উচ্চমাধ্যমিকের উচ্চতর গণিত ২য় পত্র ব্যবহারিক

1. D”PZi MwYZ e¨envwiK 2q cÎ 1
cixÿv bs 1.1 ZvwiLt
cixÿ‡bi bvgt †jLwP‡Îi mvnv‡h¨ wØgvwÎK †hvMvkÖqx †cÖvMÖvwgs G Afxó dvskb Gi gvb m‡e©v”PKiY|
kZ©vejx:
g~jZZ¡t cÖ`Ë AmgZv Ges Gi †jLwPÎ AsKb K‡i AbyKzj GjvKvi
†K․wYK we›`y¸wji ¯’vbvsK wbY©q Kwi| †K․wYK we›`y¸wji ¯’vbvs‡Ki gvb ewm‡q Gi m‡e©v”P 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. AmgZv¸wji Abyiƒc ‣iwLK mgxKiY †K †Q`K AvKv‡i cÖKvk K‡i cvB,
( )
( )
2. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I AuvwK|
3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i ( )bs †iLv¯’ ( ) I ( ); ( )bs †iLv¯’ ( ) I
( )we›`y¸wj MÖvd KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q cÖvß we›`y¸wj ms‡hvM K‡i h_vµ‡g ( ) I ( )†iLvi
†jLwPÎ AsKb Kwi|
4. ( ) we›`y AmgZv‡K wm× K‡i (∵ mZ¨)| myZivs ( ) †iLv¯’ I Gi( ) we›`yi cvk¦©¯’
mKj we›`yi †mU H AmgZvi mgvavb| Z`ªæc ( ) †iLv¯’ I Gi( ) we›`yi cvk¦©¯’ mKj we›`yi †mU
AmgZvi mgvavb| ZvQvov Øviv eySvq mgvav‡bi AbyKzj GjvKv 1g PZzf©v‡M Aew¯’Z|
5. PZzfz©R Øviv mxgve× GjvKvi we›`ymg~‡ni †mU cÖ`Ë mKj kZ©‡K wm× K‡i| AZGe, GB †ÿÎwUB mgvav‡bi
AbyKzj GjvKv|
6. PZzfz©‡Ri †K․wYK we›`yi ¯’vbvsK ( ) ( ) ( ) I ( ) Gi †Q` we›`y ( )Ges ( )
7. Afxó dvsk‡b †K․wYK we›`y¸wji gvb ewm‡q Gi m‡e©v”P gvb wbY©q Kwi|
dj msKjbt
‡K․wYK we›`y
( )
( )
( )
( )
djvdjt Gi m‡e©v”P gvb =
(0, 30)
(45, 0)
(0, 20)
(80, 0)

2. D”PZi MwYZ e¨envwiK 2q cÎ 2
cixÿv bs 1.2 ZvwiLt
cixÿ‡bi bvgt †jLwP‡Îi mvnv‡h¨ wØgvwÎK †hvMvkÖqx †cÖvMÖvwgs G Afxó dvskb Gi me©wb¤œ gvb wbY©q|
kZ©vejx:
g~jZZ¡t cÖ`Ë AmgZv Gi †jLwPÎ AsKb K‡i AbyKzj GjvKvi
†K․wYK we›`y¸wji ¯’vbvsK wbY©q Kwi| †K․wYK we›`y¸wji ¯’vbvs‡Ki gvb ewm‡q Gi me©wb¤œ 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. AmgZv¸wji Abyiƒc ‣iwLK mgxKiY
( )
( )
( )
2. GKwU QK KvM‡R ¯’vbvs‡Ki Aÿ‡iLv I
AuvwK|
3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i
( )bs †iLv¯’ ( ) I ( ); ( )bs †iLv¯’
( ) I ( ); ( )bs †iLv¯’ ( ) I ( )
we›`y¸wj MÖvd KvM‡R ¯’vcb Kwi Ges miæ †cwÝj w`‡q
cÖwZ ‡Rvov ms‡hvM K‡i h_vµ‡g ( ) ( ) I
( )†iLvi †jLwPÎ AsKb Kwi|
4. ( ) we›`y AmgZv‡K wm× K‡i bv (∵ mZ¨ bq)| myZivs ( ) †iLv¯’ I Gi
( ) we›`yi wecixZ cvk¦©¯’ mKj we›`yi †mU H AmgZvi mgvavb| Abyiƒcfv‡e ( ) †iLv¯’ I Gi( ) we›`yi
wecixZ cvk¦©¯’ mKi we›`yi †mU AmgZvi mgvavb Ges ( ) †iLv¯’ I Gi( ) we›`yi wecixZ cvk¦©¯’
mKi we›`yi †mU AmgZvi mgvavb| ZvQvov Øviv eySvq mgvav‡bi AbyKzj GjvKv
1g PZzf©v‡M Aew¯’Z|
5. I ‡iLv·qi Dci¯’ I Zv‡`i Wvb cvk¦©¯’ we›`ymg~‡ni †mU cÖ`Ë mKi kZ©‡K wm× K‡i weavh H
AÂjwUB mgvav‡bi AbyKzj GjvKv|
6. †K․wYK we›`yi ¯’vbvsK ( ) ( ) I ( ) Gi †Q` we›`y ( ), ( ) I ( ) Gi †Q` we›`y ( )Ges
( )
7. Afxó dvsk‡b †K․wYK we›`y¸wji gvb ewm‡q Gi m‡e©v”P gvb wbY©q Kwi|
dj msKjbt
‡K․wYK we›`y
( )
( )
( )
( )
djvdjt Gi me©wb¤œ gvb =
(12, 0)
(0, 22)
(18, 0)
(0, 18)
(24, 0)
(0, 6)

3. D”PZi MwYZ e¨envwiK 2q cÎ 3
cixÿv bs 1.3 ZvwiLt
cixÿ‡bi bvgt AvM©Û wP‡Î Ges RwUj msL¨v `yBwU wPwýZ K‡i G‡`i †hvMd‡ji
ciggvb (gWzjvm) I bwZ (AvM©‡g›U) wbY©q|
g~jZË¡t g~jwe›`y, -Aÿ‡K ev¯Íe Aÿ Ges -Aÿ‡K KvíwbK Aÿ a‡i I RwUj msL¨vØq‡K AvM©Û wP‡Î
wPwýZ Ki‡j g~jwe›`yi mv‡_ G‡`i ms‡hvM †iLvØq‡K mwbœwnZ evû a‡i AswKZ mvgvšÍwi‡Ki KY©wU n‡e +
Gi ciggvb Ges A‡ÿi abvZ¥K w`‡Ki mv‡_ KY©wUi Drcbœ †KvY n‡e bwZ|
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 AuvwK|
2. Aÿ I Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i Ges ‡K
( ) ( ) Øviv wb‡`©k K‡i QK KvM‡R ¯’vcb Kwi|
3. I †K mwbœwnZ evû a‡i mvgvšÍwiK AsKb Kwi Ges †hvM Kwi| Zvn‡j we›`y RwUj
msL¨v `yBwUi †hvMdj + Gi Ae¯’vb wb‡`©k K‡i| Zvn‡j, + Gi ciggvb + Ges
+ Gi bwZ .
wnmvet + = + =
dj msKjbt
we›`yi
¯’vbvsK
we›`yi
¯’vbvsK
+
we›`yi
¯’vbvsK
ciggvb wbY©q bwZ wbY©q
MÖvd n‡Z m~Î n‡Z MÖvd n‡Z
Puv`vi mvnv‡h¨
m~Î n‡Z
( ) ( ) ( )
Ni
=
GKK
√
√
√ ( )
djvdjt wb‡Y©q ciggvb = GKK Ges bwZ=
gšÍe¨t †jLwPÎ n‡Z cÖvß gvb Ges MvwYwZKfv‡e wbY©qK…Z gvb cÖvq mgvb| AZGe djvdj mwVK|
(4, 2)
(2, 8)
(2, -3)
(0, 0)

4. D”PZi MwYZ e¨envwiK 2q cÎ 4
cixÿY bs 1.4 ZvwiLt
cixÿ‡Yi bvgt AvM©Û wP‡Î Ges RwUj msL¨v `yBwU wPwýZ K‡i G‡`i
we‡qvMd‡ji ciggvb (gWzjvm) I bwZ (AvM©‡g›U) wbY©q|
g~jZË¡t g~jwe›`y, -Aÿ‡K ev¯Íe Aÿ Ges -Aÿ‡K KvíwbK Aÿ a‡i I RwUj msL¨vØq‡K AvM©Û
wP‡Î wPwýZ Ki‡j g~jwe›`yi mv‡_ G‡`i ms‡hvM †iLvØq‡K mwbœwnZ evû a‡i AswKZ mvgvšÍwi‡Ki KY©wU n‡e
Gi ciggvb Ges A‡ÿi abvZ¥K w`‡Ki mv‡_ KY©wUi Drcbœ †KvY n‡e bwZ|
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 AuvwK|
2. Aÿ I Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i Ges ‡K
( ) ( ) Øviv wb‡`©k K‡i QK KvM‡R ¯’vcb Kwi|
3. I †K mwbœwnZ evû a‡i mvgvšÍwiK AsKb Kwi Ges †hvM Kwi| Zvn‡j we›`y RwUj
msL¨v `yBwUi we‡qvMdj Gi Ae¯’vb wb‡`©k K‡i| Zvn‡j, Gi ciggvb
Ges + Gi bwZ .
wnmvet =( ) ( )=
dj msKjbt
we›`yi
¯’vbvsK
we›`yi
¯’vbvsK we›`yi
¯’vbvsK
ciggvb wbY©q bwZ wbY©q
MÖvd n‡Z m~Î n‡Z MÖvd n‡Z Puv`vi
mvnv‡h¨
m~Î n‡Z
( ) ( ) ( )
Ni
= GKK
√
√
√ ( )
djvdjt wb‡Y©q ciggvb = GKK Ges bwZ=
gšÍe¨t †jLwPÎ n‡Z cÖvß gvb Ges MvwYwZKfv‡e wbY©qK…Z gvb cÖvq mgvb| AZGe djvdj mwVK|
(0, 0)
(5, 6)
(2, 8)
(-3, 2)

5. D”PZi MwYZ e¨envwiK 2q cÎ 5
cixÿv bs 2.1 ZvwiLt
cixÿ‡Yi bvgt ‡j‡Li mvnv‡h¨ ( ) mgxKi‡Yi ev¯Íe g~‡ji Avmbœ gvb wbY©q|
g~jZË¡t †j‡Li Dci Ges Gi Rb¨
( ) hw` ( ) Ges ( ) wecixZ wPýhy³ nq Z‡e e¨ewai g‡a¨ ( ) Gi Kgc‡ÿ GKwU A_ev
we‡Rvo msL¨K g~j _vK‡e|
( ) hw` ( ) Ges ( ) GKB wPýhy³ nq Z‡e e¨ewai g‡a¨ ( ) Gi ‡Rvo msL¨K g~j _vK‡e
A_ev ‡Kvb g~j _vK‡e bv|
( ) ( ) mgxKi‡Yi †jLwPÎ Aÿ‡K †h we›`y‡Z †Q` ev ¯úk© K‡i Zvi fzR ( ) mgxKi‡Yi ev¯Íe
g~j n‡e| hw` fzR fMœvsk nq Z‡e †mB fz‡Ri Avmbœ gvb ( ) mgxKi‡Yi g~‡ji Avmbœ gvb n‡e|
( ) ( ) mgxKi‡Yi †jLwPÎ Aÿ‡K †Q` ev ¯úk© bv K‡i Z‡e ( ) mgxKi‡Yi ev¯Íe g~j g~j
_vK‡e bv|
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. ( ) mgxKi‡Y Gi K‡qKwU gvb wb‡q Gi Abyiƒc gvb †ei Kwi I
wb‡Pi QKwU •Zwi Kwi|
3. Aÿ eivei ÿz`ªZg e‡M©i evû = GKK Ges Aÿ eivei †QvU e‡M©i evû = GKK ‡¯‥j a‡i ZvwjKvfz³
we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i ( )Gi †jLwPÎ AsKb
Kwi|
4. ‡jLwPÎwU †h mKj we›`y‡Z Aÿ‡K †Q` K‡i †m mKj we›`yi fzR wbY©q K‡i cÖ`Ë mgxKi‡Yi ev¯Íe g~j wbY©q Kwi|
g~‡ji gvb wbY©qt
†jLwPÎwU Aÿ‡K ( ) Ges ( ) we›`y‡Z †Q` K‡i‡Q| AZGe ( ) Gi `yBwU g~j I | †jLwPÎ n‡Z
†`Lv hv‡”Q †h, ( ) Gi Aci g~jwU e¨ewa‡Z we`¨gvb|
e¨ewa‡Z Z…Zxq g~‡ji Avmbœ gvb wbY©qt
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
Z…Zxq g~j =
djvdjt wb‡Y©q g~j wZbwU Ges
(-3, -25)
(-2.5, -9)
(-2, 0)
(-1, 3)
(0, -4)
(1, -9)
(2, 0)
(2.5, 13.5)

6. D”PZi MwYZ e¨envwiK 2q cÎ 6
cixÿv bs 2.2 ZvwiLt
cixÿ‡bi bvgt ‡j‡Li mvnv‡h¨ ( ) mgxKi‡Yi ev¯Íe g~‡ji Avmbœ gvb wbY©q|
g~jZË¡t †j‡Li Dci Ges Gi Rb¨
( ) hw` ( ) Ges ( ) wecixZ wPýhy³ nq Z‡e e¨ewai g‡a¨ ( ) Gi Kgc‡ÿ GKwU A_ev
we‡Rvo msL¨K g~j _vK‡e|
( ) hw` ( ) Ges ( ) GKB wPýhy³ nq Z‡e e¨ewai g‡a¨ ( ) Gi ‡Rvo msL¨K g~j _vK‡e
A_ev ‡Kvb g~j _vK‡e bv|
( ) ( ) mgxKi‡Yi †jLwPÎ Aÿ‡K †h we›`y‡Z †Q` ev ¯úk© K‡i Zvi fzR ( ) mgxKi‡Yi
ev¯Íe g~j n‡e| hw` fzR fMœvsk nq Z‡e †mB fz‡Ri Avmbœ gvb ( ) mgxKi‡Yi g~‡ji Avmbœ gvb n‡e|
( ) ( ) mgxKi‡Yi †jLwPÎ Aÿ‡K †Q` ev ¯úk© bv K‡i Z‡e ( ) mgxKi‡Yi ev¯Íe g~j g~j
_vK‡e bv|
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. ( ) mgxKi‡Y Gi K‡qKwU gvb wb‡q Gi Abyiƒc gvb †ei Kwi I wb‡Pi
QKwU •Zwi Kwi|
3. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ‡¯‥j a‡i ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges ¯’vwcZ
we›`y¸wj gy³ n‡¯Í eµvKv‡i †hvM K‡i ( )Gi †jLwPÎ AsKb Kwi|
4. ‡jLwPÎwU †h mKj we›`y‡Z Aÿ‡K †Q` K‡i †m mKj we›`yi fzR wbY©q K‡i cÖ`Ë mgxKi‡Yi ev¯Íe g~j wbY©q Kwi|
g~‡ji gvb wbY©qt
†jLwPÎwU Aÿ‡K ( ) we›`y‡Z ¯úk© K‡i| ‡h‡nZz wØNvZ mgxKi‡Yi `yBwU g~j _v‡K, †m‡nZz mgxKiYwUi
mgvavb n‡e
djvdjt wb‡Y©q g~j `yBwU
(-6, 4)
(-5, 1)
-(4, 0)
(-3, 1)
(-2, 4)

7. D”PZi MwYZ e¨envwiK 2q cÎ 7
cixÿv bs ZvwiLt
cixÿ‡bi bvgt cive„‡Ëi Dc‡K‡›`ªi ¯’vbvsK ( ) Ges w`Kvÿ †iLvi mgxKiY n‡j Gi †jLwPÎ
AsKb Ki|
g~jZË¡t cive„‡Ëi Dc‡K›`ª Ges cive„‡Ëi Dci Aew¯’Z †h †Kvb we›`y n‡Z Gi w`Kv‡ÿi j¤^ `~iZ¡ n‡j,
cive„‡Ëi Dr‡Kw›`ªKZv
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 QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK|
2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i cive„‡Ëi Dc‡K›`ª ( ) QK KvM‡R ¯’vcb Kwi|
3. A‡ÿi mgvšÍivj cive„‡Ëi w`Kvÿ †iLv AvuwK|
4. Dc‡K›`ª †_‡K w`Kvÿ †iLvi Dci j¤^ AsKb K‡i we›`y‡Z †K mgwØLwÛZ Kwi& Zvn‡j
myZivs cive„‡Ëi Dci GKwU we›`y hv cive„‡Ëi kxl©|
5. ewa©Z Gi Dci †h †Kvb we›`y wb‡q H we›`y‡Z j¤^ Uvwb| †K †K›`ª K‡i Gi mgvb
e¨vmva© wb‡q GKwU e„ËPvc AvuwK| e„ËPvcwU †K I we›`y‡Z †Q` K‡i| Ges †hvM Kwi|
w`Kvÿ †iLvi Dci I j¤^Øq AuvwK| Zvn‡j Ges
myZivs cive„‡Ëi Dcwiw¯’Z `yBwU we›`y Ges .
6. ewa©Z Gi Dci h_vµ‡g we›`y wb‡q Abyiƒcfv‡e cive‡Ëi Dci I , I , I
we›`y¸wj wba©vib Kwi|
7. we›`y¸‡jv gy³ n‡¯Í eµvKv‡i †hvM K‡i cive„ËwUi †jLwPÎ AsKb Kwi|
‣ewkó¨t
i.
ii.
iii.
XX/
Y/
Y
Z A SN N1 N2 N3

8. D”PZi MwYZ e¨envwiK 2q cÎ 8
cixÿv bs ZvwiLt
cixÿ‡Yi bvgt Dce„‡Ëi GKwU Dc‡K‡›`ªi ¯’vbvsK ( ) Ges Abyiƒc w`Kvÿ †iLvi mgxKiY Ges
Dr‡Kw›`ªKZv n‡j Gi †jLwPÎ AsKb Ki|
g~jZË¡t Dce„‡Ëi GKwU Dc‡K›`ª Ges Dce„‡Ëi Dci Aew¯’Z †h †Kvb we›`y n‡Z Gi w`Kv‡ÿi j¤^ `~iZ¡ n‡j,
Dce„‡Ëi Dr‡Kw›`ªKZv †hLv‡b
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 QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AvuwK|
2. Dfq Aÿ eivei ÿz`ªZg e‡M©i evû = GKK ai cive„‡Ëi Dc‡K›`ª ( ) QK KvM‡R ¯’vcb Kwi Ges
A‡ÿi mgvšÍivj Dce„‡Ëi w`Kvÿ †iLv AvuwK|
3. Dc‡K›`ª †_‡K w`Kvÿ †iLvi Dci j¤^ AsKb Kwi| †K I we›`y‡Z Abycv‡Z AšÍwe©f³ I
ewnwe©f³ Kwi| Zvn‡j Ges . myZivs I Dce„‡Ëi Dcwiw¯’Z `yBwU we›`y|
4. Gi Dci †h †Kvb we›`y wb‡q j¤^ AvuwK| GLb †K †K›`ª K‡i Gi mgvb e¨mva© wb‡q GKwU
e„ËPvc AvuwK, hv †K h_vµ‡g I we›`y‡Z †Q` K‡i| Ges †hvM Kwi| w`Kvÿ †iLvi Dci
I j¤^Øq AuvwK| ‡`Lv hvq †h, Ges AZGe I we›`yØq Dce„‡Ëi Dci
Aew¯’Z|
5. GKB cÖwµqvq Gi Dci h_vµ‡g we›`y wb‡q I , I , I we›`y¸wj cvIqv hvq|
6. cÖvß we›`y¸wj eµvKv‡i gy³ n‡¯Í †hvM K‡i Dce„ËwUi †jLwPÎ AsKb Kwi|
‣ewkó¨t
i.
ii.
iii.
XX/
Y/
Y
Z A SN N2 N3

9. D”PZi MwYZ e¨envwiK 2q cÎ 9
cixÿY bs ZvwiLt
cixÿ‡Yi bvgt dvsk‡bi †jLwPÎ AsKb K‡i †j‡Li •ewkó¨ wbY©q|
g~jZË¡t hLb Ges
Gi †Kvb GKwU gv‡bi Rb¨ Gi A‡bK gvb _vK‡Z cv‡i, Z‡e †KejgvÎ gyL¨ gvb we‡ePbv Kiv n‡e|
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 AvuwK|
2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwi|
3. Aÿ eivei ÿz`ªZg e‡M©i evû = GKK Aÿ eivei ÿz`ªZg e‡M©i evû = a‡i ZvwjKvfz³
we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cw݇ji mvnv‡h¨ ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i ms‡hvM K‡i
Gi †jLwPÎ AsKb Kwi|
‡j‡Li •ewkó¨t
i. ‡jLwPÎwU g~j we›`yMvgx Ges †Q`nxb|
ii. †jLwPÎwU Zi½vwqZ|
iii. ‡jLwPÎwU AcÖwZmg|
XX/
Y/
Y
O(0,00
)
(1,900
)
(87,600
)
(50,300
)
(-1,-900
)
(-87,-600
)
(-50,-300
)
we:`ª: cÖ‡Z¨KwU e¨envwi‡Ki †¯‹j I Q‡Ki
g‡a¨ AmvgÄm¨ _vK‡Z cv‡i| wkÿv_x©
Zv‡`i wbR wbR wkÿ‡Ki wbKU Zv ms‡kvab
K‡i wb‡q e¨envwiK †bvUey‡Ki Rb¨ cÖ¯‘Z
Ki‡e|

10. D”PZi MwYZ e¨envwiK 2q cÎ 10
cixÿv bs ZvwiLt
cixÿ‡bi bvgt Gi Rb¨ GKB †jLwP‡Î w·KvYwgwZK dvskb I Gi wecixZ
dvskb AsKb|
g~jZË¡t ‡j‡Li Dci¯’ we›`y¸wii 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) mv‡qw›UwdK
K¨vjKz‡jUi|
Kvh©c×wZt
1. GKwU QK KvM‡R ¯’vbv‡¼i Aÿ‡iLv I AuvwK|
2. wb‡Pi ZvwjKvq Gi wfbœ wfbœ gv‡bi Rb¨ Gi cÖwZiƒcx gvb wbY©q Kwi|
3. Aÿ eivei ÿz`ªZg e‡M©i evû = , Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i
ZvwjKvfz³ we›`y¸wj QK KvM‡R ¯’vcb Kwi Ges miæ †cw݇ji mvnv‡h¨ ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i
ms‡hvM K‡i Gi †jLwPÎ AsKb Kwi|
4. ‡j‡Li Dci¯’ we›`y¸wji fzR I †KvwUi ¯’vb wewbgq K‡i cÖvß we›`y¸wj QK KvM‡R ¯’vcb Kwi
Ges miæ †cwÝwj‡ii mvnv‡h¨ ¯’vwcZ we›`y¸wj gy³ nv‡Z eµvKv‡i ms‡hvM K‡i Gi ‡jLwPÎ AsKb
Kwi|
‣ewkó¨:
i. ‡jLwPÎ g~jwe›`yMvgx bq Ges †Q`nxb|
ii. ‡jLwPÎØq Zi½vwqZ|
iii. ‡jLwPÎØq AcÖwZmg| XX/
Y/
Y
O(0,00
)
y=cosx
y=cos-1
x (0,1)
(600
,.5)
(1,0)
(.5,600
)
(1800
,-1)
(-1,1800
)

11. D”PZi MwYZ e¨envwiK 2q cÎ 11
cixÿv bs ZvwiLt
cixÿ‡bi bvgt ‣jwLK c×wZ‡Z †Kvb we›`y‡Z †Kv‡Y wµqviKZ I ei؇qi jwäi gvb I w`K
wbY©q|
g~jZË¡t Avgiv Rvwb, GKB mg‡q †Kv‡bv GKwU we›`y‡Z ci¯úi †Kv‡Y Kvh©iZ ej؇qi jwä hw`
e‡ji w`‡Ki mv‡_ †KvY Drcbœ K‡i Z‡e √ Ges
hLb
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 AvuwK|
2. Aÿ I Aÿ eivei ÿz`ªZg e‡M©i evû = ‡¯‥j a‡i n‡Z eM© evû
†K‡U †bB|
3. †iLvi we›`y‡Z ‡KvY AsKb Kwi Ges n‡Z eM©evû †K‡U
†bB|
4. Ges †K mwbœwnZ evû a‡i mvgvšÍwiK AsKb Kwi| Ges ej `yBwUi jwä n‡”Q KY©
hv e‡ji mv‡_ †KvY Drcbœ K‡i|
dj msKjbt
jwäi gvb w`K wbY©q
jwäi gvb wbY©q jwäi w`K wbY©q
MÖvd †_‡K m~Î †_‡K MÖvd †_‡K m~Î †_‡K
Ni=
√
=√
=√
=√
Puv`vi mvnv‡h¨
=
=
=
djvdjt
wb‡Y©q jwäi gvb Ges
w`K (cÖvq)
gšÍe¨t †jLwPÎ n‡Z cÖvß gvb
I MvwYwZKfv‡e wbY©xZ gvb cÖvq mgvb| AZGe djvdj mwVK|
X
X/
Y/
Y
O A
B C
P
Q R
520

M

12. D”PZi MwYZ e¨envwiK 2q cÎ 12
cixÿv bs ZvwiLt
cixÿ‡bi bvgt ‡Kvb we›`y‡Z wµqvkxj PviwU e‡ji gvb I Ges H we›`yMvgx GKwU †iLvi
mv‡_ Zv‡`i wµqv‡iLv h_vµ‡g I †KvY Drcbœ K‡i| ej¸wji jwäi gvb I w`K ‣jwLK c×wZ‡Z
wbY©q Ki|
g~jZË¡t g‡b Kwi, we›`y‡Z wµqviZ ej¸wj H we›`yMvgx †iLv Gi mv‡_ h_vµ‡g
‡KvY Drcbœ Ki‡j eivei Ges Dci j¤^ †iLv eivei ej¸wji j¤^vs‡ki mgwó h_vµ‡g
∑ Ges ∑ (awi)
Zvn‡j ej¸wji jwä √ Ges jwä †iLvi mv‡_
†KvY Drcbœ Ki‡j
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 AuvwK|
2. Aÿ I Aÿ eivei ÿz`ªZg e‡M©i evû = ‡¯‥j awi|
3. †iLvi mv‡_ Nwoi KvUvi wecixZ w`‡K
I †Kv‡Y bZ mij‡iLv n‡Z h_vµ‡g I
ejmg~n‡K wba©vwiZ †¯‥j Abymv‡i †K‡U wbB Ges w`K wPwýZ Kwi|
4. eivei ej¸‡jvi j¤^vsk wbY©q K‡i †hvM Kwi|
5. eivevi ej¸‡jvi j¤^vsk wbY©q K‡i †hvM Kwi|
6. I eivei j¤^vsk¸wji c„_K c„_K †hvMdj‡K I awi
Ges jwä √ wbY©q Kwi|
7. I ejØq‡K c~‡e©v³ †¯‥‡j I eivei h_vµ‡g
Ges ‡iLvsk Øviv mywPZ Kwi| AvqZ‡ÿÎwU c~Y© Kwi Ges
KY© AvuwK| Zvn‡j KY© Øviv jwäi gvb I w`K mywPZ n‡e|
wnmvet eivei ej¸‡jvi j¤^vs‡ki †hvMdj
=
(awi)
eivevi ej¸‡jvi j¤^vs‡ki †hvMdj
= (awi)
jwäi gvb √( ) ( ) Ges
w`K
dj msKjbt
jwäi gvb jwäi w`K
m~Î †_‡K MÖvd †_‡K m~Î †_‡K MÖvd †_‡K
Ni = Puv`vi mvnv‡h¨
djvdjt wb‡Y©q jwäi gvb Ges w`K (cÖvq)
gšÍe¨t †jLwPÎ n‡Z cÖvß gvb I MvwYwZKfv‡e wbY©xZ gvb cÖvq mgvb| AZGe djvdj mwVK|

13. D”PZi MwYZ e¨envwiK 2q cÎ 13
cixÿv bs ZvwiLt
cixÿ‡bi bvgt wg/‡m. †e‡M Lvov Dc‡ii w`‡K wbwÿß †Kvb e¯‘ mg‡q D”PZvq Ae¯’vb K‡i| I Gi cÖwZm½x
gvb wb¤œiƒct
- †jLwPÎ AsKb K‡i Zv n‡Z e„nËg D”PZv I e„nËg D”PZvq †cu․Qvi mgq wbY©q Ki‡Z n‡e|
g~jZË¡t ‡Kvb e¯‘KYv‡K ‡e‡M Lvov Dc‡ii w`‡K wb‡ÿc Ki‡j hw` e¯‘KYvwU mg‡q D”PZvq D‡V Z‡e
‡hLv‡b ga¨vKl©YRwbZ Z¡iY| m‡e©v”P D”PZv Ges m‡e©v”P D”PZvq †c․uQvi mgq |
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 AvuwK|
2. eivei Ges eivevi cwigvc Kwi|
3. Aÿ eivei ÿz`ªZg e‡M©i evû = GKK Ges Aÿ eivei ÿz`ªZg e‡M©i evû = GKK a‡i
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) we›`y¸‡jv QK
KvM‡R ¯’vcb Kwi| g‡b Kwi, we›`y¸wj h_vµ‡g
4. DcwiD³ we›`y¸wj miæ †cwÝj w`‡q ms‡hvRb K‡i - †jLwPÎ AsKb Kwi|
5. †jLwPÎ †_‡K m‡e©vPP we›`y wbY©q K‡i we›`ywU‡K Øviv m~wPZ Kwi| we›`y †_‡K A‡ÿi Dci j¤^ AvuwK|
Ges Øviv h_vµ‡g m‡e©v”P D”PZv Ges H D”PZvq †c․Qvi mgq wbY©xZ n‡e|
dj msKjbt
‡jLwP‡Î †`Lv hvq, we›`yi fzR eM© = †m‡KÛ Ges we›`yi †KvwU eM©
wgUvi| myZivs e„nËg D”PZv = wgUvi Ges e„nËg D”PZvq †c․uQvi mgq = †m‡KÛ|
m~Î e¨envi K‡i cvB;
e„nËg D”PZv wgUvi, e„nËg D”PZvq †cu․Qvi mgq
djvdjt e„nËg D”PZv wgUvi Ges e„nËg D”PZvq †cu․Qvi mgq= †m‡KÛ
gšÍe¨t †jLwPÎ n‡Z cÖvß gvb I MvwYwZKfv‡e wbY©xZ gvb cÖvq mgvb| AZGe djvdj mwVK|
XX/
Y/
Y
O
P1
P2
P3
P4
P5
P6
P7
P0P0
P
N

14. D”PZi MwYZ e¨envwiK 2q cÎ 14
cixÿY bs ZvwiLt
cixÿ‡bi bvgt mgZ¡i‡Y PjšÍ †Kvb KYv †m‡K‡Û `yiZ¡ AwZµg K‡i| Ges Gi Avbylw½K gvb wb‡Pi
mviwY‡Z †`Iqv n‡jvt
- †jLwPÎ AsKb K‡i e¯‘ KYvi Avw`‡eM I Z¡iY wbY©q Ki|
g~jZË¡t Avw`‡eM Ges mylg Z¡iY n‡j mg‡q e¯‘KYv KZ…©K AwZµvšÍ `~iZ¡
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 AvuwK|
2. eivei Ges eivei cwigvc Kwi|
3. Gi Rb¨ QK KvM‡Ri ÿz`ªZg e‡M©i evû = GKK Ges Gi Rb¨ ÿz`ªZg e‡M©i evû = GKK
†¯‥j a‡i ( ) ( ) ( ) ( ) ( ) ( ) we›`y¸‡jv QK KvM‡R ¯’vcb Kwi Ges
miæ †cwÝj w`‡q ¯’vwcZ we›`y¸wj gy³ n‡¯Í eµvKv‡i ms‡hvM K‡i - †jLwPÎ AsKb Kwi|
dj msKjbt - †jLwP‡Îi Dci Ges we›`y wbB| Gi Dci I j¤^ AvuwK|
GLv‡b, evû = wgUvi
evû = ‡m.
( )
( )
Avevi, evû wgUvi,
evû ‡m.
( )
( )
( ) ( )
( ) ( ) ( )
( )
djvdjt wb‡Y©q Avw`‡eM wgUvi/‡m. Ges Z¡iY wgUvi/‡m‡KÛ2
.
0, 0
(1, 15)
(3, 48)
(6, 130)
(8, 192)
(10, 300)
XX/
Y/
Y
O
Amgvß QK