Marius Burtea Georgeta BurteaCornelia Bascdu, Aniqoara Bl6naru, Silvia Brabeceanu, MihaelaBrdnzea, Carmen Cilinescu,Ruxandra Cristea, R5zvan Cristescu, Marijana Dragomir, Georgiana Dumiku, Beatrice Ioni!6,Marieta M5rtoiu, Adriana Moraru, Tudorila Neagoe, Paula Nica, Gabriela Nistor,Daniela Soare, Felix Tuinete, Mariana Urziceanu, Sergiu Vaicir

Teste

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MATEMATICAProbleme si exercitii),

semestrul I. mulfimi de numere

tuncfiiecualii

servicii, resurse, tehnic

CAMPION

erewnu op ltulfinru

uodBJ rnun Inpcrpsu

,o ' ,,ri= ,(,,t) (q

'o '!l= -(!r) (u

IecrpsJ rnlm seJelnd

,.\- = -q ".'q 'Qk

'D ' 9ll.

'!li = qrli (q

\ _ 1D..." cO. tDl

' '!l'lf= 9!] (e

rnlnsnpord Inlucrpsgtll)tpDt ap rlplaudot1

v n9 AJereoaD '-- - --li 't Lzl

L6srJErdorlqrB

.unsundsg.r 1$;$ucpu1

.I

a

(p

eJBnlBAe ap alsal€/""""""""" """"'rcturluetoi uiencg 'g'7

11""""""""" """'"lepueuoOxa rriencg '6'g

69""""""""" '[ nes Z ulpro ep Ileclper nc eleuorter pencg '1'g

"' eJBnlBAs op olsalV9"""""-"""' """""""osJelulaclneurouoEFlufcung'3'19s""""""""" ";'scy4euouoErrl iit"*i 'Z'i95 """"""""" """" ?cltulueEol e{cung '9'1

8r""""""""" """qplfusuodxae{ctmg'9'1Sr""""""".'' "" pcmer erlctmd 'lem1eu lueuodxe nc erslnd e{crmg 'y'1

2V""""""""" yfcurg leunesre^ql'ollqosle ul 1lcunJ '€'I0r""""""""" "'.tolglcurg eerun&uo3 '3'1

9€""""""""" "''er'pceflq glcurg'elpoe[ms ulctng'oaBcefu] $crmd 'I'l9€""""""""" """ Ul,CNns 'l

uJ,vncE rS trl,JNnx 1p1o1$e3tE""""""""""""sJEn[BAe aP 9]s3I

I € """"""""'e1erpdlq IIlEncg 'ol?ar orenmu $ueycgeoc nc g ppe.6 ep ye$enca E O ug BerB lozo1 't'ZL7,""""""""" """"'-""exeldtuoc ereurnu rclnlelcldO 'Z'ZgT,"""""""""""""""""xe1durocrgumurnrmppEnfuo3'xe1&uocJprunumtmepcugaEpsuuod'I'Z92""""""""" flXirrdl^tocuoT[rftrr\inNvgruITnl t 'z

v2""""""""" """""""""'erml€ e 3p elser02"""""""""""""""""""" """ appod Ieer rpumu mm pu4rreEol 'y'1

02""""""""" """""""""'eJEIrl€ e 3p 3ls0l9I """""",""' """"""""1GeJ lueuodxe nc polnd 'puorfer.rgumu luauodxa nc lremd '€'I2I""""""""" """" fl?olpernc lgeredg '7'1

g"""""""""" 'gpeudor4' {g'Z}a a 'leuoger.rpumu m-4ulp ,, ulpro op InlBcrpBU 'I'Is"""""""""" iITVfrUitugNnN'I

fusltnN fl(I II,{IJ,Ifl,{ 1 p101;dr3

,l}aP' 9'0-= 9ZI'0-A (c

) acareoap 'l- = Ij (q

= _f ecaJuoap 'Z= gft (e

:eldruaxg

> rp :epieloJ col nv

S .r lruEumu nc plu8a else

'p Inlecrper e$eumu ego t ulp)o ap lnloctpoY .

:i = lf (e :elduexg

) < 1t :eY{eler col nV

. \ nBS l-lt ?Z€AIOU eS

eEo else enop B aralnd mrpcrp InlecrpeJ elSeumu eg7 a lnutpto ap lnlDclpoY .

q$roa, xiEdatg , _/'rlYrurudoud

nIO 3C'INTYf,IOYU I'II-IVTU f,Uf,IAINN

{I

LL'

@ NUMEREREALE.1.1 RADICALUL DE ORDIN r DINTR-UN NUnrAn RATIONAL, n e {Z,l} ,

rnormnri.lr.

c Radicalul de ordinul 2 dintr-un numd,r real pozitiv.

Se numegte radicalul de ordinul2 dint-un num6r real pozitiv r , numirul real pozitiv a

clrui putere a doua este egali ou x.

Se noteazi 4fi rau "fi.Aulocrelatiile: G>o $i (.,6)'=x, vxe(0,+o) 9i ^[f =pl, vyelR. 16'=0.

Exemple: d J4=2;b),8=i, .).Fry=l-zl=?; O m=*o Radicalul de ordin i al unui numdrreal.

Se numegte radicalul de ordinul 3 al numdrului real r, numdrul real a cdrui putere a feia

este egal6 cu numirul x. Se noteazi {6.

Au loc relafiile: {6 e m. $i (i6)' = x, Vx G IR.

Exemple: l

a) lld =2, deoarece 23 =8.

12

mplex. .....................,........,..25,'.....,,..',.....'.,.21

,. Ecuafli bipdtrate...,.....,,....., 3 I*34

...........................67

97

363640424548565564

b) {/Ef =-L deoarece (-1)'=-t.

al-o,tx =-0,5 ,deoarece (+,s)' =4,125.fn 3 1r\' 27Jf-- =--. deoarece I -- | =--.Yo+ 4' (4/ 64

Proprietdli ale radicalilor :Radicalul produsului

a) JA =J;'Ji, a,b={o,+m);

Jrf ,r.-' r. = JA. Jrr....' Jo* a11d2t...ta1e [0,+o).

b) {ob ={;'1fb, a,b e JR;

{q.u' *4 = {i' {u'...' 414, 4,b,,...,4 . R.

2. Putereaunuiradical

a) (6f = J7, oe[o,+o), zeN;

b) (4;)^ =tr17, oeJR, rzeN.

3. Radicalulunuiraport

c)

d)

71

Mu[imi de numere

-

r*-{ffi-.:1::"1-{#s;if :4ffiG #

ffii*$,sk*FffF*$$$

\

)o)4

oPurly gp rlttlllr

Bep IFolcBJ ElBocs 0s ES 's

:€r.,gt (q :-{l_.'g,A (e

Fnl9um ezolncl€c 0s Ps 't

:rt.rv1" @ :_rs.r€l (e

rl)lErrrjn ozolncl€c es ES '€

:45t (t :gtz.ly (e

Enlgum ozelnclBc 0s 9s 'e

-/! '*A +sh:Lz-h (qJ-Ul'#l :,(r)l :_-!l (u

,',t

:6z6lnclBc es Es 'IW

{z} r [e 'e-] , ,-rol) 0*z-x 6u3x

sr\a ep ep{rpuo3 (c

r = .r erinlos eurfqo es

-.1.;.. r-rz

,i.

uruos 3p InloqEl urc

'.,i::Jo 0=t-x9+fZ) :iiipuor ound eg (q

9

-:_ax<=Z-<Y9l ?::rpuoJ aund eg (u

;-_,rJ\ (q :9.:21

*i-r*__y!_ylt:_Il'{*:[3T9]99gyr*u_wrgtyt-{

:ozaFclBc os Ps'0 > gA- s'repcsy .z-

= Il_($ * z_) = y * q*r'5| +,* = l$ -rl=,pc prlnzau .,(st_,)=

tr- u'| unrrroo., (q

'N sa'repu$v's = EA -z+r-gA*r = lf _rl*if _r!*v = o

:1epse q1du1s reru uruJoJ qns orJcs es D InxpurnN (e:'o>EA-q (q N3D

:pc tfelprv ill:s = n s

_,(f-*r)|. "(i - rl *l,z-l* o ,rrr"roJerounu nsp os

or_= fg)A_ ooor1t

,!_ f!)[ _ LZt\r ,( r_!,- _ll'

,i= li)1,= Ci 'L-=,1-l> :s=dA=lqlA (q

(e

't

ffiuM

$

(u

L

6bh z'o=

W=to'gA=C' :sz=,(s-)=_{r-f :LZ= {=J,r[ = ecltr :ue^v (e

000I i\. /z 1\ . ,

E,E/, :,1-li szJk (q loy 't]'tr-ll ,g,

'Q > D e !ft > lt rcrmle .y > q,D p)eq (qiq> p e lf> !1, rcrmle (m+.gf

= Q,, wee @JoJrJBorp€Jeo;erudruo3 .S

'U 3 q', 9ft ,, = _Jti

(qjo < g,M, o ,1t.

,lrl= _'! (e

IecrpeJ qns ap rolc€J rntm eoJelEocs .?

..u 3 s .u =, .#= :A (q

!t> l' \

:(m+.g)rg,(*-r,o] rr,ff=il (e

:azeJnclsc es ps

,L - (L\Lt- Gl/

7

't

4.

a)

S5 se determine .x € IR. pentru care sunt definite expresiile:

Jor*z: b) .,6r'*rr-:,, c) 1617* iEl.\-za) Se pune condi{ia de existenld a radicalului de ordinul 2'. 6x + 2 ) 0. Se obline succesiv:

-) I tl \6x2 -2= x ) j= x ) -l = x e i --,+* i.6 3 L3 ,/

b) Se pune condilia de existen!5 2x? +5x-3 ) 0. Se rezolvl ecua{ia de gradul doi

2x2 +5x*3= 0 oblindndu-se L= 49 $i Jq =-r, r, =1

Din tabelul de semn

-@ -3

Zx2+5x-3 l*+++0-

c) Conditiile de existenfd pentru cei doi radicali sunt: 9-r2 > 0, respectiv

xe1R", r-2*0 (condiliadeexistenldafracliei). Seobfine xe[-:,:] qi x*2, adicd

x e [-:,:] t {z} .

= g) =g\ r,l 7'

."...............-..:

\ 9 - 4j5. Ardta(i cd:

-l -.,5. iar

3rcitii -CLASAaX-a Multimi de numerc

ffiffiTN#ffi\llfl$llilillIM\ffi1. Sd se calculeze:

a) JF,6'ffi'/ft);b) {-n;W;iF, i[#'

f; tr.rJz.zs: n/o,ooo,, il;.y*or t;, aeR'. beIR*r

S[ se calculeze urm6torii radicali:

iEG; b) t[-srz; cy ift.07; ar ffi; .) m, 0 iF0,00r; el

SI se calculeze urmltorii radicali folosind proprietS(ile radicalilor:

r .r6f ; 0 ,[{V; c) , o),(rf; o6At'"

!zo

2.

a)

3.

4. S[ se calouleze urmitorii radicali folosind proprietitile radicalilor:

o S'6%'.

se obline sotu{ia x e (-.o,-3] r[*, *)

.(1,44)1

c)

arerunu ep lurlilnlN--X:gore - ,,ifrr.rc,u .-/ aprfcurg 1{1puoc ec u1 (c

x-Zl' r-Zl\ . 4_ = _i' (Itiu9 J:-!

, y.^11= (r+r)rl (r

rolc \ awruJolep es ES (q

t:t->xn4rcd Ill (lz_xl

eprsardxe eucs os pS (e 'St

9+x /\ l,= l-ln /N3xi=J (r

tv- ctl ),)iii=I:?/N=*l=tr

(e

rlnudlnur auluJopp es 9S 'lI

e1r-ur$1nut eulluJoloP os PS '9I

ffilfr- tl*lr-rl=(r)a

(a

(*)s (r

' lsqr.rl-llrr- zl* yvl (p: lsN-rl+lA-le.lz-Ezl

:,(*z*g)t lr- ile*,(ts- g'e)l

'lgA€-ls.lf -rle *le -7zl

(c

(e

(r

'6

(u

'8

(e

(c

(e

L

(l

(e

(e

'9

(q

-'tA+ -(z-r)f =(*)t ! ( 'l

"ur ?ulo.+ o el Pcnpe 0s ps

=(e-)[{r-,i)/\.il - {!l (, I

',I.Jf "rr-lr*,,1ka qecrpe; pJRI olJcs es gS

x -tl": i-rf - --:J =lx)I t

: L+xZ- zxs-f =?)g

ul aluol€A ounuJepp es 9s

l- t ,'I {,., :la:-r\-:-1

" €-r+ zri

, , J -l[ ,, ,!:j[ :r:-r/' ' c+rl rl

i-. nauad x etnrllJalop os pS 'Z Irtl

:906l'st+Wt (q

: Dft

,LZl\ @

.st-z . .z-* ,v :rr 'r G t-f ',: gt 'gA (q : g\E'gitz,

:eleJerrmu eJedtuoc es gg

/ \ / \ , te7'evrolzc esPs'rr

.o=,(r-lA * 41y-t)*,fi+g1;-o) (r|0=

,(gf*gf -n)*,(t-E--4 (l

erec qrs r eleer rororerrmu ? gcrqeuros8 ulpeur g pcqerryrr, Brperu ezelncrrJf;tJ u?f'*

,,","#t,',#.il:, II"ff * ";-*;,ffr"j';ffi ,#,r% r!' 1et

+ t) t' AV' (sft - z) t ct, (gt) t' (o) t' (q - r) r r,,' (l)r'(Et ) /' kr) r (r

ereolpcserc eurpro r{ errcs es gs 'ereolpcsercsep ,,rrs a$curg o e,e u[<- u:./ u*I'T'*"(z)t ,..,,(t )t ,,,,(gl-?)t

(,1,)7 cu:ff- u''-t(or"

:elegeode.l 1 nc ereduoc es ps oereolgcsoJcsep prns e1|curg o slss ,*6 +- U: _f pceq (q

' GA)r-(i*)r <w : (z) t -(stit 6 : (s'z) ! -kir

:roprserdxe Frfiuas ezerpn$ es ps 'ereolgcseJc pr4s ellcurg o epe U <- Ut: -/ pceq (e

ZqlA * fl t, :4gt rs Er (o :gt rs Ef (q :IEt rs g1l

( z ) ( z ) :Ilsclp€rluolPlllrnaredurocesg5

lsrr' fi)= a'lr' qQ)= v G: (tr* ?r'gn - I) = s'(e' gjr-)= r

,klt- t'gi= s '(g'+z'gh-r)=r 1p :(r+g['El)= r 'lgi'r'yz-e\=r: (z' g' - z) = s' (gll + z't + gt-) = v (q : (*'s) = s' (* * T'*) = r

:txtzei elSreoryurn q g -v ,g ay ,gvy epunflnru eqru.xe1ep es pS

',,a'oJgrz) tr ',(*-)A ',(i-) ,t ,W'?

'tI

(;

Zt _T:a- ,o (3

: yftt,,glv (c

i0> q,Or, , ,(q-).(ro-) ,ztruaq'o <D'zqep.sl' ,&rl ,ffi| t*21' ,g11' (u

(-tt, a <o,b <o;

.,r. lrEl;

r=* 15, ",600;

:L::1 :

'-. r r',|.- .-'-l'

3 = (.,J,7 -.lr),rr\

1 /a I

-.tr!J

l.:)

-I sr .0J64.

,. r.rrrnut expresiilor:

lnpare cu 1 rapoartele:

t: n ordine crescitoare

:-i: )."r(r),1(r.Jr)

'ei!.: reale a Si b care

,2

-i:-1) =0.

------------

l-ri08:CGO=.6..6*, ;

m G+ll;=7;;

e)

12.

a)

b)

13.

a)

lJr-fl.l€-..61.. *l"E-Jr001; 0 l{l-{61.1fi-.6l. .lVi"-,ffi|.Si se determine r pentru care existl expresiile:

Gi , f-2.I.-rt c) {lt*r,;

otlvt

-,

VJx -:D J-x'+lx-ro d*

Sd se determine valorile integi ale lui x pentru caxe sunt definite expresiile :

E(x)= "[-55 a,6i ; q E(x)=O:7 .ff-,E(x)= fi-^1..r, a1 E(x)= E. E.Sd se scrie frrd radicali expresiile:

:, lx'+rl-tt-{f .J6-3f , unde re(-oo,0);

., {4; -.1:=*Ji:,+r)'-J(a-2a)2, urde aeN, beZ\N.! (-3)'

15. Sd se aducd la o formd mai simplE expresiile:

a) E(r) = {r-4t.1ffi, x e(-3;2);

c)

14.

;) E(x)={IilT*J7,xe(r;2) ; d) E(x)=i(,_f,-GT,x.t;3) E(x)=lr-+l+ 't--' -zrQ-zy , xe(2;4) ; D E(x)=.,(,_sf -F-r'y, xe(:;oo) '

16. sa se determine mullimile denumere:, = l*.

x1 ffi . x);

i={,.*l \m=ro}, c={,.*l ffi=x}, a ={r.ul

17. Se se determine mullimile :

a) t=\*e N/Ja-, . x) ; b) a ={*e N/Vl2-x. x} ;

a) SI se scrie expresiile ca un cdt de radicali :

Ix-Z / aa\ . ,,'u lTl--1171-l# penfru x<-1; ii) l#,pentru

xe(-z,t); iii1.ft-r[,+r),pe'ntru x<-1'

b) Sd se determine valorile reale ale lui x pentu care au loc egalititile :

ii) S(;T=J-Gr+t '

E; ,r;zrv) {3-,=i[-]'in ce condilii funcliile f : D -+ lR si g : D' -+ lR sunt egale, daci :

c)

1t.

r)

r)

iii)

c)

6-2"11)'

rerciiii - CLASA aX-a Mu{imi de numere

arerunu op IurlilnN=.5I yqy1Q_*Iitc:axa rS euolqo.rd yf, IJVI^IAIVhI

eeurEunl alse rlfu = r\=yFur Flsurprec {e1nc1u3 (p

.., *N 3 uA,

lfape ap earcolul $111qu19 (c

'1euo{er Jplunu

nur8unl gcep {ecgusn (qle.uasqo e) ''dO '?O "dOrc rS (1uuo{uau mpu4troSlu' 7 elepmd llln4suoS (e

(,,ro1qectPer qertd5,,n Enop epunlstp olaJ€oue]u

utaruo nc'l ug crqSuqderP

-rt urnulsuoc '.N 3 ,/ oJeolel

rs (6'o)o elepundold '82(gueefepnl udula

upunu u pEeerlul eaued alse

nrms ozepcluc os ES (q

5 azo.qsuoruop es pS @ 'tZ

F&unu ozeFclsc os pS (a

r)E gcup orglro^ os pS (p

u1 eaued azo1ncluc os gS (c

t (t + j) ?c eluru es PS (q

u ,, ,r"oruoo,op os ps (z

i..r."'. urru*'g^ll^r = (r)?

J,\= = \u)v apsordxe alJ '92

't6r)g ezelncluc os ?S (a

'(6)g ezepcl?c es ES (P

lJ 9c ezer$uouep os PS (r

,p ?3r€ol€A ocglJo^ as ps (q

| - -Lt > zil pc elsJ€ es pS (e

\-+ +[?].[r]=('1,=lu\ ! eprserdxe nsp oS 'SZ

iiiil i ;j .rffbi,lir{,

fi+ u)uf z rA eA

,:l - l:!l y -!l' gf-tf '

: N3 gAe-Hl - {!:!l+ Tz-st = t (q

' -N,A, 4- r. 1u)r Er ozorsuoruep es qS

t

'6 t (yZOZ)f retftzodo-rd € rq ope op uer€ol€A ozelnclec os pS

l+tr1

'(g)g lntnrprunu e p8earlul eegud ezelnclec os pS

1+uf u[\ (1+uluf\ , = -: J : ?J ozalsuouep es ES -N="r\ I - t yrezd-

(o

(p

(c

(q

(u

I*"

= g urso:dxa oH. -tZ

z- zl' ztz+tl\CI=rG+-fi?=t (r

: s=(sr*z)zl(,-gA)i=, (e

'17,

(c

(q

(E

'02

(8

(p

(e

'6I

(r

, 2l*sl*sA u : l^+el (a

: :olrrflzodord B Jplop€ op eamole^ azeztrcerd os pS .€Z

1|o l.39l = ." ts

: tl'Z- tA =,f ,'=J

: g\9*Zrl\=,( rs Ltg+ztt=x (c

rs Ltz+vt=r(9 : tl\n-Lt=t r* tjtb+tt=x (u

:olrmz€J u1 'f iS r rolaJaunu u pcrrlsuoeE uperu rS pcnouqrJe

Brporu ozolncl€, 3s Es rode rur y ) {'x g3 etrBrB es ps.{z> q,D / girq+u} = 7 eeullllnul erd .zz

.N 3 g EcBp ecgrra^ es CS . gAr_91+ gtZ_tt = !r ersardxe ard (c

/\ ' ,\T'l - ' )= Tv' 9 19'uI 1e31se ' N 3 g', ourilrolap es ?s (q

,kf - t) = p7, *E Pc ozersuouep os PS (e

. _N 3 rzA ,19

= (a)a pc€p ocgrrol es pS

(sz)t ,r"1n lec os qs

; {e,rresqo eC '(S)S ,(V)S , (f)A rre1r,"1€c es qS

' N 3,/' {t* rz)+"'+s+€+ tA =@)s ursordxe erg

'otf,+611 (t : gA+zA (r{

(c , !.31,

(q: WJ\: JoloJe[rnu e p8ee4ug eeged ezepclec es pS

!r,tl,z-t=(,)a 1s (r-rs-)rzl=(").r ru ,ffi=(,)a,, A

=?)t

25. Se dau expresiile r(n)=[.fi'.l].[i7+ l. . .l'l*" -zr)*

r(")=[J,].[Jr]*...*[J,], r e N

a) Sdsearate cd n2 1n'+k<(ru+1)' ,Yn,k eN, k<2n.

b) SI se verifice valoarea de adevdr a afirmaliei ' [Jr' ./.-l = n, Yn,k e N, k < 2r .L' J

c) Si se demonstreze cd f (") = 2r2, Vn e N- '

C) Sd se calculeze E(9).

e) Sd se calculeze E(a9).

26. Fie expresiile A(n)==!-=*-=+-r , n>-2 siJt+Jz Jz+J: 'ln'l+,ln

111E (x) = r[*'E . *"6*"5 ..... rnlro*1ffi .

a) Sd se demonstreze cd +---- = Ji -,{k1,VteN-.Jlr-t+Jlr

SE se arate ca (J,r + t)a(n)e N,Vr e N..

SI se calcule ze peultea intreagd qi partea frac{ionard a numdrului I (l0l ) .

SE se verifice daca E(x) = vr(r00)n Vx > 0.

S[ se catculeze numdrul ,(!m)

a) Sd se demonstreze cd n +t s ^{i(r.+l)

< n +2, Vn e N-

b) sr se calculeze suma, = [rD;].' [JtC],' ["141)...' + [J2010

1013], unae [x]

este partea lnteag[ a num[rului real x . (Concurs de matematicd aplicald "A. Haimovici", 201 l,etapa Jude{eand)

28. Fie punctele O(0,0) $i P" (1,0). Pentm fiecare

valoare r e N-, construim triunghiul OP,\*,,dreptunghic in {, , cu cateta \,P,*, de lungime 1 , avAnd

interioarele disjuncte doul cdte doud (vezi figura l,''Spirala radicalilor")a) Construili punctele P2 , P3, P, (conform

al goritmului menlionat) qi calculati lungimile laturilorOP., OP3, OPo.Ce observa(i ?

b) Verificafi daci lungimea segmentului OP, este un

;iumdr ra{ional.c) Stabildi valoarea de adevdr a afirma(iei : "Lungimea segmentului O{ este egal6 cu

,[i ,vn e N"'"d) Calculatri cardinalul mullimii ,4 , rurde :

1={o eQl, .rt. lungimea segmentului OP,, pentru n e N-,n S 16} .

9r.r)= "t-z.'h;+l

Jrr I sd se calcuieze media

b)

c)

d)

e)

27.\:8r10.6.

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:rcitii -CLASAaX-a Mul{imi de numere

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