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Aa¨vq
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c„ôv
1 20 38 70 87 102 121 132 151 173 179 194 215 228 242 250 278 294

cÖ_g Aa¨vq

ev¯—e msL¨v
(Real Number) cwigvY‡K cÖZxK Z_v msL¨v AvKv‡i cÖKvk Kivi c×wZ †_‡KB MwY‡Zi DrcwË| msL¨vi BwZnvm gvbe mf¨Zvi BwZnv‡mi gZB cÖvPxb| wMÖK `vk©wbK Gwi÷U‡ji g‡Z, cÖvPxb wgk‡ii cy‡ivwnZ m¤cÖ`v‡qi MwYZ Abykxj‡bi gva¨‡g MwY‡Zi AvbyôvwbK Awf‡lK N‡U| ZvB msL¨vwfwËK MwY‡Zi m„wó hxïwLª‡÷i R‡b¥i cÖvq `yB nvRvi eQi c~‡e©| Gici bvbv RvwZ I mf¨Zvi nvZ Ny‡i Aaybv msL¨v I msL¨vixwZ GKwU mve©Rbxb iƒc aviY K‡i‡Q| ¯^vfvweK msL¨v MYbvi cÖ‡qvR‡b cÖvPxb fviZe‡l©i MwYZwe`MY me©cÖ_g k~b¨ I `kwfwËK ¯’vbxqgvb c×wZi cÖPjb K‡ib, hv msL¨v eY©bvq GKwU gvBjdjK wnmv‡e we‡ewPZ| fviZxq I Pxbv MwYZwe`MY k~b¨, FYvZ¥K, ev¯—e, c~Y© I fMœvs‡ki aviYvi we¯—…wZ NUvb hv ga¨hy‡M Aviexq MwYZwe`iv wfwË wn‡m‡e MÖnY K‡ib| `kwgK fMœvs‡ki mvnv‡h¨ msL¨v cÖKv‡ki K…wZZ¡ ga¨cÖv‡P¨i gymwjg MwYZwe`‡`i e‡j g‡b Kiv nq| Avevi ZuvivB GKv`k kZvãx‡Z me©cÖ_g exRMvwYZxq wØNvZ mgxKi‡Yi mgvavb wn‡m‡e eM©g~j AvKv‡i Ag~j` msL¨vi cÖeZ©b K‡ib| BwZnvmwe`‡`i aviYv wLª÷c~e© 50 A‡ãi KvQvKvwQ wMÖK `vk©wbKivI R¨vwgwZK A¼‡bi cÖ‡qvR‡b Ag~j` msL¨v, we‡kl K‡i `yB-Gi eM©g~‡ji cÖ‡qvRbxqZv Abyfe K‡iwQ‡jb| Ebwesk kZvãx‡Z BD‡ivcxq MwYZwe`iv ev¯—e msL¨vi cÖYvjxe× K‡i c~Y©Zv `vb K‡ib| ˆ`bw›`b cÖ‡qvR‡b ev¯—e msL¨v m¤^‡Ü wk¶v_©x‡`i my¯úó Ávb _vKv cÖ‡qvRb| G Aa¨v‡q ev¯—e msL¨v wel‡q mvgwMÖK Av‡jvPbv Kiv n‡q‡Q| Aa¨vq †k‡l wk¶v_x©ivÑ ev¯—e msL¨vi †kªwYweb¨vm Ki‡Z cvi‡e| ev¯—e msL¨v‡K `kwg‡K cÖKvk K‡i Avmbœ gvb wbY©q Ki‡Z cvi‡e| `kwgK fMœvs‡ki †kªwYweb¨vm e¨vL¨v Ki‡Z cvi‡e| Ave„Ë `kwgK fMœvsk e¨vL¨v Ki‡Z cvi‡e Ges fMœvsk‡K Ave„Ë `kwg‡K cÖKvk Ki‡Z cvi‡e| Ave„Ë `kwgK fMœvsk‡K mvaviY fMœvs‡k iƒcvš—i Ki‡Z cvi‡e| Amxg Abve„Ë `kwgK fMœvsk e¨vL¨v Ki‡Z cvi‡e| m`„k I wem`„k `kwgK fMœvsk e¨vL¨v Ki‡Z cvi‡e| Ave„Ë `kwgK fMœvs‡ki †hvM, we‡qvM, ¸Y I fvM Ki‡Z cvi‡e Ges GZ`msµvš— wewfbœ mgm¨vi mgvavb Ki‡Z cvi‡e| dg©v-1, MwYZ-9g-10g

2

MwYZ

¯^vfvweK msL¨v (Natural Number)
1, 2, 3, 4......... BZ¨vw` msL¨v¸‡jv‡K ¯^vfvweK msL¨v ev abvZ¥K ALÊ msL¨v e‡j| 2, 3, 5, 7......... BZ¨vw`

†gŠwjK msL¨v Ges 4, 6, 8, 9,......... BZ¨vw` †hŠwMK msL¨v| c~Y©msL¨v (Integers) k~b¨mn mKj abvZ¥K I FYvZ¥K ALÛ msL¨vmg~n‡K c~Y©msL¨v ejv nq| A_©vr ....... 3, 2, 1, 0, 1, 2, 3......... BZ¨vw` c~Y©msL¨v|

fMœvsk msL¨v (Fractional Number) p, q ci¯úi mn‡gŠwjK, q

0 Ges q 1 n‡j,

p AvKv‡ii msL¨v‡K fMœvsk msL¨v e‡j| †hgb : q

1 3 5 , , BZ¨vw` fMœvsk msL¨v| 2 2 3 p q n‡j fMœvsk‡K cÖK…Z fMœvsk Ges p q n‡j fMœvsk‡K AcÖK…Z fMœvsk ejv nq| †hgb :

1 1 2 1 3 4 5 5 , , , , ......... BZ¨vw` cÖK…Z fMœvsk Ges , , , , .... BZ¨vw` AcÖK…Z fMœvsk| 2 3 3 4 2 3 3 4

g~j` msL¨v (Rational Number) p I q cYmsL¨v Ges q ~ ©
3 1 3, 11 2 5.5, 5 3 0 n‡j, p q

AvKv‡ii msL¨v‡K g~j` msL¨v ejv nq| †hgb :

1.666... BZ¨vw` g~j` msL¨v| g~j` msL¨v‡K `yBwU c~Y©msL¨vi AbycvZ wnmv‡e cÖKvk

Kiv hvq| myZivs mKj c~Y©msL¨v Ges mKj fMœvsk msL¨v n‡e g~j` msL¨v|

Ag~j` msL¨v (Irrational Number)
†h msL¨v‡K p AvKv‡i cÖKvk Kiv hvq bv, †hLv‡b p, q c~Y©msL¨v Ges q q

0 , †m msL¨v‡K Ag~j` msL¨v

ejv nq| c~Y©eM© bq Giƒc †h‡Kv‡bv ¯^vfvweK msL¨vi eM©g~j GKwU Ag~j` msL¨v| †hgb :
2 1.414213....., 3 1.732....., 5 2 1.58113.... BZ¨vw` Ag~j` msL¨v| Ag~j` msL¨v‡K `yBwU

c~Y©msL¨vi AbycvZ wnmv‡e cÖKvk Kiv hvq bv| `kwgK fMœvsk msL¨v : g~j` msL¨v I Ag~j` msL¨v‡K `kwg‡K cÖKvk Kiv n‡j G‡K `kwgK fMœvsk ejv nq| †hgb,
3 3 0, 5 2 2 5, 10 3 3 3333......., 3 1 732......... BZ¨vw` `kwgK fMœvsk msL¨v| `kwgK we›`yi

ci A¼ msL¨v mgxg n‡j, G‡`i‡K mmxg `kwgK fMœvsk Ges A¼ msL¨v Amxg n‡j, G‡`i‡K Amxg `kwgK

MwYZ

3

fMœvsk

ejv

nq|

†hgb,

0 52, 3 4152

BZ¨vw`

mmxg

`kwgK

fMœvsk

Ges

1 333......., 2 123512367........... BZ¨vw` Amxg `kwgK fMœvsk msL¨v| Avevi, Amxg `kwgK fMœvsk

msL¨v¸‡jvi g‡a¨ `kwgK we›`yi ci A¼¸‡jv cybive„wË n‡j, G‡`i‡K Amxg Ave„Ë `kwgK fMœvsk Ges A¼¸‡jv cybive„wË bv n‡j G‡`i Amxg Abve„Ë `kwgK fMœvsk msL¨v ejv nq| †hgb, 1 2323........,
5 654 BZ¨vw` Amxg Ave„Ë `kwgK fMœvsk Ges 0 523050056........, 2 12340314........ BZ¨vw`

Abve„Ë `kwgK fMœvsk| ev¯—e msL¨v (Real Number) mKj g~j` msL¨v Ges Ag~j` msL¨v‡K ev¯—e msL¨v ejv nq| †hgb :
0, 1, 2, 3,..........
1 , 2 3 , 2 4 ,........ 3

2 , 3 , 5 , 6 ......
1 23, 0 415, 1 3333......., 0 62, 4 120345061.......... BZ¨vw` ev¯—e msL¨v|

abvZ¥K msL¨v (Positive Number) k~b¨ A‡c¶v eo mKj ev¯—e msL¨v‡K abvZ¥K msL¨v ejv nq| †hgb, 1, 2,
1 3 , , 2 2 2 , 0 415, 0 62, 4 120345061.............. BZ¨vw` abvZ¥K msL¨v|

FYvZ¥K msL¨v (Negative Number) k~b¨ A‡c¶v †QvU mKj ev¯—e msL¨v‡K FYvZ¥K msL¨v ejv nq| †hgb, msL¨v|
1, 2, 1 , 2 3 , 2 2, 0 415, 0 62, 4 120345061.............. BZ¨vw` FYvZ¥K

AFYvZ¥K msL¨v (Non negative Number) k~b¨mn mKj abvZ¥K msL¨v‡K AFYvZ¥K msL¨v ejv nq| †hgb, 0, 3,
1 , 0 612, 1 3, 2 120345.............. BZ¨vw` AFYvZ¥K msL¨v| 2

4

MwYZ

ev¯—e msL¨vi †kªwYweb¨vm ev¯—e msL¨v g~j` c~Y© abvZ¥K †gŠwjK 1 0 fMœvsk FYvZ¥K cÖK…Z mvaviY AcÖK…Z wgkª `kwgK mmxg Amxg Ave„Ë Ag~j` Amxg Abve„Ë

†hŠwMK

KvR :
3 7 4 †kªwYweb¨v‡m Ae¯’vb †`LvI| , 5, 7, 13 , 0, 1, 9 , 12 ,
. . 4 2 , 1 1234......, .3 2 3 5

msL¨v¸‡jv‡K ev¯—e msL¨vi

D`vniY 1| g‡b Kwi, Ges A_vr© Ges
3

3 Ges 4 Gi g‡a¨ `yBwU Ag~j` msL¨v wbY©q Ki| 3 a mgvavb : GLv‡b,

1.7320508......
2.030033000333.....

b

2.505500555.......

¯úóZ : a I b DfqB `yBwU ev¯—e msL¨v Ges DfqB 3 A‡c¶v eo Ges 4 A‡c¶v †QvU|
2.03003300333......... 4

3

2.505500555................

4

Avevi, a I b †K fMœvsk AvKv‡i cÖKvk Kiv hvq bv| a I b `yBwU wb‡Y©q Ag~j` msL¨v|

ev¯—e msL¨vi Dci †hvM I ¸Yb cÖwµqvi †gŠwjK ˆewkó¨ : 1. 2. 3. 4. a, b ev¯—e msL¨v n‡j, i a a, b ev¯—e msL¨v n‡j, i a b ev¯—e msL¨v Ges ii ab ev¯e msL¨v — b b b a a Ges ii ab ba c a b c Ges ii ab c a bc i 0 1

a, b, c ev¯—e msL¨v n‡j, i a ii a

a ev¯—e msL¨v n‡j, ev¯—e msL¨vq †Kej `yBwU msL¨v 0 I 1 we`¨gvb †hLv‡b

0

a iii a.1 1.a

MwYZ

5

5. a ev¯—e msL¨v n‡j, i a ( a ) 0 ii a

0 n‡j, a.

1 a

1

6. a, b, c ev¯—e msL¨v n‡j, a(b c ) ab ac 7. a, b ev¯—e msL¨v n‡j, a b A_ev a b A_ev a 8. a, b, c ev¯—e msL¨v Ges a b n‡j, a c b c

b

9. a, b, c ev¯—e msL¨v Ges a b n‡j, (i ) ac bc hLb c > 0 (ii ) ac cÖwZÁv : 2 GKwU Ag~j` msL¨v| Avgiv Rvwb,
1 2 4 1 2
4 2

bc n‡j, c

0

ev, 1 cÖgvY : 12 myZivs AZGe

2

1,

2

2, 2 2

4

2 Gi gvb 1 A‡c¶v eo Ges 2 A‡c¶v †QvU|
2 c~Y©msL¨v bq| 2 g~j` msL¨v nq Z‡e

2 g~j` msL¨v A_ev Ag~j` msL¨v| hw`

awi, ev, 2 ev, 2q

2

p ; †hLv‡b p I q ¯^vfvweK msL¨v I ci¯úi mn‡gŠwjK Ges q 1 q

p2 ; eM© K‡i q2 p2 ; Dfq c¶‡K q Øviv ¸Y K‡i| q

¯úóZ : 2q c~Y© msL¨v wKš‘ Ges q 1
2q Ges

p2 , c~Y©msL¨v bq, KviY p I q ¯^vfvweK msL¨v I Giv ci¯úi mn‡gŠwjK q

p2 p2 mgvb n‡Z cv‡i bv, A_©vr 2q q q p 2 Gi gvb AvKv‡ii †Kv‡bv msL¨v n‡Z cv‡i bv, A_©vr q

2

p q

2 GKwU Ag~j` msL¨v|

D`vniY 2| cÖgvY Ki †h, †Kv‡bv PviwU µwgK ¯^vfvweK msL¨vi ¸Yd‡ji mv‡_ 1 †hvM Ki‡j †hvMdj GKwU c~Y©eM© msL¨v n‡e| mgvavb : g‡b Kwi, PviwU µwgK ¯^vfvweK msL¨v h_vµ‡g x, x 1, x 2, x 3 µwgK msL¨v PviwUi ¸Yd‡ji mv‡_ 1 †hvM Ki‡j cvIqv hvq,

6

MwYZ

x x 1 x 2 x 3 x 2 3x x 2 3x 2 a (a 2) 1; x 2 3x a (a 2) 1;

1 1 a xx 3 x 1 x 2

1

a2

2a 1

a 1

2

x2

3 x 1 ; hv GKwU c~Y©eM© msL¨v|

2

†h‡Kv‡bv PviwU µwgK ¯^vfvweK msL¨vi ¸Yd‡ji mv‡_ 1 †hvM Ki‡j †hvMdj GKwU c~Y©eM© msL¨v n‡e|

KvR :

cgvY Ki †h, Ö

3 GKwU Agj` msL¨v| ~

`kwgK fMœvs‡ki †kªwYweb¨vm cÖ‡Z¨K ev¯—e msL¨v‡K `kwgK fMœvs‡k cÖKvk Kiv hvq| †hgb : 2 2 0,
2 5 0.4, 1 3 0.333....

BZ¨vw`| `kwgK fMœvsk wZb cÖKvi: mmxg `kwgK, Ave„Ë `kwgK Ges Amxg `kwgK fMœvsk| mmxg `kwgK fMœvsk : mmxg `kwg‡K `kwgK wP‡ýi Wvbw`‡K mmxg msL¨K A¼ _v‡K| †hgb : 0.12,
1.023, 7.832, 54.67, .......BZ¨vw` mmxg `kwgK fMœvsk|

AveË `kwgK fMvsk : AveË `kwg‡K `kwgK wP‡ýi Wvbw`‡Ki A¼¸‡jv ev Askwe‡kl evievi _vK‡e| „ œ „ †hgb, 3.333....., 2.454545......, 5.12765765 BZ¨vw` Ave„Ë `kwgK fMœvsk| Amxg `kwgK fMœvsk : Amxg `kwgK fMœvs‡k `kwgK wP‡ýi Wvbw`‡Ki A¼ KL‡bv †kl nq bv, A_©vr `kwgK wP‡ýi Wvbw`‡Ki A¼¸‡jv mmxg n‡e bv ev Askwe‡kl evievi Avm‡e bv| †hgb :
1.4142135......, 2.8284271....... BZ¨vw` Amxg `kwgK fMœvsk|

mmxg `kwgK I Ave„Ë `kwgK fMœvsk g~j` msL¨v Ges Amxg `kwgK fMœvsk Ag~j` msL¨v| †Kv‡bv Ag~j` msL¨vi gvb hZ `kwgK ¯’vb ch©š— B”Qv wbY©q Kiv hvq| †Kv‡bv fMœvs‡ki je I ni‡K ¯^vfvweK msL¨vq cÖKvk Ki‡Z cvi‡j, H fMœvskwU g~j` msL¨v|

KvR :
1.723, 5.2333........, 0.0025, 2.1356124......., 0.0105105........ Ges
0.450123....... fMœvsk¸‡jv‡K KviYmn †kªwYweb¨vm Ki|

MwYZ

7

Ave„Ë `kwgK fMœvsk
23 fMœvskwU‡K `kwg‡K cÖKvk Kwi| 6 23 = 6 23 3.833 6 18 50 48 20 18 20 18

j¶ Kwi, fMœvs‡ki je‡K ni w`‡q fvM K‡i `kwgK fMœvs‡k cwiYZ Kivi mgq fv‡Mi cÖwµqv †kl nq bvB| †`Lv hvq †h, fvMd‡j GKB msL¨v evievi Av‡m| GLv‡b, 3.8333..... GKwU Ave„Ë `kwgK fMœvsk| †h mKj `kwgK fMœvs‡k `kwgK we›`yi Wv‡b GKwU A¼ µgvš^‡q evievi ev GKvwaK A¼ ch©vqµ‡g evievi Av‡m, G‡`i Ave„Ë `kwgK fMœvsk ejv nq| Ave„Ë ev †cŠbtcywbK `kwgK fMœvs‡k †h Ask evievi A_©vr cybtcyb nq, G‡K Ave„Ë Ask e‡j| Ave„Ë `kwgK fMœvs‡k GKwU A¼ Ave„Ë n‡j, †m A‡¼i Dci †cŠbtcywbK we›`y Ges GKvwaK A¼ Ave„Ë n‡j, †KejgvÎ cÖ_g I †kl A‡¼i Dci †cŠbtcywbK we›`y †`Iqv nq| †hgb 2.555....... †K †jLv nq 2.5 Øviv Ges 3.124124124........ †K †jLv nq, 3.124 Øviv| `kwgK fMœvs‡k `kwgK we›`yi ci Ave„Ëvsk Qvov Ab¨ †Kv‡bv A¼ bv _vK‡j, G‡K weï× †cŠbtcywbK e‡j Ges †cŠbtcywbK `kwgK fMœvs‡k `kwgK we›`yi ci Ave„Ëvsk Qvov GK ev GKvwaK A¼ _vK‡j, G‡K wgkª †cŠbtcywbK e‡j| †hgb, 1.3 weï× †cŠbtcywbK fMœvsk Ges 4.23512 wgkª †cŠbtcywbK fMœvsk| fMœvs‡ki n‡i 2, 5 Qvov Ab¨ †Kv‡bv †gŠwjK ¸bbxqK (Drcv`K) _vK‡j, †mB ni Øviv je‡K fvM Ki‡j, KL‡bv wbt‡k‡l wefvR¨ n‡e bv| †h‡nZz ch©vqµ‡g fv‡M †k‡li A¼¸‡jv 1, 2, ......, 9 Qvov Ab¨ wKQy n‡Z cv‡i bv, †m‡nZz GK ch©v‡q fvM‡kl¸‡jv evievi GKB msL¨v n‡Z _vK‡e| Ave„Ëvs‡ki msL¨v memgq n‡i †h msL¨v _v‡K, Gi †P‡q †QvU nq| D`vniY 3| mgvavb :
3 †K `kwgK fMœvs‡k cÖKvk Ki| 11

D`vniY 4| mgvavb :

95 †K `kwgK fMœvs‡k cÖKvk Ki| 37

37 ) 95 ( 2.56756 74 210 185 250 222 280 259 210 185 250 222 28

wb‡Y©q `kwgK fMœvsk = 0.2727 ........ 0.27

wb‡Y©q `kwgK fMœvsk = 2.56756...... 2.567

8

MwYZ

Ave„Ë `kwgK‡K mvgvb¨ fMœvs‡k cwieZ©b Ave„Ë `kwg‡Ki gvb wbY©q : D`vniY 5| 0.3 †K mvgvb¨ fMœvs‡k cÖKvk Ki| mgavb : 0.3 0.3333........
0.3 10 0.333....... 10
0.333....... 1

3.333......
0.333...... 3
3

Ges

0.3 1

we‡qvM K‡i, 0.3 10 0.3 1 ev, 0.3
10 1 3 9 1 3 3 ev, 0.3 9 1 3

AZGe, 0.3 wb‡Y©q fMœvsk

D`vniY 6| 0.24 †K mvgvb¨ fMœvs‡k cÖKvk Ki| mgvavb : 0.24 = 0 .24242424........ myZivs 0.24 100 0.242424....... 100 Ges
0.24 1 0.242424....... 1
24

24.2424......
0.242424......

we‡qvM K‡i, 0.24 100 1 ev, 0.24 99 wb‡Y©q fMœvsk
8 33

24 ev, 0.24

24 99

8 33

D`vniY 7| 5.1345 †K mvgvb¨ fMœvs‡k cÖKvk Ki| mgvavb : 5.1345 myZivs Ges
5.1345345345.........
5.1345345...... 10000 5.1345345....... 10 51345.345...... 51.345...... 51345 51

5.1345 10000 5.1345 10

we‡qvM K‡i, 5.1345 9990 AZGe, 5.1345 = wb‡Y©q fMœvsk 5
51345 51 9990 51294 9990 8549 1665 5

224 1665

224 1665

MwYZ

9

D`vniY 8| 42.3478 †K mvgvb¨ fMœvs‡k cÖKvk Ki| mgvavb : 42.3478 = 42.347878........ myZivs, Ges
42.3478
42.3478
10000 100 9900 42.347878......... 10000 42.347878........ 100 42348.7878 4234.7878 423478 4234

we‡qvM K‡i, 42.3478

AZGe, 42.3478 = wb‡Y©q fMœvsk 42
287 825

423478 4234 9900

419244 9900

34937 825

42

287 825

e¨vL¨v : D`vniY 5, 6, 7 Ges 8 †_‡K †`Lv hvq †h, Ave„Ë `kwg‡K `kwgK we›`yi ci †h KqwU A¼ Av‡Q, †m KqwU k~b¨ 1 Gi Wv‡b ewm‡q cÖ_‡g Ave„Ë `kwgK‡K ¸Y Kiv n‡q‡Q| Ave„Ë `kwg‡K `kwgK we›`yi ci †h KqwU Abve„Ë A¼ Av‡Q, †m KqwU k~b¨ 1 Gi Wv‡b ewm‡q Ave„Ë `kwgK‡K ¸Y Kiv n‡q‡Q| cÖ_g ¸Ydj †_‡K wØZxq ¸Ydj we‡qvM Kiv n‡q‡Q| cÖ_g ¸Ydj †_‡K wØZxq ¸Ydj we‡qvM Kivq Wvbc‡¶ c~Y© msL¨v cvIqv †M‡Q| GLv‡b j¶Yxq †h, Ave„Ë `kwgK fMœvs‡ki `kwgK I †cŠbtcywbK we›`y DwV‡q cÖvß msL¨v †_‡K Abve„Ë As‡ki msL¨v we‡qvM Kiv n‡q‡Q| evgc‡¶ Ave„Ë `kwg‡K hZ¸‡jv Ave„Ë A¼ wQj ZZ¸‡jv 9 wj‡L Ges Zv‡`i Wv‡b `kwgK we›`yi ci hZ¸‡jv Abve„Ë A¼ wQj ZZ¸‡jv k~b¨ ewm‡q Dc‡i cÖvß we‡qvMdj‡K fvM Kiv n‡q‡Q| Ave„Ë `kwg‡K fMœvs‡k cwiYZ Kivq fMœvskwUi ni n‡jv hZ¸‡jv Ave„Ë A¼ ZZ¸‡jv 9 Ges 9 ¸‡jvi Wv‡b `kwgK we›`yi ci hZ¸‡jv Abve„Ë A¼ ZZ¸‡jv k~b¨| Avi je n‡jv Ave„Ë `kwg‡Ki `kwgK we›`y I †cŠbtcywbK we›`y DwV‡q †h msL¨v cvIqv †M‡Q, †m msL¨v †_‡K Ave„Ëvsk ev` w`‡q evwK A¼ Øviv MwVZ msL¨v we‡qvM K‡i we‡qvMdj| gš—e¨ : Ave„Ë `kwgK‡K me mgq fMœvs‡k cwiYZ Kiv hvq| mKj Ave„Ë `kwgK g~j` msL¨v|

dg©v-2 MwYZ- 9g-10g

10

MwYZ

D`vniY : 9| 5.23457 mgvavb : 5.23457 myZivs Ges
5.23457
5.23457

5.23457457457......... 100000 100
99900

523457.457457

= 523.457457
522934

we‡qvM K‡i, 5.23457 AZGe,
5.23457 =

522934 99900

261467 49950

wb‡Y©q fMœvsk

261467 49950

e¨vL¨v : `kwgK As‡k cuvPwU A¼ i‡q‡Q e‡j GLv‡b Ave„Ë `kwgK‡K cÖ_‡g 100000 (GK Gi Wv‡b cuvPwU k~b¨) Øviv ¸Y Kiv n‡q‡Q| Ave„Ë As‡ki ev‡g `kwgK As‡k `yBwU A¼ i‡q‡Q e‡j Ave„Ë `kwgK‡K 100 (GK Gi Wv‡b `yBwU k~b¨) Øviv ¸Y Kiv n‡q‡Q| cÖ_g ¸Ydj †_‡K wØZxq ¸Ydj we‡qvM Kiv n‡q‡Q| GB we‡qvMd‡ji GKw`‡K c~Y©msL¨v Ab¨w`‡K cÖ`Ë Ave„Ë `kwg‡Ki gv‡bi (100000 100) Dfq c¶‡K 99900 w`‡q fvM K‡i wb‡Y©q fMœvsk cvIqv †Mj|
99900 ¸Y|

KvR :
0.41 Ges 3.04623 †K fMœvs‡k iƒcvš—i Ki|

Ave„Ë `kwgK‡K mvgvb¨ fMœvs‡k iƒcvš—‡ii wbqg wb‡Y©q fMœvs‡ki je = cÖ`Ë `kwgK fMœvs‡ki `kwgK we›`y ev` w`‡q cÖvß msL¨v Ges Abve„Ë Ask Øviv MwVZ msL¨vi we‡qvMdj| wb‡Y©q fMœvs‡ki ni = `kwgK we›`yi c‡i Ave„Ë As‡k hZ¸‡jv A¼ Av‡Q ZZ¸‡jv bq (9) Ges Abve„Ë As‡k hZ¸‡jv A¼ Av‡Q ZZ¸‡jv k~b¨ ( 0 ) Øviv MwVZ msL¨v| GLv‡b, G wbqg mivmwi cÖ‡qvM K‡i K‡qKwU Ave„Ë `kwg‡K mvgvb¨ fMœvs‡k cwiYZ Kiv n‡jv| D`vniY 10| 45.2346 †K mvgvb¨ fMœvs‡k cÖKvk Ki| mgvavb : 45.2346 = wb‡Y©q fMœvsk 45
1172 4995
32567 32 999 21 37 32535 999 3615 111 1205 37 21 37

452346 452 9990

451894 9990

225947 4995

45

1172 4995

D`vniY 11| 32.567 †K mvgvb¨ fMœvs‡k cÖKvk Ki| mgvavb : 32.567 wb‡Y©q fMœvsk 32
32

MwYZ

11

KvR :
0.012 Ges 3.3124 †K fMœvs‡k iƒcvš—i Ki|

m`„k Ave„Ë `kwgK I wem`„k Ave„Ë `kwgK Ave„Ë `kwgK¸‡jv‡Z Abve„Ë As‡ki msL¨v mgvb n‡j Ges Ave„Ë As‡ki A¼ msL¨vI mgvb n‡j, Zv‡`i m`„k Ave„Ë `kwgK e‡j| GQvov Ab¨ Ave„Ë `kwgK¸‡jv‡K wem`„k Ave„Ë `kwgK e‡j| †hgb: 12.45 I
6.32 ; 9.453 I 125.897 m`„k Ave„Ë `kwgK| Avevi, 0.3456 I 7.45789; 6.4357 I 2.89345

wem`„k Ave„Ë `kwgK| wem`„k Ave„Ë `kwgK¸‡jv‡K m`„k Ave„Ë `kwg‡K cwieZ©‡bi wbqg †Kv‡bv Ave„Ë `kwg‡Ki Ave„Ë As‡ki A¼¸‡jv‡K evievi wjL‡j `kwg‡Ki gv‡bi †Kv‡bv cwieZ©b nq bv| †hgb,
6.4537 6.453737 6.45373 6.453737 |

GLv‡b

cÖ‡Z¨KwU

Ave„Ë

`kwgK

6.45373737......... GKwU Amxg `kwgK| cÖ‡Z¨KwU Ave„Ë `kwgK‡K mvgvb¨ fMœvs‡k cwieZ©b Ki‡j †`Lv

hv‡e cÖ‡Z¨KwU mgvb|
4537 645 63892 9900 9900 6453737 645 6453092 63892 6.453737 999900 999900 9900 6453737 64537 6389200 63892 6.453737 990000 990000 9900 6.4537

m`„k Ave„Ë `kwg‡K cwiYZ Ki‡Z n‡j msL¨v¸‡jvi g‡a¨ †h msL¨vwUi Abve„Ë As‡ki A¼ msL¨v †ewk, cÖ‡Z¨KwU Abve„Ë Ask ZZ A‡¼i Ki‡Z n‡e Ges wewfbœ msL¨vq Ave„Ë As‡ki A¼ msL¨v¸‡jvi j.mv.¸ hZ, cÖ‡Z¨KwU `kwg‡Ki Ave„Ë Ask ZZ A‡¼i Ki‡Z n‡e| D`vniY 12| 5.6, 7.345 I 10.78423 †K m`„k Ave„Ë `kwg‡K cwiYZ Ki| mgvavb : 5.6, 7.345 I 10.78423 Ave„Ë `kwg‡K Abve„Ë As‡ki A¼ msL¨v h_vµ‡g 0, 1 I 2 | GLv‡b Abve„Ë A¼ msL¨v 10.78423 `kwg‡K me‡P‡q †ewk Ges G msL¨v 2 | ZvB m`„k Ave„Ë `kwgK Ki‡Z n‡j cÖ‡Z¨KwU `kwg‡Ki Abve„Ë As‡ki A¼ msL¨v 2 n‡e| 5.6, 7.345 I 10.78423 Ave„Ë `kwg‡K Ave„Ë As‡ki msL¨v h_vµ‡g 1, 2 I 3 | 1, 2 I 3 Gi j.mv.¸ n‡jv 6 | ZvB m`„k Ave„Ë `kwgK Ki‡Z n‡j cÖ‡Z¨KwU `kwg‡Ki Ave„Ë As‡ki A¼ msL¨v 6 n‡e| myZivs 5.6 5.6666666666, 7.345 7.34545454 I 10.78423 10.78423423 wb‡Y©q m`„k Ave„Ë `kwgKmg~n h_vµ‡g 5.66666666, 7.34545454, 10.78423423

12

MwYZ

D`vniY 13| 1.7643, 3.24 I 2.78346 †K m`„k Ave„Ë `kwg‡K cwieZ©b Ki| mgvavb : 1.7643 G Abve„Ë Ask ej‡Z `kwgK we›`yi c‡ii 4 wU A¼, GLv‡b Ave„Ë Ask †bB| 3.24 G Abve„Ë As‡ki A¼ msL¨v 0 Ges Ave„Ë As‡ki A¼ msL¨v 2 , 2.78346 G Abve„Ë As‡ki A¼ msL¨v 2 Ges Ave„Ë As‡ki msL¨v 3 | GLv‡b Abve„Ë As‡ki A¼ msL¨v me‡P‡q †ewk n‡jv 4 Ges Ave„Ë As‡ki A¼ msL¨v 2 I 3 Gi j.mv.¸ n‡jv 6 | cÖ‡Z¨KwU `kwg‡Ki Abve„Ë As‡ki A¼ msL¨v n‡e 4 Ges Ave„Ë As‡ki A¼ msL¨v n‡e 6 |
1.7643000000 , 3.24 3.2424242424 I 2.78346 2.7834634634

wb‡Y©q Ave„Ë `kwgKmg~n: 1.7643000000 3.2424249424 , 2.7834634634 gš—e¨ : mmxg `kwgK fMœvsk¸‡jv‡K m`„k `kwg‡K cwiYZ Kivi Rb¨ `kwgK we›`yi me©Wv‡bi A‡¼i ci cÖ‡qvRbxq msL¨K k~b¨ ewm‡q cÖ‡Z¨KwU `kwg‡Ki `kwgK we›`yi c‡ii Abve„Ë A¼ msL¨v mgvb Kiv n‡q‡Q| Avi Ave„Ë `kwg‡K cÖ‡Z¨KwU `kwg‡Ki `kwgK we›`yi c‡ii Abve„Z A¼ msL¨v mgvb Ges Ave„Ë A¼ msL¨v mgvb Kiv n‡q‡Q Ave„Ë A¼¸‡jv e¨envi K‡i| Abve„Ë As‡ki ci †h‡Kv‡bv A¼ †_‡K ïi“ K‡i Ave„Ë Ask †bIqv hvq|

KvR :
3.467, 2.01243 Ges 7.5256 †K m`„k Ave„Ë `kwg‡K cwieZ©b Ki|

Ave„Ë `kwg‡Ki †hvM I we‡qvM
Ave„Ë `kwg‡Ki †hvM ev we‡qvM Ki‡Z n‡j Ave„Ë `kwgK¸‡jv‡K m`„k Ave„Ë `kwg‡K cwieZ©b Ki‡Z n‡e| Gici mmxg `kwg‡Ki wbq‡g †hvM ev we‡qvM Ki‡Z n‡e| mmxg `kwgK I Ave„Ë `kwgK¸‡jvi g‡a¨ †hvM ev we‡qvM Ki‡Z n‡j Ave„Ë `kwgK¸‡jv‡K m`„k Kivi mgq cÖ‡Z¨KwU Ave„Ë `kwg‡Ki Abve„Ë As‡ki A¼ msL¨v n‡e mmxg `kwg‡Ki `kwgK we›`yi c‡ii A¼ msL¨v I Ab¨vb¨ Ave„Ë `kwg‡Ki Abve„Ë As‡ki A¼ msL¨vi g‡a¨ me‡P‡q eo †h msL¨v †m msL¨vi mgvb| Avi Ave„Ë As‡ki A¼ msL¨v n‡e h_vwbq‡g cÖvß j.mv.¸ Gi mgvb Ges mmxg `kwg‡Ki †¶‡Î Ave„Ë As‡ki Rb¨ cÖ‡qvRbxq msL¨K k~b¨ emv‡Z n‡e| Gici †hvM ev we‡qvM mmxg `kwg‡Ki wbq‡g Ki‡Z n‡e| Gfv‡e cÖvß †hvMdj ev we‡qvMdj cÖK…Z †hvMdj ev we‡qvMdj n‡e bv| cÖK…Z †hvMdj ev we‡qvMdj †ei Ki‡Z n‡j †`L‡Z n‡e †h m`„kK…Z `kwgK¸‡jv †hvM ev we‡qvM Ki‡j cÖ‡Z¨KwU m`„kK…Z `kwgK¸‡jvi Ave„Ë As‡ki me©ev‡gi A¼¸‡jvi †hvM ev we‡qv‡M nv‡Z †h msL¨vwU _v‡K, Zv cÖvß †hvMdj ev we‡qvMd‡ji Ave„Ë As‡ki me©Wv‡bi A‡¼i mv‡_ †hvM ev A¼ †_‡K we‡qvM Ki‡j cÖK…Z †hvMdj ev we‡qvMdj cvIqv hv‡e| GwUB wb‡Y©q †hvMdj ev we‡qvMdj n‡e|

MwYZ

13

gš—e¨ : (K) Ave„Ë `kwgKwewkó msL¨vi †hvMdj ev we‡qvMI Ave„Ë `kwgK nq| GB †hvMdj ev we‡qvMd‡j Abve„Ë Ask Ave„Ë `kwgK¸‡jvi g‡a¨ me©v‡c¶v Abve„Ë Ask wewkó Ave„Ë `kwgKwUi Abve„Ë A¼ msL¨vi mgvb n‡e Ges Ave„Ë Ask Ave„Ë `kwgK msL¨v¸‡jvi Ave„Ë A¼ msL¨vi j.mv.¸ Gi mgvb msL¨K Ave„Ë A¼ n‡e| mmxg `kwgK _vK‡j cÖ‡Z¨KwU Ave„Ë `kwg‡Ki Abve„Ë As‡ki A¼ msL¨v n‡e mmxg `kwg‡Ki `kwgK we›`yi c‡ii A¼ msL¨v I Ab¨vb¨ Ave„Ë `kwg‡Ki Abve„Ë As‡ki A¼ msL¨vi g‡a¨ me‡P‡q eo †h msL¨v †h msL¨vi mgvb| (L) Ave„Ë `kwgK fMœvsk¸‡jv‡K mvgvb¨ fMœvs‡k cwieZ©b K‡i fMœvs‡ki wbq‡g †hvMdj ev we‡qvMdj †ei Kivi ci †hvMdj ev we‡qvMdj‡K Avevi `kwg‡K cwieZ©b K‡iI †hvM ev we‡qvM Kiv hvq| Z‡e G c×wZ‡Z †hvM ev we‡qvM Ki‡j †ewk mgq jvM‡e| D`vniY 14| 3 89, 2 178 I 5 89798 †hvM Ki| mgvavb : GLv‡b Abve„Ë As‡ki A¼ msL¨v n‡e 2 Ges Ave„Ë As‡ki A¼ n‡e 2 , 2 I 3 Gi j.mv.¸ 6 | cÖ_‡g wZbwU Ave„Ë `kwgK‡K m`„k Kiv n‡q‡Q|
3.89 2 178

= 3 89898989
2.17878787

5 89798

= 5.89798798
11 97576574

[8 8 7 2

25 , GLv‡b 2 n‡jv nv‡Zi 2 |

2

25 Gi 2 †hvM n‡q‡Q|]

11 97576576

wb‡Y©q †hvMdj 11 97576576 ev 11 97576 gš—e¨ : GB †hvMd‡j 575675 Ave„Ë Ask| wKš‘ 576 †K Ave„Ë Ask Ki‡j gv‡bi †Kv‡bv cwieZ©b nq bv| `ªóe¨ : me©Wv‡b 2 †hv‡Mi aviYv †evSvevi Rb¨ G †hvMwU Ab¨ wbq‡g Kiv n‡jv:
3.89

= = =

3 89898989 | 89

2.178
5 89798

2.17878787 87
5 89798798 | 79

11.97576576| 55

GLv‡b Ave„Ë Ask †kl nIqvi ci AviI 2 A¼ ch©š— msL¨v‡K evov‡bv n‡q‡Q| AwZwi³ A¼¸‡jv‡K GKUv Lvov †iLv Øviv Avjv`v K‡i †`Iqv n‡q‡Q| Gici †hvM Kiv n‡q‡Q| Lvov †iLvi Wv‡bi A‡¼i †hvMdj †_‡K nv‡Zi 2 G‡m Lvov †iLvi ev‡gi A‡¼i mv‡_ †hvM n‡q‡Q| Lvov †iLvi Wv‡bi A¼wU Avi †cŠbtcywbK we›`y k~b¨ nIqvi A¼wU GKB| ZvB `yBwU †hvMdjB GK|

14

MwYZ

D`vniY 15| 8 9478, 2 346 I 4 71 †hvM Ki| mgvavb : `kwgK¸‡jv‡K m`„k Ki‡Z n‡j Abve„Ë Ask 3 A‡¼i Ges Ave„Ë Ask n‡e 3 I 2 Gi j.mv.¸
6 A‡¼i|

8 947 8 2 346 4 71

= 8 947847847
2 346000000 = 4 717171717 16 011019564 1
16 011019565

8 0 1 1 10 GLv‡b wØZxq 1 n‡jv nv‡Zi 1 | 10 Gi 1 †hvM

n‡q‡Q|

wb‡Y©q †hvMdj 16 011019565

KvR : †hvM Ki : 1| 2 097

I 5 12768

2| 1 345, 0 31576 I 8 05678

D`vniY 16| 8 243 †_‡K 5 24673 we‡qvM Ki| mgvavb : GLv‡b Abve„Ë As‡ki A¼ msL¨v n‡e 2 Ges Ave„Ë As‡ki A¼ msL¨v n‡e 2 I 3 Gi j.mv.¸
6 | GLb `kwgK msL¨v `yBwU‡K m`„k K‡i we‡qvM Kiv n‡jv|

8 243 5 24673

= 8 24343434 = 5 24673673
3 †_‡K 6 we‡qvM Ki‡j nv‡Z 1

2 99669761 1 2 99669760 wb‡Y©q we‡qvMdj 2 99669760 |

wb‡Z n‡e|

gš—e¨ : †cŠbtcywbK we›`y †hLv‡b ïi“ †mLv‡b we‡qvRb msL¨v we‡qvR¨ msL¨v †_‡K †QvU n‡j me mgq me©Wv‡bi A¼ †_‡K 1 we‡qvM Ki‡Z n‡e| `ªóe¨ : me©Wv‡bi A¼ †_‡K 1 †Kb we‡qvM Kiv nq Zv †evSvevi Rb¨ wb‡P Ab¨fv‡e we‡qvM K‡i †`Lv‡bv n‡jv :
8 243 5 24673

= 8 24343434 | 34 = 5 24673673 | 67
2 99669760 | 67

wb‡Y©q we‡qvMdj 2 99669760 | 67 GLv‡b `yBwU we‡qvMdjB GK| D`vniY 17| 24 45645 †_‡K 16 437 we‡qvM Ki| mgvavb :
24 45645 16 437

= 24 45645 = 16 43743

MwYZ

15

8 01902 1 8 01901

6 †_‡K 7 we‡qvM Ki‡j nv‡Z 1

wb‡Z n‡e|

wb‡Y©q we‡qvMdj 8.01901 `ªóe¨ :
24 45645 16 437

= 24 45645 | 64 = 16 43743 | 74
8 01901 | 90

KvR : we‡qvM Ki : 1| 13 12784 †_‡K 2| 23 0394 †_‡K 9 12645

Ave„Ë `kwg‡Ki ¸Y I fvM
Ave„Ë `kwgK¸‡jv‡K fMœvs‡k cwiYZ K‡i ¸Y ev fv‡Mi KvR mgvav K‡i cÖvß fMœvskwU‡K `kwg‡K cÖKvk Ki‡jB Ave„Ë `kwgK¸‡jvi ¸Ydj ev fvMdj n‡e| mmxg `kwgK I Ave„Ë `kwg‡Ki g‡a¨ ¸Y ev fvM Ki‡Z n‡j G wbq‡gB Ki‡Z n‡e| Z‡e fv‡Mi †¶‡Î fvR¨ I fvRK `yBwUB Ave„Ë `kwgK n‡j, Dfq‡K m`„k Ave„Ë `kwgK K‡i wb‡j fv‡Mi KvR mnR nq| D`vniY 18| 4 3 †K 5 7 Øviv ¸Y Ki| mgvavb : 4 3
5 7
4 3

43 4 9
57 5 9

39 9
52 9

13 3

5 7 =

13 52 3 9

676 27

25 037

wb‡Y©q ¸Ydj 25 037 D`vniY 19| 0 28 †K 42 18 Øviv ¸Y Ki|
8 2 26 13 90 90 45 4218 42 4176 464 42 18 99 99 11 13 464 6032 = 12 185 45 11 495 wb‡Y©q ¸Ydj 12 185

mgvavb : 0 28

16

MwYZ

D`vniY 20| 2 5 4 35 1 234 mgvavb : 2 5
25 10

KZ ?

5 2 435 43 392 4 35 90 90 1234 12 1222 611 1 234 990 990 495 5 392 611 196 611 119756 13 44062... 2 90 495 8910 8910

wb‡Y©q ¸Ydj 13 44062

KvR :
1| 1 13 †K 2 6 Øviv ¸Y Ki| 2| 0 2 1 12 0 081 KZ ?

D`vniY 21| 7 32 †K 0 27 Øviv fvM Ki| mgvavb : 7 32
0 27 732 7 99 725 99 25 5 90 18 5 725 18 = 99

7 32

27 2 90 725 0 27 = 99

18 290 5 = 11 = 26.36

wb‡Y©q fvMdj 26 36 D`vniY 22| 2 2718 †K 1 912 Øviv fvM Ki| mgvavb : 2 2718
22718 2 22176 9999 9999 1912 19 1893 1 912 990 990 22716 1893 22716 990 120 1 1881 2 2718 1 912 = 9999 990 9999 1893 101 wb‡Y©q fvMdj 1 1881 D`vniY 23| 9 45 †K 2 863 Øviv fvM Ki| 945 2863 28 945 990 mgvavb : 9 45 2 863 = 100 990 100 2835 189 99 33 3 3 2 2835 10 wb‡Y©q fvMdj 3 3

gš—e¨ : Ave„Ë `kwg‡Ki ¸Ydj Ges fvMdj Ave„Ë `kwgK n‡ZI cv‡i, bvI n‡Z cv‡i|

MwYZ

17

KvR :
1| 0 6 †K 0 9 Øviv fvM Ki| 2| 0 732 †K 0 027 Øviv fvM Ki|

Amxg `kwgK
A‡bK `kwgK fMœvsk Av‡Q hv‡`i `kwgK we›`yi Wv‡bi A‡¼i †kl †bB, Avevi GK ev GKvwaK A¼ evievi ch©vqµ‡g Av‡m bv, Gme `kwgK fMœvsk Amxg `kwgK fMœvsk| †hgb, 5 134248513942307......... GKwU Amxg `kwgK msL¨v| 2 Gi eM©g~j GKwU Amxg `kwgK| GLb, 2 G eM©g~j †ei Kwi|
1 2 1 4142135........ 1 24 100 96 281 400 281 2824 11900 11296 28282 60400 56564 282841 383600 282841 2828423 10075900 8485269 28284265 159063100 141421325 17641775

Gfv‡e cÖwµqv Abš—Kvj ch©š— Pj‡jI †kl n‡e bv|
2 1 4142135...... GKwU Amxg `kwgK msL¨v|

wbw`©ó `kwgK ¯’vb ch©š— gvb Ges wbw`©ó `kwgK ¯’vb ch©š— Avmbœ gvb Amxg `kwg‡Ki gvb †Kv‡bv wbw`©ó `kwgK ¯’vb ch©š— gvb †ei Kiv Ges †Kv‡bv wbw`©ó `kwgK ¯’vb ch©š— Avmbœ gvb †ei Kiv GKB A_© bq| †hgb, 5 4325893...... `kwgKwUi ÒPvi `kwgK ¯’vb ch©š— gvbÓ n‡e 5 4325, wKš‘ 5 4325893.... `kwgKwUi ÒPvi `kwgK ¯’vb ch©š— Avmbœ gvbÓ n‡e 5 4326 | GLv‡b Ò`yB `kwgK ¯’vb ch©š— gvbÓ Ges Ò`yB `kwgK ¯’vb ch©š— Avmbœ gvbÓ GKB hv 5 43 | mmxg `kwgKI Gfv‡e Avmbœ gvb †ei Kiv hvq| gš—e¨ : hZ `kwgK ¯’vb ch©š— gvb †ei Ki‡Z ejv n‡e, ZZ `kwgK ¯’vb ch©š— †h me msL¨v _vK‡e ûeû †m msL¨v¸‡jv wjL‡Z n‡e gvÎ| Avi hZ `kwgK ¯’vb ch©š—Avmbœ gvb †ei Ki‡Z ejv n‡e, Gi cieZ©x ¯’vbwU‡Z
5, 6, 7, 8 ev 9 nq, Z‡e †kl ¯’vbwUi msL¨vi mv‡_ 1 †hvM Ki‡Z n‡e| wKš‘ hw` 1 , 2 , 3 ev 4 nq, Z‡e

†kl ¯’vbwUi msL¨v †hgb wQj †ZgbB _vK‡e, G‡¶‡Î Ò`kwgK ¯’vb ch©š— gvbÓ Ges Ò`kwgK ¯’vb ch©š— Avmbœ gvbÓ GKB| hZ `kwgK ¯’vb ch©š— †ei Ki‡Z ejv n‡e, `kwg‡Ki ci Gi †P‡qI 1 ¯’vb †ewk ch©š— `kwgK msL¨v †ei Ki‡Z n‡e| dg©v-3, MwYZ-9g-10g

18

MwYZ

D`vniY 24| 13 Gi eM©g~j †ei Ki Ges wZb `kwgK ¯’vb ch©š— Avmbœ gvb †jL| mgvavb : 3 13 3 605551........
9 66

400 396
40000 36025

7205

72105

3697500 3605525 9197500 7211101

7211101

1986399

wb‡Y©q eM©g~j 3 605551........ wb‡Y©q wZb `kwgK ¯’vb ch©š— Avmbœ gvb 3 606 D`vniY 25| 4 4623845....... `kwgKwUi 1, 2, 3, 4 I 5 `kwgK ¯’vb ch©š— gvb I Avmbœ gvb †ei Ki| mgvavb : 4 4623845 msL¨vwUi GK `kwgK ¯’vb ch©š— gvb 4 4 Ges GK `kwgK ¯’vb ch©š— Avmbœ gvb 4 5 `yB `kwgK ¯’vb ch©š— gvb 4 46 Ges `yB `kwgK ¯’vb ch©š— Avmbœ gvb 4 46 wZb `kwgK ¯’vb ch©š— gvb 4 462 Ges wZb `kwgK ¯’vb ch©š— 4 462 Pvi `kwgK ¯’vb ch©š— 4 4623 Ges Pvi `kwgK ¯’vb ch©š— Avmbœ 4 4624 cuvP `kwgK ¯’vb ch©š— gvb 4 46238 Ges cuvP `kwgK ¯’vb ch©š— Avmbœ 4 46238 |

KvR : 29 Gi eM©g~j wbY©q Ki Ges eM©g~j‡K `yB `kwgK ¯’vb ch©š— gvb Ges `yB `kwgK ¯’vb ch©š— Avmbœ gvb †jL|

MwYZ

19

Abykxjbx 1
5 7

1| 2|

cÖgvY Ki †h, (K)
1 Ges 2

(L)

(M) 10 cÖ‡Z¨‡K Ag~j` msL¨v|

(K) 0.31 Ges 0.12 Gi g‡a¨ `yBwU Ag~j` msL¨v wbY©q Ki| (L)
2 Gi g‡a¨ GKwU g~j` Ges GKwU Ag~j` msL¨v wbY©q Ki|

3|

(K) cÖgvY Ki †h, †h‡Kv‡bv we‡Rvo c~Y© msL¨vi eM© GKwU we‡Rvo msL¨v| (L) cÖgvY Ki †h, `yBwU µwgK †Rvo msL¨vi ¸Ydj 8 (AvU) Øviv wefvR¨| Ave„Ë `kwgK fMœvs‡k cÖKvk Ki : (K)
1 6

4| 5| 6|

(L)

7 11

(M) 3

2 9

(N) 3

8 15

mvgvb¨ fMœvs‡k cÖKvk Ki : (K) 0 2 (L) 0 35 (M) 0 13 (N) 3 78 (O) 6 2309 m`„k Ave„Ë `kwgK fMœvs‡k cÖKvk Ki : (K) 2 3, 5 235 (L) 7 26, 4 237 (M) 5 7, 8 34, 6 245 (N) 12 32, 2 19, 4 3256

7| 8|

†hvM Ki : (K) 0 45 0 134 (L) 2 05 8 04 7 018 (M) 0 006 0 92 0 0134 we‡qvM Ki : (K) 3 4 2 13 (L) 5 12 3 45 (M) 8 49 5 356 (N) 19 345 13 2349

9|

¸Y Ki : (K) 0 3 0 6 (L) 2 4 0 81 (M) 0 62 0 3 (N) 42 18 0 28

10| fvM Ki : (K) 0 3 0 6 (L) 0 35 1 7 (M) 2 37 0 45 (N) 1 185 0 24 11| eM©g~j wbY©q Ki (wZb `kwgK ¯’vb ch©š—) Ges `yB `kwgK ¯’vb ch©š— eM©g~j¸‡jvi Avmbœ gvb †jL : (K) 12 (L) 0 25 (M) 1 34 (N) 5 1302 12| wb‡Pi †Kvb msL¨v¸‡jv g~j` Ges †Kvb msL¨v¸‡jv Ag~j` †jL : (K) 0 4 (L) 13| mij Ki : (K) ( 0 3 0 83 ) 0 5 0 1 + 0 35 0 08 (L) [(6.27 0.5) {(0.5 0.75) 8.36}] {(0.25 0.1) (0.75 21.3) 0.5} 14| 5 I 4 `yBwU ev¯—e msL¨v| K. †KvbwU g~j` I †KvbwU Ag~j` wb‡`©k Ki| L. 5 I 4 G‡`i g‡a¨ `yBwU Ag~j` msL¨v wbY©q Ki| M. cÖgvY Ki †h, 5 GKwU Ag~j` msL¨v|
9 (M)
6 (O) 11 (N) 3
8 (P) 7

2 27 (Q) 3 (R) 5 639 3 48 7

wØZxq Aa¨vq

†mU I dvskb
(Set and Function)
†mU kãwU Avgv‡`i mycwiwPZ †hgb : wWbvi †mU, ¯^vfvweK msL¨vi †mU, g~j` msL¨vi †mU BZ¨vw`| AvaywbK nvwZqvi wn‡m‡e †m‡Ui e¨envi e¨vcK| Rvg©vb MwYZwe` RR© K¨v›Ui (1844-1918) †mU m¤ú‡K© cÖ_g aviYv e¨vL¨v K‡ib| wZwb Amxg †m‡Ui aviYv cÖ`vb K‡i MwYZ kv‡¯¿ Av‡jvob m„wó K‡ib Ges Zuvi †m‡Ui aviYv †mU ZË¡ (Set Theory) bv‡g cwiwPZ| GB Aa¨v‡q †m‡Ui aviYv †_‡K MvwYwZK I wP‡ýi gva¨‡g mgm¨v mgvavb Ges dvskb m¤ú‡K© mg¨K Ávb AR©b Kiv cÖavb j¶| Aa¨vq †k‡l wk¶v_©xiv †mU I Dc‡m‡Ui aviYv e¨vL¨v K‡i cÖZx‡Ki mvnv‡h¨ cÖKvk Ki‡Z cvi‡e| †mU cÖKv‡ki c×wZ eY©bv Ki‡Z cvi‡e| Amxg †mU e¨vL¨v Ki‡Z cvi‡e Ges mmxg I Amxg †m‡Ui cv_©K¨ wbiƒcY Ki‡Z cvi‡e| †m‡Ui ms‡hvM I †Q` e¨vL¨v Ges hvPvB Ki‡Z cvi‡e| kw³ †mU e¨vL¨v Ki‡Z cvi‡e Ges `yB I wZb m`m¨wewkó †m‡Ui kw³ †mU MVb Ki‡Z cvi‡e| µg‡Rvo I Kv‡U©mxq ¸YR e¨vL¨v Ki‡Z cvi‡e| D`vniY I †fbwP‡Îi mvnv‡h¨ †mU cÖwµqvi mnR wewa¸‡jv cÖgvY Ki‡Z cvi‡e Ges wewa¸‡jv cÖ‡qvM K‡i wewfbœ mgm¨v mgvavb Ki‡Z cvi‡e| Aš^q I dvskb e¨vL¨v Ki‡Z I MVb Ki‡Z cvi‡e| †Wv‡gb I †iÄ Kx e¨vL¨v Ki‡Z cvi‡e| dvsk‡bi †Wv‡gb I †iÄ wbY©q Ki‡Z cvi‡e| dvsk‡bi †jLwPÎ A¼b Ki‡Z cvi‡e|

†mU (Set ) ev¯—e ev wPš—v RM‡Zi my-msÁvwqZ e¯‘i mgv‡ek ev msMÖn‡K †mU e‡j| †hgb, evsjv, Bs‡iwR I MwYZ wel‡q wZbwU cvV¨eB‡qi †mU| cÖ_g `kwU we‡Rvo ¯^vfvweK msL¨vi †mU, c~Y©msL¨vi †mU, ev¯—e msL¨vi †mU BZ¨vw`| †mU‡K mvaviYZ Bs‡iwR eY©gvjvi eo nv‡Zi A¶i A, B, C ,.......... X , Y , Z Øviv cÖKvk Kiv nq| †hgb, 2, 4, 6 msL¨v wZbwUi †mU A {2, 4, 6} †m‡Ui cÖ‡Z¨K e¯‘ ev m`m¨‡K †m‡Ui Dcv`vb (element ) ejv nq| †hgb, B {a, b} n‡j, B †m‡Ui Dcv`vb a Ges b Dcv`vb cÖKv‡ki wPý ' ' .

MwYZ

21

a b

B Ges cov nq a, B Gi m`m¨ (a belongs to B) B Ges cov nq b, B Gi m`m¨ (b belongs to B )

Dc‡ii B †m‡U c Dcv`vb †bB| c B Ges cov nq c, B Gi m`m¨ bq (c does not belong to B) .

†mU cÖKv‡ki c×wZ (Method of describing Sets) : †mU‡K `yB c×wZ‡Z cÖKvk Kiv nq| h_v : (1) ZvwjKv c×wZ ( Roster Method ev Tabular Method ) Ges (2) †mU MVb c×wZ ( Set Builder Method ) (1) ZvwjKv c×wZ : G c×wZ‡Z †m‡Ui mKj Dcv`vb mywbw`©ófv‡e D‡jL K‡i wØZxq eÜbx { } Gi g‡a¨ Ave× Kiv nq Ges GKvwaK Dcv`vb _vK‡j ÕKgvÕ e¨envi K‡i Dcv`vb¸‡jv‡K Avjv`v Kiv nq| †hgb, A {a, b} B {2, 4, 6} C {wbjq, wZkv, ïåv} BZ¨vw`| (2) †mU MVb c×wZ : G c×wZ‡Z †m‡Ui mKj Dcv`vb mywbw`©ófv‡e D‡jL bv K‡i Dcv`vb wba©vi‡Yi Rb¨ mvaviY a‡g©i D‡jL _v‡K| †hgb : A {x : x ¯^vfvweK we‡Rvo msL¨v}, B {x : x beg †kªwYi cÖ_g cuvPRb wk¶v_©x} BZ¨vw`| GLv‡b, ' : ' Øviv ÔGiƒc †hbÕ ev ms‡¶‡c Ô†hbÕ ( such that ) †evSvq| †h‡nZz G c×wZ‡Z †m‡Ui Dcv`vb wba©vi‡Yi Rb¨ kZ© ev wbqg (Rule) †`Iqv _v‡K, G Rb¨ G c×wZ‡K Rule Method I ejv nq| D`vniY 1| A {7, 14, 21, 28} †mUwU‡K †mU MVb c×wZ‡Z cÖKvk Ki| mgvavb : A †m‡Ui Dcv`vbmg~n 7, 14, 21, 28 GLv‡b, cÖ‡Z¨KwU Dcv`vb 7 Øviv wefvR¨, A_©vr 7 Gi ¸wYZK Ges 28 Gi eo bq|
A {x : x, 7 Gi ¸wYZK Ges x 28} .

D`vniY 2| B {x : x, 28 Gi ¸YbxqK} †mUwU‡K ZvwjKv c×wZ‡Z cÖKvk Ki| mgvavb : GLv‡b, 28 1 28 = 2 14 = 4 7
28 Gi ¸YbxqKmg~n 1, 2, 4, 7, 14, 28

wb‡Y©q †mU B {1, 2, 4, 7, 14, 28} D`vniY 3| C {x : x abvZ¥K c~Y©msL¨v Ges x 2 18 †mUwU‡K ZvwjKv c×wZ‡Z cÖKvk Ki| mgvavb : abvZ¥K c~Y©msL¨vmg~n 1, 2, 3, 4, 5, ........... GLv‡b, x 1 n‡j, x 2 12 1

22 x 2 n‡j, x 2 x

MwYZ
22 4 32 42 52 9 16 25 hv 18 Gi †P‡q eo

3 n‡j, x 2

x

4 n‡j, x 2

x 5 n‡j, x 2

kZ©vbymv‡i MÖnY‡hvM¨ abvZ¥K c~Y©msL¨vmg~n 1, 2, 3, 4 wb‡Y©q †mU C {1, 2, 3, 4}. KvR : 1| C { 9, 6, 3, 3, 6, 9} †mUwU‡K †mU MVb c×wZ‡Z cÖKvk Ki| 2| Q { y : y c~Y© msL¨v Ges y 3
27} †mUwU‡K ZvwjKv c×wZ‡Z cÖKvk Ki|

mmxg †mU (Finite Set) : †h †m‡Ui Dcv`vb msL¨v MYbv K‡i wba©viY Kiv hvq, G‡K mgxg †mU e‡j| †hgb,
D {x, y, z}, E {3, 6, 9,........, 60}, F {x : x †gŠwjK msL¨v Ges 30 x 70} BZ¨vw` mmxg

†mU| GLv‡b, D †m‡U 3 wU Dcv`vb, E †m‡U 20 wU Dcv`vb Ges F †m‡U 9 wU Dcv`vb Av‡Q| Amxg †mU (In inite Set) : †h †m‡Ui Dcv`vb msL¨v MYbv K‡i wba©viY Kiv hvq bv, G‡K Amxg †mU e‡j| †hgb, A {x : x we‡Rvo ¯^vfvweK msL¨v}, ¯^vfvweK msL¨vi †mU N {1, 2, 3, 4, ........} , c~Y©msL¨vi †mU
Z {....... 3, 2, 1, 0, 1, 2, 3.......} , g~j` msL¨vi †mU Q

P : p I q c~Y© msL¨v Ges q q

0} ,

ev¯—e msL¨vi †mU R BZ¨vw` Amxg †mU| D`vniY 4| †`LvI †h, mKj ¯^vfvweK msL¨vi †mU GKwU Amxg †mU| mgvavb : ¯^vfvweK msL¨vi †mU N {1, 2, 3, 4, 5, 6, 7, 8, .......}
N †mU †_‡K we‡Rvo ¯^vfvweK msL¨vmg~n wb‡q MwVZ †mU A {1, 3, 5, 7, .......}

†Rvo

Ó

Ó

Ó

Ó

Ó

B {2, 4, 6, 8, .......}

3 Gi ¸wYZKmg~‡ni †mU C

{3, 6, 9, 12, .......} BZ¨vw`|

GLv‡b, N †mU †_‡K MwVZ A, B, C †mUmg~‡n Dcv`vb msL¨v MYbv K‡i wba©viY Kiv hvq bv| d‡j
A, B, C Amxg †mU|

N GKwU Amxg †mU|

KvR : wb‡Pi †mU¸‡jv †_‡K mmxg †mU I Amxg †mU †jL : 1| {3, 5, 7} 2| {1, 2, 2 2 , .......210 } 3| {3, 32 , 33 , .......} 4| {x : x c~Y©msLv Ges x 4} 5| p : p I q ci¯úi mn‡gŠwjK Ges q 1} 6| { y : y q
N Ges y 2 100

y 3}

MwYZ

23

duvKv †mU (Empty Set) : †h †m‡Ui †Kv‡bv Dcv`vb †bB G‡K duvKv †mU e‡j| duvKv †mU‡K { } e cÖKvk Kiv nq| †hgb : nwjµm ¯‹z‡ji wZbRb Qv‡Îi †mU, {x msL¨v Ges 23 x 29} BZ¨vw`|
N : 10 x 11}, {x

Øviv

N : x †gŠwjK

†fbwPÎ (Venn-Diagram) : Rb †fb (1834-1883) †m‡Ui Kvh©wewa wP‡Îi mvnv‡h¨ cÖeZ©b K‡ib| G‡Z we‡ePbvaxb †mU¸‡jv‡K mgZ‡j Aew¯’Z wewfbœ AvKv‡ii R¨vwgwZK wPÎ †hgb AvqZvKvi †¶Î, e„ËvKvi †¶Î Ges wÎfyRvKvi †¶Î e¨envi Kiv nq| Rb †f‡bi bvgvbymv‡i wPÎ¸‡jv †fb wPÎ bv‡g cwiwPZ|

Dc‡mU (Subset) : A {a, b} GKwU †mU| A †m‡Ui Dcv`vb †_‡K {a, b}, {a } { b} †mU¸‡jv MVb
Kiv hvq| Avevi, †Kv‡bv Dcv`vb bv wb‡q GLv‡b, MwVZ {a, b}, {a } { b} Dc‡m‡Ui wPý †mU MVb Ki hvq|

cÖ‡Z¨KwU A †m‡Ui Dc‡mU|
A cov nq| B, A Gi Dc‡mU A_ev B is

myZivs †Kv‡bv †mU †_‡K hZ¸‡jv †mU MVb Kiv hvq, G‡`i cÖ‡Z¨KwU †mU‡K H †m‡Ui Dc‡mU ejv nq| | hw` B †mU A Gi Dc‡mU nq Z‡e B

a Subset of A. Dc‡ii Dc‡mU¸‡jvi g‡a¨ {a, b} †mU A Gi mgvb|

cÖ‡Z¨KwU †mU wb‡Ri Dc‡mU| Avevi, †h‡Kv‡bv – 3 †mU †_‡K †h‡Kv‡bv †m‡Ui Dc‡mU|
P {1, 2, 3} Gi Q {1, 2, 3} Ges R {1, 3} `yBwU Dc‡mU| Avevi, P Q

†mU MVb Kiv hvq|

Q

P Ges R

P

cÖK…Z Dc‡mU (Proper Subset) :
†Kv‡bv †mU †_‡K MwVZ Dc‡m‡Ui g‡a¨ †h Dc‡mU¸‡jvi Dcv`vb msL¨v cÖ`Ë †m‡Ui Dcv`vb msL¨v A‡c¶v Kg G‡`i‡K cÖK…Z Dc‡mU e‡j| †hgb, A {3, 4, 5, 6} Ges B {3, 5} `yBwU †mU| GLv‡b, B Gi me Dcv`vb A †m‡U we`¨gvb
B A

Avevi, B †m‡Ui Dcv`vb msL¨v A †m‡Ui Dcv`vb msL¨vi †P‡q Kg|
B, A Gi GKwU cÖK…Z Dc‡mU Ges B

A wj‡L cÖKvk Kiv nq|

D`vniY 5| P {x, y, z} Gi Dc‡mU¸‡jv †jL Ges Dc‡mU¸‡jv †_‡K cÖK…Z Dc‡mU evQvB Ki|

24

MwYZ

mgvavb : †`Iqv Av‡Q, P {x, y, z}
P Gi Dc‡mUmg~n {x, y, z} {x, y} {x, z} { y, z} {x} {y} {z}, P Gi cÖK…Z Dc‡mUmg~n {x, y} {x, z} { y, z} {x} {y} {z}

.

†m‡Ui mgZv (Equivalent Set) :
`yB ev Z‡ZvwaK †m‡Ui Dcv`vb GKB n‡j, G‡`i‡K †m‡Ui mgZv ejv nq| †hgb : A {3, 5, 7} Ges
B {5, 3, 7} `yBwU mgvb †mU Ges A
B wPý Øviv †jLv nq|

Avevi, A {3, 5, 7} B {5, 3, 3, 7} Ges C {7, 7, 3, 5, 5} n‡j A, B I C †mU wZbwU mgZv †evSvq| A_©vr, A B C j¶Yxq, †m‡Ui Dcv`vb¸‡jvi µg e`jv‡j ev †Kv‡bv Dcv`vb cybive„wË Ki‡j †m‡Ui †Kv‡bv cwieZ©b nq bv|

†m‡Ui Aš—i (Difference of Set) : g‡b Kwi, A {1, 2, 3, 4, 5} Ges B {3, 5} | †mU A †_‡K
†mU B Gi Dcv`vb¸‡jv ev` w`‡j †h †mUwU nq Zv {1, 2, 4} Ges †jLv nq A \ B ev A B
{1, 2, 3, 4, 5} {3, 5} {1, 2, 4}

myZivs, †Kv‡bv †mU †_‡K Ab¨ GKwU †mU ev` w`‡j †h †mU MwVZ nq Zv‡K ev` †mU e‡j| D`vniY 6| P {x : x, 12 Gi ¸YbxqKmg~n} Ges Q {x : x, 3 Gi ¸wYZK Ges x 12} n‡j P Q wbY©q Ki| mgvavb : †`Iqv Av‡Q, P {x : x, 12 Gi ¸YbxqKmg~n} GLv‡b, 12 Gi ¸YbxqKmg~n 1, 2, 3, 4, 6, 12
P {1, 2, 3, 4, 6, 12}

Avevi, Q {x : x, 3 Gi ¸wYZK Ges x 12} GLv‡b, 12 ch©š— 3 Gi ¸wYZKmg~n 3, 6, 9, 12
Q {3, 6, 9, 12} {1, 2, 3, 4, 6, 12) {3, 6, 9, 12} {1, 2, 4}

P Q

wb‡Y©q †mU : {1, 2, 4}

mvwe©K †mU (Universal Set) :
Av‡jvPbv mswkó mKj †mU GKwU wbw`©ó †m‡Ui Dc‡mU| †hgb : A {x, y} †mUwU B {x, y, z} Gi GKwU Dc‡mU| GLv‡b, B †mU‡K A †m‡Ui mv‡c‡¶ mvwe©K †mU e‡j| myZivs Av‡jvPbv mswkó mKj †mU hw` GKwU wbw`©ó †m‡Ui Dc‡mU nq Z‡e H wbw`©ó †mU‡K Gi Dc‡mU¸‡jvi mv‡c‡¶ mvwe©K †mU e‡j|

MwYZ

25

mvwe©K †mU‡K mvaviYZ U Øviv cÖKvk Kiv nq| Z‡e Ab¨ cÖZx‡Ki mvnv‡h¨I mvwe©K †mU cÖKvk Kiv hvq| †hgb : mKj †Rvo ¯^vfvweK msL¨vi †mU
C {2, 4, 6........} Ges mKj ¯^vfvweK msL¨vi †mU N {1, 2, 3, 4,...........}

n‡j, C †m‡Ui mv‡c‡¶ mvwe©K †mU n‡e N .

c~iK †mU (Complement of a Set) :
U mvwe©K †mU Ges A †mUwU U Gi Dc‡mU| A †m‡Ui ewnf©~Z mKj Dcv`vb

wb‡q MwVZ †mU‡K A †m‡Ui c~iK †mU e‡j| A Gi c~iK †mU‡K Ac ev A Øviv cÖKvk Kiv nq| MvwYwZKfv‡e Ac
U\A

g‡b Kwi, P I Q `yBwU †mU Ges Q †m‡Ui †hme Dcv`vb P †m‡Ui Dcv`vb bq, H Dcv`vb¸‡jvi †mU‡K P Gi †cÖw¶‡Z Q Gi c~iK †mU ejv nq Ges †jLv nq Q c
P \ Q.

D`vniY 7| U {1, 2, 3, 4, 6, 7} A {2, 4, 6, 7} Ges B {1, 3, 5} n‡j Ac I B c wbY©q Ki| mgvavb : Ac U \ A {1, 2, 3, 4, 6, 7} \ {2, 4, 6, 7} {1, 3, 5} Ges B c U \ B {1, 2, 3, 4, 6, 7} \ {1, 3, 5} {2, 4, 6, 7} wb‡Y©q †mU Ac {1, 3, 5} Ges B c {2, 4, 6, 7} ms‡hvM †mU (Union of Sets) : `yB ev Z‡ZvwaK †m‡Ui mKj Dcv`vb wb‡q MwVZ †mU‡K ms‡hvM †mU ejv nq| g‡b Kwi, A I B `yBwU †mU| A I B †m‡Ui ms‡hvM‡K A B Øviv cÖKvk Kiv nq Ges cov nq A ms‡hvM B A_ev A Union B | †mU MVb c×wZ‡Z A B {x : x A A_ev x B} D`vniY 8| C {3, 4, 5} Ges D {4, 6, 8} n‡j, C mgvavb : †`Iqv Av‡Q, C {3, 4, 5} Ges D {4, 6, 8}
C D {3, 4, 5} {4, 6, 8} {3, 4, 5, 6, 8}
D wbY©q Ki|

†Q` †mU (Intersection of Sets): `yB ev Z‡ZvwaK †m‡Ui mvaviY Dcv`vb wb‡q MwVZ †mU‡K †Q` †mU e‡j| g‡b Kwi, A I B `yBwU †mU|
A I B Gi †Q` †mU‡K A B Øviv cÖKvk Kiv nq Ges cov nq A †Q` B ev A intersection B |

†mU MVb c×wZ‡Z A B {x : x A Ges x B} D`vniY 9| P {x
N :2 x 6} Ges Q {x N : x †Rvo msL¨v Ges x 8}

n‡j, P Q wbY©q Ki| mgvavb : †`Iqv Av‡Q, P {x dg©v-4, MwYZ-9g-10g

N :2

x

6} {3, 4, 5, 6}

MwYZ

25

mvwe©K †mU‡K mvaviYZ U Øviv cÖKvk Kiv nq| Z‡e Ab¨ cÖZx‡Ki mvnv‡h¨I mvwe©K †mU cÖKvk Kiv hvq| †hgb : mKj †Rvo ¯^vfvweK msL¨vi †mU
C {2, 4, 6........} Ges mKj ¯^vfvweK msL¨vi †mU N {1, 2, 3, 4,...........}

n‡j, C †m‡Ui mv‡c‡¶ mvwe©K †mU n‡e N .

c~iK †mU (Complement of a Set) :
U mvwe©K †mU Ges A †mUwU U Gi Dc‡mU| A †m‡Ui ewnf©~Z mKj Dcv`vb

wb‡q MwVZ †mU‡K A †m‡Ui c~iK †mU e‡j| A Gi c~iK †mU‡K Ac ev A Øviv cÖKvk Kiv nq| MvwYwZKfv‡e Ac
U\A

g‡b Kwi, P I Q `yBwU †mU Ges Q †m‡Ui †hme Dcv`vb P †m‡Ui Dcv`vb bq, H Dcv`vb¸‡jvi †mU‡K P Gi †cÖw¶‡Z Q Gi c~iK †mU ejv nq Ges †jLv nq Q c
P \ Q.

D`vniY 7| U {1, 2, 3, 4, 6, 7} A {2, 4, 6, 7} Ges B {1, 3, 5} n‡j Ac I B c wbY©q Ki| mgvavb : Ac U \ A {1, 2, 3, 4, 6, 7} \ {2, 4, 6, 7} {1, 3, 5} Ges B c U \ B {1, 2, 3, 4, 6, 7} \ {1, 3, 5} {2, 4, 6, 7} wb‡Y©q †mU Ac {1, 3, 5} Ges B c {2, 4, 6, 7} ms‡hvM †mU (Union of Sets) : `yB ev Z‡ZvwaK †m‡Ui mKj Dcv`vb wb‡q MwVZ †mU‡K ms‡hvM †mU ejv nq| g‡b Kwi, A I B `yBwU †mU| A I B †m‡Ui ms‡hvM‡K A B Øviv cÖKvk Kiv nq Ges cov nq A ms‡hvM B A_ev A Union B | †mU MVb c×wZ‡Z A B {x : x A A_ev x B} D`vniY 8| C {3, 4, 5} Ges D {4, 6, 8} n‡j, C mgvavb : †`Iqv Av‡Q, C {3, 4, 5} Ges D {4, 6, 8}
C D {3, 4, 5} {4, 6, 8} {3, 4, 5, 6, 8}
D wbY©q Ki|

†Q` †mU (Intersection of Sets): `yB ev Z‡ZvwaK †m‡Ui mvaviY Dcv`vb wb‡q MwVZ †mU‡K †Q` †mU e‡j| g‡b Kwi, A I B `yBwU †mU|
A I B Gi †Q` †mU‡K A B Øviv cÖKvk Kiv nq Ges cov nq A †Q` B ev A intersection B |

†mU MVb c×wZ‡Z A B {x : x A Ges x B} D`vniY 9| P {x
N :2 x 6} Ges Q {x N : x †Rvo msL¨v Ges x 8}

n‡j, P Q wbY©q Ki| mgvavb : †`Iqv Av‡Q, P {x
N :2 x 6} {3, 4, 5, 6}

MwYZ

27

D`vniY 11| (2x

y, 3)

(6, x

y) n‡j, ( x, y) wbY©q Ki|

mgvavb : †`Iqv Av‡Q (2x Ges x y, 3)

(6, x

y)

µg‡Rv‡oi kZ©g‡Z, 2x y 6........(1) y 3.........(2)

mgxKiY (1) I (2) †hvM K‡i cvB, 3x 9 ev x 3 mgxKiY (1) G x Gi gvb ewm‡q cvB, 6 y
( x, y) (3, 0) 6 ev y 0

Kv‡Z©mxq ¸YR (Cartesian Product) :
Iqvsmy Zuvi evwoi GKwU Kvgivi wfZ‡ii †`Iqv‡j mv`v ev bxj is Ges evB‡ii †`Iqv‡j jvj ev njy` ev meyR is Gi cÖ‡jc †`Iqvi wm×vš— wb‡jb| wfZ‡ii †`Iqv‡ji is Gi †mU A †`Iqv‡j is Gi †mU B {mv`v, bxj} Ges evB‡ii

{jvj, njy` I meyR}| Iqvsmy Zuvi Kvgivi is cÖ‡jc (mv`v, jvj), (mv`v, njy`),

(mv`v, meyR), (bxj, jvj), (bxj, njy`), (bxj, meyR) µg‡Rvo AvKv‡i w`‡Z cv‡ib| D³ µg‡Rv‡oi †mU‡K †jLv nq
A B

{(mv`v, jvj), (mv`v, njy`), (mv`v, meyR), (bxj, jvj), (bxj, njy`),(bxj, meyR)}

GwUB Kv‡Z©mxq ¸YR †mU| †mU MVb c×vwZ‡Z, A B {( x, y); x A Ges y B}
A B †K cov nq A µm B ev A cross B

D`vniY 12| P {1, 2, 3} Q {3, 4} Ges R mgvavb : †`Iqv Av‡Q, P {1, 2, 3} Q {3, 4} Ges R
P

P

Q n‡j, P R Ges R Q wbY©q Ki|

Q {1, 2, 3} {3, 4} {3}

P R {1, 2, 3} {3} {(1, 3), (2, 3), (3, 3)}

Ges R Q {3} {3, 4} {(3, 3), (3, 4)} KvR : 1| x 2 y ,1 3 1, x 3 y n‡j, (x, y) wbY©q Ki| 2
Q) R Ges (P Q) Q wbY©q Ki|

2| P {1, 2, 3}, Q {3, 4} Ges R {x, y} n‡j, (P

D`vniY 13| †h mKj ¯^vfvweK msL¨v Øviv 311 Ges 419 †K fvM Ki‡j cÖwZ †¶‡Î 23 Aewkó _v‡K G‡`i †mU wbY©q Ki| mgvavb : †h ¯^vfvweK msL¨v Øviv 311 Ges 419 †K fvM Ki‡j cÖwZ‡¶‡Î 23 Aewkó _v‡K, †m msL¨v n‡e
23 A‡c¶v eo Ges 311 23 288 Ges 419 23 396 Gi mvaviY ¸YbxqK|

28

MwYZ

g‡b Kwi, 23 A‡c¶v eo 288 Gi ¸YbxqKmg~‡ni †mU A Ges 396 Gi ¸YbxqKmg~‡ni †mU B GLv‡b, 288 1 288 2 144 3 96 4 72 6 48 8 36 9 32 12 24 16 18
A {24, 32, 36, 48, 72, 96, 144, 288}

Avevi, 396 1 396 2 198 3 132 4 99 6 66 9 44 11 36 12 33 18 22
B {33, 36, 44, 66, 99, 132, 198, 396} A B {24, 32, 36, 48, 72, 96, 144, 288} {33, 36, 44, 66, 99, 132, 198, 396} {36}

wb‡Y©q †mU {36} D`vniY 14| U {1, 2, 3, 4, 5, 6, 7, 8} A {1, 2, 6, 7} B {2, 3, 5, 6} Ges C {4, 5, 6, 7} n‡j, †`LvI †h, (i) ( A mgvavb : (i) wP‡Î GKwU AvqZ‡¶Î Øviv U Ges ci¯úi‡Q`x `yBwU e„Ë‡¶Î Øviv h_vµ‡g A, B †mU‡K wb‡`©k Kiv n‡jv| †mU
A
(A
A
B

B)

A

B Ges (ii) ( A

B)

C

(A

C)

(B

C)

Dcv`vb
B
B)
1, 2, 3, 5, 6, 7

4, 8 3, 4, 5, 8
1, 4, 7, 8

A

B
(A B) A

4, 8
B

mgvavb : (ii) wP‡Î GKwU AvqZ‡¶Î Øviv U Ges ci¯úi‡Q`x wZbwU e„Ë‡¶Î Øviv h_vµ‡g A, B, C †mU‡K wb‡`©k Kiv n‡jv| j¶ Kwi , †mU
A B ( A B)

Dcv`vb
2, 6 C 2, 4, 5, 6, 7
1, 2, 4, 5, 6, 7

A
B

C
C

2, 3, 4, 5, 6, 7 (B
C (A

(A
(A

C)
B)

C)
C)

2, 4, 5, 6, 7
(B C)

MwYZ

29

D`vniY 15| 100 Rb wk¶v_©xi g‡a¨ †Kv‡bv cix¶vq 92 Rb evsjvq 80 Rb MwY‡Z Ges 70 Rb Dfq wel‡q cvm K‡i‡Q| †fbwP‡Îi mvnv‡h¨ Z_¨¸‡jv cÖKvk Ki Ges KZRb wk¶v_©x Dfq wel‡q †dj K‡i‡Q, Zv wbY©q Ki| mgvavb : †fbwP‡Î AvqZvKvi †¶ÎwU 100 Rb wk¶v_©xi †mU U Ges evsjvq I MwY‡Z cvm wk¶v_©x‡`i †mU h_vµ‡g B I M Øviv wb‡`©k K‡i| d‡j †fbwPÎwU PviwU wb‡ñ` †m‡U wef³ n‡q‡Q, hv‡`i‡K
P, Q, R, F Øviv wPwýZ Kiv n‡jv|

GLv‡b, Dfq wel‡q cvm wk¶v_©x‡`i †mU Q B
P R

M , hvi m`m¨ msL¨v 70

ïay evsjvq cvm wk¶v_©x‡`i †mU, hvi m`m¨ msL¨v = 92 70 18 ïay MwY‡Z cvm wk¶v_©x‡`i †mU, hvi m`m¨ msL¨v = 80 70 10
Q R B M , GK Ges Dfq wel‡q cvm wk¶v_©x‡`i †mU, hvi m`m¨ msL¨v = 18 10 70 98

P
F

Dfq wel‡q †dj Kiv wk¶v_©x‡`i †mU, hvi m`m¨ msL¨v = 100 98 2 Dfq wel‡q †dj K‡i‡Q 2 Rb wk¶v_©x|

Abykxjbx 2⋅1
1| wb‡Pi †mU¸‡jv‡K ZvwjKv c×wZ‡Z cÖKvk Ki : (K) {x N : x2 9 Ges x3 130} (L) {x Z : x2 5 Ges x3 36} (M) {x N : x, 36 Gi ¸YbxqK Ges 6 Gi ¸wYZK} (N) {x N : x3 25 Ges x4 2|
264}

wb‡Pi †mU¸‡jv‡K †mU MVb c×wZ‡Z cÖKvk Ki : (K) {3, 5, 7, 9, 11} (L) {1, 2, 3, 4, 6, 9, 12, 18, 36}

(M) {4, 8, 12, 16, 20, 24, 28, 32, 36, 40} (N) { 4, 5, 6} 3|
A {2, 3, 4} B {1, 2, a} Ges C {2, a, b} n‡j, wb‡Pi †mU¸‡jv wbY©q Ki :

(K) B \ C 4|
U

(L) A

B

(M) A

C

(N) A

(B

C) (O) A

(B

C)

{1, 2, 3, 4, 5, 6, 7} B)
B)

A {1, 3, 5}

B {2, 4, 6} Ges C (ii) (B C)
B)

{3, 4, 5, 6, 7} n‡j, wbgœwjwLZ C
(A C) (B C)

†¶‡Î mZ¨Zv hvPvB Ki :
(i) ( A
(iii ) ( A

A
C

B
(A C) (B

B
C

C ) (iv) ( A

5|

Q {x, y} Ges R {m, n, } n‡j, P(Q) Ges P(R) wbY©q Ki|

30

MwYZ
B n‡j, †`LvI †h, P(C) Gi Dcv`vb msL¨v 2n ,

6|

A {a, b} B {a, b, c} Ges C

A

†hLv‡b n n‡”Q C Gi Dcv`vb msL¨v| 7| (K) ( x 1, y 2) ( y 2, 2 x 1) n‡j, x Ges y Gi gvb wbY©q Ki| (L) (ax cy, a2 c2) (0, ay cx) n‡j, (x, y) Gi gvb wbY©q Ki| (M) (6x y, 13) (1, 3x 2 y) n‡j, (x, y) wbY©q Ki| 8| (K) P {a} Q {b, c} n‡j, P Q Ges Q P wbY©q Ki| (L) A {3, 4, 5} B {4, 5, 6} Ges C {x, y} n‡j, ( A (M) P {3, 5, 7} Q {5, 7} Ges R 9|
P \ Q n‡j, (P

B) C wbY©q Ki|
Q) R wbY©q Ki|
B I A B wbY©q Ki|

A I B h_vµ‡g 35 Ges 45 Gi mKj ¸Ybxq‡Ki †mU n‡j, A

10| †h mKj ¯^vfvweK msL¨v Øviv 346 Ges 556 †K fvM Ki‡j cÖwZ‡¶‡Î 31 Aewkó _v‡K, G‡`i †mU wbY©q Ki| 11| †Kv‡bv †kªwYi 30 Rb wk¶v_©xi g‡a¨ 20 Rb dzUej Ges 15 Rb wµ‡KU †Ljv cQ›` K‡i| `yBwU †h †Kv‡bv GKwU †Ljv cQ›` K‡i Z`ªyc wk¶v_©xi msL¨v 10 ; KZRb wk¶v_©x `yBwU †LjvB cQ›` K‡i bv Zv †fb wP‡Îi mvnv‡h¨ wbY©q Ki| 12| 100 Rb wk¶v_©xi g‡a¨ †Kv‡bv cix¶vq 65% wk¶v_©x evsjvq, 48% wk¶v_©x evsjv I Bs‡iwR Dfq wel‡q cvm Ges 15% wk¶v_©x Dfq wel‡q †dj K‡i‡Q| (K) msw¶ß weeiYmn Ic‡ii Z_¨¸‡jv †fbwP‡Î cÖKvk Ki| (L) ïay evsjvq I Bs‡iwR‡Z cvm K‡i‡Q Zv‡`i msL¨v wbY©q Ki| (M) Dfq wel‡q cvm Ges Dfq wel‡q †dj msL¨vØ‡qi †gŠwjK ¸YbxqKmg~‡ni †mU `yBwUi ms‡hvM †mU wbY©q Ki|

Aš^q (Relation)
Avgiv Rvwb, evsjv‡`‡ki ivRavbx XvKv, fvi‡Zi ivRavbx w`jx Ges cvwK¯—v‡bi ivRavbx Bmjvgvev`| GLv‡b †`‡ki mv‡_ ivRavbxi GKwU Aš^q ev m¤úK© Av‡Q| G m¤úK© n‡”Q †`k-ivRavbx Aš^q| D³ m¤úK©‡K †mU AvKv‡i wbgœiƒ‡c †`Lv‡bv hvq :

Aš^q

evsjv‡`k fviZ cvwK¯—vb

XvKv w`jx Bmjvgvev`

†`k

ivRavbx

MwYZ

31

A_©vr †`k-ivRavbxi Aš^q

), (cvwK¯—vb, Bmjvgvev`)}|

hw` A I B `yBwU †mU nq Z‡e †mUØ‡qi Kv‡Z©mxq ¸YR A B †m‡Ui Aš—M©Z µg‡Rvo¸‡jvi Ak~b¨ Dc‡mU R †K A †mU n‡Z B †m‡Ui GKwU Aš^q ev m¤úK© ejv nq| GLv‡b, R †mU A B †m‡Ui GKwU Dc‡mU A_©vr, R D`vniY 15| g‡b Kwi, A {3, 5} Ges B {2, 4}
A B {3, 5} {2, 4} {(3, 2), (3, 4), (5, 2), (5, 4)}
R {(3, 2), (3, 4), (5, 2), (5, 4)}

A B

hw` x

y kZ© nq Z‡e, R {(3, 2), (5, 2), (5, 4)} y kZ© nq Z‡e, R {3, 4}

Ges hw` x

hLb A †m‡Ui GKwU Dcv`vb x I B †m‡Ui GKwU Dcv`vb y Ges (x, y) R nq, Z‡e †jLv nq x R y Ges cov nq x, y Gi mv‡_ Awš^Z ( x is related to y ) A_©vr Dcv`vb x , Dcv`vb y Gi mv‡_
R m¤úK©hy³|

Avevi, A †mU n‡Z A †m‡Ui GKwU Aš^q A_©vr R

A A n‡j, R †K A Gi Aš^q ejv nq|
A Gi m‡½ m¤úwK©Z y

myZivs A Ges B `yBwU †m‡Ui Dcv`vb¸‡jvi g‡a¨ m¤úK© †`Iqv _vK‡j x

B

wb‡q †h me µg‡Rvo (x, y) cvIqv hvq, G‡`i Ak~b¨ Dc‡mU n‡”Q GKwU Aš^q | D`vniY 16| hw` P {2, 3, 4} , Q {4, 6} Ges P I Q Gi Dcv`vb¸‡jvi g‡a¨ y we‡ePbvq _v‡K Z‡e Aš^q wbY©q Ki| mgvavb : †`Iqv Av‡Q, P {2, 3, 4} Ges Q {4, 6} cÖkœvbymv‡i, R {( x, y) : x P, y Q Ges y 2 x } GLv‡b, P Q {2, 3, 4} {4, 6} {(2, 4), (2, 6), (3, 4), (3, 6), (4, 4), (4, 6)}
R {(2, 4), (3, 6)}
2x m¤úK©

wb‡Y©q Aš^q {(2, 4), (3, 6)} D`vniY 17| hw` A {1, 2, 3} , B {0, 2, 4} Ges C I D Gi Dcv`vb¸‡jvi g‡a¨ x we‡ePbvq _v‡K, Z‡e Aš^q eY©bv Ki| mgvavb : †`Iqv Av‡Q, A {1, 2, 3} , B {0, 2, 4} cÖkœvbmv‡i, Aš^q R {( x, y) : x A, y B Ges x GLv‡b, A B {1, 2, 3}
{0, 2, 4} y 1} y 1 m¤úK©

= {(1, 0), (1, 2), (1, 4), (2, 0), (2, 2), (2, 4), (3, 0), (3, 2), (3, 4)}
R {(1, 2), (3, 4)}

32

MwYZ

KvR : hw` C {2, 5, 6} , D {4, 5} Ges C I D Gi Dcv`vb¸‡jvi g‡a¨ x y m¤úK© we‡ePbvq _v‡K Z‡e Aš^q wbY©q Ki|

dvskb (Function) : wb‡Pi A I B †m‡Ui Aš^q j¶ Kwi :

GLv‡b, hLb y x 2 , ZLb x 1 n‡j, y 3 x 2 n‡j, y 4 x 3 n‡j, y 5 A_©vr x Gi GK-GKwU gv‡bi Rb¨ y Gi gvÎ GKwU gvb cvIqv hvq Ges x I y -Gi g‡a¨ m¤úK© ˆZwi nq y x 2 Øviv| myZivs `yBwU PjK x Ges y Ggbfv‡e m¤úK©hy³ †hb x Gi †h‡Kv‡bv GKwU gv‡bi Rb¨ y Gi GKwU gvÎ gvb cvIqv hvq, Z‡e y †K x Gi dvskb ejv nq| x Gi dvskb‡K mvaviYZ y, f (x), g (x) F (x) BZ¨vw` Øviv cÖKvk Kiv nq| g‡b Kwi, y x2 2x 3 GKU dvskb| GLv‡b, x Gi †h †Kv‡bv GKwU gv‡bi Rb¨ y Gi GKwU gvÎ gvb cvIqv hv‡e| GLv‡b, x Ges y DfqB PjK Z‡e, x Gi gv‡bi Dci y Gi gvb wbf©ikxj| Kv‡RB x n‡”Q ¯^vaxb PjK Ges y n‡”Q Aaxb PjK| D`vniY 18| f ( x) x2 4x 3 n‡j, f ( 1) wbY©q Ki| mgvavb : †`Iqv Av‡Q, f ( x) x2 4x 3 f ( 1)

( 1)2 4( 1) 3 1 4 3 8

D`vniY 19| hw` g (x)

x3

ax2 3x 6 nq, Z‡e a Gi †Kvb gv‡bi Rb¨ g ( 2)

0 n‡e ?

mgvavb : †`Iqv Av‡Q, g ( x) x3 ax2 3x 6 g ( 2) ( 2)3 a( 2)2 3( 2) 6
8 4a 6 6 8 4a 4a 8

wKš‘ g ( 2) 0
4a 8 ev 4a 8 0

ev a a 2 2 n‡j, g ( 2) 0 n‡e|

MwYZ

33

†Wv‡gb (Domain) I †iÄ (Range)
†Kv‡bv Aš^‡qi µg‡Rvo¸‡jvi cª_g Dcv`vbmg~‡ni †mU‡K Gi †Wv‡gb Ges wØZxq Dcv`vbmg~‡ni †mU‡K Gi †iÄ ejv nq| g‡b Kwi, A †mU †_‡K B †m‡U R GKwU Aš^q A_©vr R
R Ges †iÄ‡K †iÄ R wj‡L cÖKvk Kiv nq|

A B . R G Aš—f©y³ µg‡Rvo¸‡jvi cÖ_g

Dcv`vb †mU n‡e R Gi †Wv‡gb Ges wØZxq Dcv`vbmg~‡ni †mU n‡e R Gi †iÄ| R Gi †Wv‡gb‡K †Wvg

D`vniY 20| Aš^q S {(2, 1), (2, 2), (3, 2), (4, 5)} Aš^qwUi †Wv‡gb I †iÄ wbY©q Ki| mgvavb : †`Iqv Av‡Q, S {(2, 1), (2, 2), (3, 2), (4, 5)}
S Aš^‡q µg‡Rvo¸‡jvi cÖ_g Dcv`vbmg~n 2, 2, 3, 4 Ges wØZxq Dcv`vbmg~n 1, 2, 2, 5 .

†Wvg S {2, 3, 4} Ges †iÄ S

(1, 2, 5}
A, y A Ges y x 1} n‡j, R †K ZvwjKv

D`vniY 21| A {0, 1, 2, 3} Ges R {(x, y) : x c×wZ‡Z cÖKvk Ki Ges †Wvg R I †iÄ R wbY©q Ki|

mgvavb : †`Iqv Av‡Q, A {0, 1, 2, 3} Ges R {( x, y) : x
R Gi ewY©Z kZ© †_‡K cvB , y

A, y

A Ges y

x 1}

x 1 x 1 Gi gvb wbY©q Kwi|

GLb, cÖ‡Z¨K x

A Gi Rb¨ y

x y

†h‡nZz 4

A Kv‡RB (3, 4)

R

R {(0, 1), (1, 2), (2, 3)}

†Wvg R {0, 1, 2} Ges †iÄ R {1, 2, 3} KvR : 1| S {( 3, 8), ( 2, 3), ( 1, 0), (0, 1), (1, 0), (2, 3)} n‡j, S Gi †Wv‡gb I †iÄ wbY©q Ki| 2| S {x, y ) : x A, y A Ges y x 1} , †hLv‡b A { 3, 2, 1, 0} | †Wvg S I †iÄ S wbY©q Ki|

dvsk‡bi †jLwPÎ (Graphs) dvsk‡bi wPÎiƒc‡K †jLwPÎ ejv nq| dvsk‡bi aviYv my¯úó Kivi †¶‡Î †jLwP‡Îi ¸i“Z¡ Acwimxg| divwm `vk©wbK I MwYZwe` †i‡b †`KvZ© me©cÖ_g exRMwYZ I R¨vwgwZi g‡a¨ m¤úK© ¯’vc‡b AMÖYx f~wgKv cvjb K‡ib| wZwb †Kv‡bv mgZ‡j ci¯úi j¤^fv‡e †Q`x `yBwU dvsk‡bi mvnv‡h¨

dg©v-5, MwYZ-9g-10g

34

MwYZ

we›`yi Ae¯’vb mywbw`©ófv‡e wbY©‡qi gva¨‡g mgZjxq R¨vwgwZ‡Z AvaywbK aviv cÖeZ©b K‡ib| wZwb ci¯úi j¤^fv‡e †Q`x mij‡iLv `yBwU‡K A¶‡iLv wn‡m‡e AvL¨vwqZ K‡ib Ges A¶Ø‡qi †Q` we›`y‡K g~jwe›`y e‡jb| †Kv‡bv mgZ‡j ci¯úi j¤^fv‡e †Q`x `yBwU mij‡iLv XOX Ges YOY AuvKv n‡jv| mgZ‡j Aew¯’Z †h‡Kv‡bv we›`yi Ae¯’vb GB †iLvØ‡qi gva¨‡g m¤ú~Y©iƒ‡c Rvbv m¤¢e| GB †iLvØ‡qi cÖ‡Z¨KwU‡K A¶ (axis) ejv nq| Abyf~wgK †iLv XOX †K x -A¶, Dj¤^ †iLv YOY †K y -A¶ Ges A¶Ø‡qi †Q`we›`y O †K g~jwe›`y ejv nq|

`yBwU A‡¶i mgZ‡j Aew¯’Z †Kv‡bv we›`y †_‡K A¶Ø‡qi j¤^ `~i‡Z¡i h_vh_ wPýhy³ msL¨v‡K H we›`yi ¯’vbv¼ ejv nq| g‡b Kwi, A¶Ø‡qi mgZ‡j Aew¯’Z P †h †Kv‡bv we›`y| P †_‡K XOX Ges YOY Gi Dci h_vµ‡g PN I PM j¤^ Uvwb| d‡j,
PM ON hv YOY n‡Z P we›`yi j¤^ `~iZ¡ Ges PN OM y hv XOX n‡Z P we›`yi j¤^ `~iZ¡| hw` PM

x Ges PN

nq, Z‡e P we›`yi ¯’vbv¼ ( x, y ) | GLv‡b, x †K fyR (abscissa) ev x ¯’vbv¼ Ges y †K †KvwU Ordinate) ev y ¯’vbv¼ ejv nq| D‡ wLZ ¯’vbv¼‡K Kv‡Z©mxq ¯’vbv¼ ejv nq| Kv‡Z©mxq ¯’vbv‡¼ mn‡RB dvsk‡bi R¨vwgwZK wPÎ †`Lv‡bv hvq| GRb¨ mvaviYZ x A¶ eivei ¯^vaxb Pj‡Ki gvb I y A¶ eivei Aaxb Pj‡Ki gvb emv‡bv nq| y = f(x) dvsk‡bi †jLwPÎ A¼‡bi Rb¨ †Wv‡gb †_‡K ¯^vaxb Pj‡Ki K‡qKwU gv‡bi Rb¨ Aaxb Pj‡Ki

Abyiƒc gvb¸‡jv †ei K‡i µg‡Rvo ˆZwi Kwi| AZtci µg‡Rvo¸‡jv x – y Z‡j ¯’vcb Kwi| cÖvß we›`y¸‡jv gy³ n‡¯Z †iLv †U‡b hy³ Kwi, hv y = f(x) dvsk‡bi †jLwPÎ| D`vniY 22| y = 2x dvsk‡bi †jLwPÎ A¼b Ki| †hLv‡b, – 3 x 3 mgvavb : – 3 x 3 †Wv‡g‡bi x-Gi K‡qKwU gv‡bi Rb¨ y Gi K‡qKwU gvb wbY©q K‡i ZvwjKv ˆZwi Kwi| x y
–3 –6 –2 –4 –1 –2

0 0

1 2

2 4

3 6

QK KvM‡R cÖwZ ¶z`ªe‡M©i evû‡K GKK a‡i, ZvwjKvq we›`y¸‡jv wPwýZ Kwi I gy³ n‡¯— †hvM Kwi|

MwYZ
1 x 1 f x

35

D`vniY 23| hw` f ( x)

3x 1 nq, Z‡e 3x 1

f

1

Gi gvb wbY©q Ki|
1

mgvavb : †`Iqv Av‡Q, f ( x)
1 f x f 1 x 1 x

3x 1 3x 1 3 x [ je I ni‡K x Øviv ¸Y K‡i ] 3 x

3

1 1 x 1 3 1 x
1

3 1 x 3 1 x

ev , f =
1

3 x 3 x [†hvRb-we‡qvRb K‡i] 3 x 3 x

= 2x =

6

3 3 wb‡Y©q gvb x x

D`vniY 24| hw` f ( y )

y3 3y2 1 1 nq, Z‡e †`LvI †h, f y (1 y ) y y3 3y2 1 y (1 y )

f (1 y )

mgvavb : †`Iqv Av‡Q, f ( y )
1 y
3

f

1 y

1 3 y

2

1

1 1 1 y y

1 3y y3 y3 y 1 y2

y2 1 3y y3 y 1 y ( y 1) 3 (1 y ) 3(1 y ) 2 1 Avevi, f (1 y ) (1 y ){1 (1 y )} 1 3 y 3 y 2 y 3 3(1 2 y y 2 ) 1 = (1 y )(1 1 y ) 1 3y 3y2 y3 3 6 y 3y2 1 = y (1 y ) 3 1 3y y (1 3 y y 3 ) = y (1 y ) y ( y 1) 1 3y y3 = y ( y 1) 1 f f (1 y ) y

1 3y y3 y3

36

MwYZ

Abykxjbx 2⋅2
1|
8 Gi ¸YbxqK †mU †KvbwU ?

(K) {8, 16, 24, ...........} (L) {1, 2, 3, 4, 8} 2|

(M) {2, 4, 8}

(N) {1, 2}

†mU C n‡Z †mU B G GKwU m¤úK© R n‡j wb‡Pi †KvbwU mwVK ? (K) R
C

(L) R

B

(M) R

C B

(N) C B

R

3|

A {6, 7, 8, 9, 10, 11, 12, 13} n‡j, wb‡Pi cÖkœ¸‡jvi DËi `vI : (i ) A †m‡Ui MVb c×wZ †KvbwU ?

4|

hw` A {3, 4}, B {2, 4} nq, Z‡e A I B Gi Dcv`vb¸‡jvi g‡a¨ x K‡i wi‡jkbwU wbY©q Ki|

y m¤úK© we‡ePbv

5|

hw` C {2, 5}, D {4, 6} Ges C I D Gi Dcv`vb¸‡jvi g‡a¨ x 1 y m¤úK©wU we‡ePbvq _v‡K Z‡e wi‡jkbwU wbY©q Ki|
1 2

6| 7| 8|

f ( x)

x 4 5 x 3 n‡j, f ( 1), f (2) Ges f

Gi gvb wbY©q Ki|
0 n‡e ?

hw` f ( y ) f ( x)

y 3 ky 3 4 y 8 nq, Z‡e k Gi †Kvb gv‡bi Rb¨ f ( 2)

x 3 6 x 2 11x 6 n‡j, f

x Gi †Kvb gv‡bi Rb¨ f ( x) 0 n‡e ?

1 1 2x 1 x2 9| hw` f ( x) nq, Z‡e Gi gvb wbY©q Ki| 1 2x 1 f 2 1 x 1 1 x2 x4 n‡j, †`LvI †h, g 2 10| g ( x) g(x2 ) 2 x x

11| wb‡Pi Aš^q¸‡jv †_‡K †Wv‡gb Ges †iÄ wbY©q Ki : (K) R {(2, 1), (2, 2), (2, 3)} (M) F (L) S {( 2, 4), ( 1, 1), (0, 0), (1, 1), (2, 4)}
5 , 2 2

1 5 , 0 , (1, 1), (1, 1), , 2 , 2 2

12| wb‡Pi Aš^q¸‡jv‡K ZvwjKv c×wZ‡Z cÖKvk Ki Ges †Wv‡gb I †iÄ wbY©q Ki : (K) R {( x, y ) : x
A, y A Ges x y 1} †hLv‡b A { 2, 1, 0, 1, 2}

(L) F {( x, y ) : x C , y C Ges x 2 y} †hLv‡b C { 1, 0, 1, 1, 3} 13| QK KvM‡R ( 3, 2), (0, 5),
1 5 , 2 6

we›`y¸‡jv ¯’vcb Ki|

14| QK KvM‡R (1, 2), ( 1, 1), (11, 7) we›`y wZbwU ¯’vcb K‡i †`LvI †h, we›`y wZbwU GKB mij‡iLvq Aew¯’Z|

MwYZ

37

(K) {x N : 6 x 13} (M) {x N : 6 x 13}

(L) {x N : 6 x 13} (N) {x N : 6 x 13} (M) {7, 11, 13} (N) {6, 9, 12} (N) A {9, 12}

(ii ) †gŠwjK msL¨v¸‡jvi †mU †KvbwU ?

(K) {6, 8, 10, 12} (L) {7, 9, 11, 13}
(iii ) 3 Gi ¸wYZK¸‡jvi †mU †KvbwU ?

(K) {6, 9}

(L) {6, 11}

(M) {9, 12}

(iv) e„nËg †Rvo msL¨vi ¸Ybxq‡Ki †mU †KvbwU ?

(K) {1, 13}

(L) {1, 2, 3, 6}

(M) {1, 3, 9}

(N) {1, 2, 3, 4, 6, 12 }

15. mvwe©K †mU U ={ x:x N Ges x we‡Rvo msL¨v} A = {x N : 2 x 7 B = { x N : 3 < x 5 Ges x 3 130} K. A †mU‡K ZvwjKv c×wZ‡Z cÖKvk Ki| L. A Ges C – B wbY©q Ki| M. B C Ges P(A C) wbY©q Ki|

38

MwYZ

Z…Zxq Aa¨vq

exRMvwYwZK ivwk
(Algebraical Expressions) exRMwY‡Z A‡bK mgm¨v mgvav‡b exRMvwYwZK m~Î e¨eüZ nq| Avevi A‡bK exRMvwYwZK ivwk we‡ lY K‡i Drcv`‡Ki gva¨‡g Dc¯’vcb Kiv n‡q _v‡K| ZvB G Aa¨v‡q exRMvwYwZK m~‡Îi mvnv‡h¨ mgm¨v mgvavb Ges ivwk‡K Drcv`‡K we‡ lY welqK welqe¯‘ wk¶v_©x Dc‡hvMx K‡i Dc¯’vcb Kiv n‡q‡Q| AwaKš‘ bvbvwea MvwYwZK mgm¨v exRMvwYwZK m~‡Îi mvnv‡h¨ Drcv`‡K we‡ lY K‡iI mgvavb Kiv hvq| c~‡e©i †kªwY‡Z exRMvwYwZK m~Îvewj I G‡`i mv‡_ m¤ú„³ Abywm×vš—¸‡jv m¤^‡Ü we¯—vwiZ Av‡jvPbv Kiv n‡q‡Q| G Aa¨v‡q H¸‡jv cybi“‡ L Kiv n‡jv Ges D`vni‡Yi gva¨‡g G‡`i KwZcq cÖ‡qvM †`Lv‡bv n‡jv| GQvovI G Aa¨v‡q eM© I N‡bi m¤cÖmviY, fvM‡kl Dccv`¨ cÖ‡qvM K‡i Drcv`‡K we‡ lY Ges ev¯—e mgm¨v mgvav‡b exRMvwYwZK m~‡Îi MVb I cÖ‡qvM m¤ú‡K© we¯—vwiZ Av‡jvPbv Kiv n‡q‡Q| Aa¨vq †k‡l wk¶v_©xiv exRMvwYwZK m~Î cÖ‡qvM K‡i eM© I N‡bi m¤cÖmviY Ki‡Z cvi‡e| fvM‡kl Dccv`¨ Kx e¨vL¨v Ki‡Z cvi‡e Ges Zv cÖ‡qvM K‡i Drcv`‡K we‡ lY Ki‡Z cvi‡e| ev¯—e mgm¨v mgvav‡bi Rb¨ exRMvwYwZK m~Î MVb Ki‡Z cvi‡e Ges m~Î cÖ‡qvM K‡i mgm¨v mgvavb Ki‡Z cvi‡e|

3⋅1 exRMvwYwZK ivwk cÖwµqv wPý Ges msL¨vwb‡`©kK A¶i cÖZxK Gi A_©‡evaK web¨vm‡K exRMvwYwZK ivwk ejv nq| †hgb,
2a 3b 4c GKwU exRMvwYwZK ivwk| exRMvwYwZK ivwk‡Z a, b, c, p, q, r, m, n, x, y, z, ..... BZ¨vw`

eY©gvjvi gva¨‡g wewfbœ Z_¨ cÖKvk Kiv nq| exRMvwYwZK ivwk msewjZ wewfbœ mgm¨v mgvav‡b GB mg¯— eY©gvjv‡K e¨envi Kiv nq| cvwUMwY‡Z ïay abvZ¥K msL¨v e¨eüZ nq, Ab¨w`‡K exRMwY‡Z k~b¨mn abvZ¥K I FYvZ¥K mKj msL¨v e¨envi Kiv nq| exRMwYZ‡K cvwUMwY‡Zi me©vqbK…Z iƒc ejv nq| exRMvwYwZK ivwk‡Z e¨eüZ msL¨v¸‡jv aªyeK (constant), G‡`i gvb wbw`©ó| exRMvwYwZK ivwk‡Z e¨eüZ A¶i cÖZxK¸‡jv PjK (variables), G‡`i gvb wbw`©ó bq, Giv wewfbœ gvb aviY Ki‡Z cv‡i|

3⋅2 exRMvwYwZK m~Îvewj exRMvwYwZK cÖZxK Øviv cÖKvwkZ †h‡Kv‡bv mvaviY wbqg ev wm×vš—‡K exRMvwYwZK m~Î ejv nq| mßg I Aóg †kªwY‡Z exRMvwYwZK m~Îvewj I GZ`msµvš— Abywm×vš—¸‡jv m¤^‡Ü Av‡jvPbv Kiv n‡q‡Q| G Aa¨v‡q H¸‡jv cybi“‡ L K‡i KwZcq cÖ‡qvM †`Lv‡bv n‡jv|

MwYZ

39

m~Î 1| (a b)2 a2 2ab b2 m~Î 2| (a b) 2 a 2 2ab b 2 gš—e¨ : m~Î 1 I m~Î 2 n‡Z †`Lv hvq †h, a2 b2 Gi mv‡_ 2ab A_ev A_©vr (a b)2 A_ev (a b)2 cvIqv hvq| m~Î 1 G b Gi ¯’‡j
{a ( b)}2 a2 2a( b) ( b)2

2ab †hvM Ki‡j GKwU c~Y©eM©,

b emv‡j m~Î 2 cvIqv hvq :

A_©vr, (a b)2 a2 2ab b2 Abywm×vš— 1| a2 Abywm×vš— 2| a2 Abywm×vš— 3| (a cÖgvY : (a b)2 b2 (a b)2 2ab

b2
b)2

(a b)2 2ab
(a b)2 4ab

a2 2ab b2

a2 2ab b2 4ab
(a b)2 4ab

Abywm×vš— 4| (a b)2 (a b)2 4ab cÖgvY : (a b)2 a2 2ab b2 a 2 2ab b2 4ab

(a b)2 4ab

Abywm×vš— 5| a 2

b2

( a b) 2 2

( a b) 2

cÖgvY : m~Î 1 I m~Î 2 n‡Z, a2 2ab b2 a2 2ab b2
2a2 2b2

(a b)2 (a b)2
(a b)2 (a b)2

†hvM K‡i, ev,

2(a2 b2 ) (a b)2 (a b)2

myZivs, (a 2 b 2 ) Abywm×vš— 6| ab

( a b) 2 2
2

( a b) 2 a b 2
2

a b 2

cÖgvY : m~Î 1 I m~Î 2 n‡Z, a2 2ab b2 (a b)2

a2 2ab b2

(a b)2 (a b)2 (a b)2

we‡hvM K‡i,

4ab

MwYZ
(iii) (a b c) 2

41

{a ( b) ( c)}2

= a2 ( b)2 ( c)2 2a( b) 2( b)( c) 2a( c) a2 b2 c2 2ab 2bc 2ac

D`vniY 1| (4x 5 y) Gi eM© KZ ? mgvavb : (4 x 5 y ) 2 (4 x) 2 2 (4 x) (5 y ) (5 y ) 2 = 16 x 2 40 xy 25 y 2 D`vniY 2| (3a 7b) Gi eM© KZ ? mgvavb : (3a 7b) 2 (3a) 2 2 (3a) (7b) (7b) 2 = 9a 2 42ab 49b 2 D`vniY 3| e‡M©i m~Î cÖ‡qvM K‡i 996 Gi eM© wbY©q Ki| mgvavb : (996) 2 (1000 4) 2 = (1000) 2 2 1000 4 (4) 2
1000000 8000 16 1000016 8000 992016

D`vniY 4| a b c d Gi eM© KZ ? mgvavb : (a b c d ) 2 {(a b) (c d )}2 = (a b) 2 2(a b)(c d ) (c d ) 2 a2 a2

2ab b 2
2ab b 2 c2

2(ac ad bc bd ) c 2
2ac 2ad d2 2bc 2bd

2cd d 2 c2 2cd d2

a2 b2

2ab 2ac 2ad

2bc 2bd

2cd

KvR : m~‡Îi mvnv‡h¨ eM© wbY©q Ki : 1| 3xy 2ax 2| 4 x 3 y 3| x 5 y 2 z
2(7 x 7 y 3z )(5 x 7 y 3 z ) (7 x 7 y 3 z ) 2 b D`vniY 5| mij Ki : (5 x 7 y 3z ) 2 mgvavb : awi, 5 x 7 y 3z cÖ`Ë ivwk = a 2
2.b.a b 2

a Ges 7 x 7 y 3 z

= a 2 2ab b 2 = ( a b) 2 = {(5 x 7 y 3z ) (7 x 7 y 3z )}2 [ a I b Gi gvb ewm‡q] = (5 x 7 y 3z 7 x 7 y 3z ) 2 = (12 x) 2 = 144x 2 dg©v-6, MwYZ-9g-10g

MwYZ
(iii) (a b c) 2

41

{a ( b) ( c)}2

= a2 ( b)2 ( c)2 2a( b) 2( b)( c) 2a( c)
= a2

b2 c2 2ab 2bc 2ac

D`vniY 1| (4x 5 y) Gi eM© KZ ? mgvavb : (4 x 5 y ) 2 (4 x) 2 2 (4 x) (5 y ) (5 y ) 2 = 16 x 2 40 xy 25 y 2 D`vniY 2| (3a 7b) Gi eM© KZ ? mgvavb : (3a 7b) 2 (3a) 2 2 (3a) (7b) (7b) 2 = 9a 2 42ab 49b 2 D`vniY 3| e‡M©i m~Î cÖ‡qvM K‡i 996 Gi eM© wbY©q Ki| mgvavb : (996) 2 (1000 4) 2 = (1000) 2 2 1000 4 (4) 2 = 1000000 8000 16 1000016 8000 = 992016 D`vniY 4| a b c d Gi eM© KZ ? mgvavb : (a b c d ) 2 {(a b) (c d )}2 = (a b) 2 2(a b)(c d ) (c d ) 2
= a2 = a2 = a2

2ab b 2
2ab b 2 b2 c2

2(ac ad bc bd ) c 2
2ac 2ad d2 2bc 2bd

2cd d 2 c2 2cd d2

2ab 2ac 2ad

2bc 2bd

2cd

KvR : m~‡Îi mvnv‡h¨ eM© wbY©q Ki : 1| 3xy 2ax 2| 4 x 3 y 3| x 5 y 2 z
2(7 x 7 y 3 z )(5 x 7 y 3 z ) (7 x 7 y 3z ) 2 b D`vniY 5| mij Ki : (5 x 7 y 3 z ) 2 mgvavb : awi, 5 x 7 y 3z cÖ`Ë ivwk = a 2
2.b.a b 2

a Ges 7 x 7 y 3 z

= a 2 2ab b 2 = ( a b) 2 = {(5 x 7 y 3z ) (7 x 7 y 3z )}2 [ a I b Gi gvb ewm‡q] = (5 x 7 y 3z 7 x 7 y 3z ) 2 = (12 x) 2 = 144x 2 dg©v-6, MwYZ-9g-10g

42

MwYZ
2 Ges xy 24 n‡j, x y Gi gvb KZ ?

D`vniY 6| x y mgvavb : ( x x y

y) 2
100

(x

y) 2
10

4 xy

(2) 2

4 24

4 96 100

D`vniY 7| hw` a 4

a 2b 2

b4

3 Ges a 2

ab b 2

3 nq, Z‡e a 2

b 2 Gi gvb KZ ?

mgvavb : a 4 a 2 b 2 b 4

(a 2 ) 2

2a 2 b 2

(b 2 ) 2

a 2b 2

= (a 2 b 2 ) 2 (ab) 2
(a 2 b 2 ab)(a 2 b 2 ab)

= (a 2 ab b 2 )(a 2 ab b 2 )
3 3(a 2 ab b 2 ) [gvb ewm‡q]
3 3 1

ev, a 2 ab b 2 GLb, a 2 ev, a 2 b 2 a2 b2 ab b 2

3 Ges a 2

ab b 2

1 †hvM K‡i cvB, 2(a 2

b2 )

4

4 2
2

2

D`vniY 8| cÖgvY Ki †h, (a b) 4 (a b) 4 mgvavb : (a b) 4 (a b) 4

8ab(a 2

b2 )

{(a b) 2 }2 {(a b) 2 }2

= {(a b) 2 (a b) 2 }{(a b) 2 (a b) 2 } = 2(a 2 b 2 ) 4ab [ (a b) 2 (a b) 2 = 8ab(a 2 b 2 )
(a b) 4 ( a b) 4 8ab(a 2 b2 ) 2(a 2 b 2 ) Ges (a b) 2 (a b) 2 4ab]

D`vniY 9| a b c 15 Ges a 2 b 2 c 2 mgvavb : GLv‡b, 2(ab bc ac)
(a b c)
2 2

83 n‡j, ab bc ac Gi gvb KZ ?
2

(a

2

b

2

c )

weKí c×wZ : Avgiv Rvwb,
(a b c) 2 (a 2 b2 c 2 ) 2(ab bc ac)

= (15) 83 = 225 83 = 142 ab bc ac 142 2 71

ev, (15) 2 83 2(ab bc ac) ev, 225 83 2(ab bc ac) ev, 2(ab bc ac) 142 ab bc ac 142 2 71

MwYZ

43

D`vniY 10| a b c

2 Ges ab bc ac 1 n‡j, (a b) 2

(b c) 2

(c a ) 2 Gi gvb KZ ?

mgvavb : (a b) 2 (b c) 2 (c a) 2 = a2
2ab b 2 b2 2bc c 2 c2 2ca a2

= (a 2 b 2 c 2 2ab 2bc 2ca) (a 2 b 2 c 2 ) = (a b c) 2 {(a b c) 2 = (2) 2 (2) 2 2 1 = 4 4 2 8 2 6 D`vniY 11| (2 x 3 y )(4 x 5 y ) †K `yBwU e‡M©i we‡qvMdjiƒ‡c cÖKvk Ki| mgvavb : awi, 2 x 3 y cÖ`Ë ivwk = ab = a Ges 4 x 5 y b
2

2(ab bc ac)}

a b 2

2

a b 2
2

2x 3y 4x 5 y 2
2

2x 3y 4x 5 y 2
2

2

[ a I b Gi gvb ewm‡q]

6x 2 y = 2

8y 2
2

2x

2(3x y ) 2(4 y x) 2 2 2 2 = (3x y ) (4 y x)

2

=

(2 x 3 y )(4 x 5 y )

(3x

y) 2

(4 y

x) 2

KvR : 1| mij Ki : (4 x 3 y ) 2 2| x y z 12 Ges x 2

2(4 x 3 y )(4 x 3 y ) (4 x 3 y ) 2 z2 50 n‡j, ( x y) 2 (y z)2 (z x) 2 Gi gvb wbY©q Ki|

y2

Abykxjbx 3⋅1
1| m~‡Îi mvnv‡h¨ eM© wbY©q Ki : (K) 2a 3b (L) 2ab 3bc (Q) 5 x 2 y (M) x 2

2 y2

(N) a

1 a

(O) 4 y 5 x

(P) ab c

(R) x 2 y 4 z

(S) 3 p 4q 5r

(T) 3b 5c 2a (U) ax by cz (Y) 997 (Z) 1007

(V) a b c d (W) 2a 3x 2 y 5 z (X) 101 2| mij Ki : (K) (2a 7) 2
2(2a 7)(2a 7) (2a 7) 2

(L) (3 x 2 y ) 2 2(3 x 2 y )(3 x 2 y ) (3x 2 y ) 2

44

MwYZ

(M) (7 p 3r 5 x) 2 2(7 p 3r 5 x)(8 p 4r 5 x) (8 p 4r 5 x) 2 (N) (2m 3n
p) 2 (2m 3n p) 2 2(2m 3n p)(2m 3n p)

(O) 6 35 6 35 2 6 35 3 65 3 65 3 65 (P) 5874 5874 3774 3774 7548 5874 (Q) (R) 3| 4| 5| 6| 7| 8| 9|
7529 7529 7519 7519 7529 7519 2345 2345 759 759 2345 759
4 Ges ab 60 n‡j, a b Gi gvb KZ ?

a b

a b 7 Ges ab 12 n‡j, a b Gi gvb KZ ?

a b 9m Ges ab 18m 2 n‡j, a b Gi gvb KZ ? x y 2 Ges xy 63 n‡j, x 2

y 2 Gi gvb KZ ?
1 x4 322 .

x 2x

1 x 2 x

4 n‡j, cÖgvY Ki †h, x 4 3 n‡j, x 2

1 Gi gvb KZ ? x2

a

1 a

2 n‡j, †`LvI †h, a 2
7 Ges a b

1 a2

a4

1 a4 b2 ) 24

10| a b

5 n‡j, cÖgvY Ki †h, 8ab(a 2

11| a b c 9 Ges ab bc ca 31 n‡j, a 2 b 2 c 2 Gi gvb wbY©q Ki| 12| a 2 b 2 c2 9 Ges ab bc ca 8 n‡j, (a b c) 2 Gi gvb KZ ?

13| a b c 6 Ges a 2 b 2 c 2 14 n‡j, (a b) 2 (b c) 2 (c a) 2 Gi gvb wbY©q Ki| 14| x y z 10 Ges xy
4 Ges z

yz

zx

31 n‡j, ( x

y) 2

(y

z)2

(z

x) 2 Gi gvb KZ ?

15| x 3, y

5 n‡j, 9 x 2 16 y 2

4z 2

24 xy 16 yz 12 zx Gi gvb wbY©q Ki|
2

16| cÖgvY Ki †h,

x 2

y

2

x 2

y

2

2

x2 2

y2

x2 2

y2

2

17| (a 2b)(3a 2c) †K `yBwU e‡M©i we‡qvMdjiƒ‡c cÖKvk Ki| 18| ( x 7)( x 9) †K `yBwU e‡M©i we‡qvMdjiƒ‡c cÖKvk Ki| 19| x 2 10 x 24 †K `yBwU e‡M©i we‡qvMdjiƒ‡c cÖKvk Ki| 20| a 4 a 2b 2 b4 8 Ges a 2 ab b 2 4 n‡j, (i ) a 2 b 2 (ii ) ab Gi gvb wbY©q Ki|

MwYZ

45

3⋅3 Nb msewjZ m~Îvewj m~Î 6| (a b) 3 cÖgvY : (a b) 3 a 3 3a 2 b 3ab 2 b 3

= a 3 b 3 3ab(a b)
(a b)(a b) 2

= (a b)(a 2 = a(a 2

2ab b 2 ) 2ab b 2 )

2ab b 2 ) b(a 2

= a 3 2a 2 b ab 2 a 2 b 2ab 2 b 3 a3 3a 2b 3ab 2 b3

= a 3 b 3 3ab(a b) Abywm×vš— 9| a 3 b 3 (a b) 3 3ab(a b) m~Î 7| (a b) 3 a3 3a 2b 3ab 2 b3

= a 3 b 3 3ab(a b) cÖgvY : (a b) 3 (a b)(a b) 2 = (a b)(a 2 2ab b 2 ) = a(a 2 2ab b 2 ) b(a 2 2ab b 2 ) = a 3 2a 2b ab 2 a 2b 2ab 2 b 3 = a 3 3a 2 b 3ab 2 b 3 = a 3 b 3 3ab(a b) j¶ Kwi : m~Î 6 G b Gi ¯’‡j
{a ( b)}3 a3 b emv‡j m~Î 7 cvIqv hvq :

( b) 3

3a( b){a ( b)}

A_©vr, (a b) 3

a 3 b 3 3ab(a b)

Abywm×vš— 10| a 3 b 3 (a b) 3 3ab(a b) m~Î 8| a 3 b 3
(a b)(a 2 ab b 2 )

cÖgvY : a 3 b 3 (a b) 3 3ab(a b)
(a b){(a b) 2
2

3ab}

= (a b)(a 2ab b 2 3ab) = (a b)(a 2 ab b 2 )

46

MwYZ

m~Î 9| a 3 b 3 (a b)(a 2 ab b 2 ) cÖgvY : a 3 b 3
( a b) 3 3ab(a b) (a b){(a b) 2 3ab}

= (a b)(a 2 2ab b 2 3ab) = (a b)(a 2 ab b 2 ) D`vniY 12| 2 x 3 y Gi Nb wbY©q Ki| mgvavb : (2 x 3 y ) 3
(2 x) 3 3(2 x) 2 3 y 3 2 x(3 y ) 2 (3 y ) 3 = 8 x 3 3 4 x 2 3 y 3 2 x 9 y 2 27 y 3 = 8 x 3 36 x 2 y 54 xy 2 27 y 3

D`vniY 13| 2 x y Gi Nb wbY©q Ki| mgvavb : (2 x y ) 3 (2 x) 3 3 (2 x) 2 y 3 2 x y 2 = 8 x 3 3 4 x 2 y 6 xy 2 y 3 = 8 x 3 12 x 2 y 6 xy 2 y 3 KvR : m~‡Îi mvnv‡h¨ Nb wbY©q Ki : 1| 3x 2 y 2| 3x 4 y 3| 397 y3 D`vniY 14| x 37 n‡j, 8 x 3 72 x 2 216 x 216 Gi gvb KZ ? mgvavb : 8 x 3 72 x 2 216 x 216 = (2 x) 3 3 (2 x) 2 6 3 2 x.(6) 2 (6) 3 = ( 2 x 6) 3 = (2 37 6) 3 = (74 6) 3 = (80) 3 = 512000 D`vniY 15| hw` x y 8 Ges xy 5 nq, Z‡e x 3 mgvavb : x 3 y3 8( x y) 2 y3 8( x y ) 2 Gi gvb KZ ?

[gvb ewm‡q]

= ( x y ) 3 3xy ( x y ) 8{( x y ) 2 4 xy} = (8) 3 3 5 8 8(8 2 4 5) = 83 15 8 8(64 20) = 83 15 8 8 84 [gvb ewm‡q]

MwYZ

47

8(8 2 15 84)

= 8(64 15 84) = 8 163 = 1304 D`vniY 16| a 2
3a 1 0 n‡j, a 3 3a 1 0 a2 1 ev, a 1 ev, a a
1 Gi gvb KZ ? a3

mgvavb : †`Iqv Av‡Q, a 2 ev, a 2 1 ev, a2 a 1 a
1 a3
1 a
3
3

3a 3

3
3

cÖ`Ë ivwk = a 3 = a =
3

3a

1 a a

3 3

1 a 1 [ a a

3]

=3 3 3 3 =0 D`vniY 17| mij Ki : (a b)(a 2 ab b 2 ) (b c)(b 2 bc c 2 ) (c a)(c 2 ca a 2 ) mgvavb : (a b)(a 2 ab b 2 ) (b c)(b 2 bc c 2 ) (c a)(c 2 ca a 2 ) = a3 b3 b3 c3 c3 a3 =0 D`vniY 18| hw` a mgvavb : †`Iqv Av‡Q, a
1 a 1 3 2
3 2 2 3

3

2 nq, Z‡e cÖgvY Ki †h, a 3 3 2

1 a3

18 3.

=
3

[je I ni‡K
2
3 2 3 2

3

2 Øviv ¸Y K‡i]

=
3

3
2

2 2
2

=

=

3

2

48

MwYZ
1 a

a

3

2

3

2

= GLb, a
3

3
1 a3

2 a 3
1 a
3
3

2
3 a

2 3
1 a a 1 a

= 2 3 = 23

32 3
3

[ a

1 a

2 3]

3

3 2 3

=8 3 3 6 3 = 24 3 6 3 = 18 3 (cÖgvwYZ) KvR : 1| x
2 n‡j, 27 x 3 54 x 2 36 x 8 Gi gvb KZ ?

2| a b 5 Ges ab 6 n‡j, a 3 b 3 4(a b) 2 Gi gvb wbY©q Ki| 3| x
5 3 n‡j, x 3
1 Gi gvb wbY©q Ki| x3

Abykxjbx 3⋅2
1| m~‡Îi mvnv‡h¨ Nb wbY©q Ki : (K) 2 x 5 (L) 2 x 2 3 y 2 (M) 4a 5 x 2 (N) 7m 2 2n (O) 403 (P)
998

(Q) 2a b 3c (R) 2 x 3 y z 2| mij Ki : (K) (4a 3b) 3 3(4a 3b) 2 (2a 3b) 3(4a 3b)(2a 3b) 2 (2a 3b) 3 (L) (2 x y ) 3 3(2 x y ) 2 (2 x y ) 3(2 x y )(2 x y ) 2 (2 x y ) 3 (M) (7 x 3b) 3 (5 x 3b) 3 6 x(7 x 3b)(5 x 3b) (N) ( x 15) 3 (16 x) 3 3( x 15)(16 x) (O) (a b c) 3 (a b c) 3 6(b c){a 2 (b c) 2 } (P) (m n) 6 (m n) 6 12mn(m 2 n 2 ) 2 (Q) ( x y )( x 2 xy y 2 ) ( y z )( y 2 yz z 2 ) ( z x)( z 2 zx x 2 )

(R) (2 x 3 y 4 z ) 3 (2 x 3 y 4 z ) 3 12 x{4 x 2 (3 y 4 z ) 2 }

MwYZ
36 n‡j, a 3

49

3| 4| 5| 6| 7| 8| 9|

a b 5 Ges ab

b 3 Gi gvb KZ ?

hw` a 3 b 3 x 19 Ges y

513 Ges a b
12 n‡j, 8 x 3

3 nq, Z‡e ab Gi gvb KZ ?
36 x 2 y 54 xy 2 27 y 3 Gi gvb wbY©q Ki|

hw` a 15 nq, Z‡e 8a 3 60a 2 150a 130 Gi gvb KZ ? a 7 Ges b 5 n‡j, (3a 5b) 3 n Ges a 3 (4b 2a) 3 b3 y3 3(a b)(3a 5b)(4b 2a) Gi gvb KZ 2 p3 3mn .

hw` a b m, a 2 b 2

p 3 nq, Z‡e †`LvI †h, m 3 xy ( x y) 2

hw` x y 1 nq, Z‡e, †`LvI †h, x 3

10| a b 3 Ges ab 2 n‡j, (K) a 2 ab b 2 Ges (L) a 3 b 3 Gi gvb wbY©q Ki| 11| a b 5 Ges ab 36 n‡j, (K) a 2 12| m 13| x
1 m a n‡j, m 3

ab b 2 Ges (L) a 3

b 3 Gi gvb wbY©q Ki|

1 Gi gvb wbY©q Ki| m3

1 x

p n‡j, x 3

1 Gi gvb wbY©q Ki| x3 1 a3 4.

14| hw` a

1 1 nq, Z‡e †`LvI †h, a 3 a

15| hw` a b c 0 nq, Z‡e †`LvI †h, (K) a 3 b 3 c 3 3abc 16| p q 17| 2 x 18| a 19| x 3
2 x

(L) q3 (b c) 2 3bc r3
1 x3

(c a ) 2 3ca

( a b) 2 3ab

1

r n‡j, †`LvI †h, p 3

3 pqr
63

3 n‡j, †`LvI †h, 8 x 3

6
1 x3

5 n‡j,

a6 1 Gi gvb wbY©q Ki| a3 3 2

18 3 n‡j, cÖgvY Ki †h, x

20| a 4 a 2 1 0 n‡j, cÖgvY Ki †h, a 3

1 a3

0

3⋅4 Drcv`‡K
†Kv‡bv ivwk `yB ev Z‡ZvwaK ivwki ¸Yd‡ji mgvb n‡j, †k‡lv³ ivwk¸‡jvi cÖ‡Z¨KwU‡K cÖ_‡gv³ ivwki Drcv`K ev ¸YbxqK ejv nq| †Kv‡bv exRMvwYwZK ivwki m¤¢ve¨ Drcv`K¸‡jv wbY©q Kivi ci ivwkwU‡K jä Drcv`K¸‡jvi ¸Ydjiƒ‡c cÖKvk Kiv‡K Drcv`‡K we‡ lY ejv nq| dg©v-7, MwYZ-9g-10g

50

MwYZ

exRMvwYwZK ivwk¸‡jv GK ev GKvwaK c`wewkó n‡Z cv‡i| †mRb¨ D³ ivwki Drcv`K¸‡jvI GK ev GKvwaK c`wewkó n‡Z cv‡i|

Drcv`K wbY©‡qi KwZcq †KŠkj :
(K) †Kv‡bv eûc`xi cÖ‡Z¨K c‡` mvaviY Drcv`K _vK‡j Zv cÖ_‡g †ei K‡i wb‡Z nq| †hgb :
(i ) 3a 2 b 6ab 2 12a 2 b 2 (ii ) 2ab( x y ) 2bc( x

3ab(a 2b 4ab) y ) 3ca( x y) (x y )(2ab 2bc 3ca)

(L) GKwU ivwk‡K c~Y© eM© AvKv‡i cÖKvk K‡i : D`vniY 1| 4 x 2 12 x 9 †K Drcv`‡K we‡ lY Ki| mgvavb : 4 x 2 12 x 9 (2 x) 2 2 2 x 3 (3) 2 = (2 x 3) 2
(2 x 3)(2 x 3)

D`vniY 2| 9 x 2 30 xy 25 y 2 †K Drcv`‡K we‡ lY Ki| mgvavb : 9 x 2 30 xy 25 y 2 = (3x) 2 2 3x 5 y (5 y ) 2 = (3 x 5 y ) 2
(3 x 5 y )(3 x 5 y ) (a b)(a b) m~Î cÖ‡qvM K‡i :

(M) GKwU ivwk‡K `yBwU e‡M©i Aš—iiƒ‡c cÖKvk K‡i Ges a 2 b 2 D`vniY 3| a 2 1 2b b 2 †K Drcv`‡K we‡ lY Ki| mgvavb : a 2 1 2b b 2 = a 2 (b 1) 2 a 2 (b 2 2b 1)

{a (b 1)}{a (b 1)}

= (a b 1)(a b 1) D`vniY 4| a 4 64b 4 †K Drcv`‡K we‡ lY Ki| mgvavb : a 4 64b 4
(a 2 ) 2 (8b 2 ) 2

= (a2)2 2 a2 8b2 (8b2)2 16a2b2 = (a2 8b2)2 (4ab)2 = (a2 8b2 4ab)(a2 8b2 4ab) = (a 2 4ab 8b 2 )(a 2 4ab 8b 2 ) KvR : Drcv`‡K we‡ lY Ki : 1| abx2 acx3 adx4 2| xa2 144xb2 3| x2 2xy 4 y 4

MwYZ

51

(N) x2 (a b)x ab ( x a)(x b) m~ÎwU e¨envi K‡i : D`vniY 5| x2 12x 35 †K Drcv`‡K we‡ lY Ki| mgvavb : x2 12x 35 x2 (5 7) x 5 7 = (x 5)(x 7) G c×wZ‡Z x2 px q AvKv‡ii eûc`xi Drcv`K wbY©q Kiv m¤¢e nq hw` `yBwU c~Y©msL¨v a I b wbY©q p Ges ab q nq| GRb¨ q Gi `yBwU ¯^wPý Drcv`K wb‡Z nq hv‡`i

Kiv hvq †hb, a b wPýhy³ n‡e|

exRMvwYwZK mgwó p nq| q 0 n‡j, a I b GKB wPýhy³ n‡e Ges q 0 n‡j, a I b wecixZ D`vniY 6| x2 5x 6 †K Drcv`‡K we‡ lY Ki| mgvavb : x2 5x 6 = (x 2)(x 3) D`vniY 7| x 2 2 x 35 †K Drcv`‡K we‡ lY Ki| mgvavb : x 2 2 x 35 x2 ( 7 5) x ( 7)( 5)

x2

( 2 3) x ( 2)( 3)

= ( x 7)(x 5) D`vniY 8| x2 x 20 †K Drcv`‡K we‡ lY Ki| mgvavb : x2 x 20 x2 (5 4) x (5)( 4)

= (x 5)(x 4) (O) ax2 bx c AvKv‡ii eûc`xi ga¨c` wefw³KiY c×wZ‡Z : ax2 bx c (rx p)(sx q) n‡e hw` ax 2 bx c rsx 2 x(rq sp ) x pq A_©vr, a rs, b rq sp Ges c pq nq|

myZivs, ac rspq (rq)( sp) Ges b rq sp AZGe, ax2 bx c AvKv‡ii eûc`xi Drcv`K wbY©q Ki‡Z n‡j ac , A_©vr, x2 Gi mnM Ges x ewR©Z c‡`i ¸Ydj‡K Ggb `yBwU Drcv`‡K cÖKvk Ki‡Z n‡e, hv‡`i exRMvwYwZK mgwó x Gi mnM b Gi mgvb nq| D`vniY 9| 12x2 35x 18 †K Drcv`‡K we‡ lY Ki| mgvavb : 12x2 35x 18

52

MwYZ

GLv‡b, 12 18 216 27 8 Ges 27 8 35
12 x 2 35 x 18 12 x 2 27 x 8 x 18

= 3x(4x 9) 2(4x 9) = (4x 9)(3x 2) D`vniY 10| 3x2 x 14 †K Drcv`‡K we‡ lY Ki| mgvavb : 3x2 x 14 3x2 7 x 6x 14 = x(3x 7) 2(3x 7) = (3x 7)(x 2) KvR : Drcv`‡K we‡ lY Ki : 1| x2 x 56

2| 16x3 46x2 15x

3| 12x2 17x 6

(P) GKwU ivwk‡K c~Y© Nb AvKv‡i cÖKvk K‡i : D`vniY 11| 8x3 36x2 y 54xy2 27 y3 †K Drcv`‡K we‡ lY Ki| mgvavb : 8x3 36x2 y 54xy2 27 y3 = (2x)3 3 (2x)2 3 y 3 2x (3 y)2 (3 y)3 = (2x 3 y)3 (Q) a 3
(2x 3 y)(2x 3 y)(2x 3 y)

b3

(a

b)( a 2

ab

b 2 ) Ges a 3 b 3 (a b)(a 2 ab b 2 ) m~Î `yBwU e¨envi K‡i:

D`vniY 12| Drcv`‡K we‡ lY Ki : (i) 8a3 27b3 (ii) a6 64 mgvavb : (i) 8a3 27b3 (2a)3 (3b)3
(2a 3b){(2a)2 2a 3b (3b)2}

= (2a 3b)(4a2 6ab 9b2)
(ii) a6 64

(a 2 ) 3 (4) 3

weKí wbqg : a 6 64

= (a2 4){(a2)2 a2 4 (4)2} = (a2 4)(a4 4a2 16) wKš‘ a Ges a4 = = = = = a6 2

(a 3 ) 2 (8) 2 8)(a 3 2 3 )(a 3 8) 23 )
(a – 2) (a2 + 2a + 4)

(a 3 (a 3

4
4a
2

a

2

2
2

2

(a 2)(a 2)
(a )
2
2 2

16

(4)

2

4a
2

2

(a
(a
(a

2
2
2

(a+2) (a2 – 2a + 4)

4)
4)
4)

2(a )(4) 4a
4a
2

2
2

(a 2)(a 2)(a 2

2a 4)(a 2 2a 4)

(2a)2

(a2

4 2a)(a2

4 2a)

(a2 2a 4)(a2 2a 4) 64 (a 2)(a 2)(a 2 2a 4)(a 2

2a 4)

MwYZ

53

KvR : Drcv`‡K we‡ lY Ki : 1| 2x4 16x 2| 8 a3 3a2b 3ab2 b3 3| (a b)3 (a b)3

(R) fMœvskmnMhy³ ivwki Drcv`K : fMœvskhy³ ivwki Drcv`K¸‡jv‡K wewfbœfv‡e cÖKvk Kiv hvq| †hgb, a3 Avevi, a3
1 27
1 27

a3

1 33

a

1 2 a 3

a 3

1 9
(1)3}

1 (27a3 1) 27

1 {(3a)3 27

1 (3a 1)(9a2 3a 1) 27

GLv‡b, wØZxq mgvav‡b PjK-msewjZ Drcv`K¸‡jv c~Y©msL¨v mnMwewkó| GB dj‡K cÖ_g mgvav‡bi g‡Zv cÖKvk Kiv hvq :
1 (3a 1)(9a2 3a 1) 27

=

1 1 (3a 1) (9a 2 3a 1) 3 9

= a

1 3

a2

a 3

1 9

D`vniY 13| x3 6x2 y 11xy 2 6 y3 †K Drcv`‡K we‡ lY Ki| mgvavb : x3 6x2 y 11xy 2 6 y3 = {x 3 3 x 2 2 y 3 x(2 y ) 2 (2 y ) 3 } xy 2 2 y 3 = (x 2 y)3 y2(x 2 y)

= ( x 2 y ){( x 2 y ) 2 = (x 2 y)(x 2 y = (x 2 y)(x 3 y)(x = (x
1 2 x 2

y 2}
y)

y)(x 2 y y)

y)(x 2 y)(x 3 y)

KvR : Drcv`‡K we‡ lY Ki : 1|
7 x 6 1 3

2| a3

1 8

3| 16 x 2 25 y 2 8 xz 10 yz

54

MwYZ

Abykxjbx 3⋅3
Drcv`‡K we‡ lY Ki (1 − 43) : 1| 3| 5| 7| 9| a2 ab ac bc
( x y )( x
9x2 24x 16

2|

ab a b 1

y ) ( x y )( y z ) ( x y )( z x) 4| ab( x y) bc( x y)

6| 8|

a4

27 a 2 1

x4 6x2 y 2

y4

(a2 b2 )( x2

y 2 ) 4abxy

4a2 12ab 9b2 4c2

10| 9x4 45a2 x2 36a4 12| 16x2 25 y 2 8xz 10 yz 14| x2 13x 36 16| a2 30a 216 18| a8 a4 2 20| x 2 37 x 650 22| 12 x 2 38 x 20 24| 4x4 27 x2 81 26| 3(a2 2a)2 22(a2 2a) 40

11| a 2 6a 8 y 2 2 y 13| 2b 2 c 2 2c 2 a 2 2a 2b 2 a 4 b 4 c 4 15| x4 x2 20 17| x6 y6 x3 y3 6 19| a2b2 8ab 105 21| 4x4 25x2 36 23| 9x2 y 2 5xy 2 14 y 2 25| ax2 (a2 1) x a

27| 14( x z)2 29( x z)(x 1) 15( x 1)2 28| (4a 3b)2 2(4a 3b)(a 2b) 35(a 2b)2 29| (a 1) x2 a2 xy (a 1) y 2 31| (a2 b2 )3 8a3b3 33| a3 6a2 12a 9 35| 8x3 12x2 6x 63 37| a3 39| 4a2
1 8

30| 24x4 3x 32| x3 3x2 3x 2 34| a3 9b3 (a b)3 36| 8a3 38| b3 27

a6 b6 27

1 1 2 4a 2 a 4a 41| ( x 5)(x 9) 15 43| ( x 1)(x 3)(x 5)(x 7) 64

40| (3a 1)3 (2a 3)3 42| ( x 2)( x 3)( x 4)( x 5) 48

44| †`LvI †h, x3 9x2 26x 24 ( x 2)(x 3)(x 4) 45| †`LvI †h, ( x 1)(x 2)(3x 1)(3x 4) (3x2 2x 1)(3x2 2x 8)

MwYZ

55

3⋅5 fvM‡kl Dccv`¨ (Remainder Theorem)
Avgiv wb‡Pi D`vniYwU j¶ Kwi :
6x2 6x
2

7 x 5 †K x 1 Øviv fvM Ki‡j fvMdj I fvM‡kl KZ ? 7 x 5 †K x 1 Øviv mvaviYfv‡e fvM Ki‡j cvB, x 1 6x2 7 x 5 6x 1 6x2 6x x 5 x 1
4

GLv‡b, x 1 fvRK, 6x2 7 x 5 fvR¨, 6 x 1 fvMdj Ges 4 fvM‡kl| Avgiv Rvwb, fvR¨ = fvRK fvMdj + fvM‡kl

GLb hw` Avgiv fvR¨‡K f (x) , fvMdj‡K h(x) , fvM‡kl‡K r I fvRK‡K ( x a) Øviv m~wPZ Kwi, Zvn‡j Dc‡ii m~Î †_‡K cvB, f ( x) ( x a) h( x) r , GB m~ÎwU a Gi mKj gv‡bi Rb¨ mZ¨|

Dfqc‡¶ x myZivs, r

a ewm‡q cvB,
0 h( a ) r r

f ( a ) ( a a ) h( a ) r

f (a)

AZGe, f (x) †K ( x a) Øviv fvM Ki‡j fvM‡kl nq f (a) . GB m~Î fvM‡kl Dccv`¨ (Remainder theorem) bv‡g cwiwPZ| A_©vr, abvZ¥K gvÎvi †Kv‡bv eûc`x f (x) †K ( x a) AvKv‡ii eûc`x Øviv fvM Ki‡j fvM‡kl KZ n‡e Zv fvM bv K‡i †ei Kivi m~ÎB n‡jv fvM‡kl Dccv`¨| fvRK eûc`x ( x a) Gi gvÎv 1, fvRK hw` fv‡R¨i Drcv`K nq, Zvn‡j fvM‡kl n‡e k~b¨| Avi hw` Drcv`K bv nq, Zvn‡j fvM‡kl _vK‡e Ges Zv n‡e Ak~b¨ †Kv‡bv msL¨v| cÖwZÁv : hw` f (x) Gi gvÎv abvZ¥K nq Ges a fvM‡kl nq f b a
0 nq, Z‡e f (x) †K (ax b) Øviv fvM Ki‡j

cÖgvY : fvRK ax b , (a 0) Gi gvÎv 1, myZivs Avgiv wjL‡Z cvwi, f ( x) (ax b) h( x) r f ( x) x b a a h( x ) r b Øviv fvM Ki‡j fvMdj nq, a h(x) Ges fvM‡kl nq r . a

a x

b a

h( x) r

†`Lv hv‡”Q †h, f (x) †K x

56

MwYZ b a f b a b a

GLv‡b, fvRK = x

myZivs fvM‡kl Dccv`¨ Abyhvqx, r

AZGe, f (x) †K (ax b) Øviv fvM Ki‡j fvM‡kl nq f

Abywm×vš— : ( x a), f ( x) Gi Drcv`K n‡e, hw` Ges †Kej hw` f (a) 0 nq| cÖgvY : awi, f (a) 0 AZGe, fvM‡kl Dccv`¨ Abyhvqx, f (x) †K ( x a) Øviv fvM Ki‡j fvM‡kl k~b¨ n‡e| A_©vr,
( x a), f ( x) Gi GKwU Drcv`K n‡e|

wecixZµ‡g, awi, ( x a), f ( x) Gi GKwU Drcv`K| AZGe, f ( x) ( x a) h( x) , †hLv‡b h(x) eûc`x| Dfqc‡¶ x f (a) f (a) a ewm‡q cvB, (a a) h(a) 0

0

myZivs, †Kv‡bv eûc`x f (x), (x a) Øviv wefvR¨ n‡e hw` Ges †Kej hw` f (a) 0 nq| GB m~Î Drcv`K Dccv`¨ (Factor theorem) bv‡g cwiwPZ| Abywm×vš— : ax b, a f b a 0 nq|

0 n‡j, ivwkwU †Kv‡bv eûc`x f (x) Gi Drcv`K n‡e, hw` Ges †Kej hw`

cÖgvY : a 0, ax b a x

b , f ( x) Gi Drcv`K n‡e, hw` Ges †Kej hw` x a b a

b = x a

b a

f (x) Gi GKwU Drcv`K nq| A_©vr, hw` Ges †Kej hw` f

0 nq| fvM‡kl Dccv‡`¨i mvnv‡h¨

Drcv`K wbY©‡qi GB c×wZ‡K k~b¨vqb c×wZI (Vanishing method) e‡j| D`vniY 1| x3 x 6 †K Drcv`‡K we‡ lY Ki| mgvavb : GLv‡b, f (x) Ges GLb, x 1, 1 ewm‡q †`wL, f (x) Gi gvb k~b¨ nq bv| wKš‘ x
2 ewm‡q †`wL, f (x) Gi gvb k~b¨ nq|

x3

x 6 GKwU eûc`x| Gi aªyec`

6 Gi Drcv`K¸‡jv n‡”Q

A_©vr, f (2) 2 3 2 6 8 2 6 0 .

MwYZ

57

myZivs, x 2 f (x) eûc`xwUi GKwU Drcv`K| f (x) x3 x 6 2x2 4 x 3x 6

x3 2x 2

= x2(x 2) 2x(x 2) 3(x 2) = (x 2)(x2 2x 3) D`vniY 2| x3 3xy2 2 y3 †K Drcv`‡K we‡ lY Ki| mgvavb : GLv‡b, x †K PjK Ges y †K aª“eK wn‡m‡e we‡ePbv Kwi| cÖ`Ë ivwk‡K x Gi eûc`x we‡ePbv K‡i awi, f (x) x3 3xy2 2 y3 3 y3 3 y3 0

Zvn‡j, f ( y)
(x

y3 3 y y 2 2 y3

y), f ( x) Gi GKwU Drcv`K|

GLb, x3 3xy2 2 y3 = x3 x2 y x2 y xy2 2xy2 2 y3 = x2( x y) xy( x y) 2 y 2( x y) = (x y)(x2 xy 2 y 2)

Avevi awi, g ( x) x2 xy 2 y 2 y2 y2 2y2 0

g ( y)
(x

y ), g ( x) Gi GKwU Drcv`K

= (x y)(x2 2xy xy 2 y2) = (x y){x(x 2 y) y(x 2 y)} = (x y)(x 2 y)(x y) = ( x y) 2 ( x 2 y)

x2

xy 2 y 2

= x 2 xy 2 xy 2 y 2 = x( x y ) 2 y ( x y ) = ( x y )( x 2 y ) x3 3 xy 2 2 y3 (x y) 2 ( x 2 y)

D`vniY 3| 54x4 27 x3a 16x 8a †K Drcv`‡K we‡ lY Ki| mgvavb : awi, f ( x) 54x4 27 x3a 16x 8a Zvn‡j, f = x 1 a 2

54

1 a 2

4

27 a
0

1 a 2

3

16

1 a 2

8a

27 4 a 8
1 a 2

27 4 a 8a 8a 8 x a A_©vr, 2x a, f ( x) Gi GKwU Drcv`K| 2

GLb, 54x4 27 x3a 16x 8a 27 x3(2x a) 8(2x a) (2x a)(27 x3 8)
(2 x a ){(3 x) 3 (2) 3 } (2x a)(3x 2)(9x2 6x 4) dg©v-8, MwYZ-9g-10g

58

MwYZ

KvR : Drcv`‡K we‡ lY Ki : 1| x3 21x 20 2| 2x3 3x2 3x 1 3| x3 6x2 11x 6

Abykxjbx 3 4
Drcv`‡K we‡ lY Ki : 1| 3| 5| 7| 9|
6x2 7x 1
6 y3

2| 4| 6| 8|

3a 3

2a 5

x3 7 xy2

x 2 5x 6

2x2 x3

x 3 2 x 2 5x 6

3x2 x3

7x 6 4 x2 x 6

a3 3a 36

10| a 4 4a 3 12| x3 3x2 4x 4 14| x3 x 24 16| 2 x 4 3x 3 3x 2 18| x6 x5 x4 x3 x2 x 20| 18x3 15x2 x 2

11| a3 a2 10a 8 13| a3 7a2b 7ab2 b3 15| x3 6x2 y 11xy2 6 y3 17| 4x4 12x3 7 x2 3x 2 19| 4x3 5x2 5x 1

3 6 ev¯—e mgm¨v mgvav‡b exRMvwYwZK m~Î MVb I cÖ‡qvM
ˆ`bw›`b Kv‡R wewfbœ mg‡q wewfbœfv‡e Avgiv ev¯—e mgm¨vi m¤§yLxb nB| GB mgm¨v¸‡jv fvlvMZfv‡e ewY©Z nq| G Aby‡”Q‡` Avgiv fvlvMZfv‡e ewY©Z ev¯—e cwi‡e‡ki wewfbœ mgm¨v mgvavbK‡í exRMvwYwZK m~Î MVb Ges Zv cÖ‡qvM Kivi c×wZ wb‡q Av‡jvPbv Kie| GB Av‡jvPbvi d‡j wk¶v_©xiv GKw`‡K †hgb ev¯—e cwi‡e‡k MwY‡Zi cÖ‡qvM m¤ú‡K© aviYv cv‡e, Ab¨w`‡K wb‡R‡`i cvwicvwk¦©K Ae¯’vq MwY‡Zi m¤ú„³Zv eyS‡Z †c‡i MwYZ wk¶vi cÖwZ AvMÖnx n‡e|

mgm¨v mgvav‡bi c×wZ :
(K) cÖ_‡gB mZK©Zvi mv‡_ mgm¨vwU ch©‡e¶Y K‡i Ges g‡bv‡hvM mnKv‡i c‡o †Kvb¸‡jv AÁvZ Ges Kx wbY©q Ki‡Z n‡e Zv wPwýZ Ki‡Z n‡e| (L) AÁvZ ivwk¸‡jvi GKwU‡K †h‡Kv‡bv PjK (awi x ) Øviv m~wPZ Ki‡Z n‡e| AZtci mgm¨vwU fv‡jvfv‡e Abyaveb K‡i Ab¨vb¨ AÁvZ ivwk¸‡jv‡KI GKB PjK x Gi gva¨‡g cÖKvk Ki‡Z n‡e| (M) mgm¨v‡K ¶z`ª ¶z`ª As‡k wef³ K‡i exRMvwYwZK ivwk Øviv cÖKvk Ki‡Z n‡e| (N) cÖ`Ë kZ© e¨envi K‡i ¶z`ª ¶z`ª Ask¸‡jv‡K GK‡Î GKwU mgxKi‡Y cÖKvk Ki‡Z n‡e| (O) mgxKiYwU mgvavb K‡i AÁvZ ivwk x Gi gvb wbY©q Ki‡Z n‡e| ev¯—e mgm¨v mgvav‡b wewfbœ m~Î e¨envi Kiv nq| m~Î¸‡jv wb‡P D‡jL Kiv n‡jv :

MwYZ

59

(1) †`q ev cÖvc¨ welqK : †`q ev cÖvc¨, A qn UvKv †hLv‡b, q = RbcÖwZ †`q ev cÖvc¨ UvKvi cwigvY

n = †jv‡Ki msL¨v

(2) mgq I KvR welqK : K‡qKRb †jvK GKwU KvR m¤úbœ Ki‡j, Kv‡Ri cwigvY, W †hLv‡b, qnx q = cÖ‡Z¨‡K GKK mg‡q Kv‡Ri †h Ask m¤úbœ K‡i,

n = KvR m¤úv`bKvixi msL¨v x = Kv‡Ri †gvU mgq
W n R‡b x mg‡q Kv‡Ri †h Ask m¤úbœ K‡i

(3) mgq I `~iZ¡ welqK : wbw`©ó mg‡q `~iZ¡, d vt . †hLv‡b, v = cÖwZ NÈvq MwZ‡eM

t = †gvU mgq

(4) bj I †PŠev”Pv welqK : wbw`©ó mg‡q †PŠev”Pvq cvwbi cwigvY, Q(t) Qo qt †hLv‡b,
Qo = b‡ji gyL Ly‡j †`Iqvi mgq †PŠev”Pvq Rgv cvwbi cwigvY|

q = cÖwZ GKK mg‡q bj w`‡q †h cvwb cÖ‡ek K‡i A_ev †ei nq| t = AwZµvš— mgq|
Q(t) t mg‡q †PŠev”Pvq cvwbi cwigvY (cvwb cÖ‡ek nIqvi k‡Z© Õ+Õ wPý Ges cvwb †ei

nIqvi k‡Z© Ô Õ wPý e¨envi Ki‡Z n‡e)| 5| kZKiv Ask welqK : p br .

†hLv‡b,

b = †gvU ivwk

r = kZKiv fMœvsk =

s 100

s%

p = kZKiv Ask = b Gi s%

6|

jvf-¶wZ welqK :
S C(I r)
C(I r)

jv‡fi †¶‡Î, S ¶wZi †¶‡Î, S

C(I

r)

60

MwYZ

†hLv‡b, S (UvKv) = weµqg~j¨
C (UvKv) = µqg~j¨
I = jvf ev gybvdv r = jvf ev ¶wZi nvi

(7) wewb‡qvM-gybvdv welqK : mij gybvdvi †¶‡Î,
I = Pnr UvKv A P I P

Pnr

P(1 nr) UvKv,

Pµe„w× gybvdvi †¶‡Î,
A P(1 r ) n

†hLv‡b,

I = n mgq c‡i gybvdv n = wbw`©ó mgq

P = g~jab r = GKK mg‡q GKK g~ja‡bi gybvdv A = n mgq c‡i gybvdvmn g~jab|

D`vniY 1| evwl©K µxov Abyôvb Kivi Rb¨ †Kv‡bv GK mwgwZi m`m¨iv 45,000 UvKvi ev‡RU Ki‡jb Ges wm×vš— wb‡jb †h, cÖ‡Z¨K m`m¨B mgvb Puv`v w`‡eb| wKš‘ 5 Rb m`m¨ Puv`v w`‡Z Am¤§wZ Rvbv‡jb| Gi d‡j cÖ‡Z¨K m`‡m¨i gv_vwcQz 15 UvKv Puv`v e„w× †cj| H mwgwZ‡Z KZRb m`m¨ wQ‡jb ? mgvavb : g‡b Kwi, mwgwZi m`m¨ msL¨v x Ges RbcÖwZ †`q Puv`vi cwigvY q UvKv| Zvn‡j. †gvU Puv`v, A qx UvKv cÖK…Zc‡¶ m`m¨ msL¨v wQj (x 5) Rb Ges Puv`v n‡jv (q 15) UvKv| Zvn‡j, †gvU Puv`v n‡jv (x 5)(q 15) cÖkœvbymv‡i, qx (x 5)(q 15)..........(i) Ges qx
45,000..........(ii)

mgxKiY (i) †_‡K cvB, qx ( x 5)(q 15) qx 5q 15x 75

ev, qx

ev, 5q 15x 75 5(3x 15) q 3x 15...........(iii)

mgxKiY (ii) G q Gi gvb ewm‡q cvB,
(3x 15) x 45000

MwYZ

61

ev, 3x2 15x

45000

ev, x2 5x 15000 [Dfqc¶‡K 3 Øviv fvM K‡i] ev, x2 5x 15000 0 ev, x2 125x 120x 15000 0 ev, x(x 125) 120(x 125) 0 ev, (x 125)(x 120) 0 myZivs, (x 125) 0 ev, x 125 A_ev ( x 120) 0 ev, x
120

†h‡nZz m`m¨ msL¨v FYvZ¥K n‡Z cv‡i bv, ZvB x Gi gvb 120 MÖnY‡hvM¨ bq| x 125

myZivs, mwgwZi m`m¨ msL¨v 125| D`vniY 2| iwdK GKwU KvR 10 w`‡b Ki‡Z cv‡i| kwdK H KvR 15 w`‡b Ki‡Z cv‡i| Zviv GK‡Î KZ w`‡b KvRwU †kl Ki‡Z cvi‡e ? mgvavb : g‡b Kwi, Zviv GK‡Î d w`‡b KvRwU †kl Ki‡Z cvi‡e| bvg KvR m¤úbœ Kivi w`b iwdK kwdK d 10 d 15

1 w`‡b cv‡i Kv‡Ri Ask
1 10

d w`‡b K‡i d 10

10
15

1 15

d 15

cÖkœvbymv‡i,

1

1 1 1 10 15 3 2 ev, d 1 30 5d ev, 1 30 30 ev, d 6 5

ev, d

myZivs, Zviv GK‡Î 6 w`‡b KvRwU †kl Ki‡Z cvi‡e|

62

MwYZ

D`vniY 3| GKRb gvwS †mªv‡Zi cÖwZK~‡j t1 NÈvq x wK.wg. †h‡Z cv‡i| †mªv‡Zi AbyK~‡j H c_ †h‡Z Zvi t2 NÈv jv‡M| †mªv‡Zi †eM I †bŠKvi †eM KZ ? mgvavb : awi, †mªv‡Zi †eM NÈvq v wK.wg. Ges w¯’i cvwb‡Z †bŠKvi †eM NÈvq u wK.wg.| Zvn‡j, †mªv‡Zi AbyK~‡j †bŠKvi Kvh©Kix †eM NÈvq (u v) wK.wg. Ges †mªv‡Zi cÖwZK~‡j †bŠKvi Kvh©Kix †eM NÈvq (u v) wK.wg.|
AwZµvš— `~iZ¡ x .......(i) [†h‡nZz, †eM = mgq t2 x Ges u v .......(ii) t1 mgxKiY (i) I (ii) †hvM K‡i cvB,

cÖkœvbymv‡i, u v

]

2u

x t1

x t2

x

1 t1

1 t2

x 1 1 2 t1 t 2 mgxKiY (i) †_‡K (ii) we‡qvM K‡i cvB,

ev, u

2v

x

1 t2 x 1 2 t2

1 t1 1 t1

ev, v

myZivs, †mªv‡Zi †eM NÈvq Ges †bŠKvi †eM NÈvq

x 1 2 t2
1 t2

1 t1

wK.wg.

x 1 2 t1

wK.wg.|

D`vniY 4| GKwU bj 12 wgwb‡U GKwU Lvwj †PŠev”Pv c~Y© Ki‡Z cv‡i| Aci GKwU bj cÖwZ wgwb‡U 14 wjUvi cvwb †ei K‡i †`q| †PŠev”PvwU Lvwj _vKv Ae¯’vq `yBwU bj GKm‡½ Ly‡j †`Iqv nq Ges †PŠev”PvwU
96 wgwb‡U c~Y© nq| †PŠev”PvwU‡Z KZ wjUvi cvwb a‡i ?

mgvavb : g‡b Kwi, cÖ_g bj Øviv cÖwZ wgwb‡U x wjUvi cvwb cÖ‡ek K‡i Ges †PŠev”PvwU‡Z †gvU y wjUvi cvwb a‡i| cÖkœvbymv‡i, cÖ_g bj Øviv 12 wgwb‡U Lvwj †PŠev”PvwU c~Y© nq y 12x .......(i)

Avevi, `yBwU bj Øviv 96 wgwb‡U Lvwj †PŠev”Pv c~Y© nq

MwYZ y 96x 96 14 .........(ii)

63

mgxKiY (i) †_‡K cvB, x

y 12

x Gi gvb mgxKiY (ii) G ewm‡q cvB, y y 96 96 14 12 ev, y 8 y 96 14 ev, 7 y 96 14

ev, y

96 14 192 7

myZivs, †PŠev”PvwU‡Z †gvU 192 wjUvi cvwb a‡i| KvR : 1| eb‡fvR‡b hvIqvi Rb¨ GKwU evm 2400 UvKvq fvov Kiv n‡jv Ges wm×vš— M„nxZ n‡jv †h, cÖ‡Z¨K hvÎx mgvb fvov w`‡e| 10 Rb hvÎx Abycw¯’Z _vKvq gv_vwcQz fvov 8 UvKv e„w× †cj| ev‡m KZRb hvÎx wM‡qwQj Ges cÖ‡Z¨‡K KZ UvKv K‡i fvov w`‡qwQj ? 2| K I L GK‡Î GKwU KvR p w`‡b Ki‡Z cv‡i| K GKv KvRwU q w`‡b Ki‡Z cv‡i| L GKvKx KZ w`‡b H KvRwU Ki‡Z cvi‡e ? 3| GK e¨w³ †mªv‡Zi cÖwZK~‡j `uvo †e‡q NÈvq 2 wK.wg. †e‡M †h‡Z cv‡i| †mªv‡Zi †eM NÈvq 3 wK.wg. n‡j, †mªv‡Zi AbyK~‡j 32 wK.wg. †h‡Z Zvi KZ mgq jvM‡e ? D`vniY 5| GKwU eB‡qi g~j¨ 24 00 UvKv| GB g~j¨ cÖK…Z g~‡j¨i 80%| evwK g~j¨ miKvi fZ©ywK w`‡q _v‡Kb| miKvi cÖwZ eB‡q KZ UvKv fZ©ywK †`b ? mgvavb : evRvi g~j¨ = cÖK…Z g~‡j¨i 80% Avgiv Rvwb, p br GLv‡b, p 24 UvKv Ges r 80%
24 b 80 100 b 30

80 100

ev, b

6 5 24 100 80 4 1

myZivs eB‡qi cÖK…Z g~j¨ 30 UvKv| fZ©ywK g~j¨ = (30 24) UvKv = 6 UvKv myZivs fZ©ywK g~j¨ 6 UvKv|

64

MwYZ

D`vniY 6| UvKvq n msL¨K Kgjv weµq Kivq r% ¶wZ nq| s% jvf Ki‡Z n‡j, UvKvq KqwU Kgjv weµq Ki‡Z n‡e ? mgvavb : µqg~j¨ 100 UvKv n‡j, r% ¶wZ‡Z weµqg~j¨ (100 r) UvKv| Zvn‡j, hLb weµqg~j¨ (100 r) UvKv, ZLb µqg~j¨ 100 UvKv
100 UvKv| 100 r Avevi, µqg~j¨ 100 UvKv n‡j, s% jv‡f weµqg~j¨ (100 s) UvKv| 100 100 s 100 µqg~j¨ UvKv n‡j, s% jv‡f weµqg~j¨ 100 r 100 100 r 100 s = UvKv| 100 r 100 s myZivs, UvKvq weµq Ki‡Z n‡e n msL¨K Kgjv 100 r 100 r msL¨K Kgjv 1 UvKvq weµq Ki‡Z n‡e n 100 s n(100 r) myZivs, UvKvq msL¨K Kgjv weµq Ki‡Z n‡e| 100 s

hLb weµqg~j¨ 1 UvKv, ZLb µqg~j¨

UvKv

D`vniY 7| kZKiv evwl©K 7 UvKv nvi gybvdvq 650 UvKvi 6 eQ‡ii gybvdv KZ ? mgvavb : Avgiv Rvwb, I s 100 7 100
Pnr .

GLv‡b, P 650 UvKv, n 6, s 7 r I

650 6

7 100

273

myZivs, gybvdv 273 UvKv| D`vniY 8| evwl©K kZKiv 6 UvKv nvi Pµe„w× gybvdvq 15000 UvKvi 3 eQ‡ii me„w×g~j I Pµe„w× gybvdv wbY©q Ki| mgvavb : Avgiv Rvwb, C †`Iqv Av‡Q,
P(1 r)n [†hLv‡b C Pµe„w×i †¶‡Î me„w×g~j] 6 , n = 3 eQi P 15000 UvKv, r 6% 100
3

6 C 15000 1 100

3 15000 1 50

3

MwYZ

65

53 50 53 53 53 = 15000 50 50 50 3 15 53 53 53 3 148877 = 25 125 25 446631 = 17865 24 25 me„w×g~j = 17865 24 UvKv

3

= 15000

Pµe„w× gybvdv = (17865 24 15000) UvKv = 2865 24 UvKv| KvR : 1| UvKvq 10 wU †jey weµq Kivq n% ¶wZ nq| z% jvf Ki‡Z n‡j, UvKvq KqwU †jey weµq Ki‡Z n‡e ? 2| evwl©K kZKiv 6
1 nvi mij gybvdvq 750 UvKvi 4 eQ‡ii me„w×g~j KZ UvKv n‡e ? 2

3| evwl©K 4 UvKv nvi Pµe„w× gybvdvq 2000 UvKvi 3 eQ‡ii me„w×g~j wbY©q Ki|

Abykxjbx 3 5
1|
x2 7 x 6 Gi Drcv`‡K we‡ wlZ iƒc wb‡Pi †KvbwU ?

(K) (x 2)(x 3) (M) ( x 1)(x 6) 2| f ( x)

(L) ( x 1)( x 8) (N) ( x 1)(x 6)

x2 4x 4 n‡j, f (2) Gi gvb wb‡Pi †KvbwU ?

(K) 4 (M) 1 3| x y
1

(L) 2 (N) 0 x y n‡j, y Gi gvb wb‡Pi †KvbwU ?

(K) (M) 1

(L) 0 (N) 2

dg©v-9, MwYZ-9g-10g

66

MwYZ

4|

x2 3x3 Gi jwNô iƒc wb‡Pi †KvbwU ? x 3x2

(K) x2 (M) 1 5|
1 x2 Gi jwNô iƒc wb‡Pi †KvbwU ? 1 x

(L) x (N) 0

(K) 1 (M) (1 x) 6|
1 {(a b)2 2

(L) x (N) (1 x)
(a b)2} Gi gvb wb‡Pi †KvbwU ?

(K) 2(a2 b2) (M) 2ab 7| x 2 x 3 n‡j, x3 8 Gi gvb KZ ? x3

(L) a2 b2 (N) 4ab

(K) 1 (M) 9 8| p4 (L) 8 (N) 16 p2 1 Gi Drcv`‡K we‡ lvwqZ iƒc wb‡Pi †KvbwU ? p 1)( p2 p 1)( p2 p 1) p 1)

(K) ( p2 (M) ( p2 9| x2 5x

(L) ( p 2 (N) ( p2

p 1)( p 2 p 1)( p2

p 1) p 1)

4 Gi Drcv`K KZ ?

(K) (x 1), (x 4) (M) ( x 2), ( x 2) 10| ( x 7)( x 5) Gi gvb KZ ? (K) x2 12x 35 (M) x2 12x 35 11|
2 9 2 9 11 11 Gi gvb KZ ? 2 9 11 (K) 1 8

(L) ( x 1), ( x 4) (N) (x 5)(x 1) (L) x2 12x 35 (N) x2 12x 35

(M) 2 12| hw` x 2 (K) 1 (M) 2
3

(L) 1 9 (N) 4

3 nq, Z‡e x2 Gi gvb KZ ?

(L) 7 4 3 (N)
2 1 3

MwYZ

67

13| f (x) x2 5x 6 Ges f (x) 0 n‡j, x = KZ ? (K) 2, 3 (M) 14| x x
5
2, 3

(L)

5, 1

(N) 1, 5

6 6x
30

x2
5x

Dc‡ii wP‡Îi me©‡gvU †¶Îdj wb‡Pi †KvbwU ? (K) x 2 5 x 30 (M) x 2 6 x 30 KZ ¸Y KvR K‡i ? (K) 2 ¸Y (M) 3 ¸Y 16| a b (K) c2 (L) x 2 (N) x 2

x 30 x 30

15| K †h KvR x w`‡b m¤úbœ Ki‡Z cv‡i, L †m KvR 3x w`‡b m¤úbœ Ki‡Z cv‡i| GKB mg‡q K, L Gi
1 ¸Y 2

(L) 2

(N) 4 ¸Y c n‡j, a 2
2ab b 2 Gi gvb c Gi gva¨‡g cÖKvk Ki‡j wb‡Pi †KvbwU n‡e ?

(L) c 2 (N) ca y 3 Gi gvb KZ ?

(M) bc 17| x y 3, xy 2 n‡j, x 3 (K) 9 (M) 19 (K) (2 x 3 y )(4 x 2 6 xy 9 y 2 ) (M) (2 x 3 y )(4 x 2 9 y 2 )

(L) 18 (N) 27 (L) (2 x 3 y )(4 x 2 6 xy 9 y 2 ) (N) (2 x 3 y )(4 x 2 9 y 2 )

18| 8 x 3 27 y 3 Gi Drcv`‡K we‡ wlZ iƒc †KvbwU ?

19| 9 x 2 16 y 2 Gi mv‡_ KZ †hvM Ki‡j †hvMdj c~Y©eM© ivwk n‡e ? (K) 6 xy (M) 24 xy 20| x y (K) x 3 (M) x 3 y3 y3

(L) 12 xy (N) 144 xy (L) x 3 y 3 12 xy 12
4 n‡j, wb‡Pi †Kvb Dw³wU mwVK ?

4 xy 3 xy

64 64

(N) x 3 y 3 12 xy 64

68

MwYZ

21| hw` x 4 (1) x 2 (K) 4 (M) 1

x 2 1 0 nq, Z‡e

1 = KZ ? x2

(L) 2 (N) 0
2

1 (2) x x

Gi gvb KZ ? (L) 3 (N) 1

(K) 4 (M) 2 (3) x 3 (K) 3 (M) 1
1 = KZ ? x3

(L) 2 (N) 0

22| K GKwU KvR p w`‡b K‡i Ges L 2 p w`‡b K‡i| Zviv GKwU KvR Avi¤¢ K‡i Ges K‡qKw`b ci K KvRwU Amgvß †i‡L P‡j †Mj| evwK KvRUzKz L r w`‡b †kl K‡i| KvRwU KZ w`‡b †kl n‡qwQj ? 23| ˆ`wbK 8 NÈv cwikÖg K‡i 50 Rb †jvK GKwU KvR 12 w`‡b Ki‡Z cv‡i| ˆ`wbK KZ NÈv cwikÖg K‡i 60 R‡b 16 w`‡b H KvRwU Ki‡Z cvi‡e ? 24| wgZv GKwU KvR x w`‡b Ki‡Z cv‡i| wiZv †m KvR y w`‡b Ki‡Z cv‡i| Zviv GK‡Î KZ w`‡b KvRwU †kl Ki‡Z cvi‡e ? 25| eb‡fvR‡b hvIqvi Rb¨ 57000 UvKvq GKwU evm fvov Kiv n‡jv Ges kZ© n‡jv †h, cÖ‡Z¨K hvÎx mgvb fvov enb Ki‡e| 5 Rb hvÎx bv hvIqvq gv_vwcQz fvov 3 UvKv e„w× †cj| ev‡m KZRb hvÎx wM‡qwQj ? 26| GKRb gvwS †mªv‡Zi cÖwZK~‡j p NÈvq d wK.wg. †h‡Z cv‡i| †mªv‡Zi AbyK~‡j H c_ †h‡Z Zvi q NÈv jv‡M| †mªv‡Zi †eM I †bŠKvi †eM KZ ? 27| GKRb gvwSi `uvo †e‡q 15 wK.wg. †h‡Z Ges †mLvb †_‡K wd‡i Avm‡Z 4 NÈv mgq jv‡M| †m †mªv‡Zi AbyK~‡j hZ¶‡Y 5 wK.wg. hvq, †mªv‡Zi cÖwZK~‡j ZZ¶‡Y 3 wK.wg. hvq| `uv‡oi †eM I †mªv‡Zi †eM wbY©q Ki| 28| GKwU †PŠev”Pvq `yBwU bj mshy³ Av‡Q| cÖ_g bj Øviv †PŠev”PvwU t1 wgwb‡U c~Y© nq Ges wØZxq bj Øviv t 2 wgwb‡U Lvwj nq| bj `yBwU GK‡Î Ly‡j w`‡j Lvwj †PŠev”PvwU KZ¶‡Y c~Y© n‡e ? (GLv‡b t1 t2 )

29| GKwU bj Øviv 12 wgwb‡U GKwU †PŠev”Pv c~Y© nq| Aci GKwU bj Øviv 1 wgwb‡U Zv †_‡K 15 wjUvi cvwb †ei K‡i †`q| †PŠev”PvwU Lvwj _vKv Ae¯’vq `yBwU bj GKm‡½ Ly‡j †`Iqv nq Ges †PŠev”PvwU 48 wgwb‡U c~Y© nq| †PŠev”PvwU‡Z KZ wjUvi cvwb a‡i ?

MwYZ

69

30| GKwU Kjg 11 UvKvq weµq Ki‡j 10% jvf nq| KjgwUi µqg~j¨ KZ ? 31| GKwU LvZv 36 UvKvq weµq Kivq hZ ¶wZ n‡jv, 72 UvKvq weµq Ki‡j Zvi wØ¸Y jvf n‡Zv, LvZvwUi µqg~j¨ KZ ? 32| K, L I M Gi g‡a¨ 260 UvKv Giƒ‡c fvM K‡i `vI †hb K Gi As‡ki 2 ¸Y, L Gi As‡ki 3 ¸Y Ges M Gi As‡ki 4 ¸Y ci¯úi mgvb nq| 33| GKwU `ªe¨ x% ¶wZ‡Z weµq Ki‡j †h g~j¨ cvIqv hvq, 3x% jv‡f weµq Ki‡j Zvi †P‡q 18 x UvKv †ewk cvIqv hvq| `ªe¨wUi µqg~j¨ KZ wQj ? 34| 300 UvKvi 4 eQ‡ii mij gybvdv I 400 UvKvi 5 eQ‡ii mij gybvdv GK‡Î 148 UvKv n‡j, kZKiv gybvdvi nvi KZ ? 35| 4% nvi gybvdvq †Kv‡bv UvKvi 2 eQ‡ii gybvdv I Pµe„w× gybvdvi cv_©K¨ 1 UvKv n‡j, g~jab KZ ? 36| †Kv‡bv Avmj 3 eQ‡i mij gybvdvmn 460 UvKv Ges 5 eQ‡i mij gybvdvmn 600 UvKv n‡j, kZKiv gybvdvi nvi KZ ? 37| kZKiv evwl©K 5 UvKv nvi mij gybvdvq KZ UvKv 13 eQ‡i me„w×g~j 985 UvKv n‡e ? 38| kZKiv evwl©K 5 UvKv nvi gybvdvq KZ UvKv 12 eQ‡i me„w×g~j 1248 UvKv n‡e ? 39| 5% nvi gybvdvq 8000 UvKvi 3 eQ‡ii mij gybvdv I Pµe„w× gybvdvi cv_©K¨ wbY©q Ki| 40| wgwói Dci g~j¨ ms‡hvRb Ki (VAT ) x% | GKRb we‡µZv f¨vUmn P UvKvi wgwó weµq Ki‡j Zuv‡K KZ f¨vU w`‡Z n‡e ? x 15, P 41.
2300 n‡j, f¨v‡Ui cwigvY KZ ?

†Kv‡bv msL¨v I H msL¨vi ¸YvZ¥K wecixZ msL¨vi mgwó 3. K. msL¨vwU‡K x Pj‡K cÖKvk K‡i Dc‡ii Z_¨‡K GKwU mgxKi‡Yi gva¨‡g cÖKvk Ki| L. 1 Gi gvb wbY©q Ki| x3 3 x M.

cÖgvY Ki x 5

1 x5

123

42.

†Kv‡bv mwgwZi m`m¨MY cÖ‡Z¨‡KB m`m¨msL¨vi 100 ¸Y Pvu`v †`Iqvi wm×vš— wb‡jb| wKš‘ 7 Rb m`m¨ Puv`v bv †`Iqvq cÖ‡Z¨‡Ki Puv`vi cwigvY c~‡e©i †P‡q 500 UvKv †e‡o †Mj| K. mwgwZi m`m¨msL¨v x Ges †gvU Puv`vi cwigvY A n‡j, G‡`i g‡a¨ m¤úK© wbY©q Ki| L. mwgwZi m`m¨ msL¨v I †gvU Puv`vi cwigvY wbY©q Ki| M. 1 †gvU Puv`vi Ask 5% nv‡i Ges Aewkó UvKv 4% nv‡i 2 eQ‡ii Rb¨ mij gybvdvq
4

wewb‡qvM Kiv n‡jv| †gvU gybvdv wbY©q Ki|

PZz_© Aa¨vq

m~PK I jMvwi`g
(Exponents and Logarithms)
A‡bK eo ev A‡bK †QvU msL¨v ev ivwk‡K m~P‡Ki mvnv‡h¨ AwZ mn‡R wj‡L cÖKvk Kiv hvq| d‡j wnmve MYbv I MvwYwZK mgm¨v mgvavb mnRZi nq| m~P‡Ki gva¨‡gB msL¨vi ˆeÁvwbK ev Av`k© iƒc cÖKvk Kiv nq| ZvB cÖ‡Z¨K wk¶v_©xi m~P‡Ki aviYv I Gi cÖ‡qvM m¤ú‡K© Ávb _vKv Avek¨K| m~PK †_‡KB jMvwi`‡gi m„wó| Avi GB jMvwi`‡gi mvnv‡h¨ msL¨v ev ivwki ¸Y, fvM I m~PK m¤úwK©Z MYbvi KvR mnR n‡q‡Q| eZ©gv‡b K¨vjKz‡jUi I Kw¤úDUvi Gi e¨envi cÖPj‡bi c~e© ch©š— ˆeÁvwbK wn‡me MYbvq jMvwi`‡gi e¨envi wQj GKgvÎ Dcvq| Z‡e GLbI G¸‡jvi weKí wnmv‡e jMvwi`‡gi e¨envi ¸i“Z¡c~Y©| G Aa¨v‡q m~PK I jMvwi`g m¤ú‡K© we¯—vwiZ Av‡jvPbv Kiv n‡q‡Q|

Aa¨vq †k‡l wk¶v_©xiv g~j` m~PK e¨vL¨v Ki‡Z cvi‡e| abvZ¥K c~Y©-mvswL¨K m~PK, k~b¨ I FYvZ¥K c~Y©-mvswL¨K m~PK e¨vL¨v I cÖ‡qvM Ki‡Z cvi‡e| m~P‡Ki wbqgvewj eY©bv I Zv cÖ‡qvM K‡i mgm¨vi mgvavb Ki‡Z cvi‡e| n Zg g~j I g~j` fMœvsk m~PK e¨vL¨v Ki‡Z cvi‡e Ges n Zg g~j‡K m~PK AvKv‡i cÖKvk Ki‡Z

cvi‡e| jMvwi`g e¨vL¨v Ki‡Z cvi‡e| jMvwi`‡gi m~Îvewj cÖgvY I cÖ‡qvM Ki‡Z cvi‡e| mvaviY jMvwi`g I ¯^vfvweK jMvwi`g e¨vL¨v Ki‡Z cvi‡e| msL¨vi ˆeÁvwbK iƒc e¨vL¨v Ki‡Z cvi‡e| mvaviY jMvwi`‡gi c~Y©K I AskK e¨vL¨v Ki‡Z cvi‡e| K¨vjKz‡jU‡ii mvnv‡h¨ mvaviY I ¯^vfvweK jMvwi`g wbY©q Ki‡Z cvi‡e|

4 1 m~PK ( Exponents or Indices) :
Avgiv lô †kªwY‡Z m~P‡Ki aviYv †c‡qwQ Ges mßg †kªwY‡Z ¸‡Yi I fv‡Mi m~PK wbqg m¤ú‡K© †R‡bwQ| m~PK I wfwË msewjZ ivwk‡K m~PKxq ivwk ejv nq|

MwYZ

71

KvR : Lvwj Ni c~iY Ki : GKB msL¨v ev ivwki µwgK ¸Y
2 3 a b 2 3 a b 2 3 3 a b b b

m~PKxq ivwk
2
3

wfwË
2 3

NvZ ev m~PK
3

a3
5
a a ..... a †K an

a †h‡Kv‡bv ev¯—e msLv n‡j, n msL¨K a Gi µwgK ¸Y, A_©vr, a

AvKv‡i †jLv nq, †hLv‡b n abvZ¥K c~Y©msL¨v| a a a a n ..... m~PK ev NvZ wfwË

n a ( n msL¨K evi a ) = a .

GLv‡b,

Avevi, wecixZµ‡g a n

a a a ........ a ( n msL¨K evi a )

m~PK kyay abvZ¥K c~Y©msL¨vB bq, FYvZ¥K c~Y©msL¨v ev abvZ¥K fMœvsk ev FYvZ¥K fMœvskI n‡Z cv‡i| A_©vr, wfwË a R (ev¯—e msL¨vi †mU) Ges m~PK n Q (gyj` msL¨vi †mU) Gi Rb¨ a n msÁvwqZ| Z‡e we‡kl †¶‡Î, n N (¯^vfvweK msL¨vi †mU) aiv nq| ZvQvov Ag~j` m~PKI n‡Z cv‡i| Z‡e Zv gva¨wgK ¯—‡ii cvV¨m~wP ewnf©~Z e‡j GLv‡b †m m¤ú‡K© Av‡jvPbv Kiv nq wb|

4 2 m~P‡Ki m~Îvewj awi, a
R; m, n N. n m~Î 1| am an am m~Î 2| n a am

am n 1 ,

hLb m

n

an

m

hLb n m

wb‡Pi Q‡Ki Lvwj Ni c~iY Ki : am, an a 0 a5 a3 (a a a8 m 5

m n m 5, n 3 a a a a5
3

n m m 3, n 5
(a a a a a)

am an

a a

a a

a) a

a3 a 5 =

a an

a a3

a3 a5

a a a a a a a a 1 1 2 5 3 a a

a n = am n a m n hLb m n am Ges n 1 , hLb n m a an m

am

72

MwYZ an bn a a a ; a 5 2]

m~Î 3| ( ab) n j¶ Kwi, (5 2) 3

(5 2) (5 2) (5 2) [ a 3
5 2 5 2 5 2 (5 5 5) ( 2 2 2) 53 2 3

mvaviYfv‡e, (ab)n

ab ab ab ....... ab [ n msL¨K ab Gi µwgK ¸Y]

(a a a ....... a) (b b b ........ b) anbn

m~Î 4|

a b
5 2

n

an bn
3

, (b
5 2

0)
5 2 5 5 5 2 2 2 53 23

j¶ Kwi,

5 2 n mvaviYfv‡e,

a b

a a a a a [ n msL¨K Gi µwgK ¸Y] ........ b b b b b n a a a ...... a a b b b ...... b b n
0)
n

m~Î 5| a 0 1, ( a Avgiv cvB, Avevi, an an a 0

an a n an

a0

a a a ..... a [je I ni Dfq‡¶‡Î n msL¨K a Gi ¸Y] a a a ..... a
1

1. n m~Î 6| a

1 an n , (a

0)

Avgiv cvB, a

a a n

an n n

[je I ni‡K a Øviv ¸Y K‡i]

n

1 a n an a n ao an

1 an

1 an

MwYZ

73

gš—e¨ :

1 an

ao an n n

ao

n

a

n

m~Î 7| a m am am

a mn am am a m ......... a m
[ n msL¨K a m Gi µwgK ¸Y]

m m ......... m m

[Nv‡Z n msL¨K m~P‡Ki †hvMdj]

an am n

a mn

a mn
52 53

D`vniY 1| gvb wbY©q Ki : (K) mgvavb : (K)

(L)
1 5
5 5

2 3

5

2 3

5

52 1 = 52 3 = 5 1 = 1 3 5 5
2 3
5

(L)

2 3

5

=

2 3

2 3

o

1.

D`vniY 2| mij Ki : (K)

54 8 16 25 125

(L)

3 2n 2n 54 23 5
3

4 2n 2n
4 5 1

2

mgvavb : (K)

5 4 8 16 2
5

54 23 2 4 2
5 4
5

54 5
3

27 2
5

125

5
20

3

54

3

27

5

2

51 2 2
(L) 3 2 n 4 2 2n

n 2

3 2

n

22 2n 2n 2
1

2

3 2n 2n

22 2n

n 2

2n

1

2n

1 2

3 2n 2n 1 1 2n 2

(3 1) 2 n 1 n 2 2 q 2 2n 1 n 2 2
1

2 2

4.

D`vniY 3| †`LvI †h, ( a p ) q r ( a q ) r p ( a r ) p mgvavb : ( a p ) q r ( a q ) r p ( a r ) p q

a p(

q r) pr

aq

(r p) pq

ar( p a pr qr q)

, [ (a m ) n

a mn ]

a pq

a qr

a pq pr qr a 0 1.

pq pr qr

dg©v-10, MwYZ-9g-10g

74

MwYZ

KvR : Lvwj Ni c~iY Ki :
(i) 3 3 3 3
3

(ii) 5

53

55

(iii) a 2 a

a

3

(iv)

4 4

1

(v) ( 5)0

4 3 n Zg g~j j¶ Kwi, 5 Avevi, 5
1 1 2 2 1 2

5 5
1 2

1 2

5 5
1 1 2 2

1 2

2

5

2

1 2

5

52
1

5
1

5 2 Gi eM© (wØZxq NvZ) = 5 Ges 5 Gi eM©g~j (wØZxq g~j) = 5 2 5 †K eM©g~‡ji wPý
1 2

Gi gva¨‡g
5
1 3

5 AvKv‡i †jLv nq|
1 3 3

Avevi, j¶ Kwi, 5 Avevi, 5
1 3

1 3

1 3

5
5

1 3

5
5

5

1 3

5
3

1 1 1 3 3 3

3

1 3

5

5
1

1 3

5.
1

5 3 Gi Nb (Z…Zxq NvZ)
1

5 Ges 5 Gi Nbg~j (Z…Zxq g~j)

53

5 3 †K Nbg~‡ji wPý

3

Gi gva¨‡g

3

5 AvKv‡i †jLv nq|

n Zg g~‡ji †¶‡Î,
1 an 1 an 1 n an . 1 an

.......

1 an

1

[ n msL¨K a n Gi µwgK ¸Y]

1

1

1

1

Avevi, a n a n a n ....... a n an 1 1 1 1 ......... n n n

[m~P‡K n msL¨K 1 Gi †hvM] n 1 n a n

a

1 n an

a.

MwYZ

75

1

1

a n Gi n Zg NvZ

a Ges a Gi n Zg g~j
1 n an

an
1 1

A_©vr,

1 an

Gi n Zg NvZ

a Ges a Gi n Zg g~j (a) n

an =

n

a | a Gi n Zg g~j‡K

n a AvKv‡i †jLv nq|

D`vniY 4| mij Ki : (K) 7 mgvavb : (K) 7
3 4

3 4

7
1 2

1 2

(L) (16)
7
5 4

3 4

(16)

1 2

(M)

10

2 3

3 4

7 = 7
3 4

1 2

3 4

(L) (16)

(16) =
3 4

1 2

(16) (16)

3 4 1 2

(16)

3 4

1 2

= (16)

1 4

(2 ) = 2 .

1 4 4

(M)

10

2 3

= 10

2 3

3 4

10
1 2

1 2

10.
1 2
2

D`vniY 5| mij Ki : (K) (12) mgvavb : (K) (12)
1 (2
2

3

54

(L) ( 3) 3
1

1 2

3

54
1 3

1 (12)
1 2

(54) 3
1
1 1 2
1

1

3)

1 2

(33 2)
1 3

(2 )
1 3

1 2 2
1 1 2 1 1 3

(33 ) 3 2 3
1

3

1 2 3
1 2

3 2

2 21

31 3
1 2

3

32 2
2 3

32 4
1 3 3

2

3 . 4

(L) ( 3) 3

1 2

2

= ( 3)( 3)( 3) = =

1 2

1 2

27 27 4

1 4

KvR : mij Ki : (i)

24 22 32

(ii)

2 3

5

2 3

5

3

1

(iii) 8 4

82

76

MwYZ

j¶Yxq :
1. a 2. a
0, a 1 k‡Z© a x

a y n‡j, x

y

0, b

0, x

0 k‡Z© a x

b x n‡j, a

b

D`vniY 6| mgvavb Ki 4 x 1 32 mgvavb : 4 x 1 32
( 2 2 ) x 1 32 , ev 2 2 x 2 2 5 x a y n‡j, x y ] 2 x 2 5, [ a ev 2 x 5 2, ev 2 x 3 3 x 2 ev

Abykxjbx 4 1 mij Ki (1 1|
33 35 36
1 1

10) : 2|
3b )
1
1

53 8 2 4 125

3|
2

73 7 3 3 34

4| x 1y

3 72 3 7 7

5|
0, y

(2

1

5 1)

1

6| (2a 9|
2n
4

7|

a 2b 1 a 2b
1

8|
9m
1 1

y 1z

z 1x , ( x

0, z

0)

4 2n 2n 2 2

3m 1 10| m m (2 )

(3 m 1 ) m

cÖgvY Ki ( 11 11|
4 1 2n 1 n n

18 ) :
2n 1

12|

2p

1

32 p q 5 p q 6 p 6 q 10 p 2 15 q aq r a2 p a b

1 50 1

a 13| am
15|
xa xb xp xq

am an xb xc
1 bc

an a xc xa
1 ca

m

1

14|

ap q a 2r xa xb

ar p a 2q b c

1 ab

1 xr xp r p q

16|

xb xc

xc xa

c a

1

p q r

17|

xq xr

q r p

1 a nq, Z‡e †`LvI †h, xyz 1

18| hw` a x

b, b y

c Ges c z

mgvavb Ki (19 19| 4 x
8

22) : 20| 22x
1

128

21|

3

x 1

3

3

2x 1

22| 2 x 21

x

3

MwYZ

77

4 4 jMvwi`g (Logarithm) m~PKxq ivwki gvb †ei Ki‡Z jMvwi`g e¨envi Kiv nq| jMvwi`g‡K ms‡¶‡c jM ( Log ) †jLv nq| eo eo msL¨v ev ivwki ¸Ydj, fvMdj BZ¨vw` log Gi mvnv‡h¨ mn‡R wbY©q Kiv hvq| Avgiv Rvwb, 2 3 8 ; GB MvwYwZK Dw³wU‡K j‡Mi gva¨‡g †jLv nq log 2 8 3 . Avevi, wecixZµ‡g, log2 8 log2 8 ax 3 n‡j, m~P‡Ki gva¨‡g †jLv hv‡e 23 3 n‡j 23
8 . GKBfv‡e, 2
3

8 ; A_©vr, 23

8 n‡j log2 8

3 Ges wecixZµ‡g,
3.

1 23

1 1 †K j‡Mi gva¨‡g †jLv hvq, log 2 8 8

N , (a 0, a 1) n‡j, x log a N †K

N Gi a wfwËK jM ejv nq|

j¶Yxq : x abvZ¥K ev FYvZ¥K hvB †nvK bv †Kb, a x me©`v abvZ¥K| ZvB ïay abvZ¥K msL¨viB j‡Mi gvb Av‡Q hv ev¯—e| k~b¨ ev FYvZ¥K msL¨vi j‡Mi ev¯—e gvb †bB| KvR 1 : j‡Mi gva¨‡g cÖKvk Ki :
(i ) 10
(ii ) 3
2
2

(iii ) 2

1 2

100 1 9 1 2

KvR 2 : duvKv RvqMv c~iY Ki : m~P‡Ki gva¨‡g j‡Mi gva¨‡g
100 1

log10 1 0 loge 1 ......
...... = ......

e0 ..... a0 1 101 10 e1 ...

(iv)

4

2

4

log10 10 1
....... = .......

....... = ...... jMvwi`‡gi m~Îvewj awi, a
0, a 1; b 0, b 1 Ges M 0, N 0.

loga a 1

m~Î 1| (K) loga 1 0, (a 0, a 1) (L) loga a 1, (a 0, a 1) cÖgvY (K) m~P‡Ki m~Î n‡Z Rvwb, a0 1 j‡Mi msÁv n‡Z cvB, loga 1 0 ( cÖgvwYZ ) (L) m~P‡Ki m~Î n‡Z Rvwb, a1 m~Î 2| loga (MN ) loga M cÖgvY : awi, loga M
M a x, N ay a j‡Mi msÁv n‡Z cvB, log a a 1 (cÖgvwYZ )| loga N y;

x, loga N

78

MwYZ

GLb, MN log a ( MN )

ax a y x ax

y

y, ev log a ( MN )

log a M

log a N [ x, y Gi gvb ewm‡q ]

loga (MN )

loga M

loga N. (cÖgvwYZ) loga N loga N loga P .........

`ªóe¨-1| loga (MNP.....) loga M `ªóe¨-2| loga (M m~Î 3| log a
M N

N) log a M

loga M log a N

cÖgvY : awi, loga M
M a x, N

x, loga N ay y;

GLb,

M N

ax ay

ax

y

loga loga M N
M N

x

y loga N (cÖgvwYZ)|

loga M

m~Î 4| loga M r

r loga M . x; M ax

cÖgvY : awi, loga M ev ( M ) r loga M r
(log a M r

( a x ) r ; ev M r rx; ev loga M r

a rx r loga M

r log a M . cÖgvwYZ ) r log a M

`ªóe¨ : (log a M ) r m~Î 5| log a M ax logb M log a b, (wfwË cwieZ©b) x, logb M
M
1 x y

cÖgvY : awi, log a M

y

M ,by y a

x

b , ev (a )

(b )

1 y y

ev b

a x y

x y

log a b logb M log a b (cÖgvwYZ)

ev, x

y log a b, ev log a M

MwYZ

79

Abywm×vš— : log a b

1 , A_ev logb a log b a

1 loga b

cÖgvY : Avgiv Rvwb, loga M
M a ewm‡q cvB, loga a

logb M loga b [m~Î 5] logb a loga b 1 , A_ev log a b log a b
1 , A_ev log a b log a b

ev, 1 log b a log a b;

log b a

1 (cÖgvwYZ)| log b a
1 (cÖgvwYZ)| logb a

ev, 1 logb a log a b ; logb a

D`vniY 7| gvb wbY©q Ki : (K) log10 100 mgvavb : (K) log10 100 log10 102
1 9 1 32

(L) log 3

1 9

(M) log 3 81

2 log10 10[ log10 M r

r log10 M ]

2 1 [ log a a 1] 2

(L) log3

log3

log3 3

2

2 log3 3 [ log a M r

r log a M ]

2 1 [ log a a 1]

2

(M) log 3 81 log 3 34
8 log
3

log 3 {( 3 ) 2 }4

log 3 ( 3 )8

3

log a M r

r log a M

8 1, [ log a a 1]
=8

D`vniY 8| (K) 5 5 Gi 5 wfwËK jM KZ ? (L) 400 Gi jM 4 ; wfwË KZ ? mgvavb : (K) 5 5 Gi 5 wfwËK jM
1 3

log5 5 5

log 5 (5 5 2 )

log5 5 2

3 log5 5,[ log a M r 2

r log a M ]

3 1,[ log a a 1] 2
3 2

80

MwYZ

(L) awi, wfwË a cÖkœg‡Z, loga 400 4 a4 400
(20)2 (2 5)
4 4

ev, a 4 ev, a 4

{(2 5)2}2

(2 5)

a

2 5

[

ax

b x n‡j, a

b]

wfwË 2 5 D`vniY 9| x Gi gvb wbY©q Ki : (L) log x 324 4 (K) log10 x 2 mgvavb : (K) log10 x x 10 ev x x
2

2 1 10 2 0 01

(L) log x 324 4 x4 324 3 3 3 3 2 2 34 2 2 ev x 4 x (3 2)
4

1 100 0 01

34 ( 2 ) 4

3 2.

D`vniY 10| cÖgvY Ki †h, 3 log10 2 log10 5 log10 40 mgvavb : evgc¶ = 3 log10 2 log10 5 log10 23 log10 5, [ log a M r r log a M ]

log10 8 log10 5 log10 (8 5),[ log n ( MN ) log10 40 log10 23 log10 5, [ log a M r r log a M ]

log a M

log a N ]

log10 8 log10 5 log10 (8 5),[ log n ( MN ) log10 40 log a M log a N ]

= Wvbc¶ D`vniY 11| mij Ki : mgvavb : log10 27 log10 27 log10 8 log10 1000 log10 1 2

log10 8 log10 1000 log10 1 2

MwYZ
1 1

81

=

log10 (33 ) 2

log10 23 log10 (103 ) 2 12 log10 10
3

3

log10 3 2 log10 23 log10 10 2 log10 12 log10 10 3 log10 (3 3 log10 2 log10 10 2 log10 (3 22 ) log10 10
3 (log10 3 2 log10 2 1) 2 , [ log10 10 1] (log10 3 2 log10 2 1) 3 = . 2

Abykxjbx 4 2
1| gvb wbY©q Ki : (K) log3 81 (O) log5 3 5. 5 2| 3| x Gi gvb wbY©q Ki : (K) log5 x
3

(L) log5 3 5

(M) log4 2

(N) log2 5 400
1 16

(L) log x 25 2

(M) log x

2

†`LvI †h, (K) 5 log10 5 log10 25 log10 125
50 log10 2 2 log10 5 log10 3 2 log10 7 147 (M) 3 log10 2 2 log10 3 log10 5 log10 360

(L) log10 4| mij Ki :

10 25 81 2 log 3 log 9 24 80 5 3 (L) log 7 7 . 7 log3 3 log 4 2

(K) 7 log10

(M) log e

a 3b 3 c3

log e

b 3c 3 d3

log e

c 3d 3 a3

3 log e b 2 c

4 5 msL¨vi ˆeÁvwbK ev Av`k© iƒc m~P‡Ki mvnv‡h¨ Avgiv A‡bK eo ev A‡bK †QvU msL¨v‡K †QvU I mnR AvKv‡i cÖKvk Ki‡Z cvwi| †hgb, Av‡jvi MwZ = 300000 wK.wg./†m. = 300000000 wgUvi/†m. = 3 100000000 wg./†m. = 3 108 wg./†m. Avevi, GKwU nvB‡Wªv‡Rb cigvYyi e¨vmva© = 0 0000000037 †m. wg. dg©v-11, MwYZ-9g-10g

82

MwYZ

=

37 †m.wg.= 37 10 10000000000

10

†m.wg.

= 3 7 10 10

10

†m.wg.= 3 7 10 9 †m.wg.|

myweavi Rb¨ A‡bK eo ev A‡bK †QvU msL¨v‡K a 10n AvKv‡i cÖKvk Kiv nq, †hLv‡b, 1 a 10 Ges n Z . †Kv‡bv msL¨vi a

10 n iƒc‡K ejv nq msL¨vwUi ˆeÁvwbK ev Av`k© iƒc|

KvR : wb‡Pi msL¨v¸‡jv‡K ˆeÁvwbK AvKv‡i cÖKvk Ki : (K) 15000 (L) 0 000512

4 6 jMvwi`g c×wZ jMvwi`g c×wZ `yB ai‡bi : (K) ¯^vfvweK jMvwi`g (Natural logarithm) : ¯‹Uj¨v‡Ûi MwYZwe` Rb †bwcqvi ( John Napier : 1550 1617 ) 1614 mv‡j e †K wfwË a‡i cÖ_g jMvwi`g m¤úwK©Z eB cÖKvk K‡ib| e GKwU Ag~j` msL¨v, e 2 71828......... | Zuvi GB jMvwi`g‡K †bwcwiqvb jMvwi`g ev e wfwËK jMvwi`g ev ¯^vfvweK jMvwi`gI ejv nq| log e x †K ln x AvKv‡iI †jLv nq| (L) mvaviY jMvwi`g ( Common Logarithm ) : Bsj¨v‡Ûi MvwYZwe` †nbwi weªMm ( Henry Briggs : 1561 1630 ) 1624 mv‡j 10 †K wfwË a‡i jMvwi`‡gi †Uwej (jM †Uwej ev jM mviwY) ˆZwi K‡ib| Zuvi GB jMvwi`g‡K weªMm jMvwi`g ev 10 wfwËK jMvwi`g ev e¨envwiK jMvwi`gI ejv nq| `ªóe¨ : jMvwi`‡gi wfwËi D‡jL bv _vK‡j ivwki (exRMwYZxq) †¶‡Î e †K Ges msL¨vi †¶‡Î 10 †K wfwË wn‡m‡e aiv nq| jM mviwY‡Z wfwË 10 ai‡Z nq|

4 7 mvaviY jMvwi`‡gi c~Y©K I AskK
(K) c~Y©K ( Characteristics ) : awi, GKwU msL¨v N †K ˆeÁvwbK AvKv‡i cÖKvk K‡i cvB,
N a 10n , †hLv‡b N log10 (a 10n )

0,1 a 10 Ges n

Z|

Dfqc‡¶ 10 wfwË‡Z jM wb‡q cvB, log10 N

log10 a log10 10n

log10 a n log10 10

= log10 N log N

n log10 a, [ log10 10 1]

wfwË 10 Dn¨ †i‡L cvB, n log a

n †K ejv nq log N Gi c~Y©K|

MwYZ

83

j¶ Kwi : QK 1
N

N Gi a 10 n AvKvi
6 237 103 6 237 102 6 237 101 6 237 100 6 237 10
1

m~PK

`kwg‡Ki ev‡gi As‡ki A¼msL¨v

c~Y©K

6237 623 7 62 37 6 237 0 6237 j¶ Kwi : QK 2
N

3 2 1 0 1

4 3 2 1 0

4 1 3 3 1 2 2 1 1 1 1 0 0 1 1

N Gi a 10 n AvKvi
6 237 10 6 237 10 6 237 10
1 2 3

m~PK

`kwg‡Ki ev‡gi As‡ki A¼msL¨v

c~Y©K

0 6237 0 06237 0 006237
QK 1 †_‡K j¶ Kwi :

1 2 3

0 1 2

(0 1) (1 1) ( 2 1)

1 2 3

cÖ`Ë msL¨vi c~Y© As‡k hZ¸‡jv A¼ _vK‡e, msL¨vwUi jMvwi`‡gi c~Y©K n‡e †mB A¼msL¨vi †P‡q 1 Kg Ges Zv n‡e abvZ¥K| QK-2 †_‡K j¶ Kwi : cÖ`Ë msL¨vi c~Y© Ask bv _vK‡j `kwgK we›`y I Gi c‡ii cÖ_g mv_©K A‡¼i gv‡S hZ¸‡jv 0 (k~b¨) _vK‡e, msL¨vwUi jMvwi`‡gi c~Y©K n‡e k~‡b¨i msL¨vi †P‡q 1 †ewk Ges Zv n‡e FYvZ¥K| `ªóe¨ 1| c~Y©K abvZ¥K ev FbvZ¥K n‡Z cv‡i, wKš‘ AskK me©`v abvZ¥K| `ªóe¨ 2| †Kv‡bv c~Y©K FbvZ¥K n‡j, c~Y©KwUi ev‡g Ô Õ wPý bv w`‡q c~Y©KwUi Dc‡i Ô Õ (evi wPý) w`‡q †jLv nq| †hgb, c~Y©K
3 †K †jLv n‡e 3 w`‡q| Zv bv n‡j AskKmn j‡Mi m¤ú~Y© AskwU FYvZ¥K eySv‡e|

D`vniY 12| wb‡Pi msL¨v¸‡jvi j‡Mi c~Y©K wbY©q Ki :
(i) 5570 (ii) 45 70 (iii) 0 4305 (iv) 0 000435

mgvavb : (i) 5570 5 570 1000 5 570 103 msL¨vwUi j‡Mi c~Y©K 3 . Ab¨fv‡e, 5570 msL¨vwU‡Z A‡¼i msL¨v 4 wU| msL¨vwUi j‡Mi c~Y©K 4 1 3 msL¨vwUi j‡Mi c~Y©K 3 .

84

MwYZ
4 570 101

(ii)

45 70

msL¨vwUi c~Y©K 1 . Ab¨fv‡e, msL¨vwUi `kwg‡Ki ev‡g, A_©vr c~Y© As‡k 2 wU A¼ Av‡Q| msL¨vwUi j‡Mi c~Y©K
2 1 1

45 70 msL¨vwUi j‡Mi c~Y©K 1

(iii) 0 4305

4 305 10
1

1

msL¨vwUi c~Y©K

Ab¨fv‡e, msL¨vwUi `kwgK we›`yi Av‡M, A_©vr c~Y© As‡k †Kv‡bv mv_©K A¼ †bB, ev k~b¨wU A¼ Av‡Q| msL¨vwUi c~Y©K A_©vr k~b¨wU 0 Av‡Q| msL¨vwUi c~Y©K
(0 1) 1 1

0 1

1 1

Ab¨fv‡e, 0 4305 msL¨vi `kwgK we›`y I Gi cieZx© 1g mv_©K A¼ 4 Gi gv‡S †Kv‡bv 0 (k~b¨) †bB,

0 4305 msL¨vwUi j‡Mi c~Y©K 1

(iv) 0 000435

4 35 10

4

msL¨vwUi j‡Mi c~Y©K

4 ev 4

Ab¨fv‡e, msL¨vwUi `kwgK we›`y I Gi cieZ©x 1g mv_©K A¼ 4 Gi gv‡S 3 wU 0 ( k~b¨ ) Av‡Q| msL¨vwUi j‡Mi c~Y©K
(3 1) 4 4

0 000435 Gi j‡Mi c~Y©K 4

(L) AskK (Mantissa) :
†Kv‡bv msL¨vi mvaviY j‡Mi AskK 1 A‡c¶v †QvU GKwU AFYvZ¥K msL¨v| GwU g~jZ: Ag~j` msL¨v| Z‡e GKwU wbw`©ó `kwgK ¯’vb ch©š— Ask‡Ki gvb †ei Kiv nq| †Kv‡bv msL¨vi j‡Mi AskK jM ZvwjKv †_‡K †ei Kiv hvq| Avevi Zv K¨vjKz‡jU‡ii mvnv‡h¨I †ei Kiv hvq| Avgiv wØZxq c×wZ‡Z, A_©vr K¨vjKz‡jU‡ii mvnv‡h¨ msL¨vi j‡Mi AskK †ei Ki‡ev| K¨vjKz‡jU‡ii mvnv‡h¨ msL¨vi mvaviY jM wbY©q : D`vniY 13| log 2717 Gi c~Y©K I AskK wbY©q Ki : mgvavb : K¨vjKz‡jUi e¨envi Kwi :
AC

log

2717

=

3 43408

log 2717 Gi c~Y©K 3 Ges AskK 43408

MwYZ

85

D`vniY 14| log 43 517 Gi c~Y©K I AskK †ei Ki| mgvavb : K¨vjKz‡jUi e¨envi Kwi :
AC

log

43 517

=

1 63866

log 43 517 Gi c~Y©K 1 Ges AskK 63866

D`vniY 15| 0 00836 Gi j‡Mi c~Y©K I AskK KZ ? mgvavb : K¨vjKz‡jUi e¨envi Kwi :
AC
log
0 00836

=

3 92221 3 92221

log 0 00836 Gi c~Y©K

3 ev 3 Ges AskK 92221

D`vniY 16| log e 10 wbY©q Ki : mgvavb : log e 10 =
=

1 log10 e

1 1 = [K¨vjKz‡jUi e¨envi K‡i] 0.43429 log10 2 71828

2.30259 (cÖvq)|

weKí : K¨vjKz‡jUi e¨envi Kwi :
AC

ln

10

=

2.30259 (cÖvq)

KvR : K¨vjKz‡jUi e¨envi K‡i wbgœwjwLZ msL¨v¸‡jvi 10 wfwËK I e wfwËK jM wbY©q Ki :
(i ) 2550 (ii) 52 143 (iii ) 0 4145 (iv) 0 0742

Abykxjbx 4 3
1| †Kvb k‡Z© a 0 K. a 2|
3

1?
L. a

0
3

0
3

M. a

0
3

N. a 1

5
6

5 Gi gvb wb‡Pi †KvbwU ?
L.
3

K. 3|

5

5

M.

5

N.

3

25

mwVK †Kvb k‡Z© og a a 1 ? K. a

0

L. a 1
2 n‡j, x Gi gvb KZ ?

M. a

0, a 1

N. a

0, a 1

4|

log x 4

5|

K. 2 L. 2 M. 4 N. 10 n GKwU msL¨v‡K a 10 AvKv‡i †jLvi Rb¨ kZ© †KvbwU ? K. 1 a 10 L. 1 a 10 M. 1 a 10 N. 1 a 10

86

MwYZ

6|

wb‡Pi Z_¨¸‡jv j¶ Ki :

i. log a ( m) p ii. 2 4

plog a m

16 Ges log 2 16 4 mgv_©K

iii. log a ( m n) log a m log a n
Ic‡ii †Kvb Z_¨¸‡jv mwVK ? K. i I ii 7| K. 3 8| L. ii I iii L. 1 M. i I iii M. 2 N. i, ii I iii N. 3

0 0035 Gi mvaviY j‡Mi c~Y©K KZ ?

0 0225 msL¨vwU we‡ePbv K‡i wb‡Pi cÖkœ¸‡jvi DËi `vI :

(1) msL¨vwUi a n AvKvi wb‡Pi †KvbwU& ? K. ( 2 5) 2 L. ( 015) 2
3

M. (1 5) 2
2

N. ( 15) 2
1

(2) msL¨vwUi ˆeÁvwbK AvKvi wb‡Pi †KvbwU ? K. 225 10 4 L. 22 5 10 (3) msL¨vwUi mvaviY j‡Mi c~Y©K KZ ? 9| L. 1 K. 2 ˆeÁvwbK iƒ‡c cÖKvk Ki : M. 2 25 10 M. 0 (M) 0 000245 N. 225 10 N. 2 (N) 37500000 (O) 0 00000014
5

(K) 6530 (L) 60 831 10| mvaviY `kwgK iƒ‡c cÖKvk Ki :

(K) 105 (L) 10 5 (M) 2 53 10 4 (N) 9 813 10 3 (O) 3 12 10 11| wb‡Pi msL¨v¸‡jvi mvaviY j‡Mi c~Y©K †ei Ki (K¨vjKz‡jUi e¨envi bv K‡i) :

(K) 4820 (L) 72 245 (M) 1 734 (N) 0 045 (O) 0 000036 12| K¨vjKz‡jUi e¨envi K‡i wb‡Pi msL¨v¸‡jvi mvaviY j‡Mi c~Y©K I AskK wbY©q Ki : (L) 63 147 (M) 1 405 (N) 0 0456 (K) 27 13| ¸Yd‡ji/fvMd‡ji mvaviY jM (Avmbœ cuvP `kwgK ¯’vb ch©š—) wbY©q Ki : (K) 5 34 8 7 wbY©q Ki : (K) log 9 (L) log 28 (M) log 42 15| †`Iqv Av‡Q, x 1000 Ges y 0 0625 K. x †K a n b n AvKv‡i cÖKvk Ki, †hLv‡b a I b †gŠwjK msL¨v| L. x I y Gi ¸Ydj‡K ˆeÁvwbK AvKv‡i cÖKvk Ki| M. xy Gi mvaviY j‡Mi c~Y©K I AskK wbY©q Ki| (L) 0 79 0 56 (M) 22 2642 3 42 (O) 0 000673 (N) 0 19926 32 4

14| hw` log 2 0 30103, log 3 0 47712 Ges log 7 0 84510 nq, Z‡e wb‡Pi ivwk¸‡jvi gvb

MwYZ

87

cÂg Aa¨vq

GK PjKwewkó mgxKiY
(Equations in One Variable)
Avgiv c~‡e©i †kªwY‡Z PjK I mgxKiY Kx Zv †R‡bwQ Ges G‡`i e¨envi wk‡LwQ| GK PjKwewkó mij mgxKi‡Yi mgvavb wk‡LwQ Ges ev¯—ewfwËK mgm¨vi mij mgxKiY MVb K‡i Zv mgvavb Kiv m¤ú‡K© mg¨K Ávb jvf K‡iwQ| G Aa¨v‡q GK PjKwewkó GKNvZ I wØNvZ mgxKiY Ges A‡f` m¤ú‡K© Av‡jvPbv Kiv n‡q‡Q Ges ev¯—ewfwËK mgm¨vi mgvav‡b G‡`i e¨envi †`Lv‡bv n‡q‡Q| Aa¨vq †k‡l wk¶v_©xiv Pj‡Ki aviYv e¨vL¨v Ki‡Z cvi‡e| mgxKiY I A‡f‡`i cv_©K¨ e¨vL¨v Ki‡Z cvi‡e| GKNvZ mgxKi‡Yi mgvavb Ki‡Z cvi‡e| ev¯—ewfwËK mgm¨vi GKNvZ mgxKiY MVb K‡i mgvavb Ki‡Z cvi‡e| wØNvZ mgxKi‡Yi mgvavb Ki‡Z cvi‡e I mgvavb †mU wbY©q Ki‡Z cvi‡e| ev¯—ewfwËK mgm¨vi wØNvZ mgxKiY MVb K‡i mgvavb Ki‡Z cvi‡e|

5 1 PjK
Avgiv Rvwb, x 3 5 GKwU mgxKiY| GwU mgvavb Ki‡Z n‡j Avgiv AÁvZ ivwk x Gi gvb †ei Kwi| GLv‡b AÁvZ ivwk x GKwU PjK| Avevi, x a 5 mgxKiYwU mgvavb Ki‡Z n‡j, Avgiv x Gi gvb wbY©q Kwi, a Gi gvb bq| GLv‡b x †K PjK I a †K aª“eK wn‡m‡e aiv nq| G‡¶‡Î x Gi gvb a Gi gva¨‡g cvIqv hv‡e| Z‡e a Gi gvb wbY©q Ki‡Z n‡j, Avgiv wjL‡ev a 5 x ; A_©vr a Gi gvb x Gi gva¨‡g cvIqv hv‡e| GLv‡b a PjK I x aª“eK wn‡m‡e we‡ewPZ| Z‡e we‡kl †Kv‡bv wb‡`©kbv bv _vK‡j cÖPwjZ ixwZ Abyhvqx x †K PjK wn‡m‡e aiv nq| mvaviYZ Bs‡iwR eY©gvjvi †QvU nv‡Zi †k‡li w`‡Ki A¶i x, y, z †K PjK wn‡m‡e Ges cÖ_g w`‡Ki A¶i a, b, c †K aª“eK wn‡m‡e e¨envi Kiv nq| †h mgxKi‡Y GKwU gvÎ AÁvZ ivwk _v‡K, Zv‡K GK PjKwewkó mgxKiY ev mij mgxKiY ejv nq| †hgb, x 3 5 mgxKi‡Y x GKwU gvÎ PjK, ZvB GwU mij mgxKiY ev GK PjKwewkó mgxKiY|

Avgiv †mU m¤ú‡K© Rvwb| hw` GKwU †mU S {x : x R, 1 x 10} nq, Z‡e x -Gi gvb 1 †_‡K 10 ch©š— †h‡Kv‡bv ev¯—e msL¨v n‡Z cv‡i| GLv‡b x GKwU PjK| Kv‡RB Avgiv ej‡Z cvwi †h, hLb †Kv‡bv A¶i cÖZxK †Kv‡bv †m‡Ui Dcv`vb †evSvq ZLb Zv‡K PjK e‡j| mgxKi‡Yi NvZ: †Kv‡bv mgxKi‡Yi Pj‡Ki m‡e©v”P NvZ‡K mgxKiYwUi NvZ e‡j| x 1 5, 2 x 1 x 5, y 7 2 y 3 mgxKiY¸‡jvi cÖ‡Z¨KwUi NvZ 1 ; G¸‡jv GK PjKwewkó GKNvZ mgxKiY|

88

MwYZ

Avevi, x2 5x 6 0, y 2 y 12, 4x2 2x 3 6x mgxKiY¸‡jvi cÖ‡Z¨KwUi NvZ 2 ; G¸‡jv GK PjKwewkó wØNvZ mgxKiY| 2x3 x2 4x 4 0 mgxKiYwU GK PjKwewkó wÎNvZ mgxKiY|

5 2 mgxKiY I A‡f` mgxKiY : mgxKi‡Y mgvb wP‡ýi `yBc‡¶ `yBwU eûc`x _v‡K, A_ev GKc‡¶ (cÖavbZ Wvbc‡¶) k~b¨ _vK‡Z cv‡i| `yB c‡¶i eûc`xi Pj‡Ki m‡e©v”P NvZ mgvb bvI n‡Z cv‡i| mgxKiY mgvavb K‡i Pj‡Ki m‡e©v”P Nv‡Zi mgvb msL¨K gvb cvIqv hv‡e| GB gvb ev gvb¸‡jv‡K ejv nq mgxKiYwUi g~j| GB g~j ev g~j¸‡jv Øviv mgxKiYwU wm× n‡e| GKvwaK g~‡ji †¶‡Î G¸‡jv mgvb ev Amgvb n‡Z cv‡i| †hgb, x 2 5x 6 0 mgxKiYwUi g~j 2, 3 | Avevi, ( x 3)2

0 mgxKi‡Y x Gi gvb 3 n‡jI Gi g~j

3, 3 |

A‡f` : mgvb wP‡ýi `yBc‡¶ mgvb NvZwewkó `yBwU eûc`x _v‡K| Pj‡Ki m‡e©v”P Nv‡Zi msL¨vi †P‡qI AwaK msL¨K gv‡bi Rb¨ A‡f`wU wm× n‡e| mgvb wP‡ýi Dfq c‡¶i g‡a¨ †Kv‡bv †f` †bB e‡jB A‡f`| †hgb, ( x 1)2 ( x 1)2 4x GKwU A‡f`; GwU x Gi mKj gv‡bi Rb¨ wm× n‡e| ZvB GB mgxKiYwU GKwU A‡f`| cÖ‡Z¨K exRMwYZxq m~Î GKwU A‡f`| †hgb, (a b) 2 a2 2ab b 2 , a 2 b 2
(a b)(a b) , (a b) 3 a2 2ab b 2 , (a b) 2

a 3 3a 2 b 3ab 2 b 3 BZ¨vw` A‡f`|

mKj mgxKiY A‡f` bq| A‡f‡` mgvb (=) wP‡ýi cwie‡Z© ' ' wPý e¨eüZ nq| Z‡e mKj A‡f`B mgxKiY e‡j A‡f‡`i †¶‡ÎI mvaviYZ mgvb wPý e¨envi Kiv nq| mgxKiY I A‡f‡`i cv_©K¨ wb‡P †`Iqv n‡jv : 1| 2| 3| 4| 5| mgxKiY mgvb wP‡ýi `yB c‡¶ `yBwU eûc`x _vK‡Z cv‡i A_ev GK c‡¶ k~b¨ _vK‡Z cv‡i| Dfq c‡¶i eûc`xi gvÎv Amgvb n‡Z cv‡i| Pj‡Ki GK ev GKvwaK gv‡bi Rb¨ mgZvwU mZ¨ nq| Pj‡Ki gv‡bi msL¨v me©vwaK gvÎvi mgvb n‡Z cv‡i| mKj mgxKiY m~Î bq| 1| A‡f` `yB c‡¶ `yBwU eûc`x _v‡K|

2| Dfq c‡¶ eûc`xi gvÎv mgvb _v‡K| 3| Pj‡Ki g~j †m‡Ui mKj gv‡bi Rb¨ mvaviYZ mgZvwU mZ¨ nq| 4| Pj‡Ki AmsL¨ gv‡bi Rb¨ mgZvwU mZ¨| 5| mKj exRMwYZxq m~ÎB A‡f`|

KvR : 1| wb‡Pi mgxKiY¸‡jvi †KvbwUi NvZ KZ I g~j KqwU ?
(i ) 3 x 1 5

(ii )

2y 5

y 1 3

3y 2

2| wZbwU A‡f` †jL|

MwYZ

89

5 3 GKNvZ mgxKi‡Yi mgvavb mgxKiY mgvav‡bi †¶‡Î K‡qKwU wbqg cÖ‡qvM Ki‡Z nq| GB wbqg¸‡jv Rvbv _vK‡j mgxKi‡Yi mgvavb wbY©q mnRZi nq| wbqg¸‡jv n‡jv : 1| mgxKi‡Yi Dfqc‡¶ GKB msL¨v ev ivwk †hvM Ki‡j c¶Øq mgvb _v‡K| 2| mgxKi‡Yi Dfqc¶ †_‡K GKB msL¨v ev ivwk we‡qvM Ki‡j c¶Øq mgvb _v‡K| 3| mgxKi‡Yi Dfqc¶‡K GKB msL¨v ev ivwk Øviv ¸Y Ki‡j c¶Øq mgvb _v‡K| 4| mgxKi‡Yi Dfqc¶‡K Ak~b¨ GKB msL¨v ev ivwk Øviv fvM Ki‡j c¶Øq mgvb _v‡K| Dc‡ii ag©¸‡jv‡K exRMwYZxq ivwki gva¨‡g cÖKvk Kiv hvq : hw` x a Ges c 0 nq Zvn‡j,
(i ) x c a c (ii ) x c x a c c b c n‡j, a b

a c (iii ) xc

ac (iv)

GQvov hw` a, b I c wZbwU ivwk nq Z‡e, a c n‡e Ges a c b n‡j, a b c n‡e| GB wbqgwU c¶vš—i wewa wn‡m‡e cwiwPZ Ges GB wewa cÖ‡qvM K‡i wewfbœ mgxKiY mgvavb Kiv nq| †Kv‡bv mgxKi‡Yi c`¸‡jv fMœvsk AvKv‡i _vK‡j, je¸‡jv‡Z Pj‡Ki NvZ 1 Ges ni¸‡jv aª“eK n‡j, †m¸‡jv GKNvZ mgxKiY|
5x 4 x 2 7 5 5 7 5x 4 x 2 5x x 4 2 mgvavb : ev, [c¶vš—i K‡i] 7 5 5 7 7 5 5 7 25x 7 x 28 10 18 x 18 ev, ev, 35 35 35 35 ev, 18x 18 ev, x 1

D`vniY 1| mgvavb Ki :

mgvavb x 1. GLb, Avgiv Ggb mgxKi‡Yi mgvavb Ki‡ev hv wØNvZ mgxKi‡Yi AvKv‡i _v‡K| G mKj mgxKiY mijxKi‡Yi gva¨‡g mgZzj mgxKi‡Y iƒcvš—i K‡i ax b AvKv‡ii GKNvZ mgxKi‡Y cwiYZ Kiv nq| Avevi, n‡i PjK _vK‡jI mijxKiY K‡i GKNvZ mgxKi‡Y iƒcvš—i Kiv nq| D`vniY 2| mgvavb Ki : ( y 1)( y 2) ( y 4)( y 2) mgvavb : ( y 1)( y 2) ( y 4)( y 2) ev, y 2 y 2 y 2 y 2 4 y 2 y 8 ev, y 2 2 y 8 ev, y 2 y 8 2 [c¶vš—i K‡i] ev, y 6 ev, y 6 mgvavb y 6 dg©v-12, MwYZ-9g-10g

90

MwYZ
6x 1 2x 4 15 7x 1 2x 1 5

D`vniY 3| mgvavb Ki I mgvavb †mU †jL : mgvavb :

6x 1 2x 4 2x 1 15 7x 1 5 6x 1 2x 1 2x 4 ev, [c¶vš—i K‡i] 15 5 7x 1 6x 1 6x 3 2x 4 4 2x 4 ev, ev, 15 7x 1 15 7 x 1 ev, 15(2x 4) 4(7 x 1) [Avo¸Yb K‡i]

ev, 30x 60 28x 4 ev, 30x 28x 60 4 ev, 2x 56, ev, x 28 mgvavb x 28 Ges mgvavb †mU S {28} D`vniY 4| mgvavb Ki :
1 1 1 1 x 3 x 4 x 2 x 5 1` 1 1 1 mgvavb : x 3 x 4 x 2 x 5 2x 7 x 4 x 3 x 5 x 2 ev, ev, 2 ( x 3)( x 4) ( x 2)( x 5) x 7 x 12

[c¶vš—i K‡i]

2x 7 x 7 x 10
2

`yB c‡¶i fMœvsk `yBwUi gvb mgvb| Avevi, `yB c‡¶i je mgvb, wKš‘ ni Amgvb| G‡¶‡Î GKgvÎ j‡ei gvb k~b¨ n‡jB `yB c¶ mgvb n‡e|
2 x 7 0 ev, 2 x 7 ev, x

7 2

x

7 2

D`vniY 5| mgvavb †mU wbY©q Ki : mgvavb : ev, ev,
2x 3 5 2 2 5
2

2x 3 5 2

weKí wbqg :
2x 3 5 2

2x 3
2x 3

[c¶vš—i K‡i]

3 2 [Dfqc¶‡K eM© K‡i]

ev, ev,

2x 3 2x 3

2 5 3

ev, 2x 3 9 ev, 2x 12 ev, x 6 cª`Ë mgxKi‡Y eM©g~‡ji wPý _vKvi Kvi‡Y ïw× cix¶v cÖ‡qvRb|

†Kv‡bv ev¯—e ivwki eM©g~j FYvZ¥K n‡Z cv‡i bv| mgxKiYwUi †Kv‡bv mgvavb †bB| mgvavb †mU : S { } ev,

MwYZ

91

cÖ`Ë mgxKiYwU‡Z x 6 ewm‡q cvB,
2 6 3 5 2 ev, 9 5 2

ev, 3 5 2 ev, 8 2 , hv Am¤¢e| mgxKiYwUi †Kv‡bv mgvavb †bB| mgvavb †mU : S { } ev, KvR : 1|
5 1 x 4 4 5 n‡j, †`LvI †h, x 6 2 5

2| mgvavb Ki I mgvavb †mU †jL :

4x 3 5

2

5 4 GKNvZ mgxKi‡Yi e¨envi ev¯—e Rxe‡b wewfbœ ai‡bi mgm¨vi mgvavb Ki‡Z nq| GB mgm¨v mgvav‡bi AwaKvsk †¶‡ÎB MvwYwZK Ávb, `¶Zv I hyw³i cÖ‡qvRb nq| ev¯—e †¶‡Î MvwYwZK Ávb I `¶Zvi cÖ‡qv‡M GKw`‡K †hgb mgm¨vi myôy mgvavb nq, Ab¨w`‡K †Zgwb cÖvZ¨wnK Rxe‡b MwY‡Zi gva¨‡g mgm¨vi mgvavb cvIqv hvq weavq, wk¶v_©xiv MwY‡Zi cÖwZ AvK…ó nq| GLv‡b cÖvZ¨wnK Rxe‡bi wewfbœ mgm¨v‡K mgxKi‡Yi gva¨‡g cÖKvk K‡i Zvi mgvavb Kiv n‡e| ev¯—ewfwËK mgm¨v mgvav‡b AÁvZ msL¨v wbY©‡qi Rb¨ Gi cwie‡Z© PjK a‡i wb‡q mgm¨vq cÖ`Ë kZ©vbymv‡i mgxKiY MVb Kiv nq| Zvici mgxKiYwU mgvavb Ki‡jB PjKwUi gvb, A_©vr AÁvZ msL¨vwU cvIqv hvq| D`vniY 6| `yB A¼wewkó †Kv‡bv msL¨vi GKK ¯’vbxq A¼wU `kK ¯’vbxq A¼ A‡c¶v 2 †ewk| A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hv‡e Zv cÖ`Ë msL¨vi wØ¸Y A‡c¶v 6 Kg n‡e| msL¨vwU wbY©q Ki| mgvavb : g‡b Kwi, `kK ¯’vbxq A¼wU x ; AZGe, GKK ¯’vbxq A¼wU n‡e x 2 . msL¨vwU 10 x ( x 2) ev, 11x 2. A¼Øq ¯’vb wewbgq Ki‡j cwiewZ©Z msL¨vwU n‡e 10( x 2) x ev, 11x 20 cÖkœg‡Z, 11x 20 2(11x 2) 6 ev, 11x 20 22 x 4 6 ev, 22 x 11x 20 6 4 [c¶vš—i K‡i] ev, 11x 22 ev, x 2 msL¨vwU 11x 2 11 2 2 24 cÖ`Ë msL¨vwU 24 .

92

MwYZ

D`vniY 7| GKwU †kªwYi cÖwZ‡e‡Â 4 Rb K‡i QvÎ emv‡j 3 wU †eÂ Lvwj _v‡K| Avevi, cÖwZ‡e‡Â 3 Rb K‡i QvÎ emv‡j 6 Rb QvÎ‡K `uvwo‡q _vK‡Z nq| H †kªwYi QvÎ msL¨v KZ ? mgvavb : g‡b Kwi, †kªwYwUi QvÎ msL¨v x . †h‡nZz cÖwZ‡e‡Â 4 Rb K‡i emv‡j 3 wU †eÂ Lvwj _v‡K, †m‡nZz H †kªwYi †e‡Âi msL¨v x 4 3

Avevi, †h‡nZz cÖwZ‡e‡Â 3 Rb K‡i emv‡j 6 Rb‡K `uvwo‡q _vK‡Z nq, †m‡nZz H †kªwYi †e‡Âi msL¨v x 6 3

†h‡nZz †e‡Âi msL¨v GKB _vK‡e, myZivs, x 6 x 3= 3 4

ev,

x 12 4

x 6 3

ev, 4x 24 3x 36 , ev, x 60

ev, 4x 3x 36 24

H †kªwYi QvÎ msL¨v 60 . D`vniY 8| Kwei mv‡ne Zuvi 56000 UvKvi wKQz UvKv evwl©K 12% gybvdvq I evwK UvKv evwl©K 10% gybvdvq wewb‡qvM Ki‡jb| GK eQi ci wZwb †gvU 6400 UvKv gybvdv †c‡jb| wZwb 12% gybvdvq KZ UvKv wewb‡qvM K‡i‡Qb ? mgvavb : g‡b Kwi, Kwei mv‡ne 12% gybvdvq x UvKv wewb‡qvM K‡i‡Qb| wZwb 10% gybvdvq wewb‡qvM K‡i‡Qb (56000 x) UvKv| GLb, x UvKvi 1 eQ‡ii gybvdv x
12x 12 UvKv, ev, UvKv| 100 100

Avevi, (56000 x) UvKvi 1 eQ‡ii gybvdv (56000 x) cÖkœg‡Z,
12x 10(56000 + 100 100 x)
= 6400

10 10(56000 UvKv, ev, 100 100

x)

UvKv|

ev, 12x 560000 10x 640000 ev, 2x 640000 560000 ev, 2x 80000 ev, x
40000

Kwei mv‡ne 12% gybvdvq 40000 UvKv wewb‡qvM K‡i‡Qb| KvR : mgxKiY MVb K‡i mgvavb Ki : 1|
3 4 fMœvskwUi je I n‡ii mv‡_ †Kvb GKB msL¨v †hvM Ki‡j fMœvskwU n‡e ? 5 5

MwYZ

93

2| `yBwU µwgK ¯^vfvweK msL¨vi e‡M©i Aš—i 151 n‡j, msL¨v `yBwU wbY©q Ki| 3| 120 wU GK UvKvi gy`ªv I `yB UvKvi gy`ªvq †gvU 180 UvKv n‡j, †Kvb cÖKv‡ii gy`ªvi msL¨v KqwU ?

Abykxjbx 5 1 mgvavb Ki (1-10) :
1| 3(5 x 3) 2( x 2) 4| 2| ay b by a a2 b2

3| ( z 1)( z 2) ( z 4)( z 2)

7x 3 2x 4 4 9 25 1 1 1 1 5| 6| 3 5 5 3 2 x 1 3x 2 5 x 4 x 1 x 4 x 2 x 3 x a x b x 3a 3b a b a b x a x b 7| 8| 0 9| 2 2 b a a b x a x b x a b a b b2 a2 10| (3 3 ) z 2 5 3 3.

mgvavb †mU wbY©q Ki (11-20) : 11| 2 x( x 3) 2 x 2 12 14| 12| 2 x
2 3x 4 3 2

13| 16|

x a x b m x

x a x c n n x m n m n x

z 2 1 1 1 2 15| 2 z 1 z 1 x x 1 x 1 1 1 1 1 2t 6 15 2t 17| 18| x 2 x 5 x 4 x 3 9 12 5t 2 2 2 2 2 2 x 2b c x 2c a x 2a b 19| 0 a b b c c a

m 4t 15 18

mgxKiY MVb K‡i mgvavb Ki (21-30) :
2 ¸Y| msL¨v `yBwUi mgwó 98 n‡j, msL¨v `yBwU wbY©q Ki| 5 21| GKwU cÖK…Z fMœvs‡ki je I n‡ii Aš—i 1 ; je †_‡K 2 we‡qvM I n‡ii mv‡_ 2 †hvM Ki‡j †h

20| GKwU msL¨v Aci GKwU msL¨vi

fMœvsk cvIqv hv‡e Zv

1 Gi mgvb| fMœvskwU wbY©q Ki| 6

22| `yB A¼wewkó GKwU msL¨vi A¼Ø‡qi mgwó 9 ; A¼ `yBwU ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hv‡e Zv cÖ`Ë msL¨v n‡Z 45 Kg n‡e| msL¨vwU KZ ? 23| `yB A¼wewkó GKwU msL¨vi `kK ¯’vbxq A¼ GKK ¯’vbxq A‡¼i wØ¸Y| †`LvI †h, msL¨vwU A¼Ø‡qi mgwói mvZ¸Y| 24| GKRb ¶z`ª e¨emvqx 5600 UvKv wewb‡qvM K‡i GK eQi ci wKQz UvKvi Dci 5% Ges Aewkó UvKvi Dci 4% jvf Ki‡jb| wZwb KZ UvKvi Dci 5% jvf Ki‡jb ?

94

MwYZ

25| GKwU j‡Â hvÎx msL¨v 47 ; gv_vwcQz †Kwe‡bi fvov †W‡Ki fvovi wØ¸Y| †W‡Ki fvov gv_vwcQz 30 UvKv Ges †gvU fvov cÖvwß 1680 UvKv n‡j, †Kwe‡bi hvÎx msL¨v KZ ? 26| 120 wU cuwPk cqmvi gy`ªv I cÂvk cqmvi gy`ªvq †gvU 35 UvKv n‡j, †Kvb cÖKv‡ii gy`ªvi msL¨v KqwU ? 27| GKwU Mvwo NÈvq 60 wK.wg. †e‡M wKQz c_ Ges NÈvq 40 wK.wg. †e‡M Aewkó c_ AwZµg Ki‡jv| MvwowU †gvU 5 NÈvq 240 wK.wg. c_ AwZµg Ki‡j, NÈvq 60 wK.wg. †e‡M KZ`~i wM‡q‡Q ?

5 5 GK PjKwewkó wØNvZ mgxKiY ax 2 bx c 0 [†hLv‡b, a, b, c aª“eK Ges a
0 ] AvKv‡ii mgxKiY‡K GK PjKwewkó wØNvZ

mgxKiY ejv nq| wØNvZ mgxKi‡Yi evgc¶ GKwU wØgvwÎK eûc`x| mgxKi‡Yi Wvbc¶ k~b¨ aiv nq|
12 eM© †m.wg. †¶Îdjwewkó GKwU AvqZvKvi‡¶‡Îi ˆ`N©¨ x †m.wg. I cÖ¯’ ( x 1) †m.wg.|
(x-1) †m.wg.

AvqZvKvi‡¶ÎwUi †¶Îdj = x( x 1) eM© †m.wg. cÖkœg‡Z, x( x 1) 12, ev x 2 x 12 0 mgxKiYwU‡Z GKwU PjK x Ges x Gi m‡e©v”P NvZ 2 | Giƒc mgxKiY n‡jv wØNvZ mgxKiY| †h mgxKi‡Y Pj‡Ki m‡e©v”P NvZ 2 , Zv‡K wØNvZ mgxKiY e‡j| Avgiv Aóg †kªwY‡Z x 2 px q Ges ax 2
12 eM© †m.wg.

x †m.wg.

bx c AvKv‡ii GK PjKwewkó wØNvZ ivwki Drcv`‡K

we‡klY K‡iwQ| GLv‡b Avgiv x 2

px q 0 Ges ax 2 bx c 0 AvKv‡ii wØNvZ mgxKi‡Yi

evgc¶‡K Drcv`‡K we‡klY K‡i Pj‡Ki gvb wbY©‡qi gva¨‡g Giƒc mgxKiY mgvavb Ki‡ev| Drcv`‡K we‡klY c×wZ‡Z ev¯—e msL¨vi GKwU ¸iyZ¡c~Y© ag© cÖ‡qvM Kiv nq| ag©wU wbgœiƒc : hw` `yBwU ivwki ¸Ydj k~b¨ nq, Z‡e ivwkØ‡qi †h‡Kv‡bvwU A_ev Dfq ivwk k~b¨ n‡e| A_©vr, `yBwU ivwk a I b Gi ¸Ydj ab 0 n‡j, a 0 ev, b 0 , A_ev a 0 Ges b 0 n‡e| D`vniY 9| mgvavb Ki : ( x 2)( x 3) 0 mgvavb : ( x 2)( x 3) 0 x 2 0 , A_ev x 3 0

x 2 0 n‡j, x

2

Avevi, x 3 0 n‡j, x 3 mgvavb x mgvavb : y 2
2 A_ev 3
3y

D`vniY 10| mgvavb †mU wbY©q Ki : y 2
3y

MwYZ

95

ev, y 2 3 y 0 [c¶vš—i K‡i Wvbc¶ k~b¨ Kiv n‡q‡Q] ev, y ( y 3) 0 3 0 y 0 , A_ev y Avevi, y 3 0 n‡j, y 3 mgvavb †mU {0, 3} D`vniY 11| mgvavb Ki I mgvavb †mU †jL : x 4 mgvavb : x 4 x 4 x ev, x( x 4) x 4

x 4 x

[Avo¸Yb K‡i]

ev, x( x 4) ( x 4) 0 [c¶vš—i K‡i] ev, ( x 4)( x 1) 0 x 4 0, A_ev x 1 0

x 4

0 n‡j, x

4

Avevi, x 1 0 n‡j, x 1 mgvavb x 1 A_ev 4 Ges mgvavb †mU {1, 4} D`vniY 12| mgvavb Ki : mgvavb : awi, x a x a
2
2

x a x a

5

x a x a

6

0

5

x a x a

6 0..........(1)

x a y x a (1) n‡Z cvB, y 2

5y 6

0

ev, y 2 2 y 3 y 6 0 ev, y ( y 2) 3( y 2) 0 ev, ( y 2)( y 3) 0 y 2 0 n‡j, y 2

A_ev y 3 0 n‡j, y 3 GLb, y x a x a
2 n‡j,

2 [ y Gi gvb ewm‡q] 1

96

MwYZ x a x a x a x a
2x 2a 3 1

ev, ev,

2 1 [†hvRb-we‡qvRb K‡i] 2 1

ev, x 3a Avevi, y 3 n‡j, ev, ev, ev, x a x a x a x a 2x 2a x a 2 1 4 2

x a x a

3 1

3 1 [†hvRb-we‡qvRb K‡i] 3 1

ev, x 2a mgvavb x 2a A_ev, 3a KvR : 1| x 2 1 0 mgxKiYwU‡K ax 2 bx c 0 mgxKi‡Yi mv‡_ Zzjbv K‡i a, b, c Gi gvb †jL| 2| ( x 1) 2
0 mgxKiYwUi NvZ KZ ? Gi g~j KqwU I Kx Kx ?

5 6 wØNvZ mgxKi‡Yi e¨envi
Avgv‡`i ˆ`bw›`b Rxe‡bi A‡bK mgm¨v mij mgxKiY I wØNvZ mgxKi‡Y iƒcvš—i K‡i mn‡R mgvavb Kiv hvq| GLv‡b, ev¯—ewfwËK mgm¨vq cÖ`Ë kZ© †_‡K wØNvZ mgxKiY MVb K‡i mgvavb Kivi †KŠkj †`Lv‡bv n‡jv| D`vniY 13| GKwU cÖK…Z fMœvs‡ki ni, je A‡c¶v 4 †ewk| fMœvskwU eM© Ki‡j †h fMœvsk cvIqv hv‡e Zvi ni, je A‡c¶v 40 †ewk n‡e| fMœvskwU wbY©q Ki| mgvavb : awi, fMœvskwU fMœvskwUi eM© = x x 4

x x 4
2

x2 ( x 4) 2

x2

x2 8 x 16

GLv‡b, je = x 2 Ges ni = x 2 8 x 16 . cÖkœg‡Z, x 2 8 x 16 x 2 40 ev, 8 x 16 40

MwYZ

97

ev, 8 x ev, 8 x

40 16

24

ev, x 3 x 4 3 4 7

x x 4
3 7

3 3 4

3 7

fMœvskwU

D`vniY 14| 50 wgUvi ˆ`N©¨ Ges 40 wgUvi cÖ¯’wewkó GKwU AvqZvKvi evMv‡bi wfZ‡ii Pviw`‡K mgvb PIov GKwU iv¯—v Av‡Q| iv¯—v ev‡` evMv‡bi †¶Îdj 1200 eM©wgUvi n‡j, iv¯—vwU KZ wgUvi PIov ? mgvavb : g‡b Kwi, iv¯—vwU x wgUvi PIov| iv¯—v ev‡` evMvbwUi ˆ`N©¨ (50 2 x) wgUvi Ges cÖ¯’ (40 2 x) wgUvi| iv¯—v ev‡` evMvbwUi †¶Îdj = (50 2 x) × (40 2 x) eM©wgUvi| cÖkœg‡Z, (50 2 x) (40 2 x) 1200 ev, 2000 80 x 100 x 4 x ev, 4 x 2 180 x 800 0 ev, x 2 45 x 200 0 [ 4 w`‡q fvM K‡i] ev, x 2 5 x 40 x 200 0 ev, x( x 5) 40( x 5) 0 ev, ( x 5)( x 40) 0 x 5 0, A_ev x 40 0 x 5 0 n‡j, x 5
2

x wg. 50 wg.

1200

x 40 0 n‡j, x

40

wKš‘ iv¯—vwUi PIov evMvbwUi cÖ¯’ 40 wgUvi †_‡KI Kg n‡e| x 40 ; x 5

iv¯—vwU 5 wgUvi PIov| D`vniY 15| kvwnK 240 UvKvq KZK¸‡jv Kjg wKbj| †m hw` H UvKvq GKwU Kjg †ewk †c‡Zv Z‡e cÖwZwU Kj‡gi `vg M‡o 1 UvKv Kg co‡Zv| †m KZ¸‡jv Kjg wKbj ? mgvavb : g‡b Kwi, kvwnK 240 UvKvq †gvU x wU Kjg wK‡bwQj| G‡Z cÖwZwU Kj‡gi `vg c‡o UvKv| †m hw` 240 UvKvq ( x 1) wU Kjg †c‡Zv Z‡e cÖwZwU Kj‡gi `vg co‡Zv dg©v-13, MwYZ-9g-10g
240 UvKv| x 1

240 x

40 wg.

98

MwYZ

cÖkœg‡Z,

240 x 1

240 240 1, ev, x x 1

240 x x

ev, 240 x ( x 1)(240 x)

[Avo¸Yb K‡i]

ev, 240 x 240 x 240 x 2 x ev, x 2 x 240 0 [c¶vš—i K‡i] ev, x 2 16 x 15 x 240 0 ev, x( x 16) 15( x 16) 0 ev, ( x 16)( x 15) 0 x 16 x 16

0 , A_ev x 15 0
0 n‡j, x 16

x 15

0 n‡j, x 15

wKš‘ Kj‡gi msL¨v x FYvZ¥K n‡Z cv‡i bv| x 16 ; x 15

kvwnK 15 wU Kjg wK‡bwQj| KvR : mgxKiY MVb K‡i mgvavb Ki : 1| GKwU ¯^vfvweK msL¨vi e‡M©i mv‡_ H msL¨vwU †hvM Ki‡j †hvMdj wVK cieZ©x ¯^vfvweK msL¨vi bq¸‡Yi mgvb n‡e| msL¨vwU KZ ? 2|
10 †m.wg. e¨vmva©wewkó GKwU e„‡Ëi †K›`ª n‡Z GKwU R¨v Gi Dci Aw¼Z j‡¤^i ˆ`N©¨ e„ËwUi Aa©-

R¨v A‡c¶v 2 †m.wg. Kg| AvbygvwbK wPÎ A¼b K‡i R¨vwUi ˆ`N©¨ wbY©q Ki| D`vniY 16| GKwU we`¨vj‡qi beg †kªwYi GKwU cix¶vq x Rb Qv‡Îi MwY‡Z cÖvß †gvU b¤^i 1950 ; GKB cix¶vq Ab¨ GKRb bZzb Qv‡Îi MwY‡Z cÖvß b¤^i 34 †hvM Kivq cÖvß b¤^‡ii Mo 1 K‡g †Mj| K. c„_Kfv‡e x Rb Qv‡Îi Ges bZzb QvÎmn mK‡ji cÖvß b¤^‡ii Mo x Gi gva¨‡g †jL| L. cÖ`Ë kZ©vbymv‡i mgxKiY MVb K‡i †`LvI †h, x 2 35 x 1950 0 M. x Gi gvb †ei K‡i `yB‡¶‡Î b¤^‡ii Mo KZ Zv wbY©q Ki| mgvavb : K. x Rb Qv‡Îi cÖvß b¤^‡ii Mo =
1950 x
1950 34 x 1 1984 x 1

bZzb Qv‡Îi b¤^imn ( x 1) Rb Qv‡Îi cÖvß b¤^‡ii Mo L. cÖkœg‡Z,
1950 1984 1 x x 1 1950 1984 ev, 1 [c¶vš—i K‡i] x x 1

MwYZ

99

ev,

1950 x 1950 1984 x 1 x( x 1) ev, x 2 x 1950 x 1984 x 1950 [Avo¸Yb K‡i]

ev, x 2 x 1950 34 x x 2 35 x 1950 0 [†`Lv‡bv n‡jv]

M. x 2 35 x 1950 0 ev, x 2 65 x 30 x 1950 0 ev, x( x 65) 30( x 65) 0 ev, ( x 65)( x 30) 0 x 65 0 , A_ev x 30 0

x 65 0 n‡j, x

65

Avevi, x 30 0 n‡j, x 30 †h‡nZz Qv‡Îi msL¨v x FYvZ¥K n‡Z cv‡i bv, myZivs , x x 30
65

cÖ_g †¶‡Î, Mo =

1950 30

65

Ges wØZxq †¶‡Î, Mo =

1984 31

64 .

Abykxjbx 5 2
1|
x †K PjK a‡i a 2 x b 0 mgxKiYwUi NvZ wb‡Pi †KvbwU ?

K. 3 2|

L. 2
4x 2ab

M. 1

N. 0
2( x 2 1)

wb‡Pi †KvbwU A‡f` ? K. ( x 1) 2 ( x 1) 2 M. (a b) 2 (a b) 2
( x 4) 2

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3|

0 mgxKi‡Yi g~j KqwU ?

K. 1 wU 4| x2 L. 2 wU L. 3,
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3, 4

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K. 3, 4

100
3x 2

MwYZ x 5 0 mgxKi‡Y x Gi mnM KZ ?

5|

K. 3 6|

L. 2 x 2 2

M. 1

N.

1

wb‡Pi mgxKiY¸‡jv j¶ Ki :
i. 2 x 3 9

ii.

1

iii. 2 x 1 5

Dc‡ii †Kvb mgxKiY¸‡jv ci¯úi mgZzj ? K. i 7|
I i i L. i i I ii i

M. i

I

ii i

N. i, ii I iii

x 2 (a b) x ab 0 mgxKi‡Yi mgvavb †mU wb‡Pi †KvbwU ?

K. {a, b} 8| cÖkœ¸‡jvi DËi `vI ?

L. {a,

b}

M. { a, b}

N. { a,

b}

`yB A¼wewkó GKwU msL¨vi `kK ¯’vbxq A¼ GKK ¯’vbxq A‡¼i wØ¸Y| GB Z‡_¨i Av‡jv‡K wb‡Pi (1) GKK ¯’vbxq A¼ x n‡j, msL¨vwU KZ ? K. 2 x K. 3x K. 18 L. 3x L. 4 x L. 20 M. 12 x M. 12 x M. 34 N. 21x N. 21x N. 36 (2) A¼Øq ¯’vb wewbgq Ki‡j msL¨vwU KZ n‡e ? (3) x 2 n‡j, g~j msL¨vi mv‡_ ¯’vb wewbgqK…Z msL¨vi cv_©K¨ KZ ?

mgvavb Ki (9 18) : 9| ( x 2)( x
3) 0

10| ( 2 x 3)( 3 x 2) 0 11| y ( y 5) 6 13| 2( z 2 9) 9 z
0
1 3 2z 1 x a 17| a x

12| ( y 5)( y 5) 24 15|

14|

x 2 4 10 x 4 5 16| x 2 10 x 4 x a x b a b 18| x b x a b a

6( x 2) x 6

4 5z 1 x b b x

2

mgvavb †mU wbY©q Ki (19 25): 19| 22| 25|
3 x 4 x 1 2

20|

x 7 x 1

2x 6 2x 1

5

21|

1 x

1 a

1 b

1 x a b

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MwYZ

101

mgxKiY MVb K‡i mgvavb Ki (26 31) : 26| `yB A¼wewkó †Kv‡bv msL¨vi A¼Ø‡qi mgwó 15 Ges G‡`i ¸Ydj 56 ; msL¨vwU wbY©q Ki| 27| GKwU AvqZvKvi N‡ii †g‡Si †¶Îdj 192 eM©wgUvi| †g‡Si ˆ`N©¨ 4 wgUvi Kgv‡j I cÖ¯’ 4 wgUvi evov‡j †¶Îdj AcwiewZ©Z _v‡K| †g‡Si ˆ`N©¨ I cÖ¯’ wbY©q Ki| 28| GKwU mg‡KvYx wÎfy‡Ri AwZfy‡Ri ˆ`N©¨ 15 †m.wg. I Aci evûØ‡qi ˆ`‡N©¨i Aš—i 3 †m.wg.| H evûØ‡qi ˆ`N©¨ wbY©q Ki| 29| GKwU wÎfy‡Ri f~wg Zvi D”PZvi wØ¸Y A‡c¶v 6 †m.wg. †ewk| wÎfyR †¶ÎwUi †¶Îdj 810 eM© †m.wg. n‡j, Gi D”PZv KZ ? 30| GKwU †kªwY‡Z hZRb QvÎ-QvÎx c‡o cÖ‡Z¨‡K Zvi mncvVxi msL¨vi mgvb UvKv Puv`v †`Iqvq †gvU
420 UvKv Puv`v DVj| H †kªwYi QvÎ-QvÎxi msL¨v KZ Ges cÖ‡Z¨‡K KZ UvKv K‡i Puv`v w`j ?

31| GKwU †kªwY‡Z hZRb QvÎ-QvÎx c‡o, cÖ‡Z¨‡K ZZ cqmvi †P‡q AviI 30 cqmv †ewk K‡i Puv`v †`Iqv‡Z †gvU 70 UvKv DVj| H †kªwYi QvÎ-QvÎxi msL¨v KZ ? 32| `yB A¼wewkó GKwU msL¨vi A¼Ø‡qi mgwó 7 ; A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq Zv cÖ`Ë msL¨v †_‡K 9 †ewk| K. PjK x Gi gva¨‡g cÖ`Ë msL¨vwU I ¯’vb wewbgqK…Z msL¨vwU †jL| L. msL¨vwU wbY©q Ki| M. cÖ`Ë msL¨vwUi A¼Øq hw` †mw›UwgUv‡i †Kv‡bv AvqZ‡¶‡Îi ˆ`N©¨ I cÖ¯’ wb‡`©k K‡i Z‡e H AvqZ‡¶ÎwUi K‡Y©i ˆ`N©¨ wbY©q Ki| KY©wU‡K †Kv‡bv e‡M©i evû a‡i eM©‡¶ÎwUi K‡Y©i ˆ`N©¨ wbY©q Ki| 33| GKwU mg‡KvYx wÎfy‡Ri f~wg I D”PZv h_vµ‡g ( x 1) †m.wg. I x †m.wg. Ges GKwU e‡M©i evûi ˆ`N©¨ wÎfyRwUi D”PZvi mgvb| Avevi, GKwU AvqZ‡¶‡Îi evûi ˆ`N©¨ ( x 3) †m.wg. I cÖ¯’ x †m.wg.| K. GKwUgvÎ wP‡Îi gva¨‡g Z_¨¸‡jv †`LvI| L. wÎfyR‡¶ÎwUi †¶Îdj 10 eM© †m.wg. n‡j, Gi D”PZv KZ ? M. wÎfyR‡¶Î, eM©‡¶Î I AvqZ‡¶‡Îi †¶Îd‡ji avivevwnK AbycvZ †ei Ki|

102

MwYZ

lô Aa¨vq

†iLv, †KvY I wÎfzR
R¨vwgwZ ev 'Geometry ' MwYZ kv‡¯¿i GKwU cÖvPxb kvLv| 'Geometry ' kãwU MªxK Geo - f~wg (earth) I metrein - cwigvc (measure) k‡ãi mgš^‡q ˆZwi| ZvB ÕR¨vwgwZÕ k‡ãi A_© Õf~wg cwigvcÕ| K…wlwfwËK

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61

¯’vb, Zj, †iLv I we›`yi aviYv

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MwYZ

103

†hgb, GKwU BU ev ev‡·i wZbwU gvÎv (ˆ`N©¨, cÖ¯’ I D”PZv) Av‡Q| GKwU †Mvj‡Ki wZbwU gvÎv Av‡Q| Gi wZb gvÎvi wfbœZv ¯úó †evSv bv †M‡jI G‡K ˆ`N©¨-cÖ¯’-D”PZv wewkó L‡Ê wef³ Kiv hvq|

Nbe¯‘i DcwifvM Zj (Surface) wb‡`©k K‡i A_©vr, cÖ‡Z¨K Nbe¯‘ GK ev GKvwaK Zj Øviv mxgve× _v‡K| †hgb, GKwU ev‡·i QqwU c„ô QqwU mgZ‡ji cÖwZiƒc| †Mvj‡Ki DcwifvMI GKwU Zj| Z‡e ev‡·i c„ôZj I †Mvj‡Ki c„ô Zj wfbœ cÖKv‡ii| cÖ_gwU mgZj (Plane) , wØZxqwU eµZj (Curved Surface) |

Zj wØgvwÎK (Two dimensional ) : Gi ïay ˆ`N©¨ I cÖ¯’ Av‡Q, †Kv‡bv D”PZv bvB| GKwU ev‡·i `yBwU gvÎv wVK †i‡L Z…Zxq gvÎv µgk n«vm K‡i k~‡b¨ cwiYZ Ki‡j, ev·wUi c„ôwe‡kl gvÎ Aewkó _v‡K| Gfv‡e Nbe¯‘ †_‡K Z‡ji aviYvq Avmv hvq| `yBwU Zj ci¯úi‡K †Q` Ki‡j GKwU †iLv (line) Drcbœ nq| †hgb, ev‡·i `yBwU c„ôZj ev‡·i GKav‡i GKwU †iLvq wgwjZ nq| GB †iLv GKwU mij‡iLv (straight line) | GKwU †jey‡K GKwU cvZjv Qzwi w`‡q KvU‡j, Qzwii mgZj †hLv‡b †jeyi eµZj‡K †Q` K‡i †mLv‡b GKwU eµ‡iLv (curved line) Drcbœ nq| †iLv GKgvwÎK (one dimensional : Gi ïay ˆ`N©¨ Av‡Q, cÖ¯’ I D”PZv †bB| ev‡·i GKwU c„ô-Z‡ji cÖ¯’ µgk n«vm †c‡q m¤ú~Y© k~b¨ n‡j, H Z‡ji GKwU †iLv gvÎ Aewkó _v‡K| Gfv‡e Z‡ji aviYv †_‡K †iLvi aviYvq Avmv hvq|

`yBwU †iLv ci¯úi †Q` Ki‡j we›`yi DrcwË nq| A_©vr, `yBwU †iLvi †Q`¯’vb we›`y ( po int) Øviv wbw`©ó nq| ev‡·i `yBwU avi †hgb, ev‡·i GK †KvYvq GKwU we›`y‡Z wgwjZ nq| we›`yi ˆ`N©¨, cÖ¯’ I D”PZv bvB, ïay Ae¯’vb Av‡Q| GKwU †iLvi ˆ`N©¨ µgk n«vm †c‡j Ae‡k‡l GKwU we›`y‡Z ch©ewmZ nq| we›`y‡K k~b¨ gvÎvi mËv (entity) e‡j MY¨ Kiv nq|

62

BDwK¬‡Wi ¯^xKvh©

Dc‡i Zj, †iLv I we›`y m¤ú‡K© †h aviYv †`Iqv n‡jv, Zv Zj, †iLv I we›`yi msÁv bq- eY©bv gvÎ| GB eY©bvq gvÎv ej‡Z ˆ`N©¨, cÖ¯’, D”PZv BZ¨vw` aviYv e¨envi Kiv n‡q‡Q, †h¸‡jv msÁvwqZ bq| BDwK¬W Zuvi ÔBwj‡g›UmÕ MÖ‡š’i cÖ_g L‡Êi ïi“‡ZB we›`y, †iLv I Z‡ji †h ÔmsÁvÕ D‡jL K‡i‡Qb Zv-I AvaywbK `„wófw½ Abymv‡i Am¤ú~Y©| BDwK¬W cÖ`Ë K‡qKwU eY©bv wbgœiƒc :

104

MwYZ

(1) hvi †Kv‡bv Ask bvB, ZvB we›`y| (2) †iLvi cÖvš— we›`y †bB| (3) hvi †Kej ˆ`N©¨ Av‡Q, wKš‘ cÖ¯’ I D”PZv bvB, ZvB †iLv| (4) †h †iLvi Dcwiw¯’Z we›`y¸‡jv GKB eive‡i _v‡K, ZvB mij‡iLv| (5) hvi †Kej ˆ`N©¨ I cÖ¯’ Av‡Q, ZvB Zj| (6) Z‡ji cÖvš— n‡jv †iLv| (7) †h Z‡ji mij‡iLv¸‡jv Zvi Ici mgfv‡e _v‡K, ZvB mgZj| j¶ Ki‡j †`Lv hvq †h, GB eY©bvq Ask, ˆ`N©¨, cÖ¯’, mgfv‡e BZ¨vw` kã¸‡jv AmsÁvwqZfv‡e MÖnY Kiv n‡q‡Q| a‡i †bqv n‡q‡Q †h, G¸‡jv m¤ú‡K© Avgv‡`i cÖv_wgK aviYv i‡q‡Q| Gme aviYvi Dci wfwË K‡i we›`y, mij‡iLv I mgZ‡ji aviYv †`Iqv n‡q‡Q| ev¯—weK c‡¶, †h‡Kv‡bv MvwYwZK Av‡jvPbvq GK ev GKvwaK cÖv_wgK aviYv ¯^xKvi K‡i wb‡Z nq| BDwK¬W G¸‡jv‡K ¯^Ztwm× ( Axioms ) e‡j AvL¨vwqZ K‡ib| BDwK¬W cÖ`Ë K‡qKwU ¯^Ztwm× : 1| †hmKj e¯‘ GKB e¯‘i mgvb, †m¸‡jv ci¯úi mgvb| 2| mgvb mgvb e¯‘i mv‡_ mgvb e¯‘ ‡hvM Kiv n‡j †hvMdj mgvb| 3| mgvb mgvb e¯‘ ‡_‡K mgvb e¯‘ we‡qvM Kiv n‡j we‡qvMdj mgvb| 4| hv ci¯ú‡ii mv‡_ wg‡j hvq, Zv ci¯úi mgvb| 5| c~Y© Zvi As‡ki †P‡q eo| AvaywbK R¨vwgwZ‡Z we›`y, mij‡iLv I mgZj‡K cÖv_wgK aviYv wn‡m‡e MÖnY K‡i Zv‡`i wKQz ˆewkó¨‡K ¯^xKvi K‡i †bIqv nq| GB ¯^xK…Z ˆewkó¨¸‡jv‡K R¨vwgwZK ¯^xKvh© ( postulate) ejv nq| ev¯—e aviYvi m‡½ m½wZ †i‡LB GB ¯^xKvh©mg~n wba©viY Kiv n‡q‡Q| BDwK¬W cÖ`Ë cvuPwU ¯^xKvh© n‡jv : ¯^xKvh© 1| GKwU we›`y †_‡K Ab¨ GKwU we›`y ch©š— GKwU mij‡iLv AuvKv hvq| ¯^xKvh© 2| LwÊZ †iLv‡K h‡_”Qfv‡e evov‡bv hvq| ¯^xKvh© 3| †h‡Kv‡bv †K›`ª I †h‡Kv‡bv e¨vmva© wb‡q e„Ë AuvKv hvq| ¯^xKvh© 4| mKj mg‡KvY ci¯úi mgvb| ¯^xKvh© 5| GKwU mij‡iLv `yBwU mij‡iLv‡K †Q` Ki‡j Ges †Q`‡Ki GKB cv‡ki Aš—t¯’ †KvYØ‡qi mgwó `yB mg‡Kv‡Yi †P‡q Kg n‡j, †iLv `yBwU‡K h‡_”Qfv‡e ewa©Z Ki‡j †hw`‡K †Kv‡Yi mgwó `yB mg‡Kv‡Yi †P‡q Kg, †mw`‡K wgwjZ nq| BDwK¬W msÁv, ¯^Ztwm× I ¯^xKvh©¸‡jvi mvnv‡h¨ hyw³g~jK bZzb cÖwZÁv cÖgvY K‡ib| wZwb msÁv, ¯^Ztwm×, ¯^xKvh© I cÖgvwYZ cÖwZÁvi mvnv‡h¨ Avevi bZzb GKwU cÖwZÁvI cÖgvY K‡ib| BDwK¬W Zvi ÔBwj‡g›UmÕ MÖ‡š’ †gvU 465wU k„•Ljve× cÖwZÁvi cÖgvY w`‡q‡Qb hv AvaywbK hyw³g~jK R¨vwgwZi wfwË| j¶ Kwi †h, BDwK¬‡Wi cÖ_g ¯^xKv‡h© wKQz Am¤ú~Y©Zv i‡q‡Q| `yBwU wfbœ we›`y w`‡q †h GKwU Abb¨ mij‡iLv A¼b Kiv hvq Zv D‡cw¶Z n‡q‡Q| cÂg ¯^xKvh© Ab¨ PviwU ¯^xKv‡h©i †P‡q RwUj| Ab¨w`‡K, cÖ_g †_‡K

MwYZ

105

PZz_© ¯^xKvh©¸‡jv G‡Zv mnR †h G¸‡jv Õ¯úóB mZ¨Õ e‡j cÖZxqgvb nq| wKš‘ G¸‡jv cÖgvY Kiv hvq bv| myZivs, Dw³¸‡jv ÕcÖgvYwenxb mZ¨Õ ev ¯^xKvh© e‡j †g‡b †bqv nq| cÂg ¯^xKvh©wU mgvš—ivj mij‡iLvi mv‡_ RwoZ weavq cieZx©‡Z Av‡jvPbv Kiv n‡e|

6 3 mgZj R¨vwgwZ c~‡e©B we›`y, mij‡iLv I mgZj R¨vwgwZi wZbwU cÖv_wgK aviYv D‡jL Kiv n‡q‡Q| G‡`i h_vh_ msÁv †`Iqv m¤¢e bv n‡jI G‡`i m¤ú‡K© Avgv‡`i ev¯—e AwfÁZvcÖm~Z aviYv n‡q‡Q| weg~Z© R¨vwgwZK aviYv wn‡m‡e ¯’vb‡K we›`ymg~‡ni †mU aiv nq Ges mij‡iLv I mgZj‡K GB mvwe©K †m‡Ui Dc‡mU we‡ePbv Kiv nq| A_©vr, ¯^xKvh© 1| RMZ (Space) mKj we›`yi †mU Ges mgZj I mij‡iLv GB †m‡Ui Dc‡mU| GB ¯^xKvh© †_‡K Avgiv j¶ Kwi †h, cÖ‡Z¨K mgZj I cÖ‡Z¨K mij‡iLv GK GKwU †mU, hvi Dcv`vb n‡”Q we›`y| R¨vwgwZK eY©bvq mvaviYZ †mU cÖZx‡Ki e¨envi cwinvi Kiv nq| †hgb, †Kv‡bv we›`y GKwU mij‡iLvi (ev mgZ‡ji) Aš—fy©³ n‡j we›`ywU H mij‡iLvq (ev mgZ‡j) Aew¯’Z A_ev, mij‡iLvwU (ev mgZjwU) H we›`y w`‡q hvq| GKBfv‡e, GKwU mij‡iLv GKwU mgZ‡ji Dc‡mU n‡j mij‡iLvwU H mgZ‡j Aew¯’Z, A_ev, mgZjwU H mij‡iLv w`‡q hvq G iKg evK¨ Øviv Zv eY©bv Kiv nq| mij‡iLv I mgZ‡ji ˆewkó¨ wn‡m‡e ¯^xKvi K‡i †bIqv nq †h, ¯^xKvh© 2| `yBwU wfbœ we›`yi Rb¨ GKwU I †Kej GKwU mij‡iLv Av‡Q, hv‡Z Dfq we›`y Aew¯’Z| ¯^xKvh© 3| GKB mij‡iLvq Aew¯’Z bq Ggb wZbwU wfbœ we›`yi Rb¨ GKwU I †Kej GKwU mgZj Av‡Q, hv‡Z we›`y wZbwU Aew¯’Z| ¯^xKvh© 4| †Kv‡bv mgZ‡ji `yBwU wfbœ we›`y w`‡q hvq Ggb mij‡iLv H mgZ‡j Aew¯’Z| ¯^xKvh© 5| (K) RM‡Z (Space) GKvwaK mgZj we`¨gvb| (L) cÖ‡Z¨K mgZ‡j GKvwaK mij‡iLv Aew¯’Z| (M) cÖ‡Z¨K mij‡iLvi we›`ymg~n Ges ev¯—e msL¨vmg~n‡K Ggbfv‡e m¤úwK©Z Kiv hvq †hb, †iLvwUi cÖ‡Z¨K we›`yi m‡½ GKwU Abb¨ ev¯—e msL¨v mswkó nq Ges cÖ‡Z¨K ev¯—e msL¨vi m‡½ †iLvwUi GKwU Abb¨ we›`y mswkó nq| gš—e¨ : ¯^xKvh© 1 †_‡K ¯^xKvh© 5 †K AvcZb ¯^xKvh© ejv nq| R¨vwgwZ‡Z `~i‡Z¡i aviYvI GKwU cÖv_wgK aviYv| G Rb¨ ¯^xKvi K‡i †bIqv nq †h, ¯^xKvh© 6| (K) P I Q we›`yhyMj GKwU Abb¨ ev¯—e msL¨v wbw`©ó K‡i _v‡K| msL¨vwU‡K P we›`y †_‡K Q we›`yi `~iZ¡ ejv nq Ges PQ Øviv m~wPZ Kiv nq| (L) P I Q wfbœ we›`y n‡j PQ msL¨vwU abvZ¥K| Ab¨_vq, PQ 0 | (M) P †_‡K Q Gi `~iZ¡ Ges Q †_‡K P Gi `~iZ¡ GKB| A_©vr PQ QP | dg©v-14, MwYZ-9g-10g

106

MwYZ
QP nIqv‡Z GB `~iZ¡‡K mvaviYZ P we›`y I Q we›`yi ga¨eZ©x `~iZ¡ ejv nq| e¨envwiKfv‡e, GB

PQ

`~iZ¡ c~e© wba©vwiZ GK‡Ki mvnv‡h¨ cwigvc Kiv nq|
¯^xKvh© 5 (M) Abyhvqx cÖ‡Z¨K mij‡iLvq Aew¯’Z we›`ymg~‡ni †mU I ev¯—e mgm¨vi †m‡Ui g‡a¨ GK-GK wgj

¯’vcb Kiv hvq| G cÖm‡½ ¯^xKvi K‡i †bIqv nq †h, ¯^xKvh© 7| †Kv‡bv mij‡iLvq Aew¯’Z we›`ymg~‡ni †mU Ges ev¯—e msL¨vi †m‡Ui g‡a¨ Ggbfv‡e GK-GK wgj ¯’vcb Kiv hvq, †hb †iLvwUi †h‡Kv‡bv we›`y P, Q Gi Rb¨ PQ wgjKi‡Yi d‡j P I Q Gi m‡½ h_vµ‡g a I b ev¯—e msL¨v mswkøó nq| GB ¯^xKv‡h© ewY©Z wgjKiY Kiv n‡j, †iLvwU GKwU msL¨v‡iLvq cwiYZ n‡q‡Q ejv nq| msL¨v‡iLvq P we›`yi m‡½ a msL¨vwU mswkøó n‡j P †K a Gi †jLwe›`y Ges a †K P Gi ¯’vbv¼ ejv nq| †Kv‡bv mij‡iLv‡K msL¨v‡iLvq cwiYZ Kivi Rb¨ cÖ_‡g †iLvwUi GKwU we›`yi ¯’vbv¼ 0 Ges Aci GKwU we›`yi ¯’vbv¼ 1 a‡i †bIqv nq| G‡Z †iLvwU‡Z GKwU GKK `~iZ¡ Ges GKwU abvZ¥K w`K wbw`©ó nq| G Rb¨ ¯^xKvi K‡i †bIqv nq †h, ¯^xKvh© 8| †h‡Kv‡bv mij‡iLv AB †K Ggbfv‡e msL¨v‡iLvq cwiYZ Kiv hvq †h, A Gi ¯’vbv¼ O Ges
B Gi ¯’vbv¼ abvZ¥K nq|

a b

nq, †hLv‡b

gš—e¨ : ¯^xKvh© 6 †K `~iZ¡ ¯^xKvh©, ¯^xKvh© 7 †K iƒjvi ¯^xKvh© Ges ¯^xKvh© 8 †K iƒjvi ¯’vcb ¯^xKvh© ejv nq| R¨vwgwZK eY©bv‡K ¯úó Kivi Rb¨ wPÎ e¨envi Kiv nq| KvM‡Ri Ici †cwÝj ev Kj‡gi m~² †duvUv w`‡q we›`yi cÖwZiƒc AuvKv nq| †mvRv iƒjvi eivei `vM †U‡b mij‡iLvi cÖwZiƒc AuvKv nq| mij‡iLvi wP‡Î `yB w`‡K ZxiwPý w`‡q †evSv‡bv nq †h, †iLvwU Dfqw`‡K mxgvnxbfv‡e we¯—…Z| ¯^xKvh© 2 Abyhvqx `yBwU wfbœ we›`y
A I B GKwU Abb¨ mij‡iLv wbw`©ó K‡i hv‡Z we›`y `yBwU Aew¯’Z nq| GB †iLv‡K AB †iLv ev BA

†iLv ejv nq| ¯^xKvh© 5 (M) Abyhvqx Giƒc cÖ‡Z¨K mij‡iLv AmsL¨ we›`y aviY K‡i| ¯^xKvh© (5) (K) Abyhvqx GKvwaK mgZj we`¨gvb| Giƒc cÖ‡Z¨K mgZ‡j AmsL¨ mij‡iLv i‡q‡Q| R¨vwgwZi †h kvLvq GKB mgZ‡j Aew¯’Z we›`y, †iLv Ges Zv‡`i m‡½ m¤úwK©Z wewfbœ R¨vwgwZK mËv m¤ú‡K© Av‡jvPbv Kiv nq, Zv‡K mgZj R¨vwgwZ
(Plane Geometry) ejv nq| G cy¯—‡K mgZj R¨wgwZB Avgv‡`i

g~j we‡eP¨ welq| myZivs, we‡kl †Kv‡bv D‡jL bv _vK‡j eyS‡Z n‡e †h, Av‡jvP¨ mKj we›`y, †iLv BZ¨vw` GKB mgZ‡j Aew¯’Z| Giƒc GKwU wbw`©ó mgZjB Av‡jvPbvi mvwe©K †mU|

MvwYwZK Dw³i cÖgvY
†h‡Kv‡bv MvwYwZK Z‡Ë¡ KwZcq cÖv_wgK aviYv, msÁv Ges ¯^xKv‡h©i Dci wfwË K‡i av‡c av‡c H ZË¡ m¤úwK©Z wewfbœ Dw³ †hŠw³Kfv‡e cÖgvY Kiv nq| Giƒc Dw³‡K mvaviYZ cÖwZÁv ejv nq| cÖwZÁvi †hŠw³KZv cÖgv‡Yi Rb¨ hyw³we`¨vi wKQz wbqg cÖ‡qvM Kiv nq| ‡hgb,

MwYZ

107

(K) Av‡ivn c×wZ (Mathematical Induction) (L) Ae‡ivn c×wZ (Mathematical Deduction) (M) we‡iva c×wZ BZ¨vw`|

we‡iva c×wZ (Proof by contradiction)
`vk©wbK Gwi÷Uj hyw³g~jK cÖgv‡Yi G c×wZwUi m~Pbv K‡ib| G c×wZi wfwË n‡jv: Ñ GKB ¸Y‡K GKB mgq ¯^xKvi I A¯^xKvi Kiv hvq bv| Ñ GKB wRwb‡li `yBwU ci¯úiwe‡ivax ¸Y _vK‡Z cv‡i bv| Ñ hv ci¯úiwe‡ivax Zv AwPš—¨bxq| Ñ †Kv‡bv e¯‘ GK mg‡q †h ¸‡Yi AwaKvix nq, †mB e¯‘ †mB GKB mg‡q †mB ¸‡Yi AbwaKvix n‡Z cv‡i bv|

64

R¨vwgwZK cÖgvY

R¨vwgwZ‡Z KZK¸‡jv cÖwZÁv‡K we‡kl ¸i“Z¡ w`‡q Dccv`¨ wn‡m‡e MÖnY Kiv nq Ges Ab¨vb¨ cÖwZÁv cÖgv‡Y µg Abyhvqx G‡`i e¨envi Kiv nq| R¨vwgwZK cÖgv‡Y wewfbœ Z_¨ wP‡Îi mvnv‡h¨ eY©bv Kiv nq| Z‡e cÖgvY Aek¨B hyw³wbf©i n‡Z n‡e| R¨vwgwZK cÖwZÁvi eY©bvq mvaviY wbe©Pb ( general enunciation) A_ev we‡kl wbe©Pb ( particular enunciation) e¨envi Kiv nq| mvaviY wbe©Pb n‡”Q wPÎwbi‡c¶ eY©bv Avi we‡kl wbe©Pb n‡”Q wPÎwbf©i

eY©bv| †Kv‡bv cÖwZÁvi mvaviY wbe©Pb †`Iqv _vK‡j cÖwZÁvi welqe¯‘ we‡kl wbe©P‡bi gva¨‡g wbw`©ó Kiv nq| G Rb¨ cÖ‡qvRbxq wPÎ A¼b Ki‡Z nq| R¨vwgwZK Dccv‡`¨i cÖgv‡Y mvaviYZ wb‡gœv³ avc¸‡jv _v‡K : (1) mvaviY wbe©Pb (2) wPÎ I we‡kl wbe©Pb (3) cÖ‡qvRbxq A¼‡bi eY©bv Ges (4) cÖgv‡Yi †hŠw³K avc¸‡jvi eY©bv| hw` †Kv‡bv cÖwZÁv mivmwifv‡e GKwU Dcvcv‡`¨i wm×vš— †_‡K cÖgvwYZ nq, Z‡e Zv‡K A‡bK mgq H Dccv‡`¨i Abywm×vš— (Corollary) wn‡m‡e D‡jL Kiv hvq| wewfbœ cÖwZÁv cÖgvY Kiv QvovI R¨vwgwZ‡Z wewfbœ wPÎ A¼b Kivi cÖ¯—vebv we‡ePbv Kiv nq| G¸‡jv‡K m¤úv`¨ ejv nq| m¤úv`¨ welqK wPÎ A¼b K‡i wPÎv¼‡bi eY©bv I †hŠw³KZv D‡jL Ki‡Z nq|

Abykxjbx 6 1
1| ¯’vb, Zj, †iLv Ges we›`yi aviYv `vI| 2| BDwK¬‡Wi cuvPwU ¯^xKvh© eY©bv Ki|

108

MwYZ

3| cuvPwU AvcZb ¯^xKvh© eY©bv Ki| 4| `~eZ¡ ¯^xKvh©wU eY©bv Ki| 5| i“jvi ¯^xKvh©wU eY©bv Ki| 6| msL¨v‡iLv eY©bv Ki| 7| i“jvi ¯’vcb ¯^xKvh©wU eY©bv Ki| 8| ci¯úi‡Q`x mij‡iLv I mgvš—ivj mij‡iLvi msÁv `vI|

†iLv, iwk¥, †iLvsk mgZjxq R¨vwgwZi ¯^xKvh© Abyhvqx mgZ‡j mij‡iLv we`¨gvb hvi cÖwZwU we›`y mgZ‡j Aew¯’Z| g‡b Kwi, mgZ‡j AB GKwU mij‡iLv Ges †iLvwUi Dci Aew¯’Z GKwU we›`y ejv nq hw` A ,
C C| C AC + CB

we›`y‡K A I B we›`yi Aš—e©Zx© = AB nq| A ,
C

I B GKB mij‡iLvi wfbœ wfbœ we›`y nq Ges

I B we›`y

wZbwU‡K mg‡iL we›`yI ejv nq| A I B Ges G‡`i Aš—e©Zx© mKj we›`yi †mU‡K A I B we›`yi ms‡hvRK †iLvsk ev ms‡¶‡c AB †iLvsk ejv nq| A I B we›`yi Aš—e©Zx© cÖ‡Z¨K we›`y‡K †iLvs‡ki Aš—t¯’ we›`y ejv nq|

†KvY mgZ‡j `yBwU iwk¥i cÖvš—we›`y GKB n‡j †KvY ‰Zwi nq| iwk¥ `yBwU‡K †Kv‡Yi evû Ges Zv‡`i mvaviY we›`y‡K kxl©we›`y e‡j| wP‡Î, OP I OQ iwk¥Øq Zv‡`i mvaviY cÖvš—we›`y O †Z
POQ Drcbœ K‡i‡Q| O we›`ywU
Q

POQ

Gi kxl©we›`y| OP Gi †h cv‡k¦© †mU‡K

Av‡Q †mB cv‡k¦© Ges

OQ Gi †h cv‡k¦© P Av‡Q †mB cv‡k¦© Aew¯’Z mKj we›`yi
POQ

Gi Af¨š—i ejv nq| †KvYwUi Af¨š—‡i

A_ev †Kv‡bv evû‡Z Aew¯’Z bq Ggb mKj we›`yi †mU‡K Gi ewnf©vM ejv nq| mij †KvY `yBwU ci¯úi wecixZ iwk¥ Zv‡`i mvaviY cÖvš—we›`y‡Z †h †KvY Drcbœ K‡i, Zv‡K mij †KvY e‡j| cv‡ki wP‡Î, AB iwk¥i cÖvßwe›`y A †_‡K AB Gi wecixZ w`‡K
AC

iwk¥ AvuKv n‡q‡Q|
BAC

AC

I AB
BAC
0

iwk¥Øq Zv‡`i mvaviY cÖvš—we›`y A †Z

Drcbœ K‡i‡Q|

†K mij †KvY e‡j| mij †Kv‡Yi cwigvc `yB mg‡KvY ev 180 |

MwYZ

109

mwbœwnZ †KvY hw` mgZ‡j `yBwU †Kv‡Yi GKB kxl©we›`y nq I Zv‡`i GKwU mvaviY iwk¥ _v‡K Ges †KvYØq mvaviY iwk¥i wecixZ cv‡k Ae¯’vb K‡i, Z‡e H †KvYØq‡K mwbœwnZ †KvY e‡j| cv‡ki wP‡Î, A we›`ywU
A we›`y
BAC I
BAC I

CAD Gi kxl©we›`y|

CAD DrcbœKvix iwk¥¸‡jvi g‡a¨ AC mvaviY
AC Gi wecixZ cv‡k Aew¯’Z|

iwk¥ | †KvY `yBwU mvaviY iwk¥
BAC Ges

CAD ci¯úi mwbœwnZ †KvY|

j¤^, mg‡KvY GKwU mij‡Kv‡Yi mgwØLÊK‡K j¤^ Ges mswkó mwbœwnZ †Kv‡Yi cÖ‡Z¨KwU‡K mg‡KvY e‡j| cv‡ki wP‡Î, we›`y‡Z
AC

BAD mij‡KvY A

iwk¥ Øviv d‡j BAC I

CAD mwbœwnZ †KvY `yBwUi

cÖ‡Z¨‡K mg‡KvY Ges BD I AC evûØq ci¯ú‡ii Dci j¤^| m~²‡KvY I ¯’~j‡KvY GK mg‡KvY †_‡K †QvU †KvY‡K m~²‡KvY Ges GK mg‡KvY †_‡K eo wKš‘ `yB mg‡KvY †_‡K †QvU †KvY‡K ¯’~j‡KvY ejv nq| wP‡Î
AOC

m~²‡KvY Ges

AOD

¯’~j‡KvY| GLv‡b

AOB GK

mg‡KvY| cÖe„× †KvY `yB mg‡KvY †_‡K eo wKš‘ Pvi mg‡KvY †_‡K †QvU †KvY‡K cÖe„×‡KvY ejv nq| wP‡Î wPwýZ AOC cÖe„×‡KvY| c~iK †KvY `yBwU †Kv‡Yi cwigv‡ci †hvMdj 1 mg‡KvY n‡j †KvY `yBwUi GKwU AciwUi c~iK †KvY| cv‡ki wP‡Î,
AOB GKwU mg‡KvY| OC iwk¥ †KvYwUi evûØ‡qi

Af¨š—‡i Aew¯’Z| Gi d‡j mgvb, A_©vr 1 mg‡KvY| †KvY|

AOC Ges

COB GB `yBwU †KvY
AOB
COB

Drcbœ n‡jv| †KvY `yBwUi cwigv‡ci †hvMdj
AOC Ges

Gi cwigv‡ci ci¯úi c~iK

110

MwYZ

m¤ú~iK †KvY `yBwU †Kv‡Yi cwigv‡ci †hvMdj 2 mg‡KvY n‡j †KvY `yBwU ci¯ci m¤ú~iK †KvY|
AB GKwU mij‡iLvi O Aš—t¯’ GKwU we›`y| OC GKwU iwk¥ hv

OA iwk¥ I OB iwk¥ †_‡K wfbœ| Gi d‡j

AOC Ges

COB AOB

GB `yBwU †KvY Drcbœ n‡jv| †KvY `yBwUi cwigv‡ci †hvMdj †Kv‡Yi cwigv‡ci mgvb, A_©vr 2 mg‡KvY, †Kbbv mij‡KvY| wecÖZxc †KvY
AOC Ges

AOB GKwU

COB ci¯úi m¤ú~iK †KvY|

†Kv‡bv †Kv‡Yi evûØ‡qi wecixZ iwk¥Øq †h †KvY ˆZwi K‡i Zv H †Kv‡Yi wecÖZxc †KvY| wP‡Î OA I OB ci¯úi wecixZ iwk¥| Avevi OC I OD ci¯úi wecixZ iwk¥| Avevi BOC I
BOD I AOC ci¯úi wecÖZxc †KvY| DOA GKwU AciwUi wecÖZxc †KvY| `yBwU

mij‡iLv †Kv‡bv we›`y‡Z ci¯úi‡K †Q` Ki‡j, †Q` we›`y‡Z `yB †Rvov wecÖZxc †KvY Drcbœ nq|

Dccv`¨ 1
GKwU mij‡iLvi GKwU we›`y‡Z Aci GKwU iwk¥ wgwjZ n‡j, †h `yBwU mwbœwnZ †KvY Drcbœ nq Zv‡`i mgwó `yB mg‡KvY| g‡b Kwi, AB mij‡iLvwUi O we›`y‡Z OC iwk¥i cÖvš—we›`y O wgwjZ n‡q‡Q| d‡j AOC I COB `yBwU mwbœwnZ †KvY Drcbœ nj| AB †iLvi Dci DO j¤^ AuvwK| mwbœwnZ †KvYØ‡qi mgwó = AOC + + COB = AOD + DOB = 2 mg‡KvY |
COB = AOD + DOC

Dccv`¨ 2
`yBwU mij‡iLv ci¯úi †Q` Ki‡j, Drcbœ wecÖZxc †KvY¸‡jv ci¯úi mgvb| g‡b Kwi, AB I CD †iLvØq ci¯úi O we›`y‡Z †Q` K‡i‡Q| d‡j
O

we›`y‡Z

AOC ,

COB ,
BOD Ges

BOD ,

AOD †KvY Drcbœ n‡q‡Q|
AOD |

AOC = wecÖZxc

COB = wecÖZxc

MwYZ

111

6 4 mgvš—ivj mij‡iLv
GKvš—i †KvY, Abyiƒc †KvY, †Q`‡Ki GKB cvk¦©¯’ Aš—t¯’ †KvY

Dc‡ii wP‡Î, AB I

CD

`yBwU mij‡iLv Ges EF mij‡iLv G‡`i‡K

P

I Q we›`y‡Z †Q` K‡i‡Q|
1, 2 ,

EF mij‡iLv AB I CD mij‡iLvØ‡qi †Q`K| †Q`KwU AB I CD mij‡iLv `yBwUi mv‡_

3 , 4 , 5 , 6 , 7 , 8 †gvU AvUwU †KvY ˆZwi K‡i‡Q| G †KvY¸‡jvi g‡a¨

(K) (L) (M) (N)

1 Ges

5 , 2 Ges 6 ,

3 Ges

7 , 4 Ges 8 ci¯úi Abyi“c †KvY|

3 Ges

6 , 4 Ges 5 n‡jv ci¯úi GKvš—i †KvY

4 , 6 Wvbcv‡ki Aš—t¯’ †KvY|
3 , 5 evgcv‡ki Aš—t¯’ †KvY|

mgZ‡j `yBwU mij‡iLv ci¯úi‡K †Q` Ki‡Z cv‡i A_ev Zviv mgvš—ivj| mij‡iLvØq ci¯úi‡Q`x nq, hw` Dfq‡iLvq Aew¯’Z GKwU mvaviY we›`y _v‡K| Ab¨_vq mij‡iLv `yBwU mgvš—ivj| j¶Yxq †h, `yBwU wfbœ mij‡iLvi me©vwaK GKwU mvaviY we›`y _vK‡Z cv‡i| GKB mgZ‡j Aew¯’Z `yBwU mij‡iLvi mgvš—ivjZv wb‡gœewY©Z wZbfv‡e msÁvwqZ Kiv hvq: (K) mij‡iLv `yBwU KLbI ci¯úi‡K †Q` K‡i bv (`yB w`‡K Amxg ch©š— ewa©Z Kiv n‡jI)| (L) GKwU mij‡iLvi cÖwZwU we›`y AciwU ‡_‡K mgvb ¶z`ªZg `~i‡Z¡ Ae¯’vb K‡i| (M) mij‡iLv `yBwU‡K Aci GKwU mij‡iLv †Q` Ki‡j hw` GKvš—i †KvY ev Abyic †KvY¸‡jv mgvb nq| ~ msÁv (K) Abymv‡i GKB mgZ‡j Aew¯’Z `yBwU mij‡iLv G‡K Aci‡K †Q` bv Ki‡j †m¸‡jv mgvš—ivj| `yBwU mgvš—ivj mij‡iLv †_‡K †h‡Kv‡bv `yBwU †iLvsk wb‡j, †iLvsk `yBwUI ci¯úi mgvš—ivj nq| msÁv (L) Abymv‡i `yBwU mgvš—ivj mij‡iLvi GKwUi †h‡Kv‡bv we›`y †_‡K AciwUi j¤^-`~iZ¡ me©`v mgvb| j¤^-`~iZ¡ ej‡Z Zv‡`i GKwUi †h‡Kv‡bv we›`y n‡Z AciwUi Dci Aw¼Z j‡¤^i ˆ`N©¨‡KB †evSvq| Avevi wecixZfv‡e, `yBwU mij‡iLvi GKwUi †h‡Kv‡bv `yBwU we›`y †_‡K AciwUi j¤^-`~iZ¡ ci¯úi mgvb n‡jI †iLvØq mgvš—ivj| GB j¤^-`~iZ¡‡K `yBwU mgvš—ivj †iLvØ‡qi `~iZ¡ ejv nq| msÁv (M) BDwK¬‡Wi cÂg ¯^xKv‡h©i mgZzj¨| R¨vwgwZK cÖgvY I A¼‡bi Rb¨ G msÁvwU AwaKZi Dc‡hvMx| j¶ Kwi, †Kv‡bv wbw`©ó mij‡iLvi Dci Aew¯’Z bq Gi~c we›`yi ga¨ w`‡q H mij‡iLvi mgvš—ivj K‡i GKwU gvÎ mij‡iLv AuvKv hvq|

112

MwYZ

Dccv`¨ 3
`yBwU mgvš—ivj mij‡iLvi GKwU †Q`K Øviv Drcbœ (K) cÖ‡Z¨K GKvš—i †KvY †Rvov mgvb n‡e| (L) †Q`‡Ki GKB cv‡ki Aš—t¯’ †KvY `yBwU ci¯úi m¤ú~iK| wP‡Î, AB ll CD Ges PQ †Q`K Zv‡`i h_vµ‡g E I F we›`y‡Z †Q` K‡i‡Q| myZivs, (K)
PEB = Abyi~c
EFD

[msÁvbymv‡i]

(L) AEF = GKvš—i EFD (M) BEF EFD = `yB mg‡KvY|
KvR : 1| mgvš—ivj mij‡iLvi weKí msÁvi mvnv‡h¨ mgvš—ivj mij‡iLv msµvš— Dccv`¨¸‡jv cÖgvY Ki|

Dccv`¨ 4
`yBwU mij‡iLv Aci GKwU mij‡iLv‡K †Q` Ki‡j hw` (K) Abyi~c †KvY¸‡jv ci¯úi mgvb nq, A_ev (L) GKvš—i †KvY¸‡jv ci¯úi mgvb nq, A_ev (M) †Q`‡Ki GKB cv‡ki Aš—t¯’ †KvYØ‡qi †hvMdj `yB mg‡Kv‡Yi mgvb nq, Z‡e H mij‡iLv `yBwU ci¯úi mgvš—ivj| wP‡Î, AB I CD †iLvØq‡K PQ †iLv h_vµ‡g E I F we›`y‡Z †Q` K‡i‡Q Ges (K) A_ev, (L) A_ev, (M)
PEB = Abyi~c

EFD

AEF = GKvš—i
BEF

EFD

EFD = `yB mg‡KvY|

myZivs, AB I CD †iLv `yBwU ci¯úi mgvš—ivj| Abywm×vš— 1| †hme mij‡iLv GKB mij‡iLvi mgvš—ivj †m¸‡jv ci¯úi mgvš—ivj|

Abykxjbx 6 2
1| †Kv‡Yi Af¨š—i I ewnf©v‡Mi msÁv `vI| 2| hw` GKB mij‡iLv¯’ wZbwU wfbœ we›`y nq, Z‡e wP‡Îi Drcbœ †KvY¸‡jvi bvgKiY Ki| 3| mwbœwnZ †Kv‡Yi msÁv `vI Ges Gi evû¸‡jv wPwýZ Ki| 4| wPÎmn msÁv `vI: wecÖZxc †KvY, c~iK †KvY, m¤ú~iK †KvY, mg‡KvY, m~²‡KvY Ges ¯’~j‡KvY|

MwYZ

113

wÎfyR wZbwU †iLvsk Øviv Ave× wPÎ GKwU wÎfyR| †iLvsk¸‡jv‡K wÎfz‡Ri evû e‡j| †h‡Kv‡bv `yBwU evûi mvaviY we›`y‡K kxl©we›`y ejv nq| wÎfz‡Ri †h‡Kv‡bv `yBwU evû kxl©we›`y‡Z †KvY Drcbœ K‡i| wÎfy‡Ri wZbwU evû I wZbwU †KvY i‡q‡Q| evû‡f‡` wÎfzR wZb cÖKvi : mgevû, mgwØevû I welgevû| Avevi †KvY‡f‡`I wÎfzR wZb cÖKvi : m~²‡KvYx, ¯’‚j‡KvYx I mg‡KvYx| wÎfz‡Ri evû wZbwUi ˆ`‡N©¨i mgwó‡K cwimxgv e‡j| wÎfz‡Ri evû¸‡jv Øviv mxgve×‡¶Î‡K wÎfzR‡¶Î e‡j| wÎfz‡Ri †h‡Kv‡bv kxl©we›`y n‡Z wecixZ evûi ga¨we›`y ch©š— Aw¼Z †iLvsk‡K ga¨gv e‡j| Avevi, †h‡Kv‡bv kxl©we›`y n‡Z wecixZ evû Gi j¤^ `~iZ¡B wÎfz‡Ri D”PZv| cv‡ki wP‡Î ABC GKwU wÎfzR| A, B, C Gi wZbwU kxl©we›`y| AB , BC , CA Gi wZbwU evû Ges Gi wZbwU †KvY mgevû wÎfyR †h wÎfz‡Ri wZbwU evû mgvb Zv mgevû wÎfyR| cv‡ki wP‡Î ABC wÎfz‡Ri AB = BC = CA | A_©vr evû wZbwUi ‰`N©¨ mgvb| ABC wÎfyRwU GKwU mgevû wÎfyR|
BAC , ABC , BCA AB , BC , CA evûi cwigv‡ci †hvMdj wÎfzRwUi cwimxgv|

mgwØevû wÎfyR †h wÎfz‡Ri `yBwU evû mgvb Zv mgwØevû wÎfyR| cv‡ki wP‡Î ABC wÎfz‡Ri AB = AC mgwØevû| welgevû wÎfyR †h wÎfz‡Ri wZbwU evûB ci¯úi Amgvb Zv welgevû wÎfyR| cv‡ki wP‡Î ABC wÎfz‡Ri AB , BC , CA evû¸‡jvi ˆ`N©¨ ci¯úi Amgvb| ABC wÎfyRwU welgevû| m~²‡KvYx wÎfyR †h wÎfy‡Ri cÖ‡Z¨KwU †KvY m~²‡KvY, Zv m~²‡KvYx wÎfyR| ABC wÎfy‡R
BAC , ABC , BCA †KvY wZbwUi cÖ‡Z¨‡K m~²‡KvY|

BC | A_©vr `yBwU evûi

‰`N©¨ mgvb, hv‡`i †Kv‡bvwUB Z…Zxq evûi mgvb bq| ABC wÎfyRwU

A_©vr cÖ‡Z¨KwU †Kv‡Yi cwigvY 900 A‡c¶v Kg| m~²‡KvYx wÎfyR| dg©v-15, MwYZ-9g-10g

ABC GKwU

114

MwYZ

mg‡KvYx wÎfyR †h wÎfy‡Ri GKwU †KvY mg‡KvY, Zv mg‡KvYx wÎfy‡R
DFE mg‡KvY, Aci †KvY `yBwU

wÎfyR| DEF
DEF I EDF

cÖ‡Z¨‡K m~²‡KvY| DEF GKwU mg‡KvYx wÎfyR| ¯’‚j‡KvYx wÎfyR †h wÎfy‡Ri GKwU †KvY ¯’ƒ~j‡KvY, Zv ¯’‚j‡KvYx wÎfyR| GHK wÎfy‡R
GKH GKwU ¯’~j‡KvY, Aci †KvY `yBwU GHK I

HGK cÖ‡Z¨‡K m~²‡KvY|

GHK GKwU ¯’‚j‡KvYx wÎfyR|

9 3 wÎfy‡Ri ewnt¯’ I Aš—t¯’ †KvY
†Kv‡bv wÎfy‡Ri GKwU evû ewa©Z Ki‡j †h †KvY Drcbœ nq Zv wÎfyRwUi GKwU ewnt¯’ †KvY| GB †Kv‡Yi mwbœwnZ †KvYwU Qvov wÎfy‡Ri Aci `yBwU †KvY‡K GB ewnt¯’ †Kv‡Yi wecixZ Aš—t¯’ †KvY e‡j| cv‡ki wP‡Î,

ABC Gi BC evû‡K D ch©š— ewa©Z Kiv n‡q‡Q|

ACD wÎfyRwUi GKwU ewnt¯’ †KvY| ABD , BAC I ACB wÎfyRwUi wZbwU Aš—t¯’ †KvY| ACB †K ACD Gi
†cÖw¶‡Z mwbœwnZ Aš—t¯’ †KvY ejv nq| cÖ‡Z¨K‡K

ABC I BAC Gi

ACD Gi wecixZ Aš—t¯’ †KvY ejv nq|

Dccv`¨ 5 wÎfy‡Ri wZb †Kv‡Yi mgwó `yB mg‡Kv‡Yi mgvb|

g‡b Kwi, ABC GKwU wÎfyR| wÎfzRwUi Abywm×vš— 1| Abywm×vš— 2| Abywm×vš— 3|
KvR :

BAC

ABC

ACB

`yB mg‡KvY|

wÎfz‡Ri GKwU evû‡K ewa©Z Ki‡j †h ewnt¯’ †KvY Drcbœ nq, Zv Gi wecixZ Aš—t¯’ †KvYØ‡qi mgwói mgvb| wÎfz‡Ri GKwU evû‡K ewa©Z Ki‡j †h ewnt¯’ †KvY Drcbœ nq, Zv Gi Aš—t¯’ wecixZ †KvY `yBwUi cÖ‡Z¨KwU A‡c¶v e„nËi| mg‡KvYx wÎfy‡Ri m~²‡KvYØq ci¯úi c~iK|

1| cÖgvY Ki †h, wÎfz‡Ri GKwU evû‡K ewa©Z Ki‡j †h ewnt¯’ †KvY Drcbœ nq, Zv Gi Aš—t¯’ wecixZ †KvY `yBwUi cÖ‡Z¨KwU A‡c¶v e„nËi|

MwYZ

115

evû I †Kv‡Yi me©mgZv : `yBwU †iLvs‡ki ˆ`N©¨ mgvb n‡j †iLvsk `yBwU me©mg| Avevi wecixZfv‡e, `yBwU †iLvsk me©mg n‡j Zv‡`i ˆ`N©¨ mgvb| `yBwU †Kv‡Yi cwigvc mgvb n‡j †KvY `yBwU me©mg| Avevi wecixZfv‡e, `yBwU †KvY me©mg n‡j Zv‡`i cwigvcI mgvb|

wÎfz‡Ri me©mgZv GKwU wÎfzR‡K Aci GKwU wÎfz‡Ri Dci ¯’vcb Ki‡j hw` wÎfzR `yBwU me©‡Zvfv‡e wg‡j hvq, Z‡e wÎfzR `yBwU me©mg nq| me©mg wÎfz‡Ri Abyi~c evû I Abyi~c †KvY¸‡jv mgvb|

ABC I DEF me©mg n‡j Ges A, B, C kxl© h_vµ‡g D, E , F kx‡l©i Dci cv‡ki wP‡Î

ABC I DEF me©mg|
DF , BC

cwZZ n‡j AB DE , AC
B E,
C F

EF Ges

A

D,

n‡e|

ABC I DEF

me©mg

†evSv‡Z ABC

DEF †jLv nq|

Dccv`¨ 6 ( evû-†KvY-evû Dccv`¨) hw` `yBwU wÎfz‡Ri GKwUi `yB evû h_vµ‡g AciwUi `yB evûi mgvb nq Ges evû `yBwUi Aš—f©y³ †KvY `yBwU ci¯úi mgvb nq, Z‡e wÎfzR `yBwU me©mg| g‡b Kwi, ABC I DEF G AB DE , AC f©y³
DF Ges Aš—

BAC = Aš—f©y³

EDF . Zvn‡j, ABC

DEF .

Dccv`¨ 7 hw` †Kv‡bv wÎfz‡Ri `yBwU evû ci¯úi mgvb nq, Z‡e G‡`i wecixZ †KvY `yBwUI ci¯úi mgvb n‡e| g‡b Kwi, ABC wÎfz‡R AB

AC | Zvn‡j,

ABC

ACB |

116

MwYZ

Dccv`¨ 8 hw` †Kv‡bv wÎfz‡Ri `yBwU †KvY ci¯úi mgvb nq, Z‡e G‡`i wecixZ evû `yBwUI ci¯úi mgvb n‡e| we‡kl wbe©Pb: g‡b Kwi, ABC wÎfz‡R | cÖgvY Ki‡Z n‡e †h, AB cÖgvY: avc (1) hw` AB h_v_©Zv

ABC

ACB

AC |

AC Ges G‡`i †KvbwUB AB Gi [mgwØevû wÎfz‡Ri f~wg msjMœ †KvYØq mgvb]
AC A_ev (ii) [ewnt¯’ †KvY Aš—t¯’ wecixZ †KvY `yBwU cÖ‡Z¨KwU
A‡c¶v e„nËi]

mgvb bv nq, Z‡e (i) AB

AB

AC n‡e|

g‡b Kwi, (i) AB

AC . AB †_‡K AC Gi mgvb DBC Gi ewnt¯’ †KvY

AD †K‡U wbB| GLb, ADC wÎfzRwU mgwØevû| myZivs
ADC ACD

ADC > ABC
ACD > ABC myZivs, ACB > ABC wKš‘ Zv cÖ`Ë kZ©we‡ivax| (2) Abyiƒcfv‡e, (ii) AB

AC n‡j †`Lv‡bv hvq †h

ABC > ACB . wKš‘ ZvI cÖ`Ë kZ©we‡ivax|
(3) myZivs, AB bv|

AC A_ev AB

AC n‡Z cv‡i

AB

AC (cÖgvwYZ)

Dccv`¨ 9 (evû-evû-evû Dccv`¨) hw` GKwU wÎfz‡Ri wZb evû Aci GKwU wÎfz‡Ri wZb evûi mgvb nq, Z‡e wÎfzR `yBwU me©mg n‡e| g‡b Kwi,
AC

ABC Ges DEF G AB DE , DF Ges BC EF . Zvn‡j,
DEF .

ABC

Dccv`¨ 10 ( †KvY-evû-†KvY Dccv`¨) hw` GKwU wÎfy‡Ri `yBwU †KvY I Zv‡`i msjMœ evû h_vµ‡g Aci GKwU wÎfy‡Ri `yBwU †KvY I Zv‡`i msjMœ evûi mgvb nq, Z‡e wÎfyR `yBwU me©mg n‡e|

MwYZ

117

g‡b Kwi,

ABC I DEF -G
Z‡e wÎfyR `yBwU

B me©mg, E,
A_©vr

C

F Ges †KvYØ‡qi msjMœ BC evû = Abyi~c

EF evû| ABC

DEF .

Dccv`¨ 11 (AwZfzR-evû Dccv`¨)
`yBwU mg‡KvYx wÎfy‡Ri AwZfyRØq mgvb n‡j Ges GKwUi GK evû AciwUi Aci GK evûi mgvb n‡j, wÎfyRØq me©mg|

ABCI DEF
ABC DEF .

mg‡KvYx

wÎfyRØ‡q

AwZfyR

AC =AwZfyR

DF Ges AB

DE .Zvn‡j,

wÎfz‡Ri evû I †Kv‡Yi g‡a¨ m¤úK© i‡q‡Q| G m¤úK© wb‡Pi Dccv`¨ 11 I Dccv`¨ 12 Gi cÖwZcv`¨ welq|

Dccv`¨ 12
†Kv‡bv wÎfz‡Ri GKwU evû Aci GKwU evû A‡c¶v e„nËi n‡j, e„nËi evûi wecixZ †KvY ¶z`ªZi evûi wecixZ †KvY A‡c¶v e„nËi| g‡b Kwi,

ABC - G AC

AB. myZivs

ABC

ACB.

Dccv`¨ 13
†Kv‡bv wÎfz‡Ri GKwU †KvY Aci GKwU †KvY A‡c¶v e„nËi n‡j, e„nËi †Kv‡Yi wecixZ evû ¶z`ªZi †Kv‡Yi wecixZ evû A‡c¶v e„nËi| we‡kl wbe©Pb: g‡b Kwi,

ABC Gi AB

ACB cÖgvY Ki‡Z n‡e †h, AC cÖgvY: avc

ABC

h_v_©Zv

(1) hw` AC evû AB evû A‡c¶v e„nËi bv nq,

[mgwØevû wÎfz‡Ri mgvb evûØ‡qi wecixZ †KvYØq mgvb]

118

MwYZ

Z‡e (i) AC (i) hw` AC

AB A_ev (ii) AC

AB n‡e|
ACB
[ ¶z`ªZi evûi wecixZ †KvY ¶z`ªZi ]

AB nq, wKš‘ kZ©vbyhvqx ABC
Zv cÖ`Ë kZ©we‡ivax| (ii) Avevi, hw`

ABC

ACB AB

AC

nq,

Z‡e

ABC

ACB n‡e|

wKš‘ ZvI cÖ`Ë kZ©we‡ivax| (2) myZivs, AC evû AB Gi mgvb ev AB †_‡K ¶z`ªZi n‡Z cv‡i bv| (cÖgvwYZ)| wÎfz‡Ri †h‡Kv‡bv `yB evûi ˆ`‡N©¨i mgwói ev Aš—‡ii mv‡_ Z…Zxq evûi ˆ`‡N©¨i m¤úK© i‡q‡Q|

AC

AB

Dccv`¨ 14 wÎfz‡Ri †h‡Kv‡bv `yB evûi ˆ`‡N©¨i mgwó Gi Z…Zxq evûi ˆ`N©¨ A‡c¶v e„nËi| g‡b Kwi, ABC GKwU wÎfzR| awi, BC wÎfzRwUi e„nËg evû| Zvn‡j, AB

AC

BC |

Abywm×vš— 1| wÎfz‡Ri †h‡Kv‡bv `yB evûi ˆ`‡N©¨i Aš—i Gi Z…Zxq evûi ˆ`N©¨ A‡c¶v ¶z`ªZi| g‡b Kwi, ABC GKwU wÎfzR| ABC Gi †h‡Kv‡bv `yB evûi ˆ`‡N©¨i Aš—i Gi Z…Zxq evûi ˆ`N©¨ A‡c¶v ¶z`ªZi| ‡hgb, AB AC BC |

Dccv`¨ 15 wÎfz‡Ri †h‡Kv‡bv `yB evûi ga¨we›`yi ms‡hvRK †iLvsk Z…Zxq evûi mgvš—ivj Ges ˆ`‡N©¨ Zvi A‡a©K| g‡b Kwi, ABC GKwU wÎfzR| D I E h_vµ‡g wÎfzRwUi AB I AC evûi ga¨we›`y| Zvn‡j, cÖgvY Ki‡Z n‡e †h DE || BC Ges DE
1 BC . 2

A¼b: D I E †hvM K‡i ewa©Z Kwi †hb EF cÖgvY: avc (1)
ADE I CEF Gi g‡a¨ AE

DE nq|

h_v_©Zv
EC ,

[ †`Iqv Av‡Q ] [A¼bvbymv‡i] [wecÖZxc †KvY]

DE EF AED CEF

MwYZ

119

ADE
ADE

CEF EFC Ges DAE ECF .

[evû-‡KvY-evû Dccv`¨] [GKvš—i †KvY]

DF || BC ev DE || BC .

(2) Avevi, DF ev DE

BC ev DE EF

BC

DE

BC ev 2 DE

BC ev DE

1 BC 2

Dccv`¨ 16 (wc_v‡Mviv‡mi Dccv`¨) mg‡KvYx wÎfz‡Ri AwZfz‡Ri Ici Aw¼Z eM©‡¶‡Îi †¶Îdj Aci `yB evûi Ici Aw¼Z eM©‡¶‡ÎØ‡qi †¶Îd‡ji mgwói mgvb| g‡b Kwi, ABC mg‡KvYx wÎfy‡Ri
AC AwZfzR| Zvn‡j, AC 2

ABC mg‡KvY Ges
BC 2 .

AB 2

Abykxjbx 6 3
1| wb‡P wZbwU evûi ˆ`N¨© †`Iqv n‡jv| †Kvb †¶‡Î wÎfyR A¼b m¤¢e? K. 5 †m. wg., 6 †m. wg. I 7 †m. wg. M. 5 †m. wg., 7 †m. wg. I 14 †m. wg. L. N. 3 †m. wg., 4 †m. wg. I 7 †m. wg. 2 †m. wg., 4 †m. wg. I 8 †m. wg.

2| wb‡Pi Z_¨¸‡jv j¶ Ki : i †h wÎfy‡Ri wZbwU †KvY mg‡KvY Zv‡K mg‡KvYx wÎfyR e‡j ii †h wÎfy‡Ri wZbwU †KvY my²‡KvY Zv‡K m~²‡KvYx wÎfyR e‡j| iii †h wÎfy‡Ri wZbwU evû mgvb Zv‡K mgevû wÎfyR e‡j wb‡Pi †KvbwU mwVK ? K. i I ii L. i I iii M. ii I iii N. i, ii I iii 3| cÖ`Ë wPÎ Abyhvqx 3 I 4 bs cÖ‡kœi DËi `vI|

GKmg‡Kv‡Yi mgvb †KvY †KvYwU? K. BOC M. COD 4| BOC Gi c~iK †Kvb †KvYwU? K. AOC M. COD

L. N. L. N.

BOD AOD BOD AOD

120

MwYZ

5| 6| 7| 8| 9|

cÖgvY Ki †h, mgevû wÎfz‡Ri evû¸‡jvi ga¨we›`ymg~n †hvM Ki‡j †h wÎfzR Drcbœ nq, Zv mgevû n‡e| cÖgvY Ki †h, mgevû wÎfz‡Ri ga¨gv wZbwU ci¯úi mgvb| cÖgvY Ki †h, wÎfz‡Ri †h‡Kv‡bv `yBwU ewnt¯’ †Kv‡Yi mgwó `yB mg‡KvY A‡c¶v e„nËi|
ABC Gi Af¨š—‡i D GKwU we›`y| cÖgvY Ki †h, AB AC BD DC.

ABC Gi BC evûi ga¨we›`y D n‡j, cÖgvY Ki †h, AB

AC

2AD.

10| cÖgvY Ki †h, wÎfz‡Ri ga¨gvÎ‡qi mgwó Zvi cwimxgv A‡c¶v ¶z`ªZi| 11| ABC mgwØevû wÎfz‡R, BA evû‡K D ch©š— Giƒcfv‡e ewa©Z Kiv nj, †hb BA AD nq| cÖgvY Ki †h, 12|
BCD GKwU mg‡KvY|
B I
C Gi mgwØLÊKØq O we›`y‡Z wgwjZ nq| ABC Gi

cÖgvY Ki †h, 13|

BOC

90

1 A. 2

ABC Gi AB I AC evû‡K ewa©Z Ki‡j B I C we›`y‡Z †h ewnt‡KvY `yBwU Drcbœ nq, Zv‡`i

mgwØLÊK `yBwU O we›`y‡Z wgwjZ n‡j, cÖgvY Ki †h,
BOC 90 1 A. 2

14| wP‡Î, †`Iqv Av‡Q, Ges
B 2 A

C = GK mg‡KvY

cÖgvY Ki †h, AB 2BC . 15| cÖgvY Ki †h, wÎfz‡Ri GKwU evû ewa©Z Ki‡j †h ewnt¯’ †KvY Drcbœ nq, Zv wecixZ Aš—t¯’ †KvYØ‡qi mgwói mgvb| 16| cÖgvY Ki †h, wÎfz‡Ri †h‡Kv‡bv `yB evûi Aš—i Zvi Z…Zxq evû A‡c¶v ¶y`ªZi| 17| wP‡Î, ABC wÎfz‡Ri
B
1 AC. 2
AC Ges

GK mg‡KvY

Ges D , AwZfzR AC Gi ga¨we›`y| cÖgvY Ki †h, BD 18|
ABC G AB

A Gi mgwØLÊK AD, BC evû‡K D we›`y‡Z †Q` K‡i|

cÖgvY Ki †h,

ADB ¯’~j‡KvY|

19| cÖgvY Ki †h, †Kv‡bv †iLvs‡ki j¤^wØLÊ‡Ki Dcwiw¯’Z †h‡Kv‡bv we›`y D³ †iLvs‡ki cÖvš— we›`yØq n‡Z mg`~ieZ©x| 20. ABC GKwU mg‡KvYx wÎfyR hvi A= GK mg‡KvY| BC evûi ga¨we›`y D. K. cÖ`Ë Z_¨ Abyhvqx ABC wÎfyRwU A¼b Ki| L. †`LvI †h, AB+AC> 2AD M. cÖgvY Ki †h, AD= 1 BC
2

mßg Aa¨vq

e¨envwiK R¨vwgwZ c~‡e©i †kÖwY‡Z R¨vwgwZi wewfbœ Dccv`¨ cÖgv‡Y I Abykxjbx‡Z wPÎ A¼‡bi cÖ‡qvRb wQj| †m me wPÎ m~²fv‡e A¼‡bi cÖ‡qvRb wQj bv| wKš‘ KL‡bv KL‡bv R¨vwgwZK wPÎ m~²fv‡e A¼‡bi cÖ‡qvRb nq| †hgb, GKRb ¯’cwZ hLb †Kv‡bv evwoi bKmv K‡ib wKsev cÖ‡KŠkjx hLb h‡š¿i wewfbœ As‡ki wPÎ Auv‡Kb| G ai‡bi R¨vwgwZK A¼‡b ïay †¯‹j I †cwÝj K¤úv‡mi mvnvh¨ †bIqv nq| B‡Zvc~‡e© †¯‹j I †cwÝj K¤úv‡mi mvnv‡h¨ wÎfzR I PZzf©yR AuvK‡Z wk‡LwQ| G Aa¨v‡q we‡kl ai‡bi wÎfzR I PZzf©yR A¼‡bi Av‡jvPbv Kiv n‡e| Aa¨vq †k‡l wk¶v_©xiv wP‡Îi mvnv‡h¨ wÎfzR I PZzf©yR e¨vL¨v Ki‡Z cvi‡e| cÖ`Ë DcvË e¨envi K‡i wÎfzR A¼b Ki‡Z cvi‡e| cÖ`Ë DcvË e¨envi K‡i mvgvš—wiK A¼b Ki‡Z cvi‡e|

7 1 wÎfzR A¼b cÖ‡Z¨K wÎfz‡Ri wZbwU evû I wZbwU †KvY i‡q‡Q| Z‡e †Kv‡bv wÎfz‡Ri AvKvi I AvK…wZ wbw`©ó Kivi Rb¨ me¸‡jv evû I †Kv‡Yi cÖ‡qvRb nq bv| †hgb, wÎfz‡Ri wZb †Kv‡Yi mgwó `yB mg‡KvY e‡j Gi †h‡Kv‡bv `yBwU †Kv‡Yi gvb †`Iqv _vK‡j Z„Zxq †KvYwUi gvb †ei Kiv hvq| Avevi, wÎfz‡Ri me©mgZv msµvš— Dccv`¨¸‡jv †_‡K †`Lv hvq †h, †Kv‡bv wÎfz‡Ri wZbwU evû I wZbwU †KvY A_©vr QqwUi g‡a¨ †KejgvÎ wbgœwjwLZ wZbwU Aci GK wÎfz‡Ri Abyiƒc wZbwUi As‡ki mgvb n‡jB wÎfzR `yBwU me©mg nq| A_©vr, G wZbwU As‡ki Øviv wbw`©ó AvKv‡ii Abb¨ wÎfzR AuvKv hvq| mßg †kÖwY‡Z Avgiv wbgœewY©Z DcvË †_‡K wÎfzR AuvK‡Z wk‡LwQ| (1) wZbwU evû

(2) `yBwU evû I Zv‡`i Aš—f©y³ †KvY

dg©v-16, MwYZ-9g-10g

122

MwYZ

(3) `yBwU †KvY I Zv‡`i msjMœ evû

(4) `yBwU †KvY I GKwUi wecixZ evû

(5) `yBwU evû I Zv‡`i GKwUi wecixZ †KvY

(6) mg‡KvYx wÎfz‡Ri AwZfyR I Aci GKwU evû

j¶Yxq †h, Dc‡ii cÖ‡Z¨K †¶‡Î wÎfz‡Ri wZbwU Ask wbw`©ó Kiv n‡q‡Q| wKš‘ †h‡Kv‡bv wZbwU Ask wbw`©ó Ki‡jB wÎfzRwU wbw`©ó nq bv| †hgb, wÎfz‡Ri wZbwU †KvY †`Iqv _vK‡j wewfbœ AvKv‡ii AmsL¨ wÎfzR AuvKv hvq (hv‡`i m`„k wÎfzR ejv hvq)|

A‡bK mgq wÎfzR AuvKvi Rb¨ Ggb wZbwU DcvË †`Iqv _v‡K, hv‡`i mvnv‡h¨ wewfbœ A¼‡bi gva¨‡g wÎfzRwU wba©viY Kiv hvq| Giƒc K‡qKwU m¤úv`¨ wb‡P eY©bv Kiv n‡jv|

MwYZ

123

m¤úv`¨ 1 wÎfz‡Ri f~wg, f~wg msjMœ GKwU †KvY I Aci `yB evûi mgwó †`Iqv Av‡Q| wÎfzRwU AuvK‡Z n‡e| g‡b Kwi, †Kv‡bv wÎfz‡Ri f~wg a , f~wg msjMœ GKwU †KvY x Ges Aci `yB evûi mgwó s †`Iqv Av‡Q| wÎfzRwU

AuvK‡Z n‡e| a A¼b : (1) †h‡Kv‡bv GKwU iwk¥ BE †_‡K f~wg a Gi mgvb K‡i BC †iLvsk †K‡U wbB| BC †iLvs‡ki B we›`y‡Z x Gi mgvb
CBF AuvwK| BDC Gi mgvb

(2) BF iwk¥ †_‡K s Gi mgvb BD Ask KvwU|
(3) C, D †hvM Kwi| C we›`y‡Z DC †iLvs‡ki †h cv‡k B we›`y Av‡Q †mB cv‡k

DCG AuvwK|
(4) CG iwk¥ BD †K A we›`y‡Z †Q` K‡i|

Zvn‡j, ABC B DwÏó wÎfzR| cÖgvY : ACD G
AC
GLb,

ADC

ACD [A¼b Abymv‡i ]

AD. x, BC a, [A¼b Abymv‡i ]

ABC G ABC

Ges BA AC

BA AD

BD

s | AZGe,

ABC B wb‡Y©q wÎfzR|

weKí c×wZ g‡b Kwi, †Kv‡bv wÎfz‡Ri f~wg a , f~wg msjMœ GKwU †KvY x Ges Aci `yB evûi mgwó s †`Iqv Av‡Q| wÎfzRwU

AuvK‡Z n‡e| A¼b : (1) †h‡Kv‡bv GKwU iwk¥ BE †_‡K f~wg a Gi mgvb K‡i
BC †iLvsk †K‡U wbB| BC †iLvs‡ki B we›`y‡Z

x Gi

mgvb

CBF AuvwK|

(2) BF iwk¥ †_‡K s Gi mgvb BD Ask KvwU|
(3) C, D †hvM Kwi| CD Gi j¤^ wØLÊK PQ AuvwK| (4) PQ iwk¥ BD iwk¥‡K A we›`y‡Z †Q` K‡i| A, C

†hvM Kwi| Zvn‡j, ABC B DwÏó wÎfzR|

124

MwYZ

cÖgvY : ACR Ges ADR G CR DR AR
ACR
ADR .

AR Ges Aš—f©y³

ARC

ARD [mg‡KvY ]

AC

AD

GLb, ABC G ABC Ges BA AC
BA AD

x, BC
BD

a, [A¼b Abymv‡i ] s . AZGe, ABC B wb‡Y©q wÎfzR|

m¤úv`¨ 2 wÎfy‡Ri f~wg, f~wg msjMœ GKwU m~²‡KvY I Aci `yB evûi Aš—i †`Iqv Av‡Q| wÎf~RwU AuvK‡Z n‡e| g‡b Kwi, †Kv‡bv wÎfz‡Ri f~wg a f~wg msjMœ m~²‡KvY n‡e| x a

x.

Ges Aci `yB evûi Aš—i d †`Iqv Av‡Q| wÎfzRwU AuvK‡Z

A¼b : (1) †h‡Kv‡bv GKwU iwk¥ BF †_‡K f~wg a Gi mgvb K‡i BC †iLvsk †K‡U wbB| BC †iLvs‡ki B we›`y‡Z x Gi mgvb
CBE AuvwK|

(2) BE iwk¥ †_‡K d Gi mgvb BD Ask †K‡U wbB| (3) C, D †hvM Kwi| DC †iLvs‡ki †h cv‡k E we›`y Av‡Q †mB cv‡k C we›`y‡Z
DCA AuvwK| CA iwk¥ BE iwk¥‡K A we›`y‡Z †Q` K‡i| Zvn‡j,

EDC Gi mgvb

ABC B DwÏó wÎfzR|

cÖgvY : A¼b Abymv‡i, ACD G
AC AD.

ADC

ACD

myZivs `yB evûi Aš—i, AB AC GLb, ABC G BC we‡kl `ªóe¨ : KvR : a, AB

AB

AD

BD ABC

d. x. myZivs, ABC B wb‡Y©q wÎfzR|

AC

d Ges

1| cÖ`Ë †KvY m~²‡KvY bv n‡j, Dc‡ii c×wZ‡Z A¼b Kiv m¤¢e bq| †Kb ? G †¶‡Î wÎfzRwU AuvKvi †Kv‡bv Dcvq †ei Ki| 2| wÎfy‡Ri f~wg, f~wg msjMœ GKwU m~²‡KvY I Aci `yB evûi Aš—i †`Iqv Av‡Q| weKí c×wZ‡Z wÎfyRwU A¼b Ki|

MwYZ

125

m¤úv`¨ 3 wÎfy‡Ri f~wg msjMœ `yBwU †KvY I cwimxgv †`Iqv Av‡Q| wÎfzRwU AuvK‡Z n‡e| g‡b Kwi, GKwU wÎfz‡Ri cwimxgv p Ges f~wg msjMœ `yBwU †KvY A¼b : (1) †h‡Kv‡bv GKwU iwk¥ DF †_‡K cwimxgv p Gi mgvb K‡i DE Ask †K‡U wbB| D I E we›`y‡Z DE †iLvs‡ki GKB cv‡k x Gi mgvb
EDL Ges

x I

y †`Iqv Av‡Q| wÎfzRwU AuvK‡Z n‡e|

y Gi mgvb

DEM AuvwK|

(2) †KvY `yBwUi wØLÊK DG I EH AuvwK|

(3) g‡b Kwi, DG I EH iwk¥Øq ci¯úi‡K A we›`y‡Z †Q` K‡i| A we›`y‡Z
DAB Ges AED Gi mgvb
EAC AuvwK|

ADE Gi mgvb

(4) AB Ges AC iwk¥Øq DE †iLvsk‡K h_vµ‡g B I C we›`y‡Z †Q` K‡i| Zvn‡j, ABC B DwÏó wÎfzR| cÖgvY : ADB G Avevi, ACE G
ADB
AEC

DAB [A¼b Abymv‡i],
EAC ; CA CE.

AB

DB.

myZivs ABC G AB BC CA DB BC CE DE
ABC ADB DAB 1 x 2 1 x 2
1 y 2

p.

x
y. myZivs

Ges

ACB

AEC

EAC

1 y 2

ABC B wb‡Y©q wÎfzR|

KvR : wÎfy‡Ri f~wg msjMœ `yBwU m~²‡KvY I cwimxgv †`Iqv Av‡Q| weKí c×wZ‡Z wÎfyRwU A¼b Ki|

D`vniY 1|GKwU wÎfzR ABC AuvK, hvi
†m.wg.|

B

60 , C

45 Ges cwimxgv AB BC CA 11

126

MwYZ

A¼b : wb‡Pi avcmg~n AbymiY Kwi: (1) †iLvsk PQ 11 †m.wg. AuvwK| (2) PQ †iLvs‡ki GKB cv‡k P Ges Q we›`y‡Z h_vµ‡g AuvwK| (3) †KvY `yBwUi wØLÊK PG I QH AuvwK| g‡b Kwi, PG I GH iwk¥Øq ci¯úi‡K A we›`y‡Z †Q` K‡i| (4) PA , QA †iLvs‡ki j¤^ mgwØLÊK AuvwK hv PQ †iLvsk‡K h_vµ‡g B I C we›`y‡Z †Q` K‡i| (5) A, B Ges A, C †hvM Kwi| Zvn‡j, ABC B DwÏó wÎfzR|
KvR : mg‡KvYx wÎfz‡Ri mg‡KvY msjMœ GKwU evû Ges AwZfzR I Aci evûi Aš—i †`Iqv Av‡Q| wÎfzRwU AuvK|

QPL

60 I

PQM

45 †KvY

Abykxjbx 7 1
1| wb‡gœ cÖ`Ë DcvË wb‡q wÎfzR A¼b Ki : K. wZbwU evûi ˆ`N©¨ h_vµ‡g 3 †m.wg., 3 5 †m.wg., 2 8 †m.wg.| L. `yBwU evûi ˆ`N©¨ 4 †m.wg., 3 †m.wg. Ges Aš—f©y³ †KvY 60 | M. `yBwU †KvY 60 I 45 Ges G‡`i msjMœ evûi ˆ`N©¨ 5 †m.wg.| N. `yBwU †KvY 60 I 45 Ges 45 †Kv‡Yi wecixZ evûi ˆ`N©¨ 5 †m.wg.| O. `yBwU evûi ˆ`N©¨ h_vµ‡g 4.5 †m.wg. I 3.5 †m.wg. Ges wØZxq evûi wecixZ †KvY 30 | P. mg‡KvYx wÎfz‡Ri AwZfzR I GKwU evûi ˆ`N©¨ h_vµ‡g 6 †m.wg. I 4 †m.wg.| 2| wb‡gœ cÖ`Ë DcvË wb‡q wÎfzR A¼b Ki : K. f~wg 3 5 †m.wg., f~wg msjMœ GKwU †KvY 60 I Aci `yB evûi mgwó 8 †m.wg | L. f~wg 4 †m.wg., f~wg msjMœ GKwU †KvY 50 I Aci `yB evûi mgwó 7.5 †m.wg | M. f~wg 4 †m.wg., f~wg msjMœ GKwU †KvY 50 I Aci `yB evûi Aš—i 1.5 †m.wg | N. f~wg 5 †m.wg., f~wg msjMœ GKwU †KvY 45 I Aci `yB evûi Aš—i 1 †m.wg | O. f~wg msjMœ †KvY `yBwU h_vµ‡g 60 I 45 I cwimxgv 12 †m.wg.| O. f~wg msjMœ †KvY `yBwU h_vµ‡g 30 I 45 I cwimxgv 10 †m.wg.| 3| GKwU wÎfy‡Ri f~wg msjMœ `yBwU †KvY Ges kxl© †_‡K f~wgi Dci Aw¼Z j‡¤^i ˆ`N©¨ †`Iqv Av‡Q| wÎfyRwU AuvK| 4| 5| 6| 7| mg‡KvYx wÎfy‡Ri AwZfyR I Aci `yB evûi mgwó †`Iqv Av‡Q| wÎfyRwU AuvK| wÎfy‡Ri f~wg msjMœ GKwU †KvY, D”PZv I Aci `yB evûi mgwó †`Iqv Av‡Q| wÎfyRwU AuvK| mgevû wÎfy‡Ri cwimxgv †`Iqv Av‡Q| wÎfyRwU AuvK|
~ wÎfz‡Ri f~wg, f~wg msjMœ GKwU ¯’j‡KvY I Aci `yB evûi Aš—i †`Iqv Av‡Q| wÎfyRwU AuvK|

MwYZ

127

7 2 PZzf©yR A¼b
Avgiv †`‡LwQ †h, wÎfz‡Ri wZbwU DcvË †`Iqv _vK‡j A‡bK †¶‡ÎB wÎfzRwU wbw`©ófv‡e AuvKv m¤¢e| wKš‘ PZzf©y‡Ri PviwU evû †`Iqv _vK‡jB GKwU wbw`©ó PZzf©yR AuvKv hvq bv| wbw`©ó PZzf©yR AuvKvi Rb¨ cuvPwU ¯^Zš¿ DcvË cÖ‡qvRb nq| wb‡gœ ewY©Z cuvPwU DcvË Rvbv _vK‡j, wbw`©ó PZzf©yR AuvKv hvq| (1) PviwU evû I GKwU †KvY (2) PviwU evû I GKwU KY© (3) wZbwU evû I `yBwU KY© (4) wZbwU evû I Zv‡`i Aš—f©y³ `yBwU †KvY (5) `yBwU evû I wZbwU †KvY| Aóg †kÖwY‡Z D‡jwLZ DcvË w`‡q PZzf©yR A¼b wel‡q Av‡jvPbv Kiv n‡q‡Q| A¼‡bi †KŠkj j¶ K‡i †`Lv hvq wKQz †¶‡Î mivmwi PZzf©yR AuvKv nq| Avevi wKQz †¶‡Î wÎfzR A¼‡bi gva¨‡g PZzf©yR AuvKv nq| †h‡nZz KY© PZzf©yR‡K `yBwU wÎfz‡R wef³ K‡i, †m‡nZz DcvË wnmv‡e GKwU ev `yBwU KY© cÖ`Ë n‡j wÎfzR A¼‡bi gva¨‡g PZzf©yR AuvKv m¤¢e nq| (1) PviwU evû I GKwU †KvY

(2) PviwU evû I GKwU KY©

(3) wZbwU evû I `yBwU KY©

(4) wZbwU evû I Zv‡`i Aš—f©y³ `yBwU †KvY

(5) `yBwU evû I wZbwU †KvY|

128

MwYZ

we‡kl ai‡bi PZzf©yR A¼‡bi Rb¨ A‡bK mgq Ggb DcvË †`Iqv _v‡K hv †_‡K wbw`©ó PZzf©yR AuvKvi Rb¨ cÖ‡qvRbxq cuvPwU ¯^Zš¿ DcvË cvIqv hvq| Zvn‡j H Dcv‡Ëi mvnv‡h¨I PZzf©yRwU AuvKv hvq| †hgb, mvgvš— wi‡Ki `yBwU msjMœ evû I Zv‡`i Aš—f©y³ †KvYwU †`Iqv _vK‡j mvgvš—wiKwU AuvKv hvq| GLv‡b wZbwU gvÎ DcvË †`Iqv Av‡Q| Avevi e‡M©i gvÎ GKwU evû †`Iqv _vK‡jB eM©wU AuvKv hvq| KviY, Zv‡Z cuvPwU DcvË, h_v e‡M©i Pvi mgvb evû I GK †KvY (mg‡KvY) wbw`©ó nq|

m¤úv`¨ 4 mvgvš—wi‡Ki `yBwU KY© I Zv‡`i Aš—f©y³ GKwU †KvY †`Iqv Av‡Q| mvgvš—wiKwU AuvK‡Z n‡e| g‡b Kwi, mvgvš—wi‡Ki KY© `yBwU a I b Ges KY©Ø‡qi Aš—f©y³ GKwU †KvY x †`Iqv Av‡Q| mvgvš—wiKwU AuvK‡Z n‡e|

A¼b : †h‡Kv‡bv iwk¥ AM †_‡K a Gi mgvb AC †iLvsk wbB|
AC Gi ga¨we›`y O wbY©q Kwi| O we›`y‡Z

x Gi mgvb

AOP AuvwK| OP Gi wecixZ iwk¥ OQ A¼b Kwi| OP I
OQ iwk¥Øq †_‡K

1 b Gi mgvb h_vµ‡g OB I OD 2

†iLvskØq wbB| A, B ; A, D ; C, B I C, D †hvM Kwi| Zvn‡j, ABCD B DwÏó mvgvš—wiK| cÖgvY : AOB I COD G OA OC Ges Aš—f©y³
AOB =Aš—f©y³

1 a, OB OD 2

1 b [A¼bvbymv‡i] 2

COD [wecªZxc †KvY]|

AZGe, AOB myZivs, AB CD Ges
ABO

COD
CDO ; wKš‘ †KvY `yBwU GKvš—i †KvY|

AB I CD mgvb I mgvš—ivj|

Abyiƒcfv‡e, AD I BC mgvb I mgvš—ivj| myZivs, ABCD GKwU mvgvš—wiK hvi KY©Øq AC I BD BO OD
AO OC 1 a 2 1 a 2 a

1 1 b b b Ges KY© `yBwUi Aš—f©y³ 2 2

AOB

x

AZGe, ABCD B wb‡Y©q mvgvš—wiK|

MwYZ

129

m¤úv`¨ 5 mvgvš—wi‡Ki `yBwU KY© I GKwU evû †`Iqv Av‡Q| mvgvš—wiKwU AuvK‡Z n‡e| g‡b Kwi mvgvš—wi‡Ki `yBwU KY© a I b Ges GKwU evû c †`Iqv Av‡Q| mvgvš—wiKwU AuvK‡Z n‡e| A¼b : a I b KY©Øq‡K mgvb `yBfv‡M wef³ Kwi| †h‡Kv‡bv iwk¥
AX †_‡K c Gi mgvb AB wbB| A I B †K †K›`ª K‡i

h_vµ‡g

a b I Gi mgvb e¨vmva© wb‡q AB Gi GKB cv‡k 2 2

`yBwU e„ËPvc AuvwK| g‡b Kwi, e„ËPvc `yBwU ci¯úi‡K O we›`y‡Z †Q` K‡i| A, O I O, B †hvM Kwi| AO †K AE eivei Ges
BO †K BF eivei ewa©Z Kwi| OE †_‡K

a 2

OC Ges OF

†_‡K

b 2

OD wbB| A, D ; D, C I B, C †hvM Kwi|

Zvn‡j, ABCD B DwÏó mvgvš—wiK| cÖgvY : AOB I COD G OA OC Ges Aš—f©y³
AOB

a ; OB 2

OD

b , [A¼bvbymv‡i] 2

AOB = Aš—fy©³
COD .

COD [wecÖZxc †KvY]

AB

CD Ges

ABO

ODC ; wKš‘ †KvY `yBwU GKvš—i †KvY|

AB I CD mgvb I mgvš—ivj|

Abyiƒcfv‡e, AD I BC mgvb I mgvš—ivj| AZGe, ABCD B wb‡Y©q mvgvš—wiK| D`vniY 1| UªvwcwRqv‡gi `yBwU mgvš—ivj evû Ges G‡`i g‡a¨ e„nËi evû msjMœ `yBwU †KvY †`Iqv Av‡Q| UªvwcwRqvgwU AuvK|

g‡b Kwi, UªvwcwRqv‡gi mgvš—ivj evûØq a Ges b , †hLv‡b a b Ges e„nËi evû a msjMœ †KvYØq I y | UªvwcwRqvgwU AuvK‡Z n‡e| A¼b : †h‡Kv‡bv iwk¥ AX †_‡K AB a wbB| B †iLvs‡ki A we›`y‡Z we›`y‡Z y Gi mgvb ABZ AuvwK| dg©v-17, MwYZ-9g-10g x Gi mgvb

x

BAY Ges B

130

MwYZ

Gevi AB †iLvsk †_‡K AE b †K‡U wbB| E we›`y‡Z BC|| AY AvuwK hv BZ iwk¥‡Z C we›`y‡Z †Q` K‡i| Gevi CD|| BA AuvwK| CD †iLvsk AY iwk¥‡K D we›`y‡Z †Q` K‡i| Zvn‡j, ABCD B DwÏó UªvwcwRqvg| cÖgvY : A¼bvbymv‡i, AB|| CD Ges AD|| EC myZivs ABCD GKwU mvgvš—wiK Ges CD PZzf©yR ABCD I AB a, CD b, AB|| CD Ges AZGe, ABCD B wb‡Y©q UªvwcwRqvg|
KvR : i¤^‡mi cwimxgv I GKwU †KvY †`Iqv Av‡Q| i¤^mwU AuvK|

AE

b . GLb,

BAD =

x,

ABC =

y (A¼b Abymv‡i)

Abykxjbx 7 2
1| mg‡KvYx wÎfy‡Ri Aci `yBwU †Kv‡Yi cwigvY †`Iqv _vK‡j wb‡gœi †Kvb †¶‡Î wÎfyR A¼b Kiv m¤¢e| K. 630 I 360 L. 300 I 700 M. 400 I 500 N. 800 I 200 2| i AvqZ GKwU mvgvš—wiK ii eM© GKwU AvqZ iii i¤^m GKwU eM© Ic‡ii Z‡_¨i Av‡jv‡K wb‡gœi †KvbwU mwVK ? K. i I ii L. i I iii M. ii I iii N. i, ii I iii cÖ`Ë wP‡Îi Av‡jv‡K 3 I 4 bs cÖ‡kœi DËi `vI

3| AOB Gi †¶Îdj KZ? K. 6 eM© GKK L. 7 eM© GKK N. 14 eM© GKK M. 12 eM© GKK 4| PZzfy©RwUi cwimxgv K. 12 GKK L. 14 GKK N. 28 GKK M. 20 GKK 5| wb‡gœ cÖ`Ë DcvË wb‡q PZzfy©R A¼b Ki : K. PviwU evûi ˆ`N©¨ 3 †m.wg., 3 5 †m.wg., 2.5 †m.wg. I 3 †m,wg. Ges GKwU †KvY 45 | L. PviwU evûi ˆ`N©¨ 3.5 †m.wg, 4 †m.wg., 2 5 †m.wg. I 3.5 †m.wg. Ges GKwU KY© 5 †m.wg.| M. wZbwU evûi ˆ`N©¨ 3 2 †m.wg., 3 †m.wg., 3 5 †m.wg. Ges `yBwU KY© 2 8 †m.wg. I 4.5 †m.wg.| N. wZbwU evûi ˆ`N©¨ 3 †m.wg., 3 5 †m.wg., 4 †m.wg. Ges `yBwU †KvY 60 I 45 |

MwYZ

131

6| wb‡gœ cÖ`Ë DcvË wb‡q mvgvš—wiK A¼b Ki : K. `yBwU K‡Y©i ˆ`N©¨ 4 †m.wg., 6 5 †m.wg. Ges G‡`i Aš—f©y³ †KvY 45 | L. `yBwU K‡Y©i ˆ`N©¨ 5 †m.wg., 6 5 †m.wg. Ges G‡`i Aš—f©y³ †KvY 30 | M. GKwU evûi ˆ`N©¨ 4 †m.wg. Ges `yBwU K‡Y©i ˆ`N©¨ 5 †m.wg., 6 5 †m.wg.| N. GKwU evûi ˆ`N©¨ 5 †m.wg. Ges `yBwU K‡Y©i ˆ`N©¨ 4.5 †m.wg., 6 †m.wg.| 7| ABCD PZzf©y‡Ri AB I BC evû Ges B, C I D †KvY †`Iqv Av‡Q| PZzf©yRwU AuvK| 8| PZzfy©‡Ri KY© `yBwUi †Q`we›`y Øviv KY© `yBwUi PviwU LwÊZ Ask Ges Zv‡`i Aš—f©y³ GKwU †KvY h_vµ‡g OA = 4 †m.wg., OB = 5 †m.wg., OC 3 5 †m.wg., OD = 4 5 †m.wg. I AOB = 80 . PZzf©yRwU AuvK| 9| i¤^‡mi GKwU evûi ˆ`N©¨ 3 5 †m.wg. I GKwU †KvY 45 ; i¤^mwU AuvK| 10| i¤^‡mi GKwU evû Ges GKwU K‡Y©i ˆ`N©¨ †`Iqv Av‡Q| i¤^mwU AuvK| 11| `yBwU K‡Y©i ˆ`N©¨ †`Iqv Av‡Q| i¤^mwU AuvK| 12| eM©‡¶‡Îi cwimxgv †`Iqv Av‡Q| eM©‡¶ÎwU AuvK| 13| RKx I Rvdiyj mv‡n‡ei emZ evwo GKB mxgv‡iLvi g‡a¨ Aew¯’Z Ges evwoi †¶Îdj mgvb| Z‡e RKxi mv‡n‡ei evwoi AvK…wZ AvqZvKvi Ges Rvdviyj mv‡n‡ei evwo mvgvš—wiK AvK…wZi| K. f~wgi ˆ`N©¨ 10 GKK Ges D”PZv 8 GKK a‡i Zv‡`i evwoi mxgv‡iLv A¼b Ki| L. †`LvI †h, RKx mv‡n‡ei evwoi mxgv‡iLv Rvdviyj mv‡n‡ei evwoi mxgv‡iLv A‡c¶v †QvU| M. RKx mv‡n‡ei evwoi ˆ`N©¨ I cÖ‡¯’i AbycvZ 4 : 3 Ges †¶Îdj 300 eM© GKK n‡j, Zv‡`i evwoi †¶ÎdjØ‡qi AbycvZ wbY©q Ki| 14| GKwU mg‡KvYx wÎfy‡Ri AwZf~R 7 †m.wg I GK evûi ˆ`N©¨ 4 †m.wg, A = 850, B = 800 Ges
C = 950

Ic‡ii Z‡_¨i Av‡jv‡K wb‡Pi cÖkœ¸‡jvi DËi `vI : K. wÎfyRwUi Aci evûi ˆ`N©¨ wbY©q Ki| L. wÎfyRwU A¼b Ki| M. wÎfyRwUi cwimxgvi mgvb cwimxgv wewkó GKwU eM© A¼b Ki| 15| ABCD PZzfy‡R©i AB = 4 †m.wg. BC = 5 †m.wg Ic‡ii Z‡_¨i Av‡jv‡K wb‡Pi cÖkœ¸‡jvi DËi `vI K. GKwU i¤^m A¼b K‡i Dnvi bvg `vI| L. cÖ`Ë Z_¨ Abyhvqx ABCD PZzfyR©wU A¼b Ki| M. cÖ`Ë PZzfy©‡Ri cwimxgvi mgvb cwimxgv wewk÷ GKwU mgevû wÎfyR A¼b Ki|

Aóg Aa¨vq

e„Ë
Avgiv †R‡bwQ †h, e„Ë GKwU mgZjxq R¨vwgwZK wPÎ hvi we›`y¸‡jv †Kv‡bv wbw`©ó we›`y †_‡K mg`~i‡Z¡ Aew¯’Z| e„Ë m¤úwK©Z wewfbœ aviYv †hgb †K›`ª, e¨vm, e¨vmva©, R¨v BZ¨vw` wel‡q Av‡jvPbv Kiv n‡q‡Q| G Aa¨v‡q mgZ‡j †Kv‡bv e„‡Ëi Pvc I ¯úk©K m¤úwK©Z cÖwZÁvi Av‡jvPbv Kiv n‡e| Aa¨vq †k‡l wk¶v_©xiv e„ËPvc, †K›`ª¯’ †KvY, e„Ë¯’ †KvY, e„‡Ë Aš—wj©wLZ PZzf©yR e¨vL¨v Ki‡Z cvi‡e| e„Ë msµvš— Dccv`¨ cÖgvY Ki‡Z cvi‡e| e„Ë m¤úwK©Z m¤úv`¨ eY©bv Ki‡Z cvi‡e|

8 1 e„Ë e„Ë GKwU mgZjxq R¨vwgwZK wPÎ hvi we›`y¸‡jv †Kv‡bv wbw`©ó we›`y †_‡K mg`~i‡Z¡ Aew¯’Z| wbw`©ó we›`ywU e„‡Ëi †K›`ª| wbw`©ó we›`y †_‡K mg`~iZ¡ eRvq †i‡L †Kv‡bv we›`y †h Ave× c_ wPwÎZ K‡i ZvB e„Ë| †K›`ª n‡Z e„Ë¯’ †Kv‡bv we›`yi `~iZ¡‡K e¨vmva© e‡j| g‡b Kwi, O mgZ‡ji †Kv‡bv wbw`©ó we›`y Ges r wbw`©ó cwigvc| mgZj¯’ †h mKj we›`y O †_‡K r `~i‡Z¡ Aew¯’Z, Zv‡`i †mU e„Ë, hvi †K›`ª O I e¨vmva© r . wP‡Î O e„‡Ëi †K›`ª, A, B I C e„Ë¯’ we›`y| OA, OB I OC Gi cÖ‡Z¨KwU e„ËwUi e¨vmva©| mgZj¯’ KwZcq we›`y‡K mge„Ë we›`y ejv nq hw` we›`y¸‡jv w`‡q GKwU e„Ë hvq A_©vr, Ggb GKwU e„Ë _v‡K hv‡Z we›`y¸‡jv Aew¯’Z nq| Dc‡ii wP‡Î A, B I C mge„Ë we›`y| r e„‡Ëi Af¨š—i I ewnf©vM hw` †Kv‡bv e„‡Ëi †K›`ª O Ges e¨vmva© r nq Z‡e O †_‡K mgZ‡ji †h mKj we›`yi `~iZ¡ r †_‡K Kg Zv‡`i †mU‡K e„ËwUi Af¨š—i Ges O †_‡K mgZ‡ji †h mKj we›`yi `~iZ¡ r †_‡K †ewk Zv‡`i †mU‡K e„ËwUi ewnf©vM ejv nh| e„‡Ëi Af¨š—i¯’ `yBwU we›`yi ms‡hvRK †iLvsk m¤ú~Y©fv‡e e„‡Ëi Af¨š—‡iB _v‡K| †Kv‡bv e„‡Ëi Af¨š—i¯’ GKwU we›`y I ewnt¯’ GKwU we›`yi ms‡hvRK †iLvsk e„ËwU‡K GKwU I †Kej GKwU we›`y‡Z †Q` K‡i| wP‡Î, P e„‡Ëi Af¨š—i¯’ GKwU we›`y Ges Q e„‡Ëi ewnt¯’ GKwU we›`y| PQ †iLvsk e„ËwU‡K †Kej R we›`y‡Z †Q` K‡i|

MwYZ

133

e„‡Ëi R¨v I e¨vm e„‡Ëi `yBwU wfbœ we›`yi ms‡hvRK †iLvsk e„ËwUi GKwU R¨v| e„‡Ëi †Kv‡bv R¨v hw` †K›`ª w`‡q hvq Z‡e R¨vwU‡K e„‡Ëi e¨vm ejv nq| A_©vr e„‡Ëi †K›`ªMvgx †h‡Kv‡bv R¨v n‡jv e¨vm| wP‡Î, AB I AC e„ËwUi `yBwU R¨v Ges e„ËwUi †K›`ª O | G‡`i g‡a¨ AC R¨vwU e¨vm; KviY R¨vwU e„ËwUi †K›`ªMvgx| OA I OC e„‡Ëi `yBwU e¨vmva©| myZivs, e„‡Ëi †K›`ª cÖ‡Z¨K e¨v‡mi ga¨we›`y| AZGe cÖ‡Z¨K e¨v‡mi ˆ`N©¨ 2r , †hLv‡b r e„ËwUi e¨vmva©|

Dccv`¨ 1| e„‡Ëi †K›`ª I e¨vm wfbœ †Kv‡bv R¨v Gi ga¨we›`yi ms‡hvRK †iLvsk H R¨v Gi Ici j¤^| g‡b Kwi, O †K›`ªwewkó ABC e„‡Ë e¨vm bq Ggb GKwU R¨v AB Ges GB R¨v Gi ga¨ we›`y M | O, M †hvM Kwi| cÖgvY Ki‡Z n‡e †h, OM †iLvsk AB R¨v Gi Dci j¤^| A¼b : O, A Ges O, B †hvM Kwi| cÖgvY : avcmg~n (1) OAM Ges OBM G
AM
OA

h_v_©Zv
[ M , AB Gi ga¨we›`y]

BM
OB

[ Df‡q GKB e„‡Ëi e¨vmva©] [ mvaviY evû ]
OBM
OMB OMB = 1 mg‡KvY|

Ges myZivs,

OM

OM

OAM
OMA

[ evû-evû-evû Dccv`¨ ]

(2) †h‡nZz †KvYØq ˆiwLK hyMj †KvY Ges Zv‡`i cwigvc mgvb, myZivs,
OMA

AZGe, OM

AB . (cÖgvwYZ)

Abywm×vš— 1| e„‡Ëi †h‡Kv‡bv R¨v Gi j¤^-wØLÊK †K›`ªMvgx| Abywm×vš— 2| †h‡Kv‡bv mij‡iLv GKwU e„Ë‡K `yB‡qi AwaK we›`y‡Z †Q` Ki‡Z cv‡i bv|
KvR : 1| Dccv`¨ 1 Gi wecixZ Dccv`¨wU wbgœiƒc: e„‡Ëi †K›`ª †_‡K e¨vm wfbœ Ab¨ †Kv‡bv R¨v Gi Ici Aw¼Z j¤^ H R¨v‡K mgwØLwÊZ K‡i Ñ cÖgvY Ki|

134

MwYZ

Dccv`¨ 2| e„‡Ëi mKj mgvb R¨v †K›`ª †_‡K mg`~ieZ©x| g‡b Kwi, O e„‡Ëi †K›`ª Ges AB I CD e„‡Ëi `yBwU mgvb R¨v| cÖgvY Ki‡Z n‡e †h, O †_‡K AB Ges CD R¨vØq mg`~ieZ©x| A¼b : O †_‡K AB Ges CD R¨v Gi Dci h_vµ‡g
OE Ges OF j¤^ AuvwK| O, A Ges O, C †hvM Kwi|

cÖgvY : avc (1) OE
I OF myZivs,
AE
AB CD.
AE BE Ges CF DF .

h_v_©Zv [ †K›`ª †_‡K e¨vm wfbœ †h‡Kv‡bv R¨v Gi Dci Aw¼Z j¤^ R¨v‡K mgwØLwÊZ K‡i ]

1 AB Ges CF 2

1 CD. 2

(2) wKš‘ AB CD
AE CF .
OCF mg‡KvYx

[ Kíbv ]

(3) GLb

OAE Ges

wÎfzRØ‡qi g‡a¨ AwZfzR OA
AE CF . OCF OAE OE OF .

AwZfzR OC Ges

[ Df‡q GKB e„‡Ëi e¨vmva ©] [ avc 2 ] [ mg‡KvYx wÎfz‡Ri AwZfzR-evû mg©mgZv Dccv`¨]

(4) wKš‘ OE Ges OF †K›`ª O †_‡K h_vµ‡g AB R¨v Ges CD R¨v Gi `~iZ¡| myZivs, AB Ges CD R¨vØq e„‡Ëi †K›`ª †_‡K mg`~ieZ©x|

Dccv`¨ 3| e„‡Ëi †K›`ª †_‡K mg`~ieZ©x mKj R¨v ci¯úi mgvb| g‡b Kwi, O e„‡Ëi †K›`ª Ges AB I CD `yBwU R¨v| O †_‡K
AB I CD Gi Dci h_vµ‡g OE I
OF

j¤^| Zvn‡j

OE I OF †K›`ª †_‡K h_vµ‡g AB I CD R¨v‡qi `~iZ¡ wb‡`©k

K‡i| OE OF n‡j cÖgvY Ki‡Z n‡e †h, AB CD. A¼b :
O, A Ges O, C †hvM Kwi|

MwYZ

135

cÖgvY : avc (1) †h‡nZz OE myZivs, OEA g‡a¨ AwZfzR
AB Ges OF CD.

h_v_©Zv [ mg‡KvY ]

OFC

GK mg‡KvY|

(2) GLb, OAE Ges OCF mg‡KvYx wÎfzRØ‡qi
OA AwZfzR OC Ges OE OF [Kíbv] OAE OCF AE CF . 1 1 (3) AE AB Ges CF CD 2 2 1 1 (4) myZivs AB CD 2 2 A_©vr., AB CD .

[Df‡q GKB e„‡Ëi e¨vmva©] [ mg‡KvYx wÎfz‡Ri AwZfzR-evû me©mgZv Dccv`¨ [ †K›`ª †_‡K e¨vm wfbœ †h‡Kv‡bv R¨v Gi Dci Aw¼Z j¤^ R¨v‡K mgwØLwÊZ K‡i ]

Abywm×vš— 1| e„‡Ëi e¨vmB e„nËg R¨v|

Abykxjbx 8 1
1| 2| cÖvgY Ki †h, †Kv‡bv e„‡Ëi `yBwU R¨v ci¯úi‡K mgwØLwÊZ Ki‡j Zv‡`i †Q`we›`y e„ËwUi †K›`ª n‡e| cÖgvY Ki †h, `yBwU mgvš—ivj R¨v Gi ga¨we›`yi ms‡hvRK mij‡iLv †K›`ªMvgx Ges R¨vØ‡qi Ici j¤^| 3| †Kv‡bv e„‡Ëi AB I AC R¨v `yBwU A we›`yMvgx e¨vmv‡a©i mv‡_ mgvb †KvY Drcbœ K‡i| cÖgvY Ki †h, AB 4|
AC.

wP‡Î O e„‡Ëi †K›`ª Ges R¨v AB cÖgvY Ki †h,
BAO CAO.

R¨v AC.

5| 6|

†Kv‡bv e„Ë GKwU mg‡KvYx wÎfz‡Ri kxl©we›`y¸‡jv w`‡q hvq| †`LvI †h, e„ËwUi †K›`ª AwZfz‡Ri ga¨we›`y| `yBwU mg‡Kw›`ªK e„‡Ëi GKwUi AB R¨v Aci e„Ë‡K C I D we›`y‡Z †Q` K‡i| cÖgvY Ki †h, AC BD. e„‡Ëi `yBwU mgvb R¨v ci¯úi‡K †Q` Ki‡j †`LvI †h, Zv‡`i GKwUi AskØq AciwUi AskØ‡qi mgvb| cÖgvY Ki †h, e„‡Ëi mgvb R¨v Gi ga¨we›`y¸‡jv mge„Ë| †`LvI †h, e¨v‡mi `yB cÖvš— †_‡K Zvi wecixZ w`‡K `yBwU mgvb R¨v A¼b Ki‡j Zviv mgvš—ivj nq|

7| 8| 9|

10| †`LvI †h, e¨v‡mi `yB cÖvš— †_‡K Zvi wecixZ w`‡K `yBwU mgvš—ivj R¨v AuvK‡j Zviv mgvb nq| 11| †`LvI †h, e„‡Ëi `yBwU R¨v Gi g‡a¨ e„nËi R¨v-wU ¶z`ªZi R¨v A‡c¶v †K‡›`ªi wbKUZi|

136

MwYZ

8 2 e„ËPvc e„‡Ëi †h‡Kv‡bv `yBwU we›`yi g‡a¨i cwiwai Ask‡K Pvc e‡j| wP‡Î
A I B `yBwU we›`yi gv‡S e„‡Ëi Ask¸‡jv j¶ Kwi| †`Lv hvq,

`yBwU As‡ki GKwU Ask †QvU, Ab¨wU Zzjbvg~jKfv‡e eo| †QvU AskwU‡K DcPvc I eowU‡K AwaPvc ejv nq| A I B GB Pv‡ci cÖvš—we›`y Ges Pv‡ci Ab¨ mKj we›`y Zvi Aš—t¯’ we›`y| Pv‡ci Aš—t¯’ we›`ywU GKwU we›`y C wbw`©ó K‡i PvcwU‡K ACB Pvc e‡j AwfwnZ Kiv nq Ges ACB cÖZxK Øviv cÖKvk Kiv nq| Avevi KL‡bv DcPvcwU AB cÖZxK Øviv cÖKvk Kiv nq| e„‡Ëi `yBwU we›`y A I B e„ËwU‡K `yBwU Pv‡c wef³ K‡i| Dfq Pv‡ci cÖvš—we›`y A I B Ges cÖvš—we›`y Qvov Pvc `yBwUi Ab¨ †Kv‡bv mvaviY we›`y †bB|

†KvY KZ©„K LwÊZ Pvc
GKwU †KvY †Kv‡bv e„‡Ë GKwU Pvc LwÊZ ev wQbœ K‡i ejv nq hw` (1) PvcwUi cÖ‡Z¨K cÖvš—we›`y †KvYwUi evû‡Z Aew¯’Z nq, (2) †KvYwUi cÖ‡Z¨K evû‡Z PvcwUi Aš—Z GKwU cÖvš—we›`y, Aew¯’Z nq Ges (3) PvcwUi Aš—t¯’ cÖ‡Z¨KwU we›`y †KvYwUi Af¨š—‡i _v‡K| wP‡Î cÖ`wk©Z †KvYwU O †Kw›`ªK e„‡Ë APB Pvc LwÊZ K‡i|

e„Ë¯’ †KvY
GKwU †Kv‡Yi kxl©we›`y †Kv‡bv e„‡Ëi GKwU we›`y n‡j Ges †KvYwUi cÖ‡Z¨K evû‡Z kxl©we›`y QvovI e„‡Ëi GKwU we›`y _vK‡j †KvYwU‡K GKwU e„Ë¯’ †KvY ev e„‡Ë Aš—wj©wLZ †KvY ejv nq| wP‡Î †KvY¸‡jv e„Ë¯’ †KvY| cÖ‡Z¨K e„Ë¯’ †KvY e„‡Ë GKwU Pvc LwÊZ K‡i| GB Pvc DcPvc, Aa©e„Ë A_ev AwaPvc n‡Z cv‡i| GKwU e„Ë¯’ †KvY e„‡Ë †h Pvc LwÊZ K‡i, †KvYwU †mB Pv‡ci Ici `Êvqgvb Ges LwÊZ Pv‡ci AbyeÜx Pv‡c Aš—wj©wLZ ejv nq| cv‡ki wP‡Î e„Ë¯’ †KvYwU APB Pv‡ci Ici `Êvqgvb Ges ACB Pv‡c Aš—wj©wLZ| j¶Yxq †h, APB I ACB G‡K Ac‡ii AbyeÜx Pvc|

MwYZ

137

gš—e¨ : e„‡Ëi †Kv‡bv Pv‡c Aš—wj©wLZ GKwU †KvY n‡”Q †mB †KvY hvi kxl©we›`y H Pv‡ci GKwU Aš—t¯’ we›`y Ges hvi GK GKwU evû H Pv‡ci GK GKwU cÖvš—we›`y w`‡q hvq| e„‡Ëi †Kv‡bv Pv‡c `Êvqgvb GKwU e„Ë¯’ †KvY n‡”Q H Pv‡ci AbyeÜx Pv‡c Aš—wj©wLZ GKwU †KvY|

†K›`ª¯’ †KvY
GKwU †Kv‡Yi kxl©we›`y †Kv‡bv e„‡Ëi †K‡›`ª Aew¯’Z n‡j, †KvYwU‡K H e„‡Ëi GKwU †K›`ª¯’ †KvY ejv nq Ges †KvYwU e„‡Ë †h Pvc LwÊZ K‡i †mB Pv‡ci Ici Zv `Êvqgvb ejv nq| cv‡ki wP‡Îi
AOB

†KvYwU GKwU †K›`ª¯’ †KvY Ges Zv APB Pv‡ci Ici `Êvqgvb| cÖ‡Z¨K †K›`ª¯’ †KvY e„‡Ë GKwU DcPvc LwÊZ K‡i| wP‡Î APB GKwU DcPvc| e„‡Ëi †Kv‡bv DcPv‡ci Ici `Êvqgvb †K›`ª¯’ †KvY ej‡Z Gi~c †KvY‡KB †evSvq hvi kxl©we›`y e„‡Ëi †K‡›`ª Aew¯’Z Ges hvi evûØq H Pv‡ci cÖvš—we›`y `yBwU w`‡q hvq|

Aa©e„‡Ëi Ici `Êvqgvb †K›`ª¯’ †KvY we‡ePbvi Rb¨ Ic‡i DwjwLZ eY©bv A_©en bq| Aa©e„‡Ëi †¶‡Î †K›`ª¯’ †KvY Ges e„Ë¯’ †KvY
BAC mg‡KvY|
BOC mij‡KvY

Dccv`¨ 4 e„‡Ëi GKB Pv‡ci Ici `Êvqgvb †K›`ª¯’ †KvY e„Ë¯’ †Kv‡Yi wØ¸Y| g‡b Kwi, O †K›`ªwewkó ABC GKwU e„Ë Ges Zvi GKB DcPvc
BC Gi Ici `Êvqgvb e„Ë¯’ BAC Ges †K›`ª¯’
2 BAC

BOC |

cÖgvY Ki‡Z n‡e †h,

BOC

A¼b: g‡b Kwi, AC †iLvsk †K›`ªMvgx bq| G †¶‡Î A we›`y w`‡q †K›`ªMvgx †iLvsk AD AuvwK| dg©v-18, MwYZ-9g-10g

138

MwYZ

cÖgvY avc (1) (2)
AOB Gi ewnt¯’ †KvY
AOB G OA OB

h_v_©Zv
BOD BAO ABO

[ewnt¯’ †KvY Aš—t¯’ wecixZ †KvYØ‡qi mgwói mgvb] [ GKB e„‡Ëi e¨vmva©]

AZGe,

BAO

ABO
BOD 2 BAO .

(3) avc (1) I (2) †_‡K (4) GKBfv‡e
AOC †_‡K

[ mgw`¦evû wÎfz‡Ri f~wg msjMœ †KvY `yBwU mgvb ]

COD

2 CAO

(5) avc (3) I (4) †_‡K
BOD COD BOC 2 BAO 2 CAO 2 BAC . [cÖgvwYZ]

A_v©r

[†hvM K‡i]

Ab¨fv‡e ejv hvq, e„‡Ëi GKB Pv‡ci Ici `Êvqgvb e„Ë¯’ †KvY †K›`ª¯’ †Kv‡Yi A‡a©K| KvR : O †K›`ª wewkó ABC e„‡Ëi AC †K›`ªMvgx n‡j Dccv`¨ 8 cÖgvY Ki|

Dccv`¨ 5 e„‡Ëi GKB Pv‡ci Dci `Êvqgvb e„Ë¯’ †KvY¸‡jv ci¯úi mgvb| g‡b Kwi, O e„‡Ëi †K›`ª Ges e„‡Ëi BCD Pv‡ci Ici `Êvqgvb
BAD I BED `yBwU e„Ë¯’ †KvY|

cÖgvY Ki‡Z n‡e †h, cÖgvY : avc

BAD

BED

A¼b : O, B Ges O, D †hvM Kwi| h_v_©Zv
BOD |
BOD 2 BAD Ges 2 BED BOD 2 BED

(1) GLv‡b BCD Pv‡ci Ici `Êvqgvb †K›`ª¯’ †KvY myZivs,
2 BAD

[GKB Pv‡ci Ici `Êvqgvb †K›`ª¯’ †KvY e„Ë¯’ †Kv‡Yi wØ¸Y ]

ev

BAD

BED

Dccv`¨ 6 Aa©e„Ë¯’ †KvY GK mg‡KvY g‡b Kwi, O †K›`ªwewkó e„‡Ë AB GKwU e¨vm Ges GKwU Aa©e„Ë¯’ †KvY| cÖgvY Ki‡Z n‡e †h,
ACB
ACB

GK mg‡KvY|

MwYZ

139

A¼b : AB Gi †h cv‡k C we›`y Aew¯’Z, Zvi wecixZ cv‡k e„‡Ëi Dci GKwU we›`y D wbB| cÖgvY : avc (1) ADB Pv‡ci Ici `Êvqgvb e„Ë¯’
ACB 1 (‡K›`ª¯’ mij †KvY 2
AOB )

h_v_©Zv

[GKB Pv‡ci Ici `Êvqgvb e„Ë¯’ †KvY †K›`ª¯’ †Kv‡Yi A‡a©K ]

(2) wKš‘ mij‡KvY
ACB

AOB `yB mg‡KvY|

1 (`yB mg‡KvY) = GK mg‡KvY| 2

Abywm×vš— 1| mg‡KvYx wÎfz‡Ri AwZfzR‡K e¨vm a‡i e„Ë A¼b Ki‡j Zv mg‡KŠwYK kxl©we›`y w`‡q hv‡e| Abywm×vš— 2| †Kv‡bv e„‡Ëi AwaPv‡c Aš—wj©wLZ †KvY m~²‡KvY| KvR : 1| cÖgvY Ki †h, †Kv‡bv e„‡Ëi DcPv‡c Aš—wj©wLZ †KvY ¯’~j‡KvY|

Abykxjbx 8 2
1|
O †K›`ªwewkó †Kv‡bv e„‡Ë ABCD GKwU Aš—wj©wLZ PZzf©yR| AB, CD KY©Øq .. we›`y‡Z †Q` Ki‡j

cÖgvY Ki †h, 2|

AOB

COD

2

AEB.

ABCD e„‡Ë AB I CD R¨v `yBwU ci¯úi E we›`y‡Z †Q` K‡i‡Q| †`LvI †h,

AED I BED

m`„k‡KvYx| 3|
O †K›`ªwewkó e„‡Ë ADB BDC

GK mg‡KvY| cÖgvY Ki †h, A I B Ges C GK

mij‡iLvq Aew¯’Z| 4|
AB I CD `yBwU R¨v e„‡Ëi Af¨š—‡i E we›`y‡Z †Q` K‡i‡Q| cÖgvY Ki †h, AB I CD PvcØq †K‡›`ª

†h `yBwU †KvY Drcbœ K‡i, Zv‡`i mgwó 5| 6|

AEC Gi wØ¸Y|

†`LvI †h, e„Ë¯’ UªvwcwRqv‡gi wZh©K evûØq ci¯úi mgvb|
AB I CD †Kv‡bv e„‡Ëi `yBwU R¨v Ges P I Q h_vµ‡g Zv‡`i Øviv wQbœ DcPvc `yBwUi ga¨we›`y|

PQ R¨v AB I CD R¨v‡K h_vµ‡g D I E we›`y‡Z †Q` K‡i| †`LvI †h, AD

AE .

8 3 e„Ë¯’ PZzf©yR e„Ëxq PZzf©yR ev e„‡Ë Aš—wj©wLZ PZzf©yR n‡jv Ggb PZzf©yR hvi PviwU kxl©we›`y e„‡Ëi Dci Aew¯’Z| G mKj PZzf©y‡Ri GKwU we‡kl ag© i‡q‡Q| welqwU Abyave‡bi Rb¨ wb‡Pi KvRwU Kwi|

140
KvR :

MwYZ

wewfbœ AvKv‡ii K‡qKwU e„Ëxq PZzf©yR ABCD AuvK| K‡qKwU wewfbœ e¨vmv‡a©i e„Ë A¼b K‡i cÖwZwUi Dci PviwU K‡i we›`y wb‡q PZzf©yR¸‡jv mn‡RB AuvKv hvq| PZzf©y‡Ri †KvY¸‡jv †g‡c wb‡Pi mviwYwU c~iY Ki| µwgK bs 1 2 3 4 5 mviwY †_‡K Kx †evSv hvq ?

A

B

C

D

A+ C

B+ D

e„Ë msµvš— Dccv`¨ Dccv`¨ 7 e„‡Ë Aš—wjwLZ PZzfy©‡Ri †h‡Kv‡bv `yBwU wecixZ †Kv‡Yi mgwó `yB mg‡KvY| g‡b Kwi, O †K›`ªwewkó GKwU e„Ë ABCD PZzf©yRwU Aš—wj©wLZ n‡q‡Q|

cÖgvY Ki‡Z n‡e †h, Ges
BAD

ABC

ADC = `yB mg‡KvY|

BCD = `yB mg‡KvY|

A¼b : O, A Ges O, C †hvM Kwi| cÖgvY : avc (1) GKB Pvc ADC Gi Dci `Êvqgvb †K›`ª¯’
AOC 2 (e„Ë¯’ ABC )

h_v_©Zv GKB Pv‡ci Dci `Êvqgvb †K›`ª¯’ †KvY e„Ë¯’ †Kv‡Yi wØ¸Y| GKB Pv‡ci Dci `Êvqgvb †K›`ª¯’ †KvY e„Ë¯’ †Kv‡Yi wØ¸Y|
ADC ) ADC

A_©vr,

AOC

2 ABC
2 (e„Ë¯’ AOC 2

(2) Avevi, GKB Pvc ABC Gi Dci `Êvqgvb †K›`ª¯’ cÖe„× †KvY
AOC

A_©vr cÖe„× †KvY
AOC

cÖe„× †KvY

AOC

2( ABC

ADC )

wKš‘

AOC + cÖe„× †KvY

AOC = Pvi mg‡KvY

MwYZ

141

2( ABC
ABC

ADC = Pvi mg‡KvY
ADC = `yB mg‡KvY|

GKBfv‡e, cÖgvY Kiv hvq †h,

BAD

BCD = `yB mg‡KvY|

Abywm×vš— 1| e„‡Ë Aš—wj©wLZ PZzf©y‡Ri GKwU evû ewa©Z Ki‡j †h ewnt¯’ †KvY Drcbœ nq Zv wecixZ Aš—t¯’ †Kv‡Yi mgvb| Abywm×vš— 2| e„‡Ë Aš—wj©wLZ mvgvš—wiK GKwU AvqZ‡¶Î|

Dccv`¨ 8 †Kv‡bv PZzf©y‡Ri `yBwU wecixZ †KvY m¤ú~iK n‡j Zvi kxl©we›`y PviwU mge„Ë nq| g‡b Kwi, ABCD PZzf©y‡R
ABC ADC = `yB mg‡KvY|

cÖgvY Ki‡Z n‡e †h, A, B, C , D we›`y PviwU mge„Ë| A¼b : †h‡nZz A, B, C we›`y wZbwU mg‡iL bq, myZivs we›`y wZbwU w`‡q hvq Giƒc GKwU I †Kej GKwU e„Ë Av‡Q| g‡b Kwi, e„ËwU
AD †iLvsk‡K E we›`y‡Z †Q` K‡i| A, E †hvM Kwi|

cÖgvY : avc A¼b Abymv‡i ABCE e„Ë¯’ PZzf©yR| myZivs
ABC AEC = `yB mg‡KvY

h_v_©Zv e„‡Ë Aš—©wjwLZ PZzfy©‡Ri `yBwU wecixZ †Kv‡Yi mgwó `yB mg‡KvY| ewnt¯’ †KvY wecixZ Aš—t¯’ †h‡Kv‡bv †Kv‡Yi †P‡q eo|

wKš‘ ABC
AEC

ADC = `yB mg‡KvY [‡`Iqv Av‡Q]
ADC

wKš‘ Zv Am¤¢e| KviY

CED Gi ewnt¯’

AEC

wecixZ Aš—t¯’ ADC myZivs E Ges D we›`yØq wfbœ n‡Z cv‡i bv|
E we›`y Aek¨B D we›`yi mv‡_ wg‡j hv‡e|

AZGe, A, B, C , D we›`y PviwU mge„Ë|

142

MwYZ

Abykxjbx 8 3
1|
ABC G
B I

C Gi mgwØLÛKØq P we›`y‡Z Ges ewnwØ©LÊKØq Q we›`y‡Z wgwjZ n‡j,

cÖgvY Ki †h, B, P, C , Q we›`y PviwU mge„Ë| 2| cÖgvY Ki †h, e„Ë¯’ PZzf©y‡Ri †h‡Kv‡bv †Kv‡Yi mgwØLÊK I Zvi wecixZ †Kv‡Yi ewnwØLÊK e„‡Ëi Ic‡i †Q` K‡i| 3|
ABCD GKwU e„Ë| CAB I CBA Gi mgwØLÊK `yBwU P we›`y‡Z Ges

DBA I

DAB

†KvYØ‡qi mgwØLÊK `yBwU Q we›`y‡Z wgwjZ n‡j, cÖgvY Ki †h, A, Q, P, B we›`y PviwU mge„Ë| 4|
O †K›`ªwewkó e„‡Ëi AB I CD R¨v `yBwU e„‡Ëi Af¨š—‡i Aew¯’Z †Kv‡bv we›`y‡Z mg‡Kv‡Y wgwjZ

n‡q‡Q| cgªvY Ki †h, 5|

AOD

BOC = `yB mg‡KvY|

mgvb mgvb f~wgi Ici Aew¯’Z †h †Kv‡bv `yBwU wÎfy‡Ri wkit‡KvYØq m¤ú~iK n‡j, cÖgvY Ki †h, Zv‡`i cwie„ËØq mgvb n‡e|
ABCD PZzf©y‡Ri wecixZ †KvYØq ci¯úi m¤ú~iK| AC †iLv hw`

6|

BAD Gi mgwØLÊK nq,

Z‡e cÖgvY Ki †h, BC CD |

8 4 e„‡Ëi †Q`K I ¯úk©K mgZ‡j GKwU e„Ë I GKwU mij‡iLvi cvi¯úwiK Ae¯’vb we‡ePbv Kwi| G‡¶‡Î wb‡Pi wP‡Îi cÖ`Ë wZbwU m¤¢vebv i‡q‡Q: (K) e„Ë I mij‡iLvi †Kv‡bv mvaviY we›`y †bB, (L) mij‡iLvwU e„Ë‡K `yBwU we›`y‡Z †Q` K‡i‡Q, (M) mij‡iLvwU e„Ë‡K GKwU we›`y‡Z ¯úk© K‡i‡Q|

(K) (L) (M) mgZ‡j GKwU e„Ë I GKwU mij‡iLvi me©vwaK `yBwU †Q`we›`y _vK‡Z cv‡i| mgZj¯’ GKwU e„Ë I GKwU mij‡iLvi hw` `yBwU †Q`we›`y _v‡K Z‡e †iLvwU‡K e„ËwUi GKwU †Q`K ejv nq Ges hw` GKwU I †Kej GKwU mvaviY we›`y _v‡K Z‡e †iLvwU‡K e„ËwUi GKwU ¯úk©K ejv nq| †k‡lv³ †¶‡Î, mvaviY we›`ywU‡K H ¯úk©‡Ki ¯úk©we›`y ejv nq| Dc‡ii wP‡Î GKwU e„Ë I GKwU mij‡iLvi cvi¯úwiK Ae¯’vb ‡`Lv‡bv n‡q‡Q| wPÎ-K G e„Ë I PQ mij‡iLvi ‡Kv‡bv mvaviY we›`y †bB, wPÎ-L G PQ mij‡iLvwU e„Ë‡K A I B `yBwU we›`y‡Z †Q` K‡i‡Q Ges wPÎ-M G PQ mij‡iLvwU e„Ë‡K A we›`y‡Z ¯úk© K‡i‡Q| PQ e„ËwUi ¯úk©K I
A GB ¯úk©‡Ki ¯úk©we›`y|

gš—e¨ : e„‡Ëi cÖ‡Z¨K †Q`‡Ki †Q`we›`yØ‡qi Aš—e©Z©x mKj we›`y e„ËwUi Af¨š—‡i _v‡K|

MwYZ

143

mvaviY ¯úk©K
GKwU mij‡iLv hw` `yBwU e„‡Ëi ¯úk©K nq, Z‡e Zv‡K e„Ë `yBwUi GKwU mvaviY ¯úk©K ejv nq|cv‡ki wPÎ¸‡jv‡Z AB Dfq e„‡Ëi mvaviY ¯úk©K| wPÎ-K I wPÎ-L G ¯úk©we›`y GKB| wPÎ-M I wPÎ-N G ¯úk©we›`y wfbœ wfbœ| `yBwU e„‡Ëi †Kv‡bv mvaviY ¯úk©‡Ki ¯úk©we›`y `yBwU wfbœ n‡j ¯úk©KwU‡K (K) mij mvaviY ¯úk©K ejv nq hw` e„Ë `yBwUi †K›`ªØq ¯úk©‡Ki GKB cv‡k¦© _v‡K Ges (L) wZh©K mvaviY ¯úk©K ejv nq hw` e„Ë `yBwUi †K›`ªØq ¯úk©‡Ki wecixZ cv‡k¦© _v‡K| wPÎ-M G ¯úk©KwU mij mvaviY ¯úk©K Ges wPÎ-N G ¯úk©KwU wZh©K mvaviY ¯úk©K| `yBwU e„‡Ëi mvaviY ¯úk©K hw` e„Ë `yBwU‡K GKB we›`y‡Z ¯úk© K‡i Z‡e H we›`y‡Z e„Ë `yBwU ci¯úi‡K ¯úk© K‡i ejv nq| Giƒc †¶‡Î, e„Ë `yBwUi Aš—t¯úk© n‡q‡Q ejv nq hw` †K›`ªØq ¯úk©‡Ki GKB cv‡k¦© _v‡KGes ewnt¯úk© n‡q‡Q ejv nq hw` †K›`ªØq ¯úk©‡Ki wecixZ cv‡k¦© _v‡K| wPÎ-K G e„Ë `yBwUi Aš—t¯úk© Ges wPÎ-L G ewnt¯úk© n‡q‡Q|

Dccv`¨ 9 e„‡Ëi †h‡Kv‡bv we›`y‡Z Aw¼Z ¯úk©K ¯úk©Kwe›`yMvgx e¨vmv‡a©i Ici j¤^| g‡b Kwi, O †K›`ªwewkó GKwU e„‡Ëi Ici¯’ P we›`y‡Z PT GKwU ¯úk©K Ges OP ¯úk© we›`yMvgx e¨vmva©| cÖgvY Ki‡Z n‡e †h,
PT OP.

A¼b : PT ¯úk©‡Ki Ici †h‡Kv‡bv GKwU we›`y Q wbB Ges O, Q †hvM Kwi| cÖgvY : †h‡nZz e„‡Ëi P we›`y‡Z PT GKwU ¯úk©K, myZivs H P we›`y e¨ZxZ PT Gi Ici¯’ Ab¨ mKj we›`y e„‡Ëi evB‡i _vK‡e| myZivs Q we›`ywU e„‡Ëi evB‡i Aew¯’Z|
OQ e„‡Ëi e¨vmva© OP Gi †P‡q eo, A_©vr, OQ OP Ges Zv ¯úk©

we›`y P e¨ZxZ PT Gi Ici¯’ Q we›`yi mKj Ae¯’v‡bi Rb¨ mZ¨| †K›`ª O †_‡K PT ¯úk©‡Ki Ici OP nj ¶y`ªZg `~iZ¡| myZivs PT OP.

144

MwYZ

Abywm×vš— 1| e„‡Ëi †Kv‡bv we›`y‡Z GKwUgvÎ ¯úk©K A¼b Kiv hvq| Abywm×vš— 2| ¯úk© we›`y‡Z ¯úk©‡Ki Ici Aw¼Z j¤^ †K›`ªMvgx| Abywm×vš— 3| e„‡Ëi †Kv‡bv we›`y w`‡q H we›`yMvgx e¨vmv‡a©i Ici Aw¼Z j¤^ D³ we›`y‡Z e„ËwUi ¯úk©K nq|

Dccv`¨ 10 e„‡Ëi ewnt¯’ †Kv‡bv we›`y †_‡K e„‡Ë `yBwU ¯úk©K Uvb‡j, H we›`y †_‡K ¯úk© we›`yØ‡qi `~iZ¡ mgvb| g‡b Kwi, O †K›`ªwewkó ABC e„‡Ëi P GKwU ewn:¯’ we›`y Ges
PA I PB iwk¥Øq e„‡Ëi A I B we›`y‡Z `yBwU ¯úk©K| cÖgvY

Ki‡Z n‡e †h, PA PB A¼b : O, A; O, B Ges O, P †hvM Kwi| cÖgvY : avc (1) †h‡nZz PA ¯úk©K Ges OA ¯úk©we›`yMvgx e¨vmva©, †m‡nZz
PA OA.
C

h_v_©Zv [¯úk©K ¯úk©Kwe›`yMvgx e¨vmv‡a©i Ici j¤^]

PAO = GK mg‡KvY|

Abyiƒ‡c

PBO = GK mg‡KvY|

PAO Ges PBO DfqB mg‡KvYx wÎfzR|

(2) GLb, PAO I Ges OA OB
PAO
PA PB

PBO mg‡KvYx wÎfzRØ‡q

AwZfzR PO = AwZfzR PO [GKB e„‡Ëi e¨vmva©]
PBO.

[mg‡KvYx wÎfz‡Ri AwZfzR- evû me©mgZv]

gš—e¨ : 1. `yBwU e„Ë ci¯úi‡K ewnt¯úk© Ki‡j, ¯úk©we›`y Qvov cÖ‡Z¨K e„‡Ëi Ab¨ mKj we›`y Aci e„‡Ëi evB‡i _vK‡e| 2. `yBwU e„Ë ci¯úi‡K Aš—t¯úk© Ki‡j, ¯úk©we›`y Qvov †QvU e„‡Ëi Ab¨ mKj we›`y eo e„ËwUi Af¨š—‡i _vK‡e|

Dccv`¨ 11 `yBwU e„Ë ci¯úi‡K ewnt¯úk© Ki‡j, Zv‡`i †K›`ªØq I ¯úk© we›`y mg‡iL| g‡b Kwi, A Ges B †K›`ªwewkó `yBwU e„Ë ci¯úi O we›`y‡Z ewnt¯úk© K‡i| cÖgvY Ki‡Z n‡e †h, A, O Ges B we›`y wZbwU mg‡iL| A¼b : †h‡nZz e„ËØq ci¯úi O we›`y‡Z ¯úk© K‡i‡Q, myZivs O we›`y‡Z Zv‡`i GKwU mvaviY ¯úk©K _vK‡e| GLb O we›`y‡Z mvaviY ¯úk©K
POQ A¼b Kwi Ges O, A I O, B †hvM Kwi|

MwYZ

145

cÖgvY A †K›`ªwewkó e„‡Ë OA ¯úk© we›`yMvgx e¨vmva© Ges POQ ¯úk©K| myZivs
POA POA = GK mg‡KvY| Z`ªyc POB = GK mg‡KvY|

POB = GK mg‡KvY + GK mg‡KvY = `yB mg‡KvY|

ev, AOB = `yB mg‡KvY A_©vr,
AOB GKwU mij‡KvY|

A, O Ges B we›`yÎq mg‡iL|

Abywm×vš— 1| Abywm×vš— 2|

`yBwU e„Ë ci¯úi‡K ewnt¯úk© Ki‡j, †K›`ªØ‡qi `~iZ¡ e„ËØ‡qi e¨vmv‡a©i mgwói mgvb| `yBwU e„Ë ci¯úi‡K Aš—t¯úk© Ki‡j, †K›`ªØ‡qi `~iZ¡ e„ËØ‡qi e¨vmv‡a©i Aš—‡ii mgvb|

KvR : 1| cÖgvY Ki †h, `yBwU e„Ë ci¯úi Aš—t¯úk© Ki‡j, Zv‡`i †K›`ªØq I ¯úk©we›`y mg‡iL n‡e|

Abykxjbx 8 4
1| 2| 3| 4| 5|
O †K›`ªwewkó GKwU e„‡Ëi ewnt¯’ †Kv‡bv we›`y P †_‡K e„‡Ë `yBwU ¯úk©K Uvbv nj| cÖgvY Ki †h,

OP mij‡iLv ¯úk©-R¨v Gi j¤^wØLÊK|

†`Iqv Av‡Q, O e„‡Ëi †K›`ª Ges PA I PB ¯úk©KØq e„Ë‡K h_vµ‡g A I B we›`y‡Z ¯úk© K‡i‡Q| cÖgvY Ki †h, PO, APB †K mgwØLwÊZ K‡i| cÖgvY Ki †h, `yBwU e„Ë GK‡Kw›`ªK n‡j Ges e„nËi e„ËwUi †Kv‡bv R¨v ¶z`ªZi e„ËwU‡K ¯úk© Ki‡j D³ R¨v ¯úk©we›`y‡Z mgwØLwÊZ nq|
AB †Kv‡bv e„‡Ëi e¨vm Ges BC e¨vmv‡a©i mgvb GKwU R¨v| hw` A I C we›`y‡Z Aw¼Z ¯úk©KØq

ci¯úi D we›`y‡Z wgwjZ nq, Z‡e cÖgvY Ki †h, ACD GKwU mgevû wÎfzR| cÖgvY Ki †h, †Kv‡bv e„‡Ëi cwiwjwLZ PZzf©y‡Ri †h‡Kv‡bv `yBwU wecixZ evû †K‡›`ª †h `yBwU †KvY aviY K‡i, Zviv ci¯úi m¤ú~iK|

8 5 e„Ë m¤úK©xq m¤úv`¨ m¤úv`¨ 1
GKwU e„Ë ev e„ËPvc †`Iqv Av‡Q, †K›`ª wbY©q Ki‡Z n‡e|
GKwU e„Ë wPÎ-1 ev e„ËPvc wPÎ-2 †`Iqv Av‡Q, e„ËwUi ev e„ËPvcwUi †K›`ª wbY©q Ki‡Z n‡e| A¼b : cÖ`Ë e„‡Ë ev e„ËPv‡c wZbwU we›`y A, B I C wbB|
A, B Ges B, C

†hvM Kwi|

AB I BC

R¨v `yBwUi

j¤^mgwØLÊK h_vµ‡g EF I GH †iLvsk `yBwU Uvwb| g‡b Kwi, Zviv ci¯úi O we›`y‡Z †Q` K‡i| myZivs, O we›`yB e„‡Ëi ev e„ËPv‡ci †K›`ª| dg©v-19, MwYZ-9g-10g

146

MwYZ

cÖgvY : EF †iLvsk AB R¨v Gi Ges GH †iLvsk BC R¨v Gi j¤^mgwØLÊK| wKš‘ EF I GH Df‡q †K›`ªMvgx Ges O Zv‡`i mvaviY †Q` we›`y| myZivs O we›`yB e„‡Ëi ev e„ËPv‡ci †K›`ª|

e„‡Ëi ¯úk©K A¼b
Avgiv †R‡bwQ †h, e„‡Ëi wfZ‡i Aew¯’Z †Kv‡bv we›`y †_‡K e„‡Ëi ¯úk©K AuvKv hvq bv| we›`ywU hw` e„‡Ëi Ici _v‡K Zvn‡j D³ we›`y‡Z e„‡Ëi GKwUgvÎ ¯úk©K A¼b Kiv hvq| ¯úk©KwU ewY©Z we›`y‡Z Aw¼Z e¨vmv‡a©i Dci j¤^ nq| myZivs, e„Ëw¯’Z †Kv‡bv we›`y‡Z e„‡Ëi ¯úk©K A¼b Ki‡Z n‡j ewY©Z we›`y‡Z e¨vmva© A¼b K‡i e¨vmv‡a©i Dci j¤^ AuvK‡Z n‡e| Avevi we›`ywU e„‡Ëi evB‡i Aew¯’Z n‡j Zv †_‡K e„‡Ë `yBwU ¯úk©K AuvKv hv‡e|

m¤úv`¨ 2 e„‡Ëi †Kv‡bv we›`y‡Z GKwU ¯úk©K AvuK‡Z n‡e| g‡b Kwi, O †K›`ªwewkó e„‡Ë A GKwU we›`y| A we›`y‡Z e„ËwU‡Z GKwU ¯úk©K AuvK‡Z n‡e| A¼b : (1) O, A †hvM Kwi| A we›`y‡Z OA Gi Dci AP j¤^ AuvwK| Zvn‡j AP wbY©q ¯úk©K| cÖgvY : OA †iLvsk A we›`yMvgx e¨vmva© Ges AP Zvi Ici j¤^| myZivs, AP †iLvB wb‡Y©q ¯úk©K| we‡kl `ªóe¨ : e„‡Ëi †Kv‡bv we›`y‡Z GKwUgvÎ ¯úk©K AuvKv nq|

m¤úv`¨ 3 e„‡Ëi ewnt¯’ †Kv‡bv we›`y †_‡K e„ËwUi ¯úk©K AuvK‡Z n‡e| g‡b Kwi, O †K›`ªwewkó e„‡Ëi P GKwU ewnt¯’ we›`y| P we›`y †_‡K H e„‡Ë ¯úk©K AuvK‡Z n‡e| A¼b : (1) P, O †hvM Kwi| PO †iLvs‡ki ga¨we›`y M wbY©q Kwi| (2) GLb M †K †K›`ª K‡i MO Gi mgvb e¨vmva© wb‡q GKwU e„Ë AvuwK| g‡b Kwi, bZzb Aw¼Z e„ËwU cÖ`Ë e„Ë‡K A I B we›`y‡Z †Q` K‡i| (3) A, P Ges B, P †hvM Kwi| Zvn‡j, AP , BP Df‡qB wb‡Y©q ¯úk©K|

MwYZ

147

cÖgvY : A, O Ges B, O †hvM Kwi| APB e„‡Ë PO e¨vm|
PAO = GK mg‡KvY [ Aa©e„Ë¯’ †KvY mg‡KvY ]

myZivs, OA †iLvsk AP †iLvs‡ki Ici j¤^| AZGe, O †Kw›`ªK e„‡Ëi A we›`y‡Z AP †iLvsk GKwU ¯úk©K| Abyiƒcfv‡e, BP †iLvskI GKwU ¯úk©K| we‡kl `ªóe¨ : e„‡Ëi ewnt¯’ †Kv‡bv we›`y †_‡K H e„‡Ë `yBwU I †Kej `yBwU ¯úk©K AuvKv hvq|

m¤úv`¨ 4
†Kv‡bv wbw`©ó wÎfz‡Ri cwie„Ë AuvK‡Z n‡e| g‡b Kwi, ABC GKwU wÎfzR| Gi cwie„Ë AuvK‡Z n‡e| A_©vr, Ggb GKwU e„Ë AuvK‡Z n‡e, hv wÎfz‡Ri wZbwU kxl©we›`y A, B I C we›`y w`‡q hvq| A¼b : (1) AB I AC †iLvs‡ki j¤^ mgwØLÊK h_vµ‡g EM I FN †iLvsk AuvwK| g‡b Kwi, Zviv ci¯úi‡K O we›`y‡Z †Q` K‡i| (2) A , O †hvM Kwi| O †K †K›`ª K‡i OA Gi mgvb e¨vmva© wb‡q GKwU e„Ë AuvwK| Zvn‡j, e„ËwU A, B I C we›`yMvgx n‡e Ges GB e„ËwUB ABC Gi wb‡Y©q cwie„Ë| cÖgvY : B, O Ges C , O †hvM Kwi| O we›`ywU AB Gi j¤^mgwØLÊK
EM Gi Ici Aew¯’Z|
OA OB , GKBfv‡e, OA OC OA OB OC

myZivs O †K †K›`ª K‡i OA Gi mgvb e¨vmva© wb‡q Aw¼Z e„ËwU
A, B I C we›`y wZbwU w`‡q hv‡e| myZivs GB e„ËwUB
ABC Gi cwie„Ë|

KvR : Ic‡ii wP‡Î GKwU m~²‡KvYx wÎfy‡Ri cwie„Ë AuvKv n‡q‡Q| ¯’zj‡KvYx Ges mg‡KvYx wÎfy‡Ri cwie„Ë A¼b Ki| j¶Yxq †h, m~²‡KvYx wÎfz‡Ri †¶‡Î cwi‡K›`ª wÎfz‡Ri Af¨š—‡i, ¯’zj‡KvYx wÎfz‡Ri †¶‡Î cwi‡K›`ª wÎfz‡Ri ewnf©v‡M Ges mg‡KvYx wÎfz‡Ri †¶‡Î cwi‡K›`ª AwZfy‡Ri Ici Aew¯’Z|

148

MwYZ

m¤úv`¨ 5
†Kv‡bv wbw`©ó wÎfz‡Ri Aš—e„©Ë AuvK‡Z n‡e| g‡b Kwi, ABC GKwU wÎfzR| Gi Aš—e„©Ë AuvK‡Z n‡e| A_©vr,
ABC

Gi wfZ‡i Ggb GKwU e„Ë AuvK‡Z n‡e, hv

BC, CA I AB evû wZbwUi cÖ‡Z¨KwU‡K ¯úk© K‡i|

A¼b :

ABC I ACB Gi mgwØLÊK h_vµ‡g BL I CM

AuvwK| g‡b Kwi, Zviv O we›`y‡Z †Q` K‡i| O †_‡K BC Gi Ici OD j¤^ AuvwK Ges g‡b Kwi, Zv BC †K D we›`y‡Z †Q` K‡i| O †K †K›`ª K‡i OD Gi mgvb e¨vmva© wb‡q GKwU e„Ë AuvwK| Zvn‡j, GB e„ËwUB wb‡Y©q Aš—e©„Ë| cÖgvY : O †_‡K AC I AB Gi Ici h_vµ‡g OE I OF j¤^ Uvwb| g‡b Kwi, j¤^Øq evûØq‡K h_vµ‡g E I F we›`y‡Z †Q` K‡i| O we›`y ABC Gi wØLÊ‡Ki Ici Aew¯’Z|
OF OD

Abyiƒcfv‡e, O we›`y
OD OE OF

ABC Gi wØLÊ‡Ki Ici Aew¯’Z e‡j OF

OD

myZivs O †K †K›`ª K‡i OD Gi mgvb e¨vmva© wb‡q e„Ë AuvK‡j Zv D, E Ges F we›`y w`‡q hv‡e| Avevi, OD, OE I OF Gi cÖvš—we›`y‡Z h_vµ‡g BC, AC I AB j¤^| myZivs e„ËwU ABC Gi wfZ‡i †_‡K Gi evû wZbwU‡K h_vµ‡g D, E I F we›`y‡Z ¯úk© K‡i| AZGe, DEF e„ËwUB ABC Gi Aš—e©„Ë n‡e|

m¤úv`¨ 6 †Kv‡bv wbw`©ó wÎfz‡Ri ewne©„Ë AuvK‡Z n‡e| g‡b Kwi, ABC GKwU wÎfzR| Gi ewne©„Ë AuvK‡Z n‡e| A_©vr, Ggb GKwU e„Ë AuvK‡Z n‡e, hv wÎfz‡Ri GKwU evû‡K Ges Aci `yB evûi ewa©Zvsk‡K ¯úk© K‡i| A¼b : AB I AC evûØq‡K h_vµg D I F ch©š— ewa©Z Kwi|
DBC I FCB Gi mgwØLÊK BM Ges CN

AuvwK| g‡b Kwi, E Zv‡`i †Q` we›`y| E †_‡K BC Gi Ici EH j¤^ AuvwK Ges g‡b Kwi Zv BC †K H we›`y‡Z †Q` K‡i| E †K †K›`ª K‡i EH Gi mgvb e¨vmva© wb‡q GKwU e„Ë AuvwK| Zvn‡j, GB e„ËwUB wb‡Y©q ewne©„Ë|

MwYZ

149

cÖgvY : E †_‡K BD I CF †iLvs‡ki Ici h_vµ‡g EG I EL j¤^ Uvwb| g‡b Kwi, j¤^Øq, †iLvskØq‡K h_vµ‡g G I L we›`y‡Z †Q` K‡i|
E we›`ywU
DBC Gi wØLÊ‡Ki Ici Aew¯’Z

EH

EG
FCB Gi wØLÊ‡Ki Ici Aew¯’Z e‡j EH

Abyiƒcfv‡e, E we›`ywU
EH EG EL

EL

myZivs E †K †K›`ª K‡i EL Gi mgvb e¨vmva© wb‡q Aw¼Z e„Ë H , G Ges L we›`y wb‡q hv‡e| Avevi, EH , EG I EL Gi cÖvš—we›`y‡Z h_vµ‡g BC, BD I CF †iLvsk wZbwU j¤^| myZivs e„ËwU †iLvsk wZbwU‡K h_vµ‡g H , G I L we›`y wZbwU‡Z ¯úk© K‡i| AZGe, HGL e„ËwUB ABC Gi ewne„Ë n‡e| gš—e¨ : †Kv‡bv wÎfz‡Ri wZbwU ewne„©Ë AuvKv hvq|
KvR : 1| wÎfz‡Ri Aci `yBwU ewne„©Ë AuvK|

Abykxjbx 8 5
1. wb‡Pi Z_¨¸‡jv j¶ Ki : i e„‡Ë ¯úk©K ¯úk© we›`yMvgx e¨vmv‡a©i Ici j¤^ ii Aa©e„Ë¯’ †KvY GK mg‡KvY iii e„‡Ëi mKj mgvb R¨v †K›`ª †_‡K mg`~ieZ©x wb‡Pi †KvbwU mwVK ? K. i I ii L. i I iii M. ii I iii N. i, ii I iii

Ic‡ii wPÎ Abyhvqx 2 I 3 bs cÖ‡kœi DËi `vI: 2.
BOD Gi cwigvY n‡e-

K.

1 2

BAC BAC

L.

1 2

BAD BAD

2

2

150

MwYZ

3. e„ËwU ABC wÎfy‡RiK. Aš—e„©Ë L. cwie„Ë M. ewnte„Ë N. Dce„Ë 4. †Kv‡bv e„‡Ëi AwaPv‡c Aš—wj©wLZ †KvYK. m~²‡KvY L. mg‡KvY M. ¯’~j †KvY N. c~iK‡KvY 5. †Kv‡bv e„‡Ë Ggb GKwU ¯úk©K AuvK †hb Zv wbw`©ó mij‡iLvi mgvš—ivj nq| 6. †Kv‡bv e„‡Ë Ggb GKwU ¯úk©K AuvK †hb Zv wbw`©ó mij‡iLvi Dci j¤^ nq| 7. †Kv‡bv e„‡Ë Ggb `yBwU ¯úk©K AuvK †hb Zv‡`i Aš—f©y³ †KvY 60° nq| 8. 3 †m.wg., 4 †m.wg. I 4 5 †m.wg. evûwewkó GKwU wÎfz‡Ri cwie„Ë AuvK Ges GB e„‡Ëi e¨vmva© wbY©q Ki| 9.
5 †m.wg evûwewkó GKwU mgevû wÎfzR ABC Gi AC evû‡K ¯úk© Kwi‡q GKwU ewne©„Ë AuvK|

10. GKwU e‡M©i Aš—e©„Ë I cwie©„Ë AuvK| 11. cÖgvY Ki †h, mgwØevû wÎf~‡Ri mgvb evûØq‡K e¨vm a‡i `yBwU e„Ë A¼b Ki‡j, Zviv f~wgi ga¨we›`y‡K ci¯úi †Q` K‡i| 12. cÖgvY Ki †h, mg‡KvYx wÎf~‡Ri AwZfy‡Ri ga¨we›`y I wecixZ kx‡l©i ms‡hvRK †iLvsk AwZfy‡Ri A‡a©K| 13.
ABC GKwU wÎfyR| AB †K e¨vm wb‡q Aw¼Z e„Ë hw` BC evû‡K D we›`y‡K †Q` K‡i, Z‡e

cÖgvY Ki †h, AC evû‡K e¨vm wb‡q Aw¼Z e„ËI D we›`y w`‡q hv‡e| 14. 15.
AB I CD GKB e„‡Ë `yBwU mgvš—ivj R¨v| cÖgvY Ki †h, Pvc AC = Pvc BD .

O †K›`ªwewkó †Kv‡bv e„‡Ëi AB I CD R¨v `yBwU e…‡Ëi Af¨š—i¯’ E we›`y‡Z †Q` Ki‡j cÖgvY Ki

†h,

AEC

1 ( BOD 2

AOC ).

16. `yBwU mgvb e¨vmwewkó e„‡Ëi mvaviY R¨v AB | B we›`y w`‡q Aw¼Z †Kv‡bv mij‡iLv hw` e„Ë `yBwUi mv‡_ P I Q we›`y‡Z wgwjZ nq, Z‡e cÖgvY Ki †h, OAQ mgwØevû| 17. O †K›`ªwewkó ABC e„‡Ë R¨v AB = x †m.wg. OD cv‡ki wPÎ Abyhvqx wb‡Pi cÖkœ¸‡jvi DËi `vI: K. e„ËwUi †¶Îdj wbY©q Ki| L. †`LvI †h, D, AB Gi ga¨we›`y|
M. OD = ( x -2) †m. wg. n‡j x Gi gvb wbY©q Ki| 2
AB

18. GKwU wÎfy‡Ri wZbwU evûi ˆ`N¨© h_vµ‡g 4 †m. wg. 5 †m. wg. I 6 †m. wg. Ic‡ii Z_¨ Abyhvqx wb‡gœi cÖkœ¸‡jvi DËi `vI: K. wÎfyRwU A¼b Ki L. wÎfyRwUi cwie„Ë A¼b Ki| M. wÎfy‡Ri cwie„‡Ëi evwn‡i †h †Kvb GKwU wbw`©ó we›`y †_‡K e„‡Ëi `yBwU ¯úk©K A¼b K‡i †`LvI †h ¯úk©KØ‡qi `~iZ¡ mgvb nq|

beg Aa¨vq

wÎ‡KvYwgwZK AbycvZ
(Trigonometric Ratios)
Avgiv cÖwZwbqZ wÎfzR, we‡kl K‡i mg‡KvYx wÎfz‡Ri e¨envi K‡i _vwK| Avgv‡`i Pvwiw`‡Ki cwi‡e‡k bvbv D`vniY †`Lv hvq †hLv‡b Kíbvq mg‡KvYx wÎfzR MVb Kiv hvq| †mB cÖvPxb hy‡M gvbyl R¨vwgwZi mvnv‡h¨ b`xi Zx‡i `uvwo‡q b`xi cÖ¯’ wbY©q Kivi †KŠkj wk‡LwQj| Mv‡Q bv D‡VI Mv‡Qi Qvqvi m‡½ jvwVi Zzjbv K‡i wbL~uyZfv‡e Mv‡Qi D”PZv gvc‡Z wk‡LwQj| GB MvwYwZK †KŠkj †kLv‡bvi Rb¨ m„wó n‡q‡Q wÎ‡KvYwgwZ bv‡g MwY‡Zi GK we‡kl kvLv| Trigonometry kãwU wMÖK kã tri(A_© wZb) gon(A_© avi) metron(A_© cwigvc) Øviv MwVZ| wÎ‡KvYwgwZ‡Z wÎfz‡Ri evû I †Kv‡Yi g‡a¨ m¤úK© wel‡q cvV`vb Kiv nq| wgki I e¨wejbxq mf¨Zvq wÎ‡KvYwgwZ e¨env‡ii wb`k©b i‡q‡Q| wgkixqiv f~wg Rwic I cÖ‡KŠkj Kv‡R Gi eûj e¨envi KiZ e‡j aviYv Kiv nq| Gi mvnv‡h¨ †R¨vwZwe©`MY c„w_ex †_‡K `~ieZx© MÖn-b¶‡Îi `~iZ¡ wbY©q Ki‡Zb| Aaybv wÎ‡KvYwgwZi e¨envi MwY‡Zi mKj kvLvq| wÎfyR msµvš— mgm¨vi mgvavb, †bwf‡Mkb BZ¨vw` †¶‡Î wÎ‡KvYwgwZi e¨vcK e¨envi n‡q _v‡K| MwY‡Zi ¸iæZ¡c~Y© †R¨vwZwe©Ávb kvLvmn K¨vjKzjv‡m Gi eûj e¨envi i‡q‡Q| Aa¨vq †k‡l wk¶v_x©ivÑ m~²‡Kv‡Yi wÎ‡KvYwgwZK AbycvZ eY©bv Ki‡Z cvi‡e| m~²‡Kv‡Yi wÎ‡KvYwgwZK AbycvZ¸‡jvi g‡a¨ cvi¯úwiK m¤úK© wbY©q Ki‡Z cvi‡e| m~²‡Kv‡Yi wÎ‡KvYwgwZK AbycvZ¸‡jvi ayªeZv hvPvB K‡i cÖgvY I MvwYwZK mgm¨v mgvavb Ki‡Z cvi‡e| R¨vwgwZK c×wZ‡Z 30 , 45 , 60 †Kv‡Yi wÎ‡KvYwgwZK Abycv‡Zi gvb wbY©q I cÖ‡qvM Ki‡Z cvi‡e|
0 I 90 †Kv‡Yi A_©c~Y© wÎ‡KvYwgwZK AbycvZ¸‡jvi gvb wbY©q K‡i cÖ‡qvM Ki‡Z cvi‡e|

wÎ‡KvYwgwZK A‡f`vewj cÖgvY Ki‡Z cvi‡e| wÎ‡KvYwgwZK A‡f`vewji cÖ‡qvM Ki‡Z cvi‡e|

91

mg‡KvYx wÎfz‡Ri evû¸‡jvi bvgKiY

Avgiv Rvwb, mg‡KvYx wÎfz‡Ri evû¸‡jv AwZfzR, f~wg I DbœwZ bv‡g AwfwnZ nq| wÎfz‡Ri Abyf~wgK Ae¯’v‡bi Rb¨ G bvgmg~n mv_©K| Avevi mg‡KvYx wÎfz‡Ri m~²‡KvYØ‡qi GKwUi mv‡c‡¶ Ae¯’v‡bi †cÖw¶‡ZI evû¸‡jvi bvgKiY Kiv nq| h_v: K. ÔAwZfzRÕ, mg‡KvYx wÎfz‡Ri e„nËg evû hv mg‡Kv‡Yi wecixZ evû L. ÔwecixZ evûÕ, hv n‡jv cÖ`Ë †Kv‡Yi mivmwi wecixZ w`‡Ki evû M. ÕmwbœwnZ evûÕ, hv cÖ`Ë †KvY m„wóKvix GKwU †iLvsk|

152

MwYZ

PON †Kv‡Yi Rb¨ AwZfzR OP , mwbœwnZ evû ON , wecixZ evû PN

OPN †Kv‡Yi Rb¨ AwZfzR OP , mwbœwnZ evû PN , wecixZ evû ON

R¨vwgwZK wP‡Îi kxl©we›`y wPwýZ Kivi Rb¨ eo nv‡Zi eY© I evû wb‡`©k Ki‡Z †QvU nv‡Zi eY© e¨envi Kiv nq| †KvY wb‡`©‡ki Rb¨ cÖvqkB wMÖK eY© e¨eüZ nq| wMÖK eY©gvjvi QqwU eûj e¨eüZ eY© n‡jv : alpha (Avjdv) beta (weUv) gamma (Mvgv) theta (w_Uv) phi (cvB) omega (I‡gMv)

cÖvPxb wMÖ‡mi weL¨vZ me MwYZwe`‡`i nvZ a‡iB R¨vwgwZ I wÎ‡KvYwgwZ‡Z wMÖK eY©¸‡jv e¨envi n‡q Avm‡Q| D`vniY 1| †Kv‡Yi Rb¨ AwZfzR, mwbœwnZ evû I wecixZ evû wPwýZ Ki|

mgvavb : (K) AwZfzR 17 GKK (L) AwZfzR p (M) AwZfzR EF wecixZ evû 8 GKK wecixZ evû r wecixZ evû EG mwbœwnZ evû 15 GKK mwbœwnZ evû q mwbœwnZ evû FG D`vniY 2| I †Kv‡Yi Rb¨ AwZfzR, mwbœwnZ evû I wecixZ evûi ˆ`N©¨ wbY©q Ki|

(K)

†Kv‡Yi Rb¨

(L)

†Kv‡Yi Rb¨

AwZfzR 25 GKK wecixZ evû 24 GKK mwbœwnZ evû 7 GKK KvR :
I

AwZfzR 25 GKK wecixZ evû 7 GKK mwbœwnZ evû 24 GKK

†Kv‡Yi Rb¨ AwZfzR, mwbœwnZ evû I wecixZ evû wb‡`©k Ki|

MwYZ

153

9 2 m`„k mg‡KvYx wÎfz‡Ri evû¸‡jvi AbycvZmg~‡ni aªyeZv KvR : wb‡Pi PviwU m`„k mg‡KvYx wÎfz‡Ri evû¸‡jvi ˆ`N©¨ †g‡c mviwYwU c~iY Ki| wÎfz‡Ri AbycvZ¸‡jv m¤ú‡K© Kx j¶ Ki ?

evûi ˆ`N©¨
BC AB AC

AbycvZ ( †Kv‡Yi mv‡c‡¶)
BC/ AC AB/ AC BC/ AB

g‡b Kwi,

XOA GKwU m~²‡KvY| OA evû‡Z †h‡Kv‡bv GKwU we›`y P wbB| P †_‡K OX evû ch©š—

PM j¤^ Uvwb| d‡j GKwU mg‡KvYx wÎfyR POM MwVZ n‡jv| GB

POM Gi PM , OM I OP

evû¸‡jvi †h wZbwU AbycvZ cvIqv hvq Zv‡`i gvb OA evû‡Z wbe©vwPZ P we›`yi Ae¯’v‡bi Ici wbf©i K‡i bv|
XOA †Kv‡Yi OA evû‡Z †h‡Kv‡bv we›`y P I P †_‡K OX evû ch©š— h_vµ‡g PM I P M1 j¤^ 1 1

A¼b Ki‡j POM I POM1 `yBwU m`„k mg‡KvYx wÎfyR MwVZ nq| 1 GLb, POM I POM1 m`„k nIqvq, 1
PM OP = P M1 OP 1 1

ev,

PM OP

=

P M1 1 ..... (i) OP 1

dg©v-20, MwYZ-9g-10g

154

MwYZ

OM OP = OM1 OP 1

ev,

OM = OP

OM1 .... (ii) OP 1
P M1 1 .... (iii) OM1

PM OM PM = ev, = P M1 OM1 OM 1

A_©vr, AbycvZmg~‡ni cÖ‡Z¨KwU aª“eK| GB AbycvZmg~n‡K wÎ‡KvYwgwZK AbycvZ e‡j|

93

m~²‡Kv‡Yi wÎ‡KvYwgwZK AbycvZ
XOA GKwU m~²‡KvY| OA evû‡Z †h‡Kv‡bv GKwU we›`y

g‡b Kwi,

P wbB| P †_‡K OX evû ch©š— PM j¤^ Uvwb| d‡j GKwU

mg‡KvYx wÎfyR POM MwVZ n‡jv| GB

POM Gi PM , OM I
XOA Gi

OP evû¸‡jvi †h QqwU AbycvZ cvIqv hvq Zv‡`i

wÎ‡KvYwgwZK AbycvZ ejv nq Ges Zv‡`i cÖ‡Z¨KwU‡K GK GKwU mywbw`©ó bv‡g bvgKiY Kiv nq|
XOA mv‡c‡¶ mg‡KvYx wÎfyR POM Gi PM wecixZ evû,

OM mwbœwnZ evû, OP AwZfyR| GLb

XOA

ai‡j,

†Kv‡Yi †h QqwU wÎ‡KvYwgwZK AbycvZ

cvIqv hvq Zv wb‡gœ eY©bv Kiv n‡jv| wPÎ †_‡K, sin =

wecixZ evû AwZfyR mwbœwnZ evû OM = cos = AwZfyR OP tan =

PM = OP

[ [ [

†Kv‡Yi mvBb (sin e) ] †Kv‡Yi †KvmvBb cos ine ] †Kv‡Yi U¨vb‡R›U tan gent ]

PM wecixZ evû = OM mwbœwnZ evû

Ges G‡`i wecixZ AbycvZ
1 [ †Kv‡Yi †Kv‡mK¨v›U cosecant ] sin 1 [ †Kv‡Yi †mK¨v›U sec ant ] sec = cos 1 [ †Kv‡Yi †KvU¨vb‡R›U cot angent ] cot = tan j¶ Kwi, sin cÖZxKwU †Kv‡Yi mvBb-Gi AbycvZ‡K †evSvq; sin I cos ec
=

Gi ¸Ydj‡K bq|

ev‡`

sin Avjv`v †Kv‡bv A_© enb K‡i bv| wÎ‡KvYwgwZK Ab¨vb¨ AbycvZ¸‡jvi †¶‡ÎI welqwU cÖ‡hvR¨|

MwYZ

155

94

wÎ‡KvYwgwZK AbycvZ¸‡jvi m¤úK©
XOA

g‡b Kwi,

GKwU m~²‡KvY|

cv‡ki wPÎ mv‡c‡¶, msÁvbyhvqx,
PM 1 OP , cos ec = = OP sin PM OM 1 OP = , sec = cos = OP cos OM PM OM 1 , cot = = tan = tan OM PM PM PM OP [je I ni‡K OP Øviv fvM K‡i] Avevi, tan = OM OM OP sin = cos sin tan = cos

sin =

Ges GKBfv‡e, cot =

cos sin

95

wÎ‡KvYwgwZK A‡f`vewj
2

(i ) (sin )

PM (cos ) = OP
2

2

OM OP

2

=

PM 2 OP2
1

OM 2 PM 2 OM 2 OP 2 = = OP 2 OP2 OP2

[ wc_v‡Mviv‡mi m~Î ]

ev, (sin )2 (cos )2 1 sin 2 cos 2 1

gš—e¨ : c~Y©msL¨v m~PK n Gi Rb¨ (sin )n †K (sin n ), (cos )n †K cosn
(ii ) sec 2 = (sec ) 2 =

BZ¨vw` †jLv nq|

OP OM

2

OP2 PM 2 OM 2 = = [ OP mg‡KvYx OM 2 OM 2

POM Gi AwZf~R e‡j]

156

MwYZ

=

PM 2 OM 2

OM 2 OM 2
2

=1

PM OM

= 1 (tan )2 = 1 tan 2

sec2

1 tan 2 tan 2 sec 2 1 1

ev, sec 2 ev, tan 2
(iii ) cosec 2
=

(cosec ) 2 =

OP PM

2

OP2 PM 2 OM 2 = [ OP mg‡KvYx PM 2 PM 2
PM 2 PM 2 OM 2 =1 PM 2

POM Gi AwZfyR e‡j]

=

OM PM

2

= 1 (cot ) 2 = 1 cot 2

cosec 2

cot 2

1

Ges cot 2

cosec 2

1

KvR t 1| wb‡Pi wÎ‡KvYwgwZK m~Î¸‡jv mn‡R g‡b ivLvi Rb¨ ZvwjKv ˆZwi Ki| cosec sec tan 1 sin 1 cos 1 cot sin cos cos sin sin 2
2

tan cot

cos
2

1
2

sec

2

1 tan

cosec

1 cot 2

D`vniY 1| tan A

4 n‡j, A †Kv‡Yi Ab¨vb¨ wÎ‡KvYwgwZK AbycvZmg~n wbY©q Ki| 3

mgvavb : †`Iqv Av‡Q, tan A

4 . 3
5

AZGe, A †Kv‡Yi wecixZ evû = 4, mwbœwnZ evû = 3 AwZfzR =
42 32 25

myZivs, sin A =

3 3 4 , cos A = , cot A = 5 4 5

MwYZ

157

5 5 , sec A = . 4 3 D`vniY 2| ABC mg‡KvYx wÎfz‡Ri cosec A =

B †KvYwU mg‡KvY| tan A 1 n‡j 2 sin A cos A 1 Gi mZ¨Zv

hvPvB Ki| mgvavb : †`Iqv Av‡Q, tan A
4 . 3

AZGe, A †Kv‡Yi wecixZ evû = 4, mwbœwnZ evû = 3 AwZfzR =
42 32 25 5
4 3 3 , cos A = , cot A = 5 4 5 5 5 cosec A = , sec A = . 4 3

myZivs, sin A =

D`vniY 2| ABC mg‡KvYx wÎfz‡Ri B †KvYwU mg‡KvY| tan A 1 n‡j 2 sin A cos A 1 Gi mZ¨Zv hvPvB Ki| mgvavb : †`Iqv Av‡Q, tan A 1 . AZGe, wecixZ evû = mwbœwnZ evû = 1 AwZfzR = 12 12 myZivs, sin A =
2

1 1 , cos A = . 2 2 1 1 1 GLb evgc¶ = 2 sin A cos A =2 =2 =1 2 2 2

= Wvbc¶ |

2 sin A cos A 1 evK¨wU mZ¨|

KvR :
1| ABC mg‡KvYx wÎfz‡Ri

C mg‡KvY, AB =29 †m.wg., BC = 21 †m.wg. Ges

ABC

n‡j,

cos 2

sin 2

Gi gvb †ei Ki|

D`vniY 3| cÖgvY Ki †h, tan mgvavb : evgc¶ = tan
=

cot

sec .cosec .

cot

sin cos cos sin 2 sin cos 2 = sin cos 1 = [ sin 2 sin cos 1 1 = sin cos = cosec sec
= sec

cos 2

1]

cosec = Wvbc¶ (cÖgvwYZ)|

158

MwYZ

sin 2 cos 2 cos 2 sin 2 1 = [ sin 2 2 2 cos sin 1 1 = 2 cos sin 2
= = sec 2

cos 2

1]

cosec 2

= Wvbc¶ (cÖgvwYZ)|

D`vniY 5| cÖgvY Ki †h, mgvavb : evgc¶ =

1 1 sin 2

1 1 cos ec 2

1

1 1 sin 2 1 = 1 sin 2
=

1 1 sin 2 1 sin 2 = 1 sin 2 = 1 = Wvbc¶ (cÖgvwYZ)|

1 1 cosec 2 1 1 1 sin 2 sin 2 1 sin 2

D`vniY 6| cÖgvY Ki : mgvavb : evgc¶ =

1 2 sin 2 A

1 2 tan 2 A

1

1 2 sin 2 A 1 = 2 sin 2 A

1 2 sin 2 A 1 = 2 sin 2 A
= = =

1 2 tan 2 A 1 sin 2 A 2 cos 2 A cos 2 A 2cos 2 A sin 2 A cos 2 A 2(1 sin 2 A) sin 2 A

2

2 2 = 2 = 1 = Wvbc¶ (cÖgvwYZ)

1 sin 2 A 1 sin 2 A sin 2 A sin 2 A

cos 2 A 2 2sin 2 A sin 2 A 1 sin 2 A 2 sin 2 A

MwYZ

159

D`vniY 7| cÖgvY Ki : mgvavb : evgc¶ =

tanA secA 1

secA 1 tanA

0

tanA secA 1 secA 1 tanA 2 2 tan A (sec A 1) = [ sec 2 A 1 tan 2 A] (secA 1) tanA

tan 2 A tan 2 A (secA 1) tanA 0 = (secA 1) tanA
= = 0 = Wvbc¶ (cÖgvwYZ)

D`vniY 8| cÖgvY Ki : mgvavb : evgc¶ =

1 sinA 1 sinA

secA tanA

1 sinA 1 sinA

=

(1 sinA)(1 sinA) [je I ni‡K (1 sinA)(1 sinA)

(1 sinA) Øviv ¸Y K‡i]

=

(1 sinA) 2 1 sin 2 A (1 sinA) 2 cos 2 A
1 sinA cosA 1 sinA Ñ cosA cosA

= = =

= secA tanA = Wvbc¶ (cÖgvwYZ)

D`vniY 9| tanA sinA a Ges tanA sinA b n‡j, cÖgvY Ki †h, a 2 b 2 mgvavb : GLv‡b cÖ`Ë, tanA sinA a Ges tanA sinA b evgc¶ = a 2 b 2 = ( tanA sinA) 2 ( tanA sinA) 2 = 4tanA sinA [ (a b) 2 (a b) 2
= 4 tan 2 A sin 2 A

4 ab .

4ab]

160

MwYZ

= 4 tan 2 A (1 cos 2 A ) = 4 tan 2 A = 4 tan 2 A = 4 ( tanA = 4 ab

tan 2 A cos 2 A sin 2 A sinA)( tanA sinA)

= Wvbc¶ (cÖgvwYZ) KvR : 1| cot 4 A cot 2 A 1 n‡j, cÖgvY Ki †h, cos 4 cos 2 A 1

2| sin 2 A sin 4 A 1 n‡j, cÖgvY Ki †h, tan 4 A tan 2 A 1 D`vniY 10| secA tanA
5 n‡j, secA tanA Gi gvb wbY©q Ki| 2

mgvavb : GLv‡b cÖ`Ë, secA tanA

5 .............(i) 2

Avgiv Rvwb, sec 2 A 1 tan 2 A ev, sec 2 A tan 2 A 1 ev, (secA tanA)(secA tanA) 1 ev,
5 (secA tanA) 1 [ (i ) n‡Z] 2

secA tanA

2 5

Abykxjbx 9 1
1| wb‡Pi MvwYwZK Dw³¸‡jvi mZ¨-wg_¨v hvPvB Ki| †Zvgvi DË‡ii c‡¶ hyw³ `vI| K. tan A Gi gvb me©`v 1 Gi †P‡q Kg L. cot A n‡jv cot I A Gi ¸Ydj M. A Gi †Kvb gv‡bi Rb¨ sec A
12 5

N. cos n‡jv cotangent Gi msw¶ß iƒc 2| 3| sin A 3 n‡j, A †Kv‡Yi Ab¨vb¨ wÎ‡KvYwgwZK AbycvZmg~n wbY©q Ki| 4

†`Iqv Av‡Q, 15 cot A 8 , sin A I sec A Gi gvb †ei Ki|

MwYZ
= 4 tan 2 A (1 cos 2 A ) = 4 tan 2 A = 4 tan 2 A = 4 ( tanA = 4 ab

161

tan 2 A cos 2 A sin 2 A sinA)( tanA sinA)

= Wvbc¶ (cÖgvwYZ) KvR : 1| cot 4 A cot 2 A 1 n‡j, cÖgvY Ki †h, cos 4 cos 2 A 1

2| sin 2 A sin 4 A 1 n‡j, cÖgvY Ki †h, tan 4 A tan 2 A 1 D`vniY 10| secA tanA
5 n‡j, secA tanA Gi gvb wbY©q Ki| 2

mgvavb : GLv‡b cÖ`Ë, secA tanA

5 .............(i) 2

Avgiv Rvwb, sec 2 A 1 tan 2 A ev, sec 2 A tan 2 A 1 ev, (secA tanA)(secA tanA) 1 ev,
5 (secA tanA) 1 [ (i ) n‡Z] 2

secA tanA

2 5

Abykxjbx 9 1
1| wb‡Pi MvwYwZK Dw³¸‡jvi mZ¨-wg_¨v hvPvB Ki| †Zvgvi DË‡ii c‡¶ hyw³ `vI| K. tan A Gi gvb me©`v 1 Gi †P‡q Kg L. cot A n‡jv cot I A Gi ¸Ydj M. A Gi †Kvb gv‡bi Rb¨ sec A
12 5

N. cos n‡jv cotangent Gi msw¶ß iƒc 2| 3| sin A 3 n‡j, A †Kv‡Yi Ab¨vb¨ wÎ‡KvYwgwZK AbycvZmg~n wbY©q Ki| 4

†`Iqv Av‡Q, 15 cot A 8 , sin A I sec A Gi gvb †ei Ki|

dg©v-21, MwYZ-9g-10g

162

MwYZ
ABC mg‡KvYx wÎfz‡Ri ABC C mg‡KvY, AB = 13 †m.wg., BC = 12 †m.wg. Ges

4|

n‡j, sin , cos I tan Gi gvb †ei Ki|
B †KvYwU mg‡KvY| tan A

5|

ABC mg‡KvYx wÎfz‡Ri

3 n‡j,

3 sin A cos A

4 Gi

mZ¨Zv hvPvB Ki| cÖgvY Ki (6 Ñ 20) : 6| 7|
(i )
(i ) (iii)

1 sec 2 A

1 cosec 2 A

1;

(ii )
(ii )

1 cos 2 A

1 cot 2 A

1 ; (iii)

1 sin 2 A

1 tan 2 A

1;

sinA cosecA

cosA 1; secA

secA cosA

tanA 1. cotA

1 1 1 2 1 sin A 1 cosec2 A

8| 9| 11| 13| 15|

(i )

tan A cot A 1 cot A 1 tan A sinA 1 cotA

sec A cos ecA 1 ;

(ii )

1 1 1 2 1 tan A 1 cot 2 A sin A.

cosA 1 tanA

sinA cosA.

10| tan A 1` sin 2 A 12| 14| 16| 18| cosecA cosecA 1

secA tanA cosecA cotA
1 1 sinA

cosecA cotA secA tanA
2sec 2 A.

cosecA cosecA 1

2sec 2 A.

1 1 sinA

1 1 2 tan 2 A. cosecA 1 cosecA 1

sinA 1 cosA 1 cosA sinA sec ) 2

2 cosec A.

tanA secA 1

secA 1 tanA

0

17| ( tan 19|

1 sin 1 sin

cotA tanB cotB tanA

cotA . tan B.

1 sinA 1 sinA

secA

tan A.

20|

secA 1 secA 1

cotA cosec A.

21| cosA sinA 22| hw` tan A

2cosA n‡j, Z‡e cÖgvY Ki †h, cos A sin A cosec 2 A sec 2 A 1 nq, Z‡e Gi gvb wbY©q Ki| cosec 2 A sec 2 A 3
4 n‡j, cos ecA cotA Gi gvb KZ ? 3

2 sin A

23| cos ecA cotA 24| cotA

b a sinA b cosA n‡j, Gi gvb wbY©q Ki| a a sinA b cosA

MwYZ

163

9 6 30 , 45 I

60

†Kv‡Yi wÎ‡KvYwgwZK AbycvZ

R¨vwgwZK Dcv‡q 30 , 45 I 60 cwigv‡ci †KvY AuvK‡Z wk‡LwQ| G mKj †Kv‡Yi wÎ‡KvYwgwZK Abycv‡Zi cÖK…Z gvb R¨vwgwZK c×wZ‡Z wbY©q Kiv hvq|
30

I

60

†Kv‡Yi wÎ‡KvYwgwZK AbycvZ
XOZ 30 Ges OZ evû‡Z P GKwU

g‡b Kwi, we›`y| PM

OX AuvwK Ges PM †K Q ch©š— ewa©Z

Kwi †hb MQ PM nq| O, Q †hvM K‡i Z ch©š— ewa©Z Kwi GLb POM I QOM Gi g‡a¨ PM
OM mvaviY evû Ges Aš—f©y³

QM ,

PMO = Aš—f©y³

QMO

90

POM

QOM

AZGe, Ges Avevi,

QOM

POM OPM
POM

30 60
QOM 30 30 60

OQM
POQ

OPQ GKwU mgevû wÎfyR|

hw` OP 2a nq, Z‡e PM mg‡KvYx OPM n‡Z cvB,
OM OP 2 PM 2

1 PQ 2

1 OP 2

a [†h‡nZz

OPQ GKwU mgevû wÎfyR ]

4a2

a2

3a .

wÎ‡KvYwgwZK AbycvZmg~n †ei Kwi : sin 30 PM a 1 , OP 2a 2 PM a tan 30 OM 3a OP 2a cosec 30 PM a

cos 30

OM OP

3a 2a

3 2

1 . 3

2 , sec 30
3.

OP OM

2a 3a

2 3

cot 30

OM PM

3a a

GKBfv‡e,

164

MwYZ

sin 60

OM OP

3a 2a

3 , cos 60 2

PM OP OP PM

a 2a

1 , tan 60 2 2a a 2 , cot 60

OM PM

3a a

3

cosec 60

OP OM

2a 3a

2 , sec 60 3

PM OM

a 3a

1 3

45

†Kv‡Yi wÎ‡KvYwgwZK AbycvZ
XOZ 45

g‡b Kwi,

Ges P, OZ Gi
OX AuvwK|

Dci¯’ GKwU we›`y| PM
OPM mg‡KvYx wÎfy‡R

POM

45

myZivs,

OPM

45
OM = a (g‡b Kwi)
2 2 2

AZGe, PM

GLb, OP 2 OM 2 PM 2 = a + a =2 a ev, OP
2a

wÎ‡KvYwgwZK Abycv‡Zi msÁv †_‡K Avgiv cvB, sin 45 cosec 45

PM OP

a 2a

1 , cos 45 2
2 , sec 45

OM OP
1 cos 45

a 2a

1 , tan 45 2
1 tan 45

PM OM 1

a 1 a

1 sin 45

2 , cot 45

9 7 c~iK †Kv‡Yi wÎ‡KvYwgwZK AbycvZ
Avgiv Rvwb †h, `yBwU m~²‡Kv‡Yi cwigv‡ci mgwó 90 n‡j, Zv‡`i GKwU‡K AciwUi c~iK †KvY ejv nq| †hgb, 30 I 60 Ges 15 I 75 ci¯úi c~iK †KvY| mvaviYfv‡e, †KvY I (90
) †KvY ci¯ú‡ii c~iK †KvY|

MwYZ

165

c~iK †Kv‡Yi wÎ‡KvYwgwZK AbycvZ g‡b Kwi,
XOY

Ges P GB †Kv‡Yi OY evûi
OX AuvwK|

Dci GKwU we›`y| PM

†h‡nZz wÎfy‡Ri wZb †Kv‡Yi mgwó `yB mg‡KvY, AZGe, POM mg‡KvYx wÎfy‡R Ges
OPM POM

PMO

90

GK mg‡KvY = 90
90
]

OPM

90

POM

[†h‡nZz

POM
) ) ) ) )

XOY

sin (90 cos (90 tan (90 cot (90 sec (90 cosec (90

OM = cos POM = cos OP PM = sin POM = sin OP OM = cot POM = cot PM PM = tan POM = tan OM OP = cosec POM = cosec PM
) OP = sec POM = sec OM
.

Dc‡ii m~Î¸‡jv wbgœwjwLZfv‡e K_vq cÖKvk Kiv hvq : c~iK †Kv‡Yi sine c~iK †Kv‡Yi cosine c~iK †Kv‡Yi tangent KvR : sec (90
)

†Kv‡Yi cosine ; †Kv‡Yi sine ; †Kv‡Yi cotangent , BZ¨vw`|
5 n‡j, cosec 3

cot

Gi gvb wbY©q Ki|

98

0

I

90

†Kv‡Yi wÎ‡KvYwgwZK AbycvZ
Gi Rb¨ wÎ‡KvYwgwZK

Avgiv mg‡KvYx wÎfz‡Ri m~¶‡KvY

AbycvZ¸‡jv wbY©q Ki‡Z wk‡LwQ| Gevi †`wL, †KvYwU µgkt †QvU Kiv n‡j wÎ‡KvYwgwZi AbycvZ¸‡jv Kxiƒc nq| †KvYwU hZB †QvU n‡Z _v‡K, wecixZ evû PN Gi ‰`N©¨ ZZB †QvU nq| P we›`ywU N we›`yi wbKUZi nq Ges Ae‡k‡l †KvYwU hLb 0 Gi Lye Kv‡Q Aew¯’Z nq,
OP cÖvq ON Gi mv‡_ wg‡j hvq|

166

MwYZ

hLb sin †KvYwU 0 Gi Lye wbK‡U Av‡m PN †iLvs‡ki ˆ`N©¨ k~‡b¨i †KvVvq †b‡g Av‡m Ges G‡¶‡Î
PN Gi gvb cÖvq k~b¨| GKB mgq, OP

†KvYwU 0 Gi Lye Kv‡Q G‡j OP Gi ˆ`N©¨ cÖvq ON

Gi ˆ`‡N©¨i mgvb nq Ges

cos

ON Gi gvb cÖvq 1. OP

wÎ‡KvYwgwZ‡Z Av‡jvPbvi myweav‡_© 0 †Kv‡Yi AeZviYv Kiv nq Ges cÖwgZ Ae¯’v‡b 0 †Kv‡Yi cÖvš—xq evû I Avw` evû GKB iwk¥ aiv nq| myZivs c~‡e©i Av‡jvPbvi m‡½ mvgÄm¨ †i‡L ejv nq †h, cos 0 sin 0 0.

1,

m~²‡KvY n‡j Avgiv †`‡LwQ tan sin , cos cot cos , sin

sec

1 , cosec cos

1 sin

,

0 †Kv‡Yi Rb¨ m¤¢ve¨ †¶‡Î G m¤úK©¸‡jv hv‡Z eRvq _v‡K †m w`‡K j¶ †i‡L msÁvwqZ Kiv nq|

tan 0 sec 0

sin 0 cos 0
1 cos 0

0 1

0

1 1. 1

0 Øviv fvM Kiv hvq bv weavq cosec 0 I cot 0 msÁvwqZ Kiv hvq bv|

Avevi, hLb

‡KvYwU 90 Gi Lye Kv‡Q, AwZfzR OP cÖvq PN Gi mgvb| myZivs, sin ‡KvYwU cÖvq 90 Gi mgvb n‡j ON k~‡b¨i KvQvKvwQ; cos
0 , sin 90

Gi gvb

cÖvq 1| Ab¨w`‡K,

Gi gvb cÖvq 0.
1.

myZivs, c~‡e© ewY©Z m~‡Îi m‡½ mvgÄm¨ †i‡L ejv nq †h, cos 90 cot 90 cos 90 sin 90 0 1 0

MwYZ

167

cosec 90

1 sin 90

1 1 1

c~‡e©i b¨vq 0 Øviv fvM Kiv hvq bv weavq tan 90 I sec 90 msÁvwqZ Kiv hvq bv| `ªóe¨ : e¨env‡ii myweav‡_© 0 , 30 , 45 , 60 I 90 †KvY¸‡jvi wÎ‡KvYwgwZK AbycvZ¸‡jvi gvb wb‡Pi Q‡K †`Lv‡bv n‡jv : †KvY AbycvZ sine cosine tangent 0

30
1 2 3 2
1 3

45
1 2
1 2
1

60
3 2 1 2

90

0 1 0 AmsÁvwqZ 1 AmsÁvwqZ

1 0 AmsÁvwqZ 0 AmsÁvwqZ 1

3

cotangent secant cosecant

3
2 3
2

1

1 3
2

2

2

2 3

j¶ Kwi : wba©vwiZ K‡qKwU †Kv‡Yi Rb¨ wÎ‡KvYwgwZK gvbmg~n g‡b ivLvi mnR Dcvq|
(i ) 0, 1, 2, 3 Ges 4 msL¨v¸‡jvi cÖ‡Z¨KwU‡K 4 Øviv fvM K‡i fvMd‡ji eM©g~j wb‡j h_vµ‡g sin 0 , sin 30 , sin 45 , sin 60 Ges sin 90 Gi gvb cvIqv hvq|
(ii ) 4, 3, 2, 1 Ges 0 msL¨v¸‡jvi cÖ‡Z¨KwU‡K 4 Øviv fvM K‡i fvMdj¸‡jvi eM©g~j wb‡j h_vµ‡g

cos 0 , cos 30 , cos 45 , cos 60 Ges cos 90 Gi gvb cvIqv hvq|
(iii ) 0, 1, 3 Ges 9 msL¨v¸‡jvi cÖ‡Z¨KwU‡K 3 Øviv fvM K‡i fvMdj¸‡jvi eM©g~j wb‡j h_vµ‡g tan 0 ,

tan 30 , tan 45 Ges tan 60

Gi gvb cvIqv hvq| (D‡jL¨ †h, tan 90 msÁvwqZ bq)|

(iv ) 9, 3, 1 Ges 0 msL¨v¸‡jvi cÖ‡Z¨KwU‡K 3 Øviv fvM K‡i fvMdj¸‡jvi eM©g~j wb‡j h_vµ‡g cot 45 , cot 60 , cot 90 Gi gvb cvIqv hvq| (D‡jL¨ †h, cot 0 msÁvwqZ bq)|

168

MwYZ

D`vniY 1| gvb wbY©q Ki : (K) (L) (M) (N) mgvavb : (K) cÖ`Ë ivwk =
1 sin 2 45 1 sin 2 45
1
=

1 sin 2 45 tan 2 45 1 sin 2 45 cot 90 tan 0 sec 30 cosec 60 sin 60 cos 30 cos 60 sin 30

1 tan 60 , 1 tan 2 60

2

sin 2 60

tan 2 45

1
1

1 2 1 2

2

2

(1) 2

[ sin 45

1 I tan 45 2

1]

1 1 2 1= 2 1= 1 1= 4 = 1 3 3 3 1 2 2 (L) cÖ`Ë ivwk = cot 90 tan 0 sec 30 cosec 60 2 2 0 =0 0 3 3
[

cot 90

0, tan 0

0, sec 30

2 , cosec 60 3

2 ] 3

(M) cÖ`Ë ivwk = sin 60 cos 30
= [ =

cos 60 sin 30

3 2

3 2

1 1 2 2 cos 30 3 , sin 30 2 cos 60 1 ] 2

sin 60

(N)

3 1 4 = =1 4 4 4 1 tan 2 60 cÖ`Ë ivwk = sin 2 60 2 1 tan 60
=

1

3

2 2

1 3 2 1 3 3 = = 4 1 3 4 2 3 1 = 4 4

3 2

2

3 4

MwYZ

169

D`vniY 2| (K) (L)
2cos(A B) 1, 2sin (A B) cosA sinA cosA sinA 1 1

3 Ges A, B m~²‡KvY n‡j, A I B Gi gvb wbY©q Ki|

3 n‡j, A Gi gvb wbY©q Ki| 3 1 tan 2 A , hw` A 1 tan 2 A
3sin 3 45 nq|

(M) cÖgvY Ki †h, cos2A (N) mgvavb Ki : 2cos 2 mgvavb : (K)

0 , †hLv‡b

m~²‡KvY|

2cos( A B ) 1
1 2
1 ] 2

ev, cos( A B )

ev, cos( A B ) cos45 [ cos45
A B 45 ..................(i ) 3
3 2

Ges 2 sin ( A B ) ev, sin ( A B) ev, sin ( A B )
A B

sin 60

[

sin 60

3 ] 2

60 ....................(ii )

(i ) I (ii ) bs †hvM K‡i cvB,
2 A 105

A

105 2

52

1 2

Avevi, (ii ) n‡Z (i ) we‡qvM K‡i cvB,
2B 15

ev,

B B

15 2 7 1 2 7 1 2

wb‡Y©q A 52

1 I B 2 dg©v-22, MwYZ-9g-10g

170

MwYZ

(L) ev, ev, ev, ev,

cosA sinA cosA sinA

1 1

3 3

cos A sin A cos A sin A cos A sin A cos A sin A
2cosA 2 2 sinA 2 3 1 cosA sinA 3

1 1

3 1 3 1

3 3

cotA cot 60
A 60

(M)

†`Iqv Av‡Q, A 45 cÖgvY Ki‡Z n‡e, cos 2 A evgc¶ = cos 2 A
= cos(2 45 ) = cos 90 = 0

1 tan 2 A 1 tan 2 A

Wvbc¶ =
=

1 tan 2 A 1 tan 2 A

1 (1) 2 1 tan 2 45 = 1 tan 2 45 1 (1) 2 0 = =0 2

evgc¶ = Wvbc¶ (cÖgvwYZ) (N) cÖ`Ë mgxKiY 2cos 2 ev, ev, ev, ev, sin 2

3sin

3

0

2(1 sin ) 3(1 sin ) 0
2(1 sin )(1 sin ) 3(1 sin ) (1 sin ){2(1 sin ) 3} 0 (1 sin ){2 sin 1} 0 0

ev, 1 sin
1

0

A_ev, 2 sin ev, 2 sin

1 0
1 1 2 sin 30

ev, sin
90

sin 90

ev, sin ev, sin ev,
30 30 .

†h‡nZz

m~²‡KvY, †m‡nZz,

MwYZ

171

Abykxjbx 9 2
1| cos = K.
1 3 3
1 n‡j cot 2

Gi gvb †Kvb&wU? L. 1

N. 2 M. 2 = 1 - cos2 2| (i) sin (ii) sec2 = 1 + tan2 (iii) cot2 = 1 - tan2 cv‡ki Z‡_¨i Av‡jv‡K wb‡gœi †KvbwU mwVK ? K. i I ii L. i I iii M. ii I iii N. i, ii I iii

wPÎ Abyhvqx 3 I 4 bs cÖ‡kœi DËi `vI| 3| sin Gi gvb †Kvb&wU ? K.
3 4 3 M. 5 4 3 4 N. 5

L.

4| cot K.
3 4 4 M. 5

Gi gvb †Kvb&wU ?
3 5 4 N. 3

L.

gvb wbY©q Ki (5-8)
1 cot 2 60 5| 1 cot 2 60

6| sec 2 60

tan 45 sin 2 60 tan 30 tan 60 .

7|

1 cos 2 60 1 cos 2 60

8|

cos 45 cot 2 60 cosec 2 30

†`LvI †h, (9-11) 9| cos 2 30 sin 2 30 cos 60 . sin 90 cos 30

10| sin 60 cos 30 11| cos 60 cos 30 12| sin 3A

cos 60 sin 30 sin 60 sin 30

cos3A. hw` A 15 nq|

172

MwYZ

13| sin 2A 14| tan 2 A 15| cos 2A

2 tanA hw` A 45 nq| 1 tan 2 A 2tanA hw` A 30 nq| 1 tan 2 A

1 tan 2 A hw` A 60 nq| 1 tan 2 A

16| 2cos(A B) 1 2sin(A B) Ges A, B m~²‡KvY n‡j †`LvI †h, A 45 , B 15 | 17| cos(A B) 1, 2sin(A B) = 18| mgvavb Ki : cosA sinA cosA sinA 3 Ges A, B m~²‡KvY n‡j, A I B Gi gvb wbY©q Ki|

3 1 3 1

19| A I B m~²‡KvY Ges cot(A B) 1, cot(A B) 20| †`LvI †h, cos 3A 4 cos 3 A 21| mgvavb Ki : sin 22| mgvavb Ki : cos 2 23| mgvavb Ki : 2sin 2 24| mgvavb Ki : tan 2 cos sin 2 3cos 3cos A hw` A 90

3 n‡j, A I B Gi gvb wbY©q Ki|
30 nq|

1 , hLb 0
2 3 5cos 0,

hLb

m~²‡KvY|

m~²‡KvY|
3 0.

(1

3 ) tan
1 4

25| gvb wbY©q Ki : 3cot2 60 + cosec230 + 5 sin2 45 – 4co2 60 26| ABC Gi B = 900, AB = 5 cm, BC = 12 cm.

K. AC Gi ˆ`N©¨ wbY©q Ki| L. C = n‡j sin + cos Gi gvb wbY©q Ki| M. †`LvI †h, sec2 +cosec2 = sec2 cosec2 27|

K. AC Gi cwigvY KZ? L. tanA + tanC Gi gvb wbY©q Ki| M. x I y Gi gvb wbY©q Ki|

`kg Aa¨vq

`~iZ¡ I D”PZv
AwZ cÖvPxb Kvj †_‡KB `~ieZx© †Kv‡bv e¯‘i `~iZ¡ I D”PZv wbY©q Ki‡Z wÎ‡KvYwgwZK Abycv‡Zi cÖ‡qvM Kiv nq| eZ©gvb hy‡M wÎ‡KvYwgwZK Abycv‡Zi e¨envi †e‡o hvIqvq Gi ¸i“Z¡ Acwimxg| †h me cvnvo, ce©Z, UvIqvi, Mv‡Qi D”PZv Ges b`-b`xi cÖ¯’ mn‡R gvcv hvq bv †m me †¶‡Î D”PZv I cÖ¯’ wÎ‡KvYwgwZi mvnv‡h¨ wbY©q Kiv hvq| G‡¶‡Î m~²‡Kv‡Yi wÎ‡KvYwgwZK Abycv‡Zi gvb †R‡b ivLv cÖ‡qvRb| Aa¨vq †k‡l wk¶v_x©ivÑ f~-†iLv, EaŸ©‡iLv, Dj¤^Zj, DbœwZ †KvY I AebwZ †KvY e¨vL¨v Ki‡Z cvi‡e| wÎ‡KvYwgwZi mvnv‡h¨ `~iZ¡ I D”PZv welqK MvwYwZK mgm¨v mgvavb Ki‡Z cvi‡e| wÎ‡KvYwgwZi mvnv‡h¨ nv‡Z-Kj‡g `~iZ¡ I D”PZv welqK wewfbœ cwigvc Ki‡Z cvi‡e| f~-†iLv, EaŸ©‡iLv Ges Dj¤^Zj : f~-†iLv n‡”Q f~wg Z‡j Aew¯’Z †h‡Kv‡bv mij‡iLv| f~-†iLv‡K kqb‡iLvI ejv nq| EaŸ©‡iLv n‡”Q f~wg Z‡ji Dci j¤^ †h‡Kv‡bv mij‡iLv| G‡K Dj¤^ †iLvI e‡j| f~wg Z‡ji Dci j¤^fv‡e Aew¯’Z ci¯úi‡”Q`x f~-†iLv I EaŸ©‡iLv GKwU Zj wbw`©ó K‡i| G Zj‡K Dj¤^ Zj e‡j| wP‡Î : f~wg Z‡ji †Kv‡bv ¯’vb C †_‡K CB `~i‡Z¡ AB D”PZv wewkó GKwU MvQ Lvov Ae¯’vq `Ûvqgvb| GLv‡b CB †iLv n‡”Q f~-†iLv, BA †iLv n‡”Q EaŸ©‡iLv Ges ABC ZjwU f~wgi Dci j¤^ hv Dj¤^ Zj| DbœwZ †KvY I AebwZ †KvY : wPÎwU j¶ Kwi, f~wgi mgvš—ivj AB GKwU mij‡iLv| O, B, P we›`y¸‡jv GKB Dj¤^ Z‡j Aew¯’Z| AB mij‡iLvi Dc‡ii P we›`ywU
AB †iLvi mv‡_
POB Drcbœ K‡i| GLv‡b, O we›`y‡Z P we›`yi POB |

DbœwZ †KvY n‡”Q

myZivs, f~Z‡ji Dc‡ii †Kvb we›`y f~wgi mgvš—ivj †iLvi mv‡_ †h †KvY Drcbœ K‡i Zv‡K DbœwZ †KvY ejv nq|

174

MwYZ

Avevi, O, A, Q we›`y¸‡jv GKB Dj¤^ Z‡j Aew¯’Z Ges Q we›`y f~-†iLvi mgvš—ivj AB †iLvi wb‡Pi w`‡K Aew¯’Z| GLv‡b, O we›`y‡Z Q we›`yi AebwZ †KvY n‡”Q m QOA . yZivs f~Z‡ji mgvš—ivj †iLvi wb‡Pi †Kv‡bv we›`y f~-†iLvi mv‡_ †h †KvY Drcbœ K‡i Zv‡K AebwZ †KvY ejv nq|

KvR : wPÎwU wPwýZ Ki Ges f~-†iLv EaŸ©‡iLv, Dj¤^Zj, DbœwZ †KvY I AebwZ †KvY wb‡`©k Ki| f~wg

we‡kl `ªóe¨ : G Aa¨v‡q mgm¨v mgvav‡bi †¶‡Î AvbygvwbK mwVK wPÎ Avek¨K| wPÎ A¼‡bi mgq wb‡Pi †KŠkj Aej¤^b Kiv `iKvi| (1) 30 †KvY A¼‡bi †¶‡Î f~wg > j¤^ n‡e| (2) 45 †KvY A¼‡bi †¶‡Î f~wg = j¤^ n‡e| (3) 60 †KvY A¼‡bi †¶‡Î f~wg < j¤^ n‡e| D`vniY 1| GKwU UvIqv‡ii cv`‡`k †_‡K 75 wgUvi `~‡i f~Zj¯’ †Kv‡bv we›`y‡Z UvIqv‡ii kx‡l©i DbœwZ
30 n‡j, UvIqv‡ii D”PZv wbY©q Ki|

mgvavb : g‡b Kwi, UvIqv‡ii D”PZv AB UvIqv‡ii cv`‡`k †_‡K BC we›`y‡Z UvIqv‡ii kxl© A we›`yi DbœwZ
ACB

h wgUvi

75 wgUvi `~‡i f~Zj¯’ C
30
75 wgUvi

mg‡KvYx ABC †_‡K cvB, tan ACB ev, tan 30 ev, h h h 1 ev, 75 3 h 75

AB BC
1 ev, 3

tan 30

3h

75 ev, h

75 3

75 3 [ni Ges je‡K 3 43.301 (cÖvq)|

3 Øviv ¸Y K‡i] ev, h

25 3

wb‡Y©q UvIqv‡ii D”PZv 43.301 wgUvi (cªvq)| D`vniY 2| GKwU Mv‡Qi D”PZv 105 wgUvi| MvQwUi kx‡l©i DbœwZ f~wgi †Kv‡bv we›`y‡Z DbœwZ †KvY 60 n‡j, MvQwUi †Mvov †_‡K f~j¯’ we›`ywUi `~iZ¡ wbY©q Ki| mgvavb : g‡b Kwi, Mv‡Qi †Mvov †_‡K f~Zj¯’ we›`ywUi `~iZ¡ BC we›`yi DbœwZ
ACB 60 x wgUvi, Mv‡Qi D”PZv AB 105 wgUvi Ges C we›`y‡Z MvQwUi kxl©

MwYZ

175

ABC †_‡K cvB, tan ACB AB ev, tan 60 0 BC 105 x ta n 60 0
105 ev, x 3

3
35 3

ev, x 3

105 ev, x

3 x 105 ev, x

105 3 ev, x 3

60.622 wgUvi (cÖvq)

MvQwUi †Mvov †_‡K f~Zj¯’ we›`ywU `~iZ¡ 60.622 wgUvi (cÖvq)|

KvR : wP‡Î AB GKwU MvQ| wP‡Î cÖ`Ë ZË¡ †_‡K 1. MvQwUi D”PZv wbY©q Ki| 2. MvQwUi cv`‡`k †_‡K f~Zj¯’ C we›`yi `~iZ¡ wbY©q Ki|

D`vniY 3| 18 wgUvi j¤^v GKwU gB GKwU †`Iqv‡ji Qv` eivei †Vm w`‡q f~wgi m‡½ 45 †KvY Drcbœ K‡i| †`IqvjwUi D”PZv wbY©q Ki| mgvavb : g‡b Kwi, †`IqvjwUi D”PZv AB h wgUvi, gBwUi ˆ`N©¨ AC 18 wgUvi Ges f~wgi m‡½ ACB 45 Drcbœ K‡i|
ABC †_‡K cvB, sin ACB

AB AC

ev, sin 45 ev,
1 2 h 18

h 18

sin 45

1 2
18 2

ev,

2h 18 ev, h

18 2

ev, ev, h

2h 18

ev, h

18 2 2

[ni Ges je‡K

2 Øviv ¸Y K‡i] ev, h

9 2

h 12.728 (cÖvq)

myZivs †`IqvjwUi D”PZv 12.728 wgUvi (cÖvq)| D`vniY 4| S‡o GKwU MvQ †n‡j co‡jv| Mv‡Qi †Mvov †_‡K 7 wgUvi D”PZvq GKwU jvwV †Vm w`‡q MvQvwU‡K †mvRv Kiv n‡jv| gvwU‡Z jvwVwUi ¯úk© we›`yi AebwZ †KvY 30 n‡j, jvwVi ˆ`N©¨ wbY©q Ki|

176

MwYZ

mgvavb : g‡b Kwi, Mv‡Qi †Mvov †_‡K AB jvwVwU †Vm w`‡q Av‡Q Ges AebwZ
ACB DBC

7 wgUvi D”PZvq

DBC

30 |

30 [GKvš—i †KvY e‡j]

ABC †_‡K cvB,

Sin ACB

AB ev, Sin30 BC

7 BC

ev,

1 2 BC

7 BC 14

Sin30

1 2

jvwVwUi ˆ`N©¨ 14 wgUvi|

KvR :
AC

wP‡Î AebwZ

CAE

60 , DbœwZ

ADB

30

36 wgUvi Ges B, C , D GKB mij‡iLvq Aew¯’Z n‡j,

AB, AD Ges CD evûi ˆ`N©¨ wbY©q Ki|

D`vniY 5| f~Zj¯’ †Kv‡bv ¯’v‡b GKwU `vjv‡bi Qv‡`i GKwU we›`yi DbœwZ †KvY 60 | H ¯’vb †_‡K 42 wgUvi wcwQ‡q †M‡j `vjv‡bi H we›`yi DbœwZ †KvY 45 nq| `vjv‡bi D”PZv wbY©q Ki| mgvavb : g‡b Kwi, `vjv‡bi D”PZv AB
ACB 60 Ges C ¯’vb †_‡K CD ADB 45 nq| h wgUvi, kx‡l©i DbœwZ 42 wgUvi wcwQ‡q †M‡j

DbœwZ

awi, BC
BD

x wgUvi|

BC

CD

x 42 wgUvi|

ABC †_‡K cvB, tan 60 0 x AB ev, BC h ............ i 3

3

h x

tan 60 0

3

Avevi, ABD †_‡K cvB, tan 45 ev, 1 ev, h h x h 3

AB BD
1 ev, h x 42

42

tan 450

42; (i) bs mgxKi‡Yi mvnv‡h¨|

MwYZ

177

ev,

3h

h

42 3 ev,

3h h

42 3 ev,

3 1h

42 3 ev, h

42 3 3 1

h 99.373 wgUvi (cÖvq) `vjvbwUi D”PZv 99.373 wgUvi (cÖvq)| D`vniY 6| GKwU LyuwU Ggb fv‡e †f‡O †Mj †h, Zvi fvOv Ask `Êvqgvb As‡ki mv‡_ 30 †KvY Drcbœ K‡i LyuwUi †Mvov †_‡K 10 wgUvi `~‡i gvwU ¯úk© K‡i| LyuwUi m¤ú~Y© ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, LyuwUi m¤ú~Y© ˆ`N©¨ AB h wgUvi| LywUwU BC x wgUvi D”PZvq †f‡O wM‡q wew”Qbœ bv n‡q fvOv Ask `Êvqgvb As‡ki mv‡_ BCD 30 Drcbœ K‡i †Mvov †_‡K BD 10 wgUvi `~‡i gvwU ¯úk© K‡i| GLv‡b, CD AC AB BC h x wgUvi BCD †_‡K cvB,

1 10 BD ev, = 3 x BC BD 1 Avevi, sin 30 ev, CD 2

ta n 30

x 10 3
10 h x

ev, h x 20 ev, h 20 x ev, h 20 10 3; [ x-Gi gvb ewm‡q ] h 37.321 (cªvq) LyuwUi ˆ`N©¨ 37.321 wgUvi (cªvq)|

KvR :
`yBwU gvBj †cv‡ói ga¨eZx© †Kv‡bv ¯’v‡bi Dc‡i GKwU †ejyb Do‡Q| †ejy‡bi ¯’v†b H gvBj †cvó `yBwUi AebwZ †KvY h_vµ‡g 30 I 60 n‡j, †ejybwUi D”PZv wgUv‡i wbY©q Ki|

Abykxjbx 10
1| K. CAD Gi cwigvb wbY©q Ki| L. AB I BC Gi ˆ`N©¨ wbY©q Ki| M. A I D Gi `~iZ¡ wbY©q Ki| 2| `ywU wK‡jvwgUvi †cv÷ A I B Gi ga¨eZ©x †Kvb ¯’v‡bi Dci O we›`y‡Z GKwU †nwjKÞvi n‡Z H wK‡jvwgUvi †cv÷Ø‡qi AebwZ †Kvb h_vµ‡g 600 Ges 300 K. msw¶ß eY©bvmn AvbycvwZK wPÎ A¼b Ki| L. †nwjKÞviwU gvwU †_‡K KZ DuPz‡Z Aew¯’Z? M. A we›`y †_‡K †nwjKÞv‡ii mivmwi `~iZ¡ wbY©q Ki| 3| Ic‡ii wP‡Î O we›`y‡Z P we›`yi DbœwZ †KvY †KvbwU? M. QOA N. POB K. QOB L. POA 4| i f~-†iLv n‡”Q f~wg Z‡j Aew¯’Z †h‡Kv‡bv mij‡iLv| ii DaŸ©‡iLv n‡”Q f~wg Z‡ji Ici j¤^ †h‡Kv‡bv mij †iLv| iii f~wg Z‡ji Dci j¤^fv‡e Aew¯’Z ci¯úi‡”Q`x f~-†iLv I DaŸ©‡iLv GKwU Zj wbw`©ó K‡i| G Zj‡K Dj¤^ Zj e‡j| dg©v-23, MwYZ-9g-10g

178

MwYZ

Ic‡ii evK¨¸‡jvi g‡a¨ †KvbwU mwVK? K. i I ii M. ii I iii cv‡ki wPÎ Abyhvqx 5-6 cÖkœ `yBwUi DËi `vI| 5| BC Gi ˆ`N©¨ n‡e Ñ
4 m 3 M. 4 2 m

L. i I iii N. i, ii I iii

K.

L. 4m N. 4 3 m L. 4m N. 4 3 m

6| AB Gi ˆ`N©¨ n‡eÑ K.
4 m 3 M. 4 2 m

7| GKwU wgbv‡ii cv`‡`k †_‡K wKQy `~‡i GKwU ¯’v‡b wgbviwUi kx‡l©i DbœwZ 30 Ges wgbviwUi D”PZv 26 wgUvi n‡j, wgbvi †_‡K H ¯’vbwUi `~iZ¡ wbY©q Ki| 8| GKwU Mv‡Qi cv`‡`k †_‡K 20 wgUvi `~‡i f~Z‡ji †Kv‡bv we›`y‡Z Mv‡Qi P~ovi DbœwZ †KvY 60 n‡j, MvQwUi D”PZv wbY©q Ki| 9| 18 wgUvi ˆ`N©¨ GKwU gB f~wgi mv‡_ 45 †KvY Drcbœ K‡i †`Iqv‡ji Qv` ¯úk© K‡i| †`IqvjwUi D”PZv wbY©q Ki| 10| GKwU N‡ii Qv‡`i †Kv‡bv we›`y‡Z H we›`y †_‡K 20 wgUvi `~‡ii f~Zj¯’ GKwU we›`yi AebwZ †KvY 30 n‡j, NiwUi D”PZv wbY©q Ki| 11| f~Z‡j †Kv‡bv ¯’v‡b GKwU ¯—‡¤¢i kx‡l©i DbœwZ 60 | H ¯’vb †_‡K 25 wgUvi wcwQ‡q †M‡j DbœwZ †KvY 30 nq| ¯—¤¢wUi D”PZv wbY©q Ki| ¯—¤¢wUi

12| †Kv‡bv ¯’vb †_‡K GKwU wgbv‡ii w`‡K 60 wgUvi GwM‡q Avm‡j wgbv‡ii kxl© we›`yi DbœwZ 45 †_‡K 60 nq| wgbviwUi D”PZv wbY©q Ki| 13| GKwU b`xi Zx‡i †Kv‡bv GK ¯’v‡b `uvwo‡q GKRb †jvK †`Lj †h, wVK †mvRv‡mvwR Aci Zx‡i Aew¯’Z GKwU UvIqv‡ii DbœwZ †KvY 60 | H ¯’vb †_‡K 32 wgUvi wcwQ‡q †M‡j DbœwZ †KvY 30 nq| UvIqv‡ii D”PZv Ges b`xi we¯—vi wbY©q Ki| 14| 64 wgUvi j¤^v GKwU LyuwU †f‡O wM‡q m¤ú~Y© wew”Qbœ bv n‡q f~wgi mv‡_ 60 Drcbœ K‡i| LyuwUwUi fvOv As‡ki ˆ`N©¨ wbY©q Ki| 15| GKwU MvQ S‡o Ggbfv‡e †f‡O †Mj †h, fvOv Ask `Êvqgvb As‡ki mv‡_ 30 †KvY K‡i Mv‡Qi †Mvov †_‡K 12 wgUvi `~‡i gvwU ¯úk© K‡i| MvQwU m¤ú~Y© ˆ`N©¨ wbY©q Ki| 16| GKwU b`xi GK Zx‡i †Kv‡bv ¯’v‡b `uvwo‡q GKRb †jvK †`L‡jv †h, wVK †mvRv‡mvwR Aci Zx‡i Aew¯’Z 150 wgUvi j¤^v GKwU Mv‡Qi kx‡l©i DbœwZ †KvY 30 | †jvKwU GKwU †bŠKv‡hv‡M MvQwU‡K j¶¨ K‡i hvÎv ïi“ Ki‡jv| wKš‘ cvwbi †mªv‡Zi Kvi‡Y †jvKwU MvQ †_‡K 10 wgUvi `~‡i Zx‡i †cuŠQj| (K) Dc‡iv³ eY©bvwU wP‡Îi gva¨‡g †`LvI| (L) b`xi we¯—vi wbY©q Ki| (M) †jvKwUi hvÎv ¯’vb †_‡K Mš—e¨ ¯’v‡bi `~iZ¡ wbY©q Ki|

GKv`k Aa¨vq

exRMwYZxq AbycvZ I mgvbycvZ
(Algebraic Ratio and Proportion)
AbycvZ I mgvbycv‡Zi aviYv _vKv Avgv‡`i Rb¨ LyeB ¸iyZ¡c~Y©| mßg †kªwY‡Z cvwUMwYZxq AbycvZ I mgvbycvZ wek`fv‡e Av‡jvPbv Kiv n‡q‡Q| G Aa¨v‡q Avgiv exRMwYZxq AbycvZ I mgvbycvZ m¤ú‡K© Av‡jvPbv Ki‡ev| Avgiv cªwZwbqZB wbg©vY mvgMÖx I wewfbœ cÖKvi Lv`¨ mvgMÖx ˆZwi‡Z, †fvM¨cY¨ Drcv`‡b, Rwg‡Z mvi cÖ‡qv‡M, †Kv‡bvI wKQzi AvKvi-AvqZb `„wób›`b Ki‡Z Ges ˆ`bw›`b Kvh©µ‡gi AviI A‡bK †¶‡Î AbycvZ I mgvbycv‡Zi aviYv cÖ‡qvM K‡i _vwK| Bnv e¨envi K‡i ˆ`bw›`b Rxe‡b A‡bK mgm¨vi mgvavb Kiv hvq| Aa¨vq †k‡l wk¶v_x©iv exRMwYZxq AbycvZ I mgvbycvZ e¨vL¨v Ki‡Z cvi‡e| mgvbycvZ msµvš— wewfbœ iƒcvš—i wewa cÖ‡qvM Ki‡Z cvi‡e| avivevwnK AbycvZ eY©bv Ki‡Z cvi‡e| ev¯—e mgm¨v mgvav‡b AbycvZ, mgvbycvZ I avivevwnK AbycvZ e¨envi Ki‡Z cvi‡e|

11 1 AbycvZ
GKB GK‡K mgRvZxq `yBwU ivwki cwigv‡Yi GKwU AciwUi KZ ¸Y ev KZ Ask Zv GKwU fMœvsk Øviv cÖKvk Kiv hvq| GB fMœvskwU‡K ivwk `yBwUi AbycvZ e‡j| `yBwU ivwk p I q Gi AbycvZ†K p : q = p wjLv nq | p I q ivwk `yBwU mgRvZxq I GKB GK‡K n‡Z q

n‡e| Abycv‡Z p †K c~e© ivwk Ges q †K DËi ivwk ejv nq| A‡bK mgq AvbygvwbK cwigvc Ki‡ZI Avgiv AbycvZ e¨envi Kwi| †hgb, mKvj 8 Uvq iv¯—vq †h msL¨K Mvox _v‡K, 10 Uvq Zvi wØ¸Y Mvox _v‡K| G †¶‡Î AbycvZ wbY©‡q Mvoxi cÖK…Z msL¨v Rvbvi cÖ‡qvRb nq bv| Avevi A‡bK mgq Avgiv e‡j _vwK, †Zvgvi N‡ii AvqZb Avgvi N‡ii AvqZ‡bi wZb¸Y n‡e| GLv‡bI N‡ii mwVK AvqZb Rvbvi cÖ‡qvRb nq bv| ev¯—e Rxe‡b GiKg A‡bK †¶‡Î Avgiv Abycv‡Zi avibv e¨envi K‡i _vwK|

11 2 mgvbycvZ hw` PviwU ivwk Giƒc nq †h, cÖ_g I wØZxq ivwki AbycvZ Z„Zxq I PZz_© ivwki Abycv‡Zi mgvb nq, Z‡e H PviwU ivwk wb‡q GKwU mgvbycvZ Drcbœ nq| a, b, c, d Giƒc PviwU ivwk n‡j Avgiv wjwL a : b = c : d | mgvbycv‡Zi PviwU ivwkB GKRvZxq nIqvi cÖ‡qvRb nq bv| cÖ‡Z¨K Abycv‡Zi ivwk `yBwU GK RvZxq n‡jB P‡j|

180

MwYZ

Dc‡ii wP‡Î, `yBwU wÎfy‡Ri f~wg h_vµ‡g a I b Ges Zv‡`i cÖ‡Z¨‡Ki D”PZv h GKK| wÎfyRØ‡qi †¶Îdj A I B eM©GKK n‡j Avgiv wjL‡Z cvwi
A B 1 ah 2 1 bh 2 a b

ev, A : B a : b

A_©vr, †¶Îdj‡qi AbycvZ f~wgØ‡qi Abycv‡Zi mgvb|

µwgK mgvbycvZx a, b, c µwgK mgvbycvZx ej‡Z †evSvq a : b = b : c. a, b, c µwgK mgvbycvZx n‡e hw` Ges †Kej hw` b2 = ac nq| µwgK mgvbycv‡Zi †¶‡Î me¸‡jv ivwk GK RvZxq n‡Z n‡e| G‡¶‡Î c †K a I b Gi Z…Zxq mgvbycvZx Ges b †K a I c Gi ga¨mgvbycvZx ejv nq| D`vniY 1| A I B wbw`©ó c_ AwZµg K‡i h_vµ‡g t1 Ges t2 wgwb‡U| A I B Gi Mo MwZ‡e‡Mi AbycvZ wbY©q Ki| mgvavb : g‡b Kwi, A I B Gi Mo MwZ‡eM cÖwZ wgwb‡U h_vµ‡g v1 wgUvi I v2 wgUvi| Zvn‡j, t1 wgwb‡U A AwZµg K‡i v1t1 wgUvi Ges t2 wgwb‡U B AwZµg K‡i v2t2 wgUvi | cÖkœvbymv‡i, v1t1, = v2t2, v1 v2 t2 t1

GLv‡b MwZ‡e‡Mi AbycvZ mg‡qi e¨¯— Abycv‡Zi mgvb| KvR : 1| 3.5 : 5.6 ‡K 1 : a Ges b : 1 AvKv‡i cÖKvk Ki| 2| x : y = 5 : 6 n‡j 3x : 5y = KZ ?

11 3 Abycv‡Zi i“cvš—i
GLv‡b Abycv‡Zi ivwk¸‡jv abvZ¥K msL¨v| (1) a : b c : d n‡j, b : a d : c [e¨¯—KiY (Invertendo) ] cÖgvY : †`Iqv Av‡Q, a c b d ad bc [Dfqc¶‡K bd Øviv ¸Y K‡i]

MwYZ

181

ev,

ad ac

(2)

(3)

(4)

(5)

bc [Dfq c¶‡K ac Øviv fvM K‡i †hLv‡b a, c Gi †KvbwUB k~b¨ bq] ac d b ev, c a A_©vr, b : a d : c a : b c : d n‡j, a : c b : d [GKvš—iKiY (alternendo) ] cÖgvY : †`Iqv Av‡Q, a c b d ad bc [ Dfqc¶‡K bd Øviv ¸Y K‡i ] ad bc ev, [Dfq c¶‡K cd Øviv fvM K‡i †hLv‡b c, d Gi †KvbwUB k~b¨ bq] cd cd a b ev, c d A_©vr, a : c b : d a b c d [ †hvRb (componendo) ] a : b c : d n‡j, b d cÖgvY : †`Iqv Av‡Q, a c b d a c 1 1 [ Dfqc‡¶ 1 †hvM K‡i ] b d a b c d A_©vr, b d a b c d [ we‡qvRb (dividendo) ] a : b c : d n‡j, b d cÖgvY : †`Iqv Av‡Q, a c b d a c 1 1 [ Dfqc¶ ‡_‡K 1 we‡qvM K‡i ] b d a b c d A_©vr, b d a b c d [ †hvRb-we‡qvRb (componendo dividendo) ] a : b c : d n‡j, a b c d cÖgvY : a : b c : d

182

MwYZ

†hvRb K‡i cvB, a b c d ..............(i) b d Avevi we‡qvRb K‡i cvB, a b c d b d b d ev, [e¨¯—KiY K‡i] ..............(ii ) a b c d a b b c d d myZivs, [(i) I (ii) ¸Y K‡i] b a b d c d a b c d A_©vr, . [ GLv‡b a b Ges c d ] a b c d a c e g a c e g (6) n‡j, cÖ‡Z¨KwU AbycvZ = . b d f h b d f h

cÖgvY : g‡b Kwi,

a b

c d

e f

g h

k.

a bk , c dk , e a c e g b d f h a b c d e f bk dk b d g h

fk , g

hk
k.

fk hk f h

k (b d f h) b d f h

wKš‘ k cÖ`Ë mgvbycv‡Zi cÖ‡Z¨KwU Abycv‡Zi mgvb|

a c e g . b d f h

KvR : 1| gvZv I Kb¨vi eZ©gvb eq‡mi mgwó s eQi| Zv‡`i eq‡mi AbycvZ t eQi c~‡e© wQj r : p, x eQi c‡i Zv‡`i eq‡mi AbycvZ KZ n‡e ? 2| GKwU j¨v¤ú‡cv÷ †_‡K p wgUvi `~‡i `uvov‡bv r wgUvi D”PZv wewkó GK e¨w³i Qvqvi ˆ`N©¨ s wgUvi| j¨v¤ú‡cv‡ói D”PZv h wgUvi n‡j, H e¨w³ j¨v¤ú‡cv÷ †_‡K KZ `~‡i `uvov‡bv wQ‡jb? D`vniY 2| wcZv I cy‡Îi eZ©gvb eq‡mi AbycvZ 7 : 2 Ges 5 eQi c‡i Zv‡`i eq‡mi AbycvZ 8 : 3 n‡e| Zv‡`i eZ©gvb eqm KZ ? mgvavb : g‡b Kwi, wcZvi eZ©gvb eqm a eQi Ges cy‡Îi eZ©gvb eqm b eQi| cÖ‡kœi cÖ_g I wØZxq kZ©vbymv‡i h_vµ‡g cvB, a 7 ...........(i) b 2 a 5 8 ...........(ii ) b 5 3

MwYZ

183

mgxKiY (i) ‡_‡K cvB, a 7b ...........(iii) 2

mgxKiY (ii) †_‡K cvB,
3(a 5) 8(b 5) ev , 3a 15 8b 40 ev , 3a 8b

25

ev , 3 ev ,

7b 8b 25 [(iii) e¨envi K‡i] 2 25

21b 16b 2
50

ev , 5b

b 10 mgxKiY (iii) G b 10 ewm‡q cvB, a

35

wcZvi eZ©gvb eqm 35 eQi Ges cy‡Îi eZ©gvb eqm 10 eQi| D`vniY 3| hw` a : b a b b : c nq, Z‡e cÖgvY Ki †h, b c
2

a2 b2 b2 c2

.

mgvavb : †`Iqv Av‡Q, a : b b2 b:c ac a b GLb, b c

2

( a b) 2 (b c) 2 a2 b2 a2 2ab b 2 2bc c 2 2ab ac a c

Ges

a2 b2 b2 c2

a 2 ac ac c 2 a (a c) c(a c) a c

ac 2bc c 2 a(a 2b c) c(a 2b c) a b b c
2

a2 b2 b2 c2

184

MwYZ

c a 2 b 2 ac bd n‡j, †`LvI †h, . 2 2 d ac bd a b a c mgvavb : g‡b Kwi, a bk Ges c dk k; b d

D`vniY 4|

a b

GLb,

a2

b2

(bk ) 2

b2

b 2 (k 2 1) b 2 (k 2 1)

k2 1 k2 1

a2 b2

(bk ) 2 b 2

ac bd Ges ac bd a2 a2 b2 b2

bk dk bd bk dk bd ac ac bd . bd

bd (k 2 1) bd (k 2 1)

k2 1 k2 1

D`vniY 5| mgvavb Ki : mgvavb : †`Iqv Av‡Q,

1 ax 1 bx 1 ax 1 bx

1, 0 b 2a 2b. 1

1 ax 1 bx 1 ax 1 bx 1 ax 1 ax

1 bx 1 bx

1 bx ev , 1 bx
1 bx ev , 1 bx

(1 ax) 2 (1 ax) 2

[ Dfq c¶‡K eM© K‡i ]

1 2ax a 2 x 2 1 2ax a 2 x 2
1 2ax a 2 x 2 1 2ax a 2 x 2 1 2ax a 2 x 2 1 2ax a 2 x 2

1 bx 1 bx ev , 1 bx 1 bx

[ †hvRb-we‡qvRb K‡i ]

2 ev , 2bx ev ,

2(1 a 2 x 2 ) 4ax
1 a2 x2 2ax

1 bx

ev , 2ax bx(1 a 2 x 2 ) ev , x{2a b(1 a 2 x 2 )} 0 nq x

0 A_ev 2a b(a a 2 x 2 ) 0

MwYZ

185

ev , b(1 a 2 x 2 ) ev , 1 a 2 x 2 ev , a 2 x 2 ev , x 2 x 2a b

2a

2a 1 b 1 2a 1 a2 b 1 2a 1 a b

wb‡Y©q mgvavb x 0 , x

1 2a 1. a b

6 x 6 mgvavb : †`Iqv Av‡Q, x 6ab ( a b) x
D`vniY 6| mgvavb Ki : A_©vr, x ev,

1 1 x 3a x 3b n‡j †`LvI †h, a b x 3a x 3b 1 1 a b [ Dfq c¶‡K abx Øviv ¸Y K‡i ]

2, a b.

6ab ( a b)

x 2b 3a a b x 3a 2b x 3a 2b x 3a a 3b ev, x 3a b a x 2a Avevi, 3b a b x 3b 2a a b ev, x 3b 2a a b x 3b 3a b x 3b a b x 3a x 3b GLb, x 3a x 3b a 3b 3a b b a b a x 3a x 3b x 3a x 3b dg©v-24, MwYZ-9g-10g

a b a b

[ †hvRb-we‡qvRb K‡i ]

[ †hvRb-we‡qvRb K‡i ]

a 3b 3a b b a a b a 3b 3a b 2(b a) b a b a 2.

2.

186

MwYZ

2p 1 x 1 0. p n‡j, cÖgvY Ki †h, p 2 1 x x 1 x 1 x mgvavb : †`Iqv Av‡Q, p 1 x 1 x 1 x 1 x 1 x 1 x p 1 [ †hvRb-we‡qvRb K‡i ] 1 x 1 x 1 x 1 x p 1
D`vniY 7|
1 x 1 x ev ,

2 1 x 2 1 x

p 1 p 1 p2 p
2 2

ev ,

1 x 1 x

p 1 p 1
[ Dfq c¶‡K eM© K‡i ]

ev ,

1 x 1 x

( p 1) 2 ( p 1)

2p 1 2p 1

ev ,

1 x 1 x 1 x 1 x

p2 p2

2 p 1 p2 2 p 1 2 p 1 p2 2p 1

[ †hvRb-we‡qvRb K‡i ]

p2 1 2p ev , p 2 1 x 2p 2p 1 0. p2 x a 3 b3 a(a b) n‡j cÖgvY Ki †h, a, b, c µwgK mgvbycvZx| D`vniY 8| a b c a3 b3 a ( a b) mgvavb : †`Iqv Av‡Q, a b c ev ,

1 x

ev , ev ,

a3 b3 a b c

a ( a b) a ( a b)

(a b)(a 2 ab b 2 ) a b c

ev ,

a 2 ab b 2 a b c

a

[Dfqc¶‡K (a + b) Øviv fvM K‡i]

ev , a 2 ab b 2 b 2 ac

a 2 ab ac

a, b, c µwgK mgvbycvZx| a b c d D`vniY 9| hw` b c d a a b mgvavb : †`Iqv Av‡Q, b c a b c d 1 ev , b c d a

nq, Z‡e cÖgvY Ki †h, c

a A_ev a b c d

0.

c d d a 1
[Dfqc¶ †_‡K 1 we‡qvM K‡i]

MwYZ

187

ev,

a b b c c d d a b c d a a c c a ev, b c d a ev, a c b c

a c d a
1 b c d

0
1 a 0

ev , (a c)

ev , (a c)

d a b c b c d a a b c)

0
0

ev , (a c)(d

nq a c 0 A_©vr a c
A_ev , a b c d 0.

D`vniY 10| hw`

x y z z

y x x

z y

Ges x, y, z mK‡j ci¯úi mgvb bv nq, Z‡e cÖgvY Ki

†h, cÖwZwU Abycv‡Zi gvb mgvavb : g‡b Kwi,

1 A_ev

1 Gi mgvb n‡e| 2

x y z

y z x

z x y

k

x k ( y z )................(i)

y k ( z x)...................(ii ) z k(x y )...................(iii)

mgxKiY (i ) †_‡K (ii ) we‡qvM K‡i cvB, x y k ( y x) ev, k ( y x)

y x

k

1

Avevi, mgxKiY (i ) , (ii ) I (iii ) †hvM K‡i cvB, x k y z 1 (x 2 (x k( y z z x x y) 2k ( x y z)

y z) y z)

1 2

cÖwZwU Abycv‡Zi gvb

1 A_ev

1 . 2

188

MwYZ

D`vniY 11| hw` ax mgvavb : g‡b Kwi, ax by

by cz k

cz nq, Z‡e †`LvI †h,

x2 yz

y2 zx

z2 xy

bc a2

ca b2

ab c2

.

x

k , y a

k , z b

k c

GLb,

x2 yz x2 yz

y2 zx y2 zx

z2 xy z2 xy

k2 a2 bc a2

bc k2 ca b2

k2 b2

ca k2 ab . 2 c

k2 c2

ab k2

bc a2

ca b2

ab c2

A_©vr,

Abykxjbx 11 1
1| 2| 3| 4| 5| 6| 7| `yBwU eM©‡¶‡Îi evûi ˆ`N©¨ h_vµ‡g a wgUvi Ges b wgUvi n‡j, Zv‡`i †¶Îd‡ji AbycvZ KZ ? GKwU e„Ë‡¶‡Îi †¶Îdj GKwU eM©‡¶‡Îi †¶Îd‡ji mgvb n‡j, Zv‡`i cwimxgvi AbycvZ wbY©q Ki| `yBwU msL¨vi AbycvZ 3 : 4 Ges Zv‡`i j.mv.¸. 180 ; msL¨v `yBwU wbY©q Ki| GKw`b †Zvgv‡`i K¬v‡m †`Lv †Mj Abycw¯’Z I Dcw¯’Z QvÎ msL¨vi AbycvZ 1 : 4 , Abycw¯’Z QvÎ msL¨v‡K †gvU QvÎ msL¨vi kZKivq cÖKvk Ki| GKwU `ªe¨ µq K‡i 28% ¶wZ‡Z weµq Kiv n‡jv| weµqg~j¨ I µqg~‡j¨i AbycvZ wbY©q Ki| wcZv I cy‡Îi eZ©gvb eq‡mi mgwó 70 eQi| Zv‡`i eq‡mi AbycvZ 7 eQi c~‡e© wQj 5 : 2| 5 eQi c‡i Zv‡`i eq‡mi AbycvZ KZ n‡e ? hw` a : b b : c nq, Z‡e cÖgvY Ki †h, a (i ) c

a2 b2 b2 c2 abc(a b c) 3 (ab bc ca) 3
1 1

(ii ) a 2 b 2 c 2

1 a3

1 b3

1 c3

a3 b3 c3
(b c) 2 c

(iii )

1

(iv ) a 2b c
1 (ii ) 3 a a x x

( a b) 2 a

8|

mgvavb Ki : (i )

1 x 1 x

a a

x x

b

(iii )

a x a x x 1 x 1

a2 a
2

x2 x
2

b , 2a b 0 Ges x 0. x
5

(iv )

x 6 x 6

(v )

ax b ax b

ax b ax b

c

MwYZ

189

1 x (vi ) 81 1 x a 9| b

3

1 x 1 x

a2 c n‡j, †`LvI †h, (i ) 2 d a b c c n‡j, †`LvI †h, d

ab b 2 ab b 2

c2 c2

cd cd

d2 d2

ac bd (ii ) ac bd

c2 c2

d2 d2

10|

a b

a 3 b3 b3 c3 (i ) 3 b c3 c3 d 3 (ii ) (a 2 b 2 c 2 )(b 2 c 2 d 2 ) (ab bc cd ) 2
11| x 12| x 13| x 14| a2 4 ab x 2a n‡j, †`LvI †h, a b x 2a
3 3

x 2b x 2b

2, a b. 3mx 2 3x m 0

m 1 m 1
2a 2a 3b 3b

3 3

m 1 n‡j, cÖgvY Ki †h, x 3 m 1
2a 2a

3b n‡j, †`LvI †h, 3bx 2 3b

4ax 3b 0.

b2

b2 c2 x y 15| b c c a

(a (a

b)2 n‡j, cÖgvY Ki †h, a, b, c µwgK mgvbycvZx| c)2

z

n‡j, cÖgvY Ki †h,

a b bz cy cx az ay bx 16| n‡j, cÖgvY Ki †h, a b c a b c b c a c a b 17| Ges a b a b b c c a x y z 19| xa yb zc ya zb xc za xb yc 1 . a b c 20| hw` (a b c) p (b c a)q (c a b)r 1 1 1 1 . q r s p cÖwZwU AbycvZ 21| hw` lx 23| hw`

a y z

x x a c z y b

b x z . c

y

x

c y

. z

0 n‡j, cÖgvY Ki †h, a

b

c.

Ges x

y

z

0 n‡j, †`LvI †h,

(a b c) s

nq, Z‡e cÖgvY Ki †h,

my

nz nq, Z‡e †`LvI †h,

x2 yz

y2 zx

z2 xy

mn l2

nl m2

lm . n2

p q

a2 a Ges 2 b b

a q p q nq, Z‡e †`LvI †h, a a q

p q . q

190

MwYZ

11 4 avivevwnK AbycvZ g‡b Ki, iwbi Avq 1000 UvKv, mwbi Avq 1500 UvKv Ges mvwgi Avq 2500 UvKv| GLv‡b, iwbi Avq : mwbi Avq = 1000 : 1500 = 2 : 3; mwbi Avq : mvwgi Avq = 1500 : 2500 = 3 : 5. myZivs iwbi Avq : mwbi Avq : mvwgi Avq = 2 : 3 : 5. `yBwU AbycvZ hw` K : L Ges L : M AvKv‡ii nq, Zvn‡j Zv‡`i‡K mvaviYZ K : L : M AvKv‡i †jLv hvq| G‡K avivevwnK AbycvZ ejv nq| †h†Kv‡bv `yBwU ev Z‡ZvwaK AbycvZ‡K GB AvKv‡i cÖKvk Kiv hvq| GLv‡b j¶Yxq †h, `yBwU AbycvZ‡K K : L : M AvKv‡i cÖKvk Ki‡Z n‡j cÖ_g AbycvZwUi DËi ivwk, wØZxq AbycvZwUi c~e© ivwki mgvb n‡Z n‡e| †hgb, 2 : 3 Ges 4 : 3 AbycvZ `yBwU K : L : M AvKv‡i cÖKvk Ki‡Z n‡j cÖ_g AbycvZwUi DËi ivwkwU‡K wØZxq AbycvZwUi c~e© ivwki mgvb Ki‡Z n‡e| A_©vr H `yBwU ivwk‡K Zv‡`i j.mv.¸. Gi mgvb Ki‡Z n‡e| GLb, 2 : 3
2 3 2 4 3 4 8 12 8 : 12

Avevi, 4 : 3

4 3

4 3 3 3

12 12 : 9 9

AZGe 2 : 3 Ges 4 : 3 AbycvZ `yBwU K : L : M AvKv‡i n‡e 8 : 12 : 9. j¶ Kwi †h, Dc‡ii D`vni‡Y mvwgi Avq hw` 1125 UvKv nq, Zvn‡j Zv‡`i Av‡qi AbycvZI 8 : 12 : 9 AvKv‡i †jLv hv‡e| D`vniY 12| K, L I M GK RvZxq ivwk Ges K : L = 3 : 4, L : M = 6 : 7 n‡j, K : L : M KZ ? mgvavb:
K L 3 4 3 3 4 3 9 L Ges 12 M 6 7 6 2 7 2 12 14

[ GLv‡b 4 I 6 Gi j. mv. ¸. 12 ]

K : L : M = 9 : 12 : 14. D`vniY 13| GKwU wÎfy‡Ri wZbwU †Kv‡Yi AbycvZ 3 : 4 : 5; †KvY wZbwU wWwMÖ‡Z cÖKvk Ki| mgvavb : wÎfy‡Ri wZb †Kv‡Yi mgwó = 180º g‡b Kwi, cÖ`Ë AbycvZ Abymv‡i †KvY wZbwU h_vµ‡g 3x, 4x Ges 5x. cÖkœvbymv‡i, 3x + 4x + 5x = 180º ev, 12x = 180º ev, x = 15º AZGe, †KvY wZbwU n‡jv 3x = 3 × 15º = 45º 4x = 4 × 15º = 60º Ges 5x = 5 × 15º = 75º D`vniY 14| hw` †Kv‡bv eM©‡¶‡Îi cÖ‡Z¨K evûi cwigvY 10% e„w× cvq, Z‡e Zvi †¶Îdj kZKiv KZ e„w× cv‡e ? mgvavb : g‡b Kwi, eM©‡¶‡Îi cÖ‡Z¨K evûi ˆ`N©¨ a wgUvi| eM©‡¶ÎwUi †¶Îdj a 2 eM©wgUvi|

10% e„w× †c‡j cÖ‡Z¨K evûi ˆ`N©¨ nq ( a a Gi 10%) wgUvi ev 1 10a wgUvi|
G‡¶‡Î, eM©‡¶ÎwUi †¶Îdj (1 10 a ) 2 eM©wgUvi ev 1 21a 2 eM©wgUvi

MwYZ

191

†¶Îdj e„w× cvq (1 21a 2 a 2 ) 0.21a 2 eM©wgUvi †¶Îdj kZKiv e„w× cv‡e

0 21a 2 100% 21% a2

KvR 1| †Zvgv‡`i †kÖwY‡Z 35 Rb QvÎ I 25 Rb QvÎx Av‡Q| eb‡fvR‡b wLPzwi LvIqvi Rb¨ cÖ‡Z¨K QvÎ I QvÎxi cÖ`Ë
Pvj I Wv‡ji AbycvZ h_vµ‡g 3 : 1 Ges 5 : 2 n‡j, ‡gvU Pvj I †gvU Wv‡ji AbycvZ †ei Ki|

11 5 mgvbycvwZK fvM
†Kv‡bv ivwk‡K wbw`©ó Abycv‡Z fvM Kiv‡K mgvbycvwZK fvM ejv nq| S †K a : b : c : d Abymv‡i fvM Ki‡Z n‡j, S †K †gvU (a + b + c +d) fvM K‡i h_vµ‡g a, b, c I d fvM wb‡Z nq| AZGe a Sa 1g Ask S Gi a b c d a b c d b Sb 2q Ask S Gi a b c d a b c d c Sc 3q Ask S Gi a b c d a b c d d Sd 4_© Ask S Gi a b c d a b c d Gfv‡e †h‡Kv‡bv ivwk‡K †h‡Kv‡bv wbw`©ó Abycv‡Z fvM Kiv hvq| D`vniY 15| wZb e¨w³i g‡a¨ 5100 UvKv Giƒ‡c fvM K‡i `vI †hb, 1g e¨w³i Ask : 2q e¨w³i Ask : 1 1 1 3q e¨w³i Ask : : nq| 2 3 9 1 1 1 1 1 1 [ 2, 3 I 9 Gi j.mv.¸. 18] mgvavb : GLv‡b : : 18 : 18 : 18 2 3 9 2 3 9 9:6:2 Abycv‡Zi ivwk¸‡jvi †hvMdj = 9 + 6 + 2 = 17. 9 1g e¨w³i Ask 5100 UvKv = 2700 UvKv 17 6 UvKv = 1800 UvKv 2q e¨w³i Ask 5100 17 2 3q e¨w³i Ask 5100 UvKv = 600 UvKv 17 AZGe wZb e¨w³ h_vK&ª‡g 2700 UvKv , 1800 UvKv Ges 600 UvKv cv‡eb|

192

MwYZ

Abykxjbx 11.2
1| a, b, c µwgK mgvbycvZx n‡j wb‡Pi †KvbwU mwVK? L. b2 = ac K. a2 = bc
M. ab = bc N. a = b=c

2| Avwid I AvwK‡ei eq‡mi AbycvZ 5:3; Avwi‡di eqm 20 eQi n‡j, KZ eQi ci Zv‡`i eq‡mi AbycvZ 7:5 n‡e? K. 5 eQi L. 6 eQi M. 8 eQi N. 10 eQi 3| wb‡Pi Z_¨¸‡jv j¶¨ Ki: i mgvbycv‡Zi PviwU ivwkB GKRvZxq nIqvi cÖ‡qvRb nq bv| ii `yBwU wÎfyR †¶‡Îi †¶Îd‡ji AbycvZ Zv‡`i f~wgØ‡qi Abycv‡Zi mgvb| iii a c a c e e g = = = n‡j G‡`i cÖwZwU Abycv‡Zi gvb b d h b b f f

g h

Dc‡ii Z_¨¸‡jvi wfwË‡Z wb‡Pi †KvbwU mwVK ? K. i I ii L. ii I iii M. i I iii N. i, ii I iii
ABC Gi †KvY¸‡jvi AbycvZ 2:3:5 Ges ABCD PZzf©~‡Ri †KvY PviwUi AbycvZ 3:4:5:6 Dc‡ii Z‡_¨i wfwË‡Z 4 I 5 bs cÖ‡kœi DËi `vI| 4| GKwU e‡M©i evûi ˆ`N©¨ wØ¸Y n‡j Dnvi †¶Îdj KZ¸Y e„w× cv‡e| K. 2 ¸Y L. 4 ¸Y M. 8 ¸Y N. 6 ¸Y 5| x : y = 7 : 5, y : z = 5 : 7 n‡j x : z = KZ ? K. 35 : 49 L. 35 : 35 M. 25 : 49 N. 49 : 25 6| GKwU Kv‡Vi cyj ˆZwii cÖv°wjZ e¨q 90,000 UvKv | wKš‘ LiP †ewk n‡q‡Q 21,600 UvKv| LiP kZKiv KZ e„w× †c‡q‡Q ?

7| 8| 9|

av‡b Pvj I Zz‡li AbycvZ 7 : 3 n‡j, G‡Z kZKiv Kx cwigvY Pvj Av‡Q ? 1 Nb †m. wg. Kv‡Vi IRb 7 †WwmMÖvg| Kv‡Vi IRb mgAvqZb cvwbi IR‡bi kZKiv KZ fvM ? K, L ,M ,N Gi g‡a¨ 300 UvKv Ggbfv‡e fvM K‡i `vI †hb, K Gi Ask : L Gi Ask = 2 : 3, L Gi Ask : M Gi Ask = 1 :2 Ges M Gi Ask : N Gi Ask = 3 : 2 nq|
2 4 5 n‡j, †K KqwU gvQ †cj? , Ges 3 5 6

10| wZbRb †R‡j 690 wU gvQ a‡i‡Q| Zv‡`i As‡ki AbycvZ

11| GKwU wÎfy‡Ri cwimxgv 45 †m. wg.| evû¸‡jvi ‰`‡N¨©i AbycvZ 3: 5: 7 n‡j , cÖ‡Z¨K evûi cwigvY wbY©q Ki|

MwYZ

193

12| 1011 UvKv‡K

3 4 6 Abycv‡Z wef³ Ki| : : 4 5 7

13| `yBwU msL¨vi AbycvZ 5 : 7 Ges Zv‡`i M. mv. ¸. 4 n‡j, msL¨v `yBwUi j. mv. ¸. KZ ? 14| wµ‡KU †Ljvq mvwKe, gykwdKzi I gvkivdx 171 ivb Ki‡jv| mvwKe I gykwdKz‡ii Ges gykwdKzi I gvkivdxi iv‡bi AbycvZ 3 : 2 n‡j †K KZ ivb K‡i‡Q ? 15| GKwU Awd‡m 2 Rb Kg©KZ©v, 7 Rb KiwYK Ges 3 Rb wcIb Av‡Q| GKRb wcIb 1 UvKv †c‡j GKRb KiwYK cvq 2 UvKv, GKRb Kg©KZ©v cvq 4 UvKv| Zv‡`i mK‡ji †gvU †eZb 150,000 UvKv n‡j, †K KZ †eZb cvq ? 16| GKwU mwgwZi ‡bZv wbe©vP‡b `yBRb cÖwZØ›`xi g‡a¨ ‡Wvbvì mv‡ne 4 : 3 †fv‡U Rqjvf Ki‡jb| hw` †gvU m`m¨ msL¨v 581 nq Ges 91 Rb m`m¨ †fvU bv w`‡q _v‡Kb, Z‡e †Wvbvì mv‡n‡ei cÖwZØ›`x KZ †fv‡Ui e¨eav‡b civwRZ n‡q‡Qb ? 17| hw` †Kv‡bv eM©‡¶‡Îi evûi cwigvY 20% e„w× cvq, Z‡e Zvi †¶Îdj kZKiv KZ e„w× cv‡e ? 18| GKwU AvqZ‡¶‡Îi ˆ`N©¨ 10% e„w× Ges cÖ¯’ 10% nªvm ‡c‡j AvqZ‡¶‡Îi ‡¶Îdj kZKiv KZ e„w× ev nªvm cv‡e ? 19| GKwU gv‡Vi Rwg‡Z †m‡Pi my‡hvM Avmvi Av‡Mi I c‡ii dj‡bi AbycvZ 4 : 7. H gv‡V †h Rwg‡Z Av‡M 304 KzB›Uvj avb dj‡Zv, †mP cvIqvi c‡i Zvi djb KZ n‡e ? 20| avb I avb †_‡K Drcbœ Pv‡ji AbycvZ 3 : 2 Ges Mg I Mg †_‡K Drcbœ mywRi AbycvvZ 4 : 3 n‡j, mgvb cwigv‡Yi avb I Mg †_‡K Drcbœ Pvj I mywRi AbycvZ †ei Ki | 21| GKwU Rwgi †¶Îdj 432 eM©wgUvi| H Rwgi ˆ`N©¨ I cÖ‡¯’i m‡½ Aci GKwU Rwgi ˆ`N©¨ I cÖ‡¯’i AbycvZ h_vµ‡g 3 : 4 Ges 2 : 5 n‡j, Aci Rwgi †¶Îdj KZ ? 22| †Rwg I wmwg GKB e¨vsK †_‡K GKB w`‡b 10% mij gybvdvq Avjv`v Avjv`v cwigvY A_© FY †bq| †Rwg 2 eQi ci gybvdv-Avm‡j hZ UvKv †kva K‡i 3 eQi ci wmwg gybvdv-Avm‡j ZZ UvKv †kva K‡i| Zv‡`i F‡Yi AbycvZ wbY©q Ki| 23| GKwU wÎfy‡Ri evû¸‡jvi AbycvZ 5:12:13 Ges cwimxgv 30 †m.wg. K. wÎfyRwU A¼b Ki Ges †KvY †f‡` wÎfyRwU Kx ai‡bi Zv wjL? L. e„nËi evû‡K ˆ`N¨© Ges ¶z`ªZi evû‡K cÖ¯’ a‡i Aw¼Z AvqZ‡¶‡Îi K‡Y©i mgvb evû wewkóe‡M©i †¶Îdj wbY©q Ki| M. D³ AvqZ‡¶‡Îi ˆ`N©¨ 10% Ges cÖ¯’ 20% e„w× †c‡j †¶Îdj kZKiv KZ e„w× cv‡e? 24| GKw`b †Kv‡bv K¬v‡m Abycw¯’Z I Dcw¯’Z wk¶v_©xi AbycvZ 1:4| K. Abycw¯’Z wk¶v_©x‡`i‡K †gvU wk¶v_©xi kZKivq cÖKvk Ki| L. 10 Rb wk¶v_©x †ewk Dcw¯’Z n‡j Abycw¯’Z I Dcw¯’Z wk¶v_©xi AbycvZ n‡Zv 1:9. †gvU wk¶v_©xi msL¨v KZ ? M. †gvU wk¶v_©xi g‡a¨ QvÎ msL¨v QvÎx msL¨vi wØ¸Y A‡c¶v 20 Rb Kg| QvÎ I QvÎxmsL¨vi AbycvZ wbY©q Ki|

dg©v-25, MwYZ-9g-10g

Øv`k Aa¨vq

`yB PjKwewkó mij mnmgxKiY
(Simple Simultaneous Equations in Two Variables)
MvwYwZK mgm¨v mgvav‡bi Rb¨ exRMwY‡Zi me‡P‡q ¸i“Z¡c~Y© welq n‡jv mgxKiY| lô I mßg †kªwY‡Z Avgiv mij mgxKi‡Yi aviYv †c‡qwQ Ges Kxfv‡e GK PjKwewkó mij mgxKiY mgvavb Ki‡Z nq Zv †R‡bwQ| Aóg †kªwY‡Z mij mgxKiY cÖwZ¯’vcb I Acbqb c×wZ‡Z Ges †jLwP‡Îi mvnv‡h¨ mgvavb K‡iwQ| Kxfv‡e ev¯—ewfwËK mgm¨vi mij mnmgxKiY MVb K‡i mgvavb Kiv nq ZvI wk‡LwQ| G Aa¨v‡q mij mnmgxKi‡Yi aviYv m¤cÖmviY Kiv n‡q‡Q I mgvav‡bi Av‡iv bZzb c×wZ m¤ú‡K© Av‡jvPbv Kiv n‡q‡Q| G QvovI G Aa¨v‡q †jLwP‡Îi mvnv‡h¨ mgvavb I ev¯—ewfwËK mgm¨vi mnmgxKiY MVb I mgvavb m¤ú‡K© we¯—vwiZ Av‡jvPbv Kiv n‡q‡Q| Aa¨vq †k‡l wk¶v_©xiv `yB PjKwewkó mij mnmgxKi‡Yi m½wZ hvPvB Ki‡Z cvi‡e| `yB PjKwewkó `yBwU mgxKi‡Yi ci¯úi wbf©ikxjZv hvPvB Ki‡Z cvi‡e| mgvav‡bi Avo¸Yb c×wZ e¨vL¨v Ki‡Z cvi‡e| ev¯—ewfwËK MvwYwZK mgm¨vi mnmgxKiY MVb K‡i mgvavb Ki‡Z cvi‡e| †jLwP‡Îi mvnv‡h¨ `yB PjKwewkó mij mnmgxKiY mgvavb Ki‡Z cvi‡e|

12 1 mij mnmgxKiY mij mnmgxKiY ej‡Z `yB PjKwewkó `yBwU mij mgxKiY‡K †evSvq hLb Zv‡`i GK‡Î Dc¯’vcb Kiv nq Ges PjK `yBwU GKB ˆewk‡ó¨i nq| Avevi Giƒc `yBwU mgxKiY‡K GK‡Î mij mgxKiY‡RvUI e‡j| Aóg †kªwY‡Z Avgiv Giƒc mgxKiY‡Rv‡Ui mgvavb K‡iwQ I ev¯—ewfwËK mgm¨vi mnmgxKiY MVb K‡i mgvavb Ki‡Z wk‡LwQ| G Aa¨v‡q G m¤ú‡K© Av‡iv we¯—vwiZ Av‡jvPbv Kiv n‡q‡Q| cÖ_‡g Avgiv 2 x y 12 mgxKiYwU we‡ePbv Kwi| GwU GKwU `yB PjKwewkó mij mgxKiY| mgxKiYwU‡Z evgc‡¶ x I y Gi Ggb gvb cvIqv hv‡e wK hv‡`i cÖ_gwU wØ¸‡Yi mv‡_ wØZxqwUi †hvMdj Wvbc‡¶i 12 Gi mgvb nq, A_©vr H gvb `yBwU Øviv mgxKiYwU wm× nq ? GLb, 2 x

y 12 mgxKiYwU †_‡K wb‡Pi QKwU c~iY Kwi :

x Gi gvb 2 0 3 5 ....

y Gi gvb

evgc¶ (2 x
4 16 0 12 6 6 10 2 .....

y ) Gi gvb Wvbc¶
12 12 12 12 12

16 12 6 2 .....

12 12 12 12 12

MwYZ

195

mgxKiYwUi AmsL¨ mgvavb Av‡Q| Zvi g‡a¨ PviwU mgvavb ( 2,16), (0,12), (3,6) I (5,2) | Avevi, Ab¨ GKwU mgxKiY x

y

3 wb‡q wb‡Pi QKwU c~iY Kwi : evgc¶ ( x
0 3 3 0 5 2 .....

x Gi gvb 2 0 3 5 ....

y Gi gvb 5 3 0 2 .....

y ) Gi gvb
3 3 3 3 3

2 5

Wvbc¶ 3

3 3 3 3

mgxKiYwUi AmsL¨ mgvavb Av‡Q| Zvi g‡a¨ PviwU mgvavb : ( 2, 5), (0, 3), (3,0) I (5, 2) hw` Av‡jvP¨ mgxKiY `yBwU‡K GK‡Î †RvU wn‡m‡e aiv nq, Z‡e GKgvÎ (5,2) Øviv Dfq mgxKiY hyMcr wm× nq| Avi Ab¨ †Kv‡bv gvb Øviv Dfq mgxKiY hyMcr wm× n‡e bv| AZGe, mgxKiY‡RvU 2 x

y 12 Ges x

y

3 Gi mgvavb : ( x, y )

(5,2)

KvR : x 2 y 1 0 I 2 x y 3 0 mgxKiYØ‡qi cÖ‡Z¨KwUi cuvPwU K‡i mgvavb †jL †hb Zb¥‡a¨ mvaviY mgvavbwUI _v‡K| 12 2 `yB PjKwewkó mij mnmgxKi‡Yi mgvavb †hvM¨Zv 2 x y 12 (K) c~‡e©i Av‡jvwPZ mgxKiY‡RvU Gi Abb¨ (GKwU gvÎ) mgvavb cvIqv †M‡Q| x y 3

Giƒc mgxKiY‡RvU‡K m½wZc~Y© ev mvgÄm¨c~Y© (Consistent ) ejv nq| mgxKiY `yBwUi x I y Gi mnM Zzjbv K‡i (mn‡Mi AbycvZ wb‡q) cvB,
2 1 1 , mgxKiY‡RvUwUi GKwU mgxKiY‡K Ab¨wUi gva¨‡g 1

cÖKvk Kiv hvq bv| G Rb¨ Giƒc mgxKiY‡K ci¯úi Awbf©ikxj ( Independent ) mgxKiY‡RvU ejv nq| m½wZc~Y© I ci¯úi Awbf©ikxj mgxKiY‡RvUi †¶‡Î AbycvZ¸‡jv mgvb bq| G‡¶‡Î aª“eKc` Zzjbv Kivi cÖ‡qvRb nq bv| (L) GLb Avgiv

2x y 6 mgxKiY‡RvUwU we‡ePbv Kwi| GB `yBwU mgxKiY mgvavb Kiv hv‡e wK ? 4 x 2 y 12

GLv‡b, 1g mgxKiYwUi Dfqc¶‡K 2 Øviv ¸Y Ki‡j 2q mgxKiYwU cvIqv hv‡e| Avevi, 2q mgxKi‡Yi Dfqc¶‡K 2 Øviv fvM Ki‡j 1g mgxKiYwU cvIqv hv‡e| A_©vr, mgxKiY `yBwU ci¯úi wbf©ikxj|

196

MwYZ

Avgiv Rvwb, 1g mgxKiYwUi AmsL¨ mgvavb Av‡Q| Kv‡RB, 2q mgxKiYwUiI H GKB AmsL¨ mgvavb Av‡Q| Giƒc mgxKiY‡RvU‡K m½wZc~Y© I ci¯úi wbf©ikxj (dependent) mgxKiY‡RvU e‡j| Giƒc mgxKiY‡Rv‡Ui AmsL¨ mgvavb Av‡Q| GLv‡b, mgxKiY `yBwUi x I y Gi mnM Ges aª“eK c` Zzjbv K‡i cvB,
2 4 1 2 6 12 1 2

A_©vr, m½wZc~Y© I ci¯úi wbf©ikxj mgxKiY‡Rv‡Ui †¶‡Î AbycvZ¸‡jv mgvb nq| (M) Gev‡i Avgiv

2 x y 12 mgxKiY‡RvUwU mgvavb Kivi †Póv Kwi| 4x 2 y 5

GLv‡b, 1g mgxKiYwUi Dfqc¶‡K 2 Øviv ¸Y K‡i cvB, 4 x 2 y 24 2q mgxKiYwU 4 x 2 y

5

we‡qvM K‡i cvB, 0 19 , hv Am¤¢e| Kv‡RB ej‡Z cvwi, G ai‡bi mgxKiY‡RvU mgvavb Kiv m¤¢e bq| Giƒc mgxKiY‡RvU Am½wZc~Y©
(inconsistent ) I ci¯úi Awbf©ikxj| Giƒc mgxKiY‡Rv‡Ui †Kv‡bv mgvavb †bB|

GLv‡b mgxKiY `yBwUi x I y Gi mnM Ges aª“eK c` Zzjbv K‡i cvB,

2 4

1 2

12 . 5

A_©vr, Am½wZc~Y© I ci¯úi Awbf©ikxj mgxKiY‡Rv‡Ui †¶‡Î Pj‡Ki mn‡Mi AbycvZ¸‡jv aª“e‡Ki Abycv‡Zi mgvb bq| mvaviYfv‡e, a1 x b1 y a2 x b2 y c1 c2

mgxKiY‡RvUwU wb‡q wb‡Pi Q‡Ki gva¨‡g `yBwU mij mgxKi‡Yi mgvavb

†hvM¨Zvi kZ© D‡jL Kiv n‡jv : mgxKiY‡RvU mnM I aªyeK c` Zzjbv
(i )

m½wZc~Y©/Am½wZc~Y© ci¯úi wbf©ikxj/ mgvavb Av‡Q Awbf©ikxj m½wZc~Y© Awbf©ikxj (KqwU)/†bB Av‡Q (GKwUgvÎ) c1 c2 c1 c2

a1 x b1 y a2 x b2 y

c1 c2 c1 c2 c1 c2

a1 a2

b1 b2

(ii )

a1 x b1 y a2 x b2 y

a1 a2 a1 a2

b1 b2 b1 b2

m½wZc~Y©

wbf©ikxj

Av‡Q (AmsL¨)

(iii )

a1 x b1 y a2 x b2 y

Am½wZc~Y©

Awbf©ikxj

†bB

MwYZ

197

GLb, hw` †Kv‡bv mgxKiY‡Rv‡U Dfq mgxKi‡Y aª“eK c` bv _v‡K, A_©vr, c1
(i ) Abyhvqx a1 a2

c2

0 nq, Z‡e Q‡Ki

b1 n‡j, mgxKiY‡RvU me©`v m½wZc~Y© I ci¯úi Awbf©ikxj| †m‡¶‡Î GKwUgvÎ b2

(Abb¨) mgvavb _vK‡e|
(ii ) I (iii ) †_‡K a1 a2 b1 n‡j, mgxKiY‡RvU m½wZc~Y© I ci¯úi wbf©ikxj| †m‡¶‡Î AmsL¨ mgvavb _vK‡e| b2

D`vniY : wb‡Pi mgxKiY‡RvU¸‡jv m½wZc~Y©/Am½wZc~Y©, wbf©ikxj/Awbf©ikxj wK bv e¨vL¨v Ki Ges G‡`i mgvav‡bi msL¨v wb‡`©k Ki| (K) x 3 y 1
2x 6 y 2

(L) 2 x 5 y 3 x 3y 1

(M) 3 x 5 y 7
6 x 106 y 15

mgvavb : (K) cÖ`Ë mgxKiY‡RvU : x 3y 2x 6 y 1 2

x Gi mnMØ‡qi AbycvZ

1 2 3 1 ,, ,, ev y ,, 6 2 1 aª“eK c`Ø‡qi AbycvZ 2 1 3 1 2 6 2

AZGe, mgxKiY‡RvUwU m½wZc~Y© I ci¯úi wbf©ikxj| mgxKiY‡RvUwUi AmsL¨ mgvavb Av‡Q| (L) cÖ`Ë mgxKiY‡RvU :
2x 5 y x 3y
2 1 5 3

3 1

x Gi mnMØ‡qi AbycvZ y ,,

,,
5 3

,,

Avgiv cvB,

2 1

mgxKiY‡RvUwU m½wZc~Y© I ci¯úi Awbf©ikxj| mgxKiY‡RvUwUi GKwUgvÎ (Abb¨) mgvavb Av‡Q| (M) cÖ`Ë mgxKiY‡RvU : 3 x 5 y
7

6 x 10 y 15

198

MwYZ

x Gi mnMØ‡qi AbycvZ

3 1 ev 6 2 5 1 ,, ,, ev y ,, 10 2 7 aª“eK c`Ø‡qi AbycvZ 15 3 6 5 10 7 15

Avgiv cvB,

mgxKiY‡RvUwU Am½wZc~Y© I ci¯úi Awbf©ikxj| mgxKiY‡RvUwUi †Kv‡bv mgvavb †bB| KvR : x 2 y 1 0, 2 x y 3 0 mgxKiY‡RvUwU m½wZc~Y© wK bv, ci¯úi wbf©ikxj wK bv hvPvB Ki Ges mgxKiY‡RvUwUi KqwU mgvavb _vK‡Z cv‡i Zv wb‡`©k Ki|

Abykxjbx 12 1 wb‡Pi mij mnmgxKiY¸‡jv m½wZc~Y©/Am½wZc~Y©, ci¯úi wbf©ikxj/Awbf©ikxj wK bv hyw³mn D‡jL Ki Ges G¸‡jvi mgvav‡bi msL¨v wb‡`©k Ki : 1| x y x 4

2| 2 x

y

3
6
0

3| x y 4 0
3 x 3 y 10 0

y 10
0

4x 2 y

4| 3 x 2 y
6x 4y

5| 3 x 2 y
9x 6 y

6| 5 x 2 y 16 0
3x 0 6 y 5 y
5

0

0

2 1

7|

1 x 2 x 2y

y
2

1

8| x 1 x 2
2y

y
0

9| x 1 x 2 y 10| ax cy 0 cx ay c2 a2 .

12 3 mij mnmgxKi‡Yi mgvavb
Avgiv ïay m½wZc~Y© I ci¯úi Awbf©ikxj mij mnmgxKi‡Yi mgvavb m¤ú‡K© Av‡jvPbv Ki‡ev| Giƒc mgxKiY‡Rv‡Ui GKwUgvÎ (Abb¨) mgvavb Av‡Q| GLv‡b, mgvav‡bi PviwU c×wZi D‡jL Kiv n‡jv :

MwYZ

199

(1) cÖwZ¯’vcb c×wZ

(2) Acbqb c×wZ

(3) Avo¸Yb c×wZ I

(4) ˆjwLK c×wZ|

Avgiv Aóg †kªwY‡Z cÖwZ¯’vcb I Acbqb c×wZ‡Z mgvavb Kxfv‡e Ki‡Z nq †R‡bwQ| G `yB c×wZi GKwU K‡i D`vniY †`Iqv n‡jv : D`vniY 1| cÖwZ¯’vcb c×wZ‡Z mgvavb Ki :
2x y 8
5 2x 3x y 2y 8..............(1) 5............( 2) 3x 2 y

mgvavb : cÖ`Ë mgxKiYØq

mgxKiY (1) n‡Z cvB, y

8 2 x.........(3)

mgxKiY (2) G y Gi gvb 8 2 x ewm‡q cvB,
3 x 2(8 2 x) 5 ev 3 x 16 4 x 5 ev 3 x 4 x 5 16 ev 7 x 21 ev x 3

x Gi gvb mgxKiY (3) G ewm‡q cvB, y 8 2 3 8 6 2

mgvavb ( x, y ) (3,2) cÖwZ¯’vcb c×wZ‡Z mgvavb : myweavgZ GKwU mgxKiY †_‡K GKwU Pj‡Ki gvb Aci Pj‡Ki gva¨‡g cÖKvk K‡i cÖvß gvb Aci mgxKi‡Y emv‡j GK PjKwewkó mgxKiY cvIqv hvq| AZtci mgxKiYwU mgvavb K‡i PjKwUi gvb cvIqv hvq| GB gvb cÖ`Ë mgxKi‡Yi †h †Kv‡bvwU‡Z emv‡bv †h‡Z cv‡i| Z‡e †hLv‡b GKwU PjK‡K Aci Pj‡Ki gva¨‡g cÖKvk Kiv n‡q‡Q †mLv‡b emv‡j mgvavb mnR nq| GLvb †_‡K Aci Pj‡Ki gvb cvIqv hvq| D`vniY 2| Acbqb c×wZ‡Z mgvavb Ki : 2 x y 8
5

3x 2 y

[`ªóe¨ : cÖwZ¯’vcb I Acbqb c×wZi cv_©K¨ †evSv‡ZB D`vniY 1 Gi mgxKiYØqB D`vniY 2 G †bqv n‡jv] mgvavb : cÖ`Ë mgxKiYØq 2 x
3x

y
2y

8..............(1)
5............( 2)

mgxKiY (1) Gi Dfqc¶‡K 2 Øviv ¸Y K‡i, 4 x 2 y 16..............(3) mgxKiY (2) 3 x 2 y 5............( 2) mgxKiY (2) I (3) †hvM K‡i cvB,

200

MwYZ

7 x 21 ev, x 3

x Gi gvb mgxKiY (1) G ewm‡q cvB,
2 3 y 8 ev, y 8 6

ev, y 2

mgvavb ( x, y )

(3,2)

Acbqb c×wZ‡Z mgvavb : myweavgZ GKwU mgxKiY‡K ev Dfq mgxKiY‡K Giƒc msL¨v w`‡q ¸Y Ki‡Z n‡e †hb ¸Y‡bi ci Dfq mgxKi‡Yi †h‡Kv‡bv GKwU Pj‡Ki mn‡Mi ciggvb mgvb nq| Gici cÖ‡qvRbgZ mgxKiY `yBwU‡K †hvM ev we‡qvM Ki‡j mnM mgvbK…Z PjKwU AcbxZ ev AcmvwiZ nq| Zvici mgxKiYwU mgvavb Ki‡j we`¨gvb PjKwUi gvb cvIqv hvq| H gvb myweavgZ cÖ`Ë mgxKiYØ‡qi †h‡Kv‡bvwU‡Z emv‡j Aci PjKwUi gvb cvIqv hvq|

(3) Avo¸Yb c×wZ :
Avo¸Yb c×wZ‡K eRª¸Yb c×wZI e‡j| wb‡Pi mgxKiY `yBwU we‡ePbv Kwi : a1 x b1 y c1 a2 x b2 y c2 0...........(1) 0..........( 2)

mgxKiY (1) †K b2 w`‡q I mgxKiY (2) †K b1 w`‡q ¸Y K‡i cvB, a1b2 x b1b2 y b2 c1 a2b1 x b1b2 y b1c2 0.........(3) 0.........( 4)

mgxKiY (3) †_‡K mgxKiY (4) we‡qvM K‡i cvB,
(a1b2 a2 b1 ) x b2 c1 b1c2 0

ev, (a1b2 a2b1 ) x b1c2 b2 c1 ev, x b1c2 b2 c1 a1b2 1 a2 b1 ..........(5)

Avevi, mgxKiY (1) †K a2 w`‡q I mgxKiY (2) †K a1 w`‡q ¸Y K‡i cvB, a1a2 x a2b1 y c1a2 a1a2 x a1b2 y c2 a1 0.........(6) 0.........(7)

mgxKiY (6) †_‡K mgxKiY (7) we‡qvM K‡i cvB,
( a2 b1 a1b2 ) y c1a2 c2 a1 0

ev, ev,

( a1b2 y

a2 b1 ) y a1b2

(c1a2 c2 a1 ) 1 a2 b1 ..........(8)

c1a2 c2 a1

(5) I (8) †_‡K cvB,

MwYZ

201

x b1c2 b2 c1 c1a2

y c2 a1 a1b2

1 a2b1

x I y Gi Giƒc m¤úK© †_‡K G‡`i gvb wbY©‡qi †KŠkj‡K Avo¸Yb c×wZ e‡j| x I y Gi DwjwLZ m¤úK© †_‡K cvB, x b1c2 b2 c1 y c1a2 c2 a1 a1b2 a1b2 1 a2 b1
, ev x

b1c2 a1b2
, ev y

b2 c1 a2b1 c1a2 a1b2 c2 a1 a2b1 c2 a1 a2b1

Avevi,

1 a2 b1

cÖ`Ë mgxKiYØ‡qi mgvavb : ( x, y ) j¶ Kwi : mgxKiY a1 x a2 x b1 y b2 y c1 c2 0 0 b1c2 x b2 c1

b1c2 a1b2

b2 c1 c1a2 , a2b1 a1b2

x I y Gi g‡a¨ m¤úK© y c1a 2 c2 a1 a1b2 1 a 2 b1 a1 a2

g‡b ivLvi wPÎ x b1 b2 c1 c2

y a1 a2

1 b1 b2

`ªóe¨ : cÖ`Ë Dfq mgxKi‡Yi aª“eK c` Wvbc‡¶ †i‡LI Avo¸Yb c×wZ cÖ‡qvM Kiv hvq| Z‡e †m‡¶‡Î wP‡ýi wKQy cwieZ©b n‡e| wKš‘ mgvavb GKB cvIqv hv‡e|

KvR : 4 x y 7 0
3x y 0 a1 x b1 y c1

mgxKiY‡RvU‡K
0

mgxKiY‡Rv‡Ui AvKv‡i cÖKvk Ki‡j
0

a2 x b2 y c2

a1 , b1 , c1 , a2 , b2 , c2 Gi gvb †ei Ki|

D`vniY 3| Avo¸Yb c×wZ‡Z mgvavb Ki : 6 x y 1
3 x 2 y 13

mgvavb : c¶vš—i cÖwµqvq cª`Ë mgxKiYØ‡qi Wvbc¶ 0 (k~b¨) K‡i cvB, 6x y 1 0 mgxKiYØq‡K h_vµ‡g a1 x b1 y 2 c1
3x 2 y 13 0

0
1

Ges a2 x b2 y c2 0 Gi mv‡_ Zzjbv K‡i cvB, a1 6, b1 1, c1

a2 dg©v-26, MwYZ-9g-10g

3, b2

2, c2

13

202

MwYZ

Avo¸Yb c×wZ‡Z cvB, x b1c2 b2 c1 c1a2 x ( 1) ( 13) 2 ( 1)

e¨vL¨v
1 c2 a1 a1b2 a2b1 y ( 1) 3 ( 13) 6 6 2 1 3 ( 1)

y

x a1 a2 b1 b2 c1 c2

y a1 a2

1 b1 b2

ev ev x 13 2

y 3 78

1 12 3

x
6 3

y

1

1 2

1 13

6 3

1 2

1 x y ev 15 75 15 x 1 15 1 ev x 15 15 15 y 1 75 Avevi, ev y 5 75 15 15 mgvavb ( x, y ) (1,5) D`vniY 4| Avo¸Yb c×wZ‡Z mgvavb Ki : 3 x 4 y 0 2x 3y 1 mgvavb : cÖ`Ë mgxKiYØq 3x 4 y 2x 3y 0 1 ev, 3x 4 y 0 0 2x 3y 1 0

Avo¸Yb c×wZ‡Z cvB, x 4 1 ( 3) 0 y 0 2 1 3 1 3 ( 3) 2 ( 4)
3 2

x
4 0 3 1

y
3 2

1
4 3

ev ev

x 4 0

y 0 3

1 9 8

x 4 x y ev 4 3 x 1 4 1 y Avevi, 3

y 3 1 1 ev, x

1 1

4

1 ev, y 1

3

mgvavb ( x, y )

(4,3)

MwYZ

203

D`vniY 5| Avo¸Yb c×wZ‡Z mgvavb Ki :

x y 8 2 3 5x 3y 4

3

mgvavb : cÖ`Ë mgxKiYØq‡K ax by c 0 AvKv‡i mvwR‡q cvB, x 2 y 3 8 8

Avevi, ev

5x 4

3y 3

3

ev

3x 2 y 6

5 x 12 y 4

ev 3 x 2 y 48 0 mgxKiYØq

ev 5 x 12 y 12 0

3 x 2 y 48 0
5 x 12 y 12 0 x y
48 12 3 5 2 12

Avo¸Yb c×wZ‡Z cvB, x 2 12 ( 12) ( 48) y ( 48) 5 12 3 1 3 ( 12) 5 2
3 5 2 12

1

x y 1 24 576 240 36 36 10 x y 1 ev 552 276 46 x y 1 ev 552 276 46 552 x 1 ev, x 12 552 46 46 276 y 1 Avevi, ev. y 6 276 46 46

ev

mgvavb : ( x, y ) (12,6) mgvav‡bi ïw× cix¶v : cÖvß x I y Gi gvb cÖ`Ë mgxKi‡Y ewm‡q cvB, 1g mgxKi‡Y, evgc¶ =

x 2

y 3

12 2

6 3

6 2

8
2q mgxKi‡Y, evgc¶ =

Wvbc¶

5x 3y 4 15 18

5 12 3 6 4 3 = Wvbc¶|

mgvavb ï× n‡q‡Q|

204

MwYZ

D`vniY 6| Avo¸Yb c×wZ‡Z mgvavb Ki : ax by mgvavb : cÖ`Ë mgxKiYØq, ax by bx ay ab ax by ab 0 0 y ( ab ) b ( ab ) a

ab

bx ay.

ev, ab x ( b ) ( ab ) ( a )( ab )

bx ay ab

1 a ( a ) b ( b) a b b a

x ab ab a b

y 1 b a

ev ev ev

x ab 2 a 2b ab 2 x ab( a b) x ab( a b) x ab( a b)

y a 2b y ab( a b) y ab( a b)

1 a2 b2
1 ( a b)( a b) (a 1 b)( a b) ab( a b) ( a b)( a b) ab a b

1 , ev x ( a b)( a b)

Avevi,

y ab( a b) ( x, y )

1 , ev y ( a b)( a b)

ab( a b) ( a b)( a b)

ab a b

ab ab , a b a b

Abykxjbx 12 2 cÖwZ¯’vcb c×wZ‡Z mgvavb Ki (1 Ñ 3) :

x y 1 2 3 x y 1 3 2 Acbqb c×wZ‡Z mgvavb Ki (4 Ñ 6) : 5| 7 x 8 y 4| 7 x 3 y 31 9
2|
9x 5 y 41
5x 4 y 3

1| 7 x 3 y 31 9 x 5 y 41

3|

x y 2 a b ax by a 2

b2

6| ax by

c
2

a x b y
0

2

c2

Avo¸Yb c`awZ‡Z mgvavb Ki (7 Ñ 15) : 7| 2 x 3 y 5 0 8| 3 x 5 y 9

9| x 2 y

7

4x 7 y 6 0

5x 3 y 1 0

2x 3y

0

MwYZ

205

10| 4 x 3 y
2x 5 a2 12

11|

7x 8y
5x 4 y

9
3
x(6
5( y 1)

12| 3 x

y 7

0

2x

y 3

13| ax by
2bx ay

b 2 14| y(3
3(3

x)
x)

y)

ab

15| ( x 7)( y 3) 7
5 x 11 y 35 0

( y 3)( x 1) 5

12 4 ˆjwLK c×wZ‡Z mgvavb
`yB PjKwewkó GKwU mij mgxKi‡Y we`¨gvb PjK x I y Gi m¤úK©‡K wP‡Îi mvnv‡h¨ cÖKvk Kiv hvq| GB wPÎ‡K H m¤ú‡K©i †jLwPÎ e‡j| G RvZxq mgxKi‡Yi †jLwP‡Î AmsL¨ we›`y _v‡K| Giƒc K‡qKwU we›`y ¯’vcb K‡i G‡`i ci¯úi mshy³ Ki‡jB †jLwPÎ cvIqv hvq| mij mnmgxKi‡Yi cÖ‡Z¨KwUi AmsL¨ mgvavb i‡q‡Q| cÖ‡Z¨KwU mgxKi‡Yi †jL GKwU mij‡iLv| mij‡iLvwUi cÖ‡Z¨KwU we›`yi ¯’vbvsK mgxKiYwU‡K wm× K‡i| †Kv‡bv †jL wbw`©ó Ki‡Z `yB ev Z‡ZvwaK we›`y †bqv Avek¨K| GLb Avgiv wb‡Pi mgxKiY‡RvUwU mgvavb Kivi †Póv Kie : 2 x y 3..........(1)
4x 2 y 6........(2)

mgxKiY (1) †_‡K cvB, y 3 2 x . mgxKiYwU‡Z x Gi K‡qKwU gvb wb‡q y Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y ( 1, 5), (0, 3) I (3, 3) | Avevi, mgxKiY (2) †_‡K cvB, 2 y 6 4 x ev, y
6 4x 2

x y

1
5

0
3

3
3

x y

2 7

0
3

6 9

mgxKiYwU‡Z x Gi K‡qKwU gvb wb‡q y Gi Abyiƒ‡c gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y ( 2, 7), (0, 3) I (6, 9) | g‡b Kwi, QK KvM‡R XOX I YOY h_vµ‡g x -A¶ I y -A¶ Ges
O g~jwe›`y|

QK KvM‡Ri Dfq A¶ eivei ¶z`ªZg eM©‡¶‡Îi cÖwZevûi ˆ`N©¨‡K GKK awi| GLb mgxKiY (1) n‡Z cÖvß ( 1, 5), (0, 3) I (3, 3) we›`yMy‡jv ¯’vcb Kwi I Zv‡`i ci¯úi mshy³ Kwi| †jLwU GKwU mij‡iLv|

206

MwYZ

Avevi, mgxKiY (2) n‡Z cÖvß ( 2, 7), (0, 3) I (6, 9) we›`y¸‡jv ¯’vcb Kwi I Zuv‡`i ci¯úi mshy³ Kwi| G‡¶‡ÎI †jLwU GKwU mij‡iLv| Z‡e j¶ Kwi, mij‡iLv `yBwU ci¯ú‡ii Dci mgvcwZZ n‡q GKwU mij‡iLvq cwiYZ n‡q‡Q| Avevi, mgxKiY (2) Gi Dfqc¶‡K 2 Øviv fvM Ki‡j mgxKiY (1) cvIqv hvq| G Kvi‡Y mgxKiYØ‡qi †jL ci¯úi mgvcwZZ n‡q‡Q| GLv‡b,
2 x y 3..........(1) 4 x 2 y 6........(2)

mgxKiY‡RvUwU m½wZc~Y© I ci¯úi wbf©ikxj| Giƒc mgxKiY‡Rv‡Ui

AmsL¨v mgvavb Av‡Q Ges mgxKiY‡RvUwUi †jL GKwU mij‡iLv| Gevi Avgiv wb‡Pi mgxKiY‡RvUwU mgvavb Kivi †Póv Kie : 2 x y 4..........(1)
4 x 2 y 12......(2)

mgxKiY (1) †_‡K cvB, y 2 x 4 . mgxKiYwU‡Z x Gi K‡qKwU gvb wb‡q y Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y ( 1, 6), (0, 4), (4, 4) | Avevi, mgxKiY (2) †_‡K cvB,
4 x 2 y 12 , ev 2 x y 6 [Dfqc¶‡K 2 Øviv fvM K‡i]

x y

1
6

0
4

4 4

ev y

2x 6

mgxKiYwU‡Z x Gi K‡qKwU gvb wb‡q y Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y (0, 6), (3, 0), (6, 6) |

x y

0

6

3 0

6 6

g‡b Kwi, QK KvM‡R XOX I YOY h_vµ‡g x -A¶ I y -A¶ Ges O g~jwe›`y| QK KvM‡Ri Dfq A¶ eivei ¶z`ªZg eM©‡¶‡Îi cÖwZevûi ˆ`N©¨‡K GKK a‡i mgxKiY (1) n‡Z cÖvß ( 1, 6), (0, 4) I (4, 4) we›`yMy‡jv ¯’vcb Kwi I Zv‡`i ci¯úi mshy³ Kwi| †jLwU GKwU mij‡iLv| Avevi, mgxKiY (2) n‡Z cÖvß (0, 6), (3, 0), (6, 6) we›`y¸‡jv ¯’vcb Kwi I G‡`i ci¯úi mshy³ Kwi| G‡¶‡ÎI †jLwU GKwU mij‡iLv|

MwYZ

207

wP‡Î j¶ Kwi, cÖ`Ë mgxKiYØ‡qi c„_Kfv‡e cÖ‡Z¨KwUi AmsL¨ mgvavb _vK‡jI †RvU wn‡m‡e Zv‡`i mvaviY mgvavb †bB| AviI j¶ Kwi †h, cÖ`Ë mgxKiY `yBwUi †jLwPÎ `yBwU ci¯úi mgvš—ivj mij‡iLv| A_©vr, †iLv `yBwU KL‡bv G‡K Aci‡K †Q` Ki‡e bv| AZGe, G‡`i †Kv‡bv mvaviY †Q` we›`y cvIqv hv‡e bv| G †¶‡Î Avgiv ewj †h, Giƒc mgxKiY‡Rv‡Ui †Kv‡bv mgvavb †bB| Avgiv Rvwb, Giƒc mgxKiY‡RvU Am½wZc~Y© I ci¯úi Awbf©ikxj| Avgiv GLb †jLwP‡Îi mvnv‡h¨ m½wZc~Y© I ci¯úi Awbf©ikxj mgxKiY‡RvU mgvavb Ki‡ev| `yB PjKwewkó `yBwU m½wZc~Y© I ci¯úi Awbf©ikxj mij mgxKi‡Yi †jL GKwU we›`y‡Z †Q` K‡i| H †Q` we›`yi ¯’vbvsK Øviv Dfq mgxKiY wm× n‡e| †Q`we›`ywUi ¯’vbvsKB n‡e mgxKiYØ‡qi mgvavb| D`vniY 7| mgvavb Ki I mgvavb †jLwP‡Î †`LvI : 2 x mgvavb : : cÖ`Ë mgxKiYØq 2 x
3x
y 8 0...........(1)

y

8

3x 2 y
2y 5 y ( 8) 3 ( 5) 2

5

0.............( 2)
1 2( 2) 3 1

Avo¸Yb c×wZ‡Z cvB, x 1 ( 5) ( 2) ( 8)

ev ev

x 5 16 y 14 1 7

y 24 10
1 7

1 4 3

x 21 x y ev 21 14 x 1 , 21 7 y Avevi, 14

ev x

21 7

3 14 7 2

1 , ev y 7

mgvavb : ( x, y ) (3,2) g‡b Kwi, XOX I YOY h_vµ‡g x -A¶ I y -A¶ Ges O g~jwe›`y| QK KvM‡Ri Dfq A¶ eivei ¶z`ªZg e‡M©i cÖwZ `yB evûi ˆ`N©¨‡K GKK a‡i (3,2) we›`ywU ¯’vcb Kwi| D`vniY 8| †jLwP‡Îi mvnv‡h¨ mgvavb Ki :
3x 5x y y 3 21

mgvavb : cÖ`Ë mgxKiYØq 3 x

y y

3..........(1) 21........(2)

5x

208

MwYZ

mgxKiY (1) †_‡K cvB, 3 x

y

3 , ev y

3x 3 x y

mgxKiYwU‡Z x Gi K‡qKwU gvb wb‡q y Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y ( 1, 6), (0, 3), (3,6) Avevi, mgxKiY (2) †_‡K cvB, 5 x y 21 , ev y
21 5 x

1 6

0 3

3 6

mgxKiYwU‡Z x Gi K‡qKwU gvb wb‡q y Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y (3,6), (4,1), (5, 4) |

x y

3 6

4 1

5

4

g‡b Kwi, XOX I YOY h_vµ‡g x -A¶ I y -A¶ Ges O g~jwe›`y| QK KvM‡Ri Dfq A¶ eivei ¶z`ªZg e‡M©i cÖwZ evûi ˆ`N©¨‡K GKK awi| GLb QK KvM‡R mgxKiY (1) n‡Z cÖvß ( 1, 6), (0, 3), (3,6) we›`yMy‡jv ¯’vcb Kwi I Zv‡`i ci¯úi mshy³ Kwi| †jLwU GKwU mij‡iLv| GKBfv‡e, mgxKiY (2) n‡Z cÖvß (3,6), (4,1), (5, 4) we›`y¸‡jv ¯’vcb Kwi I Zv‡`i ci¯úi mshy³ Kwi| G‡¶‡ÎI †jLwU GKwU mij‡iLv| g‡b Kwi, mij‡iLvØq ci¯úi P we›`y‡Z †Q` K‡i‡Q| wPÎ †_‡K †`Lv hvq, P we›`yi ¯’vbvsK (3,6) mgvavb : ( x, y ) (3,6) D`vniY 9| ˆjwLK c×wZ‡Z mgvavb Ki : mgvavb : : cÖ`Ë mgxKiYØq 2 x 5 y

2x 5 y 14 4 x 5 y 17

14...........(1) 4 x 5 y 17........(2)

mgxKiY (1) †_‡K cvB, 5 y

14 2 x , ev y

2 x 14 5

x y 3

mgxKiYwU‡Z x Gi myweavgZ K‡qKwU gvb wb‡q y Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y (3, 4),
1 , 3), ( 2, 2) | 2

4

1 2 3

2 2

x y 3
1

1 2 3

2
5

MwYZ

209

Avevi, mgxKiY (2) †_‡K cvB, 5 y 4 x 17 , ev y

4 x 17 5

mgxKiYwU‡Z x Gi myweavgZ K‡qKwU gvb wb‡q y Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi :
1 , 3 , ( 2, 5) 2 g‡b Kwi, XOX I YOY h_vµ‡g x -A¶ I y -A¶ Ges O g~jwe›`y|

mgxKiYwUi †j‡Li Dci wZbwU we›`y (3, 1),

QK KvM‡Ri Dfq A¶ eivei ¶y`ªZg e‡M©i cÖwZ `yB evûi ˆ`N©¨‡K GKK awi| GLb, QK KvM‡R mgxKiY (1) †_‡K cÖvß (3, 4),
1 , 3 I ( 2, 2) 2

we›`y¸‡jv ¯’vcb K‡i Zv‡`i cici mshy³ Kwi| †jLwU GKwU mij‡iLv| GKBfv‡e, mgxKiY (2) †_‡K cÖvß (3, 1), Kwi| †jLwU GKwU mij‡iLv| g‡b Kwi, mij‡iLvØq ci¯úi P we›`y‡Z †Q` K‡i‡Q| wP‡Î †`Lv hvq, P we›`yi ¯’vbvsK mgvavb : ( x, y )
1 , 3 2
3 x 2 8 4x
1 , 3 2

1 , 3 , ( 2, 5) we›`y¸‡jv ¯’vcb K‡i Zv‡`i cici mshy³ 2

D`vniY 10| †j‡Li mvnv‡h¨ mgvavb Ki : 3 mgvavb : cÖ`Ë mgxKiY 3 awi, y y 3 3 x 8 4x 2 3 3 x..........(1) 2 8 4 x..........( 2) 3 x 2 8 4x

Ges y

GLb, mgxKiY (1) G x Gi K‡qKwU gvb wb‡q y Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y ( 2,6), (0,3), (2,0) Avevi, mgxKiY (2) G x -Gi K‡qKwU gvb wb‡q y -Gi Abyiƒc gvb †ei Kwi I cv‡ki QKwU ˆZwi Kwi : mgxKiYwUi †j‡Li Dci wZbwU we›`y (1,4), (2,0), (3, 4) x y

x y

2
6

0 3

2 0

1 4

2 0

3

4

g‡b Kwi, XOX I YOY h_vµ‡g x -A¶ I y -A¶ Ges O g~jwe›`y| QK KvM‡Ri Dfq A¶ eivei ¶z`ªZg e‡M©i cÖwZ evûi ˆ`N©¨‡K GKK awi| dg©v-27, MwYZ-9g-10g

210

MwYZ

GLb, QK KvM‡R mgxKiY (1) †_‡K cÖvß ( 2,6), (0,3), ( 2,0) we›`y¸‡jv ¯’vcb Kwi I we›`y¸‡jv cici mshy³ Kwi| Zvn‡j, †jLwU n‡e GKwU mij‡iLv| GKBfv‡e, mgxKiY (2) †_‡K cÖvß (1,4), ( 2,0), (3, 4) we›`y¸‡jv ¯’vcb K‡i G¸‡jv cici mshy³ Kwi| Zvn‡j, †jLwU n‡e GKwU mij‡iLv| g‡b Kwi, mij‡iLvØq ci¯úi P we›`y‡Z †Q` K‡i| wP‡Î †`Lv hvq, †Q`we›`ywUi ¯’vbvsK (2,0) | mgvavb : x
KvR : 2 x y 2 , ev mgvavb : 2
3 0 mgxKi‡Yi †j‡Li Dci Q‡Ki gva¨‡g PviwU we›`y wbY©q Ki| AZ:ci QK KvM‡R

wbw`©ó ˆ`‡N©¨i GKK wb‡q we›`y¸‡jv ¯’vcb Ki I Zv‡`i ci¯úi mshy³ Ki| †jLwU wK mij‡iLv n‡q‡Q ?

Abykxjbx 12 3
†jLwP‡Îi mvnv‡h¨ mgvavb Ki : 1| 3 x 4 y 14 4x 3y 2 4| 3 x 2 y
2

2| 2 x y 1 5 x y 13 5| x y 2 2 3 2 x 3 y 13 y 3 3 3

3| 2 x 5 y 1 x 3y 2 6| 3 x

y

6

5x 3 y
7| 3 x 2 y

5

5 x 3 y 12
9| 3 x 2

4

8|

3x 4 y 1

x 2 y x 6

x 2

10| 3 x 7 3 2 x

12 5 ev¯—ewfwËK mgm¨vi mnmgxKiY MVb I mgvavb
ˆ`bw›`b Rxe‡b Ggb wKQy MvwYwZK mgm¨v Av‡Q hv mgxKiY MV‡bi gva¨‡g mgvavb Kiv mnRZi nq| G Rb¨ mgm¨vi kZ© ev kZ©vewj †_‡K `yBwU AÁvZ ivwki Rb¨ `yBwU MvwYwZK cÖZxK, cÖavbZ PjK x, y aiv nq| AÁvZ ivwk `yBwUi gvb wbY©‡qi Rb¨ `yBwU mgxKiY MVb Ki‡Z nq| MwVZ mgxKiYØq mgvavb Ki‡jB AÁvZ ivwk `yBwUi gvb cvIqv hvq| D`vniY 11| `yB A¼wewkó †Kv‡bv msL¨vi A¼Ø‡qi mgwói mv‡_ 5 †hvM Ki‡j †hvMdj n‡e msL¨vwUi `kK ¯’vbxq A‡¼i wZb¸Y| Avi msL¨vwUi A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hv‡e, Zv g~j msL¨vwU †_‡K
9 Kg n‡e| msL¨vwU wbY©q Ki|

mgvavb : g‡b Kwi, wb‡Y©q msL¨vwUi `kK ¯’vbxq A¼ x Ges GKK ¯’vbxq A¼ y | AZGe, msL¨vwU
10 x y.

MwYZ

211

1g kZ©vbymv‡i x y 5 3 x.........(1) Ges 2q kZ©vbymv‡i, 10 y x (10 x y ) 9.......( 2) mgxKiY (1) †_‡K cvB, y 3 x x 5 , ev y 2 x 5........(3) Avevi, mgxKiY (2) †_‡K cvB, 10 y y x 10 x 9 0
(3) G x Gi gvb ewm‡q cvB, y 2 4 5

ev 9 y 9 x 9 ev y ev x x 1 0

0 y -Gi

8 5 gvb ewm‡q]

ev 2 x 5 x 1 0 [ (3) n‡Z 4

3 wb‡Y©q msL¨vwU n‡e 10 x y 10 4 3 40 3 43 msL¨vwU 43

D`vniY 12| AvU eQi c~‡e© wcZvi eqm cy‡Îi eq‡mi AvU¸Y wQj| `k eQi ci wcZvi eqm cy‡Îi eq‡mi wØ¸Y n‡e| eZ©gv‡b Kvi eqm KZ ? mgvavb : g‡b Kwi, eZ©gv‡b wcZvi eqm x eQi I cy‡Îi eqm y eQi| 1g kZ©vbymv‡i x 8 8( y 8)........(1) Ges 2q kZ©vbymv‡i, x 10 2( y 10).......( 2) (1) n‡Z cvB, x 8 8 y 64 ev x 8 y 64 8 ev x 8 y 56.............(3) n‡Z cvB, x 10 2 y 20 (2) ev 8 y 56 10 2 y 20 [ (3) n‡Z x Gi gvb ewm‡q] ev 8 y 2 y 20 56 10 ev 6 y 66 ev y 11
(3) n‡Z cvB, x
8 11 56 88 56 32

eZ©gv‡b wcZvi eqm 32 eQi I cy‡Îi eqm 11 eQi| D`vniY 13| GKwU AvqZvKvi evMv‡bi cÖ‡¯’i wØ¸Y, ˆ`N©¨ A‡c¶v 10 wgUvi †ewk Ges evMvbwUi cwimxgv

100 wgUvi|
K. evMvbwUi ˆ`N©¨ x wg I cÖ¯’ y wg. a‡i mgxKiY‡RvU MVb Ki| L. evMvbwUi ˆ`N©¨ I cÖ¯’ wbY©q Ki|

212

MwYZ

M.

evMvbwUi mxgvbvi evB‡i Pviw`‡K 2 wgUvi PIov iv¯—v Av‡Q| iv¯—vwU BU w`‡q ˆZwi Ki‡Z cÖwZeM©wgUv‡i 110 00 UvKv wn‡m‡e †gvU KZ LiP n‡e ?

mgvavb : K. AvqZvKvi evMvbwUi ˆ`N©¨ x wgUvi I cÖ¯’ y wgUvi| 1g kZ©vbymv‡i 2 y x 10........(1) y wgUvi

Ges 2q kZ©vbymv‡i, 2( x y ) 100.......( 2) L. mgxKiY (1) n‡Z cvB, 2 y x 10........(1)

mgxKiY (2) n‡Z cvB, 2 x 2 y 100.......(2) [ (1) n‡Z 2 y Gi gvb ewm‡q] ev 2 x x 10 100 ev 3 x 90 ev x 30
(1) n‡Z cvB, 2 y 30 10 [ x Gi gvb ewm‡q] ev, 2 y 40 ev, y 20

x wgUvi

evMvbwUi ˆ`N©¨ 30 wgUvi I cÖ¯’ 20 wgUvi| M. iv¯—vi evB‡ii ˆ`N©¨ (30 4) wg.= 34 wg
24 wgUvi

Ges cÖ¯’ = (20 4) wg. = 24 wg. iv¯—vi †¶Îdj = iv¯—vmn evMv‡bi †¶Îdj evMv‡bi †¶Îdj
30 wgUvi 34 wgUvi

20 wgUvi

(34 24 30 20) eM©wgUvi|

(816 600) eM©wgUvi 216 eM©wgUvi| BU w`‡q iv¯—v ˆZwi Kivi LiP

216 110 UvKv 23760 UvKv|
KvR :
A
ABC wÎfz‡R B 2 x wWwMÖ, C B C n‡j, x I y Gi gvb wbY©q Ki| x

wWwMÖ,

A

y wWwMÖ Ges

Abykxjbx 12 4
1| wb‡Pi †Kvb k‡Z© ax by c Awbf©ikxj n‡e ?

0 I px qy r

0 mgxKiY‡RvUwU m½wZc~Y© I ci¯úi

MwYZ

213

K. 2|

a p

b q

L.

a p

b q

c r

M.

a p

b q

c r

N.

a p

b q

x

y

4, x

y

2 n‡j ( x, y ) Gi gvb wb‡Pi †KvbwU ?
L. (4, 2)
4 n‡j, y gvb KZ ?

K. (2, 4) 3|

M. (3. 1)

N. (1, 3)

x

y

6 I 2x

K. 2 4|

L. 4

M. 6

N. 8

wb‡Pi †KvbwUi Rb¨ cv‡ki QKwU mwVK ? K. y

x 4 y 8 Ges x

L. y

8 x

x 0 2 4 y 4 0 4 M. y 4 2 x y = KZ ?
M. 8

N. y

2x 4

5|

2x

2y
L. 4

4 n‡j, x

K. 0 6|

N. 12

wb‡Pi Z_¨¸‡jv j¶ Ki :

i. 2 x

y

0 I x 2y

0 mgxKiYØq ci¯úi wbf©ikxj|

ii. x 2 y 3 0 mgxKi‡Yi †jLwPÎ ( 3, 0) we›`yMvgx| iii. 3 x 4 y 1 mgxKi‡Yi †jLwPÎ GKwU mij‡iLv|

Dc‡ii Z‡_¨i wfwË‡Z wb‡Pi †KvbwU mwVK ? K. i I ii 7| wb‡Pi cÖkœ¸‡jvi DËi `vI : (1) NiwUi †g‡Si ˆ`N©¨ KZ wgUvi ? K. 10 K. 24 L. 8 L. 32 M. 6 M. 48 N. 4 N. 80 L. ii I iii M. i I iii N. i, ii I iii

AvqZvKvi GKwU N‡ii †g‡Si ˆ`N©¨, cÖ¯’ A‡c¶v 2 wgUvi †ewk Ges †g‡Si cwimxgv 20 wgUvi|

(2) NiwUi †g‡Si †¶Îdj KZ eM©wgUvi ?

(3) NiwUi †g‡S †gvRvBK Ki‡Z cÖwZ eM©wgUv‡i 900 UvKv wn‡m‡e †gvU KZ LiP n‡e ? L. 43200 M. 28800 N. 21600 K. 72000 mnmgxKiY MVb K‡i mgvavb Ki (8 Ñ 17) : 8| †Kv‡bv fMœvs‡ki je I n‡ii cÖ‡Z¨KwUi mv‡_ 1 †hvM Ki‡j fMœvskwU cÖ‡Z¨KwU †_‡K 5 we‡qvM Ki‡j fMœvskwU
1 n‡e| fMœvskwU wbY©q Ki| 2 4 n‡e| Avevi, je I n‡ii 5

214

MwYZ

9|

†Kv‡bv fMœvs‡ki je †_‡K 1 we‡qvM I n‡ii mv‡_ 2 †hvM Ki‡j fMœvskwU we‡qvM Ges ni †_‡K 2 we‡qvM Ki‡j fMœvskwU

1 nq| Avi je †_‡K 7 2

1 nq| fMœvskwU wbY©q Ki| 3

10| `yB A¼wewkó GKwU msL¨vi GKK ¯’vbxq A¼ `kK ¯’vbxq A‡¼i wZb¸Y A‡c¶v 1 †ewk| wKš‘ A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq, Zv A¼Ø‡qi mgwói AvU¸‡Yi mgvb| msL¨vwU KZ ? 11| `yB A¼wewkó GKwU msL¨vi A¼Ø‡qi Aš—i 4 ; msL¨vwUi A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq, Zvi I g~j msL¨vwUi †hvMdj 110 ; msL¨vwU wbY©q Ki| 12| gvZvi eZ©gvb eqm Zvi `yB Kb¨vi eq‡mi mgwói Pvi¸Y| 5 eQi ci gvZvi eqm H `yB Kb¨vi eq‡mi mgwói wØ¸Y n‡e| gvZvi eZ©gvb eqm KZ ? 13| GKwU AvqZ‡¶‡Îi ˆ`N©¨ 5 wgUvi Kg I cÖ¯’ 3 wgUvi †ewk n‡j †¶Îdj 9 eM©wgUvi Kg n‡e| Avevi ˆ`N©¨ 3 wgUvi †ewk I cÖ¯’ 2 wgUvi †ewk n‡j †¶Îdj 67 eM©wgUvi †ewk n‡e| †¶ÎwUi ˆ`N©¨ I cÖ¯’ wbY©q Ki| 14| GKwU †bŠKv `uvo †e‡q †mªv‡Zi AbyK~‡j NÈvq 15 wK.wg. hvq Ges †mªv‡Zi cÖwZK~‡j hvq NÈvq 5 wK.wg.| †bŠKvi I †mªv‡Zi †eM wbY©q Ki| 15| GKRb Mv‡g©›Um kªwgK gvwmK †eZ‡b PvKwi K‡ib| cÖwZeQi †k‡l GKwU wbw`©ó †eZbe„w× cvb| Zvi gvwmK †eZb 4 eQi ci 4500 UvKv I 8 eQi ci 5000 UvKv nq| Zuvi PvKwi ïi“i †eZb I evwl©K †eZb e„w×i cwigvY wbY©q Ki| 16| GKwU mij mgxKiY‡RvU x y 10 3x 2 y 0 K. †`LvI †h, mgxKiY‡RvUwU m½wZc~Y©| Gi KqwU mgvavb Av‡Q ? L. mgxKiY‡RvUwU mgvavb K‡i ( x, y ) wbY©q Ki| M. mgxKiYØq Øviv wb‡`©wkZ mij‡iLvØq x -A‡¶i mv‡_ †h wÎfyR MVb K‡i Zvi †¶Îdj wbY©q Ki| 17| †Kv‡bv fMœvs‡ki j‡ei mv‡_ 7 †hvM Ki‡j fMœvskwUi gvb c~Y©msL¨v 2 nq| Avevi ni n‡Z 2 we‡qvM Ki‡j fMœvskwUi gvb c~Y©msL¨v 1 nq| x a‡i mgxKiY‡RvU MVb Ki| K. fMœvskwU y L. mgxKiY‡RvUwU Avo¸Yb c×wZ‡Z mgvavb K‡i ( x, y ) wbY©q Ki| fMœvskwU KZ ? M. mgxKiY‡RvUwUi †jL A¼b K‡i ( x, y ) Gi cÖvß gv‡bi mZ¨Zv hvPvB Ki|

Î‡qv`k Aa¨vq

mmxg aviv
Finite Series cÖvZ¨wnK Rxe‡b ÕµgÕ eûj cÖPwjZ GKwU kã| †hgb- †`vKv‡bi Zv‡K †fvM¨cY¨ mvRv‡Z, bvUK I Abyôv‡bi NUbvejx mvRv‡Z, ¸`vgN‡i my›`ifv‡e `ªe¨vw` ivL‡Z µ‡gi aviYv e¨enƒZ nq| Avevi A‡bK KvR mn‡R Ges `„wób›`bfv‡e m¤úv`b Ki‡Z Avgiv eo n‡Z †QvU, wkï n‡Z e„×, nvjKv n‡Z fvix BZ¨vw` ai‡bi µg e¨envi Kwi| GB µ‡gi aviYv n‡ZB wewfbœ cÖKvi MvwYwZK avivi D™¢e n‡q‡Q| GB Aa¨v‡q Abyµg I avivi g‡a¨ m¤ú©K I GZ`msµvš— welqe¯‘ Dc¯’vcb Kiv n‡q‡Q| Aa¨vq †k‡l wk¶v_x©ivAbyµg I aviv eY©bv Ki‡Z I Zv‡`i cv_©K¨ wbiƒcY Ki‡Z cvi‡e| mgvš—i aviv e¨vL¨v Ki‡Z cvi‡e| mgvš—i avivi wbw`©óZg c` I wbw`©ó msL¨K c‡`i mgwó wbY©‡qi m~Î MVb Ki‡Z cvi‡e Ges m~Î cÖ‡qvM K‡i MvwYwZK mgm¨v mgvavb Ki‡Z cvi‡e| ¯^vfvweK msL¨vi e‡M©i I N‡bi mgwó wbY©q Ki‡Z cvi‡e| avivi wewfbœ m~Î cÖ‡qvM K‡i MvwYwZK mgm¨vi mgvavb Ki‡Z cvi‡e| ¸‡YvËi avivi wbw`©óZg c` I wbw`©ó msL¨K c‡`i mgwó wbY©‡qi m~Î MVb Ki‡Z cvi‡e Ges m~Î cÖ‡qvM K‡i MvwYwZK mgm¨vi mgvavb Ki‡Z cvi‡e|

Abyµg wb‡Pi m¤úK©wU j¶ Kwi :
1 2
3 6

4
8

5 10

n
2n

2

4

GLv‡b cÖ‡Z¨K ¯^vfvweK msL¨v n Zvi wØ¸Y msL¨v 2 n Gi mv‡_ m¤úwK©Z| A_©vr ¯^vfvweK msL¨vi †mU N {1, 2, 3,.......} †_‡K GKwU wbq‡gi gva¨‡g †hvM‡evaK †Rvo msL¨vi †mU {2, 4, 6, 8,.......} cvIqv hvq| GB mvRv‡bv †RvomsL¨vi †mUwU GKwU Abyµg| myZivs, KZK¸‡jv ivwk GKUv we‡kl wbq‡g µgvš^‡q Ggbfv‡e mvRv‡bv nq ‡h cÖ‡Z¨K ivwk Zvi c~‡e©i c` I c‡ii c‡`i mv‡_ Kxfv‡e m¤úwK©Z Zv Rvbv hvq| Gfv‡e mvRv‡bv ivwk¸‡jvi †mU‡K Abyµg (Sequence) ejv nq| Dc‡ii m¤úK©wU‡K dvskb e‡j Ges f (n) 2n wjLv nq| GB Abyµ‡gi mvaviY c` 2n . †h‡Kv‡bv Abyµ‡gi c`msL¨v Amxg| AbyµgwU mvaviY c‡`i mvnv‡h¨ wjLvi c×wZ n‡jv 2n , n 1, 2, 3,......... ev,
2n
n 1

ev,

2n .

216

MwYZ

Abyµ‡gi cÖ_g ivwk‡K cÖ_g c`, wØZxq ivwk‡K wØZxq c`, Z…Zxq ivwk‡K Z…Zxq c` BZ¨vw` ejv nq| 1, 3, 5, 7,… … Abyµ‡gi cÖ_g c` =1, wØZxq c` = 3, BZ¨vw`| wb‡P Abyµ‡gi PviwU D`vniY †`Iqv n‡jv : 1, 1, 2,
3,

3,
5,

,n, , (2n 1) , , n2 , , n , n 1

1,
1, 2

4,
2, 3

9,
3, 4

KvR : 1| wb‡P QqwU Abyµ‡gi mvaviY c` †`Iqv Av‡Q| Abyµg¸wj †jL : (i) 1 n (ii) n 1 n 1

(iii) 1n
2

(iv) 1 1 n
2

(v) ( 1) n 1 n

n 1

(vi) ( 1) n 1 n

.

2n 1

2| †Zvgiv cª‡Z¨‡K GKwU K‡i Abyµ‡gi mvaviY c` wj‡L AbyµgwU †jL|

aviv
†Kv‡bv Abyµ‡gi c`¸‡jv cici Ô+Õ 1 3 5 7 ....... GKwU aviv| 2 4 8 16 ........ GKwU aviv| Gi `yBwU c‡`i g‡a¨ m¤ú‡K©i Dci wbf©i K‡i aviv I ¸‡YvËi aviv| wPý Øviv hy³ Ki‡j GKwU aviv (Series) cvIqv hvq| †hgb, avivwUi cici `yBwU c‡`i cv_©K¨ mgvb| Avevi cici `yBwU c‡`i AbycvZ mgvb| myZivs, †h‡Kv‡bv avivi cici avivwUi ˆewkó¨| aviv¸‡jvi g‡a¨ ¸i“Z¡c~Y© `yBwU aviv n‡jv mgvš—i

mgvš—i aviv
†Kv‡bv avivi †h‡Kv‡bv c` I Zvi c~e©eZ©x c‡`i cv_©K¨ me mgq mgvb n‡j, †mB avivwU‡K mgvš—i aviv e‡j| D`vniY : 1 3 5 7 9 11 GKwU aviv| GB avivwUi cÖ_g c` 1, wØZxq c` 3 , Z…Zxq c` 5, BZ¨vw`| GLv‡b, wØZxq c` PZz_© c` cÖ_g c` = 3 1 2, Z…Zxq c` wØZxq c` = 5 3 2, Z…Zxq c` = 7 5 2, cÂg c` PZz_© c` = 9 7 2,

lô c` cÂg c` 11 9 2 myZivs, avivwU GKwU mgvš—i aviv| GB avivq cÖvß `yBwU c‡`i we‡qvMdj‡K mvaviY Aš—i ejv nq| DwjwLZ avivi mvaviY Aš—i 2. avivwUi c` msL¨v wbw`©ó| G Rb¨ GwU GKwU mmxg ev mvš—aviv (Finite Series)| D‡jL¨, mgvš—i avivi c`msL¨v wbw`©ó bv n‡j Zv‡K Amxg ev Abš—aviv (Infinite Series) e‡j| †hgb, 1 4 7 10 ... ... GKwU Amxg aviv| mgvš—i avivq mvaviYZ cÖ_g c`‡K a Øviv Ges mvaviY Aš—i‡K d Øviv cÖKvk Kiv nq| Zvn‡j msÁvbymv‡i, cÖ_g c` a n‡j, wØZxq c` a d , Z…Zxq c` a 2d , BZ¨vw`| myZivs, avivwU n‡e,

a (a d ) (a 2d ) ..... .

MwYZ

217

mgvš—i avivi mvaviY c` wbY©q g‡b Kwi, ‡h‡Kv‡bv mgvš—i avivi cÖ_g c` cª_g c` wØZxq c` Z„Zxq c` PZz_© c` .... .... .... .... a I mvaviY Aš—i

d ; Zvn‡j avivwUi

a

a (1 1) d

a d a 2d

a (2 1) d a (3 1) d

a 3d
.... .... .... ....

a (4 1) d
.... .... .... .... .... ....

n Zg c` = a (n 1)d
GB n Zg c`‡KB mgvš—i avivi mvaviY c` ejv nq| †Kv‡bv mgvš—i avivi cÖ_g c` a, mvaviY Aš—i d Rvbv _vK‡j n Zg c‡` n 1, 2, 3, 4,... ... ewm‡q ch©vqµ‡g avivwUi cÖ‡Z¨KwU c` wbY©q Kiv hvq|

g‡b Kwi, GKwU mgvš—i avivi cÖ_g c` 3 Ges mvaviY Aš—i 2| Zvn‡j avivwUi wØZxq c`

3 2 5, Z„Zxq c`

3 2 2 7, PZz_© c`

3 3 2 9, BZ¨vw`|

AZGe, avivwUi n Zg c`

3

n 1 2 2n 1 .

KvR : †Kv‡bv mgvš—i avivi cÖ_g c` 5 Ges mvaviY Aš—i 7 n‡j, avivwUi cÖ_g QqwU c`, 22Zg c`, r Zg Ges (2p+1)Zg c` wbY©q Ki|

D`vniY 1| 5 8 11 14

avivwUi †Kvb c` 383 ?

mgvavb : avivwUi cÖ_g c` a 5, mvaviY Aš—i d Bnv GKwU mgvš—i aviv| g‡b Kwi, avivwUi n Zg c` = 383 Avgiv Rvwb, n Zg c` = a (n 1)d .

8 5 11 8 3

a (n 1)d

383

ev, 5 (n 1)3 383 ev, 5 3n 3 383 ev, 3n 383 5 3 ev, 3n 381 ev, n
381 3

n 127

cÖ`Ë avivi 127 Zg c` = 383 . dg©v-28, MwYZ-9g-10g

218

MwYZ

mgvš—i avivi n msL¨K c‡`i mgwó g‡b Kwi, †h‡Kv‡bv mgvš—i avivi cÖ_g c` a , †kl c` p, mvaviY Aš—i d , c` msL¨v n Ges avivwUi n msL¨K c‡`i mgwó S n . avivwU‡K cÖ_g c` n‡Z Ges wecixZKª‡g †kl c` n‡Z wj‡L cvIqv hvq S n a ( a d ) ( a 2d ) p 2d Ges S n †hvM K‡i, 2S n ev, 2S n

p d a p

p a p

(i) (ii )

p a na

p d p p a
[

p 2d p a p ... ...

a 2d a p

(a d ) a

avivwUi c` msL¨v n ]

Sn

Avevi, n Zg c` n Sn [a {a (n 1)d }] 2 n A_©vr, S n 2a n 1 d 2 †Kv‡bv mgvš—i avivi cÖ_g c` a, †kl c` p Ges c` msL¨v n Rvbv _vK‡j, (iii ) bs m~‡Îi mvnv‡h¨ avivwUi mgwó wbY©q Kiv hvq| wKš‘ cÖ_g c` a , mvaviY Aš—i d , c` msL¨v n Rvbv _vK‡j, (iv) bs m~‡Îi mvnv‡h¨ avivwUi mgwó wbY©q Kiv hvq|

n a p 2 p a (n 1)d . p Gi gvb (iii) G ewm‡q cvB,

(iii )

cÖ_g n msL¨K ¯^vfvweK msL¨vi mgwó wbY©q g‡b Kwi, n msL¨K ¯^vfvweK msL¨vi mgwó Sn A_©vr, S n 1 2 3

(n 1) n

(i) (i) (ii )
[ n msL¨K c` ]

avivwU‡K cÖ_g c` n‡Z Ges wecixZKª‡g †kl c` n‡Z wj‡L cvIqv hvq Sn 1 2 3 n 2 (n 1) n Ges S n †hvM K‡i, 2S n

n ev, n 1 2Sn

n 2 n(n 1)

3

2

1

(n 1) (n 1) (n 1)

(n 1)

n(n 1) 2 D`vniY 2| cÖ_g 50 wU ¯^vfvweK msL¨vi ‡hvMdj wbY©q Ki| Sn mgvavb : Avgiv (iii) bs m~Î e¨envi K‡i cvB, 50(50 1) S50 25 51 1275 2 cÖ_g 50 wU ¯^vfvweK msL¨vi ‡hvMdj 1275.

(iii )

MwYZ

219

D`vniY 3| 1 2 3 4

99

KZ ?

mgvavb : avivwUi cÖ_g c` a 1, mvaviY Aš—i d Bnv GKwU mgvš—i aviv| g‡b Kwi, avivwUi n Zg c` = 99 Avgiv Rvwb, mgvš—i avivi n Zg c` = a (n 1)d ev, ev,

2 1 1 Ges †kl c` p = 99. weKí c×wZ: †h‡nZy
Sn n a p, 2 99 S99 1 99 2

a (n 1)d 99 1 (n 1)1 99 1 n 1 99 n 99

99 100 2

4950

(iv) bs m~Î n‡Z, mgvš—i avivi cÖ_g n -msL¨K c‡`i mgwón Sn {2a (n 1)d }. 2 99 myZivs, avivwUi 99 wU c‡`i mgwó S 99 {2 1 (99 1) 1} 2 99 100 99 50 4950 2 D`vniY 4| 7 12 17 mgvavb : avivwU cÖ_g c` a avivwUi 30 wU c‡`i mgwó KZ ?

99 (2 98) 2

7 , mvaviY Aš—i

d

12 7 5

Bnv GKwU mgvš—i aviv| GLv‡b c` msL¨v n 30. Avgiv Rvwb, mgvš—i avivi cÖ_g n -msL¨K c‡`i mgwó,
Sn n {2a (n 1)d}. 2

30 {2.7 (30 1)5} 15(14 29 5) 2 15(14 145) 15 159 2385 D`vniY 5| K Zvi †eZb †_‡K cÖ_g gv‡m 1200 UvKv mÂq K‡ib Ges cieZ©x gvm¸‡jvi cÖwZgv‡m Gi c~e©eZ©x gv‡mi Zzjbvq 100 UvKv †ewk mÂq K‡ib| ( i ) wZwb n Zg gv‡m KZ UvKv mÂq K‡ib ? ( ii ) Dc‡iv³ mgm¨vwU‡K n msL¨K c` ch©š— avivq cÖKvk Ki| ( iii ) wZwb cÖ_g n msL¨K gv‡m KZ UvKv mÂq K‡ib ? ( iv ) GK eQ‡i wZwb KZ UvKv mÂq K‡ib ? mgvavb : ( i ) cÖ_g gv‡m mÂq K‡ib 1200 UvKv wØZxq gv‡m mÂq K‡ib (1200 100) UvKv 1300 UvKv
Zvn‡j, 30 wU c‡`i mgwó S30 =

220

MwYZ

Z…Zxq gv‡m mÂq K‡ib (1300 100) UvKv

1400 UvKv
1500 UvKv

PZz_© gv‡m mÂq K‡ib (1400 100) UvKv avivwUi n Zg c`

myZivs, GwU GKwU mgvš—i aviv, hvi cÖ_g c` a 1200, mvaviY Aš—i d

1300 1200 100.

a (n 1)d 1200 (n 1)100 1200 100n 100 100n 1100 AZGe, wZwb n Zg gv‡m mÂq K‡ib (100n 1100) UvKv|
( ii) G‡¶‡Î n msL¨K c` ch©š—avivwU n‡e 1200 1300 1400 (100n 1100) ( iii ) wZwb cÖ_g n msL¨K gv‡m mÂq K‡ibn n {2 1200 (n 1)100} UvKv {2a (n 1)d } UvKv 2 2 n n (2400 100n 100) UvKv 2(1150 50n) UvKv 2 2 n(50n 1150) UvKv| ( iv ) Avgiv Rvwb, GK eQi = 12 gvm| G‡¶‡Î, n 12. AZGe, [ Dc‡ii (iii) n‡Z ] K GK eQ‡i mÂq K‡ib 12(50 12 1150) UvKv

12(600 1150) UvKv

12 1750 UvKv = 21000 UvKv|

Abykxjbx 13 1
1| 2 5 12 19 2| 3| 4| 5| 6| 7| 8| 9|
8 11 14 17

avivwUi mvaviY Aš—i Ges 12Zg c` wbY©q Ki| avivwUi †Kvb c` 392 ? avivwUi †Kvb c` 301 ?

4 7 10 13

†Kv‡bv mgvš—i avivi p Zg c` p2 Ges q Zg c` q2 n‡j, avivwUi ( p q) Zg c` KZ ? †Kv‡bv mgvš—i avivi m Zg c` n I n Zg c` m n‡j, ( m n )Zg c` KZ ?

1 3 5 7 8 16 24
5 11 17 23

avivwUi n c‡`i mgwó KZ ? avivwUi cÖ_g 9 wU c‡`i mgwó KZ ?
59 KZ ?

29 25 21

23 KZ ?

10| †Kv‡bv mgvš—i avivi 12 Zg c` 77 n‡j, Gi cÖ_g 23 wU c‡`i mgwó KZ ? 11| GKwU mgvš—i avivi 16 Zg c` 12| 9 7 5

20 n‡j, Gi cÖ_g 31 wU c‡`i mgwó KZ ?
144 n‡j, n Gi gvb wbY©q Ki|

avivwUi cÖ_g n msL¨K c‡`i †hvMdj

MwYZ

221

13| 2 4 6 8

avivwUi cÖ_g n msL¨K c‡`i mgwó 2550 n‡j, n Gi gvb wbY©q Ki|

14| †Kv‡bv avivi cÖ_g n msL¨K c‡`i mgwó n(n 1) n‡j, avivwU wbY©q Ki| 15| †Kv‡bv avivi cÖ_g n msL¨K c‡`i mgwó n(n 1) n‡j, avivwUi 10 wU c‡`i mgwó KZ ? 16| GKwU mgvš—i avivi cª_g 12 c‡`i mgwó 144 Ges cª_g 20 c‡`i mgwó 560 n‡j, Gi cª_g 6 c‡`i mgwó wbY©q Ki| 17| †Kv‡bv mgvš—i avivi cª_g m c‡`i mgwó n Ges cª_g n c‡`i mgwó m n‡j, Gi cª_g (m+n) c‡`I mgwó wbY©q Ki| 18| †Kv‡bv mgvš—i avivq p Zg, q Zg I r Zg c` h_vµ‡g a, b, c n‡j, †`LvI †h,

a(q r) b(r

p) c( p q)

0.
125 169 171 173 209.

19| †`LvI †h, 1 3 5 7

20| GK e¨w³ 2500 UvKvi GKwU FY wKQzmsL¨K wKw¯—‡Z cwi‡kva Ki‡Z ivRx nb| cÖ‡Z¨K wKw¯— c~‡e©i wKw¯— †_‡K 2 UvKv †ewk| hw` cÖ_g wKw¯— 1 UvKv nq, Z‡e KZ¸‡jv wKw¯—‡Z H e¨w³ Zvi FY †kva Ki‡Z cvi‡eb ?

cÖ_g n msL¨K ¯^vfvweK msL¨vi e‡M©i mgwó wbY©q g‡b Kwi, cÖ_g n msL¨K ¯^vfvweK msL¨vi e‡M©i mgwó Sn. A_©vr, S n 12 Avgiv Rvwb,

2 2 32

n2

r 3 3r 2 ev, r 3

3r 1 (r 1) 3

(r 1)3 3r 2 3r 1 Dc‡ii A‡f`wU‡Z, r 1, 2, 3, , n ewm‡q cvB, 13 2
3

03 1
3

3.12 3.2
2

3.1 1 3.2 1 3.3 1

33

23

3.32

n3
†hvM K‡i cvB,

(n 1) 3 3(12 22

3.n 2 32

3.n 1 n 2 ) 3(1 2 3
1 2 3 n 2

n3 ev, ev,

03 n3

n) (1 1 1 n(n 1)

1)

3S n 3.

3Sn

n3

n(n 1) n 2 3n(n 1) n 2

222

MwYZ

2n 3 n( 2n
2

3n 2 3n 2n
2

2n 3

3n 2 n
2

n( 2n 2 2

3n 1)

2n n 1)
2

n{2n(n 1) 1(n 1)} 2

ev,

3S n Sn

n(n 1)(2n 1) 2 n(n 1)(2n 1) 6

cÖ_g n msL¨K ¯^vfvweK msL¨vi N‡bi mgwó wbY©q g‡b Kwi, cÖ_g n msL¨K ¯^vfvweK msL¨vi N‡bi mgwó Sn. A_©vr, S n 13 23 33 Avgiv Rvwb, (r 1)2 (r 1)2 ev, (r 1) 2 r 2

n3

(r 2

2r 1) (r 2

2r 1)

4r.

r 2 (r 1) 2 4.13 4.2 3 4.33 n 2 (n 1) 2

4r.r 2

4r 3 [ Dfqc¶‡K r2 Øviv ¸Y K‡i ]

Dc‡ii A‡f`wU‡Z, r 1, 2, 3,

, n ewm‡q cvB,

2 2.12 12.0 2 32.2 2 4 2 .3 2 2 2.12 32.22

(n 1) 2 n 2

4n 3 4(13 23 33 n3 )

†hvM K‡i, (n 1) 2 .n 2 12.0 2 ev, (n 1)2.n2 ev, Sn
Sn

4Sn

n2(n 1)2 4 n(n 1) 2
2

cÖ‡qvRbxq m~Î

1 2 3
12
13

n
32 n2 n3 n(n 1) 2 n(n 1)(2n 1) 6 2 n(n 1)
2

22

23 33

MwYZ

223

we‡kl `ªóe¨: 13

23

33

n3

(1 2 3

n) 2 .

KvR : 1| cÖ_g n msL¨K ¯^vfvweK †Rvo msL¨vi mgwó wbY©q Ki| 2| cÖ_g n msL¨K ¯^vfvweK we†Rvo msL¨vi e‡M©i mgwó wbY©q Ki|

¸‡YvËi aviv
†Kv‡bv avivi †h‡Kv‡bv c` I Gi c~e©eZ©x c‡`i AbycvZ me mgq mgvb n‡j A_©vr, †h‡Kv‡bv c`‡K Gi c~e©eZ©x c` Øviv fvM K‡i fvMdj me©`v mgvb cvIqv †M‡j, †m avivwU‡K ¸‡YvËi aviv e‡j Ges fvMdj‡K mvaviY AbycvZ e‡j| †hgb, 2 4 8 16 32 avivwUi cÖ_g c` 2 , wØZxq c` 4 , Z…Zxq c` 8 , PZz_© c` 16, cÂg c` 32. GLv‡b,

8 4 2 2 , Z…Zxq c‡`i mv‡_ wØZxq c‡`i AbycvZ = 4 2 16 32 PZz_© c‡`i mv‡_ Z…Zxq c‡`i AbycvZ = 2. 2 , cÂg c‡`i mv‡_ PZz_© c‡`i AbycvZ = 8 16 myZivs, avivwU GKwU ¸‡YvËi aviv| GB avivq †h‡Kv‡bv c` I Gi c~e©eZ©x c‡`i AbycvZ me©`v mgvb| DwjwLZ avivq mvaviY AbycvZ 2 | avivwUi c` msL¨v wbw`©ó| G Rb¨ GwU GKwU ¸‡YvËi mmxg aviv| wØZxq c‡`i mv‡_ cÖ_g c‡`i AbycvZ = †fŠZ I Rxe weÁv‡bi wewfbœ †¶‡Î, e¨vsK I exgv BZ¨vw` cÖwZôv‡b Ges wewfbœ cÖKvi cÖhyw³ we`¨vq ¸‡YvËi avivi e¨vcK cÖ‡qvM Av‡Q| ¸‡YvËi avivi c` msL¨v wbw`©ó bv _vK‡j G‡K Abš— ¸‡YvËi aviv e‡j| ¸‡YvËi avivi cÖ_g c`‡K mvaviYZ a Øviv Ges mvaviY AbycvZ‡K r Øviv cÖKvk Kiv nq| Zvn‡j msÁvbymv‡i, cÖ_g c` a n‡j, wØZxq c` ar, Z…Zxq c` ar 2 , BZ¨vw`| myZivs, avivwU n‡e,

a ar ar 2

ar 3

KvR : wbgœwjwLZ †¶‡Î ¸‡YvËi aviv¸‡jv †jL : (i) cÖ_g c` 4, mvaviY AbycvZ 10 (ii) cÖ_g c` 9, mvaviY AbycvZ 1 (iii) cÖ_g c` 7, mvaviY AbycvZ 1
3

10

(iv) cÖ_g c` 3, mvaviY AbycvZ 1 (v) cÖ_g c` 1, mvaviY AbycvZ

1 2

(vi) cÖ_g c` 3, mvaviY AbycvZ 1

¸‡YvËi avivi mvaviY c` g‡b Kwi, †h‡Kv‡bv ¸‡YvËi avivi cÖ_g c` a, mvaviY AbycvZ r , Zvn‡j avivwUi cÖ_g c` Z…Zxq c` ... ... ... ...

a ar 2 ... ...

ar 1 1 ar 3 1

wØZxq c`

ar

ar 2 1

PZz_© c` ar3 ar 4 1 ... ... ... ... ... ...

n Zg c` = ar n

1

224

MwYZ

GB n Zg c`‡KB ¸‡YvËi avivi mvaviY c` ejv nq| †Kv‡bv ¸‡YvËi avivi cÖ_g c` a I mvaviY AbycvZ BZ¨vw` ewm‡q avivwUi †h‡Kv‡bv c` wbY©q Kiv hvq| r Rvbv _vK‡j n Zg c‡` ch©vqµ‡g r 1, 2, 3, D`vniY 6| 2 4 8 16 avivwUi 10 Zg c` KZ ? mgvavb : avivwUi cÖ_g c` a

2, mvaiY AbycvZ r

4 2

2.

cÖ`Ë avivwU GKwU ¸‡YvËi aviv| Avgiv Rvwb, ¸‡YvËi avivi n Zg c` = ar n avivwUi 10 Zg c` = 2 210 1 = 2 29 D`vinY 7| 128 64 32
1

1024 avivwUi mvaviY c` KZ ?

mgvavb : cÖ`Ë avivwUi cÖ_g c` a 128, mvaviY AbycvZ r Bnv GKwU ¸‡YvËi aviv| Avgiv Rvwb, ¸‡YvËi avivi mvaviY c` = ar n
1
n 1

64 128

1 . 2

1 27 1 1 myZivs, avivwUi mvaviY c` = 128 . n 1 n 1 7 2 2 2n 8 2 D`vniY 8| GKwU ¸‡YvËi avivi cÖ_g I wØZxq c` h_vµ‡g 27 Ges 9 n‡j, avivwUi cÂg c` Ges `kg c` wbY©q Ki| mgvavb : cÖ`Ë avivwUi cÖ_g c` a 27, wØZxq c` 9 9 1 Zvn‡j mvaviY AbycvZ r . 27 3 cÂg c` = ar Ges `kg c` = ar
5 1

27 27

1 3 1 3

4

27 1 27 3 33 3
3

1 3 1 36 1 . 729

9

10

1

3

6

¸‡YvËi avivi mgwó wbY©q g‡b Kwi, ¸‡YvËi avivi cÖ_g c` a, mvaviY AbycvZ r Ges c` msL¨v n. hw` n msL¨K c‡`i mgwó S n nq, Zvn‡j

Sn
Ges r.S n

a ar ar 2 ar ar 2 ar 3

ar n

2

ar n

1

(i)
[ (i) ‡K r Øviv ¸Y K‡i ] (ii)

ar n 1 ar n

we‡qvM K‡i, S n ev,

rS n

a ar n a(1 r n )

S n (1 r )

MwYZ

225

a(1 r n ) Sn , hLb r 1 1 r Avevi (ii) †_‡K (i) we‡qvM K‡i cvB, ev, S n (r 1) a(r n 1) rS n S n ar n a
A_v©r, S n

a(r n 1) , hLb r 1 . (r 1) j¶Yxq : mvaviY AbycvZ r 1 n‡j cÖ‡Z¨K c` a myZivs, G‡¶‡Î S n a a a n c` ch©š— an.
KvR : K Zvi †Q‡j‡K ¯‹z‡j †bqv-Avbvi Rb¨ GK e¨w³‡K 1jv GwcÖj †_‡K GK gv‡mi Rb¨ wb‡qvM Ki‡jb| Zvi cvwikªwgK wVK Kiv n‡jv- cÖ_g w`b GK cqmv, wØZxq w`b cÖ_g w`‡bi wØ¸Y A_©vr `yB cqmv, Z„Zxq w`b wØZxq w`‡bi wØ¸Y A_©vr Pvi cqmv| GB wbq‡g cvwikªwgK w`‡j mvßvwnK QzwUi w`bmn GK gvm ci H e¨w³ KZ UvKv cv‡eb ?

D`vniY 9| 12 24 48 mgvavb : cÖ`Ë avivwUi cÖ_g c` a avivwU GKwU ¸‡YvËi aviv| g‡b Kwi, avivwUi n Zg c` 768 Avgiv Rvwb, n Zg c` = ar n 1 ar n 1 768 ev, 12 2 n 1 768 768 ev, 2 n 1 64 12 ev, 2 n 1 26 ev, n 1 6 n 7. myZivs, avivwUi mgwó

768 avivwUi mgwó KZ ? 24 12, mvaviY AbycvZ r 12

2

1.

a(r n 1) , (r 1)

hLb r

1

12(2 7 1) 12 (128 1) 12 127 1524. 2 1 1 1 1 D`vniY 10| 1 avivwUi cÖ_g AvUwU c‡`i mgwó wbY©q Ki| 2 4 8 1 1 mgvavb : cÖ`Ë avivwUi cÖ_g c` a 1, mvaviY AbycvZ r 2 1 1 2 Bnv GKwU ¸‡YvËi aviv| GLv‡b c` msL¨v n 8. dg©v-29, MwYZ-9g-10g

226

MwYZ

Avgiv Rvwb, ¸‡YvËi avivi n -msL¨K c‡`i mgwó a(1 r n ) Sn , hLb r 1. 1 r

1 myZivs, avivwUi 8 wU c‡`i mgwó S 8

1 1

1 2 1 2

8

1

1 256 1 2

2

256 1 256

255 127 1 128 128

Abykxjbx 13⋅2
1. a, b, c I d mgvš—i avivi PviwU µwgK c` n‡j wb‡Pi †KvbwU mwVK? K. b = c d 2 b d M. c = 2

L. a =

b c 2 a c N. d = 2

2. i a+(a+d)+(d+2d).................... avivwUi cÖ_g n msL¨K c‡`i mgwó = ii 1+2+3............+n = n(n 1)(2n 1) 6 iii 1+3+5+..............+(2n-1) = n2

n {2a+(n-1)d} 2

Dc‡ii evK¨¸‡jvi †KvbwU mwVK ? K. i I ii L. i I iii M. ii I iii N. i, ii I iii wb‡Pi avivwUi wfwË‡Z 3 I 4 b¤^i cÖ‡kœi DËi `vI : log 2+ log 4 + log 8 +.......................

3. avivwUi mvaviY Aš—i †KvbwU? K. 2 M. log 2 4. avivwUi 7g c` KZ? K. log 32 M. log 128 5| 64 32 16 8 6| 7| 8|
3 9 27

L. 4 N. 2 log 2 L. log 64 N. log 256 avivwUi Aóg c` wbY©q Ki| avivwUi cÖ_g †PŠÏwU c‡`i mgwó wbY©q Ki| avivwUi †Kvb c`
1 ? 2

128 64 32
GKwU ¸‡YvËi avivi cÂg c`

2 3 8 2 Ges `kg c` n‡j, avivwUi Z…Zxq c` wbY©q Ki| 9 81

MwYZ

227

9|

1 avivwUi †Kvb c` 8 2 ? , 1, 2 , 2 10| 5 x y 135 ¸‡YvËi avivfz³ n‡j, x Ges y Gi gvb wbY©q Ki|
11| 3 x

y z 243 ¸‡YvËi avivfz³ n‡j, x, y Ges z Gi gvb wbY©q Ki| avivwUi cÖ_g mvZwU c‡`i mgwó KZ ? avivwUi (2n 1) msL¨K c‡`i mgwó wbY©q Ki| avivwUi cÖ_g `kwU c‡`i mgwó KZ ? avivwUi cÖ_g eviwU c‡`i mgwó wbY©q Ki| avivwUi n -msL¨K c‡`i mgwó
254

12|

2 4 8 16

13| 1 1 1 1 14| log 2 log 4 log 8

15| log 2 log 16 log 512 16| 2 4 8 16 17| 2 2 2 2

n‡j, n -Gi gvb KZ ?

avivwUi (2n 2) msL¨K c‡`i mgwó KZ ?

18| cÖ_g n msL¨K ¯^vfvweK msL¨vi N‡bi mgwó 441 n‡j, n Gi gvb wbY©q Ki Ges H msL¨v¸‡jvi mgwó wbY©q Ki| 19| cÖ_g n msL¨K ¯^vfvweK msL¨vi N‡bi mgwó 225 n‡j, n Gi gvb KZ ? H msL¨v¸‡jvi e‡M©i mgwó KZ ? 20| †`LvI †h, 13

23

33

103

(1 2 3

10) 2 .

13 23 33 21| 1 2 3

n3 n

210 n‡j n -Gi gvb KZ ?

22| 1 wgUvi ˆ`N©¨wewkó GKwU †jŠn `Û‡K 10 wU UzKivq wef³ Kiv n‡jv hv‡Z UzKiv¸‡jvi ˆ`N©¨ ¸‡YvËi aviv MVb K‡i| hw` e„nËg UzKivwU ¶z`ªZg UzKivi 10 ¸Y nq, Z‡e ¶z`ªZg UzKivwUi ˆ`‡N©¨i gvb Avmbœ wgwjwgUv‡i wbY©q Ki|
23| GKwU ¸‡YvËi avivi 1g c` a, mvaviY AbycvZ r, avivwUi 4_© c` -2 Ges 9g c` 8 2 K. Dc‡iv³ Z_¨¸‡jv‡K `yBwU mgxKi‡Yi gva¨‡g cÖKvk Ki| L. avivwUi 12 Zg c` wbY©q Ki| M. avivwU wbY©q K‡i cÖ_g 7wU c‡`i mgwó wbY©q Ki| 24| †Kvb avivi n Zg c` 2n-4 K. avivwU wbY©q Ki| L. avivwUi 10Zg c` Ges cÖ_g 20wU c‡`i mgwó wbY©q Ki| M. cÖvß avivwUi cÖ_g c`‡K cÖ_g c` Ges mvaviY Aš—i‡K mvaviY AbycvZ a‡i GKwU bZzb aviv ˆZwi Ki Ges m~Î cÖ‡qvM K‡i avivwUi cÖ_g 8 c‡`i mgwó wbY©q Ki|

PZz`©k Aa¨vq

AbycvZ, m`„kZv I cÖwZmgZv
`yBwU ivwki Zzjbv Kivi Rb¨ Zv‡`i AbycvZ we‡ePbv Kiv nq| AbycvZ wbY©‡qi Rb¨ ivwk `yBwU GKB GK‡K cwigvc Ki‡Z nq| G m¤ú‡K© exRMwY‡Z we¯—vwiZ Av‡jvPbv Kiv n‡q‡Q| Aa¨vq †k‡l wk¶v_©xiv Ñ
R¨vwgwZK AbycvZ m¤ú‡K© e¨vL¨v Ki‡Z cvi‡e| †iLvs‡ki Aš—we©fw³ e¨vL¨v Ki‡Z cvi‡e| AbycvZ m¤úwK©Z Dccv`¨¸‡jv hvPvB I cÖgvY Ki‡Z cvi‡e| m`„kZvi AbycvZ msµvš— Dccv`¨¸‡jv hvPvB I cÖgvY Ki‡Z cvi‡e| cÖwZmgZvi aviYv e¨vL¨v Ki‡Z cvi‡e| nv‡Z-Kj‡g ev¯—e DcKi‡Yi mvnv‡h¨ †iLv I N~Y©b cÖwZmgZv hvPvB Ki‡Z cvi‡e|

14⋅1 AbycvZ I mgvbycv‡Zi ag©
(i ) a t b x t y Ges c t d x t y n‡j, a t b c t d y t x (e¨¯—KiY) b t y (GKvš—iKiY)

(ii )

a t b b t a n‡j, a b

(iii ) a t b (iv ) a t b

x t y n‡j, b t a x t y n‡j, a t x

(v )

a t b

c t d n‡j, ad

bc (Avo¸Yb)

(vi ) a t b

x t y n‡j, a

b t b

x

y t y (†hvRb)

Ges a b t b x
(vii)
a b c a n‡j, d a

y t y (we‡qvRb) b c d (†hvRb I we‡qvRb) b c d

R¨vwgwZK mgvbycvZ
Avgiv wÎfzR‡¶‡Îi †¶Îdj wbY©q Ki‡Z wk‡LwQ| G †_‡K `yBwU cÖ‡qvRbxq Abycv‡Zi aviYv ˆZwi Kiv hvq| (1) `yBwU wÎfzR‡¶‡Îi D”PZv mgvb n‡j, Zv‡`i †¶Îdj I f~wg mgvbycvwZK|

g‡b Kwi, wÎfzR‡¶Î ABC I DEF Gi f~wg h_vµ‡g BC

d Ges Dfq †¶‡Îi D”PZv h | 1 1 myZivs, wÎfzR‡¶Î ABC Gi †¶Îdj = a h , wÎfzR‡¶Î DEF Gi †¶Îdj = d h 2 2 1 1 AZGe, wÎfzR‡¶Î ABC Gi †¶Îdj t wÎfzR‡¶Î DEF Gi †¶Îdj = a h t d h 2 2 = a t d = BC t EF |

a , EF

MwYZ

229

(2) `yBwU wÎfzR‡¶‡Îi f~wg mgvb n‡j, Zv‡`i †¶Îdj I D”PZv mgvbycvwZK|

g‡b Kwi wÎfzR‡¶Î ABC I DEF Gi D”PZv h_vµ‡g AP

k Ges Dfq‡¶‡Îi f~wg b | 1 1 myZivs, wÎfzR‡¶Î ABC Gi †¶Îdj = b h , wÎfzR‡¶Î DEF Gi †¶Îdj = b k 2 2 1 1 AZGe, wÎfzR‡¶Î ABC Gi †¶Îdj t wÎfzR‡¶Î DEF Gi †¶Îdj = b h t b k 2 2 = h t k = AP t DQ |

h , DQ

Dccv`¨ 1 wÎfz‡Ri †h‡Kv‡bv evûi mgvš—ivj mij‡iLv H wÎfz‡Ri Aci evûØq‡K ev Zv‡`i ewa©ZvskØq‡K mgvb Abycv‡Z wef³ K‡i| we‡kl wbe©Pb : ABC wÎfz‡Ri BC evûi mgvš—ivj DE †iLvsk AB I AC evûØq‡K A_ev Zv‡`i ewa©ZvskØq‡K h_vµ‡g D I E we›`y‡Z †Q` K‡i‡Q| cÖgvY Ki‡Z n‡e †h, AD t DB AE t EC . wPÎ : 1 wPÎ : 2 A¼b : B , E Ges C , D †hvM Kwi| cÖgvY : h_v_©Zv avc (1) ADE Ges BDE GKB D”PZvwewkó [GKB D”PZvwewkó wÎfzRmg~‡ni †¶Îdj f~wgi ADE AD mgvbycvwZK] BDE DB (2)Avevi, ADE Ges DEC GKB D”PZvwewkó [GKB D”PZvwewkó wÎfzRmg~‡ni †¶Îdj f~wgi ADE AE mgvbycvwZK] DEC EC [GKB f~wg DE I GKB mgvš—ivj hyM‡ji g‡a¨ (3) wKš‘ BDE DEC Aew¯’Z] ADE ADE
BDE AD (4) AZGe, DB A_©vr, AD t DB DEC AE EC AE t EC.

Abywm×vš— 1|

Abywm×vš— 2| wÎfz‡Ri †Kv‡bv evûi ga¨we›`y w`‡q Aci GK evûi mgvš—ivj †iLv Z…Zxq evû‡K mgwØLwÊZ K‡i|

ABC wÎfz‡Ri BC evûi mgvš—ivj †Kv‡bv †iLv hw` AB I AC evû‡K h_vµ‡g AB AC AB AC n‡e| D I E we›`y‡Z †Q` K‡i, Z‡e Ges AD AE BD CE

230

MwYZ

Dccv`¨ 1 Gi wecixZ cÖwZÁvI mZ¨| A_©vr †Kv‡bv mij‡iLv GKwU wÎfz‡Ri `yB evû‡K A_ev Zv‡`i ewa©ZvskØq‡K mgvb Abycv‡Z wef³ Ki‡j D³ mij‡iLv wÎfzRwUi Z…Zxq evûi mgvš—ivj n‡e| wb‡P cÖwZÁvwU cªgvY Kiv n‡jv|

Dccv`¨ 2
†Kv‡bv mij‡iLv GKwU wÎfz‡Ri `yB evû‡K A_ev Zv‡`i ewa©ZvskØq‡K mgvb Abycv‡Z wef³ Ki‡j D³ mij‡iLv wÎfzRwUi Z…Zxq evûi mgvš—ivj| we‡kl wbe©Pb : DE †iLvsk ABC wÎfz‡Ri AB I AC evûØq‡K A_ev Zv‡`i ewa©ZvskØq‡K mgvb Abycv‡Z wef³ K‡i‡Q| A_©vr, AD t DB AE t EC cÖgvY Ki‡Z n‡e †h, DE Ges BC mgvš—ivj| A¼b : B, E Ges C , D †hvM Kwi| cÖgvY : avc h_v_©Zv
ADE AD BDE DB ADE AE Ges DEC EC AD AE (2) wKš‘ DB EC ADE ADE (3) AZGe, BDE BDE BDE DEC (4) wKš‘ BDE Ges DEC GKB f~wg DE Gi GKB cv‡k©¦

[ wÎfzR `yBwU GKB D”PZvwewkó ] [ wÎfzR `yBwU GKB D”PZvwewkó ] [¯^xKvi] [(1) Ges (2) †_‡K]

(1)

Aew¯’Z| myZivs Zviv GKB mgvš—ivj hyM‡ji g‡a¨ Aew¯’Z| BC I DE mgvš—ivj|

Dccv`¨ 3 wÎfz‡Ri †h‡Kv‡bv †Kv‡Yi Aš—wØ©LÊK wecixZ evû‡K D³ †KvY msjMœ evûØ‡qi Abycv‡Z Aš—we©f³ K‡i| we‡kl wbe©Pb : g‡b Kwi, AD †iLvsk ABC Gi Aš—t¯’ A †K mgwØLwÊZ K‡i BC evû‡K D we›`y‡Z †Q` K‡i| cÖgvY Ki‡Z n‡e †h, BD t DC BA t AC A¼b : DA †iLvs‡ki mgvš—ivj K‡i C we›`y w`‡q CE †iLvsk A¼b Kwi, †hb Zv ewa©Z BA evû‡K E we›`y‡Z †Q` K‡i| cÖgvY : avc h_v_©Zv [A¼b] (1) †h‡nZz DA ll CE Ges BC I AC Zv‡`i †Q`K [Abyiƒc †KvY] AEC BAD [GKvš—i †KvY] Ges ACE CAD

MwYZ

231

(2) wKš‘ BAD
AEC

CAD ACE ;

(3) Avevi, †h‡nZz DA ll CE , (4) wKš‘ AE
BD DC
AC BA AC

AC BD DC

AE BA AE

[¯^xKvi] [ Dccv`¨ 1] [avc (2)]

Dccv`¨ 4

wÎfz‡Ri †h‡Kv‡bv evû Aci `yB evûi Abycv‡Z Aš—we©f³ n‡j, wefvM we›`y †_‡K wecixZ kxl© ch©š— Aw¼Z †iLvsk D³ kxl©‡Kv‡Yi mgwØLÊK n‡e| we‡kl wbe©Pb : g‡b Kwi, ABC wÎfz‡Ri A we›`y †_‡K Aw¼Z AD mij‡iLvsk BC evû‡K D we›`y‡Z Giƒ‡c Aš—t¯’fv‡e wef³ K‡i‡Q †h, BD t DC BA t AC cÖgvY Ki‡Z n‡e †h, AD †iLvsk BAC Gi mgwØLÊK A_©vr, A¼b : DA †iLvs‡ki mgvš—ivj K‡i C we›`y w`‡q Giƒc CE †iLvsk A¼b Kwi †hb Zv BA evûi ewa©Zvsk‡K E we›`y‡Z †Q` K‡i| cÖgvY : avc (1) BCE Gi DA ll CE
BA t AE

BAD

CAD.

(2) wKš‘ BD t DC

BA t AE AE AC AZGe ACE AEC (3) wKš‘ AEC BAD Ges ACE CAD AZGe, BAD CAD A_©vr AD †iLvsk BAC Gi mgwØLÊK|

BD t DC BA t AC BA t AC

h_v_©Zv [A¼b] [ Dccv`¨ 1] [ ¯^xKvi ] [avc 1 I avc 2 †_‡K ] [mgwØevû wÎfz‡Ri f~wg msjMœ †KvY `yBwU mgvb] [Abyiƒc †KvY ] [GKvš—i †KvY ] [avc 2 †_‡K ]

Abykxjbx 14⋅1
1| 2| 3| 4| 5| †Kv‡bv wÎfz‡Ri f~wg msjMœ †KvYØ‡qi mgwØLÊKØq wecixZ evû `yBwU‡K X I Y we›`y‡Z †Q` K‡i| XY f~wgi mgvš—ivj n‡j cÖgvY Ki †h, wÎfyRwU mgwØevû| cÖgvY Ki †h, KZK¸‡jv ci¯úi mgvš—ivj mij‡iLv‡K `yBwU mij‡iLv †Q` Ki‡j Abyiƒc Ask¸‡jv mgvbycvwZK n‡e| cÖgvY Ki †h, UªvwcwRqv‡gi KY©Øq Zv‡`i †Q`we›`y‡Z GKB Abycv‡Z wef³ nq| cÖgvY Ki †h, UªvwcwRqv‡gi wZh©K evûØ‡qi ga¨we›`yi ms‡hvRK †iLvsk mgvš—ivj evûØ‡qi mgvš—ivj| ABC wÎfz‡Ri AD I BE ga¨gvØq ci¯úi G we›`y‡Z †Q` K‡i‡Q| G we›`yi ga¨ w`‡q Aw¼Z 6 EF . DE Gi mgvš—ivj †iLvsk AC †K F we›`y‡Z †Q` K‡i| cÖgvY Ki †h, AC

232

MwYZ

6| 7|

8|

AOC BX t XC A Gi mgwØLÊK BC †K D we›`y‡Z †Q` K‡i| BC Gi mgvš—ivj †Kv‡bv †iLvsk AC †K h_vµ‡g E I F we›`y‡Z †Q` K‡i| cÖgvY Ki †h, BD t DC BE t CF ABC I DEF m`„k‡KvYx wÎfzRØ‡qi D”PZv AM I DN n‡j cÖgvY Ki †h, AM t DN AB t DE . t

ABC AOB ABC AB I

Gi BC evû¯’ †h‡Kv‡bv we›`y X Ges AX †iLv¯’ O GKwU we›`y| cÖgvY Ki †h,

Gi

14⋅2 m`„kZv ( Similarity ) mßg †kÖwY‡Z wÎfz‡Ri me©mgZv I m`„kZv wb‡q Av‡jvPbv Kiv n‡q‡Q| mvaviYfv‡e, me©mgZv m`„kZvi we‡kl iƒc| `yBwU wPÎ me©mg n‡j †m¸‡jv m`„k; Z‡e wPÎ `yBwU m`„k n‡j ‡m¸‡jv me©mg bvI n‡Z cv‡i|

m`„k‡KvYx eûfzR : mgvb msL¨K evûwewkó `yBwU eûfy‡Ri GKwUi †KvY¸‡jv hw` avivevwnKfv‡e AciwUi
†KvY¸‡jvi mgvb nq, Z‡e eûfzR `yBwU‡K m`„k‡KvYx (equiangular) ejv nq|

m`„k eûfzR : mgvb msL¨K evûwewkó `yBwU eûfz‡Ri GKwUi kxl©we›`y¸‡jv‡K hw` avivevwnKfv‡e AciwUi kxl©we›`y¸‡jvi m‡½ Ggbfv‡e wgj Kiv hvq †h, eûfzR `y&BwUi (1) Abyiƒc †KvY¸‡jv mgvb nq Ges (2) Abyiƒc evû¸‡jvi AbycvZ¸‡jv mgvb nq, Z‡e eûfzR `yBwU‡K m`„k ( Similar ) eûfzR ejv nq| Dc‡ii wP‡Î Avgiv j¶ Kwi †h, ABCD AvqZ I PQRS eM© m`„k‡KvYx| KviY, Dfq wP‡Î evûi msL¨v 4 Ges Avq‡Zi †KvY¸‡jv avivevwnKfv‡e eM©wUi †KvY¸‡jvi mgvb (me¸‡jv †KvY mg‡KvY)| wKš‘ wPÎ¸‡jvi Abyiƒc †KvY¸‡jv mgvb n‡jI Abyiƒc evû¸‡jvi AbycvZ mgvb bq| d‡j †m¸‡jv m`„k bq | wÎfz‡Ri †¶‡Î Aek¨ GiKg nq bv| `yBwU wÎfz‡Ri kxl©we›`y¸‡jvi †KvY wgjKi‡Yi d‡j m`„kZvi msÁvq D‡jwLZ kZ© `yBwUi GKwU mZ¨ n‡j AciwUI mZ¨ nq Ges wÎfzR `yBwU m`„k nq| A_©vr, m`„k wÎfzR me©`v m`„k‡KvYx Ges m`„k‡KvYx wÎfzR me©`v m`„k| `yBwU wÎfzR m`„k‡KvYx n‡j Ges G‡`i †Kv‡bv GK †Rvov Abyiƒc evû mgvb n‡j wÎfzRØq mg©mg nq| `yBwU m`„k‡KvYx wÎfz‡Ri Abyiƒc evû¸‡jvi AbycvZ aª“eK| wb‡P G msµvš— Dccv‡`¨i cÖgvY †`Iqv n‡jv|

Dccv`¨ 5
`yBwU wÎfzR m`„k‡KvYx n‡j Zv‡`i Abyiƒc evû¸‡jv mgvbycvwZK| we‡kl wbe©Pb : g‡b Kwi, ABC I DEF wÎfzRØ‡qi A D, B E Ges cÖgvY Ki‡Z n‡e †h,
C F

AB AC BC DE DF EF A¼b : ABC I DEF wÎfzRØ‡qi cÖ‡Z¨K Abyiƒc evûhyMj Amgvb we‡ePbv Kwi| AB evû‡Z P we›`y evû‡Z Q we›`y wbB †hb Ges AC AP DE Ges AQ DF nq| P I Q †hvM

K‡i A¼b m¤úbœ Kwi|

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cÖgvY avc (1) APQ I DEF Gi AP DE ,
A D

h_v_©Zv
AQ DF ,

DEF DEF ABC Ges AQP DFE ACB. A_©vr, PQ †iLvsk I BC evû‡K AB evû I AC †iLv

AZGe, APQ myZivs, APQ

[ evû-‡KvY-evûi mg©mgZv ]

†Q` Kivq Abyiƒc †KvYhyMj mgvb n‡q‡Q|
AB AC AB AC ev, . DE DF AP AQ (2) GKBfv‡e BA evû I BC evû †_‡K h_vµ‡g ED †iLvsk I EF †iLvs‡ki mgvb †iLvsk †K‡U wb‡q †`Lv‡bv BA BC hvq †h, ED EF AB BC AB AC BC A_©vr ; . DE EF DE DF EF

myZivs, PQ ll BC;

[Dccv`¨ 1]

[Dccv`¨ 1]

Dccv`¨ 5 Gi wecixZ cÖwZÁvwUI mZ¨|

Dccv`¨ 6
`yBwU wÎfz‡Ri evû¸‡jv mgvbycvwZK n‡j Abyiƒc evûi wecixZ †KvY¸‡jv ci¯úi mgvb| we‡kl wbe©Pb : g‡b Kwi,
ABC I DEF Gi AB AC BC . DE DF EF A D, B E,

C F. cÖgvY Ki‡Z n‡e †h, . A¼b: ABC I DEF Gi cÖ‡Z¨K Abyiƒc evûhyMj Amgvb we‡ePbv Kwi| AB evû‡Z P we›`y Ges AC evû‡Z Q we›`y wbB †hb AP DE Ges AQ DF nq| P I Q †hvM K‡i A¼b m¤úbœ Kwi| cÖgvY : avc

h_v_©Zv

AB AC AB AC , myZivs . DE DF AP AQ myZivs, PQ ll BC ABC APQ Ges ACB AQP ABC I APQ m`„k‡KvYx| AB BC AB BC myZivs ev, . AP PQ DE PQ

(1) †h‡nZz

[Dccv`¨ 2] [ AB †Q`K Øviv Drcbœ Abyiƒc †KvY] [ AC †Q`K Øviv Drcbœ Abyiƒc †KvY] [Dccv`¨ 5]

dg©v-30, MwYZ-9g-10g

234

MwYZ

AB BC BC [Kíbvbymv‡i] ; PQ DE EF PQ myZivs, APQ I DEF me©mg| PAQ EDF , APQ DEF , AQP ACB APQ ABC Ges AQP A D, B E, C F.

BC EF EF

[ evû-evû-evû Dccv`¨ ]
DFE

Dccv`¨ 7
`yBwU wÎfz‡Ri GKwUi GK †KvY AciwUi GK †Kv‡Yi mgvb n‡j Ges mgvb mgvb †KvY msjMœ evû¸‡jv mgvbycvwZK n‡j wÎfzRØq m`„k| we‡kl wbe©Pb : g‡b Kwi, ABC Ges
A D Ges AB DE AC DF ABC Ges DEF Ggb †h,

cÖgvY Ki‡Z n‡e †h, A¼b :

DEF m`„k|

ABC I DEF Gi cÖ‡Z¨K Abyiƒc evûhyMj Amgvb we‡ePbv Kwi| AB evû‡Z P we›`y Ges AC evû‡Z Q we›`y wbB †hb AP DE Ges AQ DF nq| P I Q †hvM K‡i A¼b

m¤úbœ Kwi| cÖgvY : avc APQ I DEF Gi AP DE , AQ DF Ges Aš—f©y³ A Aš—fy©³ D , ABC DEF
APQ E , AQP F. AB AC AB AC Avevi, †h‡nZz , myZivs . DE DF AP AQ PQ ll BC myZivs ABC APQ Ges ACB AQP A D, B E Ges C F A_©vr, ABC I DEF m`„k‡KvYx| myZivs ABC I DEF m`„k| A D,

h_v_©Zv [evû-‡KvY-evû Dccv`¨ ]

[Dccv`¨ 2]

Dccv`¨ 8
`yBwU m`„k wÎfzR‡¶‡Îi †¶ÎdjØ‡qi AbycvZ Zv‡`i †h‡Kv‡bv `yB Abyiƒc evûi Dci Aw¼Z eM©‡¶‡Îi †¶ÎdjØ‡qi Abycv‡Zi mgvb| we‡kl wbe©Pb : g‡b Kwi, ABC I DEF wÎfzRØq m`„k Ges Zv‡`i `yBwU Abyiƒc evû BC I EF . cÖgvY Ki‡Z n‡e †h, ABC t
DEF BC 2 t EF 2

MwYZ

235

A¼b : BC I EF Gi Ici h_vµ‡g AG I DH j¤^ AuvwK| g‡b Kwi, AG h, DH cÖgvY : avc
ABC

p.

h_v_©Zv (1)
ABC DEF
1 2 1 2

1 BC.h Ges 2
BC.h EF . p h.BC p.EF

DEF h = p

1 EF . p 2
BC EF
E,

(2) ABG Ges DEH wÎfzRØ‡qi B AGB DHE (= GK mg‡KvY)|
BAG ABG I EDH DEH m`„k‡KvYx, ZvB m`„k|
BC [KviY EF ABC I

(3)

h p

AB DE

DEF m`„k ]

ABC DEF

h p

BC EF

BC EF

BC EF

BC 2 EF 2

14⋅3| wbw`©ó Abycv‡Z †iLvs‡ki wefw³KiY mgZ‡j `yBwU wfbœ we›`y A I B Ges m I n †h‡Kv‡bv ¯^vfvweK msL¨v n‡j Avgiv ¯^xKvi K‡i wbB †h,
AB †iLvq Ggb Abb¨ we›`y X Av‡Q †h, X we›`ywU A I B we›`yi Aš—e©Zx© Ges AX t XB = m t n.

Ic‡ii wP‡Î, AB †iLvsk X we›`y‡Z m t n Abycv‡Z Aš—we©f³ n‡q‡Q| Zvn‡j, AX t XB = m t n.

m¤úv`¨ 1
†Kv‡bv †iLvsk‡K GKwU wbw`©ó Abycv‡Z Aš—we©f³ Ki‡Z n‡e| g‡b Kwi, AB †iLvsk‡K m t n Abycv‡Z Aš—we©f³ Ki‡Z n‡e| A¼‡bi weeiY : A we›`y‡Z †h‡Kv‡bv †KvY BAX A¼b Kwi Ges AX iwk¥ †_‡K cici AE m Ges EC n Ask †K‡U wbB| B , C †hvM Kwi| E we›`y w`‡q CB Gi mgvš—ivj ED †iLvsk A¼b Kwi hv AB †K D we›`y‡Z †Q` K‡i| Zvn‡j AB †iLvsk D we›`y‡Z m t n Abycv‡Z Aš—we©f³ n‡jv| cÖgvY : †h‡nZz DE †iLvsk ABC wÎfz‡Ri GK evû BC Gi mgvš—ivj,
AD t DB AE t EC mt n
KvR : 1| weKí c×wZ‡Z †Kv‡bv †iLvsk‡K wbw`©ó Abycv‡Z Aš—we©f³ Ki|

236

MwYZ

D`vniY 1| 7 †m.wg. ˆ`‡N©¨i GKwU †iLvsk‡K 3t2 Abycv‡Z Aš—we©f³ Ki| mgvavb : †h‡Kv‡bv GKwU iwk¥ AG AuvwK Ges AG †_‡K 7 †m.wg. mgvb †iLvsk AB wbB| A we›`y‡Z †h‡Kv‡bv †KvY BAX A¼b Kwi| AX iwk¥ †_‡K AE 3 †m.wg. †K‡U wbB Ges EX †_‡K EC 2 †m.wg. †K‡U wbB| B , C †hvM Kwi| E we›`y‡Z ACB Gi mgvb AED A¼b Kwi hvi ED †iLv AB †K D we›`y‡Z †Q` K‡i| Zvn‡j AB †iLvsk D we›`y‡Z 3 t 2 Abycv‡Z Aš—we©f³ n‡jv| D`vniY 2| GKwU wbw`©ó wÎfz‡Ri m`„k GKwU wÎfzR A¼b Ki hvi evû¸‡jv g~j wÎfz‡Ri evû¸‡jvi
3 ¸Y| 5

Abykxjbx 14⋅2
1| wb‡Pi Z_¨¸‡jv j¶¨ Ki: i `yBwU ivwki Zzjbv Kivi Rb¨ Zv‡`i AbycvZ we‡ePbv Kiv nq ii AbycvZ wbY©‡qi Rb¨ ivwk `yBwU GKB GK‡K cwigvc Ki‡Z nq iii AbycvZ wbY©‡qi †¶‡Î ivwk `ywU GKB RvZxq n‡Z nq wb‡Pi †KvbwU mwVK ? K. i I ii L. ii I iii M. i I iii N. i, ii I iii

Dc‡ii wP‡Îi Z_¨vbymv‡i (2 I 3) bs cÖ‡kœi DËi `vI: 2| ABC Gi D”PZv I f~wgi AbycvZ KZ? K. 3|
1 2 4 5 2 5 5 4

L.

M.

N.

ABD Gi †¶Îdj KZ eM© GKK? K. 6 L. 20 M. 40 N. 50 ABC- G PQ11 BC n‡j wb‡Pi †KvbwU mwVK? K. AP : PB = AQ : QC L. AB : PQ = AC : PQ M. AB : AC = PQ : BC N. PQ : BC = BP : BQ

4|

5| GKwU e‡M©i m‡e©v”P (†gvU) KZwU cÖwZmvg¨ †iLv Av‡Q? K. 10wU L. 8wU M. 6wU N. 4wU 6| cÖgvY Ki †h, `yBwU wÎfz‡Ri cÖ‡Z¨KwU hw` Aci Z…Zxq GKwU wÎfz‡Ri m`„k nq, Z‡e Zviv ci¯úi m`„k|

MwYZ

237

7|

cÖgvY Ki †h, `yBwU mg‡KvYx wÎfz‡Ri GKwUi GKwU m~²‡KvY AciwUi GKwU m~²‡Kv‡Yi mgvb n‡j, wÎfzR `yBwU m`„k n‡e| 8| cÖgvY Ki †h, mg‡KvYx wÎfz‡Ri mg‡KŠwYK kxl© †_‡K AwZfy‡Ri Dci j¤^ AuvK‡j †h `yBwU mg‡KvYx wÎfzR Drcbœ nq, Zviv ci¯úi m`„k Ges cÖ‡Z¨‡K g~j wÎfz‡Ri m`„k| 9| cv‡ki wP‡Î, B D Ges CD 4AB. cÖgvY Ki †h, BD 5 BL. 10| ABCD mvgvš—wi‡Ki A kxl© w`‡q Aw¼Z GKwU †iLvsk BC evû‡K M we›`y‡Z Ges DC evû ewa©Zvsk‡K N we›`y‡Z †Q` K‡i| cÖgvY Ki †h, BM DN GKwU aª“eK| 11| cv‡ki wP‡Î BD AC Ges
1 QC . BD 5 BL. 2 cÖgvY Ki †h, DA DC . 12| ABC I DEF Gi A D. cÖgvY Ki †h, ABC t DEF AB. AC t DE .DF . 13| ABC Gi A Gi mgwØLÊK AD, BC †K D we›`y‡Z †Q` K‡i‡Q| DA Gi mgvš—ivj CE †iLvsk ewa©Z BA evû‡K E we›`y‡Z †Q` K‡i‡Q| DQ BA 2 AQ

K. Z_¨ Abymv‡i wPÎwU A¼b Ki| L. cÖgvY Ki †h, BD t DC BA t AC M. BC Gi mgvš—ivj †Kv‡bv †iLvsk AB I AC †K h_vµ‡g P I Q we›`y‡Z †Q` Ki‡j, cÖgvY Ki †h, BD t DC BP t CQ 14| wP‡Î ABC Ges DEF `yBwU m`„k wÎfzR| K. wÎfzR `yBwUi Abyiƒc evû I Abyiƒc †KvY¸‡jvi bvg wjL| L. cÖgvY Ki †h,
ABC DEF AB 2 DE 2
3 †m.wg., EF

AC 2 DF 2

BC 2 EF 2 B 60 , BC AB 3 Ges 2 ABC 3 eM© †m.wg. nq,

M. hw` BC

8 †m.wg.,

Z‡e DEF A¼b Ki Ges Gi †¶Îdj wbY©q Ki|

14⋅4 cÖwZmgZv cÖwZmgZv GKwU cÖ‡qvRbxq R¨vwgwZK aviYv hv cÖK…wZ‡Z we`¨gvb Ges hv Avgv‡`i Kg©Kv‡Ê cÖwZwbqZ e¨envi K‡i _vwK| cÖwZmgZvi aviYv‡K wkíx, KvwiMi, wWRvBbvi, myZviiv cÖwZwbqZ e¨envi K‡i _v‡Kb| Mv‡Qi cvZv, dzj, †gŠPvK, Nievwo, †Uwej, †Pqvi mewKQzi g‡a¨ cÖwZmgZv we`¨gvb| hw` †Kv‡bv mij‡iLv eivei †Kv‡bv wPÎ fuvR Ki‡j Zvi Ask `yBwU m¤ú~Y©fv‡e wg‡j hvq †m‡¶‡Î mij‡iLvwU‡K cÖwZmvg¨ †iLv ejv nq|

238

MwYZ

Dc‡ii wPÎ¸‡jvi cÖwZwUi cÖwZmvg¨ †iLv i‡q‡Q| †k‡li wPÎwUi GKvwaK cÖwZmvg¨ †iLv i‡q‡Q|
KvR : 1| mywg KvMR †K‡U cv‡ki wP‡Îi wWRvBb ˆZwi K‡i‡Q| wP‡Î cÖwZmg †iLvmg~n wPwýZ Ki| Gi KqwU cÖwZmg †iLv i‡q‡Q ? 2| Bs‡iwR eY©gvjvi †h mKj e‡Y©i cÖwZmvg¨ †iLv i‡q‡Q †m¸‡jv wj‡L cÖwZmvg¨ †iLv wPwýZ Ki|

14⋅5 mylg eûfz‡Ri cÖwZmvg¨ †iLv eûfzR KZK¸‡jv †iLvsk Øviv Ave× wPÎ| eûfz‡Ri †iLvsk¸‡jvi ˆ`N©¨ mgvb I †KvY¸‡jv mgvb n‡j Zv‡K mylg eûfzR ejv nq| wÎfzR n‡jv me‡P‡q Kg msL¨K †iLvsk w`‡q MwVZ eûfzR| mgevû wÎfzR n‡jv wZb eûwewkó mylg eûfzR| mgevû wÎfy‡Ri evû I †KvY¸‡jv mgvb| Pvi eûwewkó mylg eûfzR n‡jv eM©‡¶Î| eM©‡¶‡ÎiI evû I †KvY¸‡jv mgvb | Abyiƒcfv‡e, mylg cÂfzR I mylg lofz‡Ri evû I †KvY¸‡jv mgvb|

mgevû wÎfzR

eM©‡¶Î

mylg cÂfzR

mylg lofzR

cÖ‡Z¨K mylg eûfzR GKwU cÖwZmg wPÎ| myZivs Zv‡`i cÖwZmvg¨ †iLvi m¤ú‡K© Rvbv Avek¨K| mylg eûfz‡Ri A‡bK evûi cvkvcvwk GKvwaK cÖwZmvg¨ †iLv i‡q‡Q| wZbwU cÖwZmvg¨ †iLv PviwU cÖwZmvg¨ †iLv cuvPwU cÖwZmvg¨ †iLv QqwU cÖwZmvg¨ †iLv

mgevû wÎfzR

eM©‡¶Î

mylg cÂfzR

mylg lofzR

cÖwZmgZvi aviYvi mv‡_ Avqbvi cÖwZdj‡bi m¤úK© i‡q‡Q| †Kv‡bv R¨vwgwZK wP‡Îi cÖwZmvg¨ †iLv ZLbB _v‡K, hLb Zvi Aa©vs‡ki cÖwZ”Qwe evwK Aa©vs‡ki mv‡_ wg‡j hvq| GRb¨ cÖwZmvg¨ †iLv wbY©‡q KvíwbK Avqbvi Ae¯’vb †iLvi mvnvh¨ ‡bqv nq| †iLv cÖwZmgZv‡K cÖwZdjb cÖwZmgZvI ejv nq|

MwYZ

239

Abykxjbx 14⋅3
1| wb‡Pi wPÎmg~‡ni †KvbwUi cÖwZmvg¨ †iLv i‡q‡Q? (K) evwoi wPÎ (L) gmwR‡`i wPÎ (M) gw›`‡ii wPÎ (M) MxR©vi wPÎ, (M) c¨v‡MvWvi wPÎ (N) cvj©v‡g›U fe‡bi wPÎ, (O) gy‡Lv‡ki wPÎ (P) ZvRgn‡ji wPÎ 2| cÖwZmvg¨ †iLv †`Iqv Av‡Q, Ab¨ dzUwK cÖ`k©b Ki :

3| cÖwZmvg¨ †iLv †`Iqv Av‡Q (W¨vkhy³ †iLv), R¨vwgwZK wPÎ m¤ú~Y© Ki Ges kbv³ Ki|

4| wb‡Pi R¨vwgwZK wP‡Î cÖwZmvg¨ †iLv wb‡`©k Ki:

5|

wb‡Pi Am¤ú~Y© R¨vwgwZK wPÎ m¤ú~Y© Ki †hb Avqbv †iLv mv‡c‡¶ cÖwZmg nq :

6|

7|

7|

wb‡Pi R¨vwgwZK wP‡Îi cÖwZmvg¨ †iLvi msL¨v wbY©q Ki: (K) mgwØevû wÎfzR (L) welgevû wÎfzR (M) eM©‡¶Î (O) mylg lofzR (P) cÂfzR (Q) e„Ë Bs‡iwR eY©gvjvi †h mKj e‡Y©i (K) AbyfzwgK Avqbv (L) Dj¤^ Avqbv (M) AbyfzwgK I Dj¤^ Dfq Avqbv mv‡c‡¶ cÖwZdjb cÖwZmgZv i‡q‡Q †m¸‡jv AuvK| cÖwZmgZv †bB Ggb wZbwU wPÎ A¼b Ki|

(N) i¤^m

14⋅6 N~Y©b cÖwZmgZv
†Kv‡bv wbw`©ó we›`yi mv‡c‡¶ N~Y©‡bi d‡j e¯‘i AvK…wZ I AvKv‡ii cwieZ©b nq bv| Z‡e e¯‘i wewfbœ As‡ki Ae¯’v‡bi cwieZ©b nq| N~Y©‡bi d‡j e¯‘i bZzb Ae¯’v‡b e¯‘i AvK…wZ I AvKvi Avw` Ae¯’v‡bi b¨vq GKB n‡j Avgiv ewj e¯‘wUi N~Y©b cÖwZmgZv i‡q‡Q| †hgb, mvB‡K‡ji PvKv, wmwjs d¨vb, eM© BZ¨vw`| GKwU wmwjs d¨v‡bi cvLv¸‡jvi N~Y©‡bi d‡j GKvwaKevi g~j Ae¯’v‡bi mv‡_ wg‡j hvq| cvLv¸‡jv Nwoi KuvUvi w`‡KI Nyi‡Z cv‡i Avevi wecixZ w`‡KI Nyi‡Z cv‡i| mvB‡K‡ji PvKv Nwoi KuvUvi w`‡KI Nyi‡Z cv‡i, Avevi wecixZ w`‡KI Nyi‡Z cv‡i| Nwoi KuvUvi w`‡K N~Y©b‡K abvZ¥K w`K wnmv‡e aiv nq|

240

MwYZ

†h we›`yi mv‡c‡¶ e¯‘wU †Nv‡i Zv n‡jv N~Y©b †K›`ª| N~Y©‡bi mgq †h cwigvY †Kv‡Y †Nv‡i Zv n‡jv N~Y©b †KvY| GKevi c~Y© N~Y©‡bi †Kv‡Yi cwigvY 360°, AaŸ© N~Y©‡bi †Kv‡Yi cwigvY 180°| wP‡Î Pvi cvLvwewkó d¨v‡bi 90° K‡i N~Y©‡bi d‡j wewfbœ Ae¯’vb ‡`Lv‡bv n‡q‡Q| j¶ Kwi, GKevi c~Y© N~Y©‡b wVK PviwU Ae¯’v‡b (90°, 180°, 270° I 360° ‡Kv‡Y N~Y©‡bi d‡j) d¨vbwU †`L‡Z ûeû GKB iKg| GRb¨ ejv nq d¨vbwUi N~Y©b cÖwZmgZvi gvÎv 4|

N~Y©b cÖwZmgZvi Ab¨ GKwU D`vniY †bqv hvq| GKwU e‡M©i KY© `yBwUi †Q`we›`y‡K N~Y©b †K›`ª awi| N~Y©b †K‡›`ªi mv‡c‡¶ eM©wUi GK-PZz_©vsk N~Y©‡bi d‡j †h‡Kv‡bv †KŠwYK wK›`yi Ae¯’vb wØZxq wP‡Îi b¨vq n‡e| Gfv‡e Pvievi GK-PZz_©vsk N~Y©‡bi d‡j eM©wU Avw` Ae¯’v‡b wd‡i Av‡m| ejv nq, e‡M©i 4 gvÎvi N~Y©b cÖwZmgZv i‡q‡Q|

j¶ Kwi, †h‡Kv‡bv wPÎ GKevi c~Y© N~Y©‡bi d‡j Avw` Ae¯’v‡b wd‡i Av‡m| ZvB †h‡Kv‡bv R¨vwgwZK wP‡Îi 1 gvÎvi N~Y©b cÖwZmgZv i‡q‡Q| N~Y©b cÖwZmgZv wbY©‡qi †¶‡Î wb‡Pi welq¸‡jv j¶ ivL‡Z n‡e: (K) N~Y©b †K›`ª (L) N~Y©b †KvY (M) N~Y©‡bi w`K (N) N~Y©b cÖwZmgZvi gvÎv|
KvR : 1| †Zvgvi Pvicv‡ki cwi‡ek †_‡K 5wU mgZjxq e¯‘i D`vniY `vI hv‡`i N~Y©b cÖwZmgZv i‡q‡Q| 2| wb‡Pi wP‡Îi N~Y©b cÖwZmgZv wbY©q Ki| (M) (N) (K) (O) (L)

14⋅7 †iLv cÖwZmgZv I N~Y©b cÖwZmgZv
Avgiv †`‡LwQ ‡h wKQz R¨vwgwZK wP‡Îi ïay †iLv cÖwZmgZv i‡q‡Q, wKQzi ïay N~Y©b cÖwZmgZv i‡q‡Q| Avevi †Kv‡bv †Kv‡bv wP‡Îi †iLv cÖwZmgZv I N~Y©b cÖwZmgZv DfqB we`¨gvb| †hgb, e‡M©i ‡hgb PviwU cÖwZmvg¨ †iLv i‡q‡Q, †Zgwb 4 gvÎvi N~Y©b cÖwZmgZv i‡q‡Q| e„Ë GKwU Av`k© cÖwZmg wPÎ| e„Ë‡K Gi †K‡›`ªi mv‡c‡¶ †h‡Kv‡bv †Kv‡Y I †h‡Kv‡bv w`‡K Nyiv‡j Gi Ae¯’v‡bi cwieZ©b j¶ Kiv hvq bv| AZGe, e„‡Ëi N~Y©b cÖwZmgZvi gvÎv Amxg| GKB mgq e„‡Ëi †K›`ªMvgx †h‡Kv‡bv †iLv Gi cÖwZmvg¨ †iLv| myZivs, e„‡Ëi AmsL¨ cÖwZmvg¨ †iLv i‡q‡Q|
KvR : 1| Bs‡iwR eY©gvjvi K‡qKwU e‡Y©i †iLv cÖwZmgZv I N~Y©b cÖwZmgZv wba©viY Ki Ges wb‡Pi mviwYwU c~iY Ki: (GKwU K‡i †`Lv‡bv n‡jv)

MwYZ

241

eY©

Z H O E C

†iLv cÖwZmgZv †bB

cÖwZmvg¨ †iLvi msL¨v 0

N~Y©b cÖwZmgZv nu¨v

N~Y©b cÖwZmgZvi gvÎv 2

Abykxjbx 14⋅4
1| wb‡Pi wP‡Îi N~Y©b cÖwZmgZv wbY©q Ki :
(M) (K) (L) (N) (O) (P)

2| GKwU †jey AvovAvwo †K‡U wP‡Îi b¨vq AvKvi cvIqv †Mj| mgZjxq wPÎwUi N~Y©b cÖwZmgZv wbY©q Ki|

3|

k~b¨¯’vb c~iY Ki : wPÎ N~Y©b †K›`ª N~Y©b cÖwZmgZvi gvÎv N~Y©b cÖwZmgZvi ‡KvY

eM© AvqZ i¤^m mgevû wÎfzR Aa©eË „ mylg cÂfzR 4| †h mKj PZzf©y‡Ri †iLv cÖwZmgZv I 1 Gi AwaK gvÎvi N~Y©b cÖwZmgZv i‡q‡Q, Zv‡`i ZvwjKv Ki| 5| 1 Gi AwaK gvÎvi N~Y©b cÖwZmgZv i‡q‡Q Giƒc wP‡Îi N~Y©b †KvY 18° n‡Z cv‡i Kx ? †Zvgvi DË‡ii c‡¶ hyw³ `vI|

dg©v-31, MwYZ-9g-10g

cÂ`k Aa¨vq

†¶Îdj m¤úwK©Z Dccv`¨ I m¤úv`¨
(Area Related Theorems and Constructions)
Avgiv Rvwb mxgve× mgZj †¶‡Îi AvK…wZ wewfbœ iKg n‡Z cv‡i| mgZj †¶Î hw` PviwU evûØviv mxgve× nq, Z‡e Zv‡K Avgiv PZzf©yR e‡j _vwK| GB PZzf©y‡Ri Avevi †kÖwY wefvM Av‡Q Ges AvK…wZ I ˆewk‡ó¨i Dci wfwË K‡i Zv‡`i bvgKiYI Kiv n‡q‡Q| GB mKj mgZj †¶‡Îi evB‡i A‡bK †¶Î Av‡Q hv‡`i evû Pv‡ii AwaK| Av‡jvwPZ G mKj †¶ÎB eûfyR‡¶Î| cÖ‡Z¨K mxgve× mgZj‡¶‡Îi wbw`©ó cwigvc Av‡Q hv‡K †¶Îdj e‡j AewnZ Kiv nq| GB mKj †¶Îdj cwigv‡ci Rb¨ mvaviYZ GK GKK evûwewkó eM©‡¶‡Îi †¶Îdj e¨envi Kiv nq Ges Zv‡`i †¶Îdj‡K eM© GKK wn‡m‡e †jLv nq| †hgb, evsjv‡`‡ki †¶Îdj 144 (cÖvq) nvRvi eM© wK‡jvwgUvi| Avgv‡`i ˆ`bw›`b Rxe‡bi cÖ‡qvRb †gUv‡Z eûfzR †¶‡Îi †¶Îdj Rvb‡Z I cwigvc Ki‡Z nq| ZvB G ¯—‡ii wk¶v_©x‡`i eûfzR †¶‡Îi †¶Îdj m¤^‡Ü mg¨K Ávb cÖ`vb Kiv AZxe ¸i“Z¡c~Y©| GLv‡b eûfzR †¶‡Îi †¶Îd‡ji aviYv Ges GZ`msµvš— KwZcq Dccv`¨ I m¤úv`¨ welqK welqe¯‘ Dc¯’vcb Kiv n‡q‡Q| Aa¨vq †k‡l wk¶v_©xiv eûfzR †¶‡Îi †¶Îd‡ji aviYv e¨vL¨v Ki‡Z cvi‡e| †¶Îdj msµvš— Dccv`¨ hvPvB I cÖgvY Ki‡Z cvi‡e| cÖ`Ë DcvË e¨envi K‡i eûfyR †¶Î A¼b I A¼‡bi h_vh©Zv hvPvB Ki‡Z cvi‡e| wÎfyR‡¶‡Îi †¶Îd‡ji mgvb PZzf©yR‡¶Î A¼b Ki‡Z cvi‡e| PZzf©yR‡¶‡Îi †¶Îd‡ji mgvb wÎfyR‡¶Î A¼b Ki‡Z cvi‡e|

15⋅1 mgZj †¶‡Îi †¶Îdj cÖ‡Z¨K mxgve× mgZj †¶‡Îi wbw`©ó †¶Îdj i‡q‡Q| GB †¶Îdj cwigv‡ci Rb¨ mvaviYZ GK GKK evûwewkó eM©‡¶‡Îi †¶Îdj‡K eM© GKK wn‡m‡e MÖnY Kiv nq| †hgb, †h eM©‡¶‡Îi GK evûi ˆ`N©¨ GK †mw›UwgUvi Zvi †¶Îdj n‡e GK eM©‡mw›UwgUvi| Avgiv Rvwb, (K)
ABCD AvqZ‡¶‡Îi

ˆ`N©¨ AB a GKK (h_v, wgUvi) cÖ¯’ BC b GKK (h_v, wgUvi) n‡j, ABCD AvqZ‡¶‡Îi †¶Îdj = ab eM© GKK (h_v, eM©wgUvi)|

MwYZ

243

(L)

ABCD eM©‡¶‡Îi evûi

ˆ`N©¨ = a GKK (h_v, wgUvi) n‡j,
ABCD eM©‡¶‡Îi †¶Îdj = a 2 eM© GKK

(h_v, eM©wgUvi)| `yBwU †¶‡Îi †¶Îdj mgvb n‡j Zv‡`i g‡a¨ Ô=Õ wPý e¨envi Kiv nq| †hgb, ABCD AvqZ‡¶‡Îi †¶Îdj = AED wÎfzR‡¶‡Îi †¶Îdj| D‡jL¨ †h,
ABC
ABC I

DEF me©mg n‡j,

DEF †jLv nq| G‡¶‡Î Aek¨B

ABC Gi †¶Îdj =

DEF Gi †¶Îdj|

wKš‘ `yBwU wÎfzR‡¶‡Îi †¶Îdj mgvb n‡jB wÎfzR `yBwU me©mg nq bv| †hgb, wP‡Î ABC Gi †¶Îdj = DBC Gi †¶Îdj| wKš‘ ABC I DBC me©mg bq|

Dccv`¨ 15⋅1
GKB f~wgi Dci Ges GKB mgvš—ivj †iLvhyM‡ji g‡a¨ Aew¯’Z mKj wÎfyR‡¶‡Îi †¶Îdj mgvb|

g‡b Kwi, ABC I DBC wÎfzR‡¶ÎØq GKB f~wg BC Gi Dci Ges GKB mgvš—ivj †iLvhyMj BC I AD
Gi g‡a¨ Aew¯’Z| cÖgvY Ki‡Z n‡e †h, †¶Î ABC Gi †¶Îdj = †¶Î DBC Gi †¶Îdj| A¼b : BC †iLvs‡ki B I C we›`y‡Z h_vµ‡g BE I CF j¤^ A¼b Kwi| Giv AD †iLvi ewa©Z Ask‡K E we›`y‡Z Ges AD †iLv‡K F we›`y‡Z †Q` K‡i| d‡j EBCF GKwU AvqZ‡¶Î ˆZwi nq| †¶Î ABC Ges AvqZ‡¶Î EBCF GKB f~wg BC Gi cÖgvY : EBCF GKwU AvqZ‡¶Î, GLb Dci Ges BC I ED mgvš—ivj †iLvs‡ki g‡a¨ Aew¯’Z| myZivs
EBCF ) Abyiƒcfv‡e,

†¶Î ABC =

1 (AvqZ‡¶Î 2

†¶Î DBC †¶‡Îi †¶Îdj =

1 (AvqZ‡¶Î EBCF ) 2

†¶Î ABC †¶Îdj =

†¶Î DBC -Gi †¶Îdj (cÖgvwYZ)|

244

MwYZ

Dccv`¨ 1
GKB f~wgi Dci Ges GKB mgvš—ivj †iLvhyM‡ji g‡a¨ Aew¯’Z mKj wÎfyR‡¶‡Îi †¶Îdj mgvb|

g‡b Kwi, ABC I DBC wÎfzR‡¶ÎØq GKB f~wg BC Gi Dci Ges GKB mgvš—ivj †iLvhyMj BC I
AD Gi g‡a¨ Aew¯’Z| cÖgvY Ki‡Z n‡e †h,

†¶Î ABC Gi †¶Îdj =

†¶Î DBC Gi

†¶Îdj| A¼b : BC †iLvs‡ki B I C we›`y‡Z h_vµ‡g BE I CF j¤^ A¼b Kwi| Giv AD †iLvi ewa©Z Ask‡K E we›`y‡Z Ges AD †iLv‡K F we›`y‡Z †Q` K‡i| d‡j EBCF GKwU AvqZ‡¶Î ˆZwi nq| cÖgvY : EBCF GKwU AvqZ‡¶Î, GLb †¶Î ABC Ges AvqZ‡¶Î EBCF GKB f~wg BC Gi Dci Ges BC I ED mgvš—ivj †iLvs‡ki g‡a¨ Aew¯’Z| myZivs
EBCF ) Abyiƒcfv‡e,

†¶Î ABC =

1 (AvqZ‡¶Î 2

†¶Î DBC †¶‡Îi †¶Îdj =

1 (AvqZ‡¶Î EBCF ) 2

†¶Î ABC †¶Îdj =

†¶Î DBC -Gi †¶Îdj (cÖgvwYZ)|

Dccv`¨ 2
GKB f~wgi Dci Ges GKB mgvš—ivj †iLvhyM‡ji g‡a¨ Aew¯’Z mvgvš—wiK‡¶Îmg~‡ni †¶Îdj mgvb|

wP‡Î, ABCD I EFGH mvgvš—wiK‡¶Î `yBwU AB I EF f~wg AB Gi Dci Ges F GKB mgvš—ivj †iLvhyMj AF I DG Gi g‡a¨ Aew¯’Z| cÖgvY Ki‡Z n‡e †h, mvgvš—wiK ABCD Gi †¶Îdj = mvgvš—wiK‡¶Î EFGH . EFGH Gi f~wg EF mgvb nq| GLb AC I EG †hvM Kwi| C I G we›`y †_‡K f~wg AF I Gi ewa©Z †iLvs‡ki Dci CL I GK j¤^ Uvwb| cÖgvY : ABC Gi †¶Îdj =
EFG Gi †¶Îdj

1 AB GL Ges 2

AB

1 EF GK . 2 EF Ges CL GK , (A¼bymv‡i)

AZGe, ABC Gi †¶Îdj = . EFG Gi †¶Îdj

MwYZ
1 1 mvgvš—wiK ABCD Gi †¶Îdj = mvgvš—wiK EFGH Gi †¶Îdj 2 2

245

mvgvš—wiK ABCD Gi †¶Îdj = mvgvš—wiK EFGH (cÖgvwYZ|

Dccv`¨ 3 (cx_v‡Mviv‡mi Dccv`¨) mg‡KvYx wÎfz‡Ri AwZfz‡Ri Ici Aw¼Z eM©‡¶‡Îi †¶Îdj Aci `yB evûi Ici Aw¼Z eM©‡¶‡ÎØ‡qi †¶Îd‡ji mgwói mgvb| we‡kl wbe©Pb : g‡b Kwi, ABC mg‡KvYx wÎfy‡Ri mg‡KvY Ges AB AwZfzR| cÖgvY Ki‡Z n‡e †h,
ACB

AB 2 BC 2 AC 2 . A¼b : AB , AC Ges BC evûi Dci h_vµ‡g ABED , ACGF Ges BCHK eM©‡¶Î A¼b Kwi| C we›`y w`‡q AD ev BE †iLvi mgvš—ivj CL †iLv AuvwK| g‡bKwi, Zv AB †K M we›`y‡Z Ges DE †K L we›`y‡Z †Q` K‡i| C I D Ges B I F †hvM Kwi|

cÖgvY : avc (1) CAD I Aš—fy©³ CAD
FAB G CA = AF , AD = AB Ges

h_v_©Zv
CAB = CAB BAD CAF
[ BAD =

CAF = 1 mg‡KvY ]

= Aš—fy©³ BAF AZGe, CAD FAB (2) wÎfzR‡¶Î CAD Ges AvqZ‡¶Î ADLM GKB f~wg AD Gi Dci Ges AD I CL mgvš—ivj †iLvØ‡qi g‡a¨ Aew¯’Z| myZivs, AvqZ‡¶Î ADLM = 2 (wÎfzR‡¶Î CAD ) (3) wÎfzR‡¶Î BAF Ges eM©‡¶Î ACGF GKB f~wg AF Gi Dci Ges AF I BG mgvš—ivj †iLvØ‡qi g‡a¨ Aew¯’Z| myZivs, eM©‡¶Î ACGF = 2 (wÎfzR‡¶Î FAB ) = 2 (wÎfzR‡¶Î CAD ) (4) AvqZ‡¶Î ADLM = eM©‡¶Î ACGF (5) Abyiƒcfv‡e C, E I A, K †hvM K‡i cÖgvY Kiv hvq †h, AvqZ‡¶Î BELM = eM©‡¶Î BCHK (6) AvqZ‡¶Î ( ADLM + BELM )= eM©‡¶Î ACGF + eM©‡¶Î BCHK ev, eM©‡¶Î ABED = eM©‡¶Î ACGF + eM©‡¶Î BCHK A_©vr, AB 2 BC 2 AC 2 [cÖgvwYZ]

[evû-†KvY-evû Dccv`¨]

[Dccv`¨ 1]

[Dccv`¨ 1] [(2) Ges (3) †_‡K ] [(4) Ges (5) †_‡K ]

246

MwYZ

m¤úv`¨ 1
Ggb GKwU mvgvš—wiK AuvK‡Z n‡e, hvi GKwU †KvY GKwU wbw`©ó †Kv‡Yi mgvb Ges hv Øviv mxgve× †¶Î GKwU wÎfzR‡¶‡Îi †¶Îd‡ji mgvb|

g‡b Kwi, ABC GKwU wbw`©ó wÎfzR‡¶Î Ges x GKwU wbw`©ó †KvY| Giƒc mvgvš—wiK AuvK‡Z n‡e, hvi GKwU †KvY x Gi mgvb Ges hv Øviv mxgve× †¶‡Îi †¶Îdj †¶Î ABC Gi †¶Îd‡ji mgvb| A¼b : BC evû‡K E we›`y‡Z mgwØLwÊ Kwi| EC †iLvs‡ki E we›`y‡Z x Gi mgvb CEF AuvwK| A we›`y w`‡q BC evûi mgvš—ivj AG iwk¥ Uvwb Ges g‡b Kwi Zv EF iwk¥‡K F we›`y‡Z †Q` K‡i| C we›`y w`‡q EF †iLvs‡ki mgvš—ivj CG iwk¥ Uvwb Ges g‡b Kwi Zv AG iwk¥‡K G we›`y‡Z †Q` K‡i| Zvn‡j, ECGF B DwÏó mvgvš—wiK| cÖgvY : A, E †hvM Kwi| GLb, †¶Î ABE Gi †¶Îdj = †¶Î AEC Gi †¶Îdj [†h‡nZz f~wg BE = f~wg EC Ges Df‡qi GKB D”PZv] †¶Î ABC Gi †¶Îdj = 2 ( †¶Î AEC Gi †¶Îdj) Avevi, mvgvš—wiK †¶Î ECGF Gi †¶Îdj 2 ( †¶Î AEC Gi †¶Îdj) [†h‡nZz, Df‡q GKB f~wg EC &Gi Dci Aew¯’Z Ges EC ll AG mvgvš—wiK †¶Î ECGF Gi †¶Îdj = †¶Î ABC Gi †¶Îdj Avevi, . CEF = x [†h‡nZz EF ll CG , A¼b Abymv‡i] mvgvš—wiK ECGF B wb‡Y©q mvgvš—wiK|

m¤úv`¨ 2
Ggb GKwU wÎfzR AuvK‡Z n‡e hv Øviv mxgve× †¶‡Îi †¶Îdj GKwU wbw`©ó PZyf©yR‡¶‡Îi †¶Îd‡ji mgvb|

g‡b Kwi, ABCD GKwU PZzf©yR‡¶Î| Giƒc GKwU wÎfzR AuvK‡Z n‡e hv Øviv mxgve× †¶‡Îi †¶Îdj ABCD PZzf©yR‡¶‡Îi †¶Îd‡ji mgvb|

MwYZ

247

A¼b : D, B †hvM Kwi| C we›`y w`‡q CE ll DB Uvwb| g‡b Kwi, Zv AB evûi ewa©Zvsk‡K E we›`y‡Z †Q` K‡i| D, E †hvM Kwi| Zvn‡j, DAE B DwÏó wÎfzR| cÖgvY : BD f~wgi Dci BDC I BDE Aew¯’Z Ges DB ll CE [A¼b Abymv‡i] †¶Î BDC Gi †¶Îdj = †¶Î BDC Gi †¶Îdj + †¶Î ABD Gi †¶Îdj| †¶Î BDE Gi †¶Îdj †¶Î ABD Gi †¶Îdj = †¶Î ADE Gi †¶Îdj| †¶Î BDE Gi †¶Îdj +

PZzf©yR‡¶Î ABCD Gi †¶Îdj = AZGe, ADE B wb‡Y©q wÎfyR|

we‡kl `ªóe¨ : Dc‡ii c×wZi mvnv‡h¨ wbw`©ó PZzf©yR‡¶‡Îi †¶Îd‡ji mgvb †¶Îdj wewkó AmsL¨ wÎfzR‡¶Î AuvKv hv‡e|

m¤úv`¨ 3
Ggb GKwU mvgvš—wiK AuvK‡Z n‡e hvi GKwU †KvY †`Iqv Av‡Q Ges Zv Øviv mxgve× †¶Î GKwU wbw`©ó PZzf©yR‡¶‡Îi †¶Îd‡ji mgvb|

g‡b Kwi, ABCD GKwU wbw`©ó PZzf©yR‡¶Î Ges x GKwU wbw`©ó †KvY| Giƒc GKwU mvgvš—wiK AuvK‡Z n‡e hvi GKwU †KvY cÖ`Ë x Gi mgvb Ges mxgve× †¶‡Îi †¶Îdj ABCD †¶‡Îi †¶Îd‡ji mgvb| A¼b : B, D †hvM Kwi| C we›`y w`‡q CF ll DB Uvwb Ges g‡b Kwi, CE, AB evûi ewa©Zvsk‡K F we›`y‡Z †Q` K‡i| AF †iLvs‡ki ga¨we›`y G wbY©q Kwi| AG †iLvs‡ki A we›`y‡Z x Gi mgvb GAK AuvwK Ges G we›`y w`‡q GH ll AK Uvwb| D we›`y w`‡q KDH ll AG Uvwb Ges g‡b Kwi, Zv AK I GH †K h_vµ‡g K I H we›`y‡Z †Q` K‡i| Zvn‡j, AGHK B DwÏó mvgvš—wiK| cÖgvY : D, F †hvM Kwi| AGHK GKwU mvgvš—wiK [A¼b Abymv‡i] †¶Î DAF Gi †¶Îdj = PZzf©yR‡&¶Î ABCD Gi †¶Îdj Ges †hLv‡b, GAK x Avevi, mvgvš—wiK †¶Î AGHK Gi †¶Îdj = wÎfzR‡¶Î DAF Gi †¶Îdj| AZGe, AGHK B wb‡Y©q mvgvš—wiK|

248

MwYZ

Abykxjbx 15
1| wÎfy‡Ri wZbwU evûi ˆ`N©¨ †`Iqv Av‡Q; wb‡Pi †Kvb †¶‡Î mg‡KvYx wÎfyR A¼b m¤¢e bq? K. 3 cm, 4 cm, 5 cm L. 6 cm, 8 cm, 10 cm M. 5 cm, 7 cm, 9 cm N. 5 cm, 12 cm, 13 cm

2|

wb‡Pi Z_¨¸‡jv j¶ Ki: i cÖ‡Z¨K mxgve× mgZj †¶‡Îi wbw`©ó †¶Îdj i‡q‡Q ii `yBwU wÎfyR †¶‡Îi †¶Îdj mgvb n‡jB wÎfyR `yBwU me©mg iii `yBwU wÎfyR me©mg n‡j Zv‡`i †¶Îdj mgvb wb‡Pi †KvbwU mwVK ? K. i I ii L. i I iii M. ii I iii N. i, ii I iii wb‡Pi wP‡Î, ABC mgevû, AD BC Ges AB=2 Z‡_¨i wfwË‡Z (3 I 4) bs cÖ‡kœi DËi `vI :

3|

4|

BD = KZ ? K. 1 M. 2 wÎfyRwUi D”PZv KZ ? K. M.
4 e. GKK 3 2 e. GKK 3

L. 2 N. 4 L.
3 e. GKK

N. 2 3 e. GKK

5| 6| 7| 8| 9|

cÖgvY Ki †h, mvgvš—wi‡Ki KY©Øq mvgvš—wiK‡¶ÎwU‡K PviwU mgvb wÎfyR‡¶‡Î wef³ K‡i| cÖgvY Ki †h, †Kv‡bv eM©‡¶Î Zvi K‡Y©i Dci Aw¼Z eM©‡¶‡Îi A‡a©K| cÖgvY Ki †h, wÎfz‡Ri †h‡Kv‡bv ga¨gv wÎfzR‡¶ÎwU‡K mgvb †¶Îdj wewkó `yBwU wÎfzR‡¶‡Î wef³ K‡i| GKwU mvgvš—wiK‡¶‡Îi Ges mgvb †¶Îdjwewkó GKwU AvqZ‡¶Î GKB f~wgi Dci Ges Gi GKB cv‡k Aew¯’Z| †`LvI †h, mvgvš—wiK‡¶ÎwUi cwimxgv AvqZ‡¶ÎwUi cwimxgv A‡c¶v e„nËi| ABC Gi AB I AC evûØ‡qi ga¨we›`y h_vµ‡g X I Y . cÖgvY Ki †h, †¶Î AXY Gi †¶Îdj = 1 (
4

†¶Î ABC GiK †¶Îdj)|

10| wP‡Î, ABCD GKwU UªvwcwRqvg| Gi AB I CD evû `yBwU mgvš—ivj| UªvwcwRqvg‡¶Î ABCD Gi †¶Îdj wbY©q Ki|

MwYZ

249

11| mvgvš—wiK ABCD Gi Af¨š—‡i P †h‡Kv‡bv GKwU we›`y| cÖgvY Ki †h, †¶Îdj + 12| †¶Î PCD Gi †¶Îdj =

†¶Î PAB Gi

1 (mvgvš—wiK‡¶Î ABCD Gi †¶Îdj) 2 ABC G BC f~wgi mgvš—ivj †h‡Kv‡bv mij‡iLv AB I AC evû‡K h_vµ‡g D I F we›`y‡Z

†Q` K‡i| cÖgvY Ki †h,
CDE . 13| ABC wÎfz‡Ri

†¶Î DBC =

†¶Î EBC Ges

†¶Î DBF =

†¶Î

A = GK mg‡KvY| D, AC Gi Dci¯’ GKwU we›`y|

cÖgvY Ki †h, BC 2 ‡`LvI †h, 4 AD 2 cÖgvY Ki †h, PB 2 16|
ABC Gi
AB 2 AC 2

AD 2
3 AB 2 .

BD 2

AC 2 .

14| ABC GKwU mgevû wÎfzR Ges AD, BC Gi Ici j¤^| 15| ABC GKwU mgwØevû mg‡KvYx wÎfzR| BC Gi AwZfzR Ges P, BC Gi Ici †h‡Kv‡bv we›`y|
PC 2 2 PA 2 .

C ¯’j‡KvY ; AD, BC Gi
BC 2

Ici j¤^| †`LvI †h,
2 BC.CD.

17|

ABC Gi
AB 2 AC 2

C m~²‡KvY ; AD, BC Gi
BC 2

Ici j¤^| †`LvI †h,
2 BC .CD.

18|

ABC Gi AD GKwU ga¨gv| †`LvI †h,
AB 2 AC 2 2( BD 2 AD 2 )

dg©v-32, MwYZ-9g-10g

lô`k Aa¨vq

cwiwgwZ
(Mensuration)
e¨envwiK cÖ‡qvR‡b, †iLvi ˆ`N©¨, Z‡ji †¶Îdj, Nbe¯‘i AvqZb BZ¨vw` cwigvc Kiv nq| G iKg †h‡Kv‡bv ivwk cwigv‡ci †¶‡Î GKB RvZxq wbw`©ó cwigv‡Yi GKwU ivwk‡K GKK wnmv‡e MÖnY Kiv nq| cwigvcK…Z ivwk Ges Giƒc wba©vwiZ GK‡Ki AbycvZB ivwkwUi cwigvc wba©viY K‡i| A_©vr cwigvc = cwigvcK…Z ivwk | GKK ivwk

wba©vwiZ GKK m¤ú‡K© cÖ‡Z¨K cwigvc GKwU msL¨v hv cwigvcK…Z ivwkwUi GKK ivwki KZ¸Y Zv wb‡`©k K‡i| †hgb, †eÂwU 5 wgUvi j¤^v| GLv‡b wgUvi GKwU wbw`©ó ˆ`N©¨ hv‡K GKK wnmv‡e aiv n‡q‡Q Ges hvi Zzjbvq †eÂwU 5 ¸Y j¤^v| Aa¨vq †k‡l wk¶v_©xiv wÎfyR‡¶Î I PZzf©yR‡¶‡Îi †¶Îd‡ji m~Î cÖ‡qvM K‡i eûfyR‡¶‡Îi †¶Îdj wbY©q Ges GZ`m¤úwK©Z mgm¨v mgvavb Ki‡Z cvi‡e| e„‡Ëi cwiwa I e„Ëvs‡ki ˆ`N©¨ wbY©q Ki‡Z cvi‡e| e„‡Ëi †¶Îdj wbY©q Ki‡Z cvi‡e| e„Ë‡¶Î I Zvi Askwe‡k‡li †¶Îdj wbY©q K‡i GZ`m¤úwK©Z mgm¨v mgvavb Ki‡Z cvi‡e| AvqZKvi Nbe¯‘, NbK I †ej‡bi †¶Îdj cwigvc Ki‡Z cvi‡e Ges G m¤úwK©Z mgm¨v mgvavb Ki‡Z cvi‡e| mylg I †hŠwMK Nbe¯‘i c„ôZ‡ji †¶Îdj cwigvc Ki‡Z cvi‡e|

16⋅1 wÎfyR‡¶‡Îi †¶Îdj c~‡e©i †kªwY‡Z Avgiv †R‡bwQ, wÎfyR‡¶‡Îi †¶Îdj =
1 2

f~wg

D”PZv

(1) mg‡KvYx wÎfyR : g‡b Kwi, ABC mg‡KvYx wÎfy‡Ri mg‡KvY msjMœ evûØq h_vµ‡g BC D”PZv we‡ePbv Ki‡j, a Ges AB b | BC †K f~wg Ges AB †K

MwYZ
1 2
1 ab 2

251

ABC Gi †¶Îdj

= =

f~wg

D”PZv

(2) wÎfyR‡¶‡Îi `yB evû I Zv‡`i Aš—f©y³ †KvY †`Iqv Av‡Q| g‡b Kwi, ABC wÎfy‡Ri evûØq BC CA b , AB c | A †_‡K BC evûi Dci AD j¤^ AuvwK| awi, D”PZv AD h | †KvY C we‡ePbv Ki‡j cvB, ev, h b
AD CA sin C

a,

sin C ev, h b sin C
1 BC AD 2 1 = a b sin C 2 1 = absin C 2 1 †¶Î ABC Gi †¶Îdj = bc sin A 2 1 = ca sin B 2
ABC Gi BC a , CA b Ges AB c|

†¶Î ABC Gi †¶Îdj =

Abyiƒcfv‡e

(3) wÎfy‡Ri wZbevû †`Iqv Av‡Q| g‡b Kwi, Gi cwimxgv 2s a b c AD BC AuvwK| awi, BD x Zvn‡j, CD a x ABD Ges ACD mg‡KvYx
AD2 AB2 BD 2 Ges AD2

AC 2 CD2

AB2

BD2 b2 AC 2 CD2
(a x)2

ev, c2 x2 ev, c 2 x 2 c2 b2

a2

2ax x 2

ev, 2ax c2 a2 b2 x a 2 b2 2a

Avevi, AD2

c2

x2

252

MwYZ

c

2

c 2 a 2 b2 2a

2

= c = = =

c2

a 2 b2 2a

c

c2

a 2 b2 2a

2ac c2 a2 b2 2ac c2 a2 b2 2a 2a {(c a)2 b2}{b2 4a2 (c a)2}

(a b c)(a b c 2b)(a b c 2a)(a b c 2c) 4a2 2s(2s 2b)(2s 2a)(2s 2c) = 4a2 4s(s a)(s b)(s c) = a2

AD

2 s(s a)(s b)(s c) a

†¶Î ABC Gi †¶Îdj = = =

1 BC AD 2 1 2 a s ( s a)( s b)(b c) a 2

s ( s a)( s b)(b c)

(4) mgevû wÎfyR : g‡b Kwi, ABC mgevû wÎfy‡Ri cÖ‡Z¨K evûi ˆ`N©¨ a
AD BC AuvwK|
BD CD a 2

ABD mg‡KvYx

BD2

AD2

AB2 a2 a 2
2

ev, AD2
AD

AB2 BD2

a2

a2 4

3a 2 4

3a 2
1 BC AD 2

†¶Î ABC Gi †¶Îdj = =

1 a 2

3a 3 2 ev, a 2 4

254

MwYZ

D`vniY 3| GKwU wÎfy‡Ri wZbwU evûi ˆ`N©¨ h_vµ‡g †m.wg., †m.wg. I †m.wg.| Gi †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, wÎfyRwUi evû¸‡jvi ˆ`N©¨ h_vµ‡g a 7 †m.wg., b 8 †m.wg. Ges c 9 †m.wg. Aa©cwimxgv s Gi †¶Îdj = a b c 2 7 8 9 †m.wg. = 12 †m.wg. 2

s(s a)(s b)(s c)

= 12(12 7)(12 8)(12 9) eM© †m.wg. = 12 5 6 7 eM© †m.wg. = 50 2 eM© †m.wg. wÎfyRwU †¶Îdj 50 2 eM© †m.wg. (cÖvq)| D`vniY 4| GKwU mgevû wÎfy‡Ri cÖ‡Z¨K evûi ˆ`N©¨ wgUvi evov‡j †¶Îdj 3 3 eM©wgUvi †e‡o hvq| wÎfyRwUi evûi ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, mgevû wÎfy‡Ri cÖ‡Z¨K evûi ˆ`N©¨ a wgUvi| Gi †¶Îdj =
3 2 a eM©wgUvi| 4

wÎfyRwUi cÖ‡Z¨K evûi ˆ`N©¨ cÖkœvbymv‡i,
3 (a 1)2 4

wgUvi evov‡j wÎfyRwUi †¶Îdj =
3 2 a 4 3 3

3 (a 1)2 eM©wgUvi| 4

ev, (a 1)2 a2 12

3 Øviv fvM K‡i] 4

ev, a2 2a 1 a2 12 ev, 2a 11 ev, a 5 5 wb‡Y©q evûi ˆ`N©¨ 5 5 wgUvi| eM© †m.wg. n‡j, mgvb mgvb D`vniY 5| GKwU mgwØevû wÎfy‡Ri f~wgi ˆ`N©¨ †m.wg.| Gi †¶Îdj evûi ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, mgwØevû wÎfy‡Ri f~wg b 60 †m.wg. Ges mgvb mgvb evûi ˆ`N©¨ a | Gi †¶Îdj = cÖkœvbymv‡i, ev, b 4a2 b2 4
1200 1200

b 4a2 b2 4 60 4a2 4

(60)2

ev, 15 4a2 3600 1200 ev, 4a2 3600
80

MwYZ

255

ev, 4a2 3600 6400 eM© K‡i ev, 4a2 10000 ev, a2 a 2500
50

wÎfyRwUi mgvb evûi ˆ`N©¨

†m.wg.|

D`vniY 6| GKwU wbw`©ó ¯’vb †_‡K `yBwU iv¯—v 120o †Kv‡Y P‡j †M‡Q| `yBRb †jvK H wbw`©ó ¯’vb †_‡K wK‡jvwgUvi I NÈvq wK‡jvwgUvi †e‡M wecixZ w`‡K iIbv n‡jv| NÈv c‡i Zv‡`i h_vµ‡g NÈvq g‡a¨ mivmwi `~iZ¡ wbY©q Ki| mgvavb : g‡b Kwi, A ¯’vb †_‡K `yBRb †jvK h_vµ‡g NÈvq wK‡jvwgUvi I NÈvq wK‡jvwgUvi †e‡M iIbv n‡q NÈv ci B I C ¯’v‡b †cuŠwQj| Zvn‡j, NÈv ci Zv‡`i g‡a¨ mivmwi `~iZ¡ n‡e BC . C †_‡K BA Gi ewa©Zvs‡ki Ici CD j¤^ Uvwb|
AB 5 10 wK‡jvwgUvi =

wK‡jvwgUvi, AC 5 8 wK‡jvwgUvi =

wK‡jvwgUvi

Ges

BAC

120o

DAC 180o 120o
ACD mg‡KvYx

60o
3 2

CD AC

sin 60o ev, CD

AC sin 60o

40

20 3

AD cos 60o ev, AD AC cos 60o AC Avevi, BCD mg‡KvYx †_‡K cvB,

Ges

40

1 2

20

BC 2

BD2

CD2

(BA

AD)2

CD2 6100

= (50 20)2 (20 3)2
BC 78 1 (cÖvq)

4900 1200

wb‡Y©q `~iZ¡ 78 1 wK‡jvwgUvi (cÖvq) GKK, GKK I GKK| e„nËi evûi wecixZ D`vniY 7| GKwU wÎfy‡Ri evû¸‡jvi ˆ`N©¨ h_vµ‡g kxl©we›`y †_‡K Aw¼Z j¤^ wÎfyRwU‡K †h `yBwU wÎfy‡R wef³ K‡i Zv‡`i †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, ABC wÎfy‡Ri BC 25 GKK, AC 20 GKK, AB 15 GKK| A kxl©we›`y †_‡K BC evûi Dci Aw¼Z j¤^ AD wÎfyR‡¶ÎwU‡K ABD I ACD †¶‡Î wef³ K‡i| awi, BD x Ges AD h
CD BC BD 25 x

ABD mg‡KvYx -G

BD2

AD2

AB2

ev, x2 h2

(15)2

256

MwYZ x2 h2

225..........(i)

Ges ACD mg‡KvYx
CD2 AD2 AC 2 ev, (25 x)2 h2 (20)2

ev, 625 50x x2 h2 400 ev, 625 50x 225 400 mgxKiY (i) Gi mvnv‡h¨ ev, 50x 450 x 9 mgxKiY (i) G x Gi gvb ewm‡q cvB,
81 h2 225 ev, h2 144

h 12

†¶Î ABD Gi †¶Îdj = Ges

1 1 BD AD 9 12 eM©GKK = 36 eM©GKK 2 2 1 1 †¶Î ACD Gi †¶Îdj = BD AD (25 9) 12 eM©GKK 2 2

=

1 16 12 eM©GKK = 96 eM©GKK 2

wb‡Y©q †¶Îdj 36 eM©GKK Ges 96 eM©GKK|

Abykxjbx 16⋅1
1| GKwU mg‡KvYx wÎfy‡Ri AwZfyR 25 wgUvi| Gi GKwU evû AciwUi
3 Ask n‡j, evû `yBwUi ˆ`N©¨ 4

2|

wbY©q Ki| wgUvi j¤^v GKwU †`Iqv‡ji mv‡_ Lvovfv‡e Av‡Q| gBwUi †Mvov †`Iqvj †_‡K KZ `~‡i miv‡j Ic‡ii cÖvš— wgUvi wb‡P bvg‡e| GKU mgwØevû wÎfy‡Ri cwimxgv †¶Îdj wbY©q Ki| GKwU wÎfy‡Ri `yBwU evûi ˆ`N©¨ wbY©q Ki| wgUvi| Gi mgvb mgvb evûi ˆ`N©¨ f~wgi
5 Ask n‡j, wÎfyRwUi 6

3|

4| 5| 6|

†m.wg.,

†m.wg. Ges cwimxgv

†m.wg.| wÎfyRwUi †¶Îdj

GKwU mgevû wÎfy‡Ri cÖ‡Z¨K evûi ˆ`N©¨ wÎfyRwUi evûi ˆ`N©¨ wbY©q Ki| GKwU wÎfy‡Ri `yB evûi ˆ`N©¨ h_vµ‡g evûØ‡qi Aš—f©y³ †KvY wbY©q Ki| GKwU mg‡KvYx wÎfy‡Ri j¤^ f~wgi

wgUvi evov‡j Gi †¶Îdj 6 3 eM©wgUvi †e‡o hvq| wgUvi, wgUvi Ges †¶Îdj eM©wgUvi n‡j,
4 Ask †_‡K 3

7|

11 Ask †_‡K 12

†m.wg. Kg Ges AwZfyR f~wgi

†m.wg. Kg| wÎfyRwUi f~wgi ˆ`N©¨ wbY©q Ki|

MwYZ

257

8| 9|

GKwU mgwØevû wÎfy‡Ri mgvb mgvb evûi ˆ`N©¨ wbY©q Ki|

wgUvi Ges †¶Îdj

eM©wgUvi n‡j, f~wgi ˆ`N©¨

GKwU wbw`©ó ¯’vb †_‡K `yBwU iv¯—v ci¯úi 135o †KvY K‡i `yBw`‡K P‡j †M‡Q| `yBRb †jvK H wbw`©ó ¯’vb †_‡K h_vµ‡g NÈvq wK‡jvwgUvi I NÈvq wK‡jvwgUvi †e‡M wecixZ gy‡L iIbv n‡jv| NÈv ci Zv‡`i g‡a¨ mivmwi `~iZ¡ wbY©q Ki| 10| GKwU mgevû wÎfy‡Ri Af¨š—i¯’ GKwU we›`y †_‡K wZbwUi Ici Aw¼Z j‡¤^i ˆ`N©¨ h_vµ‡g †m.wg., †m.wg. I †m.wg.| wÎfyRwUi evûi ˆ`N©¨ Ges †¶Îdj wbY©q Ki|

16⋅2 PZzf©yR‡¶‡Îi †¶Îdj
(1) AvqZ‡¶‡Îi †¶Îdj g‡b Kwi, ABCD AvqZ‡¶‡Îi ˆ`N©¨ AB a cÖ¯’ BC b Ges KY© AC d Avgiv Rvwb, AvqZ‡¶‡Îi KY© AvqZ‡¶ÎwU‡K mgvb `yBwU wÎfyR‡¶‡Î wef³ K‡i| AvqZ‡¶Î ABCD Gi †¶Îdj = 2 = 2 AvqZ‡¶ÎwUi cwimxgv s 2(a b) Ges ABC mg‡KvYx
AC 2 AB2 BC 2 ev, d 2 a 2 b2 d a2 b2

†¶Î ABC Gi †¶Îdj
1 a b = ab 2

(2) eM©‡¶‡Îi †¶Îdj g‡b Kwi, ABCD eM©‡¶‡Îi cÖwZ evûi ˆ`N©¨ a Ges KY© d
AC KY© eM©‡¶ÎwU‡K mgvb `yBwU wÎfyR‡¶‡Î wef³ K‡i|

eM©‡¶Î ABCD Gi †¶Îdj = 2 = 2 j¶ Kwi. eM©‡¶‡Îi cwimxgv s Ges KY© d a2 a2 2a2

†¶Î ABC Gi †¶Îdj
1 a a = a2 2

4a
2a

(3) mvgvš—wiK‡¶‡Îi †¶Îdj
(K) f~wg I D”PZv †`Iqv Av‡Q| g‡b Kwi, ABCD mvgvš—wiK‡¶‡Îi f~wg AB b Ges D”PZv DE h BD KY© mvgvš—wiK‡¶ÎwU‡K mgvb `yBwU wÎfyR‡¶‡Î wef³ K‡i| dg©v-33, MwYZ-9g-10g

258

MwYZ

mvgvš—wiK‡¶Î ABCD Gi †¶Îdj = 2 = 2

†¶Î ABD Gi †¶Îdj
1 b h 2

= bh (L) GKwU K‡Y©i ˆ`N©¨ Ges H K‡Y©i wecixZ †KŠwYK we›`y †_‡K D³ K‡Y©i Ici Aw¼Z j‡¤^i ˆ`N©¨ †`Iqv Av‡Q| g‡b Kwi, ABCD mvgvš—wiK‡¶‡Îi KY© AC d Ges Gi wecixZ †KŠwYK we›`y D †_‡K AC Gi Dci Aw¼Z j¤^ DE h | KY© AC mvgvš—wiK‡¶ÎwU‡K mgvb `yBwU wÎfyR‡¶‡Î wef³ K‡i| mvgvš—wiK‡¶Î ABCD Gi †¶Îdj = 2 = 2 = dh †¶Î ACD Gi †¶Îdj
1 d h 2

(4) i¤^‡mi †¶Îdj i¤^‡mi `yBwU KY© †`Iqv Av‡Q| g‡b Kwi, ABCD i¤^‡mi KY© AC d1 , KY© BD d Ges KY©Øq ci¯úi O we›`y‡Z †Q` K‡i| KY© AC i¤^m‡¶ÎwU‡K mgvb `yBwU wÎfyR‡¶‡Î wef³ K‡i| Avgiv Rvwb, i¤^‡mi KY©Øq ci¯úi‡K mg‡Kv‡Y mgwØLwÊZ K‡i
ACD Gi D”PZv =

d2 2

i¤^m ABCD Gi †¶Îdj = 2 = 2 =

†¶Î ACD Gi †¶Îdj d 1 d1 2 2 2

1 d1d2 2

(5) UªvwcwRqvg‡¶‡Îi †¶Îdj
UªvwcwRqvg‡¶‡Îi mgvš—ivj `yBwU evû Ges G‡`i ga¨eZ©x j¤^ `~iZ¡ †`Iqv Av‡Q| g‡b Kwi, ABCD UªvwcwRqvg‡¶‡Îi mgvš—ivj evûØ‡qi ˆ`N©¨ h_vµ‡g AB a GKK, CD b GKK Ges G‡`i ga¨eZ©x `~iZ¡ CE AF h | AC KY© UªvwcwRqvg ABCD †¶ÎwU‡K ABC I ACD †¶‡Î wef³ K‡i| UªvwcwRqvg‡¶Î ABCD Gi †¶Îdj = †¶Î ABC Gi †¶Îdj + †¶Î ACD Gi †¶Îdj = =
1 AB CE 2 1 CD AF 2

1 1 1 ah bh = h(a b) 2 2 2

MwYZ

259
3 ¸Y| Gi †¶Îdj 384 eM©wgUvi n‡j, cwimxgv I K‡Y©i 2

D`vniY 1| GKwU AvqZvKvi N‡ii ˆ`N©¨ cÖ‡¯’i

ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, AvqZvKvi N‡ii cÖ¯’ x wgUvi| N‡ii ˆ`N©¨ Ges †¶Îdj cÖkœvbymv‡i,
3x2 2
3x wgUvi 2 3x 2 x ev,

3x2 eM©wgUvi| 2

384 ev, 3x2

768 ev, x2

256

x 16 wgUvi

AvqZvKvi NiwUi ˆ`N©¨ =

3 16 wgUvi = 24 wgUvi 2

Ges cÖ¯’ = 16 wgUvi| Gi cwimxgv = 2(24 16) wgUvi = 80 wgUvi Ges K‡Y©i ˆ`N©¨ = (24)2 (16)2 wgUvi = 832 wgUvi = 28 84 wgUvi (cÖvq) wb‡Y©q cwimxgv 80 wgUvi Ges K‡Y©i ˆ`N©¨ 28 84 wgUvi (cÖvq)| D`vniY 2| GKwU AvqZ‡¶‡Îi †¶Îdj 2000 eM©wgUvi| hw` Gi ˆ`N©¨ GKwU eM©‡¶Î nZ| AvqZ‡¶ÎwUi ˆ`N©¨ I cÖ¯’ wbY©q Ki| mgvavb : g‡b Kwi, AvqZ‡¶ÎwUi ˆ`N©¨ x wgUvi Ges cÖ¯’ y wgUvi| AvqZ‡¶ÎwUi †¶Îdj = xy eM©wgUvi| cÖkœvbymv‡i, xy 2000................( 1) Ges x 10 mgxKiY ( ) G y x( x 10) y.............(2) x 10.............(3)

wgUvi Kg nZ Zvn‡j GwU

mgxKiY ( ) †_‡K cvB, y

x 10 ewm‡q cvB
0

2000 ev, x2 10x 2000

ev, x2 50x 40x 2000 0 ev, ( x 50)(x 40) 0 x 50 0 A_ev x 40 0 ev, x 50 A_ev x 40

wKš‘ ˆ`N©¨ FYvZ¥K n‡Z cv‡i bv| x 50

GLb, mgxKiY ( ) G x Gi gvb ewm‡q cvB, y 50 10 40

AvqZ‡¶ÎwUi ˆ`N©¨

wgUvi Ges cÖ¯’

wgUvi|

260

MwYZ

D`vniY 3| eM©vKvi GKwU gv‡Vi wfZ‡i Pviw`‡K wgUvi PIov GKwU iv¯—v Av‡Q| hw` iv¯—vi †¶Îdj †n±i nq, Z‡e iv¯—v ev‡` gv‡Vi wfZ‡ii †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, eM©vKvi gv‡Vi ˆ`N©¨ x wgUvi| Gi †¶Îdj x2 eM©wgUvi| gv‡Vi wfZ‡i Pviw`‡K wgUvi PIov GKwU iv¯—v Av‡Q| iv¯—v ev‡` eM©vKvi gv‡Vi ˆ`N©¨ = ( x 2 4) ev ( x 8) wgUvi| iv¯—v ev‡` eM©vKvi gv‡Vi †¶Îdj = (x 8)2 eM©wgUvi myZivs iv¯—vi †¶Îdj = {x 2 ( x 8) 2 } eM©wgUvi Avgiv Rvwb, †n±i = 10000 eM©wgUvi cÖkœvbymv‡i, x 2 ( x 8) 2 10000 ev, x 2 x 2 16 x 64 10000 ev, 16 x 10064 x 629

iv¯—vev‡` eM©vKvi gv‡Vi †¶Îdj = (629 8)2 eM©wgUvi = 385641 eM©wgUvi = 38 56 †n±i (cÖvq) wb‡Y©q †¶Îdj 38 56 †n±i (cÖvq)| eM© †m.wg. Ges GKwU KY© D`vniY 4| GKwU mvgvš—wiK‡¶‡Îi †¶Îdj †KŠwYK we›`y †_‡K D³ K‡Y©i Ici Aw¼Z j‡¤^i ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, mvgvš—wiK‡¶‡Îi GKwU KY© d Ici Aw¼Z j‡¤^i ˆ`N©¨ h †m.wg.| mvgvš—wiK‡¶ÎwUi †¶Îdj = dh eM© †m.wg. cÖkœvbymv‡i, dh 120 ev, h
120 d 120 24 5

†m.wg.| KY©wUi wecixZ

24 †m.wg. Ges Gi wecixZ †KŠwYK we›`y †_‡K K‡Y©i

wb‡Y©q K‡Y©i ˆ`N©¨ 5 †m.wg.| D`vniY 5| GKwU mvgvš—wi‡Ki evûi ˆ`N©¨ wgUvi I wgUvi Ges ¶z`ªZg KY©wU wgUvi n‡j, Aci KY©wUi ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, ABCD mvgvš—wi‡Ki AB a 12 wgUvi, AD c 8 wgUvi Ges KY© BD b 10 wgUvi| D I C †_‡K AB Gi Dci Ges AB Gi ewa©Zvs‡ki Dci DF I CE j¤^ Uvwb| A, C I B, D †hvM Kwi|

MwYZ
12 10 8 wgUvi = 15 wgUvi 2

261

ABD Gi Aa© cwimxgv s

†¶Î ABD Gi †¶Îdj =

s ( s a )( s b)( s c)

= 15(15 12)(15 10)(15 8) eM©wgUvi = 1575 eM©wgUvi = 39 68 eM©wgUvi (cÖvq)| Avevi, †¶Î ABD Gi †¶Îdj = ev, 39 68
BE 2
BE

1 AB DF 2

1 12 DF 2
AD2

ev, 6 DF
DF 2 82

39 68

DF

6 61

GLb, BCE mg‡KvYx
BC 2 CE 2
BE

(6 61)2

20 31

4 5 AZGe, AE AB

12 4 5 16 5

BCE mg‡KvYx †_‡K cvB,

AC 2
AC

AE 2 CE 2

(16 5)2 (6 61)2

315 94

17 77 (cÖvq)

wb‡Y©q K‡Y©i ˆ`N©¨ 17 77 wgUvi (cÖvq) D`vniY 6| GKwU i¤^‡mi GKwU KY© wbY©q Ki|

wgUvi Ges †¶Îdj

eM©wgUvi n‡j, Aci KY© Ges cwimxgv

mgvavb : g‡b Kwi, ABCD i¤^‡mi KY© BD d1 10 wgUvi Ges Aci KY© d2 wgUvi i¤^mwUi †¶Îdj = 1 d1d2 eM©wgUvi
2

cÖkœvbymv‡i,

1 d1d2 2

120

ev, d2 120 2 120 2 24
10 10

Avgiv Rvwb, i¤^‡mi KY©Øq ci¯úi‡K mg‡Kv‡Y mgwØLwÊZ K‡i|
OD OB 10 2

wgUvi = 5 wgUvi Ges OA OC 24 wgUvi = 12 wgUvi
2

Ges AOD mg‡KvYx -G
AD 2 OA 2 OD 2 52 (12) 2 169
AD 13

i¤^‡mi cÖwZevûi ˆ`N©¨ 13 wgUvi| i¤^‡mi cwimxgv = 4 13 wgUvi = 52 wgUvi| wb‡Y©q K‡Y©i ˆ`N©¨ 24 wgUvi Ges cwimxgv 52 wgUvi|

262

MwYZ

D`vniY 7| GKwU UªvwcwRqv‡gi mgvš—ivj evûØ‡qi ˆ`N©¨ h_vµ‡g †m.wg. I †m.wg. Ges Aci evû `yBwUi ˆ`N©¨ q_vµ‡g †m.wg. I †m.wg.| UªvwcwRqvgwUi †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, ABCD UªvwcwRqv‡gi AB 91 †m.wg., CD 51 †m.wg.| D I C †_‡K AB Gi Dci h_vµ‡g DE I CF j¤^Uvwb|
CDEF GKwU AvqZ‡¶Î|
EF CD 51 †m.wg.|

awi, AE
BF

x Ges DE
AB AF

CF

h
EF ) 91 ( x 51) 40 x

91 ( AE

ADE mg‡KvYx †_‡K cvB,
AE 2 DE 2 AD 2 ev, x2 h2
(13)2 ev, x2 h2 169.........(i)

Avevi, mg‡KvYx Gi †¶‡Î BCF BF 2 CF 2 BC 2 ev, (40 x) 2 h 2 ev, 1600 ev, 1600 ev, 1600 ev, 80x mgxKiY (
5
2

(37) 2

80x x2 h2 1369 80x 169 1396 ; ( ) bs Gi mvnv‡h¨ 169 1396 80x mgxKiY ( ) Gi gvb ewm‡q cvB,
400 x 5

) G x Gi gvb ewm‡q cvB,
163 ev, h 2

h

2

169 25 144

h 12

UªvwcwRqvg = ABCD Gi †¶Îdj =

1 ( AB CD) h 2

1 (91 51) 12 eM© †m.wg. 2 = 852 eM© †m.wg.

wb‡Y©q †¶Îdj 852 eM© †m.wg.|

16⋅3 mylg eûfz‡Ri †¶Îdj : mylg eûfz‡Ri evû¸‡jvi ˆ`N©¨ mgvb| Avevi †KvY¸‡jv mgvb| n msL¨K evûwewkó mylg eûfz‡Ri †K›`ª I kxl© we›`y¸‡jv †hvM Ki‡j n msL¨K mgwØevû wÎfzR Drcbœ K‡i| myZivs eûfz‡Ri †¶Îdj = n GKwU wÎfzR †¶‡Îi‡¶Îdj|
ABCDEF ......... GKwU mylgevû eûfzR, hvi †K›`ª .

n msL¨K evû Ges cÖwZ evûi ˆ`N©¨ a.

O, A ; O, B †hvM Kwi|

awi, AOB Gi D”PZv OA h Ges OAB θ mylg eûfz‡Ri cÖwZwU kx‡l© Drcbœ †Kv‡Yi cwigvY = 2θ

MwYZ

263

n msL¨K mylg eûfz‡Ri kxl© †Kv‡Yi mgwó = 2

n

mylg eûfz‡Ri †K‡›`ª Drcbœ †Kv‡Yi cwigvY = n msL¨K wÎfz‡Ri †Kv‡Yi mgwó 2
OAB Gi wZb‡Kv‡Yi mgwó = 2 mg‡KvY

mg‡KvY

(n 4) mg‡KvY

Giƒc n msL¨K wÎfz‡Ri †Kv‡Yi mgwó n 2 mg‡KvY
2 (n 4) mg‡KvY = n 2 mg‡KvY

ev, 2 n (2n 4) mg‡KvY ev, ev, ev,
90
2n 4 mg‡KvY 2n

1 1

2 n 2 n

mg‡KvY
90

GLb, tan

180 n h 2h a a 2

h

a tan 2

OAB Gi †¶Îdj =

1 ah 2 1 a = a tan 2 2

=

a2 tan 90 4 180 a2 cot 4 n

180 n

=

n msL¨K evûwewkó mylg eûfz‡Ri †¶Îdj =

a2 180 cot 4 n

D`vniY 8| GKwU mylg cÂfz‡Ri cÖwZevûi ˆ`N©¨ 4 †m.wg. n‡j, Gi †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, mylg cÂfz‡Ri evûi ˆ`N©¨ a 4 †m.wg. Ges evûi msL¨v n 5 a2 180 Avgiv Rvwb, mylg eûfz‡Ri †¶Îdj = cot 4 n

mylg cÂfz‡Ri †¶Îdj

=

42 180 cot 4 5

eM© †m.wg.

264

MwYZ

= 4 cot 36 eM© †m.wg. = 4 1 376 eM© †m.wg. (K¨vjKz‡jU‡ii mvnv‡h¨) = 5 506 eM© †m.wg. (cÖvq) wb‡Y©q †¶Îdj 5 506 eM© †m.wg. (cÖvq) D`vniY 9| GKwU mylg lofy‡Ri †K›`ª †_‡K †KŠwYK we›`yi `~iZ¡ wgUvi n‡j, Gi †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, ABCDEF GKwU mylg lofyR| Gi †K›`ª O †_‡K kxl©we›`y¸‡jv †hvM Kiv n‡jv| d‡j 6 wU mgvb †¶Îwewkó wÎfyR Drcbœ nq|
360 60 6 g‡b Kwi, †K›`ª O †_‡K kxl©we›`y¸‡jvi `~iZ¡ a wgUvi COD

†¶‡Î COD Gi †¶Îdj = =

1 a a sin 60 2

1 2 3 a 2 2

3 2 a 4

3 2 4 eM©wgUvi = 4 3 eM©wgUvi 4

mylg lofyR‡¶‡Îi †¶Îdj
=6 †¶Î COD Gi †¶Îdj = 6 4 3 eM©wgUvi = 24 3 eM©wgUvi|

Abykxjbx 16⋅2
1| 2| GKwU AvqZvKvi †¶‡Îi ˆ`N©¨ we¯—v‡ii wØ¸Y| Gi †¶Îdj eM©wgUvi n‡j, cwimxgv wbY©q Ki|

GKwU Rwgi ˆ`N©¨ wgUvi Ges cÖ¯’ wgUvi| H Rwgi gv‡S GKwU cyKzi Lbb Kiv n‡jv| hw` cyKz‡ii cÖ‡Z¨K cv‡oi we¯—vi wgUvi nq, Z‡e cyKz‡ii cv‡oi †¶Îdj wbY©q Ki| GKwU evMv‡bi ˆ`N©¨ wgUvi Ges cÖ¯’ wgUvi| evMv‡bi wfZ‡i mgvb cvowewkó GKwU cyKzi
1 Ask n‡j, cyKz‡ii ˆ`N©¨ I cÖ¯’ wbY©q Ki| 2

3|

Av‡Q| cyKz‡ii †¶Îdj evMv‡bi †¶Îd‡ji 4| GKwU eM©vKvi gv‡Vi evB‡i Pviw`‡K n‡j, evMv‡bi †¶Îdj wbY©q Ki|

wgUvi PIov GKwU iv¯—v Av‡Q| iv¯—vi †¶Îdj

eM©wgUvi

5|

GKwU eM©‡¶‡Îi cwimxgv GKwU AvqZ‡¶‡Îi cwimxgvi mgvb| AvqZ‡¶ÎwUi ˆ`N©¨ cÖ‡¯’i wZb¸Y Ges †¶Îdj eM©wgUvi| cÖwZwU †m.wg. eM©vKvi cv_i w`‡q eM©‡¶ÎwU euva‡Z †gvU KZwU cv_i jvM‡e|

MwYZ

265

6|

GKwU AvqZvKvi †¶‡Îi †¶Îdj eM©wgUvi| hw` Gi ˆ`N©¨ nq| AvqZvKvi †¶‡Îi ˆ`N©¨ I cÖ¯’ wbY©q Ki| GKwU mvgvš—wi‡Ki f~wg D”PZvi wbY©q Ki|
3 Ask Ges †¶Îdj 4

wgUvi Kg nq, Z‡e †¶ÎwU eM©vKvi

7|

eM©BwÂ n‡j, †¶ÎwUi f~wg I D”PZv

8|

GKwU mvgvš—wiK‡¶‡Îi †¶Îdj GKwU eM©‡¶‡Îi mgvb| mvgvš—wi‡Ki f~wg wgUvi n‡j, eM©‡¶‡Îi K‡Y©i ˆ`N©¨ wbY©q Ki| GKwU mvgvš—wi‡Ki evûi ˆ`N©¨ K‡Y©i ˆ`N©¨ wbY©q Ki| †m.wg. Ges †m.wg.| Gi ¶z`ªZg KY©wU

wgUvi Ges D”PZv

9|

†m.wg. n‡j, Aci

10| GKwU i¤^‡mi cwimxgv

†m.wg. Ges ¶z`ªZg KY©wU

†m.wg.| Gi Aci KY© Ges †¶Îdj wbY©q Ki| †m.wg. Ges Zv‡`i j¤^ `~iZ¡ †m.wg.|

11| GKwU UªvwcwRqv‡gi mgvš—ivj evû `yBwUi ˆ`‡N©¨i Aš—i UªvwcwRqvg `yBwUi mgvš—ivj evûi ˆ`N©¨ wbY©q Ki|

12| GKwU UªvwcwRqv‡gi mgvš—ivj evûØ‡qi ˆ`N©¨ h_vµ‡g †m.wg. I `yBwUi ˆ`N©¨ h_vµ‡g †m.wg. I †m.wg.| Gi †¶Îdj wbY©q Ki| 13| GKwU mylg Aófy‡Ri †K›`ª †_‡K †KŠwYK we›`yi `~iZ¡

†mw›UwgUvi Ges Aci evû

wgUvi n‡j, Gi †¶Îdj wbY©q Ki| wgUvi| evMvbwU‡K cwiPh©v Kivi Rb¨

wgUvi Ges cÖ¯’ 14| AvqZvKvi GKwU dz‡ji evMv‡bi ˆ`N©¨ wVK gvS w`‡q wgUvi PIov ˆ`N©¨ I cÖ¯’ eivei iv¯—v Av‡Q| (K) Dc‡ii Z_¨wU wP‡Îi mvnv‡h¨ msw¶ß eY©bv `vI| (L) iv¯—vi †¶Îdj wbY©q Ki| (M) iv¯—vwU cvKv Ki‡Z †m.wg. ˆ`N©¨ Ges

cÖ¯’ wewkó KqwU B‡Ui cÖ‡qvRb n‡e|

15| eûfyR wP‡Î Z_¨ Abymv‡i Gi †¶Îdj wbY©q Ki| 16| wb‡Pi wP‡Îi Z_¨ †_‡K Gi †¶Îdj wbY©q Ki|

6⋅4 e„Ë msµvš— cwigvc (1) e„‡Ëi cwiwa e„‡Ëi ˆ`N©¨‡K Zvi cwiwa ejv nq| g‡b Kwi, †Kv‡bv e„‡Ëi e¨vmva© r n‡j, Gi cwiwa c 2π r †hLv‡b π 3 14159265.......... GKwU Ag~j` msL¨v| π Gi Avmj gvb wnmv‡e 3.1416

e¨envi Kiv hvq| dg©v-34, MwYZ-9g-10g

266

MwYZ

myZivs †Kv‡bv e„‡Ëi e¨vmva© Rvbv _vK‡j π Gi Avmbœ gvb e¨envi K‡i e„‡Ëi cwiwai Avmbœ gv‡b wbY©q Kiv hvq| D`vniY 1| GKwU e„‡Ëi e¨vm 26 †m.wg. n‡j, Gi †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, e„‡Ëi e¨vmva© r e„‡Ëi e¨vm = 2r Ges cwiwa = 2π r cÖkœvbymv‡i, 2r
26 ev, r

26 2

r

13

e„‡Ëi cwiwa = 2π r = 2 3 1616 13 †m.wg. = 3 1616 13 81 64 †m.wg.(cÖvq) wb‡Y©q e„‡Ëi cwiwa 81 64 †m.wg. (cÖvq)|

(2) e„Ëvs‡ki ˆ`N©¨ g‡b Kwi, 0 †K›`ªwewkó¨ e„‡Ëi e¨vmva© r Ges AB s e„ËPvc †K‡›`ª θ o †KvY Drcbœ K‡i| e„‡Ëi cwiwa = 2π r e„‡Ëi †K‡›`ª †gvU Drcbœ †KvY = 360 o Ges Pvc s Øviv †K‡›`ª Drcbœ †Kv‡Yi wWMÖx cwigvY θ Avgiv Rvwb, e„‡Ëi †Kv‡bv Pvc Øviv Drcbœ †K›`ª¯’ †KvY H e„ËPv‡ci mgvbycvwZK| θ 360 o s ev, s 2π r π rθ 180

(3) e„Ë‡¶Î I e„ËKjv †¶Îdj :
†Kv‡bv e„Ë I Gi Af¨š—‡i ms‡hv‡M MwVZ mgZ‡ji Dc‡mUwU‡K GKwU e„Ë‡¶Î ejv nq Ges e„ËwU‡K Giƒc e„Ë‡¶‡Îi mxgv‡iLv ejv nq| e„ËKjv : GKwU Pvc I Pv‡ci cÖvš—we›`y mswkó e¨vmva© Øviv †ewóZ †¶Î‡K e„ËKjv ejv nq|
O †K›`ªwewkó e„‡Ëi cwiwai Ici A I B `yBwU we›`y n‡j AOB Gi Af¨š—‡i OA

I OB e¨vmva© Ges AB Pv‡ci ms‡hv‡M MwVZ GKwU e„ËKjv| c~‡e©i †kªwY‡Z Avgiv wk‡L G‡mwQ †h, e„‡Ëi e¨vmva© r n‡j, e„‡Ëi †¶Îdj = πr 2 Avgiv Rvwb, e„‡Ëi †Kv‡bv Pvc Øviv Drcbœ †K›`ª¯’ †KvY H e„ËPv‡ci mgvbycvwZK| myZivs G ch©v‡q Avgiv ¯^xKvi K‡i wb‡Z cvwi †h, GKB e„‡Ëi `yBwU e„Ëvsk †¶Î Ges Giv †h Pvc `yBwUi Dci `Êvqgvb G‡`i cwigvc mgvbycvwZK| g‡b Kwi, O †K›`ªwewkó e„‡Ëi e¨vmva© r
AOB e„ËKjv †¶ÎwU APB Pv‡ci Dci `Êvqgvb, hvi wWMÖx cwigvc θ | OA Dci OC j¤^ Uvwb|

e„ËKjv AOB Gi †¶Îdj = e„ËKjv AOC Gi †¶Îdj

AOB Gi cwigvc ADC Gi cwigvc

MwYZ

267

ev,

e„ËKjv AOB Gi †¶Îdj θ = o ; [ AOC 90 e„ËKjv AOC Gi †¶Îdj θ 90 o

90 o ]

ev, e„ËKjv AOB Gi †¶Îdj =

e„ËKjv ADC Gi †¶Îdj θ 90 o 1 4

= = = myZivs, e„ËKjvi †¶Îdj =

e„Ë‡¶‡Îi †¶Îdj

θ 90 o

1 πr 2 4

θ πr 2 360 o

θ πr 2 360 o

D`vniY 2| GKwU e„‡Ëi e¨vmva© 8 †m.wg. Ges GKwU e„ËPvc †K‡›`ª 56 o Drcbœ Ki‡j, e„ËPv‡ci ˆ`N©¨ Ges e„ËKjvi †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, e„‡Ëi e¨vmva© r 8 †m.wg., e„ËPv‡ci ˆ`N©¨ θ 560 o |

Ges e„ËPvc Øviv †K‡›`ª Drcbœ †KvY

Avgiv Rvwb, s

πrθ 180 o

3 1416 8 56 †m.wg. = 7 82 †m.wg. (cÖvq) 180 θ 360 o πr 2

Ges e„Ëvs‡ki †¶Îdj = =

56 3 1416 8 2 eM© †m.wg. 350

= 62 55 eM© †m.wg. (cÖvq)| D`vniY 3| GKwU e„‡Ëi e¨vm I cwiwai cv_©K¨ 90 †m.wg. n‡j, e„‡Ëi e¨vm wbY©q Ki| mgvavb : g‡b Kwi, e„‡Ëi e¨vmva© r e„‡Ëi e¨vm = 2r Ges cwiwa = 2πr cÖkœvbymv‡i, 2πr 2r 90 ev, 2r (π 1) 90 ev, r
90 2(π 1) 45 3 1416 1 21 01 (cÖvq)

wb‡Y©q e„‡Ëi e¨vmva© 21 01πr †m.wg. (cÖvq)| D`vniY 4| GKwU e„ËvKvi gv‡Vi e¨vm 124 wgUvi| gv‡Vi mxgvbv †Nu‡l 6 wgUvi PIov GKwU iv¯—v Av‡Q| iv¯—vi †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, e„ËvKvi gv‡Vi e¨vmva© r Ges iv¯—vmn e„ËvKvi gv‡Vi e¨vmva© R |

268

MwYZ

r

124 wgUvi = 62 wgUvi Ges R 2

(62 6) wgUvi = 68 wgUvi

e„ËvKvi gv‡Vi †¶Îdj = πr 2 Ges iv¯—vmn e„ËvKvi gv‡Vi †¶Îdj = πR 2 iv¯—vi †¶Îdj = iv¯—vmn gv‡Vi †¶Îdj gv‡Vi †¶Îdj = (πR 2 πr 2 ) = π ( R 2 r 2 ) = 3 1416{(68) 2 (62) 2 } eM©wgUvi = 3 1416(4624 3844) eM©wgUvi = 3 1416 780 eM©wgUvi = 2450 44 eM©wgUvi (cÖvq) wb‡Y©q iv¯—vi †¶Îdj 2450 44 eM©wgUvi (cÖvq)| KvR : GKwU e„‡Ëi cwiwa 440 wgUvi| H e„‡Ë Aš—wj©wLZ eM©‡¶‡Îi evûi ˆ`N©¨ wbY©q Ki| D`vniY 5| GKwU e„‡Ëi e¨vmva© 12 †m.wg. Ges e„ËPv‡ci ˆ`N©¨ 14 †m.wg.| e„ËPvcwU †K‡›`ª †h †KvY Drcbœ K‡i Zv wbY©q Ki| mgvavb : g‡b Kwi, e„‡Ëi e¨vmva© r 12 †m.wg., e„ËPv‡ci ˆ`N©¨ s 14 †m.wg. Ges †K‡›`ª Drcbœ †Kv‡Yi cwigvY θ | Avgiv Rvwb, S πrθ 180 o

ev, πrθ 180 o S ev, θ
180 o S πr 180 o 14 3 1416 12 66 85o (cÖvq)

wb‡Y©q †KvY 66 85o (cÖvq)| D`vniY 6| GKwU PvKvi e¨vm 4 5 wgUvi| PvKvwU 360 wgUvi c_ AwZµg Ki‡Z KZ evi Nyi‡e ? mgvavb : †`Iqv Av‡Q, PvKvi e¨vm 4 5 wgUvi PvKvwUi e¨vmva© r
4 5 wgUvi Ges cwiwa = 2πr 2

g‡b Kwi, PvKvwU 360 wgUvi c_ AwZµg Ki‡Z n evi Nyi‡e| cÖkœvbymv‡i, n 2πr 360 ev, n 360 2πr 360 2 18 2 3 1416 4 5

(cÖvq)

PvKvwU cÖvq 18 evi Nyi‡e|

MwYZ

269

D`vniY 7| 211 wgUvi 20 †m.wg. †h‡Z `yBwU PvKv h_vµ‡g 32 Ges 48 evi Nyi‡jv| PvKv `yBwUi e¨vmv‡a©i Aš—i wbY©q Ki| mgvavb : 211 wgUvi 20 †m.wg. = 21120 †m.wg. g‡b Kwi, PvKv `yBwUi e¨vmva© h_vµ‡g R I r †hLv‡b R r. PvKv `yBwUi cwiwa h_vµ‡g 2πR I 2πr Ges e¨vmv‡a©i Aš—i ( R r ) cÖkœvbymv‡i, 32 2πR 21120 ev, R
21120 32 2π 21120 105 04 (cÖvq) 32 2 3 1416

Ges 48 2πr 21120 ev, r
21120 48 2π 21120 48 2 3 1416 70 03 (cÖvq)

R r

(105 04 70 03) †m.wg. = 35 01 †m.wg. = 35 †m.wg. (cÖvq)

PvKv `yBwUi e¨vmv‡a©i Aš—i 35 wgUvi (cÖvq)| D`vniY 8| GKwU e„‡Ëi e¨vmva© 14 †m.wg.| GKwU e‡M©i †¶Îdj D³ e„‡Ëi †¶Îd‡ji mgvb| eM©‡¶ÎwUi ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, e„‡Ëi e¨vmva© r 14 †m.wg. Ges eM©‡¶ÎwUi ˆ`N©¨ a e„‡Ëi †¶Îdj πr 2 Ges eM©‡¶ÎwUi †¶Îdj = a 2 cÖkœvbymv‡i, a 2 πr 2 ev, a πr 3 1416 14 24 81 (cÖvq) wb‡Y©q ˆ`N©¨ 24 81 †m.wg. (cÖvq)| D`vniY 9| wP‡Î ABCD GKwU eM©‡¶Î hvi cÖwZevûi ˆ`N©¨ 22 wgUvi Ges AED †¶ÎwU GKwU Aa©e„Ë| m¤ú~Y© †¶ÎwUi †¶Îdj wbY©q Ki| mgvavb : g‡b Kwi, ABCD eM©‡¶ÎwUi cÖwZ evûi ˆ`N©¨ a eM©‡¶‡Îi †¶Îdj = a 2 Avevi, AED GKwU Awae„Ë
22 wgUvi = 11 wgUvi 2 1 myZivs, AED Aa©e„‡Ëi †¶Îdj = πr 2 2

Aa©e„‡Ëi e¨vmva© r

m¤ú~Y© Z‡ji †¶Îdj = ABCD eM©‡¶‡Îi †¶Îdj + AED Aa©e„‡Ëi †¶Îdj = a2
1 2 πr 2

270
1 3 1416 (11) 2 eM©wgUvi = 674 07 eM©wgUvi (cÖvq) 2

MwYZ

= {(22) 2

wb‡Y©q †¶Îdj 674 07 eM©wgUvi (cÖvq)| D`vniY 10| wP‡Î ABCD GKwU AvqZ‡¶Î hvi ˆ`N©¨ I cÖ¯’ h_vµ‡g 12 wgUvi I 10 wgUvi Ges DAE GKwU e„Ëvsk| e„Ëvsk DE Gi ˆ`N©¨ Ges m¤ú~Y© †¶‡Îi †¶Îdj wbY©q Ki| mgvavb : e„Ëvs‡ki e¨vm r e„ËPvc DE Gi ˆ`N©¨ = =
ADE e„Ëvs‡ki †¶Îdj =
AD 12 wgUvi Ges †K‡›`ª Drcbœ †KvY θ

30o

πrθ 180o 3 1416 12 30 wgUvi = 6 28 wgUvi (cÖvq) 180 θ πr 2 o 360 30 3 1416 (12)2 eM©wgUvi 360

= 37 7 eM©wgUvi (cÖvq) AvqZ‡¶Î ABCD Gi ˆ`N©¨ 12 wgUvi Ges cÖ¯’ 10 wgUvi| AvqZ‡¶ÎwUi †¶Îdj = ˆ`N©¨ cÖ¯’ = 12 10 wgUvi = 120 eM©wgUvi

m¤ú~Y© †¶‡Îi †¶Îdj = (37 7 120) eM©wgUvi = 157 7 eM©wgUvi wb‡Y©q †¶Îdj 157 7 eM©wgUvi (cÖvq)| KvR : wP‡Î Mvp wPwýZ †¶ÎwUi †¶Îdj wbY©q Ki :

Abykxjbx 16⋅3
1| 2| 3| GKwU e„ËPvc †K‡›`ª 30 †KvY Drcbœ K‡i| e„‡Ëi e¨vm 126 †m.wg. n‡j Pv‡ci ˆ`N©¨ wbY©q Ki| cÖwZ wgwb‡U 66 wgUvi †e‡M 1
1 wgwb‡U GKwU †Nvov †Kv‡bv gvV Ny‡i G‡jv| H gv‡Vi e¨vm wbY©q Ki| 2

GKwU e„Ëvs‡ki †¶Îdj 77 eM©wgUvi Ges e„‡Ëi e¨vmva© 21 wgUvi| e„ËPvcwU †K‡›`ª †h †KvY Drcbœ K‡i, Zv wbY©q Ki|

4|

GKwU e„‡Ëi e¨vmva© 14 †m.wg. Ges e„ËPvc †K‡›`ª 76 †KvY Drcbœ K‡i| e„Ëvs‡ki †¶Îdj wbY©q Ki|

MwYZ

271

5|

GKwU e„ËvKvi gvV‡K wN‡i GKwU iv¯—v Av‡Q| iv¯—vwUi wfZ‡ii cwiwa A‡c¶v evB‡ii cwiwa 44 wgUvi †ewk| iv¯—vwUi PIov wbY©q Ki| GKwU e„ËvKvi cv‡K©i e¨vm 26 wgUvi| cvK©wU‡K †eób K‡i evB‡i 2 wgUvi cÖk¯’ GKwU c_ Av‡Q| c_wUi †¶Îdj wbY©q Ki| GKwU Mvwoi mvg‡bi PvKvi e¨vm 28 †m.wg. Ges wcQ‡bi PvKvi e¨vm 35 †m.wg.| 88 wgUvi c_ †h‡Z mvg‡bi PvKv wcQ‡bi PvKv A‡c¶v KZ c~Y©msL¨K evi †ewk Nyi‡e ? GKwU e„‡Ëi cwiwa 220 wgUvi| H e„‡Ë Aš—wjwLZ eM©‡¶‡Îi evûi ˆ`N©¨ wbY©q Ki| GKwU e„‡Ëi cwiwa GKwU mgevû wÎfy‡Ri cwimxgvi mgvb| G‡`i †¶Îd‡ji AbycvZ wbY©q Ki|

6|

7|

8| 9|

10| wb‡Pi wP‡Îi Z_¨ Abyhvqx Mvp wPwýZ †¶Î¸‡jvi †¶Îdj wbY©q Ki :

6⋅5 AvqZvKvi Nbe¯‘ : wZb †Rvov mgvš—ivj AvqZvKvi mgZj ev c„ô Øviv Ave× Nbe¯‘‡K AvqZKvi Nbe¯‘ e‡j| g‡b Kwi, ABCDEFGH GKwU AvqZKvi Nbe¯‘| Gi ˆ`N©¨ AB (1) KY© wbY©q : ABCDEFGH AvqZKvi Nbe¯‘i KY© AF
ABC -G BC AB Ges AC AwZfy³|

a , cÖ¯’ BC

b , D”PZv AH

c

AC 2

AB2 BC 2

a 2 b2

Avevi, ACF G FC
AF 2
AF

AC Ges AF AwZfyR| a 2 b2 c 2

AC 2 CF 2 a 2 b2 c 2

AvqZvKvi Nbe¯‘wUi KY© = (2) mgMÖ Z‡ji †¶Îdj wbY©q : AvqZKvi Nbe¯‘wUi 6 wU Zj

a2

b2

c2

†hLv‡b, wecixZ Zj¸‡jv ci¯úi mgvb| AvqZvKvi Nb¯‘wUi mgMÖ Z‡ji †¶Îdj = 2( ABCD Z‡ji †¶Îdj + ABGH Z‡ji †¶Îdj + BCFG Z‡ji †¶Îdj)

272

MwYZ

= 2( AB AD

AB AH

BC BG)

= 2(ab ac bc) = 2(ab bc ca) (3) AvqZvKvi Nbe¯‘i AvqZb = ˆ`N©¨ = abc D`vniY 1| AvqZvKvi Nbe¯‘i ˆ`N©¨, cÖ¯’ I D”PZv h_vµ‡g, mgMÖ Z‡ji †¶Îdj, AvqZb Ges K‡Y©i ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, AvqZvKvi Nbe¯‘i ˆ`N©¨ a †m.wg.| AvqZvKvi Nbe¯‘wUi mgMÖ Z‡ji †¶Îdj = 2(ab bc ca) = 2(25 20 20 15 15 25) eM© †m.wg. = 2350 eM© †m.wg. AvqZb = abc = 25 20 15 Nb †m.wg. = 7500 Nb †m.wg. Ges K‡Y©i ˆ`N©¨ = a2 b2 c2

cÖ¯’

D”PZv

†m.wg.,

†m.wg. Ges

†m.wg.| Gi

25 †m.wg., cÖ¯’ b

20 †m.wg. Ges D”PZv c 15

= (25)2 (20)2 (15)2 †m.wg. = 625 400 225 †m.wg. = 1250 †m.wg. = 35 353 †m.wg. (cÖvq) wb‡Y©q mgMÖ Z‡ji †¶Îdj eM© †m.wg., AvqZb Nb †m.wg. Ges K‡Y©i ˆ`N©¨ 35 353 †m.wg. (cÖvq)| KvR : †Zvgvi MwYZ eB‡qi ˆ`N©¨, cÖ¯’ I D”PZv †g‡c Gi AvqZb, mgMÖ Z‡ji †¶Îdj Ges K‡Y©i ˆ`N©¨ wbY©q Ki|

6⋅6 NYK :
AvqZKvi Nbe¯‘i ˆ`N©¨, cÖ¯’ I D”PZv mgvb n‡j Zv‡K NbK ejv nq| g‡b Kwi, ABCDEFGH GKwU NbK| Gi ˆ`N©¨ = cÖ¯’ = D”PZv = a GKK (1) NYKwUi K‡Y©i ˆ`N©¨ = a2 b2 c2 = 3a2 =

3a

MwYZ

273

(2) NY‡Ki mgMÖ Z‡ji †¶Îdj = 2(a a a a a a) = 2(a2 a2 a2) = 6a2 (3) NYKwUi AvqZb = a a a = a3 D`vniY 2| GKwU NY‡Ki m¤ú~Y© c„‡ôi †¶Îdj 96 eM©wgUvi| Gi K‡Y©i ˆ`N©¨ wbY©q Ki| mgvavb : g‡b Kwi, NYKwUi avi a Gi m¤ú~Y© c„‡ôi †¶Îdj = 6a2 Ges K‡Y©i ˆ`N©¨ = 3a cÖkœvbymv‡i, 6a2 96 ev, a 2 16 a 4

NYKwUi K‡Y©i ˆ`N©¨ = 3a = 3 a = 6 928 wgUvi (cÖvq) wb‡Y©q K‡Y©i ˆ`N©¨ 6 928 wgUvi (cÖvq)| KvR : wZbwU avZe NY‡Ki avi h_vµ‡g †m.wg., †m.wg. I †m.wg.| NYK wZbwU‡K Mwj‡q GKwU

bZzb NYK ˆZwi Kiv n‡jv| bZzb NY‡Ki m¤ú~Y© c„‡ôi †¶Îdj I K‡Y©i ˆ`N©¨ wbY©q Ki|

6⋅7 †ejb :
†Kv‡bv AvqZ‡¶‡Îi †h‡Kv‡bv evû‡K A¶ a‡i AvqZ‡¶ÎwU‡K H evûi PZzw`©‡K †Nviv‡j †h Nbe¯‘i m„wó nq, Zv‡K mge„Ëf~wgK †ejb ev wmwjÊvi ejv nq| mge„Ëf~wgK †ej‡bi `yB cÖvš—‡K e„ËvKvi Zj, eµZj‡K eµc„ô ejv nq Ges mgMÖ Zj‡K c„ôZj ejv nq| AvqZ‡¶‡Îi A‡¶i mgvš—ivj N~Y©vqgvb evûwU‡K †ej‡bi m„RK ev Drcv`K †iLv e‡j|

g‡b Kwi, wPÎ K GKwU mge„&Ëf~wgK †ejb| hvi f~wgi e¨vmva© r Ges D”PZv h (1) f~wgi †¶Îdj = πr 2 (2) eµc„‡ôi †¶Îdj = f~wgi cwiwa = 2πrh (3) m¤ú~Y©Z‡ji †¶Îdj ev mgMÖZ‡ji †¶Îdj ev, c„ôZ‡ji †¶Îdj = (πr 2 2πrh πr 2) = 2πr(r h) (4) AvqZb = f~wgi †¶Îdj = πr 2h dg©v-35, MwYZ-9g-10g

D”PZv

D”PZv

274

MwYZ

D`vniY 3| GKwU mge„Ëf~wgK †ej‡bi D”PZv 10 †m.wg. Ges f~wgi e¨vmva© m¤ú~Y© c„‡ôi †¶Îdj wbY©q Ki|

†m.wg. n‡j, Gi AvqZb Ges

mgvavb : g‡b Kwi, mge„Ëf~wgK †ej‡bi D”PZv h 10 †m.wg. Ges f~wgi e¨vmva© r Gi AvqZb = πr 2h = 3 1416 72 10 = 1539 38 Nb †m.wg. (cÖvq) Ges mgMÖc„‡ôi †¶Îdj = 2πr (r h)
2 3 1416 7(7 10) eM©wgUvi (cÖvq)

747 7 eM©wgUvi (cÖvq)

KvR : GKwU AvqZvKvi KvM‡Ri cvZv gywo‡q GKwU mge„Ëf~wgK wmwjÊvi ˆZwi Ki| Gi c„ôZ‡ji †¶Îdj Ges AvqZb wbY©q Ki| D`vniY 4| XvKbvmn GKwU ev‡·i evB‡ii gvc h_vµ‡g 10 †m.wg., mgMÖ ev·wU †¶Îdj mgvavb : g‡b Kwi, ev‡·i †ea x XvKbvmn ev‡·i evB‡ii gvc h_vµ‡g 10 †m.wg., †m.wg. I †m.wg. †m.wg. I †m.wg. Ges wfZ‡ii

eM© †m.wg.| Gi †`Iqv‡ji cyi“Z¡ mgvb n‡j, ev‡·i †ea wbY©q Ki|

ev‡·i wfZ‡ii gvc h_vµ‡g a (10 2x) , b (9 2x) I c (7 2 x) †m.wg. ev‡·i wfZ‡ii mgMÖ Z‡ji †¶Îdj = 2(ab b ca) cÖkœvbymv‡i, 2(ab b ca)
262

ev, (10 2 x)(9 2 x) (9 2 x)(7 2 x) (7 2 x)(10 2 x) 131 ev, 90 38x 4x2 63 32x 4x2 70 34x 4x2 131 0 ev, 12x2 104x 92 0 ev, 3x2 26x 23 0 ev, ev, ev, ev,
23x 23 0 23( x 1) 0 23) 0 A_ev 3x 23 0 23 ev, x 1 A_ev, x 7 67 (cÖvq) 3 3x2 3x 3x( x 1) ( x 1)(3x x 1 0

wKš‘ ev‡·i †ea Gi ˆ`N©¨ ev cÖ¯’ ev D”PZvi mgvb A_ev eo n‡Z cv‡i bv| x 1

wb‡Y©q ev‡·i †ea 1 †m.wg.|

MwYZ

275

D`vniY 5| †Kv‡bv NY‡Ki c„ôZ‡ji K‡Y©i ˆ`N©¨ 8 2 †m.wg. n‡j Gi K‡Y©i ˆ`N©¨ I AvqZb wbY©q Ki| mgvavb : g‡b Kwi, NY‡Ki avi a NYKwUi c„ôZ‡ji K‡Y©i ˆ`N©¨ = K‡Y©i ˆ`N©¨ = 3a Ges AvqZb = a3 cÖkœvbymv‡i,
2a 8 2 a 8

2a

NYKwUi K‡Y©i ˆ`N©¨ = 3 8 †m.wg. = 13 856 †m.wg. (cÖvq) Ges AvqZb = 83 Nb †m.wg. = 512 Nb †m.wg.| wb‡Y©q K‡Y©i ˆ`N©¨ 13 856 †m.wg. (cÖvq) Ges AvqZb 512 Nb †m.wg.| D`vniY 6| †Kv‡bv AvqZ‡¶‡Îi ˆ`N©¨ 12 †m.wg. Ges cÖ¯’ †m.wg.| G‡K e„nËi evûi PZzw`©‡K †Nviv‡j †h Nbe¯‘ Drcbœ nq Zvi c„ôZ‡ji †¶Îdj Ges AvqZb wbY©q Ki| mgvavb : †`Iqv Av‡Q GKwU AvqZ‡¶‡Îi ˆ`N©¨ 12 †m.wg. Ges cÖ¯’ r 5 †m.wg.|

†m.wg.| G‡K e„nËi evûi PZzw`©‡K

†Nviv‡j GKwU mge„Ëf~wgK †ejb AvK…wZi Nbe¯‘ Drcbœ n‡e, hvi D”PZv h 12 †m.wg. Ges f~wgi e¨vmva© Drcbœ NY‡Ki c„ôZ‡ji †¶Îdj = 2πr(r h) = 2 3 1416 5(5 12) eM© †m.wg. = 534 071 eM© †m.wg. (cÖvq) Ges AvqZb = πr 2h = 3 1416 52 12 Nb †m.wg. = 942 48 Nb †m.wg. (cÖvq) wb‡Y©q c„ôZ‡ji †¶Îdj 534 071 eM© †m.wg. (cÖvq) Ges AvqZb 942 48 Nb †m.wg. (cÖvq)|

Abykxjbx 16⋅4
1| GKwU mvgvš—wi‡Ki `yBwU mwbœwnZ evûi ˆ`N©¨ h_vµ‡g †m.wg., †m.wg. n‡j, Gi cwimxgvi A‡a©K KZ ? (K) 12 (L) 20 (M) 24 (N) 28 GKwU mgevû wÎfy‡Ri evûi ˆ`N©¨ †m.wg. n‡j, Gi †¶Îdj KZ eM© †m.wg. ? (K) 3 3 (L) 4 3 (M) 6 3 (N) 9 3 GKwU UªvwcwRqv‡gi D”PZv †m.wg. Ges mgvš—ivj evûØ‡qi ˆ`N©¨ h_vµ‡g †m.wg. I Gi †¶Îdj KZ eM© †m.wg. ? (K) 24 (L) 64 (M) 96 (N) 504 †m.wg. n‡j,

2| 3|

276

MwYZ

4|

wb‡Pi Z_¨¸‡jv j¶ Ki : †m.wg.| (i) †m.wg. eM©vKvi cv_‡ii cwimxgv (ii) †m.wg. e¨vmv‡a©i e„ËvKvi cv‡Zi †¶Îdj 3π eM© †m.wg.| (iii) †m.wg. D”PZv Ges †m.wg. e¨vmv‡a©i †ejb AvK…wZi e¯‘i AvqZb 20π Nb †m.wg.| Dc‡ii Z‡_¨i wfwË‡Z wb‡Pi †KvbwU mwVK ? (L) i I iii (M) ii I iii (N) i ii I iii (K) i I ii wP‡Îi Z_¨ Abymv‡i wb‡Pi cÖkœ¸‡jvi DËi `vI :

5| 6| 7| 8|

ABCD AvqZ‡¶‡Îi K‡Y©i ˆ`N©¨ KZ ? (L) 14 (K) 13 ADF e„Ëvs‡ki †¶Îdj KZ ? (K) 16 (L) 32 AGB Aa©e„‡Ëi cwiwa KZ ? (K) 18 (L) 18 85 (cÖvq)

(M) 14 4 (cÖvq) (M) 64

(N) 15 (N) 128

(M) 37 7 (cÖvq) (N) 96 GKwU AvqZvKvi Nbe¯‘i ˆ`N©¨, cÖ¯’ I D”PZv h_vµ‡g 16 wgUvi, 12 wgUvi I 4 5 wgUvi| Gi c„óZ‡ji †¶Îdj, K‡Y©i ˆ`N©¨ I AvqZb wbY©q Ki| GKwU AvqZvKivi Nbe¯‘i ˆ`N©¨, cÖ¯’ I D”PZvi AbycvZ 21 : 16 : 12 Ges K‡Y©i ˆ`N©¨ 87 †m.wg. n‡j, Nb e¯‘wUi Z‡ji †¶Îdj wbY©q Ki|

9|

10| GKwU AvqZvKvi Nbe¯‘ 48 eM©wgUvi f~wgi Dci `Êvqgvb| Gi D”PZv 3 wgUvi Ges KY© 13 wgUvi| AvqZKivi Nbe¯‘i ˆ`N©¨ I cÖ¯’ wbY©q Ki| 11| GKwU AvqZvKvi Kv‡Vi ev‡·i evB‡ii gvc h_vµ‡g †m.wg., †m.wg., I †m.wg.| Gi wfZ‡ii

m¤ú~Y© c„‡ói †¶Îdj 88 eM© †m.wg.| ev·wUi Kv‡Vi cyi“Z¡ wbY©q Ki| 12| GKwU †`Iqv‡ji ˆ`N©¨ 25 wgUvi, D”PZv †m.wg., cÖ¯’ wbY©q Ki| 13| GKwU NYK AvK…wZe¯‘i c„ôZ‡ji †¶Îdj 2400 eM© †m.wg. n‡j, Gi K‡Y©i ˆ`N©¨ wbY©q Ki| 14| 12 †m.wg. D”PZvwewkó GKwU †ej‡bi f~wgi e¨vmva© †m.wg.| Gi c„ôZ‡ji †¶Îdj I AvqZb wbY©q Ki| †m.wg. Ges D”PZv wgUvi Ges cyi“Z¡ 30 †m.wg.| GKwU B‡Ui ˆ`N©¨ 10

†m.wg.| †`IqvjwU BU w`‡q ˆZwi Ki‡Z cÖ‡qvRbxq B‡Ui msL¨v

MwYZ

277

15| GKwU †ej‡bi eµZ‡ji †¶Îdj 100 eM© †m.wg. Ges AvqZb 150 Nb †m.wg.| †ej‡bi D”PZv Ges f~wgi e¨vmva© wbY©q Ki| 16| GKwU mge„Ëf~wgK wmwjÊv‡ii eµZ‡ji †¶Îdj 4400 eM© †m.wg.| Gi D”PZv 30 †m.wg. n‡j, mgMÖZj wbY©q Ki| 17| GKwU †jvnvi cvB‡ci wfZ‡ii I evB‡ii e¨vm h_vµ‡g 12 †m.wg. I 14 †m.wg. Ges cvB‡ci D”PZv wgUvi| Nb †m.wg. †jvnvi IRb 7 2 MÖvg n‡j, cvB‡ci †jvnvi IRb wbY©q Ki| wgUvi| AvqZvKvi †¶ÎwU‡K cwi‡ewóZ K‡i

18| GKwU AvqZvKvi †¶‡Îi ˆ`N©¨ 12 wgUvi Ges cÖ¯’

GKwU e„ËvKvi †¶Î Av‡Q †hLv‡b AvqZKvi †¶Î Øviv AbvwaK…Z As‡k Nvm jvMv‡bv n‡jv| (K) Dc‡ii Z‡_¨i wfwË‡Z msw¶ß eY©bvmn wPÎ AuvK| (L) e„ËvKvi †¶ÎwUi e¨vm wbY©q Ki| (M) cÖwZ eM©wgUvi Nvm jvMv‡Z 50 UvKv LiP n‡j, †gvU LiP wbY©q Ki| 19| ABC I BCD GKB f~wg BC Gi Dci Ges GKB mgvš—ivj hyMj BC I AD Gi g‡a¨ Aew¯’Z| K. Dc‡ii eY©bv Abymv‡i wPÎwU AvuK| L. cÖgvY Ki †h, M. †¶Î ABC = †¶Î BCD.

†¶‡Î ABC Gi mgvb †¶Îdj wewkó GKwU mvgvš—wiK AvuK hvi GKwU †KvY GKwU wbw`©ó †Kv‡Yi mgvb| (A¼‡bi wPý I weeiY Avek¨K)|

20| GKwU mvgvš—wiK †¶Î ABCD Ges GKwU AvqZ‡¶Î BCEF Df‡qi f~wg BC. K. GKB D”PZv we‡ePbv K‡i mvgvš—wiK †¶Î I AvqZ‡¶ÎwUi wPÎ AvuK| L. †`LvI †h, ABCD †¶ÎwUi cwimxgv BCEF †¶ÎwUi cwimxgv A‡c¶v e„nËi|

M. AvqZ‡¶ÎwUi ˆ`N©¨ I cÖ‡¯’i AbycvZ 5:3 Ges †¶ÎwUi cwimxgv 48 wgUvi n‡j, mvgvš—wiK †¶ÎwUi †¶Îdj wbY©q Ki|

mß`k Aa¨vq

cwimsL¨vb weÁvb I cÖhyw³i Dbœq‡bi AMÖhvÎvq Z_¨ Dcv‡Ëi Ae`v‡bi d‡j c„w_ex cwiYZ n‡q‡Q wek¦MÖv‡g| Z_¨ I Dcv‡Ëi `ª“Z mÂvjb I we¯—v‡ii Rb¨ m¤¢e n‡q‡Q wek¦vq‡bi| ZvB Dbœq‡bi aviv Ae¨vnZ ivLv I wek¦vq‡b AskMÖnY I Ae`vb ivL‡Z n‡j Z_¨ I DcvË m¤^‡Ü mg¨K Ávb AR©b G ¯—‡ii wk¶v_©x‡`i Rb¨ Acwinvh©| cÖvmw½Kfv‡e wk¶v_©xi Ávb AR©‡bi Pvwn`v †gUv‡bvi j‡¶¨ lô †kÖwY †_‡K Z_¨ I Dcv‡Ëi Av‡jvPbv Kiv n‡q‡Q Ges av‡c av‡c †kÖwYwfwËK welqe¯‘i web¨vm Kiv n‡q‡Q| GiB avivevwnKZvq G †kÖwY‡Z wk¶v_©xiv µg‡hvwRZ MYmsL¨v, MYmsL¨v eûfzR, AwRf †iLv, †Kw›`ªq cÖeYZv cwigv‡c msw¶ß c×wZ‡Z Mo, ga¨K I cÖPziK BZ¨vw` m¤^‡Ü Rvb‡e I wkL‡e|

Aa¨vq †k‡l wk¶v_©xivµg‡hvwRZ MYmsL¨v, MYmsL¨v eûfzR I AwRf †iLv e¨vL¨v Ki‡Z cvi‡e| MYmsL¨v eûfzR I AwRf †iLvi mvnv‡h¨ DcvË e¨vL¨v Ki‡Z cvi‡e| †Kw›`ªq cÖeYZvi cwigvc c×wZ e¨vL¨v Ki‡Z cvi‡e| †Kw›`ªq cÖeYZv cwigv‡c msw¶ß c×wZi cÖ‡qvRbxqZv e¨vL¨v Ki‡Z cvi‡e| msw¶ß c×wZi mvnv‡h¨ Mo, ga¨K I cÖPziK wbY©q Ki‡Z cvi‡e| MYmsL¨v eûfzR I AwRf †iLv †jLwP‡Îi e¨vL¨v Ki‡Z cvi‡e| Dcv‡Ëi Dc¯’vcb : Avgiv Rvwb ¸YevPK bq Ggb msL¨vm~PK Z_¨vewj cwimsL¨v‡bi DcvË| AbymÜvbvaxb DcvË cwimsL¨v‡bi KuvPvgvj| G¸‡jv Aweb¨¯—fv‡e _v‡K Ges Aweb¨¯— DcvË †_‡K mivmwi cÖ‡qvRbxq wm×v‡š— DcbxZ nIqv hvq bv| cÖ‡qvRb nq DcvË¸‡jvi web¨¯— I mviwYfz³ Kiv| Avi DcvËmg~‡ni mviwYfz³ Kiv n‡jv Dcv‡Ëi Dc¯’vcb| Av‡Mi †kÖwY‡Z Avgiv DcvËmg~n Kxfv‡e mviwYfz³ K‡i web¨¯— Ki‡Z nq Zv wk‡LwQ| Avgiv Rvwb †Kv‡bv Dcv‡Ëi mviwYfz³ Ki‡Z n‡j cÖ_‡g Zvi cwimi wba©viY Ki‡Z nq| Gici †kÖwY e¨eavb I †kÖwY msL¨v wba©viY K‡i U¨vwj wPý e¨envi K‡i MYmsL¨v wb‡ekY mviwY ˆZwi Kiv nq| GLv‡b eySvi myweav‡_© wb‡Pi D`vni‡Yi gva¨‡g MYmsL¨v wb‡ekb mviwY ˆZwi Kivi c×wZi cybiv‡jvPbv Kiv n‡jv| D`vniY 1| †Kv‡bv GK kxZ †gŠmy‡g kÖxg½‡ji Rvbyqvwi gv‡mi 31 w`‡bi ZvcgvÎv (†mjwmqvm) wb‡P †`Iqv n‡jv| ZvcgvÎvi MYmsL¨v wb‡ekb mviwY ˆZwi Ki| 14 , 14 , 14 , 13 , 12 , 13 , 10 , 10 , 11 , 12 , 11 , 10 , 9 , 8 , 9 , 11 , 10 , 10 , 8 , 9 , 7 , 6 , 6 , 6 , 6 , 7 , 8 , 9 , 9 , 8 , 7 |

MwYZ

279

mgvavb : GLv‡b ZvcgvÎv wb‡`©kK Dcv‡Ëi me‡P‡q †QvU msL¨v 6 Ges eo msL¨v 14| myZivs Dcv‡Ëi cwimi = (14 6) + 1 = 9|

GLb †kÖwY e¨eavb hw` 3 †bIqv nq Z‡e †kÖwY msL¨v n‡e

9 ev 3| 3

†kÖwY e¨eavb 3 wb‡q wZb †kÖwY‡Z DcvËmg~n web¨vm Ki‡j MYmsL¨v (NUb msL¨vI ejv nq) wb‡ekb mviwY n‡e wbgœiƒc : ZvcgvÎv (†mjwmqvm) 6 9 12 8 11 14 llll llll lll llll ll U¨vwj wPý MYmsL¨v ev NUb msL¨v 11 13 7 †gvU 31| KvR : †Zvgv‡`i †kÖwY‡Z Aa¨vqbiZ mKj wk¶v_©x‡`i `yBwU `j MVb Ki| `‡ji m`m¨‡`i IR‡bi (†KwR‡Z) MYmsL¨v wb‡ekY mviwY ˆZwi Ki|

µg‡hvwRZ MYmsL¨v ( Cumulative Frequency ) :
D`vniY 1 Gi †kÖwY 3 e¨eavb a‡i †kÖwYmsL¨v wba©viY K‡i MYmsL¨v wb‡ekY mviwY ˆZwi Kiv n‡q‡Q| D‡jwLZ Dcv‡Ëi †kÖwY msL¨v 3| cÖ_g †kÖwYi mxgv n‡jv 6 8 †m.| GB †kÖwYi MYmsL¨v 11| wØZxq †kÖwYi MYmsL¨v 13| GLb cÖ_g †kÖwYi MYmsL¨v 11 Gi mv‡_ wØZxq †kÖwYi MYmsL¨v 13 †hvM K‡i cvB 24| GB 24 n‡e wØZxq †kÖwYi µg‡hvwRZ MYmsL¨v| Avi cÖ_g †kÖwY w`‡q ïi“ nIqvq GB †kÖwYi µg‡hvwRZ MYmsL¨v n‡e 11| Avevi wØZxq †kÖwYi µg‡hvwRZ MYmsL¨v 24 Gi mv‡_ Z…Zxq †kÖwYi MYmsL¨v †hvM Ki‡j 24 + 7 = 31, hv Z…Zxq †kªwYi µg‡hvwRZ MYmsL¨v| GBfv‡e µg‡hvwRZ MYmsL¨v mviwY ˆZwi Kiv nq| Dc‡ii Av‡jvPbvi †cÖw¶‡Z D`vniY 1 Gi ZvcgvÎvi µg‡hvwRZ MYmsL¨v mviwY wbgœiƒc : ZvcgvÎv †mw›UwgUv‡i 6 9 12 8 11 14 MYmsL¨v 11 13 7 µg‡hvwRZ MYmsL¨v 11 (11 + 13) = 24 (24 + 7) = 31 8 | GB †kÖwYi wbgœmxgv 6 Ges D”Pmxgv

D`vniY 2| wb‡P 40 Rb wk¶v_©xi evwl©K cix¶vq Bs‡iwR‡Z cÖvß b¤^i †`Iqv n‡jv| cÖvß b¤^‡ii µg‡hvwRZ MYmsL¨v mviwY ˆZwi Ki| 70, 40, 35, 60, 55, 58, 45, 60, 65, 80, 70, 46, 50, 60, 65, 70, 58, 60, 48, 70, 36, 85, 60, 50, 46, 65, 55, 61, 72, 85, 90, 68, 65, 50, 40, 56, 60, 65, 46, 76|

280

MwYZ

mgvavb : Dcv‡Ëi cwiwa

= (m‡e©v”P gvb

me©wbgœgvb) + 1

= (90 35) + 1 = 55 + 1 = 56 †kÖwY e¨eavb hw` 5 aiv nq, Z‡e †kÖwY msL¨v =

56 5 = 11 2 ev 12

myZivs †kªwY e¨eavb 5 a‡i µg‡hvwRZ MYmsL¨v mviwY n‡e wbgœiƒc : cÖvß b¤^i 35 40 45 50 55 60 65 39 44 49 54 59 64 69 MYmsL¨v 2 2 5 3 5 8 6 µg‡hvwRZ MYmsL¨v 2 2+2=4 5+4=9 3 + 9 = 12 5 + 12 = 17 8 + 17 = 25 6 + 25 = 31 cÖvß b¤^i 70 75 80 85 90 95 74 79 84 89 94 99 MYmsL¨v 4 1 1 2 1 0 µg‡hvwRZ MYmsL¨v 4 + 31 = 35 1 + 35 = 36 1 + 36 = 37 2 + 37 = 39 1 + 39 = 40 0 + 40 = 40

PjK : Avgiv Rvwb msL¨vm~PK Z_¨mg~n cwimsL¨v‡bi DcvË| Dcv‡Ë e¨eüZ msL¨vmg~n n‡jv PjK| †hgb, D`vniY 1 G ZvcgvÎv wb‡`©kK msL¨v¸‡jv PjK| Z`vbyiƒc D`vniY 2 G cÖvß b¤^i¸‡jv e¨eüZ Dcv‡Ëi PjK| wewQbœ I Awew”Qbœ PjK : cwimsL¨v‡b e¨eüZ PjK `yB cÖKv‡ii nq| †hgb wewQbœ PjK I Awew”Qbœ PjK| †h Pj‡Ki gvb ïaygvÎ c~Y©msL¨v nq Zv wew”Qbœ PjK, †hgb D`vniY 2 G e¨eüZ cÖvß b¤^i| Z`vbyiƒc RbmsL¨v wb‡`©kK Dcv‡Ë c~Y©msL¨v e¨eüZ nq| ZvB RbmsL¨vg~jK Dcv‡Ëi PjK n‡”Q wew”Qbœ PjK| Avi †hmKj Pj‡Ki gvb †h‡Kv‡bv ev¯—e msL¨v n‡Z cv‡i, †m mKj PjK Awew”Qbœ PjK| †hgb D`vniY 1-G e¨eüZ ZvcgvÎv wb‡`©kK Dcv‡Ë †h‡Kv‡bv ev¯—e msL¨v n‡Z cv‡i| G Qvov eqm, D”PZv, IRb BZ¨vw` mswkó Dcv‡Ë †h‡Kv‡bv ev¯—e msL¨v e¨envi Kiv hvq| ZvB G¸‡jvi Rb¨ e¨eüZ PjK n‡”Q Awew”Qbœ PjK| Awew”Qbœ Pj‡Ki `yBwU gv‡bi ga¨eZ©x †h‡Kv‡bv msL¨vI H Pj‡Ki gvb n‡Z cv‡i| A‡bK mgq †kÖwY e¨eavb Awew”Qbœ Kivi cÖ‡qvRb nq| †kªwY e¨eavb Awew”Qbœ Kivi Rb¨ †Kv‡bv †kÖwYi D”Pmxgv Ges cieZ©x †kÖwYi wbgœmxgvi

MwYZ

281

ga¨we›`y wb‡q †mB †kÖwYi cÖK…Z D”Pmxgv Ges cieZ©x †kÖwYi cÖK…Z wbgœmxgv wba©viY Kiv nq| †hgb, D`vniY 1 G cÖ_g †kÖwYi cÖK…Z D”Pmxgv I wbgœmxgv h_vµ‡g 8 5 I 5 5 Ges wØZxq †kÖwYi D”Pmxgv I wbgœmxgv 11 5 I 8 5 BZ¨vw`| KvR : †Zvgv‡`i †kÖwYi wk¶v_©x‡`i wb‡q Abya¶© 40 R‡bi `j MVb Ki| `‡ji m`m¨‡`i IRb/D”PZv wb‡q `‡j MYmsL¨v wb‡ekY I µg‡hvwRZ MYmsL¨v mviwY ˆZwi Ki| Dcv‡Ëi †jLwPÎ : Avgiv †`‡LwQ †h, AbymÜvbvaxb msM„nxZ DcvË cwimsL¨v‡bi KuvPvgvj| G¸‡jv MYmsL¨v wb‡ekY mviwYfy³ ev µg‡hvwRZ mviwYfz³ Kiv n‡j G‡`i m¤^‡Ü mg¨K aviYv Kiv I wm×vš— †bIqv mnR nq| GB mviwYfz³ DcvËmg~n hw` †jLwP‡Îi gva¨‡g Dc¯’vcb Kiv nq, Z‡e Zv eySvi Rb¨ †hgb AviI mnR nq †Zgwb wPËvKl©K nq| G Rb¨ cwimsL¨v‡bi DcvËmg~n mviwYfz³ Kiv I †jLwP‡Îi gva¨‡g Dc¯’vcb eûj cÖPwjZ Ges e¨vcK e¨eüZ c×wZ| 8g †kÖwY ch©š— wewfbœ cÖKvi †jLwP‡Îi g‡a¨ †iLvwPÎ I AvqZ‡jL m¤^‡Ü we¯—vwiZ Av‡jvPbv Kiv n‡q‡Q Ges G¸‡jv wKfv‡e AuvK‡Z nq Zv †`Lv‡bv n‡q‡Q| GLv‡b Kxfv‡e MYmsL¨v wb‡ekY I µg‡hvwRZ MYmsL¨v mviwY †_‡K MYmsL¨v eûfzR, cvBwPÎ I AwRf †iLv AuvKv nq Zv wb‡q Av‡jvPbv Kiv n‡e| MYmsL¨v eûfzR ( Frequency Polygon ) : 8g †kÖwY‡Z Avgiv wew”Qbœ Dcv‡Ëi AvqZ‡jL AuvKv wk‡LwQ| GLv‡b Kxfv‡e cÖ_‡g Awew”Qbœ Dcv‡Ëi AvqZ‡jL Gu‡K Zvi MYmsL¨v eûfzR AuvKv nq, Zv D`vni‡Yi gva¨‡g Dc¯’vcb Kiv n‡jv| D`vniY 3| †Kvb ¯‹z‡ji 10g †kÖwYi 60 Rb wk¶v_©xi IR‡bi (wK‡jvMÖvg) MYmsL¨v wb‡ekY n‡jv wbgœiƒc : IRb (†KwR) MYmsL¨v (wk¶v_©x‡`i msL¨v) 46 5 50 51 10 55 56 20 60 61 15 65 66 10 70

(K) MYmsL¨v wb‡ek‡Yi AvqZ‡jL AuvK| (L) AvqZ‡j‡Li MYmsL¨v eûfzR AuvK| mgvavb : cÖ`Ë mviwY‡Z Dcv‡Ëi †kÖwY e¨eavb wew”Qbœ| †kÖwY e¨eavb Awew”Qbœ Kiv n‡j cÖ`Ë mviwY n‡e : †kÖwY e¨eavb IRb (†KwR) 46 51 56 61 66 50 55 60 65 70 Awew”Qbœ †kÖwYmxgv 45 5 50 5 55 5 60 5 65 5 50 5 55 5 60 5 65 5 70 5 †kªwY ga¨we›`y 48 53 58 63 68 MYmsL¨v 5 10 20 15 10

dg©v-36, MwYZ-9g-10g

282

MwYZ

(K) QK KvM‡Ri cÖwZ Ni‡K GK GKK a‡i x A¶ eivei †kÖwYmxgv Ges y A¶ eivei MYmsL¨v wb‡q AuvqZ‡jL AuvKv n‡q‡Q| x -A¶ eivei †kÖwYmxgv 45 5 †_‡K Avi¤¢ n‡q‡Q| g~jwe›`y †_‡K 45 5 ch©š— c~e©eZ©x Ni¸‡jv Av‡Q eySv‡Z fvOv wPý e¨envi Kiv n‡q‡Q| (L) AvqZ‡jL n‡Z MYmsL¨v eûfzR AuvKvi Rb¨ cÖvß AvqZ‡j‡Li AvqZmg~‡ni f~wgi mgvš—ivj wecixZ evûi ga¨we›`ymg~n wba©viY Kiv n‡q‡Q| wPwýZ ga¨we›`ymg~n †iLvsk Øviv mshy³ K‡i MYmsL¨v eûfzR AuvKv n‡q‡Q (cv‡ki wP‡Î †`Lv‡bv n‡jv)| MYmsL¨v eûfzR my›`i †`Lv‡bvi Rb¨ cÖ_g I †kl Avq‡Zi ga¨we›`yi ms‡hvM †iLvs‡ki cÖvš— we›`yØq †kÖwY e¨eavb wb‡`©kK x A‡¶i mv‡_ mshy³ Kiv n‡q‡Q| MYmsL¨v eûfzR : AwewQbœ Dcv‡Ëi †kÖwY e¨eav‡bi wecixZ MYmsL¨v wb‡`©kK we›`ymg~n‡K ch©vqµ‡g †iLvsk Øviv hy³ K‡i †h †jLwPÎ cvIqv hvq, ZvB n‡jv MYmsL¨v eûfzR| D`vniY 4| wb‡Pi MYmsL¨v wb‡ekY mviwYi eûfzR A¼b Ki| †kÖwY e¨eavb ga¨we›`y MYmsL¨v 10 20 15 8 20 30 25 10 30 40 35 15 40 50 45 30 50 60 55 45 60 70 65 41 70 80 75 15 80 90 85 7

mgvavb : x A¶ eivei QK KvM‡Ri cÖwZ `yB Ni‡K †kÖwY e¨eav‡bi 5 GKK a‡i Ges y A¶ eivei QK KvM‡Ri `yB Ni‡K MYmsL¨vi 5 GKK a‡i cÖ`Ë MYmsL¨v wb‡ek‡Yi AvqZ‡jL AuvKv n‡jv| AvqZ‡j‡Li AvqZmg~‡ni f~wgi wecixZ evûi g‡a¨ we›`y hv †kÖwYi ga¨we›`y wPwýZ Kwi| GLb wPwýZ ga¨we›`ymg~n †iLvsk Øviv mshy³ Kwi| cÖ_g †kÖwYi cÖvš— we›`y I †kl †kÖwYi cÖvš— we›`yØq‡K †kÖwY e¨eavb wb‡`©kK x A‡¶i mv‡_ mshy³ K‡i MYmsL¨v eûfzR A¼b Kiv n‡jv| KvR : †Zvgv‡`i †kÖwY‡Z Aa¨qbiZ wk¶v_©x‡`i cÖ_g mvgwqK cix¶vq evsjvq cÖvß b¤^i wb‡q MYmsL¨v eûfzR AvuK| D`vniY 5| 10g †kÖwYi 50 Rb wk¶v_©xi weÁvb wel‡q cÖvß b¤^‡ii MYmsL¨v wb‡ekY mviwY †`Iqv n‡jv| cÖ`Ë Dcv‡Ëi MYmsL¨v eûfzR AuvK (AvqZ‡jL e¨envi bv K‡i)|

MwYZ cÖvß b¤^‡ii †kÖwY e¨eavb

283

MYmsL¨v

31 40 6

41 50 8

51 60 10

61 70 12

71 80 5

81 90 7

91 100 2

mgvavb : GLv‡b cÖ`Ë DcvË wew”Qbœ| G‡¶‡Î †kÖwY e¨eav‡bi ga¨we›`y †ei K‡i mivmwi MYmsL¨v eûfzR AuvKv myweavRbK| †kÖwY e¨eavb 31 40 41 50 51 60 61 70 71 80 81 90 91 100 ga¨we›`y 40 31 45 5 55 5 65 5 75 5 85 5 95 5 2 35 5 MYmsL¨v 6 8 10 12 5 7 2 x A¶ eivei QK KvM‡Ri cÖwZ 2 Ni‡K †kÖwY e¨eav‡bi ga¨we›`yi 10

GKK a‡i Ges y A¶ eivei QK KvM‡Ri 1 Ni‡K MYmsL¨vi 1 GKK a‡i cÖ`Ë Dcv‡Ëi MYmsL¨v eûf~R AuvKv n‡jv|

KvR : 100 Rb K‡jR Qv‡Îi D”PZvi MYmsL¨v wb‡ekY †_‡K MYmsL¨v eûfzR AuvK|

D”PZv (†m.wg.) 141 150 151 160

161 170

171 180

181 190

µg‡hvwRZ MYmsL¨v †jLwPÎ ev AwRf †iLv : †Kv‡bv Dcv‡Ëi †kÖwY web¨v‡mi ci †kÖwY e¨eav‡bi D”Pmxgv x A¶ eivei Ges †kÖwYi µg‡hvwRZ MYmsL¨v y A¶ eivei ¯’vcb K‡i µg‡hvwRZ MYmsL¨vi †jLwPÎ ev

AwRf †iLv cvIqv hvq| D`vniY 6| †Kv‡bv †kªwYi 60 wk¶v_©xi 50 b¤^‡ii mvgwqKx cix¶vq cÖvß b¤^‡ii MYmsL¨v wb‡ekY mviwY n‡jv : cÖvß b¤^‡ii †kÖwY 1 10 11 20 21 30 31 40 e¨eavb MYmsL¨v 8 12 15 18 GB MYmsL¨v wb‡ek‡Yi AwRf †iLv AuvK| mgvavb : cÖ`Ë Dcv‡Ëi MYmsL¨v wb‡ek‡Yi µg‡hvwRZ MYmsL¨v mviwY n‡jv : cÖvß b¤^‡ii †kÖwY 1 10 11 20 21 30 31 40 e¨eavb MYmsL¨v 8 12 15 18 8 + 12 = 20 15 + 20 = 35 18 + 35 = 53 µg‡hvwRZ 8 MYmsL¨v 41 7 50

41 7

50

7 + 53 = 60

284

MwYZ

x -A¶ eivei QK KvM‡Ri `yB Ni‡K †kÖwY e¨eav‡bi D”Pmxgvi
GKK Ges y -A¶ eivei QK KvM‡Ri GK Ni‡K µg‡hvwRZ MYmsL¨vi 5 GKK a‡i cÖ`Ë Dcv‡Ëi µg‡hvwRZ MYmsL¨vi AwRf †iLv AuvKv n‡jv|
KvR : †Kvb GK cix¶vq MwY‡Z †Zvgv‡`i †kÖwYi 50 I Z`y‡a¶©v b¤^icÖvß wk¶v_©x‡`i b¤^‡ii µg‡hvwRZ MYmsL¨v mviwY ˆZwi Ki Ges AwRf †iLv AuvK|

†Kw›`ªq cÖeYZv : mßg I Aóg †kÖwY‡Z †Kw›`ªq cÖeYZv I Gi cwigvc mg‡Ü Av‡jvPbv Kiv n‡q‡Q| Avgiv †`‡LwQ †h, AbymÜvbvaxb Aweb¨¯— DcvËmg~n gv‡bi µgvbymv‡i mvRv‡j, DcvËmg~n gvSvgvwS †Kv‡bv gv‡bi KvQvKvwQ cwÄf~Z nq| Avevi Aweb¨¯— DcvËmg~n MYmsL¨v wb‡ekY mviwY‡Z Dc¯’vcb Kiv n‡j gvSvgvwS GKwU †kÖwY‡Z MYmsL¨vi cÖvPzh© †`Lv hvq| A_©vr, gvSvgvwS GKwU †kÖwY‡Z MYmsL¨v Lye †ewk nq| e¯‘Z DcvËmg~‡ni †Kw›`ªq gv‡bi w`‡K cywÄf~Z nIqvi GB cÖeYZvB n‡jv †Kw›`ªq cÖeYZv| †Kw›`ªq gvb GKwU msL¨v Ges GB msL¨v DcvËmg~‡ni cÖwZwbwaZ¡ K‡i| GB msL¨v Øviv †Kw›`ªq cÖeYZv cwigvc Kiv nq| mvaviYZ †Kw›`ªq cÖeYZvi cwigvc n‡jv : (1) MvwYwZK Mo (2) ga¨K (3) cÖPziK| MvwYwZK Mo : Avgiv Rvwb, DcvËmg~‡ni gv‡bi mgwó‡K hw` Zvi msL¨v Øviv fvM Kiv nq, Z‡e DcvËmg~‡ni Mo gvb cvIqv hvq| Z‡e DcvËmg~‡ni msL¨v hw` Lye †ewk nq Zvn‡j G c×wZ‡Z Mo wbY©q Kiv mgqmv‡c¶, †ek KwVb I fyj nIqvi m¤¢vebv _v‡K| G mKj †¶‡Î DcvËmg~n †kÖwY web¨v‡mi gva¨‡g mviwYe× K‡i msw¶ß c×wZ‡Z Mo wbY©q Kiv nq| D`vniY 7| wb‡P †Kv‡bv GKwU †kªwYi wk¶v_©x‡`i MwY‡Z cÖvß b¤^‡ii MYmsL¨v wb‡ekb mviwY †`Iqv n‡jv| cÖvß b¤^‡ii MvwYwZK Mo wbY©q Ki|
†kªwY e¨vwß MYmsL¨v 25 34 5 35 44 10 45 54 15 55 64 20 65 74 30 75 84 16 85 94 4

mgvavb : GLv‡b †kªwY e¨vwß †`Iqv Av‡Q weavq wk¶v_©x‡`i e¨w³MZ b¤^i KZ Zv Rvbv hvq bv| G †¶‡Î cÖ‡Z¨K †kªwYi †kªwY ga¨gvb wbY©q Kivi cÖ‡qvRb nq| †kªwY EaŸ©gvb+†kªwYi wbgœgvb †kªwY ga¨gvb = 2 hw` †kªwY ga¨gvb xi (i 1....., k ) nq Z‡e ga¨gvb msewjZ mviwY n‡e wbgœiƒc : †kªwY e¨vwß 25 35 45 55 34 44 54 64 †kªwY ga¨gvb ( xi ) 29 5 39 5 49 5 59 5 MYmsL¨v ( f i ) 5 10 15 20

( f i xi )
147 5 395 0 742 5 1190 0

MwYZ

285

65 75 85

74 84 94

69 5 79 5 89 5 †gvU f i xi

30 16 4 100

2085 0 1272 0 358 0 6190 00

wb‡Y©q MvwYwZK Mo =

1 n

k l 1

1 6190 100

= 61 9 †kÖwYweb¨vmK…Z Dcv‡Ëi MvwYwZK Mo (msw¶ß c×wZ) †kÖwYweb¨vmK…Z Dcv‡Ë MvwYwZK Mo wbY©‡qi Rb¨ msw¶ß c×wZ n‡jv mnR| msw¶ß c×wZ‡Z Mo wbY©‡qi avcmg~n Ñ 1| †kÖwYmg~‡ni ga¨gvb wbY©q Kiv 2| ga¨gvbmg~n †_‡K myweavRbK †Kvb gvb‡K AvbygvwbK Mo ( a ) aiv 3| cÖ‡Z¨K †kÖwYi ga¨gvb †_‡K AvbygvwbK Mo we‡qvM K‡i Zv‡K †kÖwY e¨wß Øviv fvM K‡i avc wePz¨wZ ga¨gvb Ñ AvbygvwbK Mo wbY©q Kiv e¨vwß avc wePz¨wZ‡K mswkó †kÖwYi MYmsL¨v Øviv ¸Y Kiv weP~¨wZi Mo wbY©q Kiv Ges Gi mv‡_ AvbygvwbK Mo †hvM K‡i Kvw•LZ Mo wbY©q Kiv|

4| 5|

msw¶ß c×wZ : G c×wZ‡Z DcvËmg~‡ni MvwYwZK Mo wbY©‡q e¨eüZ m~Î n‡jv : x a f i ui n h †hLvb, x

wb‡Y©q Mo, a

AvbygvwbK Mo, f i = i Zg †kÖwYi MYmsL¨v, ui f i = i

Zg †kÖwYi MYmsL¨v avc wePz¨wZ h = †kÖwY e¨vwß D`vniY 8| †Kv‡bv `ª‡e¨i Drcv`‡b wewfbœ ch©v‡q †h LiPmg~n (kZ UvKvq) nq Zv wb‡Pi mviwY‡Z †`Lv‡bv n‡q‡Q| msw¶ß c×wZ‡Z Mo LiP wbY©q Ki| Drcv`b LiP (kZ UvKvq) MYmsL¨v
2 6 6 10 10 14 14 18 18 22 22 26 26 30 30 34

1

9

21

47

52

36

19

3

mgvavb : msw¶ß c×wZ‡Z Abym„Z av‡ci Av‡jv‡K Mo wbY©‡qi mviwY n‡e wbgœiƒc : avc wePz¨wZ †kÖwY e¨vwß 2 6 6 10 10 14 14 18 18 22 ga¨gvb xi 4 8 12 16 20 MYmsL¨v f i ui xi h a

MYmsL¨v avc wePz¨wZ f i ui 4 27 42 47 0

a

1 9 21 47 52

4 3 2 1 0

286

MwYZ

22 26 30

Mo x

a
= 20 = 20 = 19 21

26 30 34 †gvU f i ui h n

24 28 32

36 19 3 188

1 2 3

36 38 9 37

37 4 188

79

Drcv`‡b AvbygvwbK Mo LiP 19 kZ UvKv|

¸i“Z¡ cÖ`Ë Dcv‡Ëi Mo wbY©q
A‡bK †¶‡Î AbymÜvbvaxb cwimsL¨v‡bi Pj‡Ki mvswL¨K gvb x1 , x2 ,......., xn wewfbœ KviY/¸i“Z¡/fvi Øviv cÖfvweZ n‡Z cv‡i| G mKj †¶‡Î Dcv‡Ëi gvb x1 , x2 ,......., xn Gi mv‡_ G‡`i KviY/¸i“Z¡/fvi

w1 , w2 ,....... , wn we‡ePbv K‡i MvwYwZK Mo wbY©q Ki‡Z nq| hw` n msL¨K Dcv‡Ëi gvb x1 , x2 ,......., xn n‡jv Ges G‡`i ¸i“Z¡ hw` w1 , w2 ,......., wn nq Z‡e G‡`i ¸i“Z¡ cÖ`Ë MvwYwZK Mo n‡e n xi wi xw i 1 n

wi i 1

D`nviY 9| †Kv‡bv wek¦we`¨vj‡qi K‡qKwU wefv‡Mi ¯œvZK m¤§vb †kÖwY‡Z cv‡mi nvi I wk¶v_©xi msL¨v wb‡Pi mviwY‡Z Dc¯’vcb Kiv n‡jv| D³ wek¦we`¨vj‡qi H KqwU wefv‡Mi ¯œvZK m¤§vb †kÖwY‡Z cv‡mi Mo nvi wbY©q Ki| wefv‡Mi bvg MwYZ cwimsL¨vb Bs‡iwR evsjv cÖvwYwe`¨v ivóªweÁvb cv‡ki nvi 70 80 50 90 60 85 (kZKivq) wk¶v_©xi msL¨v 80 120 100 225 135 300 mgvavb : GLv‡b cv‡mi nvi I wk¶v_©xi msL¨v †`Iqv Av‡Q| cv‡mi nv‡ii fvi n‡jv wk¶v_©xi msL¨v| hw` cv‡mi nv‡ii PjK x Ges wk¶v_©xi msL¨v PjK w aiv nq, Z‡e ¸i“Z¡ cÖ`Ë MvwYwZK Mo wbY©‡qi mviwY n‡e wbgœiƒc : xi wi wefv‡Mi bvg xi wi MwYZ 70 80 5600 cwimsL¨vb 80 120 9600 Bs‡iwR 50 100 5000

MwYZ

287

evsjv cÖvwYwe`¨v ivóªweÁvb †gvU

90 60 85

225 135 300 960

20250 8100 25500 74050

6 xi wi xw i 1 6

wi i 2

74050 960

77 14

cv‡mi Mo nvi 77 14 KvR : †Zvgv‡`i Dc‡Rjvi K‡qKwU ¯‹z‡ji Gm.Gm.wm. cv‡mi nvi I Zv‡`i msL¨v msMÖn Ki Ges cv‡mi Mo nvi wbY©q Ki|

ga¨K
8g †kÖwY‡Z Avgiv wk‡LwQ †h, †Kv‡bv cwimsL¨v‡bi DcvË¸‡jv gv‡bi µgvbymv‡i mvRv‡j †hmKj DcvË mgvb `yBfv‡M fvM K‡i †mB gvbB n‡e DcvË¸‡jvi ga¨K| Avgiv AviI †R‡bwQ †h, hw` Dcv‡Ëi msL¨v n nq Ges n hw` we‡Rvo msL¨v nq Z‡e ga¨K n‡e n 1 Zg c‡`i gvb| Avi n hw` †Rvo msL¨v nq, Z‡e n ga¨K n‡e Zg I 2
2 n 1 Zg c` `yBwUi mvswL¨K gv‡bi Mo| GLv‡b Avgiv m~Î e¨envi bv K‡i Ges 2

e¨envi K‡i Kxfv‡e ga¨K wbY©q Kiv nq Zv D`vni‡Yi gva¨‡g Dc¯’vcb Kiv n‡jv| D`vniY 10| wb‡Pi 51 Rb wk¶v_©xi D”PZi (†m.wg.) MYmsL¨v wb‡ekb mviwY †`Iqv n‡jv| ga¨K wbY©q Ki| D”PZv (†m.wg.) MYmsL¨v D”PZv †m.wg.) MYmsL¨v µg‡hvwRZ MYmsL¨v 150 4 150 4 4 155 6 155 6 10 160 12 160 12 22 165 16 165 16 38 170 8 170 8 46 175 5 175 5 51

mgvavb : ga¨K wbY©‡qi MYmsL¨v mviwY

GLv‡b n = 51 hv we‡Rvo msL¨v 51 1 ga¨K = Zg c‡`i gvb 2 = 26 Zg c‡`i gvb, = 165 wb‡Y©q ga¨K 165 †m.wg.| j¶ Kwi : 23 †_‡K 38 Zg c‡`i gvb 165| D`vniY 11 : wb‡Pi 60 Rb wk¶v_©xi MwY‡Z cÖvß b¤^‡ii MYmsL¨v wb‡ekb mvwiwY †`Iqv n‡jv| ga¨K wbY©q Ki : cÖvß b¤^i MYmsL¨v 40 2 45 4 50 4 55 3 60 7 70 10 80 16 85 6 90 4 95 3 100 1

288

MwYZ

mgvavb : ga¨K wbY©‡qi µg‡hvwRZ MYmsL¨v mviwY n‡jv : cÖvß b¤^i MYmsL¨v µg‡hvwRZ MYmsL¨v 40 2 2 45 4 6 50 4 10 55 3 13 60 7 20 70 10 30 80 16 46 85 6 52 90 4 56 95 3 59 100 1 60

GLv‡b, n = 60 hv †Rvo msL¨v| 60 60 Zg I +1 Zg c` `yBwUi gv‡bi mgwó 2 2 ga¨K = 2 30 Zg I 31 Zg c` `yBwUi gv‡bi mgwó 2
70 80 2 wb‡Y©q ga¨K 75| 150 75 2

KvR : 1| †Zvgv‡`i †kÖwYi 49 Rb wk¶v_©xi D”PZv (†m.wg.) wb‡q MYmsL¨v mviwY ˆZwi Ki Ges †Kvb m~Î e¨envi bv K‡i ga¨K wbY©q Ki| 2| c~‡e©i mgm¨v †_‡K 9 R‡bi D”PZv ev` w`‡q 40 R‡bi D”PZvi (†m.wg.) ga¨K wbY©q Ki|

†kÖwYweb¨¯— Dcv‡Ëi ga¨K wbY©q hw` †kÖwYweb¨¯— Dcv‡Ëi msL¨v nq n , Z‡e †kÖwYweb¨¯— Dcv‡Ëi c‡`i gvb ev ga¨K wbY©‡q e¨eüZ m~Î n‡jv ga¨K = L

n 2

Zg c‡`i gvb n‡”Q ga¨K| Avi

n 2

Zg

n h †hLv‡b L n‡jv †h †kÖwY‡Z Fc 2 fm ga¨K Aew¯’Z †mB †kÖwYi wbgœmxg, n MYmsL¨v, Fc ga¨K †kÖwYi c~e©eZ©x †kÖwYi †hvwRZ MYmsL¨v, f m ga¨K †kÖwYi MYmsL¨v Ges h †kÖwY e¨vwß|
D`vniY 12| wb‡Pi MYmsL¨v wb‡ekY mviwY †_‡K ga¨K wbY©q Ki : mgq (†m‡KÊ) MYmsL¨v 30 35 3 36 41 10 42 47 18 48 53 25 µg‡hvwRZ MYmsL¨v 3 13 31 56 54 59 8 60 65 6

mgvavb : ga¨K wbY©‡qi MYmsL¨v wb‡ekY mviwY : mgq (†m‡K‡Ê) MYmsL¨v †kªwY e¨wß 3 30 35 10 36 41 18 42 47 25 48 53

MwYZ

289

60

65

6 70

70

GLv‡b, n

70 ev 35| 2 2 AZGe, ga¨K n‡jv 35 Zg c‡`i gvb| 35 Zg c‡`i Ae¯’vb n‡e (48-53) †kÖwY‡Z| AZGe ga¨K †kÖwY n‡jv (48-53)| myZivs, L 48 Fc 31 fm 25 Ges h = 6|
70 Ges ga¨K = L n 2 Fc h fm

n

= 48 + (35 = 48 + 0 96 = 48 96 wb‡Y©q ga¨K 48 96

31)

6 = 48 + 4 25

6 25

KvR : †Zvgv‡`i †kÖwYi mKj wk¶v_©x‡K wb‡q 2wU `j MVb Ki| GKwU mgm¨v mgvav‡b cÖ‡Z¨‡Ki KZ mgq jv‡M (K) Zvi MYmsL¨v wb‡ekY mviwY ˆZwi Ki, (L) mviwY n‡Z ga¨K wbY©q Ki|

cÖPyiK
8g †kÖwY‡Z Avgiv wk‡LwQ †h, †Kv‡bv Dcv‡Ë †h msL¨v me©vwaK evi Dc¯’vwcZ nq, †mB msL¨vB Dcv‡Ëi cªPziK| GKwU Dcv‡Ëi GK ev GKvwaK cÖPziK _vK‡Z cv‡i| †Kv‡bv Dcv‡Ë hw` †Kv‡bv msL¨vB GKvwaKevi bv _v‡K Z‡e †mB Dcv‡Ëi †Kv‡bv cÖPziK †bB| GLv‡b Avgiv Kxfv‡e m~Î e¨envi K‡i †kÖwYweb¨¯— Dcv‡Ëi cÖPziK wbY©q Ki‡Z nq, ZvB Av‡jvPbv Kiv n‡jv|

†kÖwY web¨¯— Dcv‡Ëi cÖPziK wbY©q
†kÖwY web¨¯— Dcv‡Ëi cÖPziK wbY©‡qi m~Î n‡jv : f 1 cÖPziK = L h †hLv‡b L cÖPziK †kÖwYi A_©vr †h †kÖwY‡Z cÖPziK Aew¯’Z Zvi wbgœgvb, f f
1 2

f1 = cÖPziK †kÖwYi MYmsL¨v c~e©eZ©x †kÖwYi MYmsL¨v, f 2 cÖPziK †kÖwYi MYmsL¨v-cieZ©x †kÖwYi MYmsL¨v Ges h = †kÖwY e¨vwß| D`vniY 13| wb‡Pi MYmsL¨v wb‡ekY mviwY †_‡K cÖPziK wbY©q Ki| mgvavb : †kÖwY MYmsL¨v 4 f 31 40 1 cÖPyiK = L h f f 6 41 50 1 2 8 GLv‡b MYmsL¨v me©vwaK evi 12 Av‡Q 51 60 (71-80) †kÖwY‡Z| 12 61 70 myZivs, L 61 dg©v-37, MwYZ-9g-10g

290

MwYZ

f f d

12 8
2

4 3

12 9
2

71 80 81 90 91 100

9 7 4

10
4 10 61 4 10 7

cÖPziK = 61 +

4 3 40 = 61 + 61 5 7 7 wb‡Y©q cÖPziK 66 714

66 7 |

D`vniY 14| wb‡Pi MYmsL¨v wb‡ekY mviwY †_‡K cÖPziK wbY©q Ki : †kÖwY 41 50 51 60 61 70 71 GLv‡b, 80 L f MYmsL¨v 25 20 15 8 mgvavb : GLv‡b MYmsL¨v me©vwaK evi 25 Av‡Q (41-50) †kÖwY‡Z| myZivs, cÖPziK GB †kÖwY‡Z Av‡Q| Avgiv Rvwb, f 1 cÖPziK = L h f f
1 2

[cÖ_g †kÖwY‡Z MYmsL¨v †ewk n‡j, c~e©eZ©x †kÖwYi MYmsL¨v k~b¨] 41 25 0

1

f

25
2

20

5

cÖPziK = 41 +

25 10 25 5 25 = 41 + 10 51 30 = 49 33

8 33 |

wb‡Y©q cÖPziK 49 33 †kÖwY web¨¯— Dcv‡Ë cÖ_g †kÖwY cÖPziK †kÖwY n‡j, Zvi Av‡Mi †kÖwYi MYmsL¨v k~b¨ ai‡Z nq D`vniY 15| wb‡Pi MYmsL¨v wb‡ekY mviwYi cÖPziK wbY©q Ki : mgvavb : †kÖwY MYmsL¨v GLv‡b MYmsL¨v me©vwaK 4 10 20 evi 25 Av‡Q (41-50) †kÖwY‡Z| 16 21 30 GB †kÖwY‡Z cÖPziK we`¨gvb 20 31 40 Avgiv Rvwb, 25 41 50 f 1 cÖPziK = L h f f
1 2

MwYZ

291

GLv‡b,

L f
1

41 25 20

5

f

25 0 †kl †kÖwY cÖPziK †kÖwY n‡j, cieZ©x
2

†kÖwYi NUb msL¨v k~b¨ aiv nq h 10 5 10 AZGe, cÖPziK = 41 + 25 = 41 + 2 = 42 wb‡Y©q cÖPziK 42|

Abykxjbx 17 mwVK DË‡i wUK (√) wPý `vI : 1| wb‡Pi †KvbwU Øviv †kªwY e¨vwß †evSvq ? (K) DcvËmg~‡ni g‡a¨ e„nËg I ¶`ªZg Dcv‡Ëi e¨eavb (L) DcvËmg~‡ni g‡a¨ cÖ_g I †kl Dcv‡Ëi e¨eavb (M) cÖ‡Z¨K †kªwYi Aš—f©y³ e„nËg I ¶y`ªZg msL¨vi cv_©K¨ (N) cÖ‡Z¨K †kªwYi Aš—f©y³ e„nËg I ¶y`ªZg msL¨vi mgwó DcvËmg~n mviwYfy³ Kiv n‡j cÖwZ †kªwY‡Z hZ¸‡jv DcvË Aš—f©y³ nq Zvi wb‡`©kK wb‡Pi †KvbwU ? (K) †kªwY mxgv (L) †kªwYi ga¨we›`y (M) †kªwY msL¨v (N) †kªwYi MYmsL¨v cwimsL¨v‡bi Aweb¨š— DcvËmg~n gv‡bi µgvbymv‡i mvRv‡j DcvËmg~n gvSvgvwS †Kv‡bv gv‡bi KvQvKvwQ cwÄf~Z nq| Dcv‡Ëi GB cÖeYZv‡K ejv nq (K) cÖPziK (L) †Kw›`ªq cÖeYZv (M) Mo (N) ga¨K kxZKv‡j evsjv‡`‡ki †Kvb GKwU AÂ‡ji 10 w`‡bi ZvcgvÎvi (†mw›Uª‡MÖW) cwimsL¨vb n‡jv 10 , 9 , 8 , 6 , 11 , 12 , 7 , 13 , 14 , 5 | GB cwimsL¨v‡bi †cÖw¶‡Z (4-6) ch©š— cÖkœ¸‡jvi DËi `vI | Dc‡ii msL¨vm~PK Dcv‡Ëi cÖPziK †KvbwU ? (K) 12 (L) 5 (M) 14 (N) cÖPziK †bB Dc‡ii msL¨vm~PK Dcv‡Ëi Mo ZvcgvÎv †KvbwU ? (K) 8 (L) 8.5 (M) 9.5 (N) 9 DcvËmg~‡ni ga¨K †KvbwU ? (K) 9.5 (L) 9 (M) 8.5 (N) 8 mviwYfy³ †kªwYweb¨v¯— Dcv‡Ëi msL¨v n‡jv n , ga¨K †kªwYi wbgœmxgv L , ga¨K †kªwYi c~e©eZ©x †kªwYi µg‡hvwRZ MYmsL¨v Fc ga¨K †kªwYi MYmsL¨v fm Ges †kªwY e¨vwß h GB Z‡_¨i Av‡jv‡K wb‡Pi †KvbwU ga¨K wbY©‡qi m~Î ? (K) L n 2 Fc h fm

2| 3|

4| 5| 6| 7|

(L) L

n 2

fm

h Fm

292

MwYZ

(M) L

n Fc 2

h fm

(N) L

n 2

fn

h Fm

wb‡P †Zvgv‡`i ¯‹z‡ji 8g †kªwYi mgvcbx cix¶vq evsjvq cÖvß b¤^‡ii MYmsL¨v mviwY †`Iqv n‡jv| GB mviwY †_‡K (8 17 )ch©š— cÖ‡kœi DËi `vI : †kªwY e¨vwß 31 40 41 50 51 60 61 80 71 80 81 90 91 100 MYmsL¨v 6 12 16 24 12 8 2 µg‡hvwRZ 6 18 34 58 70 78 80 MYmsL¨v 8| 9| 10| 11| 12| 13| 14| 15| DcvËmg~‡ni KqwU †kªwY‡Z web¨¯— Kiv n‡q‡Q ? (K) 6 (L) 7 (M) 8 mviwY‡Z Dc¯’vwcZ Dcv‡Ëi †kªwY e¨vwß KZ ? (K) 5 (L) 9 (M) 10 4_© †kªwYi ga¨gvb KZ ? (K) 71.5 (L) 61.5 (M) 70.5 Dcv‡Ëi ga¨K †kªwY †KvbwU ? (K) 41 50 (L) 51 60 (M) 61 70 ga¨K †kªwYi c~e©eZ©x †kªwYi †hvwRZ MYmsL¨v KZ ? (K) 18 (L) 34 (M) 58 ga¨K †kªwYi wbgœmxgv KZ ? (K) 41 (L) 51 (M) 61 ga¨K †kªwYi MYmsL¨v KZ ? (K) 16 (L) 24 (M) 34 Dc¯’vwcZ Dcv‡Ëi ga¨K KZ ? (K) 63 (L) 63.5 (M) 65 (N) 9 (N) 15 (N) 75.6 (N) 71 80 (N) 70 (N) 71 (N) 58 (N) 65.5

16| Dc¯’vwcZ Dcv‡Ëi cÖPziK KZ ? (K) 61.4 (L) 61 (M) 70 (N) 70.4 17| †Kvb ¯‹z‡ji 10g †kªwYi 50 Rb wk¶v_©xi IRb (wK‡jvMÖvg) n‡jv : 45, 50, 55, 51, 56, 57, 56, 60, 58, 60, 61, 60, 62, 60, 63, 64, 60, 61, 63, 66, 67, 61, 70, 70, 68, 60, 63, 61, 50, 55, 57, 56, 63, 60, 62, 56, 67, 70, 69, 70, 69, 68, 70, 60, 56, 58, 61, 63, 64| (K) †kªwY e¨eavb 5 a‡i MYmsL¨v wb‡ekb mviwY ˆZwi Ki| (L) mviwY †_‡K msw¶ß c×wZ‡Z Mo wbY©q Ki| (M) MYmsL¨v wb‡ekb mviwY‡Z Dc¯’vwcZ Dcv‡Ëi MYmsL¨v eûfyR AuvK| 18| 10g †kªwYi 50 Rb wk¶v_©xi MwYZ wel‡q cÖvß b¤^‡ii MYmsL¨v wb‡ekb mviwY †`Iqv n‡jv| cÖ`Ë Dcv‡Ëi MYmsL¨v eûfyR AuvK| †kªwY e¨vwß 31 40 41 50 51 60 61 80 71 80 81 90 91 100 MYmsL¨v 6 8 10 12 5 7 2

MwYZ

293

19| †Kvb †kªwYi 60 Rb wk¶v_©xi 50 b¤^‡ii mgwqK cix¶vq cÖvß b¤^‡ii MYmsL¨v wb‡ekb mviwY n‡jv : cÖvß b¤^i MYmsL¨v 1 10 7 11 20 10 21 30 31 40 41 50 16 18 9

Dcv‡Ëi AwRf †iLv AuvK| 20| wb‡P 50 Rb wk¶v_©xi IR‡bi (†KwR) MYmsL¨v wb‡ekb mviwY †`Iqv n‡jv| ga¨K wbY©q Ki| IRb (†KvwR) MYmsL¨v e¨vwß MYmsL¨v †hvwRZ dj 45 2 45-49 4 4 50 6 50-54 8 12 55 8 60 16 55-59 10 22 65 12 60-64 20 42 70 6 65-69 12 54 70-74 6 60

21| †Zvgv‡`i †kÖwYi 60 Rb wk¶v_©xi IR‡bi (†KwR) MYmsL¨v wb‡ekb mviwY n‡jv :

(K) Dcv‡Ëi ga¨K wbY©q Ki| (L) Dcv‡Ëi cÖPziK wbY©q Ki| 22| Dcv‡Ëi †¶‡Î cÖPziK(i) †K›`ªxq cÖebZvi cwigvc : (ii) me‡P‡q †ekx evi Dc¯’vwcZ gvb (iii) me‡¶‡Î Abb¨ bvI n‡Z cv‡i Dc‡ii Z‡_¨i wfwË‡Z wb‡Pi †KvbwU mwVK? L) i I iii K) i I ii N) i, ii I iii M) ii I iii 23| †Kv‡bv we`¨vj‡qi evwl©K cix¶vq 9g †kªwYi 50 Rb wk¶v_©xi MwY‡Z cÖvß b¤^i¸‡jv wbgœiƒc:
76, 55, 97, 84, 78, 65, 92, 88, 56, 64, 98, 45, 65, 45, 85, 79, 77, 73, 73, 53, 64 87 93 93 73 68, 46, 58, 62, 34, 56, 32, 41, 67, 75, 73, 75, 69, 69, 82, 83, 89, 63, 65, 67, 57 48 39 53 62

K. cÖ`Ë Z_¨wUi aiY Kxiƒc? †Kv‡bv wb‡el‡Y GKwU †kªwYi MYmsL¨v Kx wb‡`©k K‡i ? L. Dch©y³ †kªwY e¨wß wb‡q MYmsL¨v wb‡elY ˆZwi Ki| M. msw¶ß c×wZ‡Z cÖvß b¤^‡ii Mo wbY©q Ki| 24|

K. Dc‡ii wP‡Î, cÖ_g †kªwYwUi †kªwY ga¨gvb I †kl †kªwYwUi MYmsL¨v KZ? L. wP‡Î cÖ`wk©Z Z_¨wU‡K Q‡Ki gva¨‡g cÖKvk Ki| M. ÔLÕ-As‡k cÖvß QK †_‡K wb‡elYwUi ga¨K wbY©q Ki|

294

MwYZ

DËigvjv
Abykxjbx 1
4| 5| 6| (K) 0 16 (L) 0 63 (M) 3 2 (N) 3 53 (K)
71 769 2 35 2 (L) (M) (N) 3 (O) 6 9 99 90 15 3330

(K) 2 333, 5 235 (L) 7 266, 4 237 (M) 5 777777, 8 343434 , 6 245245 (N) 12 3200, 2 1999, 4 3256

7| 8| 9|

(K) 0 589 (L) 17 1179 (M) 1 92631 (K) 1 31 (L) 1 665 (M) 3 1334 (N) 6 11062 (K) 0 2 (L) 2 (M) 0 2074 (N) 12 185

10| (K) 0 5 (L) 0 2 (M) 5 21951 (N) 4 8 11| (K) 3 4641, 3 464 (L) 0 5025, 0 503 (M) 1 1595, 1 160 (N) 2 2650, 2 265 12| (K) g~j` (L) g~j` (M) Ag~j` (N) Ag~j` (O) Ag~j` (P) Ag~j` (Q) g~j` (R) g~j` 13| (K) 9 (L) 5

Abykxjbx 2⋅1
1| (K) {4, 5} (L) { 3, 4, 5, 6} (M) {6, 12, 18, 36} 2| (K) {x N : x we‡Rvo msL¨v Ges 1 (L) {x Gi ¸YwbqK Ges x 40} (N) {x Z : x2 16 Ges x3 3| (K) {1} (L) {1, 2, 3, 4, a} 5|{{x, y}, {x}, {y}, } (N) {3, 4}
N : x, 36 Gi ¸YbxqK} (M) {x N : x, 4

216}

(M) {2} (N) {2, 3, 4, a} (O) {2} l l l l

7| (K) 2, 3 (L) (a, c) (M) (1, 5) 8| (K) {(a, b), (a, c)}, {(b, a), (c, a)} (L) {(4, x), (4, y), (5, x), (5, y)} (M) {(3, 3), (5, 3), (7,3)} 9| {1, 3, 5, 7, 9, 15, 35, 45} G&es {1, 5} 10| {35, 105} 11| Rb

Abykxjbx 2⋅2
4| {(3, 2), (4, 2)} 5| {(2, 4), (2, 6)} 6|
7, 23, 7 4 7| 2 8| 1 A_ev 2 A_ev 3 9| 16 x
1 5 , 1, , {0, 1, 1, 2, 2 2 2}

11| (K) {2}, {1, 2, 3} (L) { 2, 1, 0, 1, 3), (2, 1)} (M)

12| (K) {( 1, 2), (0, 1) , (1, 0), (2, 1)} , { 1, 0, 1, 2} , {2, 1, 0, 1} (L) {( 1, 2), (0, 0) , (1, 2)} , { 1, 0, 1} , { 2, 0, 2}

MwYZ

295

Abykxjbx 3⋅1
1| (K) 4a2 12ab 9b2 (L) 4a2b2 12ab2c 9b2c2 (M) x4 (O) 16 y 2 40xy 25x2
4x2 y2 4 (N) a2 4 y y2 2 1 a2

(P) a2b2 2abc c2 (Q) 25x4 10x2 y (S)
9 p 2 16q 2 25r 2

(R) x2 4 y 2 16z 2 4xy 16 yz 8zx (T) 9b2 25c2 4a2 30bc 20ca 12ab

24 pq 40qr 30 pr

(U) a2 x2 b2 y 2 c2 z 2 2abxy 2bcyz 2cazx

(V) a2 b2 c2 d 2 2ab 2ac 2ad 2bc 2bd 2cd (W) 4a2 9x2 4 y 2 25z 2 12ax 8ay 20az 12xy 30xz 20 yz (Y) 994009 2| (Z) 10140491 (X) 10201

(K) 16a2 (L) 36x2 (M) p2 49r 2 14rp (N) 36n2 24 pn 4 p2 (O) 100 (P) 4410000 (Q) 10 (R) 3104
3m

3|

16 4|

1 5|

6| 130 8|

1 11| 19 12| 25 4

13| 6 14| 138

15| 9 17| (2a b c)2 (b a c)2 18| (x 1)2 82 19| ( x 5)2 12 20| (i) 3 20| (ii ) 1

Abykxjbx 3⋅2
1| (K) 8x3 60x2 150x 125 (L) 8x6 36x4 y 2 54x2 y 4 27 y6 (N) 343m6 294m4n 84m2n2 8n3 (M) 64a3 240a2 x2 300ax4 125x6 (O) 65450827 (P) 994011992

(Q) 8a3 b3 27c3 12a2b 36a2c 6ab2 54ac2 9b2c 27bc2 36abc (R) 8 x 3 27 y 3 z 3 36 x 2 y 12 x 2 z 54 xy 2 27 y 2 z 6 xz 2 9 yz 2 36 xyz 2| 3| (K) 8a3 (L) 64x3 (M) 8x3 (N) 1 (O) 8(b c)3 (P) 64m3n3 (Q) 2( x 3
665 4| 54 5| 8

y3

z 3 ) (R) 64x3

6| 42880 7| 1728 10| (K) 3 (L) 9 11| (K) 133 (L) 665 14| 46 5

12| a 3 3a 13|

p3 3 p

Abykxjbx 3⋅3
1| 3|
(a b)(a c)

2|

(b 1)(a 1)

2( x y)(x

y z)

4| b(x y)(a c)

296
(3x 4)2
( x2 2xy y 2)(x2 2xy y2)

MwYZ
(a2 5a 1)(a2 5a 1)
(ax by ay by )(ax bx ay bx)

5| 7| 9|

6| 8|

(2a 3b 2c)(2a 3b 2c)

10| 9( x a)( x a)( x 2a)( x 2a) 12| (4x 5 y)(4x 5 y 2z)

11| (a y 2)(a y 4) 15| (x 2)(x 2)(x2 5) 17| ( x3 y3 3)(x3 y3 2) 19| (ab 7)(ab 15) 21| ( x 2)(x 2)(2x 3)(2x 3) 23| y 2( x 1)(9x 14) 25| (x a)(ax 1) 27| (2 z 3x 5)(10 x 7 z 3) 29| ( x ay y)(ax x y)

13| (a b c)(b c a)(c a b)(a b c) 14| ( x 4)(x 9) 16| (a 18)(a 12) 18| (a4 2)(a4 1) 20| ( x 13)(x 15) 22| (2x 5)(6x 4) 24| ( x 3)(x 3)(4x2 9) 26| (a2 2a 4)(3a2 6a 10) 28|
(3a 17b)(9a 7b)

30| 3x(2x 1)(4x2 2x 1) 32| ( x 2)(x2 x 1)

31| (a b)2(a4 2a3b 6a2b2 2ab3 b4) 33| (a 3)(a2 3a 3) 35| (2x 3)(4x2 12x 21) 37|
1 (2a 1)(4a2 8 2a 1)

34| (a b)(2a2 5ab 8b2) 36| 38|
1 (6a b)(36a 2 6ab b 2 ) 27

a2 3

b2

a4 9

a 2b 2 3

b4

39|

2a

1 2a

2a

1 2a

2

40| (a 4)(19a2 13a 7) 42| ( x2 7 x 4)(x2 7 x 18)

41| (x 6)(x 10) 43| ( x2 8x 20)(x2 8x 2)

Abykxjbx 3⋅4
1| 3| 5| 7|
(6x 1)( x 1) (x y)(x 3 y)(x 2 y)

2|

(a 1)(3a 2 3a 5)

4| (x 6)(x 1) 6| 8|
( x 3)(3x 2)
( x 1)( x 2)( x 3)

(2x 3)(x 1)
( x 2)( x 1)( x 3)

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