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'tUCl£O lfel£oe om6ema nOl£e nep60u C numume 3anJ£myro 6 Ol£aUKe 6 OTBETOB ua'tuuaJ£ MU J£6l£J£emCJ£ 'tUCl£O BJIAHK 6 3aiJaUUJ£, zo 1-12 3aiJaUUJ£M K iJPOOb. uaJ£ 12 IIoKynaTeJIb BH.n;e. 10 % OT o6oH.n;eTC.H MomeT 3aKa3aTb KynJIeHHOH SToro IIIKa<pa CTOHMOCTH nOKynKa co OTBeT: CD Ha .n;HarpaMMe Kam.n;oro 2000 ro.n;a no ro.n;a "tIHCJIeHHOCTH 0 .n;aHHbIe npHBe.n;eHbI C 2018 (B ro.n; HaCeJIeHH.H BOJIor.n;e B Ha KOHen; "tIeJI.). TbIC. 320 310 300 290 280 Onpe.n;eJIHTe, 3a C nepHO.n; 0 0 0 oo 0 0 0 0 0 0 0 0 0 0 0 0 oo 0 0 t- 0 0 t- 0 0 0 0 0 0 0 0 0 0 0 C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l C\l Ha CKOJIbKO C\l 2008 KOHn;a TbIC.H"tI ro.n;a "tIeJIOBeK no 0 "tIHCJIeHHOCTb BbIpOCJIa 2018 KOHen; C\l HaCeJIeHH.H B BOJIor.n;e ro.n;a. OTBeT: @ 2021. KorrHpoBaHHe, pacrrpOCTpaHeHHe 000 H «M3,n;aTeJIbCTBO HCrrOJIb30BaHHe «HaU;HOHaJIbHOe B KOMMepqeCKHX 06pa30BaHHe». u;eJIJIX 6e3 @ 2021. IIHCbMeHHoro qOY pa3pemeHHJI ,n;no «MIJ;HMO» rrpaBoo6JIa,n;aTeJIeH He ,n;orrYCKaeTCJI TLt1nOBbIE Er3. 64 o=J Ha KJIeT"'IaTOH 6YMare C 1 xl KJIeTKH pa3MepOM HaH,lJ;HTe qeTbIpexyrOJIbHHK. ero BAPLt1AHTbl 3K3AMEHALl.Lt10HHbIE H306pameH rrJIOm;a,n;b. OTBeT: 4 Bcero B Ha ,lJ;eJIHT Erop B qeJIOBeK, OKa3aJICH B TpeTbeM OKameTCH B rrepBOM HBaH qHCJIe TOM 1 7 rro rrO,lJ;rpyrrrrbI TPH HBaH qTO YCJIOBHH aTOM 51 TypHCTOB rpyrrrre o6pa30M H3BecTHo, qeJIOBeK rpyrrrry B rrOCa,lJ;KH KaKoBa aBTo6yce. aBTo6yce? Erop. H ,lJ;JIH TPH Toro, BepOHTHOCTb CJIyqaHHbIM aBTo6yca. qTO rrpH OTBeT: u H aH,lJ;HTe KopeHb O,2 ypaBHeHHH 5 + 4x _125. _ OTBeT: CD B BC ABCD "'IeTbIpexyrOJIbHHK = 5 D H = 27. BIIHcaHa AB oKpymHocTb, HaH,lJ;HTe qeTBepTYIO D 8, = CTOPOHY qeTbIpexyrOJIbHHKa. c OTBeT: A co Ha pHCYHKe Ha HHTepBaJIe H306pameH (-3; 8). rpaqmK B Y = KaKOH f'(x) IIPOH3BOHOH - TOqKe [-2; OTpe3Ka 3] CPYHKIJ;HH <PYHKIJ;HH f(x), OIIpeeJIeHHOH f(x) rrpHHHMaeT 3HaqeHHe? HaHMeHbllIee y OTBeT: @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H «HaIJ;HOHaJIbHOe «113,n;aTeJIbcTBo HCIIOJIb30BaHHe B KOMMepqeCKHX o6pa30BaHHe». IJ;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY )J;rro pa3pemeHHH «MIJ;HMO» IIpaBOo6JIa.,zJ;aTeJIeH He ,n;OIIYCKaeTCH BAPLt1AHT12 65 HJIHH,n;p IT] H IJ;HJIHH,n;pa 06m;He HMelOT KOHYC paBHa 60KOBOH H 27.J2. paBHa BbICOTa BbICOTY. 60KOBOH IIJIOm;a,n;b HaH,n;HTe OCHOBaHHH. pa,n;HYCY IJ;HJIHH,n;pa rrOBepXHOCTH OCHOBaHHe rrJI Om; a,n;b KOHyca. rrOBepXHOCTH OTBeT: He 3aoyiJbme em6uu 6ee nepeueemu uuempYKlfueu e [TI 3Ha"'leHHe (3 BbIpameHHSI M om6em06 1 6 eoom6em- paoomb£. 2 qacTb HaH,ZJ;HTe Ol£auK 6 om6emb£ 6b£nOl£ueuulO no 1 ): -1,5 · 5 6 OTBeT: B [ill C eMKOCTb TeJIeBH30pe rro,n;KJIIO"tJeH KOH,n;eHcaTopOM pa60TbI BbICOKOBOJIbTHOrO Ha Ha HarrpHmeHHe TeJIeBH30pa C pe3HCTOp HarrpHmeHHe TeJIeBH30pa Ha HarrpHmeHHe OTBeT ,n;aHTe B == o 34 == <1>. OM. IIocJIe KB. Bo BpeMH BbIKJIIO"tJeHHH U 3Ha"tJeHHH ,n;o IIapaJIJIeJIbHO 6 6.10 (KB) 3a BpeMH, u: aRClog 2 eCJIH KOH,n;eHcaTope, == O-t> 5.1 R U y6bIBaeT KOH,n;eHcaTope t BbIpameHHeM == corrpOTHBJIeHHeM KOH,n;eHcaTope ------- orrpe,n;eJIHeMOe C KOH,n;eHcaTopa (c), U a r,n;e rrOCJIe == 1,7 Orrpe,n;eJIHTe rrOCTOHHHaH. - BbIKJIIO"tJeHHH TeJIeBH30pa rrpOIIIJIa 51 c. 8 M KHJIOBOJIbTax. OTBeT: IIJIHTO"tJHHK [!!] B ,n;OJImeH 60JIbIIIe, ,n;eHb "tJeM KBa,n;paTHbIX 120 YJIOmHTb M B rrJIHTKH ,n;eHb ECJIH rrJIHTKH. rrJIaHHpyeT 6y,n;eT OH pa60TY 3aKOH"tJHT TO 3arrJIaHHpOBaJI, MeTpOB 2 4 Ha Ha YKJIa,n;bIBaTb ,n;HH CKOJIbKO paHbIIIe. rrJIHTo"tJHHK? YKJIa,n;bIBaTb OTBeT: HaH,ZJ;HTe TO"'lKY <PYHKIJ;HH MaKCHMYMa = Y x 3 + 2 18x + 81x + 23. 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CJIOBH.H TaKOBbI: Kam)J;bIii .HHBapb 15 Ha B03paCTaeT )J;OJIr % 110 C cpaBHeHHIO KOHD;OM npe,n;bI,n;ym;ero rO)J;a; <peBpaJI.H MJIH 1,587 C CKOJIbKO 110 1I0rameH @ 2021. pacnpOCTpaHeHHe Kam)J;oro He06xo)J;HMO rO)J;a paBHbIMH )J;BYM.H 000 H 6bIJIO py6JIeii MHJIJIHOHOB 1I0JIHOCTblO KonHpoBaHHe, HIOHb BbIIIJIaTHTb qaCTb ,n;OJIra, paBHYIO py6JIeii. «Han;HOHaJIbHOe «M3.n;aTeJlbCTBO HCnOJlb30BaHHe B KOMMepqeCKHX B3.HTO B 6aHKe, eCJIH (TO IIJIaTemaMH 06pa30BaHHe». n;eJl.HX 6e3 @ 2021. nHCbMeHHoro eCTb QOY pa3pemeHH.H qTO H3BeCTHO, 3a )J;rro )J;Ba Kpe,n;HT rO)J;a)? «MMO. npaBoo6J1a.n;aTeJleii He .n;onYCKaeTC.H 6bIJI Era. 72 [KJ HaH,ZJ;HTe Bce 3Ha"tJ:eHH.H HMeeT [KJ B 6a.rrJIbI 3a a) Mor 6) Cpe,n;HHH B) B Mor M 1 =0 M 1 mKOJIe B 0/0. H3 06eHx M 1 HaH,n;HTe 6aJIJIOB. B B 2 IIocJIe 3Toro NQ 2, mKOJIY a O,n;HH cpe,n;HHe pa3a? 10 6aJIJI 0/0, cpe,n;HHH 6aJIJI cpe,n;HHH 10 Ha BbIpOC Kam,n;bIH OKa3aJIOCb, yqam;HHC.H. qHCJIOM. NQ 1 IIHCaJIH mKOJIax. Ha HaHMeHbmee 51 n;eJIbIM TeCT mKOJIbI IIHCaJI mKOJIbI BbIpaCTH BbIpOC Kam,n;oH KOJIHqeCTBO 6bIJI TeCT 3a IIepBOHaqaJIbHbIH JIH TeCT HaTypaJIbHOe B mKOJIe B H3 TeCT. IIepemeJI TeCT, mKOJIe B 10 Ha -x CYMMapHO 6aJIJI cpe,n;HHH IIepeCqHTaHbI 0/0. 6aJIJI Cpe,n;HHH BbIpOC 10 a Ha6pa.rr TeCT, 6aJIJI 6aJIJI Ha _y2 -24) 2' IIHCaJIH yqam;HeC.H yqam;HXC.H, 6bIJIH cpe,n;HHH JIH BbIpOC 0/0, B 6aJIJI cpe,n;HHH 3HaqeHHe B NQ B M 2 mKOJIe mKOJIe 2 1? paBH.HTbC.H NQ 2 mKOJIe IIepBOHaqaJIbHOrO TaKme TaKme 6aJIJIa cpe,n;Hero M 2. mKOJIe .."mo6b£ np06epbme, pacnpOCTpaHeHHe Ka:HCOb£U 6b£l£ om6em pS£OOM 3anucan c nOMepoM coom- 3aoanus£. 6emcm6YlOUfezo @ 2021. KonHpoBaHHe, M 2 H IIHcaBmHX TeCT 2 -8(x+4)+x 2a 2 110 KpaHHeH Mepe IIHcaBmHH yqam;HHC.H, B qTO mKOJIe Kam,n;oH yqam;HXC.H, ypaBHeHHH pemeHH.H. NQ 1 mKOJIaX H3 CD ,n;Ba pOBHO +(y+4)2 BAPAHTbl aK3AMEHAU.OHHbIE CHCTeMa ROTOpbIX IIPH 2 1- = y (x+4i -y)( ( v'12-x2 { a, IIapaMeTpa TnOBbIE 000 H «HaU;HOHaJIbHOe «M3,ZJ;aTeJlbCTBO HcnOJlb30BaHHe B KOMMepqeCKHX 06pa30BaHHe.). u;eJl.HX 6e3 @ 2021. nHCbMeHHoro qOY pa3perneHH.H ):(rro «MIJ;HMO. npaBoo6J18.,ZJ;aTeJleH He ,ZJ;onYCKaeTC.H 14 BAPHAHT Om6emOM 3anJ£mylO 6 6Jl,anK,e U3Mepenuu HaJIor Ha Ha COCTaBJI.HeT 3naK, «(MunyC) npU6eoennb£- C ne nY:HCno. IIocJIe IIJIaTbI. 15 1I0JIyqHJIa 660 HaJIOra Y,l];epmaHH.H CKOJIbKO Py6JIeii. Py6JIeii KOHcTaHTHHoBHbI? MapHH IIJIaTa lfutjJpy, nucamb 3apa60THoii OT KOHcTaHTHHoBHa MapH.H 3apa60THa.H ,l];OXO,l];bI % 13 COCTaBJI.HeT ,l];OXO,l];bI nepe- coom6emcm6YlOUfe- 1 qacTL [TI eCJ£mu"If,- 3ameM coom6emcm6UU 6 K,Jl,emO"lf,K,e Eounulfb£ 06pa3lfaMU. pa60mb£, nOMepa Ka:HCoylO K,Jl,emo"lf,K,u. OmOeJl,bnOU 6 om cnpa6a o K,one"lf,naJ£ UJl,U meK,cme 6 om6ema M 1 nep60u C na"lf,unaJ£ "If,UCJl,O lfeJl,oe nOJl,e 6 OTBETOB nUlUume MU J£61£Sl,emCJ£ "If,UCJl,O BJIAHK 6 3aOanUJ£, 1,0 U 3anulUume e1,o necume 1-12 3aOanUJ£M K, Op06b. naJ£ OTBeT: Ha IT] IIo IIOKa3aHa pHCYHKe JI.H TOqKH HarJI.H,l];HOCTH Bce BO CBHHn;a 110 MeC.Hn;bI, YKa3aHbI ropH30HTaJIH CIDA. n;eHa Cpe,l];H.H.H BepTHKaJIH COe,l];HHeHbI 2017 MeC.Hn;bI - TOHHbI n;eHa 2018 H B CBHHn;a rO,l];OB. ,l];OJIJIapax OTpe3KaMH. 2600 - - r "' , J 2500 , r - , "- 'J' I i -- , , , 2400 .... JII \. I '" I , , '- ,I J ...... ,,- -- r - " 2300 '-- J T '" '- , T , , 2200 , \ [ , j \ , T T T 2100 , T ... ,. r.I -- '..... -.. .-. "' '- 2000 " --, - T \. -- 1900 t- t- . . .-4 t- t- . . .-4 . . .-4 . . .-4 t- t- . . .-4 . . .-4 t- (J) := 110 t'j . PHCYHKY t- . . .-4 t- . . .-4 S n;eHY t- . . .-4 oo . . .-4 00 00 . . .-4 . . .-4 00 . . .-4 . . .-4 00 00 . . .-4 . . .-4 S 00 . := . . 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H3 OTBeT: ,z:J:Ba [TI BIIHcaHHoro yrJIa 112° 125°. H ,n;aHTe B B qeTbIpexyrOJIbHHKa oKpymHocTb 60JIbIIIHii Haii,n;HTe H3 paBHbI OCTaBIIIHXC.H OTBeT yrJIOB. rpa,n;ycax. OTBeT: [0 Ha Ha H306pameH (-2; 11). pHCYHRe HHTepBa.rre <PYHKD;HH f(x) IIapa.rrJIeJIbHa rpaQJHR f'(x) Y KOJIHqeCTBO Haii,n;HTe -2x IIp.HMOii y IIPOH3BO,D;HOH - = y, 5 - = -<, -2 KaCaTeJIbHa.H C COBIIa,n;aeT OIIpe,D;eJIeHHOH K rpa<pHKY Heii. , J 1 \ I \ , ,0 I y=f'(x) \ I 1 I \ I , HJIH KOTOpbIX - / \ f(x), QJYHK[I;HH B TOqeK, llx \ / \ r \ ....." \ J 9 \. ./ ' OTBeT: @ 2021. KonHpoBaHHe, pacnpOCTpaHeHHe 000 H «HaU;HOHaJIbHOe «ll3,ZJ;aTeJlbCTBO HCnOJlb30BaHHe B KOMMepqeCKHX o6pa30BaHHe». u;eJl.HX 6e3 @ 2021. IIHCbMeHHoro qOY pa3perneHH.H )J;rro «Mll;HMO» npaBoo6J18.,ZJ;aTeJleH He ,ZJ;onYCKaeTC.H x BAPV1AHT14 75 =0 KOHYC BrrHCaH paBeH 19. B Pa.,n;HYC map. OCHOBaHHH 06rneM HaH,n;HTe paBeH KOHyca 06rneM mapa. pa,n;HYCY KOHyca mapa. OTBeT: He nepeneemu 3a6yobme e unempYJ(,lfueu em6uu 6ee om6emb£ HaH,ZJ;HTe ( loga 4 a 3 b ), b loga eCJIH = 1 eoom6em- 6 2 qacTL =:!:J M om6em06 pa6omb£. 6b£nonnenulO no 6nanJ(, 6 4. OTBeT: C6JIHmeHHH IIpH HeKoTopoH B rrpHMoH rrpHeMHHKOM, rro cpe,n;e perHcTpHpyeMoro H He CHrHaJIOB, 3BYKOBbIX HaBCTpeqy ,n;pyr C COBrra,n;aeT qaCTOTa ,n;pyry, qaCTOTOH f BbIpameHHeM: CJIe,n;YIOIIJ;HM orrpe,n;eJIHeTCH rrpHeMHHKa H HCTOqHHKa - (r TT ... CHrHaJIa, fo CHrHaJIa Hcxo,n;Horo c + u ., J( 0 - ,n;BHmYIIJ;HXCH 3BYKOBoro ), r rr M e = 130 C CKOpOCTb M/C CKOpOCTH ru; c-v CHrHaJIa pacrrpOCTpaHeHHH rrpHeMHHKa He M/C), a OTHOCHTeJIbHO M/C) (B C CKOpOCTH f 6y,n;eT rrpHeMHHKe B (B cpe,n;e HCTOqHHKa H MaKCHMaJIbHOH B = 15 M/C v H 9 = cpe,n;bI CHrHaJIa B KaKoH IIpH COOTBeTCTBeHHO. pacrrpOCTpaHeHHH 135 MeHee u qaCTOTa cpe,n;e CHrHaJIa rU;? OTBeT: [Q] M3 B O,n;HOH O,n;HOM Ha HarrpaBJIeHHH 114 paBHa O,n;HH KpyroBoH TOqKH KM/q, Kpyr. H 30 qepe3 rrOCJIe MHHYT CKOpOCTb CKOpOCTb OH CTapTa KM, OTBeT o,n;HOBpeMeHHO aBTOM06HJIH rrepBoro aBTOM06HJIb BTOpOH orrepemaJI aBTOM06HJIH. BToporo 25 paBHa aBTOM06HJIH. ,n;Ba CTapTOBaJIH HaH,n;HTe KOTOpOH ,n;JIHHa TpaCCbI, ,n;aHTe B KM/q. OTBeT: HaH,ZJ;HTe HaHMeHbmee 2 y=(l-x)e CPYHKIJ;HH 3HaqeHHe - X Ha [0,5; OTpe3Re 5]. OTBeT: He CD 3a6yobme nepeneemu e unempYJ(,lfueu "I£moob£ Ilpo6epbme, em6uu @ 2021. paCIIpOCTpaHeHHe om6emb£ 6 J(,a:HCOb£U 6nanJ(, om6em06 M 1 6 eoom6em- pa6omb£. 6b£nonnenulO Ob£Jl, om6em aanuean 6 empoJ(,e e nOMepOM aaoanuJl,. eoom6emem6YlOUfezo KOIIHpoBaHHe, 6ee no 000 H «M3,ZJ;aTeJlbCTBO HCIIOJlb30BaHHe «HaU;HOHaJIbHOe B KOMMepqeCKHX 06pa30BaHHe». u;eJl.HX 6e3 @ 2021. 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IIHCbMeHHoro qOY pa3peIIIeHH.H ,n;no «(MI.J;HMO» IIpaBOo6JIa,zJ;aTeJIeii He ,ZJ;OIIYCKaeTC.H Bce OTBETbI Kamoe BMrrOnHeHHOe rrpaBHnbHO BbIrronHeHHbIMH C"tIHTalOTCH "tIHCna HnH ecnH BepHO, KOHe"tlHOH 1-12 3aaHHe 1 oeHHBaeTCH 3K3aMeHyeMbIH an BepHbIH 1 OTBeT B OTBeT M aa)];aHHJI 1 725,2 1 14,5 2 9 2 2 3 12 3 2 4 0,25 4 0,22 5 5,5 5 11 6 113 6 0,75 7 2 7 7 8 60 8 45 9 324 9 -7,5 10 6250 10 1,3 11 14 11 5 12 7 12 1,2 7t a) - + 2 7t 7t k 7tk, Z, e - + 7tn, - n e Z; a) 4 13 77t. 6) 2' 157t. 97t 4' 2 13 6) 9 14 keZ; 7tk, ' meZ. , 37t 27t; . ' , 4 37t --+27tm 4 77t 4 neZ. --+27tn, - 4 14 14 15 (- 00; - 55 500 TbIC. a<-2; 1 < a < - 2 a) 19 @ 2021. paCIIpOCTpaHeHHe 6) a; 000 H 2; 00; - [- v'IO; 4]; 16 7,5 1 17 20 a=O; 2 18 a>2 19 HeT; B) - 3 ) 1a<9 a) a; 6) B) HeT; 10 232 21 «M3.n;aTeJIbCTBO HCIIOJIb30BaHHe - py6JIeii -2<a<--; 18 ( 15 4,8 17 arccos- [1; 1,5) 2]; 16 KOIIHpoBaHHe, enoro BHe 2 BapHaHT OTBeT M aa)];aHHJI 1-12 3aaHHH p06H. eCHTH"tIHOH BapHaHT 6annoM. «Han;HOHaJIbHOe B KOMMepqeCKHX o6pa30BaHHe». n;eJIHX 6e3 @ 2021. IIHCbMeHHoro QOY pa3peIIIeHHH IIO «MD;HMO» IIpaBOo6JIa..n;aTeJIeii He .n;OIIYCKaeTCH OTBETbl 185 3 BapHaHT .NQ I 4 BapHaHT OTBeT .NQ aHHJI OTBeT aa)];aHHJI I I 1 1 27 1 7 2 2 2 31 3 2,5 3 2 4 0,375 4 0,32 5 -2 5 0 6 62 6 78 7 6 7 6 8 25 8 20 9 80 9 28 10 60 10 30 11 75 11 10 12 18 12 -2 . -1-.J3; -1-.J2, a) 13 -1-.J2 6) a) -2-.J6, 6) -2+.J6 14 log2 -- 1 , 3 log2 -- 3. 6 2 13 9.J3 1 -2+.J6, , 2 6 - 4 (-00; 15 -1); 0; (0,5; 7 5 .J26 16 35 17 ( 9 ; - 1 6 - 0,5 19 ); a) 700 ( - ]; [- .J26,1 [ [4; .J25,9]; -4 .J 25, 9; .J26,1 ; 5; 0); (0; 2); (2; 15; 6) B) + <X) 6] 6M 16 ) 5 17 14 820 53 (- : ;-1,5} 18 py6JIeii (0;3 } (-1,5;0); ( ;+<X») 3 19 @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H ]; py6JIeii -0, 7; [ -6; 15 6 18 30M 14 +(0) «HaU;HOHaJIbHOe «M3.n;aTeJIbCTBO HCIIOJIb30BaHHe B KOMMepqeCKHX o6pa30BaHHe). u;eJI.HX 6e3 @ 2021. IIHCbMeHHoro a) "tIOY pa3peIIIeHH.H ,ZJ;rro 12; «MHMO)) IIpaBOo6JIa,z:J;aTeJIeii 6) 15; He B) 6 .n;OIIYCKaeTC.H Ers. 186 6 BapHaHT .NQ OTBeT aa.n;aHHJI BAa::»'-- SK3AMEHALJ.VlOHHbIE 5 BapHaHT .NQ TVinOBblE OTBeT aa.n;aHHJI 1 8 1 22,5 2 4 2 4 3 9 3 17 4 0,46 4 0,36 5 0,8 5 -4 6 37 6 53 7 -0,75 72 7 -0,2 8 135 8 9 0,4 9 10 6 10 96 11 35 11 28 12 14 12 -0,3 1 1t a) 1t + 21tk, 51t - + - 13 3 - + - 3 41tk, a) 41tk, k r,n;e E -1t; -. - 6) ' -+21tk 3 51t 71t 1t -3; E 7l; 1t -3; -3; k r,n;e ' 3 1t 3' 189 14 36 14 15 (-0,5; 84.J3 - 16 1 706 17 624,5) -7) 91(5.J2 16 378 ] (0,5; 0,5); -1 [-2;+oo); 15 --+21tk 3 13 7l; 1t 6) 1t 1t 400 py6JIe:n 23 1 080 17 18 6+2 19 a) .J57 paCIIpOCTpaHeHHe ; 6) HeT; @ 2021. KOIIHpoBaHHe, 000 000 H (21 ;+<X») 21; B) 19 2m a) HeT; - 5 2; 6) : ); 2 36; B) + 2m 182 82 «(HaU;HOHaJIbHOe «(M3,n;aTeJIbCTBO HCIIOJIb30BaHHe ( 18 py6JIe:n B KOMMepqeCKHX o6pa30BaHHe». u;eJI.HX 6e3 @ 2021. IIHCbMeHHoro qO¥ pa3pemeHH.H )J;rro «(MU;HMO» IIpaBOo6JIa,z:J;aTeJIeii He ,n;OIIYCKaeTC.H 187 Ol"BETb1 7 BapHaHT .M 8 BapHaHT aa)];aHHJI OTBeT 1 758 1 7 .M OTBeT aa)];aHHJI 2 12 2 4 3 7,5 3 13,5 4 0,24 4 0,32 5 4 5 -1 6 29 6 6 7 6 7 -3 8 315 8 176 9 2,72 9 -3 10 7 10 28 11 77 11 6 -3 12 38 12 1t a) k r,n;e 1t Z; E ) 131t --+21tk 6 ' 4 111t 1t . ; - _ ' 14 6) -1; log 0); 16 5 51t --+21tk 6 51t 131t ; _ ' Z; E 91t ; _ 4 45° ( k r,n;e 13 12 12 12 15 1t a) --+1tk 12 13 6 1tk, + - -- 4 6 2m 3 14 arctg 7 8 2 v'26 2); ( -00; ]; [- "' 4,01; 3,99; "' "' 2); [ (-2; 3,99]; v'26 ; +(0) (2; "'4,01]; [2 - - 54 17 925 py6JIeii 15 18 [ 0; 1,5); 19 a) 6) ,n;a; [2; + 00 HeT; B) - ) 16 16 18 17 78 " 19 @ 2021. pacrrpOCTpaHeHHe 000 II «M3,n;aTeJIbCTBO HCrrOJIb30BaHHe «Han;HOHaJIbHOe B KOMMep-qeCKHX o6pa30BaHHe». n;eJUIX 6e3 py6JIeii (-15.-] 18 KOIIIIpoBaHHe, 125 a) @ 2021. rrHCbMeHHoro qOY pa3pemeHH.H Ano ,n;a; «MU;HMO» rrpaBoo6JIa,ZJ;aTeJIeH 6) 8 B) HeT; He 12 ,n;orrYCKaeTC.H ' Ers. 188 10 BapHaHT OTBeT aa)];aHHJI .NQ BAPlt1AHT SK3AMEHAL.J.VlOHHbIE 9 BapHaHT .NQ TVinOBblE OTBeT aa)];aHHJI 1 4 1 8 2 200 2 9 3 4 3 2,5 4 0,44 4 0,18 5 3 5 4 6 60 6 64 7 4 7 14 8 18 8 4 9 4 9 8 10 6,5 10 9,6 11 6,4 11 22 12 -21 12 -8 1t a) k 13 6) 1t -k 6 7l. E 1t -+-k 18 ' a) 21t. ' 111t. - . _ 6) 2' 18 . 126 . 4' 771t 31t. . 24' , . 24' -; - 6) 50 py6ReH 1-H 1 a = 0; a = 3 ,n;a; (0;5] 16 694,4 = 5 .J119 15 1 -; a 2-H 17 3 000 H 06'beKT HeT; B) 43 2805 18 «H3,n;aTeJIbCTBO HCIIOJIb30BaHHe «HaU;HOHaJIbHOe B KOMMepqeCKHX o6pa30BaHHe». u;eJIHX 6e3 7 -qeROBeK, - 06'beKT 3 23 - 150 -qeROBeKa; py6ReH 4<a16 19 @ 2021. r,n;e 13 1 paCIIpOCTpaHeHHe 2 671t 111t 14 30.J2 + 16 KOIIHpoBaHHe, 24 24 [2;5) 15 a) . 24' 18 36 19 2 1t -- 3 a 24 51t --+-k, 7l; E 651t ' 14 -3; 1t - - 18 a 51t -+-k, , 4 I 791t 51t 1t . 13 1t ' -3' ' 17 k - k - 18 1t r,n;e ' 3 , 18 71t 1t --+-k 18 3' 131t. - 1t I a) @ 2021. IIHCbMeHHoro qOY pa3peIIIeHHH )J;rro ,n;a; 6) «MIJ;HMO» IIpaBOo6JIa,z:J;aTeJIeH HeT; 2220 B) He ,n;OIIYCKaeTCH 131t 4' . OTBETbl 189 11 BapHaHT eM " 12 BapHaHT 3a)J;aHHH OTBeT 1 .M 3a,IJ;aHHH OTBeT 540 1 2640 2 6 2 26 3 28 3 27 4 0,3 4 0,34 5 2 5 -2 6 6,5 6 30 7 2 7 3 8 54 8 27 9 -10 9 91 10 25 10 17 11 54 12 8 11 12 12 -9 I i 1t a) --+21tn 4 nE7l; ' 51t 13 mE7l; ' 21tn + - 6) 171t --. 6 -- , 4 4 5 8 31 .J10 14 arccos 15 15 2) 16 39 -1<a<0; < 8; -2<a<-1; 3 a) 6) HeT; @ 2021. paCIIpOCTpaHeHHe < a 000 H HeT; 1); 1,6 B) 8 a 19 676 < 0; 0 < a 4 < a < a) HeT; 6) < 5; HeT; 3; 3 < a 5 < a < B) < KOMMepqeCKHX o6pa30BaHHe». u;eJI.HX 6e3 @ 2021. IIHCbMeHHoro "tIOY pa3pemeHH.H IIO «MU;HMO» IIpaBOo6JIa,n;aTeJIeii He 4; 6 240 r «HaU;HOHaJIbHOe B 2) py6JIeii MJIH 0<a<3; > a (1; 27.J3 18 «M3,n;aTeJIbCTBO HCIIOJIb30BaHHe (-2; 17 9 17 KOIIHpoBaHHe, 140 [-2; 16 19 , -- 14 18 nEZ. , 6 13 71t 191t 6) :I a) 31t --+21tm 4 ,n;OIIYCKaeTC.H r Er3. 190 TV1nOBbIE 13 BapuaHT eM 14 BapuaHT OTBeT eM 3aaHHJI BAPV1AHTbI 3K3AMEHAL..J.V10HHbIE 3aaHHJI OTBeT 1 4320 1 18000 2 0,3 2 2420 3 13,5 3 6 4 0,4 4 0,556 5 -5 5 6 6 72,5 6 68 7 3 7 6 8 47 8 76 9 65 9 16 10 8 10 633 11 48 12 26 11 64 12 -1 1t 1t a) -+-n 13 , 1t 2 211t 6) nEZ. ' 4 + a) 231t . -, 251t . nEZ; - -, 4 13 -+1tn, 3 4 81t 101t 4 6) 3; 3' 14 6 [ lOg'5 15 5:7 17 2,58 ( 2-1 .J1O . _ ' 2 2 -4' 19 a) @ 2021. paCIIpOCTpaHeHHe 6) HeT; 000 H ] 48,5 [ -logl,25 : -1] 15 ; 16 10 17 .JI0 -1)u( 3 6 lOg'5 ; - 16 KOIIHpoBaHHe, 3 13 14 18 71t . - - -l. _ 2" -I} : 11 4,05 ..J10 18 + 1 -1 [1 . , 9 1 ..J10 . - 4. ", 9 . 2: 19 HeT; «(Han;HOHaJIbHOe «H3aTeJIbCTBO HCIIOJIb30BaHHe a) a; 6) B) HeT; 5 3 B) B KOMMep'tleCKHX o6pa30BaHHe». n;eJIHX 6e3 @ 2021. IIHCbMeHHoro QOY IIO pa3pemeHHH «(MIJ;HMO» IIp8.Boo6JIaTeJIeH He OIIYCKaeTCH 2 ) OTBETbl 191 15 BapuaHT eM eM OTBeT 3aaHHJI 16 BapuaHT OTBeT 3aaHHJI 1 84 1 13 2 485 2 960 3 26 3 31,5 4 0,0595 4 0,973 5 -2 5 -5 6 21 6 35 7 0,5 7 5,5 8 200 8 88 9 7,5 9 10 0,31 2,5 10 1,728 11 756 12 30 11 20 12 9 n a) nn, - 13n _ r.n;e 1t -3n, , - 2,8] 16 2; 3 , 13n -- -- , 3 ' 2 3 6.J3 ( -00; 15 4 - 2.J2] u [ 3, 5; 4 ) u ( 4; 4,5] u [ 4 + 2.J2; 20 18 [-3; a) 180; 6) .n;a; 22] B) 546 16 4:5:4 17 3 18 (-00; -3); 19 @ 2021. KOIIHpoBaHHe, 9n -- 14 v'5] + nEZ. , 2 141t 1:3:1 17 19 [ 3, u + 1tn - , 3 6) 4.J3 [ 3 v'5; 1t + 1tn - 13 4 14 + a) 11n _ 4 15 nEZ; 2 4 13 6) n + -n, paCilpOCTpaHeHHe 000 H «H3aTeJIbCTBO HCIIOJIb30BaHHe «Han;HOHaJIbHOe B KOMMepqeCKHX a) o6pa30BaHHe». n;eJISlX (-3; 6e3 @ 2021. IIHCbMeHHoro qOY pa3pellIeHHSI .n;a; )J;rro -1]; 6) 270; «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH 1; [3,25; B) +(0) 17 He OIIYCKaeTCSI + 00 ) Er3. 192 18 BapuaHT OTBeT 3aaHHJI eM BAPV1AHTbI 3K3AMEHAL..J.V10HHbIE 17 BapuaHT eM TV1nOBbIE 3aaHHJI OTBeT 1 6 1 196 2 1,2 2 2,5 3 53 3 240 4 0,2 4 0,875 5 1,5 5 0,75 6 13 6 165 7 6 7 3 8 10,5 8 216 9 1 9 -1 10 0,32 10 60 11 5 12 -81 11 3 12 1,5 a) -10g23; 6) 10g2 13 10g2 1 1 ,5; ,5 5M 14 a) -log 6) -10g5 5.J5 15 [- 15 O)U(2; ; 16 7 [-2; 18 19 a) @ 2021. KOIIHpoBaHHe, 12 H paCilpOCTpaHeHHe MJIH 6) a; 000 H a; -3 HJIH 13,5 17 7 18 ( -00; py6. 4) B) 19 a) H a; 3 MJIH 3] - 6) u a; py6. (0; 3] B) 10 20 «Han;HOHaJIbHOe «llaaTeJIbCTBO HCIIOJIbaOBaHHe +(0) 16 : )U[2; - [2; 3] 44 17 1O 14 2 1O ; -10g5 52; 13 B KOMMepqeCKHX 06paaOBaHHe». n;eJISlX 6ea @ 2021. IIHCbMeHHoro qOY paapellIeHHSI )J;rro «MIJ;HMO» IIpaBOo6JIa,n;aTeJIei He OIIYCKaeTCSI :-aETbl l- 19 BapHaHT OTBeT M 3aaHHJI 20 BapHaHT eM OTBeT 3aaHHJI 1 26 1 2 -11 2 9 3 20 3 6 4 0,09 4 2,5 5 -1 5 -2 6 2 6 10 7 11 7 7 8 96 8 111 9 9 9 10 10 0,006 10 120 11 7 12 -8 11 14 12 -18 34 500 n a) 4 13 6) 4 arctg -+nn, + nn r.n;e , n Z ; E 13n 4 arctg -, + 3n 1 n a) 4 13 4 15 16 12 17 5,35 -6 -6; < < a < 1 18 -2 < a < O. , 0 < a < [-3; 1) 9 17 - E 3n - 16 -2; n .J2 14 15 a r.n;e , 4 4 +(0) [-1; arctg- --, 1 + nn 4 1 11n 6) 14 arctg- -+nn, 2 5 Ii py6. MJIH 4 19 a) 6) HeT; 3 x 12 .n;a; 6 Ii x B) 9 6; 1 16 x x 12 Ii 16 Ii 4 x 17 --<a<-2. 4 12; x 4 ' -2 < < a 2. , 18 17 2<a<- 4 a) 19 @ 2021. KOIIHpoBaHHe, paCilpOCTpaHeHHe 000 H «Han;HOHaJIbHOe «HaaTeJIbCTBO HCIIOJIbaOBaHHe B KOMMepqeCKHX o6paaOBaHHe». n;eJISlX 6ea @ 2021. IIHCbMeHHoro qOY paapellIeHHSI 6) HeT; 2 )J;rro x 8 Ii x «MIJ;HMO» IIpaBOo6JIaaTeJIeH 4 B) .n;a; 4 4; 1 x 9 x 6 Ii 9 H 3 He OIIYCKaeTCSI x x 3 6; Z ; TV1nOBbIE Ers. 194 21 BapuaHT 22 BapuaHT OTBeT M 3aaHHH BAPV1AHTbI SK3AMEHAUV10HHbIE M 3aaHHH OTBeT 1 6670 1 11 2 16 2 15 3 5 3 8 4 0,26 4 0,48 5 -8,25 5 -1,8 6 86 7 -2 8 24 9 81 10 62 11 60 12 31 n a) 13 6) 103 7 7 8 39 9 -20 10 58 11 78 12 13 2n n -+-n :t- ' 2nk, + 2 4 6 3 E Z; n 7n 13n r.n;e n 11n -- 4 -2' nk 4 6) 4n k :t-+2nk, , - , 3 arCSIn .JPt17 14 15 (21tk; r.n;e + k E 5 21tk], [ 1t + 6 21tk; 1t + 21tk + 5 000 19 a) @ 2021. paCIIpOCTpaHeHHe 000 H 2: + 46 21tk] ' k r.n;e 8.J3 5 000 000 py6JIeii 4 4 6) .n;a; ' 3 17 py6JIeii 9 21tk- + 3 16 000 O<a<- 18 _2: - 3 3 17 [ 15 2.J3 "JI! arCSIn ). Z 16 KOIIHpoBaHHe, Z; 13n 5n , 3 14 E 3 13 -- , a) ' HeT; 18 a>l 19 7 «Han;HOHaJIbHOe «llaaTeJIbCTBO HCIIOJIbaOBaHHe 3' 9 B) B KOMMepqeCKHX o6paaOBaHHe». n;eJISlX 6ea a) @ 2021. IIHCbMeHHoro T.JOY paapellIeHHSI )J;rro -<a<l 3 -- -<a< .n;a; 6) B) HeT; 6 «MIJ;HMO» IIpaBOo6JIaaTeJIeH He OIIYCKaeTCSI E Z JTBETbl 195 23 BapuaHT . OTBeT 3aaHHH 26 1 24 BapuaHT eM 950 3aaHHH OTBeT 1 24,2 2 1678 2 4 3 11 3 16 4 0,25 4 0,15 5 17 5 4 6 73 6 28 7 7 7 4 8 72 8 13 9 27 9 16 10 24 10 44 11 14 11 65 12 6 12 40 a) k nk, +n -"6 13 6) Z; E 2 nn, + n 23n 3n; 6) 9n 2n; ( -3; 15 (; +(0) 16 30 17 3 (-3; 18 a) @ 2021. KOIIHpoBaHHe, paCilpOCTpaHeHHe 6) HeT; 000 H B) u [ 9; + 00 ) 17 1 18 (-00; a) 200 000 py6ReH 0); (16; HeT; 6) a; + B) 00 ) 989 1347 «Han;HOHaJIbHOe «llaaTeJIbCTBO HCIIOJIbaOBaHHe -1] 16 -1) a; 2.J2 16 19 19 Z; E 3n . arctg arccos 15 n 4' 14 14 2nn, + I4" 13 4n 6; Z; E n Z; E k nk, a) B KOMMepqeCKHX o6paaoBaHHe». n;eJISlX 6ea @ 2021. IIHCbMeHHoro qOY paapemeHHSI )J;rro «MIJ;HMO» IIpaBOo6JIaaTeJIeH He OIIYCKaeTCSI TV1nOBbIE Ers. 196 25 BapuaHT eM 26 BapuaHT OTBeT 3aaHHJI eM BAPV1AHTbl SK3AMEHAL..J.V10HHbIE 3aaHHJI OTBeT 1 19 1 188 2 4 2 9 3 4 3 3 4 0,76 4 0,28 5 3 5 87 6 15 6 17 7 6 7 13 8 4,8 8 61 9 10 9 63 10 3 10 0,87 11 13 11 13 12 -5 12 -3 +7t -3 a) 13 2 7tn, + n E Z; -+- 57t 7t 6) 3 kEZ; 3' 12 13 77t 6) 7tk 7t a) 37t . . - 12' 12' 4 1 14 ( 15 .J3333 14 J [ 1; 0 ) 15 67,5 000 411 17 -2 19 < a) a 6) ,n;a; ( -00; py6JIeii 1 HeT; @ 2021. paCIIpOCTpaHeHHe 000 H 17 < a < 9 282 -J6 18 «Han;HOHaJIbHOe B J; [3, + 000 2 26 B) «ll3aTeJIbCTBO HCIIOJIb30BaHHe arccos- 1 42 19 KOIIHpoBaHHe, - 16 -1; < 7t 11 ; + -00; 16 18 9 arccos KOMMepqeCKHX 06pa30BaHHe.). n;eJISlX 6e3 < a) @ 2021. IIHCbMeHHoro qOY pa3perneHHSI )J;rro a < HeT; «MIJ;HMO.) IIpaBOo6JIa,n;aTeJIeii -J2; 6) + 00 ) 18.J5 py6JIeii J2 < B) HeT; He a < 16 OIIYCKaeTC.H J6 DTBETbl 197 27 Ba.puaHT M 3aaHHJI 28 BapuaHT OTBeT eM 3aaHHJI OTBeT 1 145 1 180 2 650 2 44,4 3 4 3 4 4 0,2 4 0,25 5 5 5 58 6 21 6 56 7 4 7 1,6 8 72 8 32 9 12 9 64 10 17 10 20 11 11 11 15 12 -36 12 5 -+21tk, 6) --, :t- + 4 6 13 1t 1t 1t a) 231t k 21tk, E Z ; a) 4 4 21tk 14 81t 15 (O;] 15 [0; 5 1 18 (-0,5; 19 a) paCilpOCTpaHeHHe 000 1-); 000 H py6Rei (1+; 6) HeT; @ 2021. KOIIHpoBaHHe, 300 HeT; B) 11,75 «Han;HOHaJIbHOe B · , KOMMep'tleCKHX 7 17 2 (-00; o6pa30BaHHe». 6e3 - 3 ) 123 19 n;eJIHX 2] 16 18 +(0) «H3aTeJIbCTBO HCIIOJIb30BaHHe Z 3 1,5 17 E - 14 16 k ' 171t 6) -- --, 6 + 3 13 171t 151t - - a) @ 2021. IIHCbMeHHoro qOY )J;rro pa3pemeHHH HeT; «MIJ;HMO» IIpaBOo6JIaTeJIeH 6) HeT; B) He - 11 OIIYCKaeTCH Ers. 198 TV1nOBbIE 29 BapuaHT 30 BapuaHT OTBeT M aaAaHHH BAPV1AHTbl SK3AMEHAL..J.V10HHbIE M aaAaHHH OTBeT 1 37,5 1 61 2 68,2 2 420 3 2,5 3 5 4 0,3125 4 0,096 5 5,3 5 3 6 7 6 10 7 1 7 6 8 13 8 18 9 3 9 3 10 1200 10 4974 11 75 11 27 12 7 12 -1 a) 1 n -+2nn, n E 2 a) n + arccos-;- 14 arCSIn 15 3 5n 4 2 . 17 6 18 a) 19 @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 2v'6J; a; 000 H , 3 (-00; + 6) HeT; + 125 17 2v'6; + 00 ; ) 000 py6JIeii b b=-1, a) HeT; 6) a; B) > 6(3 O 10 - 15); ) 6! B) 7 «Han;HOHaJIbHOe «ll3,n;aTeJIbCTBO HCIIOJIb30BaHHe [ J; 7 18 00 0 16 19 (15; 3 5 15 [7 ! 14 46 30 - Z; - 16 7 E . , 3 6)' (1; log23) 0; (-00; k 2nk, + n + 4 6) Z; mE 4 3 arccos- 1 - 13 3 n-arccos-+2nm, 13 Z; B KOMMepqeCKHX 06pa30BaHHe». n;eJISlX 6e3 @ 2021. IIHCbMeHHoro qOY pa3pemeHH.H )J;rro «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTC.H 210) 3ET _ bl 199 31 BapuaHT . 32 BapuaHT aa,lJ;aHHJI OTBeT aa,lJ;aHHJI OTBeT 1 611 1 186 2 1500 2 56 3 3,5 3 20 4 0,995 4 0,996 5 2 5 2 6 38 6 2 eM 7 5 7 6 8 84 8 48 9 -7 9 -0,56 10 42 10 100 11 18 11 20 12 -5 12 -6 a) k EZ 21tk, + -1t ; 1t a) 2 1t 21tn, :t-+ 13 3 6) k EZ; --+21tk, EZ; n 1t --+21tm 6 51t --. 13 -1t ' ' m EZ. n EZ; 51t 3 --+21tn, , 6 14 17 15 : 9; (0;1); 127 + (27; 16 71° 17 20 71t 6) (0) -- ' 2 14 18 1 8 --<as--; 2 7 19 a) 6) a; 7 HeT; (0;1); 15 -sa<- @ 2021. 000 H (8; 17 10 + (0) 7 6 B) 12 «Han;HOHaJIbHOe «ll3aTeJIbCTBO HCnOJIb30BaHHe 2; 78° B KOMMep'tleCKHX o6pa30BaHHe. n;eJIHX 6e3 1 --<as--; a) @ 2021. nHCbMeHHoro qOY pa3pellIeHHH ,l];rro a; «MU;HMO. IIpaBOo6JIaTeJIeH 12 7 -sa<- 2 5 19 pacnpOCTpaHeHHe 11 : 16 18 KonHpoBaHHe, 6 7 15 15 231t . 6) 5 5 B) HeT; He 5 onYCKaeTCH' Er3. 200 Tlt1nOBbIE 33 BapuaHT 34 BapuaHT OTBeT M aa,l];aHHJI BAPlt1AHTbl 3K3AMEHAU.lt10HHbIE OTBeT M aa,l];aHHJI 1 31140 1 28 2 3 2 -21 3 26 3 5 4 0,01 4 0,09 5 5 5 3 6 4,5 6 7 4,5 7 8 24 8 9 9 -63 9 -22 10 2642 10 2408 11 250 11 48 12 4 12 32 n a) n 5n --+2nk 6 (-5; 3 - 3 14 0; 6 ..J134 [- ; ] 15 [2J5i ; 5) 50 16 3 16 6 17 12,5 [ -1,5; 18 19 a) 42; 6) - 0,375); 0 B) IIOJIOmHTeJIbHbIX; 17 1 029 18 0; 24 a) 6) B) 19 @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H Z ; 21t --; 2 4 2v'5J; E 2 26 arctg- - 6) -- 6 k 21tk, + - 3 , 2 ; 4; 21t + 13 , 5n -- , 3 _ 2 17n -- 14 15 a) kE'll. ' 7n 6) 6 --+21tk, 6 -+nk, 2 13 40 $ «HaU;HOHaJIbHOe «H3,n;aTeJIbCTBO HCIIOJIb30BaHHe B KOMMepqeCKHX 06pa30BaHHe». u;eJISIX 6e3 @ 2021. IIHCbMeHHoro QOY pa3perneHHH ,1];rrO 000 n ] ; 1 60; IIOJIOmHTeJIbHbIX; 36 «lVIIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH OTBETbl 201 35 BapuaHT M aa,l];aHHJI 36 BapuaHT OTBeT M aa,l];aHHJI OTBeT 1 140 1 200 2 5 2 6 3 4 3 6 4 0,9 4 0,2 5 11 5 13 6 66 6 130 7 -1 7 1 8 500 8 1372 9 -3 9 -3 10 12 10 18 11 3 11 1 12 5 12 11 7tk 7t a) k -+- 3' 6 13 1t 1t . . 1t 6 6) 6 ..J166 G } 15 16 14 (-<X>; ' 1 1 71 - (-v'2; v'2); 0; -1]; [1; v'2); (v'2; + <X» 280 16 6 --<aO. 7 9 ' a=-. 2 17 656 7' 18 250 4 3 2 < --<a 24 O. ' 7 a) 6) ,n;a; HeT; B) @ 2021. 000 H «H3,lJ;aTeJIbCTBO HCIIOJIb30BaHHe 2 «HaU;HOHaJIbHOe B 8 -<a<- 1632 19 paCIIpOCTpaHeHHe 9' 18 1 19 2 a=-. 9 -<a<- KOIIHpoBaHHe, -41t 3 15 4 17 --. 3 ..J59 (8;32) ; 141t - ' -2' kEZ; 3' . - -6' 14 27tk a) , 13 51t 6) · Z E KOMMepqeCKHX o6pa30BaHHe». u;eJISlX 6e3 a) @ 2021. IIHCbMeHHoro QOY pa3peIIIeHHSI ,1];rro ,n;a; 6) 9 HeT; B) 1711 «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He ,lJ;OIIYCKaeTCSI PEIDEHHH KPHTEPHH H OEHHBAHHH 3A,n:AHHH KOJIH-qeCTBO OT 6aJIJIOB, perneHHH OOUfue mpeoo6auu1£ 6bITb ,n;OJImHO 6bITb MorYT TeKCTa OTCYTCTBHH 8KcIIepTbI perneHHH perneHHH, IIpH JII06bIe 3a,n;aHHH K a) PemHTe 6) YKamHTe OTBeTa IIpaBHJIbHbIH IIpaBHJIbHbIH OTBeT IIPH 6aJIJIOB. co,n;epmaHHe 6e3 IIpe,n;CTaBJIeHHOrO IIporpaMM H o6ero cpe,n;Hero CCbIJIOK IIoco6HHX, yqe6HbIX HMelOHX peaJIH3aHH IIPH H ,n;OKa3aTeJIbCTBa y-qe6HHKax B co,n;epmaHeCH rocy,n;apCTBeHHYIO 06pa30BaHHH. 1 BapHaHT [ill ,n;OJImHbI 3aIIHCH IIOJIy-qeH 6a.rrJIOB. perneHHe CJIy-qaH cpOpMbI H 06ocHoBaHHo HCIIOJIb30BaTbCH HCIIOJIb30BaHHIO 06pa30BaTeJIbHbIX aKKpe,n;HTaHIO B03MomHbIe yqHTbIBaIOT. MorYT cpaKTbI, MaTeMaTH-qeCKHe peKOMeH,n;yeMbIx 0 B om6emOM: 3aIIHCH MaTeMaTH-qeCKoe He 3aIIHCH BbIIIOJIHeHHH ero KOJIH-qeCTBO TOJIbKO oco6eHHocTH Bce KOTOpOM oeHHBaeTCH IIpOBepHIOT a B perneHHe, MaKCHMaJIbHOe BbICTaBJIHeTCH OTBeT, cpOpMbI perneHHH, 3a pa36epuymb£M c IIOJIHbIM, rpaMOTHbIM, MeTo,n;bI paCCMOTpeHbI. 6bITb pa3HbIMH. 3aoauuu 6b£nol£ueuuro MaTeMaTH-qeCKH 3aBHCHT OTBeTa. IIpaBHJIbHOCTH 1(, 13-19, 3a,n;aHHH BbIIIOJIHeHHe 3a BbICTaBJIHeMbIX H IIOJIHOTbI 13-19 2sin{ ; )+sin2X=O. -x ypaBHeHHe 3Toro KOpHH ypaBHeHHH, rrpHHa,zJ;JIemaIIJ;He [ OTpe3KY 9 31t; 1t J. 2 PemeHHe. 3aIIHrneM a) Hcxo,n;Hoe B ypaBHeHHe 2 2cos +2sinx. x BH,n;e: cosx = sinx cosx.( 0; n 3Ha-qHT, HJIH cos X = 0, x oTKy,n;a = - + 2 k nk, E + cosx) Z, HJIH = O. sin x = -cos x; tg x = -1; n x=- 4 6) nEZ. +1tn, 9n 2 C IIOMOblO -qHCJIOBOH rrpHHa,zJ;JIemaIIJ;He 7n IIOJI"UUHM J QHCJIa: 9 [31t; OTpe3KY · 15n 1t 2 KOpHH, J 3n 9n · 4' 2' oT6epeM oKpymHocTH 2. 7n n n OTBeT: -+nk, a) 6 ) 7n. 15n. 9n 2' 4' 2. 3aMellallUe: MOblO OT6op rpacpHKa, @ 2021. KOIIP-:poBaHHe, kEZ; 2 paCilpOCTpaHeHHe --+nn KopHeH 000 JIHHeHHbIx «H3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe nEZ. MomeT perneHHeM H ' 2 , 4 «HaU;HOHaJIbHOe B KOMMepqeCKHX 6bITb 06ocHoBaH H H HepaBeHCTB 06pa30BaHHe». u;eJIHX 6e3 @ 2021. IIHCbMeHHoro T. JII06bIM ,n;pyrHM cIIoco6oM: C II. qOY )];rrO pa3pemeHHH «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He ,ZJ;OIIYCKaeTCH IIO- PEWEHV1 V1 KPV1TEPV1V1 3A)].AHV1L1113-19 OU.EHV1BAHV1 203 CO,ZJ;epmaHHe 060CHOBaHHO rrOJIY"tJeHbI 06ocHoBaHHo rrOJIY"tJeH OTBeTbI BepHbIe BepHbIH BaJIJILI KpHTepHJI 060HX B OTBeT B 2 rrYHKTax a, rrYHKTe MJIM rrOJIY"tJeHbI H OTBeTbI HeBepHbIe eTCH BbI"tJHCJIHTeJIbHOH H3-3a Bcex rrOCJIe,l];OBaTeJIbHOCTb BepHaH oruH6KH, ruaroB HO 060HX perueHHH aTOM rrpH rrYHKTOB: a 6 rrYHKTa PerueHHe He HH COOTBeTcTByeT H3 O,l];HOMY B rrpaBHJIbHOH a 6oKoBoe mecTHyrOJIbHOH SA pe6po paBHO rrJIOCKOCTb a) oKamHTe, 6) HaH,l];HTe KM qTO H M AB. M TOqKH H 2, paBHa IIJIOCKOCTb pe6pa D. IIpHMaH Cepe,l];HHa CO,l];epmHT AB OCHOBaHHH cTopoHa - 2 6aJLJL SC a rrepeceKaeT K. TOqKe B a TOqKa ABC rrJIOCKOCTH rreprreH,l];HKYJIHpHa SABCDEF rrHpaMH,l];e 8. 0 BbIrue rrepe"tJHCJIeHHbIX KpHTepHeB, M aKCUMaJLbHblU OD 1 HMe- rrYHKTa o6eM KD. = s CDKM. rrHpaMH,l];bI PemeHHe. IIycTb 1), a) (pHC. KY SO ABCDEF rrpHMaH SO CKOCTb SCF JIemHT B SCF ABC. 3HaqHT, 0 Cepe,l];HHa - H B KH A KM H a B ABC, so= ADM F A L TOqKa rrpHMoH rraparrJIeJIbHa @ 2021. paCIIpOCTpaHeHHe 000 H D TpeyrOJIbHHK KH rrpHMaH OTpe30K rreprreH,l];HKH OH AM HBJIHeTCH -OC PUC. JIHHHeH Cpe,l];HeH AB ADM, TpeyrOJIbHHKa HMeeM: 2 = "'64-4 OTpe3Ka a 3HaqHT, «HaU;HOHaJIbHOe <<l13,lJ;aTeJIbCTBO HCIIOJIb30BaHHe 2 CH=OC-OH=AB-OH=1,5. =4=0,5; Cepe,l];HHa OB, C B HBJIH- =2.JI5. KH , CH = B KOMMepqeCKHX OA. Tor,l];a rrpHMoH H 06pa30BaHHe». u;eJISIX 6e3 @ 2021. IIHCbMeHHoro = 2 JIHHHH Cpe,l];HHH CD. 3.JI5 1,5.2.JI5 .SO= oc - E = Me,l];HaHOH 3TOT CJIe,l];OBaTeJIbHO, SOC 2 .Jsc H HBJIHeTCH 3HaqHT, TOq- OTpe30K MH HD. CDKM. OTpe30K TpeyrOJIbHHKe a 1 PUC. ABCDEF CJIe,l];OBaTeJIbHO, qTO ,l];OKa3aHO, 2 KOIIHpoBaHHe, c KD. = 6bIJIO OH= IIycTb D BbICOTOH HBJIHeTCH rrapaJIJIeJIbHbI, AD, MKD, rrHpaMH,l];bI IIOCKOJIbKY CF H TpeyrOJIbHHKe rrJIOCKOCTH BbICOTOH B rrpHMoH rreprreH,l];HKYJIHpHa TpeyrOJIbHHKa KH OTpe30K paBH06e,l];peHHbIH 6) B rrYHKTe eTCH a, JIHHHH o6pa30M, BbICOTOH KYJIHpHa rrJIO- ABC. mecTHyrOJIbHHK OTpe3Ka Cpe,l];HHH TaKHM 3HaqHT, HBJIHIOaHCH H OTpe30K PaccMoTpHM rrpaBHJIbHbIH AB (pHC. 2). IIpHMbIe OH O. MKD. TpeyrOJIbHHKe Ka rrpaBHJIbHOrO CJIe,l];OBaTeJIbHO, KH, rrpHMaH IIoCKOJIb- n;eHTp TOqKOH rrJIOCKOCTH rrJIOCKOCTeH rrepeCeqeHHH rrJIOCKOCTH SCF. rrJIOCKOCTH qTO C COBrra,n;aeT rreprreH,l];HKYJIHpHa IIoJIyqaeM, SABCDEF. rrpaBHJIbHaH, H TOqKe B rrepeceKalOTCH rrHpaMHbI SABCDEF rrHpaMH,l];a MD H BbICOTa - mecTHyrOJIbHHKa B CF rrpHMbIe a 3HaqHT, QOY pa3perneHHSI LM «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH AOB TpeyrOJIbHHKa paCCTOHHHe ,1];rrO . 2 OT He TOqKH ,lJ;OIIYCKaeTCSI M TV1nOBbIE Ers. 204 CD IIpHMOH ,l];O OT paCCTOHHHIO paBHO L TOqKH CD IIpHMOH ,l];O BAPV1AHTbl SK3AMEHAU.V10HHbIE H 3 paBHO h paCCTOHHHH 4 Mem,l];Y AF IIpHMbIMH IlJIo8JJ;b CD. H CDM TpeyrOJIbHHKa paBHa 3 CD.-h SCDM 061>eM CDKM IIHpaMH,l];bI 3.J3 3 4 = =_.AB.AB.J3 = 282 paBeH !. KH . 3 3M =!. SCDM 3.J3 9..15 2 4 2 3 9..15 OTBeT: 4 CO,ZJ;epmaHHe HMeeTCH ,l];OKa3aTeJIbCTBO BepHoe BepHbIH -qeH HMeeTcH OTBeT B YTBepm,l];eHHH 060CHOBaHHO H a, rrYHKTa rrOJIY- 2 6 rrYHKTe ,l];OKa3aTeJIbCTBO BepHoe BaJIJILI KpHTepHJI YTBepm,l];eHHH a, rrYHKTa HJIH 1 060cHoBaHHo BepHbIH rrOJIyqeH YTBepm,l];eHHH a, rrYHKTa PemeHHe He OTBeT aTOM rrpH HH COOTBeTcTByeT B 6, rrYHKTe a rrYHKT H3 O,l];HOMY He HCrrOJIb30BaHHeM C B03MomHO, BbIrrOJIHeH KpHTepHeB, M aKCUMaJLbHblU PemMTe [!!] x HepaBeHcTBo 2 ( 1og64 3 2x - ) > 2 ( 4x log2 -12x + (X2 -12 0 BbIme rrepe-qHCJIeHHbIX 2 6aJLJL ). 9 PemeHHe. 3aIIHrneM HCXO,l];HOe x TeJIbHbIe qTO IlpH HHe x H3 Il pH BaH < x < x o6pa30M, OTBeT: (- 00; pacrrpOCTpaHeHHe 3 1,5, - 2x) OIIpe,l];eJIeHO 0 IIPH paBHO < x 2 -12 Y,l];OBJIeTBOpHeT X = )log2 (3- IIPH x 1 H > 2x) < 1,5, O. IIpHHHMaeT OTpHa- IIOJIOmHTeJIbHbIe IIpHHHMaeT 1,5, IIOJIyqaeM x BH,l]; x < 2 a OTpHaTeJIbHO, (X2 HepaBeHCTBY IIpHHHMaeT 2.J3]; 000 H ( (3 -2x); 2 BbIpameHHH perneHHe - 21og 1. HHTepBaJIa OrpaHHQeHHe TaKHM < x < 3HaqeHHe 1 HepaBeHCTBO @ 2021. KorrHpoBaHHe, < 1,5 3Toro X < log2 1 IIPH x IIPH 1 > (3 -2x) BbIpameHHe 3HaqeHHH 3HaqeHHH BH,l];e: 2 6.1og2 3aMeTHM, B HepaBeHCTBO -12 > 0, x OTKY,l];a < JII060e 3HaqHT, )log2 -12 (3 2.J3; - - > 2x) x > 2.J3. 3Haqe- o. Y qHTbI- 2.J3. - HCXO,l];HOrO x HepaBeHCTBa: < 2.J3; - 1 < x < 1,5. [1; 1,5). «HaU;HOHaJIbHOe «MaAaTeJIbCTBO HCnOJIbaOBaHHe B KOMMepqeCKHX 06paaOBaHHe». u;eJIHX 6ea @ 2021. nHCbMeHHoro qOY )];rrO paapemeHHH «MIJ;HMO» rrpaBoo6JIaAaTeJIeii He AorrYCKaeTCH PEWEHV1 V1 KPV1TEPV1V1 3A.LJ,AHV1L1113-19 OU.EHV1BAHV1 205 CoepmaHHe 060CHOBaHHO rroRyqeH BepHbIH 060cHoBaHHo rroRyqeH OTBeT, - 2.J3 BaJIJILI KpHTepHH 2 OTBeT OTRHqalOII1;HHCH OT HCKRlOqeHHeM BepHoro TOqeK 1, H/HRH 1 MJIM HeBepHbIH rroRyqeH OTBeT Bcex rrOCReOBaTeRbHOCTb BepHaH PeIIIeHHe He BbIqHCRHTeRbHoH H3-3a IIIarOB HH COOTBeTcTByeT OIIIH6KH, HO aTOM rrpH HMeeTCH peIIIeHHH H3 OHOMY 0 BbIIIIe rrepeqHCReHHbIX KpHTepHeB, M aKCUMaJLbHblU [K] oKpymHocTII Be paBH06e,n;peHHoro MeHbIIIeH Ha 60JIbIIIYlO MeHbIIIYIO AC, AD npHMbIe eCJIII C E, TOqKe a BC npHMaH II C yrJIOM AC IIpHMaH COOTBeTCTBeHHO. A BepIIIIIHbI npHMbIM BTOPllqHO B JIemaT BTOPllqHO nepeceKaeT D. TOqKe B qTO HaH,n;IITe B oKpymHocTb C. TOqKe B ABC TpeyrOJIbHIIKa oKpymHocTHX oKpymHocTb oKamIlTe, 06paaoM BHYTpeHHIIM 60JIbIIIeH II nepeceKaeT a) 6) acalOTCH npHMoyrOJIbHOrO 2 oaJLJL BE II napaJIJIeJIbHbI. oKpymHocTeH pa,n;IIYCbI 3 paBHbI 4. II PemeHHe. a) npllqeM CL IIycTb L TOqKII Tor,n;a no 06m;aH - II B 06 TeopeMe KaCaTeJIbHaH JIemaT no 3Haq1lT, npHMbIe AD yrJIbI CAD BeTCTBeHHbIe MbIX AE npHMoH 6) IIoCKOJIbKY MeHbIIIeH CEB II LCEB. c napaJIJIeJIbHbI, 3TIIX nepeCeqeHIIII npll yrOJI IIpHMoyrOJIbHbIe ACB AD npHMoH, oKpymHocTeH BE II - = IIycTb B npH- ACD TpeyrOJIbHIIKII LCEB) C ECB II no,n;0611H K03fPfPIIII;lIeHToM AC=BC=x, npHMoyrOJIbHOM 4 4 OCTPOMY 4 ACD: TpeyrOJIbHIIKe AD2 no - BE 3x no,n;06HbI 3 - 3 CD=-BC=-. Tor,n;a E ,n;lIaMeTpbI COOTBeTCTBeHHO. AD (LCAD COOT- nOCKOJIbKY paBHbI. 60JIbIIIeH II LLCB BE II = L AC. OT CTOPOHY KaCaTeJIbHOH Mem,n;y yrJIe LCAD= oKpymHocTeH, npHMoH II XOp,n;OH ,n;BYX O,n;HY = AC2 CD2. + , 36 x2 = 9x 2 + ' 16 24 x oTKy,n;a =-. 5 OTBeT:4,8. @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H «HaU;HOHaJIbHOe «H3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe B KOMMepqeCKHX o6pa30BaHHe». u;eJISIX 6e3 @ 2021. IIHCbMeHHoro QOY pa3perneHHSI ,1];rro «MIJ;HMO» IIpaBOo6JIaTeJIeH He ,ZJ;OIIYCKaeTCSI yr JIY Er3. 210 COAep3taHHe TlI1nOBblE BAPII1AHTbl 3K3AMEHAlJ.1I10HHbIE BaJIJILI KpHTepHH KpHTepHH (CM. 1 6aJIJI) 4 BepHO rrOJIyqeHbI Bce BepHo rrOJIyqeHbI TPH H3 rrepe"tlHCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 3 BepHo rrOJIyqeHbI Ba H3 rrepe"tlHCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 2 BepHo rrOJIyqeH - - - - rrepe"tlHCJIeHHbIe H3 OHH pemeHHe rrYHKTa a; 06ocHoBaHHoe pemeHHe rrYHKTa 0; B o:u;eHKa B rrpHMep PemeHHe He 1 6; rrYHKTe 06eCrre"tlHBalOID;HH 6, rrYHKTe pe3YJIbTaTbI pe3YJIbTaTOB: CJIeYIOID;HX 06ocHoBaHHoe HCKOMaH Ha HH COOTBeTcTByeT rrpebIYID;eH TO"tlHOCTb H3 OHOMY o:u;eHKH 0 BbIme rrepe"tlHCJIeHHbIX KpHTepHeB, M aKCUMa.nbllblU 7 BapuaHT 13 I a) 6) PemHTe Haii,lJ;HTe Bee 21t sin(2x+ ypaBHeHHe SToro ROpHH 1t )eos(4x+ 3 4 oa.n.n )-eOS2X= 3 ypaBHeHH.H, Sir:)' cos rrpHH8.,IJ;JIemaru;He __ 3 [ OTpe3RY 3 -21t; 1t 2 J. PemeHHe. Sin( a) 2 2x+ 1t )eos( 3 B03MO:>KHbI Ba ; )-eOS2X 4x+ sin x; ( 2; )eos( 2x+ ;) 4x+ = 1. CJIyqaH: sin(2X+ eos( B 2 2sin = 2 1t ) 3 4x+ ;) sin(2X+ eos( =1, HJIH = 1 2 1t ) 3 ;) 4x+ =-1, =-1. CJIyqae: rrepBOM 21t 2x + 1t == - 3 21tk + - 1t 2x ' == -- 2 1t ==--+1tk 12 + 21tk, x + 21tn; x==--+-n 6 ' + 1t 4x 21tn; == - 3 == -- 1t 3 nEZ. kEZ, re , 1t 4x 1t 12 2 1t IIoJI,ruaeM:J Bo X ... BTOpOM == + -- 12' 1tk rITeM E Z. CJIyqae: 2x 21t + 1t == - + -- 21tk ' 2 3 1t 4x+- 1t+21tn; == 3 CHCTeMa k perneHHH 71t 2x==--+21tk 6 21t 4x + 21tn; 3 == He ' 71t x==--+1tk 12 ' , 1t 1t r,n;e k E Z, n E Z. x==-+-n - 6 2 HMeeT. 1t IIOJIyqHJIH, "({TO == + -- 12 @ 2021. KOIIHpoBaHHe, X paCIIpOCTpaHeHHe 000 H r,n;e «HaU;HOHaJIbHOe «H3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe 1tk, B KOMMepqeCKHX k E Z. 06pa30BaHHe»). u;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY pa3perneHHH .n;rro «MIJ;HMO») IIpaBOo6JIa.,zJ;aTeJIeH He ,ZJ;OIIYCKaeTCH ;JEWEHII1s:1 111 KPII1TEPII1I11 C 6) 3AAAHII1 OlJ.EHII1BAHII1s:1 13-19 qHCJIOBOH rrOMOblO npHH8.,IJ;JIemaru;He 3 -21t; 1t 13n n =-12; 12 n -21t+ = X2 3n a) 6 Iln n n - · 2 2 k r,ZJ;e 12 E 12 Z. 12 13n ) Iln n. . - ' -12' 3aMettanue: 12 OT60p rpa<pHKa, MOblO ; - 12 - nk, + -- = -------- n OTBeT: n 21t12 Xl KOpHH, J. 2 =-21t+1t- Xl HaH,ZJ;eM oKpymHocTH [ OTpe3KY 211 12. KopHeH 6bITb MomeT JIHHeHHbIX perneHHeM COAep3taHHe 060CHOBaHHO rroRyqeHM 060cHoBaHHo rroRyqeH BepHbIH JII06bIM H H T. crroc060M: ,ZJ;pyrHM 060HX B B OTBeT 2 rrYHKTax a, rrYHKTe ... rroRyqeHM eTCH OTBeTbI HeBepHbIe BepHaH BMqHCRHTeRbHoH H3-3a marOB Bcex rrOCReOBaTeRbHOCTb omH6KH, HO 060HX pemeHHH aTOM rrpH rrYHKTOB: He HH COOTBeTcTByeT npaBHJIbHOH 2.J3, paBHa H3 OHOMY qeTbIpexyrOJIbHOH pe6po KpHTepHeB, 0 BMme rrepeqHCReHHbIX K TOqKH ,n:oKamHTe, qTO HaH,ZJ;HTe MK rrpHMbIe yrOJI AAI H BK = KD 2 oa.n.n OCHOBaHH5I cTopoHa AIDI MD DM: a , I I pe6pax rreprreH,ZJ;HKYJIHpHbI. BMK H BCC rrJIOCKOCTHMH Mem,ZJ;y AIK qTO Ha 3. paBHO M TaK, H ABCDAIBICID npH3Me 60KoBoe a COOTBeTCTBeHHO a) 6) a rrYHKTa M aKCUMa.nbllblU B 1 HMe- 0 rrYHKTa PemeHHe c:!D DDI H 2 = I 1. : C BI I Ma a) BKBI H KM = 2 2, +BK KMD rrpHMoyrOJIbHbIX TpeyrOJIbHHKOB BMD COOTBeTCTBeHHO rrOJIyqaeM, BK 2 = 2.J6 H . 3HaqHT, . rrpHMoyrOJIbHbIH, = 2.J7. , Al , , qTO , , 3aMeTHM, , qTO , , 2 =BM BM I a BKM TpeyrOJIbHHK MK H BK rrpHMbIe , , " , B rreprreH,ZJ;H- ., " " ..-:.:------------ KYJIHpHbI. A PUC. @ 2021. KOIIHpOBaHHe, paCIIpOCTpaHeHHe 000 H «HaU;HOHaJIbHOe «M3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe B KOMMepqeCKHX 06pa30BaHHe». :u;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY pa3perneHHH IIO AB OTMeqeHbI PemeHHe. KM rro- BaJIJILI MJIM H C II. KpHTepHH OTBeTbI BepHbIe 060cHoBaH HepaBeHCTB «MIJ;HMO» IIpaBOo6JIa.,zJ;aTeJIeH 1 He ,ZJ;OIIYCKaeTCH I Era. 212 YroJI 6) H BMK IIJIOCKOCTHMH Mem,ZJ;Y paBeH I BMK MK IIpHqeM IIpHMbIe IIPHMYIO IIYHKT a). IIoCTpoHM (CM. KM IIpHMOH lIeplleH,ZJ;HKYJIHPHYIO KMD I H ZKA 110 1I0,ZJ;06HbI ,ZJ;BYM MOH KM, 3HaqHT A1Z KT).LJ]. AIK MDt CJIe,ZJ;oBaTeJIbHO, IIoJIyqHJIH, qTO BMK ll3 K lIeplleH,ZJ;HKYJIHpHbI KZ 2). (pHC. M ADD IIJIOCKOCTH B I , TpeyrOJIbHHKH yrJIaM. .J3..J3 A IZ Z H DI IIpH- == 3. == Z A 1 AK a COBIIMaIOT, D lIep- Puc.2 KM. lIeH,ZJ;HKYJIHpHO MH A TOqKH BK H OTKY,ZJ;a , 110 lIepeceKalOTCH I BMK IIJIOCKOCTHMH Mem,ZJ;Y yrJIY Al ADD H BAPV1AHTbl aK3AMEHALJ.V10HHbIE ADDI. IIJIOCKOCTH a BCC H TV1nOBbIE H AKB yrOJI yroJI OTBeT: ,ZJ;ByrpaHHoro yrOJI IIJIOCKOCTH- Mem,ZJ;y yrJIa ADDI. ABK TpeyrOJIbHHKa npHMoyrOJIbHOrO 3HaqHT JIHHeHHbIH - BMK IIJIOCKOCTHMH Mem,ZJ;y BCC H = :; = = 45°. paBeH I tg AKB qTO nOJIYYIaeM, 45°. COAep3taHHe MMeeTCH OKa3aTeJIbCTBO BepHoe BepHbIH OTBeT MMeeTcH B a YTBepmeHHH rrYHKTa YTBepmeHHH rrYHKTa 060CHOBaHHO H rrOJIyqeH 2 6 rrYHKTe OKa3aTeJIbCTBO BepHoe BaJIJILI KpHTepHH a, MJIM 06ocHoBaHHo BepHbIH rrOJIyqeH YTBepmeHHH a, rrYHKTa PemeHHe He OTBeT aTOM rrpH HH COOTBeTcTByeT B He a rrYHKT H3 OHOMY 6, rrYHKTe 1 C B03MomHO, HCrrOJIb30BaHHeM BbIrrOJIHeH KpHTepHeB, 0 BbIme rrepeqHCJIeHHbIX M aKCUMa.nbllblU p x 6.5 eIIIHTe HepaBeHCTBO x + 25 -ll >025 · -, O ,5 -6.5x 2 6a.n.n +1 PemeHHe. t IIycTb = 5\ Torp;a HepaBeHCTBO 2 24t-44-5t 2 4(5t 3HaqHT t E ( 5) 1 IIoJIyqaeM: OTBeT: 5 < - x < 5 (-1; @ 2021. KonMpoBaHMe, t HJIH ; pacnpOCTpaHeHMe = 1, npHMeT +6t-1 -6t+1) -11 2 -6t+1 5t t >0. -, 6t BHP;: 5t 2 -6t+9 2 < _! > - 0 (t-3)2 0; · 4 < (t-l)(5t-1) -6t+1 0, 3. OTKY,ZJ;a x E ( -1; 0), HJIH 5 x == 3, x OTKY,ZJ;a == log5 3. log53. 0); 000 M «HaU;MOHaJIbHOe «Ma,n;aTeJIbCTBO MCnOJIbaOBaHMe B KOMMepqeCKMX o6paaoBaHMe». u;eJI,flX 6ea @ 2021. nMCbMeHHoro qOY paaperneHM,fl )];rro «MIJ;HMO» npaBoo6JIa,z:I;aTeJIeM He ,n;orrYCKaeTC,fl 1 , 111 KPII1TEPII1I11 PEWEHII1s:1 3AAAHII1 OLJ.EHII1BAHII1s:1 13-19 213 COep3taHHe 06ocHoBaHHo 2 OTBeT u 06ocHoBaHHo log BepHhIH IIOJIyqeH BaJIJILI KpHTepHH OT OTBeT, IIOJIyqeH OT JIHqalOIIJ;HHCH HCKJIlOqeHHeM BepHoro TOqKH 3, 5 1 HJIH HeBepHhIH IIOJIyqeH OTBeT Bcex IIOCJIeOBaTeJIhHOCTh BepHaH PemeHHe He BhIqHCJIHTeJIhHOH H3-3a marOB HH COOTBeTcTByeT omH6KH, HO aTOM IIPH HMeeTCH pemeHHH H3 OHOMY M aKCUMa.nbllblU Ha [K] ALB a) 6) C paBHo6e,n;peHHbIe LC qTO BbICOTa - IIJIOm;a,n;b L K, yrJIaMH M H 2 6a.n.n C IIpHMbIM IIpHMoyrOJIbHbIe C yrJIOM BHe AKC, TpeyrOJIbHHKH COOTBeTCTBeHHO. KLM. TpeyrOJIbHHKa KLM, TpeyrOJIbHHKa ABC Tpeyr"OJIbHHKa IIpHMoyrOJIbHOrO IIocTpoeHbI IIpHMbIMH ,n:oKamHTe, HaM,n;HTe BC H ABC BMC H AB AC, cTopoHax TpeyrOJIbHHKa 0 BhIme IIepeqHCJIeHHhIX KpHTepHeB, LC eCJIH = 4. PemeHHe. a) lIo 3HaqHT, XOp,n;bI AL a ,n;yrH, L.LCB lIo = L.MCB + = L.LCB 90°, a BC=a, AP CL 90° = CJIe,n;oBaL.MCL H LC 3HaQHT, AB=e - = BbICOTa PB lIJIom;a,n;b IIO B ,n;OKa3aHHOMY AP: 6HcceKTpHcbI = AC CP a) IIYHKTe PB : CB = b 6HCCeKTpHCa : AP a, PB + Tpe= AB = = L.BAL = 45°, AC OTclO,n;a CL = PA KM = KC + be b: ' = .J2 d= H .J2 d. H PAL IIo,n;o6HbI a+b . .J2 IIOJIOBlfHe b + = =d: paBHa ACL TpeyrOJIbHHKH e a+b AL KLM CM QTO IIOJIYQaeM, .J2 ero IIpOH3Be,n;eHHH CJIe,n;oBaTeJIbHO, HCKOMaH LC BbICOTbI IIJIOm;a,n;b 2 OTBeT:8. @ 2021. 000 H «HaU;HOHaJIbHOe «Ma,n;aTeJIbCTBO HCIIOJIbaOBaHHe B KOMMepqeCKHX 06paaOBaHHe». u;eJIHX 6ea @ 2021. IIHCbMeHHoro QOY paaperneHHH rro «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He = paBHa d -=8. 2 paCIIpOCTpaHeHHe e. . a KOIIHpoBaHHe, = ae TpeyrOJIbHHKa OCHOBaHHe M - a+b LACL yrJIaM. Tor,n;a P CL=d, H CBOMCTBY If = AB. H lIo be lIoCKOJIbKY Ha Tor,n;a 45°. = LACL a+b ,n;BYM L.MCB \AC=b, ABC. yrOJIbHHKa IIO + IIepeCeQeHHH OTclO,n;a K ,n;yrH KLM. lIycTb TOQKa xop,n;a3TH LCB. H = L.KCA = TpeyrOJIbHHKa 6) Ha OIIHpalOm;HeCH L.KCA L.KCL 3THMH 45°. YCJIOBHIO TeJIbHO, paBHbI. ACL 180°. = qeTbIpex3HaqHT, OKOJIO CTHrHBaeMbIe yrJIbI = 90° + BIIHcaHHbIM. oKpymHocTH TaKme BIIHcaHHbIe LACL 90° = OIIHcaHHOM C060M Mem,n;y MH L.BLA LACB LB If LACB paBHbI + qeTbIpexyrOJIbHHK yrOJIbHHKa = LACB YCJIOBHIO ,n;OIIYCKaeTCH d Er3. 214 COAep3taHHe MMeeTCH OKa3aTeJIbCTBO BepHoe BepHbIH qeH OTBeT 060CHOBaHHO B BAPII1AHTbl 3K3AMEHALJ.1I10HHbIE BaJIJILI KpHTepHH rrYHKTa a, H rrYHKTa a, H YTBepmeHHH 060CHOBaHHO rrOJIY- 3 6 rrYHKTe BepHbIH rrOJIyqeH TlI1nOBblE OTBeT B 6, rrYHKTe MJIM HMeeTCH OKa3aTeJIbCTBO BepHoe YTBepmeHHH 6 rrOJIyqeH HeBepHbIH pemeHHH rrYHKTa MMeeTCH BepHoe OKa3aTeJIbCTBO 06ocHoBaHHoM pemeHHH OTBeT YTBepmeHHH omH6KH apHcpMeTHqeCKOH H3-3a 2 06ocHoBaHHoM rrpH a, rrYHKTa MJIM rrpH THqeCKOH 6 rrYHKTa HeBepHbIH rrOJIyqeH OTBeT apHcpMe- H3-3a 1 omH6KH, MJIM 06ocHoBaHHo BepHbIH rrOJIyqeH a, rrYHKTa PemeHHe aTOM rrpH He OTBeT a rrYHKT He HH COOTBeTcTByeT B 6 rrYHKTe C HCrrOJIb30BaHHeM YTBepmeHHH BbIrrOJIHeH H3 OHOMY KpHTepHeB, o BbIme rrepeqHCJIeHHbIX M aKCUMa.nbHblU [E] AJ:reKcaH,ZJ;p Y AJ:reKcaH,ZJ;pa XOqeT B cepe,ZJ;HHe H 6bIJIO Kam,ZJ;oro me TY HaHMeHbIIIYro B ,ZJ;eHer BpeMH 6bICTpopacTyeii meJIaeMbIii KYIIHTb Ha AJ:reKcaH,ZJ;PY aKHii? IIaKeT HaqaJIe 100 CTOHJI 000 60JIee qeM Kam,ZJ;bIii Ha rO,ZJ;a Py6JIeii. aKHii IIaKeTa IIOKYIIKY He HO ,ZJ;opomaeT, OTKJI8.)J;bIBaTb HymHo IIaKeT a OTKJIa,ZJ;bIBaeT IIaKeT MeCHa B KOMIIaHHH. aKHii, IIOKYIIKY AJ:reKcaH,ZJ;p KOHe CYMMY HeKOTopoe Ha MeCHa a CYMMY, aKHii IIaKeT KYIIHTb He 3 6a.n.n 30 O,ZJ;HY 0/0. KaKYro qTo6bI MeCH, qepe3 PemeHHe. AJ:reKcaH,ZJ;p IIYCTb n-ro MeCHa 100 000 AJ:reKcaH,ZJ;pa Py6JIeii. Y 1,3 · n - B OTKJIa,ZJ;bIBaeT 1 cepe,ZJ;HHe CKOIIHTCH ,IJ:JIH Kam,ZJ;oro qTo6bI Toro x MeCHa Py6JIeii, AJ:reKcaH,ZJ;p nx a 6Y,ZJ;YT aKHH CMor 3TOM qTo6bI Heo6xo,ZJ;HMO, MeCHe, BbIIIOJIHHJIOCb x 60JIee aKHii IIaKeT KYIIHTb HepaBeHCTBO cepe,ZJ;HHe He CTOHTb 100 B K Py6JIeii. > 3 000.1, n - 1 . IIo- n 100 JIOmHM 3 000.1, n - 1 = an ,n:JIH . qTo6bI Toro AJ:reKcaH,ZJ;p CMor qepe3 HeKOTopoe BpeMH n paBHYro Heo6xo,ZJ;HMO aKHii, IIaKeT KYIIHTb HaHMeHbIIIeMY ,IJ:JIH 3Toro BbIQHCJIHM H QHCeJI H3 ,ZJ;OCTaTOQHO CpaBHHM an. n < 4 BbIIIOJIHeHO 3HaQHT, 54 HaHMeHbIIIHM II03TOMY Py6JIeii. HaHMeHbIIIaH < an, H3 a QHCeJI CYMMa, OTcro,ZJ;a . 4 n > IIPH a n n+1 10n+10 BbIIIOJIHeHO > an+l QHCJIO KOTOPYro QHCJIa. a4 HymHo IIOJIYQaeM, 54 @ 2021. paCIIpOCTpaHeHHe 925 3 = = 54 925. 4 OTKJIa,ZJ;bIBaTb AJ:reKcaH,ZJ;PY, paBHa Py6JIeii. 000 H «Ha:U;HOHa.rrbHoe «Ha,n;aTeJIbCTBO HCIIOJIbaOBaHHe B KOMMepqeCKHX 06paaOBaHHe». :u;eJIHX 6ea @ 2021. IIHCbMeHHoro q,TO an. 100000.1,3 6Y,ZJ;eT TaKHX 925 OTBeT: KOIIHpoBaHHe, an+l JIH60 60JIbIIIYro 13n - an IIPH CYMMY IIOCJIe,ZJ;OBaTeJIbHbIX ,ZJ;Ba 1,3.n an+l OTHOIIIeHHe: HX OTKJIa,ZJ;bIBaTb qOY paaperneHHH ,n;rro «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH PEWEHII1 111 KPII1TEPII1I11 3AAAHII1 OU.EHII1BAHII1 13-19 215 COep3taHHe 06ocHoBaHHo BepHo BaJIJILI KpHTepHH u rroRyqeH Ma TeMa rrOCTpoeHa 3 OTBeT BepHbIH THqeCKaH MO,ZJ;eRb, pemeHHe K CBe,ZJ;eHO HCCRe,ZJ;OBaHHIO u aTOH H MO,ZJ;eRH HeBepHbIM BepHbIM - - BepHo rroRyqeH OTBeT HO OTBeT, aTOM rrpH He 060CHOBaHO He,ZJ;OCTaTOqHO pemeHHe MO,ZJ;eRb, HH COOTBeTcTByeT H3 O,ZJ;HOMY He K CBe,ZJ;eHO pemeHHe 6bITb MomeT pemeHHe 2 omH6KH; MaTeMaTHqeCKaH MO,ZJ;eRH, PemeHHe BbIqHCRHTeRbHoM H3-3a rrOCTpoeHa aTOM pe3YRbTaT: HCCRe,ZJ;OBaHHIO KpHTepneB, 0 BhIllie rrepeqnCJIeHHbIX Ha KaKHX IIPH [-1; OTpe3Ke 1] HMeeT pOBHO TOqKY O,ll;HY f (x) cl>YHKD;HSI a 3Ha"'IeHHSlX HeC;>TpHD;aTeJIbHbIX 3 oaJtJt M aKCUMaJtbHblU Haii,l ;HTe, 1 3aBepmeHO = 3ax 4 - 8x 3 + 2 3x -7 MHHHMYMa. PemeHHe. HaM,lI;eM 3 12ax B x TOqKe TOqKOM eTCH a) T 2 IIpH 0 = 4 3ax = -4x+1==0 a 3 8x - HMeeT ,lI;Ba IIpH a E (0; 2) x 2 C « -7 MomeT x 3 == KOpHH, ,lI;Ba KOpHH: HMeeT 0, == + », « JIemHT x O. x TOqKa II03TOMY = 0 HBJIH- Ha a E @ 2021. 0 ypaB- 1 X ==-. H qHbIX 000 H 1], + 00 H TOqKH a a > Xl < eCJIM 1, 2a 3ax 4 - 3 8x + 3x 2 -7 H x a 3TO Ha Xl r,ll;e , 2a < X2 < X3; TOqKaMH - 3 BbIIIOJIHeHO [-1; OTpe3Ke 1] Toq- MHHHMYMa. IIPH Bcex HMeeT O,ll;HY a > 2. ). «HaU;HOHaJIbHOe «Ma,n;aTeJIbCTBO HCIIOJIbaOBaHHe == KOpHH: = X3 MaKCHMYMa, f (x) [ 0; 1,5) [2; [0; 1,5); paCIIpOCTpaHeHHe IIPH == eCJIH MHHHMYMa, 2+ .J 4-2a H [-1; OTpe3Ke IIPH OTBeT: TOqKY a<2. 4 pa3JIH .J 4-2a 2a <PYHKD;HH MHHHMYMa em;e 3HaqHT, TPH == X2 TOqKOM HBJIHeTCH IIoJIyqHJIH: KOIIHpoBaHHe, 1) + Ha HMeTb 2+ .J 4-2a Ka -4x MaKCHMYMa. ypaBHeHHe 2Xl TOqKa 3HaK » 6x. + u TOqKOM 4 6) 2 - 2 24x - 6x(2ax 0; == a HMeeT ypaBHeHHe HBJIHeTCH == 2 3x + 1 x OqKa 6x + MeHHeT IIpOH3BO,ll;HaH f (x) 2ax 2 24x - 3 12ax == MHHHMYMa. <l>YHKD;HH HeHHe 0 == f' (x) <PYHKD;HH: IIpOH3BO,ll;HYIO B KOMMepqeCKHX 06paaOBaHHe». u;eJIHX 6ea @ 2021. IIHCbMeHHoro qOY )];nO paapemeHHH «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH 1,5. TOqKY Er3. 216 Co.n;epmaHHe 060CHOBaHHO C rrOMOm;blO C HCKOMoro rrOMOm;blO BepHoro BaJIJILI KpHTepHH 4 3Ha"tleHHH MHomeCTBO rroy"tleHo BKIO"tIeHHeM/HcKIO"tIeHHeM TObKO TO"tleK HCKIO"tIeHHeM BKIO"tIeHHeM/HcKIO"tIeHHeM a TO"tleK TO"tlKH 1,5 = a H/HH a = a 0, a OTH"tIalO- a, 1,5 = H/HH 3Ha"tleHHH MHomeCTBO rroy"tleHo paccym,n;eHHH TObKO HCKOMoro OT m;eeCH BAPII1AHTbl 3K3AMEHAL.J.1I10HHbIE OTBeT paccym,n;eHHH BepHoro OT m;eeCH BepHbIH rroyqeH TlI1nOBblE a OTH"tIalO- a, 6bITb, MomeT TaKme, 3 2 = 2, = 2 MJIM HeBepHbIH rroy"tleH OTBeT Bce BbIrrOHeHbI marH HaH,n;eHbI BepHO PemeHHe Bce He BbI"tIHCHTebHoH H3-3a omH6KH, HO aTOM rrpH BepHO pemeHHH TPH TO"tlKH rpaHH"tIHbIe HH COOTBeTcTByeT 3Ha"tleHHH MHomeCTBa H3 O,n;HOMY KpHTepHeB, 1 a 0 BbIme rrepe"tlHCeHHbIX M aKCUMa.nbHblU I 19 I ,n:JIH Kam,n;oro a) Cym;ecTByeT CKOJIbKO 9 HaTypaJIbHOe 23 pOBHO cym;ecTByeT n rrepBbIx qHCJIO n, qTO ,n;eCHTHqHaH 3arrHCb qHCJIa nl qHCJIO n, qTO ,n;eCHTHqHaH 3arrHCb qHCJIa nl HYJIHMH? qHCeJI HaTypaJIbHbIX 3arrHCb ,n;eCHTHqHaH rrpoH3Be,n;eHHe HYJIHMH? TaKoe JIH OKaHqHBaeTCH B) HaTypaJIbHOe pOBHO Cym;ecTByeT nl qepe3 1). == TaKoe JIH OKaHqHBaeTCH 6) (11 qHCeJI HaTypaJIbHbIX 0603HaqHM n qHCJIa HaTypaJIbHOrO 4 6a.n.n nl. qHCJIa (100 100, MeHbIIIHX n, n)1 - OKaHqHBaeTCH H3 Kam,n;oro ,n;JIH 23 pOBHO KOTOpbIX HYJIHMH? PemeHHe. ,n:JIH a) n 5 KpaTHbIX 5 HbIX 3HaqHT, n 5 KpaTHbIX 5 ,n;eJIHTCH 10 BaeTCH meT H, 22 , qHCJIO HO He 1 OKaHqHBaTbCH 2 23 99. HO 23 H, , qHCeJI 000 p . qHCJIO r,n;e q, 99, a q .r, HO , He He He CJIe,n;oBaTeJIbHO, 3TO rrpOH3Be,n;eHHe ,n;eJIHTCH ero r,n;e ero CJIe,n;OBaTeJIbHO, 3arrHCb rrpH qHCJIa n > nl 100 r qHCJIO 5 Ha 5. Ha 23 . He qHCJIO Ha ,n;eJIHTCH 60JIee qeM q ,n;eJIHTCH OKaHqH- 99 n < rrpH 5. qHCJIO nl 3arrHCb nl KpaT- Ha 3TOM qHCJIO ,n;ecHTHqHaH OKaHqHBaeTCH He ,n;eJIHTCH IIpH 3HaqHT, qHCJIa Bcex Bcex rrpOH3Be,n;eHHe ,n;eJIHTCH ,n;eJIHTCH - p 3TO - 22 22 a HO , H, , KpaT- 1001 == 23 MO- He 991 .100 HYJIHMH. 6bIBaeT. Tor,n;a <tHa:U;HOHaJIbHOe B 5 10 10 Ha ,n;eJIHTCH 9 16.32==2 Ha He HYJIHMH. == ,n;o Bcex rrpOH3Be,n;eHHe q ,n;eJIHTCH Bcex rrpOH3Be,n;eHHe p==5.10.....40==5.(2.3.4.6.7.8) qHCJIO x qHCJIa x. «Ma,n;aTeJIbCTBO HCIIOJIbaOBaHHe 99 q 3TO - p 3TO - ,n;eCHTHqHaH HYJIHMH, He a He 9 qHCJIO r,n;e q, .(1.2.....19)==5 3arrHCb rrpeBOCXO,n;Hm;ee H ,n;o ,n;eHCTBHTeJIbHOrO Kam,n;oro He 1 .3, ,n;eCHTHqHaH TaKHX OT IIo3ToMY pOBHO HO , . Ha 22 10 HYJIHMH. CJIe,n;OBaTeJIbHO, paCIIpOCTpaHeHHe 1.2.... == . qHCJIO 9 10 ,n;eJIHTCH Ha a , p 40, Tor,n;a ,n;o 10 19 == == pOBHO I OT p ,n;eJIHTCH 22 @ 2021. KOIIHpoBaHHe, 5 p==5.10.....95==5 4.16.32.64.96 qHCJIO, Ha qHCeJI qHCeJI . 40. Ha n rrOJIyqaeM Ha pOBHO ,n;o ,n;eJIHTCH HaTypaJIbHbIX qHCJIO CJIe,n;oBaTeJIbHO, B) ,n:JIH JIoe 99 . 1 1 OT 40 . . OKaHqHBaeTCH HaTYPaJIbHbIX Tor,n;a CJIe,n;oBaTeJIbHO, Ha == . OT ,n;eJIHTCH nl 3arrHCb ,n:JIH HbIX He qHCJIO ,n;ecHTHqHaH 6) HO 1. 2 == qHCeJI qHCeJI 59, I n rrOJIyqaeM HaTypaJIbHbIX Ha 5. 40 HaTypaJIbHbIX ,n;eJIHTCH Ha == KOMMepqeCKHX JII06oro ,n;JIH 06paaOBaHHe)). :u;eJIHX 0603HaqHM 6ea qepe3 [X] qHCJIa HaTypaJIbHOrO @ 2021. IIHCbMeHHoro qOY paapellIeHHH ,n;rro HaH60JIbIIIee <tMIJ;HMO)) IIpaBOo6JI8.,lJ;aTeJIeH m He :u;eH JII06oro ,n;OIIYCKaeTCH Er3. 218 a) PeIIIHTe 6) ¥KamHTe 3Toro KOpHH -.J2 2x COS ypaBHeHHe (; ) + x cos BAPV1AHTbl 3K3AMEHAlJ.V10HHbIE 11 BapuaHT [ill TV1nOBbIE 1 + O. = npHHa,n;JIemaru;He ypaBHeHHH, [- OTpe3KY 51t; 7; 1 - PemeHHe. 3arrHIIIeM a) sin 3HaYIHT, x Y paBHeHHe C 6) = - ' sin x ==.J2 .J2 sin x + x KopHeH He 7 [- OTpe3KY 51t; qHCJIa qHM y OTBeT: x + .J2 )( sin x .J2) - + 21tn, n Z, E X HJIH = O. == 1t + - 4 21tm, m E Z . HMeeT. 1t oT6epeM - 2 71t KOpHH, -- 2 ]. 17n - -51t - -4' 4 n a) sin 3 : - (2 0; == oKpymHocTH 19n IIoJI = BH,lI;e: 1 x + oTKy,n;a qHCJIOBOH rrOMOID;blO npHHa,n;JIemaru;He --+2nn, 4 n 3n Z. E - 2nm, + - , m E Z · ' 4 17n 19n 6) 2 2 sin 1- B ypaBHeHHe HCXO,ll;HOe - -- ' 4 4 CO,ZJ;ep3taHHe 060CHOBaHHO rrOJIyqeHbI BepHble 060cHoBaHHo rrOJIyqeH BepHbIH OTBeTbI OTBeT BaJIJILI KpHTepHH B B 060HX 2 rrYHKTax a, rrYHKTe MJIM HMeeTCH rrYHKTa PemeHHe OTBeTbI HeBepHble rrOJIyqeHbI a H He BblqHCJIHTeJIbHOH H3-3a Bcex rrOCJIeOBaTeJIbHOCTb BepHaH omH6KH, marOB COOTBeTcTByeT HH OHOMY H3 KpHTepHeB, SABC rrpaBHJIbHOH TpeyrOJIbHOH rrHpaMH,lI;e 7. Ha AB H SC SA pe6po paBHO pe6pax AK KB 1 SM : MC : 5. : IIJIOCKOCTb rrpHqeM BC. rrpHMoH B == a) 6) ,II;oKamHTe, Mem,ll;y yrOJI @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H rrYHKTOB: cTopoHa a a «Ma,n;aTeJIbCTBO HCIIOJIbaOBaHHe 4Ha:O;HOHaJIbHOe B rrpHMoH rrapaJIJIeJIbHa rrJIOCKOCTHMH KOMMepqeCKHX a a.n blU AB OCHOBaHHH K TOqKH bll rrpHMYIO H 6, paBHa M KM 2 6a.n.n a 6oKoBoe COOTBeTCTBeHHO, H rrapaJIJIeJIbHa SA. SBC. H 06paaOBaHHe». :o;eJIHX a KCU.JVl CO,ll;epmHT 0 Bblme rrepeqHCJIeHHbIX OTMeqeHbI == rrJIOCKOCTb qTO HaH,lI;HTe 1 aTOM rrpH 6 rrYHKTa M @J HO 060HX pemeHHH 6ea @ 2021. IIHCbMeHHoro qOY paapeIIleHHH ,n;rro «MIJ;HMO» IIpaBOo6JIa.,zJ;aTeJIeH He ,n;OIIYCKaeTCH PEWEHII1s:1 111 KPII1TEPII1I11 3AAAHII1 OL.J.EHII1BAHII1s:1 13-19 219 PemeHHe. a) lIycTb lIoCKOJIbKY LM rrJIOCKOCTb BC rrpHMaH BC H a a SL: TaKHM TOQKa H H SH AH ASH rrJIOCKOCTb CJIe,ZJ;oBaTeJIbHO, BC, rrpHMoH B ASH ASH AS rrpHMYIO SA, rrpHMoH SA rrpHMoH rrJIOCKOCTHMH rrapaJIJIeJIbHOH a, BC. JIemam;eH H rrJIOCKOCTblO a H ASH, rrJIOCKOCTH B SBC. SBC paBeH ASH. yrJIY =3.J3, SH 2 ..JSB = 2 -BH SB2 = _ B2 =2 .Jl0 . KOCHHYCOB TeopeMe 2 SA LASH cos +SH 2 2SA. SH 2 -AH 31.JIO 49+40-27 = .J10 2.7.2 140 31 .J10 OTBeT: 6) arccos 140 COAep3taHHe MMeeTCH ,n;OKa3aTeJIbCTBO BepHoe BepHbIH qeH rrJIOCKOCTH rreprreH,ZJ;HKYJIHpHa HMeeM: AH 7, = B 3HaQHT, BC. cO,ZJ;epmam;eH Mem,ZJ;Y SBC H a rrapaJIJIeJIbHa Mem,ZJ;Y yrOJI c Tor,ZJ;a ABC BbICOTaMH, SBC, a BC. pe6pa rrpHMoH yr JIY o6pa30M, TpeyrOJIbHHKe lIo KL, rrpHMYIO cepe,ZJ;HHa HX rrJIOCKOCTH paBeH yrOJI rrapaJIJIeJIbHbI. TpeyrOJIbHHKOB rrJIOCKOCTb TaKHM SA H rreprreH,ZJ;HKYJIHpHa lIoCKOJIbKY HCKOMbIH - rrJIOCKOCTb H KB. cO,ZJ;epmam;aH a, HBJIHIOTCH COOTBeTCTBeHHO rrpHMbIe SA. rrpHMoH lIycTb Me,ZJ;HaHbI KL rrJIOCKOCTb rrapaJIJIeJIbHa 6) MC=AK: rrpHMbIe o6pa30M, a, s 3HaQHT, LB=SM: CJIe,ZJ;oBaTeJIbHO, L. TOqKe B rrJIOCKOCTH rrapaJIJIeJIbHa rrapaJIJIeJIbHbI, SB pe6po rrepeceKaeT MMeeTcH OTBeT B YTBepm,n;eHHH a, rrYHKTa 060CHOBaHHO H rrOJIY- 2 6 rrYHKTe ,n;OKa3aTeJIbCTBO BepHoe BaJIJILI KpHTepHH YTBepm,n;eHHH a, rrYHKTa MJIM 06ocHoBaHHo BepHbIH rrOJIyqeH YTBepm,n;eHHH a, rrYHKTa PemeHHe He OTBeT aTOM rrpH HH COOTBeTcTByeT B a rrYHKT H3 O,n;HOMY 1 6, rrYHKTe C B03MomHO, HCrrOJIb30BaHHeM BbIrrOJIHeH He 0 BbIme rrepeqHCJIeHHbIX KpHTepHeB, M aKCUMa.nbllblU PeIIIHTe > (12-6x) logo,5 HepaBeHcTBo (X2 -6x+8)+ logo,5 (x+ logo,5 2 6a.n.n 3). PemeHHe. 3arrHllIeM HCXO,ZJ;Hoe (6( logo,5 logo,5 @ 2021. KOIIHpoBaHHe, pacIIpocTpaHeHHe 000 H B HepaBeHCTBO 6 + ( logo,5 x 2 - B BH,ZJ;e: )) > logo,5 (( ) > logo,5 ( x «HaU;HOHaJIbHOe «Ha,n;aTeJIbCTBO HCIIOJIbaOBaHHe 2 - KOMMep-qecKHx 4 4 - 06paaOBaHHe». u;eJIHX 6ea x - x) )( 2 + logo,5 - @ 2021. IIHCbMeHHoro x )) + ( 2 ( logo,5 - qOY paaperneHHH x ,l];rrO ) + x + 3) logo,5 «MIJ;HMO» IIpaBOo6JIa.,zJ;aTeJIeH ; ( x + 3) . He ,n;OIIYCKaeTCH Er3. 220 HepaBeHCTBO HHMaeT -3 IIPH OIIpeeJIeHO < x < 2, < 12+x-x TV1nOBbIE II03TOMY BAPV1AHTbl 3K3AMEHAU.V10HHbIE -3 IIPH < x < 0, 2 < HepaBeHCTBO IIPH- BH: 6 -2 oTKya < OTBeT: < x 3. 4-x)(x+3); « Y"tJHTbIBaH 6 -3 orpaHH"tJeHHe 2 < 2, < x x ; 2 -x-6 -2 IIOJIy"tJaeM: < x 2. < 2). [-2; Co,n;epataHHe 060CHOBaHHO IIOJIyqeH BepHbIH 060cHoBaHHo IIOJIyqeH OTBeT, BaJIJILI KpHTepHJI 2 OTBeT OTJIHqalOID;HHCH OT HCKJIlOqeHHeM BepHoro TOqKH -2, MJIM HeBepHbIH IIOJIyqeH OTBeT Bcex IIOCJIeOBaTeJIbHOCTb BepHaH PeIIIeHHe He BbIqHCJIHTeJIbHOH H3-3a HH COOTBeTcTByeT IIIarOB 1 OIIIH6KH, HO 3TOM IIPH HMeeTCH peIIIeHHH H3 OHOMY KpHTepHeB, M aKCUMaJlbHblU 0 TO"tJKa - ,n:oKamHTe, HaHHTe B OKOJIO oIIHcaHHYIO IIepeceKaeT a) 6) BIIHcaHHoH eHTp LPOA "tJTO B oKpymHocTb APO, 6, LBAC TpeyrOJIbHHKa ABC TpeyrOJIbHHKa BO IIpHMaH oKpymHocTH. TpeyrOJIbHHKa 2 6aJlJl TO"tJKe BTOpH"tJHO P. LPAO. == IIJIOID;Mb ABC TpeyrOJIbHHK 3Toro 0 BbIllie IIepeqHCJIeHHbIX oKpymHocTH paBeH eCJIH == oIIHcaHHoH paHYC LABC 75°, == OKOJIO 60°. PemeHHe. IIoCKOJIbKY a) ABC HHK TpHcaMH oKpymHocTH, yrJIOB BHellIHHM 0 TO"tJKa yrJIOM LPOA - BIIHcaHHOH eHTp AO JIy"tJH TpeyrOJIbHHKa ABC. TpeyrOJIbHHKa AOB. LBAO == + == B YroJI POA - LBAC 1 + 2 YrJIbI H TY ABC, me PAC H PBC yry oKpymHocTH, paBHbI, TpeyrOJIb6HcceK- HBJIHIOTCH HBJIHeTCH CJIeOBaTeJIbHO, 1 LABO BO H - LABC . 2 IIOCKOJIbKY Ha OIIHpalOTCH oIIHcaHHoH OKOJIO TpeyrOJIbHHKa II03TOMY LPAO==LPAC+LOAC==LPBC+LOAC==-LABC+-LBAC. TaKHM 6) LPOA o6pa30M, R IIycTb 6 == - LPOA IIoCKOJIbKY == paHYC == @ 2021. paCIIpOCTpaHeHHe 000 H OIIHcaHHOH oKpymHocTH, LPAO, APO TpeyrOJIbHHK «H3,n;aTeJIbCTBO HCIIOJIb30BaHHe == AP == «HaU;HOHaJIbHOe B 1 1 2 2 LPAO. o P KOIIHpoBaHHe, P OHY KOMMepqeCKHX 2R sin LABP 06pa30BaHHe». u;eJIHX 6e3 OKOJIO paBHo6epeHHbIH, == 2R sin @ 2021. IIHCbMeHHoro CJIeOBaTeJIbHO, 30° QOY ABC. TpeyrOJIbHHKa == ,1];rrO pa3pemeHHH 6 . «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH - PEWEHV1 V1 KPV1TEPV1V1 3AAAHV111113-19 OU.EHV1BAHV1 TaKHM 06pa30M, IIJIOm;ab 2t APO TpeyrOJIbHHKa AP.OP.sinLAPO 2 AP paBHa .sinLACB AP 2 .sin45° 9.J2. = 222 OTBeT: 9.J2. 6) Co,n;epataHHe I1MeeTcH ,n;OKa3aTeJIbCTBO BepHoe BepHbIH qeH OTBeT 06ocHoBaHHo B YTBepm,n;eHHH rrYHKTa a, H rrYHKTa a, H 060CHOBaHHO rrOJIY- 3 6 rrYHKTe BepHbIH rrOJIyqeH BaJIJILI KpHTepHJI OTBeT B 6, rrYHKTe I1JII1 HMeeTCH ,n;OKa3aTeJIbCTBO BepHoe pemeHHH rrYHKTa I1MeeTcH BepHoe YTBepm,n;eHHH 6 rrOJIyqeH HeBepHbIH ,n;OKa3aTeJIbCTBO OTBeT YTBepm;n;eHHH omH6KH apHcpMeTHqeCKOH H3-3a 2 06ocHoBaHHoM rrpH a, rrYHKTa I1JII1 06ocHoBaHHoM rrpH pemeHHH 6 rrYHKTa HeBepHbIH rrOJIyqeH OTBeT H3-3a 1 omH6KH, apHcpMeTHqeCKOH I1JII1 06ocHoBaHHo BepHbIH rrOJIyqeH a, rrYHKTa PemeHHe 3TOM rrpH He OTBeT He a rrYHKT HH COOTBeTcTByeT B 6 rrYHKTe C HCrrOJIb30BaHHeM YTBepm,n;eHHH BbIrrOJIHeH H3 O,n;HOMY KpHTepHeB, 0 BbIme rrepeqHCJIeHHbIX M aKCUMaJlbHblU 15 [E] MecHeB). 1-ro HCJIa 14-e 110 HCJIa OJIra Ha HeKoTopMH Ha (eJIoe CpOK HCJIO TaKOBbI: B03BpaTa MeCHa HCJIO OJIr 1 % Ha B03paCTaeT 15-e HCJIO MeCHeB (CHTaHTe, OJIr 110 C cpaBHeHHIO KOHOM oKpyrJIeHHH CYMMa KpeHTa Ha OHY H TY me OJIra; MeHbrne CYMMY MeCHa. IIpebIym;ero B3HTb IIorarneHHH TO aCTb BbIIIJIaTHTb 6bITb OJImeH IIJIaHHpyeTcH IIOJIHOrO IIOCJIe He06xoHMO MeCHa Kamoro MeCHa Kamoro CKOJIbKO BbIIIJIaT 6aHKe B KpeHT 3 MeCHa; 2-ro 15-ro Ha ero Kamoro IIpebIym;ero co B3HTb IIJIaHHpyeTcH YCJIOBHH HHBapH 6aJlJl % 20 Ha KpeHTa 60JIbrne 06m;aH TO H3BeCTHO, B3HTOH CYMMbI, IIJIaTemeH BbIHCJIeHHH IIPH eCJIH KpeHT, CYMMa B KpeHT? IIpOH3BOHTCH.) He PemeHue. IIYCTb JIOBHIO, OJIr IIepe 8, paBHa 6aHKOM 110 a KpeHT COCTOHHHIO B3HTb IIJIaHHpyeTcH 15-e Ha HCJIO OJImeH Ha n IIo MeCHeB. o YMeHbrnaTbCH paBHOMepHO: 8; @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H «HaU;HOHaJIbHOe «H3aTeJIbCTBO HCIIOJIb30BaHHe B KOMMepqeCKHX 28 8(n-1) . 8 -. , n -; , n 06pa30BaHHe». u;eJIHX 6e3 o. n @ 2021. IIHCbMeHHoro QOY ,1];rrO pa3pemeHHH «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He OIIYCKaeTCH YCHYJIH TV1nOBbIE Er3. 222 IIepBoro HOCTb -qHCJIa MeCHn;a KajKOrO OJIra pa3MepOB 1-e Ha 1,01S( 1,018; CJIeOBaTeJIbHO, 8 n Bcero CJIeYIOm;HMH: 8 n ( +...++! + n n % +-. n n 8 1) 0,01(n+ 1+ ) 2 B3HTOH CYMMbI, B . KpeHT, n03TOMY n=39. =0,2; 39. 060CHOBaHHO BepHo BepHbIM rroRyqeH H HeBepHbIM BepHbIM BepHo HO Mo,n;eRH, 3TOM rrpH K CBe,n;eHO pemeHHe HCCRe,n;OBaHHIO 060cHoBaHo Mo,n;eRb, HH COOTBeTcTByeT H3 O,n;HOMY He K CBe,n;eHO pemeHHe 6bITb MomeT pemeHHe 2 omH6KH; He,n;OCTaTOqHO pemeHHe MaTeMaTHqeCKaH rrOCTpoeHa He Mo,n;eRb, BbIqHCRHTeRbHoM H3-3a OTBeT, PemeHHe 3 pe3YRbTaT: rroRyqeH OTBeT BaJlJlLI KpHTepHJI OTBeT MaTeMaTHqeCKaH rrOCTpoeHa Mo,n;eRH 3TOM n 60JIbme Co.n;epataHHe - ) ( = n 20 Ha BbInJIaT CYMMa n-1 n 2 - n n 8 0,018 +-; , 0,01(n+1) 3TOM 8 0,028 . + n 8 +0,018 OTBeT: n BbInJIaTHTb CJIeyeT 06m;aH , 6bITb 0,018(n-1) . +, . n OJIjKHbI nOCJIeOBaTeJIb- 3Ha-qHT, 1,018 2,028 ; n BbInJIaTbI O , 018 -1) n %, TaKOBa: MeCHn;a KajKOrO 1 Ha B03paCTaeT OJIr -qHCJIO BAPV1AHTbl 3K3AMEHAU.V10HHbIE HCCRe,n;OBaHHIO KpHTepHeB, 0 BbIme rrepeqHCReHHbIX M aKCUMaJlbHblU HaH,D;MTe Bce 3Ha-qeHMSI a, M3 Kam,D;OM rrpM KOTOpbIX 13xl-2x-2-a 2 x HMeeT Ba pOBHO pa3JIH-qHbIX 1 3aBepmeHO 3 6aJlJl ypaBHeHMe =0 -2x-a KOpHH. PemeHHe. KOpHHMH aeT (0; OHH 2x -2). 2 - - YpaBHeHHe @ 2021. KOIIHpoBaHHe, a 0 = 3Ha-qHT, KopeHb paCIIpOCTpaHeHHe 13xl- Oxa l1 a 2 x - 000 H = 13xl- -2 2x - He H a 0 = «HaaTeJIbCTBO HCIIOJIbaOBaHHe C HMeeT 3aaeT «HaU;HOH8.JIbHOe B KOMMep'tleCKHX 2 a - KOpHH 0 = 2 - - 2 - - a npHHHMaeT a = = KopHeH napa60JIY B To-qKe 0 H 0 a < a 6ea = x 2 @ 2021. IIHCbMeHHoro - BH (0; 3aaeT HMeeT npH 06paaOBaHHe». u;eJIHX 13xl- ypaBHeHHH 2x 2 - - a 0, = O. * a - Ha-qarrOM x 2x 2x - - BH npHHHMaeT ypaBHeHHe npH 2x JIy-q 2 i YCJIOBHe ypaBHeHHe nJIOCKOCTH Ha - 0 X < IIpH HBJIHIOTCH ypaBHeHHH BbInOJIHeHO KOTOpbIX JIH 13xl HCXOHoro -2). IIpH JIy-q l2 5x x C KOpHH Ba 2 - a - 0 > = npH B > -2, -2. - 2x. qOY ,n;rro paapemeHHH «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeii H 3a- ypaBHeHHe Ha-qarrOM a 0 He OIIYCKaeTCH To-qKe HMeeT V1 KPV1TEPV1V1 PEWEHV1 3A.n,AHV1V113-19 OU.EHV1BAHV1 KoopHHaTbI TO"tJeK 3Ha-qHT, a = x < rrapa60JIa 3Ha"tJHT, 2 x = 2x - a = x < 5x - 2x - C l1 JIy"tJOM HBJIHIOTCH perne- (X+1)(X+2)=0, { 2, - 0; C rrepeceKaeTCH rrapa60JIbI rrepece"tJeHHH a 2x 2 - TaKHM < a > a -2x, 2, - a = 5x - l1 2 x = 2, - O. < x JIy"tJOM a 2 x = 2x - 0 = < -2 < 3 3; < a -1 -1; < a a x - > x C (-1; TO"tJKax B 2x - l2 JIy"tJOM 3) H (-2; 8). 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KpHTepHJI TORbKO HRH - O. > OTBeT 6bI XOTH paCCMOTpeH 3HaqeHHH CTBO x - x 8. > paccym,n;eHHH OT = a BbIrrOJIHeHO ypaBHeHHe < BepHbIH rroRyqeH C a*-O - (X_-l)(X-2)=O,2, { 0; Co,n;epataHHe 060CHOBaHHO -2X, 2, - rrepeceKaeTCH HCXOHoe < a 2 x Bcex rrpH 2 X:: :2=X { 0; YCJIOBHe a - 0 OTBeT: x - x o6pa30M, 0; 2 x _ a rrapa60JIa CJIeOBaTeJIbHO, OT 2 x = -5X-2=X2_2X, { 2, - 0; TO"tJeK { C a CHCTeMbI: HHHMH < -2x, 5x - a KoopHHaTbI -1 2 a=x { - rrapa60JIbI rrepeCe"tJeHHH CHCTeMbI: HHHMH 13xl 223 a, H/HRH a a, 3 =-2 MHome- HRH rroRyqeHO BKRlOqeHHeM TOqeK OT OTRHqalOm;eeCH a 8, = HCKOMoro 2 -2, = MJIM HeBepHbIH rroRyqeH OTBeT Bce BbIrrORHeHbI 3a,n;aqa H RyqeH BepHO perneHHH CBe,n;eHa K (aHaRHTHqeCKH PerneHHe He BbIqHCRHTeRbHoH H3-3a rnarH B3aHMHoro HCCRe,n;OBaHHIO HO 3TOM rrpH rrapa60RbI pacrrORomeHHH HH H3 O,n;HOMY I B B rpaMMOB. <PPYKTOB 100 a) MorJIo MorJIo paBHa B) 85 JIH B r. CpeHHH Macca CpeHHH HID;HKe 100 60JIbrne JIH B 100 r? KaKYIO OKa3aTbCH H3 Kamoro KOTOpbIX <ppYKTa pa3JIH-qHoH Macca Macca <PPYKTOB, Macca Kamoro <PPYKTOB, a MaCCbI, H3 Macca cpeHHH H3 Kamoro "tJHCJIOM n;eJIbIM BbIpamaeTCH Ba 4 6aJlJl MeHbrne KOTOpbIX 100 60JIbrne KOTOpbIX Bcex paCIIpOCTpaHeHHe HID;HKe 000 H MomeT «Ha:U;HOH8.JIbHOe «H3,n;aTeJIbCTBO B 8 MeHbrne OKa3aTbCH Maccy HCIIOJIb30BaHHe MaccoH <PPYKTOB rropOBHY MeHbrne 100 H r <PPYKTOB r? HaH6oJIbrnylO @ 2021. KOIIHpoBaHHe, r. Macca 6bI XOTH 0 r. MaccoH 6) 100 paBHa 124 <PPYKTOB, eCTb HID;HKe paBHa r, paBHa 76 JIemHT HID;HKe 1 BbIrne rrepeqHCReHHbIX KpHTepHeB, M aKCUMaJlbHblU 19 BepHO rpacpHqeCKH) HRH COOTBeTcTByeT ornH6KH, KOMMepqeCKHX <PPYKT HMeTb 06pa30BaHHe». :u;eJIHX <PPYKTOB, 6e3 B @ 2021. IIHCbMeHHoro Macca H3 KOTOpbIX HID;HKe? 3TOM qOY Kamoro ,ZJ;rro pa3pemeHHH «MIJ;HMO» IIpaBOo6JIMaTeJIeii He ,n;OIIYCKaeTCH r, TV1nOBbIE Er3. 224 BAPV1AHTbl 3K3AMEHALJ.V10HHbIE PemeHHe. IIycTb a) 100 me B r k <PPYKTOB MaCCOH MaccoH pOBHO <PPYKTOB Hm;HKe (76 H 2k) - 85k HO <PPYKTbI <PPYKTOB Bce rropOBHY 6) HO IIycTb 100 r He k 5 H k + n -qHCJIa 8 B3aHMHO TaKHM 06pa30M, OTJIH-qHOH 100 OT 88 58 + H3 CJIeOBaTeJIbHO, B KOTOpbIX paBHa 100 B) Macca caMoro IIYCTb 6 6bIJIO rrYHKTe H a) 6) HeT; (k MOrJIO > + 8n n); 65 - 8 < eCTb 5 <PPYKTOB H k + n Mac- C 65. < 3Ha-qHT, 11. = 8 MeHbme <PPYKTOB, Macca Kam,n;oro < H 23 5, + 101; 101. + 3Ha-qHT, 676. < x <PPYKTa MomeT 100 MaccoH <PPYKTOB Tora r, 23n < x 8 < 25 THmeJIOrO 6bITb 676 24 <PPYKTa r, r. ECJIH B Hm;H- 101 MaccoH BbIrrOJIHeHbI. 3aa-qH YCJIOBHH x paBHa (n -1); 58 = < x 11 r, TO 676. B) HeT; n 25; caMoro 85 r, 101 + x . BaJlJlLI RpHTepHJI KpHTepHH (CM. 1 6aJIJI) 4 BepHo rrOJIyqeHbI Bce BepHo rrOJIyqeHbI TPH H3 rrepeqHCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 3 BepHo rrOJIyqeHbI ,n;Ba H3 rrepeqHCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 2 BepHo rrOJIyqeH - - 06ocHoBaHHoe - HCKOMaH - rrpHMep He H3 rrYHKTa a; pemeHHe rrYHKTa 6; B 1 o6eCrreqHBalOID;HH HH COOTBeTcTByeT pe3YJIbTaTbI B; rrYHKTe B, rrYHKTe Ha pe3YJIbTaTOB: CJIe,n;YIOID;HX pemeHHe On;eHKa B rrepeqHCJIeHHbIe O,n;HH 06ocHoBaHHoe PemeHHe rrpe,n;bI,n;yID;eH TOqHOCTb O,n;HOMY H3 On;eHKH PeIIIHTe 6) HaH,n;HTe Bce @ 2021. pacIIpocTpaHeHHe KOpHH 000 H x 24.4 ypaBHeHHe 3Toro O ,5 -11.2 x + 1 +6 «HaU;HOHaJIbHOe B KOMMepqeCKHX =0. rrpOMemYTKY rrpHHMJIemam;He ypaBHeHHH, «H3,n;aTeJIbCTBO HCIIOJIb30BaHHe - 4 6aJlJl 17 BapHaHT a) 0 BbIme rrepeqHCJIeHHbIX KpHTepHeB, M aKCUMaJlbllblU KOIIHpoBaHHe, poR- 5k. = KOJIH-qeCTBO TO OKa3aTbCH Co,n;epataHHe [!I] MaCCOH 8=t. 76 > n) <PPYKTa -qTO rrOKa3aHO, -qTO Macca IIoKameM, MaccoH 40 CPPYKTOB 1 <PPYKT MaCCOH 676 OTBeT: + m H - THmeJIOrO n Ke OKa3aTbCH r. <PPYKTOB m r, MOrJIO 100 r. 124n B + (k CJIeOBaTeJIbHO, 138 < 76, 13, 76 = He Hm;HKe 100. 408=40t; 138. = Ha m He 60JIbme Tora r. n=5t; eJIHTCH r, 0, = Hm;HKe 100 MeHbme = 60JIb- MaccoH <PPYKTOB rrpOCTbI, k=88, COH, B MaccoH <PPYKTOB 100 +124n 100m 9k 7600; = 3Ha-qHT, H MaCCOH + = r k r, Tora r. 2k) - MaCCbI, 100 60JIbme MaccoH 100 (76 . OHOH <PPYKTOB 85k IIoCKOJIbKY + 100 MeHbme Hm;HKe <PPYKTOB n H 6bITb MorYT MaCCOH B 124k + 100 MeHbme 06pa30BaHHe». u;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY ,ZJ;rro pa3pemeHHH «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH [-1; 1]. He ,n;OIIYCKaeTCH r PEWEHV1 V1 KPV1TEPV1V1 3A)J.AHV111113-19 OU.EHV1BAHV1 225 PemeHHe. IIpeo6pa3yeM a) ypaBHeHHe: x 12.4 C,n;eJIaeM 2x T or,ZJ;a peHb 2 HJIH - OTBeT: -11t+3==0, x CJIe,ZJ;OBaTeJIbHO, __ -, 0 a) -log23; <1, log21,5 < == I og2 3 - rrOJIyqeHbI 060cHoBaHHo rrOJIyqeH O. 3 t==- t==-. HJIH 3 X 1] == 2 I og2 1 ,5 . TOJIbRO rrpHHMJIemHT RO- log21,5. Co,n;epataHHe 060CHOBaHHO == HJIH , [-1; OTpe3RY rr03TOMY 6) log21,5; +3 OTRY,ZJ;a 2 -1; x -11.2 1 2 6t x 6.4 0; == 3 x 3 < +6 IIoJIHM: ==t. 1 == -log23 log21,5. 6) x 2 3aMeHY: x -22.2 OTBeTbI BepHbIe BepHbIH OTBeT OTBeTbI H3-3a BaJIJILI KpHTepHJI 060HX B B 2 rrYHKTax a, rrYHKTe llJIM u rrOJIyqeHbI HeBepHbIe HMeeTCH PellIeHHe H llIarOB HO pellIeHHH rrYHKTOB: HH COOTBeTcTByeT H3 OHOMY KpHTepHeB, 0 BbIllie rrepeqHCJIeHHbIX M aKCUMaJlbllblU OCHOBaHHeM [ill H BC 6oRoBbIe Be COOTBeTCTBeHHO HbI rrJIOCROCTb Ha a) 6) SABCD rrHpaMH,ZJ;bI 25. == HBJIHeTCH pe6pa N H a Mem,ZJ;Y == CN == Ha 8. co AB cTopoHaMH AD pe6pax Qepe3 2 H BC H N K TO"tJRH == OTMe"tJe- rrpOBe,ZJ;e- SB. pe6py rrJIOCROCTb paCCTOHHHe AK "tJTO TaR, rreprreH,ZJ;HRYJIHpHaH "tJTO 5.JI7 paBHbI 6aJlJl ABCD rrpHMoyrOJIbHHR rrHpaMH,n;bI K TO"tJRH a, ,n:oRamHTe, HaH,ZJ;HTe 1 3TOM rrpH 060HX 6 rrYHKTa He Bcex rrOCJIeOBaTeJIbHOCTb BepHaH a rrYHKTa 01lIH6KH, BbIqHCJIHTeJIbHOH rrpoxo,n;HT "tJepe3 rrpHMbIMH DS H M TO"tJRY KM. - pe6pa cepe,ZJ;HHY SB. PemeHHe. a) IIycTb HbI, 0 TO"tJRa rr03TOMY BA2 == BS 3Ha"tJHT, 2 + cepe,ZJ;HHa NK. BN OTpe3RH KD H paBHbI rrap8JIJIeJIb- S AD2 sd' == == 15 2 2.( 5.JI7)2 BSD 2 + 25 == == 850 H 850. rrpHMoyrOJIbHbIH C rrpHMbIM B IIJIOCROCTb a rrJIOCROCTb rrepeceRaeT SD, TO @ 2021. paCIIpOCTpaHeHHe BS, rr03TOMY pe6py SD. rrJIOCROCTb CJIe,ZJ;OBaTeJIbHO, BSD rro rrpHMOH, rrap8JIJIeJIbHOH OM, cO,ZJ;epmaru;eH rrpHMoH OHa rreprreH,ZJ;HRYJIHpHa rrpHMOH rrap8JIJIeJIbHa rrpHMOH H BSD: + TpeyrOJIbHHR H BD. BSD. yrJIOM KOIIHpoBaHHe, - TpeyrOJIbHHR BD2 BD rrepece"tJeHHe 0 TO"tJRa PaccMoTpHM - eCTb 000 H rro «HaU;HOHaJIbHOe <<lfa,n;aTeJIbCTBO HCIIOJIbaOBaHHe B KOMMepqeCKHX 06paaOBaHHe». u;eJIHX 6ea a A JIHHHIO cpe,ZJ;HIOIO @ 2021. IIHCbMeHHoro QOY ,1];rro paapemeHHH BSD. TpeyrOJIbHHRa «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH 15 Er3. 226 llcKoMoe 6) CKOCTH paCCTOHHHe co,n;epmeM 8M, a, rreH,n;HKYJIHpa paBHO paCCTOHHHIO KM. CJIe,n;oBaTeJIbHO, rrpHMYIO rrpoBe,n;eHHoro 8M rrJIOCKOCTH rrJIO- ,n;JIHHe paBHO rrep- a. . = 2 OTBeT: eM rrapaJIJIeJIbHOM ,n;o paCCTOHHHe 5.JI7 !8B = 8D 3TO K BAPV1AHTbl 3K3AMEHALJ.V10HHbIE rrpHMoM OT 8D rrpHMoM OT TV1nOBbIE 2 5.JI7 6) 2 Co.n;epataHHe I1MeeTcH I1MeeTcH YTBepm,lJ;eHHH ,lJ;OKa3aTeJIbCTBO BepHoe BepHbIH qeH OTBeT B 060CHOBaHHO H a, rrYHKTa rrOJIY- 2 6 rrYHKTe ,.- ,lJ;OKa3aTeJIbCTBO BepHoe BaJIJILI RpHTepHJI YTBepm,lJ;eHHH a, rrYHKTa I1JII1 060CHOBaHHO BepHbIH rrOJIyqeH a, rrYHKTa YTBepm,lJ;eHHH PerneHHe He OTBeT 3TOM rrpH HH COOTBeTcTByeT B a rrYHKT H3 O,lJ;HOMY 1 6, rrYHKTe He C HCrrOJIb30BaHHeM rrepeqHCJIeHHbIX BbIrne B03MomHO, BbIrrOJIHeH KpHTepHeB, 0 M aKCUMaAbHblU DD PellIHTe (log211- -log! x + HepaBeHcTBo xl) > 0 2 6aAA · 2 PemeHHe. 1 IIpH X == -qaCTb JIeBaH -- CMbICJI HMeeT HepaBeHCTBa 2 H paBHa HYJIIO. 1 IIpH x> YCJIOBHH HepaBeHCTBO IIOJIaeM - 2 1 (log21 logl - x l» O, 2 oTKy,n;a O<log211- x l < 1<11- 1; l x < 2. Y CJIOBHe 1 -1 CJIe,n;OBaTeJIbHO, < x < 0 2 HJIH < <x 1 - - < x < 3. 0 Y-qHTbIBaH HJIH 2 < x x >-- ' 2 Haxo,n;HM: 3. < 2 1 06FJ>e,n;HHHH MHomeCTBa, rrOJIy-qeHHbIe rrOJIy-qaeM: -- < x < QOY ,1];rro 0 HJIH 2 <x < 3. 2 OTBeT: [@ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe ; ) 0 000 H U (2; 3]. «Ha,n;aTeJIbCTBO HCIIOJIbaOBaHHe «HaU;HOHaJIbHOe B KOMMepqeCKHX o6paaOBaHHe». u;eJIHX 6ea @ 2021. IIHCbMeHHoro paapemeHHH «MIJ;HMO» IIpaBOo6JIa.,n;aTeJIeH He ,n;OIIYCKaeTCH PEWEHV1 V1 KPV1TEPV1V1 3AAAHV111113-19 OU.EHV1BAHV1 227 CoepaHHe 060CHOBaHHO rroRyqeH 060cHoBaHHo rroRyqeH H/HRH 3, BepHbIH BaJlJlLI KpHTepHJI 2 OTBeT OTRHqalOID;HHCH OTBeT, OT 1 TOqKH HCKRlOqeHHeM BepHoro -- 2 TOqKH 1 MJIM HeBepHbIH rroRyqeH PeIIIeHHe He IIIarOB Bcex rrOCReOBaTeRbHOCTb BepHaH BbIqHCRHTeRbHoH H3-3a OTBeT HH COOTBeTcTByeT OIIIH6KH, HO 3TOM rrpH HMeeTCH peIIIeHHH H3 OHOMY 0 BbIllie rrepeqHCReHHbIX KpHTepHeB, M aKCUMaJlbJ-lblU ABCD qeTbIpexyrOJIbHHK [K] AB a) 6) =CD= =BC ,n:oKamHTe, HaHHTe BIIHcaH B oKpymHocTb R pa,n;Hyca 2 6aJlJl = 27. 113BecTHo, "tITO 36. "tITO BC IIpHMbIe H AD lIapaJIJIeJIbHbI. AD. PemeHHe. OCTpbIe a) Ha palOTCH H AD BCA yrJIbI paBHbIe CAD H AB XOpbI OIlH- 1I0CKOJIbKY paBHbI, CD. 3Ha"tIHT, IIpHMbIe H BC lIapaJIJIeJIbHbI. 0603Ha"tIHM 6) BCA yrOJI a. "tIepe3 AB . lIo CHHYCOB TeopeMe 2 SIna=-=-. 2R ABC TpeyrOJIbHHK LBAC = LBCA paBH06epeHHbIH, LBAD 3Ha"tIHT, a. = ABCD qeTbIpexyrOJIbHHK 3Ha"tIHT, - LACD TaKHM 06pa30M, 110 3 = 180° = MMeeTCH LCDA = 180° - = LBAD 3a. ACD H ACB 1I0JIaeM: 6) 44. OKa3aTeRbCTBO BepHoe BepHbIH B OTBeT 060cHoBaHHo rroRyqeH BaJlJlLI KpHTepHJI rrYHKTa a, H 060CHOBaHHO rrYHKTa a, H rrpH YTBepmeHHH rrORY- 3 6 rrYHKTe BepHbIH OTBeT B 6, rrYHKTe MJIM HMeeTCH peIIIeHHH 6 rroRyqeH rrYHKTa paCIIpOCTpaHeHHe YTBepmeHHH OKa3aTeRbCTBO BepHoe @ 2021. KOIIHpoBaHHe, = oAB=(3-4sin2a)AB=44. Sina CoepaHHe qeH LCDA 1I03TOMY TpeyrOJIbHHKOB si3a AD= ; - JIH Sina 2a. = TpalleHH, LCAD - CHHYCOB AD OTBeT: + LCAD paBH06epeHHaH TeopeMe sin(1800-3a) 1I03TOMY LBAC 000 H HeBepHbIH «(HaU;HOHaJIbHOe «(HaaTeJIbCTBO HCIIOJIbaOBaHHe B KOMMepqeCKHX OTBeT H3-3a 06paaOBaHHe». u;eJIHX 6ea OIIIH6KH apHcpMeTHqeCKOH @ 2021. IIHCbMeHHoro qOY 2 060cHoBaHHoM ,ZJ;rro paapemeHHH «(MIJ;HMO» IIpaBOo6JIa,n;aTeJIeii He OIIYCKaeTCH 2a. Er3. 228 TV1nOBbIE BAPV1AHTbl 3K3AMEHAL.J.V10HHbIE OKOHl.laHUe Co,n;epataHHe MMeeTCH ,n;OKa3aTeJIbCTBO BepHoe ma6Jl. BaJIJILI KpHTepHJI YTBepm,n;eHHH a, rrYHKTa MJIM 060CHOBaHHOM rrpH THqeCKOH pemeHHH 6 rrYHKTa HeBepHbIH rrOJIyqeH apHcpMe- H3-3a OTBeT 1 omH6KH, MJIM 060cHoBaHHo BepHbIH rrOJIyqeH a, rrYHKTa PemeHHe 3TOM rrpH He OTBeT a rrYHKT He HH COOTBeTcTByeT B 6 rrYHKTe HCrrOJIb30BaHHeM C YTBepm,n;eHHH BbIrrOJIHeH H3 O,n;HOMY KpHTepHeB, [E] 25 Ha B -qeTbIpeXJIeTHHii 6H3HeC-nJIaHY MJIH py6JIeii. % 20 no IIo KpoMe npoeKT. -qHCJIO HToraM eJIoe m 3Ha-qeHHe n, y,n;BOHTCH, H 3Toro, cpa3Y -qHCJIO n py6JIeii MJIH Kam,n;oro C Ha-qarrOM cpaBHeHHIO BJIOmeHHH: npoeKT KOTOpOM npH H npoeHTOB nepBbIii, -qeTBepTbIii B B 3Ha-qeHHe BJIOmeHHH 3a H BTOpoii -qTO TaKoe, m, ro,n;a -qeTbIpe KaK BbIpaCTYT ,n;Ba TaKme eJIoe HaHMeHbmee KaK ro,n;a Haii,n;eHHoM npH a Haii,n;HTe 3a cpe,n;CTB BJIOmeHHbIMH ro,n;bI, ro,n;bI. - ,n;onOJIHHTeJIbHbIe HymHbI BO BJIOmeHHH nepBOHa-qarrbHbIe HaHMeHbmee nepBOHa-qarrbHbIe py6JIeii TpeTHii, H OCTalOTCH npoeHTbI Ha-qHCJIeHHii BJIOmeHHe BJIOmeHHbIX npHpOCT nJIaHHpyeTcH Ha-qHCJIeHHbIe nOCJIe B H ro,n;a ro,n;a. MJIH Ha-qarrbHoe npe,n;nOJIaraeT 3 6aJlJl M aKCUMaJlbHblU IIo 0 BbIme rrepeqHCJIeHHbIX paHee B MHHHMYM MHHHMYM n 3Ha-qeHHH -qeTbIpe pa3a. PemeHHe. K 3-ro 2-ro Ha-qaJIY ro,n;a ro,n;a 1,2 IIo 3-ro 36 YCJIOBHIO ro,n;a + 50. > 2,2n 4-ro n) n + n HaHMeHbmee 74,016 YCJIOBHIO > 2,2m 1,2 HMeeM ro,n;a 1,2 oTKy,n;a + 30 = + n BJIOmeHHii, MJIH a K Ha-qaJIY = 36 K Ha-qarry 2,2n. + eJIoe n pemeHHe = 7. Tor,n;a nOJIy-qHTCH Ha-qarry IIo + (30 36 K 25 1,2 nOJIy-qHTCH - + 25,984; 51,4 > 15,4 51,4 (1,2 2,2m + = + m + 51,4 MJIH. MJIH, a + m m) B KOHe 74,016 = npoeKTa 2,2m. + 100, 25,984 m > . 2,2 IIOJIy-qaeM, -qTO OTBeT: 7 m 12 H = 12 HaHMeHbmee - BepHo 3TOH - - H Mo,n;eJIH HeBepHbIH BepHbIH H3-3a HO OTBeT, paCIIpOCTpaHeHHe 000 H 3 Mo,n;eJIb, K CBe,n;eHO pemeHHe HCCJIe,n;OBaHHIO pe3YJIbTaT: rrOJIyqeH OTBeT BaJIJILI KpHTepHJI OTBeT MaTeMaTHqeCKaH rrOCTpoeHa @ 2021. KOIIHpoBaHHe, BepHbIH rrOJIyqeH pemeHHe. py6. MJIH Co,n;epataHHe 060CHOBaHHO eJIoe BbIqHCJIHTeJIbHOH «Ha:U;HOHaJIbHOe «HaaTeJIbCTBO HCIIOJIbaOBaHHe 060cHoBaHo He,n;OCTaTOqHO pemeHHe B KOMMepqeCKHX 2 omH6KH; 06paaOBaHHe». :u;eJIHX 6ea @ 2021. IIHCbMeHHoro QOY ,1];rro paapemeHHH «MIJ;HMO» IIpaBOo6JIa.,n;aTeJIeH He OIIYCKaeTCH PEWEHV1s:1 V1 KPV1TEPV1V1 3AAAHV111113-19 OLJ.EHV1BAHV1s:1 .. OKOJ-ll.laJ-lUe Co,n;epataHHe BepHO 3TOM MaTeMaTHqeCKaH IIOCTpOeHa Mo,n;eJIH, 3TOM IIPH PerneHHe He MO,n;eJIb, HH COOTBeTcTByeT BaJlJlLI KpHTepHJI perneHHe 6bITb MomeT perneHHe ma6A. H3 O,n;HOMY K CBe,n;eHO He HCCJIe,n;OBaHHIO 1 3aBeprneHO KpHTepHeB, 0 BbIrne IIepeQHCJIeHHbIX M aKCUMaAbJ-lblU 6aAA 3 a [!D HaH,lJ;HTe 3HaeHH.H Bce OrpaHHqHBalOT a, 1'13 Ram,lJ;OM IIPH MHoroyrOJIbHHK, IIJIOa,n;b 60JIee He KOToporo 2 y=alx- JIHHHH ROTOpbIX 1+l a l-2 1'1 Y=2: 0,5. PemeHHe. a 1) eCJIH JIHHHH 2 alx- = Y 1+l l-2 a Y 1'1 qmrypy, orpaHHHBaIOT =2: TO ee IIJIOru;a,n;b a 6yeT 2) a IIPH a > O 0 == OrpaHHqeHHOH cpHrYPbI, IIJIOa,n;H paBHa JIHHHH H IIapaJIJIeJIbHbI y==alxl+lal-2 JIHHHHMH Y==2:; 06pa3YIOT. He MHoroyrOJIbHHK H . a JIHHHH y==alxl+a-2 == y H ->a-2, IIPH HaHeM a6cHccbI a<4 oTKya 2 TOqeK 2 IIo3ToMY IIJIOab IIOJIyqeHHOrO I pHC.). I __ -- I , J \ / a y=-- 2 I \ \ _ / -- x J 0 \ 2 / \ TpeyrOJIbHHKa I / 1 Ixl==---. a oTKya 2 I \ JIHHHH: IIepeCeqeHHH a alxl+a-2==-, (CM. I y=alxl+lal-2 a>O a (TpeyrOJIbHHK) I y cpHrypy OrpaHHqHBalOT 2: paBHa \ / 'If a-2 ( PeIIIHM }( ; -a+2)= (2- J _ HepaBeHCTBO (2-!J < 0,5; a 2 -lOa 2 < a 16 + I I I , O. < a IIOJIyqHM 2 e YCJIOBHH yqeTOM ID a < O < a 8. < 0 < a < 4 IIOJIyqHM < 4. y 'ri - a< . a JIHHHH Y = I xl- a a 2 - Y 1'1 = IIPH HaHeM - < 2 a6cHccbI -a-2, TOqeK IIo3TOMY IIJIOab IIepeCeqeHHH 2 2 IIOJIyqeHHOrO @ 2021. 000 H B KOMMepqeCKHX -- x \ / \ / \ a \ \ I , I paBHa _a_2_ ) = ( 23aJ 2-a 06pa30BaHHe». U;eJIHX 6e3 I I I I y=alxl+lal-2 2 a Y=2: I 3 2 «HaU;HOHaJIbHOe «H3,n;aTeJIbCTBO HCIIOJIb30BaHHe \ 0 2+ ( !+ ) ( a I pHC.). I TpeyrOJIbHHKa . paCIIpOCTpaHeHHe (CM. - Ixl==-+-. a oTKya 3 KOIIHpoBaHHe, - \ / 3 JIHHHH: a alxl-a-2==-, < a oTKya \ J 4 a (TpeyrOJIbHHK) / clmrypy orpaHHHBaIOT 2: I -a-2 J @ 2021. IIHCbMeHHoro qOY ,ZJ;rro pa3pemeHHH «MIJ;HMO» IIpaBOo6JIa.n;aTeJIeii He ,n;OIIYCKaeTCH - _ I TVinOBblE Er3. 230 PemHM ( HepaBeHCTBO 3a 2 J + 2 < 2 9a 0,5; + 26a 16 + < BAPVlAHTbl 3K3AMEHAL.J.VlOHHbIE o. -a IIoJlY"tIHM 2 - 8 < a < - - . 9 4 C yqeTOM a < YCJlOBHH [- OTBeT: - nOJlY"tIHM - :) - m;eeCH B ( = - TOJIbKO a = : - a ' = 2, a BaJIJILI KpHTepHH 4 Bce ), 4 = TOqeK a -2 = H/HJIH TOqKH rpaHHqHble HO 3HaqeHHH MHomeCTBO IIOJIyqeHO HCKJIlOqeHHeM HaH,n;eHbI BepHO 2, . OTBeT paccym,n;eHHH BepHoro pemeHHH a BepHbIH HCKOMoro OT - [2; 4). u IIOJIeH IIOMOm;blO - 3 CO,ZJ;epataHHe C < a < 3 2; 060CHOBaHHO 4 2 - HeBepHO 3 2 = 3HaqeHHH MHomeCTBa orrpeeJIeHbI OTJIHqalO- a, a a 3HaqeHHH rrpOMemYTKH a, 2 HJIH HeBepHbIH IIOJIyqeH BepHo H3 [- rrpoMemyrKoB omH6KH, HO 3TOM IIPH BepHO pemeHHH 6bI XOTH paCCMOTpeH BblqHCJIHTeJIbHOH H3-3a OTBeT marH Bce BbIIIOJIHeHbI 2; :) - H3 O,n;HH [2; HJIH 4 ), H pemeHHH, CJIaeB O,n;HH BKJIIOqeHHeM/HCKJIIOqeHHeM C B03MomHO, 6w XOTH IIOJIeH 1 TOqeK, rpaHHqHbIX HJIH 3a,n;aqa C 06m;eH K CBe,n;eHa BepHO BepmHHoH PemeHHe He B3aHMHoro HCCJIe,n;OBaHHIO (aHaJIHTHqeCKH IIpHMOH H HH COOTBeTcTByeT H3 O,n;HOMY paCIIOJIOmeHHH JIyqeH ,n;BYX rpa<pHqecKH) HJIH KpHTepHeB, M aKCUMaJtbllblU M3,n;aTeJlbCTBO Ha B npHBe3J1H ECJlH py6J1eii. O,n;HaKo a) JlH HeCKOJlbKO QeHa py6J1eii, Kam,n;oii 100 MeHbIIIe KHHr C YMeHbIIIHTbCH no BbICTaBKH npHBe3J10 3K3eMnJlHpe). BbICTaBKH KOJlH"tIeCTBO "tIero MorJla KHHrH eHa OTKpbITHH ,n;o H3-3a BbIcTaBKY e,n;HHcTBeHHoM eHY 6HpKaMH co cpaBHeHHIO Hee Ha 6aJtJt 4 (Kam,n;ylO KHHry npo,n;amH - "tIHCJlO HaTyparrbHoe 6HPKY npHKJleHBalOT KHHrH «BbIro,n;HO». 10 Ha YBeJlH"tIHJlH py6J1eii, YMeHbIIIHJlOCb. KHHr C eHoii cpe,n;Heii ,n;JlH KHHrH «BbIro,n;HO» eHa cpe,n;HHH KHHr Kam,n;oii 0 Bblme IIepeqHCJIeHHbIX KHHr 6HpKOii C 6HpKOii nocJle «BbIro,n;HO» «BbIro,n;HO» OTKpbITHH ,n;o OTKpbITHH BbIcTaBKH? 6) MorJla JlH BbICTaBKH YMeHbIIIHTbCH cpe,n;HHH no co cpaBHeHHIO 6e3 KHHr eHa eHoii cpe,n;Heii «BbIro,n;HO» 6HpKH «BbIro,n;HO» 6HpKH nocJle 6e3 KHHr OTKpbITHH ,n;O OTKpbITHH BbICTaBKH? B) M3BecTHo, "tITO KHHr 6e3 6HpKOii 210 py6J1eii. C @ 2021. KOIIHpoBaHHe, nepBOHa"tlarrbHO KHHr eHa cpe,n;HHH paCIIpOCTpaHeHHe 6HpKH C - 000 H «ll3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe 90 «Ha:U;HOHaJIbHOe B KOMMepqeCKHX a 06pa30BaHHe». 6e3 @ 2021. TaKoe qOY ,l];rrO pa3pemeHHH a KHHr py6J1eii, eHa cpe,n;HHH eHa cpe,n;HHH eHa KHHr IIHCbMeHHoro py6J1b, eHbI cpe,n;HHH KOJlH"tIeCTBe :u;eJIHX 81 YBeJlH"tIeHHH py6J1eii, 110 COCTaBJlHJla KHHr COCTaBJlHJla IIocJle HaHMeHbIIIeM KaKOM Bcex eHa «BMro,n;HO COCTaBHJla «BMro,n;HO IIpH cpe,n;HHH 6HpKOii 226 py6J1eii. 6e3 6HpKH B03MomHo? «MD;HMO.) IIpaBOo6JIap;aTeJIeii He ,ZJ;OIIYCKaeTCH KHHr PEWEHVI VI KPVlTEPVIVI 3A.L1,AHVl1I113-19 OU.EHVlBAHVI tan PemeHHe. a) HO IIpe,n;nOJlOmHM, CTOHJlH 120, 94+20 Tenepb B 6) cJle Bcero eHbI 110n 116n KpaTHO 4 5, H "tIeTbIpe KHHr Jleii. IIocJle C 226 6HpKOii a KHHr 120y, o6paaoM, 80 a + y) x u JleH. no- KHHr, nocJle KHHr YBe- = 90y, 4y. II03TOMY "tIHCJlO Bcex eHa KHHr C 210 - 110 6e3 KHHr eHa KHHr 6HpKH KHHra 6HpKOii py6J1eii. 96 no - py6J1eii, 6HpKH cpe,n;HHH - 226 py6- COCTaBHJla «BbIro,n;HO» Bce py6J1eii YCJlOBHH BbInOJl- HeHbI. OTBeT: a) 6) ,n;a; B) ,n;a; 20. CO,ZJ;epataHHe BaJIJILI KpHTepHH (CM. 1 4 BepHO rrOJIyqeHbI Bce BepHo rrOJIyqeHbI TPH H3 rrepeqHCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 3 BepHo rrOJIyqeHbI ,lJ;Ba H3 rrepeqHCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 2 BepHo rrOJIyqeH - - - - rrepeqHCJIeHHbIe H3 O,lJ;HH CJIe,lJ;YIOID;HX pemeHHe rrYHKTa a; 060cHoBaHHoe pemeHHe rrYHKTa 6; HCKOMa-H On;eHKa B rrpHMep PemeHHe B He 6aJIJI) pe3YJIbTaTbI 1 06eCrreqHBalOID;HH COOTBeTcTByeT Ha 8; rrYHKTe 8, rrYHKTe KpHTepHH pe3YJIbTaTOB: 060cHoBaHHoe HH O,lJ;HOMY rrpe,lJ;bI,lJ;YID;eH TOqHOCTb H3 KpHTepHeB, On;eHKH @ 2021. paCilpOCTpaHeHHe 000 H «ll3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe «Ha:U;HOHaJIbHOe B KOMMep'tleCKHX 06pa30BaHHe». :u;eJIHX 6e3 @ 2021. IIHCbMeHHoro QOY 0 BbIme rrepeqHCJIeHHbIX M aKCU.MaJtbllblU KOIIHpoBaHHe, n 20. > o,n;Ha cpe,n;HHH eHa 6e3 Bcex 3n eCTb TO n - py6J1eii, cpe,n;HHH cpe,n;HHH KHHr eHa 6bIJlO eHa 210(n = = no 81 eHbI cpe,n;HHH py 6 20. = Tor,n;a py6J1b, py6J1eii. «BbIro,n;HO» YBeJlH"tIeHHH py6J1eii, 15 6bIJlO 90n TaKHM n 120n H 117 ypaBHeHHH: ,n;Ba H cJlY"tlai B03MomeH no KHHrH 5x, = 20. KpaTHO nepBOHa"tlaJlbHO n;eHa 90 "tITO 4n eCTb TO eCTb TO IIoKameM, IIYCTb 145x, = H Cpe,n;HHH py6- «HeBbIro,n;- 2 «BbIro,n;HbIX» KHHr. Y + 81x x) - = YBeJlH"tIeHHH IIoJlyqaeM 226(n = eHa cpe,n;HHH IIepBOHa"tlaJIbHO 30 H py6J1eii. nepBOHa"tlaJIbHO nocJle a COCTaBJlHeT 104 130, 130+104 CTaJIO py6J1eii. 30 u KHHr. BbIro,n;HbIX 120 COCTaBJlHeT oTKy,n;a n npHBe3J1H YBeJlH"tIeHHH JlH"tIeHHH py 6 JleH, nepBOHa"tlaJIbKHHr CTOHTb CTaJlH COCTaBJlHeT nYHKTa 120 KOTopbIe «BbIro,n;HbIX» KHHrH KHHr «BbIro,n;HbIX» npe,n;bI,n;ym;ero COCTaBJlHeT IIYCTb B) H3 eHa eHbI YBeJlH"tIeHHH KHHrH, TPH Cpe,n;HHH py6J1eii. eHa npHMepe KHHr HbIX» 20 IIocJle cpe,n;HHH Bcero npo,n;aBaJIHCb H py6J1eH. 2 Jleii. 94 u 57 = "tITO ,l];rrO pa3pemeHHH «MHMO» IIpaBOo6JIap;aTeJIeii 4 6aJtJt He ,lJ;OIIYCKaeTCH TVinOBblE Er3. 232 21 BapuaHT OD a) PemHTe 6) Han,D;HTe COSX+2COS( ypaBHeHHe Bce 3Toro KOpHH BAPVlAHTbl 3K3AMEHAL.J.VlOHHbIE )=.J3sin2X-l. ; 2x- rrpHHa,D;Jlemarn;He ypaBHeHHH, [- OTpe3KY 7 51t; 1t ]. - 2 PemeHHe. IIpeo6paayeM a) ypaBHeHHe: cosx+cos2x+.J3 2 cosx+2cos cos . ( 1 2 + x-I cos HJlH ) x sin2x-l; -1; == == O. 2n n x==-+nk, oTKy,n;a cosX==--, + x== -+2nk, 223 C 6) x 1 cosx==O 3Ha"tlHT, ==.J3 sin2x "tIHCJlOBOH IIOMOID;blO [- OTpe3KY rrpHHa,D;Jlemarn;He "tIHCJla 2 14n H - -- C IIOMOID;blO II. -5Jt 3 6bITb MomeT T. KOpHH, ]. 2 KopHeH cIIoco6oM: ,n;pyrHM - -, OT6op 1t 7n - - 2 3aMe1lallUe. 7 51t; 9n IIoJly"tlHM oT6epeM oKpymHocTH kEZ. r,n;e 06oCHoBaH rpa<l>HKa, H JlI06bIM JlHHeH- perneHHeM 14Jt 3 -- -- HbIX H HepaBeHCTB OTBeT: n a) + -+nk, 2 7n 9n 2n -+2nk, 6) kEZ; r,n;e 3 CO.D;epataHHe 060CHOBaHHO rrOJIyqeHbI BepHbIe 06ocHoBaHHo rrOJIyqeH BepHbIH -- ' 2 2 · ' 3 BaJIJILI KpHTepHH OTBeTbI OTBeT 14n -- -- B B 060HX 9Jt 2 2 rrYHKTax a, rrYHKTe HJIH rrOJIyqeHbI OTBeTbI HeBepHbIe HMeeTCH H PemeHHe He Bcex rrOCJIeOBaTeJIbHOCTb BepHaH a rrYHKTa BbIqHCJIHTeJIbHOH H3-3a omH6KH, marOB HO rrpH 060HX pemeHHH rrYHKTOB: 6 rrYHKTa HH COOTBeTcTByeT H3 OHOMY KpHTepHeB, TO"'lKH A a) H SC Ha OCHOBaHHH oKpymHocTH TO"tlKa IIpOTHBOIIOJlOmHbI. "tITO SM IIpHMaH C yroJl o6paayeT M C C KOHyca - cepe,n;HHa ABC IIJlOCKOCTblO 2 6aJlJl S, BepmHHon rrpH"'leM BC. TaKoH me KaK yroJl, SBC. IIJlOCKOCTblO IIpHMOH Mem,n;y SA SBC, IIJlOCKOCTblO H eCJlH AB == 6, BC 5.J2. == @ 2021. KOIIHpoBaHHe, JlemaT AB IIpHMaH HaH,n;HTe H C H oKamHTe, H 6) B A, C ,n;HaMeTpaJlbHO 0 BbIme rrepeqHCJIeHHbIX M aKCU.MaJlbHblU OD 1 3TOM pacIIpocTpaHeHHe 000 H (<li3.n;aTeJIbCTBO HCIIOJIb30BaHHe «Ha:U;HOHaJIbHOe B KOMMep'tleCKHX 06pa30BaHHe». :u;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY pa3perneHHH ,l];rrO «MIJ;HMO» IIpaBOo6JIa.n;aTeJIeH He .n;OIIYCKaeTCH == 8 PEWEHVI VI KPVlTEPVIVI 3A.L1,AHVl1I113-19 OU.EHVlBAHVI .a.: PemeHHe. a) IIpoeKHH 0 eHTp SM HBJlHeTCH OM IIpHMbIe h re - - OM SBC H STOT me , yroJl SBC. IIJlOCKOCTblO TaKoH KaK TaR me, Pac- paBHO - 2 OT BbICOTa - (TaK A KaK B IIpHMO- SO-OM SOM, TpeyroJlbHHKa , 5.J2 SA SBC. h SBC paCCTOHHHe h == - IIJlOCKOCTH o IIJlOCKOCTH STO CA). A TO"tlKH o "tIHCJlO HBJlHeTCH Jla yr OT 0 cepeHHa yrOJlbHOrO SMO. H h HCKOMoro TO"tlKH OT yroJl s SM 1- BC H IIaparrJleJlbHbI. paCCTOHHHe CTOHHHe o AB H CHHYCOM 6) IIpHMOH AB IIpHMOH Memy STO - KOHyca OM 1- BC KaK - Memy yrJloM YroJl TaK ABC K OCHOBaHHH IIJlOCKOCTb Ha OCHOBaHHH. ero HaKJlOHa yroJl S TO"tlKH KOTopaH paBHa SM AB OM == == - 2 h SO-OM 2 SM SM 3-, == -J SC 2 2 MC - .J34 == SO H -J SM == 2 - M0 2 == 5 , IIOSTOMY 15 - · .J34 h CJle.n;oBaTeJlbHO, OTBeT: HCKOMoro Clmyc arcsin 6) yrJla paBeH HMeeTCH HMeeTcH m . ,n;OKa3aTeJIbCTBO BepHoe BepHbIH · 5.J2 17 CO,ZJ;epataHHe "tIeH 3 - OTBeT B YTBepm,n;eHHH H a, rrYHKTa 060CHOBaHHO rrOJIY- 2 6 rrYHKTe ,n;OKa3aTeJIbCTBO BepHoe BaJIJIbI KpHTepHH YTBepm,n;eHHH a, rrYHKTa HJIH 060cHoBaHHo BepHbIH rrOJIyqeH a, rrYHKTa YTBepm,n;eHHH PerneHHe He OTBeT 3TOM rrpH HH COOTBeTcTByeT B a rrYHKT H3 O,n;HOMY 1 6, rrYHKTe HCrrOJIb30BaHHeM C B03MomHO, BblrrOJIHeH He KpHTepHeB, 6aJtJt M aKCU.MaJtbHblU PernHTe 41og HepaBeHcTBo 3 ( sin x )+ 81og 2 (sin ) x > 1 2 (sin 0 Bblrne rrepe"tIHCJIeHHbIX 2 . PemeHHe. IIpeo6paayeM IIycTb log2(sinx)==t, CJleOBa a H3 0 HepaBeHCTBa paCIIpOCTpaHeHHe 9t Tora TeJlbHO, @ 2021. KOIIHpoBaHHe, 9log BbIpameHHe 000 H 0 < < sin sin x x < < 0,5 +8t-1 0,5 B x > 0, sin HJlH )+8Iog t oTKya x > x < )-1 > 0 IIPH sin YCJlOBHH KOMMepqeCKHX 06pa30BaHHe». :u;eJIHX 6e3 x 1 -1 HJlH t > -. 9 . BTopoe HepaBeHCTBO HeB03MomHO, IIoJly"tlaeM: «Ha:U;HOHaJIbHOe «ll3.n;aTeJIbCTBO HCIIOJIb30BaHHe 2 (sin @ 2021. IIHCbMeHHoro QOY pa3perneHHH ,1];I O «MIJ;HMO» IIpaBOo6JIa.,n;aTeJIeH He .n;OIIYCKaeTCH > O. TVinOBblE Er3. 234 51t 1t 21tk 21tk < -+ < x HJlH 6 (21tk; OTBeT: 5 1t 21tk; + 6 IIOJIeH BepHbIH 06ocHoBaHHo IIOJIyqeH OTBeT, k r,n;e E Z. k r,ll;e Z E . BaJIJIbI KpHTepHH 2 OTJIHqalOm;HHCH OT MHome- HCKJIlOqeHHeM BepHoro 51t 21tk + - 21tk, < 1t+ x oTBeT 1t CTBa ), 21tk 1t + Co.n;epataHHe 060CHOBaHHO < 6 ], [ 21tk + 21tk -+ BAPVlAHTbl 3K3AMEHAL.J.VlOHHbIE H/HJIH 6 -+21tk, MHomeCTBa 6 1 HJIH HeBepHbIH IIOJIyqeH OTBeT PerneHHe He BbIqHCJIHTeJIbHOH H3-3a Bcex IIOCJIeOBaTeJIbHOCTb BepHaH HH COOTBeTcTByeT rnarOB ornH6KH, HO 3TOM IIPH HMeeTCH perneHHH H3 OHOMY KpHTepHeB, M aICCU.MaJtbHblU Ha [ill H AB rHnoTeHyae N M, TO"tlKH BM 1 BN = = H K H P TO"tlKa OKajKHTe, HaH,n;HTe npHMoyroJlbHoro rrpH"tIeM - BCP M "tIeTbIpexyroJlbHHK "tITO nJlOm;a,n;b ABC, TpeyroJlbHHKa BM eCJlH OTMe"tleHbI npHMoH napaJlJleJlbHa paBHo6e,n;peHHaH - ABC AB KN. OTpe3Ka - 2 6aJtJt TpeyroJlbHHKa NK npHMaH cepe,n;HHa 2 a) 6) AC H COOTBeTCTBeHHO, KN. - BC KaTeTax 0 BbIrne IIepeqHCJIeHHbIX 1 = Tpanen;HH. L.BCM H = 15° . PemeHHe. IIoCKOJlbKY a) NP npHMaH NP=-KN=BM=BN, H "tIeTbIpexyroJlbHHK 2 BNPM 3Ha"tlHT, pOM6. BC, - rrpHMoH a TaK CP KaK - K naparrJleJlb- Me,n;H8;Ha KCN, npHMoro BepIIIHHbI PM npHMaH TpeyroJlbHHKa yroJlbHoro H3 A 1 BM Ha npHMoH naparrJleJlbHa npHMO- npoBe,n;eHHaH TO yr Jla, 1 CP=-KN=BN=PM=BM. 2 CJle,n;oBaTeJlbHO, 6) BCP M "tIeTbIpexyroJlbHHK CP IIoCKOJlbKY 1 = - KN = BN PM, = 2 L.PCM CM 3Ha"tlHT, 6HcceKTpHca - LABC LKNC = LABC BC =CN @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H BC = +BN = B KOMMep'tleCKHX L.CMP = =.J3 Tpanen;HH. CP M = L.BCM c paBHo6e,n;peHHbIH, n03TOMY . Tor,n;a L.BCP = CN 30°, «Ha:U;HOHaJIbHOe (dI3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe L.M paBHo6e,n;peHHaH TpeyroJlbHHK BCP. yrJla = N B = 2L.BCM KN cos = 30° 2.15° = = .J3 2. 30° , =.J3 , 2 AC +1, 06pa30BaHHe». :u;eJIHX 6e3 = BCtg30° @ 2021. IIHCbMeHHoro .r1 = QOY pa3perneHHH ,l];rrO . «MHMO» IIpaBOo6JIa.,n;aTeJIeH He ,ZJ;onYCKaeTCH PEWEHVI VI KPVlTEPVIVI 3A.L1,AHVll/113-19 OU.EHVlBAHVI 235 CJIe)J;OBaTeJIbHO, 8 2.J3 OTBeT: =.!AC ABC =.!(.J23 .BC 2 .J3 +1). 2.J3 +1 +3 = . .J3 3 3 + . 3 Co.n;epataHHe HMeeTCH ,n;OKa3aTeJIbCTBO BepHoe BepHbIH qeH OTBeT 060cHoBaHHo B rrYHKTa a, H rrYHKTa a, H YTBepm,n;eHHH 060CHOBaHHO rrOJIY- 3 6 rrYHKTe BepHbIH rrOJIyqeH BaJIJILI KpHTepHH OTBeT B 6, rrYHKTe HJIH HMeeTCH ,n;OKa3aTeJIbCTBO BepHoe pemeHHH rrYHKTa HMeeTcH BepHoe YTBepm,n;eHHH 6 rrOJIyqeH HeBepHbIH ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH omH6KH apHcpMeTHqeCKOH H3-3a OTBeT 2 060cHoBaHHoM rrpH a, rrYHKTa HJIH 060cHoBaHHoM rrpH 6 rrYHKTa pemeHHH HeBepHbIH rrOJIyqeH OTBeT H3-3a 1 omH6KH, apHcpMeTHqeCKOH HJIH 060cHoBaHHo BepHbIH rrOJIyqeH a, rrYHKTa PemeHHe 3TOM rrpH He OTBeT a rrYHKT He HH COOTBeTcTByeT B 6 rrYHKTe HCrrOJIb30BaHHeM C YTBepm,n;eHHH BbIrrOJIHeH H3 O,n;HOMY 0 BbIme rrepeqHCJIeHHbIX KpHTepHeB, M aICCU.MaJtbHblU IIJIaHHPyeTcH [ill rO)J;a. ITO Kam)J;oro C cpaBHeHHIO TOJIbKO B JIbrOTHbIH BbI)J;aTb B cepe)J;HHe ITO 3-ro KOHe )J;OJIr rO)J;a. 1-ro BbIITJIa"tlHBaeT 8 ITpeBbICHT ITPH 20 0/0 BbIITJIa"tlHBaeT ITepBOHa"tlarrbHOMY. ITOrarnaH CYMMbI, Kpe)J;HTa, "tIeTbIpe Ha aaeMID;HK paBHbIM O)J;HHaKOBbIe paaMep Ha BoapacTaeT rO)J;OB HeHaMeHHO )J;OJIr HaHMeHbrnHH aaeMID;HKa 2-ro H py6JIeH MHJIJIHOHOB )J;OJIr Kpe)J;HTa KOHe aaeMID;HK rO)J;OB "tIHCJIO eJIoe OCTaBJIHH HaH)J;HTe aaeMID;HKa BbIITJIaT B Kpe)J;HTY, 4-ro H ITOJIHOCTblO. )J;eHcTBHH rO)J;a Ha"tlarrOM ITpOeHTbI Ha Kpe)J;HT 3 6aJtJt BeCb o6ID;aH KOTOpOM CYMMa Py6JIeH. MJIH PemeHue. 060aHa"tlHM "tIHBaeT paaMep PaccMoTpHM BoapacTeT )J;OJIr 3-ro H )J;HHe )J;O rO)J;OB. OH rO)J;a TO ITOrarneH, 0,48 aa ITOrarneHHe 4-ro 4-ro Kpe)J;HTa Bcero 0,28. ITO eCTb 8. )J;Ba rO)J;a. 060aHa"tlHM IIocJIe BbIITJIaTbI B aa Kpe)J;HTa 1,28. paBeH "tIepea B 1,2(1,28 x 3-ro B x). )J;Ba rO)J;a rO)J;OB rO)J;a. B )J;OJIr 4-ro KOHe 1,2(1,28 paBHa 2-ro - 1,28 H - )J;OJIr ITO rO)J;a B CYMMbI BeCb x) 3-ro cepe)J;HHe paBeH rO)J;a BbIITJIa- aaeMID;HK BbIITJIa"tlHBaeMOH paaMep KOHe BbIITJIaTa ITOCJIe)J;HHH H CJIe)J;YIOID;He "tIepea - 1-ro KOHe a x, KOHe B 6bITb )J;OJImeH YCJIOBHIO paBHa 3Ha"tlHT, 144 1,2(1,28 H o6ID;HH BbIITJIaT paaMep x) - paBeH x, 2,2x o 48 + = 72 , 1,448, = 8 > 8 8 = . no 55 ' 948 8 = 36 = 220 94 8 55 94 x > 440 55' 8 YCJIOBIUO . 55 @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H «Ha:U;HOHaJIbHOe «M3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe B KOMMep'tleCKHX 06pa30BaHHe». :u;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY ,l];rrO pa3pemeHHH «MD;HMO» IIpaBOo6)1a.,ZJ;aTeJIei He cepe- ,ZJ;OIIYCKaeTCH x. Er3. 236 S IIpH 5 == OTBeT: 3TO 5 HepaBeHCTBO a BepHO, COep3taHHe BepHo 4 == OHO BAPVlAHTbl 3K3AMEHAL.J.VlOHHbIE KaK HeBepHO, H npH S. MeHbrnHX py6J1eH. MJlH 060CHOBaHHO S npH TVinOBblE BepHbIM rroRyqeH 3 OTBeT MaTeMaTHqeCKaH rrOCTpoeHa BaJIJILI KpHTepHH K CBe,n;eHO pemeHHe Mo,n;eRb, HCCRe,n;OBaHHIO u STOH - - H Mo,n;eRH, HeBepHbIM BepHbIM BepHo H3-3a OTBeT HO OTBeT, Mo,n;eRH, He omH6KH; 060cHoBaHo He,n;OCTaTOQHO pemeHHe STOM rrpH PemeHHe 2 BbIqHCRHTeRbHoM MaTeMaTHqeCKaH rrOCTpoeHa STOM pe3YRbTaT: rrOReH Mo,n;eRb, HH COOTBeTcTByeT H3 O,n;HOMY He K CBe,n;eHO pemeHHe 6bITb MomeT pemeHHe HCCRe,n;OBaHHIO Bce 3Ha-qeHH5I a, ( 0, == ==a perneHHH. rneCTb pOBHO 3a) x + - 3 6aJtJt ypaBHeHHH CHCTeMa KOTOpbIX 2)(y + -ax ay { Ixyl HMeeT' H3 Kam,ll;OM IIPH 0 BbIme rrepeqHCReHHbIX KpHTepHeB, M aICCU.MaJtb1-lblU HaH,lI;HTe 1 3aBepmeHO PemeHHe. 1I3 BToporo ECJlH a CHCTeMbI ypaBHeHHH 0, == TO nOJly"tlaeM O. > a o. CHCTeMY y-x=o, { 3Ha"tlHT, > a nOJly"tlaeM: IIOJly"tlHM x=y=O. OTKY,lI;a 0, == xy CHCTeMY ( a(y {I xy - I == x) 2)(y + 3a) x + - == 0, a, 2 y-x==-- oTKy,n;a HJlH a Ha eM Koop,n;HHaTHoH ECJlH MbI H npHMbIe nOJly"tla- rHnep- ,n;Be rneCTb TO COBna,n;aIOT, npHMbIe ,n;e + a. xy== nJlOCKOCTH napaJIJleJlbHbIe ,n;Be 60JlbI. y-x==-3a, pa3- 2 perneHHH JlH"tIHbIX IIo3ToMY HeB03MomHO. - * 3a, a a"* OTKY,lI;a . a IIpH 3TOM YCJlOBHH rHnep60Jla y == - rrepeceKa- x eT :3TO H3 Kam,n;ylO ,n;aeT "tIeTbIpe @ 2021. KOIIHpoBaHHe, npHMbIX pacIIpocTpaHeHHe B ,n;BYX pa3J1H"tIHbIX 000 H perneHHH «Han;HOHaJIbHOe «M3,n;aTeJIbCTBO HCIIOJIb30BaHHe B TO"tlKax. pa3J1H"tIHbIX KOMMepqeCKHX ,n;aHHoH 06pa30BaHHe). n;eJIHX 6e3 CHCTeMbI. @ 2021. IIHCbMeHHoro qOY pa3perneHHH ,l];rro «MD;HMO) IIpaBOo6JIa,n;aTeJIeii He ,n;OIIYCKaeTCH VI KPVlTEPVIVI PEWEHVls:I 3AAAHVl1I113-19 OL.J.EHVlBAHVls:I 237 a Ee nOJly"tlalOTcH perneHHH )J;Ba npH rHnep6oJlOH npHMbIX nepeCe"tleHHH ==-- y X B )J;BYX pa3J1H"tIHbIX TY npHMYIO, "tITO 6J1Hme "tITO K JlH TO"tlKax. OT )J;aJIbrne 3Toro HymHo, Ha"tlaJla JlH rHnep60Jla H Koop)J;HHaT, Koop)J;HHaT. Ha"tlarry "tITo6bI 3Toro He 06HX HMeJla "tITo6bI HymHo, )J;Bam)J;bI H3 nepeCeKaJla TO"tleK C npHMoH, KBa,n;paTHbIx )J;BYX ypaB- HeHHH 2 x -!x+a 0 == x H 2 -3ax+a 0 == a HMeJlO O)J;HO ypaBHeHHH 3THX TbI )J;Ba pOBHO 6bITb )J;OJlmHbI 0, > a "tITO K npHXO)J;HM 3 (a 4 3HaKOB. 2 ( 9a - ) 4a < 0 HCKpHMHHaH- · HepaBeHCTBY )(9a-4) -1 0, > 1. a> HJlH a <- OTKY)J;a ) 4a - KopHeH. IIoJly"tlaeM: HMeJlO He )J;pyroe npOTHBOnOJlOmHbIX ( :2 Y"tIHTbIBaH, a KOpHH, pa3J1H"tIHbIX 9 OTBeT: 0 4 < a < 1. > a -; 9 Co.n;epataHHe 060CHOBaHHO C BepHoro IIOMOm;blO ( a = 1, a = : ' a paccym,n;eHHH ), = HO BKJIlOqeHHeM Bce HeBepHO a TOqeK TOqKH rpaHHqHbIe orrpeeJIeHbI 3HaqeHHH MHomeCTBO IIOJIyqeHO TOJIbKO HaH,n;eHbI BepHO pemeHHH 4 OTBeT HCKOMoro OT OTJIHqalOm;eeCH B BepHbIH IIOJIyqeH BaJIJILI KpHTepHH = 1 H/HJIH a 3 =- 9 3HaqeHHH MHomeCTBa 3HaqeHHH rrpOMemYTKH a, 4 a a, 2 HJIH IIOJIyqeH HeBepHbIH BbIIIOJIHeHbI BepHO OTBeT Bce marH omH6KH, HO 3TOM IIPH BepHO pemeHHH 6bI XOTH paCCMOTpeH (- : ) rrpOMemYTKa BbIqHCJIHTeJIbHOH H3-3a <X.J; H (1; + <X.J O,n;HH ), H3 CJIyqaeB C B03MomHO, H pemeHHH, BKJIIOqeHHeM IIOJIyqeHbI ,n;Ba TOqeK, rpaHHqHbIX 1 HJIH 3a,n;aqa H (aHaJIHTHqeCKH IIpHMbIX PemeHHe K CBe,n;eHa BepHo He COOTBeTcTByeT B3aHMHoro HCCJIe,n;OBaHHIO HJIH HH rHIIep60JI paCIIOJIOmeHHH rpacpHqeCKH) O,n;HOMY H3 KpHTepHeB, M aICCU.MaJtbHblU @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H «HaU;HOHaJIbHOe (dla.n;aTeJIbCTBO HCIIOJIbaOBaHHe B KOMMep'tleCKHX o6paaOBaHHe». u;eJISlX 6ea @ 2021. IIHCbMeHHoro QOY pa3pemeHHSI 0 BbIme IIepeqHCJIeHHbIX ,1];IIO «MD;HMO» IIpaBOo6JIa.n;aTeJIeii 4 6aJtJt He .n;OIIYCKaeTCSI Er3. 238 M3BecTHo, "tITO 2, ,ZJ;OCTOHHCTBO 3a Kam,ZJ;YIO Bce MOrJlH n MOrJlH H JlemaJlo 10 6e3 C,ZJ;a"tlH H3 Kam,ZJ;aH MOHeT, c,ZJ;eJlaJIa Bce CBOH TOJlbKO 3THMH MOrJla KOTOpbIX HMeTb paCnJlaTHBIIIHCb nOKynKH, MOHeTaMH, 3TOM npH nOTpaTHB nOKynKH COCTOHTb nOKynKH COCTOHTb H3 6J10I(HOTa H3 "tIaIIIKH 56 3a py6J1eii H 29 3a pyrIKH py6J1eii, 14? ee Bce 20 3a nHpomKa py6J1eii, AHH n eCJlH aJIb60M TOJlbKO KynHJla 10 3a "tIaH py6J1eii, 15 3a CbIpKa py6J1eii 19? == nHTHpy6J1eBbIx 3a 85 py6J1eii KOJlH"tIeCTBO HaHMeHbIIIee eCJlH n AHH py6J1eii. OT,ZJ;eJlbHO ee Bce == JlH KaKoe B) HJlH BAPVlAHTbl 3K3AMEHAL.J.VlOHHbIE KOIIIeJlbKa. H3 JlH eCJlH 6) KOIIIeJlbKe 5 nOKynKY MOHeTbI a) B TVinOBblE n H 6bITb MOrJlO MOHeT B KOIIIeJlbKe, 24? == - PemeHHe. a) HanpHMep, JleBbIe MOHeTbI (8 H 2 ,ZJ;Byxpy6J1eBbIe 6) IIpe,ZJ;noJlomHM, a. 3a py6J1eBbIx. a HeTY, C060B. HO, B 3a H nepBOM 7 ,ZJ;Be Mepe MOHeTaMH. Bo BTOpOM B) l BbIX, 37 k IIpHMep THpy6J1eBbIx == aJIb60M m H 3l - 3a 6bI - - - a) 24- == 8. JlH60 Bce Kpaii- no 6 Ha6paTb HeJlb3H 22 HJlH 11 :3TO MOHeT. TaKme py6J1eii. MOrJla He C,ZJ;eJlaTb l 15, l- 48 m, 7 == H m B) 7. == 2 2l- - 24 nOTpaTHB 2k Tor,ZJ;a 2m "tIHCJlO + 5l l He"tleTHO. + COOTBeTCTBeHHO 10m + YKa3aHHbIe 85 == He AHH k MOHeTbI: 10m == l H == 37 Ha ,ZJ;Byxpy6J1e- k + l + - 3l. m == 6) HeT; Bce rrOJIeHbI rrOJIeHbI H 5 rrOJIyqeHbI rrOJIeH ,lJ;Ba H3 H3 O,lJ;HH HCKOMaH 3anJlaTHTb 7 pOBHO B rrepeqHCJIeHHbIX CJIe,lJ;YIOm;HX pemeHHe rrYHKTa 0; 6a.rrJI) Ha 1 Ha 1 4 pe3YJIbTaTbI 6a.rrJI) 6a.rrJI) pe3YJIbTaTOB 3 pe3YJIbTaTOB 2 1 06eCrreqHBalOm;HH COOTBeTcTByeT 1 8; rrYHKTe 8, rrYHKTe Ha pe3YJIbTaTOB: a; B BaJIJILI KpHTepHH (CM. KpHTepHH (CM. KpHTepHH rrepeqHCJIeHHbIX rrYHKTa On;eHKa He (CM. pemeHHe 06ocHoBaHHoe rrpHMep TPH H3 HH O,lJ;HOMY rrpe,lJ;bI,lJ;ym;eH TOqHOCTb H3 On;eHKH @ 2021. paCIIpOCTpaHeHHe 000 H «H3,ZJ;aTeJIbCTBO HCIIOJIb30BaHHe «Ha:U;HOHaJIbHOe B KOMMepqeCKHX o6pa30BaHHe». :u;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY 0 BbIme rrepeqHCJIeHHbIX KpHTepHeB, ,rr;rro pa3pemeHHH «MD;HMO» IIpaBOo6JIa.,ZJ;aTeJIeH BbI- 8. KpHTepHH rrepeqHCJIeHHbIe 24. 3Ha"tlHT, 3 1, paBHOM ,ZJ;eJlHTCH MOrJla 85 8m H IIpH H "tITO nOKa3bIBaeT, 5l + M aICCU.MaJtbHblU KOIIHpoBaHHe, 6bITb MOrJlH He MOHeTa, MeHee MeHee AHH py6J1eii, ,ZJ;ecHTHpy6J1eBbIx. CJle,ZJ;oBaTeJlbHO, 28 H 22 34, paBHO ,ZJ;a; 06ocHoBaHHoe PemeHHe He "tITO 85 3a CO,ZJ;epataHHe - MOHeT. MOHeT. OTBeT: BepHo BepHo BepHo BepHo He nHpomOK nOJlyrIHJlOCb nOKa3bIBaIOT, KynHJla Ha 37 pameHHe 3anJlaTHJla Tor,ZJ;a k ,ZJ;eJlHTCH CJle,ZJ;oBaTeJlb- 8 OHH MOHeT. ,ZJ;eCHTHpy6J1eBaH OCTaBIIIHecH 10 py6J1eii cno- 06pa30M. nOJly"tlaeM 3l - 8 nHpomOK MO- BbIIIIe MOHeT. 60Jlee He ,ZJ;BYX- npOTHBOpe"tlHIO. npOTHBOpe"tlHH AHH IIYCTb nHTHpy6J1eBbIx OTclO,ZJ;a MOHeT, 6 JlH60 3, JlH60 5 nHTHpy6J1eBYIO YKa3aHHbIX 6bIJla MOHeTbI. K OHa Tpe6yeMbIM 3a JlH60 HHX JlH60 2, 11 "tIaH "tIaIIIKY JlH60 O,ZJ;HY Tpex H3 O,ZJ;HHM JlH60 JlH60 6bI XOTH MomHO 3anJlaTHJla cpe,ZJ;H nOCKOJlbKY IIoJly"tleaHbIe 3anJlaTHTb py6J1eii 3anJlaTHJla OHa cJly"tlae HeB03MomHO, nOKynKH OHa 3Ha"tlHT, nHTHpy6J1eBbIe IIpHIIIJlH MOHeTY, 6bIJla 3a o6pa30M. nHTHpy6J1eBbIe, 2 JlH60 H 3 MOHeTY, 3 ,ZJ;Byxpy6nHTHpy6J1eBbIe MOHeT Tpe6yeMbIM nOKynKH 3anJlaTHJla OHa cJly"tlae c,ZJ;eJlaJIa 10 ,ZJ;ecHTHpy6J1eBbIx ,ZJ;ecHTHpy6J1eBYIO 1 py"tlKY ,ZJ;OJlmHa CbIpOK CbIpOK 3a ,ZJ;ecHTHpy6J1eBYIO AHH OCTaBIIIHeCH ,ZJ;Byxpy6J1eBbIe. Heii a 5 6J10KHOT 3a AHH "tITO 1 CbIpOK Ha6paTb 3Ha"tlHT, "tIaii 3a MOHeT), (6 MOHeT). JlH60 3anJlaTHJla OHa 3anJlaTHB 4 oaJtJt He ,ZJ;OIIYCKaeTCH nH- PEWEHLt1 Lt1 KPLt1TEPLt1Lt1 3AAAHLt1 OU.EHLt1BAHLt1 13-19 239 27 BapuaHT lliJ a) PemHTe 6) Haii,n;HTe (2 ypaBHeHHe Bce 2 cos 3Toro KOpHH 3sin x + x 3) - ( .J2 .log2 cos ) x 0 = [-5n; OTpeaKY IIpHHa,n;JIemaID;He ypaBHeHHH, . 3n - ] . PemeHHe. IIpeo6paayeM a) ypaBHeHHe: (2 2 -2sin 2 (2sin PernaH .J2 ( log2 3HaQHT, IIepBoe cos x ) 0 = x + 3sin ( .J2 cosx) .log2 ).log2 x-3sinx+l 2 2 sin HJIH x 1 x + = 0 = 0; = (.J2 cosx) 3 sin - O. x cos 1, = x oTKy,n;a + = BTopoe sinx IIOJIyqaeM ypaBHeHHe, 0, > x Haii,n;eM 6) =-+21tk, x IIOJIyqaeM: k r,n;e < -51t HepaBeHCTBY y,n;OBJIeTBOpHIOID;He 21tk -51t -31t, < < HepaBeHCTBa BToporo HepaBeHCTBa: p eTbero 3aMe1lallUe. He p OT6op k Haxo,n;HM 21tk -31t < C -2, = k aBeHCTBa: a = -2 YCJIOBHe H x a k -+21tk, 6) 060cHoBaHHo rrOJIyqeH = - - 41t OTBeT OTBeTbI H3-3a -31t . 171t . 4 -- . 6 H JII06bIM H cIIoco6oM: ,n;pyrHM T. 171t 151t -- ' 6 060HX B B C II. -4' 4 BaJIJILI KpHTepHH OTBeTbI BepHbIH < 231t = -- COep3taHHe BepHble 21tk 4 oKpymHocTH Z; + --. 231t E 4 6 rrOJIyqeHbI - 151t = 06ocHoBaH e,n;HHHQHOii 1t + 060CHOBaHHO < x=---41t=-4 3TOM IIPH =--41t 4 x , IIoJIyqaeM: 1t -2 = 6bITb MomeT IIOMOID;blO a) -+21tk, 2 rrYHKTax a, rrYHKTe HJIH rrOJIeHbI HeBepHble HMeeTCH rrYHKTa PemeHHe H He BblqHCJIHTeJIbHOH Bcex rrOCJIeOBaTeJIbHOCTb BepHaH a omH6KH, marOB HO 060HX pemeHHH rrpH 0 rrYHKTa COOTBeTcTByeT @ 2021. paCIIpOCTpaHeHHe 000 H 1 3TOM rrYHKTOB: HH OHOMY H3 «Ha.n;aTeJIbCTBO HCIIOJIbaOBaHHe «Ha:U;HOHaJIbHOe B KOMMepqeCKHX o6paaOBaHHe». :u;eJIHX 6ea @ 2021. IIHCbMeHHoro qOY paaperneHHH 0 Bblme rrepeqHCJIeHHbIX KpHTepHeB, M aICCUMaJtbHblU KOIIHpoBaHHe, . 6 6 1t OTBeT: -51t HJIH 1t k KopHeii rpaQ.:>HKa, MOID;blO -31t. < 4 IIepBoro T Z 1t + - 1t Ma YtIHTbIBaH =-. < x 1t + -- 4 Ma E 1 Slnx HJIH k Z. E 1t Ma O. 6 KOpHH, -51t 1 = r,n;e 2 1t COS X > 21tk, + - 4 . PernaH COS YCJIOBHH IIPH 1t .J2 IIOJIyqaeM ypaBHeHHe, -3) x ,l];rro «MD;HMO» IIpaBOo6JI8.,lJ;aTeJIeH oaJtJt 2 He .n;OIIYCKaeTCH IIO- Era. 240 [K] B o6pa3yrom;aH HJIHH,n;pe M3BecTHo, HOBaHHH. a) 6) C TO"tlKa - OCHOBaHHH ,D:oKamHTe, HaH,n;HTe BbI6paHbI HJIHH,n;pa l LACB "tITO "tITO CC rrpH"tIeM ' yrOJI paCCTOHHHe AB B AC a Ha oKpymHocTH oKpymHocTH AC a ,n;pyroro - oc- ,n;HaMeTp 6. == l BC 60°. paBeH AC rrpHMoH ,n;o CC , H l C, H HJIHH,n;pa, == rrpHMbIMH TO"tlKH OT l 45°, == Mem,n;y o6pa3yrom;aH 3.J2 - Ha OCHOBaHHH. B A, TO"tlKH BAPlIIAHTbl aK3AMEHALJ.lIIOHHbIE rrJIOCKOCTH rreprreH,n;HKYJIHpHa oCHoBaHHH H3 o,n;Horo TlIInOBblE . l PemeHHe. IIycTb a) BBl Mem,n;y yrOJI 3Ha"tlHT, rreprreH,n;HKYJIHpHa BlC H l BC paBeH BlC AB rrpHMbIM BBlClC AClB yrJIY OCHOBaHHH ,n;HaMeTp rrpHMaH l l C 1 BC, rrpHMOH o6pa30M, a , rr03TOMY rrpHMoyrOJIbHHK, rr03TO- HJIHH,n;pa, TaKHM ABB - . rraparrJIeJIbHaH , l BBl. H rrJIOCKOCTH rreprreH,n;HKYJIHpHa l Tor,n;a HJIHH,n;pa. Ha orrHpaeTCH rrpHMOH. OH MY AC rrpHMbIMH ABC YrOJI o6pa3yrom;aH - Bl rrpHMaH 3Ha"tlHT, ABlC yroJI l rrpHMoH. B rrpHMoyrOJIbHOM & 4 BC == 3Ha"tlHT, 6) AB == 3.J2, == LA4& BC 2 cc;f + cq + 2 == BC 3.J6 == . ero I7S 3" 6, . 2 AB ==" o6pa30M, ABC TpeyrOJIbHHK h, BbICOTe I A4 TaKHM BBl. H CJIe,n;oBaTeJIbHO, BBlC. paBHO CO,ll;epmaHHe HMeeTCH .n;OKa3aTeJIbCTBO BepHoe BepHbIH HMeeTcH 2 AB == rrpoBe,n;eHHoH 2 Bq + rn 6" == 2, K h . l rHrroTeHY3e. AB B4 . == rrpHMoyrOJIbHbIH, IIoJIyqaeM: == I7S 6 1,5" · A4 B OTBeT YTBepm.n;eHHH 060CHOBaHHO H a, nYHKTa nOJIY- 2 6 nYHKTe .n;OKa3aTeJIbCTBO BepHoe BaJIJILI KpHTepHJI YTBepm.n;eHHH a, nYHKTa HJIH 06ocHoBaHHo BepHbIH nOJIY"tIeH a, nYHKTa YTBepm.n;eHHH PemeHHe He OTBeT 3TOM npH HH COOTBeTcTByeT B a nYHKT H3 O.n;HOMY 1 6, nYHKTe He C B03MomHO, HCnOJIb30BaHHeM BbInOJIHeH KpHTepHeB, OU 2 6aJlJl 1 log5( : )-log5(X+2) +2 HepaBeHcTBo 0 BbIme nepe"tlHCJIeHHbIX M aKCUMaJlb1tblU PeIIIHTe AB rrpHMaH 1,5.J6. OTBeT: "tIeH l rrpHMbIM paccToHHHe I ==" ABlC + BBf 2 rreprreH,n;HKYJIHpHa HCKOMoe B4 AB == rrJIOCKOCTH rreprreH,n;HKYJIHpHa rro3ToMY A& 60°. == AB IIpHMaH HKe TpeyrOJIbH < lOg5( XX: ). PemeHHe. JIeBaH 3TOM "tIaCTb YCJIOBHH HMeeT HepaBeHCTBa HepaBeHCTBO CMbICJI -2 rrpH < x < -1,5 HJIH x> 0, rr03TOMY BH,n;: rrpHHHMaeT 2x+3 x+1 < . - x(x+2) 2X2 x +3x x 2 2 , +3x+2 < . , - x @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H «HaU;HOHaJIbHOe «H3.n;aTeJIbCTBO HCIIOJIb30BaHHe 2 B KOMMepqeCKHX (x+2) x o6pa30BaHHe». u;eJIHX 6e3 2 (x+2) @ 2021. IIHCbMeHHoro qOY )];rro pa3pemeHHH «MIJ;HMO» IIpaBOo6JIa,n;aTeJIeH He .n;OIIYCKaeTCH rrpH II PEWEHlII KPlIITEPlIIlII 3AAAHlIIVl13-19 OU.EHlIIBAHlII 241 2 x -2 < 0; (X-v'2)(X+v'2) OTKy,n;a O<x - v'2 < x v'2. < Y"tIHTbIBaH O. < -2 OrpaHH"tIeHHe < -1,5 < x 0, > x HJIH rrOJIy"tlaeM: v'2. < OTBeT: (0; v'2] . CO,ll;epmaHHe 060CHOBaHHO IIOJIy"tleH 060cHoBaHHo IIOJIY"tleH BepHbIH BaJIJILI KpHTepHJI 2 OTBeT OTJIH"tIalOID;HHCH OTBeT, OT HCKJIlO"tIeHHeM BepHoro .J2, TO"tlKH HJIH 1 HeBepHbIH IIOJIyqeH OTBeT Bcex IIOCJIe)J;OBaTeJIbHOCTb BepHaH PemeHHe He BbI"tIHCJIHTeJIbHOH H3-3a marOB HH COOTBeTcTByeT omH6KH, HO 3TOM IIPH HMeeTCH pemeHHH H3 O)J;HOMY KpHTepHeB, M aKCUMaJlb1tblU 00 OKpymHocTb a) 6) C BC 9, == ,D:oKamHTe, HaH,n;HTe 01 eHTpOM ABCD. TparreHH AB "tITO == C AD, H KacaeTCH a 60KOBOH TaKme CD BC, CTOpOH 2 6aJlJl H AB CTOpOHbI AD. MaBecTHo, 15. == 0102 rrpHMaH BC 02 eHTpOM AD == "tITO OCHOBaHHH KacaeTCH OKpymHocTb 8, CD 4, 0 BbIme IIepe"tlHCJIeHHbIX OCHOBaHHHM rrapaJIJIeJIbHa ABCD. TparreHH 0102. PemeHHe. TO"tlKa a) 3Ha"tlHT, 01 ABCD. AHaJIOrH"tIHO, HeH JIHHHH TparreHH ° ° 1 ABCD, BaHHHM 6) - cepe,n;HHa BC AB, rrpHMbIX TO"tlKa HAD, rr03TOMY 6HcceKTpHcaMH LAOIB eCTb AB yrJIOB 0 90 == rrpHMoyrOJIbHOrO ero cpe,n;rrpH- OCHO- + CD. ce- A AOI LABO:t = D BOI H 3Ha"tlHT, COOTBeTCTBeHHO. LABC LBAD; KOI = - 90 0 , rrpoBe,n;eHHaH Me,n;HaHa, K rHrroTeHyae AOIB. C0 TO"tlKH - ABC H CJIe,n;oBaTeJIbHO, TpeyrOJIbHHK rHrroTeHyae L a paBHoy,n;arreHa JIy"tlH DAB TpeyrOJIbHHKa AHaJIOrH"tIHO, K C TparreHH AB, 01 LBAO:t TO Ha rraparrJIeJIbHa CTOpOHbI CD. CTOpOHbI HBJIHIOTCH B ABCD. TparreHH K JIemHT JIHHHIO 0102 rrpHMaH BC. Tparre- CJIe,n;oBaTeJIbHO, cpe,n;HIOIO aHa"tlHT, IIycTb pe,n;HHa OT co,n;epmHT 2 a ABCD. H JIHHHH 02 TO"tlKa AD rrpHMbIX cpe,n;HeH Ha JIemHT HH MaH OT paBHoy,n;aJIeHa 01 TO"tlKa K, 2 D rrpHMoyrOJIbHbIH, 02 01' H L L02 a JIemaT Ha - cpe,n;HeH Me,n;HaHa, JIHHHH rrpoBe,n;eHHaH ABCD. TparreHH 3Ha"tlHT, 0 10 OTBeT: == KL-K0 AD+BC 1 CD AB ' -L02 == == ---- 5. 222 6) 5. @ 2021. KonHpoBaHHe, 2 pacnpOCTpaHeHHe 000 H «ll3.n;aTeJIbCTBO HCnOJIb30BaHHe «HaU;HOHaJIbHOe B KOMMepqeCKHX 06pa30BaHHe». u;eJIHX 6e3 @ 2021. nHCbMeHHoro qOY )];rrO pa3pemeHHH «MIJ;HMO» npaBoo6JIa.n;aTeJIeH He .n;onYCKaeTCH Er3. 242 CO,ll;epmaHHe HMeeTCH BepHbIH qeH OKa3aTeJIbCTBO BepHoe OTBeT 06OCHOBaHHO B BAPlIIAHTbl 3K3AMEHAL..J.lIIOHHbIE BaJIJILI KpHTepHJI rrYHKTa a, H rrYHKTa a, H YTBepmeHHH 060CHOBaHHO rrOJIY- 3 6 rrYHKTe BepHbIH rrOJIyqeH TlIInOBblE B OTBeT 6, rrYHKTe HJIH HMeeTCH OKa3aTeJIbCTBO BepHoe YTBepmeHHH 6 rrOJIyqeH HeBepHbIH pemeHHH rrYHKTa HMeeTcH BepHoe OKa3aTeJIbCTBO 06ocHoBaHHoM pemeHHH OTBeT a, rrYHKTa YTBepmeHHH omH6KH apH<pMeTHqeCKOH H3-3a 2 06ocHoBaHHoM rrpH HJIH npH TH"IIeCKOH 6 rrYHKTa HeBepHbIH rrOJIyqeH OTBeT apH<pMe- H3-3a 1 omH6KH, HJIH 06ocHoBaHHo BepHbI rrOJIyqeH a, nYHKTa PemeHHe 3TOM IIPH He B OTBeT a rrYHKT He HH COOTBeTcTByeT 6 rrYHKTe HCrrOJIb30BaHHeM C YTBepmeHHH BbIrrOJIHeH H3 OHOMY KpHTepHeB, o BbIme rrepeqHCJIeHHbIX M aKCUMaJlbHblU 15 [ill B3HTb llJIaHHpyeTcH MapTa 6aHKe B Kpe,n;HT 26 Ha 3 6aJlJl YCJIOBHH MeCHeB. ero B03BpaTa TaKOBbI: 1-ro "({HCJIa 2-ro 14-e 110 15-ro "({HCJIa "({HCJIO Kam,n;oro Kam,n;oro % 3 Ha B03paCTaeT lIorameHHH 15-e aa ,n;OJIra 1-ro 110 C cpaBHeHHIO B3HTb B ,n;OJIr KOHOM IIpe,n;bI,n;Yl.QerO 6bITb IIOJIHOCTblO 06I.QaH eCJIH Kpe,n;HT, "({aCTb ,n;OJIra; 6bITb ,n;OJImeH Ha 40 TbICHq MeCHa; ,n;OJImeH Kpe,n;HT 1924 BbIIIJIaTHTb 25-ii 110 "({HCJIO MeCHa COCTaBHT He06xo,n;HMO MeCHa C MeCHa Py6JIeii K 26-ro 15-MY "({HCJIY KaKYIO CYMMY llJIaHHpyeTcH MeHbme ero ,n;OJIr MeCHa; IIpe,n;bI,n;Yl.QerO co MeCHa Kam,n;oro IIorameH. BbIIIJIaT CYMMa 1I0CJIe 1I0JIHOrO Py6JIeii? TbICH"({H PemeHHe. IIYCTb (B ,n;o CYMMa Kpe,n;HTa Py6JIeii) TbIC. HYJIH A A; HOCTb "({HCJIa 1,03A; CJIe,n;OBaTeJIbHO, Bcero 1,78A 1 pacrrpOCTpaHeHHe Kam,n;oro ,n;OJIr MeCHa 80; (B - ,n;OJIr - 110 6aHKoM lIepe,n; ,n;OJImeH YMeHbmaTbCH Py6JIeii) 40; + 3 Ha 1,03(A 0,03(A 960) 3H-a"({HT, IIOCJIe,n;OBaTeJIbTaKOBa: "({HCJIO 1000). - 6bITb ,n;OJImHbI - 0/0, 1-e Ha 960); - O. 1000; - COCTOHHHIO 1,03(A TbIC. A 960; B03paCTaeT 40); 40) A ...; CJIe,n;YIOI.QHMH: 40; + +1000+1,03(A-IOOO)=1,78A-390 2 390 - = 300 TbICHQ «Ha.n;aTeJIbCTBO HCnOJIbaOBaHHe = A 2314; 1,03(A - 1000). KOMMep-qeCKHX (TblC. py6JIeH), 1300. B paBHa Kpe,n;HT, 1 MJIH 300 TbIC. py6JIeii. py6JIeii. «HaU;HOHaJIbHOe B = B3HTb IIJIaHHpyeTcH KOTOpyro 000 H 1, 78A 1924; MHJIJIHOH @ 2021. KonHpoBaHHe, - 0,03(A 2A-960 CYMMa, OTBeT: YCJIOBHIO BbIIIJIaTHTb CJIe,n;yeT 3HaQHT, 1,03(A 40; 25.0,03. oTKy,n;a "({HCJIO Py6JIeii) TbIC. BbIIIJIaTbI + - MeCHa (B ,n;OJIra A 40; - Kam,n;oro pa3MepOB 0,03A IIo Py6JIeii. TbICH"({ 15-e Ha 06pa30M: CJIe,n;YIOI.QHM IIepBoro A paBHa COCTOHHHIO 110 06paaOBaHHe». u;eJIHX 6ea @ 2021. nHCbMeHHoro qOY ,[(rro paapemeHHH «MIJ;HMO» npaBoo6JIa.n;aTeJIeH He .n;onYCKaeTCH PEWEHV1 V1 KPV1TEPV1V1 3AAAHV1 OU.EHV1BAHV1 13-19 243 CO,IJ;ep3taHHe 06ocHoBaHHo BepHo H MoeRH HeBepHbIM BepHbIM - - BepHo HO MoeRH, PerneHHe He 2 06ocHoBaHo HeOCTaTOHO perneHHe 3TOM npH HCCReOBaHHIO ornH6KH; MaTeMaTHeCKaH nOCTpoeHa 3TOM K CBeeHO perneHHe BbIHcRHTeRbHoM H3-3a OTBeT, MoeRb, pe3YRbTaT: nORyqeH OTBeT 3 OTBeT MaTeMaTHeCKaH nOCTpoeHa 3TOM BepHbIM nORyeH BaJIJILI KpHTepHJI MoeRb, perneHHe HH COOTBeTcTByeT H3 OHOMY K CBeeHO perneHHe 6bITb He MomeT HCCReOBaHHIO 1 3aBeprneHO KpHTepHeB, HaH,D;HTe 3HaQeHHSI Bce a, rrapaMeTpa { HMeeT qeTbIpe pOBHO 4 x x y2 + 2 H3 Ram,D;OM rrpH + y a2 = Ia = ypaBHeHHH CHCTeMa ROTOpbIX 3 6aJlJl MaKCUMaJlbHblU [ill 0 BbIrne nepeHCReHHbIX , 11 + pemeHHH. PemeHHe. Mcxo,n;HaH CHCTeMa { TaKHM meHHH x +(la+ll-x2 = y o6paaoM, la+ll t 2 -x pOBHO , Tor,n;a, X 4 - -la+ll y 2 -2Ia+llx 8TO KOpHH. 2 2Ia+llx +2a+l = 0, 2 -x ypaBHeHHH 6HKBa,n;paTHoe Kor,n;a 4 2 { ; paaJIHqHbIX qeTbIpe a = 2x HMeeT 2 CHCTeMa Hcxo,n;HaH TOJIbKO H Tor,n;a 4 ypaBHeHHH: CHCTeMe paBHOCHJIbHa . HMeeT paaJIHqHbIX qeTbIpe pOBHO pe- ypaBHeHHe +2a+l = 0 Kor,n;a BbIIIOJIHHeTCH, KBa,zJ;paTHoe ypaBHe- HHe 2 2t HMeeT 1-.J2 a < QTo6bI HbIH ( a + 1)2 - 8 2a ( + - QTo6bI t 0,5 6bITb 1- ,n;HCKpHMHHaHT a 2 2a -1 - > 0 , 6bIJIH o,n;Horo cBo6o,n;- aHaKa, IIOJIOmHTeJIbHbIM: TO OTpHaTeJIbHbIM, a > HJIH (-0,5; pacnpOCTpaHeHHe 0; > + 1 > 0 , 1-.J2); 000 H 1 +.J2 (1 + K03<P<PHHeHT IIOJIOmHTeJIbHbIMH, -1 . HMeeT pOBHO qeTbIpe pemeHHH . .J2; + 00 «HaHoHa.rrbHoe B a *- eCTb ypaBHeHHH CHCTeMa «H3,ZJ;aTeJIbCTBO HCnOJIb30BaHHe 6bIJIH ypaBHeHHH Hcxo,n;HaH .J2 @ 2021. KonHpoBaHHe, KBa,n;paTHoro o6paaoM, OTBeT: ero KOpHH, 0,5. ,n;OJImeH < a < ) ypaBHeHHH 6bITb ,n;OJImeH ypaBHeHHH KOpHH TaKHM - 1 KBa,n;paTHoro 2a IIPH ,n;Ba . IIOJIyqeHHOrO 3Toro a> oTKy,n;a HMeJIO ypaBHeHHe 1+.J2 a> HJIH KOpHH qJIeH 0 IIOJIOmHTeJIbHbIM: 4 oTKy,n;a = KOpHH. KBa,n;paTHoe IIOJIyqeHHOe 6bITb ,n;OJImeH IIOJIOmHTeJIbHbIX ,n;Ba pOBHO QTo6bI -2Ia+llt+2a+l KOMMepqeCKHX ) . 06pa30BaHHe». eJIHX 6e3 @ 2021. nHCbMeHHoro qOY )];nO pa3pemeHHH «MIJ;HMO» npaBoo6JIa,z:J;aTeJIeH He ,ZJ;onYCKaeTCH IIPH Er3. 244 CO,ll;epmaHHe 060CHOBaHHO C BepHbIH nOJIyqeH nOMOIIJ;blO BAPlIIAHTbl 3K3AMEHAL..J.lIIOHHbIE BaJIJILI KpHTepHJI 4 OTBeT TO"tleK BKJIlO"tIeHHeM TOJIbKO 3Ha"tleHHH MHomeCTBO nOJIyqeHo paccymeHHH BepHoro HCKOMoro OT lOIIJ;eeCH TlIInOBblE a = -0,5, a OTJIH"tIa- a, 1-.J2, == 3 H/HJIH a==l+.J2 B ( == HHH HaHeHbI BepHO pemeHHH a 1-.J2, a -0,5, == a Bce 1 == TO"tlKH rpaHH"tIHbIe .J2), + HO HeBepHO 3Ha"tleHHH MHomeCTBa a 3Ha"tle- npOMemYTKH onpeeJIeHbI a, 2 MJIM HeBepHbIH nOJIy"tleH BepHo 1-.J2) 00; 6bI (1 H omil6KH, HO 3TOM npH BepHO pemeHHH XOTH paCCMOTpeH (- Ka marH BbI"tIHCJIHTeJIbHOH H3-3a OTBeT Bce BbInOJIHeHbI H3 OHH .J2; + ), + 00 CJIy-qaeB H pemeHHH B03MomHO, Ba nOJIy-qeHbI BKJIlO"tIeHHeM C npOMemYTTO"tleK, rpaHH"tIHbIX 1 MJIM 3a)J;a"tla H K CBeeHa BepHO HCCJIeOBaHHIO (aHaJIHTH"tIeCKH npHMbIX PemeHHe He B3aHMHoro HH COOTBeTcTByeT pacnOJIOmeHHH oKpymHocTH rpa(}>H"tIeCKH) HJIH H3 OHOMY o BbIme nepe"tlHCJIeHHbIX KpHTepHeB, M aKCUMaJlb1tblU Ha [ill 12 HalIHcaHO ,n;OCKe HaHMeHbillHX ceMH H3 pa3JIHqHbIX HHX 8, paBHO a Cpe,n;Hee qHCeJI. HaTyparrbHbIX apHcpMeTHqeCKOe apHcpMeTHqeCKOe cpe,n;Hee 4 6aJlJl HaH60JIbillHX ceMH 16. paBHO a) MomeT 6) MomeT B) HaH,n;HTe HaH60JIbillee JIH JIH H3 3THX apHcpMeTHqeCKOe cpe,n;Hee HaHMeHbillee qHCeJI ,n;BeHa,n;aTH Bcex 3HaqeHHe 18? paBHHTbCH qHCeJI ,n;BeHa,n;aTH II? paBHHTbCH apHcpMeTHqeCKOrO cpe,n;Hero Bcex qHCeJI. ,n;BeHa,n;aTH PemeHHe. HaH60JIbillee ECJIH a) He 18 60JIbille ille HJIH 6) 16 15 + + cpe,n;Hee B C> CTBO B) C 36, == 76. 3TO A + B YCJIOBHIO 3HaqHT, B + C B + C 56, == MeHb- qHCeJI H ce,n;bMoro C. paBHa 11. paBHO A + B + C 112, illeCToro CYMMa qHCeJI ,n;eCHTH == A, paBHa Tor,n;a IIpe,n;lIo- IIOJIyqaeM: 132, == BbIIIOJIHHTbCH ,n;OJImHO IIOCKOJIbKY HymHo (A HalIHcaHHbIe < +2 0f3 +5 ag, HepaBeH- < lltl, 5a6 @ 2021. pacnpOCTpaHeHHe 000 H 2 ar +5 +5ar < < +35 «H3,ZJ;aTeJIbCTBO HCnOJIb30BaHHe ag, a6 +6 2ag +2ag «HaU;HOHaJIbHOe B l +3 llt2, < + C) - KOMMepqeCKHX a < ag, < 2 < a a 3 ar+ llt2, ar l l2 < +1 06pa30BaHHe». 6e3 , == 168 , ..., B. - a 2 l2 < +4 llto, ag, IIpHqeM , @ 2021. ar+ llto' 4 < l tl' oTKy,n;a 5B+35 +2llt2; nHCbMeHHoro a . < +2lltl +2llto u;eJIHX < ... B B. paBHbI < HaHMeHb- qHCeJI: B) +(B ,n;OCKe, 6y,n;eT apHcpMeTHqeCKOe 3HaqeHHe Ha ar+ Cpe,n;Hee Bcex + HaH60JIbillee HaHTH qHCJIa, 112. == CYMMa == a KonHpoBaHHe, qHCeJI apHcpMeTHqeCKOe cpe,n;Hee HaH60JIbillHX IIHTH HeB03MomHO, A + B + C Tor,n;a 105, HX HaH60JIbillHX ceMH CYMMa a qHCeJI Bcex HaHMeHbilleH 6y,n;eT Kor,n;a IIYCTb TO == 2,5B. IIo illHM, 12 + CYMMa == oTKy,n;a 13 + B, a paBHa apHcpMeTHqeCKOe A + B 56, qHCeJI == 18, paBHO 14 HaHMeHbillHX IIHTH CYMMa qTO JIOmHM, qHCJIO + 15. paBHO IIYCTb BeJIHqHHe IIO 17 + qOY ,[(rro pa3pemeHHH < 2C. «MIJ;HMO» npaBoo6JIa,ZJ;aTeJIeH He ,ZJ;onYCKaeTCH II PEWEHlII KPlIITEPlIIlII 7B 3HaqHT, IIoKameM, HbI B qTO 27. = 2B+2C-35 < A+B+C 168-B 12 B 8, 10, o6pa30M, TaKHM =11 ' 245 B =189; qHCJIO 6, 3, 2, qHCJIa H 3A)J,AHlIIVl13-19 OU.EHlIIBAHlII MomeT . 27. paBHHTbCH 14, 13, 27 < 15, 16, HaHMeHbrnee HaIIpHMep, 19, TO 18, 17, 3HaqeHHe eCJIH Ha apHcpMeTHqeCKOrO cpe,n;Hero paBHO 75. a) 6) HeT; 11,75. B) HeT; CO,ll;ep3taHHe BaJIJILI KpHTepHJI (CM. KpHTepHH 1 6aJIJI) 4 BepHo nOJIy"tleHbI Bce BepHo nOJIy"tleHbI TPH H3 nepe"tlHCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 3 BepHo nOJIy"tleHbI ,n;Ba H3 nepe"tlHCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 2 BepHo nOJIyqeH - - nepe"tlHCJIeHHbIe H3 O,n;HH CJIe,n;YIOW;HX perneHHe nYHKTa a; 06ocHoBaHHoe perneHHe nYHKTa 6; - HCKOMaH - npHMep B o:u;eHKa B He pe3YJIbTaTbI 1 8; nYHKTe .. 06eCne"tlHBalOW;HH 8, nYHKTe Ha pe3YJIbTaTOB: 06ocHoBaHHoe PerneHHe HH COOTBeTcTByeT npe,n;bI,n;yw;eH TO"tlHOCTb H3 O,n;HOMY KpHTepHeB, o:u;eHKH 0 BbIrne nepe"tlHCJIeHHbIX M aKCUMaJlb1tblU a) PemHTe 6) Haii,n;HTe 4. ypaBHeHHe Bce cOSX 16 3Toro ROpHH 9. - COSX 4 2 + ypaBHeHHH, O. = rrpHHa,n;JIemaru;He [- OTpe3RY PemeHHe. IIycTb a) t 1 = t HJIH - = cosx 4 Tor,n;a , 4 6aJlJl 31 BapuaHT t BbIIIOJIHeHbI 12 OTBeT: OU HaIIHca- ,n;OCKe 3a,n;aqH YCJIOBHH 3aIIHrneTCH ypaBHeHHe B 4 t BH,n;e 2 21t; 9 t + 2 - ; - = 0, ] . oTKy,n;a 2. = 4 1 IIpH t IIpH t=2 = 4 C 6) 4 rrOJIyqaeM: cOSX 4 cosx qHCJIOBOH rrpHHa,n;JIemaru;He = -1, x oTRy,n;a -1t = 1 =2; cosx= 21t; ; - ] + k EZ 21tk, . 1t 2' + x= oTRy,n;a oT6epeM oKpymHocTH [- OTpe3RY cosx ; 4 rrOJIyqaeM: IIOMOID;blO 1 = "3+ 21tn nEZ. , KOpHH, . 51t IIoJIyqHM qHCJIa: 3aMellaHue. HbIX OT6op C H HepaBeHCTB OTBeT: a) @ 2021. KonHpoBaHHe,' KopHeH CIIoco6oM:' ,n;pyrHM pacnpOCTpaHeHHe T. 000 6bITb MomeT 060CHOBaH rpacpHKa, JII06bIM H JIHHeH- perneHHeM 1t II. 2 k 51t 1t EZ; + -+21tn, n 3 «M3.n;aTeJIbCTBO HCnOJIb30BaHHe -21t -1t · IIOMOID;blO -1t+21tk, H -1t -3; «HaU;HOHaJIbHOe B KOMMepqeCKHX EZ; 06pa30BaHHe». u;eJIHX 6e3 6) --. ' -1t. 3 @ 2021. nHCbMeHHoro qOY ,IUIO pa3pemeHHH «MIJ;HMO» npaBoo6JIa,z:J;aTeJIeH He .n;onYCKaeTCH Er3. 246 CO,ll;ep3taHHe 060CHOBaHHO rrOJIyqeHbI BepHbIe 060cHoBaHHo rrOJIyqeH BepHbIH TlIInOBblE BAPlIIAHTbl 3K3AMEHALJ.lIIOHHbIE BaJIJILI KpHTepHJI OTBeTbI B OTBeT B 060HX 2 rrYHKTax a, rrYHKTe HJIH HMeeTCH H PemeHHe omH6KH, marOB HO pemeHHH - rrYHKTOB HH COOTBeTcTByeT H3 OHOMY 0 BbIme rrpHBeeHHbIx KpHTepHeB, Ha SA pe6pe OTMeqeHa )UD H TOqKa qeTbIpexyrOJIbHOH IIpaBHJIbHOH M, IIpHqeM SABCD IIHpaMH,lJ;bI MA SM: 5 = 1. : P TOqKH - ABCD OCHOBaHHeM C Q H 2 6aJlJl M aKCUMaJlbHblU I14l 1 3TOM rrpH 060HX 6 rrYHKTa He Bcex rrOCJIeOBaTeJIbHOCTb BepHaH a rrYHKTa BbIqHCJIHTeJIbHOH H3-3a OTBeTbI HeBepHbIe rrOJIyqeHbI BC pe6ep Cepe,lJ;HHbI COOTBeTCTBeHHO. a) ,D:oKamHTe, TpalIeHeH. 6) HaH,lJ;HTe qTO CeqeHHe o6eMoB OTHorneHHe pa36HBaeT MPQ IIJIOCKOCTblO IIHpaMH,lJ;bI Ha MHororpaHHHKOB, paBH06e,lJ;peHHOH HBJIHeTCH MPQ IIJIOCKOCTb KOTopbIe IIHpaMH,lJ;Y. PemeHHe. SN: NB IIO N IIycTb a) 5 = LSAB 3HaqHT, MN *AB. MN OTpe3KH CeqeHHe II03TOMY MN TpaneHH PaCCTOHHHe 1 : o6eM II lIaparrJIeJIbHbI, AB. IIpHMOH H He paBHbI, M PQ 5: M BN P 3TO - TOqKH IIOCKOJIbKY MQ=NP, II03TOMY A SABCD H3 C 6, IIJIOMH o6eMoBIIHpaMH,lJ; MABPQ IIHpaMH,lJ;bI BNP. OCHOBaHHeM BNP a B V. paBeH qeTbIpexyrOJIbHOH IIJIOCKOCTH ,lJ;O K OTHOCHTCH MBNP H OCHOBaHHeM OT paCCTOHHHIO BNP TpeyrOJIbHHKOB C H ASBC SBC paBHO 5: 72, A TOqKH OTHOCHTCH KaK TO eCTb 5V paBeH - 144 ABPQ IIpHMoyrOJIbHHKa OT paBHbI, MBNP M OTHorneHHe IIHpaMH,lJ;bI PaccToHHHe MN aparrJIeJIbHbI HpaMH,lJ;bI IIHpaMH,lJ;bI KaK 3HaqHT, IIJIOMb H S HHMH. Mem,lJ;y IIJIOCKOCTblO COCTOHT TOqKH OT IIJIOCKOCTH 12. II H NBP MAQ H LMAQ=LNBP, MNPQ paBH06e,lJ;peHHaH. o6eM IIycTb 6) AMQBNP IIHTHrpaHHHK H TpeyrOJIbHOH ABPQ ,lJ;O yrJIY qTO PQ. TpalIeHH 3TOH H lIaparrJIeJIbHa PQ H SMN H AB IIpHMbIe TaKme IIHpaMH,lJ;bI TpeyrOJIbHHKH MA=NB, QA=PB, 3HaqHT, a SB, lIo,lJ;o6HbI pe6pe Ha SAB CTopOHaM LSMN, PQ = IIpHMaH 3HaqHT, a TOqKa TpeyrOJIbHHKH IIpOlIopHOHarrbHbIM ,lJ;BYM H 1. : TaKaH - M ,lJ;O 1: 6, COCTaBJIHeT IIOJIOBHHY ABCD IIJIOCKOCTH IIJIOMH OTHOCHTCH K KBMpaTa OT paCCTOHHHIO ABCD. TOqKH V ,lJ;O 3TOH IIJIOCKOCTH TmmM KaK II03TOMY 06'beM o6pa30M, o6eM MABPQ IIHpaMH,lJ;bI 17V AMQBNP paBeH TO , paBeH eCTb . 12 06'beMOB OTHomeHHe 144 AMQBNP MHororpaHHHKOB OTBeT: 6) @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 17: H CDSNPQM paBHO 17: 127. 127. 000 H «HaU;HOHaJIbHOe «H3,lJ;aTeJIbCTBO HCIIOJIb30BaHHe B KOMMepqeCKHX 06pa30BaHHe). u;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY ,[(rro pa3pemeHHH «MIJ;HMO) npaBoo6JIa,lJ;aTeJIeH He ,lJ;OnYCKaeTCH S PEWEHlII II KPlIITEPlIIlII 3AAAHlIIVl13-19 OU.EHlIIBAHlII 247 COep3taHHe MMeeTCH ,n;OKa3aTeJIbCTBO BepHOe BepHbIH "tIeH MMeeTcH OTBeT B YTBepm,n;eHHH 060CHOBaHHO H a, rrYHKTa rrOJIY- 2 6 rrYHKTe ,n;OKa3aTeJIbCTBO BepHoe BaJIJILI KpHTepHJI YTBepm,n;eHHH a, rrYHKTa MJIM 060CHOBaHHO rrYHKTa BepHbIH rrOJIyqeH a, PerneHHe 3TOM rrpH He OTBeT a rrYHKT He HH COOTBeTcTByeT B 1 6 rrYHKTe HCrrOJIb30BaHHeM C YTBepmeHHSI BbIrrOJIHeH H3 O,n;HOMY 0 BbIrne rrpHBe,n;eHHbIX KpHTepHeB, M aKCUMaJtbltblU PemHTe I15l log HepaBeHCTBO 4 a x 8 > + log3 ( ;7 ) log3 2 oaJtJt · 2 log3 x -log3 X X a PemeHHe. IIycTb t = log a x, Tora HepaBeHCTBO 4 t 8 > t t 3 - 2 t t -4t+4 2 t( OTKya IIpH IIpH IIpH t < 0; t t < 0 t = 2 IIOJIy"tlHM: t > 3 IIOJIHM: PemeHHe = t (t-2)2 t( t logax logax logax> HCXOHoro 9; 1); t( t - 3) t( t 0 . , 3) - 0. > , 3) 3) - 3) - t > 3. 2; IIOJIHM: (0; 8 > t( < = 0, 2, 3, oTKya 0 < x oTKya x = 9. oTKya x > 27. < 1. HepaBeHCTBa: O<x<l; OTBeT: 4t-12 - 3t - >0' - 2 ' - t t . + - BH: IIpHMeT (27; (0). + COep3taHHe 060CHOBaHHO rrOJIyqeH BepHbIH 060cHoBaHHo rrOJIyqeH OTBeT, x>27. x=9; BaJIJILI KpHTepHJI 2 OTBeT OTJIHqalOID;HHCH OT HCKJIlOqeHHeM BepHoro 9, TOqKH MJIM rrOJIyqeH BepHaH PerneHHe HeBepHbIH OTBeT Bcex rrOCJIe,n;OBaTeJIbHOCTb He BbIqHCJIHTeJIbHOH H3-3a COOTBeTcTByeT HH rnarOB O,n;HOMY 1 ornH6KH, HO rrpH HMeeTCSI 3TOM perneHHH H3 KpHTepHeB, M aKCUMaJtbHblU @ 2021. KonHpoBaHHe, pacnpOCTpaHeHHe 000 H «M3.n;aTeJIbCTBO HCnOJIb30BaHHe «Han;HOHaJIbHOe B KOMMepqeCKHX 06pa30BaHHe». n;eJIHX 6e3 @ 2021. nHCbMeHHoro qOY 0 BhIme rrepeqHCJIeHHhIX )];nO pa3pemeHHH <cMnHMO. npaBo06JIa.n;aTeJIeii oaJtJt 2 Be AonYCKaeTCH Er3. 248 ABCD B TpaneHH lliJ B3HJIH a) ,D:oKamHTe, 6) HaM,n;HTe "liTO BM "liTO yrJIbI BC B ,n;Ba H yrOJI BCD CDM BC. OCHOBaHHH MeHbIIIe pa3a BAM BAPlIIAHTbl 3K3AMEHALJ.lIIOHHbIE BHYTPH TpaneHH npHMbIe. CM. == ABC, yrOJI npHMoM ,n;o AD OCHOBaHHe M TaK, TO"llKY TlIInOBblE eCJIH paBHO 64 paBeH 0 a , OT paCCTOHHHe M TO"llKH AD. CTopOHe PemeHHe. lIycTb a) Tor,n;a no,n;o6HH TpeyrOJIbHHKH BL CL 2, H cepe,n;HHHbIe eHTp BLC nepneH,n;HKY OKOJIO onHcaHHoM BM == CM H lIycTb - CHM LBMC == OTBeT: 3Toro TpeyrOJIbHHKa 3TOM pa,n;HYCbI BC, BC. 90°. lIo 3Ha"llHT, BHM TpeyrOJIbHHKH n03TOMY BnHcaHHoro LABC = LLBC yrJIa 180° = L LBLC - - LBCL = 71°. 71 0. 6) COep3taHHe MMeeTC.H ,n;OKa3aTeJIbCTBO BepHoe BepHbIH qeH cepe,n;HHHbIH - npHMoyrOJIbHbIe, oTRy,n;a - oKpymHocTH, MH CBOMCTBY 45°, - - oKpymHoCTH. Tor,n;a paBHo6e,n;peHHbIe LBC LBLC= JIHpbI CTopOHe c - cepe,n;HHa K nepneH,n;HKYJIHp H C K03cpcpHHeHToM no,n;o6HbI A H D 2AD. 3Ha"llHT, cepe,n;HHbI H AM DM TaKHM o6pa30M, B TpeyrOJIbHHKe a M BLC, == KaK H B L. TO"llKe B nepeceKalOTCH ALD H BC nOCKOJIbKY CD H COOTBeTCTBeHHO. n03TOMY 6) AB npHMbIe OTBeT 06ocHoBaHHo B YTBepm,n;eHH.H rrYHKTa a, H 060CHOBaHHO rrYHKTa a, H rrpH rrOJIY- 3 6 rrYHKTe BepHbIH rrOJIyqeH BaJIJILI KpHTepHJI OTBeT B 6, rrYHKTe MJIM 2 HMeeTC.H ,n;OKa3aTeJIbCTBO BepHoe perneHHH rrYHKTa MMeeTC.H BepHoe 6 rrOJIyqeH YTBepm,n;eHH.H HeBepHbIH ,n;OKa3aTeJIbCTBO YTBepm,n;eHH.H ornH6KH apHcpMeTHqeCKOM H3-3a OTBeT 06ocHoBaHHoM a, rrYHKTa MJIM 06ocHoBaHHoM rrpH 6 rrYHKTa perneHHH HeBepHbIM rrOJIyqeH OTBeT H3-3a 1 ornH6KH, apHcpMeTH"LIeCKOH MJIM 06ocHoBaHHo rrYHKTa PerneHHe BepHbIH rrOJIyqeH a, aTOM rrpH He rrYHKT COOTBeTcTByeT OTBeT a HH He B 6 rrYHKTe C HCrrOJIb30BaHHeM YTBepm,n;eHH.H BbIrrOJIHeH O,n;HOMY H3 KpHTepHeB, M aKCUMaJlbHblU @ 2021. KOIIHpoBaHHe, paCIIpOCTpaHeHHe 000 H «HaU;HOHaJIbHOe «H3.n;aTeJIbCTBO HCIIOJIb30BaHHe B KOMMepqeCKHX o6pa30BaHHe». u;eJIHX 6e3 @ 2021. IIHCbMeHHoro qOY pa3perneHHH 0 BbIrne rrepeqHCJIeHHbIX )];rro «MII;HMO» IIpaBOo6JIa,z:J;aTeJIeH 3 6aJlJl He .n;OIIYCKaeTCH PEWEHV1 [E] 3AAAHV1 KPV1TEPV1V1'OU.EHV1BAHV1 V1 B HIOJIe 2022 Y CJIOBHH ero 249 B3HTb I[JIaHHpyeTcH ro,n;a 6aHKe B Kpe,n;HT Ha 000 300 CYMMY py6JIeii. TaKOBbI: B03BpaTa Kam,n;bIii 13-19 HHBapb % r Ha YBeJIHqHBaeTCH ,n;OJIr IIO cpaBHeHHIO C BbIIIJIaTHTb O,n;HHM KOHOM IIpe,n;bI,n;yero ro,n;a; cpeBparrH C HIOHb IIO Heo6xo,n;HMO ro,n;a Kam,n;oro IIJIaTemOM qaCTb ,n;OJIra. Haii,n;HTe B IIepBbIii eCJIH r, 6y,n;eT ro,n; qTO H3BeCTHO, 6y,n;eT py6JIeii, Kpe,n;HT 160 BbIIIJIaqeHO 000 IIorarneH IIOJIHOCTblO a 3a BTOpoii BO ro,n; IIpHqeM py6JIeii. ro,n;a, ,n;Ba 240 - 000 PemeHHe. r Ha 1 IIYCTb k HIOJIb ,ZJ;OJImeH = + IIo -. YMeHbmaTbCH 2 15k : k= -8k-12=O; 060CHOBaHHO - BepHo 160000k - 240000 - = H rroRyqeH OTBeT HO OTBeT, Mo,n;eRH, PerneHHe He 3 Mo,n;eRb, K CBe,n;eHO perneHHe BbIqHCRHTeRbHoM HCCRe,n;OBaHHIO 3TOM 060cHoBaHo Mo,n;eRb, HH COOTBeTcTByeT H3 O,n;HOMY He K CBe,n;eHO perneHHe 6bITb MomeT perneHHe 2 ornH6KH; He,n;OCTaTOqHO perneHHe MaTeMaTHqeCKa.H rrpH BaJlJlLI KpHTepHH pe3YRbTaT: H3-3a rrOCTpoeHa HCCRe,n;OBaHHIO Haii.n;HTe Bee 3HaqeHHH a, rrpH KpHTepHeB, . J 5 -7 0 BbIrne rrpHBe,n;eHHbIX oln( x 9x 2 a - 2 KO ) = TOpbIX . J 5 -7 3 6aJlJl ypaBHeHHe x oln( 3x + a) KopeHb. O,n;HH pOBHO H3 Kam.n;OM 1 3aBeprneHO M aKCUMaJlbHblU HMeeT 0, OTBeT MaTeMaTHqeCKa.H rrOCTpoeHa Mo,n;eRH 3TOM BepHbIM rroRyqeH HeBepHbIM BepHbIM - 2 r=20. ; CO,IJ;ep3taHHe 3TOM COCTOHHHIO IIO 20. OTBeT: BepHo py6JIHX) (B o6pa30M: 300000k 160000, - 6aHKoM IIepe,n; ,n;OJIr CJIe,n;YIOHM 300000k 300000, oTKy.n;a YCJIOBHIO 100 PemeHHe. 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