8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
1/101
www. mxahme`mdagacaurmat.dcjospjt . r j
Yarnaetm
221-122
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
2/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
1 W]DNMGP]C NNN (72p) Yarnaeta 221
1. Wm gjesn`mr iueg na { }; 1 i S , ( )4
1 x
i x x
8+
.
=p a) W sm gacgucmzm `mrnvata iuegnmn i . =p d) W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn i . =p g)W sm `mhjestrmzm g ( ) 0 pmetru jrngm 1 i x x 3 .
4. Wm gjesn`mr iueg na ( )4 , 2
; , .1, 2
x x m x i i x
x x
+ 8
+ 5
=p a) W sm aratm g iueg na i a`hntm prnhntnvm pm .
=p d) W sm gacgucmzm2
1
( ) . x i x `x
=p g)W sm `mtmrhnem vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx a oraingucun iueg nmn\ X ( ) ( ); 2
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
3/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
4 W]DNMGP]C NNN (72p) Yarnaeta 224 1. Wm gjesn`mr iueg na ; i , ( ) . x x i x m m8
=p a) W sm gacgucmzm2
( ) (2)cnh
x
i x i x
.
=p d)W sm aratm g iueg na i mstm grmsgtjarm pm .
=p g)W sm gacgucmzm (2) (1) ... (422>), ue`m ; , ( ) ( ) ( )W o o o o o x i x i x 8 + + + 8 . 4. Wm gjesn`mr iueg nncm , ; , i I ( ) n ( ) ( 1) x x i x xm I x x m8 8 .
=p a) W sm vmrningm g iueg na I mstm j prnhntnv a iueg nmn i . =p d)W sm gacgucmzm arna supraimmn pcaem `mtmrhneatm `m orainguc iuegnmn i , axa Jx n `rmptmcm 2 x 8 n 1. x 8
=p g)W sm `mhjestrmzm g ( )4
41
( ) ( ) ( ) 14, pmetru jrngm 1
( )
x i t i t i t x`t x
x i t
+8 5 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
4/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
7 W]DNMGP]C NNN (72p) Yarnaeta 227
1. Wm gjesn`mr iueg na ( ); 2, i + , ce( ) x i x x
8 .
=p a) W sm vmrningm g ( ) 4 ce' ,4
x i x
x x
8 pmetru jrngm ( )2< x + .
=p d) W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn i. =p g)W sm `mhjestrmzm g = 77 = .
4. Wm gjesn`mr iueg na ; i , ( ) , 14 , 1
xm m x i x
x x
8
+ 5.
=p a) W sm aratm g iueg na i a`hntm prnhntnvm pm .
=pd) W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn
\ X; 2,4o , ( ) ( )o x i x8 , \ X2,4 x .
=p g)W sm gacgucmzm2
4
( ) x i x`xm .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
5/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
0 W]DNMGP]C NNN (72p) Yarnaeta 220 1. Wm gjesn`mr iueg na ; i , ( ) x i x x m8 + .
=p a) W sm gacgucmzm ( ), i x x . =p d) W sm aratm g i mstm `msgrmsgtjarm pm ( X,2 n grmsgtjarm pm \ )2, + .
=p g)W sm `mtmrhnem mguana asnhptjtmn jdcngm gtrm + ca orainguc iueg nmn i . 4. Wm gjesn`mr iueg na ;o , ( ) 7 4( 1) 7 1o x x x8 + .
=p a) W sm gacgucmzm1
2
( )o x `x .
=p d) W sm `mtmrhnem euhruc rmac 1a 5 astimc egt ( )( )71
9
a x ao x x m x m 8 .
=p g)W sm gacgucmzm( )1
4 422>
2
7 7 ( ) x o x `x+ .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
6/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
= W]DNMGP]C NNN (72p) Yarnaeta 22= 1. Wm gjesn`mr iueg na ; i , ( ) 422> 422>( 1) 1 i x x x8 .
=p a) W sm gacgucmzm (2) (2) i i + . =p d) W sm sgrnm mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )1
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
7/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
9 W]DNMGP]C NNN (72p) Yarnaeta 229
1. Wm gjesn`mr iueg na \ ); 2, i + , ( ) 11 4
x x i x
x x
+8 +
+ +.
=p a) W sm gacgucmzmcnh ( ) x
i x+
.
=p d) W sm vmrningm g 4 41 1( ) ( 1) ( 4) i x
x x 8 +
+ +, jrngarm ar in 2 x .
=p g)W sm `mhjestrmzm g \ )1 ( ) 4, pmetru jrngm 2,4
i x x + .
4. Wm gjesn`mr iueg na ( ) 4; , 1 x i i x x m 8 + + . =p a) W sm aratm g jrngm prnhntnv a iueg nmn i mstm grmsgtjarm pm .
=p d) W sm gacgucmzm ( )1
2
x i x `x .
=p g)W sm `mhjestrmzm g ( )
1
ce 17
m i x`x m x 8 + .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
8/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
? W]DNMGP]C NNN (72p) Yarnaeta 22? 1. Wm gjesn`mr iueg na ; i , ( ) 4 x i x m x8 + .
=p a) W sm gacgucmzm1
( ) (1)cnh
1 x i x i
x.
=p d)W sm `mhjestrmzm g iueg na i eu arm asnhptjt gtrm + . =p g)W sm `mhjestrmzm g iueg na i mstm gjevmx pm .
4. Wm gjesn`mr iueg na \ ); 1, i + , 1( )(1 ce )
i x x x
8+
.
=p a) W sm gacgucmzm1
'( )m
i x `x .
=p d) W sm aratm g jrngm prnhntnv a iueg nmn i mstm grmsgtjarm pm \ )1, + .
=p g)W sm `mtmrhnem euhruc rmac ( )41,a m astimc egt arna supraimmn pcaem, `mtmrhneatm `m orainguc
iueg nmn i , axa Jx , `rmptmcm `m mguann 4 n x a x m8 8 , s inm moac gu7
ce .4
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
9/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
W]DNMGP]C NNN (72p) Yarnaeta 226
1. Wm gjesn`mr iueg na ( ) { }; 2, S i m+ , 1 ce( ) 81 ce
x i x
x
+.
=p a) W sm gacgucmzm ( )1
cnh x
i x
.
=p d) W sm vmrningm g 4
4( ) ,(1 ce )
i x x x
8 jrngarm ar in ( ) { }2< S x m + .
=p g)W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm + ca orainguc iueg nmn i .
4. Wm gjesn`mr iueg nncm ( ), ; 2, , i o + ( ) 1 n ( ) x i x m o x x
8 8 .
=p a) W sm `mtmrhnem hucnhma prnhntnvmcjr iuegnmn i o+ .
=p d) W sm aratm g 4 0 4
4 4
1
1 ( ( ) ( ))
4m m
i x o x `x +
+ 8 .
=p g) Ijcjsne` mvmetuac iaptuc g 4 44ad a d + , pmetru jrngm ,a d , s sm `mhjestrmzm g 4 0 4
1
1 1
0 x m mm `x
x
+ .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
10/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>> W]DNMGP]C NNN (72p) Yarnaeta 22> 1. Wm gjesn`mr iueg na ; , i 4( ) ( ), ue`m , , x i x m ax dx g a d g8 + + .
=p a) Rmetru 1, 2a d g8 8 8 , s sm gacgucmzmcnh ( ) x
i x+
.
=p d) W sm vmrningm g (2) (2) i i d 8 . =p g)W sm `mtmrhnem , ,a d g astimc egt (2) 2, (2) 1 i i 8 8 n (2) 0 i 8 .
4. Wm gjesn`mr netmoracmcm1
2
1
1
e
e x
N `x x
+8
+, *pmetru jrngm e .
=p a) W sm gacgucmzm1 N . =p d)Ijcjsne`, mvmetuac, iaptuc g 4 x x pmetru jrngm \ X2,1 x , s sm `mhjestrmzm g 4 1 N N .
=p g)W sm `mhjestrmzm g *11
+4ce4, pmetru jrngm1e e
N N ee
+ + 8 +
.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
11/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
12 W]DNMGP]C NNN (72p) Yarnaeta 212
1. Wm gjesn`mr iueg na ; i ,4
4
, 1( )
, 1
x x x i x
x x x
8
+ 3.
=p a) W sm stu`nmzm gjetneuntatma iuegnmn i e puegtuc 2 1 x 8 . =p d) W sm gacgucmzm (2) (4) i i + . =p g) W sm `mhjestrmzm g iueg na i mstm gjegav pm ( )
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
12/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
11 W]DNMGP]C NNN (72p) Yarnaeta 211
1. Wm gjesn`mr iueg na ( ); 2, i + , 4 41 1
( )( 1)
i x x x
8 ++
.
=p a) W sm vmrningm g ( )( )
7 74 4
' ,
1
i x
x x
8
+
jrngarm ar in ( )2, x .
=p d) W sm `mhjestrmzm g iueg na i mstm `msgrmsgtjarm pm netmrvacuc( )2, .+ =p g)W sm gacgucmzm ( )7cnh
x x i x
+ .
4. Wm gjesn`mr iueg na ( ); 2, i + , ce( ) + x i x x x
8 .
=p a) W sm gacgucmzm1
ce( ( ) )
m x
i x `x x
.
=p d) W sm vmrningm g 4
1
( )4
m m i x `x 8 .
=p g)W sm aratm g nruc garm arm tmrhmeuc omemrac ( )1
( ) , 1
e
e
m
e
m
N i x x `x e
+
8
mstm j prjormsnm arnthmtng
gu ra na 1.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
13/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
14 W]DNMGP]C NNN (72p) Yarnaeta 214 1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) 4ce i x x x8 .
=p a) W sm gacgucmzm ( )( ), 2, i x x + . =p d) W sm `mhjestrmzm g iueg na i mstm gjevmx pm netmrvacuc( )2, + .
=p g)W sm aratm g ( )4
ce ,0
m i x jrngarm ar in ( )2, x + .
4. Wm gjesn`mr iueg nncm \ X; 2,1h i , 4 4 4( ) ( 1) +1, ue`mh i x h x h h x h8 + + .=p a) W sm gacgucmzm 1 ( ) i x `x .
=p d) W sm gacgucmzm1
22
( ) xm i x `x.
=p g)W sm `mtmrhnem *h astimc egt1
2
7( )
4h i x `x 8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
14/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
W]DNMGP]C NNN (72p) Yarnaeta 217
1. Wm gjesn`mr iueg na { }; 1 , i S ( )1
xm i x
x8
+.
=p a) W sm vmrningm g ( )
4( ) ,
1
x xm i x
x
8
+ jrngarm ar in } S 1 x .
=p d) W sm `mtmrhnem mguana asnhptjtmn gtrm ca orainguc iueg nmn . i =p g) W sm `mhjestrmzm g ( ) 1 i x , pmetru jrngm 1 x 5 .
4. Rmetru inmgarme sm gjesn`mr 4
cem e
em
x N `x
x8 .
=p a) W sm vmrningm g 2 1 N 8 .=p d) W sm gacgucmzm1 N .=p g) Ijcjsne`, mvmetuac, iaptuc g 1 ce 4 x jrngarm ar in 4, x m m , s sm `mhjestrmzm g
14 11 4 ,1
e e
e
+
+ pmetru jrngm e .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
15/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
10 W]DNMGP]C NNN (72p) Yarnaeta 210
1. Wm gjesn`mr iueg na ( ); 2, , i + ce( ) x i x x
8 .
=p a) W sm gacgucmzm ( ) i m . =p d)W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm sprm + a oraingucun iueg nmn i . =p g)W sm `mhjestrmzm g , pmetru jrngm 2m x x m x 5 .
4. Wm gjesn`mr iueg na \ X; 0, 0 , i 4( ) 19 i x x8 .
=p a) W sm gacgucmzm0
4
2
( ) i x `x .
=p d) W sm vmrningm g =
=
2( ) x
`x i x
8 .
=p g)W sm `mhjestrmzm g 2
2 ( ) 6h
i x `x , jrngarm ar in\ X2,4h .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
16/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
1= W]DNMGP]C NNN (72p) Yarnaeta 21= 1. Wm gjesn`mr iueg nncm ;e i , 2 1( ) 1 n ( ) ( ), pmetru jrngm . x e e i x m i x i x e+ 8 8
=p a) W mtmrhnem 1 ( ), i x x .=p d) W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm + a oraingucun iueg nmn 2 i .
=p g)W sm gacgucmzm 4 42( ) 1cnh
x i x x
x+ .
4. Wm gjesn`mr iueg na ; , i 4( ) 1 x i x m x8 + .
=p a) W sm vmrningm g 1
42
( ) 1.
1
i x`x m
x8
+
=p d) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn ( ) ( ); , xo o x xm i x 8 ,axa Jx n `rmptmcm `m mguann 2 x 8 n 1 x 8 .
=p g) W sm gacgucmzm ( )1
4
1
1 x i x `x+ .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
17/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
19 W]DNMGP]C NNN (72p) Yarnaeta 219
1. Wm gjesn`mr iueg na4
1, 2; , ( ) , ue`m
, 2
xm x i i x a
x x a x
3 8
+ + .
=p a) W sm `mtmrhnema astimc egt iuegna i s inm gjetneu e puegtuc 2 2 x 8 .
=p d) W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc 11< 1 Am
, puegt garm apar nem
oraingucun iuegnmn . i =p g)W sm aratm g iueg na ' i mstm grmsgtjarm pm ( )2
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
18/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
1? W]DNMGP]C NNN (72p) Yarnaeta 21?
1. Wm gjesn`mr iueg na ; i * , 4( ) xm
i x x
8 .
=p a) W sm gacgucmzm ( ), i x x .
=p d) W sm `mhjestrmzm g iueg na i mstm `msgrmsgtjarm pm ( X2, 4 . =p g)W sm aratm g 7 44 7m m .
4. Wm gjesn`mr iueg na ( ); 2, i + , ( ) ce i x x x8 .
=p a) W sm gacgucmzm4
4
1
( ( ) ce ) x i x x `x + .
=p d) W sm `mhjestrmzm g jrngm prnhntnv I a iueg nmn i mstm gjegav pm netmrvacuc(1, )+ .=p g)W sm gacgucmzm arna supraimmn pcaem guprnesm etrm orainguc iuegnmn \ X; 1, , ( ) ( )f m f x i x x 8 + ,
axa Jx n `rmptmcm 1 x 8 n x m8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
19/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
16 W]DNMGP]C NNN (72p) Yarnaeta 216 1. Wm gjesn`mr iueg na ; i , ( ) ( )4 4 ( ) 1 1 i x x x8 + + .
=p a) W sm vmrningm g ( ) 0 i x x 8 , pmetru jrngm x .
=p d) W sm gacgucmzm4
( )cnh
x
i x
x+.
=p g)W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn ( ) ( )( )'
; , i x
o o x i x
8 .
4. Wm gjesn`mr iueg na ( ) ( ); 2< , ce x i i x m x+ 8 + .=p a) tnne` g ( ); 2
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
20/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
1> W]DNMGP]C NNN (72p) Yarnaeta 21>
1. Wm gjesn`mr iueg na ( ); 2, i + , 4ce
( ) x
i x x
8 .
=p a) W sm gacgucmzm ( )( ), 2, i x x + .
=p d) W sm gacgucmzmcnh ( ) x i x+ . =p g)W sm `mhjestrmzm g 12 ( )
4 i x
m3 , pmetru jrngm ), x m + .
4. Wm gjesn`mr iueg na ( ); 2, i + , 4 41 1
( )( 1)
i x x x
8 +
.
=p a) W sm gacgucmzm 41
1( )
( 1)
m
x i x `x x
+ +
.
=p d) W sm aratm g jrngm prnhntnv a iueg nmn i mstm grmsgtjarm pm ( )2, + .
=p g)W sm vmrningm g 4
144 ( ) ( )61
i x i x `x 8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
21/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
42 W]DNMGP]C NNN (72p) Yarnaeta 242
1. Wm gjesn`mr iueg na \ X; 2,1 i , ( )4
xm i x
x8
+.
=p a) W sm gacgucmzm \ X( ), 2,1 i x x .
=p d) W sm aratm g i mstm iuegnm grmsgtjarm pm \ X2
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
22/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
41 W]DNMGP]C NNN (72p) Yarnaeta 241
1. Wm gjesn`mr iueg na { }; 1 i S ,4 4
( )1
x x i x
x
+ +8 .
=p a) W sm vmrningm g ( )
4
44 7
( ) ,1
x x i x
x
8 pmetru jrngm { } S 1 x .
=p d) W sm `mtmrhnem mguana asnhptjtmn jdcngm gtrm + ca orainguc iueg nmn i .
=p g)W sm aratm g ( ) 1 6 i x i x
, jrngarm ar in 1 x 5 .
4. Wm gjesn`mr iueg na ; i , ( ) 7 7 x x i x 8 + .
=p a) W sm gacgucmzm 1
1
( ) i x `x .
=p d) W sm gacgucmzm vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn\ X; 2,1 , ( ) 7 xo o x 8 .
=p g)W sm aratm g jrngm prnhntnv I a iueg nmn i mstm gjegav pm ( X,2 n gjevmx pm \ )2, + .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
23/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
44 W]DNMGP]C NNN (72p) Yarnaeta 244 1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) ce i x x m x8 .
=p a) W sm gacgucmzm ( )( ), 2, i x x + .
=p d) W sm gacgucmzm ( )cnh( ) x m
i x
i x .
=p g)W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn i.
4. Wm gjesn`mr iueg na \ ); 4, i + , 1 1 ( )1
i x x x
8 + .
=p a) W sm gacgucmzm4
1( )
1
m
i x `x x
.
=p d) W sm aratm g jrngm prnhntnv I a iueg nmn i mstm gjegav pm \ )4
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
24/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
47 W]DNMGP]C NNN (72p) Yarnaeta 247
1. Wm gjesn`mr iueg na ; , i ( )4( ) 4 1 x i x x x m8 + .=p a) W sm gacgucmzm ( ), i x x . =p d) W sm `mtmrhnem puegtmcm `m mxtrmh acm iuegnmn i .
=p g)W sm gacgucmzm ( )cnh 1( ) x
i x x
i x+
.
4. Wm gjesn`mr iueg nncm \ ), ; 1, i I + , ( ) 1ce i x x x
8 + n ( ) ( )1 ce 1I x x x x8 + + .
=p a) W sm aratm g iueg na I mstm j prnhntnv a iueg nmn i .
=p d) W sm gacgucmzm ( )4
1
x i m `x.
=p g) W sm `mhjestrmzm g ( ) ( )44
1
7ce 4 1( ) 4 i x I x `x 8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
25/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
40 W]DNMGP]C NNN (72p) Yarnaeta 240
1. Wm gjesn`mr iueg na ( ); 2, i + ,0
( ) ce0 x
i x x8 .
=p a)W sm gacgucmzm ( )( ), 2, i x x + .
=p d) W sm `mtmrhnem puegtuc `m mxtrmh ac iuegnmn i. =p g)W sm `mhjestrmzm g
4 1ce
0 x
x
, pmetru jrngm ( )2, x + .
4. Inm4
1
e xe N x m `x8 , pmetru e .
=p a) W sm gacgucmzm2 N . =p d) W sm aratm g 41 N m8 .=p g) W sm `mhjestrmzm g ( ) ( )111 4 1ee ee N N m m+++ + 8 , pmetru jrngm e .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
26/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
4= W]DNMGP]C NNN (72p) Yarnaeta 24= 1. Wm gjesn`mr iueg na ; , i ( ) x i x m x8 .
=p a) W sm gacgucmzm ( ), i x x . =p d)W sm `mhjestrmzm g ( ) 1, i x pmetru jrngm x .
=p g)W sm sgrnm mguana asnhptjtmn jdcngm gtrm ca orainguc iueg nmn i . 4. Wm gjesn`mr iueg na ; , i ( ) 7 4 , i x x hx ex p8 + + + ue`m , ,h e p .
=p a) Rmetru 2, 7, 4h e p8 8 8 , s sm gacgucmzm1
2
( ) i x `x .
=p d) W sm `mtmrhnem , , ,h e p tnne` g ( 1) (1) 2 i i 8 8 n1
1
( ) 0 i x `x 8 .
=p g)W sm gacgucmzm 02
1cnh ( ) .
x
x i t `t
x+
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
27/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
49 W]DNMGP]C NNN (72p) Yarnaeta 249 1. Wm gjesn`mr iueg na ; , i ( ) 1 x i x m x8 .
=p a) W sm gacgucmzm `mrnvata iuegnmn i .=p d)W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn i .
=p g)W sm aratm g 4 4
4, x x
m m x x+ + + pmetru jrngm x . 4. Wm gjesn`mr iueg nncm ( ), ; 2, , i o + ( ) 1 ce i x x8 + n ( ) ceo x x x8 .
=p a) W sm aratm g o mstm j prnhntnv a iueg nmn i .
=p d) W sm gacgucmzm ( ) ( )1
m
i x o x `x .
=p g)W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmno , axa Jx n `rmptmcm `m mguann1 x 8 n x m8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
28/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
4? W]DNMGP]C NNN (72p) Yarnaeta 24?
1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) ce x i x x
8 .
=p a) W sm gacgucmzm ( ) ( ), 2< i x x + .=p d)W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn i . =p g)W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm ca orainguc iueg nmn i .
4. Wm gjesn`mr iueg na ; i , ( ) 1220 422> x i x x8 + . =p a) W sm `mtmrhnem ( ) i x `x . =p d) W sm aratm g jrngm prnhntnv a iueg nmn i mstm iuegnm grmsgtjarm pm .
=p g)W sm gacgucmzm ( )1
4
2
x i x `x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
29/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
46 W]DNMGP]C NNN (72p) Yarnaeta 246
1. Wm gjesn`mr iueg na ; i , ( )1
1, 1
ce , 1
xm x i x m
x x
8
5.
=p a) W sm stu`nmzm gjetneuntatma iuegnmn i e puegtuc 2 1 x 8
. =p d)W sm `mtmrhnem mguana asnhptjtmn gtrm ca orainguc iueg nmn i . =p g)W sm aratm g iueg na i mstm gjegav pm ( )1, + .
4. Wm gjesn`mr iueg na ; , i ( )4
44 1
1
x x i x
x
+ +8
+.
=p a) W sm `mtmrhnem ( ) ( )4 1 x i x `x+ .
=p d) W sm vmrningm g ( ) ( )1
2
ce 4 . i x `x m8
=p g)W sm aratm g ( ) ( )1
2
( 1) i x i x m `x m m 8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
30/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
4> W]DNMGP]C NNN (72p) Yarnaeta 24> 1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) ce i x x x8 .
=p a) W sm aratm g ( ) ( )1 1 1 i i 8 .=p d) W sm `mtmrhnem puegtuc `m mxtrmh ac iuegnmn i .
=p g)W sm gacgucmzm ( )cnh x
i x x x+
4. Wm gjesn`mr netmoracmcm1
21
xm N `x
x8
+ n
1
21
x xm K `x
x8
+.
=p a) W sm vmrningm g 1 N K m+ 8 .
=p d) ]tncnze`, mvmetuac, nemoacntatma 1 xm x + , a`mvrat pmetru jrngm x , s sm aratm g 14
K .
=p g)W sm `mhjestrmzm g ( )
1
42
44 1
xm m N `x
x8 +
+.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
31/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
72 W]DNMGP]C NNN (72p) Yarnaeta 272 1. Wm gjesn`mr iueg na ; i , ( ) 4 x i x x m8 + .
=p a) W sm gacgucmzm ( ) ( )2
2cnh
x
i x i
x.
=p d) W sm aratm g iueg na i mstm gjevmx pm . =p g)W sm rmzjcvm e hucnhma euhmrmcjr rmacm mguana ( ) ( ) ( ) 7 x i x i x i x m + 8 .
4. Rmetru jrngm euh r eaturac e sm gjesn`mr ( )1
2
1 e
e N x x `x8 + .
=p a) W sm gacgucmzm1 N . =p d) ]tncnze` iaptuc g ( ) ( ) 11 1e e x x ++ + , pmetru jrngm e n \ X2,1 x , s sm aratm g
422> 4226 N N .
=p g) Ijcjsne`, mvmetuac, n`metntatma ( ) ( ) ( )11 1 1e e e x x x x+
+ 8 + + , a`mvrat pmetru jrngm e n
x , s sm aratm g ( )( )
14 11 4
e
e e N e e
+
+8+ +
.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
32/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
71 W]DNMGP]C NNN (72p) Yarnaeta 271 1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) 4 ce i x x x8 .
=p a) W sm aratm g ( ) ( )4ce 1 i x x x 8 + , jrngarm ar in ( )2, x + .
=p d) W sm gacgucmzm ( )cnhce x
i x
x x+
.
=p g)W sm `mhjestrmzm g ( ) 14
i xm
, pmetru jrngm 2 x 5 .
4. Wm gjesn`mr iueg na ; , i ( ) x i x xm8 .
=p a) W sm `mtmrhnem ( )1
2
x i x m `x.
=p d) W sm aratm g ( )1
2
4 1. i x `x m 8
=p g)W sm gacgucmzm ( )4
4
1
i x `x x
.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
33/101
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
34/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
77 W]DNMGP]C NNN (72p) Yarnaeta 277
1. Wm gjesn`mr iueg na \ ); 2, i + , ( ) 41 x
x
m i x
x m8
+.
=p a)W sm vmrningm g ( ) ( )
( )44 1 x
x
m x i x
x m
8
+, pmetru jrngm \ )2, x + .
=p d)W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm + ca orainguc iueg nmn i .
=p g)W sm aratm g ( ) 11 ,1
m i x
m
+ jrngarm ar in 2 x .
4. Rmetru jrngm euh r eaturac emeuc e sm gjesn`mr,1
21
e
e x
N `x x
8+
.
=p a) W sm gacgucmzm1 N .
=p d) W sm aratm g 11
1e e N N
e+ + 8 +
, jrngarm ar in e .
=p g) ]tncnze`, mvmetuac, nemoacntatma4 1
e ee x x x
x
+, a`mvrat pmetru jrngm \ X2,1 x n e , s sm
`mhjestrmzm g 422>1
4212 14
N .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
35/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
70 W]DNMGP]C NNN (72p) Yarnaeta 270 1. Wm gjesn`mr iueg na ; i , ( ) 4( 4 7) x i x x x m8 + + .
=p a) W sm gacgucmzm ( ), i x x .
=p d) W sm `mtmrhnem( ) ( )
2
2cnh x
i x i
x . =p g)W sm `mhjestrmzm g iueg na i mstm grmsgtjarm pm .
4. Wm gjesn`mr iueg nncm ( ), ; 2, , i o + ( ) 4 ce i x x x x8 + n ( ) 4 ce 1o x x x8 + + . =p a)W sm aratm g i mstm j prnhntnv a iueg nmno.
=p d) W sm gacgucmzm ( ) ( )1
m
i x o x `x .
=p g)W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn i , axa Jx n `rmptmcm `mmguann 1 x 8 n x m8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
36/101
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
37/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
79 W]DNMGP]C NNN (72p) Yarnaeta 279
1. Wm gjesn`mr iueg na ; i , ( ) ( )4 7 7 x i x x x m8 .=p a) W sm gacgucmzm ( ), i x x .=p d)W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm sprm ca orainguc iueg nmn i . =p g) W sm aratm g taeometa ca orainguc iueg nmn , i `us e puegtuc `m gjjr`jeatm ( )4, ( 4) i , mstm
paracmc gu axa Jx .
4. Wm gjesn`mr iueg na ; i `at prne ( )4, 2
1, 2 x x x
i xm x
+ 38
+ .
=p a) W sm aratm g iueg na i a`hntm prnhntnvm pm .
=p d) W sm gacgucmzm ( )1
1
i x `x .
=p g)W sm `mhjestrmzm g ( )
14
2 4
m x i x `x 8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
38/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
7? W]DNMGP]C NNN (72p) Yarnaeta 27?
1. Wm gjesn`mr iueg na \ ); 1, , i + ( ) cece
x x i x
x x8
+.
=p a) W sm gacgucmzm ( )1
cnh x
i x
.
=p d) W sm aratm g ( ) ( )( )44 ce 1
ce x i x
x x 8
+, jrngarm ar in \ )1, x + .
=p g)W sm `mtmrhnem mguana asnhptjtmn gtrm + ca orainguc iueg nmn \ ); 1, ,o + ( ) ( )
( )( )41 i x
o x i x
8+
.
4. Wm gjesn`mr iueg nncm , ; , i o ( ) ( )4ce 1 i x x8 + n ( ) 44
1
xo x
x8
+.
=p a) W sm vmrningm g ( )1
2
ce 4 i x `x 8 .
=p d) W sm `mhjestrmzm g ( ) ( ) .o x `x i x8 + G
=p g)W sm gacgucmzm ( )( )
4
41
o x`x
i x.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
39/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
76 W]DNMGP]C NNN (72p) Yarnaeta 276
1. Wm gjesn`mr iueg na ; i , ( )4
41
1
x i x
x8
+.
=p a) W sm aratm g ( )
( )4
4
0,
1
x i x
x
8
+
jrngarm ar in x .
=p d)W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn i .
=p g) tnne` g ; ,o ( ) ( ) 1o x i x i x
8 +
, s sm `mtmrhnem
( ) ( ) ( ) ( )4 7 422> 4212422>2
cnh x
o x o x o x o x x
x
+ + + + +.
4. Wm gjesn`mr 4
cem
ee
m
N x x `x8 , pmetru jrngm .e
=p a) W sm gacgucmzm2 N . =p d) W sm aratm g 1e e N N + , jrngarm ar in e .
=p g)W sm `mhjestrmzm g arm cjg rmcana ( )4 4
1
4 1
4 4
e
e e
m m e N N
8 , pmetru jrngm .e
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
40/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
7> W]DNMGP]C NNN (72p) Yarnaeta 27> 1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) ce 1 i x x x8 + .
=p a) W sm gacgucmzm ( ) ( ), 2< i x x + .=p d)W sm `mtmrhnem puegtuc `m mxtrmh ac iuegnmn i .
=p g)W sm aratm g ( )4 4 2m i .4. Wm gjesn`mr iueg na ; , i ( )
1, 1
1, 1
x x i x
x x
8
+ 3.
=p a) W sm gacgucmzm ( )4
1
i x `x .
=p d) W sm `mtmrhnem ( )2,1a astimc egt ( ) 1a
a
i x `x 8 .
=p g)W sm gacgucmzm ( )1
2
x x i m `x.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
41/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
02 W]DNMGP]C NNN (72p) Yarnaeta 202
1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) 4 41
i x x x
8 .
=p a) W sm gacgucmzm ( ) i x , pmetru ( )2, x + .
=p d) W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )1
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
42/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
01 W]DNMGP]C NNN (72p) Yarnaeta 201
1. Inm iuegna ( ); 1, i + , ( ) 4 11
x i x
x8 .
=p a) W sm gacgucmzm ( )( ), 1< i x x +
=p d) W sm vmrningm g ( ) ( )44
cnh 14 x
i x i
x8 .
=p g)W sm aratm g iueg na i mstm `msgrmsgtjarm pm netmrvacuc( )1, + .
4. Wm gjesn`mr iueg nncm ( ), ; 2, i o + , ( ) 1 x i x x
+8 n ( ) 1 ce
0o x x8 .
=p a) W sm aratm g ( )0
1
ce 0 4 i x `x 8 + .
=p d) W sm vmrningm g ( )0
1
7ce 0
0o x ` x 8 .
=p g)W sm gacgucmzm ( ) ( )4 41
m i x o x `x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
43/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
04 W]DNMGP]C NNN (72p) Yarnaeta 204 1. Wm gjesn`mr iueg na ; i , ( ) 4212 4212 x i x x8 + .
=p a) W sm `mtmrhnem ( ), i x x .=p d)W sm `mhjestrmzm g iueg na i mstm gjevmx pm .
=p g)W sm gacgucmzm ( ) ( )2
2cnh x
i x i x
.
4. Wm gjesn`mr iueg nncm ( ), ; 2, i o + , ( )( )4
1
1 i x
x x8
+ n ( ) 1o x
x8 .
=p a) W sm vmrningm g ( )1
1m
o x ` x 8 .
=p d) Ijcjsne` n`metntatma ( ) ( ) 4 1 x
i x o x x
8 +
, a`mvrat pmetru jrngm 2 x 5 , s sm gacgucmzm ( )1
m
i x `x .
=p g)]tncnze` nemoacntatma ( ) 41
4 i x
x , a`mvrat pmetru jrngm \ X1, x m , s sm aratm g 4 1 1ce
4m m
m+ + .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
44/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
07 W]DNMGP]C NNN (72p) Yarnaeta 207
1. Wm gjesn`mr iueg na ; i , ( )4
41
1
x x i x
x x
+8
+ +.
=p a) W sm `mtmrhnem mguana asnhptjtmn gtrm ca orainguc iueg nmn i .
=p d) W sm aratm g ( ) ( )( )
4
44
4 1
1
x i x
x x
8
+ +, pmetru jrngm x .
=p g)W sm `mhjestrmzm g jrngarm ar in x avmh ( ) ( )0 44 47
i x i x + .
4. Wm gjesn`mr iueg na ( ); 2, , i + ( ) 1 i x x x
8 .
=p a) W sm gacgucmzm ( )1
m
i x `x .
=p d) W sm aratm g jrngm prnhntnv a iueg nmn i mstm gjevmx pm netmrvacuc( )2,+
. =p g)W sm `mhjestrmzm g vjcuhmcm gjrpurncjr jd neutm prne rjta na e kuruc axmnJx , a oraingmcjr
iueg nncjr \ X, ; 1, ,o f m ( ) ( )o x i x8 n ( ) 1f x i x
8
suet moacm.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
45/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
00 W]DNMGP]C NNN (72p) Yarnaeta 200 1. Wm gjesn`mr iueg na ; i , ( ) 4 x i x x m8 + .
=p a) W sm vmrningm g ( )2 1 i 8 . =p d)W sm aratm g iueg na i mstm gjevmx pm .
=p g)W sm gacgucmzm ( )cnh x x i x
m+
.
4. Wm gjesn`mr iueg na ; i , ( ) x i x m x8 .
=p a) W sm vmrningm g ( )1
2
74
i x `x m8 .
=p d) W sm gacgucmzm ( )1
2
x i x `x .
=p g)W sm aratm g `ag ;I mstm j prnhntnv a iueg nmn i , atuegn ( ) ( ) ( )4
ce 4 1m
m
i x `x I I x
8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
46/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
0= W]DNMGP]C NNN (72p) Yarnaeta 20= 1. Wm gjesn`mr iueg nncm , ; i o , ( ) ( )1 x i x x m8 n ( ) xo x x m8 .
=p a) W sm vmrningm g ( ) ( ) i x o x 8 pmetru jrngm x .=p d)W sm `mtmrhnem mguana asnhptjtmn sprm ca orainguc iueg nmno .
=p g)@ag N mstm ue netmrvac, s sm `mhjestrmzm g iueg na o mstm grmsgtjarm pm N `ag n euhan`ag iueg na i mstm gjevmx pm N .
4. Wm gjesn`mr iueg nncm \ ), ; 1, i o + , ( ) ce x i x x
8 n ( ) 41 ce x
o x x
8 .
=p a) W sm aratm g iueg na i mstm j prnhntnv a iueg nmno .
=p d) W sm gacgucmzm ( ) ( )1
m
i x o x `x .
=p g)W sm `mtmrhnem euhruc rmac ( )1
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
47/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
W]DNMGP]C NNN (72p) Yarnaeta 209
1. Wm gjesn`mr iueg na ( ) ( ) ( )4 1, 2
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
48/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
0? W]DNMGP]C NNN (72p) Yarnaeta 20? 1. Wm gjesn`mr iueg na \ ); 1, i + , ( ) 4ce i x x x8 .
=p a) W sm gacgucmzm ( ) \ ), 1< i x x + .
=p d) W sm `mhjestrmzm g 4212 1ce422> 4
.
=p g) Ijcjsne` iaptuc g 41 4 x x , jrngarm ar in 1, 4 x , s sm `mhjestrmzm nemoacntatma4 4ce x x x , pmetru jrngm 1, 4 x .
4. Rmetru inmgarme sm gjesn`mr 7
44
.1
e
e x
N `x x
8
=p a) W sm aratm g 21 7
ce4 4
N 8 .
=p d) W sm gacgucmzm1 N .=p g) W sm `mhjestrmzm g
1 1
47 4
,1
e e
e e N N e
+ +
+ 8+
jrngarm ar in e .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
49/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
06 W]DNMGP]C NNN (72p) Yarnaeta 206
1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) 1 11
i x x x
8 +
.
=p a) W sm aratm g ( )( )4 4
1 1
1 i x
x x 8
+, pmetru jrngm 2 x 5 .
=p d) W sm `mhjestrmzm g ( )1 1 ,1
i x x x
+
jrngarm ar in ( )1< x + .
=p g)W sm gacgucmzm ( ) 1cnh x
x i x i x+
.
4. Wm gjesn`mr ( )
7
41
1
1e e
N `x x x
8+
, ue`m e .
=p a) W sm vmrningm g 2 47 1
7 N N
+ 8 .
=p d) ]tncnze` n`metntatma ( ) 441 1
11
x x x x x
8 ++
a`mvrat pmetru jrngm 2 x , s sm `mtmrhnem 1 N .
=p g)W sm aratm g 41
1e e N N
e+ 3 , jrngarm ar in e , 4e .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
50/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
0> W]DNMGP]C NNN (72p) Yarnaeta 20> 1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) ( )4 ce i x x x8 .
=p a) W sm gacgucmzm ( ) ( ), 2, i x x .
=p d) W sm gacgucmzm ( ) ( )1
1cnh
1 x
i x i
x.
=p g)W sm aratm g iueg na i mstm grmsgtjarm pm ( )2, + .
4. Wm gjesn`mr iueg nncm ( ), ; 2, i o + , ( ) ce i x x x8 + n ( ) 44
xo x
x
+8 .
=p a) W sm aratm g iueg na i mstm j prnhntnv a iueg nmno .
=p d) W sm gacgucmzm ( ) ( )0
1
i x o x `x .
=p g)W sm `mhjestrmzm g ( ) ( )0
1
1o x i x `x 8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
51/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=2 W]DNMGP]C NNN (72p) Yarnaeta 2=2
1. Wm gjesn`mr iueg na ; i , ( )1 , 2
, 2 x x x
i xm x
+ 8
3.
=p a) W sm stu`nmzm gjetneuntatma iuegnmn i e puegtuc 2 2 x 8 .=p d)W sm `mtmrhnem mguana asnhptjtmn gtrm ca orainguc iueg nmn i .=p g)W sm `mhjestrmzm g iueg na i mstm gjegav pm netmrvacuc( )2, + .
4. Wm gjesn`mr iueg nncm , ; i o , ( )4 x i x m8 n ( )o x x8 .
=p a) W sm `mtmrhnem ( ) \ ), 2< i x `x x .
=p d) W sm gacgucmzm ( ) ( )1
2
i x o x `x .
=p g)W sm vmrningm g ( ) ( )1
=2 >>
2
1122m
i x o x `x
8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
52/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=1 W]DNMGP]C NNN (72p) Yarnaeta 2=1
1. Wm gjesn`mr iueg na ; i , ( ) i x 84 7 , 1
ce , 1
x x
x x
+
5.
=p a) W sm stu`nmzm gjetneuntatma iuegnmn i e puegtuc 2 1 x 8 .
=p d) W sm gacgucmzm ( )cnh x
i x x+
.
=p g)W sm `mtmrhnem ( ) ( )4 422>
422>
...cnh
x x x
x
i m i m i m
x+
+ + + .
4. Wm gjesn`mr iueg nncm , ; i I , ( ) 4 4 x i x m x x8 + + n ( )7
4 17
x xI x m x8 + + + .
=p a) W sm aratm g iueg na I mstm j prnhntnv a iueg nmn i .
=p d) W sm gacgucmzm ( )1
2
i x `x .
=p g)W sm gacgucmzm arna supraimmn pcaem hronentm `m orainguc iuegnmn \ X; 2,1 ,f
( ) ( ) 4 4
1 x i x x x
f xm
8
+ , axa Jx n `rmptmcm `m mguann 2 n 1 x x8 8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
53/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=4 W]DNMGP]C NNN (72p) Yarnaeta 2=4
1. Wm gjesn`mr iueg na ; i , ( ) i x 89 , 0
, 0
ax x
x x
3, ue`m a mstm parahmtru rmac.
=p a) W sm `mtmrhnem vacjarma rmac a cun a astimc egt iueg na i s inm gjetneu e puegtuc 2 0 x 8 .
=p d) W sm gacgucmzm ( )> i . =p g) W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )>,7 A .
4. Rmetru jrngarm e sm gjesn`mr iueg nncm \ ); 2,e i , ( )2 1 i x 8 n ( ) ( )12
x
e e i x i t `t + 8 .
=p a) W sm `mtmrhnem ( )1 i x , ue`m \ )2 , x .
=p d) W sm `mhjestrmzm g ( )4
11
1ce
0
m m i x x `x
+ 8 .
=p g) W sm gacgucmzm vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn
\ X; 2,1o
, \ X4( ) ( ), 2,1o x i x x8
.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
54/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=7 W]DNMGP]C NNN (72p) Yarnaeta 2=7
=p 1. a)W sm gacgucmzm4
41
7 4 1cnh
7 0 1 x x x
x x
+.
=p d)W sm `mtmrhnem netmrvacmcm `m gjevmxntatmn netmrvacmcm `m gjegavntatm acm iuegnmn ; i ,
( )0 4
9 16 14 i x x x x8 + + . =p g)Wm gjesn`mr iueg na ( ); 2,o + , ( ) ( )4 1 ceo x x x8 . W sm `mhjestrmzm g ( ) 2,o x jrngarm
ar in ( )2< x + .
4. Wm gjesn`mr iueg na ; i , ( )1, 2
1, 2
1
x x i x
x x x
+ 3
8
+
=p a) W sm `mhjestrmzm g iueg na i a`hntm prnhntnvm pm .
=p d) W sm gacgucmzm ( )1
2
i x `x .
=p g) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn ( ) ( )4; ,o o x x i x 8 ,axa Jx n `rmptmcm `m mguann 1 x 8 n 4 x 8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
55/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=0 W]DNMGP]C NNN (72p) Yarnaeta 2=0
1. Wm gjesn`mr iueg nncm , ; i o , ( )4
41
1
x i x
x8
+ n ( ) 1 x
xo x
m8 .
=p a) W sm vmrningm g ( ) ( )
4
4cnh 2
4 x
o x o
x
8 .
=p d) W sm `mtmrhnem gjjr`jeatmcm puegtucun `m mxtrmh ac iueg nmn i . =p g)W sm `mhjestrmzm g ( ) ( ) 4
11o x i x
m + , jrngarm ar in x .
4. Wm gjesn`mr iueg nncm \ ), ; 2< i o + , ( ) 11
i x x
8+
n ( ) 44
11
xo x
x8 +
+.
=p a) W sm vmrningm g ( )1
2
ce 4 i x `x 8 .
=p d) W sm gacgucmzm ( )1
2
o x `x .
=p g)W sm aratm g mxnst ( )2 2
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
56/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
== W]DNMGP]C NNN (72p) Yarnaeta 2==
1. Wm gjesn`mr iueg na ; i , ( ) 7 1 , 14 , 1
x x i x
ax x
+ 8
+ 5.
=p a) W sm `mtmrhnem vacjarma parahmtrucun rmaca astimc egt iueg na i s inm gjetneu e puegtuc 2 1 x 8 .
=p d)W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm ca orainguc iueg nmn i .=p g)W sm gacgucmzm ( )( )( )cnh 1
x i x x
.
4. Wm gjesn`mr iueg na \ ); 2,I + , ( ) 1 11 4
I x x x
8 + +
.
=p a) W sm `mtmrhnem iuegna \ ); 2, i + astimc egt iueg na I s inm j prnhntnv pmetru iueg na i .=p d) W sm `mhjestrmzm g iueg na I mstm `msgrmsgtjarm pm \ )2, + .
=p g) W sm `mhjestrmzm g 1
2
1 1( )
9 4I x ` x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
57/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=9 W]DNMGP]C NNN (72p) Yarnaeta 2=9 1. Wm gjesn`mr iueg na ; i , ( ) 1 x i x m x8 .
=p a) W sm gacgucmzm ( ), i x x .
=p d) W sm gacgucmzm ( )
( )cnh
x
i x
i x+
.
=p g)W sm aratm g 422> 42124212 422>m m+ + .
4. Wm gjesn`mr iueg nncm \ ) ( )7
, ; 2, ,1
x i o i x
x+ 8
+ n ( ) ( )"o x i x8 .
=p a) W sm gacgucmzm ( )4
2
1 ( ) x i x `x+ .
=p d) W sm gacgucmzm1
2
( )o x `x .
=p g)W sm `mtmrhnem prnhntnva iuegnmno a grmn asnhptjt sprm + mstm `rmapta `m mguanm 4 . y x8
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
58/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=? W]DNMGP]C NNN (72p) Yarnaeta 2=? 1. Wm gjesn`mr iueg na ; i , ( ) 1 x i x m mx8 .
=p a) W sm gacgucmzm ( ), i x x .=p d) W sm aratm g iueg na i mstm gjevmx pm . =p g)W sm `mtmrhnem gjjr`jeatmcm puegtucun `m netmrsmgnm `netrm taeometa ca orainguc iuegnmn i e puegtuc
( )2,2J n `rmapta `m mguanm 1 x 8 .
4. Wm gjesn`mr iueg na ; i ,
( )7 , 2
, 2
x x i x
x x x
8
+ 5.
=p a) W sm aratm g iueg na i a`hntm prnhntnvm pm .
=p d) W sm gacgucmzm ( )1
1
. i x `x
=p g)W sm `mhjestrmzm g `ag ( ) ( ) ,d g
a d
i x `x i x `x8
ue`m a,d,g suet euhmrm rmacmn iuegna ;I
mstm j prnhntnv a iueg nmn , i atuegn euhmrmcm ( ) ( ) ( ), ,I a I d I g suet tmrhmen gjesmgutnvn an uemnprjormsnn arnthmtngm.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
59/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=6 W]DNMGP]C NNN (72p) Yarnaeta 2=6 1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) ce i x x x8 .
=p a) W sm gacgucmzm ( )( ), 2, i x x + .=p d) W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn i . =p
g)W sm `mhjestrmzm g ( )1 ce , jrngarm ar in 2, x x x + +
.
=p 4. a)W sm gacgucmzm( )4
27
1
cnh .1
x
x
t t `t
x+
+ +
+
=p d) Wm gjesn`mr iueg na ( ); 2 , , i + ( ) 41
. i x x
8 W sm `mtmrhnem prnhntnva ( ); 2,I + a
iueg nmn , i garm vmrning rmcana (1) 2.I 8 =p g)W sm `mtmrhnem euhruc rmac pjzntnv a tnne` g vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx ,
a oraingucun iueg nmn \ X; 2,1 i , ( ) 4 i x ax8 mstm moac gu= .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
60/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
=> W]DNMGP]C NNN (72p) Yarnaeta 2=>
1. Wm gjesn`mr iueg na { }; 1 i S , ( ) 11
x i x
x
+8 .
=p a) W sm gacgucmzm { }( ), S 1 i x x .
=p d) W sm gacgucmzm ( ) ( ) 11
cnh 1 x
i x i
x
+ . =p g)W sm `mtmrhnem asnhptjta jrnzjetac gtrm + ca orainguc iueg nmn i .
4. Rmetru jrngm euh r eaturac emeuc e sm gjesn`mr \ X; 2,1e i , ( ) e xe i x x m8 n ( )1
2e e N i x `x8 .
=p a) W sm vmrningm g ( )1
12
14
xm i x `x 8 .
=p d) W sm gacgucmzm1 N .=p g)W sm `mhjestrmzm g 1e e N eN m+ 8 , jrngarm ar in , 4e e .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
61/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
92 W]DNMGP]C NNN (72p) Yarnaeta 292 =p 1. a)W sm stu`nmzm gjetneuntatma iuegnmn ; i , ( )
1 , 1
4 1 , 1
x x i x
x x
+ 38
e puegtuc 2 1 x 8 .
=p d) W sm gacgucmzm `mrnvata iuegnmn ;o , ( ) 7 44 1= 40 1o x x x x8 + .
=p g)W sm `mtmrhnem euhruc rmac pjzntnv a astimc egt4 4
cnh 74 x a
x a x a
8 .
4. Rmetru inmgarme sm gjesn`mr iueg nncm \ X; 1, 4e i , ( )1 1 1 1
.1 4e
i x x x x x e
8 + + + ++ + +
=p a) W sm gacgucmzm4
21
( ) . i x `x
=p d) Rmetru e s sm gacgucmzm arna supraimmn pcaem `mtmrhneatm `m orainguc iuegnmn e i , axa Jxn `rmptmcm 1, 4 x x8 8 .
=p g) tnne` g I mstm j prnhntnv a iueg nmn 1 i , s sm aratm g iueg na \ X; 1, 4 ,O =
( ) ( )
9
O x I x x8
mstm
grmsgtjarm.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
62/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
91 W]DNMGP]C NNN (72p) Yarnaeta 291 1. Wm gjesn`mr iueg na ; i , ( ) 4 ce 4 x i x x8 .
=p a) W sm gacgucmzm ( ), i x x .
=p d) W sm gacgucmzm ( ) ( )7
7cnh
7 x
i x i
x.
=p g)W sm `mtmrhnem puegtuc `m mxtrmh ac iuegnmn i .=p 4. a) W sm `mtmrhnem prnhntnvmcm iuegnmn ; i , ( ) x i x m8 . =p d) W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn
\ X; 1, ,o m ( ) ce xo x x
8 .
=p g) W sm gacgucmzm( )
7
1
14
`x x x +
.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
63/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
94 W]DNMGP]C NNN (72p) Yarnaeta 294
1. Wm gjesn`mr iueg na { }; S 7 i , ( ) 17
x i x
x
+8 .
=p a) W sm gacgucmzm ( ) }, S 7 . i x x
=p d) W sm gacgucmzm 0( ) (0)
cnh 0 x i x i
x .
=p g)W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm + ca orainguc iueg nmn i .
4. Wm gjesn`mr iueg na \ ); 2, i + , ( ) 11
i x x
8+
.
=p a) W sm gacgucmzm1
2
( ) i x `x .
=p d) W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn \ X; 2,4 ,f ( ) ( ).f x i x8
=p g) W sm aratm g `ag , 2a 5 , atuegn ( )1
1 1 .4 1
a
a i x `xa a
+
+ +
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
64/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
97 W]DNMGP]C NNN (72p) Yarnaeta 297
1. Wm gjesn`mr iueg na \ ); 1 , i + , ( ) 1 x x i x m x
8 + .
=p a) W sm gacgucmzm \ )( ), 1 , i x x + .
=p d) W sm stu`nmzm hjejtjena iueg nmn i pm \ )1, + .=p g) W sm sgrnm mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )1 , A m .
4. Wm gjesn`mr iueg na ( ) 4= , 1
; ,7 1 , 1
x x i i x
x x
+ 3 8
+ .
=p a) W sm `mhjestrmzm g iueg na i a`hntm prnhntnvm.
=p d) W sm gacgucmzm ( )4
7
. i x `x
=p g) W sm aratm g, pmetru jrngm \ )1,h arna supraimmn pcaem `mtmrhneatm `m orainguc iuegnmn i , axa
Jx n `rmptmcm `m mguann n 1 x h x h8 8 + mstm gmc pune=0
.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
65/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
90 W]DNMGP]C NNN (72p) Yarnaeta 290
1. Wm gjesn`mr iueg nncm \ ), ; 2 , i f + , ( )4 1
x i x
x8
+ n ( ) ( )4f x i x8 .
=p a) W sm vmrningm g ( )
( )4
4
4,
1
xf x
x
8
+
jrngarm ar in 2 x .
=p d) W sm `mtmrhnem mguana asnhptjtmn gtrm + ca orainguc iueg nmn i .=p g) W sm `mhjestrmzm g iueg na f mstm grmsgtjarm pm netmrvacuc \ )2< .+
4. Wm gjesn`mr iueg na \ ); 2, i + , ( ) 1 1 11 7
i x x x
8 ++ +
.
=p a) W sm aratm g ( )( ) ( )1
2
441 7
7 x x i x `x+ + 8 .
=p d) W sm gacgucmzm ( )1
2
i x `x .
=p g) W sm `mtmrhnem euhruc rmac pjzntnvl astimc egt arna supraimmn pcaem `mtmrhneatm `m orainguciueg nmn i , axa Jx n `rmptmcm `m mguann 2 n x x l 8 8 s inm moac gu cel l + .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
66/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
9= W]DNMGP]C NNN (72p) Yarnaeta 29=
1. Wm gjesn`mr iueg na ; i , ( ) 44
1
x i x
x8
+.
=p a) W sm gacgucmzm ( ), i x x .
=pd) W sm `mtmrhnem puegtmcm `m mxtrmh acm iuegnmn i .
=p g) W sm `mhjestrmzm g ( ) ( )7 4 i x i x+ , pmetru jrngm x .4. Wm gjesn`mr iueg na ; i , ( ) 4 i x x8 + .
=p a) W sm gacgucmzm ( )1
2
i x `x .
=p d) W sm gacgucmzm ( )1
2
xm i x `x.
=p g) W sm `mtmrhnem euhruc rmac p astimc egt vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx , a
oraingucun iuegnmn \ X ( ) ( ); 2,1 ,f f x i px 8 , pmetru jrngm \ X2,1 x s inm hnenh.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
67/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
99 W]DNMGP]C NNN (72p) Yarnaeta 299
1. Wm gjesn`mr iueg na ; , i ( )
4 7, 2
47
, 24
x x
x i x x x
+
+8
+ 3
.
=p a) W sm stu`nmzm gjetneuntatma iuegnmn i e puegtuc 2 2 x 8 . =p d) W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm + ca orainguc iueg nmn i .
=p g) W sm aratm g ( ) \ )7 , 4 , jrngarm ar in 2-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
9? W]DNMGP]C NNN (72p) Yarnaeta 29? 1. Wm gjesn`mr iueg nncm , ; , i o ( ) 7 47 0 i x x x8 + n ( ) 7 4= 6 0o x x x x8 + .
=p a) W sm gacgucmzm ( ) ( ), i x o x x .
=p d) W sm gacgucmzm ( )
( )4cnh
x
i x
o x.
=p g)W sm `mhjestrmzm g ( ) 2 i x , ( )jrngarm ar in 2 , . x +
4. Wm gjesn`mr iueg nncm ( ), ; 2 , , i I ( ) 1 x x i x m x
8 + n ( ) ce xI x m x x8 + .
=p a) W sm `mhjestrmzm g iueg na I mstm j prnhntnv pmetru iueg na i .
=p d) W sm gacgucmzm ( )( )4
1
ce x I x x x `x + .
=p g)W sm `mtmrhnem parahmtruc rmach astimc egt arna supraimmn pcaem guprnesm etrm orainguc iuegnmn i ,axa Jx n `rmptmcm `m mguann 1 x 8 n x m8 s inm moac gu 4hm .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
69/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
96 W]DNMGP]C NNN (72p) Yarnaeta 296 1. Wm gjesn`mr iueg na ; i , ( ) 7 7 i x x x8 + .
=p a) W sm gacgucmzm ( ), . i x x =p d) W sm aratm g iueg na i mstm grmsgtjarm pm .
=p g)W sm gacgucmzm 7 ( )cnh x i x x .
4. Wm gjesn`mr iueg na ; i , ( ) ( X
( )
1, ,1
4ce 4, 1 ,
x x
x i x x x
+
8 +
.
=p a) W sm `mhjestrmzm g iueg na i a`hntm prnhntnvm pm .
=p d) W sm gacgucmzm1
2
( 4) ( ) x i x `x .
=p g)W sm gacgucmzm ( )( )1
1cnh 4
x
x i t `t
x++ .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
70/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
9> W]DNMGP]C NNN (72p) Yarnaeta 29>
1. Wm gjesn`mr iueg na ( ); 2 , i + , ( )4
ce4
x i x x8 + .
=p a) W sm gacgucmzm ( )( ), 2< i x x + .
=p d) W sm gacgucmzm ( ) ( )1
1cnh .1 x
i x i x
=p g)W sm `mtmrhnem netmrvacmcm `m gjevmxntatmn netmrvacmcm `m gjegavntatm acm iuegnmn i .4. Wm gjesn`mr iueg na \ ); 2, i + , ( ) ( )1 ,e i x x e 8 + .
=p a) Rmetru 4e 8 s sm gacgucmzm ( )4
1
i x `x .
=p d) Rmetru 1e 8 s sm `mtmrhnem \ )2
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
71/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?2 W]DNMGP]C NNN (72p) Yarnaeta 2?2 1. Wm gjesn`mr iueg na ( ); 2, i + , ( ) i x x x8 + .
=p a) W sm gacgucmzm ( )( ), 2, i x x + . =p d) W sm aratm g iueg na i mstm grmsgtjarm pm ( )2, + .
=p g)W sm `mtmrhnem gjjr`jeatmcm puegtucun oraingucun iueg nmn , i e garm taeometa ca oraingarm paeta moac gu
74
.
4. Wm gjesn`mr iueg na \ ); 2< i + , ( ) 1 1 .1 4
i x x x
8 ++ +
=p a) W sm vmrningm g ( )( ) ( ) 41 4 7 , 2 x x i x `x x x x+ + 8 + + G .
=p d) W sm gacgucmzm ( )1
2
i x `x .
=p g)W sm gacgucmzm vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn
\ X; 2 ,1 ,f ( ) ( ) ( )11
1f x i x i x
x8 +
+.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
72/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?1 W]DNMGP]C NNN (72p) Yarnaeta 2?1 1. Rmetru jrngm e sm gjesn`mr iueg nncm ( ); 2,e i , ( )2 ce i x x8 n ( ) ( )1e e i x i x8 .
=p a) W sm `mtmrhnem iuegna 1 i . =p d) W sm `mtmrhnem mguana asnhptjtmn gtrm + ca orainguc iueg nmn 4 i .
=p g)W sm aratm g ( ) ( )2 11 1 i x
i x , jrngarm ar in ( )2, x + .
4. Wm gjesn`mr iueg na ; i , ( ) 44
1
x i x
x8
+.
=p a) W sm gacgucmzm ( )1
2
m
i x `x .
=p d) W sm `mhjestrmzm g jrngm prnhntnv a iueg nmn i mstm iuegnm grmsgtjarm pm netmrvacuc( )2 , + .
=p g) W sm `mhjestrmzm g ( ) ( ) ( ) ( )1 4 7 0
2 1 4 7
i x `x i x `x i x `x i x `x+ 5 + .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
73/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?4 W]DNMGP]C NNN (72p) Yarnaeta 2?4
1. Wm gjesn`mr iueg na ; i , ( ) 7 7 i x x x
8 + .
=p a) W sm gacgucmzm ( ), i x x .
=p d) W sm gacgucmzm ( ) ( )1 1cnh 1 x i x i x . =p g)W sm `mtmrhnem netmrvacmcm `m hjejtjenm acm iuegnmn i .
4. Wm gjesn`mr iueg na \ X; 2,1 i , ( ) 44 i x x x8 . =p a) W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn i .
=p d) W sm gacgucmzm1
2
( ) i x `x .
=p g)W sm gacgucmzm2
42
( )
cnh
x
x
i t `t
x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
74/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?7 W]DNMGP]C NNN (72p) Yarnaeta 2?7
1. Wm gjesn`mr iueg na ; i , ( )
4
4
4
7, 1
1 , ue`m4
, 1
4
x x
x i x a x a
x
x
+
+8 +
5
+
.
=p a) W sm `mtmrhnem euhruc rmaca astimc egt iueg na i s inm gjetneu e puegtuc 2 1 x 8 . =p d) W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm ca oraingucun iuegnmn i . =p g) W sm `mtmrhnem euhruc rmaca astimc egt paeta taeometmn ca oraing e puegtuc ( )( )4< 4 i s inm moac gu 1.
4. Wm gjesn`mr iueg na ; i , ( )4 x i x m8 .
=p a) W sm vmrningm g ( )1
2
1 i x `x m8 .
=p d) W sm gacgucmzm ( )1
2
x i x `x .
=p g)W sm `mhjestrmzm g ( )1
2
1 i x `x m .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
75/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?0 W]DNMGP]C NNN (72p) Yarnaeta 2?0
1. Wm gjesn`mr iueg nncm { }, ; 1, 4 i f S , ( ) ( )( )1 4 i x x x8 n ( ) ( )
( )' i x
f x i x
8 .
=p a) W sm aratm g ( ) 1 1
1 4
f x
x x
8 +
.
=p d) W sm `mhjestrmzm g iueg na f mstm `msgrmsgtjarm pm ( )
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
76/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?= W]DNMGP]C NNN (72p) Yarnaeta 2?=
1. Wm gjesn`mr iueg na ; i , ( ) 41
, 21
4 1, 2
x i x x
x x
8 +
+ 5
.
=p a) W sm stu`nmzm gjetneuntatma iuegnmn i e puegtuc 2 2 x 8
. =p d) W sm `mhjestrmzm g iueg na i mstm grmsgtjarm pm netmrvacuc( ), 2 .
=p g)W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc 11,4
A
.
4. Rmetru inmgarm *e sm gjesn`mr iueg na ;e i , ( )( )4
1
1e e
i x x
8
+
.
=p a) W sm vmrningm g ( )11
1 1m
i x `x 8 .
=p d) W sm `mtmrhnem prnhntnvaO a iueg nmn ; ,o ( ) ( )41o x
i x8 , garm vmrning rmcana ( ) 171 1=O
8 .
=p g)W sm gacgucmzm ( )1
2
,e x i x `x ue`m , 4e e .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
77/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?9 W]DNMGP]C NNN (72p) Yarnaeta 2?9
1. Wm gjesn`mr iueg na ( ) ( ) 1; 2, , x i i x x
+ 8 .
=p a) W sm vmrningm g ( ) 1
4
x i x
x x
+ 8 , pmetru jrngm ( )2< x + .
=p d) W sm aratm g 422> 4211 4212 4212 . =p g)W sm aratm g iueg na i eu arm asnhptjt gtrm + .
4. Wm gjesn`mr iueg na ( )( )
4 4, 1; ,
1 ce , 1
x x x i i x
x x x
+ 3 8
+ .
=p a) W sm aratm g iueg na i a`hntm prnhntnvm pm .
=p d) W sm vmrningm g ( )1
2
?
9 i x `x 8 .
=p g)W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmn Jx , a oraingucun iueg nmn
\ X ( ) ( ); 1< ,1
i xf m f x x
8+
.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
78/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?? W]DNMGP]C NNN (72p) Yarnaeta 2?? 1. Wm gjesn`mr iueg na ( ); 2, i + T , ( ) ( )7 ce i x x x8 .
=p a) W sm gacgucmzm ( ) ( ), 2, i x x + .
=p d) W sm gacgucmzm1
( ) (1)cnh
1 x
i x i
x.
=p g)W sm `mhjestrmzm g iueg na i mstm gjevmx pm ( )2, .+
4. Wm gjesn`mr iueg nncm , ;I i T T , ( ) xI x x m8 n ( ) ( )1 x i x x m8 + . =p a)W sm vmrningm g iueg na I mstm j prnhntnv a iueg nmn i .=p d) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmnI , axa Jx n `rmptmcm `m mguann
2 x 8 n 1 x 8 .
=p g)W sm gacgucmzm ( ) ( )1
2
1 x
I x i x`x
m +.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
79/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?6? W]DNMGP]C NNN (72p) Yarnaeta 2?6 ?6
1. Wm gjesn`mr iueg na ; i T T , ( )4
4 1
x i x
x8
+.
=p a) W sm vmrningm g ( )
( )4
4
42
1
x i x
x
8
+
pmetru jrngm x T .
=p d) W sm `mtmrhnem mguana asnhptjtmn gtrm + ca orainguc iueg nmn i .=p g)W sm aratm g ( ) ( )7 74226 422> i i .
4. Wm gjesn`mr iueg nncm \ X, ; 2,1 i o T , ( ) 4 x i x 8 n ( ) xo x x m8 .
=p a) W sm `mtmrhnem ( ) i x `x . =p d)W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmno , axa Jx n `rmptmcm `m
mguann 2 x 8 n 1 x 8 .
=p g)W sm gacgucmzm 22
( )
cnh
x
x
i t `t
x.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
80/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
?> W]DNMGP]C NNN (72p) Yarnaeta 2?> 1. Wm gjesn`mr iueg na ; i T T , ( ) 4 7 x x i x 8 + .
=p a) W sm gacgucmzm ( ), i x x . =p d)W sm `mtmrhnem asnhptjta sprm a iueg nmn i .
=p g)W sm aratm g iueg na i mstm gjevmx pm . 4. Rmetru inmgarm *e sm gjesn`mr iueg nncm \ X; 2,1e i T , ( ) 1
e
e x
i x x
8+
.
=p a) W sm gacgucmzm ( ) ( )
14
42
1 x i x `x+ .
=p d) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn 1 i , axa Jx n `rmptmcm `m mguann2 x 8 n 1 x 8 .
=p g)W sm aratm g ( )1
422>
2
ce 4 i x `x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
81/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
62 W]DNMGP]C NNN (72p) Yarnaeta 262
1. Wm gjesn`mr iueg na { }; 1 i S T T , ( ) 111
i x x x
8 + + .
=p a) W sm vmrningm g ( )( )
4
44
1
x x i x
x
8 pmetru jrngm { }1 x S T .
=p d) W sm `mtmrhnem mguana asnhptjtmn jdcngm gtrm + ca orainguc iueg nmn i . =p g)W sm `mhjestrmzm g ( ) 0 i x , pmetru jrngm ( )1< x + .
4. Rmetru inmgarme sm gjesn`mr iueg nncm ;e i T T , ( )1
x
e ex
m i x
m8
+.
=p a) W sm gacgucmzm ( )2 i x `x , x T . =p d) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn 1 i , axa Jx n `rmptmcm `m mguann
2 x 8 n 1 x 8 .
=p g) W sm aratm g ( ) ( )
1 1
12 2 ,e e
i x `x i x `x+
pmetru jrngm e E
.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
82/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
61 W]DNMGP]C NNN (72p) Yarnaeta 261 1. Wm gjesn`mr iueg na ( ); 2, i + T , ( ) 74 7 i x x x8 .
=p a) W sm vmrningm g ( )7 4
1 1 i x
x x 8 , pmetru jrngm 2 x 5 .
=p d) W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )1< 1 A .=p g)W sm aratm g ( ) 1 i x , pmetru jrngm 2 x 5 .
4. Wm gjesn`mr iueg na ;a i T T , ( ) 1a i x ax8 + , ue`m a T . =p a) W sm `mtmrhnema T astimc egt iueg na ;I T T , ( ) 4 1I x x x8 + + s inm j prnhntnv a iueg nmn a i .
=p d) W sm gacgucmzm ( )1
12
xm i x `x .
=p g)W sm `mhjestrmzm g ( )1
4
2
1
0a i x `x pmetru jrngm a T .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
83/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
64 W]DNMGP]C NNN (72p) Yarnaeta 264 1. Wm gjesn`mr iueg na ( ); 2< i + T , ( ) ( )7 i x x x8 .
=p a) W sm vmrningm g ( ) 7 7 ,4
x i x
x 8 pmetru jrngm 2 x 5 .
=p d) W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )1< 4 A . =p g)W sm `mhjestrmzm g 4 7 x
x+ pmetru jrngm 2 x 5 .
4. Rmetru inmgarm *e sm gjesn`mr iueg nncm \ X; 2,1e i T , ( ) e x
e i x m8 .=p a) W sm `mtmrhnem ( )1 i x `x .
=p d) W sm gacgucmzm ( )1
12
x i x `x .
=p g) W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn\ X; 2,1o T , ( ) ( )
7o x x i x8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
84/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
67 W]DNMGP]C NNN (72p) Yarnaeta 267
1. Wm gjesn`mr iueg na ; i T T , ( ) 174
x x i x
8 .
=p a) W sm gacgucmzm ( ) i x , ue`m x T .
=p d) W sm gacgucmzm2
( ) (2)cnh x
i x i x
.
=p g)W sm `mhjestrmzm g iueg na i mstm grmsgtjarm pm T .
4. Wm gjesn`mr iueg na ( ); 2, i + T , ( ) 1 i x x x
8 + .
=p a) W sm `mtmrhnem ( ) i x `x , ue`m 2 x 5 . =p d) W sm gacgucmzm vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn
\ X; 1, 4 ,o `minent prne ( ) ( )o x i x8 , \ X1, 4 x .
=p g)W sm gacgucmzm ( )1
cem
i x x `x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
85/101
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
86/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
6= W]DNMGP]C NNN (72p) Yarnaeta 26=
1. Wm gjesn`mr iueg na ( ); 2, i + T , ( )4 1 x
i x x
+8 .
=p a) W sm vmrningm g ( )4
41 x
i x x
8 , pmetru jrngm 2 x 5 .
=p d) W sm `mtmrhnem mguana asnhptjtmn jdcngm ca orainguc iuegnmn i . =p g)W sm aratm g iueg na i mstm gjevmx pm (2, )+ .
4. Wm gjesn`mr iueg nncm \ X, ; 2,1 i o T , ( ) x i x m8 n ( ) x xo x m m8 + . =p a) W sm `mtmrhnem ( ) i x `x . =p d) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn \ X; 2,1f T , `minent prne
( ) ( )f x x i x8 , axa Jx n `rmptmcm `m mguann 2 x 8 n 1 x 8 .=p g) W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmnJx , a oraingucun iueg nmno.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
87/101
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
88/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
6? W]DNMGP]C NNN (72p) Yarnaeta 26? 1. Wm gjesn`mr iueg na ( ) ( )4; , 4 1 x i i x x x m 8 + + .
=p a) W sm vmrningm g ( ) ( )( )1 7 x i x x x m 8 + + , jrngarm ar in x . =p d) W sm `mtmrhnem mguana asnhptjtmn gtrm ca orainguc iueg nmn i .=p g)W sm `mhjestrmzm g ( ) ( ) 7
64 0 i i m
+ .
4. Wm gjesn`mr iueg na ( )4 7 4, 1
; ,ce , 1
x x x i i x
x x
+ 8
5 .
=p a) W sm aratm g iueg na i a`hntm prnhntnvm. =p d) W sm `mhjestrmzm g jrngm prnhntnv a iueg nmn i mstm gjevmx pm ( )1
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
89/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
66 W]DNMGP]C NNN (72p) Yarnaeta 266 1. Wm gjesn`mr iueg na ; i T T , ( ) 7 7 1 i x x x8 + .
=p a) W sm gacgucmzm ( )1 i . =p d) W sm `mtmrhnem netmrvacmcm `m gjevmxntatmn netmrvacmcm `m gjegavntatm acm iuegnmn i .
=p g)W sm aratm g ( ) 7 i x , pmetru jrngm 4 x . 4. Wm gjesn`mr iueg nncm ( ), ; 2< i I + T , ( ) 4
11 i x
x8 n ( ) 1I x x
x8 + .
=p a)W sm vmrningm g iueg na I mstm j prnhntnv a iueg nmn i . =p d) W sm gacgucmzm arna supraimmn pcaem guprnesm etrm orainguc iuegnmn i , axa Jx n `rmptmcm `m mguann
1 x 8 n 4 x 8 .
=p g) W sm gacgucmzm ( )1
cem
I x x x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
90/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
6> W]DNMGP]C NNN (72p) Yarnaeta 26> 1. Inm iuegna ; i T T , ( ) 7 44 7 1 i x x x8 + .
=p a) W sm gacgucmzm ( )1 i . =p d) W sm `mtmrhnem netmrvacmcm `m gjegavntatmn netmrvacmcm `m gjevmxntatm acm iuegnmn i .
=p g)W sm aratm g ( ) 2 i x , pmetru jrngm 14
x .
4. Wm gjesn`mr iueg nncm \ X, ; 2,1 i o T , ( ) x i x m8 n ( ) 1 xo x m 8 . =p a) W sm `mtmrhnem hucnhma prnhntnvmcjr iuegnmn i . =p d) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn \ X; 2,1f T , ( ) ( )f x x i x8 ,
axa Jx n `rmptmcm `m mguann 2 x 8 n 1 x 8 .
=p g)W sm aratm g ( ) ( )( )
14
2
2o x i x `x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
91/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>2 W]DNMGP]C NNN (72p) Yarnaeta 2>2 1. Wm gjesn`mr iueg na ( ); 2, i + T , ( ) 4 ce i x x x8 .
=p a) W sm vmrningm g ( ) 1 x i x x
8 , pmetru jrngm 2 x 5 .
=p d) W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )1
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
92/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>1 W]DNMGP]C NNN (72p) Yarnaeta 2>1 1. Wm gjesn`mr iueg na ( ); 2, i + T , ( ) 4 ce i x x x x8 .
=p a) W sm gacgucmzm ( ) ( ), 2, i x x + . =p d) W sm aratm g iueg na i mstm gjevmx pm ( )2, + . =p g) W sm aratm g ( ) 2, i x jrngarm ar in 2 x 5 .
4. Rmetru inmgarm *e sm gjesn`mr iueg nncm \ X; 2, 4e i T , ( ) ( )4 e
e i x x8 . =p a) W sm `mtmrhnem ( )1 i x `x , ue`m \ X2, 4 x .
=p d) W sm gacgucmzm arna supraimmn pcaem guprnesm etrm orainguc iuegnmn \ X; 2, 4 ,o ( ) ( )1 xo x i x m8 ,axa Jx n `rmptmcm `m mguann 2 x 8 n 4 x 8 .
=p g)W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn = i .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
93/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>4 W]DNMGP]C NNN (72p) Yarnaeta 2>4
1. Wm gjesn`mr iueg na ( ); 2, i + T , ( ) xm
i x x
8 .
=p a) W sm vmrningm g ( ) ( )4
1,
xm x i x
x
8 pmetru jrngm 2 x 5 .
=p d) W sm `mtmrhnem asnhptjta vmrtngac ca orainguc iueg nmn i . =p g)W sm `mhjestrmzm g , xm mx pmetru jrngm 2 x 5 .
4. Inm iuegna \ X; 1, 4 i T , ( ) 4 i x x x
8 + .
=p a) W sm `mtmrhnem hucnhma prnhntnvmcjr iuegnmn i . =p d) W sm gacgucmzm vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn i .
=p g)W sm gacgucmzm ( )4
1
ce i x x `x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
94/101
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
95/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>0 W]DNMGP]C NNN (72p) Yarnaeta 2>0 1. Wm gjesn`mr iueg na ; i T T , ( ) x i x x m8 .
=p a) W sm vmrningm g ( ) ( )1 , x i x x m 8 + pmetru jrngm x T . =p d) W sm `mtmrhnem netmrvacmcm `m gjevmxntatmn netmrvacmcm `m gjegavntatm acm iuegnmn i . =p g)W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm ca orainguc iueg nmn i .
4. Rmetru inmgarm *e sm gjesn`mr iueg nncm \ X; 2,1e i T , ( )4
1
e
e x x
i x x
+ +8
+.
=p a) W sm `mtmrhnem ( )1 x i x `x .
=p d) W sm gacgucmzm ( )1
42
i x `x .
=p g)W sm aratm g arna supraimmn pcaem, guprnesm etrm orainguc iuegnmn 4226 i n axa Jx n `rmptmcm2 x 8 n 1 x 8 , mstm han hng sau moac gu 4.
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
96/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>= W]DNMGP]C NNN (72p) Yarnaeta 2>=
1. Wm gjesn`mr iueg na ( ); 1, i + T , ( )4
1 x
i x x
8 .
=p a) W sm vmrningm g ( )( )
4
44
,
1
x x i x
x
8 pmetru jrngm 1 x 5 .
=p d) W sm `mtmrhnem mguana asnhptjtmn jdcngm gtrm + ca orainguc iueg nmn i . =p g)W sm aratm g ( ) ( )7 74 7 i i .
4. Wm gjesn`mr iueg nncm \ X, ; 2,1 i o T , ( ) 1 i x x8 n ( ) 1o x x8 . =p a) W sm `mtmrhnem ( ) i x `x . =p d)W sm gacgucmzm arna supraimmn pcaem guprnesm etrm orainguc iuegnmno , axa Jx n `rmptmcm `m mguann
2 x 8 n 1 x 8 .
=p g) W sm gacgucmzm ( )1
1
ce
m
i x x `x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
97/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>9 W]DNMGP]C NNN (72p) Yarnaeta 2>9
1. Wm gjesn`mr iueg na ( ); 2, , i + ( ) ce x i x x
8 .
=p a)W sm vmrningm g ( ) 41 ce
, x
i x x
8 pmetru jrngm 2 x 5 .
=p d) W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm + ca orainguc iueg nmn i . =p g)W sm aratm g ( ) ( )4226 422> i i .
4. Wm gjesn`mr iueg na \ X; 2,1 , i ( ) i x x8 . =p a) W sm `mtmrhnem ( ) i x `x .
=p d) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn \ X; 2,1 ,o ( ) ( )4
4 1
i xo x
x8
+,
axa Jx n `rmptmcm `m mguann 2 x 8 n 1 x 8 .=p g)W sm gacgucmzm vjcuhuc gjrpucun jdneut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn
\ X; 2,1 ,f ( ) ( )4 x
f x m i x8 , ue`m \ X2,1 x .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
98/101
Hnenstmruc M`uganmn, Gmrgmtrnnn PnemrmtucunGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>? W]DNMGP]C NNN (72p) Yarnaeta 2>?
1. Wm gjesn`mr iueg na ; i T T , ( )4
41
1
x x i x
x x
+8
+ +.
=p a) W sm vmrningm g ( )
( )
4
44
4 4
1
x i x
x x
8
+ +
pmetru jrngm x T .
=p d) W sm `mtmrhnem mguana asnhptjtmn jrnzjetacm g trm + ca orainguc iueg nmn i . =p g)W sm aratm g ( ) ( )7 7422> 4212 i i .
4. Inm iuegna \ X; 1, , i m ( ) ce i x x8 . =p a) W sm `mtmrhnem ( ) i x `x . =p d)W sm gacgucmzm arna supraimmn pcaem guprnesm etrm orainguc iuegnmn i , axa Jx n `rmptmcm `m mguann
1 x 8 n x m8 .
=p g)W sm aratm g ( )1
m x mm i x `x m m .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
99/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>6 W]DNMGP]C NNN (72p) Yarnaeta 2>6
1. Wm gjesn`mr iueg na ( ); 1, i + T , ( )1
xm i x
x8 .
=p a) W sm vmrningm g ( ) ( )
( )4
4
1
xm x i x
x
8 , pmetru jrngm 1 x 5 .
=p d) W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )44< A m . =p g)W sm `mhjestrmzm g ( ) 4 i x m , pmetru jrngm 1 x 5 .
4. Rmetru inmgarm *e sm gjesn`mr iueg nncm \ X; 1, 0 ,e i ( ) 0ee i x x x8 + .
=p a) W sm vmrningm g ( )0
11
10 = .
7 i x `x 8
=p d) W sm gacgucmzm
( )
0
4
41
4.
x`x
i x
+
=p g)W sm `mtmrhnem vjcuhuc gjrpucun jd neut prne rjta na e kuruc axmnJx , a oraingucun iueg nmn
\ X; 1, 0 ,o ( )( )41
o x i x
8 .
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
100/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
DAGACA]TMAP 422>-HAPMHAPNG - Rrjda @, HP4, prjoraha H4
>> W]DNMGP]C NNN (72p) Yarnaeta 2>> 1. Wm gjesn`mr iueg na ( ); 2, i + T , ( ) 74 7 i x x x8 + .
=p a)W sm vmrningm g ( )7 4
11 i x
x 8 , pmetru jrngm 2 x 5 .
=p d) W sm `mtmrhnem mguana taeometmn ca orainguc iuegnmn i e puegtuc ( )1
8/11/2019 Matematica M2 - Subiectul III cu 100 de variante din 2009
101/101
Hnenstmruc M`uganmn, Gmrgmtrnnn NejvrnnGmetruc Eanjeac pmetru Gurrngucuhn Mvacuarm e evhetuc Rrmuenvmrsntar
122 W]DNMGP]C NNN (72p) Yarnaeta 122
1. Wm gjesn`mr iueg na ( ); 2, i + T , ( )4 1 x
i x x
+8 .
=p a) W sm vmrningm g ( )4
41 x
i x x
8 , pmetru jrngm 2 x 5 .
=p d) W sm `mtmrhnem mguana asnhptjtmn jdcngm gtrm + ca orainguc iueg nmn i . =p g)W sm aratm g iueg na i mstm gjevmx pm ( )2, + .
4. Rmetru inmgarm e sm gjesn`mr iueg nncm \ X; 2,1e i T , ( ) ( )1 1e xe i x x m+8 + . =p a) W sm `mtmrhnem ( )2 m x i x `x . =p d) W sm `mtmrhnem arna supraimmn pcaem guprnesm etrm orainguc iuegnmn 1 i , axa Jx n `rmptmcm `m
mguann 2 x 8 n 1 x 8 .
=p g) W sm aratm g ( ) ( ) ( )1 1 1
4226 4212 422>
2 2 2
4 i x `x i x `x i x `x+ .
Top Related