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Autori: Prof. Prof. Prof. Prof. Prof. Prof. Prof. Prof.

gr. gr. gr. gr. gr. gr. dr. gr.

I, I, I, I, I, I,

Florica Banu &RVWHO&KLWHú Ovidiu Cojocaru Mihai Contanu Sergiu Marinescu ,RQ5RúX 'XPLWUX6YXOHVFX

I, Marilena Stoica

Coordonator: Prof Univ. dr. &RQVWDQWLQ1L

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11

5.1. Tehnici de testare – Itemi obiectivi Descriere>,1HDFúX$6WRLFDSúLXUPWRDUHOH@ 7HVWHOH GH SURJUHV úFRODU ± úL vQ VSHFLDO FHOH VWDQGDUGL]DWH ± FXSULQG LWHPLRELHFWLYLLWHPLFDUHUHDOL]HD]RVWUXFWXUDUHDVDUFLQLORUSURSXVHHOHYLORU vQFRQFRUGDQ FXRELHFWLYHOHSHFDUHWHVWHOHúLOHDVXPÌQFDWHJRULDLWHPLORU RELHFWLYL VXQW LQFOXúL LWHPL FX DOHJHUH GXDO DGHYUDWIDOV LWHPL GH WLS SHUHFKHúLLWHPLFXDOHJHUHPXOWLSO &RQVWUXLUHD XQRU LWHPL GH R FDOLWDWH VXSHULRDU FRUHF L GLQ SXQFW GH YHGHUH WHKQLF úL DGHFYD L RELHFWLYHORU SUHFRQL]DWH HVWH R DGHYUDW DUW (OHPHQWHOHVSHFLILFHDOHDFHVWHLDUWHDXXQIXQGDPHQWWHRUHWLFFHVHED]HD] vQSULPXOUkQG SH FXQRDúWHUHD SULQFLSLLORU úL WHKQLFLORU GH SURLHFWDUH D DFHVWRU LWHPL SUHFXP úL SH YDORULILFDUHD úL SRWHQ DUHD DYDQWDMHORU SH FDUH OH RIHU evaluatorului. 7UVWXUDSULQFLSDODLWHPLORURELHFWLYLRFRQVWLWXLHDúDFXPVXJHUHD] úL GHQXPLUHD ORU obiectivitatea ULGLFDW vQ PVXUDUHDHYDOXDUHD UH]XOWDWHORU vQY ULL FKLDU GDF DFHVWHD VH VLWXHD] GH RELFHL vQ ]RQD LQIHULRDU D GRPHQLXOXL FRJQLWLY 3HQWUX DFHDVW FDWHJRULH GH LWHPL QX HVWH QHFHVDU R VFKHP GH QRWDUH GHWDOLDW YH]L LWHPL GH WLS HVHX SXQFWDMXO FRUHVSXQ]WRU DFRUGkQGXVH VDX QX vQ IXQF LH GH PDUFDUHD UVSXQVXOXL FRUHFW OD LWHP ,WHPLL RELHFWLYL SRW IL IRORVL L SHQWUX RULFH GLVFLSOLQ – cu un grad de utilitate GLIHULW vQ IXQF LH GH VFRSXO WHVWXOXL RELHFWLYHOH úL FRQ LQXWXULOH PVXUDWH ± FHHDFHOHRIHUXQDYDQWDMGHRVHELWDVXSUDFHORUODO LLWHPL 5.1.1. Tehnica alegerii duale 7HKQLFDVH FDUDFWHUL]HD] SULQ VROLFLWDUHD HOHYLORU GH D DVRFLD XQXO VDX PDLPXOWHHQXQ XULFXXQDGLQFRPSRQHQWHOHXQRUFXSOXULGHDOWHUQDWLYHGXDOH FXP DU IL DGHYUDWIDOV FRUHFWJUHúLW GDQX DFRUGGH]DFRUG HQXQ IDFWXDO HQXQ GHRSLQLHHWF 8WLOL]DUHDWHKQLFLLGHWHVWDUHFXUVSXQVGXDO D &XQRDúWHUHDXQRUQR LXQLUHJXOLGHFDOFXO Exemplu (i) : &LWHúWH FX DWHQ LH DILUPD LLOH GH PDL MRV ÌQ FD]XO vQ FDUH DSUHFLH]L F DILUPD LD HVWH DGHYUDW vQFHUFXLHúWH OLWHUD $ ÌQ FD] FRQWUDU vQFHUFXLHúWH

12


OLWHUD ) úL vQORFXLHúWH vQ VSD LXO OLEHU FXYLQWHOH VXEOLQLDWH FX DFHOHD FDUH IDF DILUPD LDDGHYUDW A. F. ( )UDF LLOH ]HFLPDOH SHULRGLFH UHSUH]LQW QXPHUH LUD LRQDOH. A. F. ( ) 2. Triunghiul isoscel are toate bisectoarele congruente. A. F. ( ) 3. Tetraedrul regulat este o SLUDPLG WULXQJKLXODU UHJXODW 5VSXQVUD LRQDOHHFKLODWHUDO$ Exemplu (ii): &LWHúWH FX DWHQ LH DILUPD LLOH XUPWRDUH ÌQ FD]XO vQ FDUH DSUHFLH]L F DILUPD LD HVWH DGHYUDW vQFHUFXLHúWH OLWHUD $ ÌQ FD] FRQWUDU vQFHUFXLHúWH litera F. $ ) 6HJPHQWXO GHWHUPLQDW GH RULFDUH GRX YkUIXUL QHDOWXUDWH DOH XQXLSROLJRQFRQYH[VHQXPHúWHGLDJRQDO $)2ULFHSULVPSDWUXODWHUDUHED]DSWUDW A F 3. a n –1=(a–1)(a n–1 +a n–2 +...+a+1), (∀)n∈N * úLD∈R. 5VSXQV$)$ Exemplu (iii): &LWHúWH FX DWHQ LH ILHFDUH GLQWUH vQWUHEULOH GH PDL MRV 'DF DSUHFLH]L FUVSXQVXOFRUHFWHVWH'$vQFHUFXLHúWHOLWHUD'ÌQFD]FRQWUDUvQFHUFXLHúWH litera N. D N 1. Este 51% din 45 mai mare decât 20? D N 2. Este 50% din 6/4 egal cu 3/2? D 1'DFGLQWUXQQXPUHVWHHVWHQXPUXOPDLPLFGHFkW" 5VSXQV'11 E &DSDFLWDWHDGHDLGHQWLILFDUHOD LDGHWLSFDX]HIHFW Exemplu: )LHFDUH GLQ HQXQ XULOH GH PDL MRV HVWH FRPSXV GLQ GRX SURSR]L LL ILHFDUHDGHYUDW6DUFLQDWDHVWHVDSUHFLH]LGDFSURSR]L LDDGRXDH[SOLF GH FH HVWH DGHYUDW SULPD ÌQ FD]XO DFHVWD vQFHUFXLHúWH '$ ÌQ FD] FRQWUDU vQFHUFXLHúWH18


13

'$ 18 3DUDQWH]HOH VXQW IRDUWH LPSRUWDQWH SHQWUX XQ H[HUFL LX GHRDUHFHVWDELOHVFRUGLQHDHIHFWXULLRSHUD LLORU '$ 18 3WUDWXO HVWH SROLJRQ UHJXODW SHQWUX F DUH ODWXULOH FRQ JUXHQWHúLXQJKLXULOHFRQJUXHQWH '$ 18 5HFLSURFD WHRUHPHL OXL 3LWDJRUD QX WUHEXLH GHPRQVWUDW SHQWUXFHDVHDSOLFIRDUWHUDU DA NU 4. ƒ(x)=2x 2 –x+7>0, (∀)x∈R SHQWUX F RUGRQDWD YkUIXOXL SD UDEROHLHVWHSR]LWLY 5VSXQV1X'D1X ; 4. Nu. $YDQWDMHúLOLPLWHDOHXWLOL]ULLWHKQLFLLDOHJHULLGXDOH Principalul avantaj legat de utilizarea acestei tehnici este acela al DERUGULL vQWUXQ LQWHUYDO GH WLPS UHGXV D XQXL YROXP PDUH GH UH]XOWDWH DOH vQY ULL'HRELFHLFRPSOH[LWDWHDDFHVWRULWHPLHVWHUHGXVVDXPHGLH Una dintre cele mai întemeiate critici aduse acestei tehnici este aceea FLGHQWLILFDUHDXQXLHQXQ FDILLQGLQFRUHFWQHDGHYUDWHWFQXLPSOLFvQPRG QHFHVDUFXQRDúWHUHDGHFWUHVXELHFWDDOWHUQDWLYHLDGHYUDWH

5HFRPDQGULSHQWUXFRQVWUXLUHDLWHPLORUXWLOL]kQGWHKQLFDDOHJHULLGXDOH a) 9RU IL HYLWDWH HQXQXULOH FX FDUDFWHU IRDUWH JHQHUDO DWXQFL FkQG VH VROLFLWDSUHFLHUHDORUGUHSWDGHYUDWHVDXIDOVH. Exemplu: $ ) 5H]XOWDWXO vQPXO LULL D RULFURU GRX QXPHUH HVWH PDL PDUH decât oricare dintre factori. Â úLÂ b) 9RUILHYLWDWHHQXQXULQHUHOHYDQWHGLQSXQFWGHYHGHUHPDWHPDWLF Exemplu: $ ) 3ULPD FDUWH GH PDWHPDWLF D IRVW WLSULW vQ OLPED URPkQ vQ anul 1837. c) 9RUILHYLWDWHHQXQXULDFURUVWUXFWXUSRDWHJHQHUDDPELJXLWLVDX GLILFXOWLGHvQHOHJHUH.


14

Exemple : 1. $ ) 8Q QXPU QDWXUDO FDUH QX VH SRDWH VFULH VXE IRUPD N HVWH QXPULPSDUN∈N). Reformulare: $)8QQXPUFDUHVHVFULHVXEIRUPDNHVWHQXPULPSDUN∈N). 2. A F Cel mai mDUHQXPUQDWXUDOIRUPDWGLQSDWUXFLIUHGLIHULWHHVWH Reformulare: $ ) &HO PDL PDUH QXPU QDWXUDO IRUPDW GLQ SDWUX FLIUH RULFDUH GRX diferite este 9876. d) 9RUILHYLWDWHHQXQXULOHOXQJLFRPSOH[HFXDPQXQWHGDWHLQXWLOH. e) 9D IL HYLWDW LQWURGXFHUHD D GRX VDX D PDL PXOWRU LGHL vQWUXQ HQXQ (FXH[FHSLDVLWXDLLORUvQFDUHVHXUPUHúWHFXQRDúWHUHDVDXvQHOHJHUHDXQRU UHODLLFDX]HIHFW).

Exemple de itemi 1. 'LVFLSOLQD &ODVD &DSLWROXO $ULWPHWLF &ODVD D 9D 2SHUD LL FX numere naturale 2ELHFWLYXO (OHYXO YD IL FDSDELO V DSOLFH FRUHFW RUGLQHD HIHFWXULL RSHUD LLORU (QXQ ÌQ FD]XO vQ FDUH DSUHFLH]L F HJDOLWDWHD HVWH DGHYUDW vQFHUFXLHúWHOLWHUD$ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD)úLVFULHvQVSD LXOOLEHU VXEOLQLDWUH]XOWDWXOFDUHIDFHDILUPD LDDGHYUDW A. F. ______ 1. (5 2 –2 4 ) 2 =81. A. F. ______ 2. 64:8:2 3 =64. 5VSXQV$)5H]XOWDWXOHVWH 2. 'LVFLSOLQD&ODVD&DSLWROXO$ULWPHWLF&ODVDD9D0XO LPL 2ELHFWLYXO (OHYXO YD IL FDSDELO V UHFXQRDVF GDF XQ HOHPHQW GDW DSDU LQHVDXQXXQHLPXO LPL


15

(QXQ 3ULYL L GLDJUDPHOH GH PDL MRV 'DF UVSXQVXO IRUPXODW HVWH FRUHFW vQFHUFXLHúWH $ GDF HVWH IDOV vQFHUFXLHúWH ) 'DF D L vQFHUFXLW ) VFULH LvQVSD LXOOLEHUVXEOLQLDWUH]XOWDWXOFRUHFW A. F. ____________ 1. A={1;2;5;7;8} A. F. ____________ 2. B={2;3;4;5} A. F. ____________ 3. C={5;4;6;7} A. F. ____________ 4. B∩C={2;3;4;5;6;7} A. F. ____________ 5. A–C={1;7;8} A. F. ____________ 6. (A∪B)\C={6} A. F. ____________ 7. A∩B∩C={5} A. F. ____________ 8. (C–A)∩B={4}. 5VSXQV$GHYUDWH 3. 'LVFLSOLQD&ODVD&DSLWROXO$ULWPHWLF&ODVDD9D'LYL]LELOLWDWH 2ELHFWLYXO(OHYXOYDILFDSDELOVXWLOL]H]HFRUHFWFXDQWLILFDWRUXO universal “oricare”. 34n − 1 , n∈N'DFDSUHFLH]LFSURSR]L LDHVWH (QXQ )LHQXPUXO3 = 2 DGHYUDWvQFHUFXLHúWHOLWHUD$ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD) A F 1. (∀) n∈N, P∈N A F 2. (∀) n∈NFLIUDXQLW LORUOXL3HVWH A F 3. (∀) n∈NFLIUD]HFLORUOXL3HVWHSDU A F 4. (∀) n∈NQXPUXO3HVWHGLYL]LELOFX 5VSXQV$GHYUDWHúL 4. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,D'LYL]LELOLWDWH 2ELHFWLYXO(OHYXOYDILFDSDELOVUHFXQRDVFQXPHUHSULPH (QXQ 'DFDSUHFLH]LFDILUPD LDHVWHDGHYUDWvQFHUFXLHúWHOLWHUD$ ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD) 1) HVWHQXPUSULP A F 2) HVWHQXPUSULP A F 3) HVWHQXPUSULP A F 4) 1,2,3,5,7 sunt toate numere prime A F 5) 2,3,5,7,11 sunt toate numere prime A F 6) 3,5,7,9,11 sunt toate numere prime A F 5VSXQV$GHYUDW

16


5. 'LVFLSOLQD&ODVD&DSLWROXO$ULWPHWLF&ODVDD9,D3URFHQWH 2ELHFWLYXO (OHYXO YD IL FDSDELO V DIOH S GLQWUXQ QXPU UD LRQDO pozitiv. (QXQ 'DF DSUHFLH]L F UH]XOWDWXO HVWH DGHYUDW vQFHUFXLHúWH OLWHUD $ ÌQ FD] FRQWUDU vQFHUFXLHúWH OLWHUD ) 'DF DL vQFHUFXLW ) VFULH vQ VSD LXO subliniat rezultatul corect. A. F. ______ 1. 25% din 250=62,5. $ ) BBBBBB 'DF R IDPLOLH GHSXQH OD EDQF 521 FX R GREkQG IL[ DQXDO GH SULPHúWH GXS XQ DQ suma de 10 miliane de lei. 5VSXQV$)5VSXQVFRUHFW521 6. 'LVFLSOLQD&ODVD&DSLWROXO*HRPHWULH&ODVDD9,D6XPDPVXULORU unghiurilor unui triunghi. 2ELHFWLYXO (OHYXO YD IL FDSDELO V DSOLFH WHRUHPD UHIHULWRDUH OD VXPD PVXULORUXQJKLXULORUXQXLWULXQJKL (QXQ 'DF DSUHFLH]L F UH]XOWDWXO HVWH DGHYUDW vQFHUFXLHúWH OLWHUD $ ÌQ FD] FRQWUDU vQFHUFXLHúWH OLWHUD ) 'DF DL vQFHUFXLW ) VFULH vQ VSD LXO subliniat rezultatul corect. $)BBBBBB'DFWULXQJKLXO$%&GUHSWXQJKLFvQ$DUHXQJKLXO% GHPVXU o DWXQFLXQJKLXO&PVRDU o . $ ) BBBBBB 'DF vQ WULXQJKLXO $%& [$( úL [BF sunt bisectoare interioare, m(∠BAE)=30 o úLP∠ABF)=20 o , atunci m(∠ACB)=90 o . 5VSXQV$)5VSXQVFRUHFWP∠ACB)=80 o 7. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,D0XO LPL 2ELHFWLYXO(OHYXOYDILFDSDELOVUHFXQRDVFUHOD LLvQWUHPXO LPL (QXQ 'DFDSUHFLH]LFDILUPD LDHVWHDGHYUDWvQFHUFXLHúWHOLWHUD$ ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD) 'RXPXO LPL$úL%VXQWHJDOHGDFDXDFHODúL QXPUGHHOHPHQWH A F 'RXPXO LPL$úL%VXQWHJDOHGDFDXDFHOHDúL elemente. A F 3) {0}⊂N A F


17

4) 0∈N A F 5) {1,2,3}⊆{1,2,3} A F 6) {0}=∅ A F 7) {1,2}⊂{1,5,7} A F 8) {{1}}={1} A F 5VSXQV$GHYUDWH 8. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,,D5GFLQDSWUDW 2ELHFWLYXO(OHYXOYDILFDSDELOVDSOLFHWHRULDUHIHULWRDUHODQR LXQHDGH SWUDWSHUIHFW (QXQ 'DF DSUHFLH]L F QXPUXO GDW HVWH SWUDW SHUIHFW vQFHUFXLHúWH OLWHUD$ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD) A F 1. n= 81 ⋅ 8 ⋅ 1250 A F 2. a= 5n + 2 , n∈N A F 3. b=5 2n+1 ⋅10⋅2 4n+1 , n∈N A F 4. m=2 1997 . 5VSXQV$GHYUDWH 9. Disciplina / Clasa / Capitolul: Geometrie / Clasa a VII-a / Patrulatere 2ELHFWLYXO (OHYXO YD IL FDSDELO V FXQRDVF SURSULHW LOH SDWUXODWHUHORU studiate. (QXQ &LWHúWH FX DWHQ LH DILUPD LLOH GH PDL MRV ÌQ FD]XO vQ FDUH DSUHFLH]L F DILUPD LD HVWH DGHYUDW vQFHUFXLHúWH OLWHUD $ vQ FD] FRQWUDU vQFHUFXLHúWH) 1. Trapezul este un patrulater convex. A F 2. Rombul este un poligon regulat. A F 3WUDWXOHVWHXQURPE A F 3WUDWXOHVWHXQWUDSH] A F 'UHSWXQJKLXOHVWHXQSWUDW A F 6. Dreptunghiul este un paralelogram. A F 5VSXQV$GHYUDWH 10. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,,D1XPHUHUHDOH 2ELHFWLYXO(OHYXOYDILFDSDELOVFRPSDUHGRXQXPHUHUHDOH (QXQ 'DF DSUHFLH]L F SURSR]L LD GLQ GUHDSWD QXPUXOXL GH RUGLQH HVWHDGHYUDWvQFHUFXLHúWHOLWHUD$ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD)


18

1. a=b unde a= 4 − 3 úLE= 9 − 2

A

F

2. x0, ∀n≥ úL OLP a n DWXQFL H[LVW Q 0 ∈N* astfel încât

A

F

A

F

Q

∀m>n≥n 0 avem : a m ≤a n . D Q 'DFD n >0, ∀n≥úLOLP Q =A0, y n ! úL OLP ( x n +y n )=+∞, atunci Q→ ∞

OLP (x n · y n )=+∞.

Q→ ∞

A F

'DF úLUXULOH UHDOH

( x n ) n≥1 , ( y n ) n≥1 , (z n ) n≥1

YHULILF UHOD LD

OLP ( x n · y n ·zn )=+∞, atunci OLP (x n + y n +zn )=+∞.

Q→ ∞

Q→ ∞

A F

'DF OLP a n =+∞, atunci ∃n 0 ∈N* astfel încât a n ≤a n+1 , ∀n≥n 0 .

A F

'DF OLP b n =–∞, atunci ∃n 1 ∈N* astfel încât b n+1 ≤b n , ∀n≥n 1 .

Q→ ∞ Q→ ∞

*KLG GH HYDOXDUH OD 0DWHPDWLF A F

A F A F

25

8. DDF (a n ) n ≥1 , (b n ) n ≥1 VXQW GRX úLUXUL UHDOH (b n ) n ≥1 VWULFW FUHVFWRU úL D D −D QHPUJLQLWúL∃ OLP Q = l ∈ #, atunci OLP Q Q = l. Q→ ∞ E Q→ ∞ E Q Q − EQ 9. 2ULFHSXQFWGHGLVFRQWLQXLWDWHDOXQHLIXQF LLPRQRWRQHHVWHGHVSH D,  1 sin , x ≠ 0 QX HVWH PRQRWRQ SH QLFL R )XQF LD ƒ:R→R, f ( x ) =  x  0, x = 0

YHFLQWDWHDRULJLQLL A F 11. 'HWHUPLQD L XQ LQWHUYDO ,⊂R SH FDUH IXQF LD ƒ de la punctul 10 este PRQRWRQ 5VSXQV )DOV&RQWUDH[HPSOXx n =–2, y n =1/n, ∀n≥1. $GHYUDW)LH9∈U(+∞) ⇒ ∃ε>0, (ε,+∞)⊂V, ∃n 1 ∈N*, ∀n≥n 1 , x n >ε. Fie n 2 =max{n 0 ,n 1 `úLQ≥n 2 ⇒ y n ≥x n >ε deci OLP y n =+∞. Q→ ∞

$SOLFPLQHJDOLWDWHDPHGLLORUúLH[HUFL LXOx n +y n ≥2 [Q Â \ Q →+∞, deci x n +y n →+∞$ILUPD LDHVWHDGHYUDW )DOV&RQWUDH[HPSOXx n QúLy n =1/n, ∀n≥1. $GHYUDW$SOLFPLQHJDOLWDWHDPHGLLORU x n +y n +zn ≥3 [Q Â \ Q Â ]Q deci (x n +y n +zn )→+∞. )DOV&RQWUDH[HPSOX± 2 , 10 2 –1, ..., 10 n , 10 n –1, ... )DOV&RQWUDH[HPSOX±±± ± 2 , –(10 2 –1), ..., –10 n , –(10 n – 1), ... )DOV&RQWUDH[HPSOXD n =(–1) n úLE n =n, ∀n≥1. $GHYUDW 10. x 0 HVWH SXQFWXO GH GLVFRQWLQXLWDWH GH VSH D D ,,D GHFL ƒ nu este PRQRWRQSHQLFLRYHFLQWDWHDRULJLQLL  2 1 11. I=  ,   5π 2π  27. 'LVFLSOLQD&ODVD&DSLWROXO$QDOL]PDWHPDWLF&ODVDD;,,D Integrale definite. 2ELHFWLYXO(OHYXOYDILFDSDELOVFDOFXOH]HLQWHJUDOHIRORVLQGVXEVWLWX LL trigonometrice. (QXQ 'DFDSUHFLH]LFDILUPD LDHVWHDGHYUDWvQFHUFXLHúWHOLWHUD$ ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD)


26

(3 − x ) 7 ∫0 (3 + x) 9 dx este: 3

Valoarea integralei A F

1.

A F

2.

1 16 1 48

5H]ROYDUH)DFHPVXEVWLWX LD[ FRVW ϕ(t), ϕ:[0, π/2]→[0,3] este GHULYDELOFXGHULYDWDFRQWLQX 1 ϕ / (t)=-3sint, ϕ(0)=3, ϕ(π/2)=0 deci I= ⋅ 3

π /2

∫ 0

π

1 1 t t t 2 tg 15 (tg ) / dt = ⋅ tg 16 = 2 2 48 2 0 48

5VSXQV$GHYUDW 28. 'LVFLSOLQD&ODVD&DSLWROXO$QDOL]PDWHPDWLF&ODVDD;,,D Integrale definite. 2ELHFWLYXO(OHYXOYDILFDSDELOVGHULYH]HRIXQF LHGHILQLWSULQWUR LQWHJUDO (QXQ 'DFDSUHFLH]LFDILUPD LDHVWHDGHYUDWvQFHUFXLHúWHOLWHUD$ ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD) x2 sin t 6HFRQVLGHUIXQF LDI 1, π →R, f(x)= ∫ dt , atunci: t 1

[

]

$)IHVWHVWULFWPRQRWRQ $)IHVWHQHPRQRWRQ sin t 5H]ROYDUH)XQF LD g (t ) = HVWHFRQWLQXSHLQWHUYDOXO 1, π t

[

] deci

HVWHLQWHJUDELOúLDGPLWHSULPLWLYSH*$SOLFkQGWHRUHPDOXL/HLEQL]1HZWRQ RE LQHP f(x)=G(x2 * FDUHHVWHGHULYDELO sin x 2 2 sin x 2 / / 2 f (x)=2xG (x )=2x 2 = >0, ∀x∈[1, π ) x x 5VSXQV$GHYUDW 29. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD;,,D*UXSXUL 2ELHFWLYXO(OHYXOYDILFDSDELOVUHFXQRDVFFRPXWDWLYLWDWHDJUXSXOXL simetric. (QXQ 'DFDSUHFLH]LFDILUPD LDHVWHDGHYUDWvQFHUFXLHúWHOLWHUD$ ÌQFD]FD]FRQWUDUvQFHUFXLHúWHOLWHUD)


27

Fie (S n ,° JUXSXOFRPXWDWLYDOSHUPXWULORUGHJUDGQDWXQFL A F 1. n≥3 A F 2. n≤2 5VSXQV$GHYUDW 30. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD;,,D,QHOHúLFRUSXUL 2ELHFWLYXO(OHYXOYDILFDSDELOVGHWHUPLQHGLYL]RULLSURSULLDLXQXL SROLQRPGDWFXFRHILFLHQ LGLQWUXQFRUSILQLW (QXQ 'DFDSUHFLH]LFDILUPD LDHVWHDGHYUDWvQFHUFXLHúWHOLWHUD$ ÌQFD]FRQWUDUvQFHUFXLHúWHOLWHUD) Polinomul f = X + ∈ Z [X ] este. A F 1. reductibil A F 2. ireductibil. 5VSXQV$GHYUDW 5.1.2. Tehnica perechilor 'HVFULHUHúLXWLOL]DUH [,1HDFúX$6WRLFD] 7HKQLFD SHUHFKLORU VROLFLW GLQ SDUWHD HOHYLORU VWDELOLUHD XQRU FRUHVSRQGHQ HvQWUHFXYLQWHSURSR]L LLIUD]HQXPHUHOLWHUHVDXDOWHFDWHJRULL GHVLPEROXULGLVWULEXLWHSHGRXFRORDQH (OHPHQWHOH GLQ SULPD FRORDQ GHQXPLWH SUHPLVH FRQVWLWXLH HQXQ XO LWHPXOXL LDU FHOH GLQ FRORDQD D GRXD UHSUH]LQW UVSXQVXULOH &ULWHULXO VDX FULWHULLOH SH ED]D FURUD VH VWDELOHúWH UVSXQVXO FRUHFW VXQW HQXQ DWH H[SOLFLWDWHvQLQVWUXF LXQLOHFDUHSUHFHGFRORDQHOHGHSUHPLVHúLGHUVSXQVXUL 8WLOL]DUHD VH OLPLWHD] OD PVXUDUHD DELOLW LL GH D LGHQWLILFD UHOD LD H[LVWHQWvQWUHGRXFDWHJRULL WHUPHQL±GHILQL LL reguli – exemple simboluri – concepte PHWRGH±H[HPSOLILFUL &HULQ HGHSURLHFWDUH D VLQFOXGXQQXPULQHJDOGHUVSXQVXULúLGHSUHPLVHLDUHOHYLLV ILH LQVWUXL L F ILHFDUH UVSXQV SRDWH IL IRORVLW R GDW GH PDL PXOWH RUL VDX QLFLRGDW


28

E OLVWD UVSXQVXULORU V ILH DQJDMDW vQWUR RUGLQH ORJLF GH H[HPSOX RUGLQHDDOIDEHWLFSHQWUXUVSXQVXULFDUHSUHVXSXQH[SULPDUHDvQFXYLQWHVDX RUGLQHD FUHVFWRDUH GHVFUHVFWRDUH SHQWUX UVSXQVXUL QXPHULFH $FHDVW FHULQ YL]HD] HOLPLQDUHD IXUQL]ULL RULFURU LQGLFLL FDUH DU SXWHD FRQGXFH HOHYXOVSUHÄJKLFLUHD³UVSXQVXOXLFRUHFW F WRDWHSUHPLVHOHúLUVSXQVXULOHXQXLLWHPGHDFHODúLWLSVILHSODVDWH SHDFHHDúLSDJLQ

$YDQWDMHúLOLPLWH Tehnica permite abordarea unui foarte important volum de rezultate de vQY DUH vQWUXQ LQWHUYDO GH WLPS UHGXV FX XWLOL]DUHD HILFLHQW D VSD LXOXL SH IRLOH GH WHVW FkW úL FX XWLOL]DUHD HILFLHQW D WLPSXOXL SURIHVRUXOXL OD QRWDUH 8úXULQ DFRQVWUXF LHLLWHPLORUHVWHGHDVHPHQHDXQDYDQWDMIUHFYHQWPHQ LRQDW vQ OHJWXU FX DFHDVW WHKQLF FX WRDWH F HVWH SUREDELO PDL FRUHFW V VSXQHPFHVWHPDLXúRUGHFRQVWUXLWXQLWHPGHFDOLWDWHVODEGHFkWXQXOGH EXQFDOLWDWH7HKQLFDQXSRDWHILXWLOL]DWSHQWUXDERUGDUHDXQRUUH]XOWDWHGH vQY DUH FRPSOH[H ILLQG GH DVHPHQHD GLILFLO vQ XQHOH FD]XUL V FRQVWUXLP OLVWHGHSUHPLVHVDXGHUVSXQVXULFDUHVILHRPRJHQH Exemple de itemi 1. 'LVFLSOLQD&ODVD&DSLWROXO$ULWPHWLF&ODVDD9D0XO LPL 2ELHFWLYXO (OHYXO YD IL FDSDELO V UHFXQRDVF GHILQL LD RSHUD LLORU FX PXO LPL (QXQ ÌQ SWUDWXO GLQ VWkQJD ILHFUXL UkQG VFULH L FLIUD FRUHVSXQ]WRDUH PXO LPLLUHVSHFWLYH A∪B

1

{xx∈$úLx∈B}

A∩B

2

{xx∉A sau x∉B}

A–B

3

{xx∈A sau x∈B}

4

{xx∉$úLx∈B}

5

{xx∈$úLx∉B}

5VSXQV$∪B→3; A∩B→1; A–B→5.


29

2. 'LVFLSOLQD&ODVD&DSLWROXO$ULWPHWLF&ODVDD9D'LYL]LELOLWDWH 2ELHFWLYXO (OHYXO YD IL FDSDELO V UHFXQRDVF GLYL]RULL XQXL QXPU natural. (QXQ 6FULH LvQSWUDWHOHOLEHUH GLQ GUHDSWD ILHFUXL QXPU GH WUHL FLIUH divizorul admis din coloana din stânga. În cazul în care nici unul dintre QXPHUHOHGLQVWkQJDQXHVWHGLYL]RUPDUFD LvQFVX VHPQXOî 2 121 3 122 5 123 125 127 5VSXQV→x; 122→2; 123→3; 125→5; 127→x. 3. 'LVFLSOLQD &DSLWROXO &ODVD $ULWPHWLF &ODVD D 9D 1XPHUH UD LRQDOH 2ELHFWLYXO (OHYXO YD IL FDSDELO V XWLOL]H]H VLPSOLILFDUHD VDX DPSOLILFDUHDIUDF LLORU (QXQ ÌQVFULH vQ VSD LXO GLQ ID D ILHFUXL QXPU GLQ FRORDQD $ OLWHUD GLQ FRORDQD%FDUHLQGLFXQQXPUHJDOFXQXPUXOGLQFRORDQD$ A B 720 1 ______ 1. M 480 3 2 462  2 ______ 2. N    5 1155 2541 7 ______ 3. P 7623 11 n n −1 7 ⋅ 11 21 ______ 4. n −1 Q n 14 7 ⋅ 11 1998 1997 1996 2 ⋅ 3 ⋅5 18 ______ 5. 1996 1997 1998 R 25 2 ⋅ 3 ⋅5 22 S 55 5VSXQV→Q; 2→S; 3→M; 4→P; 5→N.


30

4. Disciplina / Clasa / Capitolul: Geometrie / Clasa a VI-a / Triunghiul 2ELHFWLYXO(OHYXOYDILFDSDELO V LGHQWLILFH WLSXUL GH WULXQJKLXUL vQY DWH LQkQGFRQWúLGHFODVLILFDUHDWULXQJKLXULORUGXSODWXULGDUúLGXSXQJKLXUL (QXQ ÌQFRORDQD$DYH LGHVHQDWHFkWHYDWLSXULGHWULXQJKLXUL ÌQVFULH L vQ VSD LXO GLQ ID D ILHFUHL ILJXUL OLWHUD VDX OLWHUHOH GLQ FRORDQD % FDUH corespund tipului de triunghi desenat.

1 2 __

3

A B D±DVFX LWXQJKLF b – obtuzunghic c – dreptunghic d – isoscel e – echilateral f – scalen

4

5VSXQV→c,d; 2→b,d; 3→f; 4→c. 5. Disciplina / Clasa / Capitolul: Geometrie / Clasa a VI-a / Clasificarea triunghiurilor. 2ELHFWLYXO(OHYXOYDILFDSDELOVFODVLILFHWULXQJKLXULOHGXSGLIHULWHFULWHULL (QXQ ÌQ FRORDQD $ DYH L VSHFLILFDWH WLSXUL GH WULXQJKLXUL ÌQVFULH L vQ VSD LXO GLQ ID D ILHFUXL WLS GLQ FRORDQD $ OLWHUD GLQ FRORDQD % FDUH FRUHVSXQGHWLSXOXLGHWULXQJKLPHQ LRQDWvQFRORDQD$ A B BBB7ULXQJKLXOFXGRXODWXULFRQJUXHQWH D .∆ dr. isoscel ___ 2. Triunghiul cu un unghi drept E. ∆ scalen ___ 3. Triunghiul cu toate laturile congruente F. ∆ obtuzunghic BBB7ULXQJKLXOFXXQXQJKLGUHSWúLGRX G. ∆ echilateral laturi congruente H. ∆ isoscel ___ 5. Triunghiul cu un unghi obtuz I. ∆ dreptunghic 5VSXQV→H; 2→I; 3→G; 4→D; 5→F.


31

6. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,D3URSRU LL 2ELHFWLYXO (OHYXO YD IL FDSDELO V XWLOL]H]H SURSULHWDWHD IXQGDPHQWDO D SURSRU LHL (QXQ ÌQVFULH vQ VSD LXO GLQ VWkQJD QXPUXOXL GH RUGLQH GLQ SULPD FRORDQOLWHUDGLQDGRXDFRORDQFDUHFRUHVSXQGHUH]XOWDWXOXLFRUHFW I II 15 x 3 A _____ 1. = 2 5 13 3 4 _____ 2. = B 5 x 1 1 x 1 C _____ 3. = 2 8 2 1 2 2 = 2x D 2 _____ 4. 15 − 2 ⋅ 3 36 2 15 :(2 7 ⋅ 7 + 2 7 ) 32 _____ 5. = E 4 3 x +1 6 F 5 3 G 4 5VSXQV 1 F

2 G

3 E

4 B

5 D

7. Disciplina / Clasa / Capitolul: Geometrie / Clasa a VII-a / Patrulatere 2ELHFWLYXO (OHYXO YD IL FDSDELO V FXQRDVF SURSULHW LOH SDWUXODWHUHORU vQY DWH (QXQ 3HSULPDOLQLHDWDEHOXOXLGHPDLMRVVXQWGHVHQDWHFkWHYDSDWUXODWHUH LDU SH SULPD FRORDQ VXQW VFULVH SURSULHW L DOH DFHVWRUD 0DUFKHD] FX [ FVX HOH FRUHVSXQ]WRDUHSURSULHW LORUDGHYUDWHSHQWUXILHFDUHSDWUXODWHU Figura Trapez Paralelogram Dreptunghi Romb 3WUDW Proprietatea 'RXODWXULRSXVHVXQWSDUDOHOH /DWXULOHRSXVHVXQWGRXFkWH GRXSDUDOHOH


32

/DWXULOHRSXVHVXQWGRXFkWH GRXFRQJUXHQWH Toate laturile congruente Unghiurile opuse sunt congruente Unghiurile sunt drepte Diagonalele sunt congruente Diagonalele sunt perpendiculare 'LDJRQDOHOHVHvQMXPW HVF 8. 'LVFLSOLQD&ODVD&DSLWROXO0DWHPDWLF&ODVDD9,,D5HFDSLWXODUHILQDO 2ELHFWLYXO (OHYXO YD IL FDSDELO V UHFXQRDVF IRUPXOH VDX UHOD LL vQY DWH (QXQ ÌQVFULH vQ VSD LXO OLEHU GLQ VWkQJD QXPHUHORU GH RUGLQH DOH HQXQ XULORUFRUHVSXQ]WRDUHFRORDQHL0OLWHUDGLQFRORDQD1FDUHFRUHVSXQGH formulei corecte. M N % E × K

1. Aria triunghiului

A.

2. Perimetrul dreptunghiului 3. Aria trapezului

B. 2ab/(a+b) C. 2(L+A)

0HGLDDULWPHWLFDGRXQXPHUH 5. Media geometricDGRXQXPHUH 5VSXQV

D. E.

a⋅b

E×K

F. (a+b)/2 1 E

2 C

3 A

4 F

5 D

9. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,,D1XPHUHUHDOH 2ELHFWLYXO(OHYXOYDILFDSDELOVUH]ROYHH[HUFL LLFXQXPHUHUHDOH (QXQ ÌQVFULH vQ VSD LXO OLEHU GLQ VWkQJD QXPHUHORU GH RUGLQH DOH HQXQ XULORU FRUHVSXQ]WRDUH FRORDQHL 0 OLWHUD GLQ FRORDQD 1 FDUH FR UHVSXQGHQXPUXOXLFRUHFW. M N 1 2 -1 5 ____ 1. (- ):(- ) A 2 3 2 1 ____ 2. 75:( 48 − 2 27 ) B 3


33

( )

6 1 : 2 3 : 27 − 5 5 2 5   1 ____ 4.  + − ⋅ 2  8 18 32  ____ 3.

5VSXQV 1 B

2 A

3 D

C

0

D

−

2 15

E

−

1 12

4 E

10. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,,,D1XPHUHUHDOH 2ELHFWLYXO (OHYXO YD IL FDSDELO V UHFXQRDVF HOHPHQWHOH PXO LPLORU N, Z, Q, R, R–Q, R*. (QXQ ÌQVFULH vQ VSD LXO GLQ ID D ILHFUXL QXPU GLQ FRORDQD $ OLWHUD GLQ FRORDQD % FDUH FRUHVSXQGH PXO LPLL GLQ FDUH IDFH SDUWH QXPUXO PHQ LRQDW vQ coloana A: A B ___ 1. x=2,(3) D. x∈N* ___ 2. x=–5 E. x∈R–Q ___ 3. x=0 F. x∈Z–N G. x∉R ___ 4. x= − ___ 5. x=π ___ 6. x=–11

H. x∈Q–Z, x pozitiv I. x nul J. x∈Q–Z, x negativ.

5VSXQV 1 H

2 J

3 I

4 5 G E

6 F

11. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,,,D)XQF LDOLQLDU 2ELHFWLYXO (OHYXO YD IL FDSDELO V LGHQWLILFH SXQFWH FDUH DSDU LQ JUDILFXOXLXQHLIXQF LLGDWH (QXQ ÌQVFULH vQ VSD LXO GLQ ID D ILHFUXL QXPU GLQ FRORDQD $ OLWHUD GLQ FRORDQD%FDUHLQGLFSXQFWXOFHDSDU LQHJUDILFXOXLIXQF LHLGLQFRORDQD$ A B ____ 1. f:R→R, f(x)=x+1 M(2;0) ____ 2. g:R→R, g(x)=2x-1 N(-1;3)


34

____ 3. h:R→R, h(x)=-

1 x+1 2

____ 4. i:R→R, i(x)=7

P(0; -1) Q(2;3) R(1;7)

5VSXQV 1 Q

2 P

3 4 M R

12. Disciplina / Clasa / Capitolul: Geometrie / Clasa a VIII-a / Corpuri rotunde. 2ELHFWLYXO (OHYXO YD IL FDSDELO V LGHQWLILFH IRUPXOHOH SHQWUX DIODUHD ariei totale a unor corpuri geometrice. (QXQ ÌQVFULHvQVSD LXOGLQID DILHFUXLQXPUGLQFRORDQD$OLWHUDGLQ FRORDQD%FDUHLQGLFIRUPXODFRUHFW A B ____ 1. Cilindru circular drept M AWRWDO =4πR 2 ____ 2. Con circular drept N A ODWHUDO =πG(R+r) ____ 3. Trunchi de con circular drept P A ODWHUDO =2πRh ____ 4. Sfera Q A WRWDO =πh(R+r) R A ODWHUDO =πRG 5VSXQV 1 2 3 4 P R N M 13. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD,;D)XQF LL 2ELHFWLYXO(OHYXOYDILFDSDELOVLGHQWLILFHGRPHQLXOPD[LPGHGHILQL LH DOXQHLIXQF LLGDWH [ − [ + , unde I⊂R. (QXQ )LHIXQF LDƒ:I→RGHILQLWSULQƒ(x)= [− ÌQVFULH vQ VSD LXO GLQ VWkQJD QXPUXOXL GH RUGLQH GLQ SULPD FRORDQ OLWHUDGLQDGRXDFRORDQFDUHFRUHVSXQGHPXO LPLLFRUHFWH I II BBB'RPHQLXOPD[LPGHGHILQL LH A. {-4,-1} BBB0XO LPHDQXPHUHORUx∈Z astfel încât B. {0,1,3,4,5,8} ƒ(x)∈Z C. R\{2}


35

BBB0XO LPHDQXPHUHORUx∈N astfel încât ƒ(x)∈N BBB0XO LPHDQXPHUHORUx∈N astfel încât ƒ(x)∈Z BBB0XO LPHDQXPHUHORUx∈Z-N astfel încât ƒ(x)∈Z

D. {4,5,8} E. x∈{–4,–1,0,1,3,4,5,8} F. {2;3} G. {3,4,5,8}

5VSXQV& ; 2.E ; 3.G ; 4.B ; 5.A. 14. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD,;D)XQF LL 2ELHFWLYXO(OHYXOYDILFDSDELOVDSOLFHSURSULHW LOHIXQF LLORUvQY DWH (QXQ 3H SULPD OLQLH D WDEHOXOXL GH PDL MRV VXQW HQXPHUDWH IXQF LL LDU SH SULPD FRORDQ VXQW VFULVH SURSULHW L DOH DFHVWRUD 0DUFKHD] FX î FVX HOHFRUHVSXQ]WRDUHSURSULHW LORUDGHYUDWHSHQWUXILHFDUHIXQF LH Proprietatea

)XQF LD ƒ:R→R ƒ:R→R ƒ:(0,∞)→R ƒ:(–∞,0)→R ƒ:[–2,5]→R ƒ(x)=5 ƒ(x)=ax, ƒ(x)=ax+b, ƒ(x)=ax+b, ƒ(x)=|x| a>0

a0, b≠0

)XQF LDHVWHFUHVFWRDUH )XQF LDHVWHVWULFWFUHVFWRDUH )XQF LDHVWHGHVFUHVFWRDUH )XQF LDHVWHVWULFWGHVFUHV FWRDUH )XQF LDHVWHPRQRWRQ )XQF LDHVWHLQMHFWLY )XQF LDHVWHVXUMHFWLY )XQF LDHVWHELMHFWLY )XQF LDHVWHSDU )XQF LDHVWHLPSDU )XQF LDQXHVWHQLFLSDU QLFLLPSDU 15. 'LVFLSOLQD &ODVD &DSLWROXO $OJHEU &ODVD D ;,,D 3ROLQRDPH FX FRHILFLHQ LFRPSOHFúL 2ELHFWLYXO (OHYXO YD IL FDSDELO V VWDELOHDVF SROLQRDPHOH LUHGXFWLELOH GLQWUROLVWGDWGHSROLQRDPHXWLOL]kQGSULQFLSLLFXQRVFXWHGHLUHGXFWLELOLWDWH

36


(QXQ ÌQVFULH vQ VSD LXO GLQ VWkQJD ILHFUHL SURSULHW L SROLQRPXO GLQ D GRXDFRORDQFHFRUHVSXQGHDFHVWXLD ______ 1. polinomul este ireductibil în Z[X] f=X 2 -2 ______ 2. polinomul este ireductibil în Q[X] g=X 2 +X+1 ______ 3. polinomul este ireductibil în R[X] h=X 3 +X ______ 4. polinomul este ireductibil în C[X] p=X 4 +X 3 +X 2 +X+1 q=2X+3 5VSXQVIJST 2. f, g, p, q. 3. g, q. 4. q. Singurele polinoame ireductibile din C[X] sunt cele de gradul întâi. Polinoamele ireductibile din R[X] sunt cele de forma mX+n, m∈R ∗ , n∈R úLD; 2 +bX+c, a∈R ∗ , b,c∈5úLE 2 -4ac1; B. a∈(1,2); C. a>2. 5VSXQV$GHYUDWHVXQWDILUPD LLOH$úL% 50  1 1−    2 1 1 1 1 HFXD LDGDWQXDUHVROX LL π π b) Â VLQ [ + Â FRV [ = ⇔ sinx·cos +sin ·cosx=1 ⇔ π π π  π   ⇔ sin  x +  =1 ⇔ x+ ∈  + 2 kπ k ∈ Z  ⇔ x ∈  + 2kπ k ∈ Z  sau  4 2  4  D (FXD LDVLQx+cosx=

∈Im f . 6. 'LVFLSOLQD&ODVD&DSLWROXO0DWHPDWLF&ODVDD,;D)XQF LL 2ELHFWLYXO(OHYXOVúWLHFRQGL LLOHGHLQYHUVDELOLWDWHDXQHLIXQF LLUHDOH GHYDULDELOUHDO (QXQ &RPSOHWHD] VSD LLOH SXQFWDWH DVWIHO vQFkW V RE LL R DILUPD LH DGHYUDW 2IXQF LHƒ:R→RHVWHLQYHUVDELOGDFúLQXPDLGDFHDHVWH Rezolvare: ƒLQYHUVDELO⇔ ƒELMHFWLY 7. 'LVFLSOLQD &ODVD &DSLWROXO 0DWHPDWLF &ODVD D ;D )XQF LD H[SRQHQ LDOúLIXQF LDORJDULWPLF 2ELHFWLYXO (OHYXO YD IL FDSDELO V GHILQHDVF IXQF LD PRQRWRQ úL V R DSOLFHvQUH]ROYDUHDXQRUHFXD LLH[SRQHQ LDOH (QXQ &RPSOHWHD] VSD LLOH SXQFWDWH DVWIHO vQFkW V RE LL DILUPD LL DGHYUDWH


61

2IXQF LHƒ:R→RHVWHVWULFWFUHVFWRDUHGDF∀x 1 ,x 2 ∈R cu x 1 ;@Â VH FRQVLGHU SROLQRDPHOH ~ ƒ ; úL g=X 3 'HWHUPLQD L IXQF LLOH SROLQRPLDOH f úL g~ asociate polinoamelor ƒ ~ úLg5HPDUFD LFƒúLg sunt .............. iar f úL g~ sunt .............. ~ ~ ~ Rezolvare: f :Z 2 →Z 2 ; g~ :Z 2 →Z 2 . f ( 0 )= 0 = g~ ( 0 ); f ( 1 )= 1 = g~ ( 1 ), deci ƒ ~ úLg sunt distincte iar f úL g~ sunt egale. 5.2.2ÌQWUHEULVWUXFWXUDWH Descriere [Stoica A. (coordonator), 1996]: 2vQWUHEDUHVWUXFWXUDWHVWHIRUPDWGLQPDLPXOWH VXEvQWUHEUL ± GH WLS RELHFWLY VDX VHPLRELHFWLY ± OHJDWH vQWUH HOH SULQWUXQ HOHPHQW FRPXQ ([LVW XQ VSD LX JRO vQWUH WHKQLFLOH GH HYDOXDUH FX UVSXQV OLEHU úL FHOH FX UVSXQV limitat impuse de itemii obiectivi. Acest gol poate fi acoperit prin utilizarea vQWUHEULORU VWUXFWXUDWH 6FKHPDWLF PRGXO GH SUH]HQWDUH D XQHL vQWUHEUL VWUXFWXUDWHDUDWDVWIHO

Material / Stimul (texte, date, diagrame, grafice etc.)  6XEvQWUHEUL  Date suplimentare  6XEvQWUHEUL

&HULQ HGHSURLHFWDUH : ±vQWUHEDUHDWUHEXLHVFHDUUVSXQVXULVLPSOHODvQFHSXWúLVFUHDVF GLILFXOWDWHD DFHVWRUD VSUH VIkUúLW *UDGXO GH GLILFXOWDWH SRDWH IL vQ JHQHUDO asociat cu lungimea itemului; ± ILHFDUH VXEvQWUHEDUH QX YD GHSLQGH GH UVSXQVXO FRUHFW OD VXEvQWUHEDUHDSUHFHGHQW ±VXEvQWUHEULOHWUHEXLHVILHvQFRQFRUGDQ FXPDWHULDOHOHVWLPXOLL ±ILHFDUHVXEvQWUHEDUHWHVWHD]XQXOVDXPDLPXOWHRELHFWLYH


65

± XQ VSD LX YD IL OVDW SH IRDLD SH FDUH HVWH VFULV vQWUHEDUHD FRUHVSXQ]WRUOXQJLPLLILHFUXLUVSXQV $YDQWDMHúLOLPLWH Avantaje: ÌQWUHEULOHVWUXFWXUDWHSHUPLW ± WUDQVIRUPDUHD XQXL LWHP FRPSOH[ vQWUR VXLW GH LWHPL RELHFWLYL sau semiobiectivi; ± VWUXFWXUDUHD VXEvQWUHEULORU VH IDFH DVWIHO vQFkW V WHVWH]H R YDULHWDWHGHFXQRúWLQ HSULFHSHULúLFDSDFLW L – construirea proJUHVLYDXQHLGLILFXOW LúLDXQHLFRPSOH[LW LGRULWH ±FHUHUHDXQXLQXPUGHVXEvQWUHEULOHJDWHSULQWURWHPFRPXQ – utilizarea unor materiale auxiliare (grafice, diagrame etc.). Limite: – materialele auxiliare sunt relativ dificil de proiectat; ± UVSXQVXO OD R VXEvQWUHEDUH GHSLQGH XQHRUL GH UVSXQVXO OD VXEvQWUHEULOHSUHFHGHQWH ±SURLHFWDUHDXQHLvQWUHEULVWUXFWXUDWHQHFHVLWPDLPXOWWLPS

Exemple de itemi 1. 'LVFLSOLQD&ODVD&DSLWROXO$ULWPHWLF&ODVDD9D)UDF LL 2ELHFWLYXO(OHYXOVILHFDSDELOVVFULH VXE IRUP LUHGXFWLELO R IUDF LH GDW (QXQ 6HFRQVLGHUIUDF LD

111111111 . 135802469

$UWD L F QXPUWRUXO IUDF LHL VH GLYLGH FX úL FX LDU QXPLWRUXOVHGLYLGHFXúLFX 6FULH LIUDF LDVXEIRUPLUHGXFWLELO 9 111111111 37 ⋅ 9 ⋅ 333667 5VSXQV$YHP = = . 135802469 37 ⋅ 11 ⋅ 333667 11 2. 'LVFLSOLQD&ODVD&DSLWROXO$ULWPHWLF&ODVDD9D0XO LPL 2ELHFWLYXO(OHYXOVILHFDSDELOVHQXPHUHHOHPHQWHOHXQHLPXO LPL (QXQ 6HGDXPXO LPLOH$ {x / x ∈ N ,3 ≤ x ≤ 8}


66

B= {3,4,5,6,7} D 6XQWHJDOHPXO LPLOH" E &kWHVXEPXO LPLDUH%" F &DUHHVWHFDUGLQDOXOPXO LPLL$" 5VSXQVD 1XGHRDUHFH$ {3,4,5,6,7,8} ≠ {3,4,5,6,7} =B. b) 2 5 =32. c) 6. 3. Disciplina / Clasa / Capitolul: Geometrie / Clasa a VI-a / Triunghiul. 2ELHFWLYXO(OHYXOVFDOFXOH]HPVXULOHXQJKLXULORUXQXLWULXQJKL (QXQ )LH XQ WULXQJKL FX GRX ODWXUL FRQJUXHQWH úL PVXUD XQXL XQJKL o de 70 . a) Ce fel de triunghi este? E &DOFXOD LPVXULOHXQJKLXULORUWULXQJKLXOXL F 6XQW PDL PXOWH YDULDQWH OD UH]ROYDUHD SXQFWXOXL E " 'DF GD UH]ROYD LOH 5VSXQVD 7ULXQJKLLVRVFHOE o , 40 o , 70 o ; c) 55 o , 70 o , 55 o . 4. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,D1XPHUHvQWUHJL 2ELHFWLYXO (OHYXO YD IL FDSDELO V FDOFXOH]H YDORDUHD DEVROXW D XQRU numere întregi. (QXQ 6HGDXQXPHUHOH$ 2 123 + 2 123 − 382 : 381

(

(

B= 4 82 + 4 82 − 3123

) ): 3

122

.

D 1XPHUHOH$úL%VXQWSR]LWLYH" E &DOFXOD L$ F &DOFXOD L% 5VSXQVD 'DE $ F % 5. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,,D1XPHUHUHDOH 2ELHFWLYXO (OHYXO YD IL FDSDELO V UHFXQRDVF HOHPHQWHOH PXO LPLORU GH QXPHUHLQFOXVHvQPXO LPHDQXPHUHORUUHDOH (QXQ )LH$ −2; 0; 3,(5); π + 1; 5; 1,2; -4 ; 4 ; 3

{

d)

}

$IOD LD $∩N; b) A∩Z; c) A∩Q; d)A-R. 5VSXQV D {0, 2, 5}; b) {-2, 0, 2, 5}; c) {-2, 0, 1,2; 2, 3,(5); 5}; −4 .


67

6. 'LVFLSOLQD&ODVD&DSLWROXO*HRPHWULH&ODVDD9,,D5HOD LLPHWULFH 2ELHFWLYXO(OHYXOYDILFDSDELOVUH]ROYHFRUHFWXQWULXQJKLGUHSWXQJKLF (QXQ 6H G XQ WULXQJKL GUHSWXQJKLF FX FDWHWHOH E F [ úL ipotenuza a=x+4. D &DOFXOHD]OXQJLPLOHODWXULORUWULXQJKLXOXL E 'DF[ FDOFXOHD]SHULPHWUXO F &DOFXOHD]DULDWULXQJKLXOXL 5VXQVD D E F E 3 F $ 7. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD9,,,D3ROLQRDPH 2ELHFWLYXO (OHYXO YD IL FDSDELO V DSOLFH WHRUHPD vPSU LULL FX UHVW OD polinoame. (QXQ 6HFRQVLGHUSROLQRPXO3; ; 4 -4X 3 -3X 2 +2X+7. D 'HWHUPLQD LUHVWXOvPSU LULLOXL3; OD; E $UWD LF3D >0, ∀a∈R; F 'HVFRPSXQH LvQIDFWRULLUHGXFWLELOL3; 5VSXQVD 3 5; b) P(a)=(a-1) 2 (2a+1) 2 +6>0, ∀a∈R. c) P(X)-10=(2X-3)(X+1)(2X2 -X+1). 8. Disciplina / Clasa / Capitolul: Geometrie / Clasa a VIII-a / Paralelism în VSD LX 2ELHFWLYXO (OHYXO YD IL FDSDELO V IRORVHDVF WUDQ]LWLYLWDWHD UHOD LHL GH paralelism între drepte. (QXQ 3DUDOHORJUDPXO$%&'úLWUDSH]XO$%()$%|| EF) sunt situate în SODQHGLIHULWH$UDWF a) Punctele C, D, F, E sunt coplanare; E 'UHSWHOH&(úL)'VXQWFRQFXUHQWH Rezolvare: a) Din ABCD paralelogram ⇒ AB ||&'úLGLQ$%()WUDSH]⇒ AB||EF, deci CD||()UH]XOW&'()FRSODQDUH b) Din CD||()úL&'≠()UH]XOW&')(WUDSH]GHFL&(úL')VXQWGUHSWH concurente. 9. 'LVFLSOLQD&ODVD&DSLWROXO$OJHEU&ODVDD,;D)XQF LL 2ELHFWLYXO(OHYXOYDILFDSDELOVDSOLFHSURSULHW LDOHIXQF LLORU


68

(QXQ )LHIXQF LDƒ:R→R cu proprietatea ƒDƒDƒ=ƒúLPXO LPHD M={x∈R  (ƒDƒ)(x)=x}. D 6VHDUDWHF0≠∅úLƒ(R)=M. E 'DFJ0→M, g(x)=ƒ(x), ∀x∈0DWXQFLJHVWHELMHF LH F ' XQ H[HPSOX GH IXQF LH ƒ FDUH YHULILF LSRWH]D úL FDUH QX HVWHELMHF LHDSRLGHWHUPLQD LPXO LPHD0úLUHVWULF LDJDIXQF LHLƒ la M. Rezolvare: a) Fie x∈R, (ƒDƒ)(ƒ(x))=ƒ(x) ⇒ ƒ(x)∈M ⇒ M≠∅ úL ƒ(R)⊂M. Fie x∈M atunci ƒ(ƒ(x))=x=ƒ(y) deci x∈ƒ(R DGLF0⊂ƒ(R); M=ƒ(R). b) Fie g(x 1 )=g(x 2 ) ⇒ ƒ(x 1 )=ƒ(x 2 ) ⇒ ƒ(ƒ(x 1 ))=ƒ(ƒ(x 2 )) ⇒ x 1 =x 2 , deci g este LQMHFWLY)LHx∈M ⇒ x=ƒ(ƒ(x))=ƒ(y)=g(y), y∈0GHFLJHVWHVXUMHFWLY c) ƒ:R→R, ƒ(x)=0, ∀x∈R, ƒDƒDƒ=ƒ, ƒQXHVWHELMHFWLY 0 ^`úLJ0→0J HVWHELMHFWLY 10. 'LVFLSOLQD&ODVD$OJHEU&ODVDD,;D)XQF LDGHJUDGXODOGRLOHD 2ELHFWLYXO(OHYXOYDILFDSDELOVGHWHUPLQHPXO LPLvQFRQGL LLGDWH (QXQ )LH0RPXO LPHGHQXPHUHUHDOHFXSURSULHW LOH a) Z⊂M; b) x, y∈M ⇒ x+y∈0úL[\∈M, c) a= 2 + 3 ∈M. Se cer: 'HWHUPLQRHFXD LHFXFRHILFLHQ LvQWUHJLFDUHDUHUGFLQDa. 'HPRQVWUHD]F 3 − 2 ∈M, 2 3 ∈0úL 2 2 ∈M. Rezolvare. 5LGLFkQG OD SWUDW RE LQHP a 2 = 5 + 2 6 ⇔ a 2 -5= 2 6 5LGLFP OD 2

SWUDWúLRE LQHPa 4 -10a +1=0. 1 1 1 ⇒ = 10a − a 3 ; 3 − 2 = = 10a − a 3 . 3 a a 10a − a )RORVLQG SURSULHW LOH D E F úL XOWLPD UHOD LH RE LQHP F 3 − 2 ∈M; (-1)( 2 + 3 )= − 2 − 3 ∈M. Deci ( 2 + 3 )+( 3 − 2 )= 2 3 ∈0 úL 3 − 2 )+( − 2 − 3 )=- 2 2 ∈M ⇒ (-1)(- 2 2 )= 2 2 ∈M. 2) a(a 3 -10a)=-1 ⇒ a =

11. 'LVFLSOLQD&ODVD$OJHEU&ODVDD,;D)XQF LL 2ELHFWLYXO(OHYXOYDILFDSDELOVDSOLFHFRPSXQHUHDIXQF LLORU


69

(QXQ 6H FRQVLGHU IXQF LD VWULFW FUHVFWRDUH I5→5 FDUH YHULILF UHOD LDIDfDf)(x)=x3 [&DOFXOHD]I 5H]ROYDUH'DFI 0. PA QA

G ′D = λ úL DSOLFkQG G ′A


70

Folosim

(

) (

2 a 4 + b4 ≥ a 2 + b2

inegalitatea:

)

2

≥

(

)

2 a 2 + b 2 ≥ (a + b)

2

⇒

a 2 + b2 ≥

1 2

;

1 1 ⇒ a 4 + b4 ≥ . 4 8

13. 'LVFLSOLQD&ODVD$OJHEU&ODVDD,;D,QGXF LHPDWHPDWLF 2ELHFWLYXO (OHYXO YD IL FDSDELO V DSOLFH R YDULDQW D SULQFLSLXOXL LQGXF LHLPDWHPDWLFH 1 (QXQ D 'XQH[HPSOXGHQXPULUD LRQDO z pentru care z + ∈Q . z 1 1 E 'DFz ∈ R − Q úL z + ∈Q atunci z n + n ∈ Q, ∀n ≥ 1. z z 1 F 'DFzHVWHQXPULUD LRQDOúLH[LVW n ∈ N * astfel încât z n + n VILH z 1 LUD LRQDODWXQFL z + HVWHLUD LRQDO z 1 5H]ROYDUH D 5H]ROYP HFXD LD z + =3 ⇒ z 2 − 3z + 1 = 0 ⇒ z 3+ 5 1 3± 5 . 1XPUXO z = z1,2 = ∈ R − Q úL z + ∈Q . 2 2 z E 6HGHPRQVWUHD]FXDMXWRUXOYDULDQWHLD,,DDLQGXF LHLPDWHPDWLFH 1 Pentru n=1 avem z + ∈Q . z 1 3UHVXSXQHPF z m + m ∈ Q, 1 ≤ m ≤ n . z 1 1 1  1    z n +1 + n +1 =  z n + n   z +  −  z n −1 + n −1  ∈ Q .  z  z z  z  1 1 F 'DF z + ∈Q FRQIRUP SXQFWXOXL E UH]XOW z n + n ∈ Q, ∀n ≥ 1, z z 1 FRQWUDGLF LH'HFL z + ∈ R − Q . z 14. Disciplina /&ODVD$QDOL]PDWHPDWLF&ODVDD;,D/LPLWHGHIXQF LL 2ELHFWLYXO (OHYXO YD IL FDSDELO V FXQRDVF GDF R IXQF LH GDW HVWH PUJLQLW V FDOFXOH]H OLPLWD OD VWkQJD úL OLPLWD OD GUHDSWD vQWUXQ SXQFW SHQWUXIXQF LDGDW (QXQ )LHƒ:R→RFXSURSULHW LOH a) ƒPUJLQLWSHR. b) ƒDUHOLPLWODVWkQJDvQx 0 ILQLW


71

c) ƒQXDUHOLPLWODGUHDSWDvQx 0 =0. 6WDELOHúWH FDUH GLQWUH XUPWRDUHOH IXQF LL DX WRDWH SURSULHW LOH HQXQ DWH 1 x≤0  x,   cos , x < 0 1) f ( x ) =  2) f ( x ) =  x 1 cos , 0 x >   1 , x≥0 x  1, x ≤ 0 −1, x ≤ 0  3) f ( x ) =  4) f ( x ) =  1 1, x > 0 sin x , x > 0 5VSXQV 15. 'LVFLSOLQD&ODVD&DSLWROXO$QDOL]DPDWHPDWLF&ODVDD;,D/LPLWH GHIXQF LL 2ELHFWLYXO(OHYXOVFDOFXOH]HOLPLWHGHIXQF LLvQ∞, –∞. 1   a) lim n ⋅   b) lim x ⋅   n →+ ∞ → +∞ x n  x (QXQ &DOFXOHD]     c) lim x ⋅   d) lim x ⋅  . → −∞ x → x x x Rezolvare: a) ∀n≥2; a n =0 deci OLP a n =0. Q

1 b) OLP x   =0. [ →+∞  x  1 1 c) –1<   ≤ 'DFx>0 atunci 1–x 0  D 'HWHUPLQPXO LPHDSXQFWHORUGHFRQWLQXLWDWHDOHOXLƒ. b) 'HWHUPLQPXO LPHDSXQFWHORUGHGHULYDELOLWDWHDOHOXLƒúLFDOFXOD L ƒ'(x). c) ƒ are puncte critice ? d) ƒ are puncte de extrem local ? 5VSXQVD R*.

−1, x < 0 b) R úLƒ'(x)=  1, x > 0

c) ƒ'(x)≠0, ∀x∈R* deci ƒ nu are puncte critice. d) Da : x 0 HVWHSXQFWGHPLQLPJOREDOGHFLúLORFDODOIXQF LHLƒ. 17. 'LVFLSOLQD&ODVD&DSLWROXO$QDOL]DPDWHPDWLF&ODVDD;,D/LPLWH GHIXQF LL 2ELHFWLYXO (OHYXO YD IL FDSDELO V GHWHUPLQH PXO LPHD SXQFWHORU GH FRQWLQXLWDWHDOHXQHLIXQF LLGDWH  x, x < 0 (QXQ 6HFRQVLGHUIXQF LDƒ:R*→R prin ƒ(x)=  . 2 x , x > 0 &DUHHVWHPXO LPHDSXQFWHORUGHFRQWLQXLWDWHDOXLƒ? 2) ƒ(–1)·ƒ(1)=–2