Hsc math practical 2nd paper2015 wg0

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)

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|
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 ‡iLvÎ‡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)

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)

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|
Kvh©c×wZt
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)

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 eëwai g‡a¨ ( ) Gi Kgc‡ÿ GKwU A_ev
we‡Rvo msL¨K g~j _vK‡e|
( ) hw` ( ) Ges ( ) GKB wPýhy³ nq Z‡e eëwai 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|
Kvh©c×wZt
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 eëwa‡Z we`¨gvb|
eëwa‡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)

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|
Kvh©c×wZt
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)

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
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

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

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
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 †cwÝ‡ji 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 eënvwi‡Ki †¯‹j I Q‡Ki
g‡a¨ AmvgÄm¨ _vK‡Z cv‡i| wkÿv_x©
Zv‡ì wbR wbR wkÿ‡Ki wbKU Zv ms‡kvab
K‡i wb‡q eënvwiK †bvUey‡Ki Rb¨ cÖ¯‘Z
Ki‡e|

cixÿv bs ZvwiLt
cixÿ‡bi bvgt Gi Rb¨ GKB †jLwP‡Î wÎ‡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|
K¨vjKz‡jUi|
Kvh©c×wZt
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 †cwÝ‡ji 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
)

cixÿv bs ZvwiLt
cixÿ‡bi bvgt ‣jwLK c×wZ‡Z †Kvb we›`y‡Z †Kv‡Y wµqviKZ I eiØ‡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 ejØ‡qi jwä hw`
e‡ji w`‡Ki mv‡_ †KvY Drcbœ K‡i Z‡e √ Ges
hLb
Kvh©c×wZt
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

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
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|

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 |
Kvh©c×wZt
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

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¡
K¨vjKz‡jUi|
Kvh©c×wZt
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

Hsc math practical 2nd paper2015 wg0

More Related Content

What's hot

Viewers also liked

Similar to Hsc math practical 2nd paper2015 wg0

Recently uploaded

Hsc math practical 2nd paper2015 wg0