matemáticas anaya 3º eso solucionario tema 3

7/22/2019 Matemticas Anaya 3 ESO Solucionario Tema 3

1/29

_icuokilas n cns notkvkhnhas ha onhn apernja?

Qlkhnh ?. Vrieraskilas

VEKLN 1;

VNRN AGVATNR

Qln prieraskl nsigbrisn

_upl qua tkalas uln midn ha pnpac ha 6,>8 gg ha erisir. Onhn vaz qua cn pckaenssa hupckon su erisir. Ounlhi mns maomi saks i skata hibcaoas, yn li puahas hibcnr-cn gs, pari kgnekln qua puhkarns mnoarci hkaz, vaklta a, klocusi, okloualtn vaoas.^u erisir oraas qua ccaenrn n nconlznr asa pnpac9

Oigpruabn qua oil >6 hibcaoas suparnrns ac erisir hac ckbri gs eirhi ha cnbkbckitaon. [, gs nsigbrisi, oil 22 hibcaoas ibtalhrns ul erisir gnyir qua cnncturn ha cn tirra Akjjac (?28 g).

Vnrn ranckznr tus ocoucis, utkckzn acjnotir oilstnltaal cn oncoucnhirn. Raouarhn4

_k tkalas oncoucnhirn oil VNLXNCCN_ALOKCCN, oil cn saoualokn4

2 * * 7 7 7

sa ibtkalal cis rasuctnhis 8, =, >:,

Onhn vaz qua hns n cn taocn 7, sa guctkpckon pir 2 (sa hupckon) ac lgari quamny al cn pnltnccn.

_k ajaotns 2* *6,>8 7 7 7, ibtkalas, al gkcgatris, ac erisir nconlznhi >828? l vaoas

pir ac pnpac trns l hibcaoas.

Oil oncoucnhirn ha VNLXNCCNHA_ORKVXKZN, cn saoualokn as4

2 7S* 6,>8 7 7 7

Cn midn tkala ul erisir ha 6,>8 gg. Nc hibcnrcn >6 vaoas, ac erisir sarn ha4

2>6 6,>8 gg 7 > 628 6,>8 gg 7 >8?,?: gg 7 >8,??: og

Nc mnoar 22 hibcaoas, talhrngis4

222 6,>8 7 8 >;8?68 6,>8 7 1=5262,1: gg 7 1=5,2621: g

Oultis hibcaoas laoasktnrns pnrn suparnr cn ncturn hac Avarast (= =8= g)9

2: hibcaoas.

Ouc sarn ac erisir sk ci puhkarns hibcnr 16 vaoas9 Oigprnci oil cn hkstnlokn hacn Xkarrn nc _ic (>16 gkccilas ha `kcgatris).

>15 :2: 666 `g < hkstnlokn ha cn Xkarrn nc _ic

Ve. >


2/29



VEKLN :6

Qln notkvkhnh

N ouc ha cns suoaskilas ha cn haraomn oirraspilha astn tirra9

n) >, 1, ;, >?, >5,

b) >, 8, ;, >:, 21, ?:,

o) 2, 8, =, >:, ?2, :8,

h) >, ?, ;, 25, =>, 28?,

a) >, >, 2, ?, 1, =,

j) >56, >26, 56, 26, ?6, =6,

e) >, ?, :, =, >:, >=, ?:,

Oirraspilha n cn suoaskl n).

>Nvarkeun ac orktarki oil ac qua sa mn jirgnhi onhn uln ha cns suoaskilas ha nrrkbn ynnha tras trgklis gs n onhn uln.

n) Orktarki4 onhn trgkli sa ibtkala sugnlhi 8 nc nltarkir.

2>, 21, 2;,

b) Orktarki4 cis trgklis sil cis ounhrnhis ha cis lgaris lnturncas.

8;, :8, =>,

o) Orktarki4 onhn trgkli sa ibtkala guctkpckonlhi ac nltarkir pir 2, i bkal, sil cns suoask-vns pitalokns ha 24 2>, 22, 2?,

>2=, 21:, 1>2,

h) Orktarki4 onhn trgkli sa ibtkala guctkpckonlhi ac nltarkir pir ?.

52;, 2 >=5, : 1:>,

a) Orktarki4 onhn trgkli sa ibtkala sugnlhi cis his nltarkiras.

>?, 2>, ?8,

j) Orktarki4 onhn trgkli sa ibtkala rastnlhi 16 nc nltarkir.

>?6, >=6, 2?6,

e) Orktarki4 cis trgklis pnras sa ibtkalal sugnlhi 2 nc nltarkir, y cis trgklis kgpnras saibtkalal guctkpckonlhi ac nltarkir pir 2.

?=, 5:, 5=,

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3/29



2Jirgn okloi suoaskilas oil orktarkis skgkcnras n cis nltarkiras. Al ncel onsi, klvl-tnta ac orktarki.

Raspuastn nbkartn.Adagpci4

n) Orktarki4 ibtalagis onhn trgkli guctkpckonlhi ac nltarkir pir 2.

?, :, >2, 28, 8=,

b) Orktarki4 ibtalagis onhn trgkli sugnlhi >,1 nc trgkli nltarkir.

>0 2,10 80 1,10 50 =,10

o) Orktarki4 ibtalagis cis trgklis pnras guctkpckonlhi ac nltarkir pir ?, y cis kgpnras,

sugnlhi ? nc nltarkir. >, ?, :, >=, >1, 81, 8=,

h) Orktarki4 cis trgklis sil cis oubis ha cis lgaris lnturncas.

>, =, 25, :8, >21, 2>:,

a) Orktarki4 ibtalagis onhn trgkli rastnlhi = hac nltarkir.

>66, ;2, =8, 5:, :=, :6,

?Klhkon ouc as cn racnoklo

2o>

7o

?o2

7 ha cn suoaskl o) ha nrrkbn.

Cn racnokl as 2.

8Astnbcaoa cn racnokl (oiokalta) altra onhn his trgklis oilsaoutkvis ha cn suoaskl h)qua npnraoa nrrkbn.

Cn racnokl as ?.

Ve. 2


4/29



VEKLN :>

:Asorkba cis okloi prkgaris trgklis ha4

el7 l? ml7 l

2 ?l+ 5 kl7l ?l+ 8

el4 >, =, 25, :8, >21,

ml4 1, 1, 5, >>, >5,

kl421

, >:

, 6, >=

, 2;

,

5Jirgn uln suoaskl raourralta oil astis hntis4

d>7 2 d27 ? dl7dl 2+dl >

2, ?, 1, =, >?, 2>, ?8, 11,

=Klvaltn itrns his suoaskilas raourraltas oil hntis hkstkltis n cis nltarkiras.

Raspuastn nbkartn. Vir adagpci4

n) n>7 ?, n27 1, nl7 2nl >+ nl 2

_uoaskl4 ?, 1, >?, ?>, 51, >=>,

b)b>7 >, b27 ?, bl7 bl >+ (bl 2)2

_uoaskl4 >, ?, 8, >?, 2;, >;=, > 6?;,

;Asorkba cis ountri prkgaris trgklis ha cns suoaskilas qua tkalal pir trgkli eala-rnc4

n) nl7 ? + 1(l >) b)bl7 ? >2 l >

o) ol7 (l >)(l 2) h)hl7 l2 l

n) ?, =, >?, >=, b) ?, ?2

,?8

,?=

,

o) 6, 6, 2, :, h) 6, 2, :, >2,

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>6Hasoubra cn cay ha raourralokn y nnha ul luavi trgkli n onhn uln ha cns skeukal-tas suoaskilas4

n) >, 8, 1, ;, >8, 2?, (Hkjaralokn)

b) >, 2, ?, :, >>, 26, (Racnokiln onhn acagalti oil cis tras nltarkiras)

o) >0 20 >,10 >,510 (_agksugn)

h) >, 2, 2, >, >/2, >/2, >, (Oiokalta)

n) Luavi trgkli4 ?5

Cay ha raourralokn4 nl7 nl 2 nl >

b) Luavi trgkli4 ?5

Cay ha raourralokn4 bl7 bl >+ bl 2+ bl ?

o) Luavi trgkli4 >,:21

Cay ha raourralokn4 ol7ol >+ ol 2

2

h) Luavi trgkli4 2

Cay ha raourralokn4 hl7hl >hl 2

>>Oilstruya uln suoaskl ouyn cay ha raourralokn san nl7 nl >+ l. (Hnca nc prkgartrgkli ac vncir qua qukarns).

Raspuastn nbkartn.

Adagpci4 ?, 1, =, >2, >5, 2?, ?6,

>2n) Oigpruabn qua ac trgkli ealarnc ha cn suoaskl >, >, >, >, >, >, as sl7(>)l.

b) Mnccn ac trgkli ealarnc ha astns suoaskilas4

nl = >, >, >, >, >, >,

bl = >, 2, ?, 8, 1, :,

n) s>7 (>)>7 >

s27 (>)27 >

s?7 (>)?7 >

s87 (>)87 >

Cis trgklis sl oil l pnr sil >, y ounlhi l as kgpnr sil keuncas n >. Oiklokhaoil cis trgklis ha cn suoaskl hasorktn.

b)nl7 (>)

l+ >

bl7 (>)l+ > l

Ve. 2


6/29



VEKLN :2

Oil oncoucnhirn

Nnha ountri trgklis n onhn uln ha astns suoaskilas. _k haokgis qua al n) cn hkjaraloknas ?, ouc sar cn hkjaralokn al cns hags9

n) 2, 1, =, >>, >8, >5, b) >26, >86, >:6, >=6, 266, 226,

o) ;, 5, 1, ?, >, >, ?, 1, h) 1,=?0 1,=50 1,;>0 1,;10 1,;;0 :,6?0

n) 26, 2?, 2:, 2;, hkjaralokn4 ? b) 286, 2:6, 2=6, ?66, hkjaralokn4 26

o) 5, ;, >>, >?, hkjaralokn4 2 h) :,650 5,>>0 :,>10 :,>;0 hkjaralokn4 6,68

>Ac prkgar trgkli ha uln prieraskl nrktgtkon as s>7 1 y cn hkjaralokn as h7 2,1.

Asorkba sus hkaz prkgaris trgklis.Mnz ci gksgi pnrn itrn ouyi prkgar trgkli as t>7 26 y ouyn hkjaralokn as h7 ?.

Vrieraskl sl4 10 5,10 >60 >2,10 >10 >5,10 260 22,10 210 25,10

Vrieraskl tl4 26, >5, >8, >>, =, 1, 2, >, 8, 5,

2Oncoucn, pnrn cns prieraskilas ha nrrkbn4

b?: o?> h>666

b)b>7 >26 y h7 26 = bl7 b>+ (l >) h7 >26 + (l >) 26 7

7 >26 + 26l 26 7 >66 + 26l

Ns4 b?:7 >66 + 26 ?: 7 =26

o) o>7 ; y h7 2 = ol7 ; + (l >) (2) 7 ; 2l+ 2 7 >> 2l

Ns4 o?>7 >> 2 ?> 7 1>

h)h>7 1,=? y h7 6,68 = hl7 1,=? + (l >) 6,68 7 1,=? + 6,68l 6,68 7

7 1,5; + 6,68l

Ns4 h>6667 1,5; + 6,68 > 666 7 81,5;

?Mnccn ac trgkli ealarnc ha cns prieraskilas b), o) y h). (Kltaltn mnoarci skl npckonr cnjrgucn, skgpcagalta rnzilnlhi).

bl7 >66 + 26 l ol7 >> 2 l hl7 1,5; + 6,68 l

8n) _k his trgklis ha uln prieraskl sil4

s>7 : y s?7 ;

nvarkeun ac vncir ha cn hkjaralokn, h.

b) Mnccn ac trgkli ealarnc ha cn prieraskl, sl.

n) h7 >,1

b) sl7 : + >,1 (l >) 7 : + >,1l >,1 7 8,1 + >,1l

Ve. >


7/29



VEKLN :?

1Mnccn cn sugn ha tihis cis lgaris kgpnras galiras qua >66.

Ac trgkli ealarnc ha cis lgaris kgpnras as nl7 2l >. Ac ctkgi kgpnr galir qua>66 as ;;, qua rasuctn sar n16. Ns, cn sugn as4

_167(n>+ n1i) 16

27

(> + ;;) 16

27 2166

:n) _k n>7 1 y n27 5, oncoucn n86 y _86.

b) _k b>7 1 y b27 >2, oncoucn _?2.

n) n>7 1 y h7 2 = nl7 1 + (l >) 2 7 ? + 2l

Cuaei4 n867 ? + 2 86 7 =? y _867(1 + =?) 86

27 >5:6

b)b>7 1 y h7 5 = bl7 1 + (l >) 5 7 2 + 5l

Ns4 b?27 2 + 5 ?2 7 222

Cuaei4 _?27(1 + 222) ?2

27 ?:?2

5_k ac prkgar trgkli ha uln prieraskl as o>7 >5 y ac quklti as o17 ;, mnccn cn sugn_26.

Oigi o>7 >5 y o17 ; = o>7 >5 y h7 2

Ns4 ol7 >5 + (l >)(2) 7 >; 2l0 o267 >; 2 26 7 2>

Cuaei4 _267(>5 2>) 26

27 86

=Cis prkgaris trgklis ha uln prieraskl nrktgtkon sil n>7 8, n27 5. Mnccn astnsugn4

n>6+ n>>+ n>2+ + n>;+ n26

Oigi n>7 8 y n17 5, talagis qua cn hkjaralokn ha astn prieraskl as h7 ?.

Lis pkhal cn sugn ha cis trgklis hac hokgi nc vkeskgi. Ci qua vngis n mnoar as oncoucnr_26 y rastnrca _;4

_267(n>+ n26) 26

27

(n>+ n>+ >; ?) 26

27 (8 + 8 + 15) 26

27 :16

_;7(8 + 8 + = ?) ;

27 >88

Vir tnlti, cn sugn pahkhn as4

:16 >88 7 16:

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8/29



VEKLN :1

>Oilstruya uln prieraskl eaigtrkon ouyi prkgar trgkli as >21 y ouyn rnzl as

6,8.

>210 160 260 =0 ?,20 >,2=0 6,1>20

2Ha uln prieraskl eaigtrkon oilioagis n>7 6,:21 y n?7 6,;. Mnccn r y cis saksprkgaris trgklis.

6,; 7 6,:21r2 = r27 >,88 = r7 >,2

Vir tnlti, mny his prieraskilas4

r7 >,2

6,:210 6,510 6,;0 >,6=0 >,2;:0 >,11120

r7 >,2

6,:210 6,510 6,;0 >,6=0 >,2;:0 >,11120

?Al uln prieraskl eaigtrkon ha trgklis pisktkvis, n>7 2 y n?7 :. Mnccn nl, n>>y n>2.

: 7 2 r2 = r27 ? = r7 ?

Oigi as uln prieraskl ha trgklis pisktkvis, cn rnzl tngbkl ci as.r7 ?

nl7 2 (? )l >

n>>7 2 (? )>6

7 2 ?17 8=:

n>27 2 (? )>>

7 2 ?1 ? 7 8=:?

8Al uln prieraskl eaigtrkon, ac prkgar trgkli as n> 7 1 y cn rnzl as r7 >,8.Nvarkeun, oil nyuhn ha cn oncoucnhirn, ouc as ac trgkli gs nvnlznhi ouyi vncir as

kljarkir n > 666 666.

n?57 ;>>>25,5=>

n?=7 > 25115=,=;?

= As n?5.

1Al uln prieraskl eaigtrkon, n>7 > 666 y r7 6,=. Nvarkeun, oil cn oncoucnhirn,ouc as ac trgkli gs nvnlznhi ouyi vncir as gnyir qua >.

n?>7 >,2?5

n?27 6,;;

= As n?>.

Ve. >


9/29



VEKLN ::

:_keukalhi ac prioahkgkalti utkckznhi pnrn mnccnr _l, oncoucn4

? + : + >2 + 28 + 8= + ;: + >;2 + ?=8

? + : + >2 + 28 + 8= + ;: + >;2 + ?=8 7? 2= ?

2 >7 5:1

5Oultis halnrkis sa ccav, al titnc, ac oalturkl hac pribcagn rasuacti 8 ha cn peklnnltarkir9

_>:7> 2>: >

2 >7 :11?1 halnrkis

= Oncoucn cn sugn ha cis hkaz prkgaris trgklis ha uln prieraskl eaigtrkon quaougpcn n>7 =,>;2 y r7 2,1.

_>67=,>;2 2,1>6 =,>;2

2,1 >7 12655,=52

;Ajaotn cn sugn > + ? + ; + 25 + + ?5.

> + ? + ; + + ?57> ?= >

? >7 ?2=6

Ve. >


10/29



VEKLN :5

>6Al uln prieraskl eaigtrkon, n>7 = y r7 6,51. Oncoucn cn sugn ha sus kljklktis

trgklis.

_F

7=

> 6,517

=

6,217 ?2

>>Al uln prieraskl eaigtrkon, n>7 ?6 y r7 6,2. Oncoucn cn sugn ha tihis sustrgklis.

_F

7?6

> (6,2)7

?6

>,27 21

>2Al uln prieraskl eaigtrkon, su ounrti trgkli as n87 >6 y ac saxti as n:7 6,8.Mnccn4 cn rnzl, r 0 ac prkgar trgkli, n>0 ac iotnvi trgkli, n=0 cn sugn ha cisiomi prkgaris trgklis, _=0 y cn sugn ha sus kljklktis trgklis, _.

n:7 n8 r2 = 6,8 7 >6 r2 = r27 6,68 = r7 6,2

r7 6,2 = >6 7 n> 6,2? = >6 7 n> 6,66= = n>7 >216

n=7 n> 6,25 = n=7 >216 6,2

5 = n=7 6,6>:

_=7> 216 > 216 6,2=

> 6,27 >1:2,8;:

_F

7> 216

> 6,27 > 1:2,1

r7 6,2 = >6 7 n> (6,2)? = n>7 >216

n=7 > 216 (6,2)57 6,6>:

_=7>216 (> 216) (6,2)=

> 6,27 >68>,::8

_F

7> 216

> (6,2)7

> 216

>,27 > 68>,:

)

Ve. >


11/29

_icuokilas n Adarokokis y pribcagns?


VEKLN :=

Vrnotkon

_uoaskilas4 jirgnokl, trgkli ealarnc

> Asorkba cis okloi prkgaris trgklis ha cns skeukaltas suoaskilas4

n) Onhn trgkli sa ibtkala sugnlhi 5 nc nltarkir. Ac prkgari as >6.

b) Ac prkgar trgkli as 6,>. Cis hags sa ibtkalal guctkpckonlhi ac nltarkir pir 2.

o) Ac prkgari as 20 ac saeulhi, 8, y cis skeukaltas, cn sagksugn ha cis his nltarkiras.

n) >6, ?, 8, >>, >=, b) 6,>0 6,20 6,80 6,=0 >,:0

o) 20 80 ?0 ?,10 ?,210

2 Asorkba cis trgklis n>6 y n21 ha cns skeukaltas suoaskilas4

n) nl7 ?l > b)bl7l2+ >

2 o) ol7 (>)

l+ >l

h)hl7 > +(>)l

>6 a) al7 l (l >) j )jl7

l 2l+ 2

n)

n>67 2;

n217 58 b)

b>67>6>2

7 16,1

b217:282

7 ?>2 o)

o>67 > +>>6

7 >>>6

o217 > +>

217 28

21

h)

h>67 >,>

h217 6,; a)

a>67 >6 ; 7 ;6

a217 21 28 7 :66 j)

j>67=>2

7 2?

j2172?25

? Asorkba cis okloi prkgaris trgklis ha cn skeukalta suoaskl4

n>7 > nl7 2nl >+ ?>, 1, >?, 2;, :>,

8 Nvarkeun ac orktarki oil ac qua sa mn jirgnhi onhn uln ha cns skeukaltas suoa-skilas4

n) >>, ;, 5, 1, b) >2

, >8

, >=

, >>:

, o) 2,10 2,;0 ?,?0 ?,50

h) >, >2

, >?

, >8

, a) =, >2, >=, 25, j) 6, ?, =, >1,

n) Rastnlhi 2 ulkhnhas nc trgkli nltarkir4 nl7 >> (l >)2 7 >? 2l

b) Guctkpckonlhi pir >2

ac trgkli nltarkir4 nl7 >2l

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o) _ugnlhi 6,8 nc trgkli nltarkir4 nl7 2,1 + (l >) 6,8 7 2,> + 6,8l

h) Hkvkhkalhi > pir l, cuenr qua ioupn ac trgkli4 nl7>l

a) Guctkpckonlhi pir >,1 ac trgkli nltarkir4 nl7 = >,1l >

j) Rastnlhi > n cis ounhrnhis ha cis lgaris lnturncas4 nl7 l2 >

1 Mnccn ac trgkli ealarnc ha astns suoaskilas4

n) >2, >8, >:, >=, b) >2

, 2?

, ?8

, 81

,

o) >, ?, ;, 25, h) > 2, 2 ?, ? 8, 8 1,

n) nl7 >6 + 2l b)nl7l

l+ >o) nl7 ?

l > h)nl7 l (l+ >)

: Buson uln cay ha raourralokn pnrn hajklkr cns skeukaltas suoaskilas4

n) =, >6, 2, =, >6, b) ?, 2, 2?

, >?

, >2

,

n) n>7 = n27 >6 nl7 nl > nl 2 b)n>7 ? n27 2 nl7nl >nl 2

Vrieraskilas nrktgtkons

5 Asorkba cis okloi prkgaris trgklis y n26 ha cns skeukaltas prieraskilas nrkt-gtkons4

n) n>7 >,10 h7 2 b)n>7 ?20 h7 1

o) n>7 10 h7 6,1 h)n>7 ?0 h7 8

n) >,10 ?,10 1,10 5,10 ;,10 n267 >,1 + >; 2 7 ?;,1

b) ?2, 25, 22, >5, >20 n267 ?2 + >; (1) 7 :?

o) 10 1,10 :0 :,10 50 n267 1 + >; 6,1 7 >8,1h) ?, 5, >>, >1, >;0 n267 ? + >; (8) 7 5;

= Mnccn, al onhn onsi, ac trgkli ealarnc y oncoucn, haspus, n164

n) 21, >=, >>, 8, b) >?, >>, ;, 5,

o) >,80 >,;0 2,80 2,;0 h) ?, =, >?, >=,

n) n>7 210 h7 50 nl7 21 + (l >)(5) 7 ?2 5l0 n167 ?>=

b)n>7 >?0 h7 20 nl7 >? + (l >)2 7 >1 + 2l0 n167 =1

o) n>7 >,80 h7 6,10 nl7 >,8 + (l >)6,1 7 6,; + 6,1l0 n167 21,;

h)n>7 ?0 h7 10 nl7 ? + (l >)(1) 7 2 1l0 n167 28=

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; Oncoucn cn sugn ha cis vaklta prkgaris trgklis ha cns skeukaltas prieraskilasnrktgtkons4

n) n>7 10 h7 2 b)n>7 >0 n27 5

o) Cis lgaris pnras. h) Cis gctkpcis ha ?.

n) n267 1 + >; 2 7 8?0 _267(1 + 8?) 26

27 8=6

b)h7 5 (>) 7 :0 n267 > + >; (:) 7 >>10 _267P> + (>>1)W 26

27 > >:6

o) h7 2, n>7 2, n267 2 + >; 2 7 860 _267(2 + 86) 26

27 826

h)n>7 ?, h7 ?, n267 ? + >; ? 7 :60 _267(? + :6) 26

27 :?6

Vrieraskilas eaigtrkons

>6 Asorkba cis okloi prkgaris trgklis ha cns skeukaltas prieraskilas eaigtrkons4

n) n>7 6,?0 r7 2 b)n>7 ?0 r7>2

o) n>7 2660 r7 6,> h)n>7>

=>0 r7 ?

n) 6,?0 6,:0 >,20 2,80 8,=0 b) ?, ?

2

, ?

8

, ?

=

, ?

>:

,

o) 2660 260 20 6,20 6,620 h) >=>

, >25

, >;

, >?

, >,

>> Mnccn, al onhn uln ha cns suoaskilas skeukaltas, ac trgkli ealarnc4

n) 260 =0 ?,20 >,2=0 b) 86, 26, >6, 1,

o) :0 ;0 >?,10 26,210 h) 6,8=0 8,=0 8=0 8=60

n) nl7 26 6,8l > b)nl7 86 >2

l >

o) nl7 : (>,1)l >

h)nl7 6,8= >6l >

>2 Oncoucn cn sugn ha cis hkaz prkgaris trgklis ha cns prieraskilas eaigtrkonsskeukaltas4

n) n>7 10 r7 >,2 b)n>7 10 r7 2

n) _>671 >,2>6 1

>,2 >7 >2;,= b)_>67

1 (2)>6 12 >

7 > 561

>? Mnccn cn sugn ha cis kljklktis trgklis ha cns prieraskilas eaigtrkons sk-eukaltas4

n) n>7 80 r7 >? b)n>7 >50 r7 6,;1

n) _F7n>

> r7

8> (>/?)

7 : b)_F7>5

> 6,;17 ?86

Ve. ?


14/29



VEKLN :;

Npckon ci npralhkhi

>8 Khaltkjkon cns prieraskilas nrktgtkons, cns eaigtrkons y cns qua li sil pri-eraskilas. Ibtl ac trgkli ealarnc ha onhn uln4

n) >, ;=

, 18

, >>=

, b)>, 2, ?, 8,

o) 6,20 6,620 6,6620 h) 2, ?2

, 8?

, 18

,

n) Vrieraskl nrktgtkon, h7>=

. Xrgkli ealarnc4 nl7 > + (l >)>=

7>5

+>=

l

b) Li as prieraskl. Xrgkli ealarnc4 nl7 l

o) Vrieraskl eaigtrkon, r7 6,>.

Xrgkli ealarnc4 nl7 6,2 (6,>)l >

h) Li as prieraskl.

Cis lugarnhiras 2, ?, 8, 1, jirgnl uln prieraskl nrktgtkon ouyi trgkli eala-rnc as l+ >.

Cis haligklnhiras >, 2, ?, 8, jirgnl uln prieraskl nrktgtkon ha trgkli eala-rnc l.

Xrgkli ealarnc ha cn suoaskl4 nl7l+ >l

>1 Mnccn ac prkgar trgkli y ac trgkli ealarnc ha cns skeukaltas prieraskilasnrktgtkons4

n) h7 10 n=7 ?5 b)n>>7 >50 h7 2

Raouarhn qua n=7 n>+ 5h 0 sustktuya y mnccn n>.

n) n=7 n>+ 5h = ?5 7 n>+ 5 1 = n>7 2

nl7 2 + (

l >) 1 7 ? + 1

l

b)n>>7 n>+ >6h = >5 7 n>+ >6 2 = n>7 ?

nl7 ? + (l >)2 = nl7 1 + 2l

>: Mnccn cn hkjaralokn y ac prkgar trgkli ha cns prieraskilas nrktgtkons skeukaltas4

n) n27 >=0 n57 >5 b)n87 >10 n>27 ?;

Xal al oualtn qua n57 n2+ 1h.

n) n57 n2+ 1h = >5 7 >= + 1h = h7 5

n>7 n2 h = n>7 >= (5) 7 21b)n>27 n8+ =h = ?; 7 >1 + =h = h7 ?

n87 n>+ ?h = >1 7 n>+ ; = n>7 :

Ve. >


15/29



>5 ^u cuenr ioupn ul trgkli ouyi vncir as 1: al cn prieraskl nrktgtkon ha-jklkhn pir n>7 = y h7 ?9

1: 7 = + (l

>) ?=

1: 7 1 + ?l=

l

7 >5

>= Oncoucn cn rnzl y ac prkgar trgkli ha cns prieraskilas eaigtrkons skeukal-tas4

n) n17>

=>0 n?7

>;

b)n27 6,:0 n87 2,8

n) n17n?r2 =

>=>

7 >;

r2 = r27>;

= r7 >?

Mny his sicuokilas4

_k r7 >?

4 n>7 >

_k r7 >?

4 n>7 >

b)n87 n2 r2 = 2,8 7 6,: r2 = r7 2

Mny his sicuokilas4

_k r7 24 6,?0 6,:0 >,20 2,80 8,=0

_k r7 24 6,?0 6,:0 >,20 2,80 8,=0

>; Mnccn ac prkgar trgkli y asorkba ac trgkli ealarnc ha cns skeukaltas priera-skilas4

n) n?7 ?0 r7>

>6 b)n87 26,210 r7 >,1

n) n?7 n>r2 = ? 7 n> >>6

2 = n>7 ?660 nl7 ?66 >>6

l >

b)n87 n>r? = 26,21 7 n>(>,1)

? = n>7 :0 nl7 : (>,1)l >

26 Oncoucn cn sugn ha cis okloi prkgaris trgklis ha uln prieraskl eaigtrkon alcn qua n>7 > 666 y n87 =. _a puaha mnccnr cn sugn ha sus kljklktis trgklis9

n87 n>r? = = 7 > 666 r? = r7 ? =

> 6667 2

>67 >

1

_17n>r

1 n>r >

7

> 666 >1 1 > 666

>1

>

7 > 28;,:

_a puaha mnccnr cn sugn ha sus kljklktis trgklis, pirqua cn rnzl ast oigpralhkhn

altra > y >.

_F7n>

> r7 > 666

> >/17 >216

Ve. 2


16/29



2> Oncoucn cns skeukaltas sugns4

n) 2 + 22+ 2?+ + 2>6.

b) Cis lgaris pnras galiras qua > 666.o) Cis lgaris kgpnras galiras qua > 666.

n) Xalagis uln prieraskl eaigtrkon oil n>7 2 y r7 2. Lis pkhal oncoucnr _>64

_>67n> r

l n>r >

7 2 2>6 2

2 >7 2

>> 2>

7 2 68:

b) Cis lgaris pnras jirgnl uln prieraskl nrktgtkon oil n>7 2 y h7 2.

Vnrn sugnr cis lgaris pnras galiras qua > 666, mny qua oncoucnr _8;;, puas4

nl7 2l= ;;= 7 2 8;;

Vir tnlti4

_8;;7(n>+ nl) l

27 (2 + ;;=) 8;;

27 28; 166

o) Vnrn cis kgpnras4

n>7 >

h7 2

nl7 2l >

Ns, talagis qua oncoucnr _166, puas4

;;; 7 2l > = l7 166

_1667(n>+ nl) l

27 (> + ;;;) 166

27 216 666

Rasuacva pribcagns

22 Al ul tantri, cn prkgarn jkcn hkstn hac asoalnrki 8,1 g, y cn iotnvn, ;,51 g.

n) Ouc as cn hkstnlokn altra his jkcns9

b) N qu hkstnlokn hac asoalnrki ast cn jkcn >59

n) n=7 n>+ 5h = ;,51 7 8,1 + 5h = h7 6,51 g

Cn hkstnlokn altra his jkcns as 6,51 g.

b)n>57 n>+ >: h7 8,1 + >: 6,51 7 >:,1 g ast cn jkcn >5.

2? Vnrn prapnrnr uln onrrarn, ul hapirtkstn oigkalzn oirrkalhi ? `g y nugaltn>,1 `g su ra-oirrkhi onhn hn. Oultis hns tkala qua altralnr pnrn ccaenr n mnoarul raoirrkhi ha >1 `g9

nl7 n>+ (l >)h = >1 7 ? + (l >) >,1 = >1 7 >,1 + >,1l

l7 ; hns

Ve. ?


17/29



28 Al ac ni >;=: jua vksti ac oigatn Mnccayhasha cn Xkarrn, n cn qua sa noarononhn 5: nis. Astn arn cn ounrtn vaz qua lis vksktnbn hasha qua ac nstrligi Mnccayci hasoubrk.

n) Al qu ni jua hasoubkarti9

b) Oulhi sar vksti al ac skeci ]]K9

n) n87 n>+ ?h = >;=: 7 n>+ ? 5: = n>7 >51=. Jua hasoubkarti al >51=.

b)n17 >;=: + 5: 7 26:2. _a var al 26:2.

21 Cn hisks ha ul gahkongalti as >66 ge ac prkgar hn y 1 ge galis onhn uliha cis skeukaltas. Ac trntngkalti hurn >2 hns. Oultis gkckerngis tkala qua ti-gnr ac aljargi hurnlta tihi ac trntngkalti9

n>27 n>+ >>h = n>27 >66 + >> (1) 7 81

_>27(n>+ n>2) >2

27 (>66 + 81) >2

27 =56 ge

2: Ql tkpi ha bnotarkn sa raprihuoa pir bkpnrtkokl onhn ounrti ha mirn. Oul-tns bnotarkns mnbr haspus ha : mirns9

Cn raprihuookl ha cns bnotarkns as uln prieraskl eaigtrkon ha r7 2. Xrgkli eala-rnc4 nl7 2

l >.

Oigi : 8 7 28 ounrtis ha mirn, oncoucngis n287 228 >4

n287 =?== :6= bnotarkns mnbr haspus ha : mirns.

25 Cn pibcnokl ha ul okarti pns nugaltn pir trgkli gahki ul 2,1% nlunc. _kcn pibcnokl notunc as ha ? gkccilas, ouc sar haltri ha >6 nis9

n>67 ? >,621;7 ? 58: 1=; haltri ha >6 nis.

2= Qln gqukln alvnsnhirn pkarha onhn ni ul >1% ha su vncir. _k mn oistnhi26 666 , ouc sar su vncir haltri ha 1 nis9

n17 n> r8 = n17 26666 (> 6,>1)

87 >6 886sar su vncir haltri ha 1 nis.

2; Qln bicn qua ruahn pir ul pcnli klocklnhi raoirra > g hurnlta ac prkgar sa-

eulhi, 8 g hurnlta ac saeulhi, 5 g hurnlta ac taroari, y ns hurnlta >6 saeulhis.^u hkstnlokn mn raoirrkhi al titnc9

>, 8, 5, as uln prieraskl nrktgtkon oil h7 ?.

n>67 n>+ ; ? = n>67 > + ; ? 7 2= g raoirra al >6 s.

?6 Oncoucn ac lgari ha bciquas laoasnrkis pnrn oilstrukr uln tirra oigi cn hacn jkeurn ha cn pekln :6, pari qua talen 16 pksis.

Cis bciquas ha cn tirra astl al prieraskl nrktgtkon oil h7 84 >, 1, ;, >?,

Mny qua oncoucnr cn sugn ha 16 trgklis4

n167 n>+ 8;h = n167 > + 8; 8 7 >;5

_167(n>+ n16) 16

27 (> + >;5) 16

27 8 ;16 bciquas.

Ve. 8


18/29



VEKLN 56

?> Qln pacitn ona hasha uln okartn ncturn y rabitn nsoalhkalhi cis ?/8 ha cn nctu-

rn nltarkir. Haspus ha hnr al ac suaci pir taroarn vaz, nconlzn 18 og. Hasha quncturn sa had onar9 Oncoucn cn hkstnlokn raoirrkhn mnstn qua sa pnrn.

Xalagis uln prieraskl eaigtrkon ha cn qua oilioagis su rnzl, r7?8

.

Lis pkhal su prkgar trgkli, snbkalhi qua n?7 18.

nl7 n> rl >= 18 7 n>?8

2= n>7 ;:

Cn pacitn sa cnlzn hasha ;: og ha ncturn.

Nmirn lis pkhal cn sugn ha cns hkstnlokns0 as haokr, _F

4

_F

7 ;:> ?/8

7 ?=8 og

?2 Rasuacti al ac ckbri hac ncugli.

?? Oncoucn cn jrnookl ealarntrkz ha astis lgaris utkckznlhi ac gtihi hac adar-okoki nltarkir4

n) 5,)

? b) ?,1)

8 o) 6,)

2?

n) 5,?)

7 5,???? 7 5 + 6,? + 6,6? + 6,66? +

_ugn ha cis kljklktis trgklis ha cn prieraskl?

>6,

?>66

,?

> 666

_F

7?/>6

> >/>67

?;

7>?

5,?)

7 5 +>?

722?

b) ?,18)

7 ?,18888 7 ?,1 + 6,68 + 6,668 + 6,6668 + 7

7 ?1>6

+ 8>66

+ 8> 666

+ ?>6 666

+

_F

7 8/>66> >/>6

7 86;66

7 281

?,18)

7 ?1>6

+ 281

7 ?>;;6

o) 6,2?)

7 6,2?2?2?2? 72?

>66+

2?>6 666

+?

> 666 666+

_F

72?/>66

> >/>67

2?;;

6,2?

)

7

2?

;;

Ve. >


19/29



?8 Cns ahnhas ha 1 margnlis astl al prieraskl nrktgtkon y sugnl ?1 nis. Acgnyir tkala >? nis. Ouc as cn ahnh ha onhn uli9

_nbagis qua _17 ?1 y qua n

17 >?.

_17(n>+ n1) 1

2= ?1 7

n>+ >?

2 1 = n>+ >? 7 >8 = n>7 >

Nhags4

n17 n>+ 8 h= >? 7 > + 8h= h7 ?

Vir tnlti, cns ahnhas ha cis margnlis sil >, 8, 5, >6 y >? nis.

?1 Qln rnln hn snctis al clan raotn mnokn hacnlta, y onhn vaz snctn cis 2/? hac snctinltarkir. ^ukara ntrnvasnr uln omnron okroucnr ha 1 g ha rnhki, raoirrkalhi su hk-gatri. _u prkgar sncti as ha 2 g. Vnsnr pir ac oaltri ha cn omnron9 Ccaenr nc itricnhi ha cn omnron9

Cis snctis jirgnl uln prieraskl eaigtrkon, oil n>7 2 y r72?

.

_k cn rnln snctn kljklktngalta, al titnc raoirrarn4

_F

7 2> 2/?

7 : g

Vir tnlti, s pnsnrn hac oaltri ha cn omnron (1 g), pari li ccaenr lulon nc itri cnhi (>; g).

?: Vnrn nhirlnr ul pnsai sa oicionl n ci cnrei ha su clan oaltrnc uln jkcn ha dnr-hklarns maxneilncas, rihanhns ha bnchisns ha cn gksgn jirgn, oigi guastrn cn jk-eurn. Oultns bnchisns sa laoasktnrl pnrn pilar 21 dnrhklarns9

Zangis ul hkbudi4

Ns, as jokc altalhar qua mnoal jnctn4

Vnrn > dnrhklarn, > 8 + 2 bnchisns.

Vnrn 2 dnrhklarns, 2 8 + 2 bnchisns. Vnrn ? dnrhklarns, ? 8 + 2 bnchisns.

Vnrn l dnrhklarns, l 8 + 2 bnchisns.

Ve. 2


20/29



As uln prieraskl nrktgtkon oil n>7 : y h7 8, yn qua4

nl7 : + (l >)8 7 8l+ 2

Vir tnlti, pnrn 21 dnrhklarns mnoal jnctn4 n217 8 21 + 2 7 >62 bnchisns.

Kltars oigpuasti

?5 Rasuacti al ac ckbri hac ncugli.

?= Hapisktngis : 666 nc 1% nlunc, nc oigkalzi ha ul okarti ni. Nvarkeun aconpktnc hkspilkbca nc jklnc ha onhn ni, hurnlta : nis.

_a trntn ha uln prieraskl eaigtrkon, hilha n>7 : 666 y r7 >,61. Lis astl praeul-tnlhi pir n2, n?, n8, n1, n: y n5.

_u trgkli ealarnc as4

nl7 n> rl >= nl7 : 666 >,61

l >

Vir tnlti4

n27 : 666 >,61 7 : ?66

n?7 : 666 >,6127 : :>1

n87 : ;81,51

n175 2;?,68

n:7 5 :15,:;

n57 = 686,15

?; Hapisktngis al ul bnloi > 666 nc 2,1% sagastrnc nc oigkalzi ha ul okartini. Nvarkeun ac onpktnc hkspilkbca nc jklnc ha onhn sagastra, hurnlta ? nis, sk lisnongis lklel hklari.

As uln prieraskl eaigtrkon ha rnzl > + 2,1>667 >,621.? nis sil : sagastras. _us trgklis sil4

> 666 >,6210 > 666 >,62120 > 666 >,621?0 > 666 >,62180 > 666 >,62110> 666 >,621: = > 6210 > 616,:?0 > 65:,=;0 > >6?,=>0 > >?>,8>0 > >1=,:;

86 Rasuacti al ac ckbri hac ncugli.

Ve. ?


21/29



VEKLN 5>

8> Nc oigkalzi ha onhn ni klerasngis : 666 nc 1% nlunc. Ha qu onpktnc

hkspilhragis nc jklnc hac saxti ni9 >.ar 2. ?.ar 8. 1. :.

: 666 : 666 >,61:

: 666 : 666 >,611

: 666 : 666 >,618

: 666 : 666 >,61?

: 666 : 666 >,612

: 666 : 666 >,61

Cis sugnlhis jirgnl pnrta ha uln prieraskl eaigtrkon ha rnzl r 7 >,61, oiln>7 : 666 >,61. Ci qua lis astl pkhkalhi as _:4

_:7n: r n>

r >7 : 666 >,61

5 : 666 >,616,61

7 82 =12,61

Astn as cn okjrn qua raokbkragis nc jklnc hac :.ni.

82 Qln parsiln klkokn ul pcnl ha palskilas klerasnlhi, nc prklokpki ha onhn ni,? 666nc :% nlunc.

^u onpktnc talhr haltri ha >6 nis9

_a trntn ha uln prieraskl eaigtrkon oil n>7 ? 666 >,6: y r7 >,6:.

Ci qua lis astl pkhkalhi as _>64

_>67n>6 r n>

r >7 ? 666 >,6:

>> ? 666 >,6:6,6:

7 8> ;>8,;?

Vribcagns +

8? Ql oigaroknlta raokba ul pahkhi ha 26 ondns ha lnrnldns, 5 ha ocnsa axtrn y >?ha uln onckhnh kljarkir. ^ukara axpilarcns nc pbckoi jirgnlhi uln pkrgkha ha bnsaounhrnhn oil >2 lnrnldns ha cnhi al cn bnsa, ha jirgn qua cns lnrnldns vkskbcas sanlha cn ocnsa axtrn.

_k al onhn ondn mny ncrahahir ha 86 lnrnldns, talhr sujkokaltas lnrnldns pnrn acci9

Ac prkgar pksi as ul ounhrnhi ha >2 lnrnldns ha cnhi.

Cns qua sa val sil4 >2 8 8 7 88

Al ac 2.pksi sa val >> 8 8 7 86.

Al ac ?.arpksi sa val >6 8 8 7 ?:.

Cns lnrnldns vkskbcas sil 88 + 86 + ?: + + 8 + >.

Ve. >


22/29



Cis >> prkgaris pksis jirgnl uln prieraskl nrktgtkon ouyi trgkli ealarnc asnl7 88 (l >) 8 7 8= 8l y ouyn sugn as4

_>>7

88 + 8

2 >> 7 2:8Nnhkgis cn qua oiriln cn pkrgkha y sil 2:1 cns lnrnldns vkskbcas.

Cns 5 ondns ha lnrnldns axtrns oiltkalal 5 86 7 2=6 lnrnldns, qua sil sujkokaltas pnrncn pnrta vkskbca.

Cns lnrnldns qua li sa val sil4

>62+ ;2+ =2+ 52+ :2+ 12+ 82+ ?2+ 22+ >27 ?=1

Oigi tkala >? 86 7 126, tngbkl sil sujkokaltas.

88 Ql nerkouctir haba aomnr ul oubi ha neun n onhn uli ha cis vaklta rbicas quamny al su muarti. Astis astl ncklanhis n hkstnlokns raeucnras ha : g n ci cnrei haul ongkli, y cn hkstnlokn hac prkgar rbic n cn jualta as ha >2 g.

n) _k onhn vaz ccavn ul oubi, qu hkstnlokn mnbr raoirrkhi mnstn raenr cis 26 rbi-cas y hadnr ac oubi al su piskokl klkoknc, dulti n cn jualta9

b) [ sk ccavnrn his oubis al onhn vknda9

n) Vnrn raenr ac prkgari y hadnr ac oubi hilha astnbn, raoirra >2 gatris ha khn y >2 ga-tris ha vuactn = n>7 28.

Vnrn raenr ac saeulhi y hadnr ac oubi al cn jualta, raoirra ?: gatris = n27 ?:.

Vnrn raenr ac taroari y hadnr ac oubi al cn jualta, raoirra 8= gatris = n?7 8=.

As uln prieraskl nrktgtkon oil h7 >2.

nl7 n>+ (l >) h7 28 + (l >) >2 7 28 + >2l >2 7 >2l+ >2

nl7 >2l+ >2

n267 >2 26 + >2 7 212

Al titnc raoirrar4

_267(n

>+ n

26) 26

2 7 (28 + 212) 262 7 2 5:6 g

b) Vnrn raenr cis rbicas >.y 2., raoirra (hadnlhi ac oubi al cn jualta) ?: g4 b>7 ?:.

Vnrn raenr cis rbicas ?.y 8., raoirra :6 g = b27 :6.

As uln prieraskl nrktgtkon oil h7 28.

bl7 ?: + (l >) 28 7 ?: + 28l 28 7 28l+ >2

b>67 286 + >2 7 212

Al titnc raoirrar4

_>67 (b>+ b>6) >6

27 (?: + 212) >6

27 > 886 g

Ve. 2


23/29



81 ^uaragis oilstrukr ul bitaccari oigi ac ha cn jkeurn, al ac qua onhn bitaccnioupn his oachkccns. Ibsarvn qua al asta onbal luava bitaccns.

Oultns bicns y oultis pncis sil laoasnrkis pnrn mnoar uli al ac qua quapnl hi-oa bitaccns9

Vnrn ac prkgar pksi sa laoasktnl 28 bicns y 8: pncis.

Vnrn his pksis sa laoasktnl ?: bicns y 51 pncis.

Cns bicns jirgnl uln prieraskl nrktgtkon oil n>7 28 y h7 >2. Ac trgkli ealarncas nl7 28 + (l >) >2 7 >2 + >2l.

Cis pncis jirgnl uln prieraskl nrktgtkon oil4

n>7 8: y h7 2; = nl7 8: + (l >) 2; = nl7 >5 + 2;l

Vnrn >2 bitaccns laoasktngis ul pksi gs. Vir tnlti4

n87 >2 + >2 8 7 :6 bicnsn87 >5 + 2; 8 7 >?? pncis

8: Ac hn > ha okarti gas, ul ngkei ca pripusi n itri ul trnti4

Onhn hn ha asta gas t ga hns >66 666 y yi hupckoi ac hklari qua mny al astnondn (ul oltkgi), qua, n jkl ha gas, ta pihrs ccavnr. Ac itri, haspus ha palsnr yaomnr oualtns oil cn oncoucnhirn, oiltast rkalhi4 Vir qu li ga ci pripilas hal-tri ha ul ni, axnotngalta9

Kltaltn nvarkeunr al qu jaomn puhi talar cuenr astn oilvarsnokl y dustkjkon cn ras-puastn.

Arn ac hn uli ha jabrari ha ul ni nltarkir n ul bkskasti. As haokr, ac gas notunc tkala 2=hns y ac hac ni qua vkala, 2;.

Ns, asta ni cns oualtns sncal oigi skeua4

Qln npirtnokl ha >66666nc hn supila >66666 2= 7 2=66666.

Hibcnlhi onhn hn uln onltkhnh klkoknc ha 6,6>, sa ibtkala4

6,6> 22=7 2 :=8?18,1:, onltkhnh kljarkir n cn prkgarn.

_kl agbnrei, jabrari hac ni qua vkala talhr ul hn gs (2;)4

Qln npirtnokl ha >66666nc hn supila >66666 2; 7 2;66666. Hibcnlhi onhn hn uln onltkhnh klkoknc ha 6,6>, sa ibtkala4

6,6> 22;7 1?:= 56; , onltkhnh guy suparkir n cn nltarkir.

Ve. ?


24/29



85 Astis ounhrnhis sa mnl ibtalkhi ulkalhi cis pultis gahkis ha his cnhis oil-tkeuis4

= og

n) Mnccn cns rans ha cis saks prkgaris ounhrnhis ha astn suoaskl. Ouc sar su tr-gkli ealarnc9

b) Asorkba cn suoaskl jirgnhn pir cns cilektuhas ha cis cnhis.o) Oncoucn cn sugn ha cns rans ha cis kljklktis ounhrnhis ealarnhis ha asn jirgn.

n) Ibsarvngis qua ac ran ha onhn ounhrnhi as cn gktnh hac ran hac ounhrnhi nltarkir.Vir tnlti, cn suoaskl ha cns rans as4

n>7 :8 og2, n27 ?2 og

2, n?7 >: og2, n87 = og

2, n17 8 og2, n:7 2 og

2,

As uln prieraskl eaigtrkon ha rnzl r7 >2

. Ac trgkli ealarnc as4

nl7 :8

>

2

l >7 2: >

2l >

7 2:

2l >

7 2: (l >)7 2: l+ >7 25 l

nl7 25 l

b) Ac cnhi ha ul ounhrnhi as keunc n cn rnz ounhrnhn ha su ran. Vir tnlti, cn suoaskl ha

cns cilektuhas ha cis cnhis sar4 :8, ?2, >:, =, 8, 2,

As haokr4 =, 82, 8, 22, 2, 2,

o) Oigi n>7 :8 y r7>2

, talagis qua4 _F7n>

> r7 :8

> >2

7 :8

>2

7 >2= og2

Raflaxkiln sibra cn tairn

8= Al cn prieraskl 2,12

, 21=

, >21?2

, sa puaha mnccnr cn sugn ha sus kljklktis

trgklis9 Dustkjonci.

Li sa puaha mnccnr cn sugn ha cis kljklktis trgklis ha asn prieraskl eaigtrkon pir-

qua su rnzl as 18

, qua as gnyir qua >.

Ve. 8


25/29



8; _k al uln prieraskl nrktgtkon n2+ n>?7 ?2, pihagis snbar oulti vnca n=+n59 Vir qu9

n=+

n5 sugn ci gksgi qua

n2+

n>?7 ?2, pirqua4

n2+ n>?7 (n>+ h) + (n>+ >2h) 7 2n>+ >?h

n=+ n57 (n>+ 5h) + (n>+ :h) 7 2n>+ >?h

16 Auockhas, al sus Acagaltis, utkckzn cn skeukalta jrgucn pnrn cns prieraskilaseaigtrkons4

nl+ > n>n>+ n2+ + nl

7n2 n>

n>

N pnrtkr ha accn, ibtl cn jrgucn ha cn sugn ha cis trgklis ha uln prieraskl

eaigtrkon, tnc oigi cn magis astuhknhi al astn ulkhnh.

nl+ > n>n>+ n2+ + nl

7n2 n>

n>= n>(nl+ > n>) 7 _(n2 n>) =

= _7n>(nl+ > n>)

n2 n>7

n>(nlr n>)

n>r n>7

n>(nlr n>)

n>(r >) = _7

nlr n>r >

Ve. 1


26/29

_icuokilas n [ pnrn targklnr?


VEKLN 52

Caa y oigpralhaQln suoaskl jngisn

Cns nbadns gnomi lnoal ha muavis li jartkckznhis0 as haokr, tkalal gnhra pari li pnhra.

Cns nbadns magbrn lnoal ha muavis jartkckznhis.

Ac skeukalta asquagn lis pargkta ibsarvnr ac lgari ha nltapnsnhis ha uln nbadn gn-omi al cns hkstkltns ealarnokilas4

G

M G M G M M G M

G

G >

G MM

G

M

M

M

M

M

>

2

?

1

=

Oultis nltaoasiras tkala uln nbadn gnomi al cn hokgn ealarnokl ha nltapnsnhis9Ouc as cn cay ha jirgnokl ha cn suoaskl ibtalkhn4 >, >, 2, ?, 1, 9

Raouarhns ogi sa ccngn astn suoaskl9

Ac lgari ha nltapnsnhi al onhn uln ha cns hkaz prkgarns ealarnokilas as4

> 2 ? 1 = >? 2> ?8 11 =;

Cay ha jirgnokl4 Onhn trgkli sa ibtkala sugnlhi cis his qua ca praoahal4

n>7 >0 n27 20 nl7 nl >+ nl 2

_uoaskl ha Jkbilnook.

Ve. >


27/29

_icuokilas n [ pnrn targklnr?


VEKLN 5?

Oildaturn y ealarnckznIB_ARZN4 >? 7 > = >27 >2

>?+ 2? 7 ; = ?27 (> + 2)2

>?+ 2?+ ??7 ?: = :27 (> + 2 + ?)2

MNTQLNOILDAXQRN4 Vuahas prahaokr ac vncir ha cns skeukaltas axpraskilas9

>?+ 2? + ??+ 8?7 9 >?+ 2?+ ??+ 8?+ 1?7 9 Oigprubnci!

EALARNCKTNXQ_OILOCQ_KILA_4

Ouc sarn ac vncir ha >?+ 2?+ ??+ + >6?9

Acnbirn uln jrgucn qua ta pargktn oncoucnr4

_l7 >?+ 2?+ ??+ + l ? ouncqukarn qua san ac trgkli lnturnc l.

Cn ibsarvnokl ha cis prkgaris onsis sagbrnr cn sispaomn ha qua sa ougpca ci skeukalta4

>?+ 2?+ ??+ 8?7 > + = + 25 + :8 7 >66

(> + 2 + ? + 8)27 >627 >66

_a oiljkrgn ci supuasti. [ pihagis saeukr oigpribnlhi4

>?+ 2?+ ??+ 8?+ 1?7 > + = + 25 + :8 + >21 7 221

(> + 2 + ? + 8 + 1)27 >127 221

>?+ 2?+ ??+ 8?+ + l?7 (> + 2 + ? + 8 + + l)27 (> + l) l2 2

As haokr, cn sugn ha cis oubis ha cis l prkgaris lgaris lnturncas as keunc nc ounhrnhi hacn sugn ha hkomis lgaris.

Cn hagistrnokl, npckonlhi ac gtihi ha klhuookl, sa puaha aloiltrnr al ac ckbri hkektnc.

Ve. >


28/29

_icuokilas n cn Nutiavncunokl?


VEKLN 5?

_nbas ibtalar ac trgkli ealarnc ha uln suoaskl y utkckznrci pnrn oncoucnr ul trgklioilorati9 Vuahas hajklkr uln suoaskl gahknlta uln cay ha raourralokn9

>Mnccn ac trgkli ealarnc ha cns skeukaltas suoaskilas4

n) >2

, 2?

, ?8

, 81

, b) > ?, 2 8, ? 1, 8 :,

n) nl7l

l+ > b)nl7 l(l+ 2)

2Oncoucn ac trgkli >6 ha cns suoaskilas skeukaltas4

nl7 1 2l bl7 > +(>)l

2

n>67 1 2 >6 7 >10 b>67 > +(>)>6

27 > + >

27 ?

2

?Hala, pir raourralokn, cn suoaskl 1, >>, 2?, 85,

n>7 10 nl7 2nl >+ >

Khaltkjkons prieraskilas nrktgtkons9 Asorkbas su trgkli ealarnc y oncoucns cn sugn hal trgklis9

8Asorkba ac trgkli ealarnc y oncoucn cn sugn ha cis 26 prkgaris trgklis ha cn suoa-skl 1, 5, ;, >>,

As uln prieraskl nrktgtkon4 n>7 10 h7 2

nl7 1 + (l >)2 = nl7 ? + 2l

n267 ? + 2 26 7 8? = _2671 + 8?

2 26 7 8=6

1Mnccn cn hkjaralokn y ac prkgar trgkli ha uln prieraskl nrktgtkon al cn qua n?7 =y n=7 ??.

n=7 n?+ 1h= ?? 7 = + 1h= 21 7 1h= h7 1

:Ouc ha astns suoaskilas as uln prieraskl nrktgtkon9

n) >6, 21, ?1, 86, b) >, ?, :, >6

o) 1, ;2

, 8, 52

, h) 2, 8, =, >:,

Cn o), yn qua as cn lkon qua ougpca4

;

2 1 7 >

20 8 ;

27 >

20 5

2 8 7 >

2

Ve. >


29/29

_icuokilas n cn Nutiavncunokl?_nbas ibtalar ac trgkli ealarnc ha uln prieraskl eaigtrkon, cn sugn ha l trgk-lis i, sk juasa piskbca, cn ha sus kljklktis trgklis9

5Asorkba ac trgkli ealarnc y oncoucn cn sugn ha cis iomi prkgaris trgklis ha cn suoa-skl4

6,>0 6,20 6,80 6,=0 6,>:0

_a puaha mnccnr cn sugn ha sus kllktis trgklis9

As uln prieraskl eaigtrkon4 n>7 6,>0 r7 2

nl7 6,> 2l >= n=7 6,> 2

57 >2,=

_=7n=r n>r >

7 >2,= 2 6,>2 >

7 21,1

Li sa puaha mnccnr cn sugn ha sus kllktis trgklis, pirquar< >.

_nbas rasicvar pribcagns al cis qua talens qua raoilioar ul tkpi u itri ha prieraskl9

=Qln agprasn ijraoa n ul agpcanhi ul suachi ha > 666 y uln subkhn ha >66 ncni. Itrn ca ijraoa ac gksgi suachi oil uln subkhn hac >6% nlunc.

Rnziln ouc ha cns his as gadir, oigpnrnlhi ac suachi haltri ha >6 nis.

Al ac prkgar onsi talagis uln prieraskl nrktgtkon.

n>7 > 6660 h7 >66 = n>67 > 666 + ; >66 7 > ;66

Al ac saeulhi onsi as uln prieraskl eaigtrkon.

n>7 > 6660 r7 >,> = n>67 > 666 >,>;7 2 ?15,;

As gadir cn saeulhn ijartn.

;Vnrn rihnr ul nluloki sa mn oiltrntnhi n ul ernl lgari ha parsilns, qua habaloicionrsa al 1> cns. Onhn cn tkala his parsilns gs qua cn nltarkir y al cn cn 2:tkala qua mnbar 15 parsilns.

Nvarkeun oultns parsilns mny al cn prkgarn cn, oultns al cn ctkgn y ac lgarititnc ha parsilns qua kltarvkalal al ac nluloki.

As uln prieraskl nrktgtkon ha cn qua snbagis l7 1>, h7 2 y n2:7 15.

n2:7 n>+ 21h= 15 7 n>+ 21 2 = n>7 5

n1>7 n>+ 16h= n1>7 5 + 16 2 7 >65

_1>75 + >65

2 1> 7 2 ;65 parsilns al titnc.

Ve. 2

matemáticas anaya 3º eso solucionario tema 3

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