÝhm§Mm dmna Ho$bm Amho. Xþgar JmoîQ> åhUOo XÿaMo A§Va XmI{dÊ`mgmR>r aofm§Mr bm§~r OmñV R>odbr Amho Am{U H$_r A§VamgmR>r bhmZ aofm. åhUOoM {VZo ZH$mem à_mU~ÜX H$mT>bm Amho.

AmVm Vwåhr Imbrb àým§Mr CÎmao XoD$ eH$mb :

(i) amKdMr emim Ë`mÀ`m KamnmgyZ {H$Vr A§Vamda Amho?

(ii) H$moUmMr emim KamnmgyZ Odi Amho, amKdMr H$s _rZmMr?

(iii) _mJm©darb _hËdmÀ`m IyUm H$moUË`m AmhoV?

KaXþH$mZemimañVmVbmd

BpñnVi

EH$ km

AmH¥$Vr 7.3

`mdê$Z Amåhmbm g_OVo H$s R>am{dH$ {MÝho/IwUm§Mm dmna Am{U A§Vam§Mm CëboI `m_wio ZH$mem dmMZmbm gmono Pmbo. O{_Zrda Agbobo Iao A§Va d ZH$memV Agbobo A§Va ho EH$m `mo½` à_mUmV AgVo. ho EH$ {ZpûMV à_mU KoD$Z Ho$bo OmVo. ZH$mem H$mT>Vm§Zm H$moUVo à_mU ¿`mdo ho OmUyZ KoUo Amdí`H$ Amho, åhUOo IamoIa Agbobo A§Va ZH$mem_Üò 1mm AWdm 1cm à_mUmV Xe©{dbo Amho. `mMm AW© Agm hmoVmo H$s Oa EImXr ì`ŠVr ZH$mem H$mT>V Agob Va Ë`mZo ho {ZpíM¨V Ho$bo nm{hOo H$s ZH$memV 1cm A§Va ho 1km AWdm 10km A§Va Xe©{dVo. ho à_mU ZH$mem ZH$memà_mUo ~XbVo. na§Vy EH$mM ZH$memV àË`jmV à_mU ~XbV Zmhr. CXmhaUmW©, ^maVmÀ`m ZH$memgmo~V {X„rMm ZH$mem nhm.

Vwåhmbm Ago AmT>iob H$s Ooìhm EH$mM AmH$mamMo XmoZ ZH$meo H$mT>bo OmVmV Ë`mV A§Va Am{U à_mU doJdoJio AgVo. Vo åhUOo {X„rÀ`m ZH$memV 1 cm ho H$_r A§Va Xe©{dVo Va VoM à_mU ^maVmÀ`m ZH$memV OmñV A§Va Xe©{dVo.

EImXo {R>H$mU OmñV {dpñVU© Agob Va Ë`mgmR>r H$mT>bm OmUmam ZH$mem bhmZ AgVmo, VgoM OmñV A§Va 1 cm à_mUmV Xe©{dbo OmVo.

`mdê$Z òUmam gmam§e AmnU Agm H$mTy> eH$Vmo H$s

1. EH$ ZH$mem hm EH$ {d{eîQ> dñVy qH$dm {R>H$mUmMo Xþgè`m dñVy / {R>H$mUmÀ`m ñWmZmer g§~§Y Xe©{dVmo.

2. doJdoJù`m dñVy§ / ñWmZm§Zm Xe©{dÊ`mgmR>r doJdoJù`m g§Ho$Vm§Mm/ {MÝhm§Mm dmna Ho$bm OmVmo.

116 J{UV

3. ZH$mem_Üò H$moUVmhr g§X^© AWdm ÑîQ>rH$moZ AgV Zmhr. åhUOoM nhmUmè`mÀ`m Odi AgUmar dñVy hr Xe©{dVmZm Xÿa AgUmè`m dñVyÀ`m AmH$mamMr Xe©{dbr OmVo. CXmhaUmgmR>r AmH¥$Vr 7.4 nhm.

AmH¥$Vr 7.4 4. ZH$memgmR>r Agbobo à_mU ho R>am{dH$ ZH$memgmR>r {ZpûMV AgVo. H$mJXmdarb A§Va ho

Iè`m A§Vamer Ë`m à_mUmV H$_r Ho$bo OmVo.

ho H$am 1. Imbr {Xbobm EH$m ZH$mem nhm (AmH¥$Vr 7.5).

ñ_emZ J«§Wmb` Ý`m`mb` nmo{bg ñQ>oeZ gamoda

~g ñWmZH$

CÚmZ

Ob H|$Ð CÚmZ ZXr

H$m°boO nmoñQ>Am°{\$g

g_mO ^dZ

hm°pñnQ>bàmW{_H$ emim_mÜ`{_H$emim

A{¾em_H$ H|$Ð

AmH¥$Vr 7.5 (a) Imbr gm§{JVë`mà_mUo ZH$memV a§J âm.

nmUr - {Zim, Ap½Zem_H$ H|$Ð - bmb, dmMZmb` - Zm[a¨Jr

emim - {ndim, CÚmZ - {hadm, g_mO ^dZ - Jwbm~r

hm°pñnQ>b - Om§îm, ñ_emZ - Vn{H$ar.

(b) Xþgam añVm Am{U _w»` añVm {OWo N>oXVmo Ë`m {R>H$mUr {haì`m a§JmV X hr IyU H$am, Á`m {R>H$mUr {Vgam añVm ZXrbm {_iVmo VoWo H$mi`m a§JmV Y, VgoM _w»` añVm Am{U n{hbm añVm N>oXVmo Ë`m {R>H$mUr bmb a§JmV Z hr IyU H$am.

(c) H$m°boOnmgyZ gamodamH$S>o OmUmam OdiMm _mJ© {H$a{_Or Vm§~S>`m a§JmV XmIdm.

2. Vw_À`m KamnmgyZ emion`ªV OmÊ`mÀ`m _mJm©Mm ZH$mem _hËdmÀ`m g§Ho$V/{MÝhmgh Xe©dm.


ñdmÜ`m` 7.2

1. EH$m ehamMm {Xbobm ZH$mem nhm.

ñ_emZ

J«§Wmb`

añVm A

añVm

DañVm

añVm

~mOmanoR>

nmo{bg ñQ>oeZ

Zohé amoS>

emór amoS>

_.Jm§Yr amoS>

gamoda

~gñWmZH$

CÚmZ

Ob H|$ÐH$m°boO

g_mO ^dZ

hm°pñnQ>b

àmW{_H

$ emim

_mÜ`{_H$emim

A{¾em_H$ H|$Ð

Imbrb CÎmao Úm.

(a) ZH$memV a§J âm : nmUr - {Zim, Ap½Zem_H$ H|$Ð - bmb, dmMZmb` - Zm[a¨Jr

emim - {ndim, CÚmZ - {hadm, g_mO ^dZ - Jwbm~r, H$m°boO - A~mobr,

hm°pñnQ>b - Om§îm, ñ_emZ - Vn{H$ar.

(b) Zohê$ _mJ© Am{U _mJ© C' {OWo {_iVmV {VWo {hadm 'X', Jm§Yr _mJ© d amoS> 'A' {OWo {_iVmV VoWo {hadm 'Y' {bhm.

(c) dmMZmb`mnmgyZ ~g ñWmZH$mH$S>o OmUmam OdiMm _mJ© bmb a§JmZo Xe©dm.

(d) ZH$memV nyd}H$S>o CÚmZ Amho H$s ~mOmanoR>?

(e) ZH$memV X{jUoH$S>o àmW{_H$ emim Amho H$s _mÜ`{_H$ emim?

2. `mo½` à_mU KoD$Z Am{U {d{dY dñVygmR>r `mo½` g§Ho$V KoD$Z Vw_À`m dJ© ImobrMm ZH$mem H$mT>m.

3. `mo½` à_mU KoD$Z Am{U IoimMo _¡XmZ, _w»` B_maV, ~JrMm BË`mXr AZoH$ {R>H$mUo Xe©{dÊ`mgmR>r `mo½` g§Ho$VmMm dmna H$ê$Z emioÀ`m AmdmamMm ZH$mem H$mT>m.

4. Vw_À`m {_Ìmbm Vw_À`m Kar nmohmoMÊ`mgmR>r H$moUVrhr AS>MU CX²^dUma Zmhr Aem àH$maMm `mo½` gyMZmgh ZH$mem H$mT>m.

118 J{UV

7.4 n¥îR>^mJ, H$S>m Am{U {eamoq~Xÿ

Imbrb KZmH¥$Vt nhm.

{eamoq~Xÿ (Vertex)

n¥ð>^mJ (Face)

H$S>m (Edge)

_bm H$moUVmhr {eamoq~Xÿ Zmhrd gnmQ> n¥ð>^mJ nU Zmhr. Va _r H$moU?

H$moS>o

àËòH$ KZmH¥$Vr hr ~hþ^wOmH¥$Vr joÌm§nmgyZ ~Zbr Amho. Ë`m§Zm AmnU Ë`m KZmH¥$VrMo n¥îR>^mJ (faces) åhUVmo, ho n¥îR>^mJ Á`m EH$m aofoV {_iVmV Ë`m H$S>m (edges) hmo`. H$S>m Á`m {R>H$mUr {_iVmV Vmo {eamoq~Xÿ (vertex) Aem KZmH¥$Vr§Zm ~hþn¥îR>Ho$ (Polyhedron) åhUVmV.

hr ~hþn¥ð>Ho$ AmhoV hr ~hþn¥ð>Ho$ ZmhrV

~hþn¥îR>Ho$ hr ~hþn¥îR>H$ Zgboë`m KZmH¥$VrnmgyZ doJir H$er? Imbrb AmH¥$VtMo bjnyd©H$ {ZarjU H$am. nwT>o {Xboë`m {VÝhr KZm~m~V Vwåhr OmUm.


dH«$n¥ð>^mJ nmíd©n¥ð>^mJ

nm`m

nm`mnmíd©n¥ð>^mJ

nm`m

Jmob e§Hy$ d¥Îm{M{V

~{hd©H«$ ~hþn¥îR>Ho$ : ~{hd©H«$ ~hþ^wOmH¥$VtMr g§H$ënZm g§H$ënZm AmR>dm. hrM H$ënZm ~mø ~hþn¥îR>H$mZm g_ê$n Amho.

~{hd©H«$ ~hþn¥ð>Ho$ AmhoV. hr ~{hd©H«$ ~hþn¥ð>Ho$ ZmhrV.

{Z`{_V (gwg_) ~hþn¥îR>Ho$ : EH$ ~hþn¥îR>H$ Voìhm {Z`{_V ~hþn¥îR>H (gwg_ ~hþn¥îR>H$)(Regular Polyhedron) åhUVm òB©b Ooìhm Ë`mMo gd© n¥îR>^mJ {Z`{_V ~hþ^wOmH¥$Vr§Zr ~Zbbo AgyZ àËòH$ {eamoq~Xÿer {_iUmè`m n¥îR>^mJm§Mr g§»`m g_mZ Agob.

ho {Z`{_V ~hþn¥îR>H$ Amho. Ë`mMo gd© n¥îR>^mJ EH$ê$n AgyZ Vo {Z`{_V ~hþ^wOmH¥$Vr AmhoV. Ë`mÀ`m {eamoq~XÿMr aMZm g_mZ n¥îR>^mJm§À`m g§»`m§Zr Pmbr Amho.

ho {Z`{_V ~hþn¥îR>H$ Zmhr. gd© ~mOy EH$ê$n AmhoV na§Vy {eamoq~XÿMr aMZm g_mZ n¥îR>^mJm§Zr Pmbr Zmhr. A q~XÿV 3 n¥îR>^mJ {_iVmV Am{U B q~XÿV 4 n¥îR>^mJ {_iVmV.

Amnë`m g^modVr AmT>iUmè`m ~hþn¥îR>H$m§À`m JQ>mVrb bmobH$ Am{U {na°{_S²>g _hËdmMo AmhoV.

ho bmobH$ AmhoV. ho {na°{_S²>g AmhoV.

120 J{UV

AmnU bmobH$ ~hþn¥îR>H$ Amho Ago åhUVmo H$maU Ë`mMm nm`m d darb ~mOy (_mWm) EH$ê$n ~hþ^wOmH¥$Vr AgVmV Am{U BVa nmíd©n¥îR>^mJ ho AmH$mamZo Am`V AgVmV.

Ë`mMà_mUo {na°{_S> ho XoIrb ~hþn¥îR>H$ AgyZ Ë`mMm nm`m ~hþ^wOmH¥$Vr AgVmo. (Ë`mbm {H$Vrhr ~mOy Agy eH$VmV). `mMo nmûd©n¥îR>^mJ {ÌH$moUmH¥$Vr AgyZ EH$M g_mB©H$ {eamoq~Xÿ AgVmo. (Oa ~hþ^wOmH¥$VrÀ`m gnmQ> ~mOydarb q~Xÿ gmoSy>Z Ë`m§Mo gd© H$monao EH$Ì OmoS>ë`mg Vwåhmbm {na°{_S>Mr à{VH¥$Vr {_iVo).

bmobH$ AWdm {na°{_S>Mo Zmd ho Ë`mÀ`m nm`mÀ`m Zmdmdê$Z Cëbo{Ibo OmVo. åhUOo fQ>H$moZmH¥$Vr bmobH$ `mMm nm`m fQ>H$moZmH¥$Vr AgVmo. {ÌH$moUmH¥$Vr {na°{_S> åhUOo Á`mMm nm`m {ÌH$moUmH¥$Vr AgVmo. Am`VmH¥$Vr bmobH$ åhUOo H$m`? Mm¡agmH¥$Vr {na°{_S> åhUOo H$m`? `mMm AW© ñnîQ> Amho H$s Á`mMm nm`m àËòH$s AZwH«$_o Am`VmH¥$Vr Am{U Mm¡agmH¥$Vr Amho.

ho H$am

Imbr VŠË`mV {Xboë`m ~hþ^wOmH¥$VtgmR>r n¥îR>^mJ, {eamoq~Xÿ Am{U H$S>m `m§Mr g§»`m {bhm. (`mV 'V' åhUOo {eamoq~XÿMr g§»`m, 'F' åhUOo n¥îR>^mJm§Mr g§»`m Am{U 'E' hr H$S>m§Mr g§»`m)

KZmH¥$Vr F V E F+V E+2

KZm`V

{ÌH$moUmH¥$Vr {na°{_S>

{ÌH$moUmH¥$Vr bmobH$

Mm¡agmH¥$Vr nm`mMm {na°{_S>

Mm¡agmH¥$Vr nm`mMm bmobH$

eodQ>À`m XmoZ ñV§^mdê$Z Vwåhr H$m` AZw_mZ H$mT>mb? àËòH$ KQ>ZoV

F + V = E + 2, åhUOoM F + V – E = 2 ho òVo H$m? `m g§~§Ymbm `wbaMo gyÌ

åhUVmV. gd© ~hþn¥îR>H$mgmR>r ho gyÌ gË` Amho ho {gÜX hmoVo.

{dMma H$am, MMm© H$am Am{U {bhm.

EImÚm KZmH¥$VrMr F,V Am{U E `m§Mo H$mhr ^mJ H$mnyZ H$mT>bm Va H$m` hmoB©b? (`mgmR>r _D$ßbmñQ>rH$Mm KZ ¿`m Am{U Ë`mMm H$monam H$mnyZ Vnmgm.


ñdmÜ`m` 7.3

1. EH$m ~hþn¥îR>H$mbm ho n¥îR>^mJ Agy eH$Vrb H$m?

(i) 3 {ÌH$moU

(ii) 4 {ÌH$moU

(iii) EH$ Mm¡ag Am{U Mma {ÌH$moU

2. H$moUË`mhr {Xboë`m g§»ò BVŠ`m n¥îR>^mJm§Mm ~hþn¥îR>H$ Agy eHo$b H$m? (gyMZm : {na°{_S>Mm {dMma H$am)

3. Imbrbn¡H$s H$moUVo bmobH$ AmhoV?

{Iim Q>moH$ Z H$mT>bobr nopÝgb

S>~mH$mJXmdarb dOZ

4. (i) bmobH$ Am{U d¥Îm{MVr g_mZ H$go?

(ii) {na°{_S> Am{U e§Hy$$ g_mZ H$go ?

5. Mm¡agmH¥$Vr bmobH$ hm KZmer gmaIm AgVmo H$m? {ddaU H$am.

6. Imbrb KZmH¥$VrgmR>r `wbaMo gwÌ nS>Vmim.

122 J{UV

7. `wbaÀ`m gwÌmMm Cn`moJ H$ê$Z _mhrV Zgbobo nX H$mT>m.

n¥îR>^mJ ? 5 20

{eamoq~Xÿ 6 ? 12

H$S>m 12 9 ?

8. EH$m ~hþn¥îR>H$mbm 10 n¥îR>^mJ, 20 H$S>m Am{U 15 {eamoq~Xÿ Agy eH$Vrb H$m?


1. {Û{_Vr` Am{U {Ì{_Vr` dñVy AmoiIUo.

2. EImÚm dñVy_Üò gm_mdboë`m {d{dY AmH¥$Ë`m/AmH$ma AmoiIUo.

3. {Ì{_Vr` dñVy doJdoJù`m ñWmZmdê$Z nm{hë`mg doJdoJio Ñî` {XgVo.

4. ZH$mem hm {MÌmnojm doJim AgVmo.

5. ZH$mem, EImXr dñVy/ ñWmZ ho Xþgar dñVy/ñWmZ m§Mm g§~§Y Xe©{dUmam _mJ© gy{MV H$aVmo.

6. {d{dY dñVy/ñWmZ Xe©{dÊ`mgmR>r g§Ho$Vm§Mm dmna Ho$bm OmVmo.

7. ZH$mem_Üò g§X^© AWdm à_mU~ÜXVm ZgVo.

8. {d{eîQ> ZH$memgmR>r _mnZmMo à_mU {ZpûMV AgVo.

9. H$moUË`mhr ~hþn¥îR>H$m§gmR>r F + V – E = 2. mV 'F' n¥îR>^mJm§Mr g§»`m, 'V' H$S>m§Mr g§»`m Am{U 'E' {eamoq~XÿMr g§»`m. `m g§~§Ymbm `wbaMo gyÌ åhUVmV.

8.1.JwUmoÎma Am{U eoH$S>odmar `m§Mr AmR>dU H$aUo

Amåhmbm _mhrV Amho H$s JwUmoÎma åhUOo XmoZ n[a_mUm§Mr VwbZm. EH$m Q>monbrV XmoZ àH$mamMr \$io AmhoV, g_Om 20 g\$aM§X Am{U 5 g§Ìr. Va g§Í`m§Mr g§»`m Am{U g\$aM§Xm§Mr g§»`m `m§Mo JwUmoÎma = 5 : 20.

AnyUmªH$m§Mm dmna H$ê$Z Aerhr VwbZm H$ê$ eH$Vmo,

520

14

=

g§Í`m§Mr g§»`m g\$aM§Xm§À`m g§» oÀ`m 520

14

= Amho. JwUmoÎmamÀ`m ñdê$nmV, ho 1 : 4 nU

dmMVm§Zm ""1 bm "4'', qH$dm ""EH$mg Mma'' Ago dmMVmV.

AWdm

g\$aM§Xm§Mr g§»`m Am{U g§Í`m§Mr g§»`m `m§Mo JwUmoÎma = 205

41

= Á`mMm AW© g\$aM§Xm§Mr

g§»`m g§Í`m§À`m g§»òÀ`m 4 nQ> Amho. hr VwbZm AmnU eoH$S>odmarMm dmna H$ê$Z XoIrb H$ê$ eH$Vmo.

25 \$imn¡H$s 5 g§Ìr AmhoV åhUyZ g§Í`m§Mr eoH$S>odmar 5

2544

20100

20× = = % . N>oX

100 H$ê$Z KoVbm.

AWdm

EH$_mZ (Unitary) nÜXVrZo 25 \$imn¡H$s g§Í`m§Mr g§»`m 5 Amho. åhUyZ 100 \$imn¡H$s g§Í`m§Mr g§»`m

= × =525

100 20

Q>monbr_Üò \$ŠV g\$aM§X d g§Ìr AmhoV.

åhUyZ g\$aM§Xm§Mr eoH$S>odmar + g§Í`m§Mr eoH$S>odmar = 100.

AWdm, g\$aM§Xm§Mr eoH$S>odmar + 20 = 100

AWdm, g\$aM§Xm§Mr eoH$S>odmar = 100 - 20 = 80

`mdê$Z Q>monbrV 20% g§Ìr Am{U 80% g\$aM§X AmhoV.

àH$aU

8 à_mUm§Mr VwbZm

124 J{UV

CXmhaU 1 : B`Îmm VII Mr ghb Am`mo{OV H$aÊ`mMr `moOZm EH$m emioZo Ho$br. EHy$U {dÚmÏ`mªÀ`m g§»òn¡H$s 60% _wbr AmhoV åhUOo Ë`m 18 AmhoV. ghbrMo {R>H$mU emionmgyZ 55 {H$._r. A§Vamda Amho Am{U dmhVyH$ H§$nZr Xa {H$ _rbm ` 12 IM© bmdVo. AënmonhmamMm EHy$U IM© ` 4280

òUma Amho. _J Vwåhr gm§Jy eH$mb H$m?

1. _wbtÀ`m g§»òMo _wbm§À`m g§»òer Agbobo JwUmoÎma {H$Vr?

2. ghbr_Üò {dÚmÏ`mªgmo~V XmoZ {ejH$ gh^mJr Pmë`mg àËòH$s òUmam IM© {H$Vr?

3. Ë`m§À`m àdmgmVrb Wm§~Ê`mMo n{hbo {R>H$mU emionmgyZ 22 {H$._r. A§Vamda Amho. Va 55 {H$._r. EHy$U A§VamÀ`m ho A§Va eoH$S>m {H$Vr? àdmg H$aÊ`mMo A§Va eoH$S>m {H$Vr {eëbH$ am{hbo?

CH$b :

1. _wbtÀ`m g§»òMo _wbm§À`m g§»òer Agbobo JwUmoÎma H$mT>Ê`mgmR>r A{e_m Am{U Om°Z Imbrb CÎmao KoD$Z Ambo.

A{e_mZo `m nÜXVrZo Ho$bo.

{dÚmÏ`mªMr EHy$U g§»`m x _mZy. Ë`mVrb 60% _wbr AmhoV. åhUyZ x Mo 60% = 18 60

10018× =x AWdm x = 18 × 100

60 = 30

EHy$U {dÚmWu = 30

AWdm

Om°ZZo EH$_mZ nÜXV dmnabr

100 {dÚmÏ`mªn¡H$s 60 _wbr AmhoV. 10060

{dÚmÏ`mªn¡H$s EH$ _wbJr Amho.

18 _wbr AgVrb Va EHy$U {dÚmWu {H$Vr?

EHy$U {dÚmWu = 10060

× 18 = 30

åhUyZ, _wbm§Mr g§»`m = 30 – 18 = 12. `mdê$Z _wbtÀ`m g§»òMo _wbm§À`m g§»òer Agbobo JwUmoÎma 18 : 12 AWdm 18

1232

= .

1812

32

= ho 3 : 2 Ago {b{hbo OmVo d dmMVmZm 3 bm 2 (VrZmg XmoZ) Ago dmMVmV.

2. àËòH$mgmR>r òUmam IM©

dmhVyH$ IM© = Om`Mo d `m`Mo A§Va × Xa

= (55 × 2) × 12 = 110 × 12 = 1320

EHy$U IM© = AënmonhmamMm IM© + dmhVyH$sMm IM©

= ` 4280 + ` 1320 = ` 5600

ghbrbm OmUmè`m§Mr EHy$U g§»`m = 18 _wbr + 12 _wbo + 2 {ejH$ = 32 ì`ŠVr.

à_mUm§Mr VwbZm 125

àËòH$mgmR>r òUmam IM© H$mT>Ê`mgmR>r A{e_m Am{U Om°Z `mZr EH$_mZ nÜXVrMm dmna Ho$bm.

32 ì`ŠVtgmR>r òUmam EHy$U IM© ` 5600.

EH$m ì`ŠVrgmR>r òUmam IM© = ` 560032

= ` 175

3. Wm§~boë`m n{hë`m {R>H$mUmMo A§Va = 22 {H$._r.

A§VamMr eoH$S>odmar H$mT>Ê`mH$arVm;

A{e_mZo hr nÜXV dmnabr :

2255

2255

100100

40= × = %

{VZo JwUmoÎmambm 100100

1= Zo JwUyZ

Ë`mMo eoH$S>odmarV ê$nm§Va Ho$bo.

AWdm

Om°ZZo EH$_mZ nÜXV dmnabr.

55 {H$._r. n¡H$s, 22 {H$._r Mm àdmg Ho$bm Amho.

1 {H$._r. n¡H$s, 2255

100× km àdmg Ho$bm.

100 {H$._r. n¡H$s, 2255

100× {H$._r. àdmg Ho$bm.

åhUOoM EHy$U A§Vamn¡H$s 40%. A§VamMm àdmg nyU© Ho$bm.

XmoKm§Zr H$mT>bobo CÎma g_mZ hmoVo. emioVyZ {ZKmë`mZ§Va Ë`m§À`m àW_ Wm§~Ê`mÀ`m {R>H$mUmMo A§Va ho EHy$U àdmgmn¡H$s 40% A§Vamda Agob. åhUyZ àdmg H$amd`mÀ`m am{hboë`m A§VamMr eoH$S>odmar = 100% – 40% = 60%.

à`ËZ H$am.

EH$m àmW{_H$ emioV {eH$Umè`m {dÚmÏ`mªÀ`m nmbH$m§Zm {dMmaÊ`mV Ambo H$s Ë`m§Zr Amnë`m

_wbm§-_wbtÀ`m J¥hnmR>mgmR>r {H$Vr Vmg IM© H$ê$Z _XV H$aVmV. 90 nmbH$ 112 Vmg Vo 1

12 Vmg

_XV H$aVmV. ~mOyÀ`m AmH¥$Vr_Üò nmbH$ _XV H$arV Agboë`m doioMr {d^mJUr {Xbr Amho.

20% nmbH$ {Xdgmbm 112 Vmgnojm OmñV doi _XV H$aVmV. 30% nmbH$ 11

2 Vmg Vo 11

2 Vmg

n`ªV _XV H$aVmV, 50% nmbH$ A{O~mV _XV H$aV ZmhrV. `mdê$Z nwT>rb àým§Mr CÎmao Úm:

Vmg Vo Vmgnojm OmñV_XV H$aVmV.Vmg n`ªV _XV H$aVmV

A{O~mV _XVH$aV ZmhrV.

126 J{UV

(i) {H$Vr nmbH$m§Mr _m{hVr g§J«hrV Ho$br?

(ii) {H$Vr OUm§Zr AmnU _XV H$arV Zmhr Ago gm§{JVbo ?

(iii) {H$Vr OUm§Zr AmnU 112 Vmgmnojm OmñV doi _XV H$aVmo Ago gm§{JVbo ?

ñdmÜ`m` 8.1

1. ImbrbgmR>r JwUmoÎma H$mT>m.

(a) gm`H$bMm doJ 15 {H$._r. Xa Vmg Am{U ñHy$Q>aMm doJ 30 {H$._r. Xa Vmg.

(b) 5 _r. bm 10 {H$._r.

(c) 50 n¡go bm ` 5.

2. Imbrb JwUmoÎmam§Mo eoH$S>odmarV ê$nm§Va H$am. (a) 3 : 4 (b) 2 : 3

3. 25 {dÚmÏ`m©n¡H$s 72% {dÚmÏ`mªZm J{UV {df` AmdS>Vmo Va {H$Vr {dÚmÏ`mªZm J{UV AmdS>V Zmhr?

4. EH$m \w$Q>~m°b g§KmZo EHy$U Ioiboë`m gm_Ý`mn¡H$s 10 gm_Zo qOH$bo. Oa Ë`m§Zr qOH$boë`m gm_Ý`m§Mr eoH$S>odmar 40 Agob, Va Ë`m§Zr EHy$U {H$Vr gm_Zo Ioibo?

5. M_obrZo Amnë`mH$S>rb EHy$U a¸$_oÀ`m 75% aŠH$_ IM© Ho$ë`mZ§Va {VÀ`mnmer ` 600 {eëbH$ am{hbo. Va àma§^r {VÀ`mnmer Agbobr aŠH$_ {H$Vr?

6. EH$m ehamVrb 60% bmoH$m§Zm {H«$Ho$Q> AmdS>Vo, 30% bmoH$m§Zm \w$Q>~m°b Am{U Caboë`m bmoH$m§Zm BVa Ioi AmdS>VmV. Va BVa Ioi AmdS>Umè`m bmoH$m§Mo eoH$S>m à_mU {H$Vr? Oa EHy$U bmoH$m§Mr g§»`m 50 bmI Agob, Va àËòH$ Ioi AmdS>Umè`m§Mr AMyH$ g§»`m H$mT>m.

8.2. eoH$S>m dmT> AWdm eoH$S>m KQ> H$mT>Uo.

X¡Z§{XZ OrdZmV AmnUmbm dma§dma Aem àH$maMr _m{hVr AmT>iV AgVo.

(i) N>mnrb qH$_Vrda 25% gyQ> (ii) noQ´>mobÀ`m qH$_VrV 10% dmT>.

Aem àH$maMr AmUIr H$mhr CXmhaUo nmhÿ.

CXmhaU 2 : _mJrb dfu EH$m ñHy$Q>aMr qH$_V ` 34,000 hmoVr. `mdfu Ë`mV 20% dmT> Pmbr; Va ñHy$Q>aMr AmVmMr qH$_V {H$Vr?


CH$b:

A{_VmÀ`m åhUÊ`mZwgma Vr àW_ qH$_VrV Pmbobr dmT> H$mTy> BpÀN>Vo Or ` 34,000 Mr 20% Amho Am{U Ë`mZ§Va ZdrZ qH$_V.

` 34,000 Mo 20% = 20100

34000×

= ` 6,800

AmVmMr (ZdrZ) qH$_V = OwZr qH$_V + dmT>

= ` 34,000 + ` 6800

= ` 40,800

AWdm

gw{ZVm EH$_mZ nÜXVrMm dmna H$aVo.

20% dmT> `mMm AW© ` 100 Mo ` 120

Pmbo. åhUyZ ` 34,000 bm Pmbobr dmT> {H$Vr?

dmT>bobr qH$_V= ` 120100

34000×

= ` 40,800

`mMà_mUo qH$_Vr_Üò Pmbobr eoH$S>m KQ> hr AmnU EHy$U qH$_VrV {H$Vr KQ> Pmbr ho _yi qH$_VrVyZ dOm~mH$s H$éZ H$mTy> eH$Vmo.

g_Om, {dH«$s dmT>{dÊ`mgmR>r ñHy$Q>aÀ`m qH$_VrV 5% KQ> Ho$br Va ñHy$Q>aMr qH$_V H$mTy>.

ñHy$Q>aMr qH$_V= `34,000

KQ> = ` 34,000 Mo 5%

= 5100

34000× = `1700

ZdrZ qH$_V = OwZr qH$_V- KQ>

= ` 34000 – ` 1700 = ` 32300

`m àH$aUmÀ`m nwT>rb ^mJmV XoIrb `mMm dmna H$ê$.

8.3. gyQ> (Discount)

EImÚm dñVyÀ`m N>mnrb qH$_VrV Ho$bobr KQ>

(qH$_V H$_r) åhUOo gyQ> hmo`.

{JèhmB©H$m§Zm _mb IaoXr H$aÊ`mgmR>r AmH${f©V

H$aÊ`mH$[aVm AWdm _mbmMr {dH«$s dmT>{dÊ`mH$[aVm

gyQ> {Xbr OmVo.

N>mnrb qH$_Vr_YyZ {dH«$sMr qH$_V dOm

H$ê$Z AmnUmbm gyQ> {H$Vr {Xbr ho H$mT>Vm òVo.

åhUyZ gyQ> = N>mnrb qH$_V - {dH«$sMr qH$_V.

25% gyQ> gob

{dH«$s{dH«$s

50% gyQ>

128 J{UV

CXmhaU 3 : EH$m dñVyMr N>mnrb qH$_V ` 840 Amho Am{U Vr ` 714 bm {dH$br. Va gyQ> {H$Vr {Xbr d % gyQ> {H$Vr?

CH$b : gyQ> = N>mnrb qH$_V - {dH«$sMr qH$_V.

= ` 840 – ` 714 = ` 126

gyQ> Zoh_r N>mnrb qH$_Vrda {Xbr OmVo. åhUyZ N>mnrb qH$_V hr _yi qH$_V åhUyZ AmnU dmnam`bm hdm.

N>mnrb qH$_V ` 840 AgVmZm EHy$U gyQ> ` 126 Amho,

Va N>mnrb qH$_V ` 100 AgVmZm gyQ> {H$Vr Agob?

gyQ> = 126840

100 15× =% %

eoH$S> gyQ> Xa {Xbm Agob Va Vwåhr EHy$U gyQ> H$mTy> eH$Vm.

CXmhaU 4 : EH$m \«$m°H$Mr `mXrVrb qH$_V ` 220 Amho, Ë`mda 20% gyQ> Omhra Ho$br Amho. Va gwQ>rMr EHy$U aŠH$_ Am{U {dH«$sMr qH$_V {H$Vr?

CH$b : `mXrVrb qH$_V `mMm AW© N>mnrb qH$_V. Ë`mda 20% gyQ> `mMm AW©, ` 100 (N>m. qH$.) da {_iUmar gyQ> ` 20.

EH$_mZ nÜXVrZwgma ` 1 da {_iUmar gyQ> ` 20

100.

` 220 da, gyQ> = 20100

× ` 220 = ` 44

{dH«$sMr qH$_V = (` 220 – ` 44) = ` 176

aohmZmZo {dH«$sMr qH$_V `m à_mUo H$mT>br -

` 220

20% gwQ>

20% gyQ> `mMm AW© N>mnrb qH$_V ` 100 AgVm§Zm ` 20 gyQ.>

Voìhm {dH«$sMr qH$_V ` 80. EH$_mZ nÜXVrMm dmna H$ê$Z,

Ooìhm N>mnrb qH$_V ` 100, Va {dH«$sMr qH$_V ` 80;

Oa N>mnrb qH$_V ` 1 Va {dH«$sMr qH$_V ` 80

100Ooìhm N>mnrb qH$_V ` 220 Amho Voìhm,

{dH«$sMr qH$_V = 80

100 × ` 220 = ` 176

Oar _r gyQ>> {H$Vr Vo H$mT>bo Zmhr, Var _r WoQ> {dH«$sMr qH$_V H$mTy> eH$Vmo.


à`ËZ H$am

1. EH$m XþH$mZmV 20% gyQ> {Xbr OmVo. Imbrb dñVw§Mr {dH«$sMr qH$_V {H$Vr Agob? (a) S´>og, N>mnrb qH$_V ` 120

(b) ~yQ> OmoS>, N>mnrb qH$_V ` 750

(c) ~°J, N>mnrb qH$_V ` 250

2. EH$m Q>o~bMr N>mnrb qH$_V ` 15,000 AgyZ Vo ` 14,400 bm CnbãY Amho. Va EHy$U {Xbobr gyQ> Am{U gyQ> Xa {H$Vr?

3. EH$ H$nmQ> 5% gyQ> XoD$Z ` 5225 bm {dH$bo Va Ë`mMr N>mnrb qH$_V H$mT>m.

8.3.1 eoH$S>odmarVrb A§XmO:

Vw_Mo XþH$mZmVrb ~rb ` 577.80 Pmbo Am{U XþH$mZXma Vwåhmbm 15% gyQ> XoVmo. Va XþH$mZXmambm Úmd`mÀ`m aŠH$_oMm A§XmO H$gm H$amb?

(i) ` 577.80 `m aŠH$_oMo ~rb. OdiÀ`m XeH$mV g_m`moOrV H$ê$, Vr åhUOo, ` 580.

(ii) Ë`mMo 10% åhUOoM ` 10100

580× = ` 58

(iii) darb a¸$_oMo AY} ` 12

58× = ` 29

(iv) (ii) d (iii) _Yrb a¸$_oMr ~oarO Ho$ë`mg ` 87 hmoVo.

åhUyZ ~rbMr a¸$_ ` 87 AWdm ` 85 Zo H$_r Ho$ë`mg Ë`mMr aŠH$_ A§XmOo ` 495 hmoB©b.

1. `mM ~rbMr a¸$_ 20% à_mUmV A§XmO H$am.

2. ` 375 Mo 15% H$mT>m`Mm à`ËZ H$am.

8.4. IaoXr {dH«$sbm g§~§YrV qH$_Vr (Z\$m Am{U VmoQ>m)

emioÀ`m _oimì`m_Üò _r "Z{e~mÀ`m {MÇ>tMm' ñQ>m°b Mmby R>odVo. àËòH$ {MÇ>rgmR>r _r ` 10 Mr AmH$maUr H$aVo na§Vy Ë`mda Úmd`mMr àËòH$ dñVy ` 5 À`m qH$_VrMr R>odVo.

åhUOo Vy 100% Z\$m {_i{dUma Amhog.

Zmhr, ~{jgmÀ`m dñVwbm _r AmdaU KmbyZ {MH$Q>nÅ>r bmdUma Ë`m_wio àËòH$s ` 3 IM© hmoUma. åhUyZ _r IM© Ho$bobr aŠH$_ ` 8 Agob. Ë`m_wio _bm {_iUmam Z\$m ` 2 \$ŠV åhUOoM, 2

8100× = 25% \$ŠV.

130 J{UV

H$mhrdoim dñVy IaoXr H$aVm§Zm AWdm {dH$Ê`mnydu Ë`m dñVyda H$mhr a¸$_ IM© H$amdr bmJVo. Vr a¸$_ Ë`m dñVyÀ`m _yi qH$_Vr_Üò {_idmdr bmJVo.Aem IMm©bm BVa IM© daIM© AWdm daH$S> ewëH$ åhQ>bo OmVo. `mV XþéñVrMm, _OwarMm, dmhVwH$sMm IM© `m§Mm g_mdoe hmoVmo.

8.4.1 IaoXrMr qH$_V/{dH«$sMr qH$_V, % Z\$m/% VmoQ>m H$mT>Uo.

CXmhaU 5 : gmohZZo EH$ OwZm {\«$O ` 2500 bm IaoXr Ho$bm. Ë`mÀ`m XþéñVrgmR>r Ë`mZo ` 500 IM© H$ê$Z Vmo ` 3300 bm {dH$bm. Va Ë`mbm Pmbobm eoH$S>m Z\$m AWdm VmoQ>m {H$Vr Vo H$mT>m.

CH$b : IaoXrMr qH$_V (I. qH$.) = 2500 + 500 (daH$S> IM© Omo I. {H$§. _Üò {_i{dbm)

= ` 3000

{dH«$sMr qH$_V ({d.{H$§) = Voìhm ` 3300

Á`m AWu {d.{H$ > I.{H$§., Voìhm Ë`mbm Z\$m Pmbm = ` 3300 – ` 3000 = ` 300

` 3000 da Pmbobm Z\$m ` 300, Va ` 100 bm {H$Vr Z\$m Pmbm.

Z\$m = 3003000

100 303

10× = =% % % P PCP

% = ×100

Q>rn : IaoXrMr qH$_V = Cost Price = CP

{dH«$sMr qH$_V =Selling price = SP

Z\$m/bm^ /\$m`Xm = Profit = P

VmoQ>m/ hmZr/ZwH$gmZ = Loss = L

vvà`ËZ H$am

1. {dH«$sMr qH$_V, (SP) 5% Z\$m KoD$Z {Z[ûMV Ho$br Amho Va {dH«$sMr qH$_V H$mT>m.

(a) EH$m gm`H$bMr qH$_V ` 700 d daH$S> IM© ` 50 Amho. (b) ~mJoVrb JdV N>mQ>m`Mo `§Ì ` 1150 bm IaoXr Ho$bo d dmhVyH$

IM© ` 50 Ambm. (c) ` 560 bm EH$ n§Im {dH$V KoD$Z Ë`mÀ`m XþéñVrgmR>r ` 40 IM©

Ho$bo.

CXmhaU 6 : EH$m XþH$mZXmamZo ` 10 bm EH$ à_mUo 200 ~ë~ IaoXr Ho$bo. Ë`mVrb 5 ~ë~ ~mX Pmbo Ë`m_wio Vo ~mOwbm H$mT>bo. {eëbH$ ~ë~ àËòH$s ` 12 bm EH$ `mà_mUo {dH$bo. Va Pmbobm % Z\$m AWdm % VmoQ>m H$mT>m.


CH$b : 200 ~ë~Mr IaoXrMr qH$_V

= 200 × ` 10 = ` 2000

5 ~ë~ ~mX Pmbo.

{eëbH$ ~ë~Mr g§»`m = 200 – 5

= 195

ho àËòH$s ` 12 bm EH$ à_mUo {dH$bo.

{d{H«$ qH$_V ` 12

IaoXr qH$_V ` 10

195 ~ë~Mr {d.qH$. = ` 195 × 12 = ` 2340

`mV Ë`mbm ZŠH$sM Z\$m Pmbm ({d.{H$§ > I.{H$§)

Z\$m = ` 2340 – ` 2000 = ` 340

` 2000 da ` 340 Z\$m, Va ` 100 bm {H$Vr?

% Z\$m = 3402000

100 17× = % .

CXmhaU 7 : àËòH$s ` 1200 à_mUo {_ZyZo XmoZ n§Io {dH$V KoVbo. Ë`mVrb EH$ n§Im {VZo 5% VmoQ>çmZo d Xþgam n§Im 10% Zâ`mZo {dH$bm. Va àËòH$ n§»`mMr {dH«$sMr qH$_V H$mT>m. Ë`mM

à_mUo {Vbm `m ì`dhmamV Pmbobm EHy$U Z\$m qH$dm VmoQ>m H$mT>m.

CH$b : n§»`mMr IaoXrMr qH$_V = ` 1200 àËòH$s. EH$

n§Im 5% VmoQ>çmZo {dH$bm.

åhUyZ Oa CP ` 1200 Va SP = `95

1001200× = ` 1140

Xþgam n§Im 10% Zâ`mZo {dH$bm. åhUOo CP ` 100 Va SP = ` 110.

åhUOo Oa CP ` 1200, Va SP = ` 110100

1200× = ` 1320.

åhUOo `mV EHy$U VmoQ>m Pmbm H$s Z\$m?

`mgmR>r Amnë`mbm EHy$U C.P d EHy$U S.P H$mT>m`bm hdm, Á`m_wio EHy$U VmoQ>m Pmbm H$s Z\$m ho gm§JVm òB©b.

EHy$U C.P= ` 1200 + ` 1200 = ` 2400

EHy$U S.P = ` 1140 + ` 1320 = ` 2460

EHy$U S.P > EHy$U C.P Z\$m = (` 2460 - ` 2400) = ` 60

132 J{UV

à`ËZ H$am

1. EH$m XþH$mZXmamZo XmoZ XyaXe©Z g§M (TV) àËòH$s `10000 à_mUo IaoXr Ho$bo. Ë`mVrb EH$ g§M 10% Zâ`mZo {dH$bm d Xþgam g§M 10% VmoQ>çmZo {dH$bm. Va `m ì`dhmamV Ë`mbm Pmbobm EHy$U Z\$m qH$dm VmoQ>m H$mT>m.

8.5 {dH«$sH$a/ _wë`md{Y©V H$a/ dñVy godm H$a(Sales Tax/Value Added Tax/Goods and Service Tax)

{ejH$m§Zr dJm©_Üò EH$ ~rb XmI{dbo Á`mV Imbrb {ef©H$ Xe©{dbr hmoVm.

~rb H«$_m§H$ VmarInXmWmªMr `mXr

A.H«$ Vnerb à_mU Xa aŠH$_{~bMr a¸$_ + ST (5%)

EHy$U

ST åhUOo {dH«$sH$a (Sales Tax) dñVy IaoXr H$aVmZm AmnU XoVmo.

{dH«$sH$a hm {dH«$s Ho$boë`m nXmWm©da gaH$maZo AmH$mabobm AgVmo.

XþH$mZXma Vmo J«mhH$mH$Sy>Z dgyb H$ê$Z gaH$mabm O_m H$aVmo. `mMr AmH$maUr dñVy§À`m {dH«$s Ho$boë`m [H§$_Vrda Ho$br OmVo Am{U {~bÀ`m aŠH$_oV Vr {_i{dbr OmVo. hëbr qH$_Vr_Üò {_i{dboë`m H$ambm _wë`md{Y©V H$a (Value Added Tax) åhUyZ AmoiIbo OmVo.

CXmhaU 8 : ({dH«$sH$a H$mT>Uo) : EH$m XþH$mZmV amoba ñHo$Q>g²À`m OmoS>rMr qH$_V ` 450 Amho. Ë`mda 5% {dH«$sH$a AmH$mabm OmVmo. Va ~rbMr a¸$_ H$mT>m.

CH$b : Oa ` 100 da ` 5 H$a AmH$mabm,

Va ` 450 da AmH$mabm OmUmam H$a = ` 5

100450× = ` 22.50

~rbMr a¸$_ = dñVyMr qH$_V + {dH«$s H$a

= ` 450 + ` 22.50 = ` 472.50

CXmhaU 9 : (_wë`md{Y©V H$a (VAT)) : d{hXmZo ` 3300 XoD$Z EH$ EAa Hw$ba IaoXr Ho$bm. `m qH$_VrV 10% H$a g_m{dîQ> Amho. Va _wë`md{Y©V H$a {_idm`À`m AmYrMr åhUOoM EAaHw$baMr _yi qH$_V H$mT>m.


CH$b : dñVyÀ`m qH$_VrV VAT {_i{dbobm Amho. åhUOoM 10%

VAT Mm AW© VAT Z {_i{dVm dñVyMr qH$_V ` 100 Am{U VAT

{_idyZ hmoUmar qH$_V ` 110. VAT {_idyZ dñVyMr qH$_V ` 110 AgVmZm

_yi qH$_V ` 100. Oa H$a {_idyZ Pmbobr qH$_V ` 3300 Agob Va _yi

qH$_V = ` 100110

3300× = ` 3000

CXmhaU 10 : gbr_Zo EH$ dñVy ` 784 bm {dH$V KoVbr Á`mV 12% dñVy godm H$a (GST) {_i{dbobm Amho. Va dñVy godm H$a (GST) {_i{dÊ`mnydu Ë`m dñVyMr qH$_V {H$Vr?

CH$b : dñVyMr _yi qH$_V ` 100 _mZy. GST = 12%

GST {_idyZ dñVyMr qH$_V ` (100 + 12) = `112

Oa {d.qH$ ` 112 Va _yi qH$_V = ` 100

Va {d.qH$ ` 784 Va _yi qH$_V = `100112

784× = `700.


1. EH$ g§»`m XmoZ nQ>rZo dmT>{dbr AgVm Ë`m g§»òMr 100% dmT> hmoVo. Oa Ë`mM g§»òMo AY} Ho$bo AgVm eoH$S>m {H$Vr H$_r hmoB©b?

2. ` 2400 nojm ` 2000 ho eoH$S>m {H$VrZo H$_r? ` 2400 ho ` 2000 nojm OmñV Agbobo eoH$S>m à_mU ho darb eoH$S>m à_mUmer g_mZ Amho H$m?

ñdmÜ`m` 8.2

1. EH$m _mUgmbm Ë`m§À`m nJmamV 10% dmT> {_imbr. Ë`mMm ZdrZ nJma ` 1,54,000 Amho. Va Ë`mMm AJmoXaMm nJma {H$Vr hmoVm?

2. a{ddmaÀ`m {Xder àmUrg§J«hmb`mbm 845 bmoH$m§Zr ôQ> {Xbr. gmo_dmar 169 bmoH$m§Zr ôQ> {Xbr. Va gmo_dmar ôQ> XoUmè`m bmoH$m§À`m g§»òV eoH$S>m {H$Vr KQ> Pmbr?

3. EH$m XþH$mZXmamZo 80 dñVy ` 2400 bm IaoXr Ho$ë`m Am{U Ë`m 16% Zâ`mZo {dH$ë`m. Va EH$m dñVyMr {dH«$sMr qH$_V H$mT>m.

134 J{UV

4. EH$m dñVyMr IaoXrMr qH$_V ` 15,500. Ë`mÀ`m XþéñVrgmR>r ` 450 IM© Ho$bo. Vr dñVy 15% Z\$m KoD$Z {dH$br. Va {dH«$sMr qH$_V H$mT>m.

5. EH$ pìhgrAma (VCR) Am{U EH$ Q>r.ìhr. àËòH$s ` 8000 à_mUo IaoXr Ho$bo. XþH$mZXmamZo pìhgrAma 4% VmoQ>çmZo {dH$bm Am{U Q>r.ìhr 8% Zâ`mZo {dH$bm. Va `m g§nyU© ì`dhmamV Pmbobm eoH$S>m Z\$m AWdm VmoQ>m H$mT>m.

6. EH$m XþH$mZmV gob Mmby AgyZ {VWo àËòH$ dñVyÀ`m N>mnrb qH$_Vrda 10% gyQ> Omhra Ho$br Amho. EH$m {JèhmB©H$mZo ` 1450 bm EH$ à_mUo EH$ OrZ n±Q> d ` 850 bm EH$ à_mUo XmoZ eQ>©g² {dH$V KoVbo. Va Ë`mZo XþH$mZXmambm gyQ> dOm H$ê$Z {H$Vr a¸$_ Úmdr?

7. EH$m Jdù`mZo àËòH$s ` 20000 à_mUo XmoZ åher {dH$ë`m EH$m ì`dhmamV Ë`mbm 5% Z\$m Pmbm Va Xþgè`mV 10% VmoQ>m Pmbm. Va g§nyU© ì`dhmamV Pmbobm Z\$m AWdm VmoQ>m {H$Vr Vo H$mT>m. (Q>rn : AmYr I. qH§$. H$mT>m).

8. EH$m Q>r. ìhr. Mr qH$_V ` 13000 Amho. Ë`mda 12% XamZo {dH«$s H$a AmH$mabm OmVmo. {dZmoXZo Vmo IaoXr Ho$ë`mg Ë`mbm Úmdr bmJUmar a¸$_ H$mT>m.

9. EH$m gob_Üò 20% gyQ> {Xbr OmV hmoVr. VoWyZ AéUZo EH$ ñHo$Q>g²Mr OmoS>r {dH$V KoVbr. Ë`mZo {VWo ` 1600

{Xbo, Va ñHo$Q>gÀ`m OmoS>rMr N>mnrb qH$_V H$mT>m.

10. _r EH$ hoAa-S´>mà {dH$V KoVbm. Á`mMr qH$_V 8% ìh°Q> {_idyZ ` 5400 Amho. Va ìh°Q> Z {_i{dVm Ë`m dñVyMr qH$_V {H$Vr Vo H$mT>m.

11. EH$ dñVy 18% GST g{hV ` 1239 bm IaoXr Ho$br. Va GST {_i{dÊ`mnydu Ë`m dñVyMr qH$_V {H$Vr hmoVr?

8.6 MH«$dmT>ì`mO :

""~±Ho$_Üò _wXV R>odrdarb, (Fixed Deposit) ì`mO Xa dfm©bm 9% AWdm ~MV ImË`mda {_iUmao ì`mO 5% Xa dfm©bm'' Aer {dYmZo Vwåhr EoH$br AgVrb AWdm dmMbr AgVrb.

~±H$ AWdm nmoîQ> Am°\$sg `m gma»`m g§ñWm_Üò R>odboë`m R>odrda Ë`m§Zr {Xbobr OmXmMr a¸$_ åhUOo ì`mO. bmoH$m§Zr H$O© KoVbobr a¸$_ Agob Va Ë`mda Vo XoIrb ì`mO âVmV. `m AmYr AmnU gai ì`mO H$go H$mT>mdo ho {eH$bmo AmhmoV.

CXmhaU 10 : ` 10,000 ho Xa gmb 15% ì`mO XamZo 2 dfm©gmR>r H$Om©D$ KoVbo. `m a¸$_oda 2 dfm©À`m eodQ>r Úmdo bmJUmao gai ì`mO Am{U naV H$amd`mMr EHy$U a¸$_ {H$Vr Vo H$mT>m.


CH$b : ` 100 bm 1 dfm©gmR>r Úmdo bmJUmao ì`mO ` 15.

åhUyZ ` 10,000 darb 1 dfm©Mo ì`mO = ` 15

10010000 1500× = = ` 1500

2 dfm©Mo gai ì`mO = 1500 × ` 2 = ` 3000

XmoZ dfmªÀ`m eodQ>r Úmdr bmJUmar amg = _wÔb + ì`mO = ` 10,000 + ` 3000 = ` 13,000

à`ËZ H$am` 15,000 Mo Xa gmb 5% XamZo 2 dfm©Mo hmoUmao gai ì`mO Am{U amg H$mT>m.

_mÂ`m d{S>bm§Zr 3 dfm©À`m _wXVrZo nmoîQ> Am°\$sg_Üò H$mhr a¸$_ R>odbr Amho. Xa dfu _mJrb dfm©nojm `mdfm©À`m a¸$_oV dmT> hmoVo.

~±H$_Üò Am_Mr Wmo‹S>r a¸$_ Amho. Xa dfu Ë`m_Üò ì`mO {_i{dbo OmVo `mMr Zm|X nmg~wH$_Üò XmI{dbr OmVo. ì`mOmMr a¸$_ Xadfu gmaIr AgV Zmhr, Vr àËòH$ dfu dmT>Vo.

gm_mÝ`nUo {Xbo OmUmao ì`mO AWdm AmH$maUr Ho$bobo ì`mO ho gai ì`mO AgV Zmhr.ì`mOmMr AmH$maUr hr _mJrb dfm©À`m {e„H$ a¸$_oda Ho$br OmVo. `mbmM ì`mOmghrV a¸$_oda ì`mO AmH$maUr AWdm MH«$dmT> ì`mO (compound interest) (CI) åhUVmV.

EH$ CXmhaU ¿`m d dfm© dfm©Mo hmoUmao ì`mO H$mT>m. Xa dfu a¸$_ AWdm _wÔb ~XbV OmVo.

MH«$dmT>ì`mO H$mT>Uo

{hZmZo Xa gmb 8% MH«$dmT>ì`mOXamZo 2 dfmªÀ`m _wXVrgmR>r ` 20,000 H$O© KoVbo. Va XmoZ dfm©À`m eodQ>r hmoUmao MH«$dm‹T> ì`mO (CI) d Úmdr bmJUmar aŠH$_ {H$Vr H$mT>m.

Añb_Zo {ejH$m§Zm {dMmabo H$s Ë`m§Zr ì`mOmMr AmH$maUr Xa dfu doJir H$amd`mMr H$m. {ejH$m§Zr Ë`m§Zm "hmo`' åhUyZ gm§{JVbo Am{U nwT>rb nm`è`m§Mm Cn`moJ H$aÊ`mg gm§{JVbo.

1. àW_ EH$ dfm©Mo gai ì`mO (SI) H$mT>m. n{hë`m dfm©Mo _wÔb P1 _mZy òWo P1 = ` 20,000.

SI1 = Xa gmb 8% XamZo n{hë`m dfm©Mo SI1 = ` 20000 8

100×

= ` 1600

2. AmVm Úmd`mMr AWdm {_idm`Mr amg H$mT>m. Vr Xþgè`m dfm©Mr _wÔb Agob.

n{hë`m dfmªÀ`m eodQ>r hmoUmar amg = P1 + SI1

= ` 20,000 + ` 1600

= ` 21,600 = P2 (Xþgè`m dfm©Mr _wÔb)

136 J{UV

3. AmVm nwÝhm `m aŠH$_oda hmoUmao gai ì`mO H$mT>m.

SI2 = Xa gmb 8% XamZo Xþgè`m dfm©Mo SI = ` 21 600 8100

, × = ` 1728

4. Xþgè`m dfmªÀ`m eodQ>r Úmd`mMr AWdm {_imd`mMr a¸$_ H$mT>m.

Xþgè`m dfm©À`m eodQ>r hmoUmar amg = P2 + SI2

= ` 21,600 + ` 1728

= ` 23,328

XmoZ dfm©Mo EHy$U ì`mO = ` 1600 + ` 1728 = ` 3328

[aVmZo {dMmabo H$s hr a¸$_ gaiì`mOmnojm doJir Amho H$m?

{ejH$m§Zr {VbmM 2 dfmªMo gaiì`mO H$mTy>Z nhmÊ`mg gm§{JVbo.

2 dfm©Mo SI = ` 20000 8 2

100× ×

= ` 3200.

[aVmZo gm§{JVbo Ooìhm MH«$dmT>ì`mO AmH$mabo Voìhm {hZmbm `128 ì`mO OmñV Úmdo bmJbo.

AmVm AmnU gaiì`mO Am{U MH«$dmT>ì`mO `mVrb \$aH$ nmhÿ. _wÔb `100 KoD$Z àma§^ H$ê$. VŠVm nyU© H$aÊ`mMm à`ËZ H$ê$.

gai ì`mOmZo MH«$dmT>ì`mOmZo

`mMm AW©

O_m Pmboë`m

ì`mOmda Vwåhr

ì`mO XoVm.

n{hbo df©_wÔb

10% Zo ì`mO

` 100.00

` 10.00

` 100.00

` 10.00

dfm©eodQ>r amg ` 110.00 ` 110.00

Xþgao df©_wÔb

10% Zo ì`mO

` 100.00

` 10.00

` 110.00

` 11.00

dfm©eodQ>r amg ` (110+10) = ` 120.00 ` 121.00

{Vgao df©_wÔb

10% Zo ì`mO

` 100.00

` 10.00

` 121.00

` 12.10

dfm©eodQ>r amg ` (120+10) = ` 130.00 ` 133.10

bjmV ¿`m 3 dfm©_Üò,

gaiì`mOmZo {_i{dbobo ì`mO = ` (130 - 100) = ` 30

MH«$dmT> ì`mOmZo {_i{dbobo ì`mO = ` (133.10 - 100) = ` 33.10

ho gwÜXm Ü`mZmV R>odm H$s gaiì`mOm_Üò _wÔb aŠH$_ ~XbV Zmhr, na§Vy MH«$dmT>ì`mOm_Üò Vr dfm©-dfm©bm ~XbV AgVo.


8.7 MH«$dmT>ì`mO H$mT>Ê`mMo gwÌ

Pw~oXmZo {ejH$m§Zm {dMmabo H$s ""MH«$dmT>ì`mO H$mT>Ê`mgmR>r EImXr gmonr nÜXV Amho H$m?'' {ejH$ åhUmbo ""MH«$dmT>ì`mO H$mT>Ê`mgmR>r bhmZ nÜXV Amho. Vr H$mT>Ê`mMm à`ËZ H$ê$.''

g_Om P1 `m a¸$_o da Xa gmb R% ì`mOXamZo MH«$dmT>ì`mOmZo ì`mOmMr AmH$maUr Ho$br OmVo.

g_Om P1 = ` 5000 Am{U R = 5% Xagmb. `m nydu {Xboë`m nm`è`m§Mm dmna H$ê$Z,

1. SI1 = ` 5000 5 1100

× ×AWdm SI1 =

P R1 1100× ×

åhUyZ, A1 = ` 5000 + 5000 5 1

100× × AWdm A P SI P PR

1 1 1 11

100= + = +

` 5000 1 5100 2+

= P P R P1 21100

+

=

2. SI2 = ` 5000 1 5100

5 1100

+

× ×AWdm SI P R

22 1100

=× ×

= 5000 5

1001 5

100× +

= P R R1 1

100 100+

×

= P R R1

1001

100× +

A2 = ` 5000 1 5100

5000 5100

1 5100

+

+ × +

` A2 = `P2 + SI2

= ` 5000 1 5100

1 5100

+

+

P R P R R1

11100 100

1100

+

+×

× +

= ` 5000 1 5100

2

3+

= P = P R R1 1

1001

100+

+

= P R P1

2

31100

+

=

138 J{UV

`mà_mUo 'n' dfm©À`m eodQ>r hmoUmar amg

A P R

n

n

= +

1 1

100

qH$dm AmnU Ago hr åhUy eH$Vmo A = P A P Rn

n

= +

1 1

100

åhUyZ Pw~oXm åhUbr, `mMm dmna H$ê$Z Amåhmbm n dfm©À`m eodQ>r Úmdr bmJUmar amg H$mT>Ê`mMo gyÌ {_imbo. nU ho MH«$dmT>ì`mO H$mT>Ê`mMo gyÌ Zmhr.

AéUm bJoMM åhUmbr CI = A – P ho Amåhmbm _mhrV Amho, Ë`mdê$Z Amåhmbm MH«$dm‹T>ì`mO H$mT>Uo EH$X_ gmono hmoB©b.

CXmhaU 11 : ` 12,600 Mo 2 dfm©Mo Xa gmb 10% XamZo dm{f©H$ AmH$maUrZo hmoUmao MH«$dmT> ì`mO H$mT>m.

CH$b : Amnë`mbm _mhrV Amho A P R n

= +

1100

`mV _wÔb (P) = ` 12,600, ì`mO Xa (R) = 10,

dfmªMr g§»`m (n) = 2

A = ` 12600 1 10100

2

+

= ` 12600 1110

2

= ` 12600 × 1110

1110

× = ` 15246

CI = A – P = ` 15246 – ` 12600 = `2646

8.8 dm{f©H$ AmH$maUr AWdm AY© dm{f©H$ (ghm_mhr) AmH$maUr ì`mO Xa.

ì`mOmÀ`m XamZ§Va "dm{f©H$ AmH$maUr' hm CëboI Ho$bm OmVmo. ho OmUyZ ¿`m`Mo Amho H$m`? Ë`mbm AW© Amho H$maU ì`mOmMr AmH$maUr AY©dm{f©H$ AWdm {V_mhr nU Ho$br OmVo. ` 100 da dm{f©H$ AmH$maUr Am{U AY©dm{f©H$ ì`mO AmH$maUr H$ê$Z ì`mO H$mT>ë`mg H$m` hmoVo Vo nmhÿ.

ì`mOmMr dm{f©H$ AmH$maUr Ho$br OmV Zmhr Voìhm _wXV Am{U ì`mO Xa.

Á`m _wXVrÀ`m nwV©VoZ§Va àËòH$ doir _wÔbmMo hmoUmao ì`mO {_idyZ ZdrZ amg ~Z{dbr OmVo Ë`m _wXVrbm ê$nm§VaU _wXV åhUVmV. Oa ì`mOmMr AmH$maUr AY©dm{f©H$ Ho$br Va EH$m dfm©V XmoZ ê$nm§VaU _wXVr AgVmV. Oa ì`mO AmH$maUr {V_mhr Ho$br Va EH$m dfm©V ê$nm§VaU _wXVr 4 AgVrb.

à`ËZ H$am `. 8000 Mo 2 dfm©Mo Xa gmb

5% XamZo dm{f©H$ AmH$maUrZo hmoUmao MH«$dmT>ì`mO gwÌmZo H$mT>m.


P = ` 100 Mr 10% XamZo dm{f©H$ AmH$maUr

P = ` 100 Mr 10% XamZo ghm_mhr AmH$maUr

_wXV 1 df© KoD$Z_wXV 6 _{hZo AWdm

12 df©

1 = ` 100 10 1100

× × = ` 101 = `

100 10 12

100

× × = ` 5. ì`mO Xa AYm© hmoVmo.

A = `100 + ` 10 = ` 110 A = 100 + 5 = 105 nwT>rb ghm _{hÝ`mgmR>r P = ` 105.

åhUyZ, I = ` 105 10 1

2100

× × = ` 5.25

Am{U A = ` 105 + ` 5.25 = ` 110.25

Vwåhr nm{hbo H$m, Oa ì`mOmMr AmH$maUr AY© dm{f©H$ AWdm ghm_mhr hmoV Agob Va Amåhr ì`mO XmoZ doim H$mT>Vmo åhUyZ _wXV XþßnQ> hmoVo Am{U ì`mO Xa AYm© hmoVmo.

à`ËZ H$amàËòH$mMm H$mbmdYr d ì`mO Xa H$mT>m.

1. _wXV 112

df© d Xa gmb 8% XamZo a¸$_ H$Om©D$ KoVbr Á`mMr ì`mO AmH$maUr AY©

dm{f©H$ hmoVo.

2. _wXV 2 df© d Xa gmb 8% XamZo a¸$_ H$Om©D$ KoVbr Á`mMr ì`mO AmH$maUr AY© dm{f©H$ hmoVo.


_wXV EH$ df©, ì`mO Xa 16% Xa gmb `m à_mUo a¸$_ H$Om©D$ KoVbr. ì`mOmMr AmH$maUr {V_mhr Ho$br OmVo Va EH$m dfm©V {H$Vr doim ì`mO AmH$maUr Ho$br OmB©b?

CXmhaU 12 : ` 12,000 Mo H$O© Xa gmb 10% XamZo 112 dfm©Z§Va naV H$amd`mMo Pmë`mg

{H$Vr a¸$_ âmdr bmJob? ì`mOmMr AmH$aUr Xa ghm_mhr Ho$br OmVo.

140 J{UV

CH$b :

n{hë`m ghm_mhrgmR>r _wÔb = ` 12,000 n{hë`m ghm_mhrgmR>r _wÔb = ` 12,000

112 dfm©_Üò EHy$U 3 ghm_mhr òVmV.

åhUyZ ì`mOmMr AmH$maUr 3 doim hmoVo.

ì`mO Xa = 10% Mo AY}

= 5% ghm_mhrbm

A P R n

= +

1100

= ` 12000 1 5100

3

+

= ` 12000 × 2120

2120

2120

× ×

= ` 13891.50

_wXV = 6 _{hZo = 6

12 df© = 11

2 df© Xa = 10%

I = ` 12000 10 12

100

× × = ` 600

A = P + I = ` 12,000 + ` 600

= ` 12600. ho nwT>rb ghm_mhrMo _wÔb.

I = ` 12600 10 1

2100

× × = ` 630

{Vgè`m ghm_mhrMr _wÔb = ` 12600 + ` 630

= ` 13,230

I = ` 13230 10 1

2100

× × = ` 661.50

A = P + I = 13230 + 661.50 = 13,891.50

à`ËZ H$am ImbrbgmR>r amg H$mT>m

1. ` 2,400 Mr 2 dfm©À`m eodQ>r Xa gmb 5% XamZo dm{f©H$ MH«$dmT>ì`mO AmH$maUr H$ê$Z.

2. ` 1,800 Mr EH$ dfmªÀ`m eodQ>rMr Xa gmb 8% XamZo {V_mhr MH«$dmT>ì`mO AmH$maUr H$ê$Z.

CXmhaU 13 : ` 10,000 Mr a¸$_ 1 df© 3 _{hÝ`mgmR>r Xa gmb 8112% XamZo dm{f©H$ ì`mO

AmH$maUrZwgma Jw§V{dbr Va {_iUmao MH«$dmT>ì`mO H$mT>m.

CH$b : _`warZo àW_ _wXV dfm©_Üò ~XbyZ KoVbr.

1 df© 3 _{hZo = 1 312

114

= df© = 1 312

114

= df©.

_`warZo {Xboë`m qH$_Vr gwÌmV _m§S>ë`m. Ë`m `m à_mUo Amë`m

A = ` 10000 1 17200

114

+


AmVm {Vbm g_OoZm. {VZo {ejH$m§Zm {dMmabo H$s KmVm§H AnyUmªH$mV Agob Va J{UV H$go gmoS>dmdo?

{ejH$m§Zr {Vbm gyMZm {Xë`m.

àW_ nyU© H$mbmdYrMr amg H$mT>, `m J{UVmV 1 dfm©gmR>r. Vr amg nwT>rb H$mbmdYrMo _wÔb _mZyZ Ë`mda 1 312

114

= dfm©Mo gaiì`mO H$mT>m>.

A = ` 10000 1 17200

+

= ` 10000 × 217200 = ` 10,850

AmVm hr amg nwT>rb 1 312

114

= dfm©gmR>r _wÔb Agob. åhUyZ ` 10,850 Mo 1 312

114

= dfm©Mo hmoUmao gaiì`mO H$mTy>.

SI = ` 10850 14

17

100 2

× ×

×

= ` 10850 1 17800

× × = ` 230.56

n{hë`m dfm©Mo ì`mO = ` 10,850 - ` 10,000 = ` 850.

nwT>rb 1 312

114

= dfm©Mo ì`mO = ` 230.56.

åhUyZ EHy$U MH«$dmT>ì`mO = ` 850 + ` 230.56 = ` 1080.56.

8.9 MH«$dmT>ì`mOmÀ`m gwÌmMo Cn`moJ

H$mhr àg§Jr AmnU MH«$dmT>ì`mO H$mT>Ê`mÀ`m `m gwÌmMm dmna BVa H$mhr {R>H$mUr H$aVmo Vo åhUOo,

(i) bmoH$g§»òV hmoUmar dmT> (AWdm KQ>) H$mT>Ê`mgmR>r.

(ii) gwú_O§Vy_Üò dmT> hmoÊ`mMm Xa _mhrV Agë`mg hmoUmar dmT> H$mT>Ê`mgmR>r.

(iii) dfm©À`m _Yrb H$mbmV dñVyMr qH$_V dmT>ë`mg qH$dm H$_r Pmë`mg dñVyMr qH$_V H$mT>Ê`mgmR>r.

CXmhaU 14 : 1997 _Üò EH$m ehamMr bmoH$g§»`m 20,000 hmoVr. Ë`mV Xa dfu 5% dmT> hmoVo. Va gZ 2000 À`m eodQ>r Pmbobr bmoH$g§»`m H$mT>m.

CH$b : Xadfu bmoH$g§»òV 5% dmT> hmoVo : Ë`m_wio Xa dfm©À`m àma§^r ZdrZ bmoH$g§»òMm AmH$S>m Agob. åhUyZ AmnU åhUy eH$Vmo H$s hr dmT> MH«$dmT>rÀ`m ñdê$nmV Amho.

142 J{UV

1998 À`m àma§^rMr bmoH$g§»`m = 20,000 (hr AmnU n{hë`m dfm©Mr _wÔb _mZy)

5% Zo dmT>= 5100

× 20,000 = 1000

1999 Mr bmoH$g§»`m = 20,000 + 1000 = 21,000 ho Xþgè`m dfm©Mo _wÔb åhUyZ ¿`m.

5% Zo dmT> = 5100

× 21,000 = 1050

2000Mr bmoH$g§»`m= 21,000 + 1050 = 22,050 ho {Vgè`m dfm©Mo _wÔb åhUyZ ¿`m.

5% Zo dmT>= 5100

× 22,050 = 1102.5

2000 À`m eodQ>r hmoUmar bmoH$g§»`m = 22050 + 1102.5 = 23152.5

AWdm

2000 À`m eodQ>r hmoUmar bmoH$g§»`m = 20000 1 5100

3

+

= 20000 2120

2120

2120

× × × = 23152.5

åhUyZ A§XmOo bmoH$g§»`m = 23153

AéUmZo {dMmabo Oa H$_r AWdm KQ> Pmë`mg H$m` H$amdo. {ejH$m§Zr Imbrb CXmhaU {Xbo.

CXmhaU 15 : EH$ Q>r. ìhr ` 21000 XoD$Z {dH$V KoVbm. EH$ dfm©Z§Va Ë`mÀ`m qH$_VrV 5% KQ> Pmbr (KQ> AWdm Ad_yë`Z åhUOo dmnam_wio AWdm dñVy dmnaÊ`mMm H$mbmdYr dmT>ë`mZo Ë`mÀ`m qH$_VrV hmoUmar KgaU) Va EH$ dfm©Z§Va Q>r.ìhr. Mr qH$_V {H$Vr Agob Vo H$mT>m.

CH$b : _wÔb = ` 21,000

KQ> = Xa dfu ` 21000 Mo 5%

= ` 21000 5 1

100× ×

= ` 1050

EH$ dfm©À`m eodQ>r hmoUmar qH$_V = ` 21000 – ` 1050 = ` 19,950

n`m©`r nÜXV : AmnU hr qH$_V WoQ> Imbrb à_mUo H$mTy> eH$Vmo.

EH$ dfm©À`m eodQ>r hmoUmar qH$_V = ` 21000 1 5100

−

= ` 21000 1920

× = ` 19,950


à`ËZ H$am

1. `10,500 qH$_V Agboë`m EH$m `§ÌmMr 5% Zo qH$_VrV KQ> hmoVo. Va EH$ dfm©Z§Va Ë`mMr hmoUmar qH$_V H$mT>m.

2. EH$m ehamMr gÜ`mMr bmoH$g§»`m 12 bmI Amho. bmoH$g§»`m dmT>rMo à_mUo 4% Agob Va 2 dfm©Z§VaMr bmoH$g§»`m H$mT>m.

ñdmÜ`m` 8.3

1. ImbrbgmR>r MH«$dmT>ì`mO d amg H$mT>m.

(a) ` 10,800 Mo 3 dfm©Mo Xa gmb 12112% XamZo dm{f©H$ AmH$maUrZo.

(b) ` 18000 Mo 2112 dfmªMo Xa gmb 10% XamZo dm{f©H$ AmH$maUrZo.

(c) ` 62,500 Mo 1112 dfmªMo Xa gmb 8% XamZo AY© dm{f©H$ AmH$aUrZo.

(d) ` 8000 Mo 1 dfm©Mo Xa gmb 9% XamZo AY© dm{f©H$ AmH$aUrZo. (gai ì`mOmMo gwÌ dmnê$Z dfm©-dfmªMo ì`mO H$mTy>Z nS>Vmim H$ê$ eH$Vm)

(e) ` 10,000 Mo 1 dfm©Mo Xa gmb 8% XamZo AY© dm{f©H$ AmH$aUrZo.

2. H$_bmZo ñHy$Q>a KoÊ`mgmR>r ~±Ho$H$Sy>Z 26,400 H$Om©D$ KoVbo. Á`mda ~±H$ Xa gmb 15% XamZo dm{f©H$ ì`mO AmH$maUr H$aVo. Va {Vbm 2 df© 4 _{hÝ`mÀ`m eodQ>r nyU© H$O© ì`mOmgh naV H$aÊ`mgmR>r EHy$U {H$Vr aŠH$_ âmdr bmJob.

(gyMZm : àW_ dm{f©H$ AmH$maUr H$ê$Z 2 dfm©Mr amg H$mT>m Am{U Ë`m amerda 4

12 dfm©Mo gai ì`mO H$mT>m)

3. \$m°~rZmZo Xa gmb 12% XamZo ` 12,500 Mr aŠH$_ 3 dfm©gmR>r gaiì`mOmZo H$Om©D$ KoVbr. amYmZo {VVH$sM a¸$_ {VVŠ`mM _wwXVrgmR>r 10% Xa gmb XamZo MH«$dmT>ì`mOmÀ`m dm{f©H$ AmH$maUrZo H$Om©D$ KoVbr. Va H$moUr OmñV ì`mO {Xbo nm{hOo d {H$Vr?

4. O_eoQ>OrH$Sy>Z _r ` 12,000 Xagmb 6% XamZo 2 df© _wXVrgmR>r gaiì`mOmZo H$Om©D$ KoVbo Oa _r hrM aŠH$_ MH«$dmT> ì`mOmZo H$Om©D$ KoVbr AgVr, Va _bm OmñVrMr {H$Vr a¸$_ Úmdr bmJbr AgVr?

5. dmgwXodZo ` 60,000 Xa gmb 12% XamZo ghm_mhr ì`mO AmH$maUrÀ`m VËdmda Jw§V{dbo. Va Ë`mbm (i) 6 _{hÝ`mZ§Va, (ii) 1 dfm©Z§Va {H$Vr aŠH$_ naV {_iob?

144 J{UV

6. Aar\$Zo ` 80,000 Mo H$O© ~±Ho$VyZ KoVbo. ì`mO Xa Xa gmb 10% Amho. Oa Ë`mZo 112

dfm©Z§Va nyU© aŠH$_ ì`mOmgh naV Ho$br. Oa ì`mO AmH$maUr

(i) dm{f©H$ Ho$br (ii) AY© dm{f©H$ Ho$br. Va âboë`m aŠH$_oVrb \$aH$ H$mT>m.

7. _m[a`mZo 8000 EH$m CÚmoJmV Jw§V{dbo. {Vbm Xa gmb 5% XamZo dm{f©H$ AmH$maUrZo MH«$dmT> ì`mO {Xbo OmUma Amho; Va (i) {VÀ`m ImË`mda Xþgè`m dfm©À`m eodQ>r {H$Vr aŠH$_ O_m Agob Vo H$mT>m. (ii) {Vgè`m dfm©Mo ì`mO {H$Vr Vo H$mT>m.

8. ` 10,000 Mr 112 dfm©Mr Xa gmb 10% XamZo ì`mOmMr AmH$maUr AY© dm{f©H$ Agob Va

hmoUmar amg Am{U MH«$dmT>ì`mO H$mT>m. Oa ì`mOmMr AmH$maUr dm{f©H$ Ho$br Va `mda {_iUmao ì`mO OmñV Agob H$m?

9. am_Zo ` 4096 Xa gmb 12112 % ì`mO XamZo 18 _{hÝ`mgmR>r {Xbo MH«$dmT> ì`mOmMr AmH$aUr

ghm_mhr hmoVo Va _wXVrZ§Va Ë`mbm {H$Vr a¸$_ naV {_iob?

10. EH$m {R>H$mUmMr bmoH$g§»`m Xa gmb 5% XamZo dmTy>Z 2003 _Üò Vr 54,000 …Pmbr Va,

(i) 2001 _Üò bmoH$g§»`m {H$Vr hmoVr?

(ii) 2005 _Üò bmoH$g§»`m {H$Vr Agob? 11. EH$ à`moJemioV EH$m à`moJmÀ`mdoir gwú_ {OdmUy§Mr g§»`m Xa Vmgmbm 2.5% à_mUo dmT>V

hmoVr. Á`m {OdmUy§Mr g§»`m àma§^r 5,06,000 hmoVr Ë`m§Mr g§»`m nwT>rb XmoZ VmgmÀ`m eodQ>r {H$Vr Agob Vo H$mT>m.

12. EH$ ñHy$Q>a ` 42,000 bm {dH$V KoVbr. {VÀ`m _yi qH$_VrV Xa

dfm©bm 8% KQ> hmoVo, Va EH$ dfm©Z§Va hmoUmar ñHy$Q>aMr qH$_V

H$mT>m.

Amåhr H$m` MMm© Ho$br?

1. gyQ> åhUOo N>mnrb qH$_Vrda H$_r Ho$bobr aŠH$_.

gyQ> = N>mnrb qH$_V - {dH«$sMr qH$_V.

2. gyQ>Mm eoH$S>m Xa {Xbm Agob Va gyQ>Mr a¸$_ H$mT>Vm òVo.

gyQ> = N>mnrb qH$_V × eoH$S>m gyQ> Xa.

3. dñVy IaoXr Ho$ë`mZ§Va Ambobm OmñVrMm IM© AmnU dñVyÀ`m _yi qH$_VrV {_i{dVmo.Ë`mbm AmnU daH$S> IM© AWdm BVa IM© AWdm daIM© åhUVmo.

I. {H$§. (CP) = IaoXrMr _yi qH$_V + daH$S> IM©.


4. {dH«$s H$a hm dñVyÀ`m {dH«$sda gaH$maH$Sy>Z AmH$mabm OmVmo Am{U ~rbÀ`m aŠH$_oV {_i{dbm OmVmo.

{dH«$sH$a = {~bÀ`m a¸$_odarb % H$a.

5. GST ho dñVy Am{U god H$a `mda AmYm[aV Amho. hm dñVy§Mm nyadR>m qH$dm godm qH$dm XmoÝhrda AmH$mabm OmVmo.

6. _mJrb dfm©À`m amerda (A = P + I) AmH$maUr H$ê$Z H$mT>boë`m ì`mOmbm MH«$dmT> ì`mO åhUVmV.

7. (i) MH«$dmT>ì`mOmMr AmH$maUr dm{f©H$ Ho$ë`mg hmoUmar amg

P R n

1100

+

; P ho _wÔb, R hm ì`mO Xa, n hr _wXV

(ii) ì`mOmMr AmH$maUr AY©dm{f©H$ Ho$ë`mg hmoUmar amg

= P 1200

2

+

R n

R2 = AY©dm{f©H$ Xa

(2n = AY© dfmªMr g§»`m)

9.1.n[aMÀmnU Aä`mg Ho$bm AmhmV H$s gnmQ> ~§XrñV AmH¥$VrMr n[a{_Vr åhUOo Ë`mÀ`m gd© ~mOy§À`m bm§~rMr ~oarO Am{U joÌ\$i åhUOo AmH¥$VrZo ì`mnbobr OmJm hmo`. AmnU {ÌH$moU, Am`V, dVw©i Aem AZoH$ gnmQ> AmH¥$VtMr n[a{_Vr Am{U joÌ\$i H$mTy> eH$Vmo. VgoM Am`VmH$ma AmH$mamÀ`m nm`dmQ>> (nXnW) Am{U H$S>m `m§Mo joÌ\$i H$mT>Uo, ho hr AmnU {eH$bmo AmhmoV.

`m àH$aUm_Üò AmnU Mm¡H$moZmgma»`m gnmQ> Am{U {g_rV AmH¥$VtMr n[a{_Vr Am{U joÌ\$i H$mT>Uo `m g§~§YrMr J{UVo gmoS>{dÊ`mMm à`ËZ H$ê$.

VgoM AmnU KZ, KZm`V Am{U d¥Îm{MVr `mgma»`m KZmH¥$VtMo n¥îR>\$i Am{U KZ\$i `m{df`r {eH$Uma AmhmoV.

9.2.ñ_aU H$am_mJrb Aä`mgmMr COiUr H$aÊ`mgmR>r H$mhr CXmhaUo KoD$. {Xbobr AmH¥$Vr Am`VmH¥$Vr ~mJoMr (AmH¥$Vr 9.1) AgyZ Ë`mMr bm§~r d é§Xr AZwH«$_o 30m Am{U 20m Amho.

(i) ~mJoÀ`m ~mhoarb Hw§$nUmMr EHy$U bm§~r {H$Vr Agob? Oa AmnUmg Hw§$nUmMr bm§~r H$mT>md`mMr Agob Va Ë`m ~mJoMr n[a{_Vr H$mT>Uo Amdí`H$ Amho. Vr åhUOo 100 m. (Vnmgm).

(ii) m ~mJogmR>r EHy$U {H$Vr O_rZ ì`mnbr Amho? `m O{_ZrMo _moO_mn H$aÊ`mgmR>r AmnUmg ~mJoMo joÌ\$i _mhrV Agbo nm{hOo Am{U Vo joÌ\$i åhUOoM 600 Mm¡ag _rQ>a (m2) (H$go?)‹

(iii) ~mJoÀ`m AmVrb n[a{_Vrbm Agbobr nm`dmQ> 1 m é§XrMr AgyZ Ë`m nm`dmQ>oda {g_|Q> Kmbm`Mo Amho. Oa 4m2 gmR>r EH$ {g_|Q>Mr {nedr Amdí`H$ Agob Va ~mJoVrb g§nyU© nm`QdmQ>oda {g_|Q> KmbÊ`mgmR>r {H$Vr {neì`m {g_|Q> bmJob?

1 m

2 m1.5 m

30 m

20 m

àH$aU

9 joÌ_mnZ

AmH¥$Vr 9.1

joÌ_mnZ 147

bmJUmè`m EHy$U {g_|Q>À`m {neì`m =nm`dmQo>Mo (nXnWMo) EHy$U joÌ\$i

EH$m {nedrdmQ>o Ho$boë`m ^mJmMo {g_|Q>rH$aU

{g_|Q>rH$aU Ho$boë`m nm`dmQ>Mo joÌ\$i = ~mJoMo joÌ\$i - {g_|Q>rH$aU Z Ho$boë`m ~mJoMo joÌ\$i.

nm`dmQ> 1m. é§X Amho. åhUyZ (30 – 2) × (20 – 2) m2. ho {g_|Q>rH$aU Z Ho$boë`m Am`VmMo joÌ\$i Amho.

Vo åhUOoM, 28 × 18 m2

åhUyZ, bmJUmè`m EHy$U {g_|Q>À`m {neì`m = ________

(iv) AmH¥$Vr 9.1 _Üò XmI{dë`mà_mUo 1.5m × 2m _mnZmMo \w$bm§Mo XmoZ Am`VmH¥$Vr dm\o$ ~mJoV AmhoV Am{U am{hboë`m ^mJmda JdVmMo AmdaU Amho. Va JdVmMo AmdaU Agboë`m ^mJmMo joÌ\$i H$mT>m.

Am`VmH$ma dmâ`m§Mo joÌ\$i = ....................

{g_|Q>rH$aU Z Ho$boë`m ^mJmMo joÌ\$i = .................... .

JdVmMo AmdaU Agboë`m ^mJmMo joÌ\$i = ....................

Amnë`mbm \$ŠV Am`VM Zìho Va BVa ^m¡{_VrH$ AmH¥$Ë`m§Mr {d{eîQ> _mnZo {Xbr AgVm Ë`m§Mo joÌ\$i H$mTy> eH$Vmo. AmR>dÊ`mMm à`ËZ H$am Am{U OmoS>çm Owidm.

AmH¥$Vr AmH$ma joÌ\$i

Am`V

Mm¡ag

{ÌH$moU

a × a

b × h

pb2

148 J{UV

g_m§Va^wO Mm¡H$moZ

dVw©i

12

b × h

a × b

darb àËòH$ AmH¥$VrMr n[a{_Vr gwÌmV {bhÊ`mMm à`ËZ H$ê$ eH$mb H$m?

à`ËZ H$am

(a) Mm¡H$Q>rV {Xboë`m joÌ\$imdê$Z AmH¥$VrÀ`m OmoS>`m Owidm.

14 cm 14 cm

14 cm

7 cm

7 cm

7 cm

14 cm

7 cm11 cm9 c

m7 cm7 cm 14 cm2

77 cm2

98 cm2

(b) àËòH$ AmH¥$VrMr n[a{_Vr {H$Vr Vo {bhm.

ñdmÜ`m` 9.1

1. AmH¥$VrV Xe©{dë`mà_mUo Mm¡ag Am{U Am`VmH¥$Vr

eoVm§Mr _mno {Xbr AmhoV. Oa Ë`m§Mr n[a{_Vr g_mZ

Agob Va H$moUË`m eoVmMo joÌ\$i A{YH$ Amho? 60 m 80 m

2. lr_Vr H$m¡{eH$ `m§À`mH$S>o AmH¥$VrV Xe©{dë`mà_mUo

Mm¡agmH¥$Vr _mnZmMr EH$ OmJm Amho. Va ~mOyÀ`m

OmJoÀ`m _Ymo_Y Ë`m§Zm KamMo ~m§YH$m_ H$am`Mo Amho.

{e„H$ OmJoV ~mJ V`ma H$aÊ`mgmR>r ` 55 Xa m2

à_mUo {H$Vr IM© òB©b?25m

25m

20m

15m Ka

~mJ

joÌ_mnZ 149

3. AmH¥$VrV Xe©{dë`mà_mUo ~mJoMm AmH$ma _Yo

Am`VmH$ma AgyZ Ë`mÀ`m XmoÝhr Q>moH$mbm AY©JmobmH$ma

^mJ AmhoV. Va Ë`m ~mJoMo joÌ\$i Am{U n[a{_Vr

H$mT>m. (Am`VmMr bm§~r Amho 20 – (3.5+3.5)m.)

7 m

20 m

4. g_m§Va^wO Mm¡H$moZmH¥$Vr EH$m \$aerMr nm`mMr ~mOy 24cm d g§JV C§Mr 10cm Amho. Va, 1080 m2 joÌ\$i AgUmè`m OmJo_Üò Aem EHy$U {H$Vr \$aem ~gy eH$Vrb? (Oa Amdí`H$Vm Agob Va H$mon§ao âÊ`mgmR>r JaOoZwgma \$aerMo ^mJ H$ê$ eH$Vm.)

5. O{_Zrda {dIwaboë`m doJdoJù`m AmH$mamÀ`m AÝZmÀ`m VwH$S>`m§^modVr EH$ _w§Jr \o$è`m _maV Amho. Va H$moUË`m AmH$mamÀ`m AÝZmÀ`m VwH$S>`m^modVr _w§Jrbm A{YH$ doi \o$ar _mamdr bmJob? bjmV R>odm H$s, dVw©imMr n[a{_Vr C = 2pr , òWo r hr dVw©imMr {ÌÁ`m Amho.

1.5 cm

2.8 cm2.8 cm 2 cm

2 cm

2.8 cm

9.3.g_b§~ Mm¡H$moZmMo joÌ\$i (Area of trapezium)

_w»` añË`mÀ`m eoOmar ZP_mMm ñdV…Mm ^yI§S> (plot) Amho (AmH¥$Vr 9.2). Vmo eoOmar AgUmè`m BVa ^yI§S>mgmaIm Am`VmH¥$Vr Zmhr na§Vy Ë`m ^yI§S>mÀ`m {déÜX ~mOy§Mr EH$ OmoS>r g_m§Va Amho. Ë`m_wio ZP_mbm {_imbobm ^yI§S> hm g_b§~ Mm¡H$moZmÀ`m AmH$mamMm Amho. Va Vwåhr Ë`mMo joÌ\$i H$mTy> eH$mb H$m?

20 m

12 m

20 m

12 m

20 m

30 m

10 m


AmH¥$Vr 9.3 _Üò XmI{dë`mà_mUo Ë`m ^yI§S>mÀ`m {eamoq~Xÿbm Zm§do Úm.

AmH¥$Vr 9.3 _Üò XmI{dë`mà_mUo EC || AB AgyZ EH$m ~mOybm Am`VmH$ma AmH$ma Am{U Xþgè`m ~mOybm {ÌH$moUr AmH$ma Ago Ë`mMo XmoZ ^mJ Ho$bo AmhoV. (VgoM ∠C hm H$mQ>H$moZ Amho).

Δ ECD Mo joÌ\$i = 12

12

12 10h c× = × × = 60m2.

Am`V ABCE Mo joÌ\$i = h × a = 12 × 20 = 240m2.

150 J{UV

g_b§~ Mm¡H$moZ ABDE Mo joÌ\$i =Δ ECD Mo joÌ\$i + Am`V ABCE Mo joÌ\$i = 60 + 240 = 300 m2. òWo XmoÝhr joÌ\$i EH$Ì Ho$bo AgVm g_b§~ Mm¡H$moZmMo joÌ\$i {_iVo. åhUOoM,

ABDE Mo joÌ\$i = 12 2h c h a h c a× + × = +

= hc a h c a a+

=+ +

22 2

= hb a+( )

2 = C§Mr ×

(g_m§Va ~mOy§Mr ~oarO)2

h, b Am{U a À`m qH$_Vr _m§S>ë`m AgVm, hb a+( )

=2

300 m2. qH$_V {_iVo.

à`ËZ H$am

1. ZP_mÀ`m ~{hUrH$S>o gwÜXm g_b§~ Mm¡H$moZmÀ`m AmH$mamMm EH$ ^yI§S> Amho. AmH¥$Vr 9.4 _Üò Xe©{dë`mà_mUo Ë`m ^yI§S>mMo VrZ ^mJ Ho$bo. Va g_b§~ Mm¡H$moZ WXYZ Mo joÌ\$i

= ha b+( )

2 {gÜX H$am.

2. Oa h = 10 cm, c = 6 cm ., b = 12 cm Am{U d = 4 cm Agob Va àËòH$ mJmMr doJir qH$_V H$mT>m Am{U WXYZ À`m joÌ\$im_Üò {_idm.

VgoM h, a Am{U b `m qH$_Vr ha b+( )

2 _Üò

_m§Sy>Z nS>Vmim nhm.AmH¥$Vr 9.4

ho H$am

1. AmH¥$VrV XmI{dë`mà_mUo EH$m AmboI nonaÀ`m VwH$S>`mda WXYZ g_b§~ Mm¡H$moZ aMyZ Vmo H$mnm (AmH¥$Vr 9.5 nhm).

AmH¥$Vr 9.5

2. XY ~mOy Xþ_Sy>Z Ë`mMm _Ü`q~Xÿ H$mT>m d Ë`mbm A ho Zmd Úm (AmH¥$Vr 9.6 ).

AmH¥$Vr 9.6

joÌ_mnZ 151

3. WXYZ `m g_b§~ Mm¡H$moZmMo ZA À`m aofoV XmoZ ^mJ H$am. AmH¥$Vr 9.7 _Üò Xe©{dë`mà_mUo AX da AY òB©b Agm DZYA R>odm. `mVrb _moR>çm {ÌH$moUmÀ`m nm`mMr bm§~r {H$Vr Agob? VgoM `m {ÌH$moUmMo joÌ\$i H$mT>Ê`mMo gyÌ amerÀ`m ñdê$nmV (AmH¥$Vr 9.7) {bhm.

AmH¥$Vr 9.7

4. g_b§~ Mm¡H$moZ WXYZ Mo joÌ\$i Am{U `m {ÌH$moUmMo joÌ\$i g_mZ Amho. (H$go?) {ÌH$moUmÀ`m joÌ\$imÀ`m amerdéZ g_b§~ Mm¡H$moZmMo joÌ\$i H$mT>>m d Ë`mMo gwÌ amer ñdê$nmV {bhm.

Oa AmnUmg g_b§~ Mm¡H$moZmMo joÌ\$i H$mT>md`mMo Agob Va g_m§Va ~mOy§Mr bm§~r Am{U Ë`m XmoZ g_m§Va ~mOy_Yrb b§~m§Va _m{hV Agbo nm{hOo. g_b§~ Mm¡H$moZmMo joÌ\$i ho XmoZ g_m§Va ~mOy§À`m bm§~tMr ~oarO Am{U b§~m§Va `m§À`m JwUmH$mamÀ`m {Zå_o AgVo.

à`ËZ H$am

Imbrb g_b§~ Mm¡H$mo§ZmMo joÌ\$i H$mT>m (AmH¥$Vr 9.8 nhm).

7 cm

3 cm

9 cm

10 cm6 cm

5 cm

AmH¥$Vr 9.8

ho H$am

n[a{_Vr {^ÝZ na§Vy joÌ\$i g_mZ AgUmè`m g_m§Va^wO Mm¡H$moZmMr aMZm hr B`Îmm 7 drÀ`m dJm©V {eH$bmo AmhmoV. AerM {H«$`m g_b§~ Mm¡H$moZmbmgwÜXm hmoD$ eH$Vo H$m?

n[a{_Vr {^ÝZ nU g_mZ joÌ\$i AgUmao Imbrb g_b§~ Mm¡H$moZ Vnmgm (AmH¥$Vr 9.9 nhm).

AmH¥$Vr 9.9

152 J{UV

AmnUmg _mhrV Amho H$s, gd© EH$ê$n AgUmè`m AmH¥$Ë`m g_joÌ AgVmV. g_joÌ AgUmè`m AmH¥$Ë`m EH$én AgVmV H$m? `m gd© AmH¥$Ë`m EH$ê$n AmhoV H$m?

{^ÝZ joÌ\$i nU g_mZ n[a{_Vr AgUmao H$_rVH$_r VrZ g_b§~ Mm¡H$moZ AmboIH$mJXmda H$mT>m.

9.4.gm_mÝ` Mm¡H$moZmMo joÌ\$i (Area of a General Quadrilateral)EH$m gm_mÝ` Mm¡H$moZm_Üò EH$m H$Um©_wio XmoZ {ÌH$moU V`ma hmoVmV. `mM {ÌH$moUrH$aUmÀ`m _XVrZo gm_mÝ` Mm¡H$moZmMo joÌ\$i H$mT>Vm òVo. AmH¥$Vr 9.10 Mm Aä`mg H$am.

ABCD Mm¡H$moZmMo joÌ\$i = (D ABC Mo joÌ\$i ) + (D ADC Mo joÌ\$i)

= 12

121 2AC AC×

+ ×

h h

= 12 1 2AC × +( )h h

= 12 1 2d h h+( )

òWo 'd' åhUOo AC H$Um©Mr bm§~r hmo`.

CXmhaU 1 : AmH¥$Vr 9.11 _Üò Xe©{dboë`m PQRS Mm¡H$moZmMo joÌ\$i H$mT>m.CH$b : òWo d = 5.5 cm, h1 = 2.5cm, h2 = 1.5 cm,

2.5

cm

1.5

cm

5.5 cm

AmH¥$Vr 9.11

PQRS Mo joÌ\$i = 12

d (h1 + h2)

= 12

× 5.5 × (2.5 + 1.5) cm2

= 12

× 5.5 × 4 cm2 = 11 cm2

à`ËZ H$am

AmnUmg _m{hVM Amho H$s, g_m§Va^wO Mm¡H$moZ hm gwÜXm EH$ Mm¡H$moZM Amho. AmVm òWo g_m§Va^wO Mm¡H$moZmMo joÌ\$i H$mT>Ê`mgmR>r Ë`mMr XmoZ {ÌH$moUm_Üò {d^mJUr H$am. Ë`m§Mo joÌ\$i H$mT>m. VgoM g_m§Va^wO Mm¡H$moZmMo XoIrb joÌ\$i H$mT>m. Vwåhmbm _mhrV Agboë`m gwÌmer Vo EH$ê$n hmoVo. H$m? (AmH¥$Vr 9.12).

AmH¥$Vr 9.12

AmH¥$Vr 9.10

joÌ_mnZ 153

9.4.1. {deof Mm¡H$moZm§Mo joÌ\$i (Area of Special Quadrilateral)

{ÌH$moUrH$aUmMm Cn`moJ H$ê$Z g_^wO Mm¡H$moZmMo joÌ\$i H$mT>Ê`mMo gwÌ V`ma H$ê$ eH$Vmo. AmH¥$Vr 9.13 _Üò ABCD hm g_^wO Mm¡H$moZ (rhombus) Amho. åhUyZ Ë`mMo H$U© nañnam§Zm Xþ^mJVmV.

ABCD g_^wO Mm¡H$moZmMo joÌ\$i = (D ACD Mo joÌ\$i) + (D ABC Mo joÌ\$i)

= 12

12

× ×

+ × ×

AC OD AC OB

AmH¥$Vr 9.13

= 12

× AC × (OD + OB)

= 12

× AC × BD = 12

d1 ×d2 = (AC = d1 Am{U BD = d2) )

åhUOoM, g_^wO Mm¡H$moZmMo joÌ\$i ho Ë`mÀ`m XmoZ H$UmªÀ`m bm§~rÀ`m JwUmH$mamÀ`m {Zå_o AgVo.

CXmhaU 2 : 10 cm Am{U 8.2 cm H$Um©Mr bm§~r AgUmè`m g_^wO Mm¡H$moZmMo joÌ\$i H$mT>m. CH$b : g_^wO Mm¡H$moZmMo joÌ\$i =

12

× d1 × d2 òWo d1 , d2 H$Um©Mr bm§~r

= 12

× 10 × 8.2 cm2 = 41 cm2.

{dMma H$am, MMm© H$am Am{U {bhm.

g_m§Va^wO Mm¡H$moZm_Üò EH$ H$U© aMbm AgVm Ë`mMr XmoZ EH$ê$n {ÌH$moUm_Üò {d^mJUr hmoVo. Va AmnU g_b§~ Mm¡H$moZmMr XmoZ EH$ê$n {ÌH$moUm_Üò {d^mJUr H$ê$ eH$Vmo H$m?

à`ËZ H$am

{Xboë`m Mm¡H$moZm§Mo joÌ\$i H$mT>m (AmH¥$Vr 9.4).

4 cm4 cm 8 cm

5 cm

2 cm

5 cm

4 cm

4 cm

7 cm

3 cm

6 cm

5 cm

4 cm

6 cm

4 cm

AmH¥$Vr 9.14

154 J{UV

9.5. ~hþ^wOmH¥$VrMo joÌ\$i (Area of polygon)

AmnU Mm¡H$moZmMo {ÌH$moUm_Üò ^mJ H$ê$Z Ë`mMo joÌ\$i H$mT>bo Amho. `mM nÜXVrZo AmnUmg ~hþ^wOmH¥$VrMo joÌ\$i H$mT>md`mMo Amho. Imbrb n§M~hþ^wOmH¥$VrMo {ZarjU H$am? (AmH¥$Vr 9.15 d 9.16).

ABCDE `m n§M~hþ^wOmH¥$Vr_Üò XmoZ H$U© AC d AD aMbo AgVm Ë`mMo VrZ {d^mJ hmoVmV.

åhUyZ ABCDE Mo joÌ\$i = DABC Mo joÌ\$i

+ DACD Mo joÌ\$i

+ DAED Mo joÌ\$i

ABCDE `m n§M~hþ^wOm§H¥$Vr_Üò AD hm EH$ H$U© Am{U BF Am{U CG b§~ AgyZ Ë`mMo Mma {d^mJ hmoVmV.

åhUyZ ABCDE Mo joÌ\$i = H$mQ>H$moZ DAFB Mo joÌ\$i + g_b§~ Mm¡H$moZ BFGC Mo joÌ\$i + H$mQ>H$moZ DCGD Mo joÌ\$i

+ DAED Mo joÌ\$i.

(BFGC `m g_b§~ Mm¡H$moZmÀ`m g_m§Va ~mOy AmoiIm)

à`ËZ H$am

(i) Imbrb ~hþ^wOmH¥$VrMo (AmH¥$Vr 9.17) ^mJ H$am ({ÌH$moU Am{U g_b§~ Mm¡H$moZm_Üò) Am{U Ë`mMo joÌ\$i H$mT>m.

AmH¥$Vr 9.17

EFGHI `m ~hþ^wOmH¥$Vr_Üò FI Va MNOPQR `m ~hþ^wOmH¥$Vr_Üò NQ hm H$U© Amho.

AmH¥$Vr 9.15

AmH¥$Vr 9.16

joÌ_mnZ 155

(ii) Imbrb AmH¥$VrV Xe©{dë`mà_mUo ABCDE `m ~hþ^wOmH¥$VrMo {d^mJ Ho$bo AmhoV (AmH¥$Vr 9.18).

Oa AD = 8 cm, AH = 6 cm, AG = 4 cm, AF = 3 cm, Am{U b§~ BF = 2 cm, CH = 3 cm., EG = 2.5 cm. Agob Va Ë`mMo joÌ\$i H$mT>m.

ABCDE ~hþ^wOmH¥$VrMo joÌ\$i = DAFB ~ Mo joÌ\$i + .......

Δ AFB Mo joÌ\$i = 12

× AF × BF = 12

× 3 × 2 = ....

g_b§~ Mm¡H$moZ FBCH Mo joÌ\$i = FH × BF CH+( )2

= 32 3

2×

+( ) [FH = AH–AF] = ΔCHD Mo joÌ\$i

= 12

×HD× CH = ....; ΔADE Mo joÌ\$i

= 12

× AD × GE =...............

åhUyZ, ABCDE `m ~hþ^wOmH¥$VrMo joÌ\$i = ...........

AmH¥$Vr 9.18

(iii) MP = 9 cm,, MD = 7 cm,

MC = 6 cm, MB = 4 cm, MA = 2 cm,

Va MNOPQR ~hþ^wOmH¥$VrMo joÌ\$i H$mT>m (AmH¥$Vr 9.19).

NA, OC, QD Am{U RB ho H$U© MP da b§~ AmhoV.

AmH¥$Vr 9.192

cm

2.5

cm

2.5

cm 3 cm

CXmhaU 1 : EH$m g_b§~ Mm¡H$moZmH¥$Vr eoVmMo joÌ\$i 480 m2 Amho. Oa XmoZ g_m§Va ~mOy_Yrb A§Va 15m Am{U EH$ g_m§Va ~mOy 20m. Agob Va Xþgar g_m§Va ~mOy H$mT>m.

CH$b : g_b§~ Mm¡H$moZmMr EH$ g_m§Va ~mOy = a = 20 m.

Va Xþgar g_m§Va ~mOy b Amho. C§Mr h = 15 cm.

g_b§~ Mm¡H$moZmMo joÌ\$i = 480 m2 Amho.

g_b§~ Mm¡H$moZmMo joÌ\$i = 12 h (a + b)

\ 480 = 12 × 15 × (20 + b) qH$dm 480 2

15× = 20 + b

qH$dm 64 = 20 + b qH$dm b = 44 m

åhUyZ, g_b§~ Mm¡H$moZmMr Xþgar g_m§Va ~mOy 44 m Amho.

156 J{UV

CXmhaU 2 : Oa g_^wO Mm¡H$moZmMo joÌ\$i 240 cm2 Am{U EH$ H$U© 16 cm Agob Va Xþgam H$U© H$mT>m.

CH$b : EH$ H$Um©Mr bm§~r = d1 = 16 cm

Am{U Xþgè`m H$Um©Mr bm§~r = d2

. g_^yO Mm¡H$moZmMo joÌ\$i = 12 × d1 ×

d2 = 240

12

× 16 × d2 = 240 \ d2 = 30 cm

åhUyZ, Xþgè`m H$Um©Mr bm§~r 30 cm Amho.

CXmhaU 3 : MNOPQR `m fQ>^wOmH¥$VrÀ`m ~mOy 5 cm. AmhoV (AmH¥$Vr 9.20). A_Z Am{U [aYr_m `m ~hþ^wOmH¥$VrMo XmoZ doJdoJù`m [aVrZo ^mJ H$aVmV (AmH¥$Vr 9.21).

XmoÝhr nÜXVtMm Cn`moJ H$ê$Z fQ>^wOmH¥$VrMo joÌ\$i H$mT>m.

AmH¥$Vr 9.21A_ZMr nÜXV[aYr_mMr nÜXV

AmH¥$Vr 9.208 cm

5 cm11 cm

CH$b : A_ZMr nÜXV : MNOPQR hr EH$ fQ>^wOmH¥$Vr AgyZ oWo NQ _wio XmoZ EH$ê$n g_b§~ Mm¡H$moZmV {d^mJUr hmoVo. mgmR>r Vwåhr H$mJX Xþ_Sy>Z nmhÿ eH$Vm (AmH¥$Vr 9.22).

AmVm, MNQR `m g_b§~ Mm¡H$moZmMo joÌ\$i = 411 5

2×

+( ) = 2 × 16 = 32 cm2.

åhUyZ MNOPQR `m fQ>^wOmH¥$VrMo joÌ\$i = 2 × 32 = 64 cm2.

4 cmAmH¥$Vr 9.22

[aYr_mMr nÜXV : DMNO Am{U DRPQ ho EH$ê$n [ÌH$moU AgyZ Ë`m§Mr C§Mr 3cm Amho (AmH¥$Vr 9.23). ho nS>VmiÊ`mgmR>r Vwåhr Ë`mMà_mUo XmoZ {ÌH$moU H$mnyZ Ë`m OmJoda R>odm.

DMNO Mo joÌ\$i = 12

× 8 × 3 = 12cm2.

= DRPQ Mo joÌ\$i

Am`V MQPR Mo joÌ\$i 8 × 5 = 40cm2.

AmVm MNOPQR fQ>^wOmH¥$VrMo joÌ\$i = 40+12+12 = 64cm2.

AmH¥$Vr 9.23

3 cm

5 cm

3 cm

joÌ_mnZ 157

13 m

24 m

8 m

ZXr

añVm

100 m

ñdmÜ`m` 9.2

1. EH$m Q>o~bmMm darb n¥îR>^mJ g_b§~ Mm¡H$moZmÀ`m AmH$mamMm Amho. Oa Ë`mÀ`m g_m§Va ~mOy 1 m Am{U 1.2 m bm§~ Am{U Ë`m_Yrb b§~m§Va 0.8 m Agob, Va Ë`m g_b§~ Mm¡H$moZmMo joÌ\$i H$mT>m.

2. EH$m g_b§~ Mm¡H$moZmMo joÌ\$i 34 cm2 Am{U Ë`mMr C§Mr 4 cm AgyZ EH$ g_m§Va ~mOy 10 cm Amho. Va Xþgè`m g_m§Va ~mOyMr bm§~r H$mT>m.

3. ABCD g_b§~ Mm¡H$moZmÀ`m AmH$mamÀ`m eoVmÀ`m Hw§$nUmMr bm§~r 120m Amho. Oa BC = 48m, CD = 17m. Am{U AD = 40m. Va eoVmMo joÌ\$i H$mT>m. òWo AB ~mOy hr g_m§Va AgUmè`m ~mOy AD Am{U BC bm b§~ Amho.

4. EH$m Mm¡H$moZmH¥$Vr eoVmÀ`m H$Um©Mr bm§~r 24m Amho Am{U Ë`mda Ë`mÀ`m {déÜX {eamoq~XÿVyZ 8m Am{U 13m bm§~rMo b§~ H$mT>bo AmhoV. Va Ë`m eoVmMo joÌ\$i H$mT>m.

5. g_^wO Mm¡H$moZmMo H$U© 7.5 cm Am{U 12 cm AgVrb Va Ë`mMo joÌ\$i H$mT>m.

6. EH$m g_^wO Mm¡H$moZmMr ~mOy 5 cm Am{U Ë`mMr C§Mr 4.8 cm. Agob, Va Ë`mMo joÌ\$i H$mT>m. Oa Ë`m Mm¡H$moZmMm EH$ H$U© 8 cm Agob Va Xþgè`m H$Um©Mr bm§~r H$mT>m.

7. EH$m B_maVrÀ`m hm°b_Üò g_^wO Mm¡H$moZmÀ`m AmH$mamÀ`m 3000 \$aí`m ~gdm`À`m AmhoV Am{U àËòH$ \$aerMm H$U© 45 cm Am{U 30cm bm§~rMm Amho. Oa àËòH$ \$aerÀ`m nm°bregmR>r Xa m2 bm ` 4 IM© òV Agob Va EHy$U IM© {H$Vr hmoB©b?

8. _mohZZo g_b§~ Mm¡H$moZmÀ`m AmH$mamMr EH$ eoVO_rZ {dH$V KoVbr. Ë`m O{_{ZÀ`m EH$m ~mOybm ZXr d Xþgè`m~mOybm añVm AgyZ Vo EH$_oH$mbm g_m§Va AmhoV. ZXrÀ`m ~mOyZo Agboë`m O{_ZrMr bm§~r añË`mÀ`m ~mOyZo Agboë`m O{_{ZÀ`m XþßnQ> Amho.Oa O{_{ZMo joÌ\$i

1 m

1.2 cm

0.8 m

158 J{UV

10500 m2 Am{U XmoZ g_m§Va ~mOyMo b§~m§Va 100 m Agob Va ZXrÀ`m g§~§YrV AgUmè`m ~mOyMr bm§~r H$mT>m.

9. AmH¥$VrV Xe©{dë`mà_mUo EH$m ßb°Q>\$m°_©Mm n¥îR>^mJ gwg_ AîQ>^wOmH¥$VrÀ`m AmH$mamV da dmT>{dbm Amho. Va `m AîQ>^wOmH¥$Vr n¥îR>^mJmMo joÌ\$i H$mT>m.

11 m4 m

5 cm

10. {Xboë`m AmH¥$VrV Xe©{dë`mà_mUo n§M^wOmH¥$Vr AmH$mamMr EH$ ~mJ Amho. Ë`m ~mJoMo joÌ\$i emoYÊ`mgmR>r Á`moVr Am{U H${dVmZo XmoZ doJdoJù`m nÜXVtMm dmna Ho$bm Amho.

15 m

15 m 15 m30 m

Á`moVrMr nÜXV H${dVmMr nÜXV

XmoÝhr nÜXVrZo ~mJoMo joÌ\$i H$mT>m. `m ì`{V[aŠV joÌ\$i H$mT>Ê`mMr Vwåhr AmUIrZ H$moUVrhr nÜXV V`ma H$ê$ eH$mb H$m?

11. AmVrb _mno 16 cm × 20 cm Am{U ~mhoarb _mno 24 cm × 28 cm. AgUmar EH$ g§b½Z {MÌ Mm¡H$Q> (\«o$_) Amho. Oa àËòH$ ^mJmMr OmS>r g_mZ Agob Va Mm¡H$Q>rÀ`m àËòH$ ^mJmMo joÌ\$i H$mT>m.

28 cm16 cm

20 c

m

24 cm

9.6. KZmH¥$VtMo AmH$ma (solid shapes)

Vwåhr _mJrb B`ÎmoV {Û{_Vr` AmH¥$Ë`m `m {Ì{_Vr` AmH$mamÀ`m n¥îR>^mJmV Agy eH$VmV ho {eH$bm AmhmV. Aem KZmH¥$VtMo {ZarjU H$am H$s Á`m§Mr AmnU MMm© AmVmn`ªV Ho$br Amho.

{eamoq~Xÿ

d¥Îm{MVr ({gb|S>a)

n¥îR>^mJ

e§Hy$H$S>m

KZ

KZm`V

{na°_rS>

joÌ_mnZ 159

AmH¥$Vr 9.24

XmoZ qH$dm XmoZnojm A{YH$ g_mZ (EH$ê$n) n¥îR>^mJmÀ`m AmH$mamMo {ZarjU H$am.

Ë`mZm Zmdo Úm Am{U H$moUVm AmH$ma ârd EH$ê$n n¥îR>^mJ Xe©{dVmo?

à`ËZ H$am

gm~U (soaps), IoiUr (toys), noñQ> (pastes), ImÚnXmW© (snacks) B. nXmW© KZ, KZm`V, d¥Îm{MVrÀ`m AmH$mam_Üò Amd¥Îm (packing) Ho$bobo AgVmV. Ago H$mhr ~m°Šg Jmoim H$am (AmH¥$Vr 9.25).

AmH¥$Vr 9.25

gd© ghm n¥ð>^mJ Am`VmH$maAgyZ {déÜX n¥ð>^mJ g_mZ AmhoV.åhUyZ VrZ OmoS>rMo n¥ð>^mJ g_mZ AmhoV.

KZm`VmH$ma ~m°Šg KZmH$ma ~m°Šg

gd© ghm n¥îR>^mJ Mm¡agmH¥$Vr AgyZ g_mZAmhoV.

EH$ dH«$ n¥îR>^mJ Am{U

XmoZ dVw©imH$ma n¥îR>^mJ

g_mZ AmhoV

dH«$n¥ð>^mJ

d¥Îm{MVrH$ma ~m°Šg

dV©imH$ma nm`m Am{U darb

_mWm g_mZ AmhoV.

AmVm EH$mM doir EH$mM àH$maMo ~m°Šg ¿`m. Ë`mbm AgUmao gd© n¥îR>^mJ H$mnm. àËòH$ n¥îQ>^mJmÀ`m AmH$mamMo {ZarjU H$am Am{U EH$ê$n n¥îR>^mJ EH$Ì R>odm. Ë`m Z§Va ~m°ŠgÀ`m n¥îR>^mJm§Mr g§»`m _moOm. Vwåhr Ho$bobo {ZarjU Z_yX H$am.

160 J{UV

AmH¥$Vr 9.26hr b§~ d¥Îm{MVr Amho.

Imbrb KQ>H$mMo AdbmoH$Z Ho$bo Amho H$m?

d¥Îm{MVrbm EH$ê$n dVw©imH$ma n¥îR>^mJ AgyZ Ë`m nañnam§Zm g_m§Va AmhoV (AmH¥$Vr 9.26). {ZarjU H$am H$s, d¥Îm{MVrÀ`m n¥îR>^mJmÀ`m _Ü`mZm OmoS>Umar aofm hr Ë`mÀ`m nm`mbm g_m§Va Amho. Aem àH$maÀ`m d¥Îm{MVrbm b§~ dVw©imH$ma d¥Îm{MVr åhUVmV. Oar {d{dY AmH$mamÀ`m d¥Îm{MVr Agë`m Var AmVm Amåhr b§~ dVw©imH$ma d¥Îm{MVrMm Aä`mg H$ê$`m (AmH¥$Vr 9.27).

AmH¥$Vr 9.27hr b§~ d¥Îm{MVr Zmhr


òWo Xe©{dboë`m KZmH¥$Vrbm d¥Îm{MVr åhUUo MwH$sMo H$m Amho?

9.7. KZ, KZm`V Am{U d¥Îm{MVrMo n¥îR>\$i (surface area of cube, cuboid and cylinder)

B_«mZ, _moZrH$m Am{U Ognmb `mZr g_mZ C§Mr Agboë`m AZwH«$_o KZm`V, KZ Am{U d¥Îm{MVrÀ`m AmH$mamÀ`m ~m°ŠgZm a§J {Xbm.

AmH¥$Vr 9.28

`m {VKmn¡H$s H$moUr A{YH$ ^mJ a§J{dbm ho {Z{üV H$aÊ`mMm à`ËZ Ë`m§Zr Ho$bm. h[aZo gwM{dbo H$s àËòH$ ~m°ŠgMo n¥ð>\$i H$mT>ë`mg Ë`m§Zm _XV hmoB©b.

g§nyU© n¥îR>\$i H$mT>Ê`mgmR>r àËòH$ n¥îR>^mJmMo joÌ\$i H$mTy>Z Ë`mMr ~oarO H$am. KZmÀ`m n¥îR>^mJmÀ`m joÌ\$imMr ~oarO åhUOoM, Ë`m KZmMo dH«$n¥îR\$i hmo`. ho g_OÊ`mgmR>r àËòH$ AmH$ma H«$_mH«$_mZo KoD$.

joÌ_mnZ 161

9.7.1 KZm`V (cuboid)

EH$m KZm`V ~m°Šgbm H$mnyZ Vmo CKS>m Am{U gnmQ> H$ê$Z R>odm (AmH¥$Vr 9.29). VgoM Ë`mMo gai ^mJ AmnU AmH¥$VrV nmhÿ eH$Vmo (AmH¥$Vr 9.30).

àËòH$ ~mOyMo n[a_mU (_moO_mn) {bhm. AmnUmg _m{hVM Amho H$s. KZm`Vmbm OmoS>`mMo OwiUmao VrZ g_mZ n¥îR>^mJ AgVmV. àËòH$ n¥îR>^mJmMo joÌ\$i H$mT>Ê`mgmR>r H$moUVr nÜXV Vwåhr dmnê$ eH$mb?

Ë`m ~m°ŠgÀ`m àËòH$ n¥îR>^mJmMo EHy$U joÌ\$i H$mT>m. Amåhr nmhÿ eH$Vmo H$s, KZm`VmMo g§nyU© joÌ\$i = joÌ\$i I + joÌ\$i II + joÌ\$i 1II + joÌ\$i 1V + joÌ\$i V + joÌ\$i VI Amho.

= h × 1 + b × 1 + b × h + l × h + b × h + 1 × b

åhUyZ, g§nyU© joÌ\$i = 2(h × 1 + b × h + b × 1).

òWo h, l Am{U b ho AZwH«$_o KZm`Vm§Mr C§Mr, bm§~r Am{U é§Xr Amho.

g_Om, Oa darbà_mUo Oa C§Mr, bm§~r Am{U é§Xr AZwH«$_o 20cm, 15cm Am{U 10cm Agob Va,

g§nyU© joÌ\$i = 2 (20 × 15 + 20 × 10 + 10 × 15)

= 2 ( 300 + 200 + 150)

= 1300 m2

à`ËZ H$am

{Xboë`m KZmH¥$VtMo g§nyU© joÌ\$i H$mT>m (AmH¥$Vr 9.31)

AmH¥$Vr 9.314 cm6 cm 4 cm

2 cm

4 cm

10 cm

AmH¥$Vr 9.29 AmH¥$Vr 9.30noñ

Q>

162 J{UV

KZm`VmMo nmíd©n¥îR>\$i H$mT>Ê`mgmR>r ~mOy§À`m

n¥îR>^mJmMm dmna H$am. (òWo darb Am{U Imbrb n¥îR>^mJ

Z_yX Z H$aVm) CXmhaUmW©, EH$m KZmH¥$VrÀ`m ImobrÀ`m

Mmahr ~mOy§Mo EHy$U joÌ\$i AmnU ~gboë`m Ë`mM ImobrÀ`m

nmíd©n¥îR>\$im_Üò AgVo (AmH¥$Vr 9.32) åhUyZ KZmH¥$VrMo

nmíd©n¥îR>\$i ho 2(h × l + b × h) qH$dm 2h (l + b) Amho.

à`ËZ H$am

1. EH$m Mm°H$boQ>r a§JmÀ`m H$mJXmÀ`m nÅ>rMm dmna H$ê$Z KZm`VmH¥$Vr S>ñQ>a (duster) À`m nmíd©n¥îR>\$imbm H$S>oZo Vo KÅ> ~g{dbo. AmVm Vmo H$mJX H$mT>m Am{U Ë`m H$mJXmMo joÌ\$i

H$mT>m. Vo Ë`m S>ñQ>aMo nmíd©n¥îR>\$i Amho H$m?

2. Vw_À`m dJ©ImobrMr bm§~r, é§Xr Am{U C§Mr _moOm Am{U emoYm.

(a) {I‹S>H$sMo Am{U XadmOmMo EHy$U joÌ\$i dJiyZ.

(b) `m ImobrMo nmíd©n¥îR>\$i H$mT>m.

(c) nm§T>am a§J XoÊ`mgmR>r ImobrMo g§nyU© joÌ\$i H$mT>m.


1. KZm`VmMo g§nyU© n¥îR>\$i = nmíd©n¥îR>\$i + (2 × nm`mMo joÌ\$i) Ago gm§Jy eH$Vmo H$m?

2. Oa EH$m KZm`VmMr bm§~r Am{U C§Mr ~Xbë`mg (AmH¥$Vr 9.33 (i)) AmUIrZ EH$ KZmH¥$Vr V`ma

hmoVo. (AmH¥$Vr 9.33 (ii)) Va òWo nmíd©n¥îR>\$i ~Xby eHo$b H$m? AmH¥$Vr 9.33

9.7.2 KZ (cube)

ho H$am

EH$m AmboI H$mJXmda Imbrbà_mUo Z_yZm aMm Am{U Ë`mbm H$mnyZ ~mhoa H$mT>m. (AmH¥$Vr 9.34

(i)) AmnUmg _m{hVM Amho H$s, `màH$maMm Z_yZm åhUOoM EH$ KZmMr OmbmH¥$Vr Amho) Ë`mÀ`m

A§Ë` aofodê$Z Ë`mbm Xþ_S>m (fold) (AmH¥$Vr 9.34 (ii)) Am{U Ë`mÀ`m H$S>m§Zm {MH$Q>nÅ>r (tape) bmdm H$s, Ë`m_wio Vmo EH$ KZ V`ma hmoB©b (AmH¥$Vr 9.34 (iii)).

AmH¥$Vr 9.32Imbrb ^mJ

darb ^mJ

joÌ_mnZ 163

AmH¥$Vr 9.32Imbrb ^mJ

darb ^mJ

AmH¥$Vr 9.34

AmH¥$Vr 9.35

(a) KZmMr bm§~r, é§Xr Am{U C§Mr {H$Vr Agob? {ZarjU H$am H$s, KZmMo gd© n¥îR>^mJ Mm¡agmÀ`m AmH$mamMo AmhoV H$m? `mdê$ZM g_OVo H$s, KZmMr bm§~r, é§Xr Am{U C§Mr g_mZ AgVo (AmH¥$Vr 9.35 (i)).

(b) àËòH$ n¥îR>^mJmMo joÌ\$i {bhm. Vo g_mZ AmhoV H$m?

(c) `m KZmMo g§nyU© joÌ\$i {bhm.

(d) Oa KZmMr àËòH$ ~mOy l Agob Va àËòH$ n¥îR>^mJmMo joÌ\$i H$m` Agob? (AmH¥$Vr 9.35 (ii)).

l ~mOy AgUmè`m KZmMo g§nyU© joÌ\$i ho 6l2 Ago AmnU åhUy eH$Vmo H$m?

à`ËZ H$am

A `m KZmMo n¥îR>\$i H$mT>m Am{U B `m KZmMo nmíd©n¥îR>\$i H$mT>m (AmH¥$Vr 9.36).

AmH¥$Vr 9.3610 cm 10 cm

8 cm

8 cm

8 cm

10 c

m

164 J{UV


(i) àËòH$s b ~mOy AgUmao XmoZ KZ EH$_oH$mZm OmoS>bo Am{U

Ë`mnmgyZ, EH$ KZm`V {Z_m©U Ho$bm (AmH¥$Vr 9.37). `m KZmMo n¥îR>\$i {H$Vr Agob? Vo 12b2 Amho H$m? VrZ KZ EH$Ì OmoSy>Z hmoUmè`m EH$m KZm`VmMo n¥îR>\$i 18b2 Amho H$m? H$maU H$m`?

AmH¥$Vr 9.37

(ii) gdm©VH$_r n¥îR>\$i AgUmam KZm`V V`ma H$aÊ`mgmR>r Vwåhr g_mZ bm§~r AgUmè`m 12 KZm§Mr H$em àH$mao aMZm H$amb?

(iii) KZmÀ`m n¥îR>\$imbm a§J {Xë`mZ§Va g_mZ _mnmMo EHy$U 64 KZ H$mnyZ ~mhoa H$mT>m (AmH¥$Vr 9.38). Ë`mn¡H$s {H$Vr n¥îR>^mJm§Zm a§J bmJbm Zmhr? Ë`mn¡H$s {H$Vr KZm§Mm 1 n¥îR>^mJ a§J{dbm Amho? 2 n¥îR>^mJ a§J{dbo AmhoV? 3 n¥îR>^mJ a§J{dbo AmhoV?

AmH¥$Vr 9.38

9.7.3 d¥Îm{MVr (cylinders)AmnU A{YH$ à_mUmV d¥Îm{MVrZm b§~ dVw©imH$ma d¥Îm{MVr_Üò {ZarjU Ho$bo Amho. CXmhaUmW©, nÍ`mMm S>~m (tin), JmobmH$ma Im§~ (round pillars), Q>çy~bmB©Q> (tubelight), nmÊ`mMo nmB©n (water pipes) BË`mXr.

joÌ_mnZ 165

ho H$am

(i) EH$ d¥Îm{MVr ho AmH$mamMm S>~m qH$dm ~m°Šg KoD$Z Ë`mÀ`m nm`mH$S>rb ~mOyImbr AmboImMm H$mJX R>odm Am{U Ë`m _mnmZo Vmo H$mJX H$mnm (AmH¥$Vr 9.39 (i)). Ë`m d¥Îm{MVrÀ`m C§MrÀ`m é§Xrer g_mZ AgUmè`m bm§~rÀ`m AmH$mamMm AmUIrZ EH$ AmboImMm H$mJX ¿`m. hmM H$mJX Ë`m S>ã`mbm Jw§S>mim (ZH$mo Agboë`m AmboImMm H$mJX H$mnyZ Q>mH$m)(AmH¥$Vr 9.39 (ii)).

d¥Îm{MVrÀ`m ñdê$nmV (AmH¥$Vr 9.39 (iv)) V`ma H$aÊ`mgmR>r Ë`m AmboIÀ`m H$mJXmÀ`m VwH$S>`mZm (AmH¥$Vr 9.39 (iii)) EH$Ì OmoS>m. S>ã`mÀ`m H$S>obm bmdboë`m AmboImÀ`m AmH$mamMm H$mJX Agob H$m?

AmH¥$Vr 9.39

åhUOoM, hm Va Am`VmH$ma Amho. Ooìhm d¥Îm{MVrMo ^mJ AmnU EH$Ì H$ê$Z Ë`mbm nÅ>rZo OmoS>bo Voìhm dVw©imMm n[aK hm Am`VmH$ma bm§~rÀ`m nÅ>rer g_mZ hmoB©b. Am`VmH$ma nÅ>rMr bm§~r (l) Am{U é§Xr (h), VgoM dVw©imH$ma nm`mMr {ÌÁ`m (r) `m§Mr Zm|X H$am. Oa 2pr = nÅ>rMr bm§~r Agob, Va Am`VmH$ma nÅ>rMo joÌ\$i 2prh Amho, Ago Vnmgm. Mm¡agmH¥$Vr H$mJXmdarb d¥Îm{MVrÀ`m ñdê$nmVrb {H$Vr Mm¡ag EH$H$ dmnaÊ`mV Ambo Vo _moOm. Va ho A§XmOo 2pr(r + h) er g_mZ hmoB©b.

(ii) d¥Îm{MVrMo g§nyU© joÌ\$i Xþgè`m ~mOyZo 2pr(r + h) hmoB©b Agm AmnU {ZîH$f© H$mT>bm. Imbrb à_mUo d¥Îm{MVrÀ`m H$mnboë`m ^mJm§Mr H$ënZm H$am (AmH¥$Vr 9.40).

AmH¥$Vr 9.40

joÌ\$i

joÌ\$i

joÌ\$i

166 J{UV

Q>rn : Oa p Mr qH$_V Z_yX Ho$br Zgob

Va AmnU π =227 KoVmo.

d¥Îm{MVrMo dH«$n¥îR>\$i 2prh Amho.

d¥Îm{MVrMo g§nyU© n¥îR>\$i = pr2 + 2prh + pr2

= 2pr2 + 2prh

= 2pr (r + h)

AmnU AmnU ho H$am

Imbrb d¥Îm{MVr§Mo g§nyU© n¥îR>\$i H$mT>m (AmH¥$Vr 9.41).

AmH¥$Vr 9.41

8 cm2 m

2 m14 cm


d¥Îm{MVrMo dH«$n¥îR>\$i ho d¥Îm{MVrÀ`m nm`mMm narK X d¥Îm{MVrMr C§Mr AgVo ho Z_yX H$am. Ë`mM à_mUo KZm`VmMo dH«$n¥îR>\$i ho Ë`mÀ`m nm`mMr n[a{_Vr X KZm`VmMr C§Mr, Ago AmnU {bhÿ eH$Vmo H$m?

CXmhaU 4 : KZm`VmH$ma AmH$mamMr EH$ _mem§Mr noQ>r Amho. {OMr _mno ~mhoê$Z 80 cm × 30cm × 40cm AmhoV. AmnUmg Ë`m noQ>rMm nm`m (Vi). ~mOy§Mo n¥îR>^mJ Am{U nmR>r_mJMm n¥îR>^mJ a§JrV H$mJXmZo PmH$m`Mm Amho. Va Amdí`H$ AgUmè`m a§JrV H$mJXmMo joÌ\$i H$mT>m.

CH$i : _memÀ`m noQ>rMr bm§~r = l = 80cm .

_memÀ`m noQ>rMr é§Xr = b = 30cm .

_memÀ`m noQ>rMr C§Mr = h = 40cm.

nm`mMo joÌ\$i = l × b = 80 × 30 = 2400 cm2.

~mOyÀ`m n¥îR>^mJmMo joÌ\$i = b × h = 30 × 40

= 1200 cm2

_mJrb ^m½mmÀ`m n¥îR>^mJmMo joÌ\$i = l × h = 80 × 30 = 3200 cm2.

Amdí`H$ joÌ\$i = nm`mMo joÌ\$i + nmR>r_mJmÀ`m n¥îR>^mJmMo joÌ\$i

joÌ_mnZ 167

+ (2 × ~mOyÀ`m n¥îR>^mJmMo joÌ\$i)

= 2400 + 3200 + (2 × 1200)

= 8000 cm2.

åhUyZ, Amdí`H$ AgUmè`m a§JrV H$mJXmMo joÌ\$i 8000cm2.

CXmhaU 5 : EH$m KZm`VmÀ`m ImobrMr AmVrb _mno 12m × 8m × 4m AmhoV. Oa a§JH$m_ H$aÊ`mMm IM© ` 5 Xa m2 Agob Va ImobrÀ`m Mmahr ~mOy a§J{dÊ`mgmR>r òUmam IM© {H$Vr? VgoM Ë`m ImobrÀ`m Mma ~mOy gmo~V ImobrÀ`m N>VmbmgwÜXm a§JH$m_ H$aÊ`mMm EHy$U IM© H$mT>m.

CH$b : ImobrMr bm§~r = l = 12m.

ImobrMr é§Xr = b = 8m.

ImobrMr C§Mr = h = 4m.

ImobrÀ`m Mmahr q^VrMo joÌ\$i = nm`mMr n[a_rVr × ImobrMr C§Mr

= 2 (l + b) × h = 2 (12 + 8) × 4

= 2 × 20 × 4 = 160 m2.

Xa m2 gmR>r a§JH$m_ H$aÊ`mMm IM© = ` 5

åhUyZ, Ë`m ImobrÀ`m Mma hr q^VrZm a§JH$m_ H$aÊ`mMm EHy$U IM© = ` (160 × 5)

= ` 800.

N>VmMo joÌ\$i = 12 × 8 = 96 m2

N>Vmbm a§JH$m_ H$aÊ`mMm EHy$U IM© = ` (96 × 5) = ` 480.

åhUyZ Imobrbm a§JH$m_ H$aÊ`mMm EHy$U IM© = ` (800 + 480) = ` 1280.

CXmhaU 6 : EH$m B_maVr_Üò d¥Îm{MVr AmH$mamMo EHy$U 24 Im§~ AmhoV. àËòH$ Im§~m§Mr C§Mr 4m Am{U {ÌÁ`m 28 cm Amho. Oa Im§~m§Zm a§JH$m_ H$amd`mMm Agob Va ` 8 Xa m2 à_mUo IM© òV Agob Va gd© Im§~m§À`m dH«$n¥îR>\$imbm a§J bmdÊ`mMm EHy$U {H$Vr IM© òB©b?

CH$b : d¥Îm{MVrÀ`m Im§~mMr {ÌÁ`m r = 28 cm

= 0.28 m.

C§Mr h = 4 m

d¥Îm{MVrMo dH«$n¥îR>\$i = 2prh

168 J{UV

Im§~mMo dH«$n¥îR>\$i = 2 227

0 28 4 7 04× × × =. . m2.

24 Im§~m§Mo dH«$n¥îR>\$i = 7.04 × 24 = 168.96 m2.

1 m2 joÌ\$imbm a§JH$m_ H$aÊ`mgmR>r òUmam IM© = ` 8

åhUyZ, a§JH$m_mMm EHy$U IM© = 168.96 × 8 = ` 1351.68

CXmhaU 7 : Oa d¥Îm{MVrMo EHy$U n¥îR>\$i 968 cm2 Am{U Ë`mMr {ÌÁ`m 7 cm Agob Va Ë`m d¥Îm{MVrMr C§Mr H$mT>m.

CH$b : d¥Îm{MVrMr C§Mr = h ; {ÌÁ`m = r = 7cm EHy$U n¥îR>\$i = 2pr (h + r)

åhUyZ, 2 227

7 7 968× × × +( ) =h

h = 15 cm Va d¥Îm{MVrMr C§Mr h = 15 cm Amho.

ñdmÜ`m` 9.3

1. ~mOyÀ`m AmH¥$Vr_Üò Xe©{dë`mà_mUo XmoZ KZm`VmH$ma noQ>`m AmhoV. mn¡H$s H$moUË`m noQ>rbm V`ma H$aÊ`mgmR>r H$_r à_mUmV gm{hË` bmJob?

2. 80 cm × 48 cm × 24 cm. _mnmÀ`m gyQ>Ho$gda OmS>H$mnS>r AmdaU Kmbmd`mMo Amho. Va Aem 100 gyQ>Ho$gZm AmdaU KmbÊ`mgmR>r 96 cm é§Xr AgUmao {H$Vr _rQ>a H$mnS> bmJob?

3. 600 cm2 nmíd©n¥îR>\$i AgUmè`m KZmMr ~mOy H$mT>m.

4. éŠgmaZo 1 m × 2 m × 1.5 m _mnmÀ`m H$nmQ>mbm ~mhoê$Z a§J XoÊ`mMo R>a{dbo. Va H$nmQ>mMm Vi^mJ gmoSy>Z {VZo H$nmQ>mÀ`m EHy$U a§J{dboë`m ^mJmMo n¥îR>\$i {H$Vr? 2m1m

1.5m

5. S>°{Z`b KZm`VmH$maÀ`m 15m, 10m Am{U 7m AZwH«$_o bm§~r, é§Xr Am{U C§Mr AgUmè`m {^§VrZm Am{U N>Vmbm a§J XoV Amho. Oa 100m2 joÌ\$i OmJm a§J{dÊ`mg EH$ S>~m a§J bmJV Agob, Va Ë`m nyU© Imobrbm a§J XoÊ`mgmR>r EHy$U {H$Vr S>ã`m§Mr Amdí`H$Vm ^mgob?

joÌ_mnZ 169

6. {Xboë`m XmoZ KZmH¥$VrVrb gmå` Am{U \$aH$ `m§Mo {díbofU H$am. H$moUË`m KZmH¥$VrMo nmíd©n¥îR>\$i A{YH$ Amho.

7 cm 7 cm

7 cm

7 cm

7 cm

7. {ÌÁ`m 7m Am{U C§Mr 3m AgUmar YmVyÀ`m nÍ`mMr Q>mH$s ~Z{dÊ`mg EHy$U {H$Vr _mnmÀ`m nÍ`mMr Amdí`H$Vm Agob?

8. EH$m nmoH$i d¥Îm{MÎmrMo nmíd©n¥îR>\$i 4224cm2 Amho. Ë`mMm EH$ VwH$S>m Ë`m d¥Îm{MVrMr C§Mr Yê$Z H$mnbm d 33cm é§Xr Agbobm EH$ Am`V V`ma Ho$bm. Va Ë`m Am`VmH$ma VwH$S>`mMr n[a{_Vr H$mT>m.

9. EH$ añVm gnmQ>rH$aU H$aÊ`mgmR>r amoS>amobaÀ`m (Road Roller) MmH$mMo 750 nyU© \o$ao bmJVmV. Oa amoS>amobaÀ`m MmH$mMm ì`mg 84cm Am{U bm§~r 1m Agob, Va Ë`m añË`mMo joÌ\$i H$mT>m.

10. XþYmMr nmdS>a V`ma H$aUmè`m EH$m H§$nZrZo 14cm ì`mg AgUmam Am{U 20cm C§Mr AgUmè`m d¥Îm{MVrÀ`m S>ã`m_Üò XÿY nmdS>a aUm Ho$br. H§$nZrZo Ë`m S>ã`mÀ`m g^modVr bo~b (label) {MH$Q>{dbm Amho (AmH¥$VrV Xe©{dë`mà_mUo). Oa bo~b da Am{U Imbr 2cm ^mJ OmJm gmoSy>Z _Ymo_Y bmdbm Agob, Va Ë`m bo~bMo EHy$U joÌ\$i {H$Vr?

20 cm

2 cm

14 cm

XþYmMr nmdS>a

9.8 KZ, KZm`V Am{U d¥Îm{MVr `m§Mo KZ\$i (volume of cube, cuboid and cylinder)

{Ì{_Vr` dñVyZo ì`mnboë`m OmJoÀ`m amerbm Ë`mMo KZ\$i åhUVmV. Vw_À`m AmOy~mOybm AgUmè`m dñVy§À`m KZ\$imMr VwbZm H$aÊ`mMm à`ËZ H$am. CXmhaUmW© : Imobr_Üò Agboë`m H$nmQ>mÀ`m KZ\$imnojm ImobrMo KZ\$i A{YH$ AgVo. Ë`mMà_mUo noÝgrbÀ`m noQ>rMo KZ\$i ho Ë`m_Yrb Agboë`m a~a Am{U noZmÀ`m KZ\$imÀ`mnojm A{YH$ AgVo. Va Aem dñVy§Mo qH$dm Ë`m_Yrb dñVyMo KZ\$i Vwåhr H$mTy> eH$Vm H$m?

170 J{UV

bjmV R>odm H$s, EImÚm joÌmMo (OmJoMo) joÌ\$i H$mT>Ê`mgmR>r Mm¡ag EH$H$mMm dmna H$aVmo. Va KZmH¥$VrMo KZ\$i ho KZ EH$H$mMm dmnê$ H$ê$Z gm§JVmo. H$maU ho ì`ŠV H$aÊ`mgmR>r KZmH¥$Vr hm `mo½` AmH$ma Amho. (Ogo Mm¡ag hm OmJoMo joÌ\$i _mnZmMm `mo½` AmH$ma Amho.)

Ooìhm EImÚm joÌ\$i H$mT>m`Mo Agob, Voìhm OmJoMo Mm¡ag EH$H$_Üò Amåhr ^mJ H$aVmo.VgoM, EH$mÚm KZmH¥$VrMo KZ\$i H$mT>Ê`mg {VMo KZ EH$H$m_Üò ^mJ H$aUo Amdí`H$ AgVo.

{ZarjU H$am H$s, ~mOybm AgUmè`m àËòH$ dñVyMo KZ\$i 8 KZ EH$H$ Amho. (AmH¥$Vr 9.42)

Amåhr Ago gm§Jy eH$Vmo H$s, {Xboë`m dñVyÀ`m KZ\$imMo _mnZ H$mT>Ê`mgmR>r Ë`m dñVy_Yrb KZmMr g§»`m AmoiIUo KZ\$imMo _mnZ Ho$ìhmhr KZ EH$H$m _ÜòM _moOVmV Vo åhUOo.

AmH¥${V 9.42

1 KZ cm = 1 cm × 1 cm × 1 cm = 1 cm3

= 10 mm × 10 mm × 10 mm = ........... mm3

1 KZ m = 1 m × 1 m × 1 m = 1 m3 1 KZ mm = 1 mm. × 1 mm. × 1 mm. = ........... mm3

= 0.1 cm × 0.1 cm × 0.1 cm = ........... cm3

AmVm AmnU d¥Îm{MVr, KZ Am{U KZm`VmMo KZ\$i H$mT>Ê`mMr gyÌo nmhÿ. Ë`mgmR>r EH$mZ§Va EH$ dñVy KoD$.

9.8.1 KZm`V (cuboid)EH$mM g_mZ AmH$mamMo 36 KZ ¿`m. (åhUOoM, àËòH$ KZmMr bm§~r g_mZ Amho) Ë`m§Mr

KZm`VmÀ`m ñdê$nmV _m§S>Ur H$am. Vwåhr Ë`m§Mr AZoH$ àH$mam_Üò _m§S>Ur H$ê$ eH$Vm. Imbrb VŠVm nhm Am{U _moH$ù`m OmJm âm.

KZm`V bm§~r l é§Xr b C§Mr h l × b × h = V

(i) 12 EH$H$ 3 EH$H$ 12 3 1 12 × 3 × 1 = 36

(ii)

joÌ_mnZ 171

(iii)

(iv)

Vwåhr `m§Mo H$m` {ZarjU Ho$bo?

KZm`VmMo ñdê$n {Z_m©U hmoÊ`mgmR>r AmnU 36 KZmMm dmna Ho$bm Am{U Ë`mVrb àËòH$ KZm`VmMo KZ\$i ho bm§~r, é§Xr Am{U C§Mr `m§À`m JwUmH$mamer g_mZ Pmbo. darb CXmhaUmdê$Z, KZm`VmMo KZ\$i = l × b × h Ago AmnU gm§Jy eH$Vmo. AmnU Ago hr gm§Jy eH$Vmo H$s, KZm`VmÀ`m nm`mMo

joÌ\$i = l × b Amho. \ KZm`VmMo KZ\$i = nm`mMo joÌ\$i × C§Mr

AmH¥${V 9.43

ho H$am

EH$ H$mJXmMm VwH$S>m ¿`m. Ë`mMo joÌ\$i _moOm. AemM EH$m H$mJXmÀ`m {T>Jmè`mVyZ KZm`VmÀ`m AmH$mamMo H$mhr g_mZ AmH$mamÀ`m H$mJXmMo {ZdS> H$am (AmH¥$Vr 9.43). `m {T>Jmè`mMr C§Mr _moOm. KZm`VmMo KZ\$i H$mT>VmZm àW_ Ë`m H$mJXmÀ`m WamMr C§Mr Am{U VwH$S>çmMr joÌ\$i `m§Mm JwUmH$ma H$mT>m.

Aem àH$maMr nÜXV KZnXmWm©Mo KZ\$i H$mT>Ê`mMo {ZîH$f© `m {Xboë`m H¥$VrZo ñnîQ> H$aÊ`mMr H$ënZm {_iVo. (Ooìhm KZnXmWm©Mm nm`m Am{U darb ~mOy EH$ê$n AgVo. EH$_oH$m§er g_m§Va AgVmV Am{U Ë`mÀ`m H$S>m nm`mda b§~ AgVmV). `mM nÜXVrZo H$mhr dñVy§Mo KZ\$i H$mT>Vm òVo. Agm {dMma H$ê$ eH$mb H$m?

AmH¥$Vr 9.43

à`ËZ H$am

Imbrb KZm`VmMo KZ\$i H$mT>m.(i)

AmH¥$Vr 9.44

2 cm

3 cm

3 cm24 m8 cm 2

172 J{UV

9.8.2 KZ (cube)KZ hm EH$ KZm`VmMm EH$ _w»` KQ>H$ Amho OoWo l = b = h åhUyZ.

KZmMo KZ\$i l × l × l = l3

à`ËZ H$am

Imbrb KZmMo KZ\$i H$mT>m (i) ~mOy 4 cm AgVmZm. (ii) ~mOy 1.5 m AgVmZm.

ho H$am

EH$m KZm`VmMo ñdê$n {Z_m©U H$aÊ`mgmR>r EHy$U 64 KZmMm Cn`moJ H$ê$Z aMZm H$am. àËòH$ _m§S>boë`m aMZoMo dH«$n¥îR>\$i H$mT>m. {Z_m©U Pmboë`m KZ AmH$mamÀ`m dñVyMo KZ\$i Am{U dH«$n¥îR>\$i g_mZ Agy eHo$b H$m?


EH$ H§$nZr {~ñH$sQ>m§Mr {dH«$s H$aVo. Ë`m§Zm AmdaU KmbÊ`mgmR>r KZm`VmÀ`m AmH$mamMo ~m°Šg dmnaVmV. Ë`m_Üò A ~m°Šg → 3 cm × 8 cm × 20 cm, d B ~m°Šg →4 cm × 12 cm × 10 cm AmH$mamMo AmhoV. H§$nZrÀ`m Am{W©H$ÑîQ>rZo H$moUË`m ~m°ŠgMm AmH$ma `mo½` Agob? H$m? Ë`m H§$nZrÀ`m Am{W©H$ÑîQ>rZo ~m°ŠgMm BVa AmH$ma Ë`m§À`m KZ\$imÀ`m g_mZ hmoD$ eHo$b Agm gëbm Vwåhr H§$nZrbm XoD$ eH$mb H$m?

9.8.3 d¥Îm{MVr (cylinder)

AmnUmg _mhrVM Amho H$s, KZm`VmMo KZ\$i H$mT>VmZm Ë`mÀ`m nm`mMo joÌ\$i (VimMo joÌ\$i) Am{U C§Mr `m§Mm JwUmH$ma H$mT>Uo Amdí`H$ Amho. AemM àH$mao AmnU d¥Îm{MVrMo KZ\$i H$mTy> eH$Vmo H$m?KZm`VmgmaIoM AgUmao d¥Îm{MVrMm _mWm Am{U nm`m EH$ê$n Am{U EH$_oH$m§bm g_m§Va AgVmo. KZm`Vm à_mUo d¥Îm{MVrMo dH«$n¥îR>\$i Ë`mÀ`m nm`mer b§~ AgVo.åhUyZ, KZm`VmMo KZ\$i = nm`mMo joÌ\$i × C§Mr = l × b × h = lbhd¥Îm{MVrMo KZ\$i = nm`mMo joÌ\$i × C§Mr = pr2 × h = pr2 h

KZm`V d¥Îm{MVr

nm`mMo joÌ\$i

joÌ_mnZ 173

à`ËZ H$am

Imbrb d¥Îm{MVtMo KZ\$i H$mT>m

(i)

10 cm

7 cm(ii)

250 m

2 m

2

9.9 KZ\$i Am{U j_Vm (volume and capacity)

`m XmoZ eãXm_Üò A{YH$ Agm \$aH$ AmT>iyZ òV Zmhr.

(a) nXmWm©Zo EHy$U ì`mnbobr OmJm åhUOoM KZ\$i hmo`.

(b) nmÌmV Agboë`m dñVyMo EHy$U à_mU åhUOoM j_Vm hmo`.

{Q>n : EH$m nÍ`mÀ`m S>ã`m_Üò 100 cm3 nmUr gm_mdV Agob Va nmÊ`mMr j_Vm 100 cm3 Amho.

j_Vm hr {bQ>a_Üò gwÜXm _moObr OmVo. {bQ>a Am{U cm3 Mm g§~§Y åhUOoM 1 mL = 1 cm3,

1L = 1000 cm3

åhUyZ 1 m3 = 1000000 cm3 = 1000L

CXmhaU 8 : EH$m KZm`VmMo KZ\$i 275 cm3 Amho Am{U Ë`mÀ`m nm`mMo joÌ\$i 25 cm2

Agob Va Ë`m KZm`VmMr C§Mr H$mT>m.

CH$b : KZm`VmMo KZ\$i = nm`mMo joÌ\$i × C§Mr

Va KZm`VmMr C§Mr = KZm`VmMo KZ\$i

nm`mMo joÌ\$i

27525

= 11cm

KZm`VmMr C§Mr = 11cm

CXmhaU 9 : EH$ JmoXm_ (godown) KZm`VmÀ`m AmH$mam_Üò AgyZ Ë`mMr _mno 60m × 40m × 30m AmhoV. EH$m bhmZ KZm`VmH$ma noQ>rMo KZ\$i 0.8 m3 Amho. Va Aem EHy$U {H$Vr noQ>çm Ë`m JmoXm_m_Üò _mdVrb?

174 J{UV

CH$b : EH$m ~m°ŠgMo KZ\$i = 0.8 m3

JmoXm_mMo KZ\$i = 60 × 40 × 30 = 72000m3

JmoXm_m_Üò g§J«hrV Ho$bobo EHy$U ~m°Šg = JmoXm_mMo KZ\$i

EH$m ~m°ŠgMo KZ\$i

= 60 × 40 × 30 0.8

= 90,000

åhUyZ, Ë`m JmoXm_m_Üò EHy$U g§J«hrV Ho$bobr KZm`VmÀ`m ~m°ŠgMr g§»`m 90,000 Amho.

CXmhaU 10 : 14 cm é§Xr AgUmè`m EH$m Am`VmH$ma H$mJXmbm Ë`mÀ`m é§XrH$Sy>Z Jw§S>mibo Am{U 20 cm {ÌÁ`m AgUmar EH$ d¥Îm{MVr V`ma Pmbr. Va Ë`m d¥Îm{MVrMo KZ\$i H$mT>m. (AmH¥$Vr 9.45). (p Mr qH$_V π =

227

¿`m)

CH$b : Am`Vmbm Ë`mÀ`m é§XrH$Sy>Z Jw§S>miyZ EH$ d¥Îm{MVr V`ma Ho$br. Ë`m H$mJXmMr é§Xr hr d¥Îm{MVrMr C§Mr Amho åhUyZ d¥Îm{MVrMr {ÌÁ`m 20 cm Amho. .

14 cm

20 cm

14 cm

AmH¥${V 9.45 d¥Îm{MVrMr C§Mr h = 14 cm

{ÌÁ`m = r = 20 cm

d¥Îm{MVrMo KZ\$i V = pr2h

22 20 20 147

× × × = 17600 cm3

åhUyZ, d¥Îm{MVrMo KZ\$i ho 17600 cm3 Amho.

CXmhaU 11 : 11 cm × 4 cm Mm EH$ Am`VmH$ma H$mJX Xþ_Sy>Z Ë`mMo A{Vì`mnZ (overlapping) Z H$aVm 4 cm C§Mr AgUmam EH$ d¥Îm{MVr V`ma hmoVo. Va Ë`m d¥Îm{MVrMo KZ\$i H$mT>m.CH$b : H$mJXmMr bm§~r hr Ë`m d¥Îm{MVrÀ`m nm`mMm n[aK Amho Am{U H$mJXmMr é§Xr hr

d¥Îm{MVrMr C§Mr Amho.

joÌ_mnZ 175

AmVm, d¥Îm{MVrMr {ÌÁ`m = r Am{U C§Mr = h

d¥Îm{MVrÀ`m nm`mMm n[aK = 2pr = 11

2 227

11× × =r

∴r = 74

cm

åhUyZ, d¥Îm{MVrMo KZ\$i V = pr2h

= × × ×227

74

74

4 cm3 = 38.5 cm3

åhUyZ, d¥Îm{MVrMo KZ\$i 38.5 cm3 Amho.

ñdmÜ`m` 9.4

1. EH$ d¥Îm{MVr AmH$mamMr Q>mH$s Amho. Va H$moUË`m pñWVrV (doir) Ë`m d¥Îm{MVrMo n¥îR>\$i Am{U H$moUË`m pñWVrV {VMo KZ\$i emoYyZ H$mT>mb.

(a) {VMr YmaU j_Vm OmUyZ KoÊ`mgmR>r. (b) {Vbm {Jbmdm H$aÊ`mgmR>r bmJUmè`m {g_|Q>À`m nmoË`mMr

g§»`m H$mT>Ê`mgmR>r. (c) {VÀ`mVrb nmUr ^ê$Z H$mT>Ê`mgmR>r bmJUmè`m bhmZmV

bhmZ Q>mH$s¨Mr g§»`m H$mT>Ê`mgmR>r.

2. A `m d¥Îm{MVrMm ì`mg 7 cm Am{U C§Mr 14 cm Amho. B m d¥Îm{MVrMm ì`mg 14 cm Am{U C§Mr 7 cm Amho. Va H$moUVo hr J{UV Z gmoS>{dVm H$moUË`m d¥Îm{MVrMo KZ\$i A{YH$ Amho `mMm {dMma H$ê$ eH$mb H$m? `m d¥Îm{MVrMo KZ\$i H$mTy>Z Vo nS>Vmim. H$moUË`m d¥Îm{MVrMo KZ\$i A{YH$ VgoM n¥îR>\$i A{YH$ Amho Vo Vnmgm.

7 cm

14 cm

14 cm

7 cm

3. EH$m KZm`VmMo KZ\$i 900 cm3 Amho. Oa Ë`mÀ`m nm`mMo joÌ\$i 180 cm2 Agob Va

C§Mr H$mT>m.

4. EH$m KZm`VmMr _mno 60 cm × 54 cm × 30 cm AmhoV. {Xboë`m Ë`m KZm`Vm_Üò

6 cm. Mr ~mOy AgUmao EHy$U {H$Vr bhmZ KZ gm_mdbo OmVrb?

176 J{UV

5. EH$m d¥Îm{MVrMo KZ\$i 1.54 m3 Am{U nm`mMm ì`mg 140 cm Agob Va Ë`m d¥Îm{MVrMr

C§Mr H$mT>m.

6. 7 m bm§~r Am{U 1.5 m {ÌÁ`m AgUmar EH$ XþYmMr Q>mH$s d¥Îm{MVr AmH$mamV Amho. Ë`m nyU© Q>mH$s_Üò EHy$U {H$Vr {bQ>a XÿY gm_mdbo OmB©b?

7. Oa EH$m KZmMr àËòH$ H$S>m XþßnQ> Ho$br Va

(i) {H$Vr nQ>rZo Ë`mMo n¥îR>\$i dmT>ob? (ii) {H$Vr nQ>rZo Ë`mMo KZ\$i dmT>ob?

8. Xa {_{ZQ>mbm 60 {bQ>a à_mUo EH$m nmUr g§J«h H$aÊ`mÀ`m Q>mH$sV nmUr gmoS>bo Amho. Ë`m Q>mH$sMo KZ\$i 108 m3 Amho. Va Vr Q>mH$s âÊ`mg EHy$U {H$Vr Vmg bmJVrb?


1. joÌ\$i

(i) g_b§~ Mm¡H$moZ = g_m§Va ~mOy§À`m ~oaOoÀ`m {Zå_o × Ë`m§À`m_Yrb b§~mVa

(ii) g_ wO Mm¡H$moZ = H$UmªÀ`m JwUmH$mamÀ`m {Zå_o

2. KZmH¥$VrMo n¥îR>\$i åhUOo gd© n¥îR>^mJm§À`m joÌ\$im§Mr ~oarO.

3. KZm`VMo n¥îR>\$i : 2(lb + bh + hl)

KZmMo n¥îR>\$i = 6l2

d¥Îm{MVrMo = 2pr(r + h)

4. KZmH¥$VmZo ì`mnbobo EHy$U joÌ åhUOoM KZ\$i hmo`

5. KZ\$i

KZm`V = l × b × h

KZ = l3

d¥Îm{MVr : pr2h

6. (i) 1cm3 = 1 mL

(ii) 1L = 1000 cm3.

(iii) 1m3 = 1000000 cm3 = 1000L

KZ Am{U KZ_yi

ñdmÜ`m` 1.1

1. (ii) Am{U (iv)

2. (i) 3 (ii) 2 (iii) 3 (iv) 5 (v) 10

3. (i) 3 (ii) 2 (iii) 5 (iv) 3 (v) 11

4. 20 KZm`V

ñdmÜ`m` 1.2

1. (i) 4 (ii) 8 (iii) 22 (iv) 30 (v) 25

(vi) 24 (vii) 48 (viii) 36 (ix) 56 (x) 45

2. (i) MyH$ (ii) ~amo~a (iii) MyH$ (iv) MyH$ (v) MyH$

(vi) MyH$ (vii) ~amo~a

3. 11,17,23, 32.

KmVm§H$ Am{U KmV

ñdmÜ`m` 2.1

1. (i) 19 (ii)

116 (iii) 32

2. (i) ( )

3

1

4− (ii) 6

1

2 (iii) (5)4 (iv)

( )2

13

(v) ( )3

114−

3. (i) 5 (ii) 12 (iii) 29 (iv) 1 (v) 81

16

4. (i) 250 (ii) 160

CÎmao

178 J{UV

5. m = 2 6. (i) – 1 (ii) 512125

7. (i) 4625t

2 (ii) 55

ñdmÜ`m` 2.2

1. (i) 8.5 × 10–12 (ii) 9.42 × 10–12 (iii) 6.02 × 1015

(iv) 8.37 × 10–9 (v) 3.186 × 1010

2. (i) 0.00000302 (ii) 45000 (iii) 0.00000003

(iv) 1000100000 (v) 5800000000000 (vi) 3614920

3. (i) 1 × 10–6 (ii) 1.6 × 10–19 (iii) 5 ×10–7

(iv) 1.275 × 10–5 (v) 7 × 10–2

4. 1.0008 × 102

_m{hVr hmVmiUo

ñdmÜ`m` 3.1

1. (b), (d). `m gd© KQ>Zm_Üò _m{hVr hr g§^mJloUr_Üò {d^mJy eH$Vo.

2.

XþH$mZXma JUVr IwUm g§»`m

W 28

M 15

B 5

G 12

CÎmao 179

3.

loUr JUVr IwUm dma§dmaVm

800-810 3

810-820 2

820-830 1

830-840 9

840-850 5

850-860 1

860-870 3

870-880 1

880-890 1

890-900 4

EHy$U 30

4. (i) 830 - 840

(ii) 10

(iii) 20

5. (i) 4-5 Vmg

(ii) 34

(iii) 14

nJma (ê$. _Üò)→

H$m_

Jmam§M

r g§»

`m→

180 J{UV

ñdmÜ`m` 3.2

1. (i) 200 (ii) bKw g§JrV

(iii) emñÌr` - 100, bKw emñÌr` - 200, bKw - 400, OmZnX- 300

2. (i) {hdmim (ii) {hdmim - 150°, nmdgmim- 120°, CÝhmim - 90°

(iii)

CÝhmim90°nmdgmim 120°

{hdmim 150°

3.

{ndim30°

bmb 60° {Zim

180°{hadm 90°

4. (i) qhXr (ii) 30 JwU (iii) hmo`

5.

H$ÝZS>20°

qhXr 200°

V{_i 35°

_amR>r 45°

B§J«Or 60°

ñdmÜ`m` 3.3

1. (a) {ZînÎmr → A, B, C, D (b) HT, HH, TH, TT (òWo HT åhUOo n{hë`m ZmÊ`mMr N>mn ~mOy da Am{U Xþgè`m

ZmÊ`mMr H$mQ>m ~mOy da.)

2. EH$m KQ>ZoÀ`m {_iÊ`mÀ`m {ZînÎmr.

(i) (a) 2, 3, 5 (b) 1,4,6

(ii) (a) 6 (b) 1,2,3,4,5

3. (a) 15

(b) 113

(c) 47

CÎmao 181

4. (i) 110

(ii) 12

(iii) 25

(iv)9

10

5. {hadm ^mJ {_iÊ`mMr g§^mì`Vm = 35

{Zim ^mJ Zgbobo ^mJ {_iÊ`mMr g§^mì`Vm = 45

6. A{d^mÁ` g§»`m {_iÊ`mMr g§^mì`Vm = 12

A{d^mÁ` g§»`m ZgUmè`m g§»`m {_iÊ`mMr g§^mì`Vm = 12

5 nojr _moR>r g§»`m {_iÊ`mMr g§^mì`Vm = 16

5 nojm _moR>r g§»`m Z {_iÊ`mMr g§^mì`Vm = 56

g_ Am{U ì`ñVà_mU

ñdmÜ`m` 4.1

1. Zmhr

2. bmb a§JmMo ^mJ 1 4 7 12 20

_yi {_lUmMm ^mJ 8 32 56 96 160

3. 24 ^mJ 4. 700 ~mQ>ë`m

5. 10-4 cm ; 2 cm 6. 21 m

7. 2.25 × 107 H«°$ñQ>b, 5.4 × 108 H«°$ñQ>b 8. 4 cm.

9. (i) 6 m. (ii) 8 m. (iii) 75 cm. .

10. 168 km.

ñdmÜ`m` 4.2

1. (i) (iv) (v) 2. 4→ 25,000; 5→20,000; 8→12,500; 10→ 10,000

20 → 5,000 {dOoË`mbm {Xbobr hr a¸$_ {dOoË`m§Mm g§»òÀ`m ì`ñVà_mUmV Amho.

182 J{UV

3. 8 → 450 10 → 360 12 → 300 (i) hmo` (ii) 240 (iii) 9

4. 6 5. 4 6. 3 {Xdg 7. 15 noQ>çm

8. 49 `§Ìo 9. 112 Vmg

10. (i) 6 {Xdg (ii) 6 ì`ŠVr

11. 40 {_{ZQ>o

AmboIm§Mm n[aM`

ñdmÜ`m` 5.1

1. (a) 36.5° C (b) 12 _Ü`mÝh (c) 1 p.m, 2 p.m

(d) 36.5° C. x- Ajmda 1 p.m. Am{U 2 p.m. _Yrb q~Xÿ hm 1 p.m. Am{U 2 p.m. Xe©{dUmè`m XmoZ q~XÿnmgyZ g_A§Vamda Amho. åhUyZ Vmo 1.30 p.m. Zo Xe©{dVm òVmo. Ë`mMà_mUo y -Ajmdarb 36°C d 37°C _Yrb q~Xÿ 36.5°C Xe©{dVmo.

(e) 9a.m. Vo 10a.m.; 10a.m. Vo 11a.m. , 2 p.m. VoÀ 3p.m.

2. (a) (i) ` 4 H$moQ>r (ii) ` 8 H$moQ>r

(b) (i) ` 7 H$moQ>r (ii) ` 8.5 H$moQ>r (A§XmOo)

(c) ` 4 H$moQ>r (d) 2005

3. (a) (i) 7cm (ii) 9 cm

(b) (i) 7 cm (ii) 10 cm

(c) 2 cm (d) 3cm

(e) Xþgam AmR>dS>m (f) n{hbm AmR>dS>m

(g) Xþgè`m AmR>dS>çmÀ`m eodQ>r.

4. (a) _§Jidma,ewH«$dma, a{ddma (b) 35°c (c) 15°c (d) Jwédma

6. (a) 4 EH$H$ = 1 Vmg

(b) Vmg (c) 22 km

(d) hmo`; ho AmboImÀ`m (10am - 10.30am) AmS>ì`m ^mJmZo Xe©{dbo OmVo.

(e) 8am Am{U 9 am _Üò

7. (iii) ho eŠ` Zmhr.

CÎmao 183

ñdmÜ`m` 5.2

1. (a) Am{U (b) _Üò AgUmao q~Xÿ EH$m aofoV òVmV. (c) _Yrb q~Xÿ EH$m aofoda òV ZmhrV.

2. aofm x - Ajmbm (5,0) Am{U y - Ajmbm (0, 5) _Üò N>oXVo.

3. 0(0, 0), A(2, 0), B (2, 3), C (0, 3), P (4, 3), Q (6, 1), R (6, 5), S (4, 7), K (10, 5), L (7, 7), M (10, 8).

4. (i) ~amo~a (ii) MyH$ (iii) ~amo~a

ñdmÜ`m` 5.3

1. (b) (i) 20 km (ii) 7.30 a.m

(c) (i) hmo` (ii) ` 200 (iii) ` 3500

2. (a) hmo` (ii) Zmhr

Ad`drH$aU

ñdmÜ`m` 6.1

1. (i) 12 (ii) 2y (iii) 14pq (iv) 1 (v) 6ab

(vi) 4x (vii) 10 (viii) x2y2

2. (i) 7(x – 6) (ii) 6(p – 2q)

(iii) 7a(a + 2) (iv) 4z (–4 + 5z2)

(v) 10lm(2l + 3a) (vi) 5xy(x – 3y)

(vii) 5(2a2 – 3b2 + 4c2) (viii) 4a(–a + b + c)

(ix) xyz (x + y +z ) (x) xy(ax + by + cz)

3. (i) ( x + 8) (x + y) (ii) (3x + 1) (5y – 2) (iii) (a + b) (x – y)

(iv) (5p + 3) (3q + 5) (v) (z – 7) (1 – xy)

184 J{UV

ñdmÜ`m` 6.2

1. (i) (a +4)2 (ii) (p – 5)2 (iii) (5m + 3)2

(iv) (7y + 6z)2 (v) 4(x –1)2 (vi) (11b – 4 c)2

(vii) (l – m)2 (viii) (a2 + b2)2

2. (i) (2p – 3q) (2p + 3q) (ii) 7(3a – 4b) (3a + 4b)

(iii) (7x – 6)(7x+6) (iv) 16x3(x – 3) (x + 3)

(v) 4lm (vi) (3xy – 4) (3x + 4)

(vii) (x – y – z)(x – y + z) (viii) (5a – 2b + 7c) (5a + 2b – 7c)

3. (i) x(ax + b) (ii) 7(p2 + 3q2)

(iii) 2x(x2 + y2 + z2) (iv) (m2 + n2) (a + b)

(v) (l + 1)(m + 1) (vi) (y + 9) (y + z)

4. (i) (a – b) (a + b) (a2 + b2) (ii) (p – 3)(p + 3) (p2 + 9)

(iii) (x – y – z)(x + y + z)[x2 + (y + z)2] (iv) z(2x – z) (2x2 – 2xz + z2)

(v) (a – b)2 (a + b)2

5. (i) (p + 2) (p + 4) (ii) (q – 3) (q – 7) (iii) (p + 8) (p –2)

ñdmÜ`m` 6.3

1. (i) 3

2x (ii) – 4y (iii) 6pqr

(iv) ( )1 5 63

x − (v) –2a2b2

2. (i) ( )1 5 63

x − (ii) 3y4 – 4y2 + 5 (iii) 2 (x + y + z)

(iv) ( )21 2 32

x x+ + (v) q3 – p3

3. (i) 2x – 5 (ii) 5 (iii) 6y (iv) xy (v) 10abc

CÎmao 185

4. (i) 5(3x + 5) (ii) 2y(x + 5) (iii) ( )12

r p + q t

(iv) 4(y2 + 5y + 3) (v) (x + 2)(x + 3)

5. (i) y + 2 (ii) m – 16 (iii) 5(p – 4)

(iv) 2z (z – 2) (v) ( )52

q p q−

(vi) 3(3x – 4y) (vii) 3y (5y –7)

ñdmÜ`m` 6.4

1. (i) 4 (x – 5)= 4x – 20 2. x (3x + 2) = 3x2 + 2x

3. 2x + 3y = 2x + 3y 4. x + 2x + 3x = 6x

5. 5y + 2y + y – 7y – xy 6. 3x + 2x = 5x

7. (2x)2 + 4(2x) + 7 = 4x2 + 8x + 7 8. (2x)2 + 5x = 4x2 + 5x

9. (3x + 2)2 = 9x2 + 12x + 4

10. (a) (–3)2 + 5(–3) + 4 = 9 – 15 + 4 = – 2

(b) (–3)2 – 5(–3) + 4 = 9 + 15 + 4 = 28

(c) (–3)2 + 5(–3) = 9 – 15 = –6

11. (y – 3)2 = y2–6y + 9 12. (z + 5)2 = z2 + 10z + 25

13. (2a + 3b)(a – b) = 2a2 + ab – 3b2 14. (a + 4) (a + 2) = a2 + 6a + 8

15. (a – 4)(a – 2) = a2 –6a + 8 16. 2

3 13

xx

2

=

17. 2 2

2 2 2 2

3 1 3 1 113 3 3 3

+ = + = +x xx x x x

18. 3 33 2 3 2

=+ +x x

x x

19. 3 34 3 4 3

=+ +x x

20. 4 5 4 5 51

4 4 4 4+ = + = +x xx x x x

21. 7 5 7 5 7 15 5 5 5+ = + = +x x x

186 J{UV

KZmH¥$VtMo Ñí`rH$aU

ñdmÜ`m` 7.1

1. (a) →(iii) → (iv) (b) →(i) → (v) (c) →(iv) → (ii)

(d) →(v) → (iii) (e) →(ii) → (i)

2. (a) (i) →g_moa, (ii) →~mOy , (iii) →_mWm

(b) (i) →~mOy , (ii) →g_moa, (iii) →_mWm

(c) (i) →g_moa (ii) →~mOy (iii) →_mWm

(d) (i) →g_moa (ii) →~mOy , (iii) →_mWm

3. (a) (i) →_mWm, (ii) →g_moa, (iii) →~mOy

(b) (i) →~mOy, (ii) →g_moa (iii) →_mWm

(c) (i) →_mWm,, (ii) →~mOy , (iii) →g_moa

(d) (i) →~mOy (ii) →g_moa, (iii) →_mWm

(e) (i) →g_moa, (ii) →_mWm (iii) →~mOy

ñdmÜ`m` 7.3

1. (i) Zmhr (ii) hmo` (iii) hmo`

2. eŠ` Amho. \$ŠV Oa n¥îR>^mJm§Mr g§»`m 4 er g_mZ qH$dm 4 nojm OmñV Agob Va.

3. (ii) Am{U (iv) \$ŠV

4. (i) EH$ bmobH$ hm d¥Îm{MVr AgVmo Ooìhm Ë`mÀ`m nm`mÀ`m ~mOy§Mr g§»`m OmñVrV OmñV (Iyn) AgVo.

(ii) EH$ {na°{_S> hm e§Hy$ AgVmo Ooìhm Ë`mÀ`m nm`mÀ`m ~mOy§Mr g§»`m OmñVrV OmñV AgVo.

5. Zmhr. Vmo KZm`V gwÜXm Agy eH$Vmo.

7. n¥îR>^mJ-8, {eamoq~Xÿ - 6, H$S>m - 30.

8. Zmhr.

CÎmao 187

à_mUm§Mr VwbZm

ñdmÜ`m` 8.1

1. (a) 1 : 2 (b) 1 : 2000 (c) 1 : 10

2. (a) 75% (b) 66 2/3 %

3. 7 {dÚmWu 4. 25 gm_Zo 5. 2400

6. 10% {H«$Ho$Q>→ 30 bj→ 15 BVa Ioi →5

ñdmÜ`m` 8.2

1. `. 1, 40,000 2. 80% 3. ` 34.80 4. ` 18,342.50

5. 2% Mm Z\$m 6. ` 2,835 7. ` 1,269.84 8. ` 14,560

9. ` 2,000 10. ` 5,000 11. ` 1,050

ñdmÜ`m` 8.3

1. (a) amg = `. 15,377; MH«$dmT>ì`mO = `. 4577.34

(b) amg =` 22,869; ì`mO = ` 4869

(c) amgÀ =` 70,304; ì`mO = ` 7,804

(d) amg =` 8,736.20 ; ì`mO = ` 736.20

(e) amg =` 10,816; ì`mO = ` 816

2. ` 36,659.70

3. \$~rZm ` 362.50 OmñV âVo.

4. ` 43.20

5. (i) ` 63,600 (ii) ` 67,416

6. (i) ` 92,400 (ii) ` 92,610

7. (i) ` 8,820 (ii) ` 441

8. amg = ` 11,576.25; ì`mO = ` 1,576.25 hmo`

9. ` 4, 913 10. (i) gw_mao 48,980 (ii) gw_mao 59,535

11. ` 5,31,616 (A§XmOo) 12. ` 38,640

188 J{UV

joÌ_mnZ

ñdmÜ`m` 9.1

1. (a) 2. ` 17, 875

3. joÌ\$i = 129.5 m2.; n[a{_Vr = 48 m.

4. 45000 \$aí`m. 5. (b)

ñdmÜ`m` 9.2

1. 0.88 sq. m. 2. 7 cm.. 3. 660 m2

4. 252 sq. m. 5. 45 sq. cm.

6. 24 sq. m. 6. cm. 7. ` 810

8. 140 m. 9. 119 sq. m.

10. Á`mo{VMr nÜXV dmnê$Z, joÌ\$i = ( )1 152 30 152 2

× × × + = 337.5 m2.

H${dVmMr nÜXV dmnê$Z, joÌ\$i = 1 15 15 15 152

× × + × = 337.5 m2.

11. 80 cm2., 96 cm2., 80 cm2, 96 cm2.

ñdmÜ`m` 9.3

1. (a) 2. 144 m. 3. 10 cm.

4. 11 m2. 5. 5 H°$Z

6. g_ê$nVm - XmoÝhtMr C§Mr g_mZ Amho.

\$aH$ - EH$ d¥Îm{MVr Amho, Xþgar KZ Amho. KZmMo nmíd©n¥îR>\$i OmñV Amho.

7. 440 m2. 8. 322 cm. 9. 1980 m2. 10. 704 cm2 .

ñdmÜ`m` 9.4

1. (a) KZ\$i (b) n¥îR>\$i (c) KZ\$i

2. B d¥Îm{MVrMo KZ\$i OmñV Amho, B d¥Îm{MVrMo nmíd©n¥îR>\$i OmñV Amho.

3. 5 cm. 4. 450 5. 1 m.. 6. 49500 l.

7. (i) 4 doim (ii) 8 doim 8. 30 Vmg

CÎmao 189

ho \$ŠV IoimgmR>r

1. nm`WmJmoagMr {ÌHo$ g§~§Yr A{YH$ H$mhr -

nm`WmJmoagMr {ÌHo$ Oer 2m, m2 –1, m2 + 1 {b{hÊ`mMr EH$ nÜXV Amåhr nm{hbr Amho.

nm`WmJmoagMr {ÌHo$ a, b, c åhUOo a2 + b2 = c2 . Oa Amåhr XmoZ Z¡g{J©H$ g§»`m m

Am{U n (m>n) dmnaë`m Am{U a = m2 – n2 , b = 2mn, c = m2 + n2 Va Amåhr nmhÿ eH$Vmo H$s c2 = a2 + b2.m Am{U n À`m doJdoJù`m qH$_VrgmR>r Oa m>n Agob Va Z¡g{J©H$ g§»`m a, b Am{U c `m nm`WmJmoagÀ`m {ÌHo$ ñdê$nmV emoYy eH$Vmo.

CXmhaUmW© : m = 2, n = 1 ¿`m.

åhUOo a = m2 – n2 = 3, b = 2mn = 4, c = m2 + n2 = 5 hr nm`WmJmoagMr {ÌHo$ AmhoV. (VnmgyZ nhm).

m = 3, n = 2 gmR>r Amåhmbm {_iob.

a = 5, b = 12, c = 13. Or nm`WmJmoagMr {ÌHo$ AmhoV.

AmUIr H$mhr m d n À`m qH$_Vr KoD$Z Aer nm`WmJmoagMr nwÝhm {ÌHo$ emoYm.

2. Ooìhm nmUr JmoRy>Z ~\©$ ~ZVmo, Voìhm Ë`mMo KZ\$i 4% Zo dmT>Vo. Va 221 cm3 BVŠ`m KZ\$imMm ~\©$ V`ma H$aÊ`mg {H$Vr KZ\$imMo nmUr ¿`mdo bmJob?

3. Oa MhmMr qH$_V 20% Zo dmT>br, Va nydu BVH$mM IM© R>odÊ`mg MhmMm dmna {H$Vr Q>ŠŠ`mZo H$_r H$amdm bmJob?

4. EH$ nwañH$ma XoÊ`mMr gwê$dmV 1958 _Üò Pmbr. Ë`mdoir nwañH$ma qOH$Ê`mÀ`m 28loUr (categories) hmoË`m. 1993 _Üò `m loUr 93 hmoË`m.

(i) 1958 _Üò {Xbobo nwañH$ma ho 1993 _Üò {Xboë`m nwañH$mamÀ`m {H$Vr Q>ŠHo$ hmoVo?

(ii) 1993 _Üò {Xbobo nwañH$ma ho 1958 _Üò {Xboë`m nwañH$mamÀ`m {H$Vr Q>ŠHo$ hmoVo?

5. EH$m _Y_mí`m§À`m nmoù`mVrb EHy$U _Y_mem§n¡H$s EH$ nM_m§e (15 ) _Y_mem H$X§~m PmS>mÀ`m

\w$bm§À`mda ~gë`m, EHy$U _Y_mem§n¡H$s EH$ {ÌVr`m§e (13 ) _Y_mem {gqbYrar PmS>mÀ`m

\w$bm§À`mda ~gë`m Am{U `m XmoÝhrVrb \$aH$m§À`m {VßnQ> _Y_mem Hw$Q>Om PmS>mÀ`m \w$bm§À`mda ~gë`m. Am{U EHy$U _Y_mem§n¡H$s Xhm _Y_mem H$moR>ohr ~gë`m ZmhrV. Va nmoù`mVrb _Y_mem§Mr EHy$U g§»`m {H$Vr? (bjmV ¿`m, H$X§~m, {gqbYrar Am{U Hw$Q>Om hr \w$bo òUmar PmS>o AmhoV. ho CXmhaU àmMrZ ^maVr` ~rOJ{UVÀ`m nwñVH$mVyZ KoVbo Amho.).

190 J{UV

6. EH$m Mm¡agmMo joÌ\$i H$mT>VmZm eoIaZo Mm¡agmMo joÌ\$i H$mT>Ê`mMo gwÌ dmnabo Am{U Ë`mMm {_Ì _mê$\$Zo Mm¡agmMr n[a{_Vr H$mT>Ê`mMo gwÌ dmnabo. g§»`mË_H$[aË`m XmoÝhr CÎmao g_mZ hmoVr Va Ë`m§Zr joÌ\$i H$mT>boë`m Mm¡agmÀ`m ~mOyÀ`m EH$H$mMr g§»`m gm§Jm.

7. EH$m Mm¡agmMo joÌ\$i ho g§»`mË_H$[aË`m Ë`mÀ`m ~mOyÀ`m ghmnQ>rZo bhmZ Amho. Ago AgUmao H$mhr Mm¡ag emoYm d Ë`m§Mr `mXr H$am.

8. EH$m dVw©imH$ma d¥V{MVrMo KZ\$i ho g§»`mË_H$[aË`m {VÀ`m dH«$n¥îR>\$imBVHo$ AgUo eŠ` Amho H$m? Oa hmo` Va Ho$ìhm Vo gm§Jm.

9. {bbmZo Amnë`m dmT>{Xder H$mhr _¡{ÌUtZm MhmnmZmgmR>r Kar ~mobmdbo. {VÀ`m AmB©Zo Ë`m _¡{ÌUtZm ImÊ`mgmR>r H$mhr nwè`m Agbobr VmQ>o Q>o~bda R>odbr. Oa {bbmÀ`m AmB©Zo àËòH$ VmQ>mV 4 nwè`m R>odë`m Va EH$ VmQ> [aH$m_o amhVo. Am{U Oa àËòH$ VmQ>mV 3 nwè`m R>odë`m Va EH$ nwar {eëbH$ amhVo. Va Q>o~bmda R>odbobr VmQ>o d nwè`m `m§Mr g§»`m H$mT>m.

10. EH$ g§»`m Aer Amho H$s {VMm KZ d Vr g§»`m g_mZ AmhoV, _mÌ Ë`m g§»òMm dJ© doJir g§»`m òVo. Va Vr g§»`m H$moUVr?

11. 1 Vo 20 _Yrb g§»`m§Mr AmS>dr aMZm Aer H$am H$s XmoZ H«$_dma g§»`m§Mr ~oarO nyU© dJ© hmoB©b.

CÎmao

2. 12122 cm3. 3.

216 %3 (i) 34.5% (ii) 289%

5. 150

6. 4 EH$Ho$

7. ~mOy =1,2,3,4,5 EH$Ho$.

8. hmo`, Ooìhm {ÌÁ`m = 2 EH$H$.

9. nwè`m§Mr g§»`m = 16, VmQ>m§Mr g§»`m = 5

10. - 1

11. EH$ nÜXV - 1, 3, 6, 19, 17, 8 (1 + 3 = 4, 3 + 6 = 9 B). hoM Xþgè`m H$mhr nÜXVtZr H$aÊ`mMm à`ËZ H$am.

Documents

H$Zm©Q>H$ gaH$ma J{UV 8 - Kar textbooks/class8/8th...KZ Am{U KZ_yi 3 VŠVm - 2 Amåhr g_ AmhmoV. åhUyZ Am_Mo KZ g_ AmhoV. Amåhr {df_ AmhmoV. åhUyZ Am_Mo KZ {df_ AmhoV. g »`m KZ