8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
1/101
www.cxfkcbdcnflfjfurcft.njh`spht .rh
Qfrefbtc
996-699
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
2/101
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
3/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.
> \SNECLTSJ E (
3p 6.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef > 6 6x ; .3p >.\sc frftc lvclthruj v f n l; + +
, ubdc , < , >f e g n e g l e g; ; + ; +
cstc lhjebefr lu vclthruj
4 1d e g; +
3p 4 = ... > 69> >> 3 > 9x x x x+ + ; , ubdc 6 >,x x subt shjueejc clufece
>< 9x x f + ; .
3p 1.\sc dckhbstrczc rcjfef > >lhs lhs 6N L+ ; b treub`meuj FNLdrcptub`mel b F.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
4/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.< \SNECLTSJ E (> < jh`
bb + .
3p >.Aec publtcjc ( ... lhs605+ + + , teebd lhs 59 9; .
3p 1.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef( )
> 3
4,4>>
xx
;
.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
5/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.4 \SNECLTSJ E (.\sc rcprczebtc `rfael aublef 7 , ( ) 4a a x x ; .
3p >6 >x x+ , teebd l 6x e >x subt shjueejc clufece>
> 6 9x x ; .
3p 4.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >j` ( 63 ) >x x ; .
3p 3.\sc dctcrkebc vfjhfrcf bukrujue rcfj fteebd lpubltuj (4, )L f sc fajpc drcfptf dctcrkebftdcpubltcjc ( 3, >)F e ( (seb 43 seb6
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
6/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.3 \SNECLTSJ E ( 6a a x x ; .
3p >.\sc dctcrkebc x pcbtru lfrc3
> 6x
.
3p >6 >C x x; + , ubdc 6x e >x subt shjueejc clufece>
> < < 9x x ; .
3p 4.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >3jh` (> 6) 6x x ; .
3p 3.\sc dctcrkebc clufef drcptce lfrc trclc preb publtcjc ()N .
3p 1.Treub`meuj FNLfrc 69FN FL; ; e ( ) 6>9k F ; . \sc lfjlujczc jub`ekcf jfturee NL.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
7/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.1 \SNECLTSJ E (v fe g; +
e > .\sc rczhjvc b kujekcf bukcrcjhr bfturfjc ebclufef > < 3x : .
3p 6
> 6
b b
bC
++ +
;
+
cstc bukr bfturfj, hrelfrc fr ae b .
3p 4.\sc dctcrkebc clufef drcptce lfrc trclc preb publtuj (6, 6)F e cstc pcrpcbdelujfrpc drcfptf dcclufec 6 9x y+ + ; .
3p 3.Treub`meuj FNLfrc 9k FLN ; . \sc lfjlujczc jub`ekcf jfturee FN .
3p 1.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef 6> (9,>3)x x ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
8/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.0 \SNECLTSJ E (( 6) , 4 >v f e g v e g; + ; +
subt lhjebefre.
3p 5\ ; + + + + + .
3p 4.Treub`meuj drcptub`mel FNLfrc ( ) 59 , 0F x x; + e { }1
Z 6 |6
N xx
;
. \sc dctcrkebc F N .
3p 1.\sc frftc lvrauj pfrfnhjce fshleftc aublece >7 , ( ) >( 6) >a a x kx k x k ; + + + + , lu 9k sc fajpc drcfptf dc clufec 6y x; + .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
9/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.= \SNECLTSJ E ( > ) 7 > > >C ; + .
3p >.\sc dctcrkebc bukcrcjc rcfjcxey deb prh`rcsef fretkctel ,6, ,3,0,5,66,....x y .
3p )N . \sc dctcrkebc clufef kcdefthfrce sc`kcbtujueFN .
3p 4.\sc rczhjvc b kujekcf bukcrcjhr bfturfjc clufef ( )( )6 > >69b b b+ + ; .
3p 3.\sc dctcrkebc ebtcrvfjcjc dc khbhthbec fjc aublece >7 , ( ) < 4a a x x x ; + + .
3p 1.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef 6 < >x x+ + ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
10/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.5 \SNECLTSJ E (.\sc frftc l < 60 jh` 4
>6=
>
xx
+ ;
.
3p 1.\sc lfjlujczc jub`ekcf jfturee FLf treub`meujue FNL, teebd l 6>, ( ) 19 , ( ) 03NL k F k N; ; ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
11/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.69 \SNECLTSJ E (3% deb >99=.
3p >.\sc rczhjvc sestckuj < 6
x y
x y
+ ;
;, ,x y .
3p )F e ( 6,6)N .
3p 4.\sc lfjlujczc seb 43 lhs6> 1 > = 9x x + ; .
3p 1.\sc lfjlujczc vfjhfrcf cxprcsece 6 >
> 6
,x x
Cx x
; + ubdc 6 >,x x subt shjueejc clufece>
1 1 9x x + ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
12/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.66 \SNECLTSJ E (,.\sc rczhjvc b kujekcf bukcrcjhr rcfjc ebclufef > 6 < ) 6x x ; .
3p 1.\sc dctcrkebc lhhrdhbftcjc publtujue dc kebek fj aublece 7a , >( ) 1 =a x x x; + .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
13/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.6> \SNECLTSJ E (
3p 6.\sc dctcrkebc k teebd lvclthree >f e g; +
e ( >) >n k e g; +
vcreaelc`fjetftcf >f n;
.
3p >.\sc dctcrkebc aublef 7 , ( )a a x fx n ; + teebd lpubltcjc (6,>), ( 6,1)F N fpfreb `rfaelujueaublece a.
3p 14bb : .3p 4.Treub`meuj FNLfrc jfturejc 3, 6FL NL FN; ; ; . \sc lfjlujczc seb sebN L+ .3p 3.\sc rczhjvc b kujekcf bukcrcjhr rcfjc ebclufef > 1 3 9x x + .3p 1.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef j` ( 6) >j`( 6) 6x x+ ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
14/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.6< \SNECLTSJ E (
x y
x y
+ ; + ;
, ,x y .
3p >.\sc dctcrkebc f pcbtru lfrc drcptcjc 6d 7 > 1 9x y+ ; e > 7d > 3 9x fy+ + ; subtpcrpcbdelujfrc.
3p
< 4x ; .
3p 4.Treub`meuj FNLfrc ( ) 59k F ; , 5FN ; e 6>FL ; . \sc lfjlujczc lhs lhsN L+ .
3p 3.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef < < 6 6x x ; .
3p 1.\sc dctcrkebc vfjhrejc }Z 6k fstacj blt clufef >( 6) 6 9k x kx k + + ; sfenshjuee rcfjc.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
15/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.64 \SNECLTSJ E (, 6 6x .
3p
66 6
(9,6>3)4
xx
;
.
3p 1.\sc dctcrkebc f fstacj blt vrauj pfrfnhjce fshleftc aublece 7a , >( ) 6 9x y+ ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
16/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.
63 \SNECLTSJ E ( 9< > 0
x y
x y
;+ ;
, ,x y .
3p >.Aec publtcjc ( 1,4), (6,9)F N e ( 6,3)L . \sc lfjlujczc lhhrdhbftcjc lcbtrujue dc `rcutftc fj treub`meujueFNL.
3p
> >995x
.
3p 4.\sc lfjlujczc seb6>9 lhs639\ ; + .3p 3.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >6 > 6 >,x x subt shjueejc clufece>
< 3 9x x ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
17/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.61 \SNECLTSJ E ( 6 .\sc lfjlujczc6 6
6> 6 >
+ +
.
3p > < 3 9x x .
3p 1.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >3jh` ( 6) 9x x+ + ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
18/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.60 \SNECLTSJ E ()F N fpfreb`rfaelujue aublece.
3p >.\sc lfjlujczc 4 >v f n l; +
, ubdc 3 0 , > < , 3 3f e g n e g l e g; ; + ; +
.
3p 6
x xC
x x; + , ubdc 6 >,x x subt shjueejc clufece
>1 4 9x x + ; .
3p 3.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef j` (> 4 4) 6x x+ + ; .
3p 1.\sc lfjlujczc > < > < > > + .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
19/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.6= \SNECLTSJ E (
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
20/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.65 \SNECLTSJ E ( 6. \sc dctcrkebc fstacj blt vclthree 6 >( v e g v e g ; + + + ; +
saec lhjebefre.
3p n ; e 3
FN FL; ; e ( ) x x+ ; .
3p 1.\sc rczhjvc sestckuj>
y x
y x x
; +
; +, ,x y .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
21/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>9 \SNECLTSJ E (9
3p 6.\sc dctcrkebc publtcjc dc ebtcrsclec fjc `rfaelujue aublece >7 , ( ) 1 3a a x x x ; + lu fxcjcdc lhhrdhbftc.
3p >.\sc dctcrkebc jub`ekcf jfturee NLf treub`meujue FNLteebd l 4FN FL; ; e ( ) 6>9k F ; .
3p 1>x x x x x x+ + + + + ; .
3p 4.\sc dctcrkebc rfef e prekuj tcrkcb fj ubce prh`rcsee fretkctelc 6( )b bf teebd l6 > < 4
>6
4>
f f f
f f f
+ + ;
+ + ;.
3p 3.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef 6 6 6< (9,(
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
22/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>6 \SNECLTSJ E (6
3p 6.\sc rczhjvc b kujekcf bukcrcjhr bfturfjc bcbujc clufef ( )( )6 > >4b b b+ + ; .
3p >.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >6
69699
x; .
3p 6, >, =x x x + + subt tcrkcbe lhbscluteve fe ubce
prh`rcsee `chkctrelc.
3p 1.\sc dctcrkebc k teebd lclufef > (
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
23/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>> \SNECLTSJ E (>
3p 6.\sc dctcrkebc f fstacj blt drcptcjc 6d 7 > > 9fx y+ + ; e > 7< 6 9d x y + ; saec pfrfjcjc.
3p >.\sc lfjlujczc vfjhfrcf cxprcsece C ; 6 >
> 6
x x
x x+ , ubdc 6 >,x x subt shjueejc clufece
>3 3 9x x + ; .
3p
4
x
+ ;
.
3p 4.\sc dctcrkebc prekuj tcrkcb e rfef ubce prh`rcsee `chkctrelc 6( )b bn , teebd l> 4
6 9
n n
n n
+ ;
+ ;.
3p 3.\sc rczhjvc b kujekcf bukcrcjhr btrc`e ebclufef > >9x .
3p 1.Treub`meuj FNLfrc >, 4FL FN; ; e ( ) 19k F ; . \sc lfjlujczc jub`ekcf jfturee NL.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
24/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu.
Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.
>< \SNECLTSJ E ( < (> )(6,3)
(4 ) 1 .
3p >.\sc lfjlujczc pcrekctruj treub`meujue FNLb lfrc ( ) 59 , >9k F NL ; ; e ( ) )N . \sc dctcrkebc lhhrdhbftcjc publtujue N , teebd l F cstc kegjhluj
sc`kcbtujue NN .
3p 4.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >jh` (6 6) 6x + ; .
3p 3.\sc dctcrkebc bukruj rcfj xdeb rcjfef dc c`fjetftc > 3 = 66 ... 633x+ + + + + ; .
3p 1.\sc dctcrkebc aublef dc `rfduj fj dhejcf >7 , ( )a a x fx nx f ; + + , 9f teebd lfrcvfjhfrcf kfxekc`fjlu 6> , hnebutb publtuj dc fnsles > .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
25/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>4 \SNECLTSJ E (4
3p 6. \sc dctcrkebc f , teebd ldrcptcjc 6 7 > 9d x y+ ; e > 7 > 6 9d x fy + ; subt pcrpcbdelujfrc.
3p >.\sc dctcrkebc kujekcf { }* | 6
6
y x
y x x
;
; +, ,x y .
3p 4.\sc lfjlujczc \ ;< 6
seb6>9 lhs6>9> >
+
.
3p 3.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef j` j`( =9x+ + + + + ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
26/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>3 \SNECLTSJ E (3
3p 6. \sc dctcrkebc clufef drcptce lfrc trclc preb publtcjc (6,9)F e (9,6)N .3p >.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef ( )
46 61x ; .
3p < > 9x x + .3p 4.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef
f f f
f f f
+ ;
;.
3p 1.Treub`meuj FNL frc ( ) 59 , 1, 6>k F FL NL ; ; ; . \sc lfjlujczc jub`ekcf jfturee FN.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
27/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>1 \SNECLTSJ E ( 6 9x x+ .3p 4.\sc dctcrkebc clufef drcptce lfrc trclc preb publtcjc ( )6, >F e ( )>, 6N .
3p 3.\sc rczhjvc b clufef 5 4 < < 9x x + ; .
3p 1.Treub`meuj FNL frc ( ) 59k F ; , 1FN ; e < 1NL; . \sc lfjlujczc fref treub`meujue FNL .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
28/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>0 \SNECLTSJ E (0
3p 6.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >3 3x ; .
3p >.\sc lfjlujczc seb6>9 . 3p F e ( )>,6 .N
3p 4.\ sc lfjlujczc prhnfnejetftcf lf, fjc`bd ub cjckcbt deb kujekcf { }9, 6, >,
4 < 9x x + ; .
3p 3.\sc dctcrkebc prekuj tcrkcb 6f fj ubce prh`rcsee fretkctelc ( ) 6b bf teebd l6 3
6 >9
f f
f f
+ ;
+ ;.
3p 1.\sc rczhjvc b ebclufef > < > 6x x x+ + 2 .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
29/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>= \SNECLTSJ E (=
3p 6.\c lhbsedcrvclthree 6 < 4v e g; +
e > 4 v v+
.
3p >.Treub`meuj FNLfrc ( ) NL ; . \sc lfjlujczc jub`ekcf jfturee .FL
3p < 6996 > > > > .\ ; + + + + +
3p 4.\sc rczhjvc b clufef ( )>jh` > 6 9x + ; .
3p 3.teebd l 6 >,x x subt shjueejc clufece>
0 < 9,x x+ + ; ssc lfjlujczc> >6 >
6 6.
x x+
3p 1.\sc dctcrkebc vfjhrejc pfrfkctrujue rcfj k fstacj blt >> 3 9,x x k + 2 pcbtru hrelc .x
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
30/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.>5 \SNECLTSJ E (5
3p 6.\sc rczhjvc b ebclufef > 1 3 9.x x + 3p >.\sc rczhjvc b clufef ( )n e g;
subt lhjebefre.
3p 1.Treub`meuj FNLfrc ( ) 59k F ; , ( ) 19k N ; e 5.FL ; \sc lfjlujczc jub`ekcf bjekee FD
f treub`meujue FNL.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
31/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.x ; .
3p .
3p 4.\sc dctcrkebc vfjhfrcf kebekf aublece ( ) >7 , > > < 55 699
+ + +
+ + +
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
32/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.x + ; .
3p 3.\sc dctcrkebc rfef ubce prh`rcsee fretkctelc ( )6b b
f
teebd l> 3
< 5
>1
,> , 6,> .F N L
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
33/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc. \SNECLTSJ E ( > > > 6jh` jh` < jh` 4 jh` .\sc lfjlujczc fj leblejcf tcrkcb fj ubce prh`rcsee `chkctrelc ( ) 6b bn teebd l 66
, 4FL FN; ; e ( ) 19k F ; . \sc lfjlujczc jub`ekcf jfturee .NL
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
34/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc. = 9x x ; .3p ,NL ; ( ) 43k F ; e ( ) 19 .k N ; \sc lfjlujczc jub`ekcf jfturee .FL
3p 4.\c lhbsedcrvclthree 4
f n +
3p 3.\sc rczhjvc b clufef ( )6>
jh` > >
6 < < 3
< < 3
B ; +
cstc bfturfj.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
35/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc. .f n
3p >.\sc frftc l
jh` 61 cstc bukr bfturfj.
3p > 6
>6
lhs lhs lhs>
fn L nl F lf N f n l + + ; + + ,
ubdc , ,f n l subt jub`ekejc jfturejhr ,NL FL, rcspcltev .FN
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
36/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc. 1.a ;
3p jh` < > >6 > 6C x x; + , ubdc 6x e >x subt shjueejc clufece>
3 6 9.x x + ;
3p 3. \sc dctcrkebc jub`ekcf jfturee FN f treub`meujue FNLteebd l =, ( ) 43NL k F; ; e
( ) 693 .k N ;
3p 1.Aec treub`meuj FNLhfrclfrc e H ub publt frnetrfr deb pjfb. \sc dckhbstrczc l
.FN FL HN HL ;
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
37/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc. >, 4NL FL; ; e ( ) j` 1j` 3 9.x x + ;
3p 1.\sc dctcrkebc x W pcbtru lfrc ( )( )> >6 3 9.x x+
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
38/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc. < < 4+ +
+ + +
.
3p 1.Clufef > < 9x x ; frc shjueejc 6 >,x x . \sc lfjlujczc6 >
> 6
x x
x x+ .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
39/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc. 4 9,x x k+ + 2 hrelfrc fr ae x .
3p 3.\sc dctcrkebc aublef ( )7 ,a a x fx n ; + , lu ,f n ,fstacj blt publtcjc6 6
,> >
F
e
( )9,
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
40/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc. ,6x .
3p 3.\sc rczhjvc b ebclufef6
6.> > 6 9k x x + 2 , hrelfrc fr ae .x
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
41/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.49 \SNECLTSJ E ( < 0+ + cstc bfturfj.
3p 7 , < 6a a x x kx ; + + strcflpreb
publtuj ( ), 3 .F k
3p 1.\sc rczhjvc b ebclufef>
6.6
x
x
+2
+
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
42/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.46 \SNECLTSJ E (9 jce rcprczebt >3% debtr-h sukdc nfbe. \sc dctcrkebc sukf dc nfbe.3p >.Aec vclthree < 3f e g; +
e >n e g; +
. \sc lfjlujczc
6 6.
< >f n +
3p .x x+ ;
3p 3.\sc rczhjvc b ebclufef> 6
>.7 , > 6, .a a x x kx k ; + + \sc dctcrkebc k fstacj blt shjueejc 6 >,x x fjc
clufece ( ) 9a x ; svcreaelc rcjfef > >6 > 64x x+ 2 .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
43/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.4> \SNECLTSJ E (
3p 6.\sc lfjlujczc fref treub`meujue FNL, teebd l ( ) 59k F ; , 6>NL ; e 1FL ; .
3p >.\sc lfjlujczc < 6 < < 6 . +
3p > 0 5 9x x ; .3p 4.\sc dctcrkebc aublef ( )7 ,a a x fx n ; + , lu ,f n , fstacj blt `rfaeluj ce sebtcrscltczc
fxf Hx b publtuj ( )>,9F e fxf Hy b publtuj ( )9, 4 .N
3p 3.\sc rczhjvc b ebclufef>
69.
< >x x:
+ +
3p 1. Aec treub`meuj FNL lu vraurejc ( ) ( ) ( )6,< , 9, < , 6,> .F N L \ sc slrec clufef kcdefbce ,FK
ubdc K cstc kegjhluj jfturee .NL
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
44/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.4< \SNECLTSJ E (>jh` ( 4) FN NL LF o FN+ + ;
.
3p 4.\sc rczhjvc b clufef ( )>>
66 .x ;
3p 3.\sc dctcrkebc pfrfkctruj rcfj k fstacj blt clufef >> 9x kx k + ; sfenshjuee rcfjc c`fjc.
3p 1.\sc lfjlujczc sukf > 556 < < < .+ + + +
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
45/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.44 \SNECLTSJ E ( > >FN NL LD DF o FN+ + + ;
.3p >.\sc rczhjvc b clufef 0 4.x ; 3p 9 seb6>9 .+ 3p 4.\sc dctcrkebc vfjhfrcf kfxekf aublece ( ) >7 , 3.a a x x ; +
3p 3.Aec aublef ( )7 , < >.a a x x ; \sc dctcrkebc lhhrdhbftcjc ubue publt dc pc `rfaeluj aublecea
pcbtru lfrc fnslesf cstc c`fjlu hrdhbftf.
3p 1.Duplc Furcj f lmcjtuet >9% debtr-h sukdc nfbe e bl
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
46/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.43 \SNECLTSJ E (% deb 6999. 3p >.\sc rczhjvc b clufef < 6 9.x + ;
3p 9 , 4KB KY k BKY BY; ; ; . \ sc lfjlujczc jub`ekcf
bjekee ( )K^ ^ BY f treub`meujue .KBY
3p 4.\sc dctcrkebc *f fstacj blt drcptcjc ' 7 > 3 9d x y + ; e '' 7 4 6 9d fx y + ; saecpcrpcbdelujfrc.
3p 3.\sc rczhjvc b ebclufef>995
9.>99=x
:
3p 1.Aec aublef ( ) ( )> >7 , 6 4, .a a x k x kx k ; + + \sc dctcrkebc k fstacj blt ( 6) 6a ;
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
47/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.41 \SNECLTSJ E (9% , prcuj ubce kfebe dc spjft cstc ==9 jce. Lfrc f ahst prcuj bfebtc dcrcdulcrc8
3p >.\sc dctcrkebc tcrkcbuj 6=f fj ubce prh`rcsee fretkctelc ( ) 6b bf teebd l> =
< 69
>4.
4=
f f
f f
+ ;
+ ;
3p < 4 6 9x x+ ; , lu shjueejc 6 >,x x . \sc lfjlujczc > >6 >
6 6.
x x+
3p 4.Trfpczuj eshslcj ( )FNLD FD NL; frc 69 e ( ) 19 .FD k FDL; ; \sc lfjlujczc destfbf dc jfpubltuj F jf drcfptf LD .
3p 3.\sc dctcrkebc f fstacj blt publtuj ( )6, 6K sfpfrebdrcptce 7 > < 9d fx y+ + ; .
3p 1.\sc rczhjvc b clufef 6 6x x+ ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
48/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.40 \SNECLTSJ E (.\sc lfjlujczc destfbf debtrc publtcjc ( )6,6F e ( )>, > .N
3p a a x kx ; + s trcflpreb publtuj
( )4,1 .F
3p 4.Drcptub`meuj FNLD frc 6> e ( ) > < 9kx kx+ ; sfenshjuee rcfjc c`fjc.
3p 1.\sc rczhjvc b clufef 4 4 3 > .x x+ ;
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
49/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.4= \SNECLTSJ E (,
4 < 9.x x + ;
3p >.\sc lfjlujczc sukf 6 > < 49.+ + + +
3p 4 6 9x kx + ; sfenshjuee rcfjc.3p 4.\sc lfjlujczc destfbf dc jf publtuj ( )6,>F jf drcfptf 7 6 9.d x y+ + ;
3p 3.\sc rczhjvc b clufef >0 = 0 0 9.x x + ;
3p 1.\sc lfjlujczc6
lhs6
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
50/101
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
51/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.39 \SNECLTSJ E (99 6>=b+ ; + .
3p >.\sc dctcrkebc k fstacj blt clufef > 6 9x k x k ; sfenshjuee rcfjc c`fjc.
3p F , N , .
3p 3. \ sc dctcrkebc x fstacj blt bukcrcjc > < >, 1 3x , x x s aec tcrkcbe lhbscluteve fe ubceprh`rcsee fretkctelc.
3p 1.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >j` 3j` 1 9x x ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
52/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.36 \SNECLTSJ E (f fj ubce prh`rcsee fretkctelc ( ) 6b bf teebd l 69 69f ; e 63 63f ; .
3p >.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef> \SNECLTSJ E (
3p 6.\sc rczhjvc b clufef >64
jh` >x ; .
3p >. \sc rczhjvc b ebclufef >< 5 9x .3p 6a ; e ( )< 6a ; .
3p 4. \sc dctcrkebc kujekcf { 6 6F x x; .3p 3. Aec publtuj K kegjhluj sc`kcbtujue FN , efr H ub publt hfrclfrc deb pjfb. \sc dckhbstrczc l
6( )
>HK HF HN; +
.
3p 1. Treub`meuj FNLfrc h( ) 59k F ; , h( )
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
54/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.
3< \SNECLTSJ E (7 , ( ) < 3 6a a x x x ; + .
3p F x x; + : .
3p 4. \sc dctcrkebc k teebd l`rfaeluj aublece>
7 , ( )< 3
ka a x x ; , lhbebc publtuj
F k
.
3p 3. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef > 6 > > > > > =51x x x + + ; .3p 1. \sc frftc l, dflb treub`meuj FNLfrc jhl c`fjetftcf seb >seb lhsF N L; , ftuble _ [ _ [FN FL .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
55/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.34 \SNECLTSJ E ( 3 9x .3p >. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef ( )>6
>
jh` 6 6x + ; .
3p < >9956 > > > ... >\ ; + + + + + jf < sc hnebc rcstuj 9.3p 4. \sc slrec clufef drcptce lfrc lhbebc publtcjc ( )F e ( )6,3N .
3p 3. \sc rczhjvc sestckuj dc clufee6
>< 6
y x
y x x
;
; +, ,x y .
3p 1. Treub`meuj FNLfrc h( ) 59k F ; , NL; 69 e lhs 9,1L ; . \sc lfjlujczc pcrekctruj treub`meujue FNL.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
56/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.33 \SNECLTSJ E (+ .
3p >. \sc dctcrkebc fj dhejcf tcrkcb fj prh`rcsece `chkctrelc ( )6b b
n
lfrc frc rfef >q ; e = ;>31n .
3p ,x x fjc clufece ( )> 3 < 9x k x k+ + ; , teebd l 6 > 4x x+ ; .
3p 4. \sc rczhjvc b clufef 6< < 69=x x+ + ; .3p 3. \c lhbsedcrpubltcjc ( )3,=F , ( )>,N f e ( , >)L n . \sc dctcrkebc ,f n pcbtru lfrc publtuj N
cstc kegjhluj sc`kcbtujue FL.
3p 1. \sc lfjlujczch h
h h
seb6
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
57/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.31 \SNECLTSJ E (
b b ; .
3p >. \sc lfjlujczc sukf prekejhr 69 tcrkcbe fe ubce prh`rcsee fretkctelc 6( )b bf teebd l 3 64f; e 63 44f ; .
3p x; saec shjuec f clufece ( )> >6 > < 9k x kx+ ; .3p 4. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef 6 < 6x x; .3p 3. Treub`meuj FNLfrc < 3FL , FN ; e 19k( F) ; . \sc dctcrkebc jub`ekcf jftureeNL.
3p 1. Aec publtcjc 6 >F , e 6 4N , . \sc slrec clufef kcdefthfrce sc`kcbtujue FN .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
58/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.30 \SNECLTSJ E ( > > < 9x kx k + + ; frc dhushjuee rcfjc c`fjc.3p >. \sc lfjlujczc sukf prekejhr 1 tcrkcbe fe ubce prh`rcsee `chkctrelc ( )
6b bn
, lfrc frc tcrkcbe phzeteve,
6; = 9x x + ; .3p 4.\sc dckhbstrczc lb hrelc treub`me FNL lu h( ) 59k F ; frc jhl c`fjetftcf > >seb seb 6N L+ ; .
3p 3. \sc lfjlujczc6 6 6 6 6
...6 > > < < 4 665 6>9 6>9 6>6
+ + + + +
+ + + + +
.
3p 1. Aec FNLD ub pfrfjcjh`rfk dc lcbtru H e Y ub publt hfrclfrc deb pjfbuj pfrfjcjh`rfkujue . \sc
dckhbstrczc l 4YF YN YL YD YH+ + + ;
.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
59/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.3= \SNECLTSJ E (> 0 5 9x x+ : .3p >.Aec aublef 7 , ( ) < ... >9a a a a+ + + + .
3p >
n l
f
, ubdc f, n, lsubt
jub`ekejc jfturejhrNL,FL, rcspcltevFN.
3p 4. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef 61 4x x+ ; .3p 3. Aec publtcjc ( )>, > < >990 >99=
...> < < 4 >99= >995
+ + + .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
60/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.35 \SNECLTSJ E (.\sc dctcrkebc kujekcfF ;{ }, 64 9N x x nx f; + + ; e{ },3F N ; .
3p 1. Aec treub`meujFNL. Yubltcjc , ,K B Y subt kegjhflcjc jfturejhr treub`meujue e H cstc ub publthfrclfrc deb pjfbuj treub`meujue. \sc dckhbstrczc lHF HN HL HY HK HB + + ; + +
.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
61/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.19 \SNECLTSJ E ( 6. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >3 3 >x x+ ; .
3p > 9x x k+ ; sfenshjuee rcfjc.3p 4. Aec aublef 7 , ( ) < >a a x x ; . \sc dctcrkebc lhhrdhbftcjc publtujue dc pc rcprczcbtfrcf
`rfaelf aublece a pcbtru lfrc fnslesf cstc c`fjlu hrdhbftf.
3p 3. Aec FNLD ub pfrfjcjh`rfk, efr H ub publt hfrclfrc deb pjfbuj pfrfjcjh`rfkujue. \sc frftc l
HF HL HN HD+ ; +
.
3p 1. \sc dckhbstrczc lbtr-ub treub`me hfrclfrc FNLfrc jhl rcjfef lhs lhsf L l F n+ ; , ubdc f, n, l
subt jub`ekejc jfturejhrNL,FL, rcspcltevFN.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
62/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.16 \SNECLTSJ E ( < 6b b+ + ; .
3p >.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef ( )>x + ; .3p >seb seb sebN L F+ ; cstc drcptub`mel.
3p 3. \sc dctcrkebc ,f n teebd l { }> < 9F x x fx; + + ; , { }> 6 9N x x nx f; + + + ; e{ }6,
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
63/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.1> \SNECLTSJ E (
3p 6. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef ( )>4jh` 6 9x + ; .3p >. Clufef ( ) ( )>6 > 9k x k x k ; lu { }6k frc shjueejc 6 >,x x . \sc dctcrkebc { }6k
fstacj blt 6 > 6 > >x x x x+ ; .
3p
kF
fpfrebc `rfaelujue aublece 7 , ( )a a x x k ; + .
3p 4. teebd lFNLcstc ub treub`me drcptub`mel lu h( ) 59k F ; , ssc dckhbstrczc l
( ) ( )> >
seb seb lhs lhsN L N L+ + cstc bukr btrc`.
3p 3. Aec pftrujftcruj lhbvcxFNLD . Dflpubltcjc ,K B subt kegjhflcjc jfturejhr FN , rcspcltev LD , ssc
dckhbstrczc l ( )6>
KB FD NL; + .
3p 1.\sc dctcrkebc *b pcbtru lfrc6 6 6
... 6 < > 6b b
+ + + ;
+ + +
.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
64/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.1< \SNECLTSJ E (. \sc rczhjvc b kujekcf bukcrcjhr rcfjc ebclufef> 4 3
< 5 6=x
+ .
3p 6 9x x k+ + ; sfenshjuee rcfjc destebltc.
3p 4. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef ( )>6x x + ; .
3p 3. Yubltcjc ( )>,9F , ( )4,9N e (9,1)L subt vraurejc ubue treub`me. \sc dctcrkebc jub`ekcf kcdefbce
lhrcspubzthfrc jftureeNL.
3p 1. \sc dckhbstrczc lb hrelc treub`me FNLcstc fdcvrftc`fjetftcf lhs lhsf N n F l + ; , ubdc f,
n, lsubt jub`ekejc jfturejhrNL,FL, rcspcltevFN.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
65/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.14 \SNECLTSJ E (. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef > 63 6>3x+ ; .3p > 9x fx f ; lu f , ssc dctcrkebc lcfjfjtshjuec >x .
3p 4. \sc lfjlujczc sukf > < 56 > > > ... >\ ; + + + + + .3p 3. Aec , ,K B Y kegjhflcjc jfturejhr ,NL FL , rcspcltev FN fjc treub`meujue FNL . \sc frftc l
9FK NB LY+ + ;
.
3p 1.\sc lfjlujczc vfjhfrcf cxprcseceseb6>9 seb639
lhs6>9 lhs639C
;
+
.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
66/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.13 \SNECLTSJ E (9b b+ + ; .
3p >.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef>
> 6x x
; .
3p 1 r e g; + +
, > ,x x fjc clufece ( ) ( )> > 4 > 9kx k x k+ + + ;
svcreaelc rcjfef6 >
6 6>
x x+ ; .
3p 1.\sc frftc l b hrelc treub`me FNL, lfrc frc ( ) 59k F ; , frc jhl c`fjetftcf
( )( ) >seb seb lhs lhs > seb sebl N n L l N n L f N L + + ; , ubdc f, n, lsubt jub`ekejc jfturejhrNL,FL,rcspcltevFN.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
67/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.11 \SNECLTSJ E ( < 4 6> + .
3p >. \c lhbsedcraubleejc ( ) ( )> >, 7 , >, > < 3a ` a x x x ` x x x ; + ; + + . \sc dctcrkebc shjueejc
rcfjc fjc clufece ( ) ( )a x ` x; .
3p 3>, 34n n; ; .
3p 4. \sc rczhjvc b clufef>
>6
< 4
x
x
+
;
+
.
3p 3. \sc lfjlujczc fref ptrftujue FNLD teebd l ( )6,>N e ( )>,
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
68/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.10 \SNECLTSJ E (,6995 .3p >. \sc rczhjvc b clufef ( )>jh` < > 9x ; .
3p > > ... > 69> 6 > 9k x k x k+ + ; vcreael
rcjfef ( )6 > 6 >6 3
>
x x x x+ ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
69/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.1= \SNECLTSJ E ( 4r e g
; + , < >r e g
; + . \sc dctcrkebc , , fstacj
blt 6 > r r r ;
.
3p >. \sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef > 6 >> < > 6 9x x+ + ; .
3p < 56 < < < ... ,x x fjc clufece ( )
>6 > 9kx k x+ + ; ,
vcreaelrcjfef 6 > 6 > lhsf n L; , ftuble n l; , ubdc fe nsubtjub`ekejc jfturejhr NL, rcspcltev FL.
3p 1. Sb hneclt lhst>399 jce. Hnecltuj s-f ecatebet lu 69% deb prc, fphe s-f ecatebet lu 6>% deb bhuj prc.\sc lfjlujczc vfjhfrcf ujtekujue prc.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
70/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.15 \SNECLTSJ E (49 dc pf`ebe, f dhuf ze .\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef >< > > 6 6 9x k x k ; vcreael
rcjfef 6 > 6 >>
995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.
09 \SNECLTSJ E ( < 4 699
.
3p >.\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef 3jh`
> 4x
; .
3p 4
>
y x
y x x
; +
; , ,x y .
3p 4. Aec aublef 7 , ( ) < 1 >a a x x k ; + . \sc dctcrkebc k fstacj blt publtuj (4 3, )F k k sfpfreb`rfaelujue aublece a.
3p 3.Sb treub`me FNLfrc ( ) ( ) 19 , ND ; , ssclfjlujczc jub`ekcf jfturee FL.
3p 1.\sc lfjlujczc vfjhfrcf cxprcseceseb639 seb6>9
seb639 seb6>9
+
.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
72/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.
06 \SNECLTSJ E (| 1 0 9F x x x; ; .3p >. \sc dctcrkebc ,f n fstacj blt ( ) ( ) > x ; .3p 3. \sc dctcrkebc k pcbtru lfrc vrauj pfrfnhjce aublece ( ) ( )>7 , > 4 < 1 0a a x x k x k ; + + +
cstc setuft pc fxf Hx .
3p 1. b treub`meuj FNLsc tec l ,FD NL D NL , ( ) ( )19 ,
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
73/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.
\SNECLTSJ E (
3p 6.
\sc rczhjvc b kujekcf bukcrcjhr rcfjc clufef 6x ; .
3p >. Aec aublef 7 , ( ) > 4a a x x ; + . \sc dctcrkebc lhhrdhbftcjc ubue publt fj `rfaelujue aublece apcbtru lfrc hrdhbftf cstc c`fjlu dunjuj fnslesce.
3p > >1 < 6 > 9k x k x k+ ; .
3p 4. Sbuj deb ub`meurejc ubue trfpcz eshslcj cstc dc 43 , efr bjekcf trfpczujue frc jub ekcf > . \scdctcrkebc sukf jub`ekejhr jfturejhr bcpfrfjcjc fjc trfpczujue.
3p 3. H pcrshfbf lmcjtuet >9% debtr-h suke bl619 jce deb cf. \sc dctcrkebc sukf ebeefj, teebd le-f rkfs 03% deb flcfstf.
3p 1. Aec publtcjc ( )6,3F e ( )>,>N . \sc dctcrkebc clufef kcdefthfrce sc`kcbtujue FN .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
74/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.0< \SNECLTSJ E (jh` < > >x+ ; .3p >. \sc lfjlujczc sukf prekcjhr 69 bukcrc bfturfjc ekpfrc.
3p 4 63 9x x ; lu shjueejc 6 >,x x . \sc lfjlujczc> >6 >x x+ .
3p 4. H pcrshfbfrc ub sfjfreu dc 6999 jce. Lfrc vf ae sfjfreuj pcrshfbce duph kelhrfrc lu 3%, urkftdch krerc lu 3%8
3p 3. \sc dctcrkebc lhhrdhbftcjc publtujue b lfrc drcfptf dctcrkebftdc publtcjc ( )F e ( )>,=N
ebtcrscltcfzfxf Hy .
3p 1. b treub`meuj drcptub`mel FNL ( )( )59k F ; kcdefbf lhrcspubzthfrc ephtcbuzce frc jub`ekcf dc3 lk, efr lftctf FN cstc dc 3 lk. \sc lfjlujczc jub`ekcf jfturee FL.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
75/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.04 \SNECLTSJ E (x x ; .
3p 7 , ( ) >a a x x x ; + .
3p 4. Aec aublef 7 , ( ) < >a a x x ; + . \sc lfjlujczc ( ) ( ) ( ) ( )6 > < ... 69a a a a+ + + + .
3p 3. teebd lclufef drcptce dctcrkebftdc publtcjc (>,>)F e (
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
76/101
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
77/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.01 \SNECLTSJ E (x + ; .
3p < 3 , 6 >6 >
6 6C x x
x x; + , ubdc 6 >,x x subt shjueejc clufece
>0 9x x ; .
3p 1.\sc lfjlujczc>
seb6>9 > lhs6>9 > seb 43 > lhs19
995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.00 \SNECLTSJ E (f; e rfef >q; . \sc dctcrkebc sukf 1 =f f .3p >.\sc lfjlujczc prcuj ubue prhdus duph slukperc lu 3% , teebd lprcuj ebeefj crf dc 6>9 jce.3p > 3 > 9x x+ 2 .3p 4.Treub`meuj FNLfrc 59k( F ) ; , 3 69FN , NL; . \sc lfjlujczc fref treub`meujue FNL.
3p 3.b treub`meuj FNL, publtcjc K , B, Ysubt kegjhflcjc jfturejhr NL, FL, rcspcltev FN. \sc frftc l
> > > 9FK NB LY + + ; .
3p 1.\sc rczhjvc b clufef>
4> 5 6995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.0= \SNECLTSJ E ( jh` 6
0+ + .
3p >.Aec 6( )b bf h prh`rcsec fretkctellu 0 6661, >9.f f; ; \sc dctcrkebc sukf prekejhr >9 dc tcrkcbefe prh`rcsece fretkctelc.
3p ( ) 3 4a x x x; + .
3p 4.\c lhbsedcrtreub`meuj FNL, lu vclthree dc phzeec < 4 , 3 , >F N Lr e g r e r g; + ; ;
. \sc
dctcrkebc vclthruj dc phzeec fj lcbtrujue dc `rcutftc fj treub`meujue FNL.
3p 3.\sc rczhjvc b clufef 6 >< < < 660x x x+ ++ + ; .
3p 1.Treub`meuj FNLfrc 0, =, 6>FN NL FL; ; ; . \sc lfjlujczclhs
lhs
F
N
.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
80/101
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
81/101
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
82/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.=6 \SNECLTSJ E (( ) 7"> < 0 3, "p x x x x x+ ; + . \sc dctcrkebc vfjhfrcf dc fdcvr fprhphzeece ( .\sc slrec rcjfeejc jue Qetc pcbtru clufef >> 0 9x x ; .3p >6f r+ .
3p 4.b treub`meuj FNLsc lubhsl 3v e g; +
dupvclthree f e g; +
e >
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
83/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.=> \SNECLTSJ E (
3p 6.\sc dctcrkebc clufef drcptce lfrc trclc preb publtcjc (.\sc lfjlujczc pcrekctruj treub`meujueFNL lfrc frc ( ) 59 , ( ) 43k F k N; ; e 3 >NL; .
3p
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
84/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.=< \SNECLTSJ E (> , > , > , > , > .3p >.\sc lfjlujczc destfbf debtrc publtcjc ( )F e (9,3)N .
3p 3
, ,4 6
x yx y
x y
;
+ ; .
3p 4.\sc lfjlujczc vfjhfrcf cxprcsece 6 > 6 >>C x x x x; + , ubdc 6 >,x x subt shjueejc clufece>
> < 6 9x x ; .
3p 3.\sc rczhjvc b clufef >0) lhs639 seb 43 lhs43> >
+ .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
85/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.=4 \SNECLTSJ E ( 963 3 36>3
.
3p >.Aec ( ) 6b bf h prh`rcsec fretkctellu 6 3f ; e rfef >r ; . \sc lfjlujczc ( )>
6 0f f+ .
3p
6 6C
x x; + , ubdc 6 >,x x subt shjueejc clufece
>< > 0 9x x + ; .
3p 4.\sc rczhjvc b clufef > Nr e g; +
e
>Lr g;
. \sc dctcrkebc vclthruj dc phzeec fj lcbtrujue dc `rcutftc fj treub`meujue FNL.
3p 1.b treub`meuj FNLsc lubhsl =,
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
86/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.=3 \SNECLTSJ E ( > >x ; .
3p >.\sc slrec clufef drcptce lfrc trclc preb publtuj ( )6,4F e frc pfbtf6
>k ; .
3p < 1
, ,> 4
x yx y
x y
;
+ ; .
3p 4.\sc studeczc sckbuj aublece 7a , >( ) > 3 >a x x x; + .
3p 3.\sc rczhjvc b clufef> 1 6
>61
x x ; .
3p 1.Aec treub`meuj FNL. teebd l seb seb seb 9f F n N l L ; , ssc frftc ltreub`meuj FNLcstcdrcptub`mel, ubdc f, n, lsubt jub`ekejc jfturejhrNL,FL, rcspcltevFN.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
87/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.=1 \SNECLTSJ E ( 4 6
16 6> > 7 > >0 >
+
.
3p >.Aec ( ) 6b bn h prh`rcsec `chkctrel, lu 3 >0n ; e lu rfef6
< 6a x x x; .
3p 4.\sc rczhjvc b clufef ( )3jh` > >x ; .
3p 3.\c lhbsedcrpubltcjc ,F N lfrc fu lf vclthre dc phzeec 4 , > 3F Nr e g r e g; + ; +
. \sc dctcrkebc
vclthruj dc phzeec fj publtujue _ [K FN , teebd l >FK
KN; .
3p 1.b treub`meuj FNLsc lubhsl 4,
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
88/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.=0 \SNECLTSJ E (.\sc lfjlujczc 3 3 36
jh` >3 jh` 3 jh`6>3
+ .
3p 6
x yx y
x y
+ ;
; .
3p 4.Aec ( )6b b
f
h prh`rcsec fretkctellu 6 4f ; e rfef 6y x; + .
3p 1.\sc rczhjvc b clufef < < 4 6x x+ ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
89/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.== \SNECLTSJ E (.\sc rcprczebtc `rfael aublef 7a , ( ) > 1a x x; + .
3p ( ) >a x x kx; + + , ubdc kcstc ub bukr rcfj. \sc dctcrkebc kteebd lvfjhfrcf kebekf aublecea cstc c`fjlu >.
3p 4.\sc rczhjvc b clufef6 6
> < 6 94 >
x x
+ ;
.
3p 3.\sc dctcrkebc bukcrcjc rcfjc fe nteebd lpubltcjc ( ),6F f e ( )6,N f n+ fpfreb drcptce dc clufec
> 4 9x y+ + ; .
3p 1.\sc lfjlujczc fref treub`meujueFNLteebd l >, 4FN FL; ; e ( )
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
90/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.=5 \SNECLTSJ E ( 3 1 9x x + ; .3p >.\c lhbsedcrvclthree > , < , 3u e g v e w g; + ; ;
. \sc lfjlujczc vclthruj >u v w
.
3p q ; . \sc dctcrkebc sukf prekejhr 1 tcrkcbe fe
prh`rcsece `chkctrelc.
3p 4.\sc rczhjvc b ebclufef > 1 9x + .
3p 3.\sc lfjlujczcseb6
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
91/101
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
92/101
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
93/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.5> \SNECLTSJ E (
3p 6.b rcpcruj lfrtczefb xHy , ssc lfjlujczc destfbf debtrc publtcjc ( )>,3F e ( )4,9N .3p >.\sc lfjlujczc prhnfnejetftcf lf, fjc`bd ub cjckcbt deb kujekcf {9,>,4,1,=,69}F ; , flcstf saec
shjuec f clufece > 1 = 9x x + ; .
3p 6a x x; .
3p 4.\sc dcf cxckpju dc h aublec ( )7 ,a a x fx n ; + , lu ,f n , lrcslthfrc, f lrce
rcprczcbtfrc `rfaelstrcflpreb publtuj ( )>,9K .
3p 3.\c lhbsedcrtreub`meuj FNLlu 0, =, 5FN NL FL; ; ; . \sc lfjlujczc6 6 6
lhs lhs lhsF N Lf n l
+ , ubdc f, n, lsubt jub`ekejc jfturejhrNL,FL, rcspcltevFN.
3p 1.\sc rczhjvc b clufef 6 > < 49< < < 995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.5< \SNECLTSJ E (.\sc studeczc sckbuj aublece 7a , ( ) < 1a x x; .
3p
6jh` =jh` = >> 45 .
3p 4.\sc rczhjvc b clufef < 3 > >x ; .3p 3.b rcpcruj lfrtczefb xHy sc lhbsedcrpubltcjc ( ) ( ) ( )1,9 , 9, < , >,3F N L . \sc lfjlujczc fref
treub`meujue FNL.3p 1.\sc frftc ldflb treub`meuj FNLfvck lhs lhs lhsn N l L f F + ; , ftuble treub`meuj cstc
drcptub`mel, ubdc f, n, lsubt jub`ekejc jfturejhrNL,FL, rcspcltevFN.
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
95/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.54 \SNECLTSJ E (.Jfturejc treub`meujue FNLsubt 1, 0, =NL FL FN; ; ; . \sc lfjlujczc lhs lhsN L .3p a x x; +
lu fxcjc dc lhhrdhbftc Hxe Hy.
3p 4.\sc dctcrkebc clufef drcptce lfrc trclc preb publtuj ( )>,
> 6, ,
< 3
x yx y
x x y
+ ;
+ ; .
3p 1.\sc rczhjvc b clufef >5 >
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
96/101
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
97/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.51 \SNECLTSJ E (
x yx y
x y
; + ;
.
3p >.Treub`meuj FNLfrc ( ) 59 , 1, 1 > 6x x .
3p 1.b rcpcruj lfrtczefb xHy sc lhbsedcrpubltcjc (
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
98/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.50 \SNECLTSJ E (.\sc slrec rcjfeejc jue Qetc pcbtru clufef >4 (< ) 1 > 9x k x k + + ; , ubdc k .
3p 4x x ; .3p 3.\sc dctcrkebc aublef 7a , ( )a x fx n; + , , , 9f n f teebd l`rfaeluj su lhbebc
publtcjc (>, seb69 > < seb 19 > seb
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
99/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.5= \SNECLTSJ E (, 3 9b b .
3p >.\sc rcprczebtc `rfael aublef 7a , ( ) > > 6a x x x; + + .
3p 0x +
trclc preb publtuj ( ) >seb 6>9 lhs 639+ .
3p 1.\sc rczhjvc b clufef >> >> jh` 3 jh` > 9x x + ; .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
100/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
NFLFJFS^CFT >995-KFTCKFTELYrhnf D, KT995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.55 \SNECLTSJ E (.\sc dctcrkebc publtcjc dc ebtcrsclec fjc rcprczcbtree `rfaelc f aublece 7a ,
( ) > 3 4a x x x; + lu fxf Hx .
3p , 4), ( 6,9), (4,9)F N L . \sc lfjlujczc pcrekctrujtreub`meujue FNL.
3p 3.\sc rczhjvc b ebclufef < < 6x x .
3p 1.\sc rczhjvc b clufef < < >> > 0 6x x x x + + ; + .
8/11/2019 Matematica M3 si M4 - Subiectul I cu 100 variante din 2009
101/101
Kebestcruj Cdulfece, Lcrlctree e EbhvreeLcbtruj Bfehbfj pcbtru Lurrelujuk e Cvfjufrc b bvkbtuj Yrcubevcrsetfr
C]FKCBSJ DC NFLFJFS^CFT >995Yrhnslresjf KFTCKFTEL Yrhnf D
Aejecrf vhlfehbfj, prhaejuj pcdf`h`el, spclefjezfrcf bvthr-cdulfthfrc. Thftc sunecltcjc subt hnje`fthree. Tekpuj cacltev dc julru cstc dc < hrc. \c flhrd69 publtc deb haeleu. Jf thftc sunecltcjc sc lcr rczhjvre lhkpjctc.699 \SNECLTSJ E (.\sc lfjlujczc 0 jh` jh` < jh` 65
+ .
3p 33
x x+
;
.
3p 3.\sc rczhjvc sestckuj
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