dvskb
dvskb wK Avm‡jB Lye KwVb wRwbm ?
mswewae× mZK©xKiY t
ÔdvskbÕ msµvšÍ Avgv‡`i Av‡jvPbv¸‡jv G‡Kev‡iB ˆeVKx Xs Gi| Avu‡Zj m¤cÖ`v‡qi () Rb¨ GB Sheet wU iwPZ bq; GwU
GKvšÍB, Avgiv hviv mvaviY QvÎ/QvÎx (hviv fvZ‡K fvZB ewj, KwVb K‡i ÔAbœÕ ewj bv) Zv‡`i Rb¨ iwPZ|
GKUz fq †`LvB, †Kgb?
ÔdvskbÕ Kx? †KZvex fvlvq Gi DËi njt Òhw` †Kvb Aš^‡qi gva¨‡g cÖ_g †m‡Ui cÖ‡Z¨K Dcv`v‡bi Rb¨ wØZxq †mU †_‡K Abb¨
Dcv`vb cvIqv hvq Ges hw` cÖ_g †m‡Ui cÖ‡Z¨KwU Dcv`vb cÖwZ”Qwe wewkó nq, Z‡e H Aš^q‡K ejv nq cÖ_g †mU †_‡K wØZxq †m‡U
GKwU dvskbÓ|
fq jvM‡jv? Gfv‡e DËi Ki‡j Avu‡Zj m¤cÖ`vq nqZ Lywk nb, wKš‘ Avgiv hviv mvaviY QvÎ/QvÎx Zviv Lywk n‡Z cvwi bv| KviY
Dc‡iv³ msÁvi gv_vgyÛy wKQyB cÖvq †evSv †Mj bv| wKš‘ fq cvIqvi wKQy †bB| KviY Avwg GiKg †KZvex c×wZ‡Z dvskb †evSv‡Z
hvw”Q bv| Avgvi K_vevZ©v¸‡jv n‡e G‡Kev‡iB mvaviY| †Zv n‡q hvK ... ...
Ôm¤úK©ÕGi mv‡_ m¤úK© Av‡Q †Zv?
Ôm¤úK©Õ ev ÔAš^qÕ m¤ú‡K© †Zvgv‡`i aviYv bv _vKvi K_v bq| ZeyI Avevi GKUz g‡b Kwi‡q w`w”Q| ÔAš^qÕ ev Ôm¤úK©Õ-Gi msÁv
†`qvi wKQyB ‡bB| ÔAš^qÕ gv‡b Ôm¤úK©Õ Avi Ôm¤úK©Õ gv‡b ÔAš^qÕ| eySv †Mj bv? Av”Qv Z‡e bx‡Pi mgxKiYwU †Lqvj Ki t
y = 2x + 18
aiv hvK, GB mgxKiYwU w`‡q gvwmK †gvevBj wej wn‡me Kiv n‡e| GLv‡b x Øviv ÔK_vi cwigvYÕ A_v©r GK gv‡m †gvU hZ wgwbU
K_v ejv n‡q‡Q Zv wb‡`©k Kiv n‡”Q Ges y Øviv eySv‡bv n‡”Q gvwmK Ô†gvevBj wejÕ|
hw` cÖwZ wgwb‡Ui PvR© nq 2 UvKv K‡i Ges Ò¸Y¸YÓ mvwf©‡mi Rb¨ gvwmK 18 UvKv †K‡U †bqv nq, Z‡e GB mgxKi‡Yi mvnv‡h¨
†gvevBj wej wnmve Kiv hv‡e| A_©vr, GLv‡b mgxKiYwU Øviv ÔK_vi cwigvY ( x ) Õ Ges Ô‡gvevBj wej ( y) Õ Gi g‡a¨ m¤úK© ev Aš^q
¯’vcb Kiv n‡q‡Q|
G‡ÿ‡Î AviI GKUz wRwbm g‡b †i‡Lv| Zv nj, Ô†gvevBj wejÕ ‡K Z_v y Gi gvb ¸‡jv‡K ejv nq, ÔK_vi cwigvYÕ Z_v x Gi
gv‡bi cÖwZ”Qwe | †hgb, x = 1 n‡j, y = 20 | A_©vr, GLv‡b 1 Gi cÖwZ”Qwe nj 20 | GKB fv‡e x = 10 n‡j y = 38 |
A_©vr 10 Gi cÖwZ”Qwe nj 38 BZ¨vw`| Avi GLv‡b x †K ejv nq ¯^vaxb PjK Ges y †K ejv nq Aaxb PjK (A_©vr, hvi gvb Avgiv
wb‡R †_‡K emve †mwU ¯^vaxb PjK Ges Ab¨wU n‡e Aaxb PjK)|
GLb, m¤úK© †Zv KZ iKg-B n‡Z cv‡i| wewfbœ ai‡bi Ôm¤úK©Õ ev ÔAš^qÕ ¸‡jvi g‡a¨ we‡kl ai‡bi wKQy Ôm¤úK©Õ †K dvskb e‡j|
wKš‘ cÖkœ nj †mB we‡kl ai‡bi m¤úK©¸‡jv Kx?
m¤úK©Uv nj t ¯^vaxb Pj‡Ki cÖwZwU ev¯Íe gv‡bi Rb¨ hw` Aaxb Pj‡Ki †KejgvÎ GKwU K‡i ev¯‘e gvb cvIqv hvq, Z‡e H Aaxb
PjK‡K ¯^vaxb Pj‡Ki dvskb e‡j| †hgb, GLv‡b Ô†gvevBj wej ( y) Õ nj ÔK_vi cwigvY ( x ) Õ Gi dvskb| KviY GLv‡b
y = 2x + 18 mgxKiY †_‡K x Gi cÖwZwU ev¯Íe gv‡bi Rb¨ y Gi †Kej GKwU K‡i ev¯Íe gvb cvIqv hv‡”Q| †hgb,
x = 0 n‡j, y = 18
x = 1 n‡j, y = 20
.................... (i )
x = 5 n‡j, y = 28
BZ¨vw` BZ¨vw`|
ZvB ejv hvq, y n‡”Q x Gi dvskb Ges wjLv nq y = f ( x ) ; A_©vr y = 2 x + 18 bv wj‡L wjLv †h‡Z cv‡i, f ( x ) = 2x + 18 |
y n‡”Q x Gi dvskb- GB K_vi G‡Kev‡i mvaviY A_© nj y, x Gi Dci wbf©ikxj|
wKš‘ hw` Avgiv wjwL, y 2 = 2x + 18 (GUv g‡b Ki Ab¨ ‡h‡Kvb GKwU m¤úK©)| G‡ÿ‡Î,
x = 9 n‡j, y =  6
.................... (ii )
x = 23 n‡j, y =  8
x = 41 n‡j, y =  10
BZ¨vw` BZ¨vw`|
cwieZ©‡bi cªZ¨‡q wbišÍi c_Pjv...
1
D™¢vm GKv‡WwgK GÛ GWwgkb †Kqvi
dvskb
A_©vr, x Gi cÖwZwU ev¯Íe gv‡bi Rb¨ y Gi `ywU K‡i (GKwUi †ekx) ev¯Íe gvb cvIqv hv‡”Q| ZvB G‡ÿ‡Î y , x Gi dvskb bq|
GwU †Mj dvsk‡bi GKwU msÁv| G ch©v‡q dvskb Kx- Zv bv eyS‡jI mgm¨v ‡bB, Avwg Ab¨ AviI `yÕfv‡e dvsk‡bi msÁv w`e Ges
wZbwU msÁv †h Avm‡j GKB wRwbm Zv †`wL‡q w`e |
Gevi Avwg dvskb eySv‡bvi Rb¨ wP‡Îi mvnvh¨ wbe| wb‡Pi wPÎ `yÕwU †Lqvj Kit
a
b
f
g
c
h
d
g
h
f
a
b
wPÎ- 1
wPÎ- 2
wPÎ `yÕwU‡Z wKQy m¤úK© †`Lv‡bv Av‡Q| †hgb wPÎ-1 G a Gi mv‡_ f ; b Gi mv‡_ h Ges c Gi mv‡_ g m¤úK©hy³ | GLv‡b wPÎ1 Gi m¤úK©‡K dvskb ejv hv‡e; wKš‘ wPÎ- 2 Gi m¤úK©‡K dvskb ejv hv‡e bv|
KviY Kx †mUvB †Zv eje| Zvi Av‡M AviI K‡qKUv wPÎ †Lqvj Kiv hvK|
mšÍvb
gv
a
b
c
d
e
mšÍvb
a
b
c
wPÎ-3
mšÍvb
d
e
f
g
mšÍvb
gv
a
b
m
g
e
c
t
wPÎ-4
gv
m
a
b
c
d
gv
n
p
wPÎ- 6
mšÍvb
wPÎ-5
gv
a
b
c
d
n
wPÎ- 7
mšÍvb
gv
p
c
d
e
f
wPÎ- 8
Dc‡ii wPθ‡jv‡Z wewfbœ ai‡bi m¤úK© †`Lv‡bv Av‡Q| G‡`i †KvbUv‡K dvskb ejv hv‡e Avi †KvbUv‡K ejv hv‡e bv- †`wL Zv mn‡R
†ei Kiv hvq wKbv? mn‡RB †ei Kivi Rb¨ cÖwZwU wP‡Îi evg cv‡ki †mU‡K (evg cv‡ki †Mvjøv ev Oval Gi m`m¨‡`i‡K) Avgiv aie
ÔmšÍvbÕ- †`i †mU Ges Wvb cv‡ki †mU‡K aie ÔgvÕ †`i †mU| Zxi wPý¸wj Ò†Kvb mšÍvbÓ, Ò†Kvb gvÓ Gi mv‡_ m¤úK© hy³ Zv cÖKvk
Ki‡Q Ges Zxi wP‡ýi w`K n‡e mšÍvb †_‡K gv‡qi w`‡K|
G‡Kev‡iB mvaviY wKQy cÖkœt
Av”Qv, †Zvgv‡K hw` cÖkœ Kiv nq- (i ) GKB gv‡qi GKvwaK mšÍvb _vK‡Z cv‡i wKbv? DËi LyeB mnRt n¨v, Aek¨B _vK‡Z cv‡i| wKš‘
hw` cÖkœ Kiv nq, (ii ) GKB mšÍv‡bi GKvwaK gv _vK‡Z cv‡i wKbv? (GLv‡b Aek¨ Avwg mr gv ev cvZv‡bv gvÕi K_v ejwQ bv)| DËi
n‡e- bv, _vK‡Z cv‡i bv| hw` ewj (iii ) gv Qvov †Kvb mšÍvb Rb¥ wb‡Z cv‡i wKbv? DËi mnRt bv, n‡Z cv‡i bv| wKš‘ hw` ewj(iv ) mšÍvbwenxb †Kvb gv n‡Z cv‡i wK bv? DËi n‡e- nu¨v, Aek¨B n‡Z cv‡i| GB PviwU cÖ‡kœi DËi hw` Zzwg w`‡Z cvi, Zvn‡j
dvskb †evSv †Zvgvi Rb¨ †gv‡UB KwVb n‡e bv| Pj †`Lv hvK ...
evgcv‡ki c„ôvi wPθwj †`‡L wb‡R‡K cÖkœ PviwU Ki| ev¯Í‡e hv NU‡Z cv‡i, wP‡Î hw` Zv N‡U _v‡K ( †hgb- GKB gv‡qi GKvwaK
mšÍvb _vK‡Z cv‡i) Z‡e Zv dvskb| Avi wP‡Î hw` Am¤¢e wKQy †`Lv‡bv n‡q _v‡K (†hgb, GKB mšÍv‡bi GKvwaK gv Av‡Q wKsev gv
Qvov mšÍvb-GiKg D™¢U wKQy) Z‡e Zv dvskb n‡e bv|
wPÎ- 1 G cÖwZwU mšÍv‡bi Avjv`v Avjv`v gv Av‡Q| Zv †Zv n‡ZB cv‡i| Kv‡RB GwU dvskb|
wPÎ-2 G †`Lv hv‡”Q GKB mšÍv‡bi `ywU gv, (†hgb a Gi gv n‡”Q d I g ) hv m¤¢e bq| ZvB GwU dvskb bq|
wPÎ-3 G †`Lv hv‡”Q me mšÍv‡bi GKwU K‡i gv| Zvn‡j †Zv GwU dvskb, bv wK? wKš‘ Avm‡j GwU dvskb bq| KviY, GKwU mšÍvb ( b )
†K †`Lv hv‡”Q gv QvovB Rb¥ wb‡q‡Q| †mUv †Zv Avi m¤¢e bq| ZvB GwU dvskb bq|
cwieZ©‡bi cªZ¨‡q wbišÍi c_Pjv...
2
D™¢vm GKv‡WwgK GÛ GWwgkb †Kqvi
dvskb
dvsk‡bi kZ© wK Zvn‡j Clear n‡q‡Q?
kZ©-1 t cÖwZwU mšÍv‡bi gv _vK‡Z n‡e, wKš‘ cÖwZwU gv‡qi mšÍvb _vK‡Z n‡e Ggb †Kvb K_v †bB|
kZ©-2 t GKB gv‡qi GKvwaK mšÍvb _vK‡Z cv‡i, wKš‘ GKB mšÍv‡bi GKvwaK gv _vKv m¤¢e bq|
Gevi Zvn‡j e‡jv †Zv wPÎ-4 dvskb wK bv? Confused ? Confused nIqvi wKQy †bB| GwU Aek¨B dvskb| KviY cÖwZwU
mšÍv‡biB GKwU K‡i gv Av‡Q| So no mgm¨v| wKš‘ mgm¨v nj GKwU gv (g ) Gi †Kvb mšÍvb †bB| Av‡i, GwU †Zv n‡ZB cv‡i|
KviY gv Qvov mšÍvb n‡Z cv‡i bv, wKš‘ mšÍvbnxb gv _vK‡ZB cv‡ib| ZvB GwU dvskb|
wPÎ-5 †Lqvj Ki| c Ges t `yÕR‡biB GKwU K‡i gv| wKš‘ e Gi gv `ywU (a , b) | Ab¨ mevB dvsk‡bi kZ© cvjb Ki‡jI †h †Kvb
GKwU m`m¨ hw` kZ© f½ K‡i (†hgb, G‡ÿ‡Î a kZ© f½ K‡i‡Q) Z‡e Zv dvskb n‡e bv| Gi mv‡_ gvbeRxe‡bi GKwU ¸iæZ¡c~Y©
Philosophy mv`„k¨c~Y©| †mwU n‡jv Òag© wPi K‡ó iwÿZ nq, wKš‘ GKw`‡bi AmveavbZvq webó nqÓ|
- ew¼g P›`ª P‡Ævcva¨vq (wele„ÿ)
ZË¡Uv nq‡Zv A‡b‡KB eyS‡Z cviQ bv| eyS‡Z †Póv Ki| cvi‡e|
Av”Qv hv ‡nvK, wPÎ-6 Gi Kx Ae¯’v? GwU wKš‘ dvskb| KviY me mšÍv‡bi gv Av‡Q Ges †Kv‡bv mšÍv‡bi GKvwaK gv †bB, hv k‡Z©i m‡½
mvgÄm¨c~Y©| Z‡e wPÎ-6 G GKRb gv ( m ) Gi `ywU mšÍvb Av‡Q| Zv‡Z †Zv †Kvb mgm¨v †bB| bv wK?
Gevi †Zvgiv ej wPÎ-7 Ges wPÎ-8 Gi g‡a¨ †Kvb&wU dvskb Ges †KvbwU dvskb bq| cviv hv‡”Q? Avkv Kwi, cviv hv‡”Q| bv cvi‡j
Avgv‡K †gvevB‡j GKUv call w`I| [ ( 01819)10–(1710650 )8 ]
†Wv‡gb, †Kv-†Wv‡gb Ges †iÄ t
Gch©v‡q Avgiv †Wv‡gb, †Kv-†Wv‡gb Ges †iÄ m¤ú‡K© aviYv jv‡fi †Póv Kie| Dc‡ii D`vniY¸wj‡Z evg cv‡ki †Mvjøvi
(Oval ) †fZ‡i wKQy gvb Ges Wvb cv‡ki †Mvjøvi †fZ‡i wKQy gvb †`qv n‡q‡Q Ges Zxi wPý w`‡q †Kvb gv‡bi mv‡_ †Kvb gvb m¤úK©hy³
Zv †`Lv‡bv n‡q‡Q| evg cv‡ki Oval Gi †h gvb¸‡jv i‡q‡Q †m¸‡jv nj ¯^vaxb Pj‡Ki †mU hv‡`i‡K Avgiv eje †Wv‡gb| (KwVb K‡i bv
ej‡j †Zv Avevi Avgv‡`i †c‡Ui fvZ nRg nq bv)| Avi Wv‡bi gvb ¸‡jv nj Aaxb Pj‡Ki †mU hv‡`i‡K Avgiv eje †Kv-†Wv‡gb |
Dc‡ii D`vniY¸wj‡Z ÔmšÍvbÕ †`i‡K a‡iwQ Avgiv Ô¯^vaxb PjKÕ (†Wv‡gb) Avi ÔgvÕ †`i‡K a‡iwQ ÔAaxb PjKÕ (†Kv-†Wv‡gb)|
Zvn‡j †iÄ Kv‡`i‡K ejv n‡e? DËi njt †Kv-†Wv‡gb †m‡Ui †h mKj Dcv`vb †Wv‡gb †m‡Ui †Kvb bv †Kvb Dcv`v‡bi mv‡_ m¤úK©hy³
Zv‡`i‡K †iÄ e‡j| KwVb n‡q †Mj, ZvB bv? wVK Av‡Q, Pj mnRfv‡e †`Lv hvK| Dc‡ii D`vniY¸wj‡Z mKj gv‡qivB (mšÍvbwewkó
Ges mšÍvbwenxb ) †Kv-†Wv‡gb, wKš‘ †KejgvÎ mšÍvbwewkó gv‡qivB †iÄ| A_©vr,
Ò mKj †iÄ-B †Kv-†Wv‡gb, wKš‘ mKj †Kv-†Wv‡gb †iÄ bv-I n‡Z cv‡i|Ó
Pj, `yÕGKUv D`vniY †`Lv hvKmšÍvb
gv
m
n
a
b
c
d
o
p
wPÎ- 9
†Wv‡gb ={a, b, c, d}
†Kv-†Wv‡gb ={m, n, o, p}
†iÄ ={m, n, o}
cwieZ©‡bi cªZ¨‡q wbišÍi c_Pjv...
mšÍvb
gv
a
b
c
d
j
k
mšÍvb
gv
p
c
d
e
f
l
wPÎ- 10
†Wv‡gb ={a, b, c, d}
†Kv-†Wv‡gb ={j, k, l}
†iÄ ={k}
3
wPÎ- 11
G‡ÿ‡Î?
Av‡i, GwU‡Zv dvskb-B bq|
D™¢vm GKv‡WwgK GÛ GWwgkb †Kqvi
dvskb
Zvn‡j Avgiv `yBfv‡e (mgxKiY n‡Z Ges wPÎ n‡Z) dvskb †K msÁvwqZ Kijvg| Giv wK Avjv`v? Aek¨B bq|
KviY, (i ) bs I (ii ) bs mgxKiY‡K ( cÖ_g I wØZxq c„ôvq D‡jøwLZ) wP‡Î cÖKvk Ki‡j cvB†Wv‡gb
†Kv-†Wv‡gb
0
1
5
18
20
28
..
.
wPÎ-12
†Wv‡gb
†Kv-†Wv‡gb
+6
−6
+8
−8
9
..
.
...
23
(i ) bs mgxKiY Gi wPÎ
wPÎ-13
..
(ii ) bs mgxKiY Gi wPÎ
(i ) bs mgxKiY dvskb wQj wKš‘ (ii ) bs mgxKiY dvskb wQj bv| wPÎ †_‡KI Zv-B †`Lv hv‡”Q|
Gevi Avgiv Z…Zxq Av‡iK fv‡e dvsk‡bi msÁv w`e| †mwU nj µg‡Rvo Øviv| wb‡Pi µg‡Rv‡oi †mU(Aš^q) `ywU †Lqvj KiA = (1,2), (9,5), (2,7), (− 1,0)
B = (1,2), (9,5), (1,7), (− 1,0)
Avgiv Rvwb, cÖwZ †Rvovi cÖ_g Dcv`vb nj †Wv‡gb (¯^vaxb PjK) Ges wØZxq Dcv`b nj †iÄ (Aaxb PjK)| †hgb, G‡ÿ‡ÎA Gi Rb¨ †Wv‡g‡bi †mU nj 1,9,2,−1 Ges †i‡Äi †mU nj 2,5,7,0|
B Gi Rb¨ †Wv‡g‡bi †mU nj 1,9,−1 Ges †i‡Äi †mU nj 2,5,7,0| wPÎ Gu‡K †`L †Zv †Kgb Av‡m?
†Wv‡gb
†iÄ
1
9
2
−1
2
5
7
0
wPÎ-14 (Aš^q A Gi wPÎ )
†Wv‡gb
†iÄ
1
9
2
5
7
0
−1
[ B Gi †ÿ‡Î †Wv‡g‡bi †m‡U 1 `yBevi wjLvi
`iKvi †bB| KviY †mU wjLvi mgq Avgiv
repetition avoid Kwi]
wPÎ-15 (Aš^q B Gi wPÎ )
wPÎ Abymv‡i, A Aš^qwU dvskb wb‡`©k K‡i wKš‘ B dvskb bq| KviY, B Gi †ÿ‡Î †Wv‡gb Gi GKwU m`m¨ {1} Gi Rb¨ †iÄ
Gi `yBwU m`m¨ {2,7} Av‡Q| (A_©vr, GKB mšÍv‡bi `yBwU gv| Zv wK m¤¢e?)| wK? †evSv †Mj †Zv?
Zvn‡j dvsk‡bi bZzb GKwU msÁv cvIqv †Mj t Ò†Kvb Aš^‡qi cÖwZwU µg‡Rv‡oi †Wv‡g‡bi cÖwZwU Dcv`vb hw` wfbœ wfbœ nq, Z‡e
†mB Aš^q‡K dvskb e‡j|Ó Z‡e †iÄ GKB n‡jI mgm¨v †bB, wfbœ wfbœ n‡jI mgm¨v †bB | A_©vr, kZ© nj †Wv‡g‡bi †Kvb Dcv`vb
repeat n‡j dvskb n‡e bv|
C = (1,5), (2,5), (3,5), (4,5)
Last Av‡iKUv example †`LGUv wKš‘ dvskb| KviY †Wv‡gb mevB wfbœ wfbœ| Avi wPÎ n‡Z †`LmšÍvb
gv
1
2
3
4
5
cwieZ©‡bi cªZ¨‡q wbišÍi c_Pjv...
[G‡ÿ‡Î GKRb gv‡qi PviwU mšÍvb| †m †Zv n‡ZB cv‡i| ZvB GwU GKwU dvskb]
4
D™¢vm GKv‡WwgK GÛ GWwgkb †Kqvi
dvskb
GK-GK dvskb Ges mvwe©K ev me©MÖvnx dvskb t GLb hw` †Zvgiv wR‡Ám Ki GK-GK dvskb, me©MÖvnx ev mvwe©K dvskb Gme Kx? Zvn‡j GK K_vq DËi w`‡q †`qv hv‡e|
GK-GK dvskb t hw` †Kvb gv‡qi GKwUi †ekx mšÍvb bv _v‡K (Z‡e gv wbtmšÍvb n‡Z cv‡ib) Z‡e Zv n‡e GK GK dvskb|
mšÍvb
gv
mšÍvb
gv
mšÍvb
gv
a
h
a
a
d
d
e
g
b
b
b
f
c
e
c
c
k
g
wPÎ-16
wPÎ-17
wPÎ-18
†hgb- wPÎ-16 I wPÎ-17 GK-GK dvskb wb‡`©k Ki‡Q| Zvn‡j wPÎ-18 †`‡L Kx g‡b n‡”Q? GK-GK? Av‡i; GwU †Zv dvskbB bq!!!
me©MÖvnx ev mvwe©K dvskb t hw` †Kvb gv-B wbtmšÍvb bv _v‡Kb A_©vr, me gv‡qi-B hw` GK ev GKvwaK mšÍvb _v‡K Z‡e Zv n‡e mvwe©K
ev me©MÖvnx dvskb | ‡hgbmšÍvb
gv
mšÍvb
gv
mšÍvb
gv
a
b
c
d
a
b
c
h
y
wPÎ-19
d
e
f
g
e
a
b
t
g
wPÎ-20
GwU mvwe©K dvskb
(KviY wbtmšÍvb †Kvb gv †bB)
GLb bx‡Pi wPÎ ¸‡jv jÿ¨ KimšÍvb
a
b
d
f
wPÎ-21
GwU mvwe©K dvskb bq
(KviY GKRb wbtmšÍvb gv Av‡Qb)
gv
k
m
s
t
GwU wK mvwe©K dvskb?
Av‡i, GwUI †Zv dvskb-B bq|
mšÍvb
gv
a
m
n
f
g
wPÎ-22
wPÎ-23
GK-GK wKš‘ mvwe©K bq
(e¨vL¨v †`qvi wK `iKvi Av‡Q?)
GK-GK Ges mvwe©K dvskb (KviY,
†Kvb gv‡qi-B GKwUi †ekx mšÍvb
†bB Ges wbtmšÍvb †Kvb gv †bB)
mšÍvb
gv
mšÍvb
gv
m
n
o
p
c
e
f
g
a
b
d
wPÎ-24
d
mvwe©K wKš‘ GK-GK bq
wPÎ-25
GwU mvwe©KI bq, GK-GKI bq
ev¯Íe D`vniY t
Pj Avgiv dvsk‡bi `yÕGKwU ev¯Íe D`vniY †Lqvj Kwi| f~wg †_‡K hw` †Kvb e¯‘‡K Lvov Dc‡ii w`‡K wb‡ÿc Kiv nq Z‡e D”PZv (h)
1
I mgq (t) Gi g‡a¨ m¤úK© nj t h = ut − gt 2 ; G‡ÿ‡Î t Gi cÖwZwU ev¯Íe gv‡bi Rb¨ h Gi †Kej GKwU K‡i ev¯Íe gvb
2
cvIqv hvq| ZvB h nj t Gi dvskb| wKš‘ wecixZfv‡e t wKš‘ h Gi dvskb bq| †Kb ej‡Zv ?
GiKg dvsk‡bi Av‡iv 10wU ev¯Íe D`vniY wPš‘v K‡i ‡ei Ki‡e| KviY wkÿvi D‡Ïk¨B nj wPšÍv Ki‡Z wkLv‡bv| hZÿY Zzwg wPšÍv
Ki‡Z wkL‡e bv, ey‡S wb‡e †h †Zvgvi wkÿv Am¤ú~Y© Ges e¨_©|
cwieZ©‡bi cªZ¨‡q wbišÍi c_Pjv...
5
D™¢vm GKv‡WwgK GÛ GWwgkb †Kqvi
dvskb
fq †nvK `~i t
G ch©v‡q AviI wKQy wRwbm, †h¸‡jv †`L‡j †Zvgiv fq cvI, †m¸‡jv mnRfv‡e cÖKv‡ki †Póv Kie| †hgb- †Zvgv‡`i MwYZ eB‡Z
cv‡e, F t A → B ; GLv‡b F Øviv eySv‡”Q dvsk‡bi bvg| (dvsk‡bi bvg F Qvov Ab¨ wKQyI n‡Z cv‡i)| Avi A w`‡q eySv‡”Q F
dvskbwUi †Wv‡g‡bi †mU Ges B w`‡q eySv‡”Q †Kv-†Wv‡g‡bi †mU| Zvn‡j ej‡Zv, F t R → R gv‡b wK? gv‡b n‡jv- GB dvsk‡bi
†Wv‡gb nj ev¯Íe msL¨v Ges †Kv-†Wv‡gbI nj ev¯Íe msL¨v| Avevi F t R → R + w`‡q eySvq GB dvsk‡bi †Wv‡gb nj †h †Kvb
ev¯Íe msL¨v Ges †Kv-†Wv‡gb nj †Kej AFYvZ¥K ev¯Íe msL¨v| D`vniY ¯^iƒc wb‡Pi AsKwU †Lqvj Ki t
1
# F t A → B Ges F( x ) =
; cÖ`Ë F dvskbwUi †Wv‡gb wbY©q Ki|
x−2
GB AsKwUi DËi nj t F Gi †Wv‡gb A = R − {2} ; A_©vr 2 Qvov mKj ev¯Íe msL¨vB nj ewY©Z F dvsk‡bi †Wv‡gb|
1
1
Gi KviY nj, F( x ) =
†_‡K †Lqvj Ki, x = 2 n‡j F( x ) Gi gvb Awb‡Y©q n‡q hvq, KviY nj Awb‡Y©q| A_©vr, ¯^vaxb
x−2
0
Pj‡Ki gvb 2 n‡j Aaxb Pj‡Ki A_©vr F( x ) Gi ev¯Íe gvb cvIqv hvq bv| 2 Qvov x Gi Ab¨ †h †Kvb ev¯Íe gv‡bi Rb¨B F( x )
Gi †Kvb bv †Kvb ev¯Íe gvb cvIqv hvq| ZvB cÖ`Ë F dvskbwUi †Wv‡gb nj t A = R − {2} |
A_©vr, †Kvb ev¯Íe msL¨v hw` F( x ) Gi gvb wbY©‡q Awb‡Y©q ev KvíwbK Ae¯’v ˆZwi K‡i, (†hgb Av‡Mi D`vni‡Y 2 nj †mB culprit)
Z‡e †mB culprit ev‡` R Gi mKj gvb-B nj F Gi †Wv‡gb|
†hgb, F( x ) = 2x + 18 ; GB dvsk‡bi †Wv‡gb nj R A_©vr mKj ev¯Íe msL¨v| KviY R -Gi †Kvb gvbB GLv‡b culprit bq|
wKš‘ F( x ) = 5 − x n‡j, F - Gi †Wv‡gb n‡e Ò R − {5 Gi †P‡q eo x Gi mKj gvb}Ó ev Ab¨fv‡e ejv hvq, F Gi †Wv‡gb
n‡e 5 ev 5 Gi †P‡q †QvU mKj ev¯Íe gvb| KviY, x Gi gvb 5 Gi eo n‡j †hgb, 5.1 ev 6 , ev 10 BZ¨vw` †h †KvbwU n‡jB
F( x ) Gi gvb KvíwbK n‡q hvq| ZvB 5 Gi †P‡q eo ‡h‡Kvb gvbB GLv‡b culprit.
ZvB Avkv Kwi, GLb †_‡K culprit LuyR‡Z mgm¨v n‡e bv| Avi mgm¨v n‡j Av‡iv †ekx wPšÍv Ki, AviI †ekx †Póv Ki| culprit
LyuR‡Z †kLvUv Lye important. †Kb Rvb? AsK Kivi Rb¨ Ges GKB mv‡_ GB †`kUv‡K my›`i Kivi Rb¨| KviY GB †`‡ki culprit
¸‡jv‡K wVKg‡Zv wPb‡Z cvi‡j we‡k¦i `iev‡i Avgv‡`i mevi gv_v wbPz †_‡K AviI wbPz K‡i w`‡q †`kUv nq‡Zv cvuPevi `ybx©wZ‡Z
P¨vw¤úqb n‡Zv bv| ZvB Pj Avgiv As‡Ki, mgv‡Ri Ges wb‡R‡`i g‡bi ga¨Kvi culprit ¸‡jv‡K wPwb, †m¸‡jv‡K aŸsm K‡i †`kUv‡K
my›`i K‡i M‡o Zywj|
my›`i evsjv‡`‡ki cÖZ¨vkv Ges cÖZ¨q wb‡q Pj Avgiv kc_ †bB Ò wcÖq g„wËKv, wcÖq Rb¥f~wg, wcÖqZg ¯^‡`kZzwg fq †c‡qv bv, gv †Mv, Zzwg fq †c‡qv bv|
Avgiv †R‡M AvwQ GB a~mi ivw·Z;
†Zvgv‡KB G‡b w`‡ev c„w_exi me‡P‡q D¾¡j w`b|Ó
[wkíx t gvngy`y¾vgvb evey]
Gevi Avgiv gv_v Zz‡j `uvove, `uvov‡ZB n‡e|
gvngy`yj nvmvb †mvnvM
D‡`¨v³v,
MÖæc
cwiPvjK, D™¢vm-D‡b¥l wkÿv cwievi
B. Sc.(Engr.) in EEE, BUET
SSC – 5th Stand (D.B.- 1998)
HSC – 4th Stand (D.B.- 2000)
sohag@onnorokom.com
cwieZ©‡bi cªZ¨‡q wbišÍi c_Pjv...
6
D™¢vm GKv‡WwgK GÛ GWwgkb †Kqvi