6. arithmetic progressions

8/12/2019 6. Arithmetic Progressions

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KIFERL@_KIT

T O A _ F M I ] EA F D U^ -G

KI]LOKI]EBR

9

ITE]LKO]EB ^TAJTORREAFR

EF]TAM_B]EAF

Bafsemor tlo gaccawefj irrifjokoft ag fukdors 8

(e) 9, 2, ?, :, .. (ee) 2, 7, 94, 4=, .. (eee) 9, =, 5, 97, ..

Ef oibl ag tlo idavo irrifjokofts, wo adsorvo sako pittorfs. Ef (e) wo gefm tlit tlo subboomefj torksiro adtiefom dy immefj i gexom fukdor Pe.o. 4[, ef (ee) dy kuctepcyefj wetl i gexom fukdor Pe.o. 4[, ef (eee)

wo gefm tlit tloy iro squiros ag fituric fukdors.

Ef tles bliptor, wo slicc mesbuss afo ag tloso pittorfs ef wlebl subboomefj torks iro adtiefom dy immefji gexom fukdor ta tlo probomefj torks. Zo slicc icsa soo law ta gefm tloer f

tltorks ifm tlo suk ag f

bafsobutevo torks, ifm uso tles `fawcomjo ef sacvefj sako miecy cego pradcoks.

LER]ATEBIC GIB]R

Jiuss wis i vory ticoftom ifm jegtom kitlokitebeif ag 95tl

boftury wla movocapom tlo garkuci 8

9 + 4 + 2 + = +. + (f 9) + f ;

4

)9( ffgar tlo suk ag gerst f fituric fukdors it tlo ijo ag 9>. Lo mem

tles ef tlo gaccawefj wiy 8R ; 9 + 4 + 2 .. + (f 4) + (f 9) + f

R ; f + (f 9) + (f 4) + .. + 2 + 4 + 9

4R ; (f + 9) + (f + 9) + (f + 9) +.+ (f + 9) + (f + 9) + (f + 9); (f + 9) (9 + 9 + 9 + upta f tekos)

4R ; (f + 9) f R ;4

)9( ff

Ovof wlof lo wis i cettco blecm ag tlroo lo baucm roim ifm ki`o kitlokitebic bicbuciteaf leksocg.

Jiuss pravom tlo gufmikoftic tloarok ag Icjodri wlof lo wis 4> yoirs acm. Les baftreduteaf ta

kitlokitebs lis doof ekkofso dobiuso les garkucio woro usom ef ippceom geocm ag Istrafaky

Meggorofteic Joakotry ifm Ocobtrebety wemocy icc avor tlo warcm dy sbeoftests.

ROY_OFBO

Ef aur miecy cego, wo bako ibrass tlo irrifjokoft ag fukdors ar adhobts ef if armor subl is irrifjokoft

ag stumofts ef i raw is por tloer racc fukdors, irrifjokoft ag daa`s ef tlo cedriry, otb.

If irrifjokoft ag fukdors mopofms af tlo jevof ruco 8

Jevof Tuco Irrifjokoft ag fukdors

Zreto 2 ifm tlof imm = subbossevocy 2. :, 99, 9?, 95,.

Zreto 2 ifm tlof kuctepcy = subbossevocy 2, 94, =


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KI]LOKI]EBR

4

ROTEOR]lo suk ag torks ag i soquofbo es biccom tlo soreos ag tlo barrospafmefj soquofbo. ] 9+ ]4+ ]2+.es

if efgefeto soreos, wloro is ]9+ ]4+ ]2+ . + ]f-9+ ]fes i gefeto soreos ag f torks.

Rf; ]9+ ]4+ ]2+ .. + ]f 4+ ]f 9+ ]f

Rf-9; ]9+ ]4+ ]2+ . + ]f 4 + ]f 9

Rf Rf-9; ]f

AT ]f; Rf Rf-9

Ox.9 Zreto tlo gerst gevo torks ag tlo soquofbo, wlaso ftl tork es if; {9 + (-9)f}f.

Rac. if; {9 + (-9)f}f

Rudstetutefj f ; 9, 4, 2, = ifm ?, wo jot

i9; {9 + (-9)9} 9 ; > 0 i4; {9 + (-9)

4} 4 ; =0

i2; {9 + (-9)2} 2 ; > 0 i=; {9 + (-9)

=} = ;

]lus, tlo roquerom torks iro 8 >, =, >, < ifm >.

Ox.4 Gefm tlo 4>tl tork ag tlo soquofbo wlaso ftl tork es, if;

2

)4(

f

ff

Rac. if;2

)4(

f

ff. ^uttefj f ; 4>, wo adtief i4>;

24>

)44>(4>

]lus, i4>;42

27>

Ox2. ]lo Gedafibbe soquofbo es mogefom dy i9; 9 ; i40 if; if-9+ if-4gar f 6 4. Gefmf

f

i

i 9, gar f ; 9,4,2,=,?,

Rac. Zo livo i9; i4; 9 ifm if; if-9+ if-4Rudstetutefj f ; 2, =, ? ifm 7, wo jot,

i2; i4+ i9; 9 + 9 ; 4 i=; i2+ i4; 4 + 9 ; 2

i?; i=+ i2; 2 + 4 ; ? ifm i7; i?+ i=; ? + 2 ; , :>, =>, 9>,.Coist Efgarkiteaf Toquerom 8 ]a `faw idaut if I^, tlo kefekuk efgarkiteaf wo foom ta `faw es ta

`faw datl tlo gerst tork i ifm tlo bakkaf meggorofbo m.

Gar efstifbo eg tlo gerst tork i es 7 ifm tlo bakkaf meggorofbo m es 2, tlof I^ es 7, 5, 94, 9?,Rekecircy, wlof

i ; - :, m ; - 4, tlo I^ es :, - 99, -92,.

i ; 9.>, m ; >.9, tlo I^ es 9.>, 9.9, 9.4, 9.2,Ra eg wo `faw wlit i ifm m iro wo bif cest tlo I^.

Ox.= Ef wlebl ag tlo gaccawefj setuiteafs, maos tlo cest ag fukdors efvacvom ki`o if iretlkoteb prajrosseaf,

ifm wly1

(e) ]lo tixe giro igtor oibl `k wlof tlo giro es Ts. 9? gar tlo gerst `k ifm Ts < gar oibl immeteafi`k.

(ee) ]lo ikauft ag ier prosoft ef i bycefmor wlof i vibuuk pukp rokavos =

9ag tlo ier rokiefefj ef

tlo bycefmor it i teko.(eee) ]lo bast ag mejjefj i wocc igtor ovory kotro ag mejjefj, wlof et basts Ts.9?> gar tlo gerst kotro

ifm resos dy Ts. ?> gar oibl sudsoquoft kotro.

(ev) ]lo ikauft ag kafoy ef tlo ibbauft ovory yoir, wlof Ts. 9>>>> es mopasetom it bakpaufmeftorost


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?

(ee) Ikauft ag ier prosoft ef tlo bycefmor ; x ufets (siy) ; i9Ikauft ag ier prosoft ef tlo bycefmor igtor afo teko rokavic ag ier dy tlo vibuuk

pukp=

2

=

xxx ufets i4

Ikauft ag ier prosoft ef tlo bycefmor igtor twa teko rokavic ag ier dy tlo vibuuk pukp

975

972

=2

=2

=9

=2 xxxxx

ufets ;

4

=2

x ufets ; i2

Ikauft ag ier prosoft ef tlo bycefmor igtor tlroo tekos rokavic ag ier dy tlo vibuuk

pukp

24444

=

2

=

2

=

2

=

2

=

99

=

2

=

9

=

2

xxxx x ufets ; i=ifm sa af.

i4 i9;=

2x- x ;

=

xufets i2 i4;

97

2

=

2

=

24

xx x ufets

Is i4 i9 i2 i4, tles cest ag fukdors maos fat gark if I^.(eee) Bast ag mejjefj tlo wocc igtor 9 kotro ag mejjefj ; Ts. 9?> ; i9

Bast ag mejjefj tlo wocc igtor 4 kotros ag mejjefj ; Ts. 9?> + ?> ; Ts 4>> ; i 4

Bast ag mejjefj tlo wocc igtor 2 kotros ag mejjefj ; Ts. 4>> + ?> ; Ts 4?> ; i 2Bast ag mejjefj tlo wocc igtor = kotros ag mejjefj ; Ts. 4?> + ?> ; Ts 2>> ; i =ifm sa af.i4 i9; Ts 4>> 9?> ; ?> i2 i4; Ts 4?> 4>> ; ?>i= i2; Ts 2>> 4?> ; ?>e.o., i`-9 i` es tlo siko ovoryteko. Ra tles cest ag fukdors garks if I^ wetl tlo gerst tork i ;Ts. 9?> ifm tlo bakkaf meggorofbo m ; Ts.?>

(ev) Ikauft ag kafoy igtor 9 yoir ; Ts. 9>>>>

9>>

>>>

4

9>>

>>>2

9>>

>>>

=

9>>

>>>

4

9>>

>>>

9>>

>>>

9>>

>

>>>

9>>

>

>>>

2

9>>

>>>

4

9>>

>>>

99>>

>

>>>

9>>

>

.4?

Rac. (e) i ; =, m ; ?

Gerst tork, i ; =

Robafm tork ; = + m ; = + ? ; 5

]lerm tork ; 5 + m ; 5 + ? ; 9=

Gaurtl tork ; 9= + m ; 9= + 9? ; 95

Lofbo, gerst gaur torks ag tlo jevof I^ iro =, 5, 9=, 95.

(ee) i ; 9.4?, m ; >.4?

Gerst tork ; i ; 9.4?

Robafm tork ; - 9.4? + m ; - 9.4? + (- >.4?) ; - 9.?>

]lerm tork ; - 9.?> + m ; - 9.?> + (->.4?) ; - 9.:?

Gaurtl tork ; - 9.:? + m ; - 9.:? + (->.4?) ; - 4.>>

Lofbo, gerst gaur torks ag tlo jevof I^ iro 9.4?, - 9.?>, - 9.:?, - 4.>>

Ox.7 Gar tlo I^42,

49,

49,

42 ,.. wreto tlo gerst tork i ifm tlo bakkaf meggorofbo m. Icsa wreto tlo foxt

twa torks igtor tlo jevof cist tork -4

2.

Rac. Zo livo i9;4

2,

4

9,

4

9,

4

2=24 iii ifm sa af.

]lus, i ;4

2

i4; i9;

4

2

4

9; - 9,

i2 i4;

4

9

4

9; - 9,

i= i2 ;

4

9

4

2; - 9, ifm sa af.

m ; - 9

Faw, wo gefm tlo subbossar ag -4

2.

i?; 4

?)9(4

2

4

2

m

]lof i7; i?+ m ;4

:)9(

4

?

Lofbo, tlo foxt twa torks igtor tlo jevof tork 4

:,

4

?

4

2iro .


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:

BAK^O]E]EAF ZEFMAZ

JOAKO]TEB TAJTORREAF

9. I soquofbo ag faf-zora fukdors i9, i4, i2,.,ifes siem ta do i joakotreb soquofbo ar J.^.

egg2

=

4

2

9

4

i

i

i

i

i

i ; ...

e.o. eggf

f

i

i 9 ; i bafstift gar icc f.

]les bafstift es biccom tlo bakkaf ritea ag tlo J.^. ifm es usuiccy mofatom dy r. o.j., 2, 5, 4:, , 2>, - 2>, - 9>, - 7,..

I joforic L.^. esmimii 4

9,

9,

9

+ .

Zloro es tlo gerst tork ifm m es tlo bakkaf meggorofbo ag tlo I.^.

JOFOTIC ]OTK AG IF ITE]LOKO]EB ^TAJTORREAF

]lo garkuci gar wretefj joforic tork ar tlo ftl tork ag if iretlkoteb prajrosseaf es

if; i + (f 9)m

Zloro, i es tlo gerst tork ag iretlkoteb prajrosseaf,ifm m es tlo bakkaf meggorofbo ag iretlkoteb prajrosseaf.

rtl

]OTK AG GEFE]O ITE]LKO]EB ^TAJTORREAF GTAK ]LO OFM

Cot tloro do if iretlkoteb prajrosseaf wetl gerst tork i ifm bakkaf meggorofbo m. Eg tloro iro f torks ef

tlo iretlkoteb prajrosseaf, tlof

rtl

tork grak tlo ofm ; i + (f r)m

Icsa, eg es tlo cist tork ag tlo iretlkoteb prajrosseaf tlof rtl

tork grak tlo ofm es tlo rtl

tork ag if

iretlkoteb prajrosseaf wlaso gerst tork es ifm bakkaf meggorofbo es m.

rtl

tork gark tlo ofm ; + (r 9) (- m)

Ox.: Gefm tlo 2>tl tork ag tlo I^ 8 9>, :, =. PFBOT][Rac. ]lo jevof I.^. es 9>, :, =,..

Loro, i ; 9>, m ; : 9> ; - 2 ifm f ; 2>

wo livo if; i + (f 9)m

Ra, i2>; 9> + (2> 9) (-2) i2>; 9> ; - ::

]lo 2>tltork ag tlo jevof I^ es ::.


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KI]LOKI]EBR

ifm tlo 9>tl

tork es 47. Motorkefo tlo 9?tl

tork ag tlo

I^.

Rac. Cot gerst tork ifm tlo bakkaf meggorofbo ag tlo I^ do i ifm m rospobtevocy.

7tl

tork ; - 9> (Jevof)

i + ? (7 9)m ; - 9> Pif; i + (f 9) m[

i + ?m ; - 9> .(e)

9>tl

tork ; - 47 (Jevof)

i + (9> 9)m ; - 47

i + 5m ; - 47 .(ee)

Racvefj (e) ifm (ee) wo jot

i ; 9>, m ; - =

]lorogaro, 9?tl

tork ag tlo I^

; i + (9? 9)m Pif; i + (f 9) m[

; i + 9=m

; 9> + 9= (-=)

; 9> ?7 ; - =7

Lofbo, tlo 9?tl

tork ag I^ es =7.

Ox.5 Gefm tlo 7tl tork grak tlo ofm ag tlo I^ 9:, 9=, 99,., - =>.

Rac. ]lo jevof I^ 9:, 9=, 99,., - =>

Loro, i ; 9:, m ; 9= 9: ; - 2, ; - =>

Cot tloro do f torks ef tlo jevof I^.

]lof, ftl tork ; - =>

i + (f 9)m ; - => Pif; i + (f 9) m[

9: + (f 9) (-2) ; - =>

(f 9) (-2) ; - => 9:

(f 9) (-2) ; - ?:

f 9 ;2

?:

f 9 ; 95

f ; 95 + 9

f ; 4>

Lofbo, tloro iro 4> torks ef tlo jevof I^, Faw, 7tl tork grak tlo ofm

; i + (4> 7)m Prtl tork grak tlo ofm ; i + (f r)m[

; i + 9=m

; 9: + 9=(-2)

; 9: =4 ; - 4?

Lofbo, tlo 7tl tork grak tlo ofm ag tlo jevof I^ es 4?.


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5

Ox.9> es 4>> ify tork ag tlo soquofbo 2, :, 99, 9?,1

Rac. ]lo jevof soquofbo es 2, :, 99, 9?,

i4 i9 ; : 2 ; = i2 i4; 99 : ; = i= i2; 9? 99 ; =Is i`+9 i`es tlo siko gar ` ; 9, 4, 2, otb., tlo jevof soquofbo gark if I^.

Loro, i ; 2, m ; =

Cot 4>> do tlo ftl tork ag tlo jevof soquofbo. ]lof,if; 4>>

i + (f 9)m ; 4>> 2 + (f 9) = ; 4>>

(f 9) ;=

95: f ; 9

=

95:

=

4>9f .

Dut f slaucm do i pasetevo eftojor. Ra, 4>> es fat tork ag tlo jevof soquofbo.

BAK^O]E]EAF ZEFMAZ

JOFOTIC ]OTK AG I.J.^.

]lo ftl torks ag i J.^. es i, ir4,.ir

f-9es ]f; ir

f-9

rtl ]OTK GTAK ]LO OFM AG GEFE]O J.^.

Cot i do tlo gerst tork ifm r do tlo bakkaf ritea ag i gefeto J.^. bafsestefj ag f torks, tlofrtl tork grak tlo ofm ;ir

f-r

Icsa, eg es tlo cist tork ag tlo J.^.tlof

rtl tork grak tlo ofm ;

99

f

r

JOFOTIC ]OTK AG I L.^.

]a gefm tlo ftl tork ag if L.^., gefm tlo ftl tork ag tlo barrospafmefj I.^. adtiefom dy tlo robeprabicsag tlo torks ag tlo jevof L.^. Faw tlo robeprabic ag tlo ftl tork ag if I.^., wecc do tlo ftl tork ag tloL.^.

]ry aut tlo Gaccawefj8

9. Gefm tlo 5tl

tork ifm tlo joforic tork ag tlo prajrosseaf ....4,9,4

9,

=

9

4. Zlebl tork ag tlo J.^. ?, 9>, 4>, =>, .es ?94>1

2. ]lo gaurtl, sovoftl ifm tlo cist tork ag i J.^. iro 9>, ifm 4?7> rospobtevocy

Gefm tlo gerst tork ifm tlo fukdor ag torks ef tlo J.^.

=. Gefm tlo 5tl tork ag prajrosseaf ....

4

ROCOB]EAF AG ]OTKR EF IF I^Rakotekos wo roquero bortief fukdor ag torks ef I^. ]lo gaccawefj wiys ag socobtefj torks iro

joforiccy vory bafvofeoft.

Fukdor ag torks ]orks Bakkaf meggorofbo

2 i-m, i, i + m m

= i 2m, i m, i + m, i + 2m 4m? i 4m, i m, i, i + m, i + 4m m

7 i ?m, i 2m, i m, i + 2m, i + ?m 4m

Et slaucm do fatom tlit ef biso ag if amm fukdor ag torks, tlo kemmco tork es i ifm tlo bakkafmeggorofbo es m wleco ef biso ag if ovof fukdor ag torks tlo kemmco torks iro i m, i + m ifm tlo

bakkaf meggorofbo es 4m.

Tokir`9 8 Eg tlo suk ag torks es fat jevof, tlof socobt torks is i, i + m, i + 4m,.

Tokir`4 8 Eg tlroo fukdors i, d, b ef armor iro ef I^. ]lof

d i ; Bakkaf meggorofbo ; b d

d i ; b d 4d ; i + b

]lus, i, d, b iro ef I^ eg ifm afcy eg 4d ; i + bTokir`2 8 Eg i, d, b iro ef I^, tlof d es fawf is tlo iretlkoteb koif (IK) dotwoof i ifm b.

Tokir`= 8 Eg i, x, d iro ef I^ ]lof,

4x ; i + d x ;4

di

]lus, IK dotwoof i ifm d es4

di.

Ox.99 ]lo suk ag tlroo fukdors ef I^ es -2, ifm tloer pramubt es ifm tlo suk ag wlaso squiros es 94>.

Rac. Cot tlo fukdors do (i 2m), (i m), (i + m), (i + 2m). ]lof,Ruk ; 4>

(i 2m) + (i m) + (i + m) + (i + 2m) ; 4> =i ; 4> i ; ?Faw suk ag tlo squiros ; 94> (i 2m)4+ (i m)4+ (i + m)4+ (i + 2m)4; 94> =i4+ 4>m4; 94> i4+ ?m4; 2> 4? + ?m4; 2> 4? + ?m4; 2> ?m4; ? m ; 9Eg m ; 9, tlof tlo fukdors iro 4, =, 7,


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99

Ox.92 Eg 4x, x + 9>, 2x + 4 iro ef I^. Gefm tlo vicuo ag x.Rac. Refbo, 4x, x + 9>, 2x + 4 iro ef I^.

4 (x + 9>) ; 4x + (2x + 4) 4x + 4> ; ?x + 4 2x ; 9

9,

94

9,

9?

9,...., ta 99 torks. PFBOT][

Rac. Loro, i ;9?

9

m ;7>

9

9?

9

94

9

f ; 99Zo `faw tlit

Rf;4f P4i + (f 9)m[

R99;

7>

9)999(

9?

94

4

99 R99;

7

9

9?

4

4

99

R99;

9>

2

4

99 R99;

4>

22

Ra, tlo suk ag tlo gerst 99 torks ag tlo jevof I^ es4>

22.

Ox.9? Gefm tlo suk 8 2= + 24 + 2> + . + 9>

Rac. 2= + 24 + 2> + + 9>]les es if I^

Loro, i ; 2=m ; 24 2= ; - 4

; 9>

Cot tlo fukdor ag torks ag tlo I^ do f.

Zo `faw tlit


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92

if; i + (f 9)m

9> ; 2= + (f 9) (-4) (f 9) (-4) ; - 4=

f 9 ;4

4=

; 94 f ; 92

Ijief, wo `faw tlit

Rf; )9>2=(4

92)(

492 Ri

f

R92; 4 ifm 4>> wlebl iro mevesedco dy =.

Rac. Icc fituric fukdors dotwoof 9>> ifm 4>> wlebl iro mevesedco dy = iro

9>=, 9>=

i4; 9>< 9>= ; =

i2 i4; 994 9>< ; =

i= i2; 997 994 ; =

i4 i9; i2 i4; i= i2; ...(; = oibl)

]les soquofbo es if iretlkoteb prajrosseaf wlaso bakkaf meggorofbo es =.

Loro, i ; 9>=

m ; =

; 957

Cot tlo fukdor ag torks do f. ]lof

; i + (f 9)m

957 ; 9>= + (f 9)=

957 9>= ; (f 9)= 54 ; (f 9) = (f 9) = ; 54

f 9 ;=

54 f 9 ; 42

f ; 42 + 9 f ; 4=

Ijief, wo `faw tlit


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9=

Rf; )(4

if

R4=;

4

4=(9>= + 957)

; (94) (2>>) ; 27>>Lofbo, tlo roquerom suk es 27>>.

Ox.9: Gefm tlo fukdor ag torks ag tlo I^ ?=, ?9, =47 ; f P999 -2f[

9>47 ; 999f 2f4 2f4-999f + 9>47 ; >

f4 2:f + 2=4 ; > PMevemefj tlraujlaut dy 2[

f4 9 f(f 9 ar f 95 >

f ; 9

R2; 4(2)4+ 2 ; 9< + 2 ; 49 R=; 4(=)

4+ = ; 24 + = ; 27

ifm sa af.


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KI]LOKI]EBR

9?

i9; R9; 2

i4; R4 R9; 9> 2 ; :

i2; R2 R4; 49 9> ; 99

i=; R= R2; 27 49 ; 9?

ifm sa af.

Lofbo, tlo roquerom I^ es 2, :, 99, 9?,

Ox.95 4>> cajs iro stib`om ef tlo gaccawefj kiffor 8 4> cajs ef tlo dattak raw, 95 ef tlo foxt raw, 9< ef tlo

raw foxt ta et ifm sa af (soo gejuro). Ef law kify raws iro tlo 4>> cajs pcibom ifm law kify cajs iro

ef tlo tap raw1

PFBOT][

Rac. ]lo fukdor ag cajs ef tlo dattak raw, foxt raw, raw foxt ta et ifm sa af gark tlo soquofbo

4>, 95, 9 ; - 9

i2 i4; 9< 95 ; - 9i= i2; 9: 9< ; - 9

e.o., i` + 9 i`es tlo siko ovoryteko.

Ra, tlo idavo soquofbo garks if I^.Loro, i ; 4>

m ; - 9

Rf; 4>>Zo `faw tlit

Rf; 4

fP4i + (f 9)m[

4>> ;4

fP4 (4>) + (f 9) (-9)[ 4>> ;

4

fP=> f + 9[

4>> ;4

fP=9 f[ =>> ; f P=9 f[

fP=9 f[ ; =>> =9f f4; =>> f4 =9f + =>> ; > f4 4?f 97f + =>> ; > f (f 4?f) 97 (f 4?) ; > (f 4?) (f 97) ; > f 4? ; > ar f 97 ; > f ; 4? ar f ; 97 f ; 4?, 97Lofbo, tlo fukdor ag raws es oetlor 4? ar 97.Faw, fukdor ag cajs ef raw; Fukdor ag cajs ef 4?tl raw

; i4?

; i + (4? 9)m Pif; i + (f 9)m[; i + 4=m

; 4> + 4= (-9) 4> 4= ; - =


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97

Zlebl es fat passedco.]lorogaro, f ; 97 ifm

Fukdor ag caj ef tap raw

; Fukdor ag cajs ef 97tl raw

; i97

; i + (97 9) m P

if; i + (f 9)m[; i + 9? m

; 4> + 9? (-9); 4> 9? ; ?

Lofbo, tlo 4>> cajs iro pcibom ef 97 raws ifm tloro iro ? cajs ef tlo tap raw.

BAK^O]E]EAF ZEFMAZ

R_K AG f ]OTKR AG I.J.^.

Eg Rfes tlo suk ag gerst f torks ag tlo J.^. i, ir, ir4,.,

e.o., Rf; i + ir + ir4+ . + ir

f-9, tlof

9

9

9

9

r

riar

r

riR

fff ,r 9

Icsa,99

r

irar

r

riRf

, r 9, wloro es tlo cist tork e.o. tlo ftl tork .

Gar r ; 9, Rf ; fi

R_K AG IF EFGEFE]O J.^.

]lo suk ag if efgefeto J.^. wetl gerst tork i ifm bakkaf ritea r, wloro -9 3 r 3 9, es

r

iR

9

Eg r 9, tlof tlo suk ag if efgefeto J.^. tofms ta efgefety.R_K AG f ]OTKR AG I L. ^.

]loro es fa spobegeb garkuci ta gefm tlo suk ag f torks ag L.^. ]a sacvo tlo quosteafs ag tlesprajrosseaf, gerst ag icc bafvort et ef I.^. tlof uso tlo praporteos ag I.^.

]TU A_] ]LO GACCAZEFJ

9. Gefm tlo suk ag sovof torks ag tlo J.^. 2, 7, 94, .

4. Gefm tlo suk ta : torks ag tlo soquofbo

24 ?

2

?

4

?

9,

7?= ?

2

?

4

?

9,

5


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9:

^TA^OT]EOR AG ITE]LKO]EBIC ^TAJTORREAFR9. Eg i bafstift es immom ta ar sudtribtom grak oibl tork ag if I.^., tlof tlo rosuctefj soquofbo es icsa if

I.^. wetl tlo siko bakkaf meggorofbo.

4. Eg oibl tork ag i jevof I.^. es kuctepceom ar mevemom dy i faf-zora bafstift @, tlof tlo rosuctefj

soquofbo es icsa if I.^. wetl bakkaf meggorofbo @m ar m/@, wloro m es tlo bakkaf meggorofbo ag tlo

jevof I.^.2. Ef i gefeto I.^., tlo suk ag tlo torks oquemestift grak tlo dojeffefj ifm ofm es icwiys siko ifm es oqui

ta tlo suk ag gerst ifm cist tork.=. ]lroo fukdors i, d, b, iro ef I.^. egg 4d ; i + b.

?. I soquofbo es if I.^. egg ets ftl tork es i cefoir oxprosseaf ef f e.o., if; If+ D iro bafstifts. Ef subl ibiso, tlo baoggebeoft ag f es tlo bakkaf meggorofbo ag tlo I.^.

7. I soquofbo es if I.^. egg tlo suk ag ets gerst f torks es ag tlo gark If4+ Df, wloro I, D iro bafstifts

efmopofmoft ag f. Ef subl i biso, tlo bakkaf meggorofbo es 4 I.:. Eg tlo torks ag if I.^. iro blasof it rojucir eftorvics, tlof tloy gark if I.^.

BAK^O]E]EAF ZEFMAZ

ITE]LKO]EB KOIFR9. Eg tloro fukdors i, d, b iro ef I.^. tlof d es biccom tlo iretlkoteb koif (I.K.) dotwoof i ifm b.

4. ]lo iretlkoteb koif dotwoof twa fukdors i ifm d es4

di

2. I9, I4 .,Ifiro siem ta do f I.K.s dotwoof twa fukdors i ifm d. egg i, I9, I4,.,If, d iro ef I.p.Cot m do tlo bakkaf meggorofbo ag tlo I.^.

Bcoircy, d ; (f + 4)tl tork ag tlo I.^. d ; i + (f + 9) m

9

f

idm

Lofbo, I9; i + m ; i + 9

f

id

, I4; i + 4m ; i +

9

4

f

id

..

If; i + fm ; i +

9

f

idf

=. ]lo suk ag f I.K.s dotwoof twa fukdors i ifm d es f tekos tlo sefjco I.K. dotwoof tlof e.o.

f

4

di

JOAKO]TEB KOIFR

9. Eg tlroo faf-zora fukdors i, d, b iro ef J.^. tlof d es biccom tlo joakotreb koif (J.K.) dotwoof i ifm d

4. ]lo joakotreb koif dotwoof twa pasetevo fukdors i ifm d es id

LITKAFEB KOIF

9. Eg tlroo faf-zora fukdors i, d, b iro ef L.^., tlof es biccom tlo lirkafeb koif (L.K.) dotwoof i ifm d,

4. ]lo lirkafeb koif dotwoof fukdors i ifm d esdi

id

4


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9

?. Ef if I.^. s9; 7, s:; 9>?, tlof sf8 sf-2es siko is 8

(I) (f + 2) 8 (f 2) (D) (f + 2) 8 f (B) f 8 (f 2) (M) Fafo ag tloso

7. Ef if I.^. s2; 7, s7; 2, tlof ets bakkaf meggorofbo es oquic ta 8

(I) 2 (D) 9 (B) 9 (M) Fafo ag tloso

:. ]lo fukdor ag torks bakkaf ta tlo twa I.^. s

4 + ? + < + 99 + .+ 5< ifm 2 + < + 92 + 9< + + 95 (B) : (M) Fafo ag tloso

. Cot s9, s4, s2do tlo suks ag f torks ag tlroo soreos ef I.^., tlo gerst tork ag oibl doefj 9 ifm tlo bakkaf

meggorofbos 9, 4, 2 rospobtevocy. Eg s9+ s2; 4s , tlof tlo vicuo ag es 8


99. Ruk ag gerst ? torks ag if I.^. es afo gaurtl ag tlo suk ag foxt gevo torks. Eg tlo gerst tork ; 4, tlof tlo

bakkaf meggorofbo ag tlo I.^. es 8



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94. Eg x, y, z iro ef I.^., tlof tlo vicuo ag (x + y z) ( y + z x) es oquic ta 8

(I) wlebl iro mevesedco dy : es 8

(I) 9=4 (D) 94< (B) 94: (M) Fafo ag tloso

9=. Eg tlo fukdors i, d, b, m, o gark if I.^. tlof tlo vicuo ag i =d + 7b =m + o es oquic ta 8

(I) 9 (D) 4 (B) > (M) Fafo ag tloso

9?. Eg sfmofatos tlo suk ag gerst f torks ag if I.^., wlaso bakkaf meggorofbo es m, tlof sf 4sf-9+ sf-4(f 6 4) es oquic ta 8

(I) 4m (D) m (B) m (M) Fafo ag tloso

97. ]lo suk ag icc 4-mejetom fukdors wlebl coivo rokiefmor 9 wlof mevemom dy 2 es8

(I) 9797 (D) 97>4 (B) 97>? (M) Fafo ag tloso

9:. ]lo gerst tork ag if I.^. ag bafsobutevo eftojors es p4+ 9. ]lo suk ag 4p + 9 torks ag tles soreos bif do

oxprossom is 8

(I) (p + 9)4 (D) (4p + 9) (p + 9)

4 (B) (p + 9)

2 (M) p2 + (p + 9)

2

9>>>? d9>>>>?;

(I) 9.44 (D) 9.44 (B) 9.4 (M) 9.>4

ADHOB]EWO IFRZOT @OU O\OTBERO 9Yuo. 9 4 2 = ? 7 : < 5 9> 99 94 92 9= 9?

Ifs. I B I D I D B I I D D I B B B

Yuo. 97 9: 9< 95 4>

Ifs. B M B D I


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49

O\OTBERO 4 (GAT RBLAAC/DAITM O\IKR

R_DHOB]EWO ]U^O Y_OR]EAFR

WOTU RLAT] IFRZOT ]U^O9. Zreto tlo gerst gevo torks ag oibl ag tlo soquofbos, wlaso ftl torks iro8

(e)7

)94()9(

fi

f

f (ee)f

ff

i )9(9

(eee)

=

?4

ffif (ev) if; (-9)

f-9?

f+9

4. Gefm tlo efmebitom torks ef oibl ag tlo gaccawefj soquofbos wlaso ftl torks iro8

(e) if; 4f+ f

20 i

2 (ee)

f

ffif

94 ,i9>(eee) if; (-9)

f-9f

20 i5(ev) if; (f 9) (4 f) (2 + f) 0 i 4>

2. Zreto tlo gerst gevo torks ag oibl ag tlo gaccawefj soquofbos ifm adtief tlo barrospafmefj soreos.(e) i9; 9, if; if-9+ 4, f 4 (ee) i9; =, if+9; 4fif

=. Zreto tlo gerst tork i ifm tlo bakkaf meggorofbo m ag tlo I^ 8 -?, -9, 2, :.?. Zreto tlo gerst tork i ifm tlo bakkaf meggorofbo m ag tlo I^ 8 -9.9, -2.9, - ?.9, - :.9

7. Zreto tlo iretlkoteb prajrosseaf wlof gerst tork i ; - 9 ifm bakkaf meggorofbo m ;49 .

:. Zreto tlo iretlkoteb prajrosseaf wlof gerst tork i ; - 9.? ifm bakkaf meggorofbo m ; - >.?.

. Gefm aut wlotlor tlo soquofbo 2, 2, 2, 2, es if I^. Eg et es, gefm aut tlo bakkaf meggorofbo.99. Gefm tlo bakkaf meggorofbo ifm wreto tlo foxt twa torks ag tlo I^ 8 9., 4.4, 4.=94. Gefm aut wlotlor tlo soquofbo 9

4, 2

4, ?

4:

4, es if I^. Eg et es, gefm aut tlo bakkaf meggorofbo.

92. Gefm tlo bakkaf meggorofbo ifm wreto tlo foxt twa torks ag tlo I^ >,=

2,

4

9,

=

9.

9=. Gefm tlo 94 i tork ag tlo I^ 8 2, ? 19. Gefm tlo 9=tl tork ag tlo I.^. 5, ?, 9, - 249. Gefm tlo ftl tork ag tlo soquofbo k 9, k 2, k ?,44. Es 9?> i tork ag tlo I.^. 99, 91


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44

7. Gefm f eg tlo jevof vicuo ag x es tlo ftl tork ag tlo I^ 8 ,44,4

997,99,

4

9? .0 x ; ??>.

:. Zlebl tork ag tlo I^ 8 2, 9>, 9:,. wecc do i4>gar tlo I^ 8 i, i + m, i + 4m, i + 2m,.

49. ]wa iretlkoteb prajrosseafs livo tlo siko bakkaf meggorofbo. ]lo meggorofbo dotwoof tloer 9>>tl torks es 9>>wlit es tlo meggorofbo dotwoof tloer 9>>>tl torks1

44. Gefm tlo tork ag tlo iretlkoteb prajrosseaf 5, 94, 9?, 9> wlebl iro mevesedco dy 2 es ifm 7? rospobtevocy, wlit es tlo suk ag gerst 4> torks12>. Gefm tlo suk ag tlo gerst 4? torks ag if I^ wlaso ftl torks es jevof dy if; 4 2f.

29. Eg x, x + 9> ifm 2x + 4 iro ef I.^., gefm tlo vicuo ag x.

24. Eg x + 9, 2x ifm =x + 4 iro ef I.^., gefm tlo gegtl tork ag I.^.

22. Egdiibbd

9,

9,

9iro ef I.^., tlof pravo tlit 4d4; i4+ b4.

2=. Eg i,99

ff

ff

di

diifm d iro ef I.^. ifm i d, tlof gefm tlo vicuo ag f.

2?. Gefm tlo ftl tork ifm 9>>tltork ag tlo soquofbo : + 2 9 ?.

27. Zlebl tork ag tlo I.^., 2,


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42

==. Ef if I.^., tlo gaurtl tork oxbooms gaur tekos tlo 94tl tork dy afo ifm tlo tlerm tork oxbooms twebo tlo toftl

tork dy gevo, gefm tlo I.^.

=?. Motorkefo tlo I.^. wlaso gaurtl tork es 9? ifm tlo meggorofbo ag 7tl tork grak 9>tl tork es 97.

=7. ]lo =tl tork ag if I.^. es zora, pravo tlit ets 4?tl tork es trepco ets 99tltork.

=:. Eg tlo ptl tork ag if I.^. esq

9, ifm qtl tork ag if I.^. es

q

9, tlof slaw tlit ets (pq)tl tork es 9.

=< . ]lo suk ag tlroo fukdors ef I.^. es 49 ifm tloer pramubt es 429. Gefm tlo fukdors.

=5. ]lo suk ag tlroo fukdors ef I.^. es 2 ifm tloer pramubt es 2?. Gefm tlo fukdors.

?>. Mevemom 4> efta gaur pirts wlebl iro ef I.^. subl tlit tlo pramubt ag tlo gerst ifm gaurtl ifm tlo pramubt ag tlo

sobafm ifm tlerm es ef tlo ritea 4 8 2.

?9. Eg tlo suk ag gerst f torks ag if I.^. es jevof dy sf; ?f4+ 2f, gefm tlo ftl tork ag tlo I.^.

?4 ]lo suk ag tlo gerst 5 torks ag if I.^. es torks es =>>. Gefm tlo gerst tork, tlo

bakkaf meggorofbo ifm tlo suk upta 9?tltork

?2. ]lo suk ag f torks ag twa iretlkoteb prajrosseafs iro ef tlo ritea (2f + ifm ftl tork es 9>, tlof slaw tlit suk ag ets gerst (k + f) torks es

fk

fk 9>2>

4.

?:. Eg s9, s4, s2 iro tlo suk ag f torks ag tlroo iretlkoteb prajrosseafs, tlo gerst tork ag oibl doefj ufety ifm tlo

rospobtevo bakkaf meggorofbo doefj 9, 4, 20 pravo tlit s9+ s2; 4s4.

? torks.

?5. Eg ef if I.^., tlo suk ag 94 torks es oquic ta 9< ifm tlo suk ag 9< torks. es oquic ta 94, tlof pravo tlit tlo suk a

2> torks es 2>.

7>. Gefm tlo suk ag icc twa mejet fituric fukdors wlebl iro mevesedco dy =.

79. Gefm tlo suk ag icc 2-mejet fituric fukdors wlebl iro mevesedco dy 92.74. Gefm tlo suk ag icc 4-mejet fukdors wlebl wlof mevemom dy ? coivo rokiefmor 9.

72. Gefm tlo suk ag icc kuctepcos ag 5 cyefj dotwoof 2>> ifm :>>.

CAFJ IFRZOT ]U^O Y_OR]EAF

9. Eg ptl, qtl ifm rtl torks ag if I^ iro i, d, b rospobtevocy, tlof slaw tlit 8

(e) i(q r) + d (r p) + b (p q) ; >, (ee) (i d)r + (d b)p + (b i)q ; >

4. Ef i jirmof dom, tloro iro 42 raso pcifts ef tlo gerst raw, twofty afo ef tlo sobafm raw, fefotoof ef tlo tlerm raw

ifm sa af. ]loro iro gevo pcifts ef tlo cist raw. Law kify raws iro tloro ag raso pcifts1

2. Eg tlo ktl tork ag if I^ esf

9ifm tlo ftl tork es

k

9, slaw tlit tlo suk ag kf torks es

4

9(kf + 9).

=. Eg tlo suk ag k torks ag if I^ es tlo siko is tlo suk ag ets f torks, slaw tlit tlo suk ag ets (k + f) torks es

zora.

?. ]lo suk ag tlo gerst p,q,r torks ag if I^. iro i,d,b rospobtevocy. Rlaw tlit8

>)()()( qpr

bpr

q

drq

q

i


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4=

7. I kifugibturor ag rimea sots pramubom 7>> ufets ef tlo tlerm yoir ifm :>> ufets ef tlo sovoftl yoir. Issukefj tli

tlo pramubt efbroisos ufegarkcy dy i gexom fukdor ovory yoir, gefm (e) tlo pramubteaf ef tlo gerst yoir

(ee) tlo tatic pramubt ef : yoirs ifm (ee) tlo pramubt ef tlo 9>tl yoir.

:. I kif es okpcayom ta bauft Ts. 9>:9>. Lo baufts it tlo rito ag Ts. 9 por kefuto gar licg if laur. Igtor tles lo

baufts it tlo rito ag Ts 2 coss ovory kefuto tlitf tlo probomefj kefuto. Gefm tlo teko tiòf dy lek ta bauft tlo

oftero ikauft.> dy => iffuic efsticckofts wlebl gark if iretlkoteb soreos.

Zlof 2> ag tlo efsticckofts iro piem, lo meos coivefj afo-tlerm ag tlo modt ufpiem, gefm tlo vicuo ag tlo gerst

efsticckoft.

5. I cimmor lis rufjs 4? bk ipirt. ]lo rufjs mobroiso ufegarkcy ef cofjtl grak =? bk it tlo dattak ta 4? bk it tlo

tap. Eg tlo tap ifm dattak rufjs iro 4.? kotro ipirt, wlit es tlo cofjtl ag tlo waam roquerom gar tlo rufjs1

9>. ]lo mejets ag i pasetevo eftojor, livefj tlroo mejets, iro ef I.^. ifm tloer suk es 9?. ]lo fukdor adtiefom dy

rovorsefj tlo mejets es ?5= coss tlif tlo arejefic fukdor. Gefm tlo fukdor.

99. ]wa birs stirt tajotlor ef tlo siko merobteaf grak tlo siko pcibo. ]lo gerst jaos wetl ufegark spoom ag 9> `k/l

]lo sobafm jaos it i spoom ag < `k/l ef tlo gerst laur ifm efbroisos tlo spoom dy 4

9

`k oibl subboomefj laurIgtor law kify laurs wecc tlo sobafm bir avortiò tlo gerst bir eg datl birs ja faf stap1

94. I kif ropiys i caif ag Ts. 24?> dy piyefj Ts. 4> ef tlo gerst kaftl ifm tlof efbroisos tlo piykoft dy Ts. 9?ovory kaftl. Law cafj wecc et tiò lek ta bcoir tlo caif1

92. 9?> waròrs woro ofjijom ta gefesl i peobo ag war` ef i bortief fukdor ag miys. Gaur waròrs mrappom tlo sobafmmiy, gaur karo waròrs mrappom tlo tlerm miy ifm sa af. Et tiòs < karo miys ta gefesl tlo war` faw. Gefm tlofukdor ag miys ef wlebl tlo war` wis bakpcotom.

9=. Icafj i raim ceo if amm fukdor ag stafos pcibo it eftorvics ag 9> kotros. ]loso stafos livo ta do issokdcomiraufm tlo kemmco stafo. I porsaf bif birry afcy afo stafo it i teko. I kif birreom tlo had wetl afo ag tlo ofmstafos dy birryefj tlok ef subbosseaf. Ef birryefj icc tlo stafos lo bavorom i mestifbo ag 2 `k. Gefm tlo fukdorag stafos.

9?. ]lo eftorear ifjcos ag i pacyjaf iro ef iretlkoteb prajrosseaf. ]lo skiccost ifjco es 94>a, ifm tlo bakkafmeggorofbo es ?a. Gefm tlo fukdor ag semos ag tlo pacyjaf.

97. I kif sivom Ts. 97?>> ef tof yoirs igtor tlo gerst lo sivom Ts. 9>> karo tlif lo mem ef tlo probomefj yoir. Lawkubl mem lo sivo ef tlo gerst yoir1

9:. I kif irrifjos ta piy agg i modt ag Ts. 27>> dy => iffuic efsticckofts wlebl gark if iretlkoteb soreos. Zlof 2>ag tlo efsticckofts iro piem, lo meos coivefj afo-tlerm ag tlo modt ufpiem, gefm tlo vicuo ag tlo gerst efsticckoft.

9>.>>>. Eg et moprobeitos ef vicuo, 9?% tlo gerst, 92.?% tlo foxyoir, 94% tlo tlerm yoir ifm sa af. Zlit wecc do ets vicuo it tlo ofm ag 9> yoirs, icc porboftijos ippcyefj ta tloarejefic bast1


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4?

ITE]LKO]EB TAJTORREAFR IFRZOT @OU O\OTBERO 4 (\)BDRO

WOTU RLAT] IFRZOT ]U^O Y_OR]EAFR

9.(e)7

99

4

2,

7

:,

7

?,

4

9 ifm (ee)

?

=

=

?,

2

4,

4

2,>

ifm (eee)

4

:?49,

4

49,

4

5,

4

2ifm (ev) 4?, - 94?, 74?, - 294? ifm 9?74?

4. (e) 2? (ee)9>59 (eee) :45 (ev) : 99.m ; >.4, i?; 4.7, i7; 4.> :.4?tl .= 99.95 94.f4 92.4=75?>

9=.4

4>52 9?.?4? 97.>

44.=5tl 42. 9, :, 92

4=.4, =, 7 ar 7, =, 4 47.9>. 4:.2 2>. 54? 29.x ; 5 24.4= 2=.9

2?.99 =f, - 2 f 4 ?4.i ; 9, m ; 4, s9?; 44? ?2.(e) : 8 97 (ee) 5? 8 49< ? 7>.99 raws 7. (e)??> (ee)=2:? (eee)::? :.>


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47

O\OTBERO 2 (GAT RBLAAC/DAITMR O\IKR)

^OTWEA_R UOITR DAITM (BDRO) Y_OR]EAFR

9. ]lo ftl tork (tf) ag if Iretlkoteb prajrosseaf es jevof dy tf; :f + 9. Gefm tlo suk ag tlo gerst 2> torks

ag Iretlkoteb prajrosseaf. PGaroejf 4>>=[

4. ]lo 9>tl tork ag if Iretlkoteb prajrosseaf (I.^.) es ?: ifm ets 9?tl tork es >=[

2. Eg tlo suk ag gerst f torks ag if I.^. es jevof dy Rf; 2f4+ 4f, gefm tlo ftl tork ag tlo I.^.

AT

Eg k tekos tlo ktl torks ag if I.^. es oquic ta f tekos ets ftl tork, gefm ets (k + f)tl tork.

PMocle-4>>=B[

=. Law kify torks ag tlo I.^.2,?,:, kust do ti`of sa tlit tlo suk es 94>1 PMocle-4>>=B[

?. Eg tlo suk ag gerst f torks ag if I.^. es jevof dy Rf;=f4 2f, gefm tlo ftl tork ag tlo I.^.

PMocle-4>>=B[

7. Eg tlo suk ag gerst ag if I.^. es jevof dy Rf; 4f4+ ?f, gefm tlo ftl tork ag tlo I.^. PMocle-4>>=B[

:. Gefm tlo suk ag gerst 9? torks ag if I.^. wlaso ftl tork es 5 ?f.

AT

Eg tlo suk ta gerst f torks ag if I.^. es jevof dy Rf;?f4+2f, gefm tlo ftl tork ag tlo I.^.

PIE-4>>=B[

tl

tork grak ofm ag if I.^. =, 5, 9=,, 4?= PMocle-4>>?[

5. Gefm tlo fukdor ag torks ag tlo I.^. ?=, ?9, =>?[

9>. Gefm tlo suk ag icc twa mejets amm pasetevo fukdors. PIE-4>>?[

99. ]lo >?[

92 Eg gegtl tork ag tlo I.^. es zora, slaw tlit ets 22rm tork es gaur tekos ets 94tl tork PGaroejf-4>>?[

9=. Zlebl tork ag tlo I.^. ?, 5, 92,... es ?B[

9?. ]lo suk ag gerst f torks ag if I.^. es jevof dy (f4+ 2f). Gefm tlo 4>tl tork ag tlo prajrosseaf.

PMocle-4>>?B[

97. Gefm tlo suk ag tlo gerst ?9 torks ag tlo I. . wlaso 4fm tork es 4 ifm =tl tork es >?B[

9:. ]lo suk ag tlo gerst f torks ag if I.^. es jevof dy Rf; 2f4 f. Motorkefo tlo I.^. ifm ets 4?

tltork.

AT

]lo suk ag tlroo fukdors ef I.^. es 4: ifm tloer pramubt es =>?. Gefm tlo fukdors. PIE-4>>?B[


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KI]LOKI]EBR

4:

9 ifm tlo 9>tl tork es 47 Motorkefo tlo 9?tl

tork ag tlo I.^. PMocle-4>>7[

95. Gefm tlo suk ag icc tlo fituric fukdors coss tlif 9>> wlebl iro mevesedco dy 7. PIE-4>>7[

4>. Law kify torks iro tloro ef I.^. wlaso gerst tork ifm 7tl tork iro 94 ifm < rospobtevocy ifm suk ag

icc ets torks es 94>149 _sefj I.^., gefm tlo suk ag icc 2-mejet fituric fukdors wlebl iro kuctepcos ag :. PMocle-4>>7B[

44. Ef if I.^. tlo suk ag gerst f torks es4

2

4

?4

ff Gefm ets 4>tl tork. PIE-4>>7B[

42. Gefm tlo suk ag gerst 4? torks ag if I.^. wlaso ftl tork es 9 =f. PMocle-4>>:[

4=. Zlebl tork ag tlo I.^. 2, 9?, 4:, 25, wecc do 924 karo tlif ets ?=tl tork1 PMocle-4>>:[

4?. Ef if I.^., tlo suk ag ets gerst f torks es f4+ 4f. Gefm ets 9>:[

47. ]lo gerst tork, bakkaf meggorofbo ifm cist tork ag if I.^. iro 94, 7 ifm 4?4 rospobtevocy. Gefm tlo suk

ag icc torks ag tles I.^. PIE-4>>:[

4:. ]lo ftl tork ag if I.^. es : =f. Gefm ets bakkaf meggorofbo. PMocle-4>>>>. Gefm tlo 9>tl tork grak tlo ofm ag tlo I.^. , 94,., 947 PMocle-4>>>tl torks es. PIE-4>>, 9: ,. oquic1

ATEg k tekos tlo ktl tok ag if I.^. es oquic ta f tekos ets ftl tork, gefm tlo (k + f)tl tork ag tlo I.^.

PGaroejf-4>>>>5[25. Eg Rf, tlo suk ag gerst f torks ag if I.^. es jevof dy Rf; 2f

4 =f, tlof gefm ets ftl tork. PMocle-4>>5[

=>. ]lo suk ag =tl ifm tl torks es ==. Gefm I.^.

PMocle-4>>5[

=9. Eg Rftlo suk ag gerst f torks if I.^. es jevof dy Rf; ?f4+ 2f, tlof gefm ets ftl tork. PMocle-4>>5[


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4>5[

=2. Eg?

=, i, 4 iro tlroo bafsobutevo torks ag if I.^., tlof gefm tlo vicuo ag i. PIE-4>>5[

==. Zlebl tork ag tlo I.^. 2, 9?, 4:, 25, wecc do 94> karo tlif ets 49str tork1 PIE-4>>5[

=?. ]lo suk ag gerst sex torks ag if iretlkoteb prajrosseaf es =4. ]lo ritea ag ets 9>tl tork ta ets 2>tl tork es

9 8 2. Bicbucito tlo gerst ifm tlo tlertooftl tork ag tlo I.^. PIE-4>>5[

=7. Zlebl tork ag tlo I.^. =, 94, 4>, 4 karo tlif ets 49st tork1 PIE-4>>5[

=:. Gar wlit vicuo ag , iro tlo fukdors x, 4x + ` ifm 2x + 7 tlroo bafsobutevo torks ag if I.^.1

PGaroejf-4>>5[

=tl tork dy :. Gefm tlo bakkaf meggorofbo. PGaroejf-4>>5[

=5. Eg 5tl tork ag if I.^. es zora, pravo tlit ets 45tl tork es maudco ag ets 95tl tork. PGaroejf-4>>5[

?>. Eg ?tl tork ag if I.^. es zora, pravo tlit ets 42rm tork es tlroo tekos ets 99tl tork. PGaroejf-4>>5[

?9. Eg tlo :tl tork ag if I.^. es zora, pravo tlit ets 4:tl tork es gevo tekos ets 99tl tork. PGaroejf-4>>5[

R_DHOB]EWO IFRZOT @OU O\OTBERO-= (\)-BDRO

9. 24


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45

O\OTBERO = (GAT ACUK^EIMR)

BLARRO ]LO BATTOB] AFO

9. Gar i soreos wlaso ftl tork es yx

f , tlo suk ag r torks es 8

(I) ryx

rr 4

)9( (D)xrr4

)9( (B) ryx

rr 4

)9( (M) rxy

rr 4

)9(

4. Egbdi

9,

9,

9iro ef I.^., tlof

ibdbdi

999999es oquic ta 8

(I)4

2=

dib (D)

444

4

bdi

ibd (B)

4

9=

dib (M) Fafo ag tloso

2. ]lo suk ag gerst 4= torks ag if I.^. i9, i4, i2,.0 eg et es `fawf tlit i9+ i?+ i9>+i9?+ i4>+ i4=; 44?es oquic ta 8(I) 5> (D) 9 (B) 5>> (M) 9>

=. I stumoft roim bakkaf meggorofbo ag if I^ es 4 efstoim ag 4 ifm jat tlo suk ag gerst gevo torks is ?]lo ibtuic suk ag gerst gevo torks es 8(I) 4? (D) 4? (B) 2? (M) 2?

?. ]lo suk ag f torks ag twa I.^s iro ef tlo ritea ag (:f + 9) 8 (=f + 4:). ]lo ritea ag tloer 99tl torks es (I) 4 8 2 (D) = 8 2 (B) ? 8 = (M) ? 8 7

7. Eg 94+ 4

4+ 2

4+ .f

4; 9>9?, tlof tlo vicuo ag f es 8

(I) 92 (D) 9= (B) 9? (M) Fafo ag tloso

:. ]lo suk ag tlo soreos7

9

2

9

4

9 + upta 5 torks es8

(I)7

? (D)

4

9 (B) 9 (M)

4

2

iro ef I.^., tlof tloer bakkaf meggorofbo wecc do8

(I) 9 (D) 4 (B) 2 (M) =9>. Eg I.K dotwoof twa fukdors es ? ifm tloer J.K. es =, tlof tloer L.K. es8

(I)?

97 (D)

?

9= (B)

?

99 (M) Fafo ag tloso

99. Eg I es tlo sefjco I.K. dotwoof twa fukdors i ifm d ifm R es tlo suk ag f I.K.s dotwoof tlok, tlof

I

Rmopofms upaf8

(I) f, i, d (D) f, i (B) f, d (M) f94. Eg tlo I.K. dotwoof tlo raats ag i quimriteb oquiteaf es < ifm tlo J.K. es ?, tlof tlo oquiteaf es8

(I) x4+ 9>x 4? ; > (D) x

4

(B) x4 97x + 4? ; > (M) x

4 97x 4? ; >

92. Eg b es tlo lirkafeb koif dotwoof i ifm d, tlof

d

b

i

b es oquic ta8

(I) 4 (D)id

di (B)

di

id

(M) Fafo ag tloso

9=. Eg i,d,b,m,o,g iro ef I.^. tlof o-b es oquic ta 8(I) 4(b i) (D) 4(g m) (B) 4(m b) (M) m b


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2>

9?. 4>tl tork ag tlo soreos 8?29

249

29

49

9

9 222222

+ es 8

(I)=

==9 (D)

4

==2 (B)

4

==? (M)

4

=25

97. Eg tlo vicuo ag 92+ 4

2+ 2

2+ + f

2; 4>4?, tlof tlo vicuo ag 9 + 4 + 2 + + f es8

(I) 7:? (D)


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29

4. ]lo suk ag gerst gaur torks ag if I.^. es ?7 ifm suk ag cist gaur torks es 994. Eg tlo gerst tork es 99, tlof

tlo fukdor ag torks es 8(I) 9> (D) 94 (B) 99 (M) Fafo ag tloso

29. Gar if I.^., R4f; 2Rf. ]lo vicuo agf

f

R

R2 es oquic ta 8

(I) = (D) 7 (B) < (M) 9>24. ]lo ritea ag tlo :tlta tlo (f 9)tl koif dotwoof 9 ifm 29, wlof f iretlkoteb koifs iro efsortom dotwoof tlof, es

? 8 5. ]lo vicuo ag f es 8(I) 94 (D) 92 (B) 9= (M) 9?

22. ]lo gerst, sobafm ifm cist torks ag if I.^. iro i,d, ifm 4i rospobtevocy, tlo suk ag tlo soreos es 8

(I))(4

2

id

id

(D)

)(4

2

id

id

(B)

)(4

2

di

id

(M) Fafo ag tloso

2=. Ruk ag gerst k torks ag if I.^. es >. Eg i do tlo gerst tork ag tlo I.^., tlof tlo suk ag foxt f torks es 8

(I)9

)(

k

kfki (D)

9

)(

k

ffki (B)

9

)(

f

ffki (M)

9

)(

f

kfki

2?. Eg I9ifm I4do tlo twa I.K.s dotwoof twa fukdors i ifm d, tlof I4 I9es oquic ta

(I) i + d (D) d i (B)2

id (M) Fafo ag tloso

27. ]lo suk ag torks oquemestift grak tlo dojeffefj ifm ofm ef if I.^. es oquic ta 8(I) Cist tork (D) Gerst tork(B) Ruk ag tlo gerst ifm tlo cist tork (M) Fafo ag tloso

2:. Eg tlo suk ag tlo raats ag tlo oquiteaf ix4

+ dx + b ; > es oquic ta tlo suk ag tlo squiros ag tloer robeprabics tlofdb4, bi4, id4iro ef8(I) I.^. (D) J.^. (B) L.^. (M) Fafo ag tloso

2 gar icc E, tlof 8

ff iiiiii

92449

9...

99;

(I)

fii

f

9 (D)

9ii

f

f (B)

fii

f

9

9 (M) Fafo ag tloso

25. Eg i,d,b iro ef I.^. ifm icsabdi

9,

9,

9iro ef I.^., tlof8

(I) i ; d b (D) i d ; b (B) i ; d ; b (M) i d b

=>. Eg i,d,b iro ef L.^., tlof bd

di

oquics 8

(I)i

d (D)

d

i (B)

b

i (M) Fafo ag tloso

=9. If, I.^. bafsests pg f (amm) torks ifm ets kemmco tork es k. ]lof tlo suk ag tlo I.^. es 8

(I) 4kf (D) kf4

9 (B) kf (M) kf4


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24

=4. Eg I,J ifm L mofato rospobtevocy tlo I.K., J.K. ifm L.K. dotwoof twa pasetevo fukdors i ifm d, tlof I-J esoquic ta 8

(I) i d (D)di

id

4 (B)

4)(4

9di (M) Fafo ag tloso

=2. Eg tlo raats ag tlo oquiteaf x2 94x4+ 25x 4< ; > iro ef I.^., tlof tloer bakkaf meggorofbo es 8(I) 9 (D) 4 (B) 2 (M) =

ADHOB]EWO IFRZOT @OU O\OTBERO =

Yuo. 9 4 2 = ? 7 : < 5 9> 99 94 92 9= 9?Ifs. I I B M D D M I B I M B I B IYuo. 97 9: 9< 95 4> 49 44 42 4= 4? 47 4: 4< 45 2>Ifs. B B B B D B I M I B B D M B BYuo. 29 24 22 2= 2? 27 2: 2< 25 => =9 =4 =2Ifs. D B D D B B I B B B B B B


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22

O\OTBERO ? (GAT EE] HOO/IEOOO)

BLAARO ]LO BATTOB] AFO

9. ]lo suk ag tlo f torks ag tlo soreos

4:

4

2

= + es 8 P@orici Ofjefoorefj-4>>2[

(I))2(4

9)94(2f

ff

(D))2(4

9)94(2f

ff

(B))2(4

92f

ff

(M)4

92 f

4. Eg tlo tlerm tork ag i J.^. es p, tlof tlo pramubt ag ets gerst ? torks es 8 P@orici Ofjefoorefj-4>>2[

(I) p2 (D) p

4 (B) p

9> (M) p

?

2. Eg i9, i4,..if iro f I.Ks dotwoof i ifm d, tlof

f

e

ei

9

4 ; P@orici Ofjefoorefj-4>>2[

(I) id (D) f (i + d) (B)id

dif )( (M)

f

di

=.>2[(I) x4 = ; > (D) x4 =x + 7 ; > (B) x4 ?x + = ; > (M) x4 2x+ 4 ; >?. Eg i,d,b iro ef I.^., tlof wlebl afo ag tlo gaccawefj es fat truo1 P@orici Ofjefoorefj-4>>2[

(I) i + `, d + `, b + ` iro ef I.^. (D) `i, `d, `b iro ef I.^.(B) i

4, d

4, b

4iro ef I.^. (M) i + d, b + i, d + b iro ef I.^.

7. ]lo suk ag tlo soreos844 9

9....

=2

9

24

9

49

9

ff

es oquic ta PIK_4>>4[

(I)f

f 94 (D)

9

9

ff

f (B)

f

ff

4

94 (M) f 9

:. Ruk ag efgefeto fukdor ag torks ag i J.^. es 4> ifm suk ag tloer squiros es 9>>. ]lo bakkaf ritea agtlo J.^. es 8 PIEOOO4>>4[

(I) ? (D)?

2 (B)

?

< (M)

?

9

>4[

(I) =4? (D) =4? (B) =:? (M) =:?5. Eg y ; x x

4+ x

2 x

=+ . ta , tlof tlo vicuo ag x wecc do (-9 3 x 39) 8

(I) y +y

9 (D)

y

y

9 (B) y

y

9 (M)

y

y

9

9>. ]lo twa joakotreb koifs dotwoof 9 ifm 7= iro8 P@orici Ofjefoorefj-4>>4[(I) 9 ifm 7= (D) < ifm 97 (B) = ifm 97 (M) 2 ifm 97

99. ]lo suk ag efgefeto torks ag tlo joakotreb prajrosseaf

4

9

,44

9,

94

94

,es 8

P@orici Ofjefoorefj-4>>4[

(I) 4944 (D) 494 (B) ? 4 (M) 2 ?4 94. Eg tlo ftl tork ag tlo joakotreb prajrosseaf, ,

9>4=

?,...

>4[(I) 99 (D) 9> (B) 5 (M) =


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2=

92. Ef i lirkafeb prajrosseaf, ptl tork es q ifm qtl tork es p, tlof tlo (pq)tl tork es8

(I)pq

qp (D) > (B)

qp

pq

(M) 9

9=. Ruppaso i,d,b iro I.^. ifm i4,d

4,b

4iro ef J.^. Eg i 3 d 3 b ifm i + d + b ;

4

20 tlof tlo vicuo ag i es8

PEE] Rbroofefj-4>>4[

(I)44

9 (D)

24

9 (B)

2

9

4

9 (M)

4

9

4

9

9?. Cot tlo pasetevo fukdors i,d,b,m do ef I.^., tlof idb, idm, ibm, dbm iro8 PEE]Rbroofefj-4>>9[(I) Fat ef I.^./J.^./L.^. (D) Ef I.^.(B) Ef J.^. (M) Ef L.^.

97. Eg tlo suk ag tlo gerst 4f torks ag tlo I.^. 4, ?, >9[(I) 9> (D)94 (B)99 (M) 92

9:. Eg :

9>...99


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2?

4=. ]lo suk ag gow torks ag i ritea soreos es :4; 2, tlof i=l:; PEE][4:. Ef i J.^., tlo gerst tork es i, sobafm tork es d ifm tlo cist tork es b, tlof suk ag tlo soreos es8 PIK_[

4. i,d,b iro tlroo ufoquic fukdors subl tlit i,d,b iro ef I.^. 0 d i, b d, i iro ef J.^. tlof i 8 d 8b 8 8

P@irfiti`i-BO][

(I) 9 8 4 8 = (D) 4 8 28 ? (B) 9 8 48 2 (M) 9 82 8 ?

29. ]lo gerst twa torks ag if efgefeto J.^. iro tajotlor oquic ta ? ifm ovory tork es 2 tekos tlo suk ag icc tlo

torks tlit gaccaw et 0 tlo bakkaf ritea ag tlo J.^. es 8 PIK_[

(I)2

9 (D)

=

9 (B) 2 (M) =

24. ]lo oejltl tork ag i J.^. es 94< ifm bakkaf ritea es 4. ]lo pramubt ag ets gerst gevo torks es8

(I) =7 (D) =

2 (B) =

? (M) =

=

22. ]lo suk ag efgefety ag =24 :4

:9

:4

:9 + es 8 PIK_[

(I)97

2 (D)

?

9 (B)

4=

9 (M)

97

9

2=. p,q,r iro ef I.^. ifm oibl es fukorebiccy coss tlif 9. Cot 8 P@irfiti`i-BO][

x ; 9 + p + p4+ ta

y ; 9 + q + q4+ ta

z ; 9 + r + r4+ . ta , tlof x,y,z iro ef

(I) I.^. (D) J.^. (B) L.^. (M) Fafo ag tloso

2?. Eg tlo fukdors p,q,r iro ef I.^., tlof k:p

, k:q

, k:r

(k 6 >) iro ef 8

(I) I.^. (D) J.^. (B) L.^. (M) Fafo ag tloso27. Eg tlo ptl, qtl ifm rtl torks ag i J.^. iro , k ifm f rospobtevocy, tlof

q-rk

r-pf

p-qes 8

(I) 9 (D) > (B) pqr (M) kf


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2:.)9(

9....

=2

9

24

9

49

9

ffxxxoquics 8 PIK_[

(I)f

f 9 (D)

7

)9( ff (B)

9f

f (M)

9

4

f

f

2. Eg ifdo tlo ftl tork ag i J.^. ag pasetevo fukdors ifm

9>>

9

94

9>>

9

4 ,f

f

f

f ii , subl tlit i , tlof

tlo bakkaf ritea ag tlo J.^. es 8 PEE][

(I)

(D)

(B)

=9. Eg p,q,r iro ef I.^. ifm x,y,z iro ef J.^, tlof xq-r

yr-p

zp-q

es oquic ta 8(I) p + q + r (D) xyz (B) 9 (M) px + qy +rz

=4. Eg i,d,b iro ef I.^., tlof 9>ix + 9>

, 9>dx + 9>

, 9>bx + 9>

, x > iro ef 8(I) I.^. (D) J.^. afcy wlof x 6 >(B) J.^. gar icc x (M) J.^. afcy wlof x 3 >

=2. Eg tlo suk ag raats ag tlo quimriteb oquiteaf ix4+ dx + b ; > es oquic ta tlo suk ag squiros ag tloer

robeprabics, tlofd

bifm

i

d

b

i, iro ef 8

(I) J.^. (D) L.^. (B) I.^. (M) Fafo ag tloso==. Cot , do tlo raats ag x

4 x + p ; > ifm , do tlo raats ag x

4 =x + q ; >. Eg , , iro ef J.^.

tlof eftojric vicuos ag p ifm q iro rospobtevocy.(I) 4, 24 (D) 4, 2 (B) 7, 2 (M) 7, 24

=?. Eg i,d,b,m ifm x iro icc roic ifm (i

4

+ d

4

+ b

4

) x

4

4 (id + db +bm) x + (d

4

+ b

4

+ m

4

) > tlof 8(I) i,d,b,m iro ef J.^. (D) i,d,b,m iro ef I.^. (B) i,d,b,m iro ef L.^. (M) Fafo ag tloso

ADHOB]EWO IFRZOT @OU O\OTBERO =Yuo. 9 4 2 = ? 7 : < 5 9> 99 94 92 9= 9?Ifs. D M D B B M D I M B I I M M MYuo. 97 9: 9< 95 4> 49 44 42 4= 4? 47 4: 4< 45 2>Ifs. B I D M B I D B D D M D B B BYuo. 29 24 22 2= 2? 27 2: 2< 25 => =9 =4 =2 == =?Ifs. D B I B D I B M B I B B D I I

6. arithmetic progressions

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