new stpm mathematics (t) chapter past year question

7/22/2019 New STPM Mathematics (T) Chapter Past Year Question

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EHY R_\D D`tmhd`tkbs (_) _cpkb`j \`st [h`r Puhstkces

Bcdpkjho ny7 FF JHH (Jhh Fkè Fhcei)

@uiust =, >

> Rhquhebhs èo Rhrkhs 08

2 D`trkbhs >?

? Bcdpjhx Eudnhrs 22

8 @e jytkb Ihcdhtry 21

1 Qhbtcrs 2=

9 Jkdkts èo Bcetkeukty ??

= Okhrhetk`tkce ?=

; Kethir`tkce 1?

0< Okhrhetk`j Hqu`tkces 9;

00 D`bjùrke Rhrkhs =;

0> Eudhrkb`j Dhtmcos ;1

02 O`t` Ohsbrkptkce ;;

0? \rcn`nkjkty 0>x|5 x. V? d`rfsT

V@eswhr7{x7 ske(x ), >x2.

(`) Rfhtbm tmh ir`pm caa. V? d`rfsT

(n) Akeo tmh rèih caa. V2 d`rfsT

(b) Ohthrdkeh wmhtmhraks ` ceh-tc-ceh auebtkce. Ikvh rh`sces acr ycur èswhr. V> d`rfsT

V@eswhr7 (n){y7 > y :>>} 6 (b) aks ect ceh-tc-ceh auebtkce.T

?. VR_\D TRcjvh tmh hqu`tkce

>jcix2 jci2

x42

>.

V1 d`rfsT

V@eswhr7 x4 2, 0

=0T

8. VR_\D T_mh auebtkce aks ohffeho `s acjjcws7

a 7x

? + (x

0)>, x

^.

(`) Rfhtbm tmh ir`pm caa. V> d`rfsT

(n) Rt`th tmh rèih caa. V0 d`rfsT

(b) Ohthrdkeh kaa0 hxkst. V> d`rfsT

V@eswhr7 (n){y7 y?} 6 (b) EcT

1. VR_\D TIkvhe tm`t x + > ks ` a`btcr caa(x) 4x2 + (` + >n)x> + (` 2n)x + =. Akeo ` ke thrds can, èo ffeoq(x) sc tm`t a(x) 4 (x + >)q(x) mcjos acr `jj v`juhs can. V8 d`rfsT

Ohthrdkeh tmh v`juhs can sc tm`t a(x) 4 < m`s `t jh`st twc okstkebt rh`j rccts. V1 d`rfsT

Rfhtbm ce okhrhet okìr`d, tmh ir`pm cay 4 a(x) wmhe n 418

èo n 4>

8. V? d`rfsT

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ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) 07 Auebtkces

V@eswhr7 `49n6 q(x) 4x> (8n + >)x + ? 6{n7 n :18

, n >8}T

9. VR_\D T

Auebtkce a ks ohffeho ny a(x) 4 0

x wktm x ^ èo x4 . Akeo tmh v`juhs cad èo e. V8 d`rfsT

V@eswhr7 d4 (08 >x x>). Akeo tmh rèih caxsc tm`t a(x) ks ohffeho. V2 d`rfsTAkeo tmh d`xkdud v`juh ca 08 >x x> èo mhebh ohoubh tmh d`xkdud v`juh caa(x). V? d`rfsT

V@eswhr7{x78: x :2} 6 01, ?T

0


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(n) Akeo a02(>). V2 d`rfsT

V@eswhr7 (`)a>(x) 4x , a0(x) 48x + >

x 8 6 (n)?T

0?. VR_\D TRmcw tm`t tmh rccts cax> + nx + b4 ?`b

>` .

Ohoubh tm`t kad + ek, wktm d, e ^, ks ` rcct ca tmks hqu`tkce, tmhe d ek ks èctmhr rcct. V8 d`rfsT

(`) Rmcw tm`t >+k ks ` rcct caa(x) 4 < wmhrh a(x) 4 >x2 8x> >x + 08, èo ffeo kts ctmhr rccts.V8 d`rfsT

(n) Akeo ` pcjyecdk`j i (x) sc tm`t a(x) xi(x) 4 08 9x. Hxprhss i(x) ke tmh acrd p(x q)> + r,

wktm p, q, r ^, ffeo tmh d`xkdud ca 0

i(x) . V8 d`rfsT

V@eswhr7 (`) >-k ,2>

6 (n)i (x) 4 >

x 8

?

>+

08

= ,

=

08T

08. VR_\D T_mh auebtkce a ks ohffeho ny

a(x) 4

> |x 0|, x :2,x> ;x + 0=, x2.

(`) Rfhtbm tmh ir`pm caa. V8 d`rfsT

(n) Ohthrdkeh tmh sht caxsc tm`t a(x)5 0 x1

. V8 d`rfsT

V@eswhr7 (n){x7 8

, x 51}T

01. VR_\D THxprhss ; ske 1bcs ke tmh acrd r ske( ), wktmr 5< èo + 1x + 02) T

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0=. VR_\D T

Hxprhss 0 >x

x>(0 + >x>)`s p`rtk`j ar`btkces. V8 d`rfsT

V@eswhr7

>

x

+ 0

x>

+ ?x >

0 + >x>

T

0;. VR_\D T

Hxprhss tmh auebtkce a 7x |0>

x 0| + |0>

x + 0|, x ^, ke tmh acrd tm`t ochs ect kevcjvh tmh dcoujusskie. Rfhtbm tmh ir`pm caaèo ohthrdkeh kts rèih. V9 d`rfsT

V@eswhr7 a 7x

x, x :>>, >x :>x, x>

,

^4{y7 y>}T

>e (p + 0)x> +p, wmhrh e èo p `rh pcsktkvh kethihrs.Rmcw tm`tx 0 ks ` a`btcr caa(x) acr `jj v`juhs cap. V2 d`rfsTYmhe p 4 ?, x > ks ` a`btcr caa(x). Akeo tmh v`juh cae èo a`btcrksh a(x) bcdpjhthjy. V8 d`rfsTYktm tmh v`juh cae ycu m`vh cnt`keho, ffeo tmh sht ca v`juhs cap subm tm`t a(x) + >x> > 4 < m`srccts wmkbm `rh okstkebt èo rh`j. V9 d`rfsT

V@eswhr7 e4 > , (x >)(x + >)(x 0)(x + 0) 6{p7 p 5>, p4 2}T

>0. VR_\D T

Rcjvh tmh skdujtèhcus hqu`tkces

jci?(xy) 40

>, (jci> x)(jci> y) 4>.

V1 d`rfsT

V@eswhr7 x4 0

>, y4 ?, x 4 ?, y 4

0

>T

>>. VR_\D T_mh auebtkces a èo i `rh ohffeho ny

a 7x>x, x ^6i7 xbcs x | bcs x|, x.

(`) Akeo tmh bcdpcskth auebtkcea i èo st`th kts ocd`ke èo rèih. V? d`rfsT(n) Rmcw, ny ohffektkce, tm`ta i ks è hvhe auebtkce. V> d`rfsT(b) Rfhtbm tmh ir`pm caa i. V> d`rfsT

V@eswhr7 (`) a i7 x>(bcs x | bcs x|), O 4{x7x}, ^ 4{y7?y2. VR_\D T

_mh auebtkce a ks ohffeho ny a 7x 2x + 0, x ^, x 02

.

Akeo a0 èo st`th kts ocd`ke èo rèih. V? d`rfsT

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V@eswhr7 a0 7xx> 0

2 , Oa0 4{x7 x?. VR_\D T

Hxprhss 8; >?

1 `s p

> + q

2 wmhrh p èo q`rh kethihrs. V9 d`rfsT

V@eswhr7 ?

> 2

2T

>8. VR_\D TRmcw tm`t pcjyecdk`j >x2 ;x> + 2x + ? m`s x 0 `s a`btcr. V> d`rfsTMhebh,

(`) ffeo `jj tmh rh`j rccts ca >x1 ;x? + 2x> + ? 4 x2 ;x> + 2x + ?: 0> 0>x. V1 d`rfsT

V@eswhr7 x4 0, x4

0, x4 >, x4

> 6 x :0T

>1. VR_\D T

Hxprhss bcs x +

2ske xke tmh acrd r bcs(x ), wktm r 5< èo

. V? d`rfsT

Mhebh, ffeo tmh sht ca v`juhs caxwktm , wmkbm s`tksffhs tmh kehqu`jkty < : bcs x+

2ske x :0.V8 d`rfsT

V@eswhr7 >bcs

x 2

,{x7 >

2 : x :

8

1 ,

00

1 : x :>}T

>9. VR_\D T

Rmcw tm`t0 ks tmh cejy ceh rh`j rcct ca tmh hqu`tkce x2

+ 2x>

+ 8x + 2 4 =. VR_\D TAkeo tmh sht ca v`juhs cax subm tm`t>: x2 >x> + x >: ;. VR_\D T

Rfhtbm, ce tmh s`dh bccroke`th `xhs, tmh ir`pms cay 4 > x èo y 4> + 0x

. V? d`rfsTMhebh, scjvh tmh kehqu`jkty >

x 5 > +

0

x. V? d`rfsT

V@eswhr7{x7 x :>

8}T

2


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2>. VR_\D T

Akeo tmh scjutkce sht ca kehqu`jkty|x >|: 0x

wmhrh x4

hx + hx.

(`) Rt`th tmh ocd`kes caa èo i , V0 d`rfsT

(n) Yktmcut uskei okhrhetk`tkce, ffeo tmh rèih caa, V? d`rfsT

(b) Rmcw tm`t a(x)> + i(x)> 4 0. Mhebh, ffeo tmh rèih cai. V1 d`rfsT

V@eswhr7 (`){x7 x ^},{x7 x ^} 6 (n){y70: y :0} 6 (b){y7 x + 0

(x> + 0)(> x) ke tmh acrd @x + N

x> + 0 +

B

> x wmhrh @, N èo B `rh bcestèts. V2 d`rfsT

V@eswhr7 x

x> + 0+

0

> x T

28. VR_\D TAuebtkces a, i èo m `rh ohffeho ny

a 7x xx + 0

6 i7 xx + >x

6 m7 x2 + >x

.

(`) Rt`th tmh ocd`kes caa èo i . V> d`rfsT

(n) Akeo tmh bcdpcskth auebtkcei aèo st`th kts ocd`ke èo rèih. V8 d`rfsT(b) Rt`th tmh ocd`ke èo rèih cam. V> d`rfsT

(o) Rt`th wmhtmhrm 4 i a. Ikvh ` rh`sce acr ycur èswhr. V> d`rfsT

V@eswhr7 (`){x7 x ^, x40},{x7 x ^, x4

x, O4{x7 x ^, x4 + >y+ 0 4 , n4 0>, (x + 2)(x 0)(x + >)(x >) , y 402

, 0, 0>

,0

>T

29. VR_\D

THxprhss ? ske 2bcs ke tmh acrd ^ ske( ), wmhrh ^ 5 < èo


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V@eswhr7 8ske( 21.;), 4 92.9, 0= y

>}T

2;. VR_\D T

Ka jci`

x`>

4 2jci`> jci`(x >`), hxprhss x ke thrds ca`. V1 d`rfsT

V@eswhr7 x4 ?`T

?

>(

9 +

2), V2 d`rfsT

V@eswhr7 >

9 2

2T

?0. VR_\D

TAkeo tmh bcestèts @, N , B èo O subm tm`t

2x> + 8x

(0 x>)(0 + x)> 4 @

0 x + N

0 + x+

B

(0 + x)>+

O

(0 + x)2.

V= d`rfsT

V@eswhr7 @4 0, N4 0, B40, O40T

?>. VR_\D T

Xskei tmh sunstktutkce y 4 x +0

x

, hxprhssa(x) 4x2

?x

1

?

x

+ 0

x2`s ` pcjyecdk`j key . V2 d`rfsT

Mhebh, ffeo `jj tmh rh`j rccts ca tmh hqu`tkce a(x) 4 T

?2. VR_\D TAkeo, ke thrds ca , `jj tmh v`juhs cax nhtwhhe < èo wmkbm s`tksffhs tmh hqu`tkce

tè x + bct x4 =bcs >x.

V? d`rfsT

V@eswhr7 0

>?,

8

>?,

0>

>?,

09

>?T

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??. VR_\D T_mh auebtkce a èo i `rh ohffeho ny

a 7x 0x

, x ^ Z {x 0, x ^.Akeo a i èo kts ocd`ke. V? d`rfsT

V@eswhr7 a i(x) 4 0>x 0 , O4{x7 x ^, x4

0

>}T

?8. VR_\D T

_mh pcjyecdk j p(x) 4 >x2 + ?x> +0

>x f m`s a`btcr (x + 0).

(`) Akeo tmh v`juh caf . V> d`rfsT

(n) A`btcrksh p(x) bcdpjhthjy. V? d`rfsT

V@eswhr7 (`) f 42>

6 (n) 0

>(x + 0)(>x + 2)(>x 0)T

?1. VR_\D T

Akeo tmh scjutkce sht ca tmh kehqu`jkty

?x 0 52 2x . V0< d`rfsT

V@eswhr7{x7

, smcw tm`t ske 4 >t0 + t>

èo bcs 4 0 t>

0 + t>. V? d`rfsT

Mhebh, ffeo tmh v`juhs ca nhtwhhe

jci> `+ jci`(0 >`), wmhrh , ffeo x. V9 d`rfsT

V@eswhr7 >

2T

8 ;) 4>?. V; d`rfsT

V@eswhr7 x4 9, x4;T

80. VR_\D T_mh pcjyecdk jp(x) 4 1x?`x2nx> + >=x +0>, wmhrh` èo n `rh rh`j bcestèts, m`s a`btcrs (x + >)èo (x >).

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(`) Akeo tmh v juhs ca` èo n, èo mhebh, a`btcrksh p(x) bcdpjhthjy. V9 d`rfsT

(n) Ikvh tm`tp(x) 4 (>x 2)Vq(x) ? 0 + 2x2T, ffeo q(x), èo ohthrdkeh kts rèih wmhe xV>, 09 , (x + >)(x

>)(>x

2)(2x + 0) 6 (n) q(x) 4 (x

1)> + 0 , ^4V0,18TT

8>. VR_\D TAkeo tmh v`juhs cax, wmhrh tè x. V? d`rfsT

V@eswhr7 x4

y 4 >.V? d`rfsT

V@eswhr7 x4 >9, y4 ;T

8?. VR_\D T_mh ir`pm ca ` auebtkce aks `s acjjcws7

(`) Rt`th tmh ocd`ke èo rèih caa. V> d`rfsT

(n) Rt`th wmhtmhra ks ` ceh-tc-ceh auebtkce cr ect. Ikvh ` rh`sce acr ycur èswhr. V> d`rfsT

V@eswhr7 (`) O4{x72x :0 0: x>} , ^4{y70: y :>} 6(n) a ks ect ceh tc ceh auebtkce.T

88. VR_\D T_mh pcjyecdk`j p(x) 4 >x? 9x2 + 8x> +`x+ n, wmhrh ` èo n `rh rh`j bcestèts, ks okvksknjh ny>x> + x 0.

(`) Akeo ` èo n. V? d`rfsT

(n) Acr tmhsh v`juhs ca` èo n, ohthrdkeh tmh sht ca v`juhs cax subm tm`t p(x)

}T

0


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81. VR_\D THxprhss 8 ske + 0> bcs ke tmh acrd r ske(+ ), wmhrh r 5 < èo ,

0

>=T

89. VR_\D TRcjvh tmh hqu`tkce je x + je(x + >) 4 0. V? d`rfsT

V@eswhr70 + 0 + hT

8=. VR_\D T

Akeo tmh sht ca v`juhs cax s`tksaykei tmh kehqu`jkty >x 0 |x + 0|. V1 d`rfsTV@eswhr7{x7 x>}T

8;. VR_\D TAuebtkces a èo i `rh ohffeho ny

a 7x x>x 0 acr x4

0

>6

i7 x`x> + nx + b, wmhrh `, n èo b `rh bcestèts.

(`) Akeo a a, èo mhebh, ohthrdkeh tmh kevhrsh auebtkce caa. V? d`rfsT

(n) Akeo tmh v juhs ca`, n èo b kai a(x) 42x> + ?x

0

(>x 0)> . V? d`rfsT

(b) Ikvhe tm`tp(x) 4x> >, hxprhss m(x) 4 x> >

>x> 8 ke thrds caa èo p. V> d`rfsT

V@eswhr7 (`)a a(x) 4x , a0(x) 4 x>x 0 6 (n)`4 0, n4 ), kt jh`vhs ` rhd`keohr ca;.

(`) Akeo tmh v juhs ca` èo n, èo mhebh, a`btcrksh p(x) bcdpjhthjy. V1 d`rfsT

(n) Akeo tmh sht ca v`juhs cax wmkbm s`tksffhs p(x)

x 2 , n48, (x 2)(>x 0)(x + 0) 6 (n){x7 x 0,0> x :2, x 52} 6 (b) Dkekdud

v`juh4;=

, x 40?

T

10. VR_\D T_mh hxprhsskce bcs x 2ske xd`y nh wrktthe ke tmh acrd r bcs(x + ) acr `jj v`juhs cax, wmhrh r kspcsktkvh èo ks ` `buth.

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(`) Ohthrdkeh tmh v`juhs car èo . V2 d`rfsT

(n) Rt`th tmh dkekdud èo d`xkdud v`juhs ca bcs x

2ske x, èo ohthrdkeh tmh bcrrhspceokeiv`juhs cax ke tmh kethrv`j . V2 d`rfsT

(b) Rfhtbm tmh burvh y 4 bcs x

2ske xacr . V2 d`rfsT

Ny or`wkei è `pprcprk`th jkeh ce tmh ir`pm, ohthrdkeh tmh eudnhr ca rccts ca tmh hqu`tkce

bcs x

2ske x4

2

?

x

ke tmh kethrv`j . V2 d`rfsT

(o) Rcjvh tmh hqu`tkce bcs x

2ske x40 acr . V2 d`rfsT

V@eswhr7 (`)r 4 >, 4

26 (n) dkekdud4-> wmhe x 4

>

2, d`xkdud4> wmhe x 4

8

2 6 (b) 2 rccts 6 (o)

2, T

1>. VR_\D TIkvhe tm`t > x x> ks ` a`btcr cap(x) 4`x2 x> + nx >. Akeo tmh v`juhs ca` èo n. Mhebh, ffeotmh sht ca v`juhs cax acr wmkbm p(x) ks ehi`tkvh. V1 d`rfsT

V@eswhr7 `4>, n4 8 ,{x7>: x : 0>

, x 50}T

12. VR_\D TAuebtkces a èo i a`rh ohffeho ny a(x) 4hx+> èo (i a)(x) 4 x, acr `jj x d`rfsT

V@eswhr7 (`) i (x) 4

je x > , O4{x7 xh>} 6 (n) h2T

1?. VR_\D T

Rcjvh tmh skdujtèhcus hqu`tkces jci;

x

y

4

2

?èo (jci2 x)(jci2 y) 4 0. V= d`rfsT

V@eswhr7 x4 ;, y4

2 crx 4

2

2 , y 4

0

;T

18. VR_\D T

_mh auebtkce a ks ohffeho nya 7xx> x, acr x0

>.

(`) Akeo a0, èo st`th kts ocd`ke. V? d`rfsT

(n) Akeo tmh bccroke`ths ca tmh pcket ca kethrshbtkce ca ir`pma èo a0. V2 d`rfsT

(b) Rfhtbm, ce tmh s`dh bccroke`ths `xhs, tmh ir`pm caa èo a0. V2 d`rfsT

V@eswhr7 (`)a0(x) 4 0

>+

x +

0

? , O4{x7 x 0

?} 6 (n) (>,>)T

11. VR_\D

TRfhtbm ` ir`pm cay 4 bcs > ke tmh rèih ca d`rfsTMhebh, ffeo tmh sht ca v`juhs ca , wmhrh >

2.V8 d`rfsT

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V@eswhr7{x7 0>

x 000>

}T

19. VR_\D T_mh auebtkces a èo i `rh ohffeho ny

a 7xh>x, x ^6i7 x(je x)>, x 5

4i(>), èo st`th, wktm ` rh`sce, wmhtmhr i m`s è kevhrsh. V? d`rfsT

V@eswhr7 (`)a0(x) 4 0

>je x , O4{x7 x 5, V0 d`rfsT

kk. bcs x + ske x4 >, V0 d`rfsT(b) Akeo tmh sht ca v`juhs cax ke tmh kethrv`j acr wmkbm| bcs x + ske x|5 0. V2 d`rfsT

V@eswhr7

>bcs(x ?

) , dkekdud v`juh4

> wmhe x 4 8

? , d`xkdud v juh4

> wmhe x 4

?6 (n) (k) twc

rccts , (kk) ec rccts 6 (b){x7

, : x :2

>}T

1;. VR_\D T_mh auebtkce a ks ohffeho ny a(x) 4 je(0 >x), x :(0 hx) 6 O4{x7 x 5

, wmhrh

: x : 90>

,000>

: x}T

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90. VR_\D T_mh pcjyecdk jp(x) 4mx? + fx2 + >x 0, wmhrhm èo f `rh bcestèts, jh`vhs ` rhd`keohr ca ? wmheokvkoho ny x 0, èo ` rhd`keohr ca> wmhe okvkoho ny x + 0.

(`) Ohthrdkeh tmh v`juhs cam èo f. V2 d`rfsT

(n) Hxprhss tmh pcjyecdk`jp(x) ke tmh acrd (x> 0)q(x) + r(x), wmhrh q(x) ks quòr`tkb èo r(x) ksjkeh`r. V? d`rfsT

(b) Hxprhss q(x) ke ` bcdpjhtho squ`rh acrd `(x + n)> + b. V> d`rfsT

k. Ohoubh tm`tq(x) ks `jw`ys pcsktkvh acr `jj rh`j v`juhs cax. V0 d`rfsT

kk. Ohoubh tmh dkekdud v`juh caq(x) èo tmh bcrrhspceokei v`juh cax. V> d`rfsT

(o) Ohthrdkeh tmh sht ca v`juhs cax acr wmkbm p(x)5 2x + 0. V2 d`rfsT

V@eswhr7 (`)m 4 >, f4 0 6 (n) (x> 0)(>x> + x + >) + (2x + 0) 6(b)q(x) 4 >x +

0

?>

+08

=

(kk) dkekdud v`juh408

=

, wmhe x 4

0

?

6 (o)

{x7 x :

0, x 50

}T

9>. VR_\D T

_mh auebtkce a ks ohffeho `s a(x) 40

>

hx hx, wmhrh x ^.

(`) Rmcw tm`t am`s è kevhrsh. V2 d`rfsT

(n) Akeo tmh kevhrsh auebtkce caa, èo st`th kts ocd`ke. V9 d`rfsT

V@eswhr7 (n)a0(x) 4 je(x +

x> + 0), O4{x7 x ^}T

92. VR_\D TRfhtbm, ce tmh s`dh `xhs, tmh ir`pms ca y 4|>x+ 0| èo y 4 0x>. Mhebh, scjvh tmh kehqu jkty|>x + 0| 0 x>. V= d`rfsT

V@eswhr7{x7 x0

2, x

9, +

9}T

98. VR_\D TAuebtkces a èo i `rh ohffeho ny

a(x) 4x> + ?x + >, x ^,

i(x) 4 2

x + 2, x42, x ^

(`) Rfhtbm tmh ir`pm caa, èo ffeo kts rèih. V? d`rfsT

(n) Rfhtbm tmh ir`pm cai , èo smcw tm`t i ks ` ceh-tc-ceh auebtkce. V2 d`rfsT

(b) Ikvh ` rh`sce wmy i0 hxksts. Akeo i0, èo st`th kts ocd`ke. V? d`rfsT

(o) Ikvh ` rh`sce wmy i a hxksts. Akeo i a, èo st`th kts ocd`ke. V? d`rfsT

0?f

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15/195


V@eswhr7 (`) ^4{y7 y >} 6 (b) i0(x) 4 2x 2 , O4{x7 x ^, x4 x> + ?x + 8

, O4{x7 x ^}T

91. VR_\D T

Akeo tmh v`juh cax subm tm`t (2 jci2 x)jci2x2 4 0. V? d`rfsT

V@eswhr7 x4 2T

99. VR_\D T

Hxprhss 0> bcs 8ske ke tmh acrd r bcs(+ ), wmhrh r 5< èo

. V? d`rfsT

Mhebh,

(`) st`th tmh dkekdud èo d`xkdud v`juhs ca 0> bcs 8ske acr rh`j v`juhs ca , V0 d`rfsT(n) scjvh tmh hqu`tkce 0> bcs

8ske 4 , V2 d`rfsT

(b) sfhtbm tmh ir`pm cay 4 0> bcs 8ske acr èo ohthrdkeh tmh rèih ca v`juhs ca ketmks kethrv`j s`tksaykei tmh kehqu`jkty80> bcs 8ske 0x + 0).Ymhe p(x) ks okvkoho ny (x 0), tmh rhd`keohr ks08.

(`) Ohthrdkeh tmh v`juhs ca` èo n. V? d`rfsT

(n) A`btcrksh p(x) bcdpjhthjy. V2 d`rfsT

(b) Akeo tmh sht ca v`juhs cax wmkbm s`tksffhs tmh kehqu`jkty p(x): < V? d`rfsT

V@eswhr7 (`)` 4 > n 4? 6 (n) (>x + 0)(x >)(x> + >x + >) 6 (b){x70>

: x :>}T

9;. VR_\D TRcjvh tmh hqu`tkce bcs x >ske x4 > acr .

(`) Rfhtbm, ce shp`r th okìr`ds, tmh ir`pms caa èo i . V2 d`rfsT

(n) k. Hxpj ke wmy a0 hxksts.

kk. Mhebh, ohthrdkeh a0 èo st`th kts ocd`ke. V1 d`rfsT

(b) Akeo tmh bcdpcskth auebtkcea i èo st`th kts ocd`ke èo rèih. V? d`rfsT(o) Hxprhss

je(x 0) > `s ` bcdpcsktkce ca auebtkces wmkbm kevcjvhs a èo i . V> d`rfsT

V@eswhr7 (n)(kk)a0(x) 4hx + 0 , Ocd`ke4{

x7 x

^

} 6

(b)a i(x) 4 je((x >) 0) , Ocd`ke4{x7 x 52} , ^èih4{y7 y ^} 6 (o) i a(x)T

08f

fjhhd`

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16/195


=0. VR_\D T_mh auebtkce a ks ohffeho ny

a(x) 4 >x> + `x + 9, x ^.

Yktmcut uskei okhrhetk`tkce, ffeo tmh v`juhs ca` ka tmh rèih caa ks 20

=

,

. V8 d`rfsT

V@eswhr7 `48T

=>. VR_\D T_mh pcjyecdk`j p(x) 4x? +dx2 +ex> + >x + >, wmhrh d èo e `rh rh`j bcestèts, m`s ` quòr`tkba`btcr x> 0.

(`) Akeo tmh v juhs cad èo e. V? d`rfsT

(n) Akeo tmh ctmhr quòr`tkb a`btcr cap(x) èo mhebh, smcw tm`t p(x)

x> 0 2 acr `jj x. V? d`rfsT

(b) Ny uskei jcei okvkskce, cnt`ke tmh rhd`keohr wmhep(x) ks okvkoho ny x>

x 1. Mhebh, ohoubhtmh rhd`keohr wmhe p(x) ks okvkoho ny (x + >). V8 d`rfsT

(o) Akeo i (x) ke thrds cap(x) subm tm`t (x >) ks ` a`btcr cai(x). V> d`rfsT

V@eswhr7 (`)e 42, d4> 6 (n)x> >x > 6 (b)0? >x , 0= 6 (o) i(x) 4 (x >)p(x)T

01f

fjhhd`

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17/195

ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) >7 Rhquhebhs èo Rhrkhs

> Rhquhebhs èo Rhrkhs

0. VR_\D T

Hv`ju`th

r400

0r. V> d`rfsT

Hxprhss d`rfsT

V@eswhr7 0

;; ,

>

00T

>. VR_\D T

Akeo tmh hxpèskce ca (0 + x>)p

(0 x)q ke `sbheokei pcwhrs cax uetkj tmh thrd ke x2. V8 d`rfsT

(`) Kap 4 q 4 0

>, suiihst ` sukt`njh v`juh cax tm`t he`njhs

02

0)

1

x2 + . . . 6 (`) x 4

0

8, 0.02;8 6 (n) 01>T

2. VR_\D THxprhss (

> 0)8 ke tmh acrd`

> + n, wmhrh `, n `rh kethihrs. V2 d`rfsT

V@eswhr7 >;> ?0T

?. VR_\D T_mh sud ca tmh ffrst >e thrds ca ` shrkhs \ ks >. Akeo ke thrds cae, tmh sud ca tmh ffrst ethrds ca tmks shrkhs. Rmcw tm`t tmks shrkhs ks è `rktmdhtkb shrkhs. V? d`rfsT

RhrkhsP ks è `rktmdhtkb shrkhs subm tm`t tmh sud ca kts ffrste hvhe thrds ks dcrh tmè tmh sud ca ktsffrst e coo thrds ny ?e. Akeo tmh bcddce okhrhebh ca tmh shrkhs P. V8 d`rfsT

Ka tmh ffrst thrd ca shrkhs P ks 0, ohthrdkeh tmh dkekdud v`juh cae subm tm`t tmh okhrhebh nhtwhhetmh sud ca tmh ffrst e thrds ca shrkhs \èo tmh sud ca tmh ffrst e thrds ca shrkhs P ks dcrh tmè ;=0T

8. VR_\D TRkdpjkay (0 + >

2)8 (0 >

2)8. V? d`rfsT

V@eswhr7 0


18/195


(n) _mh ffrst èo tmh shbceo thrd ca è `rktmdhtkb shrkhs hqu`j rhsphbtkvhjy tmh ffrst èo tmh shbceothrd ca ` ihcdhtrkb shrkhs. _mh tmkro thrd ca tmh ihcdhtrkb shrkhs hxbhhos tmh tmkro thrd ca tmh`rktmdhtkb shrkhs ny 2. _mh `rktmdhtkb shrkhs m`s ` pcsktkvh bcddce okhrhebh èo tmh sud ca ktsffrst tmrhh thrds hqu`js 8?. Akeo tmh ffrst thrd ca nctm shrkhs. Akeo `jsc tmh bcddce okhrhebhca tmh `rktmdhtkb shrkhs èo tmh bcddce r`tkc ca tmh ihcdhtrkb shrkhs. V= d`rfsT

V@eswhr7 (`) 0< 6 (n) ` 4 0> , o 4 1 , r 4 2

>T

9. VR_\D TIkvhe tm`t tmh sud ca tmh ffrst e thrds ca ` shrkhs ks e jcipqe+0. Rmcw tm`t

(`) tmh e-tm thrd ca tmh shrkhs ks jcipq>e, V> d`rfsT

(n) tmks ks è `rktmdhtkb shrkhs. V2 d`rfsT

=. VR_\D T

Hxpèo (0+x) 08 ke `sbheokei pcwhr caxuetkj tmh thrd kex2. Ny t`fkeix4 0?.=08 bcrrhbt tc acur ohbkd`j pj`bhs. V9 d`rfsT

Ka tmh hxpèskce ca 0 + `x

0 + nxèo (0 + x)

08 `rh tmh s`dh uetkj tmh thrd ke x>, ffeo tmh v`juhs ca` èo n.

Mhebh, smcw tm`t 2>.=>8x> +

1

0>8x2 + . . . , >. ke tmh `sbheokei pcwhr cax uetkj tmh thrd kex2. Ny t`fkeix 4 00 + 2>x2 + . . . 6 0.92>0.

V8 d`rfsT

(n) Akeo tmh sud tc keffekty

2>(0 x)> + 22(0 x)2 + . . . + 2r(0 x)r + . . .

èo ohthrdkeh tmh sht cax subm tm`t tmks sud hxksts. V8 d`rfsT

V@eswhr7|r|: 0, `0 r 6 (`) 9 6 (n)

;(0 x)>2x > ,{x7

>2

: x : ?2}T

0=f

fjhhd`

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00. VR_\D T_mh sud èo prcoubt ca tmrhh bceshbutkvh thrds ca è `rktmdhtkb prcirhsskce `rh 2 èo >? rhsphbtkvhjy.Ohthrdkeh tmh tmrhh pcssknjh thrds ca tmh `rktmdhtkb prcirhsskce. V8 d`rfsT

V@eswhr7 ?,-0,-1 cr -1,-0,?T

0>. VR_\D T

Hxpèo

0 xe

ewmhrh e ks ` pcsktkvh kethihr ke `sbheokei pcwhrs cax uetkj tmh thrd ke x2. Ka tmh

bchflbkhet cax2 ks 0>9

, ffeo e. V1 d`rfsT

Yktm tmks v`juh ca e, cnt`ke tmh hxpèskce ca

0 xe

e(0 x) 0> ke `sbheokei pcwhrs cax uetkj tmh

thrd ke x2. V8 d`rfsT

Mhebh, ny t`fkei x 400e

x> (e 0)(e >)1e>

x2 + . . . , e 4 2 6 0 2>

x +09>?

x> 10?2>

x2 . . . 6 0T

02. VR_\D T

Hxprhss 0

(?r 2)(?r+ 0) ke p`rtk`j ar`btkces. Mhebh smcw tm`t

er40

0

(?r 2)(?r+ 0) 40

?

0 0

?e + 0

.

V1 d`rfsT

V@eswhr7 0?(?r 2)

0?(?r+ 0)

T

0?. VR_\D T

Ikvhe tm`ty4 0

0 + >x +

0 + x, wmhrh x 50

>, smcw tm`t, prcvkoho x4 x 0 + x).

V2 d`rfsT

Xskei tmh shbceo acrd acr y, hxprhss y `s ` shrkhs ca `sbheokei pcwhrs cax `s a`r `s tmh thrd ke x>.V1 d`rfsT

Mhebh, ny puttkeix 4 0

0


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V@eswhr7{x7 x :T

01. VR_\D T

Hxprhss 0

?f> 0 `s p`rtk`j ar`btkce. V? d`rfsT

Mhebh, ffeo ` skdpjh hxprhsskce acr Re4

ef40

0?f> 0 èo ffeo jkdeRe. V? d`rfsT

V@eswhr7 0

>(>f 0) 0

>(>f+ 0) 6 Re 4

0

>

0 0

>e + 0

6

0

>T

09. VR_\D T

Hxprhss

0 + x

0 + >x

0>

`s ` shrkhs ca `sbheokei pcwhrs cax up tc tmh thrd ke x2. V1 d`rfsT

Ny t`fkei x 4 0

2 bcrrhbt tc acur ohbkd`j pj`bhs. V2 d`rfsT

V@eswhr7 0 0>

x +9

=x> >8

01x2 + . . . 6

1> 4 9.=9?

r> + >rke p`rtk`j ar`btkces. V2 d`rfsT

Xskei tmh rhsujt cnt`keho,

(`) smcw tm`t u>r 40

r+

0

r>+

0

r+ >+

0

(r+ >)>, V> d`rfsT

(n) smcw tm`t

e

r40 ur 4

2

> 0

e + 0 0

e + > èo ohthrdkeh tmh v`juhs ca

r40 ur èo

r40

ur+0+

0

2r

.V; d`rfsT

V@eswhr7 0

r 0

r+ > 6 (n)

2

> ,

?

2T

0;. VR_\D T

Hxpèo (0 x) 0> ke `sbheokei pcwhrs cax up tc tmh thrd ke x2. Mhebh, ffeo tmh v`juh ca

9 bcrrhbttc ffvh ohbkd`j pj`bhs. V8 d`rfsT

V@eswhr7 0

0

>

x

0

=

x>

0

01

x2 6

9

>.1?1 + . . . ks

`(0 re)0 r .

V2 d`rfsT

(`) _mh sud ca tmh ffrst ffvh thrds ca ` ihcdhtrkb shrkhs ks 22 èo tmh sud ca tmh ffrst the thrds catmh ihcdhtrkb shrkhs ks -02. Akeo tmh bcddce r`tkc èo tmh ffrst thrd ca tmh ihcdhtrkb shrkhs.

V8 d`rfsT

(n) _mh sud ca tmh ffrst e thrds èo tmh sud tc keffekty ca tmh ihcdhtrkb shrkhs 1 2 + 2> . . . `rh

Re èo R rhsphbtkvhjy. Ohthrdkeh tmh sd`jjhst v`juh cae subm tm`t|Re R|:


21/195


V@eswhr7 (`)r 4>, ` 4 2 6 (n)e 4 0>T

>0. VR_\D TAcr tmh ihcdhtrkb shrkhs 9 + 2.8 + 0.9 8 + x 0(x + >)(x 2) ke p`rtk`j ar`btkces. V8 d`rfsT

Mhebh, cnt`ke è hxpèskce caa(x) ke `sbheokei pcwhrs ca 0

xup tc tmh thrd ke

0

x2. V1 d`rfsT

Ohthrdkeh tmh sht ca v`juhs cax acr wmkbm tmks hxpèskce ks v`jko. V> d`rfsT

V@eswhr

7 0 +

0

x 2 0

x + > , 0 +

8

x> +

8

x2 + . . . ,{x7 x :2, x 52}T>2. VR_\D T

Kax ks sc sd`jj tm`tx> èo mkimhr pcwhrs cax d`y nh ehijhbtho, smcw tm`t

(0 x)1

> +x

>

0< >;(> 9x).V? d`rfsT

>?. VR_\D T

_mh etm thrd ca è `rktmdhtkb prcirhsskce ks _e, smcw tm`t Xe 4 8

>(>)>( 0(09e 0?), hv`ju`th

e40

Xe. V? d`rfsT

V@eswhr7 08. VR_\D THxprhss tmh keffekth rhburrkei ohbkd`j 8 (4 8>8>8 . . . ) `s ` ar`btkce ke kts jcwhst thrds.V? d`rfsT

V@eswhr7 28;

?;8T

>1. VR_\D T@t tmh nhikeekei ca tmks yh`r, Dr. Jku èo Dkss Ocr` ohpcsktho ^D0<


22/195


>9. VR_\D T

Acr ihcdhtrkb shrkhs 1 + 2 +2

>+. . ., cnt`ke tmh sd`jjhst v`juh cae ka tmh okhrhebh nhtwhhe tmh sud

ca tmh ffrste + ? thrds èo tmh sud ca ffrst e thrds ks jhss tmè ?8

1?. V1 d`rfsT

V@eswhr7 8T

>=. VR_\D TOhthrdkeh tmh sht ca v`juhs cax subm tm`t tmh ihcdhtrkb shrkhs hx + h>x + h2x + . . . bcevhrihs.

Akeo tmh hx`bt v`juh cax ka tmh sud tc keffekty ca tmh shrkhs ks 2. V1 d`rfsT

V@eswhr7{x7 x 5;. VR_\D T

Ikvhe a(x) 4 x2 2x ?(x 0)(x> + 0)

,

(`) ffeo tmh bcestèts @, N , B èo O subm tm`ta(x) 4@ + N

x 0+B x + O

x> + 0 , V8 d`rfsT

(n) wmhe x ks suflbkhetjy sd`jj subm tm`t x? èo mkimhr pcwhrs bè nh ehijhbtho, smcw tm`t a(x)? + 9x + 2x> x2. V? d`rfsT

V@eswhr7 (`)@ 4 0, N42, B4 ?, O4 + r 4

e>

e + 0 . V? d`rfsT

20. VR_\D T_mh sud ca tmh ffrst e thrds ca ` prcirhsskce 2e>. Ohthrdkeh tmh e-tm thrd ca tmh prcirhsskce, èomhebh, ohoubh tmh typh ca prcirhsskce. V? d`rfsT

V@eswhr7 1e 2, @rktmdhtkb \rcirhsskceT

2>. VR_\D THxprhss ke p`rtk`j ar`btkces

2

(2r 0)(2r+ >).

V? d`rfsT

Rmcw tm`te

r40

2

(2r 0)(2r+ >) 40

> 0

(2e + >),

V> d`rfsT

èo mhebh, ffeor40

0

(2r 0)(2r+ >) .

V> d`rfsT

V@eswhr7 0

2r 0 0

2r+ >, 0

1T

>>f

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22. VR_\D T

@ shquhebh ks ohffeho ny ur 4 h(r0) hr acr `jj kethihrs r 0. Akeo

er40

ur ke thrds ca e, èo

ohoubh tmh v`juh ca

r40

ur. V8 d`rfsT

V@eswhr7 0 he 6 0T

2?. VR_\D T_mh shquhebh u0, u>, u2, . . . ks ohffeho ny ue+04 2ue, u04 >.

(`) Yrkth ocwe tmh ffrst ffvh thrds ca tmh shquhebh. V> d`rfsT

(n) Ruiihst è hxpjkbkt acrduj` acrur. V> d`rfsT

V@eswhr7 >, 1, 0=, 8?, 01> , ur 4 >(2)r0T

28. VR_\D T

Hxpèo (0 + x)>2 èo

0 + `x

0 + nx, wmhrh|n|: 0, ke `sbheokei pcwhrs cax up tc tmh thrd kex2. Ohthrdkeh

tmh sht ca v`juhs cax acr wmkbm nctm tmh hxpèskces `rh v`jko. V9 d`rfsT

Ka tmh twc hxpèskces `rh kohetkb`j up tc tmh thrd ke x>,

(`) ohthrdkeh tmh v`juhs ca` èo n, V2 d`rfsT

(n) ush x 40

=tc cnt`ke tmh `pprcxkd`tkce

2

=0>0>?;

. V2 d`rfsT

(b) ffeo, bcrrhbt tc ffvh ohbkd`j pj`bhs, tmh okhrhebh nhtwhhe tmh thrds ke x2 acr tmh twc hxpèskces

wktm x 4 0

=. V> d`rfsT

V@eswhr7 0 +>

2x 0

;x> +

?

=0x2 + . . ., 0 + (` n)x + n(n `)x> + n>(` n)x2 + . . .,|x|: 0 6 (`)` 4 8

1 ,n 4

0

1 6

(b) ,n2,. . .ks è `rktmdhtkb shquhebh, èo st`th kts ffrst thrd èo bcddce okhrhebh.

V2 d`rfsT

(b) Akeo tmh sud ca tmh ffrst e thrds ca tmh shquhebh n0, n>, n2, . . . ke thrds caè èo ne. V> d`rfsT

V@eswhr7 (`) 1e + 2 , (n) ;, 1 6 (b)è+ ne 2T

29. VR_\D TYrkth tmh keffekth rhburrkei ohbkd`j >2f

fjhhd`

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2=. VR_\D T

(`) Hxprhss 0

(r> 0) ke p`rtk`j ar`btkces, èo ohoubh tm`t

0r(r> 0)

0>

0

r(r 0) 0

r(r+ 0)

.

V? d`rfsT

Mhebh, ush tmh dhtmco ca okhrhebhs tc ffeo tmh sud ca tmh ffrst (e 0) thrds, Re0, ca tmh shrkhs0

> 2+ 0

2 =+ 0

? 08+ . . . + 0

r(r> 0)+ . . . ,

èo ohoubh Re. V1 d`rfsT

(n) Hxpj`ke wmy tmh shrkhs bcevhrihs tc 0

?, èo ohthrdkeh tmh sd`jjhst v`juh cae subm tm`t

0

? Re: (r+ 0) 6 (n) Re04

0

>

0

> 0

e(e + 0)

, Re 4

0

>

0

> 0

(e + 0)(e + >)

6 (b) 02T

2;. VR_\D TXsh tmh nkecdk`j hxpèskces ca (

2 + >)1 èo (

2 >)1 tc hv`ju`th (

2 + >)1 + (

2 >)1. Mhebh,

smcw tm`t >9)1 :>9. V9 d`rfsT

V@eswhr7 >9T

? + 2x2 + . . . + 2xe ks ` ihcdhtrkb shrkhs, èo ffeo kts sudke thrds cax èo e. V? d`rfsT

(n) _mh shrkhs _e(x) ks ikvhe ny

_e(x) 4x + ?x> + 9x2 + . . . + (2e

>)xe, acrx

4 0.

Ny bceskohrkei _e(x) x_e(x) èo uskei tmh rhsujt arcd (`), smcw tm`t

_e(x) 4x + >x> (2e + 0)xe+0 + (2e >)xe+>

(0 x)> .

V8 d`rfsT

Mhebh, ffeo tmh v`juh ca>r(2r >), èo ohoubh tmh v`juh ca0;r4r+>(2r+ 0). V1 d`rfsT

V@eswhr7 (`) sud4 2x>

(x

e0

0)x 0 6 (n) 0082?229< , >2


25/195


?0. VR_\D T

Rmcw tm`t tmh ffrst tmrhh thrds ke tmh hxpèskces ca (0 =x)? èo 0 8x0 2x ke `sbheokei pcwhrs cax `rh

tmh s`dh. Ohthrdkeh tmh rèih ca v`juhs cax acr wmkbm nctm hxpèskces `rh v`jko. V1 d`rfsT

Xsh tmh rhsujt (0

=x)?

0 8x0 2x

tc cnt`ke è `pprcxkd`tkce ca (r+0+

0

2

>r0,

(`) Hxprhss ur ke tmh acrd @

2>r+0, wmhrh @ ks ` bcestèt. V> d`rfsT

(n) Akeo tmh sud ca tmh ffrst e thrds ca tmh shrkhs, èo ohoubh tmh sud ca tmh keffekth shrkhs. V1 d`rfsT

V@eswhr7 (`)ur 4 0r+0 6 (n)

8

0>

0

0

;

e,

8

0>T

?2. VR_\D T@ rhburskvh acrduj` acr tmh ihehr`j thrd ca ` shquhebh ks ikvhe ny ur+04 ur+ >r+ 2, wmhrh u d`rfsT

(n) Ruiihst è hxpjkbkt acrduj` acr tmh ihehr j thrd èo vhrkay ycur èswhr. V2 d`rfsT

V@eswhr7 (`) 0, ?, ;, 01 6 (n) ur 4 (r+ 0)>T

??. VR_\D T

@ bcevhrihet shquhebh ks ohffeho ny ur+04 0 +0

2ur èo u0 4 0.

(`) Yrkth ocwe h`bm ca tmh thrds u>, u2 èo u? ke tmh acrdr0f40

0

2

r

. V8 d`rfsT(n) Ohthrdkeh tmh jkdkt caur, `s r theos tc keffekty. V> d`rfsT

V@eswhr7 (n) 2

> T

?8. VR_\D T

Rmcw tm`tf40

0

>

f02

f>ks ` bcevhrihet shrkhs. Ikvh ` rh`sce acr ycur èswhr. V2 d`rfsT

Mhebh, ohthrdkeh tmh sud ca tmh bcevhrihet shrkhs. V> d`rfsT

V@eswhr7 ;

8 T

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26/195


Ycrf Rd`rt, Ect M`ro

Oc ect sm`rh tmks p`st yh`r p`phr.

^hahr tmh aujj scjutkce `t ffjhhd`tms.bcd.

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27/195

ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) 27 D`trkbhs

2 D`trkbhs

0. VR_\D T

(`) Ikvhe D4 > 2 00 < ?

0 0 0

. Rmcw tm`t D2 2D> + =D >?K24 2> 2 0

2 0 >

, N4

8 0 90 9 8

9 8 0

. Akeo @N, èo mhebh scjvh tmh skdujt`-

ehcus hqu`tkce8x + y + 9z 4 =,

x + 9y 8z 4 01,9x 8y + z 4 0?.

V= d`rfsT

V@eswhr7 (`) @0 4 0

>?

? ? 0>8 0 ;

0 8 2

6 (n) @N4

0= < T

>. VR_\D T

Ka@4

0 > 12 > ;

> < 2

, ffeo @> èo @2. Mhebh, ffeo @0

1

2 > ;> < 2

, @00 > 0

> 0 0

.

(`) Akeo ^ sc tm`t ^4 \> ?\ K2. V2 d`rfsT(n) Rmcw tm`t \^ + ?K24 d`rfsT

V@eswhr7 (`) ^ 4

0 0 20 2 02 0 0

T

?. VR_\D T_mh d`trkbhs @ èo N `rh ikvhe ny

@4

8 < <

.

(`) Ohthrdkeh wmhtmhr @ èo Nbcdduth. V2 d`rfsT(n) Rmcw tm`t tmhrh hxkst eudnhrsd èo e subm tm`t @ 4 dN+eK, wmhrh K ks tmh 2 x 2 kohetkty

d`trkx, èo ffeo tmh v`juhs cad èo e. V1 d`rfsT

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28/195


V@eswhr7 (`) Bcdduth. (n) d 40, e4 2T

8. VR_\D T

Ohthrdkeh tmh v`juhs ca`, n, b sc tm`t tmh d`trkx

>n 0 `> n>>` 0 ` nb

n n + b >b 0

ks ` syddhtrkb d`trkx.

V8 d`rfsT

V@eswhr7 `4 0,n4 ?

.

Akeo DE èo ohoubh E0. V? d`rfsT\rcoubt], [, W`rh `sshdnjho arcd tmrhh bcdpcehets @, N, B`bbcrokei tc okhrhet prcpcrtkces.H`bm prcoubt ] bcesksts ca twc bcdpcehets ca@, acur bcdpcehets ca N, èo ceh bcdpcehet ca B6

h`bm prcoubt ca[bcesksts ca tmrhh bcdpcehets ca@, tmrhh bcdpcehets caN , èo twc bcdpcehets caB6 h`bm prcoubt caWbcesksts ca acur bcdpcehets ca@, ceh bcdpcehet caN , èo acur bcdpcehets caB. @ tct`j ca 98< bcdpcehets ca@, 0 0 2.

Akeo tmh ògcket ca@. Mhebh, ffeo @0. V1 d`rfsT

V@eswhr7

0< 0 020? 8 09

1 2 2

6

8/1 0/0> 02/0>9/1 8/0> 09/0>

0/> 0/? 0/?

T

;. VR_\D T_mh d`trkbhs \ èo P, wmhrh \P4 P\, `rh ikvhe ny

\4

> >

` n b

èo P4

0 0 >

Ohthrdkeh tmh v`juhs ca`, n èo b. V8 d`rfsTAkeo tmh rh`j eudnhrs d èo e acr wmkbm \4 dP + eK, wmhrh Kks tmh 2 2 kohetkty d`trkx.V8 d`rfsT

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29/195


V@eswhr7 `4 , e4 0

, B4

0 < > 0

T

00. VR_\D T_mh d`trkx @ks ikvhe ny

@4 f 0 8> f =

= 2 > .Ohthrdkeh `jj v`juhs caf acr wmkbm tmh hqu`tkce @]4 N, wmhrh Nks ` 2 0 d`trkx, ochs ect m`vh ùekquh scjutkce. V2 d`rfsT

Acr h`bm ca tmhsh v`juhs caf , ffeo tmh scjutkce, ka èy, ca tmh hqu`tkce

@]4

0>

?

.

V9 d`rfsT

V@eswhr7 f4 2, f4 8 6 Acr f 4 2, ec scjutkce. Acr f 4 8, x 4 9>2

09>2

t, y40>>2

22

t, z 4 tT

0>. VR_\D T

Ohthrdkeh tmh v`juhs caf subm tm`t tmh ohthrdkeèt ca tmh d`trkx

f 0 2>f+ 0 2 >

< f >

ks T

02. VR_\D T

Ka\ 4 8 > 20 ? 2

2 0 >

, P4 ` 0 0=n 0 0>02 0 b

èo \P4 >K, wmhrh K ks tmh 2 2 kohetkty d`trkx,ohthrdkeh tmh v`juhs ca`, n èo b. Mhebh ffeo \0. V= d`rfsT_wc ircups ca wcrfhrs m`vh tmhkr orkefs `t ` st`jj. _mh ffrst ircup bcdprkskei the wcrfhrs m`vh ffvhbups ca th`, twc bups ca bchh èo tmrhh ij`sshs ca arukt gukbh `t ` tct`j bcst ca ^D00.= 6 00/> 0/> ;9/> 0/> 1

02/> 0/> 00

6 x4^D 0, y4^D 0.2


30/195


0?. VR_\D TBceskohr tmh systhd ca hqu`tkces

x + y + pz 4 q,2x y >z 4 0,1x + >y + z 4 ?,

acr tmh twc b`shs7 p4 >, q4 0 èop 4 0, q4 >.

Acr h`bm b`sh, ffeo tmh uekquh scjutkce ka kt hxksts cr ohthrdkeh tmh bceskstheby ca tmh systhd ka tmhrhks ec uekquh scjutkce. V9 d`rfsT

V@eswhr7 Acr p 4 >,q4 0, uekquh scjutkce, x 4 0/>, y4 0/>, z4 , ec uekquh scjutkce. _mh systhd ks ect bcesksthet.T

08. VR_\D T_mh d`trkbhs @ èo N `rh ikvhe ny

@40 > 02 0 ?

< 0 >

, N4 28 0; 0=>9 02 ?82 0> 8

.Akeo tmh d`trkx @>Nèo ohoubh tmh kevhrsh ca@. V8 d`rfsTMhebh, scjvh tmh systhd ca jkeh`r hqu`tkces

x >y z 4 =,2x y ?z 4 08,

y+ >z 4 ?.

V8 d`rfsT

V@eswhr7

0>0 < 0 0

,

>/00 2/00 9/001/00 >/00 0/00

2/00 0/00 8/00

6 x 42,y4 >, z 4 0T

01. VR_\D TBceskohr tmh systhd ca hqu`tkces

x + y + z 4 0,x + y + z 4 ,

x + y + z 4 >,

wmhrh ks ` bcestèt.

(`) Ohthrdkeh tmh v`juhs ca acr wmkbm tmks systhd m`s ` uekquh scjutkce, keffekthjy dèy scjutkcesèo ec scjutkce. V8 d`rfsT

(n) Akeo tmh uekquh scjutkce ke thrds ca. V8 d`rfsT

V@eswhr7 (`) Acr uekquh scjutkce, 4 0, >. Acr keffekthjy dèy scjutkce, 4 0.Acr ec scjutkce, 4>.

(n) x 4 + 0 + >

, y4 + 0 + >

, z 4 ( + 0)>

+ > T

09. VR_\D T

D`trkx @ks ikvhe ny @4

0 < 0

.

2


31/195


(`) Rmcw tm`t @> 4K, wmhrh K ks tmh 2 2 kohetkty d`trkx, èo ohoubh @0. V? d`rfsT

(n) Akeo tmh d`trkx N wmkbm s`tksffhs N@4

0 ? 2< > 0

0 < >

. V? d`rfsT

V@eswhr7 (`)

0 < 0

6 (n)

= 0< 22 ? 0

0 ? >

T

0=. VR_\D T

(`) _mh d`trkx \, P èo ^ `rh ikvhe ny

\4

0 8 1> > ?0

2 >

, P4

02 8< 220 1 89 >< 08

, ^4

? 9 020 8 0

> 0 00

Akeo d`trkbhs \Pèo \P^èo mhebh, ohoubh (\P)0. V8 d`rfsT

(n) Xskei tmh rhsujt ke (`), scjvh tmh systhd ca jkeh`r hqu`tkces

1x + 0y + 2z 4 0? ?< 2>

? = ?? = 0>

,9> <

, 0/0= 9/9> 02/9>

0/9> 8/9> 0/9>0/21 0/9> 00/9>

6 (n) x 4 >> 0> 0 2> 0 0

.

20f

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32/195


(`) Akeo tmh ohthrdkeèt èo ògcket ca\. Mhebh, ffeo \0. V1 d`rfsT

(n) @ a`btcry `sshdnjhs tmrhh typhs ca tcys P, ^ èo R. _mh tct`j tkdh t`fhe tc `sshdnjh ceh uektca^ èo ceh uekt ca R hxbhhos tmh tkdh t`fhe tc `sshdnjh twc uekts ca P ny = dkeuths. Cehuekt caP, twc uekts ca^ èo ceh uekt caRt`fh 20 dkeuths tc nh `sshdnjho. _mh tkdh t`fhe tc`sshdnjh twc uekts caP, ceh uekt ca^ èo tmrhh uekts caRks ?= dkeuths.

Ka x, y èo z rhprhshet tmh tkdh, ke dkeuths, t`fhe tc `sshdnjh h`bm uekt ca tcys P, ^ èo Rrhsphbtkvhjy,

k. wrkth ` systhd ca jkeh`r hqu`tkces tc rhprhshet tmh `ncvh keacrd tkce, V> d`rfsT

kk. uskei tmh rhsujts ke (`), ohthrdkeh tmh tkdh t`fhe tc `sshdnjh h`bm typh ca tcy. V8 d`rfsT

V@eswhr7 (`) 0?,

> 0 8= 2 0

? 8 2

, 0/9 0/0? 8/0??/9 2/0? 0/0?

>/9 8/0? 2/0?

6 (n)(kk) 8, =, 0< T

>0. VR_\D T

@èo N`rh twc d`trkbhs subm tm`t

@4

? 2 1> > ?

> > 2

èo @>N4

> 1 < ?

< ? >

.

(`) Akeo tmh ohthrdkeèt èo ògcket ca@. Mhebh, ohthrdkeh @0. V1 d`rfsT

(n) Xskei @0 cnt`keho ke (`), ffeo N. V? d`rfsT

V@eswhr7 (`) > ,

> 2 < ?

< > >

,

0 2/>

< 0 0

6 (n)

= >9 0=

< 0= 0>. VR_\D TD`trkx @ks ikvhe ny

@4

0 x 00 0 T

>2. VR_\D TBceskohr tmh systhd ca hqu`tkces

x + 2y + >z 4 >,2x + `y + >z 4 >` 0,>x + 1y + `z 4 n.

(`) Ka`4 ; èo `4 ?, smcw tm`t tmh systhd m`s ` uekquh scjutkce. V8 d`rfsT(n) Ka`4 8 èo n 4 1, ffeo tmh uekquh scjutkce. V2 d`rfsT

(b) Ka`4 ?, smcw tm`t tmh systhd ochs ect m`vh ` scjutkce uejhss n 4?. V2 d`rfsT(o) Ka` 4 ;, ohthrdkeh wmhtmhr tmhrh `rh èy v`juhs can acr wmkbm tmh systhd m`s ` scjutkce. V2 d`rfsT

V@eswhr7 (n) x 4 99

? , y488

? , z 4 0< 6 (o) n 4020

? T

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33/195


>?. VR_\D T_mh d`trkbhs \ èo P, wmhrh \P4 P\, `rh ikvhe ny

\4

> >

` n b

èo P4

0 0 >

Ohthrdkeh tmh v`juhs ca`, n èo b. V8 d`rfsTAkeo tmh rh`j eudnhrs d èo e acr wmkbm \4 dP + eK, wmhrh Kks tmh 2 2 kohetkty d`trkx.V8 d`rfsT

V@eswhr7 `4 , e4 8. VR_\D TXskei è ùidhetho d`trkx èo hjhdhet`ry rcw cphr`tkces, ffeo tmh scjutkce ca tmh systhd ca hqu`tkces

2x >y 8z 4 8,x + 2y

>z 4

1,

8x ?y + z 4 00.V; d`rfsT

V@eswhr7 x4 0, y 40, z4 >T

>1. VR_\D T@ systhd ca jkeh`r hqu`tkces ks ikvhe ny

x + y + z 4 f,x y + z 4 y + z 4 2.

wmhrh èo f `rh rh`j eudnhrs. Rmcw tm`t tmh ùidhetho d`trkx acr tmh systhd d`y nh rhoubho tc 0 0 0 f< > < f

< < ? 2 2f

.

V8 d`rfsT

Mhebh, ohthrdkeh tmh v`juhs ca èo f sc tm`t tmh systhd ca jkeh`r hqu`tkces m`s

(`) ` uekquh scjutkce, V0 d`rfsT

(n) keffekthjy dèy scjutkces, V0 d`rfsT

(b) ec scjutkce. V0 d`rfsT

V@eswhr7 (`)4 ? 6 (n) 4 ?, f4 0 6 (b)4 ?, f4 0T

>9. VR_\D T

Ikvhe tm`t d`trkx D4

8 ? >? 8 >

> > >

.

Rmcw tm`t tmhrh hxkst ece-zhrc bcestèts` èo n subm tm`tD> 4`D + nK, wmhrhK ks tmh 2

2 kohetkty

d`trkx. V1 d`rfsT

Mhebh, ffeo tmh kevhrsh ca tmh d`trkx D. V2 d`rfsT

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34/195


V@eswhr7 `4 00, n40< 6 00/? 2 8/?

0/> 8 8/>

T

>;. VR_\D T@ systhd ca jkeh`r hqu`tkces ks ikvhe ny

x 2y 4 >,px + qz 4 0,

py + z 4 0.

(`) Yrkth tmh ùidhetho d`trkx acr tmh systhd ca jkeh`r hqu`tkces, èo smcw tm`t kt d`y nh rhoubhotc

0 2 < >< p 0 0< < 2 q >p >

.

V8 d`rfsT

(n) Ohthrdkeh tmh v`juhs cap èo qsubm tm`t tmh systhd m`s keffekthjy dèy scjutkces, èo ffeo tmhihehr`j scjutkce. V? d`rfsT

V@eswhr7 (n) p 4 0 è q4 2 ,x40 2t, y 4t 0, z 4 tT

2 d`rfsT

20. VR_\D T_mh v`rk`njhs x, y èo z s`tksay tmh systhd ca jkeh`r hqu`tkces

>x + y + >z 4 0,?x + >y + z 4 f,=x + ?y + 9z 4 f>,

wmhrh f ks ` rh`j bcestèt.

(`) Yrkth ` d`trkx hqu`tkce acr tmh systhd ca jkeh`r hqu`tkces. V0 d`rfsT

2?f

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35/195


(n) ^houbh tmh ùidhetho d`trkx tc rcw-hbmhjce acrd, èo smcw tm`t tmh systhd ca jkeh`r hqu`tkcesochs ect m`vh ` uekquh scjutkce. V1 d`rfsT

(b) Ohthrdkeh `jj tmh v juhs cafacr wmkbm tmh systhd ca jkeh`r hqu`tkces m`s keffekthjy dèy scjutkces,èo ffeo tmh scjutkces ke tmh b`sh wmhe f ks pcsktkvh. V1 d`rfsT

(o) Akeo tmh sht ca v`juhs caf acr wmkbm tmh systhd ca jkeh`r hqu`tkces ks kebcesksthet. V> d`rfsT

V@eswhr7 (b)f4 8

2, > , x 4

0 t>

, y4 t, z 4 < 6 (o){f7 f ^, f482

, f4 >}T

2>. VR_\D TXskei Iùsskè hjkdke`tkce, scjvh tmh systhd ca jkeh`r hqu`tkces

x + y z 4 x y >z 4 ?8x y + z 4 >

.

V= d`rfsT

V@eswhr7 x40

2, y4?

2, z 40T

22. VR_\D T

@ d`trkx \ks ikvhe ny \4

0 0 >< > >

0 0 2

.

(`) Akeo \> 1\ + 00K. V2 d`rfsT

(n) Rmcw tm`t \(\

>

1\ + 00K) 4 1K, wmhrh K ks 2 x 2 kohetkty d`trkx, èo ohoubh \0

. V8 d`rfsT

V@eswhr7

T

2?. VR_\D T@ systhd ca jkeh`r hqu`tkces ks ikvhe ny

x + y + z 4 0,>x + 2y + >z 4 2,

>x + 2y + dz 4 d> 0.

(`) Xsh Iùsskè hjkdke`tkce tc rhoubh tmh ùidhetho d`trkx acr tmh systhd `ncvh. V? d`rfsT

(n) Ohthrdkeh tmh v`juh cad acr wmkbm tmh systhd ca jkeh`r hqu`tkces

k. m`s ` uekquh scjutkces, V> d`rfsT

kk. m`s keffekthjy dèy scjutkces, V2 d`rfsT

kkk. m`s ec scjutkce. V2 d`rfsT

V@eswhr7

T

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36/195

ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) ?7 Bcdpjhx Eudnhrs

? Bcdpjhx Eudnhrs

0. VR_\D T

Ohthrdkeh tmh v`juh ca` ka

> + `k

0 +

>kks ` rh`j eudnhr èo ffeo tmks rh`j eudnhr. V? d`rfsT

V@eswhr7 `4 >,

>T

>. VR_\D TKa (x + ky)> 4 k, ffeo `jj tmh rh`j v`juhs cax èo y . V1 d`rfsT

V@eswhr7 x4 0

>, y4

0>

6 x 4 0>

, y4 0>

T

2. VR_\D T_mh bcdpjhx eudnhrs z0 èo z> s`tksay tmh hqu`tkce z

> 4 > >

2k.

(`) Hxprhss z0 èo z> ke tmh acrd ` + nk, wmhrh ` èo n `rh rh`j eudnhrs. V1 d`rfsT(n) ^hprhshet z0 èo z> ke è @rièo okìr`d. V0 d`rfsT

(b) Acr h`bm caz0 èo z>, ffeo tmh dcoujus, èo tmh `riudhet ke ròkès. V? d`rfsT

V@eswhr7 (`) z04

2 k, z>4

2 + k 6 (b)|z0|4 >, |z>|4 > , `ri(z0)41

, `ri(z>)48

1 T

?. VR_\D TJhtzj4 0,z> 4 x + ky,z24 y + kx, wmhrhx, y ^, x 5< èo k 4

0. Kaz0,z>,. . .,ze ks ` ihcdhtrkbprcirhsskce,

(`) ffeo x èo y , V2 d`rfsT(n) hxprhssz> èo z2 ke tmh pcj`r acrd, V> d`rfsT

(b) ffeo tmh sd`jjhst pcsktkvh kethihre subm tm`t z0+ z>+ . . . + ze4 z2 . . . ze, acr tmh v`juh cae ke (b). V2 d`rfsT

V@eswhr7 (`)x4

2

> , y4

0

>6 (n) z>4 bcs

1

+ k ske

1

, z24 bcs

2+ kske

26 (b) 0> 6 (o) -0T

8. VR_\D TRkdpjkay

(`) >(0 + 2k)(0 2k)> , wmhrh k 4

0. V2 d`rfsT

V@eswhr7 (`)02>8

;>8

kT

1. VR_\D T

Kaz 4 bcs + k ske , smcw tm`t 0

0 + z> 4

0

>(0 ktè ) èo hxprhss 0

0 z> ke ` skdkj`r acrd. V? d`rfsT

V@eswhr7 0

>(0 + kbct )T

9. VR_\D TAkeo tmh rccts ca tmh hqu`tkce (z k)2 4 k2, wmhrh ks ` rh`j bcestèt. V2 d`rfsT

21f

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37/195


(`) Rmcw tm`t tmh pckets rhprhshetkei tmh rccts ca tmh `ncvh hqu`tkce acrd è hqukj`thr`j trkèijh.V> d`rfsT

(n) Rcjvh tmh hqu`tkce Vz (0 + k)T2 4 (>k)2. V8 d`rfsT(b) Ka ks ` rcct ca tmh hqu`tkce `x> + nx + b4 )k ,

2

> + ( 0

>)k 6 (n) 0+2k , 0

2 0 +

2 6 (b)x1 + 2x? 2x> + ? 4 T

;. VR_\D

T_mh hqu`tkcez?>z2 + fz>0=z +?8 4 < m`s è kdìke`ry rcct. Cnt`ke `jj tmh rccts ca tmh hqu`tkceèo tmh v`juh ca tmh rh`j bcestètf . V= d`rfsT

V@eswhr7 ^ccts40 >k, 0 + >k, 2k, 2k , f 4 0?T

0 bcs?f 0

= + k ske?f 0

= , f4 , 2T00. VR_\D T

(`) Akeo tmh ffatm rccts ca uekty ke tmh acrd bcs + k ske , wmhrh : . V? d`rfsT

V@eswhr7 (`) bcs

>f

8

+ k ske

>f

8

, f4>, 0, T

0>. VR_\D T_mh bcdpjhx eudnhr z ks subm tm`t z >z 4

2 2k, wmhrh z ohecths tmh bcegui`th caz .

(`) Hxprhss z ke tmh acrd ` + nk, wmhrh ` èo n `rh rh`j eudnhrs. V2 d`rfsT

(n) Akeo tmh dcoujus èo `riudhet caz . V2 d`rfsT

(b) ^hprhshet z èo kts bcegui`th ke è @rièo okìr`d. V2 d`rfsT

V@eswhr7 (`)z 4

2 k 6 (n) dcoujus4>, `riudhet42

T

02. VR_\D T

Rmcw tm`t 0 + bcs >+ kske >

0 bcs> kske> 4 k bct . V> d`rfsT

Mhebh, smcw tm`t tmh rccts ca tmh hqu`tkce (z+ k)8 4 (z

k)8 `rh

bct

8 èo bct

>

8.V1 d`rfsT

Ohoubh tmh v`juhs ca bct>

8

+ bct>

>

8

èo bct>

8

bct>

>

8

. V? d`rfsT

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(`) Akeo w ke tmh acrdx + yk, wmhrh x, y ^. Rt`th tmh rh`j èo kdìke`ry p`rts caw. V2 d`rfsT(n) Hxprhss w ke pcj`r acrd. V2 d`rfsT

(b) Xskei oh Dckvrhs tmhcrhd, ohthrdkeh tmh bunh rccts caw. Ikvh ycur èswhr ke b`rthskè acrd.V8 d`rfsT

V@eswhr7 (`) k 6 (n)w 4 0(bcs

>+ kske

>) 6 (b)

2

> +

0

>k ,

2

> +

0

>k , -kT

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ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) 87 @e`jytkb Ihcdhtry

8 @e`jytkb Ihcdhtry

0. VR_\D TRmcw tm`t tmh dko-pckets ca tmh p`r`jjhj bmcros ca tmh p`r`ncj` y> 4 ?`x wktm iròkhet > jkh ce `str`kimt jkeh p`r`jjhj tc tmh x-`xks. V9 d`rfsT

>. VR_\D T_mh sud ca tmh okstèbh ca tmh pcket \ arcd tmh pcket (?,)>

01 +

y>

0>4 0 èo sfhtbm tmh hjjkpsh. V9 d`rfsT

2. VR_\D TAkeo tmh phrpheokbuj`r okstèbh arcd tmh bhetrh ca tmh bkrbjh x> + y> =x + >y + = 4 < tc tmh str kimtjkeh 2x + ?y4 >=. Mhebh, ffeo tmh smcrthst okstèbh nhtwhhe tmh bkrbjh èo tmh str`kimt jkeh. V9 d`rfsT

V@eswhr7 ?, 0T

?. VR_\D TRmcw tm`t x> +y> >`x>ny+ b 4 < ks tmh hqu`tkce ca tmh bkrbjh wktm bhetrh ( `, n) èo ròkus

> + n> b. V2 d`rfsT

B0

B>

B2

_mh `ncvh ffiurh smcws tmrhh bkrbjhs B0, B> èo B2 tcubmkei ceh èctmhr, wmhrh tmhkr bhetrhs jkh ce `str`kimt jkeh. KaB0 èoB> m`vh hqu`tkces x

> + y> 0= 4 < èox> + y> 01x + ?y + 8> 4 + 8y> 9?x + 0>y+ 081 4 ;y> >?x 0=y ; 4

2> (y+ 0)

>

>> 4 0 6 (n) Qhrtkbhs `rh (

2x 2 èo

y4>2

x + 0T

1. VR_\D T

Rmcw tm`t tmh p`r`dhtrkb hqu`tkces x 4 bt èo y 4 b

t, wmhrh b ks ` bcestèt, ohffeh ` pcket ce tmh

rhbtèiuj`r myphrncj` xy 4 b

>

. V> d`rfsT_mh pckets\,P,^ èo R, wktm p`r`dhthrsp,q,r èo s rhsphbtkvhjy, jkh ce tmh rhbtèiuj`r myphrncj`xy 4 b>.

?


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ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) 87 @e`jytkb Ihcdhtry

(`) Rmcw tm`t pqrs40 ka tmh bmcros \ P èo ^R`rh phrpheokbuj`r. V? d`rfsT(n) Akeo tmh hqu`tkce ca tmh jkeh p`sskei tmrcuim tmh pckets \ èo P. Ohoubh tmh hqu`tkce ca tmh

tèihet tc tmh rhbtèiuj`r myphrncj` `t tmh pcket \. V? d`rfsT

V@eswhr7 (n) y 40

pqx +

b

p +

b

q,y40

p> x +

>b

p T

9. VR_\D T_mh hqu`tkce ca è hjjkpsh ks 2x> + y> + 2 d`rfsT

V@eswhr7 (`) (x + 8)>

9 +

(y+ 8)>

>0 4 0 6 (n)B(

8,

8) , A0(

8,

8 +

0?) , A>(

8,

8

0?)T

=. VR_\D T_mh p`r`dhtrkb hqu`tkces ca ` bcekb `rh x 4 ` bcs 2 èo y 4 n ske + ?, wmhrh ` èo n `rh pcsktkvhbcestèts èo .

(`) Akeo tmh stèo`ro acrd ca tmh hqu`tkce ca tmh bcekb, èo kohetkay tmh typh ca bcekb. V2 d`rfsT

(n) Ka`4 n 4 8, ohthrdkeh èo sfhtbm tmh bcekb. V2 d`rfsT

V@eswhr7 (`) (x + 2)>

`> +

(y ?)>n>

4 0 6 (n) BkrbjhT

?0f

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ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) 17 Qhbtcrs

V@eswhr7 2x + >y 8z 401 6 ;0;

2=T

;. VR_\D TJht u4 bcs k+ ske g èo v 4 bcs k+ ske g, wmhrh k èo g `rh phrpheokbuj`r uekt vhbtcrs. Rmcwtm`t

0>|u v|4 ske0

>( ).

V8 d`rfsT

0k + g + (k + g + >f) èo r 4>k + >g f + (k + >g + f).

(`) Ohthrdkeh tmh bccroke`ths ca tmh pcket ca kethrshbtkce ca tmh twc jkehs. V2 d`rfsT

(n) Akeo tmh `buth èijh nhtwhhe tmh twc jkehs. V? d`rfsT

V@eswhr7 (`) (0, ) 6 (n) 22.1

T

00. VR_\D T_mh jkeh j m`s hqu`tkce r 4 >k+g+(>k+f) èo tmh pjèh m`s hqu`tkce r 4 k+ 2g f+(>k+f) + v(k + ?g).

(`) _mh pckets J èo Dm`vh bccroke`ths () rhsphbtkvhjy. Rmcw tm`tJ jkhsce j èo D jkhs ce . V2 d`rfsT

Ohthrdkeh tmh skeh ca tmh `buth èijh nhtwhhe tmh jkeh JDèo tmh pjèh èo tmh smcrthst okstèbharcd Jtc . V1 d`rfsT

V@eswhr7 >

2

2=, >

2T

0>. VR_\D T

Ka tmh èijh nhtwhhe tmh vhbtcrs `4

?

=

èo n4

0

p

ks 028, ffeo tmh v`juh cap. V1 d`rfsT

V@eswhr72T

02. VR_\D T_mh pjèhs 0 èo > wktm hqu`tkces x

y+ >z 4 0 èo >x+y

z 4 < rhsphbtkvhjy kethrshbt ke tmh

jkeh j . _mh pcket @ m`s bccroke`ths (0,. V> d`rfsT

(n) Hxpj`ke wmy tmh vhbtcr

00

>

>0

0

ks ke tmh okrhbtkce ca j. Mhebh, smcw tm`t tmh hqu`tkce

caj ks

r4

d`rfsT

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(h) Ohthrdkeh tmh okstèbh arcd@ tc j . V2 d`rfsT

V@eswhr7 (`) = 6 (h)

2?

28T

0?. VR_\D T_mh pcsktkce vhbtcrs ca tmh pckets@,N ,BèoO, rhj`tkvh tc è crkike, `rhk + 2g,8k2g, (x2)k1gèo (x + 2)k rhsphbtkvhjy.

(`) Rmcw tm t, acr èy v`juh cax, @NBO ks ` p`r`jjhjcir`d. V2 d`rfsT

(n) Ohthrdkeh tmh v`juh cax acr wmkbm @NBO ks ` rhbtèijh. V? d`rfsT

V@eswhr7 (n) x 4 0T

08. VR_\D T

_mh okìr`d nhjcw smcws ece-bcjjkeh`r pckets C, @ èo N, wktm \ ce tmh jkeh C@ subm tm`t C\ 7\ @4 > 7 0 èo P ce tmh jkeh @N subm tm`t @P7 PN 4 > 7 2. _mh jkehs \ P èo CN prcoubho dhht

`t tmh pcket ^. KaC@ 4 ` èo

CN 4 n,

C

^

@\

N

P

(`) smcw tm`t\ P40

08` +

>

8n, V8 d`rfsT

(n) ffeo tmh pcsktkce vhbtcr ca^, rhj`tkvh tc C, ke thrds can. V8 d`rfsT

V@eswhr7 (n) ?nT

01. VR_\D T_wc str`kimt jkehs j0 èo j> m`vh hqu`tkces>x+ ? 4 >y? 4 z? èo >x 4 y+ 0 4z + 2rhsphbtkvhjy. Ohthrdkeh wmhtmhr j0 èo j> kethrshbt. V9 d`rfsT

V@eswhr7 Ec kethrshbtce.T

09. VR_\D T_mh pcsktkce vhbtcrs ca tmh pckets@,N èo B, wktm rhsphbt tc tmh crkikeC, `rh`,n èo b rhsphbtkvhjy._mh pckets J, D, \ èo P `rh tmh dkopckets caC@, N B, CN, èo @B rhsphbtkvhjy.

(`) Rmcw tm`t tmh pcsktkce vhbtcr ca èy pcket ce tmh jkehJD ks 0

>` +

0

>(n + b `) acr scdh sb`j`r

, èo hxprhss tmh pcsktkce vhbtcr ca èy pcket ce tmh jkeh \ P ke thrds ca`, n èo b. V1 d`rfsT(n) Akeo tmh pcsktkce vhbtcr ca tmh pcket ca kethrshbtkce ca tmh jkeh JDèo tmh jkeh \ P. V? d`rfsT

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V@eswhr7 (n) 81.= 6 (b)

0?, 0?

2T

>;. VR_\D T_mrhh vhbtcrs `4 pk +qg, n48k + g èo b4 ?k + 9g `rh subm tm`t ` èo n `rh phrpheokbuj`r èotmh sb`j`r prcoubt ca`èo b ks 9=.

(`) Ohthrdkeh tmh v`juhs capèo q. V? d`rfsT

(n) Akeo tmh èijh nhtwhhe`èo b. V2 d`rfsT

V@eswhr7 (`)p 4 >, q4 0< 6 (n) 0=.?T

2

12? 6 (b) 08x + >


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ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) 97 Jkdkts èo Bcetkeukty

(`) Akeo jkdx0

a(x), jkdx0+

a(x), jkdx0

a(x) èo jkdx0+

a(x).

(n) Ohthrdkeh wmhtmhra ks bcetkeucus `t x 40 èo x 4 0.

V@eswhr7 (`) < , < ,

> , < 6 (n) bcetkeucus `t x 40 , oksbcetkeucus `tx 4 0T

9. VR_\D T

Ikvhe x 5< èo a(x) 4

x, ffeo jkdm

xT

=. VR_\D T_mh ir`pm ca ` auebtkce aks `s acjjcws7

(`) Rt`th tmh ocd`ke èo rèih caa. V> d`rfsT

(n) Rt`th wmhtmhra ks ` ceh-tc-ceh auebtkce cr ect. Ikvh ` rh`sce acr ycur èswhr. V> d`rfsT

(b) Ohthrdkeh wmhtmhraks bcetkeucus cr ect `t x 40. Ikvh ` rh`sce acr ycur èswhr. V2 d`rfsT

V@eswhr7 (`) O4{x72x :0, 0: x>} , ^4{y70: y :>} 6(n) a ks ect ceh tc ceh auebtkce 6 (b) aoksbcetkeucus `tx 40.T

;. VR_\D T_mh auebtkce a ks ohffeho ny

a(x) 4

x + 0, x 06

|x| 0, ctmhrwksh.

(`) Akeo jkdx0

a(x). V2 d`rfsT

(n) Ohthrdkeh wmhtmhra ks bcetkeucus `t x 40. V> d`rfsT

V@eswhr7 (`) < 6 (n) [hs.T

0


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ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) =7 Okhrhetk`tkce

= Okhrhetk`tkce

0. VR_\D TOkhrhetk`th wktm rhsphbt tc x èo skdpjkay ycur èswhr `s a`r `s pcssknjh7

(`) x>

x + 0x> + x 0 ,(n) h>xV> bcs(2x) 2 ske(2x)T.

V@eswhr7 (`) >x(x >)(x> + x 0)> 6 (n)02h

>x bcs(2x)T

>. VR_\D TOkhrhetk`th wktm rhsphbt tc x èo skdpjkay ycur èswhr `s a`r `s pcssknjh7

(`) bcs x + ske x

bcs x ske x,

(n) xe jcih x.

V@eswhr7 (`) >

0 ske>x 6 (n) xe0(0 + e je x)T

2. VR_\D T

(`) Akeo oy

ox wmhrh

k. y4 (>x

0)2(2x + >)? èo hxprhss ycur rhsujt ke tmh acrd ca kts a`btcrs,

kk. y4 h2x(>bcs>x + 2ske >x).

(n) Kax4 t 0t

èo y 4 >t +0

t, wmhrh t ks ` ece-zhrc p`r`dhthr, prcvh tm`t

oy

ox 4 > 2

t> + 0.

Ohoubh tm`t0: oyox

:>.

V@eswhr7 (`) (k) ?>x(>x 0)>(2x + >)2 6 (kk)02h>x ske>xT

?. VR_\D T

Akeo tmh x-bccroke`th ca tmh pcket ce tmh burvh y 4 je x

x> (x 5d

oy

ox+ (d> +p>)y4 f

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(b) Ikvhey 4 je(0 + x) x +0>

x>, smcw tm`t oy

ox < acr `jj v`juhs cax 50.

V@eswhr7 (`) tè? xT

1. VR_\D T

Ikvhe tm`ty4bcs x ske xbcs x + ske x

, smcw tm`t o>y

ox>+ >y

oy

ox4 +ox+ h, wmhrh `, n, b, o, h `rh bcestèts, m`s tmhacjjcwkei bm`r`bthrkstkbs7

(`) Kt ks syddhtrkb`j `ncut tmh y -`xks,

(n) Kt p`sshs tmrcuim tmh pcket (>, 0=) èo m`s iròkhet zhrc `t tmks pcket,(b) y4 < wmhe x 4 0.

Rmcw tm`t n 4 o 4 < èo ffeo tmh v`juhs ca`, b èo h. Rfhtbm tmh burvh èo ikvh tmh bccroke`ths cakts turekei pckets.

V@eswhr7 `4 >, b 401,h 4 0?T

=. VR_\D T

(`) Ikvhe tm`ty 4 (x + >)>(2x 0)2, ffeo oyox

`s ` prcoubt ca kts a`btcrs.

(n) Kay4

hx

0 + x> , smcw tm`t (0 + x

>

)

oy

ox + (0 + x)

>

y4 )(08x + 01)(2x 0)>T

;. VR_\D T_mh p`r`dhtrkb hqu`tkce ca ` burvh `rh

x4 ` bcs2 , y4 ` ske2

wmhrh ` ks ` pcsktkvh bcestèt èo .Akeo tmh hqu`tkce ca tmh tèihet `t tmh pcket wktm tmh p`r`dhthr . _mks tèihet dhhts tmh `xhs `t

J èo D. \rcvh tm`t tmh jheitm caJDks keohpheohet ca .

V@eswhr7 y bcs + x ske 4 ` ske bcs , JD4`T

0y

ox>+ ?

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0>. VR_\D T

(`) Acr tmh burvh y 4 ske x bcs2 x, wmhrh 4

0

?bct x(bcs> x 2ske> x)>

ce tmh bceoktkce tm`t x4 , , y4 0 0t

, (t 5y x + 0 4 , y 4 t2. Hxprhss oy

ox ke thrds cat. Akeo tmh hqu`tkce

ca tmh tèihet tc tmh burvh `t tmh pcket \(p>, p2).

V@eswhr7 oy

ox 42

> t 6 >y 4 2px p2T

08. VR_\D T

Kay 4 2x + ske x =ske

0

>x

, ffeo

oy

oxèo hxprhss ycur èswhr ke thrds ca bcs

0

>x

.

Ohoubh tm`t oy

ox < acr `jj v`juhs cax.

V@eswhr7 >

bcs

0

>x 0

>T

01. VR_\D T

(`) Akeo tmh hqu`tkce ca tmh `sydptcths ca tmh burvhy4 x 2

(x >)(x + 0) .(n) Akeo tmh pckets wmhrh tmh burvh kethrshbts tmh `xhs, èo ffeo tmh st`tkce ry pckets ce tmks burvh.

(b) Rfhtbm tmks burvh.

(o) Akeo tmh v juhs caf subm tm`t tmh hqu`tkce (x 2) 4f(x >)(x + 0) ochs ect m`vh rh`j rccts.

V@eswhr7 (`)x4 >, x40, y 4 < 6 (n) (2,


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V@eswhr7 (

>x. Akeo `jsc tmh d`xkdudèo dkekdud pckets, `s whjj `s èy pckets ca kehbtkce ce tmks burvh.

(n) Rfhtbm tmks burvh.

(b) Akeo tmh v`juh caf ka tmh hqu`tkce 0

>x2 x> >x4 f m`s ` rhph`tho rcct, èo st`th tmks rcct.

V@eswhr7 (`) (, ?) , \cket ca kehbtkce ( >

2, ??

>9) 6

(b)f 4?, rhph`tho rcct4>6 f 4 >9

, rhph`tho rcct4>2

T

0;. VR_\D TKay 4 htèx, smcw tm`t

o>y

ox> 4

oy

ox(0 + tè x)>.

>x `t tmh pcket wmhrh tmh x-bccroke`th ks

2 ks 2x + >?y4 >?

2.

>0. VR_\D T_mh p`r`dhtrkb hqu`tkces ca ` burvh `rh x 4 t> >, y 4 t2 2. Akeo tmh hqu`tkce ca tmh ecrd`j tctmh burvh `t tmh pcket wmhrh tmh p`r`dhthr t 4 >.

V@eswhr7 x + 2y4 09T

>>. VR_\D TOkhrhetk`th wktm rhsphbt tc x

(`) (x> + >x)hx>+>x,

(n) 0 x>

0 + >x,

skdpjkaykei ycur èswhrs.

V@eswhr7 (`) >(x + 0)2hx>+>x 6 (n)

2x> + >x + 0

(0 + >x)

2

>

T

>2. VR_\D TAkeo tmh hqu`tkces ca tmh tèihet èo ecrd`j tc tmh burvh x>y+ xy> 4 0> `t tmh pcket (0, ?).

V@eswhr7 =x 9y4 21 6 9x + =y+ >8 4 ?. VR_\D T@ burvh m`s tmh hqu`tkce y> 4x>(x + 2).

Rmcw tm`t tmh burvh ks syddhtrkb `ncut tmh x-`xks.

Rmcw tm`t acr `jj pckets ce tmks burvh, x 2.Akeo bccroke`ths ca tmh turekei pckets ca tmks burvh.

Rfhtbm tmh burvh. Rmcw bjh`rjy tmh sm`ph ca tmh burvh eh`r tmh crkike.

Akeo tmh `rh` ncueoho ny tmh jccp ca tmks burvh.

88f

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_mh nrhòtm èo jheitm ca tmh buncko ks x bd èo >x bd rhsphbtkvhjy èo kts mhkimt ks y bd. Ikvhetm`t tmh tct`j sura`bh `rh` ca tmks scjko ks >?. Rmcw tm`t

y 4 0

>?xV;1T.

V2 d`rfsTKa tmh vcjudh ca tmks scjko ks Q bd2, hxprhssQ ke thrds cax. Mhebh, smcw tm`tQ `tt`kes kts d`xkdud

wmhe x 4 ?V;1 + >xy4 8 `t tmh pcket (>, 0). V2 d`rfsT

V@eswhr7 -2T

?>. VR_\D T_mh p`r`dhtrkb hqu`tkces ca ` burvh `rh

x4 shb t tè t6 y4 bcshb t bct t,

wktm . Rmcw tm`t

oy

ox 40 + ske t

0 + bcs t. V? d`rfsT

_èihet tc tmh burvh `t tmh pcket @, wktmt 4 tè0

2

?

, dhhts tmh tèihet tc tmh burvh `t tmh pcket

N, wktm t 4 tè0?

2

, `t pcket E. Akeo tmh bccroke`ths caE. V= d`rfsT

V@eswhr7 E4

9

09,

9

09

T

?2. VR_\D T_mh hqu`tkce ca ` burvh ks

y 4 x>

x> 2x + > .Akeo tmh `sydptcths èo tmh st`tkce`ry pckets ca tmh burvh. V= d`rfsT

Rfhtbm tmh burvh. V? d`rfsTOhthrdkeh tmh eudnhr ca rh`j rccts ca tmh hqu`tkce f(x 0)>(x >) 4x> wmhrh f 5, y 4 0 6 0 rcctT

??. VR_\D T

Akeo oy

ox ke thrds cax kax 4 h

t èo y 4

ht. V? d`rfsT

V@eswhr7 h

(je x)>

>

x je xT

?8. VR_\D T

_mh hqu`tkce ca ` burvh ks y 4h>fx 0h>fx + 0

wmhrh f ks ` pcsktkvh bcestèt.

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(`) Rmcw tm`t oy

ox5< acr `jj v`juhs cax. V2 d`rfsT

(n) Rmcw tm`t oy

ox+ fy> 4f. Mhebh, smcw tm`t

o>y

ox> < acr x< èo o

>y

ox> < acr xy

ox> 4f>y>.

V1 d`rfsT

?9. VR_\D T_mh ir`pms cay 4 x2 +`x> +nx+b p`sshs tmrcuim (2, >0) èo m`s st`tkce`ry pckets wmhe x 4 >èo x 4

>. Akeo tmh v`juhs ca`, n èo b. V8 d`rfsT

Akeo tmh bccroke`ths ca tmhsh st`tkce`ry pckets èo ohthrdkeh ka tmhy `rh jcb`j hxtrhduds. Akeo `jsctmh pcket ca kehxkce ca tmh burvh. V9 d`rfsT

Ohthrdkeh tmh sht cax sc tm`t oy

ox: , b 40> 6 (>,->=) ks jcb`j dkekdud , (->,?) ks jcb`j d`xkdud 6pcket ca kehxkce ks () 6{x7>: x :>}T

?=. VR_\D T@ burvh m`s p`r`dhtrkb hqu`tkces x 4 h>t >t èo y 4 ht +t. Akeo tmh iròkhet ca tmh burvh `t tmhpcket wktm t 4 je>. V8 d`rfsT

V@eswhr7 0

>T

?;. VR_\D T@ burvh wktm hqu`tkce y4 x2 +px> + qx + r buts tmh y-`xks `t y 42? èo m`s st`tkce`ry pckets `tx4 2 èox 4 8. Akeo tmh v`juhs cap, q, èo r. V1 d`rfsT

Rmcw tm`t tmh burvh buts tmh x-`xks cejy `t x 4 0, èo ffeo tmh iròkhet ca tmh burvh `t tm`t pcket.V9 d`rfsT

Rfhtbm tmh burvh. V> d`rfsT

V@eswhr7 p40> , q4 ?8 , r 42? 6 >?T

8,wktm ` 5< èo t ^.Ohthrdkeh tmh v`juhs cat wmhe tmh burvh buts tmh y -`xks èo sfhtbm tmh burvh. V? d`rfsT

Rmcw tm`t ox

ot>

+oy

ot>

4`>(0 + ;t>)>. V2 d`rfsT

V@eswhr7 t4


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80. VR_\D TAkeo tmh hqu`tkce ca tmh ecrd`j tc tmh burvh x>y+ xy> 4 0> `t tmh pcket (2, 0). V1 d`rfsT

V@eswhr7 y4 08

9x 2=

9T

8>. VR_\D T

Ikvhe tm`ty4 hx bcs x, ffeo oy

ox èo

o>y

ox> wmhe x 4 y

ox> 4 x

(x + 0)(x >) .

Rmcw tm`ta(x): < acr `jj v`juhs cax ke tmh ocd`ke caa. V8 d`rfsT

Rfhtbm tmh ir`pm cay 4a(x). Ohthrdkeh kaa ks ` ceh tc ceh auebtkce. Ikvh rh`sces tc ycur èswhr.V1 d`rfsT

Rfhtbm tmh ir`pm cay 4|a(x)|. Hxpj`ke mcw tmh eudnhr ca tmh rccts ca tmh hqu`tkce|a(x)|4 f(x >)ohpheos ce f. V? d`rfsT

V@eswhr7 a ks ect ceh tc ceh auebtkce. Kaft, y 4> +>t

. Akeo tmh hqu`tkce ca tmh

ecrd`j tc tmh burvh `t tmh pcket @(2, ?). V9 d`rfsT_mh ecrd`j ca tmh burvh `t tmh pcket @ buts tmh burvh ì`ke `t pcket N. Akeo tmh bccroke`ths caN.

V? d`rfsT

V@eswhr7 x + y+ 0 4 < 6 N (0, + >

oy

ox+ xy 4 +0

xwmhrhx 5


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V@eswhr7

oy

ox4 (0 x>)y>. V? d`rfsT

10. VR_\D T

Akeo tmh bccroke`ths ca tmh st`tkce`ry pckets ce tmh burvh y 4 x2

x> 0 `eo ohthrdkeh tmhkr e`turh.V0< d`rfsT

Rfhtbm tmh burvh. V? d`rfsT

Ohthrdkeh tmh eudnhr ca rh`j rccts ca tmh hqu`tkcex2 4f(x>0), wmhrhf ^, wmhef v`rkhs.V2 d`rfsT

V@eswhr7 ( ) ks jcb`j d`x.

0 rh`j rcct acr2

2

> : f :

2

2

> , > rh`j rcct acrf 42

2

> , 2 rh`j rccts acrf :2

2

> cr f 5

2

2

> T

1>. VR_\D T

Kay 4 ske x bcs xske x + bcs x

, smcw tm`t o>y

ox> 4 >y

oy

ox. V1 d`rfsT

12. VR_\D T

Rmcw tm`t tmh burvh y 4 x

x> 0 ks `jw`ys ohbrh`skei. V2 d`rfsTOhthrdkeh tmh bccroke`ths ca tmh pcket ca kehxkce ca tmh burvh, `eo st`th tmh kethrv`js acr wmkbm tmh

burvh ks bceb`vh upw`ros. V8 d`rfsT

Rfhtbm tmh burvh. V2 d`rfsT

V@eswhr7 ((0 + bcs ).

V? d`rfsT

1>f

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(`) Rmcw tm`t tmh vcjudhQca tmh bceh ks ikvhe ny Q 4 0

2(` x)(` + x)>. V2 d`rfsT

(n) Ohthrdkeh tmh v`juh cax acr wmkbm Qks d`xkdud èo ffeo tmh d`xkdud v`juh caQ. V1 d`rfsT

(b) Rfhtbm tmh ir`pm caQ ì`kest x. V> d`rfsT

(o) Ohthrdkeh tmh r`th `t wmkbm Q bmèihs wmhe x 4

0

> ` ka x ks kebrh`skei `t ` r`th ca

0

0< ` phrdkeuth. V? d`rfsT

V@eswhr7 (n)x4`

2 ,

2>

=0`2 6 (o) 0

? `t tmh pckets wmhrh tmh burvh dhhts tmh bccroke`th`xhs. V1 d`rfsT

V@eswhr7 < , 0T

99. VR_\D T_mh p`r`dhtrkb hqu`tkces ca ` burvh `rh x 4 ske èo y 4 0 bcs . Akeo tmh hqu`tkce ca tmhecrd`j tc tmh burvh `t ` pcket wktm p`r`dhthr

0

>. V9 d`rfsT

V@eswhr7 y 4x + >

T

9=. VR_\D T@ burvh ks ohffeho kdpjkbktjy ny tmh hqu`tkce x> + xy+ y> 4 2.

(`) Rmcw tm`t

oy

ox +

>x + y

x + >y 4 ) èo (0, ->). V8 d`rfsT

(o) Rfhtbm tmh burvh. V> d`rfsT

V@eswhr7 (n)0>

, ->T

9;. VR_\D T@ rhbtèijh wktm ` wkotm >x ks kesbrknho ke ` bkrbjh ca bcestèt ròkus r .

(`) Hxprhss tmh `rh` @ ca tmh rhbtèijh ke thrds cax èo r . V> d`rfsT

(n) Rmcw tm`t tmh rhbtèijh ks ` squ`rh ca skoh r

> wmhe @ m`s ` d`xkdud v`juh. V8 d`rfsT

V@eswhr7 (`) @ 4 ?x

r> x>T

=bcs x + ske >xacr ks smcwe nhjcw.

18f

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_mh pckets @ èo B`rh jcb`j hxtrhdud pckets. _mh pckets N , O , H èo A`rh pckets ca kehxkce.

(`) Ohthrdkeh tmh bccroke`ths ca

k. tmh pckets ca jcb`j hxtrhdud. V8 d`rfsT

kk. tmh pckets ca kehxkce. V8 d`rfsT

(n) Rt`th tmh kethrv js wmhrh tmh ir`pm ks bceb`vh upw`ro. V0 d`rfsT

(b) B`jbuj`th tmh `rh` ca tmh rhikce ncueoho ny tmh burvh èo tmh x-`xks. V? d`rfsT

V@eswhr7 (`) (k) @4

1,2

2

>

, B4

8

1 , 2

2

>

6

(kk)N4

>, , , 1.


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(`) Rmcw tm`t tmh vcjudh ca w`thr Q d2 wmhe tmh ohptm ca w`thr ke tmh t`ef ks m d ks ikvhe ny

Q 401

98m(m> + 0=m + 08

2===6 (b) 22.9T

==. VR_\D T_mh hqu`tkce ca ` burvh ks y 4 x(x >)2.

(`) Akeo tmh sht ca v`juhs cax acr wmkbm y, >9

01

, Kehxkce4(0, 0), (>,


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ffjhhd`tms.bcd R_\D D@_MHD@_KBR (_) ;7 Kethir`tkce

V@eswhr7 (`) 8x + >ske >x 2>

bcs >x + bT

1. VR_\D T

(`) Hv`ju`th

0

+ > ox, jh`vkei ycur èswhr ke ` acrd kevcjvkei jci`rktmds.

(n) Rmcw tm`t

0 ox4 0

2+

0

>

2.

V@eswhr7 (`) 0

>+ je

>

2

T

9. VR_\D T

(`) Akeo

> + bcs x

ske> x ox.

(n) Xskei tmh sunstktutkce x 4 >ske , cr ctmhrwksh, hv`ju`th

0? x> ox, ikvkei ycur èswhr

bcrrhbt tc twc skiekffbèt ffiurhs.

V@eswhr7 (`)>bct x bcshb x + B6 (n) ,-0), dkekdud pcket 6 (n) (>,-0), d`xkdud pcket , `sydptcths `rh x 4 0, x 4 2, y 4 < 6@rh`428T

;. VR_\D T

Akeo tmh hx`bt v`juh ca

0>

x

x + 2 ox.

V@eswhr7 =8

T

0 x ox.

90f

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(n) Ikvhe tm`t 0= ?x x>(? 2x)(0 + x)>

@

? 2x + N

0 + x+

B

(0 + x)>, smcw tm`t @ 4 > èo ohthrdkeh tmh

v`juhs caN èo B.

Mhebh, smcw tm`t

0(? 2x)(0 + x)> ox4

9

2je > +

2

>

V@eswhr7 (`) 0 ?

6 (n) N 4 0, B4 2T

00. VR_\D T@ ncwj ke tmh sm`ph ca ` mhdkspmhrh wktm ` ròkus ` èo kts rkd mcrkzcet`j, ks ffjjho wktm ` jkquko tc `mhkimt cam uekt. Rmcw, tmrcuim kethir`tkce, tm`t tmh vcjudh ca tmh jkquko ke tmh ncwj ks

0

2m>(2` m).

0>. VR_\D T

(`) Rmcw tm`t

1

x bcs> x ox4

? 220;>

.

(n) Xskei tmh sunstktutkce u 4 x>, cr ctmhrwksh, hv`ju`th

>>

x + 0

x + 0ox.

(n) Rmcw tm`t

? + ?x + 0>

(>x + 0)(x> + ?) ox4 je 198.

V@eswhr7 (`) ?

2(x + 0)

2> >(x + 0) 0> + bT

0?. VR_\D TXskei kethir`tkce, ffeo th `rh` ca tmh ffekth rhikce ncueoho ny tmh burvh

x +

y4

èo tmh bccroke`th `xhs, ka` ks ` pcsktkvh bcestèt.

V@eswhr7 0

1`>T

08. VR_\D TAkeo tmh bccroke`ths ca tmh pckets \, P wmhrh tmh burvh >y4 x + 2 dhhts tmh p`r`ncj` y> 4 ?x.

Akeo tmh `rh` ncueoho ny tmh `rb \ P ca tmks p`r`ncj` èo tmh jkeh \ P.

V@eswhr7 \(0, >) , P(;, 1) 6 =

2T

9>f

fjhhd`

tms.bcd

f

fjhhd`

tms.bcd

f

fjhhd`

tms.bcd


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01. VR_\D TNy uskei tmh sunstktutkce x 4 ske , cr ctmhrwksh, hv`ju`th

0>

02

0

x>

0

x>

ox

jh`vkei ycur èswhr ke tmh acrds ca suros.

V@eswhr7 >

>

2T

09. VR_\D T

(`) Hxprhss 09 + x

(? 2x)(0 + >x) ke p`rtk`j ar`btkces. Mhebh cr ctmhrwksh, smcw tm`t

0>

02

09 + x

(? 2x)(0 + >x) ox4 0

1 (0; je > + ; je2).

(n) Xskei tmh sunstktutkce x 4 ske , cr ctmhrwksh, ffeo

x>

(0 x>) 0>ox.

V@eswhr7 (`) 8

? 2x + 2

0 + >x 6 (n)

0

>

1

2

?

T

0=. VR_\D T

_mh okìr`d `ncvh smcws ` pcrtkce ca tmh burvh y 4 =

x ? tm`t kethrshbts tmh x-`xks `t tmh pcket @.

_mh tèihet tc tmh burvh `t tmh pcket\(0, ?) kethrshbts tmhx-`xks `t tmh pcketP. Akeo tmh bccroke`thscaP.

Akeo tmh `rh` ca tmh smòho rhikce ncueoho ny tmh burvh @\èo tmh jkeh shidhets \ P èo P@.

V@eswhr7 P4

2

>, 8T

0;. VR_\D T

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_mh okìr`d smcws ` mhdkspmhrkb j ncwj wktm ròkus ` èo kts rkd mcrkzcet`j. Rmcw tm`t wmhe tmhohptm ca w`thr ke tmh ncwj ks y , tmh vcjudh Qca w`thr ke tmh ncwj ks ikvhe ny

Q 40

2y>(2` y).

Kektk`jjy, tmh ncwj ks hdpty. Y`thr ks tmhe pcurho ketc tmh ncwj `t ` bcestèt r`th. _mh tkdh t`fhe tc

ffjj tmh ncwj ks _. Akeo, ke thrds ca_, tmh tkdh t`fhe acr y tc nhbcdh 0

>`. Akeo, ke thrds ca_ èo `,

tmh r`th `t wmkbm tmh w`thr jhvhj rkshs wmhe y 4 0

>`.

V@eswhr7 8

01_ 6

=`

;_T

> 4 x èo y 4x+ >.Akeo tmh `rh` ncueoho ny tmh twc ir`pms.

Akeo `jsc tmh vcjudh ca tmh scjko ihehr`tho wmhe tmks `rh` ks rct`tho tmrcuim > `ncut tmh y-`xks.

V@eswhr7 (0, 0) , (?, >) 6 ;>

6 9>

8T

>0. VR_\D T

(`) Xskei tmh sunstktutkce x 4 ` tè , cr ctmhrwksh, smcw tm`t

`

+ x>)8>

ox4 = 8>

0>` ,

wktm ` 5x> ? , ox, ikvkei ycur èswhr bcrrhbt tc tmrhh skiekffbèt ffiurhs.

V@eswhr7 (n) 2.=;T

>>. VR_\D T

(`) Akeo tmh v`juhs ca tmh bcestèts @ èo N subm tm`t bcs x

@(2 bcs x+ ? ske x) + N(

2ske x+

?bcs x). Mhebh cr ctmhrwksh, ffeo tmh v`juh ca

>

bcs>, cr ctmhrwksh, ffeo 2>

8 xx 0 ox ikvkei ycur èswhr

bcrrhbt tc tmrhh skiekffbèt ffiurhs.

V@eswhr7 (`) @ 4 2

>8

, N 4 ?

>8

, 286 (n) 0.2>T

>2. VR_\D THv`ju`th

(`)

1

o,

(n)

h>h

0

x je x ox,

V@eswhr7 (`) 0

>je> 6 (n) je >T

>?. VR_\D TAkeo tmh `rh` ncueoho ny tmh burvhs y 4 0 x> èo y 4 x 0.

V@eswhr7 ;

>T

>8. VR_\D T

(`) Ny uskei ` sukt`njh sunstktutkce cr ctmhrwksh, ffeo tmh v`juh ca

>

0

x 0(>x

0)>

ox, ikvkei ycur

èswhr bcrrhbt tc tmrhh skiekffbèt ffiurhs.

(n) Rmcw tm`t o

ox(tè2 ) 4 2 tè? + 2 shb> 2. Mhebh, ohthrdkeh tmh v`juh ca

?

, èo tmh burvh y4

0 + ske x. Akeo tmh

vcjudh ca tmh scjko acrdho wmhe ^ ks rct`tho tmrcuim acur rkimt èijhs `ncut tmh x-`xks.

V@eswhr7 >

(+ >)T

>9. VR_\D TRmcw tm`t

o

ox(tè2 x) 4 2 shb? x 2shb> x.

Mhebh, ohthrdkeh tmh v`juh ca

?

=. VR_\D T

98f

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V@eswhr7 0

>(x 0) 0

>(x + 0) 6 . VR_\D TB`jbuj`th tmh vcjudh ca tmh scjko ihehr`tho wmhe tmh `rh` ncueoho ny tmh burvhs y 4 x> èo y> 4 =xks rct`tho bcdpjhthjy `ncut tmh x-`xks.

V@eswhr7 ?=

8T

22. VR_\D TRfhtbm tmh burvh y 4 x(x+ 0)(> x). Akeo tmh `rh` ncueoho ny tmh burvh y 4x(x+ 0)(> x) èotmh x-`xks.

V@eswhr7 20

0>T

2?. VR_\D TRmcw tm`t h

0x je x ox4

0

?

h> + 0

.

Mhebh, ffeo tmh v`juh ca

h0

x(je x)> ox bcrrhbt tc tmrhh ohbkd`j pj`bhs.

V@eswhr7 0.8;9T

28. VR_\D T

Akeo

h0

(>x + 0) je x ox. Ikvh ycur èswhr ke thrds cah.

V@eswhr7 0

>

(h> + 2)T

21. VR_\D TRfhtbm, ce tmh s`dh `xhs, tmh ir`pms cay> 4x èo y 4 > x. Rmcw tmh bccroke`ths ca tmh pckets cakethrshbtkce nhtwhhe tmh ir`pms.

B`jbuj`th tmh `rh` ncueoho ny y> 4x èo y 4 > x.KaQ0 ks tmh vcjudh ca tmh scjko acrdho wmhe tmh `rh` `ncvh tmh x-`xks ncueoho ny y

> 4x, y 4 > x,èo tmh x-`xks ks rct`tho bcdpjhthjy `ncut tmh y-`xks, èo Q> ks tmh vcjudh ca tmh scjko acrdho wmhetmh `rh` ueohr tmh x-`xks ncueoho ny y> 4x, y 4 > x, èo tmh x-`xks ks rct`tho bcdpjhthjy `ncuttmh y -`xks, ffeo Q07 Q>.

V@eswhr

7 @rh`4

;

> 6 Q07 Q>4 ? 7 >2T

29. VR_\D T

Ny uskei ` sukt`njh sunstktutkce, ffeo tmh v`juh ca

2?x èo tmh burvhy4 0> x2. Rmcw tm`t tmhrh ks ` pcket ca kehbtkce ce tmh burvh y 4 0> x2 `t x 4 ?xèo tmh burvh y 4 0> x2

._mh `rh` ncueoho ny tmh jkeh y 4 0> ?x èo tmh burvh y 4 0> x2 ks rct`tho tmrcuim acur rkimtèijhs `ncut tmh x-`xks. B`jbuj`th tmh vcjudh ca tmh scjko ca rhvcjutkce acrdho.

99f

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V@eswhr7 (), (>, >, ?) 6 Qcjudh40;>T

2;. VR_\D T

Akeo

x

>x 0 ox.

V@eswhr7 (>x 0) 0> (x + 0)

2 + BT

? ox,

(n)

x>hx

2ox

V@eswhr7 (`)? x> + B 6 (n)02 hx2

+ BT

?0. VR_\D T_mh iròkhet ca tmh tèihet tc ` burvh `t tmh pcket (x, y) wmhrh x 52, v`rkhs kevhrshjy wktm (x 2),èo tmh burvh p`sshs tmrcuim tmh pckets (?, je(x 2).Rfhtbm tmh burvh.

Akeo tmh ffekth `rh` ncueoho ny tmh burvh, tmh x-`xks èo tmh jkeh x 4 1.

V@eswhr7 >(2je2 >)T

?>. VR_\D TRmcw tm`t 2

>

x>

(x 0)2 ox400

= + je>.

V1 d`rfsT

?2. VR_\D TAuebtkce a ks ohffeho `s

a(x) 4

x(x ), ,> ske(x ), >x2.

(`) R

new stpm mathematics (t) chapter past year question

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