Mayius Burtea Georgeta BurteaGeta Bercaru, Cristina Bocan, Daniela Dincd, Latta Durnitru, Ionu{ Georgescu,
Titi Hanghiuc, Simona Ionescu, Roxana Kifor, Paula Nica, Ilie Pertea,
Mihai Popeang6, Daniela Podumneac6, Diana Radu, Carmen Rusu, Dorin Rusu,
Loredana Taga
Au$lisrul qgstsr a fost aprobat prin OMEN nr. 3022108.01.2018
a
sr)
I
T
I
CL.ASA a Xl-a
MATEMATICAProblemeTeste
a r OO
exercrtrr)
sisteme de ecualii liniarefuncfii derivabilestudiul func1iilor cu ajutorulderivatelor
profilul tehnic
CAMPIONBucureqti 2018
v
dueulurJalaO
{t'r}t3>ilnes lr+w'il - gl= (f')PP Pc ulo^V
a$n1og
,d :> tu eurruelep os PS 'Z
rr 'glrqesJe,\ul elso tr sacl4Bru
=9-9=17)leP:rueuriqgEirq?sJsAUI O1SO etululu o
a{n1og'ellqesJe.\ul luns
netu pcep euruuelop 0s 9s 'I
= yy rcfience etinlos o
= yy rcrlence etinlos o
l'llIqBSJoAuI also tr Poefl ':r = ,-(,-v)
(e
runr (l) " w, g 'y Pceo" .
rueruolo Ituoultu else 'IY ePun
e lntueuelduroJ'Y r esndsuerl
runlpu tect4eur oleluelualg
(r') r"PF'rpun ' r.---'. =,Y "t
:ee1u1r1e8e col erv .asD^u! D)lrlDM lnfi)lD)
rrsa (3,) 'w-y ocl4eruo .,ur ar,(erunu es 8' €ecIJl€I tr .
'"1=y.g=g.y Wcv!
er3ure6 '(:n)'lt, )v etl .)ltail4 'toziatg
;U].\IJI gJIUIVIAI
$S
86"" "" eger8oryqrgsr""""""""' """""""""'rurlsNndsvu rs [lvJrcNl21""""""""' """ YJIIYNETVIAI YZI-IYNV'EAIJV-InJIdVIAU EI^[E-lgOUd99""""""""' """"'roltticunj e pcgur8 ee;eluezerdeg 'g
t9""""""""' ropdcun; Frpnls ur enop € relelrJop inlog .Z
09""""""""' """"""""""rop(cun; Inrpnts u! rgtul rol€^rrep Inlo6 'I09 rololelrrop puolnlu nc -ropricun; tnlpnts 11 yn1o1rde3
89""""""""' " 1e1rdsog.1 rn1 epp8eg 'L
09""""""""' "'11icunj rourl ? rop Inurpro op Btu^rrecJ '9
LV""""""""' esnduoc ropdcurg BeJeArJeC 'E
2n""""""""' allqu^rrap yicurg nc lferedg ',6e""""""""' erelueureie topricutg alele^rrog 'tSt""""""""' ........' elerelel elelrro(l 'eleJrnurluoc rS alulryrqe,trreo .Z
I €""""""""' """"" Eculeruoe8 a.reterd-re1uy .1cund un_rlur rricury reun ei€^rro1l .I
ITaTIUVAIuAA IIjSNnC 1 1n1011de3
12""""""""' """"""""" prqe31y'aAIIylnJIdvSau Ehtilfsodd22""""""""' """ ssneD rnl epolofitr 'Z't8l """"""""' "" roruer3 rnl epolelN 'I'tL1""""""""' oJelurl Joleuo]srs e oJe,\lozoJ ep apole14tr .V
n1""""""""" """"'€leoolr1eru Brrtrog 'elncsounceu € llntu Ier n, o-rerurl Ilienoo op outolsrs .,01""""""""' """"" o1l?oJrrluru rriencg 'Z
s """""""""'""'elrq€sro^ur ocrrlBl^l .I9"""""' "'auvlNt-l IIJVnJE EO:Ih[EJ_SI5 t p1o11de3
SNIIUdIIJ
ffi:Ii?:i;:" {.-ie
MATRICE INVERSABILE iN
27
Breviar teoretic
r Fie A e M,,(c). trlatrice a A senumeEte matrice inversabill dacS existi B e M,,('iJ), astfel
ircdt A'B = B'A= 1,.
o Matricea B se numeqte inversa matricei I gi se noteazd B = A't '
o o matrice A e M *(c) este inversabild dac[ 9i nurnai dac5 det(l)+ 0.
Calculul matricei inverser Are loc egalitatea:
A ' = -)--. l'. unde ,4. este matricea adjunctd a matricei I .
det (l)Elementeie matricei adjuncte ,4- sunt complemenlii algebrici ai elementelor matricei
transpuse ,l . complementul algebric d,, al elementului a, este dat de relalia 5,, = (_1)'t' A,,.
unde A, este minorul elementului dr.
. DacI A, B e M,, (C) sunt inversabile au loc relaliile:
a)
a
(o')-' = o, b) (aa)'=s-t.trrDacd A este inversabil5, atunci:
o solu{ia ecua{iei AX = B este X = A-1 'B;
o solulia ecualiei XA= B este X = B'A t.6060636672
75
l. sa se derenninc daci ,u,ri..r. I = [] :)
, = [], :')
sunt inversabile.Solu{ie
o matrice este inversabild dacd are detenninantul diferit de zero.
ob{inem: det(z) = 5 - 6 = -t *0, det (B) : 3 - 3 = 0, det(c) = 0, det(D) = 0' Rezulta cd
matricea I este inversabild, iar matriceie B, c, D nu sunt inversabile.
z. Si se determi ne nt e* pentru care marricea , =(' il .r,. irversabild.[r, 8,
Solufie
Avem cd det(,a) =16-m2. Din condi{ia det(z)+ 0 se obtrine c6 nt2 -16 * 0' deci
m+*4 saum€A\{-4,4}.
Determinanli
" ^.ffib,ffiffi
(tI
c =12
[:
1 l\ (tr 11, o=lztt) [:
0 1)
I rlI12)
riuEuILuJalaOoreig^ul op Iiplpm ed uircraxe rS euelqor6 .€-IX B vSvTC - yCIJyI IAJyI I
soclJ}€ru 3l!I '8
r/ ole3ul€(u PJeplsuoc os ',
gr rnl €sro^ul deuriu-repg (q
:fg \3 SV [e1nc1e3 (e
)0)I
, 0 l= r' : elecrJ]Btu olJ '9
0 r).IE SSJOAIII
I)I
I i= F' eeruBru Pc du1ErY '9
t)(t I +'rrutuadgllqesrenurerse I r [-n I
lz z+x
0 r)t o
l=r' . z)
\ gY)' rgJI lUnS 3lecl.
; I)| 7l=V
-t 0 L)
'Y-'I=,(Y*'t) $ (y+'I)=,(v-'t) gcgtltrzoterfulerptsesceulc .21 =(r*rt) (r-rt)
'1e;trse'rS (r +' t). {v - z t) = rr -, I wq
.
rv -, I = ro -, I =, I :Arsecrns rueav
a;fn1og.olrqesrelurluns
U+rI l* y_rIeloJrrteu gc etem es ps 'b = zv gc eelulerrdord n, (x) ,w. v eecr4errr greprsuos as .s
't*'z\n(!'--l=,,\r )
erfnlos nc 0 < Z+wS_ eMZ , w u171nper8 ep erienceur Etlnzea .0, (g_ *Z){t_w)V_V=V
pc aurfqo as '11= xA,O* (r)lrp ulq .E_ utz+xZ_ ,x(t_t)= (z)r"p euriqo eg
'o * (y)rcp pcep plrqesrolur olse y e;,curelN
a;fn1og"!r
= r ectJo
'rY *evlrv)4eurpc fe1p.rV
(q
(e
:aleculelu el.{ ',
€+rr),*, l=,t)
=V(z o o'\ _ (z o o)
lz- t ,- li =,r,nt'lz- t ,- I
[o I- t) ('o r- t)'r=l' 'l=,,,
.u=lo Il -. lo Il
tr rl p rl- = "e 'o = lo ,l=
''eI rlc -t
I
lr rl
lr rlI l= "olorl '
.,-=[ li-=,r,r=li l='r.,-=|j l-=,,
eecrJlaru e;ec n-qued ,V,> w euruuelop es BS .?
arJcs os plcunlpu uecrJl€htr
:Joloculsru olesJo^ur azelnclm es ps
' 'lto {t 9l ,,.t+D t)' \t D)
ruolap agsg8 epole,r rulued tS
mlruleruered ellrol€^ {eUV 'g
)t ( nsoc purs-)t u '[ru,. ,*r.J t't.
3 (q {l :)[s aJLII€ru e4urP erec rfugY
[: i)=r '(: :)=,epcr:1eru e4urP erec {egY
T
(e
L
hrl -=.? 't=l l=.?l0 €l
:rcuqe81e rdueueldruoc
(r r r)*p1nr1n3'l L I l=r',:r.ur^v'ElrqBsro^urelso u lrop
.z =(r)opeurlqoes (q
l.o t t)
[,: ';)= ", q* = u esre^,r Be,rr,tu,,, '[,1 ';)= o arsa prcunrpe sorrr]er^r
(z r)'z="9'E-=t'g'l-= z:q'z-t'g :rcrrqe8lerdueueyduo3
[, ,)=, ,gc eutiqo eS 'EIIqBsre^uI erse V \xap 1= t-2.7 = (p,)tep rue.,ry (e
alinJog
(t e) :l l=v\I Z)
(too) 'l r t rl=r (q
[r t t) 'I(e
'g
)
w:::. !"i;ijii w,;!,";/ t;*,p"se ;ii:i
1. Afla1i care dintre matricele urmdtoare este inversabili:
(t2) (23) (ti) ,=l:j, :.l, u=fr'; il\3 o) \4 6) \, t) [o z t) ls ts _to]
2. Aflali care dintre matrice sunt inversabile qi determinali inversele lor:
", f , ,..l, b) c :), .r[, _r\ ( i 2\
[s 4) \2 6) tl ,,J' o'[-, 'i'
= ' , ,'=(' -')J;r(r) (-3 z)'
101 i) . Calcul5m
I t .i
I ^^ lrol. = IJ. d., = -1. .l = *1,r -' It tl
l: tlr.r. a.. =l l-')
Ir rl
: -1 o)ILI 't'r r) I I
)l
2)
; j *" inversabild pentru
r e l. se obline cd
mt - 5m +2 > 0 cu solufia
55 se arate ci matricele
,', - .1 ) gi. astfel,
-,i r ;i (1.+A)-t=Ir-A.
(cosa sina)t)
[-rin, "o"o )'
(2.
8. Fie matric." , =
[?
(t I 0\ (t -2 7\ (3 -l t)ol r r r l; erlo t -21; r,rl-: s -2
I
[o r r,J [o o r,J [r -2 r)3. Afla{i valorile parametrului a real pentru care fiecare din matricele de mai jos este inversabili
gi pentru valorile gisite determina(i inversa matricei:
(a ,) ,.r, o*,), .,[l :, 1'l, ,,li 1 ;.], .,['l' ";' llu' [o t)' o'[, o )' ''[; : ;,J' ''[; ], ;) "l ; i ;l
(3 o o) (z I 3)
{. Fie matric rte, .t=lz , o l,
u =lo , , I
[t 2 t) [o o r.J
a) Arltati cd matricele sunt inversabile;
b) Calculali A-' ,B-', (m)-' 9i ardta{i ce: (.lA)-' = 3-t ' tr-t '
(t a+t 2 )5. Ar6ta{i c6matricea l =1 1 I b+l l este inversabildpentru orice a,b e,4. qi determinali
[r 1 t)inversa ei.
(t o -,) (t o ')6.Fiematricele:l=10 I 0l.B=10 101
[o o rJ [o o rJ
a) Calculati AB qi BA;
b) Determina{i inversa lui B '
(r7. Se considera matricele I = I ^
\0
30\I
4 I l. Si se calculeze determinantul inversei matricei date.
0 t)
o -r\ (1 z)
, , 1,, =
I , -1 l.
Sunt matricele AB qi BA inversabile?
' tr 0)
::.'. ftare Determinanli
i.
t1ueulurJalaOBereip,ru3 ep riplrun ed rrircrexe rS auelqor4 .p-IX e VSy-IJ - yJIIVNiIJVI I
mrrslu &=12 e3rutrued (q
!,-8' Is ,-u Iieuguereq @
F elscuterutS g= z etg '92
W utP g'Y eIecI.IlBIu old 'W,
(r, €)ppl0 t Zl=.Vpcel.'EZ
[o o IJ
e- ' ('f + f,)rep i
= ('1 + r-,')leP
"(E) tltr 3 fr secl4s{u eld 'ZZ
,ttlni =(Q'o)r-w Pc fulgrv
(r- ,\rierp:nl' 'l=g gxe1
\I I,/
,= (q'o)W eeclJlelu PP eS
alse / BocIJtrBtu Ec IislPJV
'.'It-yl= rY g) Ietqry
0 r)r f Ol=V uertJlEuetg -t
0 z)
'r-g Il€uluuoleC
zlrqesre^ul tr €eclJlelu elsg
-ZI)- z Il=u EerlrteruelJ
I
:.- z l)',,((r)x) $ePc1u3
=, Inl olIJolBA rieururreleq
Y =(s)x'(v)x P'f€rerv
= l' ?3cIJl€Iu gloplsuoc eS
're es,re,\ur rfeuruueleprspyrqus-re,\urolso gr+'7 eecrrlerupcrielgry ''o= rg lgJuJie-Irsper,leluo g oljl .r,I
'pllqssJelulelse'€7b +yd = ,_(rt-tr ) o_rurn.r1ued,y.:>b,d eurlureleposes (q
iTtu = ,v erec n.rtruod x > ll./ ourruJolop es es (e
(ot t \
I t ,)=' eoclrleul ereplsuor es '9I
(o , z\r-g Pcli€]Prv ''I-tr=s ls lf z r l=r/ olecrrleruerc .sI
[t z t)
PIrqsSJeAuI else
.8''-n t=
'.Y = ,-V gcputtls w
',-tr, 1ie1nc1e3\. - ,Y )Y er tlElPrv
tr :secrJlPru or{ .nl
IEor InJler.uered {eUV .I I
: Y eecr4eru elc.ol
= tr eocrrlEtu eld .6
(q
(e
(q
(e
'tz
(q
(e
'v7,
'@*'ll=,(v-'t):lc{errsuoueg (q :e1=,v pcrielp-ry (e
ig s-)I - l= r Erruleur orJ .cll'fr 8- )
(Z \ t-t
I i '* l=,, [0 t )'7=/r,p:pcpurrlS x)u,w elrrole^ lieuv. (t+u'u I-ru)
\ r,, , )=' Earr'uur'u orl 'Zl
'leoJ r uJsoJecuo
:rrc=(,rs+v,
(r r o)lo r ol:[o o z)
(* , ,lt > u'l x I- * l=v
eecrrleru erec rrlued rir
(e x z)
olu,l
t)
;)D
riegy' y: r.a
(;i:
'6I
(:
(;
ta
'81.(r o).v = ,V 'le
,)= ,, pc puttlS eleer ereurnu g ts
Z*ZT I-MLLt I lut
. (qI}BUV 'i
\Z
l1i.: = l..i-t=A'0 t)
-l = ,-{'.
18. Se considerd matricea ^=l: ]l qi mul1im ea G:{x(")/ s,x(a) = I,+a'A}'
[3 3)
a) Ardtali ca x(,t)'x(n)= x(a+b+4ab);
b) Determinali valorile lui a e i*, pentru care X (a) este inversabild'
c) calculali (x(r))-'.(t 2 -3)
19. Fiematricea l=11 2 -3lgipentruunarealfixatfie B=aA+I:'
[,24)a) Este matricea I inversabild?
b) Determina{i .B-1.
(2. o l)| , ol.Fie matricea A=10
[r o 2)
Aratali cd A2 =4A-3Ir;Aratali cd matricea I este inversabild qi determinali I ''
Seddmatrice a M(a,b)=(':u b,l,n,z,eq'
\ b a-b)(t I )
D""a B=[l -r)u'at*icd
M(a,b)=a]z+bB ei B':=21';
Ardtali cd M'\a.bt=.lr:* -U)
(o 1 1)
Fiematricea AeU.(R',,t=lt O ,lCalculali detAEi At qiardta{ic6:
[r r oJ
1
det(,4-' + /,) = -dst( I + 1,).
(r 0 0)
13. oura l' =l2 i o I o.,..rina1i matricea ,4.
[, 2 t)
24. Fie matricele A,B dinM, (R) astfel incat ^,
=(t, l'l ar*,*' AB + 1BA)-,
2s. Fie aeR simatricer. r=[::;; :"JJ ,:t::;; i::la) Determina{i A-t Si B-t ;
b) Pentru ce a e R matricea -B este inversa hi A?
,: . _j- este inversabild
;nind cd: det A = 2,
1- I
: r::r ersabild.
l.:: rnversabild gi
20.
a)
b)
21.
a)
b)
'r't
.., tr.ilc Determinanli
(ueuIuJalaOOIareip.r.u; ep !-ulpn ed rrircrexe rS etuelqor4 .€-IX B VSV.I:) * y)IJV1IEIVIAI
a$n1og
)=[: :) " [: :)tu e1tence eAIozoJ es PS
0 z)o ' l= *,qr plrnzeut ,-lI z)rt€)I
) Ll=n, oullqoes
r o t)(roo)l, t ol=treecu,rJttzt)= J elSs rerlencoellnloS
z )q-( r z-')!=r-.,lr-[z- | )r-
"*- [: l)=o u*o
a$n1og
.: : ,l) =-l::):ey{unce ellozoJ 3s PS
s-), I
= _Y :elsr tctiuttre
'9 'i= "9 '[: '^)=, ,\8 t)
:Arseccns uro^v .ptrq,Sre^ur elso y ,ep.t = 8-6 = (r)pp ,u,ito .s [t !J: , ,,, (q
(r r) (r r)(r o'\
l'-'.i=[,t,Jl,-r):*atsctatlcncaerinlos
(r'_ o,)=
, o,,, [.] ?)= ., ,,.,
gpunlpeBoJul?W .l=rrg,O=rzg,Z-=.rg.l=,,g lunsrcrrqe8lel4ue*e1arro3 [l l)=r,
rsr=(r.)rrp..,n[l o) (t r) (t o) -.. Ir o) \' ' ' [z t)=Y ""'t'^ [, )',1, ;)=r poptlnzes
[' ",)=n
"" (E
tl i)=('J Il (,:,,r"r"s .[r: e-)=,
r. r* [,t_
t;)=., ,t=2,s,8_=,,e ,r_=,
(q
{')=r? tl (p\€/ (z t)
.{'.' ' 'l= , [f r] (q\r v) (r t)
rY
alinlog
.(r z') (r r) .l l=l l.x \0 L) \0 t)
{r 'l= " [, ol\r El lr t)
:ol€oculsru olniencs $u,r1ozeg
flfYflf,turyru rrJ,vncm
nI-'relse ro BSroAur rs gpqesrazrur alse
' ,-(,r)= ,(,-Y) :gc {e1gry 'oleal oror.unu eluetuole nc rop Inurpro op glrqusro^ur oJuletu o tr olC .gZ
'plrq€sro^ur elsa gr pcsp retunu tS gcep plrqusre^ur etso tr gc rialsuoureg (q:trg = gtr pc rlelgry (r
'g +v : gy tgttn 1e;lse (;r) z4 > g,y all .17,
, .t, BsJolrlr riuuruuelep rS ppqesrea.ur
frt*t e ,, ) alse u BerrJrBr.u ecrielp.ry'y-=r'q'r, 'l eg iq+l Dq l=H
eerurur.uelJ .gz
[ ,, qD ,D+l)
(c
(e
't
L
Y-'I (q i yg= -y (e
(t r r)
l, , 'l=zBerrrl,uerc '62
fr r t)
(e
(r
:ec elEJe es ps
= _\' olsa rerfunce e{n1og
. .,.,. '[o t] = , ,rno
[z t)r-t_(r r'l 1.t z,-
'- =[o ,-.,J [o t
=
- :.h 'l-=r?'l-=
(r r) rLUI\V I l=ValJ\0 t-)
.;: ai r'l3tricea I este
r\ ersabrlA.
':. .eale. .{rdta(i cd:
I
I -- I
( -r 0\ (-t 1)i) t'. ,=[ ,' ,J or","ca det( A)=-t, ''=[o
,,J' 4 =1, 4, =0'
n,, =-r. 6,,=-t. r.=[], l,) ,' , =-(1, l,)=[,' ?) t"'*'"sistemuruieste
(t o\ f-l o\ (-t o)
"=[, ,.J [ ' ,.J=[-, t)
r) o,.u ,=[l 1). ""..cd det( A)=2.'n=(:;) ^.=(], ,'l ,'
, '=:(:, ;)Soru{ia ecuariei este x = i[r, ;) ii) = iti)
2, Sd se rezolve ecualiile:
(t 2\ (-t z :)-r) t,,,J "=[ , ;;),
Solufie(r z)
.r) o".a I = [i i)rr,"u^ cd det(l) =
,,_-t( I -2)=lf-, ,).' =T[-, rJ-][ 2 -t)
Solulia ecualieieste X = 1[ -'- 3\. 2
ot)t.1 t)
3) Matricea r=[i : i] ""
det(z)=l,deciesteinversab,d.
(t o o.1 (rSeobline 'o=1, ' o
I' o. =1,
[: t t) lo
(2 ,o)r, _,l-r , ,l.l o ,Rezutti"aX=l + ,,1[; ;[z o 1/'
3. Si se rezolve ecualia matriceal5:
(t ') , [3 2]-f r 'l[: s,] '^ \+ t) lo t)Solufie
2l-r 3
4020
(t z :)b) ,lo I rl=
[o o r.J
2\/-r z :) r(t o 3) (t-,.J[ ' , ,J=;[-, : :J=[-'
o\,l
t.1lt)
(ti-r
-2 -'), -, l.0 1)
-2 -l\ (r, -, I Ei a-'=l o
o t) [o('t -1 -1-r\ I -I l-l 5 -1-ll=l
, J l4 -8 -'3
' (2 -2 -l
-z)t.t)
t :\{= laredet(,1)=1q10 l/ \ /
'. D-:: = t. lututri."u adjunctd
rsacild. Avem succesiv:
'3 -t\i: .-{ ^ l. Solulia-8 3)
L1,10 Determinanli