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.

X X/

Y/

Y

Z A S N 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.

X X/

Y/

Y

Z A S N

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|

X X/

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|

X X/

Y/

Y

O(0,00)

y=cosx

y=cos-1x (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|

X X/

Y/

Y

O

P1

P2

P3

P4 P5

P6

P7

P0 P0

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)

X X/

Y/

Y

O

Amgvß QK

Hsc math practical 2nd paper2015 wg0

Education

Transcript of Hsc math practical 2nd paper2015 wg0