D”PZi MwYZ e¨envwiK 1g cÎ

cixÿv bs 1.1 ZvwiLt

cixÿ‡bi bvgt Ges we› y؇qi ms‡hvRK †iLvsk‡K Abycv‡Z AšÍwe©f³Kvix we› yi ¯’vbvsK

wbY©q|

g~jZZ¡t hw` Gi Dci we› ywUi Rb¨ nq †hb Kv‡Z©mxq

Z‡j I ywU we› y nq Ges -A‡ÿi mgvšÍivj

nq Z‡e we› y †K Abycv‡Z fvM Ki‡e| Avevi

I we› y yBwUi ms‡hvM †iLvsk‡K we› ywU

Abycv‡Z AšÍwe ©f³ Ki‡j, we› yi ¯’vbv¼

(

)

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi

(iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi|

Kvh©c×wZt

1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv

I AuvwK|

2. Dfq Aÿ eivei ÿz ªZg e‡M©i evû = GKK

a‡i Ges we› y yBwU emvB|

awi we› yØq I | miæ †cwÝj w`‡q ms‡hvM K‡i

‡iLvsk †jLwP‡Î Dc ’vcb Kwi|

3. we› y w`‡q -A‡ÿi mgvšÍivj †iLvi Dci

GKK I GKK a‡i I yBwU we› y wbB|

4. †hvM Kwi Ges †iLv AsKb Kwi hv †K we› y‡Z †Q` K‡i|

5. we› yi ¯’vbvsK wPwýZ Kwi|

dj msKjbt

we› yi ¯’vbvsK

MÖvd n‡Z cÖvß gvb m~Î n‡Z cÖvß gvb

(30 Ni, 10 Ni)= (

)

djvdjt cÖ`Ë we› y؇qi ms‡hvM †iLvsk‡K Abycv‡Z AšÍwef³Kvix we› yi ¯’vbvsK

gšÍe¨: MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbwg©Z gvb GKB| AZGe djvdj mwVK|

A(3,5)

C(6,2)

B(7,1)

P Q G

X O X/

Y

Y/

P A

B

C

Q

1

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs 1.2 ZvwiLt

cixÿ‡bi bvgt wÎfz‡Ri kxl©we› y Ges wÎfzRwU †ÿÎdj wbY©q|

g~jZË¡t wÎfz‡Ri kxl©Îq Ges n‡j, wÎfzRwUi †ÿÎdj

|

| eM© GKK|

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK

K¨vjKz‡jUi|

Kvh©c×wZt

1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK|

2. Dfq Aÿ eivei ÿz ªZg

e‡M©i evû = GKK a‡i

Ges

we› y wZbwU emvB|

miæ †cwÝj w`‡q

ms‡hvM K‡i wÎfzRwU A¼b

Kwi|

3. we› y w`‡q -A‡ÿi

mgvšÍivj †iLv AvuwK|

4. I n‡Z Gi

Ici h_vµ‡g I j¤ AvuwK|

wnmvet we› yi ¯’vbv¼ ( Ni, Ni)=

we› yi ¯’vbv¼ ( Ni, Ni)=

wÎfz‡Ri †ÿÎdj

MÖvd n‡Z cÖvß gvb m~Î n‡Z cÖvß gvb

= UªwcwRqvg Gi †ÿÎdj+wÎfzR

Gi †ÿÎdj wÎfzR Gi †ÿÎdj =

eM©GKK

|

| |

|

eM©GKK

djvdjt wÎfz‡Ri †ÿÎdj eM©GKK

gšÍe¨t MÖvd †_‡K cÖvß gvb I MvwYwZKfv‡e wbwg©Z gvb GKB| AZGe djvdj mwVK|

B(-9, 1)

C(-3, -1) M(5, -1)

L(5, 1)

A(5, 6)

P

Q

X X/

Y/

Y

O

2

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs 1.3 ZvwiLt

cixÿ‡bi bvgt mij‡iLvi †jLwPÎ AsKb|

g~jZË¡t I m¤wjZ mKj †hvMvkÖqx mgxKiY GKwU mij‡iLv wb‡ ©k K‡i hLb I

DfqB GKB mv‡_ k~b¨ bq|

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK

K¨vjKz‡jUi|

Kvh©c×wZt

1. cÖ`Ë mgxKiY n‡Z cvB-

mgxKiYwU‡Z Gi K‡qKwU gvb wb‡q Gi Abyiƒc gvb †ei Kwi I wb‡Pi QKwU •Zix Kwi:

2. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK|

3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‹j a‡i I we› y¸wj ¯’vcb

Kwi Ges miæ †cwÝj w`‡q ms‡hvM K‡i mij‡iLvi †jLwPÎ AsKb Kwi|

‡jLwP‡Îi •ewkó¨t

i) cÖ`Ë mij‡iLvi Xvj-†Q` AvK…wZ

I

e‡j †iLvwU

Aÿ‡K abvZ¥K w`‡K

GKK ~‡i †Q` Ki‡e|

ii)

e‡j †iLvwU -A‡ÿi abvZ¥K w`‡Ki mv‡_ m~ÿè‡KvY Drcbœ K‡i|

C(-7, -2)

B(-2, 1)

A(3, 4)

3

X/ X

Y

Y/

O

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿv bs 2 ZvwiLt

cixÿ‡bi bvgt mgxKi‡Yi (e„Ë) †jLwPÎ AsKb|

g~jZË¡t mgxKiY GKwU e„‡Ëi mgxKiY hvi †K‡› ªi ¯’vbvsK Ges

e¨vmva© |

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii)

mv‡qw›UwdK K¨vjKz‡jUi|

Kvh©c×wZt

1. e„ˇK

e„‡Ëi mv‡_ Zzjbv K‡i cvB,

e„‡Ëi †K› ª Ges e¨vmva© =

2. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK|

3. Dfq Aÿ eivei ÿz ªZg e‡M©i evû = GKK ‡¯‹j a‡i e„‡Ëi †K› ª we› yi Ae ’vb wPwýZ

Kwi|

4. we› y‡K †K› ª K‡i GKK e¨vmva© wb‡q ‡cwÝj K¤úvm Gi mvnv‡h¨ e„Ë AsKb Kwi|

‡j‡Li •ewkó¨t

i. †jLwU GKwU e„Ë|

ii. †jLwU Awew”Qbœ|

iii. e„ËwU g~jwe› yMvgx|

iv) e„ËwU Aÿ‡K GKwU we› y‡Z Ges Aÿ‡K GKwU we› y‡Z †Q` K‡i|

4

C(-3, 4)

X X/

Y/

Y

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs 3.1 ZvwiLt

cixÿ‡bi bvgt GKwU wÎfz‡Ri evû¸wj h_vµ‡g †m.wg., †m.wg. Ges †m.wg. n‡j H wÎfz‡Ri e„nËg I

ÿz`ªZg †KvY wbY©q| g~jZË¡t g‡b Kwi GKwU wÎfzR hvi wZbwU evû h_vµ‡g †m.wg., †m.wg. Ges

†m.wg. ‡Z e„nËg evû †m.wg. Gi wecixZ †KvY e„nËg †KvY Ges ÿz ªZg evû

†m.wg. Gi wecixZ †KvY ÿz`ªZg †KvY| Zvn‡j cÖ`Ë Dcv‡Ëi mvnv‡h¨ AsKb K‡i Puv`vi mvnv‡h¨

e„nËg I ÿz`ªZg †Kvb wbb©q Kwi Ges m~Î

I

†_‡K cÖvß gv‡bi mv‡_

mZ¨Zv hvPvB Kwi|

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj

K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi|

Kvh©c×wZt

1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I A¼b Kwi|

2. Dfq Aÿ eivei ÿz ªZg e‡M©i evû = GKK ‡¯‹j awi|

3. MÖvd †ccv‡i eivei ÿz`ªZg e‡M©i evûi mgvb K‡i e„nËg evû ‡m.wg

†K‡U †bB|

4. ‡K †K› ª K‡i ÿz`ªZg e‡M©i evûi mgvb e¨vmva© wb‡q GKwU e„ËPvc AvuwK Ges †K

†K› ª K‡i ÿz ªZg e‡M©i evûi mgvb e¨vmva© wb‡q AviI GKwU e„ËPvc AvuwK| e„ËPvcØq

ci¯úi we› y‡Z †Q` K‡i| Ges †hvM Kwi| Zvn‡j †Z

‡m.wg., ‡m.wg Ges ‡m.wg m~wPZ K‡i|

5. Puv`vi mvnv‡h¨ e„nËg †KvY I ÿz`ªZg †KvY wbY©q Kwi|

wnmvet

dj msKjbt

e„nËg †KvY wbY©q ÿz`ªZg †KvY wbY©q

†m.wg.

†m.wg.

†m.wg.

MÖvc †_‡K cÖvß gvb m~Î †_‡K cÖvß gvb MÖvc †_‡K cÖvß gvb m~Î †_‡K cÖvß gvb

djvdjt wb‡Y©q e„nËg †KvY Ges ÿz`ªZg †KvY

gšÍe¨t MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbYx©Z gvb cÖvq mgvb| AZGe djvdj mwVK|

5

X X/

Y/

Y

A

B

C

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs 3.2 ZvwiLt

cixÿ‡Yi bvgt GKwU wÎfz‡Ri †KvY¸wj n‡j wÎfzRwUi evû¸wji AbycvZ wbY©q|

g~jZË¡t g‡b Kwi, Gi †KvY¸wj I

Gi wecixZ evû¸wj

h_vµ‡g | Zvn‡j cÖ`Ë DcvË n‡Z MÖv‡di mvnv‡h¨ Ges

m~‡Îi mvnv‡h¨ I

Gi AbycvZ wbY©q Kwi|

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj

K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi|

Kvh©c×wZt

1. GKwU MÖvd †ccv‡i ’vbvs‡Ki Aÿ‡iLv I AvuwK|

2. Dfq Aÿ eivei ÿz ªZg e‡M©i evû = †m.wg. †¯‹j awi|

3. MÖvd †ccv‡i eivei ‡m.wg †K‡U †bB|

4. Puv`vi mvnv‡h¨ we› y‡Z I we› y‡Z

AsKb Kwi| I

‡iLv ci¯úi‡K we› y‡Z †Q` K‡i|

5. MÖvd †_‡K Puv`vi mvnv‡h¨ Ges †cwÝj K¤úv‡mi mvnv‡h¨ I evûi •`N©¨ †g‡c eivei ewm‡q

h_vµ‡g I evû؇qi •`N©¨ wbY©q Kwi|

wnmvet Avgiv Rvwb, -‡Z

Avevi

dj msKjbt

djvdjt wb‡Y©q AbycvZ

gšÍe¨t MÖvd †_‡K cÖvß gvb Ges MvwYwZKfv‡e wbYx©Z gvb cÖvq mgvb| AZGe djvdj mwVK|

wbY©q

MÖvd †_‡K cÖvß AbycvZ M~Î †_‡K cÖvß AbycvZ

6

X/ X

Y

Y/

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs 4.1 ZvwiLt

cixÿ‡Yi bvgt dvsk‡bi Ges iƒcvšÍwiZ

‡jLwPÎ AsKb K‡i †j‡Li •ewkó¨

wbY©q|

g~jZË¡t GKwU cive„‡Ëi mgxKiY hvi kxl©we› y g~jwe› y‡Z Ges Aÿ -Aÿ|

Gi †jL

wb‡Ri mgvšÍiv‡j GKK ev‡g mwi‡q w`‡q cive„‡Ëi †jL cvIqv hvq hvi kxl©we› y |

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii)

mv‡qw›UwdK K¨vjKz‡jUi|

Kvh©c×wZt

1. GKwU MÖvd †ccv‡i ’vbvs‡Ki Aÿ‡iLv I AvuwK|

2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit

3. Dfq Aÿ eivei ÿz ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we› y¸wj QK KvM‡R ¯’vcb Kwi Ges

miæ †cwÝj w`‡q ’vwcZ we› y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi|

4. †jLwUi cÖwZwU we› y‡K e‡M©i evûi mgvb A_©vr GKK evg w`‡K mwi‡q Gi

‡jL AsKb Kwi|

‰ewkó¨t

i. †jLwPÎ yBwU cive„Ë| Gi kxl©we› y

Gi kxl© we› y

ii. Gi †jL A‡ÿi mv‡c‡ÿ, I

Gi †jL †iLvi mv‡c‡ÿ cÖwZmg|

iii. Dfq eµ‡iLvB wb‡Pi w`‡K †Lvjv AvK…wZi cive„Ë|

7

(-3, -9)

(-1, -1)

0, 0

(1, -1)

(2, -4)

(3, -9) (-6, -9)

(-5, -4)

(-4, -1)

*-3, 0) X/ X

Y

Y/

(2,-1)

(-2,-4) (-4,-1)

(0,-9)

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs 4.2 ZvwiLt

cixÿ‡Yi bvgt GKB †jLwPÎ dvsk‡bi I Zvi wecixZ dvsk‡bi †jLwPÎ AsKb I †j‡Li

•ewkó¨ wbY©q|

g~jZË¡t †j‡Li Dci ’ we› y¸wji fzR I †KvwUi ¯’vb wewbgq K‡i

Gi

†jLwPÎ AsKb Kiv hvq A_ev, ‡iLvi mv‡c‡ÿ Gi cÖwZ”Qwe AsKb K‡i

Gi ‡jL cvIqv hvq|

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) ‡cwÝj K¤úvm (vii)

mv‡qw›UwdK K¨vjKz‡jUi|

Kvh©c×wZt

1. GKwU MÖvd †ccv‡i ’vbvs‡Ki Aÿ‡iLv I AvuwK|

2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit

-6

-7

3. Dfq Aÿ eivei ÿz ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we› y¸wj QK KvM‡R ¯’vcb Kwi Ges

miæ †cwÝj w`‡q ’vwcZ we› y¸wj †hvM K‡i Gi †jL AsKb Kwi|

4. GB †¯‹‡j we› y¸wj ¯’vcb Kwi Ges miæ †cwÝj w`‡q ’vwcZ we› y¸wj †hvM K‡i

Gi †jL AsKb Kwi|

‣ewkó¨t

i. †jLwPÎ yBwU mij †iLv wb‡ ©k K‡i|

ii. ‡jLwPÎØq GKwU we› y‡Z †Q` K‡i|

iii. ‡jLwPÎØq †iLvi mv‡c‡ÿ ci¯ú‡ii cÖwZ”Qwe|

8

(-6, -7)

(0, 5)

(1, 7)

(2, 9)

(-7, -6)

(5, 0) (7, 1)

(9, 2)

X/ X

Y

Y/

(0,0)

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs ZvwiLt

cixÿ‡Yi bvgt w·KvYwgwZK dvsk‡bi †jL AsKb K‡i †j‡Li •ewkó¨ wbY©q hLb

g~jZË¡t myZivs GKwU ch©vqx

(Periodic) dvskb hvi ch©vq , ZvB cÖwZ e¨ewa‡Z Gi †jLwPÎ AweKj GKiƒc n‡e|

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) Puv`v (vii) ‡cwÝj

K¤úvm (viii) mv‡qw›UwdK K¨vjKz‡jUi|

Kvh©c×wZ:

1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK|

2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit

3. -Aÿ eivei ÿz`ªZg e‡M©i evû = Ges -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i

ZvwjKvfz³ we› y¸wj QK KvM‡R ’vcb Kwi Ges miæ †cwÝj w`‡q ’vwcZ we› y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i

Gi †jL AsKb Kwi|

‣ewkó¨t

i. †jLwPÎ Awew”Qbœ|

ii. Bnv †XD‡qi gZ|

iii. Bnv g~jwe› yMvgx

iv. GKwU ch©vqe„Ë dvskb hvi ch©vq

9

X/ X

Y

Y/

O(0,0)

(900,1)

(-1800,0)

(-900,1)

(1800,0)

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs ZvwiLt

cixÿ‡Yi bvgt ciggvb dvsk‡bi †jL AsKb K‡i †j‡Li •ewkó¨ wbY©q|

g~jZË¡t mgxKi‡Y Gi mKj ev¯Íe gv‡bi Rb¨ Gi gvb AFbvZ¥K|

, hLb

= , hLb

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK

K¨vjKz‡jUi|

Kvh©c×wZ:

1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK|

2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit

3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we› y¸wj QK KvM‡R ’vcb Kwi Ges

miæ †cwÝj w`‡q ’vwcZ we› y¸wj gy³ n‡¯Í †hvM K‡i Gi †jL AsKb Kwi|

‣ewkó¨t

i. †jLwPÎwU

†iLvi mv‡c‡ÿ cÖwZmg

ii. ‡jLwPÎwU 1g PZzf©vM I 2q PZzf©v‡M Amxg

ch©šÍ we Í…Z|

iii. †jLwPÎwU g~jwe› y‡Z †Q` K‡i bv|

iv. †jLwPÎwU A‡ÿi abvZ¥K w`‡K we`¨gvb|

10

(-2, 5)

(-1, 3)

(0, 1)

(0.5, 0)

(1, 1)

(2, 3)

(3, 5)

X/ X

Y

Y/

O

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs ZvwiLt

cixÿ‡Yi bvgt we› yi mwbœK‡U dvsk‡bi †jL‡K Avmbœfv‡e H we› y‡Z ¯úk©‡Ki †jL

Øviv ¯’vbxqfv‡e cÖwZ¯’vcb|

g~jZË¡t we› yi mwbœK‡U dvskmb‡K Avmbœfv‡e H we› y‡Z ¯úk©K Øviv ¯’vbwqfv‡e cÖwZ¯’vcb Kivi

m~Î, cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK

K¨vjKz‡jUi|

Kvh©c×wZt

1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK|

2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit

3. -Aÿ eivei ÿz ªZg N‡ii evû = Ges -Aÿ eivei ÿz ªZg N‡ii evû = GKK †¯‹j a‡i

ZvwjKvfz³ we› y¸wj QK KvM‡R ’vcb Kwi Ges miæ †cwÝj w`‡q ’vwcZ we› y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i

Gi †jL AsKb Kwi|

4. we› y‡Z ¯úk©K AsKb Kwi|

wnmvet

n‡Z cvB,

djvdjt we› yi mwbœK‡U

dvsk‡bi

†jL‡K Avmbœfv‡e H we› y‡Z ¯úk©K

Gi †jL Øviv ¯’vbxqfv‡e

cÖwZ ’vcb Kiv nj|

11

(-90, -1) (-60, -0.87)

(-30, -0.5)

(30, 0.5)

(60, 0.87) (90, 1)

X/ X

Y

Y/

O(0,0)

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs ZvwiLt

cixÿ‡Yi bvgt dvsk‡bi Rb¨ we› y‡Z I wbY©q, ‡hLv‡b Ges †jLwP‡Î

I cÖ`k©b|

g~jZË¡t ¯vaxb PjK I Aaxb Pj‡Ki AšÍi‡Ki ga¨Kvi m¤úK© Ges ¯vaxb Pj‡Ki AwZÿz ª

cwieZ©b Gi Rb¨ Aaxb Pj‡Ki AwZ ÿz`ª cwieZ©b cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK

K¨vjKz‡jUi|

Kvh©c×wZt

1. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AvuwK|

2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwit

3. -Aÿ I -Aÿ eivei ÿz ªZg e‡M©i evû = GKK †¯‹j a‡i ZvwjKvfz³ we› y¸wj QK KvM‡R ’vcb Kwi

Ges miæ †cwÝj w`‡q ’vwcZ we› y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i Gi †jL AsKb Kwi|

4. we› y‡Z ¯úk©K AsKb Kwi| mij‡iLv‡K ¯úk©KwU Ges dvskbwU h_vµ‡g I we› y‡Z

†Q` K‡i|

wnmvet

n‡Z cvB,

myZivs

Ges

wPÎ n‡Z cvB,

djvdjt I ‡jLwP‡Î cÖ`k©b Kiv n‡jv|

12

X/ X

Y

Y/

O(0,0)

A(2,4)

N

P

Q(3,9)

dy

y

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs ZvwiLt

cixÿ‡Yi bvgt QqwU †KvwU e¨envi K‡i ∫

Gi gvb wbY©q|

g~jZË¡t g‡b Kwi †ÿÎdj ∫

QqwU †KvwUi Rb¨

e¨envi K‡i ∫

Gi gvb wbY©q Kwi|

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) † ‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK K¨vjKz‡jUi|

Kvh©c×wZt

1. e¨ewa‡Z mg ~ieZx© wU †KvwU Gi Rb¨ GB e¨ewai wb¤œcÖvšÍ I

DaŸ©cÖv‡šÍi we‡qvMdj‡K Øviv fvM K‡i cÖ‡Z¨K ÿz ª As‡ki †`N© Gi gvb wbY©q Kwi|

2. Gi gvb n‡Z m~Î e¨envi K‡i wbY©q Kwi †hLv‡b

3. ‡_‡K wbY©q Kwi|

4. Aÿ eivei ÿz ªZg e‡M©i evû = GKK I -Aÿ eivei ÿz ªZg e‡M©i evû = GKK a‡i

ZvwjKvfz³ we› y¸wj QK KvM‡R ¯’vcb Kwi| miæ †cwÝj w`‡q we› y¸wj ms‡hvRb K‡i †jLwPÎwU AsKb Kwi|

5. cÖvß QqwU †KvwU‡K A‡ÿi mwnZ †¯‹‡ji mvn‡h¨ mshy³ K‡i wU UªvwcwRqvg AvKv‡i cÖKvk Kwi|

wnmvet

(

)

= (

)

= eM© GKK (cÖvq)

djvdj:

wb‡Y©q †ÿÎdj ∫

= eM©GKK cÖvq|

gšÍe¨t Gi gvb hZ †ekx n‡e Gi gvb KZ

ÿz`ª n‡e Ges Gi gvb AwaKZi ï× n‡e|

13

(0, 1) (0.16, 1.03) (0.32, 1.11)

(0.48, 1.26)

(0.64, 1.51)

(0.8 1.9)

X

Y

Y

O

y0 y1 y2 y3 y4 y5

x0 x1 x2 x3 x4 x5

D”PZi MwYZ e¨envwiK 1g cÎ

cixÿY bs ZvwiLt

cixÿ‡Yi bvgt cuvPwU †KvwU e¨envi K‡i ∫

Gi gvb wbY©q|

g~jZË¡t g‡b Kwi, †ÿÎdj ∫

cuvPwU †KvwUi Rb¨

e¨envi K‡i Gi gvb wbY©q Kwi|

cÖ‡qvRbxq DcKiYt (i) †cwÝj (ii) †¯‹j (iii) MÖvd †ccvi (iv) B‡iRvi (v) kvc©bvi (vi) mv‡qw›UwdK

K¨vjKz‡jUi|

Kvh©c×wZt

1. e¨ewa‡Z mg`~ieZx© wU †KvwU Gi Rb¨ GB e¨ewai wb¤œcÖvšÍ I DaŸ© cÖv‡šÍi

we‡qvMdj‡K Øviv fvM K‡i cÖ‡Z¨K ÿz ª As‡ki •`N©¨ Gi gvb wbY©q Kwi|

2. Gi gvb n‡Z m~Î e¨envi& K‡i wbY©q Kwi †hLv‡b

3.

‡_‡K Gi gvb wbY©q Kwit

4. Aÿ eivei ÿz ªZg e‡M©i evû = GKK I -Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i

ZvwjKvfz³ we› y¸wj QK KvM‡R ¯’vcb Kwi| miæ †cwÝj w`‡q we› y¸wj ms‡hvRb K‡i †jLwPÎwU AsKb Kwi|

5. cÖvß cuvPwU †KvwU‡K A‡ÿi mwnZ †¯‹‡ji mvn‡h¨ mshy³ K‡i wU UªvwcwRqvg AvKv‡i cÖKvk Kwi|

wnmvet

(

)

= (

)

=

= eM© GKK (cÖvq)

djvdjt

wb‡Y©q †ÿÎdj ∫

= eM©GKK (cÖvq)|

gšÍe¨t Gi gvb hZ †ekx n‡e Gi gvb KZ

ÿz`ª n‡e Ges Gi gvb AwaKZi ï× n‡e|

14

X O

y0

x0

y1

x1

y2

x2

y3

x3

y4

x4

Y

(0,1)

(.25,.9412)

(.50,.80)

(.75,.64)

(1,.50)