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M1C8

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Atikur Rahman
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0% found this document useful (0 votes)
101 views14 pages

M1C8

Uploaded by

Atikur Rahman
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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(For HSC & Pre-Admission)

D”PZi MwYZ cª_g cÎ


Aa¨vq-08 : dvskb I dvsk‡bi †jLwPÎ

mvwe©K e¨e¯’vcbvq K…ZÁZv


D™¢vm g¨v_ wUg D™¢vm-D‡b¥l-DËiY
wkÿv cwiev‡ii mKj m`m¨
cÖ”Q`
†gvt ivwKe †nv‡mb cÖKvkbvq
D™¢vm GKv‡WwgK GÛ GWwgkb †Kqvi
Aÿi web¨vm
cÖKvkKvj
AvjvDwÏb, dqmvj I AvivdvZ
cÖ_g cÖKvk: Rvbyqvwi, 2023 Bs
Aby‡cÖiYv I mn‡hvwMZvq me©‡kl ms¯‹iY: †m‡Þ¤^i, 2023 Bs
gvngy`yj nvmvb †mvnvM AbjvBb cwi‡ekK
gynv¤§` Aveyj nvmvb wjUb rokomari.com

19 5 / -
)
gvÎ

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k Z cuP v b e Ÿ B U

KwcivBU © D™¢vm
mg¯Í AwaKvi msiw¶Z| GB eB‡qi †Kv‡bv AskB cÖwZôv‡bi wjwLZ AbygwZ
e¨ZxZ d‡UvKwc, †iKwW©s, ˆe`y¨wZK ev hvwš¿K c×wZmn †Kv‡bv Dcv‡q
cybiærcv`b ev cÖwZwjwc, weZiY ev †cÖiY Kiv hv‡e bv| GB kZ© jw•NZ n‡j
Dchy³ AvBwb e¨e¯’v MÖnY Kiv n‡e|
wcÖq wk¶v_©x eÜziv,
†Zvgiv wkÿv Rxe‡bi GKwU ¸iæZ¡c~Y© av‡c c`vc©Y K‡i‡Qv| gva¨wg‡Ki covïbv †_‡K
D”P-gva¨wg‡Ki covïbvi avuP wfbœ Ges e¨vcK| gva¨wgK ch©šÍ †hLv‡b Ô†evW© eBÕ-B wQj me, †mLv‡b
D”P-gva¨wg‡K welqwfwËK wbw`©ó †Kv‡bv eB †bB| wKš‘ evRv‡i †evW© Aby‡gvw`Z wewfbœ †jL‡Ki
A‡bK eB cvIqv hvq| G Kvi‡YB wk¶v_©xiv cvV¨eB evQvB‡qi †¶‡Î wØavq †fv‡M| GQvov
gva¨wg‡Ki Zzjbvq D”P-gva¨wg‡K wm‡jevm wekvj nIqv m‡Ë¡I cÖ¯ÍzwZi Rb¨ LyeB Kg mgq cvIqv
hvq| Rxe‡bi Ab¨Zg ¸iæZ¡c~Y© GB av‡ci ïiæ‡ZB wØav-Ø›Ø †_‡K gyw³ w`‡Z Avgv‡`i GB Parallel
Text| D”P-gva¨wgK chv©‡q wk¶v_©x‡`i nZvkvi GKwU gyL¨ KviY _v‡K cvV¨eB‡qi ZvwË¡K Av‡jvPbv
eyS‡Z bv cviv| GRb¨ wk¶v_©x‡`i gv‡S ey‡S ey‡S covi cÖwZ Abxnv ˆZwi nq| ZviB dj¯^iƒc
wk¶v_©xiv HSC I wek^we`¨vjq fwZ© cix¶vq fv‡jv djvdj Ki‡Z e¨_© nq|
†Zvgv‡`i †jLvcov‡K AviI mnR I cÖvYešÍ K‡i †Zvjvi welqwU gv_vq †i‡L Avgv‡`i Parallel
Text eB¸‡jv mvRv‡bv n‡q‡Q mnR-mvejxj fvlvq, AmsL¨ ev¯Íe D`vniY, Mí, KvUz©b Avi wPÎ
w`‡q| cÖwZwU UwcK wb‡q Av‡jvPbvi c‡iB i‡q‡Q MvwYwZK D`vniY; hv Uwc‡Ki ev¯Íe cÖ‡qvM Ges
MvwYwZK mgm¨v mgvavb m¤ú‡K© aviYv †`Iqvi cvkvcvwk cieZ©x UwcK¸‡jv eyS‡ZI mvnvh¨ Ki‡e|
†Zvgv‡`i †evSvi myweavi Rb¨ ¸iæZ¡c~Y© msÁv, ˆewkó¨, cv_©K¨ BZ¨vw` wb‡`©k‡Ki gva¨‡g Avjv`v Kiv
n‡q‡Q| GQvovI †hme wel‡q mvaviYZ fzj nq, †mme welq ÔmZK©ZviÕ gva¨‡g †`Lv‡bv n‡q‡Q|
Z‡e ïay eyS‡Z cvivUvB wKš‘ h‡_ó bq, Zvi cvkvcvwk `iKvi ch©vß Abykxjb| Avi GB welqwU AviI
mnR Ki‡Z cÖwZwU Aa¨v‡qi K‡qKwU UwcK †k‡l hy³ Kiv n‡q‡Q ÔUwcKwfwËK weMZ eQ‡ii cÖkœ I
mgvavbÕ| hvi g‡a¨ i‡q‡Q weMZ †evW© cixÿvi cÖ‡kœi cvkvcvwk ey‡qU, iæ‡qU, Kz‡qU, Pz‡qU I XvKv
wek¦we`¨vjqmn wewfbœ wek¦we`¨vj‡qi fwZ© cix¶vi cÖkœ I mgvavb| Gfv‡e av‡c av‡c Abykxjb Kivi
d‡j †Zvgiv †evW© cix¶vi kZfvM cÖ¯‘wZi cvkvcvwk fwZ© cixÿvi cÖ¯‘wZI wb‡Z cvi‡e GLb †_‡KB|
GQvovI Aa¨vq †k‡l i‡q‡Q Ô¸iæZ¡c~Y© cÖ¨vK&wUm cÖe‡jgÕ I ÔMvwYwZK mgm¨vewjÕ hv Abykxj‡bi
gva¨‡g †Zvgv‡`i cÖ¯ÍzwZ c~Y©v½ n‡e|
Avkv KiwQ, Avgv‡`i GB Parallel Text GKB mv‡_ D”P-gva¨wg‡K †Zvgv‡`i †ewmK MV‡b mnvqZv
K‡i, HSC cix¶vq A+ wbwðZ Ki‡e Ges fwel¨‡Z wek¦we`¨vjq fwZ©hy‡×i Rb¨ cÖ¯ÍzZ ivL‡e|
†Zvgv‡`i mvwe©K mvdj¨ I D¾¡j fwel¨Z Kvgbvq-

D™¢vm g¨v_ wUg


µ. µ.
bs welqe¯‘ c„ôv bs welqe¯‘ c„ôv
01 †mU I Zvi cÖKvi‡f` 02 23 Aš^q I dvsk‡bi iƒcvšÍi 61

02 e¨ewa 04 24 †jLwPÎ ¯’vbvšÍiKiY 62

03 mmxg e¨ewa 06 25 †jLwP‡Îi AvKviMZ cwieZ©b 64

04 Amxg e¨ewa 07 26 †jLwP‡Îi cÖwZdjb 68

05 †mU g¨vwcs I Kv‡Z©mxq ¸YR 08 27 †jLwP‡Îi cÖwZmgZv 70

06 Aš^q, dvskb I G‡`i wPwýZKiY 09 28 GK bR‡i †jLwP‡Îi ¯’vbvšÍi, AvKviMZ cwieZ©b 72

07 g¨vwcs Gi mvnv‡h¨ dvsk‡bi ¯úó aviYv 11 29 eM©g~j msµvšÍ wewfbœ dvskb 72

08 †Wv‡gb, †Kv‡Wv‡gb I †iÄ 12 30 )


g~j` dvskb f(x)= Q(x)
P(x)
) 80

09 dvsk‡b PjK I aªæe‡Ki f‚wgKv 14 31 n-Zg g~j msµvšÍ dvskb 88

10 dvsk‡bi †jLwPÎ I dvsk‡bi cÖKvk 19 32 ciggvb dvskb 89

11 Piecewise Function 22 33 m~PKxq dvskb (y=ax; a>0,a≠1) 91

12 dvsk‡bi gvb wbY©q msµvšÍ mgm¨vewj 23 34 jMvwi`wgK dvskb 95

13 GK-GK dvskb I Many-One function 27 35 Piecewise dvsk‡bi †jLwPÎ 98

14 mvwe©K dvskb 32 36 †Wv‡gb †iÄ wbY©q msµvšÍ mgm¨vewj 100

15 cÖwZm½ dvskb 34 37 aªæeK dvskb 106

16 GK-GK dvskb I mvwe©K dvskb msµvšÍ mgm¨vewj 36 38 A‡f` dvskb 107

17 wecixZ dvskb I wecixZ Aš^q 37 39 ms‡hvwRZ dvskb 108

18 wecixZ dvskb msµvšÍ mgm¨vewj 43 40 ms‡hvwRZ dvskb msµvšÍ mgm¨vewj 110

19 †Wv‡gb-†iÄ wbY©q c×wZ msµvšÍ Av‡jvPbv 48 cÖkœgvjv-08

20 wØNvZ eûc`x dvsk‡bi †jLwPÎ 55 42 Brainstorming Question 115

21 †jLwP‡Îi mvnv‡h¨ wØNvZ mgxKi‡Yi g~‡ji cÖK…wZ wbY©q 59 43 GK‡Î me ¸iæZ¡c~Y© m~Î 116

22 wØNvZ eûc`x dvsk‡bi m‡ev©”P I me©wb¤œ gvb 60 44 ¸iæZ¡c~Y© cª¨vK&wUm cÖe‡jg 119
cvi¯úwiK mn‡hvwMZv-B cv‡i
c„w_ex‡K AviI my›`i Ki‡Z...
mywcÖq wk¶v_©x,
Avkv Kwi Gev‡ii ÒHSC Parallel TextÓ †Zvgv‡`i Kv‡Q
AZx‡Zi †P‡q Av‡iv †ewk DcKvix wn‡m‡e we‡ewPZ n‡e
BbkvAvjøvn&| eBwU m¤ú~Y© ÎæwUgy³ ivL‡Z Avgiv †Póvi †Kv‡bv
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D‡jøwLZ B-†gBj G AewnZ Ki‡j K…ZÁ _vK‡ev Ges Avgiv
Zv cieZ©x ms¯‹i‡Y ms‡kvab K‡i wbe BbkvAvjøvn&|
Email : solutionpt.udvash@gmail.com
Email-G wbgœwjwLZ welq¸‡jv D‡jøL Ki‡Z n‡e:
(i) ÒHSC Parallel TextÓ Gi wel‡qi bvg, fvm©b
(evsjv/Bswjk), (ii) c„ôv b¤^i (iii) cÖkœ b¤^i (iv) fzjUv Kx
(v) Kx nIqv DwPr e‡j †Zvgvi g‡b nq
D`vniY: ÒHSC Parallel TextÓ Math 1st Paper,
Chapter-08, Bangla Version, Page-26, Question-05,
†`Iqv Av‡Q, [x] wKš‘ n‡e [2y]
fzj QvovI gvb Dbœq‡b †h‡Kvb civgk© AvšÍwiKfv‡e MÖnY Kiv
n‡e| cwi‡k‡l gnvb Avjøvni wbKU †Zvgv‡`i mvdj¨ Kvgbv
KiwQ|
ïf Kvgbvq
D™¢vm g¨v_ wUg
,

: ,

: , ,

, ?
( ): ?

(Rene Descartes)
(Leibnitz) ( )
(Euler)
Element of Algebra - y=f x
(Newton) Euler
(Fourier)

(Dirichlet)

Dirichlet

1
,

- /

(Set and Its Types)


- ,
?
,

: (Set)

: A , ,A= -
, , B , ,B=
,

, , ,
,

: ,
U

2
, 3
2 1 , A

(Subset) (Superset)
A= - , , C= ,C A
, C A ,A C

⊆ “⊇” ,C⊆A A ⊇ C.
,C A C A ⊂ .

,
( )
1 ( 0)

n , (
)

: (Power Set) P(A)

3
A B A B
(common element) ,

A ,

A , ,
U( ) A

:
A A ,A ,A =A =U A = U\A

, A A (
) ,A ∪A =U A A
A ∩ A = ϕ.
, ,P= , , } ,P A

U
(finite set) ”

ℕ = 1, 2, 3, 4, 5, … ? ,
(infinite set)

,
,
ℝ (ℕ ,
; ℕ⊆ℝ

(Interval)
, , ,
Age limitation ,
12 , 50
, x 12 12 50
, , 12 ≤ x < 50
,x 12 50 x A
x A :

,A
? : A = x: x ∈ ℝ+ , 12 ≤ x < 50

4
, A

12 50 ,

{} ,A 12 50 {}
() []
:
12 ≤ x < 50
? , , 12
50 A
[ )
, A = 12, 50
,

() [] convention

A,
A = x: x ∈ ℝ, 14 ≤ x ≤ 30 ,

,
[] , A = 14, 30
A , A = x: x ∈ ℝ, 16 < x < 20
, ()
, A = 16, 20
() ][ : 2, 5 2, 5 2, 5 2, 5

: a b (a < b)

: (i) ( ] (ii) ) [
: (i) [ (ii) ]

,5 P ,P ?
: P = 5, ∞

5
? P , ‘∞’ ]
, ?
‘∞’ , ‘∞’
, ‘∞’
‘∞’ , ‘ ∞’

:
(1) (Finite interval)
(2) (Infinite interval)

(Finite Interval)
a b (a < b) a b
:

(i) (Open Interval):


(a, b) :
A = 2, 5 = 2, 5 , set builder method ,A 2 5
, 2 5
, A = x: x ∈ ℝ, 2 < x < 5

, A = x: x ∈ ℝ, a < x < b

(ii) - (Closed-Open Interval):


, ,
; -
: D = 33, 37 = 33, 37 = x: x ∈ ℝ, 33 ≤ x < 37
:

, D = x: x ∈ ℝ, a ≤ x < b

6
(iii) - (Open-Closed Interval):
, ,
; -
: C = 9, 14 = 9, 14 = x: x ∈ ℝ, 9 < x ≤ 14
:

, C = x: x ∈ ℝ, a < x ≤ b

(iv) (Closed Interval):


,
: B = 20, 25 = x: x ∈ ℝ, 20 ≤ x ≤ 25
:

, B = x: x ∈ ℝ, a ≤ x ≤ b

(Infinite Interval)

(i) :
a
: P = 10, ∞ = 10, ∞ = x: x ∈ ℝ, x > 10
:

(ii) :
a :
Q= ∞, 51 = ∞, 51 = x: x ∈ ℝ, x < 51
:

(iii) :
: R = 3, ∞ = 3, ∞ = x: x ∈ ℝ, x ≥ 3
:

(iv) :
a
:S= ∞, 100 = ∞, 100 = x: x ∈ ℝ, x ≤ 100
:

7
(v) :
∞ ∞ , ℝ
,R= ∞, ∞ = ∞, ∞ = x: x ∈ ℝ, ∞ < x < ∞
:

(Set Mapping and Cartesian Product)


: , ,
14, 15, 16 , A= , ,
B = 14, 15, 16 ,

(set
mapping)
A B
A B ,n A =n B
- ,

, ,

:
R = ( , 14), ( , 15), ( , 16)
,R ( , 14), ( , 15) ( , 16)
(A B)

(Ordered Pair)

:
() ,

8
, A B ?
,
: ,A= , , ,
B = 14, 15, 16 R ,
= × = ( , 14), ( , 15), ( , 16), ( , 14), ( , 15), ( , 16), ( , 14), ( , 15), ( , 16)}

, , ,
, R ,
R = x, y : x, y ∈ A × B; y x }= ( , 14), ( , 15), ( , 16) .

A B , A
m B n
, ?

, ?

, (Relation, Function and their identification)


ক , A = 1, 2, 3, 4 এ ং B = 1, 2, 8, 27, 64, 25
Aএ উ দ কঘ ক Bএ উ দ ও ত দ য ক, P ে ে দখ :

ি -

9
,P= 1, 1 , 2, 8 , 3, 27 , 4, 64 ,P⊆ A×B
আ য , A = 1, 4, 9 এ ং C = 3, 2, 1, 0,1, 2, 3
Aএ উ দ ক � ক Cএ উ দ ও ত দ য ক, Q ে ে দখ :

ি -
,Q= 1, 1 , 1, 1 , 4, 2 , 4, 2 , 9, 3 , 9, 3 ,Q⊆A×C
,
input output input , output

অ : দ ত ই ক অ ,AওB ক ই A×B দ ত
অ উ কA কB এক অ ।

: ে - P ত দখ ,A ত উ দ B এক উ দ কত। আ
ে - Q ত দখ ,A এক উ দ C ই উ দ কত।
, input output
:P input - output

, , 3, 5 = 5, 3 , ,
5, 3 = 3, 5 [ 5, 3 3, 5 ] ,
? ,

: a, b = c, d , a = c b = d 3≠5 3, 5 ≠ 5,3

10

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