Prova de Matemática Vestibular da FUVEST_ 2004_Fase 2. Resolução e comentário pela Profa. Maria Antônia Conceição Gouveia.

Transcrição

1 Prova de Matemática Vestibular da FUVEST_ 004_Fase. Resolução e comentário pela Profa. Maria Antônia Conceição Gouveia. Q~PHUR GH JRV PDUFDGRV QRV MRJRV GD SULPHLUD URGDGD GH XP FDPSHRQDWR GH IXWHER IRL H D VHJXQGD URGDGD VHUmR UHDL]DGRV PDLV MRJRV 4XD GHYH VHU R Q~PHUR WRWD GH JRV PDUFDGRV QHVVD URGDGD SDUD TXH D PpGLD GH JRV QDV GXDV URGDGDV VHMD VXSHULRU j PpGLD REWLGD QD SULPHLUD URGDGD" 3ULPHLUD URGDGD: pGLD GH JRV, HJXQGD URGDGD *RV PDUFDGRV 0pGLD GH JRV QDV GXDV URGDGDV x + 5 x + 5,.,5 33 x x 8 5(6367 JRV 7UrV FLGDGHV %H&VLWXDPVH DR RQJR GH XPD HVWUDGD UHWD % VLWXDVH HQWUH H&HDGLVWkQFLD GH %D&pLJXD D GRLV WHUoRV GD GLVWkQFLD GH DWp % 8P HQFRQWUR IRL PDUFDGR SRU PRUDGRUHV XP GH FDGD FLGDGH HP XP SRQWR 3 GD HVWUDGD RFDL]DGR HQWUH DV FLGDGHV % H & H j GLVWkQFLD GH NP GH 6DEHQGRVH TXH 3 HVWi NP PDLV SUyLPR GH & GR TXH GH % GHWHUPLQDU D GLVWkQFLD TXH R PRUDGRU GH % GHYHUi SHUFRUUHU DWp R SRQWR GH HQFRQWUR 6H D GLVWkQFLD GH % D & p LJXD D GRLV WHUoRV GD GLVWkQFLD GH DWp % BC 3BC AB AB 3 3 & % QDLVDQGR R JUiILFR UHSUHVHQWDWLYR GD VLWXDomRSUREHPD FRQFXtPRV TXH 3(x + 0) + x x x x 30 x 40

2 5HVSRVWD PRUDGRU % GHYHUi SHUFRUUHU DWp R SRQWR GH HQFRQWUR NP 8P WULkQJXR %& WHP DGRV GH FRPSULPHQWRV % %& H & 6HMDP 0HRVSRQWRV GH AB WDLV TXH CM p D ELVVHWUL] UHDWLYD DR kqjxr AĈB H CN p D DWXUD UHDWLYD DR DGR AB 'HWHUPLQDU R FRPSULPHQWR GH MN Sta & % 0 Sendo CM D ELVVHWUL] LQWHUQD UHDWLYD DR kqjxr AĈB BM 0-4BM BM BM 5 - BM 6 3 Sendo o ângulo C Bˆ A, oposto ao menor lado do triângulo, é agudo. Utilizando a relação métrica num triângulo qualquer, relativa a um ângulo agudo AC² BC² + AB² - AB.BN vem: BN 0BN 37 BN. 0 Pela figura percebemos que BN BM + MN MN BN BM MN HVSRVWD MN 30 &RQVLGHUH D HTXDomR ]ð ] z RQGH p XP Q~PHUR UHD H z LQGLFD R FRQMXJDGR GR Q~PHUR FRPSHR ] D 'HWHUPLQDU RV YDRUHV GH SDUD RV TXDLV D HTXDomR WHP TXDWUR UDt]HV GLVWLQWDV E 5HSUHVHQWDU QR SDQR FRPSHR DV UDt]HV GHVVD HTXDomR TXDQGR

3 D 6HQGR ] E FL FRP E H F UHDLV H ]ð ] z WHPRV, ]ð Eð Fð EFL,, ] EFL,,, EFL EE ± FFL 6XEVWLWXLQGR RV YDRUHV,,, H,,, QD HTXDomR ]ð D] D z Eð Fð EFL EFL EE ± FFL Eð Fð EFL E ±EF±F FL E ±EFL Eð Fð EFL E ±EFL EðFð E ± E H EF F EF ± F 'H EF F EF ± F E ± F F RXE &RQVLGHUDQGR F HP Eð Fð E ± E Eð E ± E EE +) 0 E RXE ] RX ] 0, FRP &RQVLGHUDQGR E HP Eð Fð E ±E F F + F ± ±. 6HQGR F XP Q~PHUR UHD HQWmR ] L RX ] + L ƒ V TXDWUR VRXo}HV VmR ], ] 0 ] L FRP H] + LFRP 5HVSRVWD H E 6HQGR ], ] 0 ] LH] + L DILR GH XP Q~PHUR FRPSHR p R SRQWR GHWHUPLQDGR SHR SDU RUGHQDGR FXMDV FRRUGHQDGDV VmR D SDUWH UHD H D SDUWH LPDJLQiULD UHVSHFWLYDPHQWH GR Q~PHUR HP TXHVWmR 5HVSRVWD,P ] ] ] ] 5H] ]

4 SURGXWR GH GXDV GDV UDt]HV GR SRLQ{PLR S ±P pljxd D ± 'HWHUPLQDU D R YDRU GH P E DV UDt]HV GH S D &RQVLGHUHPRV H FRPR UDt]HV GR SRLQ{PLR S ±P VHQGR GDGR GR SUREHPD H 'H S P P + P + P 5HVSRVWD P E 'H P S ± 6HQGR + H FRPR ± RX + 5HVSRVWD V UDt]HV GH S VmR + H ILJXUD DEDLR UHSUHVHQWD GXDV SRLDV FLUFXDUHV & H & GH UDLRV 5 FP H 5 FP DSRLDGDV HP XPD VXSHUItFLH SDQD HP 3 H 3 UHVSHFWLYDPHQWH 8PD FRUUHLD HQYRYH DV SRLDV VHP IRJD 6DEHQGRVH TXH D GLVWkQFLD HQWUH RV SRQWRV 3 H 3 p FP GHWHUPLQDU R FRPSULPHQWR GD FRUUHLD

5 P 4 M A 3 / 3 3 C P 3 / 3 3 / B P N P 3 GLVWkQFLD 3 3 FP p D GLVWkQFLD HQWUH RV FHQWURV GH & H& & 5 5 R WULkQJXR UHWkQJXR %& WJ%Ö %Ö 33 0 HR DUFR FRPSULPHQWR GR DUFR 33 0 p π π H R FRPSULPHQWR GR DUFR 303 π π 40, p π π FRPSULPHQWR GD SRLD p + + FP ( + π ) FP ( + π )FP D ILJXUD DEDLR RV SRQWRV % H & VmR YpUWLFHV GH XP WULkQJXR UHWkQJXR VHQGR %Ö R kqjxr UHWR 6DEHQGRVH TXH % SHUWHQFH j UHWD ±\ H3 pr FHQWUR GD FLUFXQIHUrQFLD LQVFULWD QR WULkQJXR %& GHWHUPLQDU DV FRRUGHQDGDV D GR YpUWLFH % E GR YpUWLFH &

6 C H P G Y X_ A B y E y D β y y F Analisando a figura vemos que A distância PE, do ponto P(3,4), centro da circunferência, à reta AB é o raio da mesma circunferência: r + A interseção da circunferência com a reta de equação x-y0 é dada pela solução do sistema; \ \ + \ \ \ + + \ y² - 4y (y-)² 0 y E ( 4,) (, ponto B pertence à reta x y 0 BE aplicando o teorema de Pitágoras no triângulo EDB: 4y ² + y ² 5 y (,, a) Resposta: Pela figura e de (, H (,, FRQFXtPRV TXH R SRQWR % ponto B (6,3) pertence à reta BC. A reta BC é perpendicular à reta x y 0 a equação da reta BC é da forma y -x + a a a 5 a equação da reta BC é y -x +5. (,,, A reta AC é da forma bx y 0 e sua distância ao ponto P(3,4) é: E PH E ( E + ) E E ( + ) E + ± 9b² - 4b + 6 5b² + 5 4b² -4b + 0 b ± ± ± b E RX E. Sendo x y 0 a equação da reta AB, podemos escreve-la na sua forma

7 reduzida com \ HTXDomR GD UHWD & p \ WJ. Como >β (,9 WJ β > WJ β E E b) 5HVSRVWD SRQWR & p GDGR HQWmR SHD LQWHUVHomR GRV JUiILFRV GDV HTXDo}HV (,,, H (,9 \ + + H \ \ C (,). D ILJXUD DEDLR FDGD XPD GDV TXDWUR FLUFXQIHUrQFLDV HWHUQDV WHP PHVPR UDLR U H FDGD XPD GHDV p WDQJHQWH jv RXWUDV GXDV H j FLUFXQIHUrQFLD LQWHUQD & 6H R UDLR GH & p LJXD D GHWHUPLQDU D R YDRU GH U E D iuhd GD UHJLmR KDFKXUDGD ) ) ' ' & U U % U &RQVLGHUHPRV % & H ' FRPR FHQWURV GDV FLUFXQIHUrQFLDV HWHUQDV WDQJHQWHV GXDV D GXDV H DR PHVPR WHPSR WDQJHQWHV j FLUFXQIHUrQFLD GH UDLR H FHQWUR HP ž

8 5HVRYHQGR R WULkQJXR UHWkQJXR % SHD DSLFDomR GR WHRUHPD GH 3LWiJRUDV Uð Uð Uð Uð U Uð U ± + + U + + D 5HVSRVWD YDRU GH U p + QDLVDQGR D ILJXUD FRQVWUXtGD SDUD D LQWHUSUHWDomR GR SUREHPD FRQFXtPRV TXH D iuhd 6SLQWDGD GH D]X FRUUHVSRQGH j VRPD GDV iuhdv GH XP FtUFXR GH UDLR U FRP XP FtUFXR GH UDLR 6 πuð π π π π ππ ππ DGR GR TXDGUDGR %&' p U E iuhd SHGLGD 6 6 %&' ± 6 Uð ππ + ] ππ ( + ) ππ 6 π π Resposta: 6 π π 6HMD P XP Q~PHUR UHD H VHMDP IHJIXQo}HV UHDLV GHILQLGDV SRU I ± H J P P D (VERoDU QR SDQR FDUWHVLDQR UHSUHVHQWDGR DEDLR RV JUiILFRV GH IHGHJ TXDQGR P HP E 'HWHUPLQDU DV UDt]HV GH I J TXDQGR P F 'HWHUPLQDU HP IXQomR GH P R Q~PHUR GH UDt]HV GD HTXDomR I J D 6HQGR I ± WHPRV D FRQVLGHUDU GXDV SRVVLELLGDGHV I ± FXMD UDL] p I ± ð FXMD UDL] p J 6HP J P P 6HP J

9 5HVSRVWD f(x) x² - x + g(x) x/4 + / g(x) x+ E 6H J P P H P J HVWH FDVR VH I J ±, 6H ± ð ð RX,, 6H ± ð ð 5HVSRVWDVV UDt]HV GD HTXDomR I F J VmR H g(x) mx+m f(x) x² - x + g(x) x/ + y 0,75(x+) - g(x) x/ + y 0,(x+) Para m 0, g(x) mx+m 0 x - g(x) mx+m

10 'HWHUPLQHPRV DV UDt]HV GH J PP PP TXDTXHU TXH VHMD R YDRU DWULEXtGR D P FRQILUPDGR SHR JUiILFR J PP p XP IHLH GH LQILQLWDV UHWDV SDVVDQGR SHR SRQWR 'HVWH IHLH HLVWH XPD TXH SDVVD SHR SRQWR GHWHUPLQDGR SRU I I p XP SRQWR GR JUiILFR GH I 3DUD J PP P 5HVSRVWD QDLVDQGR R JUiILFR H HYDQGR HP FRQWD TXH P FRQFXtPRV TXH 3DUD P HLVWHP GXDV UDt]HV GLIHUHQWHV 3DUDP HLVWHP TXDWUR UDt]HV GLIHUHQWHV 3DUD P! HLVWHP WUrV UDt]HV GLIHUHQWHV R VyLGR 6 UHSUHVHQWDGR QD ILJXUD DR DGR D EDVH %&' p XP UHWkQJXR GH DGRV % H' DV IDFHV %() H '&() VmR WUDSp]LRV DV IDFHV ') H %&( VmR WULkQJXRV HT LiWHURV H R VHJPHQWR () WHP FRPSULPHQWR 'HWHUPLQDU HP IXQomR GH R YRXPH GH 6 ' ) ( & % ( ) & % 0, B ' - / B 6HFFLRQDQGR R VyLGR VHJXQGR RV SDQRV )+0 H (, SHUSHQGLFXDUHV DR SDQR %& GHWHUPLQDPRV R SULVPD WULDQJXDU )0+,( GH EDVH )+0 H DWXUD )( H DV +

11 SLUkPLGHV )+0' H (,%& HTXLYDHQWHV GH EDVHV UHWDQJXDUHV H DJUXUDV UHVSHFWLYDPHQWH )- H ( GH PHGLGDV LJXDLV,8P GRV GDGRV GR SUREHPD p TXH R WULkQJXR '( p HT LiWHUR RJR VXD DWXUD () R WULkQJXR UHWkQJXR )-/ SHD DSLFDomR GR WHRUHPD GH 3LWiJRUDV )- 9RXPH GDV SLUkPLGHV, 9RXPH GR SULVPD,, 6RPDQGR RV YRXPHV, H,, +