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C, D, Al , Bl.
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a) MomeT JIH y MHillH 6bITb 30 MOHeT?
6) KaKoe HaH60JIbillee KOJIH"tIeCTBO MOHeT MomeT 6bITb y MHillH?
B) KaKaH HaH60JIbillaH CYMMa py6JIeH MomeT 6bITb y MHillH?
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B OTBeTe 3aIIHmHTe MeHbmHH H3 KopHeH.
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B) KaKaH HaH60JlbrnaH CYMMa py6J1eH MO.iKeT 6bITb y KOJlH?
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coom6emcm6YlOUf-ezo aaoanUJ£.
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BAPV1AHT 7 43
,l{ 1£1£ aanucu pemeuuu u om6em06 ua aaiJauuJ£ 13-19 ucnol£bayume BJIAHK
OTBETOB M 2. 3anumume cua"l£al£a uOMep 6b£nOI£UJ£eMOZO aaiJauuJ£ (13, 14
u m. iJ.), a aameM nOl£uoe 06ocuo6auuoe pemeuue u om6em. Om6emb£ aanUCb£-
6aume "l£emKO U paa60p"l£U60.
OU a) PemHTe ypaBHeHHe sin ( 2x +21t
) COS ( 4x +1t
) -
cos 2x =
Sir: ).
3 3cos
__
3
6) Haii,ZJ;HTe Bce KOpHH :3TOrO ypaBHeHH.H, rrpHHa,ZJ;JIemam;He OTpe3KY [ -21t;3
2
1t
J.
oB rrpaBHJIbHOH qeTbIpexyrOJIbHOH rrpH3Me ABCDA1B1C1D 1 CTopOHa OCHOBaHH.H AB
paBHa 2.J3, a 60KOBoe pe6po AAI paBHO 3. Ha pe6pax AIDI H DDI OTMeqeHbI
COOTBeTCTBeHHO TOqKH K H M TaK, qTO AIK == KDI ,
a DM: MDI
= 2 : 1.
a) ,n:oKamHTe, qTO rrpHMbIe MK H BK rreprreH,n;HKYJIHpHbI.
6) HaH,n;HTe yrOJI Mem,n;y rrJIOCKOCTHMH BMK H BCCI
.
� P6.5
x-11
>0 25�
emHTe HepaBeHCTBO25x+0,5 -6.5x +1
-, ·
oHa CTopOHax AC, AB H BC rrpHMoyrOJIbHOrO TpeyrOJIbHHKa ABC C rrpHMbIM yr JIOM C BHe
TpeyrOJIbHHKa ABC rrOCTpoeHbI paBHo6e,n;peHHbIe rrpHMoyrOJIbHbIe TpeyrOJIbHHKH AKC,
ALB H BMC C rrpHMbIMH yrJIaMH K, L H M COOTBeTCTBeHHO.
a) ,n:oKamHTe, qTO LC -
BbICOTa TpeyrOJIbHHKa KLM.
6) HaH,n;HTe rrJIOm;a,n;b TpeyrOJIbHHKa KLM, eCJIH LC == 4.
[ill A.rreKcaH,n;p XOqeT KyrrHTb rraKeT aKD;HH 6bIcTpopacTym;eH KOMrraHHH. B HaqaJIe ro,n;a
y AJIeKcaH,n;pa He 6bIJIO ,n;eHer Ha rroKyrrKY aKD;HH, a rraKeT CTOHJI 100 000 py6JIeH.B cepe,n;HHe Kam,n;oro MeCHD;a A.rreKcaH,n;p OTKJIa,n;bIBaeT Ha rroKyrrKY rraKeTa aKD;HH O,n;HY
H TY me CYMMY, a B KOHD;e MeCHD;a rraKeT ,n;opomaeT, HO He 60JIee qeM Ha 30 0/0. KaKYro
HaHMeHbmyro CYMMY HymHo OTKJIa,n;bIBaTb A.rreKcaH,n;py Kam,n;bIH MeCHD;, qTo6bI qepe3
HeKOTopoe BpeMH KyrrHTb meJIaeMbIH rraKeT aKD;HH?
o HaH,ZJ;HTe, rrpH KaKHX HeOTpHD;aTeJIbHbIX 3HaqeHH.HX a CPYHKD;H.H f (x) = 3ax4
- 8x3
+ 3x2
-7
Ha OTpe3Ke [-1; 1] HMeeT pOBHO O,n;HY TOqKY MHHHMYMa.
@ 2021. 000 «M3.n;aTeJIbCTBO «HaU;HOHa.rrbHoe 06pa30BaHHe». @ 2021. qOY ,n;rro «MIJ;HMO»KonHpoBaHHe, pacnpOCTpaHeHHe H HCnOJIb30BaHHe B KOMMepqeCKHX u;eJI.HX 6e3 nHCbMeHHoro pa3pemeHH.H npaBoo6JIa.n;aTeJIei He .n;onYCKaeTC.H
-
44 Ers. TV1nOBbIE SK3AMEHAU.V10HHbIE BAPV1AHTbl
O!J ,n;JUI Kam,ll;OrO HaTypaJIbHOrO �nrcJIa n 0603HaYIHM YIepe3 n! rrpOH3Be,ll;eHHe rrepBblX n
HaTYPaJIbHbIX qHCeJI (I! = 1).
a) Cym;ecTByeT JIH TaKoe HaTYPaJIbHOe qHCJIO n, qTO �eCHTHqHaH 3aIIHCb qHCJIa n!
OKaHqHBaeTCH pOBHO 9 HYJIHMH?
6) Cym;ecTByeT JIH TaKoe HaTYPaJIbHOe qHCJIO n, qTO �eCHTHqHaH 3aIIHCb qHCJIa n!
OKaHqHBaeTCH pOBHO 23 HYJIHMH?
B) CKOJIbKO cym;ecTByeT HaTypaJIbHbIX qHCeJI n, MeHbillHX 100, �JIH KajK�OrO H3 KOTOpbIX
�eCHTHqHaH 3aIIHCb qHCJIa n! ·
(100-
n)! OKaHqHBaeTCH pOBHO 23 HYJIHMH?
CDnp06epbme, "l£m06b£ Ka:HCOb£U om6em 6b£l£ aanucan pJ£OOM c nOMepOM coom-
6emcm6YlO�ezo aaoanuJ£.
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BAPHAHT 8
Om6emOM I\, 3aiJaJ£UJ£M 1-12 J£6nJ£emCJ£ lfenoe "l£ucno unu 1\,0J£e"l£J£aJ£ iJecJ£mu"I£-
J£aJ£ iJP06b. 3anumume "l£ucno 6 none om6ema 6 mel\,cme pa60mb£, 3ameM nepe-
J£eCume e1,o 6 BJIAHK OTBETOB M 1 cnpa6a om J£OMepa coom6emcm6yro�e-
1,0 3aiJaJ£UJ£, J£a"l£UJ£aJ£ c nep60u I\,nemo"l£l\,u. Ka3ICiJyro lfucpPY, 3J£al\, « MUJ£YC)
U 3anJ£myro numume 6 omiJenbJ£OU I\,nemo"l£l\,e 6 coom6emcm6UU c npu6eiJeJ£J£b£-MU 6 6naJ£l\,e 06pa3lfaMU. EiJUJ£Ulfb£ U3MepeJ£UU nucamb J£e J£Y3ICJ£O.
qacTh 1
=0B npOIIIJIOM ro,n;y BO BpeM.H KOHcpepeHD;HH B cpe,n;HeM 3a ,n;eHb pacxo,n;OBaJIOCb
80 rraKeTHKOB qaH. B 3TOM ro,n;y opraHH3aTopbI peillHJIH KyrrHTb qaii C 3arraCOM B 5 0/0
rro cpaBHeHHIO K pacxo,n;y rrpOillJIOrO ro,n;a. KOHcpepeHn;HH ,n;JIHTCH 4 ,n;HH. B rraqKe qaH
50 rraKeTHKOB. KaKoe HaHMeHbillee KOJIHqeCTBO TaKHX rraqeK qaH Ha,n;o KyrrHTb?
OTBeT:
IT] MOru;HOCTb OTOrrHTeJIH B aBTOM06HJIe perYJIHpyeTcH ,n;orrOJIHHTeJIbHbIM corrpOTHBJIeHHeM.
TIpH 3TOM MeHHeTCH CHJIa TOKa B 3JIeKTpHqeCKOii n;errH 3JIeKTpo,n;BHraTeJIH: qeM
MeHbille corrpOTHBJIeHHe, TeM 60JIbille CHJIa TOKa H TeM 6bIcTpee Bparu;aeTCH BeHTHJIHTOp
OTOrrHTeJIH. Ha pHCYHKe rrOKa3aHa 3aBHCHMOCTb CHJIbI TOKa OT BeJIHqHHbI
corrpOTHBJIeHHH. Ha ropH30HTaJIbHOii OCH OTMeqeHO corrpOTHBJIeHHe B OMax,
Ha BepTHKaJIbHOii OCH--
CHJIa TOKa B aMrrepax. Orrpe,n;eJIHTe, Ha CKOJIbKO aMrrep
YMeHbillHJIaCb CHJIa TOKa B n;errH rrpH YBeJIHqeHHH corrpOTHBJIeHHH C 1 OMa ,n;o 2,5 OMa.
y
12
10
8
6
4
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KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIa.,n;aTeJIeH He ,ZJ;OIIYCKaeTCH
46 Ers. TVinOBblE SK3AMEHALJ.VlOHHbIE BAPVlAHTbl
OJHa KJIeTqaTOH 6YMare C pa3MepOM KJIeTKH 1 x 1 H306pameHa TpaIIeD;HH.
HaH,lJ;HTe ee IIJIOID;Mb.
OTBeT:
� HaH�HTe BepOHTHOCTb Toro, �TO c�y�aHHo BM6paHHoe HaTypanbHoe �HC�O H3 �Hce�
OT 1 ,lJ;O 25 (BKJIlOqHTeJIbHo) 6Y,lJ;eT ,lJ;eJIHTbCH Ha 3.
OTBeT:
(1
J3X+5
OJ HaH�HTe KopeHb ypaBHeHHH5
= 0,04.
OTBeT:
OTBeT:
16' PMHYC oKpymHocTH, oIIHcaHHoH OKOJIO TpeyrOJIbHHKa ABC
�paBeH 2.J3. HaH,lJ;HTe AB, eCJIH yrOJI ACB paBeH 120
0
.
[TIHa pHCYHKe H306pameH rpacpHK y
= f' (x)-
IIpOH3BO�HOH CPYHKD;HH f(x), oIIpe�e�eHHoHHa HHTepBaJIe (-8; 3). B KaKoH TOqKe OTpe3Ka [-5; 0] <PYHKD;HH f(x) IIpHHHMaeT
HaH60JIbmee 3HaqeHHe?
y��I I I
I I I I
Y= f'(x)
r \/ \ I ,
\,
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-
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1\ /� � �\ / \ I
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3APV1AHT 8 47
�
==u
LJD
O!J
B cocY,lJ;, HMelOm;HH CPOPMY rrpaBHJIbHOH TpeyrOJIbHOH rrpH3Mbl, HaJIHJIH
1100 CM3
BO,lJ;bl H rrOJIHOCTblO B Hee rrorpY3HJIH ,lJ;eTaJIb. TIp� aTOM
ypOBeHb jKH,lJ;KOCTH B cocY,lJ;e rrO,lJ;HHJICH C OTMeTKH 25 CM ,lJ;O OTMeTKH
29 CM. qeMY paBeH 06'beM ,lJ;eTaJIH? OTBeT Bblpa3HTe B CM3
.
OTBeT:
He aa6yobme nepeneemu Bee omBemb£ B 6nanK omBemoB M 1 B eoomBem-
emBUU e unempYK�ueu no Bb£nonnenulO pa6omb£.
qacTh 2
1t . 71t
(21t
)HaH,lJ;HTe 3HaqeHHe BblpajKeHHH 2.J6cos-Sln-tg --
.
4 6 3
OTBeT:
He3aBHCHMoe areHTCTBO HaMepeHO BBeCTH peHTHHr R HOBOCTHblX H3,lJ;aHHH Ha OCHOBe
rrOKa3aTeJIeH HHcpopMaTHBHocTH In, orrepaTHBHOCTH Op H 06'beKTHBHoCTH Tr
rry6JIHKan;HH, a TaKjKe KaqeCTBa Q caHTa. KajK,lJ;blH OT,lJ;eJIbHbIH rrOKa3aTeJIb-
n;eJIoe
qHCJIO OT 0 ,lJ;O 4.
CocTaBHTeJIH peHTHHra CqHTaIOT, qTO HH<popMaTHBHocTb rry6JIHKan;HH n;eHHTCH BABoe,
a 06'beKTHBHOCTb -
BTpoe ,lJ;opOjKe, qeM orrepaTHBHOCTb H KaqeCTBO caHTa, TO eCTb
R=2In+Op+3Tr+ Q
.
A
HaH,lJ;HTe, KaKHM ,lJ;OJIjKHO 6blTb qHCJIO A, qTo6bl H3,lJ;aHHe, Y KOToporo Bce rrOKa3aTeJIH
MaKCHMaJIbHbl, rrOJIyqHJIO peHTHHr 1.
OTBeT:
,n:opora MejK,lJ;y rrYHKTaMH A H B COCTOHT H3 rrO,lJ;'beMa H crrYCKa, a ee ,lJ;JIHHa paBHa
36 KM. TIYTb H3 A B B 3aHHJI y TypHCTa 10 qaCOB, H3 KOTOpblX 2 qaca ymJIO Ha crrYCK.
HaH,lJ;HTe CKOpOCTb TYPHCTa Ha crrYCKe, eCJIH OHa 60JIbme CKOpOCTH Ha rrO,lJ;'beMe
Ha 3 KM/q. OTBeT ,lJ;aHTe B KM/q.
OTBeT:
� Haii,n;HTe HaH60JIbmee 3HaYIeHHe CPYHKD;HH y=71n(x+5)-7x+l0 Ha OTpe3Ke [-4,5; 0].
OTBeT:
CDHe aa6yobme nepeneemu Bee omBemb£ B 6nanK omBemoB M 1 B eoomBem-
emBUU e unempYK�ueu no Bb£nonnenulO pa6omb£.
npoBepbme, l£moob£ Ka3ICob£u omBem Ob£l£ 3anuean B empoKe e nOMepoM
eoomBemcmBYlO�ezo 3aOanUJL.
@ 2021. 000 «M3.n;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BHHHe». @ 2021. qOY ,n;rro «MU;HMO»KorrHpoBaHHe, pacrrpOCTpaHeHHe H HCrrOJIb30BaHHe B KOMMepqeCKHX u;eJI.HX 6e3 -'-CbMeHHoro pa3perneHH.H rrpaBoo6JIa.n;aTeJIeH He .n;orrYCKaeTC.H
48 Er3. TV1nOBbIE 3K3AMEHALJ.V10HHbIE BAPV1AHTbI
,l{ 1£1£ aanucu pemeuuu u om6em06 ua aaiJauu1£ 13-19 �cnol£bayume BJIAHK
OTBETOB M 2. 3anumume cua"l£al£a uOMep 6b£nOI£U1£eMOZO aaiJauu1£ (13, 14
u m. iJ.), a aameM nOl£uoe 06ocuo6auuoe pemeuue u om6em. Om6emb£ aanucbl,-
6aume "l£emKO u paa60p"l£U60.
� a) PemHTe ypaBHeHHe cos2x-sin2x=cosx+sinx+1.
6) HaH,lI;HTe Bce KOpHH :3TOrO ypaBHeHHSI, rrpHH8.,ZI;JIemaru;He OTpe3KY [-5
2
1t
; -1t ] .
ODB rrpaBHJIbHOH qeTbIpexyrOJIbHOH rrpH3Me ABCDA1B1C1D 1 CTopOHa OCHOBaHHSI AB
paBHa 3, a 60KOBoe pe6po AAI paBHO .J3. Ha pe6pax CID IH DDI OTMeqeHbl
COOTBeTCTBeHHO TOqKH K H M TaK, qTO DIK = KCI ,
a DM: MDI
= 1 : 3.
a) ,n:oKamHTe, qTO npHMbIe MK H BK nepneH�HKYJIHpHbI.
6) HaH�HTe yrOJI Mem�y nJIOCKOCTHMH BMK H ABBI.
[!!] PemHTe HepaBeHCTBO Ig4(x2 _4)2 -lg2(x
2 _4)4 > 192.
ODHa cTopoHax AC, AB H BC npHMoyrOJIbHOrO TpeyrOJIbHHKa ABC C rrpHMbIM yr JIOM C BHe
TpeyrOJIbHHKa ABC nocTpoeHbI paBHo6e�peHHbIe npHMoyrOJIbHbIe TpeyrOJIbHHKH AKC,
ALB H BMC C npHMbIMH yrJIaMH K, L H M COOTBeTCTBeHHO.
a) ,n:OKamHTe, qTO LC -
BbICOTa TpeyrOJIbHHKa KLM.
6) HaH�JfTe nJIOrn;Mb TpeyrOJIbHHKa KLM, eCJIH LC = 6 .
[E] CepreH XOqeT KynHTb naKeT aKD;HH 6bIcTpopacTyrn;eH KOMnaHHH. B HaqaJIe ro�a
y CepreH He 6bIJIO �eHer Ha nOKynKY aKD;HH, a naKeT CTOHJI 160 000 py6JIeH. B cepe�HHe
Kam�oro MeCHD;a CepreH OTKJIa�bIBaeT Ha nOKynKY naKeTa aKD;HH O�HY H TY me CYMMY,
a B KOHD;e MeCHD;a rraKeT �opomaeT, HO He 60JIee qeM Ha 25 0/0. KaKYIO HaHMeHbIIIYIO
CYMMY HymHo OTKJIa�bIBaTb CeprelO Kam�bIH MeCHD;, qTo6bI qepe3 HeKOTopoe BpeMH
KynHTb meJIaeMbIH naKeT aKD;HH?
� HaH,lI;HTe, rrpH KaKHX HerrOJIOmHTeJIbHbIX 3HaqeHHSlX a IPYHKD;HSI rex) = ax4
+4x3 -3X2-5
Ha OTpe3Ke [-2; 2] HMeeT �Be TOqKH MaKCHMYMa.
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BAPV1AHT 8 49
D!J )J;JI.H Kam,n;oro HaTypaJIbHOrO YIHCJIa n 0603HaYIHM YIepe3 n! rrpoH3Be,n;eHHe rrepBbIx n
HaTypaJIbHbIX qHCeJI (11 == 1).
a) Cym;ecTByeT JIH TaKoe HaTypaJIbHOe qHCJIO n, qTO ,n;eCHTHqHaH 3aIIHCb qHCJIa n!
OKaHqHBaeTCH pOBHO 10 HYJIHMH?
6) Cym;ecTByeT JIH TaKoe HaTypaJIbHOe qHCJIO n, qTO ,n;eCHTHqHaH 3aIIHCb qHCJIa n!
OKaHqHBaeTCH pOBHO 17 HYJIHMH?
B) CKOJIbKO cym;ecTByeT HaTypaJIbHbIX qHCeJI n, MeHbillHX 75, ,n;JIH Kam,n;oro H3 KOTOpbIX
,n;eCHTHqHaH 3aIIHCb qHCJIa n!. (75-
n)! OKaHqHBaeTCH pOBHO 17 HYJIHMH?
,!f\\�
n p06epbme, 1.£m06b£ Ka3ICob£u om6em 6b£1£ aanucan pROOM c nOMepoM coom-
6emcm6ywUfezo aaoanuR.
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BAPHAHT 9
Om8emOM K 3aiJanUJLM 1-12 JL8l£JLemCJL lfel£oe "1£UCl£O Ul£U KOne"1£naJL iJecJLmu"1£-
naJL iJpo6b. 3anumume "1£UCl£O 8 nOl£e om8ema 8 meKcme pa60mb£, 3ameM nepe-
necume ezo 8 BJIAHK OTBETOB M 1 cnpa8a om nOMepa coom8emcm8YlO�e-
zo 3aiJanUJL, na"1£unaJL C nep80u Kl£emO"1£KU. KamiJylO lfutjJpy, 3naK «(MunyC)
U 3anJLmylO numume 8 omiJel£bnou Kl£emO"1£Ke 8 coom8emcm8UU C npu8eiJennb£-MU 8 6l£anKe 06pa3lfaMU. EiJunulfb£ U3Mepenuu nucamb ne nymno.
qacTL 1
co �JIH IIOKpaCKH 1 KB. M IIOTOJIKa B cpe.n;HeM Tpe6yeTcH 120 r KpaCKH. KpacKa IIpo.n;aeTcHB 6aHKax IIO 1,5 Kr. Erop pemHJI KynHTb KpaCKY C 3aIIaCOM B 10
%K cpe.n;HeMY pacxo.n;y.
KaKoe HaHMeHbmee KOJIH"IleCTBO 6aHoK KpaCKH HymHo KYIIHTb .n;JIH IIOKpaCKH IIOTOJIKa
IIJIOm;a.n;blO 42 KB. M?
OTBeT:
OJ Kor.n;a CaMOJIeT HaxO.n;HTCH B ropH30HTaJIbHOM IIOJIeTe, IIo.n;'beMHaH CHJIa, .n;eHcTBYIOm;aHHa KpbIJIbH, 3aBHCHT OT CKOpOCTH. Ha pHCYHKe H306pameHa 3Ta 3aBHCHMOCTb .n;JIH
HeKOToporo CaMOJIeTa. Ha ropH30HTaJIbHOH OCH OTMeqeHa CKOpOCTb B KHJIOMeTpax B qac,
Ha BepTHKaJIbHOH OCH--
IIo.n;'beMHaH CHJIa B TOHHax CHJIM. OIIpe.n;eJIHTe no rpa�HKY,
Ha CKOJIbKO KHJIOMeTpOB B qac YBeJIHqHJIaCb CKOpOCTb IIOJIeTa IIPH YBeJIHqeHHH IIo.n;'beMHOH
CHJIbI C 1 TOHHbI .n;o 4 TOHH.
y
5
4
3
2
1
o 100 200 300 400 500 x
OTBeT:
coHa KJIeTTIaTOM 6YMare c pa3MepOM KJIeTKH 1 xl H306pameH
paBHocTopOHHHH TpeyrOJIbHHK. HaH.n;HTe pa.n;HYC OIIHcaHHOH
OKOJIO Hero oKpymHoCTH.
OTBeT:
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BAPV1AHT 9 51
[TIB Kacpe Ha O�HOH IIOJIKe B CJIyqaHHOM IIopH�Ke CTOHT 50 qaHHbIX qameK: 30 3eJIeHbIX,
10 KpaCHbIX H 10 CHHHX. Ha �pyroH IIOJIKe B CJIyqaHHOM IIopH�Ke CTOHT 50 6JIIO�e�:
30 3eJIeHbIX, 10 KpaCHbIX H 10. CHHHX. HaH�HTe BepOHTHOCTb Toro, qTO CJIyqaHHO
BbI6paHHbIe qamKa H 6JIIO��e 6Y�YT O�HHaKOBoro �BeTa.
OTBeT:
CD HaM,lI;HTe KopeHb ypaBHeHH.H (2x -11)2 = ( 2x -1 )2.
OTBeT:
161 HaH�HTe IIJIOm;a�b IIpHMoyrOJIbHOrO TpeyrOJIbHHKa, eCJIH ero KaTeT H rHIIOTeHY3a paBHbI�
COOTBeTCTBeHHO 15 H 17.
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= f(x), orrpe,ll;eJIeHHOM Ha HHTepBarre (-9; 2).
OIIpe�eJIHTe KOJIH"lleCTBO �eJIbIX TOqeK, B KOTOpbIX IIpOH3BO�HaH CPYHK�HH OTpH�aTeJIbHa.
��I I I
I I I
Y= f (x)
I ",
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�HJIHH�pa paBeH 5. Y BToporo �HJIHH�pa
BbICOTa B 2,5 pa3a MeHbme, a pa�HYC
OCHOBaHHH B 3 pa3a 60JIbme, qeM y IIepBoro.
HaH�HTe 06'beM BToporo �HJIHH�pa.
...------.... ---------
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He aa6yiJbme nepeneemu 8ee om8emb£ 8 6l£anl\, om8em08 M 1 8 eoom8em-
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�o KOHeqHOrO 06'beMa V2 (B JI). Pa6oTa, COBeprnaeMaH BO�OH IIPH cmaTHH B03�yxa,
VI �mBbIqHCJIHeTCH IIO cpopMYJIe A = avTlog 2 -, r�e a = 8,5 IIOCTOHHHaH,
V2 MOJIb· K
T = 300 K -
TeMIIepaTypa B03�yxa. HaH�HTe, KaKOH 06'beM V2 6y�eT 3aHHMaTb B03�YX
B KOJIOKOJIe, eCJIH IIPH cmaTHH B03�yxa 6bIJIa COBeprneHa pa60Ta B 25 500 �m. OTBeT
�aHTe B JIHTpax.
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42-IIpo�eHTHoro paCTBopa O�Horo H Toro me Bem;eCTBa. CKOJIbKO KHJIOrpaMMOB
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paCTBOp Bem;ecTBa?
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cm8UU c uncmpYI\,",ueii no 8b£nOJ£nenUlO pa60mb£.
npo6epbme, I£moob£ l\,a3ICob£u om6em Ob£l£ aanucan 6 cmpol\,e c nOMepOM
coom6emcm6YlO"Ilfezo aaoanult.
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OTMeqeHa
TOqKa M TaK, qTO CM: MCI
= 1 : 2. IIJIOCKOCTb a IIaparrJIeJIbHa IIpHMOH HlElH IIpOXO�HT qepe3 TOqKH M H A.
a) �oKamHTe, qTO CeqeHHe IIpH3MbI ABCDEFGHAlBlClDlElFlGlHl IIJIOCKOCTblO a-
paBHo6e�peHHaH TpaIIe�HH.
6) HaH�HTe 06'beM IIHpaMH�bI, BepmHHoH KOTOpOH HBJIHeTCH TOqKa Fl , a OCHOBaHHeM
-
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PemHTe HepaBeHCTBO 9. 2 Iog 3(5-x) + 2l+log3x - 2 Iog 3(5x-x2 )
< 18.
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TpaIIe�HH ABCD, pa36HBaeT ee Ha �Be TpaIIe�HH, B Kam�ylO H3 KOTOpbIX MomHO
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6) 1I3BecTHo, qTO pa�HYC 3THX oKpymHocTeH paBeH 4, a MeHbmee OCHOBaHHe BC
HCXO�HOH TpaIIe�HH paBHO 14. HaH�HTe pa�HYC oKpymHocTH, KacalOm;eHcH 60KOBOH
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6) MomeT JIH O�HHM H3 9THX qHCeJI 6bITb qHCJIO 66?
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BpeMH B ceKYH�ax, rrpome�mee
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,
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r,ZJ;e a = 13,5)J;m
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COBepmeHa pa60Ta B 16 200 )J;m. OTBeT ,n;aHTe B aTMoccpepax.
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QeM BTopaH Tpy6a 3arrOJIHHeT pe3epByap 06'beMOM 396 JIHTpOB. IIepBaH Tpy6a rrporrYCKaeT
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8
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H B3aHMHO IIepIIeH�HKY JI5IpHbIe o6pa3YIOIIJ;He.
6) HaM�HTe paCCT05lHHe OT IIJIOCKOCTH CeqeHH5I �o n;eHTpa OCHOBaHH5I KOHyca.
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6) MomeT JIH O�HHM H3 3THX QHCeJI 6bITb QHCJIO 33?
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BAPHAHT 11
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BAPV1AHT 11 59
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06pa30M �eJIHT Ha TPH no�rpynnbI no 7 qeJIOBeK �JIH nOCa,zJ;KH B TPH MHKpoaBTo6yca.
KaKoBa BepOHTHOCTb Toro, qTO meHH H Cama CJIyqaHHO OKamYTCH B O�HOM
MHKpoaBTo6yce?
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= 64 .
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paBHa 37. HaH�HTe pa,n;HYC oKpymHoCTH.
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= f'(x)-
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60 Er3. TV1nOBbIE 3K3AMEHALJ.V10HHbIE BAPV1AHTbl
[TI �HJIHH,n;p H KOHYC HMelOT 06m;He OCHOBaHHe H BbICOTY . 06�eM
IJ;HJIHH,n;pa paBeH 162. Haft,n;HTe o6�eM KOHyca.
OTBeT:
He 3aoyiJbme nepeueemu 6ee om6emb£ 6 Ol£aUK om6em06 M 1 6 eoom6em-
em6uu e uuempYK-q,ueu no 6b£nOl£ueuulO paoomb£.
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�JIH CMaTWBaHHH Ka6eJIH Ha 3aBo,n;e HCnOJIb3YIOT JIe6e,n;KY, KOTopaH paBHOYCKopeHHO
HaMaTbIBaeT Ka6eJIb Ha KaTYllIKY. YroJI, Ha KOTOpbIH nOBopa"tJHBaeTCH KaTYllIKa,
�t2H3MeHHeTCH co BpeMeHeM no 3aKOHY <p
== rot + -
, r,n;e t-
BpeMH B MHHYTax,2
ro == 60° /MHH Ha"tJarrbHaH yrJIOBaH CKOpOCTb Bpam;eHHH KaTYllIKH, a � == 6° /MHH2
-
yrJIOBOe YCKopeHHe, C KOTOpbIM HaMaTbIBaeTCH Ka6eJIb. Pa6o"tJHH ,n;OJImeH npOBepHTb Xo,n;
ero HaMOTRH He II03me Toro MOMeHTa, Ror,n;a yrOJI HaMOTRH <p ,n;OCTHrHeT 3375°.
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npOBepHTb ee pa60TY. OTBeT BbIpa3HTe B MHHYTax.
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pe3epByap o,n;Ha BTopaH TPy6a?
OTBeT:
� Haii,D;HTe TO"'lKY MHHHMYMa CPYHKIJ;HH Y= x
2- 28x + 961nx - 5.
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BAPV1AHT11 61
,l1l£st aanucu pemeuuu u Om6em06 ua aaiJauust 13-19 ucnol£bayume BJIAHK
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6aume "ltemKO u paaoop'tu60.
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7
2
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[J!]B IIpaBHJIbHOH TpeyrOJIbHOH IIHpaMH,ZJ;e SABC CTopOHa OCHOBaHHH AB paBHa 6, a 6oKoBoe
pe6po SA paBHO 7. Ha pe6pax AB H SC OTMe"({eHbI TO"({KH K H M COOTBeTCTBeHHO,
IIpH"({eM AK : KB == SM : MC == 1 : 5. IIJIOCKOCTb a CO,ZJ;epmHT IIPHMYIO KM H lIapaJIJIeJIbHa
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a) �OKamHTe, "({TO IIJIOCKOCTb a lIaparrJIeJIbHa IIpHMOH SA.
6) HaH,ZJ;HTe yrOJI Mem,ZJ;y IIJIOCKOCTHMH a H SBC.
� PellIHTe HepaBeHCTBO logo,5 (12 - 6x ) > logo,5 ( x2
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I 17 I
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62 Er3. TLt1nOBbIE 3K3AMEHALJ.Lt10HHbIE BAPLt1AHTbl
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B .H�HKe JIemHT 76 <PPYKTOB, Macca Kam�oro H3 KOTOpbIX BbIpamaeTC.H n;eJIbIM QHCJIOM
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paBHa 100 r?
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BAPHAHT 12
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320
300
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OTBeT:
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BC = 5 H �D = 27. HaH,lJ;HTe qeTBepTYIO CTOPOHY
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BAPLt1AHT12 65
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rrOBepXHOCTH IJ;HJIHH,n;pa paBHa 27.J2. HaH,n;HTe rrJI Om; a,n;b
60KOBOH rrOBepXHOCTH KOHyca.
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6OM. Bo BpeMH
pa60TbI TeJIeBH30pa HarrpHmeHHe Ha KOH,n;eHcaTope Uo
== 34 KB. IIocJIe BbIKJIIO"tJeHHH
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u:orrpe,n;eJIHeMOe BbIpameHHeM t == aRClog 2
�(c), r,n;e a == 1,7
-
rrOCTOHHHaH. Orrpe,n;eJIHTeU
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OTBeT ,n;aHTe B KHJIOBOJIbTax.
OTBeT:
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OTBeT:
� HaH,ZJ;HTe TO"'lKY MaKCHMYMa <PYHKIJ;HH Y= x
3 + 18x 2+ 81x + 23.
CD
OTBeT:
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OTBETOB M 2. 3anumume cna"l£al£a nOMep 6b£nOl£nJ£eMOZO 3aiJanUJ£ (13, 14
U m. iJ.), a 3ameM nOl£noe o6ocno6annoe pemenue u om6em. Om6emb£ 3anUCb£-
6aume "l£emKO U pa36op"l£U60.
a) PemHTe ypaBHeHHe 2Sin2x-3.J3sin( � +x )-5=0.6) YKamHTe KOpHH 3Toro ypaBHeHHH, npHH3.,lJ;JIemaru;He OTpe3KY [-
5
2
1t
;-
1t ].
B IIpaBHJIbHOH "tJeTbIpexyrOJIbHOH IIHpaMH)J;e SABCD CTopOHa OCHOBaHHH AB paBHa 4,
a 6oKoBoe pe6po SA paBHO 5. Ha pe6pe SC OTMe"tJeHa To"tJKa K, IIpH"tJeM SK : KC == 1 : 3.
IIJIOCKOCTb a CO)J;epmHT TO"tJKY K H lIapaJIJIeJIbHa IIJIOCKOCTH SAD.
a) ,lJ:oKamHTe, "tJTO ce"tJeHHe IIHpaMH)J;bI SABCD IIJIOCKOCTblO a-
TpalIen;HH.
6) HaH)J;HTe o6�eM IIHpaMH)J;bI, BepmHHoH KOTOpOH HBJIHeTCH To"tJKa S, a OCHOBaHHeM-
ce"tJeHHe IIHpaMH)J;bI SABCD IIJIOCKOCTblO a.
15 I PemHTe HepaBeHCTBO log2 (18-9x )-log2 (x +2) > log2 (X2 -6x+8).
�
lliJ
TO"tJKa 0 -
n;eHTp BIIHcaHHoH B TpeyrOJIbHHK ABC oKpymHoCTH. IIpHMaH BO BTOpH"tJHO
lIepeceKaeT OIIHcaHHYIO OKOJIO 3Toro TpeyrOJIbHHKa oKpymHocTb B To"tJKe E.
a) ,lJ:oKamHTe, "tJTO LEOC == LECO.
6) HaH)J;HTe IIJIOm;Mb TpeyrOJIbHHKa ACE, eCJIH pa)J;HYC olIHcaHHoH OKOJIO
TpeyrOJIbHHKa ABC oKpymHocTH paBeH 6.J3, LABC == 600
.
15 HHBapH IIJIaHHpyeTcH B3HTb Kpe)J;HT B 6aHKe Ha 49 MeCHn;eB. YCJIOBHH ero B03BpaTa
TaKOBbI:
1-ro "tJHCJIa Kam)J;oro MeCHn;a )J;OJIr B03paCTaeT Ha 1%
IIO cpaBHeHHIO C KOHn;OM
IIpe)J;bI)J;ym;ero MeCHn;a;
co 2-ro IIO 14-e "tJHCJIO Kam)J;oro MeCHn;a Heo6xo)J;HMO BbIIIJIaTHTb "tJaCTb )J;OJIra;
15-ro "tJHCJIa Kam)J;oro MeCHn;a )J;OJIr )J;OJImeH 6bITb Ha O)J;HY H TY me CYMMY MeHbIIIe
)J;OJIra Ha 15-e "tJHCJIO IIpe)J;bI)J;ym;ero MeCHn;a.
KaKYIO CYMMY IIJIaHHpyeTcH B3HTb B Kpe)J;HT, eCJIH 06m;aH CYMMa BbIIIJIaT IIOCJIe IIOJIHOrO
ero norameHHH COCTaBHT 2 MJIH py6JIeH?
(C"tJHTaiiTe, "tJTO oKpyrJIeHHH IIPH BbI"tJHCJIeHHH IIJIaTemeH He IIPOH3BO)J;HTCH.)
DD HaH,ZJ;HTe Bce 3Ha"'leHHH a, npH Kam)J;OM H3 KOTOpbIX ypaBHeHHe
Ix- 6 1+a-6=0
x2
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HMeeT pOBHO )J;Ba pa3JIH"tJHbIX KOpHH.
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B .Hm;HKe JIemHT 58 oBom;eii, Macca Kam,ZJ;oro H3 KOTOpbIX BbIpamaeTC.H n;eJIbIM "tIHCJIOM
rpaMMOB. B .Hm;HKe eCTb XOT.H 6bI ,ZJ;Ba OBOm;a pa3JIH"tIHOii MaCCbI, a cpe,ZJ;H.H.H Macca Bcex
oBom;eii paBHa 1000 r. Cpe,ZJ;H.H.H Macca oBom;eii, Macca Kam,ZJ;oro H3 KOTOpbIX
MeHbIIIe 1000 r, paBHa 976 r. Cpe,ZJ;H.H.H Macca oBom;eii, Macca Kam,ZJ;oro H3 KOTOpbIX
60JIbIIIe 1000 r, paBHa 1036 r.
a) MorJIo JIH B .Hm;HKe OKa3aTbC.H IIOPOBHY oBom;eii Maccoii MeHbIIIe 1000 r H oBom;eii
Maccoii 60JIbIIIe 1000 r?
6) MorJIo JIH B .Hm;HKe OKa3aTbC.H pOBHO 12 oBom;eii, Macca Kam,ZJ;oro H3 KOTOpbIX
paBHa 1000 r?
B) KaKYIO HaHMeHbIIIYIO Maccy MomeT HMeTb OBOm; B 3TOM .Hm;HKe?
np06epbme, "l£mOOb£ KamiJb£u om6em Ob£l£ 3anucan PJLiJOM c nOMepOM coom-
6emcm6YlOUfezo 3aiJanUJL.
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BAPHAHT 13
Om6emOM K 3aiJaUUJ£M 1-12 J£6l£J£emCJ£ lfel£oe "f.£UCl£O Ul£U KOUe"f.£UaJ£ iJecJ£mu"f.£-
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zo 3aiJaUUJ£, ua"f.£uuaJ£ C nep60ii Kl£emO"f.£KU. Ka:HCiJylO lfutjJpy, 3uaK «MUUYC)
U 3anJ£mylO numume 6 omiJel£buoii Kl£emO"f.£Ke 6 coom6emcm6UU C npu6eiJeuub£-MU 6 6l£aUKe 06pa3lfaMU. EiJuuulfb£ U3Mepeuuii nucamb ue UY:HCUO.
qacTb 1
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OTBeT:
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6HpmH BO Bce ToprOBbIe �HH MaH 2019 ro�a. lIo ropH30HTaJIH YKa3aHbI qHCJIa MeCHn;a,
no BepTHKaJIH-
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15,0
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14,31 2 3 5 6 7 8 9 10 12 13 14 15 16 17 19 20 21 22 23 24 26 27 28 29 30 31
Onpe�eJIHTe no �HarpaMMe, Ha CKOJIbKO �OJIJIapOB n;eHa TpOHCKOH YHn;HH cepe6pa 6 MaH
6bIJIa BbIllIe, qeM 26 MaH.
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BAPLt1AHT13 69
oHa KJIeTQaTOH 6YMare C pa3MepOM
TpaIIeD;HH. Haii,lJ;HTe ee IIJIOm;a,lJ;b.
KJIeTKH lxl H306pameHa
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6 OqKOB ».
OTBeT:
co HaH,ZJ;HTe KopeHb ypaBHeHHH .J 11- 5x = 1- x ·
ECJIH ypaBHeHHe HMeeT 60JIee O,lJ;HOrO KOpHH, B OTBeTe 3aIIHllIHTe HaH60JIbllIHii
H3 KopHeii.
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= f'(x)-
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CPYHKD;HH f(x) IIapaJIJIeJIbHa IIpHMOii y= x + 18 HJIH COBIIa,lJ;aeT C Heii.
y �"
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:-0 Era. TV1nOBbIE aK3AMEHAl..t�OHHbIE BAPV1AHTbl
8 KoHYC BIIHcaH B map. Pa,n;HYC OCHOBaHHH KOHyca paBeH Pa.,ZJ;HYcy mapa. 06�eM mapa
paBeH 188. HaH)J;HTe 06�eM KOHyca.
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f( v) =
to(ru;), r,D;e c
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OTBeT:
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CKOpOCTb O)J;Horo M3 HHX Ha 10 KM/q 60JIbme CKOpOCTH )J;Pyroro?
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00 HaH,D;HTe HaH60JIbmee 3Ha"tJ:eHHe QJYHRU;HH y=(x2+22x-22)e2 - X
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6aume 'temJ(,o u pa36op'tu60.
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x + 5) = -2 .
2
6) Hd,ZJ;HTe Bce ROpHH 3Toro ypaBHeHHH, IIpHHa,zJ;JIe.mam;He OTpe3RY [5n; l�n 1
�B IIpaBHJIbHOH TpeyrOJIbHOH yce"tJ:eHHoH IIHPaMH,ZJ;e ABCA1B1C 1 IIJIOm;a,zJ;b HH.mHero
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pe6po AC IIpOBe)J;eHa IIJIOCKOCTb a, KOTOpa.H lIepeceKaeT pe6po BBI B TOqKe K H )J;eJIHT
IIHpaMH)J;Y Ha )J;Ba MHororpaHHHKa paBHoro 06rneMa.
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HaBHraTop npe)J;JIaraeT Kam)J;OMY H3 HHX 5 paBHo�eHHbIX MaprnpYTOB, H aBTOM06HJIHCTbI
BbI6HpalOT MaprnpYT CJIy"tlaHHbIM 06pa30M. HaH)J;HTe BepOHTHOCTb Toro, "tiT 0
aBTOM06HJIHCTbI BbI6epYT O)J;HH H TOT me MaprnpYT.
OTBeT:
5 1HaH)J;HTe KopeHb ypaBHeHHH 3-x == 5-.
9 3
OTBeT:
OCHOBaHHH Tpane�HH paBHbI 15 H 26. HaH�HTe 60JIbrnHH
H3 OTpe3KOB, Ha KOTopbIe )J;eJIHT cpe)J;HIOIO JIHHHIO aTOH
Tpane�HH o)J;Ha H3 ee )J;HarOHaJIeH.
OTBeT:
A B
MaTepHaJIbHaH TO"tlKa )J;BHmeTcH npHMOJIHHeHHO no 3aKoHY x(t) == t2
- 9t -
22, r)J;e x-
paCCTOHHHe OT TO"tlKH OTC"tIeTa B MeTpax, t-
BpeMH B ceKYH�ax, nporne)J;rnee C Ha"tlaJIa
)J;BHmeHHH. B KaKoH MOMeHT BpeMeHH (B ceKYH)J;ax) ee CKOpOCTb 6bIJIa paBHa 3 M/C?
OTBeT:
@ 2021. 000 «M3,ZJ;aTeJlbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe)). @ 2021. qOY .D;rro «MIJ;HMO))KOIIHpoBaHHe, paCilpOCTpaHeHHe H HCIIOJlb30BaHHe B KOMMepqeCKHX u;eJl.HX 6e3 IIHCbMeHHoro pa3pemeHH.H IIpaBOo6J18.,lJ;aTeJleH He ,ZJ;OIIYCKaeTC.H
BAPV1AHT17 89
==u
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�
[ill
00
CD
OT TpeyroJlbHoH IIHpaMH)J;bI, 06'beM KOTOpOH paBeH 42, OTce-qeHa TpeyroJlbHa.H IIHpaMH)J;a
IIJlOCKOCTbIO, IIpoxo)J;.HID;eH -qepe3 BepmHHY IIHpaMH)J;bI H cpe)J;HIOIO JlHHHIO OCHOBaHH.H.
HaH)J;HTe 06'beM oTce-qeHHoH TpeyroJlbHoH IIHpaMH)J;bI.
OTBeT:
He aaoyiJbme nepeneemu 6ee om6emb£ 6 Ol£an" om6em06 M 1 6 eoom6em-
em6uu e unempY"ltueu no 6b£nOl£nenulO paoomb£.
qacTL 2
log3 4
HaH)J;HTe 3Ha-qeHHe BbIpameHH.H + log2 0,5 .
log3 2
OTBeT:
PaCCTO.HHHe OT Ha6J1IO)J;aTeJl.H, Haxo)J;.HID;eroc.H Ha He60JlbmOH BbICOTe h KM Ha)J; 3e MJleH ,
)J;O Ha6J1IO)J;aeMoH HM JlHHHH ropH30HTa BbIqHCJI.HeTC.H IIO qJopMYJle I = ..J 2Rh,
r)J;e R = 6400 KM pa)J;HYC 3eMJlH. C KaKoH BbICOTbI ropH30HT BH)J;eH
Ha paCCTO.HHHH 64 KM? OTBeT BbIpa3HTe B KHJlOMeTpax.
OTBeT:
CMemaB 31-IIpO�eHTHbIH H 57-IIpO�eHTHbIH paCTBopbI KHCJlOTbI H )J;06aBHB 10 Kr -qHCTOH
BO)J;bI, IIOJly-qHJlH 22-IIpO�eHTHbIH paCTBOp KHCJlOTbI. ECJlH 6bI BMeCTO 10 Kr BO)J;bI
)J;06aBHJlH 10 Kr 50-IIpO�eHTHoro paCTBopa TOH me KHCJlOTbI, TO IIOJly-qHJlH 6bI
47-IIpO�eHTHbIH paCTBOp KHCJlOTbI. CKOJlbKO KHJlOrpaMMOB 31-IIpO�eHTHoro paCTBopa
HCIIOJlb30BanH )J;JI.H IIoJlyqeHH.H CMeCH?
OTBeT:
HaH)J;HTe HaHMeHbmee
Ha OTpe3Ke [0; ; ].
y= - 9 - 8.J3n + 24.J3x - 48.J3 sin x3Ha-qeHHe qJYHK�HH
OTBeT:
He aaoyiJbme nepeneemu 6ee om6emb£ 6 ol£an" om6em06 M 1 6 eoom6em-
em6uu e unempY"ltueu no 6b£nOl£nenulO paoomb£.
np06epbme, "'mOOb£ Ka3ICiJb£u om6em Ob£l£ 3anucan 6 cmpOKe c nOMepOM
coom6emcm6YJOUfezo 3aiJanuJt.
@ 2021. 000 «(M3,ZJ;aTeJIbCTBO «(HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY AnO «(MU;HMO»KOIIHpoBaHHe, paCilpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX
- 6e3 IIHCbMeHHoro pa3perneHHH IIpaBOo6JI8.,lJ;aTeJIei He ,ZJ;OIIYCKaeTCH
90 Er3. TV1nOBbIE 3K3AMEHALl.V10HHbIE BAPV1AHTbl
lfJl,s£ aanucu pemenuu U om6em06 na aaoanus£ 13-19 UCnOJl,bayume BJIAHK
OTBETOB :M 2. 3anumume cna"l£aJl,a nOMep 6b£nOJl,nS£eMOZO aaoanus£ (13, 14
u m. 0.), a aameM nOJl,noe o6ocno6annoe pemenue u om6em. Om6emb£ aanUCb£-
6aume "l£emKO u paa60p"l£U60.
[1!] a) PernHTe ypaBHeHHe 24. 4x
- O ,5 -11. 2x+l + 6 = o.
6) HaH,l];HTe Bce KOpHH 3Toro ypaBHeHHH, IIpHHa,l];JIemarn;He IIpOMemYTKY [-1; 1].
[K]OCHOBaHHeM IIHpaMH,l];bl SABCD HBJIHeTCH IIpHMoyrOJIbHHK ABCD co CTopOHaMH AB = 15
H BC = 25. Bce 6oKoBble pe6pa IIHpaMH,l];bl paBHbl 5.JI7 Ha pe6pax AD H BC OTMe"tleHbl
COOTBeTCTBeHHO TO"tlKH K H N TaK, "tITO AK = CN = 8. Qepe3 TO"tlKH K H N IIpOBe,l];eHa
IIJIOCKOCTb a, IIeplIeH,l];HKYJIHpHaH pe6py SB.
a) ,n:oKamHTe, "tITO IIJIOCKOCTb a IIpOXO,l];HT "tIepe3 TO"tlKY M -
Cepe,l];HHY pe6pa SB.
6) HaH,l];HTe paCCTOHHHe Mem,l];Y IIpHMblMH DS H KM.
� PernHTe HepaBeHCTBO �x+ � ·
log! (log211-xl» O.
2
11"61 qeTblpexyrOJIbHHK ABCD BIIHcaH B oKpymHocTb pa,l];Hyca R ='27. M3BecTHo, "tITO
� AB =BC =CD =36.
lliJ
a) ,n:oKamHTe, "tITO IIpHMble BC H AD IIapaJIJIeJIbHbl.
6) HaH,l];HTe AD.
lIo 6H3HeC-IIJIaHY "tIeTblpeXJIeTHHH IIpOeKT rrpe,l];IIOJIaraeT Ha"tlaJIbHOe BJIOmeHHe-
25 MJIH py6JIeH. lIo HToraM Kam,l];OrO rO,l];a IIJIaHHpyeTcH IIpHpOCT BJIOmeHHblX Cpe,l];CTB
Ha 20%
IIO cpaBHeHHIO C Ha"tlaJIOM rO,l];a. Ha"tlHCJIeHHble IIpo:u;eHTbl OCTalOTCH BJIOmeHHblMH
B IIpOeKT. KpoMe 3Toro, cpa3Y IIOCJIe Ha"tlHCJIeHHH IIpo:u;eHTOB HymHbl ,l];OIIOJIHHTeJIbHble
BJIOmeHHH: n;eJIoe "tIHCJIO n MJIH py6JIeH H B IIepBblH, H BO BTOpOH rO,l];bl, a TaKme n;eJIoe
"tIHCJIO m MJIH py6JIeH H B TpeTHH, H B qeTBepTblH rO,l];bl. HaH,l];HTe HaHMeHbIIIee
3Ha"tleHHe n, IIPH KOTOpOM IIepBOHa"tlaJIbHble BJIOmeHHH 3a ,l];Ba rO,l];a KaK MHHHMYM
y,l];BOHTCH, H HaHMeHbIIIee 3Ha"tleHHe m, TaKoe, qTO IIPH HaH,l];eHHOM paHee 3Ha"tleHHH n
IIepBOHa"tlaJIbHble BJIOmeHHH 3a "tIeTblpe rO,l];a BblpaCTYT KaK MHHHMYM B "tIeTblpe pa3a.
@ 2021. 000 (<lI3.n;aTeJIbcTBo «Ha:U;HOHaJIbHOe o6pa30BaHHe». @ 2021. qOY ,l];rrO «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX :u;eJI.HX 6e3 IIHCbMeHHoro pa3pemeHH.H IIpaBOo6JIa.n;aTeJIeH He .n;OIIYCKaeTC.H
3APV1AHT17 91
a
� HaH,lI;HTe Bce 3Ha-qeHH.H a, IIPH Kam,ll;OM H3 KOTOpbIX JIHHHH y=alx- 2 1+l a l-2 H
Y=2OrpaHH"tIHBalOT MHoroyrOJIbHHK, nJIOrn;a,lI;b KOToporo He 60JIee 0,5.
� M3,lI;aTeJIbCTBO Ha BbICTaBKY npHBe3JIO HeCKOJIbKO KHHr ,lI;JI.H npO,lI;amH (Kam,lI;YIO KHHry
npHBe3JIH B e,lI;HHCTBeHHOM 3K3eMnJI.Hpe). �eHa Kam,lI;OH KHHrH-
HaTypaJIbHOe "tIHCJIO
py6JIeH. ECJIH n;eHa KHHrH MeHbIIIe 100 py6JIeH, Ha Hee npHKJIeHBalOT 6HPKY «BbIrO,lI;HO » .
O,lI;HaKO ,lI;0 OTKpbITH.H BbICTaBKH n;eHY Kam,lI;OH KHHrH YBeJIH"tIHJIH Ha 10 py6JIeH,H3-3a "tIero KOJIH"tIeCTBO KHHr C 6HpKaMH «BbIrO,lI;HO» YMeHbIIIHJIOCb.
a) MorJIa JIH YMeHbIIIHTbC.H Cpe,lI;H.H.H n;eHa KHHr C 6HpKOH «BbIrO,lI;HO» nOCJIe OTKpbITH.H
BbICTaBKH no cpaBHeHHIO co Cpe,lI;HeH n;eHoH KHHr C 6HpKOH «BbIrO,lI;HO» ,lI;0 OTKpbITH.H
BbIcTaBKH?
6) MorJIa JIH YMeHbIIIHTbC.H Cpe,lI;H.H.H n;eHa KHHr 6e3 6HpKH «BbIrO,lI;HO» nOCJIe OTKpbITH.H
BbICTaBKH no cpaBHeHHIO co Cpe,lI;HeH n;eHoH KHHr 6e3 6HpKH «BbIrO,lI;HO» ,lI;0 OTKpbITH.H
BbICTaBKH?
B) M3BeCTHO, "tITO nepBOHa"tlaJIbHO Cpe,lI;H.H.H n;eHa Bcex KHHr COCTaBJI.HJIa 110 Py6JIeii,
Cpe,lI;H.H.H n;eHa KHHr C 6HpKOH «B@rO,lI;HO» COCTaBJI.HJIa 81 Py6JIb, a Cpe,lI;H.H.H n;eHa
KHHr 6e3 6HpKH- 226 py6JIeH. IIocJIe YBeJIH"tIeHH.H n;eHbI Cpe,lI;H.H.H n;eHa KHHr
C 6HpKOH «B@rO,lI;HO» COCTaBHJIa 90 py6JIeH, a Cpe,lI;H.H.H n;eHa KHHr 6e3 6HpKH210 py6JIeH. IIpH KaKOM HaHMeHbIIIeM KOJIH"tIeCTBe KHHr TaKoe B03MomHo?
CDnp06epbme, "l£m06b£ Ka3ICob£u om6em 6b£1£ aanucan pJ£OOM c nOMepoM coom-
6emcm6YlOUfezo aaoanuJ£.
@ 2021. 000 «1I3.n;aTeJIbCTBO «Ha:U;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY ,l];rrO «MIJ;HMO»
KonHpoBaHHe, pacnpOCTpaHeHHe H HCnOJIb30BaHHe B KOMMepqeCKHX :u;eJUIX 6e3 nHCbMeHHoro pa3pemeHHfl rrpaBoo6JIa,n;aTeJIeH He )J;orrYCKaeTCfl
BAPHAHT 18
Om6emOM l\, aaiJanUJ£M 1-12 J£6nJ£emCJ£ lfenoe "l£ucno unu l\,one"l£naJ£ iJecJ£mu"I£-
naJ£ iJP06b. 3anulUume "l£ucno 6 none om6ema 6 mel\,cme pa60mb£, aameM nepe-
necume e1,o 6 BJIAHK OTBETOB :M 1 cnpa6a om nOMepa �00m6emcm6yroUfe-
1,0 aaiJanuJ£, na"l£unaJ£ c nep60u l\,nemO"l£l\,u. Ka3ICiJyro lfutjJpy, anal\, «MunyC»
u aan.1l,myro nUlUume 6 omiJenbnou l\,nemO"l£l\,e 6 coom6emcm6UU c npu6eiJennb£-MU 6 6nanl\,e 06paalfaMU. EiJunulfb£ uaMepenuu nucamb ne nY3ICno.
qacTb 1
coPOCT "tIeJIOBeKa 6 CPYTOB 5 �IOHMOB. BbIpa3HTe ero pOCT B caHTHMeTpax, eCJIH 1 CPYT paBeH
12 �IOHMaM. C"tIHTaHTe, "tITO 1 �IOHM paBeH 2,54 CM. Pe3YJIbTaT oKpyrJIHTe �O n;eJIoro
"tIHCJIa caHTHMeTpOB.
OTBeT:
OJHa pHCYHKe TO"tlKaMH nOKa3aHO aTMoccpepHoe �aBJIeHHe B HOpHJIbCKe C 1 no 3 Mas
2019 ro�a. lIo ropH30HTaJIH YKa3aHbI MOMeHTbI H3MepeHHH, no BepTHKaJIH
aTMoccpepHoe �aBJIeHHe B MHJIJIHMeTpax pTYTHoro cToJI6a. ,n:JIS HarJIS�HOCTH TO"tlKH
coe�HHeHbI OTpe3KaMH.
758
753
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,-4 � t- o � � � C'\I ,-4 � t- o � � � C'\I ,-4 � t- o � � � C'\I,-4 ,-4 ,-4 ,-4 C'\I ,-4 ,-4 ,-4 ,-4 C'\I ,-4 ,-4 ,-4 ,-4 C'\I
1 MaH 2MaH 3MaH
Onpe�eJIHTe no PHCYHKY, Ha CKOJIbKO BbIpOCJIO aTMoccpepHoe �aBJIeHHe 3a 24 "tIaca
C 19 "tIaCOB 1 Mas �o 19 "tIaCOB 2 Mas.
OTBeT:
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BAPV1AHT18 93
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<PHrYPbI.
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,1];Ba aBTOM06HJIHCTa, He3aBHCHMO ,lJ;pyr OT ,lJ;pyra, BbIe3malOT H3 rrYHKTa A B rrYHKT B.
HaBHraTop rrpe,lJ;JIaraeT Kam,lJ;OMY H3 HHX 8 paBHOn;eHHbIX MaprnpYTOB, H aBTOM06HJIHCTbI
BbI6HpalOT MaprnpYT CJIyqaMHbIM o6pa30M. HaM,lJ;HTe BepO.HTHOCTb Toro, "tiT 0
aBTOM06HJIHCTbI BbI6epYT pa3JIH"tIHbIe MaprnpYTbI.
OTBeT:
7 5
[TI HaH)];HTe KopeHb ypaBHeHH.H 79
x = 56
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OTBeT:
IIp.HMa.H, rrpOBe,lJ;eHHa.H rrapaJIJIeJIbHO 60KOBOM CTopOHe
Tparren;HH "tIepe3 KOHen; MeHbrnero OCHOBaHH.H,
paBHoro 41, OTceKaeT TpeyrOJIbHHK, rrepHMeTp KOToporo
paBeH 83. HaM,lJ;HTe rrepHMeTp Tparren;HH.
OTBeT: A E B
Ma TepHaJIbHa.H TO"'IKa )];BHmeTC.H np.HMOJIHHeHHO no 3aKOHY x ( t ) = .!.. t3
- 6t + 20, r)];e x-
3
paCCTO.HHHe OT TO"tlKH OTC"tIeTa B MeTpax, t-
BpeM.H B ceKYH,lJ;aX, rrporne,lJ;rnee C Ha"tlaJIa
,lJ;BHmeJIH.H. B KaKoM MOMeHT BpeMeHH (B ceKYH,lJ;ax) ee CKOpOCTb 6bIJIa paBHa 3 M/C?
OTBeT:
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94 Er3. TV1nOBbIE 3K3AMEHALl.V10HHbIE BAPV1AHTbl
CD
CD
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paBHbI 12, a BbICOTa paBHa 6.J3.
OTBeT:
He aa6yobme nepeneemu 6ee om6emb£ 6 61£anK om6em06 M 1 6 eoom6em-
em6uu e unempYKlfueu no 6b£nol£nenuro pa60mb£.
qacTb 2
Hu log7 2
I 10aH,n;HTe 3HaqeHHe BbIpameHH.H-
og5 .
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OTBeT:
,n:BHra.HCb co CKOpOCTblO V = 5 M/C, TpaKTOp Tarn;HT caHH C CHJIOM F = 100 KH,
HanpaBJIeHHOM no,n; OCTpbIM yrJIOM a K ropH30HTY. MOrn;HOCTb, pa3BHBaeMa.H TpaKTopOM,
BbIqHCJI.HeTC.H no cpopMYJIe N = Fvcosa. HaM,n;HTe, npH KaKOM yrJIe a (B rpa,n;ycax) 3Ta
KH.M
MOrn;HOCTb 6y,n;eT paBHa 250 KBT (KBT-
3TO ).C
OTBeT:
,n:Ba qeJIOBeKa OTnpaBJI.HIOTC.H H3 o,n;Horo ,n;OMa Ha nporYJIKY ,n;o onYIIIKH JIeca,
Haxo,n;.Hrn;eMC.H B 3,5 KM OT ,n;OMa. O,n;HH H,n;eT co CKOpOCTblO 3,6 KM/q, a ,n;pyroM-
co CKOpOCTblO 4,8 KM/q. ,n:OM,n;.H ,n;o onYIIIKH, BTOpOM C TOM me CKOpOCTblO B03Bparn;aeTC.H
06paTHO. Ha KaKOM paCCTO.HHHH OT ,n;oMa npOH30M,n;eT HX BCTpeqa? OTBeT ,n;aMTe
B KHJIOMeTpax.
OTBeT:
DD Han,ll;HTe TO"IKY MHHHMYMa CPYHKD;HH Y= (3
-
2x)cosx + 2sinx + 4, npHHa,IJ;JIemam;yro
npOMemYTKY (0; ; ).
CD
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He aa6yobme nepeneemu 6ee om6emb£ 6 61£anK om6em06 M 1 6 eoom6em-
em6uu e unempYKlfueu no 6b£nol£nenuro pa60mb£.
llp06epbme, "l£mOOb£ Ka3ICOb£U om6em Ob£Jl, aanueau 6 empOKe e UOMepOM
eoom6emem6YlOzq,ezo aaoaUU1L.
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�PV1AHT 1 8 95
lfl£J£ aanucu pezuenuu u om6em06 na aaoanuJ£ 13-19 ucnol£bayume BJIAHK
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u m. 0.), a aameM nOl£noe o6ocno6annoe pezuenue u om6em. Om6emb£ aanUCb£-
6aume "l£emKO U paa60p"l£U60.
=:!D a) PerrHiTe ypaBHeHlie 4.25x + O ,5 -60.5
x - 1+1 = O.
6) HaH,l];HTe Bce KOpHH 3Toro ypaBHeHHH, IIpHHa,l];JIemam;He IIpOMemYTKY [-3; -1].
�OCHOBaHHeM IIHpaMH,l];bI T ABCD HBJIHeTCH IIpHMoyrOJIbHHK ABCD co CTopOHaMH AB = 26
H BC = 18. Bce 6oKoBble pe6pa IIHpaMH,l];bl paBHbl 10.J5. Ha pe6pax AB H CD OTMe"tleHbl
COOTBeTCTBeHHO TO"tlKH N H M TaK, "tITO BN = DM = 12. Qepe3 TO"tlKH N H M IIpOBe,l];eHa
IIJIOCKOCTb a, IIeplIeH,l];HKYJIHpHaH pe6py T A.
a) ,1];oKamHTe, "tITO IIJIOCKOCTb a IIpOXO,l];HT "tIepe3 TO"tlKY K -
Cepe,l];HHY pe6pa T A.
6) HaH,l];HTe paCCTOHHHe Mem,l];Y IIpHMblMH TC H KN.
� PeIIIliTe HepaBeHCTBO v' x + 3 .log 1 (log311 + xl) < 0 .
3
'16l qeTblpexyrOJIbHHK ABCD BIIHcaH B oKpymHocTb pa,l];Hyca R = 12. M3BecTHo, "tITO
� AB=BC=CD= 18.
[E]
a) ,1];oKamHTe, "tITO IIpHMble BC H AD IIapaJIJIeJIbHbl.
6) HaH,l];HTe AD.
lIo 6H3HeC-IIJIaHY "tIeTbIpeXJIeTHHH IIpOeKT IIpe,l];IIOJIaraeT Ha"tlaJIbHOe BJIOmeHHe-
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Ha 15%
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"tIHCJIO m MJIH py6JIeH H B TpeTHH, H B "tIeTBepTblH rO,l];bl. HaH,l];HTe HaHMeHbIIIee
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npHBe3JIH B e,n;HHCTBeHHOM 3K3eMnJI.Hpe). �eHa Kam,n;oii KHHrH-
HaTyparrbHoe qHCJIO
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cpe,n;H.H.H �eHa KHHr C 6HPKOii «BbIro,n;HO» COCTaBJI.HJIa 67 py6JIeii, a cpe,n;H.H.H �eHa
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BAPHAHT 19
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CD
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f(x) == 0 Ha OTpe3Ke [-2; 5].
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em6uu e unempYKlfueu no 6b£nOl£nenulO paoomb£.
qacTL 2
1 2 6- - -
[TI HaH,n;HTe 3HaYIeHHe BbIpameHHSI 2,5 7 .2 7 .10 7·
OTBeT:
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IIO,n; yrJIOM 2a ,n;pyr K ,n;pyry. 8HeprHH (B ,n;moYJIHx), BbI,n;eJIHIOm;aHCH IIPH HX a6COJIIOTHO
HeYlIpyroM coy,n;apeHHH, BbIqHCJIHeTCH IIO <popMYJIe Q == mv2
sin2
a, r,n;e m-
Macca
B KHJIOrpaMMax, v-
CKOpOCTb B M/c. HaH,n;HTe, IIO,n; KaKHM HaHMeHbmHM yrJIOM 2a
(B rpa,n;ycax) ,n;OJImHbI ,n;BHraTbCH TeJIa, qTo6bI B pe3YJIbTaTe coy,n;apeHHH BbI,n;eJIHJIOCb
3HeprHH He MeHee 750 ,n;moYJIeH.
OTBeT:
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3.AP�AHT 20 105
=:!D Hair,ZJ;HTe
[12 14
]13;
13·
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CD
�Be 6pHra,n;bI, COCTO.Hm;He H3 pa6o"t.J:Hx o�HHaKoBoH KBaJIH<pHKaD;HH, O�HOBpeMeHHO
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- 15 pa6o"t.J:Hx. Qepe3 5 �HeH IIOCJIe Ha"t.J:aJIa pa60TbI B IIepBYIO 6pHra�y
IIepemJIH 7 pa6o"t.J:Hx H3 BTOpOH 6pHra�bI. B HTore o6a 3aKa3a 6bIJIH BbIIIOJIHeHbI
O�HOBpeMeHHO. HaH�HTe, CKOJIbKO �HeH IIoTpe6oBaJIOCb Ha BbIIIOJIHeHHe 3aKa30B.
OTBeT:
HaHMeHbmee <PYHKD;HH y= 4x
2- 12x + 4lnx - 10 Ha OTpe3Ke3Ha"t.J:eHHe
OTBeT:
He aa6yobme nepenecmu 6ce om6emb£ 6 6nanK om6em06 M 1 6 coom6em-
cm6UU c uncmpYKlfueu no 6b£nonnenulO pa60mb£.
np06epbme, "'moob£ Ka3ICOb£ii om6em Ob£l£ aanucan 6 cmpOKe c nOMepOM
coom6emcm6YlO�ezo aaoanUJ£.
lfn1L aanucu pemenuii u om6em06 na aaoanu1L 13-19 ucnonbayiime B.ffAHK
OTBETOB M 2. 3anumume cna'£ana nOMep 6b£nOnn1LeMOZO aaoanu1L (13, 14
u m. 0.), a aameM nOl£noe ooocno6annoe pemenue u om6em. Om6emb£ aanUCb£-
6aiime '£emKO U paa6op'£u60.
c:!!J a) PemHTe yp8.BHeHHe 125sin2x =CJ5)5sin2x .0,2.
[J!]
6) HaH�HTe Bce KOpHH 3Toro ypaBHeHH.H, IIpHHa,zJ;JIemam;He IIpOMemYTKY [-31t; -21t].
OCHOBaHHe IIHpaMH�bI SABC -
paBHocTopOHHHH TpeyrOJIbHHK ABC. BOKoBoe pe6po SA
IIepIIeH�HKYJI.HpHO IIJIOCKOCTH OCHOBaHH.H, TO"t.J:KH M H N -
cepe�HHbI pe6ep BC H AB
COOTBeTCTBeHHO, IIpH"t.J:eM SN = AM.
a) �oKamHTe, "t.J:TO yrOJI Mem�y IIp.HMbIMH AM H SN paBeH 60°.
6) HaH�HTe paCCTO.HHHe Mem�y 3THMH IIp.HMbIMH, eCJIH BC = 6.
[ill PemHTe Hep8.BeHCTBO log! (5 - 5x ) <
log! ( x2
- 3x + 2 ) + log4 ( x + 4 ).4 4
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106 Era. T�nOBbIE aK3AMEHAU�OHHbIE BAP�AHTbl
�
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B TpeyrOJIbHHRe ABC Bce CTOpOHbI pa3JIHqHbI. ITpHMaH, co�epmam;aH BbICOTY BH
TpeyrOJIbHHRa ABC, BTOPHqHO nepeceRaeT onHcaHHYIO OROJIO 3Toro TpeyrOJIbHHRa
oRpymHocTb B TOqRe K. OTpe30R BN -
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a) �ORamHTe, qTO AC H KN naparrJIeJIbHbI.
6) HaH�HTe paCCTOHHHe OT TOqRH N �o npHMoH AC, eCJIH pa�HYC onHcaHHoH OROJIO
TpeyrOJIbHHRa ABC oRpymHocTH paBeH 6.J6, L.BAC == 30°, LABC == 105°.
ITo 6H3HeC-nJIaHY qeTbIpeXJIeTHHH npoeRT npe�nOJIaraeT HaqarrbHOe BJIOmeHHe-
10 MJIH py6JIeH. ITo HToraM Ram�oro ro�a nJIaHHpyeTcH npHpOCT BJIOmeHHbIX cpe�CTB
Ha 12%
no cpaBHeHHIO C HaqaJIOM ro�a. HaqHCJIeHHbIe npo�eHTbI OCTalOTCH BJIOmeHHbIMH
B npoeRT. KpoMe 3Toro, cpa3Y nOCJIe HaqHCJIeHHH npo�eHTOB HymHbI �OnOJIHHTeJIbHbIe
BJIOmeHHH: �eJIoe qHCJIO n MJIH py6JIeH H B nepBbIH, H BO BTOpOH rO�bI, a TaRme �eJIoe
qHCJIO m MJIH py6JIeH H B TpeTHH, H B qeTBepTbIH rO�bI. HaH�HTe HaHMeHbmee
3HaqeHHe n, npH ROTOpOM nepBOHaqaJIbHbIe BJIOmeHHH 3a �Ba ro�a BbIpaCTYT RaR
MHHHMYM B nOJITOpa pa3a, H HaHMeHbmee 3HaqeHHe m, TaRoe, qTO npH HaH�eHHoM
paHee 3HaqeHHH n nepBOHaqaJIbHbIe BJIOmeHHH 3a qeTbIpe ro�a RaR MHHHMYM YTPOHTCH.
lliJ HaH,ZJ;HTe Bce 3HaQeHH.H a, IIPH Kam,ZJ;OM H3 KOTOpbIX CHCTeMa YP8.BHeHHH
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== ( a + 2) x2
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00
CD
HMeeT pOBHO qeTbIpe pa3JIHqHbIX pemeHHH.
CTopOHa RBa,n;paTa Ha 2 CM �JIHHHee mHpHHbI npHMoyrOJIbHHRa, nJIOm;a,n;H 3THX <pHryp
paBHbI, a Bce �JIHHbI CTOpOH-
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6) MomeT JIH �JIHHa npHMoyrOJIbHHRa 6bITb paBHoH 9?
B) HaH�HTe Bce B03MomHbIe BapHaHTbI TaRHX nap npHMoyrOJIbHHROB H RBa,n;paTOB.
B OTBeTe YRamHTe �JIHHbI HX CTOpOH.
np06epbme, '£m06bt Ka3ICObtii om6em 6btl£ aanucan pnooM c nOMepoM coom-
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BAPHAHT 21
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1,0 aaoanulL, na�unaIL C nep60ii l\,l£emO�l\,U. Ka3ICoylO lfuq,Py, anal\, «MunyC)
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qacTL 1
meJIe3HO�OpOjKHbIii 6HJIeT �JIH B3pOCJIOro CTOHT 580 py6JIeii. CTOHMOCTb 6HJIeTa �JIH
illKOJIbHHKa COCTaBJIHeT 50%
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H3 17 illKOJIbHHKOB H 3 B3pOCJIbIX. CKOJIbKO py6JIeii CTOHT 6HJIeTbI Ha BCIO rpYIIIIY?
OTBeT:
Ha �HarpaMMe IIOKa3aHa cpe�HHH TeMlIepaTypa B03�yxa B Ka3aHH 3a KajK�bIii MeCH�
2017 ro�a. lIo ropH30HTaJIH YKa3bIBalOTCH MeCH�bI, IIO BepTHKaJIH-
TeMlIepaTypa
B rpMycax D:eJIbCHH.
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TeMlIepaTypa 3a MeCH� B Ka3aHH 6bIJIa HaH6oJIbilleii. 3alIHillHTe B OTBeTe 3HaqeHHe
cpe�Heii TeMlIepaTypbI B 3TOT MeCHD; B rpa�ycax D:eJIbCHH.
OTBeT:
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108 Er3. TV1nOBbIE 3K3AMEHALJ.V10HHbIE BAPV1AHTbl
oHa KJIeTQaTOH 6YMare H306pameH Kpyr IIJIOrn;a,u;bIO
HaH�HTe IIJIOID;a�b 3aKpameHHoro ceKTopa.
60.
OTBeT:
CD rrepe� HaqaJIOM IIepBOrO Typa qeMIIHOHaTa IIO TeHHHCY yqaCTHHKOB pa36HBalOTHa HrpOBbIe II ap bI CJIyqaHHbIM o6pa30M C IIOMOID;blO mpe6H.H. Bcero B qeMIIHOHaTe
yqaCTByeT 51 ClIopTCMeH, cpe�H KOTOpbIX 14 ClIopTCMeHOB H3 POCCHH, B TOM qHCJIe T.
HaH�HTe BepO.HTHOCTb Toro, qTO B IIepBOM Type T. 6y�eT HrpaTb C KaKHM-JIH60
ClIopTCMeHOM H3 POCCHH.
OTBeT:
4 5
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OTBeT:
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, yrOJI ABD
paBeH 36°. HaH�HTe yrOJI CAD. OTBeT �aHTe B rpa�ycax.
OTBeT:
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BAPV1AHT 21 109
�Ha pHcYHKe H306pameHbI rpacpHK CPYHK[J;HH Y
= f(x) H KaCaTeJIbHaH K HeMY B TO"'lKe
C a6CD;HCCOH XO. HaH�HTe 3HaqeHHe rrpOH3BO�HOH <PYHKD;HH f(x) B TOqKe XO.
y.
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8C == 15. HaH�HTe �RHHY OTpe3Ka BD.
s
OTBeT:
B
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[TI Haii,D;HTe 3Ha"'leHHe BbIpameHHH (274
t : ( 92
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OTBeT:
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[ill
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a BeJIOCHrre,n; rrpH 3TO�
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13 060poTax rre,n;arreH, eCJIH Ha 60JIbrnOH 3Be3,n;O"tlKe 40 3y6beB, Ha MarroH -
15, a ,n;HaMeTp
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OTBeT:
MMeeTCH ,n;Ba cocy,n;a. IIepBbIH co,n;epmHT 55 Kr, a BTOpOH- 20 Kr paCTBopOB KHCJIOTbI
pa3JIH"tIHOH KOH�eHTpa�HH. ECJIH 3TH paCTBopbI CMernaTb, TO rrOJIY"tlHTCH paCTBOp,
co,n;epmam;HH 68%
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TO rrOJIyrIHTCH paCTBOp, co,n;epmam;HH 75%
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co,n;epmHTCH B rrepBOM cocy,n;e?
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4
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7
2
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cepe,n;HHa BC.
a) ,IJ:oKamHTe, "tITO rrpHMaH SM 06pa3yeT C rrJIOCKOCTblO ABC TaKOH me yrOJI, KaK
H rrpHMaH AB C rrJIOCKOCTblO SBC.
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H SC = 5.J2.
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paBH06e)J;peHHaH TpaIIe�HH.
6) Haii)J;HTe IIJIOID;a)J;b TpeyrOJIbHHKa ABC, eCJIH BM == 1 H LBCM == 15°.
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BbIIIJIaT 3aeMID;HKa IIpeBbICHT 8 MJIH py6JIeii.
=::ill Haii.n;HTe Bce 3HaQeHHSI a, IIPH Ram.n;OM H3 ROTOpbIX CHCTeMa ypaBHeHHii
{( ay-ax+2)(y-x+ 3a) == 0,
Ixyl == a
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HMeeT pOBHO rneCTb perneHHii.
M3BeCTHO, qTO B KOrneJIbKe JIemaJIO n MOHeT, Kam)J;aH H3 KOTOpbIX MOrJIa HMeTb
)J;OCTOHHCTBO 2, 5 HJIH 10 py6JIeii. AHH c)J;eJIaJIa Bce CBOH IIOKYIIKH, paCIIJIaTHBrnHCb
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Bce MOHeTbI H3 KOrneJIbKa.
a) MOrJIH JIH Bce ee IIOKYIIKH COCTOHTb H3 6JIOKHOTa 3a 56 py6JIeii H pyqKH 3a 29 py6JIeii,eCJIH n == 14?
6) MOrJIH JIH Bce ee IIOKYIIKH COCTOHTb H3 qarnKH qaH 3a 10 py6JIeii, CbIpKa 3a 15 py6JIeiiH IIHpomKa 3a 20 py6JIeii, eCJIH n == 19?
B) KaKoe HaHMeHbrnee KOJIHqeCTBO IIHTHpy6JIeBbIX MOHeT MOrJIO 6bITb B KOrneJIbKe,
eCJIH AHH KYIIHJIa TOJIbKO aJIb60M 3a 85 py6JIeii H n == 24?
np08epbme, "l£m06b£ Ka3ICob£u om8em 6b£1£ 3anucan pJtOOM c nOMepoM coom-
8emcm8YlOUfezo 3aoanuJt.
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CD
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BAPHAHT 22
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120 Er3. TlI1nOBblE 3K3AMEHAL..J.1I10HHbIE BAPlI1AHTbI
I1nJt aanucu pemeuuii u Om6em06 ua aaiJauuJt 13-19 ucnonbayiime BJIAHK
OTBETOB M 2. 3anumume cua-rtana uOMep 6b£nonuJteMOZO aaiJauuJt (13, 14
u m. iJ.), a aameM nonuoe Ooocuo6auuoe pemeuue u om6em. Om6emb£ aanUCb£-
6aiime -rtemKO u paaoop-rtu60.
[!!] a) PemHTe ypaBHeHHe ((0,04tnxfoSX = 5-.J3 s
inx.
6) HaH,lI;HTe Bce KOpHH 3Toro ypaBHeHH.H, IIpHHa,zJ;JIemam;He IIpOMemYTKY [5
2
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a) IIocTpoHTe Ce"tleHHe Ky6a nJIOCKOCTblO, npoxO,ZJ;Hm;eH "tIepe3 TO"tlKH B, Al H DI.
6) HaH,ZJ;HTe yrOJI Mem,ZJ;y nJIOCKOCTHMH BAIC IH BAlDI.
15 I PemHTe HepaBeHcTBo log2 ( x2
- 2 ) -log2 X < log2 ( X -
:2 ),
lliJ OKpymHocTb C n;eHTpoM B TO"tlKe 0 nepeceKaeT Kam,ZJ;YIO H3 CTOpOH Tpanen;HH ABCD
B ,ZJ;BYX TO"tlKax. qeTbIpe nOJIY"tIHBmHeCH XOP,ZJ;bI oKpymHocTH paBHbI.
a) ,lJ;oKamHTe, "tITO 6HcceKTpHcbI Bcex yrJIOB Tpanen;HH nepeceKalOTcH B O,ZJ;HOH TO"tlKe.
6) HaH,ZJ;HTe BbICOTY Tpanen;HH, eCJIH oKpymHocTb nepeceKaeT 60KOBYIO CTOPOHY AB
B TOqKaX K H L TaK, "tITO AK = 19, KL = 12, LB = 3.
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no cpaBHeHHIO C KOHn;OM
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me CYMMY MeHbme ,ZJ;OJIra I-Ia 15-e "tIHCJIO npe,ZJ;bI,ZJ;ym;ero MeCHn;a;
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HaH,ZJ;HTe r, eCJIH H3BeCTHO, "tITO 06m;aH CYMMa BbInJIaT nOCJIe nOJIHOrO norameHH.H
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�1AHT 23 121
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{( a + 1)( x
2+ y2 )+( a -1)x +( a + 1)y +2 == 0,
xy-1 == x- y
HMeeT pOBHO "tIeTbIpe pa3JIH"tIHbIX pemeHHH.
�Bce "tIJIeHbI KOHe"tlHOM IIOCJIe�OBaTeJIbHOCTH HBJIHIOTCH HaTypaJIbHbIMH "tIHCJIaMH. Kam�bIM
"tIJIeH aTOM IIOCJIe�OBaTeJIbHOCTH, Ha"tlHHaH co BToporo, JIH60 B 12 pa3 60JIbme, JIH60
B 12 pa3 MeHbme IIpe�bI�ym;ero. CYMMa Bcex "tIJIeHOB IIOCJIe�OBaTeJIbHOCTH paBHa 8750.
a) MomeT JIH IIOCJIe�OBaTeJIbHOCTb COCTOHTb H3 �BYX "tIJIeHoB?
6) MomeT JIH IIOCJIe�OBaTeJIbHOCTb COCTOHTb H3 Tpex "tIJIeHOB?
B) KaKoe HaH60JIbmee KOJIH"tIeCTBO "tIJIeHOB MomeT 6bITb B IIOCJIe�OBaTeJIbHOcTH?
.J)llp06epbme, "l£mOObl, Ka3ICObl,ii om6em Obl,n aanucau pJtOOM C uOMepoM coom-
6emcm6YlOUfezo aaoauuJt.
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BAPHAHT 24
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Kecume ezo 6 BJIAHK OTBETOB M 1 cnpa6a om KOMepa coom6emcm6YlO�e-
zo aaiJaKuJt, Ka"ltUKaJt c nep60u I(,nemo"ltl(,u. Ka3ICiJylO lfutjJpy, aKal(, «(MUKYC)
u aanJtmylO nUlUume 6 omiJenbKou I(,nemo"ltl(,e 6 coom6emcm6UU c npu6eiJeKKbt-MU 6 OnaKl(,e oopaalfaMU. EiJUKUlfbt uaMepeKUU nucamb Ke KY3ICKO.
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Bcex co06ru;eHHH?
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2017 H 2018 rO�OB. IIo ropH30HTaJIH YKa3aHbI MeCHIJ;bI, 110 BepTHKaJIH-
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rrepBblH aBTOM06HJIb, a qepe3 qac rrOCJIe 3Toro HaBCTpeqy eMY H3 ropo�a B BbleXaJI
co CKOpOCTblO 80 KM/q BTOpOH aBTOM06HJIb. HaH�HTe CKOpOCTb rrepBoro aBTOM06HJIH,
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OTBeT:
CDHe aaoyobme nepeKecmu 6ce om6embt, 6 onaKK om6em06 N 1 6 coom6em-
cm6UU c UKcmpYK"",ueu no 6bt,nonKeKUlO paoombt,.
np06epbme, "l£moobt Ka3ICObt,ii om6em 6bt,n aanucan 6 cmpOKe c nOMepOM
coom6emcm6YlOUfezo aaoanUJL.
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��,'A�T 24 125
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6aume 1.£emKO U paaOOp1.£U60.
(sinx
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6) HaH,lI;HTe Bce KOpHH 3Toro ypaBHeHH.H, npHHa,zJ;JIemaID;He npOMemYTKY [2n; 7; ].
=:!!J ,n;aH Ky6 ABCDA1B1C1D 1.
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B1C 1 , AD.
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a) ,n;oKamHTe, qTO 6HcceKTpHcbI Bcex yrJIOB TpaIIe�HH IIepeceKalOTCH B O,l];HOH TOqKe.
6) HaH,l];HTe BbICOTY TpaIIe�HH, eCJIH oKpymHocTb IIepeceKaeT 60KOBYlO CTOPOHY AB
B TOqKaX K H L TaK, qTO AK == 13, KL == 6, LB == 1.
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{a( X2 + y2 )-ax+( a-3)y+1 = 0,
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y-x
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Bce qJIeHbI KOHeqHOM IIOCJIe�OBaTeJIbHOCTH .HBJI.HIOTC.H HaTypaJIbHbIMH qHCJIaMH. Kam�bIM
qJIeH aTOM IIOCJIe�OBaTeJIbHOCTH, HaqHHa.H co BToporo, JIH60 B 14 pa3 60JIbrne, JIH60
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a) MomeT JIH IIOCJIe�OBaTeJIbHOCTb COCTO.HTb H3 �BYX qJIeHoB?
6) MomeT JIH IIOCJIe�OBaTeJIbHOCTb COCTO.HTb H3 Tpex qJIeHOB?
B) KaKoe HaH60JIbrnee KOJIHqeCTBO qJIeHOB MomeT 6bITb B IIOCJIe�OBaTeJIbHOCTH?
llp06epbme, "l£mOObl, Ka3ICObl,ii om6em Obl,n aanucaK pJtOOM c KOMepOM coom-
6emcm6YlO�ezo aaoaKuJt.
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BAPHAHT 25
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1,0 aaiJaKuJt, Ka"ltUKaJt c nep60u I(,nemo"ltl(,u. Ka3ICiJylO lfutjJpy, aKal(, «(MUKYC)
u aanJtmylO numume 6 omiJenbKou I(,nemo"ltl(,e 6 coom6emcm6uu c npu6eiJeKKbt-MU 6 OnaKl(,e oopaalfaMU. EiJUKUlfbt uaMepeKUU nucamb Ke KY3ICKO.
qacTL 1
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6) HaH�HTe IIJIOm;a�b qeTbIpexyrOJIbHHKa ABCD, eCJIH H3BeCTHO, qTO AB = 6
H AH = 2-15.
[K]31 �eKa6pH 2014 ro�a MHXaHJI B3HJI B 6aHKe HeKOTOpylO CYMMY B Kpe�HT IIO� 100/0
rO�OBbIX. CxeMa BbIIIJIaTbI Kpe�HTa CJIe�YIOm;aH: 31 �eKa6pH Kam�oro CJIe�YIOm;ero
ro�a 6aHK HaqHCJIHeT IIpo�eHTbI Ha OCTaBmYIOCH CYMMY �OJIra (TO eCTb YBeJIHqHBaeT
�OJIr Ha 10 %), 3aTeM MHXaHJI IIepeBO�HT B 6aHK 2928200 py6JIeH. KaKYIO CYMMY
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(TO eCTb 3a qeTbIpe ro�a)?
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f(x) = x2
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136 Ers. TLt1nOBbIE SK3AMEHAU.Lt10HHbIE BAPLt1AHTb
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IIMeeTc.H 8 KapTOqeK. Ha HHX 3anHCbIBalOT no O,ll;HOMY Kam,ll;Oe H3 qHCeJI -1, 3, 4, -5.
7, -9, -10, 11. KapTOqKH nepeBOpaqHBalOT H nepeMeIIIHBalOT. Ha HX qHCTbIX CTOpOHax
3aHOBO nHIIIYT no O,ll;HOMY Kam,ll;Oe H3 qHCeJI -1, 3, 4, -5, 7, -9, -10, 11. IIocJIe 3Tom
qHCJIa Ha Kam,ll;OH KapTOqKe CKJIa,z:t;bIBaIOT, a nOJIyqeHHbIe BoceMb CYMM nepeMHomalOT.
a) MomeT JIH B pe3YJIbTaTe nOJIyqHTbC.H O?
6) MomeT JIH B pe3YJIbTaTe nOJIyqHTbC.H I?
B) KaKoe HaHMeHbIIIee n;eJIoe HeOTpHn;aTeJIbHOe qHCJIO MomeT B pe3YJIbTaTe nOJIyq:HTbC.H?
llp06epbme, -."mo6bf, Ka:»cobf,ii om6em 6Wl£ aanucaK pSLOOM c KOMepOM coom-
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BAPHAHT 27
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IIo ropH30HTaHH YKa3aHbI MeCH�bI, no BepTHKaRH �eHa TOHHbI �HHKa
B �ORRapax CillA. �RH HarRH�HOCTH TOqKH coe�HHeHbI OTpe3KaMH.
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138 Ers. TLt1nOBbIE SK3AMEHAU.Lt10HHbIE BAPLt1AHT�
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3TOH Tparre�HH.
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4B rpyrrrre TypHCTOB 20 qeJIOBeK. Mx BepTOJIeTOM �OCTaBJI.HIOT B TPY�Ho�ocTyrrHbIH
paHoH, rrepeB03.H rro 4 qeJIOBeKa 3a peHC. rrOp.H�OK, B KOTOpOM BepTOJIeT rrepeB03HT
TypHCTOB, CJIyq:aeH. HaH�HTe BepO.HTHOCTb Toro, qTO TypHCT B., BXO�.Hm;HH B COCTaB
rpyrrrrbI, rrOJIeTHT rrepBbIM peiiCOM BepTOJIeTa.
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IT] rrJIOm;a�b rrapaJIJIeJIOrpaMMa ABCD paBHa 28.
TOqKa E -
cepe�HHa CTOpOHbI AD. HaH�HTe rrJIOm;a,n;b
Tparre�HH BCDE.
B C
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OJHa pHCYHKe H306pameH rpaqmK y
= rex)-
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Ha HHTepBaJIe (- 4; 8). HaH�HTe TOqKY 3KCTpeMYMa CPYHK�HH f(x), rrpHHa�JIemam;ylO
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f \4
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BAPLt1AHT 27 139
==u Qepe3 cpe�HIOIO JIHHHIO OCHOBaHHH TpeyrOJIbHOM rrpH3MbI
rrpoBe�eHa rrJIOCKOCTb, rrapaJIJIeJIbHaH 60KOBOMY pe6py. IIJIOm;a,n;b
60KOBOM rroBepxHocTH OTCeqeHHOM TpeyrOJIbHOM rrpH3MbI
paBHa 36. HaM�HTe rrJIOm;a�b 60KOBOM rrOBepXHOCTH HCXO�HOM
rrpH3MbI.
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He aaoyobme nepeneemu 6ee Om6embf, 6 OJ£anJ(, om6em06 M 1 6 eoom6em-
em6uu e unempYJ(,lfueu no 6bf,nOJ£nenUlO paOOmbf,.
qacTh 2
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�
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rrO�KJIlOqHTb HarpY3KY C corrpOTHBJIeHHeM R (B OM). HarrpHmeHHe (B B) Ha 3TOM HarpY3Ke
�RBbIqHCJIHeTCH rro cpopMYJIe U ==
. IIpH KaKOM 3HaqeHHH corrpOTHBJIeHHH HarpY3KHR+r
HarrpHmeHHe Ha HeM 6y�eT paBHO 170 B? OTBeT �aMTe B OMax.
OTBeT:
3aKa3 Ha H3rOTOBJIeHHe 209 �eTaJIeM rrepBbIM pa60qHM BbIrrOJIHHeT Ha 8 qaCOB 6bIcTpee,qeM BTOpOM. CKOJIbKO �eTaJIeM 3a qac H3rOTaBJIHBaeT BTOPOM pa60qHM, eCJIH H3BeCTHO,
qTO rrepBbIM 3a qac H3rOTaBJIHBaeT Ha 8 �eTaJIeM 60JIbme?
OTBeT:
c:!D HaHAHTe HaHMeHbmee 3Ha�eHHe «PYHK[J;HH Y= 9x -In ( x + 5 )9 Ha OTpe3Ke [- 4,5; 0].
OTBeT:
Q)He aaoyobme nepeneemu 6ee Om6embf, 6 OJ£anJ(, om6em06 M 1 6 eoom6em-
em6uu e unempYJ(,lfueu no 6bf,nOJ£nenUlO paOOmbf,.
npo6epbme, -."moow J(,a:HCiJwu om6em OWJl, aanuean 6 empOKe e nOMepOM
eoom6emem6YlOUfezo aaiJanult.
/{J£1£ aanueu pemenuu u om6em06 na aaoanu1£ 13-19 UenOJ£bayume B.nAHK
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u m. 0.), a aameM nOJ£noe oooeno6annoe pemenue u om6em. Om6embf, aanUebf,-
6aume -."emJ(,o u paaoop-",u60.
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x+3sinx-3).log 2 (.J2cosx)=O.6) HaM�HTe Bce KOpHH 3Toro ypaBHeHHH, rrpHHa,n;JIemam;He OTpe3KY [-5n;
- 3n ] .
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140 Era. T�nOBbIE aK3AMEHAU.�OHHbIE BAP�AHTbl
ODB n;HJIHH�pe o6pa3yro:rn;aH IIepIIeH�HKYJIHpHa IIJIOCKOCTH OCHOBaHHH. Ha oKpymHocTH
o�Horo H3 oCHoBaHHii n;HJIHH�pa BbI6paHbI TOqKH A, B H C, a Ha oKpymHocTH �pyroro
OCHOBaHHH-
TOqKa C1 , IIpHqeM CC
1
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o6pa3yro:rn;aH n;HJIHH�pa, a AC -
�HaMeTp
OCHOBaHHH. M3BecTHo, qTO LACB = 45°, AB = 3.J2, CC1
= 6.
a) �oKamHTe, qTO yrOJI Mem�y IIpHMbIMH AC1
H BC paBeH 60°.
6) Haii�HTe paCCTOHHHe OT TOqKH B �o IIpHMOii AC1
.
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).
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AB TpaIIen;HH ABCD. OKpymHocTb C n;eHTpOM 02 KacaeTCH CTOpOH BC, CD H AD.
M3BecTHo, qTO AB = 9, BC = 8, CD = 4, AD = 15.
a) �oKamHTe, qTO IIpHMaH 0102 IIapaJIJIeJIbHa OCHOBaHHHM TpaIIen;HH ABCD.
6) Haii�HTe 0102.
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K 15-MY qHCJIY 26-ro MeCHn;a Kpe�HT �OJImeH 6bITb IIOJIHOCTbro IIorameH.
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ero IIorameHHH COCTaBHT 1924 TbICHqH py6JIeii?
Haii�HTe Bce 3HaqeHHH IIapaMeTpa a, IIPH Kam�OM H3 KOTOpbIX CHCTeMa ypaBHeHHii
{x
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Ha �OCKe HaIIHcaHO 12 pa3JIHqHbIX HaTypaJIbHbIX qHCeJI. Cpe�Hee apH<pMeTHqeCKOe
ceMH HaHMeHbIIIHX H3 HHX paBHO 8, a cpe�Hee apH<pMeTHqeCKOe ceMH HaH60JIbmHX
paBHO 16.
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6) MomeT JIH cpe�Hee apH<pMeTHqeCKOe Bcex �BeH�n;aTH qHCeJI paBHHTbCH II?
B) Haii�HTe HaHMeHbmee 3HaqeHHe cpe�Hero apH<pMeTHqeCKOrO Bcex �BeH�n;aTH qHCeJI.
np06epbme, "l£m06b£ Ka3ICOb£u om6em 6b£1£ aanucaK pROOM C KOMepoM coom-
6emcm6YlOUfezo aaoaKuR.
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==0
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BAPHAHT 28
Om6emOM K aaiJaKuRM 1-12 R6nRemCR lfenoe 1.£ucno unu KOKe1.£KaR iJecRmU1.£-
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nepeKecume ezo 6 BJIAHK OTBETOB M 1 cnpa6a om KOMepa coom6emcm6Y-
lOUfezo aaiJaKuR, Ka1.£UKaR c nep60ii Knem01.£KU. Ka3ICiJylO lfucpPY, aKaK «MUKYC)
U aanRmylO numume 6 omiJenbKoii Knem01.£Ke 6 coom6emcm6UU C npu6eiJeKKb£-MU 6 6naKKe o6paalfaMU. EiJUKUlfb£ uaMepeKuii nucamb Ke KY3ICKO.
qacTb 1
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O�Horo HOMepa mypHaJIa- 36 py6JIeii. 3a ITOJIro�a AHH KYITHJIa 25 HOMepOB mypHaJIa.
Ha CKOJIbKO py6JIeii MeHbme OHa 6bI ITOTpaTHJIa, eCJIH 6bI ITO�ITHCaJIaCb Ha mYPHaJI?
OTBeT:
Ha �HarpaMMe ITOKa3aH KYPC IOmHoa<PPHKaHCKoro paH�a, YCTaHOBJIeHHblii
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QHCJIa MeCHn;a, ITO BepTHKaJIH-
n;eHa 10 IOmHoa<PPHKaHcKHx paH�OB B py6JIHX.
48
47
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1 2 5 6 7 8 9 12 13 14 15 16 19 20 21 22 23 26 27 28 29 30
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C 13 ITO 23 MapTa 2018 ro�a.
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H306pameH TpeyrOJIbHHK ABC. HaH�HTe �JIHHY ero
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BOTBeT:
A
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y
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1 2 x
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3APV1AHT 28 143
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2
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c
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A H B
IT]Ha pHCYHKe H306pameH rpaqmK Y
= f'(x) -
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y== f(x) napaJIJIeJIbHa npHMon y
= -x + 2 HJIH cOBnMaeT CHen.
y
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He aaoyiJbme nepenecmu 6ce om6emb£ 6 onanK om6em06 M 1 6 coom6em-
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08UZH yrJIOBaH CKOpOCTb
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oKpyrJIHTe ,n;o �eJIoro "£IHCJIa 060pOTOB.
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,n;opome, 06m;aH CTOHMOCTb rroKyrrKH BbIpOCJIa 6bI Ha 130 %. ECJIH 6bI rrJIaTOK CTOHJI
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154 Ers. Tlt1nOBbIE SK3AMEHAUlt10HHbIE BAPlt1AHTbl
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u m. 0.), a aameM nOl£noe ooocno6annoe pemenue u om6em. Om6emb£ aanUCb£-
6aume "ltemJ\,o u paaOOp"ltU60.
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6) YKamHTe KOpHH 3Toro ypaBHeHHH, KOTopbIe 60JIbme 0,3.
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6) MomeT JIH nOJIyqeHHaH CYMMa 6bITb 60JIbme 1 000 OOO?
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BAPHAHT 31
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B KM/q.
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158 Ers. Tlt1nOBbIE SK3AMEHALl.lt10HHbIE BAPlt1AHTbl
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+ 2 = O.
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; ] .
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BAPHAHT 32
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B 3JIeKTpHqeCKOH n;errH cpoHapHKa rra,ll;aeT. Ha pHCYHKe rrOKa3aHa 3aBHCHMOCTb
HarrpHmeHHH B n;errH OT BpeMeHH pa60TbI cpoHapHKa. Ha ropH30HTaJIbHOH OCH OTMeqaeTCH
BpeMH pa60TbI cpOHapHKa B qacax, Ha BepTHKaJIbHOH OCH-
HarrpHmeHHe B BOJIbTax.
Orrpe,ll;eJIHTe rro PHCYHKY, qepe3 CKOJIbKO qaCOB pa60TbI cpOHapHKa HarrpHmeHHe
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BAPIltAHT 32 161
�B cpe�HeM H3 3000 ca�OBbIX HaCOCOB, IIOCTYIIHBIIIHX B IIpo�amy, 12 IIO�TeKaIOT.
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He IIO�TeKaeT..
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A B
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162 Er3. TV1nOBbIE 3K3AMEHAUV10HHbIE BAPV1AHTbl
8
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B p03eTKY 3JIeKTpOCeTH IIO�KJIlOqeHbI IIpH6opbI, o6m;ee COlIpOTHBJIeHHe KOTOpbIX
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3JIeKTpOo6orpeBaTeJIb. OlIpe�eJIHTe HaHMeHbIIIee B03MomHoe COlIpOTHBJIeHHe R2
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IIpOBO�HHKOB C COlIpOTHBJIeHHHMH Rl H R2
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a �JIH HOpMaJIbHOrO CPYHKD;HoHHpoBaHHH 3JIeKTpOCeTH
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o6m;ee
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KOTopaH, npH6bIB B IIYHKT B, TOTqaC IIOBepHYJIa o6paTHo H B03BpaTHJIaCb B A.
K 3TOMY BpeMeHH IIJIOT IIpOIIJIbIJI 44 KM. HaM�HTe CKOpOCTb MOTOpHOM JIO�KH
B HelIo�BHmHoH Bo�e, eCJIH CKOpOCTb TeqeHHH peKH paBHa 4 KM/q. OTBeT �aMTe
B KM/q.
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12 I Haii,ll;HTe TOqKY MHHHMYMa c}JYHKD;HH Y= 5x -
51n(x + 7) + 7.
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�.��AHT 32 163
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u m. iJ.), a aameM nOl£noe o6ocno6annoe pemenue u om6em. Om6emb£ aanUCb£-
6aume '£eml\,O U paa6op'£u60.
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H AD COOTBeTCTBeHHO.
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pa36HBaeT IIHpaMH�Y.
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Ha ,n;ocKe?
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5,557 KM. Ha CKOJIbKO KHJIOMeTpOB 60JIbIIIe rrOJIyqHJIOCb 6bI y rrepeBO�qHKOB, eCJIH
6bI OHH HCrrOJIb30BaJIH He rrOqTOBOe JIbe, a MopCKoe?
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166 Era. Tlt1nOBbIE aK3AMEHAUlt10HHbIE BAPlt1AHTbl
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�5X7
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2
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y
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co CKOpOCTblO 50 KM/q BbIeXaJI rrepBbIH aBTOM06MJIb, a qepe3 �Ba qaca nOCJIe 3Toro
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B KMJIOMeTpax.
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18 -
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CD
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168 Era. TVinOBblE 3K3AMEHALJ,VlOHHbIE BAPVlAHTbl
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"tITO KB==4. Qepe3 TO"tlKH K H CI IIpOBe�eHa IIJIOCKOCTb a, rrapaJIJIeJIbHaH IIpHMOH BDl.
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C pe6poM AIBI.
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ro�a;
C cpeBpaJIH IIO HIOHb Kam�oro ro�a Heo6xo�HMO BbIIIJIaTHTb O�HHM IIJIaTemOM "tIaCTb
�OJIra.
HaH�HTe r, eCJIH H3BeCTHO, "tITO Kpe�HT 6y�eT IIOJIHOCTblO lIorarneH 3a �Ba ro�a, IIpH"tIeMB lIepBbIH ro� 6y�eT BbIIIJIa"tleHO 320 000 py6JIeH, a BO BTOpOH ro�
- 450 000 py6JIeH.
I 18 I HaH)J;HTe Bce 3Ha-qeHH.S1 a, npH Kam)J; OM H3 KOT OpbIX ypaBHeHHe
ax + .J 5 - 4x -
x2
== 3a + 3
�
CD
HMeeT e�HHcTBeHHbIH KopeHb.
Ha �OCKe HarrHcaHO 60JIee 35, HO MeHee 49 n;eJIbIX "tIHCeJI. Cpe�Hee apHcpMeTH"tIeCKOe3THX "tIHCeJI paBHO 5, cpe�Hee apHcpMeTH"tIeCKOe Bcex IIOJIOmHTeJIbHbIX H3 HHX paBHO 14,a cpe�Hee apHcpMeTH"tIeCKOe Bcex OTpHn;aTeJIbHbIX H3 HHX paBHO -7.
a) CKOJIbKO "tIHCeJI HalIHcaHO Ha �ocKe?
6) KaKHx "tIHCeJI HalIHcaHO 60JIbrne: IIOJIOmHTeJIbHbIX HJIH OTpHn;aTeJIbHbIX?
B) KaKoe HaH60JIbmee KOJIH"tIeCTBO IIOJIOmHTeJIbHbIX "tIHCeJI MomeT 6bITb cpe�H HHX?
llp08epbme, "l.£m06b£ Ka:JICiJb£ii om8em 6b£1£ aanucan PJLiJOM c nOMepoM coom-
8emcm8YlOUfezo aaiJanuJL.
@ 2021. 000 «M3,n;aTeJIbCTBO «Ha:U;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY ,[(IIO «MU;HMO »
KorrHpoBaHHe, pacrrpOCTpaHeHHe H HCrrOJIb30BaHHe B KOMMepqeCKHX :u;eJISIX 6e3 rrHCbMeHHoro pa3perneHHSI rrpaBoo6JIa,n;aTeJIeH He ,n;orrYCKaeTCH
BAPHAHT 34
Om8emOM I\, aaiJanul£M 1-12 1£8l£l£emCl£ lfel£oe 1.£UCl£O Ul£U I\,One1.£nal£ iJecl£mU1.£-
nal£ iJP06b. 3anumume 1.£UCl£O 8 nOl£e om8ema 8 mel\,cme pa60mb£, aameM nepe-
necume ezo 8 BJIAHK OTBETOB .M 1 cnpa8a om nOMepa coom8emcm8YlOUfe-
zo aaiJanul£, na1.£Unal£ C nep80u l\,l£em01.£I\,U. Ka3ICiJylO lfutjJpy, anal\, «MunyC)
U aanl£mylO numume 8 omiJel£bnou l\,l£em01.£l\,e 6 coom8emcm8UU C npu8eiJennb£-MU 8 6l£anl\,e o6paalfaMU. EiJunulfb£ uaMepenuu nucamb ne nY3ICno.
qacTL 1
==0qaHHbIe KJIMlIepbI
-
caMbIe 6bIcTpbIe IIapYCHbIe Kopa6JIM. HeKoTopbIe M3 HMX MOrJIM
pa3BMBaTb CKOpOCTb �o 20 Y3JIOB. IIepeBe�MTe B KMJIOMeTpbI B qac CKOPOC"Tb KJIMlIepa,
KOTOpbIH �eJIaeT 15 Y3JIOB. 1 Y3eJI paBHHeTCH 1 MOpCKOH MMJIe B qac. 1 MopCKaH MMJIH
paBHHeTCH 1852 MeTpaM. Pe3YJIbTaT oKpyrJIMTe �o n;eJIoro qMCJIa KMJIOMeTpOB B qac.
OTBeT:
CDHa pMCYHKe IIOKa3aHO M3MeHeHMe TeMlIepaTypbI B03�yxa Ha IIpOTHmeHMM Tpex CYTOK.
IIo ropM30HTaJIM YKa3bIBaeTCH �aTa M BpeMH, IIO BepTMKaJIM-
3HaqeHMe TeMlIepaTypbI
B rpa�ycax �eJIbCMH. OlIpe�eJIMTe IIO PMCYHKY HaMMeHbmylO TeMlIepaTYPY B03�yxa
24 HHBapH. OTBeT �aHTe B rpa�ycax �eJIbCMH.
t,OC
-8
-10
-12
-14
-16
-18
-20
-22
-24
0:00 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00
22 HHBapH 23 HHBapH 24 HHBapH
OTBeT:
@ 2021. 000 «M3.n;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. QOY ,1];rrO «MIJ;HMO»KonHpoBaHHe, pacnpOCTpaHeHHe H HCnOJIb30BaHHe B KOMMepqeCKHX u;eJI,flX 6e3 nHCbMeHHoro pa3perneHH,fl npaBoo6JIa,n;aTeJIeH He .n;onYCKaeTC,fl
170 Era. TV1nOBbIE aK3AMEHALJ.V10HHbIE BAPV1AHTbl
IT]Ha KJIeT"tIaTOM 6YMare C pa3MepoM KJIeTKH 1 x 1 H306pameH
"tIeTbIpexyrOJIbHHK ABCD. HaM)J;HTe )J;HarOHaJIb BD.
OTBeT:
IT]KaKOBa BepOHTHOCTb Toro, "tITO llOCJIe)J;HHe )J;Be �HcppbI TeJIecpoHHoro HOMepa CJIy"tlaMHoroa60HeHTa B CYMMe )J;alOT 10?
OTBeT:
�7X9
CD Haii,ZJ;HTe KopeHb ypaBHeHHSI3
= 2.
OTBeT:
IT]4
B TpeyrOJIbHHKe ABC yrOJI C paBeH 90°, AB == 82, tg A == -. HaM)J;HTe BbICOTY CH.5
OTBeT:
CDHa pHCYHKe H306pameH rpaq,HK HeKoTopoM CPYHKD;HH Y
= t(x). O,ZJ;Ha H3 nepBoo6pa3HbIXaTOM cpYHK�HH paBHa
( )1 3 2
F x ==-x -x +2x-3.3
HaM)J;HTe llJIOm;a)J;b 3aIIITpHxoBaHHoM cpHrYPbI.
o 3 x
OTBeT:
� 2021. 000 «Ha,n;aTeJIbCTBO «HaU;HOHaJIbHOe o6paaOBaHHe». @ 2021. qOY Ano «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe If HCIIOJIbaOBaHHe B KOMMepqeCKHX U;eJIHX 6ea IIHCbMeHHoro paapemeHHH IIpaBOo6JIa,n;aTeJIei He ,n;OIIYCKaeTCH
BAPV1AHT 34 171
=0HaH.n;HTe o6rneM MHororpaHHHKa, BeprnHHaMH KOToporo HBJIHIOTCH TO"tlKH A, C, A
l , Bl
npaBHJIbHOH TpeyrOJIbHOH rrpH3MbI ABCAlBlC l , rrJIOm;a.n;b OCHOBaHHH KOTOpOH paBHa 9,
a 60KOBoe pe6po paBHO 3.
OTBeT:
He aa6yiJbme nepeneemu 6ee om6emb£ 6 61£anK om6em06 M 1 6 eoom6em-
em6uu e unempYKlfueu no 6b£nol£nenuro pa60mb£.
qacTL 2
191 HaH.n;HTe 3Ha"tleHHe BbIpameHHH� cos21°.cos69°
-11sin42°
OTBeT:
�BbICOTa .n;epeBHHHoro CTeJIJIama .n;JIH KHHr paBHa h = (a + b)n + a MHJIJIHMeTpOB, r.n;e
a-
TOJIm;HHa O.n;HOH .n;OCKH (B MM), b -
BbICOTa O.n;HOH rrOJIKH (B MHJIJIHMeTpax), n-
"tIHCJIO TaKHX rrOJIOK. HaH.n;HTe BbICOTY KHHmHoro CTeJIJIama H3 7 rrOJIOK, eCJIH a = 21 MM,
b = 320 MM. OTBeT BbIpa3HTe B MHJIJIHMeTpax.
OTBeT:
lliJ TIYTemeCTBeHHHR rreperrJIbIJI OReaH Ha SlXTe co Cpe,ll;HeM CROpOCTblO 26 RM/"'I. 06paTHoOH JIeTeJI Ha CaMOJIeTe co CKOpOCTblO 312 KM/"tl. HaH.n;HTe cpe.n;HIOIO CKOpOCTb
rrYTerneCTBeHHHKa Ha rrpOTHmeHHH Bcero rrYTH. OTBeT .n;aHTe B KHJIOMeTpax B "tIac.
OTBeT:
O!J HaM,lI;HTe HaH60JIbmee 3Ha"'leHHe <PYHRD;HH Y= 5cosx -9x + 27 Ha OTpe3Re [0;
3
2
1t
J.OTBeT:
CDHe aa6yiJbme nepeneemu 6ee om6embt 6 61£anK om6em06 M 1 6 eoom6em-
em6uu e unempYKlfueu no 6b£nol£nenuro pa60mbt.
np06epbme, '£m06bt Ka:HCiJbtii om6em 6btn aanucaK 6 cmpOKe c KOMepOM
coom6emcm6YJOUfezo aaiJaKUJ£.
@ 2021. 000 «Ha,n;aTeJIbCTBO «HaU;HOHaJIbHOe o6paaOBaHlIe». @ 2021. 1:J:OY Ano «MIJ;HMO»KOIIHpoBaHHe, pacnpOCTpaHeHHe H HCnOJIbaOBaHHe B KOMMepqeCKHX u;eJIHX 6ea nHCbMeHHoro paapemeHHH npaBoo6JIa,n;aTeJIeH He ,n;onYCKaeTCH
172 Era. TV1nOBbIE aK3AMEHAL.J.V10HHbIE BAPV1AHTbl
1(1£1£ aanucu pemenuu U om6em06 na aaiJanu1£ 13-19 ucnonbayume BJIAHK
OTBETOB M 2. 3anumume cna'tana nOMep 6b£nonn1£eM020 aaiJanu1£ (13, 14
u m. iJ.), a aameM nonnoe o6ocno6annoe pemenue u om6em. Om6emb£ aanUCb£-
6aume 'temKo U paa6op'tu60.
[!!] a) PeIIIHTe ypaBHeHHe (2X2- 5x -12)(2cosx + 1) = O.
6) HaH,lI;HTe Bce ROpHH 3Toro ypaBHeHHH, npHHa,n;JIemam;He npOMemYTRY [- ; ; 1t 1
O!JB nHpaMH,lI;e ABCD pe6pa DA, DB H DC nonapHO nepneH,lI;HRYJIHpHbI, aAB = BC =AC = 14.
a) ,1];oKa.iKHTe, "tITO 3Ta nHpaMH.n;a npaBHJIbHaH.
6) Ha pe6pax DA H DC OTMe"tleHbI TO"tlKH M H N COOTBeTCTBeHHO, npH"tIeM
DM : MA = DN : NC = 6 : 1. Hai1:.n;HTe nJIOm;a.n;b Ce"tleHHH MNB.
DD PeIIIHTe HepaBeHCTBO 2 .16-x
-17. 4-x
+ 8 < 0.
�
[J!J
B TpeyrOJIbHHKe ABC H3BeCTHO, "tITO L.BAC = 60°, LABC = 45°. ITpo.n;OJI.iKeHHH BbICOT
TpeyrOJIbHHKa ABC nepeceKalOT onHcaHHYIO OKOJIO Hero OKPY.iKHOCTb B TO"tlKaX M, N, P.
a) ,1];oKa.iKHTe, "tITO TpeyrOJIbHHK MNP npHMoyrOJIbHbIi1:.
6) Hai1:.n;HTe nJIOm;a.n;b TpeyrOJIbHHKa MNP, eCJIH H3BeCTHO, "tITO BC = 10.
B HIOJIe 2022 ro.n;a nJIaHHpyeTcH B3HTb Kpe.n;HT B 6aHKe Ha CYMMY 545 000 py6JIei1:.Y CJIOBHH ero B03BpaTa TaKOBbI:
Ka.iK.n;bIi1: HHBapb .n;OJIr YBeJIH"tIHBaeTCH Ha 40 % no cpaBHeHHIO C KOH�OM npe.n;bI.n;ym;ero
ro.n;a;
C cpeBpaJIH no HIOHb Ka.iK.n;oro ro.n;a Heo6xo.n;HMO BbInJIaTHTb O.n;HHM nJIaTe.iKOM "tIaCTb
.n;OJIra.
CKOJIbKO py6JIei1: 6y.n;eT BbInJIa"tleHO 6aHKY, eCJIH H3BeCTHO, "tITO Kpe.n;HT 6y.n;eT nOJIHOCTblO
norarneH TpeMH paBHbIMH nJIaTe.iKaMH (TO eCTb 3a TPH ro.n;a)?
@ 2021. 000 «Ha�aTeJIbCTBO «HaU;HOHaJIbHOe 06paaOBaHHe». @ 2021. qOY AnO «MIJ;HMO»KOIIHpoBaHHe, pacIIpocTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX U;eJIHX 6ea IIHCbMeHHoro paaperneHHH IIpaBOo6JIa�aTeJIeH He �OIIYCKaeTCH
3APV1AHT34 173
� Haii.n;HTe Bee 3Ha-qeHHSI a, IIPH Ram.n; OM H3 ROTO pbIX ypaBHeHHe
10a+ .J -7+8x-x2
=ax+3
HMeeT e.n;HHCTBeHHbIH KopeHb.
�Ha .n;OCKe HaIIHcaHO 60JIee 55, HO MeHee 65 �eJIbIX "tIHCeJI. Cpe.n;Hee apHcpMeTH"tIeCKOe
3THX "tIHCeJI paBHO 7, cpe.n;Hee apHcpMeTH"tIeCKOe Bcex IIOJIO.iKHTeJIbHbIX H3 HHX paBHO 15,
a cpe.n;Hee apHcpMeTH"tIeCKOe Bcex OTpH�aTeJIbHbIX H3 HHX paBHO -5.
a) CKOJIbKO "tIHCeJI HaIIHcaHO Ha .n;ocKe?
6) KaKHx "tIHCeJI HaIIHcaHO 60JIbIIIe: IIOJIO.iKHTeJIbHbIX HJIH OTpH�aTeJIbHbIX?
B) KaKoe HaH60JIbIIIee KOJIH"tIeCTBO IIOJIO.iKHTeJIbHbIX "tIHCeJI MO.iKeT 6bITb cpe.n;H HHX?
CDnp06epbme, "l£m06b£ Ka3ICiJb£ii om6em 6b£n 3anucan pstiJOM c nOMepOM coom-
6emcm6yroUfezo 3aiJanUst.
@ 2021. 000 «Ha�aTeJIbCTBO «HaU;HOHaJIbHOe o6paaOBaHHe». @ 2021. qOY Ano «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX u;eJI.HX 6ea IIHCbMeHHoro paaperneHH.H IIpaBOo6JIa�aTeJIeH He �OIIYCKaeTC.H
[TI
CD
CD
BAPHAHT 35
Om6emOM K 3aiJanUgM 1-12 g6ngemCg lfenoe 1.£ucno unu KOne1.£nag iJecgmu1.£-
nag iJP06b. 3anumume 1.£ucno 6 none om6ema 6 meKcme pa60mbt, 3ameM nepe-
necume ezo 6 BJIAHK OTBETOB M 1 cnpa6a om nOMepa coom6emcm6yroUfe-
zo 3aiJanUg, na1.£unag c nep60u Knem01.£KU. Ka3ICiJyro lfutjJpy, 3naK «(MunyC)
u 3angmyro numume 6 omiJenbnou Knem01.£Ke 6 coom6emcm6UU c npu6eiJennbt-MU 6 6nanKe 06pa3lfaMU. EiJunUlfbt U3Mepenuu nucamb ne nY3ICno.
qacTL 1
ITpH3epaMH ropo,n;cKOH OJIHMnHa,n;bI no MaTeMaTHKe CTaJIH 7 yqeHHKOB, qTO COCTaBHJIO
5 % OT qHCJIa yqaCTHHKOB. CKOJIbKO qeJIOBeK yqaCTBOBaJIO B OJIHMnHa,n;e?
OTBeT:
Ha pHCYHKe nOKa3aHO H3MeHeHHe TeMnepaTypbI B03,n;yxa Ha npOTHmeHHH Tpex CYTOK.
ITo ropH30HTaJIH YKa3bIBalOTCH ,n;aTa H BpeMH, no BepTHKaJIH-
3HaqeHHe TeMnepaTypbI
B rpa,n;ycax IJ:eJIbCHH. Onpe,n;eJIHTe no PHCYHKY HaHMeHbllIYIO TeMnepaTYPY B03,n;yxa
28 MaH. OTBeT ,n;aHTe B rpa,n;ycax IJ:eJIbCHH.
t,OC
18
16
14
12
10
8
6
4
2
0:00 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00 6:00 12:00 18:00 0:00
28 MaH 29 MaH 30 MaH
OTBeT:
Ha KJIeTqaTOH 6YMare C pa3MepOM KJIeTKH 1 x 1 H306pameH
TpeyrOJIbHHK ABC. HaH,lI;HTe ,n;JIHHY ero 6HcceKTpHcbI,
npoBe,n;eHHoH H3 BepllIHHbI B. B
c
OTBeT:
@ 2021. 000 «Ha,n;aTeJIbCTBO «HaU;HOHaJIbHOe 06paaOBaHHe». @ 2021. qOY .IJ;rrO «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX u;eJIHX 6ea IIHCbMeHHoro paapemeHHH IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH
3APV1AHT 35 175
�3a KpyrJlbIH CTOJI Ha 21 CTYJl B cJly-qaHHoM IIopH,n;Ke paccamHBalOTCH 19 MaJlbqHKOB
H 2 ,n;eBOqKH. HaH,n;HTe BepOHTHOCTb Toro, qTO ,n;eBOqKH He OKajEYTCH Ha coce,n;HHX
MeCTax.
OTBeT:
=u HaH,lI;HTe RopeHb ypaBHeHH5I (x- 5)3 = 216.
OTBeT:
==u qeTbIpexyroJlbHHK ABCD BIIHcaH B oKpymHoCTb. YroJl ABD
paBeH 250
, yroJl CAD paBeH 410
. HaH,n;HTe yroJl ABC. OTBeT
,n;aHTe B rpa,n;ycax.
B
OTBeT:
IT]Ha pHCYHKe H306pameH rpacpHK cpYHKD;HH Y
= f(x). .Ha OCH a6CD;HCC OTMeqeHbI TOqKH
-
2, -1, 1, 4. B KaKoH H3 STHX TOqeK 3HaqeHHe IIpOH3BO,n;HOH HaH6oJlbllIee? B OTBeTe
YKamHTe STY TOqKY.
-2 -1 1 4 x
OTBeT:
@ 2021. 000 «H3�aTeJIbCTBO «HaU;1I0HaJIbHOe o6pa30BaHlle». @ 2021. QOY ,1J;rro «MU;HMO»KOIIllpoBaHlle, pacIIpocTpaHeHlle II IICIIOJIb30BaHlle B KOMMepqeCKIIX u;eJIHX 6e3 IIIICbMeHHoro paaperneHIIH IIpaBOo6JIap;aTeJIeif He �OIIYCKaeTCH
176 Era. T�nOBbIE aK3AMEHAU.�OHHbIE BAP�AHTbl
IT] IIpHMoyrOJIbHbIH rrapaJIJIeJIerrHrre.n; orrHcaH OROJIO
�HJIHH.n;pa, pa.n;HYc oCHoBa�HH H BbICOTa ROToporo
paBHbI 5. HaH.n;HTe 06'beM rrapaJIJIeJIerrHrre.n;a.I
I
I
I
I
I .,J---- _
I� ---- �------�
""'--,.�
,. ........-
----
OTBeT:
He aaoyobme nepeneemu 6ee om6emb£ 6 Ol£anK om6em06 M 1 6 eoom6em-
em6uu e unempYKlfueu no 6b£nol£nenuro paoomb£.
qacTL 2
� Hu
3102
2 13n3
102
· 2 13n
�aH.n;HTe 3HaqeHHe BbIpameHHH " � cos
8-
" � SIn
8.
OTBeT:
DQ:JB xO,ZJ;e
?acna,ZJ;apa,ZJ;HoaKTHBHoro H30Tona ero Macca YMeHbmaeTCSl no 3aKOHY
m ==
mo .2 T, r.n;e mo-
HaqaJIbHaH Macca H30Torra, t-
BpeMH, rrporne.n;rnee OT HaqaJIbHOrO
MOMeHTa, T -
rrepHO.n; rrOJIypacrra.n;a. B HaqaJIbHbIH MOMeHT BpeMeHH Macca H30Torra
44 Mr. IIepHo.n; ero rrOJIypacrra.n;a COCTaBJIHeT 6 MHH. HaH.n;HTe, qepe3 CROJIbRO MHHYT
Macca H30Torra 6y.n;eT paBHa .11 Mr.
OTBeT:
11 MOTopHaH JIo.n;Ra rrpornJIa rrpOTHB TeqeHHH peRH 112 RM H BepHYJIaCb B rrYHRT
OTrrpaBJIeHHH, 3aTpaTHB Ha o6paTHbIH rrYTb Ha 6 qaCOB MeHbrne. HaH.n;HTe CROpOCTb
TeqeHHH, eCJIH CROpOCTb JIO.n;RH B Herro.n;BHmHoH Bo.n;e paBHa 11 RM/q. OTBeT .n;aHTe
B RM/q.
OTBeT:
[!!] HaH,ZJ;HTe TOTIKY MHHHMYMa «PYHKIJ;HH Y= (x
2- 7x + 7)e
x- 17
.
OTBeT:
CDHe aa6yobme nepeneemu 6ee om6emb£ 6 Ol£anK om6em06 M 1 6 eoom6em-
em6uu e unempYKlfueu no 6b£nol£nenuro paoomb£.
npo6epbme, l£mo6b£ Ka3ICob£u om6em Ob£l£ aanucan 6 empOKe c nOMepOM
coom6emcm6YJOUfezo aaoanUJ£.
@ 2021. 000 «H3�aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY AnO «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 IIHCbMeHHoro pa3perneHHH IIpaBOo6JIa�aTeJIeH He �OIIYCKaeTCH
BAPiI1AHT 35 177
lfl£g aanucu pemenuu u om6em06 na aaiJanug 13-19 ucnol£bayume BJIAHK
OTBETOB M 2. 3anumume cna'tal£a nOMep 6b£nOl£ngeMOZO aaiJanug (13, 14
u m. iJ.), a aameM nOl£noe o6ocno6annoe pemenue u om6em. Om6emb£ aanUCb£-
6aume 'temKo u paa6op'tu60.
OD a) PemHTe ypaBHeHHe cos4x + cos2x = O.
6) Haii.n;HTe Bce ROpHH 3Toro ypaBHeHH5I, npHHa,ZJ;JIemaru;He OTpe3RY [-1t; : J.
[illB IIHpaMH)J;e ABCD pe6pa DA, DB H DC IIOIIapHO IIepIIeH)J;HKYJIHpHbI,
a AB=BC=AC=9�.
a) ,n:oKamHTe, qTO STa IIHpaMH)J;a IIpaBHJIbHaH.
6) Ha pe6pax DA H DC OTMeqeHbI TOqKH M H N COOTBeTCTBeHHO, IIpHqeM
DM: MA = DN: NC = 2 : 7. Hai1:)J;HTe IIJIOm;a)J;b CeqeHHH MNB.
(2
)2
2
c:.!!JPemHTe HepaBeHCTBO log2 x
- 21og 2x + 361og 2
x + 45 < 181og 2 x.
[J!]
D!J
OKpymHocTb C �eHTpOM 0, BIIHcaHHaH B TpeyrOJIbHHK ABC, KacaeTCH ero CTOpOH AB,
AC H BC B TOqKaX Cl , Bl H Al COOTBeTCTBeHHO. BHcceKTpHca yrJIa A IIepeceKaeT STY
oKpymHocTb B TOqKe Q, JIemam;ei1: BHYTPH TpeyrOJIbHHKa ABlC l.
a) ,n:OKamHTe, qTO ClQ-
6HcceKTpHca yrJIa AClB l.
6) HaH.n;HTe paCCTOHHHe OT TOqKH 0)J;0 �eHTpa oKpymHocTH, BIIHCaHHOi1: B TpeyrOJIbHHKAClB l ,
eCJIH H3BeCTHO, qTO BC = 15, AB = 13, AC = 14.
B HIOJIe 2022 rO)J;a IIJIaHHpyeTcH B3HTb Kpe)J;HT B 6aHKe Ha CYMMY 928 200 py6JIei1:.Y CJIOBHH ero B03BpaTa TaKOBbI:
Kam)J;bIi1: HHBapb )J;OJIr YBeJIHqHBaeTCH Ha 10 % IIO cpaBHeHHIO C KOH�OM IIpe)J;bI)J;ym;ero
rO)J;a;
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)J;OJIra.
CKOJIbKO py6JIei1: 6Y)J;eT BbIIIJIaqeHO 6aHKY, eCJIH H3BeCTHO, qTO Kpe)J;HT 6Y)J;eT IIOJIHOCTblO
IIorameH qeTbIpbMH paBHbIMH IIJIaTemaMH (TO eCTb 3a qeTbIpe rO)J;a)?
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178 Ers. TiI1nOBblE SK3AMEHALl.iI10HHbIE BAPiI1AHTbl
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[ill
CD
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HMeeT pOBHO O�HH KopeHb Ha OTpe3Ke [0; 1 ].
Mama H HaTama �eJIarrH cpoTorpacpHH HeCKOJIbKO �HeH no�pH�. B nepBbIH �eHb Mama
c�eJIarra m cpoTorpacpHH, a HaTama -
n cpoTorpacpHH. B Ka.iK�bIH CJIe�YIOm;HH �eHb
Ka.iK�aH H3 �eBOqeK �eJIarra Ha O�HY cpoTorpacpHIO 60JIbme, qeM B npe�bI�ym;HH �eHb.
H3BecTHo, qTO HaTama 3a BCe BpeMH c�eJIarra CYMMapHO Ha 935 cpoTorpacpHH 60JIbme,
qeM Mama, H qTO cpoTorpacpHpoBarrH OHH 60JIbme O�Horo ,ZI;HH.
a) MorJIH JIH OHH cpoTorpacpHpoBaTb B TeqeHHe 5 �HeH?
6) MorJIH JIH OHH cpoTorpacpHpoBaTb B TeqeHHe 6 �HeH?
B) KaKoe HaH60JIbmee cYMMapHoe qHCJIO cpoTorpacpHH MOrJIa C�eJIaTb HaTama 3a Bce
�HH cpoTorpacpHpoBaHHH, eCJIH H3BecTHo, qTO B nOCJIe�HHH �eHb Mama c�eJIarra
MeHbme 50 cpoTorpacpHH?
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6emcm6yroUfezo aaiJanug.
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BAPHAHT 36
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nast iJP06b. 3anuULume 1.£ucno 6 none om6ema 6 meKcme pa60mb£, aameM nepe-
necume ezo 6 BJIAHK OTBETOB M 1 cnpa6a om nOMepa coom6emcm6yroUfe-
zo aaiJanust, na1.£unast c nep60u Knem01.£KU. Ka3ICiJyro lfutjJpy, anaK «MunyC)
u aanstmyro nUULume 6 omiJenbnou Knem01.£Ke 6 coom6emcm6uu c npu6eiJennb£-MU 6 6nanKe 06paalfaMu. EiJunulfb£ uaMepenuu nucamb ne nY3ICno.
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10%
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(B TbICH"tIaX TOHH) 3a 2016 ro.n;. Cpe.n;H rrpe.n;CTaBJIeHHbIX CTpaH IIepBoe MeCTO IIO BbIIIJIaBKe
aJIIOMHHHH 3aHHMaJIa POCCHH, .n;eCHToe MeCTO 3aHHMaJI KaTap. KaKoe MeCTO 3aHHMaJIa
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180 Ers. TVinOBblE SK3AMEHALJ.VlOHHbIE BAPVlAHTbl
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CD
c
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1 x 1 H306pameH TpeyrOJIbHHK ABC. HaH�HTe
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paBeH 72°, yrOJI CAD paBeH 58°. HaH�HTe yrOJI ABC. OTBeT
�aHTe B rpa,n;ycax.
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B
Ha pHCYHKe H306pameH rpacpHK cpYHK�HH y= f(x). Ha OCH a6C�HCC OTMeqeHbI TOqKH
-2, 1, 3, 4. B KaKoH H3 3THX TOqeK 3HaqeHHe ITpOH3BO�HOH HaH6oJIbIIIee? B OTBeTe
YKamHTe 3TY TOqKY.
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3APV1AHT 36 181
==u IIpHMoyrOJIbHblH rrapaJIJIeJIerrHrre� orrHcaH OROJIO
�HJIHH�pa, pa�HYC OCHOBaHHH H BbICOTa ROToporo
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m ==
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O!J MOTopHaH JIO�Ra rrpOIIIJIa rrpOTHB TeqeHHH peRH 143 RM H BepHYJIaCb B rrYHRT
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B RM/q.
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2- 26x + 26)e
x- 17
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OTBeT:
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npo8epbme, 'tmo6bl, Ka3ICob£u om8em 6b£l£ aanucan B cmpoKe c uOMepoM
coom8emcmBYlOUfezo aaiJanuJt.
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u m. d.), a aameM nOl£noe o6ocno8annoe pemenue u om8em. Om8emb£ aanUCb£-
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6) HaH.n;HTe Bce KOpHH 3Toro ypaBHeHHSI, IIpHHa.n;Jlemanwe OTpe3KY [-5n;- 4n J.
@]B IIHpaMH.n;e ABCD pe6pa DA, DB H DC IIOilapHO IIepIIeH.n;HKYJlSlpHbI, aAB = BC =AC = 10.
a) �OKa.iKHTe, qTO 3Ta IIHpaMH,ZJ;a IIpaBHJIbHaH.
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3
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2
(22-x2- 1 )
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BAPVlAHT 36 183
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[ill
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HMeeT pOBHO O�HH KopeHb Ha OTpe3Ke [0; 1 ] .
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c�eJIaJIa m cpoTorpacpHH, a HaTama -
n cpoTorpacpHH. B KajK�bIH CJIe�YlOm;HH �eHb
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H3BeCTHO, "({TO HaTama 3a BCe BpeMH c�eJIaJIa CYMMapHO Ha 1131 cpoTorpacpHIO 60JIbme,
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a) MorJIH JIH OHH cpOTorpacpHpOBaTb B Te"({eHHe 13 �HeH?
6) MorJIH JIH OHH cpoTorpacpHpoBaTb B Te"({eHHe 12 �HeH?
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"tIHCna HnH KOHe"tlHOH �eCHTH"tIHOH �p06H.
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1t
a) -
12+ 1tk, r,n;e k E Z;
6 ) _
131t.
1t 111t
12'
-
12;
12
45°
( -1; 0); log 5 3
8
54 925 py6JIeii
14
15
16
17
18
19
[ 0; 1,5); [2; + 00 )
a) ,n;a; 6) HeT; B) 16
@ 2021. 000 «M3,n;aTeJIbCTBO «Han;HOHaJIbHOe o6pa30BaHHe». @ 2021. qOY Ano «MU;HMO»KOIIIIpoBaHHe, pacrrpOCTpaHeHHe II HCrrOJIb30BaHHe B KOMMep-qeCKHX n;eJUIX 6e3 rrHCbMeHHoro pa3pemeHH.H rrpaBoo6JIa,ZJ;aTeJIeH He ,n;orrYCKaeTC.H
188 Ers. TVinOBblE SK3AMEHAL.J.VlOHHbIE BAPlt1AHT�
BapHaHT 9
.NQ aa)];aHHJI OTBeT
1 4
2 200
3 4
4 0,44
5 3
6 60
7 4
8 18
9 4
10 6,5
11 6,4
12 -21
1t 1t 1t 1t 1t
a) -k -+-k --+-k r,n;e6
'
18 3' 18 3'
k E 7l.,
136)
131t. 21t. 111t._
1t.
-
18'
-3'-
18'
2'
71t 1t 51t. .
---
18'
-
3'
18
14 36 + 30.J2
15 [2;5)
16 1
17 126 694,4 py6ReH
181 1
a � -3; a = -
-; a = 0; a = -; a � 33 3
19 a) ,n;a; 6) HeT; B) 2805
BapHaHT 10
.NQ aa)];aHHJI OTBeT
1 8
2 9
3 2,5
4 0,18
5 4
6 64
7 14
8 4 I
I
9 8
10 9,6
11 22
12 -8
1t
k51t 1t 51t 1t
a)-
, -+-k, --+-k, r,n;e4 24 2 24 2
k E 7l;
13
6)651t
.
111t.
671t. 31t.
771t.
131t.
24' 4' 24',
24' 4'791t-
24
145 .J119
13
15 (0;5]16 50
1-H 06'beKT - 7 -qeROBeK,
17 2-H 06'beKT - 23 -qeROBeKa;
43 150 py6ReH
18 4<a�16
19 a) ,n;a; 6) HeT; B) 2220
@ 2021. 000 «H3,n;aTeJIbCTBO «HaU;HOHaJIbHOe o6pa30BaHHe». @ 2021. qOY )J;rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 IIHCbMeHHoro pa3peIIIeHHH IIpaBOo6JIa,z:J;aTeJIeH He ,n;OIIYCKaeTCH
OTBETbl 189
BapHaHT 11
eM 3a)J;aHHH OTBeT
1 540
2 6
3 28
4 0,3
"5 2
I
i 6 6,5
7 2
8 54
9 -10
10 25
11 54
12 8
a)1t
nE7l;--+21tn4
'
1331t
mE7l;--+21tm4
'
191t 171t
6)--. --
4,
4
1431 .J10
arccos
140
15 [-2; 2)
16 9�
17 39
18-2<a<-1; -1<a<0; 0<a<3;
3 < a < 8; a > 8
19 a) HeT; 6) HeT; B) 676 r
BapHaHT 12
.M 3a,IJ;aHHH OTBeT
1 2640
2 26
3 27
4 0,34
5 -2
6 30
7 3
8 27
9 91
10 17
11 12
12 -9
a)51t
nEZ.:I - + 21tn,
6,
13
6)71t
--
6
145�
8
15 (-2; 1); (1; 2)
16 27.J3
17 1,6 MJIH py6JIeii
18a < 0; 0 < a < 3; 3 < a < 4;
4 < a < 5; 5 < a < 6
19 a) HeT; 6) HeT; B) 240 r
@ 2021. 000 «M3,n;aTeJIbCTBO «HaU;HOHaJIbHOe o6pa30BaHHe». @ 2021. "tIOY �IIO «MU;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJI.HX 6e3 IIHCbMeHHoro pa3pemeHH.H IIpaBOo6JIa,n;aTeJIeii He ,n;OIIYCKaeTC.H
190 Er3. TV1nOBbIE 3K3AMEHAL..J.V10HHbIE BAPV1AHTbI
BapuaHT 13
eMOTBeT
3a�aHHJI
1 4320
2 0,3
3 13,5
4 0,4
5 -5
6 72,5
7 3
8 47
9 65
10 8
11 48
12 26
1t 1t
nEZ.a) -+-n
4 2' ,
13
6)211t
.
231t 251t.
-
-, -,
4 4 4
14 13�
15 [- lOg�'5
6; lOg�'5
6
]16 5:7
17 2,58
(2�-1
._
.J1O
-1)u( _
.JI0 -l.-I}
182
'
2 2"
3 1.
-4' 2:
19 a) HeT; 6) HeT; B) 3
BapuaHT 14
eM 3a�aHHJI OTBeT
1
2
3
4
5
6
7
8
9
10
11
12
18000
2420
6
0,556
6
68
6
76
16
633
64
-1
13
1t
a) + -+1tn, nEZ;3
101t 81t.
71t
6)-
3;-
3' 3
48,5
[ -logl,25 : ; -1]
14
15
16
17
10 : 11
4,05
18..J10 + 1
-
9
.
..J10 -1.
[ 1 4. 2 ),
9",
19 a) �a; 6) HeT; B) 5
@ 2021. 000 «H3�aTeJIbCTBO «(Han;HOHaJIbHOe o6pa30BaHHe». @ 2021. QOY �IIO «(MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMep'tleCKHX n;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIp8.Boo6JI�aTeJIeH He �OIIYCKaeTCH
OTBETbl 191
BapuaHT 15
eM 3a�aHHJI OTBeT
1 84
2 485
3 26
4 0,0595
5 -2
6 21
7 0,5
8 200
9 7,5
10 0,31
11 20
12 9
n n
nEZ;a) nn, - + -n, r.n;e
134 2
_
13n, -3n, _
11n6)
4 4
14 4.J3
15 [ 3 - v'5; 2,8] u [ 3, 2; 3 + v'5]
16 1:3:1
17 20
18 [-3; 22]
19 a) .n;a; 6) 180; B) 546
BapuaHT 16
eMOTBeT
3a�aHHJI
1 13
2 960
3 31,5
4 0,973
5 -5
6 35
7 5,5
8 88
9 2,5
10 1,728
11 756
12 30
1t 1t
a) + - + 1tn,
- + 1tn,
nEZ.
3 2,
13
141t 9n 13n
6)-- -- --
3,
2'
3
14 6.J3
15 ( -00; 4 -
2.J2] u [ 3, 5; 4 ) u ( 4; 4,5] u [ 4 + 2.J2; + 00 )
16 4:5:4
17 3
18 (-00; -3); (-3; -1]; 1; [3,25; +(0)
19 a) .n;a; 6) 270; B) 17
@ 2021. 000 «H3�aTeJIbCTBO «Han;HOHaJIbHOe o6pa30BaHHe». @ 2021. qOY )J;rro «MIJ;HMO»KOIIHpoBaHHe, paCilpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX n;eJISlX 6e3 IIHCbMeHHoro pa3pellIeHHSI IIpaBOo6JIa,n;aTeJIeH He �OIIYCKaeTCSI
192 Er3. TV1nOBbIE 3K3AMEHAL..J.V10HHbIE BAPV1AHTbI
BapuaHT 17
eM 3a�aHHJI OTBeT
1 6
2 1,2
3 53
4 0,2
5 1,5
6 13
7 6
8 10,5
9 1
10 0,32
11 5
12 -81
13a) -10g23; 10g2 1 ,5;
6) 10g2 1 ,5
145M
2
15 [- � ; O)U(2; 3]
16 44
17 7 H 12 MJIH py6.
18 [-2;-
: )U[2; 4)
19 a) �a; 6) �a; B) 20
BapuaHT 18
eM 3a�aHHJI OTBeT
1 196
2 2,5
3 240
4 0,875
5 0,75
6 165
7 3
8 216
9 -1
10 60
11 3
12 1,5
13a) -log 52; -10g5 1O ;
6) -10g5 1O
14 5.J5
15 [2; +(0) HJIH -3
16 13,5
17 7 H 3 MJIH py6.
18 ( -00;-
3] u (0; 3]
19 a) �a; 6) �a; B) 10
@ 2021. 000 «lla�aTeJIbCTBO «Han;HOHaJIbHOe 06paaOBaHHe». @ 2021. qOY )J;rro «MIJ;HMO»KOIIHpoBaHHe, paCilpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX n;eJISlX 6ea IIHCbMeHHoro paapellIeHHSI IIpaBOo6JIa,n;aTeJIei He �OIIYCKaeTCSI
:-aETbl l�-�
BapHaHT 19
M 3a�aHHJI OTBeT
1 26
2 -11
3 20
4 0,09�
-15
6 2
7 11
8 96
9 9
10 0,006
11 7
12 -8
n
arctg 4 + nn ,a) -+nn, r.n;e n E Z ;
134
6)13n
arctg 4 + 3n-,
4
14 1
15 [-1; +(0)
16 12
17 5,35
a < -6; -6 < a < -2;
18 1-2 < a < O. 0 < a <
-
,
4
19a) HeT; 6) .n;a; B) 9 x 16 Ii 12 x 12;
3 x 12 Ii 6 x 6; 1 x 16 Ii 4 x 4
BapHaHT 20
eM 3a�aHHJI OTBeT
1 34 500
2 9
3 6
4 2,5
5 -2
6 10
7 7
8 111
9 10
10 120
11 14
12 -18
n 1a) -+nn, arctg- + nn , r.n;e n E Z ;
13 4 4
6)11n 1
--, arctg-- 3n
4 4
14 .J2
15 [-3; 1)
16 9
17 2 Ii 5 MJIH py6.
17--<a<-2. -2 < a < 2.
4' ,
1817
2<a<-4
19a) HeT; 6) .n;a; B) 4 x 9 Ii 6 x 6;
2 x 8 Ii 4 x 4; 1 x 9 H 3 x 3
@ 2021. 000 «Ha�aTeJIbCTBO «Han;HOHaJIbHOe o6paaOBaHHe». @ 2021. qOY )J;rro «MIJ;HMO»KOIIHpoBaHHe, paCilpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX n;eJISlX 6ea IIHCbMeHHoro paapellIeHHSI IIpaBOo6JIa�aTeJIeH He �OIIYCKaeTCSI
194 Ers. TV1nOBbIE SK3AMEHAUV10HHbIE BAPV1AHTbI
BapuaHT 21
M 3a�aHHH OTBeT
1 6670
2 16
3 5
4 0,26
5 -8,25
6 86
7 -2
8 24
9 81
10 62
11 60
12 31
n n 2na) -+-n :t- + 2nk, r.n;e n E Z;
134 2
'
3
6)13n 7n 11n
--
,
-2'--
4 4
314 arCSIn
.JPt17
15 (21tk; � + 21tk], [5
6
1t
+ 21tk; 1t + 21tk ).r.n;e k E Z
162.J3 + 3
3
17 5 000 000 py6JIeii
4a>l18 O<a<-
9'
19 a) .n;a; 6) HeT; B) 7
BapuaHT 22
M 3a�aHHH OTBeT
1 11
2 15
3 8
4 0,48
5 -1,8
6 103
7 7
8 39
9 -20
10 58
11 78
12 13
a) nk ,
n
:t-+2nk, k E Z;
133
6) 4n,
5n ,
13n-
3
14 "JI!arCSIn -
46
15 [_2: + 21tk- 2: + 21tk] r.n;e k E Z
3'
3'
16 8.J3
17 5 000 000 py6JIeii
4
� �18 -<a< -- -<a<l9 3' 3
19 a) .n;a; 6) HeT; B) 6
@ 2021. 000 «lla�aTeJIbCTBO «Han;HOHaJIbHOe o6paaOBaHHe». @ 2021. T.JOY )J;rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX n;eJISlX 6ea IIHCbMeHHoro paapellIeHHSI IIpaBOo6JIa�aTeJIeH He �OIIYCKaeTCSI
JTBETbl 195
BapuaHT 23
.� 3a�aHHH OTBeT
1 26 950
2 1678
3 11
4 0,25
5 17
6 73
7 7
8 72
9 27
10 24
11 14
12 6
a) nk, k E Z;
+n 2 n E Z;13 -"6+ nn,
6)23n
4n3n; 6;
14 arccos �15 (�; +(0)
16 30
17 3
18 (-3; -1)
19 a) HeT; 6) �a; B) 1347
BapuaHT 24
eM 3a�aHHH OTBeT
1 24,2
2 4
3 16
4 0,15
5 4
6 28
7 4
8 13
9 16
10 44
11 65
12 40
a) nk, k E Z;
n
13 I4"+ 2nn, n E Z;
6) 2n;9n
3n.
4'
14 arctg 2.J2
15 ( -3; -1] u [ 9; + 00 )
16 16
17 1 200 000 py6ReH
18 (-00; 0); (16; + 00 )
19 a) HeT; 6) �a; B) 989
@ 2021. 000 «lla�aTeJIbCTBO «Han;HOHaJIbHOe o6paaoBaHHe». @ 2021. qOY )J;rro «MIJ;HMO»KOIIHpoBaHHe, paCilpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX n;eJISlX 6ea IIHCbMeHHoro paapemeHHSI IIpaBOo6JIa�aTeJIeH He �OIIYCKaeTCSI
196 Ers. TV1nOBbIE SK3AMEHAL..J.V10HHbIE BAPV1AHTbl
BapuaHT 25
eM 3a�aHHJI OTBeT
1 19
2 4
3 4
4 0,76
5 3
6 15
7 6
8 4,8
9 10
10 3
11 13
12 -5
a) +7t 2 n E Z;-3
+ 7tn,
13
6)77t
3
141
arccos
.J3333
15 ( -00; � J [ 1; ; + 00)16 67,5
17 411 000 py6JIeii
18 -2 < a < -1; 1 < a < 2
19 a) ,n;a; 6) HeT; B) 26
BapuaHT 26
eM 3a�aHHJI OTBeT
1 188
2 9
3 3
4 0,28
5 87
6 17
7 13
8 61
9 63
10 0,87
11 13
12 -3
a)7t 7tk
-+- kEZ;
1312 3'
6)7t 57t 37t
. .-
12' 12' 4
149
7t-
arccos-
11
15 ( -00; 1 J; [3, + 00 )
16 42 + 18.J5
17 9 282 000 py6JIeii
18 -J6 < a < -J2; J2 < a < J6
19 a) HeT; 6) HeT; B) 16
@ 2021. 000 «ll3�aTeJIbCTBO «Han;HOHaJIbHOe 06pa30BaHHe.). @ 2021. qOY )J;rro «MIJ;HMO.)KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX n;eJISlX 6e3 IIHCbMeHHoro pa3perneHHSI IIpaBOo6JIa,n;aTeJIeii He �OIIYCKaeTC.H
DTBETbl 197
Ba.puaHT 27
M 3a�aHHJI OTBeT
1 145
2 650
3 4
4 0,2
5 5
6 21
7 4
8 72
9 12
10 17
11 11
12 -36
a)1t 1t
-+21tk, :t- + 21tk, k E Z ;
136 4
6)231t 151t 171t
--, --,--
6 4 4
14 1,5�
15 (O;�]
16 5
17 1 300 000 py6Rei
18 (-0,5; 1-�); (1+�; +(0)
19 a) HeT; 6) HeT; B) 11,75
BapuaHT 28
eM 3a�aHHJI OTBeT
1 180
2 44,4
3 4
4 0,25
5 58
6 56
7 1,6
8 32
9 64
10 20
11 15
12 5
a)1t
- - + 21tk k E Z ·
3' ,
13
6)171t-
3
14 8�1t
15 [0; 2]
16 7
17 2
18 (-00; - 3 )
19123
a) HeT; 6) HeT; B)-
11
@ 2021. 000 «H3�aTeJIbCTBO «Han;HOHaJIbHOe o6pa30BaHHe». @ 2021. qOY )J;rro «MIJ;HMO»KOIIHpoBaHHe, paCilpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMep'tleCKHX n;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JI�aTeJIeH He �OIIYCKaeTCH
198 Ers. TV1nOBbIE SK3AMEHAL..J.V10HHbIE BAPV1AHTbl
BapuaHT 29
M aaAaHHH OTBeT
1 37,5
2 68,2
3 2,5
4 0,3125
5 5,3
6 7
7 1
8 13
9 3
10 1200
11 75
12 7
n
a) -+2nn, n E Z;2
3n-arccos-+2nm, mE Z;
134
3arccos- + n + 2nk, k E Z;
4
6)3 5n
n + arccos-;-4 2
14 .�arCSIn -
46
15 0; (1; log23)
16 30
17 6
18(-00; 7 -
2v'6J; [7 + 2v'6; 15);
(15; + 00 )
19 a) �a; 6) HeT; B) 6!7
BapuaHT 30
M aaAaHHH OTBeT
1 61
2 420
3 5
4 0,096
5 3
6 10
7 6
8 18
9 3
10 4974
11 27
12 -1
1 1a) -
.
, ,
133 3�
6)' !3
14 5
15 (-00; 0 J; [ � ; + 00 )16 7
17 125 000 py6JIeii
18 b=-1, b > O
19 a) HeT; 6) �a; B) 6(310
- 210)
@ 2021. 000 «ll3,n;aTeJIbCTBO «Han;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY )J;rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX n;eJISlX 6e3 IIHCbMeHHoro pa3pemeHH.H IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTC.H
_ 3ET bl 199
BapuaHT 31
.� aa,lJ;aHHJI OTBeT
1 611
2 1500
3 3,5
4 0,995
5 2
6 38
7 5
8 84
9 -7
10 42
11 18
12 -5
a) -1t + 21tk, k EZ ;
1t
13:t-+ 21tn, n EZ;
3
6)51t
--.-1t
3'
14 17 : 127
15 (0;1); 9; (27; + (0)
16 71°
17 20
15 1 8 1518 --<as--; -sa<-
7 2 7 7
19 a) �a; 6) HeT; B) 6
BapuaHT 32
eM aa,lJ;aHHJI OTBeT
1 186
2 56
3 20
4 0,996
5 2
6 2
7 6
8 48
9 -0,56
10 100
11 20
12 -6
a)1t
--+21tk, k EZ;2
1t
--+21tm m EZ.6
' ,
1351t
--+21tn, n EZ;6
6)71t 231t
.--
2'
6
14 7 : 11
15 (0;1); 2; (8; + (0)
16 78°
17 10
12 1 7 1218 --<as--; -sa<-
5 2 5 5
19 a) �a; 6) HeT; B) 5
@ 2021. 000 «ll3�aTeJIbCTBO «Han;HOHaJIbHOe o6pa30BaHHe�. @ 2021. qOY ,l];rro «MU;HMO.KonHpoBaHHe, pacnpOCTpaHeHHe H HCnOJIb30BaHHe B KOMMep'tleCKHX n;eJIHX 6e3 nHCbMeHHoro pa3pellIeHHH IIpaBOo6JI�aTeJIeH He �onYCKaeTCH'
200 Er3. Tlt1nOBbIE 3K3AMEHAU.lt10HHbIE BAPlt1AHTbl
BapuaHT 33
M aa,l];aHHJI OTBeT
1 31140
2 3
3 26
4 0,01
5 5
6 4,5
7 4,5
8 24
9 -63
10 2642
11 250
12 4
a)n n
-+nk, --+21tk,2 6
135n
kE'll.--+2nk6
' ,
6)7n 17n 5n
--
,--
,--
2 6 2
14�26
arctg-4
15 (-5;-
2v'5J; 0; [2J5i 5)
16 6�
17 12,5
18 [ -1,5; -
0,375); 0
19 a) 42; 6) IIOJIOmHTeJIbHbIX; B) 24
BapuaHT 34
M aa,l];aHHJI OTBeT
1 28
2 -21
3 5
4 0,09
5 3
6 $ 40
7 6
8 9
9 -22
10 2408
11 48
12 32
a) _
3; 4;
2
21t13 + - + 21tk, k E Z ;
3
3 21t6) --; -
2 3
14 6 ..J134
15 [- � ; ; ]
1650�
3
17 1 029 000
18 0; n ; 1 ]a) 60;
19 6) IIOJIOmHTeJIbHbIX;
B) 36
@ 2021. 000 «H3,n;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. QOY ,1];rrO «lVIIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJISIX 6e3 IIHCbMeHHoro pa3perneHHH IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH
OTBETbl 201
BapuaHT 35
M aa,l];aHHJI OTBeT
1 140
2 5
3 4
4 0,9
5 11
6 66
7 -1
8 500
9 -3
10 12
11 3
12 5
a)7t 7tk
k Z·
-+- E ,
136 3'
6)51t 1t 1t 1t
. . .-
-6' -2'-
6'
6
14 ..J166
15 G ; � } (8;32)
16 4
17 1 1 71 280
6 9--<a�O. a=-.
7'
7'18
3 24-<a<-
2 7
19 a) ,n;a; 6) HeT; B) 1632
BapuaHT 36
M aa,l];aHHJI OTBeT
1 200
2 6
3 6
4 0,2
5 13
6 130
7 1
8 1372
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9
2
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2cos2
x +2sinx. cosx = 0; cosx.( sinx + cosx) = O.
n
3Ha-qHT, HJIH cos X = 0, oTKy,n;a x = - + nk, k E Z, HJIH sin x = -cos x; tg x = -1;2
n
x=-
4+1tn, nEZ.
6) C IIOMOblO -qHCJIOBOH oKpymHocTH oT6epeM KOpHH,
rrpHHa,zJ;JIemaIIJ;He OTpe3KY [31t;9
2
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MOblO rpacpHKa, perneHHeM JIHHeHHbIx HepaBeHCTB H T. II.
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a 6oKoBoe pe6po SA paBHO 8. TOqKa M -
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rrJIOCKOCTb a B TOqKe K.
a) oKamHTe, qTO KM = KD.
6) HaH,l];HTe o6eM rrHpaMH,l];bI CDKM.
OD
PemeHHe.
a) IIycTb rrpHMbIe CF H MD rrepeceKalOTCH B TOqKe H
(pHC. 1), a SO -
BbICOTa rrHpaMHbI SABCDEF. IIoCKOJIb-
KY rrHpaMH,l];a SABCDEF rrpaBHJIbHaH, n;eHTp rrpaBHJIbHOrO
mecTHyrOJIbHHKa ABCDEF COBrra,n;aeT C TOqKOH O. 3HaqHT,
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CKOCTb SCF rreprreH,l];HKYJIHpHa rrJIOCKOCTH ABC.
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rrepeCeqeHHH rrJIOCKOCTeH SCF H a, rreprreH,l];HKYJIHpHa
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B TpeyrOJIbHHKe MKD.
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(pHC. 2). IIpHMbIe AB H CF rrapaJIJIeJIbHbI, a TOq-
Ka 0 Cepe,l];HHa OTpe3Ka AD, CJIe,l];OBaTeJIbHO, OTpe30K
OH -
Cpe,l];HHH JIHHHH TpeyrOJIbHHKa ADM H MH = HD.
TaKHM o6pa30M, OTpe30K KH HBJIHeTCH Me,l];HaHOH
H BbICOTOH B TpeyrOJIbHHKe MKD, 3HaqHT, 3TOT TpeyrOJIbHHK
paBH06e,l];peHHbIH H KM = KD.
6) B rrYHKTe a 6bIJIO ,l];OKa3aHO, qTO rrpHMaH KH rreprreH,l];H-
KYJIHpHa rrJIOCKOCTH ABC, CJIe,l];OBaTeJIbHO, OTpe30K KH HBJIH-
eTCH BbICOTOH rrHpaMH,l];bI CDKM.
s
A D
B c
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OH=2 =4=0,5; CH=OC-OH=AB-OH=1,5.
B TpeyrOJIbHHKe SOC HMeeM:
so= .Jsc 2-OC
2= "'64-4 =2.JI5. KH =
CH.SO=
1,5.2.JI5=
3.JI5.,
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4
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4
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2
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x2
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3HaqeHHH IIPH x < 1.
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-12 OTpHaTeJIbHO, a 3HaqHT, JII060e 3Haqe-
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BaH OrpaHHQeHHe x < 1,5, IIOJIyqaeM x < - 2.J3.
TaKHM o6pa30M, perneHHe HCXO,l];HOrO HepaBeHCTBa: x < - 2.J3; 1 < x < 1,5.
OTBeT: (- 00;-
2.J3]; [1; 1,5).
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paBH06e,n;peHHoro npHMoyrOJIbHOrO TpeyrOJIbHIIKa ABC C npHMbIM yrJIOM C JIemaT
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a) oKamIlTe, qTO npHMbIe AD II BE napaJIJIeJIbHbI.
6) HaH,n;IITe AC, eCJIII pa,n;IIYCbI oKpymHocTeH paBHbI 3 II 4.
PemeHHe.
a) IIycTb CL -
06m;aH KaCaTeJIbHaH ,n;BYX oKpymHocTeH,
npllqeM TOqKII L II B JIemaT no O,n;HY CTOPOHY OT npHMoH AC.
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LCAD= LLCB = LCEB.
3Haq1lT, npHMbIe AD II BE napaJIJIeJIbHbI, nOCKOJIbKY COOT-
BeTCTBeHHbIe yrJIbI CAD II CEB npll nepeCeqeHIIII 3TIIX npH-
MbIX npHMoH AE paBHbI.
6) IIoCKOJIbKY yrOJI ACB npHMoH, AD II BE -
,n;lIaMeTpbI
MeHbIIIeH II 60JIbIIIeH oKpymHocTeH COOTBeTCTBeHHO.
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4 4
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9x2
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5
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2
1t
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3
1t
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x; sin ( 2x+ 2; )eos( 4x+ ; ) = 1.
B03MO:>KHbI Ba CJIyqaH:
sin(2X+2
3
1t
) =1,HJIH
eos( 4x+ ; ) = 1
B rrepBOM CJIyqae:
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2x + - == - + 21tk3 2
'
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4x + - == 21tn;3
1t
2x == -- + 21tk,6
1t
4x == -- + 21tn;3
1t
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12'M
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2x + - == -- + 21tk3 2
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4x+- == 1t+21tn;3
CHCTeMa perneHHH He HMeeT.
71t2x==--+21tk
6'
21t4x == - + 21tn;
3
1t
IIOJIyqHJIH, "({TO X == -- + 1tk, r,n;e k E Z.12
sin(2X+2
3
1t
) =-1,
eos( 4x+ ; ) =-1.
1t
x ==--+1tk12
'
,
1t 1t
x==--+-n
12 2
re kEZ, nEZ.
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6 )13n
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.
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H ADDI. Al K DIIIJIOCKOCTH BMK H ADD
I lIepeceKalOTCH 110 IIpH-
MOH KM, IIpHqeM IIpHMbIe MK H BK lIeplleH,ZJ;HKYJIHpHbI
(CM. IIYHKT a). IIoCTpoHM IIPHMYIO KZ B IIJIOCKOCTH ADDI ,
lIeplleH,ZJ;HKYJIHPHYIO IIpHMOH KM (pHC. 2). TpeyrOJIbHHKH
KMDI
H ZKA 1I0,ZJ;06HbI 110 ,ZJ;BYM yrJIaM.
A Z KT) .J3..J33HaqHT
1 .LJ].
OTKY,ZJ;a A Z == == 3.,
AIK MDtI
1
CJIe,ZJ;oBaTeJIbHO, TOqKH A H Z COBIIMaIOT, a AK lIep-
lIeH,ZJ;HKYJIHpHO KM.
IIoJIyqHJIH, qTO yrOJI AKB - JIHHeHHbIH yrOJI ,ZJ;ByrpaHHoro yrJIa Mem,ZJ;y IIJIOCKOCTH-
MH BMK H ADDI.
ll3 npHMoyrOJIbHOrO TpeyrOJIbHHKa ABK nOJIYYIaeM, qTO tg AKB = ::; = �� = 1,
M
Z
A D
Puc.2
a 3HaqHT yroJI Mem,ZJ;y IIJIOCKOCTHMH BMK H BCCI paBeH 45°.
OTBeT: 45°.
COAep3taHHe KpHTepHH BaJIJILI
MMeeTCH BepHoe �OKa3aTeJIbCTBO YTBepm�eHHH rrYHKTa a H 060CHOBaHHO rrOJIyqeH2
BepHbIH OTBeT B rrYHKTe 6
MMeeTcH BepHoe �OKa3aTeJIbCTBO YTBepm�eHHH rrYHKTa a,
MJIM1
06ocHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6, B03MomHO, C HCrrOJIb30BaHHeM
YTBepm�eHHH rrYHKTa a, rrpH aTOM rrYHKT a He BbIrrOJIHeH
PemeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aKCUMa.nbllblU 6a.n.n 2
� p6.5
x-ll
>025�
eIIIHTe HepaBeHCTBO25
x + O ,5 -6.5x +1-,
·
PemeHHe.
24t-44-5t2
+6t-1>0.
4(5t 2 -6t+1)-,
3HaqHT t E ( � ; 5) HJIH t = 3.
1
IIoJIyqaeM:- < 5
x< 1, OTKY,ZJ;a x E ( -1; 0),
5
IIycTb t = 5\ Torp;a HepaBeHCTBO npHMeT BHP;:
6t -11 _! > 05t
2-6t+1 4
- ·
t2
-6t+9< 0;
(t-3)2< 0,
5t2
-6t+1 (t-l)(5t-1)
HJIH 5x
== 3, OTKY,ZJ;a x == log5 3.
OTBeT: (-1; 0); log53.
@ 2021. 000 «Ma,n;aTeJIbCTBO «HaU;MOHaJIbHOe o6paaoBaHMe». @ 2021. qOY )];rro «MIJ;HMO»KonMpoBaHMe, pacnpOCTpaHeHMe M MCnOJIbaOBaHMe B KOMMepqeCKMX u;eJI,flX 6ea nMCbMeHHoro paaperneHM,fl npaBoo6JIa,z:I;aTeJIeM He ,n;orrYCKaeTC,fl
PEWEHII1s:1 111 KPII1TEPII1I11 OLJ.EHII1BAHII1s:1 3AAAHII1� 13-19 213
CO�ep3taHHe KpHTepHH BaJIJILI
06ocHoBaHHo IIOJIyqeH BepHhIH OTBeT 2
06ocHoBaHHo IIOJIyqeHu
OTBeT, OT JIHqalOIIJ;HHCH OT BepHoro HCKJIlOqeHHeM TOqKH
log 5 3,
HJIH 1
IIOJIyqeH HeBepHhIH OTBeT H3-3a BhIqHCJIHTeJIhHOH omH6KH, HO IIPH aTOM HMeeTCH
BepHaH IIOCJIe�OBaTeJIhHOCTh Bcex marOB pemeHHH
PemeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, IIepeqHCJIeHHhIX BhIme 0
M aKCUMa.nbllblU 6a.n.n 2
Ha cTopoHax AC, AB H BC IIpHMoyrOJIbHOrO Tpeyr"OJIbHHKa ABC C IIpHMbIM yrJIOM C BHe
TpeyrOJIbHHKa ABC IIocTpoeHbI paBHo6e,n;peHHbIe IIpHMoyrOJIbHbIe TpeyrOJIbHHKH AKC,
ALB H BMC C IIpHMbIMH yrJIaMH K, L H M COOTBeTCTBeHHO.
a) ,n:oKamHTe, qTO LC -
BbICOTa TpeyrOJIbHHKa KLM.
6) HaM,n;HTe IIJIOm;a,n;b TpeyrOJIbHHKa KLM, eCJIH LC = 4.
a) lIo YCJIOBHIO LACB + L.BLA = 90° + 90° = 180°.
3HaqHT, qeTbIpexyrOJIbHHK LACB BIIHcaHHbIM.
XOp,n;bI AL If LB OIIHcaHHOM OKOJIO qeTbIpex-
yrOJIbHHKa LACB oKpymHocTH paBHbI. 3HaqHT,
paBHbI Mem,n;y C060M CTHrHBaeMbIe 3THMH xop,n;a-
MH ,n;yrH, a TaKme OIIHpalOm;HeCH Ha 3TH ,n;yrH
BIIHcaHHbIe yrJIbI ACL H LCB. Tor,n;a
LACL = L.LCB = 45°.
lIo YCJIOBHIO L.KCA = L.MCB = 45°. CJIe,n;oBa-
TeJIbHO, L.KCL = L.KCA + LACL = 90° H L.MCL =
= L.MCB + L.LCB = 90°, a 3HaQHT, LC -
BbICOTa
TpeyrOJIbHHKa KLM.
6) lIycTb BC=a, \AC=b, AB=e H CL=d, P -
TOQKa IIepeCeQeHHH CL H AB. Tor,n;a IIO ,n;OKa3aHHOMY B IIYHKTe a) CP 6HCCeKTpHCa Tpe-
yrOJIbHHKa ABC. lIo CBOMCTBY 6HcceKTpHcbI AP: P B = AC : CB = b : a, AP + P B = AB = e.
be ae
OTclO,n;a AP = If P B =.
a+b a+b
lIoCKOJIbKY LACL = L.BAL = 45°, IIOJIYQaeM, QTO TpeyrOJIbHHKH ACL H PAL IIo,n;o6HbIAC CL be e a+b
IIO ,n;BYM yrJIaM. OTclO,n;aPA
=
AL'
b:a+b
=d:.J2
H d=.J2
.
lIJIom;a,n;b TpeyrOJIbHHKa KLM paBHa IIOJIOBlfHe IIpOH3Be,n;eHHH ero BbICOTbI LC = d
a bHa OCHOBaHHe KM = KC + CM =
.J2+
.J2= d. CJIe,n;oBaTeJIbHO, HCKOMaH IIJIOm;a,n;b paBHa
[K]
PemeHHe.
d2
-=8.2
OTBeT:8.
K
M
@ 2021. 000 «Ma,n;aTeJIbCTBO «HaU;HOHaJIbHOe 06paaOBaHHe». @ 2021. QOY �rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX u;eJIHX 6ea IIHCbMeHHoro paaperneHHH IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH
214 Er3. TlI1nOBblE 3K3AMEHALJ.1I10HHbIE BAPII1AHTbl
COAep3taHHe KpHTepHH BaJIJILI
MMeeTCH BepHoe �OKa3aTeJIbCTBO YTBepm�eHHH rrYHKTa a, H 060CHOBaHHO rrOJIY-
qeH BepHbIH OTBeT B rrYHKTe 63
060CHOBaHHO rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6,
MJIM
HMeeTCH BepHoe �OKa3aTeJIbCTBO YTBepm�eHHH rrYHKTa a, H rrpH 06ocHoBaHHoM
pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a apHcpMeTHqeCKOH omH6KH
2
MMeeTCH BepHoe �OKa3aTeJIbCTBO YTBepm�eHHH rrYHKTa a,
MJIM
rrpH 06ocHoBaHHoM pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a apHcpMe-
THqeCKOH omH6KH,
MJIM
06ocHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6 C HCrrOJIb30BaHHeM YTBepm�eHHH
rrYHKTa a, rrpH aTOM rrYHKT a He BbIrrOJIHeH
1
PemeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme
M aKCUMa.nbHblU 6a.n.n
o
3
[E] AJ:reKcaH,ZJ;p XOqeT KYIIHTb IIaKeT aK�Hii 6bICTpopacTy�eii KOMIIaHHH. B HaqaJIe rO,ZJ;a
Y AJ:reKcaH,ZJ;pa He 6bIJIO ,ZJ;eHer Ha IIOKYIIKY aK�Hii, a IIaKeT CTOHJI 100 000 Py6JIeii.B cepe,ZJ;HHe Kam,ZJ;oro MeCH�a AJ:reKcaH,ZJ;p OTKJIa,ZJ;bIBaeT Ha IIOKYIIKY IIaKeTa aK�Hii O,ZJ;HY
H TY me CYMMY, a B KOH�e MeCH�a IIaKeT ,ZJ;opomaeT, HO He 60JIee qeM Ha 30 0/0. KaKYro
HaHMeHbIIIYro CYMMY HymHo OTKJI8.)J;bIBaTb AJ:reKcaH,ZJ;PY Kam,ZJ;bIii MeCH�, qTo6bI qepe3
HeKOTopoe BpeMH KYIIHTb meJIaeMbIii IIaKeT aK�Hii?
PemeHHe.
IIYCTb AJ:reKcaH,ZJ;p OTKJIa,ZJ;bIBaeT B cepe,ZJ;HHe Kam,ZJ;oro MeCH�a x Py6JIeii. K cepe,ZJ;HHe
n-ro MeCH�a Y AJ:reKcaH,ZJ;pa CKOIIHTCH nx Py6JIeii, a aK�HH 6Y,ZJ;YT CTOHTb He 60JIee
100 000 ·
1,3n
- 1Py6JIeii. ,IJ:JIH Toro qTo6bI AJ:reKcaH,ZJ;p CMor KYIIHTb IIaKeT aK�Hii
100 000.1, 3n - 1
B 3TOM MeCH�e, Heo6xo,ZJ;HMO, qTo6bI BbIIIOJIHHJIOCb HepaBeHCTBO x >. IIo-
n
100 000.1, 3n - 1
JIOmHM an=
. ,n:JIH Toro qTo6bI AJ:reKcaH,ZJ;p CMor qepe3 HeKOTopoe BpeMHn
KYIIHTb IIaKeT aK�Hii, Heo6xo,ZJ;HMO H ,ZJ;OCTaTOQHO OTKJIa,ZJ;bIBaTb CYMMY 60JIbIIIYro JIH60
paBHYro HaHMeHbIIIeMY H3 QHCeJI an. CpaBHHM ,ZJ;Ba IIOCJIe,ZJ;OBaTeJIbHbIX TaKHX QHCJIa.
an+l 1,3.n 13n,IJ:JIH 3Toro BbIQHCJIHM HX OTHOIIIeHHe:
-
. OTcro,ZJ;a IIOJIYQaeM, q,TO
an n+1 10n+10
IIPH n < 4 BbIIIOJIHeHO an+l < an, a IIPH n > 4 BbIIIOJIHeHO an+l > an.
100000.1,33
3HaQHT, HaHMeHbIIIHM H3 QHCeJI a 6Y,ZJ;eT QHCJIO a4= = 54 925.
n
4
II03TOMY HaHMeHbIIIaH CYMMa, KOTOPYro HymHo OTKJIa,ZJ;bIBaTb AJ:reKcaH,ZJ;PY, paBHa
54 925 Py6JIeii.
OTBeT: 54 925 Py6JIeii.
@ 2021. 000 «Ha,n;aTeJIbCTBO «Ha:U;HOHa.rrbHoe 06paaOBaHHe». @ 2021. qOY ,n;rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX :u;eJIHX 6ea IIHCbMeHHoro paaperneHHH IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH
PEWEHII1 111 KPII1TEPII1I11 OU.EHII1BAHII1 3AAAHII1 13-19 215
COep3taHHe KpHTepHH BaJIJILI
06ocHoBaHHo rroRyqeHu
3BepHbIH OTBeT
BepHo rrOCTpoeHa Ma TeMa THqeCKaH MO,ZJ;eRb, pemeHHe CBe,ZJ;eHO K HCCRe,ZJ;OBaHHIOu
aTOH MO,ZJ;eRH H rroRyqeH pe3YRbTaT:2
-
HeBepHbIM OTBeT H3-3a BbIqHCRHTeRbHoM omH6KH;-
BepHbIM OTBeT, HO pemeHHe He,ZJ;OCTaTOqHO 060CHOBaHO
BepHo rrOCTpoeHa MaTeMaTHqeCKaH MO,ZJ;eRb, pemeHHe CBe,ZJ;eHO K HCCRe,ZJ;OBaHHIO1
aTOM MO,ZJ;eRH, rrpH aTOM pemeHHe MomeT 6bITb He 3aBepmeHO
PemeHHe He COOTBeTcTByeT HH O,ZJ;HOMY H3 KpHTepneB, rrepeqnCJIeHHbIX BhIllie 0
M aKCUMaJtbHblU oaJtJt 3
Haii,ll;HTe, IIPH KaKHX HeC;>TpHD;aTeJIbHbIX 3Ha"'IeHHSlX a cl>YHKD;HSI f (x) = 3ax4
- 8x3
+ 3x2
-7
Ha OTpe3Ke [-1; 1] HMeeT pOBHO O,ll;HY TOqKY MHHHMYMa.
PemeHHe.
HaM,lI;eM IIpOH3BO,ll;HYIO <PYHKD;HH: f' (x) == 12ax3
- 24x2
+ 6x.
12ax3
- 24x2
+ 6x == 0; 6x(2ax2
-4x + 1) == O.
B TOqKe x == 0 IIpOH3BO,ll;HaH MeHHeT 3HaK C «-
» Ha « + », II03TOMY TOqKa x = 0 HBJIH-
eTCH TOqKOM MHHHMYMa.
<l>YHKD;HH f (x) = 3ax4
- 8x3
+ 3x2
-7 MomeT HMeTb em;e TOqKY MHHHMYMa, eCJIH ypaB-
HeHHe 2ax2
-4x+1==0 HMeeT ,lI;Ba KOpHH, a 3HaqHT, IIPH a<2.
1a) IIpH a = 0 ypaBHeHHe HMeeT ,lI;Ba KOpHH: x == 0 H X ==-.
4
T1
u
OqKa x ==
4HBJIHeTCH TOqKOM MaKCHMYMa.
6) IIpH a E (0; 2) ypaBHeHHe HMeeT TPH pa3JIH qHbIX KOpHH:
2- .J 4-2a 2+ .J 4-2aXl
== 0, X2== H X3
=, r,ll;e Xl < X2 < X3;
2a 2a
TOqKa x2
HBJIHeTCH TOqKOM MaKCHMYMa, a TOqKH Xl H x3
-
TOqKaMH MHHHMYMa. Toq-
2+ .J 4-2aKa x
3JIemHT Ha OTpe3Ke [-1; 1], eCJIM < 1, a 3TO BbIIIOJIHeHO IIPH Bcex a > 1,5.
2a
IIoJIyqHJIH: <PYHKD;HH f (x) == 3ax4
- 8x3
+ 3x2
-7 Ha OTpe3Ke [-1; 1] HMeeT O,ll;HY TOqKY
MHHHMYMa IIPH a E [ 0; 1,5) H a > 2.
OTBeT: [0; 1,5); [2; + 00 ).
@ 2021. 000 «Ma,n;aTeJIbCTBO «HaU;HOHaJIbHOe 06paaOBaHHe». @ 2021. qOY )];nO «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX u;eJIHX 6ea IIHCbMeHHoro paapemeHHH IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH
216 Er3. TlI1nOBblE 3K3AMEHAL.J.1I10HHbIE BAPII1AHTbl
Co.n;epmaHHe KpHTepHH BaJIJILI
060CHOBaHHO rroyqeH BepHbIH OTBeT 4
C rrOMOm;blO BepHoro paccym,n;eHHH rroy"tleHo MHomeCTBO 3Ha"tleHHH a, OTH"tIalO-
3m;eeCH OT HCKOMoro TObKO BKIO"tIeHHeM/HcKIO"tIeHHeM TO"tleK a = 1,5 H/HH a = 2
C rrOMOm;blO BepHoro paccym,n;eHHH rroy"tleHo MHomeCTBO 3Ha"tleHHH a, OTH"tIalO-
m;eeCH OT HCKOMoro TObKO HCKIO"tIeHHeM TO"tlKH a = 0, a TaKme, MomeT 6bITb,
BKIO"tIeHHeM/HcKIO"tIeHHeM TO"tleK a = 1,5 H/HH a = 2,2
MJIM
rroy"tleH HeBepHbIH OTBeT H3-3a BbI"tIHCHTebHoH omH6KH, HO rrpH aTOM BepHO
BbIrrOHeHbI Bce marH pemeHHH
BepHO HaH,n;eHbI Bce TPH rpaHH"tIHbIe TO"tlKH MHomeCTBa 3Ha"tleHHH a 1
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepe"tlHCeHHbIX BbIme 0
M aKCUMa.nbHblU 6a.n.n 4
I 19 I ,n:JIH Kam,n;oro HaTypaJIbHOrO qHCJIa n 0603HaqHM qepe3 nl rrpoH3Be,n;eHHe rrepBbIx n
HaTypaJIbHbIX qHCeJI (11 == 1).
a) Cym;ecTByeT JIH TaKoe HaTypaJIbHOe qHCJIO n, qTO ,n;eCHTHqHaH 3arrHCb qHCJIa nl
OKaHqHBaeTCH pOBHO 9 HYJIHMH?
6) Cym;ecTByeT JIH TaKoe HaTypaJIbHOe qHCJIO n, qTO ,n;eCHTHqHaH 3arrHCb qHCJIa nl
OKaHqHBaeTCH pOBHO 23 HYJIHMH?
B) CKOJIbKO cym;ecTByeT HaTypaJIbHbIX qHCeJI n, MeHbIIIHX 100, ,n;JIH Kam,n;oro H3 KOTOpbIX
,n;eCHTHqHaH 3arrHCb qHCJIa nl. (100-
n)1 OKaHqHBaeTCH pOBHO 23 HYJIHMH?
PemeHHe.
a) ,n:JIH n == 40 rrOJIyqaeM n I == 1. 2 .
. . .
. 40 ==
p.
q, r,n;e qHCJIO p-
3TO rrpOH3Be,n;eHHe Bcex
KpaTHbIX 5 HaTypaJIbHbIX qHCeJI OT 1 ,n;o 40, a q-
3TO rrpOH3Be,n;eHHe Bcex He KpaT-
HbIX 5 HaTypaJIbHbIX qHCeJI OT 1 ,n;o 40. Tor,n;a qHCJIO p==5.10.....40==5.(2.3.4.6.7.8),n;eJIHTCH Ha 59, HO He ,n;eJIHTCH Ha 5 10
, a qHCJIO q ,n;eJIHTCH Ha 16.32==29
, HO He ,n;eJIHTCH
Ha 5. 3HaqHT, qHCJIO nl ,n;eJIHTCH Ha 10 9,
HO He ,n;eJIHTCH Ha 10 10, H, CJIe,n;oBaTeJIbHO, ero
,n;ecHTHqHaH 3arrHCb OKaHqHBaeTCH pOBHO 9 HYJIHMH.
6) ,n:JIH n == 99 rrOJIyqaeM n I == 1.2.... . 99 ==
p.
q, r,n;e qHCJIO p-
3TO rrpOH3Be,n;eHHe Bcex
KpaTHbIX 5 HaTypaJIbHbIX qHCeJI OT 1 ,n;o 99, a q-
3TO rrpOH3Be,n;eHHe Bcex He KpaT-
HbIX 5 HaTYPaJIbHbIX qHCeJI OT 1 ,n;o 99.
Tor,n;a qHCJIO p==5.10.....95==519
.(1.2.....19)==522
.r, r,n;e qHCJIO r He ,n;eJIHTCH Ha 5.
CJIe,n;oBaTeJIbHO, qHCJIO p ,n;eJIHTCH Ha 522
, HO He ,n;eJIHTCH Ha 523
. IIpH 3TOM qHCJIO q
,n;eJIHTCH Ha 4.16.32.64.96 == 222
.3, HO He ,n;eJIHTCH Ha 5. 3HaqHT, qHCJIO nl ,n;eJIHTCH
Ha 10 22,
HO He ,n;eJIHTCH Ha 10 23, H, CJIe,n;OBaTeJIbHO, ero ,n;ecHTHqHaH 3arrHCb OKaHqH-
BaeTCH pOBHO 22 HYJIHMH. IIo3ToMY ,n;eCHTHqHaH 3arrHCb qHCJIa nl rrpH n < 99 He MO-
meT OKaHqHBaTbCH pOBHO 23 HYJIHMH, a rrpH n > 100 qHCJIO ,n;eJIHTCH Ha 1001 == 991 .100
H, CJIe,n;OBaTeJIbHO, ,n;eCHTHqHaH 3arrHCb qHCJIa nl OKaHqHBaeTCH 60JIee qeM 23 HYJIHMH.
CJIe,n;oBaTeJIbHO, TaKHX qHCeJI He 6bIBaeT.
B) ,n:JIH Kam,n;oro ,n;eHCTBHTeJIbHOrO qHCJIa x 0603HaqHM qepe3 [X] HaH60JIbIIIee :u;e-
JIoe qHCJIO, He rrpeBOCXO,n;Hm;ee x. Tor,n;a ,n;JIH JII06oro HaTypaJIbHOrO qHCJIa m H JII06oro
@ 2021. 000 «Ma,n;aTeJIbCTBO <tHa:U;HOHaJIbHOe 06paaOBaHHe)). @ 2021. qOY ,n;rro <tMIJ;HMO))KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX :u;eJIHX 6ea IIHCbMeHHoro paapellIeHHH IIpaBOo6JI8.,lJ;aTeJIeH He ,n;OIIYCKaeTCH
218 Er3. TV1nOBbIE 3K3AMEHAlJ.V10HHbIE BAPV1AHTbl
BapuaHT 11
[ill a) PeIIIHTe ypaBHeHHe COS 2x -.J2 cos ( ; + x ) + 1 = O.
6) ¥KamHTe KOpHH 3Toro ypaBHeHHH, npHHa,n;JIemaru;He OTpe3KY [- 51t; -
7; 1PemeHHe.
a) 3arrHIIIeM HCXO,ll;HOe ypaBHeHHe B BH,lI;e:
1- 2 sin2
x + .J2 sin x + 1 == 0; (2 sin x + .J2 ) ( sin x-
.J2) == O.
3HaYIHT, sin x = -
' oTKy,n;a x = -
: + 21tn, n E Z, HJIH X = -
3
4
1t
+ 21tm, m E Z .
Y paBHeHHe sin x ==.J2 KopHeH He HMeeT.
6) C rrOMOID;blO qHCJIOBOH oKpymHocTH oT6epeM KOpHH,
npHHa,n;JIemaru;He OTpe3KY [- 51t; -
7
2
1t
].19n 17n
IIoJI y qHM qHCJIa - -
-4' 4
71t--
2
-51t
19n6) -
4'
n
--+2nn,4
17n--
4
3nn E Z. - - + 2nm, m E Z ·
,
4'
OTBeT: a)
CO,ZJ;ep3taHHe KpHTepHH BaJIJILI
060CHOBaHHO rrOJIyqeHbI BepHble OTBeTbI B 060HX rrYHKTax 2
060cHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe a,
MJIM
rrOJIyqeHbI HeBepHble OTBeTbI H3-3a BblqHCJIHTeJIbHOH omH6KH, HO rrpH aTOM 1
HMeeTCH BepHaH rrOCJIeOBaTeJIbHOCTb Bcex marOB pemeHHH 060HX rrYHKTOB:
rrYHKTa a H rrYHKTa 6
PemeHHe He COOTBeTcTByeT HH OHOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX Bblme 0
M a KCU.JVl a.n bll blU 6a.n.n 2
@JB rrpaBHJIbHOH TpeyrOJIbHOH rrHpaMH,lI;e SABC cTopoHa OCHOBaHHH AB paBHa 6, a 6oKoBoe
pe6po SA paBHO 7. Ha pe6pax AB H SC OTMeqeHbI TOqKH K H M COOTBeTCTBeHHO,
rrpHqeM AK : KB == SM : MC == 1 : 5. IIJIOCKOCTb a CO,ll;epmHT rrpHMYIO KM H rrapaJIJIeJIbHa
rrpHMoH BC.
a) ,II;oKamHTe, qTO rrJIOCKOCTb a rrapaJIJIeJIbHa rrpHMoH SA.
6) HaH,lI;HTe yrOJI Mem,ll;y rrJIOCKOCTHMH a H SBC.
@ 2021. 000 «Ma,n;aTeJIbCTBO 4Ha:O;HOHaJIbHOe 06paaOBaHHe». @ 2021. qOY ,n;rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX :o;eJIHX 6ea IIHCbMeHHoro paapeIIleHHH IIpaBOo6JIa.,zJ;aTeJIeH He ,n;OIIYCKaeTCH
PEWEHII1s:1 111 KPII1TEPII1I11 OL.J.EHII1BAHII1s:1 3AAAHII1� 13-19 219
a) lIycTb rrJIOCKOCTb a rrepeceKaeT pe6po SB B TOqKe L.
lIoCKOJIbKY rrpHMaH BC rrapaJIJIeJIbHa rrJIOCKOCTH a, rrpHMbIe
LM H BC rrapaJIJIeJIbHbI, a 3HaQHT,
SL: LB=SM: MC=AK: KB.
CJIe,ZJ;oBaTeJIbHO, rrpHMbIe KL H SA rrapaJIJIeJIbHbI.
TaKHM o6pa30M, rrJIOCKOCTb a, cO,ZJ;epmam;aH rrpHMYIO KL,
rrapaJIJIeJIbHa rrpHMoH SA.
6) lIycTb TOQKa H -
cepe,ZJ;HHa pe6pa BC. Tor,ZJ;a
Me,ZJ;HaHbI AH H SH TpeyrOJIbHHKOB ABC H SBC
COOTBeTCTBeHHO HBJIHIOTCH HX BbICOTaMH, a 3HaQHT,
rrJIOCKOCTb ASH rreprreH,ZJ;HKYJIHpHa rrpHMoH BC.
CJIe,ZJ;oBaTeJIbHO, rrJIOCKOCTb ASH rreprreH,ZJ;HKYJIHpHa rrJIOCKOCTH a, rrapaJIJIeJIbHOH
rrpHMoH BC, H rrJIOCKOCTH SBC, cO,ZJ;epmam;eH rrpHMYIO BC.
lIoCKOJIbKY rrJIOCKOCTb a rrapaJIJIeJIbHa rrpHMoH SA, JIemam;eH B rrJIOCKOCTH ASH,
HCKOMbIH yrOJI paBeH yr JIY Mem,ZJ;Y rrpHMoH SA H rrJIOCKOCTblO SBC.
TaKHM o6pa30M, yrOJI Mem,ZJ;Y rrJIOCKOCTHMH a H SBC paBeH yrJIY ASH.
B TpeyrOJIbHHKe ASH HMeeM:
AS = 7, AH =3.J3, SH = ..JSB 2-BH
2= � SB2 _ B�2 =2 .Jl0 .
PemeHHe.
s
c
B
lIo TeopeMe KOCHHYCOB
SA2
+SH2
-AH2
cos LASH =
2SA. SH
49+40-27
2.7.2 .J10
31 .J10OTBeT: 6) arccos
140
31.JIO
140
COAep3taHHe KpHTepHH BaJIJILI
MMeeTCH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a, H 060CHOBaHHO rrOJIY-2
qeH BepHbIH OTBeT B rrYHKTe 6
MMeeTcH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a,
MJIM1
06ocHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6, B03MomHO, C HCrrOJIb30BaHHeM
YTBepm,n;eHHH rrYHKTa a, rrpH aTOM rrYHKT a He BbIrrOJIHeH
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aKCUMa.nbllblU 6a.n.n 2
�PeIIIHTe HepaBeHcTBo logo,5 (12-6x) > logo,5 (X2 -6x+8)+ logo,5 (x+ 3).
PemeHHe.
3arrHllIeM HCXO,ZJ;Hoe HepaBeHCTBO B BH,ZJ;e:
logo,5 ( 6 ( 2 -
x ) ) > logo,5 ( ( 4 -
x ) ( 2 -
x ) ) + logo,5 ( x + 3) ;
logo,5 6 + logo,5 ( 2 -
x ) > logo,5 ( 4 - x) + logo,5 ( 2 -
x ) + logo,5 ( x + 3) .
@ 2021. 000 «Ha,n;aTeJIbCTBO «HaU;HOHaJIbHOe 06paaOBaHHe». @ 2021. qOY ,l];rrO «MIJ;HMO»KOIIHpoBaHHe, pacIIpocTpaHeHHe H HCIIOJIbaOBaHHe B KOMMep-qecKHx u;eJIHX 6ea IIHCbMeHHoro paaperneHHH IIpaBOo6JIa.,zJ;aTeJIeH He ,n;OIIYCKaeTCH
220 Er3. TV1nOBbIE 3K3AMEHAU.V10HHbIE BAPV1AHTbl
HepaBeHCTBO OIIpe�eJIeHO IIPH -3 < x < 2, II03TOMY IIPH -3 < x < 2 HepaBeHCTBO IIPH-
HHMaeT BH�:
6 « 4-x)(x+3); 6 < 12+x-x2
; x2
-x-6 < 0,
oTKy�a -2 < x < 3. Y"tJHTbIBaH orpaHH"tJeHHe -3 < x < 2, IIOJIy"tJaeM: -2 < x < 2.
OTBeT: [-2; 2).
Co,n;epataHHe KpHTepHJI BaJIJILI
060CHOBaHHO IIOJIyqeH BepHbIH OTBeT 2
060cHoBaHHo IIOJIyqeH OTBeT, OTJIHqalOID;HHCH OT BepHoro HCKJIlOqeHHeM TOqKH -2,
MJIM1
IIOJIyqeH HeBepHbIH OTBeT H3-3a BbIqHCJIHTeJIbHOH OIIIH6KH, HO IIPH 3TOM HMeeTCH
BepHaH IIOCJIe�OBaTeJIbHOCTb Bcex IIIarOB peIIIeHHH
PeIIIeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, IIepeqHCJIeHHbIX BbIllie 0
M aKCUMaJlbHblU 6aJlJl 2
�TO"tJKa 0 -
�eHTp BIIHcaHHoH B TpeyrOJIbHHK ABC oKpymHocTH. IIpHMaH BO BTOpH"tJHO
IIepeceKaeT oIIHcaHHYIO OKOJIO 3Toro TpeyrOJIbHHKa oKpymHocTb B TO"tJKe P.
a) ,n:oKamHTe, "tJTO LPOA == LPAO.
6) HaH�HTe IIJIOID;Mb TpeyrOJIbHHKa APO, eCJIH pa�HYC oIIHcaHHoH OKOJIO
TpeyrOJIbHHKa ABC oKpymHocTH paBeH 6, LBAC == 75°, LABC == 60°.
PemeHHe.
1 1LPOA == LBAO + LABO == - LBAC + - LABC .
2 2
a) IIoCKOJIbKY TO"tJKa 0 -
�eHTp BIIHcaHHOH B TpeyrOJIb-
HHK ABC oKpymHocTH, JIy"tJH AO H BO HBJIHIOTCH 6HcceK-
TpHcaMH yrJIOB TpeyrOJIbHHKa ABC. YroJI POA HBJIHeTCH
BHellIHHM yrJIOM TpeyrOJIbHHKa AOB. CJIe�OBaTeJIbHO,
YrJIbI PAC H PBC paBHbI, IIOCKOJIbKY OIIHpalOTCH Ha O�HY
H TY me �yry oKpymHocTH, oIIHcaHHoH OKOJIO TpeyrOJIbHHKa
ABC, II03TOMY
P
1 1LPAO==LPAC+LOAC==LPBC+LOAC==-LABC+-LBAC.
2 2
TaKHM o6pa30M, LPOA == LPAO.
6) IIycTb R == 6 -
pa�HYC oKpymHocTH, OIIHcaHHOH OKOJIO TpeyrOJIbHHKa ABC.
IIoCKOJIbKY LPOA == LPAO, TpeyrOJIbHHK APO paBHo6e�peHHbIH, CJIe�OBaTeJIbHO,
o P == AP == 2R sin LABP == 2R sin 30° == 6 .
@ 2021. 000 «H3,n;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. QOY ,1];rrO «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH
PEWEHV1� V1 KPV1TEPV1V1 OU.EHV1BAHV1� 3AAAHV111113-19 2 �t-�
TaKHM 06pa30M, IIJIOm;a�b TpeyrOJIbHHKa APO paBHa
AP.OP.sinLAPO AP2
.sinLACB AP2
.sin45°= 9.J2.
222
OTBeT: 6) 9.J2.
Co,n;epataHHe KpHTepHJI BaJIJILI
I1MeeTcH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a, H 060CHOBaHHO rrOJIY-
qeH BepHbIH OTBeT B rrYHKTe 6
06ocHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6,
I1JII1
HMeeTCH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a, H rrpH 06ocHoBaHHoM
pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a apHcpMeTHqeCKOH omH6KH
I1MeeTcH BepHoe ,n;OKa3aTeJIbCTBO YTBepm;n;eHHH rrYHKTa a,
I1JII1
rrpH 06ocHoBaHHoM pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a
apHcpMeTHqeCKOH omH6KH, 1
I1JII1
06ocHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6 C HCrrOJIb30BaHHeM YTBepm,n;eHHH
rrYHKTa a, rrpH 3TOM rrYHKT a He BbIrrOJIHeH
3
2
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aKCUMaJlbHblU 6aJlJl 3
[E]15 HHBapH IIJIaHHpyeTcH B3HTb Kpe�HT B 6aHKe Ha HeKoTopMH CpOK (�eJIoe �HCJIO
MecH�eB). YCJIOBHH ero B03BpaTa TaKOBbI:
1-ro �HCJIa Kam�oro MeCH�a �OJIr B03paCTaeT Ha 1 % 110 cpaBHeHHIO C KOH�OM
IIpe�bI�ym;ero MeCH�a;
co 2-ro 110 14-e �HCJIO Kam�oro MeCH�a He06xo�HMO BbIIIJIaTHTb �aCTb �OJIra;
15-ro �HCJIa Kam�oro MeCH�a �OJIr �OJImeH 6bITb Ha O�HY H TY me CYMMY MeHbrne
�OJIra Ha 15-e �HCJIO IIpe�bI�ym;ero MeCH�a.
Ha CKOJIbKO MeCH�eB IIJIaHHpyeTcH B3HTb Kpe�HT, eCJIH H3BeCTHO, �TO 06m;aH CYMMa
BbIIIJIaT IIOCJIe IIOJIHOrO IIorarneHHH Kpe�HTa Ha 20%
60JIbrne CYMMbI, B3HTOH B Kpe�HT?
(C�HTaHTe, �TO oKpyrJIeHHH IIPH BbI�HCJIeHHH IIJIaTemeH He IIpOH3BO�HTCH.)
PemeHue.
IIYCTb CYMMa Kpe�HTa paBHa 8, a Kpe�HT IIJIaHHpyeTcH B3HTb Ha n MeCH�eB. IIo YC-
JIOBHIO, �OJIr IIepe� 6aHKOM 110 COCTOHHHIO Ha 15-e �HCJIO �OJImeH YMeHbrnaTbCH �o HYJIH
paBHOMepHO:
8(n-1)8;
28 8
n
. -.
, , -; o.nn
@ 2021. 000 «H3�aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. QOY ,1];rrO «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIa,n;aTeJIeH He �OIIYCKaeTCH
222 Er3. TV1nOBbIE 3K3AMEHAU.V10HHbIE BAPV1AHTbl
IIepBoro -qHCJIa KajK�OrO MeCHn;a �OJIr B03paCTaeT Ha 1 %, 3Ha-qHT, nOCJIe�OBaTeJIb-
HOCTb pa3MepOB �OJIra Ha 1-e -qHCJIO KajK�OrO MeCHn;a TaKOBa:
1,01S( n -1)1,018; ;
n
2,028 1,018.
,
n n
CJIe�OBaTeJIbHO, BbInJIaTbI �OJIjKHbI 6bITb CJIe�YIOm;HMH:
O 0188
. 0,018(n-1) 8.
, +, +,
n n n
Bcero CJIe�yeT BbInJIaTHTb
8 +0,018 (n
+
n-1 +...+�+! ) = 8 ( 1+0,01(n+ 1)
) .
n n n n 2
0,028 8+-;
n n
0,018 8+-.
n n
06m;aH CYMMa BbInJIaT Ha 20%
60JIbme CYMMbI, B3HTOH B Kpe�HT, n03TOMY
0,01(n+1)=0,2; n=39.
2
OTBeT: 39.
Co.n;epataHHe KpHTepHJI BaJlJlLI
060CHOBaHHO rroRyqeH BepHbIM OTBeT 3
BepHo rrOCTpoeHa MaTeMaTHqeCKaH Mo,n;eRb, pemeHHe CBe,n;eHO K HCCRe,n;OBaHHIO
3TOM Mo,n;eRH H rroRyqeH pe3YRbTaT:2
-
HeBepHbIM OTBeT H3-3a BbIqHCRHTeRbHoM omH6KH;-
BepHbIM OTBeT, HO pemeHHe He,n;OCTaTOqHO 060cHoBaHo
BepHo rrOCTpoeHa MaTeMaTHqeCKaH Mo,n;eRb, pemeHHe CBe,n;eHO K HCCRe,n;OBaHHIO1
3TOM Mo,n;eRH, rrpH 3TOM pemeHHe MomeT 6bITb He 3aBepmeHO
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCReHHbIX BbIme 0
M aKCUMaJlbHblU 6aJlJl 3
� HaH,D;MTe Bce 3Ha-qeHMSI a, rrpM Kam,D;OM M3 KOTOpbIX ypaBHeHMe
13xl-2x-2-a=0
x2
-2x-a
HMeeT pOBHO �Ba pa3JIH-qHbIX KOpHH.
PemeHHe.
KOpHHMH HCXO�Horo ypaBHeHHH HBJIHIOTCH KOpHH ypaBHeHHH 13xl- 2x - 2 -
a = 0,
�JIH KOTOpbIX BbInOJIHeHO YCJIOBHe i 2- 2x -
a * O.
IIpH X < 0 ypaBHeHHe 13xl- 2x - 2 -
a = 0 npHHHMaeT BH�- 5x - 2 -
a = 0 H 3a-
�aeT Ha nJIOCKOCTH Oxa JIy-q l1 C Ha-qarrOM B To-qKe (0; -2). IIpH x > 0 ypaBHeHHe
13xl- 2x - 2 -
a = 0 npHHHMaeT BH� x- 2 -
a = 0 H 3a�aeT JIy-q l2 C Ha-qarrOM B To-qKe
(0; -2). 3Ha-qHT, ypaBHeHHe 13xl- 2x - 2 -
a = 0 HMeeT �Ba KOpHH npH a > -2, HMeeT
O�HH KopeHb npH a = -2 H He HMeeT KopHeH npH a < -2.
YpaBHeHHe x2
- 2x -
a = 0 3a�aeT napa60JIY a = x2
- 2x.
@ 2021. 000 «Ha�aTeJIbCTBO «HaU;HOH8.JIbHOe 06paaOBaHHe». @ 2021. qOY ,n;rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMep'tleCKHX u;eJIHX 6ea IIHCbMeHHoro paapemeHHH IIpaBOo6JIa,n;aTeJIeii He �OIIYCKaeTCH
PEWEHV1� V1 KPV1TEPV1V1 OU.EHV1BAHV1� 3A.n,AHV1V113-19 223
Koop�HHaTbI TO"tJeK rrepeCe"tJeHHH rrapa60JIbI a = x2
- 2x C JIy"tJOM l1 HBJIHIOTCH perne-
HHHMH CHCTeMbI:
{a=x
2
-2x,
{-5X-2=X2_2X,
{(X+1)(X+2)=0,
a = - 5x - 2, a = - 5x - 2, a = - 5x - 2,x < 0; x < 0; x < O.
3Ha-qHT, rrapa60JIa a = x2
- 2x rrepeceKaeTCH C JIy"tJOM l1 B TO"tJKax (-1; 3) H (-2; 8).
Koop�HHaTbI TO"tJeK rrepece"tJeHHH rrapa60JIbI a = x2
- 2x C JIy"tJOM l2 HBJIHIOTCH perne-
HHHMH CHCTeMbI:
{a
_
x2
-2x,
{X::::2=X
2
-2X,
{(X_-l)(X-2)=O,
a - x- 2, a - x
- 2, a - x- 2,
x > 0; x > 0; x > O.
3Ha"tJHT, rrapa60JIa a = x2
- 2x rrepeceKaeTCH C JIy"tJOM l2 B TO"tJKax (1; -1) H (2; 0).
CJIe�OBaTeJIbHO, YCJIOBHe x2
- 2x - a*-O BbIrrOJIHeHO �JIH KopHeH ypaBHeHHH
13xl- 2x - 2 -
a = 0 rrpH Bcex a, KpOMe a = -1, a = 0, a = 3 H a = 8.
TaKHM o6pa30M, HCXO�Hoe ypaBHeHHe HMeeT pOBHO �Ba KOpHH rrpH -2 < a < -1;
-1 < a < 0; 0 < a < 3; 3 < a < 8; a > 8.
OTBeT: -2 < a < -1; -1 < a < 0; 0 < a < 3; 3 < a < 8; a > 8.
Co,n;epataHHe KpHTepHJI BaJIJILI
060CHOBaHHO rroRyqeH BepHbIH OTBeT 4
C rrOMOm;blO BepHoro paccym,n;eHHH rroRyqeHO MHomeCTBO 3HaqeHHH a,3
OT RHqalOm;eeCH OT HCKOMoro TORbKO BKRlOqeHHeM TOqKH a =-2
BepHo paCCMOTpeH XOTH 6bI O,n;HH H3 cRyqaeB perneHHH, H rroRyqeHO HRH MHome-
CTBO 3HaqeHHH a, OTRHqalOm;eeCH OT HCKOMoro TORbKO BKRlOqeHHeM TOqeK a = 8,
a=3 H/HRH a = -2, HRH MHomeCTBO 3HaqeHHH a, OTRHqalOm;eeCH OT HCKOMoro
TORbKO BKRlOqeHHeM TOqeK a = 0, a = -1 H/HRH a = -2, 2
MJIM
rroRyqeH HeBepHbIH OTBeT H3-3a BbIqHCRHTeRbHoH ornH6KH, HO rrpH 3TOM BepHO
BbIrrORHeHbI Bce rnarH perneHHH
3a,n;aqa BepHO CBe,n;eHa K HCCRe,n;OBaHHIO B3aHMHoro pacrrORomeHHH rrapa60RbI1
H RyqeH (aHaRHTHqeCKH HRH rpacpHqeCKH)
PerneHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCReHHbIX BbIrne 0
M aKCUMaJlbHblU 6aJlJl 4
19 IB HID;HKe JIemHT 76 <PPYKTOB, Macca Kam�oro H3 KOTOpbIX BbIpamaeTCH n;eJIbIM "tJHCJIOM
rpaMMOB. B HID;HKe eCTb XOTH 6bI �Ba <ppYKTa pa3JIH-qHoH MaCCbI, a cpe�HHH Macca Bcex
<PPYKTOB paBHa 100 r. Cpe�HHH Macca <PPYKTOB, Macca Kam�oro H3 KOTOpbIX MeHbrne
100 r, paBHa 85 r. Cpe�HHH Macca <PPYKTOB, Macca Kam�oro H3 KOTOpbIX 60JIbrne 100 r,
paBHa 124 r.
a) MorJIo JIH B HID;HKe OKa3aTbCH rropOBHY <PPYKTOB MaccoH MeHbrne 100 r H <PPYKTOB
MaccoH 60JIbrne 100 r?
6) MorJIo JIH B HID;HKe OKa3aTbCH MeHbrne 8 <PPYKTOB, Macca Kam�oro H3 KOTOpbIX
paBHa 100 r?
B) KaKYIO HaH6oJIbrnylO Maccy MomeT HMeTb <PPYKT B 3TOM HID;HKe?
@ 2021. 000 «H3,n;aTeJIbCTBO «Ha:U;HOH8.JIbHOe 06pa30BaHHe». @ 2021. qOY ,ZJ;rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX :u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIMaTeJIeii He ,n;OIIYCKaeTCH
224 Er3. TV1nOBbIE 3K3AMEHALJ.V10HHbIE BAPV1AHTbl
PemeHHe.
a) IIycTb B Hm;HKe k <PPYKTOB MaCCOH MeHbme 100 r, k <PPYKTOB MaccoH 60JIb-
me 100 r H (76-
2k) <PPYKTOB MaccoH pOBHO 100 r. Tor�a
85k + 124k + 100 . (76-
2k) = 7600; 9k = 0,
HO Bce <PPYKTbI He MorYT 6bITb O�HOH MaCCbI, 3Ha-qHT, B Hm;HKe He MOrJIO OKa3aTbCH
rropOBHY <PPYKTOB MaCCOH MeHbme 100 r H <PPYKTOB MaccoH 60JIbme 100 r.
6) IIycTb B Hm;HKe k <PPYKTOB MaCCOH MeHbme 100 r, m <PPYKTOB MaCCOH poR-
HO 100 r H n <PPYKTOB MaccoH 60JIbme 100 r. Tor�a
85k + 100m +124n = 100. (k + m + n); 8n = 5k.
IIoCKOJIbKY -qHCJIa 5 H 8 B3aHMHO rrpOCTbI,
k=88, n=5t; 408=40t; 8=t.
TaKHM 06pa30M, k + n = 88 + 58 = 138. CJIe�OBaTeJIbHO, KOJIH-qeCTBO <PPYKTOB C Mac-
COH, OTJIH-qHOH OT 100 r, �eJIHTCH Ha 13, H 138 < 76, TO eCTb 8 < 5 H k + n < 65. 3Ha-qHT,
m = 76 -
(k + n) > 76 - 65 = 11.
CJIe�OBaTeJIbHO, B Hm;HKe He MOrJIO OKa3aTbCH MeHbme 8 <PPYKTOB, Macca Kam,n;oro
H3 KOTOpbIX paBHa 100 r.
B) IIYCTb Macca caMoro THmeJIOrO <PPYKTa paBHa x r, Tor�a
124n > x + 101 . (n -1); x < 23n + 101.
B rrYHKTe 6 6bIJIO rrOKa3aHO, -qTO n = 58 H 8 < 5, 3Ha-qHT,
n < 25; x < 23 25 + 101; x < 676.
IIoKameM, -qTO Macca caMoro THmeJIOrO <PPYKTa MomeT 6bITb 676 r. ECJIH B Hm;H-
Ke 40 CPPYKTOB MaccoH 85 r, 11 <PPYKTOB MaccoH 100 r, 24 <PPYKTa MaccoH 101 r
H 1 <PPYKT MaCCOH 676 r, TO YCJIOBHH 3a�a-qH BbIrrOJIHeHbI.
OTBeT: a) HeT; 6) HeT; B) 676.
Co,n;epataHHe RpHTepHJI BaJlJlLI
BepHo rrOJIyqeHbI Bce rrepeqHCJIeHHbIe (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTbI 4
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 O,n;HH H3 CJIe,n;YIOID;HX pe3YJIbTaTOB:- 06ocHoBaHHoe pemeHHe rrYHKTa a;
- 06ocHoBaHHoe pemeHHe rrYHKTa 6; 1
-
HCKOMaH On;eHKa B rrYHKTe B;
-
rrpHMep B rrYHKTe B, o6eCrreqHBalOID;HH TOqHOCTb rrpe,n;bI,n;yID;eH On;eHKH
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aKCUMaJlbllblU 6aJlJl 4
BapHaHT 17
[!I] a) PeIIIHTe ypaBHeHHe 24.4x - O ,5 -11.2
x + 1+6 =0.
6) HaH,n;HTe Bce KOpHH 3Toro ypaBHeHHH, rrpHHMJIemam;He rrpOMemYTKY [-1; 1].
@ 2021. 000 «H3,n;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY ,ZJ;rro «MIJ;HMO»
KOIIHpoBaHHe, pacIIpocTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH
PEWEHV1� V1 KPV1TEPV1V1 OU.EHV1BAHV1� 3A)J.AHV111113-19 225
PemeHHe.
a) IIpeo6pa3yeM ypaBHeHHe:
12.4x
-22.2x
+6 == 0; 6.4x
-11.2x
+3 == O.
1 3C,n;eJIaeM 3aMeHY: 2 x
==t. IIoJI�HM: 6t2
-11t+3==0, OTRY,ZJ;a t==- HJIH t==-.
3 2
T 2x1
2x
__
3I 3 I 1 5or,ZJ;a == - HJIH -, CJIe,ZJ;OBaTeJIbHO, x ==
-
og2 , HJIH X ==
og2 , .
3 2
6) -log23 < -1; 0 < log21,5 <1, rr03TOMY OTpe3RY [-1; 1] rrpHHMJIemHT TOJIbRO RO-
peHb log21,5.
OTBeT: a) -log23; log21,5; 6) log21,5.
Co,n;epataHHe KpHTepHJI BaJIJILI
060CHOBaHHO rrOJIyqeHbI BepHbIe OTBeTbI B 060HX rrYHKTax 2
060cHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe a,
llJIMu
01lIH6KH, rrpH 1rrOJIyqeHbI HeBepHbIe OTBeTbI H3-3a BbIqHCJIHTeJIbHOH HO 3TOM
HMeeTCH BepHaH rrOCJIe�OBaTeJIbHOCTb Bcex llIarOB pellIeHHH 060HX rrYHKTOB:
rrYHKTa a H rrYHKTa 6
PellIeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIllie 0
M aKCUMaJlbllblU 6aJlJl 2
[illOCHOBaHHeM rrHpaMH,ZJ;bI SABCD HBJIHeTCH rrpHMoyrOJIbHHR ABCD co cTopoHaMH AB == 15
H BC == 25. B�e 6oRoBbIe pe6pa rrHpaMH,n;bI paBHbI 5.JI7 Ha pe6pax AD H BC OTMe"tJe-
HbI COOTBeTCTBeHHO TO"tJRH K H N TaR, "tJTO AK == CN == 8. Qepe3 TO"tJRH K H N rrpOBe,ZJ;e-
Ha rrJIOCROCTb a, rreprreH,ZJ;HRYJIHpHaH pe6py SB.
a) ,n:oRamHTe, "tJTO rrJIOCROCTb a rrpoxo,n;HT "tJepe3 TO"tJRY M -
cepe,ZJ;HHY pe6pa SB.
6) HaH,ZJ;HTe paCCTOHHHe Mem,ZJ;Y rrpHMbIMH DS H KM.
PemeHHe.
a) IIycTb TO"tJRa 0 -
rrepece"tJeHHe BD H NK. OTpe3RH BN H KD paBHbI H rrap8JIJIeJIb-
HbI, rr03TOMY TO"tJRa 0 -
cepe,ZJ;HHa BD. S
PaccMoTpHM TpeyrOJIbHHR BSD:
BD2 == BA2 + AD2 == 152
+ 252
== 850 H
BS2
+ sd' == 2.( 5.JI7)2 == 850.
3Ha"tJHT, TpeyrOJIbHHR BSD rrpHMoyrOJIbHbIH C rrpHMbIM
yrJIOM BSD.
IIJIOCROCTb a rreprreH,ZJ;HRYJIHpHa pe6py BS, rr03TOMY OHa
rrap8JIJIeJIbHa rrpHMOH SD. CJIe,ZJ;OBaTeJIbHO, rrJIOCROCTb a
rrepeceRaeT rrJIOCROCTb BSD rro rrpHMOH, rrap8JIJIeJIbHOH
rrpHMOH SD, TO eCTb rro rrpHMoH OM, cO,ZJ;epmaru;eH cpe,ZJ;HIOIO JIHHHIO TpeyrOJIbHHRa BSD.
B
A
@ 2021. 000 <<lfa,n;aTeJIbCTBO «HaU;HOHaJIbHOe 06paaOBaHHe». @ 2021. QOY ,1];rro «MIJ;HMO»
KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX u;eJIHX 6ea IIHCbMeHHoro paapemeHHH IIpaBOo6JIa,n;aTeJIeH He ,n;OIIYCKaeTCH
226 Er3. TV1nOBbIE 3K3AMEHALJ.V10HHbIE BAPV1AHTbl
6) llcKoMoe paCCTOHHHe paBHO paCCTOHHHIO OT rrpHMoM 8D ,n;o rrapaJIJIeJIbHOM eM rrJIO-
CKOCTH a, co,n;epm�eM rrpHMYIO KM. CJIe,n;oBaTeJIbHO, 3TO paCCTOHHHe paBHO ,n;JIHHe rrep-
rreH,n;HKYJIHpa 8M, rrpoBe,n;eHHoro OT rrpHMoM 8D K rrJIOCKOCTH a.
8M = !8B =
5.JI7.
2 2
OTBeT: 6)5.JI7
2
Co.n;epataHHe RpHTepHJI BaJIJILI
I1MeeTcH BepHoe ,lJ;OKa3aTeJIbCTBO YTBepm,lJ;eHHH rrYHKTa a, H 060CHOBaHHO rrOJIY-2
qeH BepHbIH OTBeT B rrYHKTe 6,.-
I1MeeTcH BepHoe ,lJ;OKa3aTeJIbCTBO YTBepm,lJ;eHHH rrYHKTa a,
I1JII11
060CHOBaHHO rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6, B03MomHO, C HCrrOJIb30BaHHeM
YTBepm,lJ;eHHH rrYHKTa a, rrpH 3TOM rrYHKT a He BbIrrOJIHeH
PerneHHe He COOTBeTcTByeT HH O,lJ;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIrne 0
M aKCUMaAbHblU 6aAA 2
DDPellIHTe HepaBeHcTBo � x + � -log! (log211- xl) > 0 ·
2
PemeHHe.
1IIpH X == -- JIeBaH -qaCTb HepaBeHCTBa HMeeT CMbICJI H paBHa HYJIIO.
2
1IIpH YCJIOBHH x> -
2IIOJI�aeM HepaBeHCTBO
logl (log21 1 -
x l» O,2
oTKy,n;a
O<log211- x l < 1; 1<11- x l < 2.
CJIe,n;OBaTeJIbHO,1
-1 < x < 0 HJIH 2 <x < 3. Y-qHTbIBaH Y CJIOBHe x >--
2'
1- - < x < 0 HJIH 2 < x < 3.
2
Haxo,n;HM:
106FJ>e,n;HHHH rrOJIy-qeHHbIe MHomeCTBa, rrOJIy-qaeM:
-- < x < 0 HJIH 2 <x < 3.2
OTBeT: [- � ; 0 ) U (2; 3].
@ 2021. 000 «Ha,n;aTeJIbCTBO «HaU;HOHaJIbHOe o6paaOBaHHe». @ 2021. QOY ,1];rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX u;eJIHX 6ea IIHCbMeHHoro paapemeHHH IIpaBOo6JIa.,n;aTeJIeH He ,n;OIIYCKaeTCH
PEWEHV1� V1 KPV1TEPV1V1 OU.EHV1BAHV1� 3AAAHV111113-19 227
Co�ep�aHHe KpHTepHJI BaJlJlLI
060CHOBaHHO rroRyqeH BepHbIH OTBeT 2
060cHoBaHHo rroRyqeH OTBeT, OTRHqalOID;HHCH OT BepHoro HCKRlOqeHHeM TOqKH1
--
2
H/HRH TOqKH 3,1
MJIM
rroRyqeH HeBepHbIH OTBeT H3-3a BbIqHCRHTeRbHoH OIIIH6KH, HO rrpH 3TOM HMeeTCH
BepHaH rrOCRe�OBaTeRbHOCTb Bcex IIIarOB peIIIeHHH
PeIIIeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, rrepeqHCReHHbIX BbIllie 0
M aKCUMaJlbJ-lblU 6aJlJl 2
qeTbIpexyrOJIbHHK ABCD BIIHcaH B oKpymHocTb pa,n;Hyca R = 27.
AB =BC =CD= 36.
a) ,n:oKamHTe, "tITO IIpHMbIe BC H AD lIapaJIJIeJIbHbI.
6) HaH�HTe AD.
[K]
PemeHHe.
a) OCTpbIe yrJIbI BCA H CAD paBHbI, 1I0CKOJIbKY OIlH-
palOTCH Ha paBHbIe XOp�bI AB H CD. 3Ha"tIHT, IIpHMbIe BC
H AD lIapaJIJIeJIbHbI.
6) 0603Ha"tIHM yrOJI BCA "tIepe3 a. lIo TeopeMe CHHYCOB
.AB 2
SIna=-=-.
2R 3
113BecTHo, "tITO
TpeyrOJIbHHK ABC paBH06e�peHHbIH, 1I03TOMY
LBAC = LBCA = a. 3Ha"tIHT, LBAD = LBAC + LCAD = 2a.
qeTbIpexyrOJIbHHK ABCD -
paBH06e�peHHaH Tpalle�HH, 1I03TOMY LCDA = LBAD = 2a.
3Ha"tIHT,
LACD = 180° - LCAD - LCDA = 180° - 3a.
TaKHM 06pa30M, 110 TeopeMe CHHYCOB �JIH TpeyrOJIbHHKOB ACD H ACB 1I0JI�aeM:
AD
� ; AD= si�3a oAB=(3-4sin2a)AB=44.sin(1800-3a) Sina Sina
OTBeT: 6) 44.
Co�ep�aHHe KpHTepHJI BaJlJlLI
MMeeTCH BepHoe �OKa3aTeRbCTBO YTBepm�eHHH rrYHKTa a, H 060CHOBaHHO rrORY-3
qeH BepHbIH OTBeT B rrYHKTe 6
060cHoBaHHo rroRyqeH BepHbIH OTBeT B rrYHKTe 6,
MJIM2
HMeeTCH BepHoe �OKa3aTeRbCTBO YTBepm�eHHH rrYHKTa a, H rrpH 060cHoBaHHoM
peIIIeHHH rrYHKTa 6 rroRyqeH HeBepHbIH OTBeT H3-3a apHcpMeTHqeCKOH OIIIH6KH
@ 2021. 000 «(Ha�aTeJIbCTBO «(HaU;HOHaJIbHOe 06paaOBaHHe». @ 2021. qOY ,ZJ;rro «(MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX u;eJIHX 6ea IIHCbMeHHoro paapemeHHH IIpaBOo6JIa,n;aTeJIeii He �OIIYCKaeTCH
228 Er3. TV1nOBbIE 3K3AMEHAL.J.V10HHbIE BAPV1AHTbl
OKOHl.laHUe ma6Jl.
Co,n;epataHHe KpHTepHJI BaJIJILI
MMeeTCH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a,
MJIM
rrpH 060CHOBaHHOM pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a apHcpMe-
THqeCKOH omH6KH, 1
MJIM
060cHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6 C HCrrOJIb30BaHHeM YTBepm,n;eHHH
rrYHKTa a, rrpH 3TOM rrYHKT a He BbIrrOJIHeH
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aKCUMaJlbHblU 6aJlJl 3
[E]IIo 6H3HeC-nJIaHY -qeTbIpeXJIeTHHii npoeKT npe,n;nOJIaraeT Ha-qarrbHoe BJIOmeHHe
-
25 MJIH py6JIeii. IIo HToraM Kam,n;oro ro,n;a nJIaHHpyeTcH npHpOCT BJIOmeHHbIX cpe,n;CTB
Ha 20%
no cpaBHeHHIO C Ha-qarrOM ro,n;a. Ha-qHCJIeHHbIe npo�eHTbI OCTalOTCH BJIOmeHHbIMH
B npoeKT. KpoMe 3Toro, cpa3Y nOCJIe Ha-qHCJIeHHii npo�eHTOB HymHbI ,n;onOJIHHTeJIbHbIe
BJIOmeHHH: �eJIoe -qHCJIO n MJIH py6JIeii H B nepBbIii, H BO BTOpoii ro,n;bI, a TaKme �eJIoe
-qHCJIO m MJIH py6JIeii H B TpeTHii, H B -qeTBepTbIii ro,n;bI. Haii,n;HTe HaHMeHbmee
3Ha-qeHHe n, npH KOTOpOM nepBOHa-qarrbHbIe BJIOmeHHH 3a ,n;Ba ro,n;a KaK MHHHMYM
y,n;BOHTCH, H HaHMeHbmee 3Ha-qeHHe m, TaKoe, -qTO npH Haii,n;eHHoM paHee 3Ha-qeHHH n
nepBOHa-qarrbHbIe BJIOmeHHH 3a -qeTbIpe ro,n;a BbIpaCTYT KaK MHHHMYM B -qeTbIpe pa3a.
PemeHHe.
K Ha-qaJIY 2-ro ro,n;a nOJIy-qHTCH 1,2 25 + n = 30 + n MJIH BJIOmeHHii, a K Ha-qaJIY
3-ro ro,n;a-
1,2 (30 + n) + n = 36 + 2,2n.
IIo YCJIOBHIO 36 + 2,2n > 50. HaHMeHbmee �eJIoe pemeHHe n = 7. Tor,n;a K Ha-qarry
3-ro ro,n;a nOJIy-qHTCH
36 + 15,4 = 51,4 MJIH.
K Ha-qarry 4-ro ro,n;a HMeeM 1,2 51,4 + m MJIH, a B KOH�e npoeKTa
1,2 (1,2 51,4 + m) + m = 74,016 + 2,2m.
IIo YCJIOBHIO 74,016 + 2,2m > 100,
25,984oTKy,n;a 2,2m > 25,984; m >
.
2,2
IIOJIy-qaeM, -qTO m = 12 -
HaHMeHbmee �eJIoe pemeHHe.
OTBeT: 7 H 12 MJIH py6.
Co,n;epataHHe KpHTepHJI BaJIJILI
060CHOBaHHO rrOJIyqeH BepHbIH OTBeT 3
BepHo rrOCTpoeHa MaTeMaTHqeCKaH Mo,n;eJIb, pemeHHe CBe,n;eHO K HCCJIe,n;OBaHHIO
3TOH Mo,n;eJIH H rrOJIyqeH pe3YJIbTaT:2
-
HeBepHbIH OTBeT H3-3a BbIqHCJIHTeJIbHOH omH6KH;-
BepHbIH OTBeT, HO pemeHHe He,n;OCTaTOqHO 060cHoBaHo
@ 2021. 000 «Ha�aTeJIbCTBO «Ha:U;HOHaJIbHOe 06paaOBaHHe». @ 2021. QOY ,1];rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX :u;eJIHX 6ea IIHCbMeHHoro paapemeHHH IIpaBOo6JIa.,n;aTeJIeH He �OIIYCKaeTCH
PEWEHV1s:1 V1 KPV1TEPV1V1 OLJ.EHV1BAHV1s:1 3AAAHV111113-19 ..
OKOJ-ll.laJ-lUe ma6A.
Co,n;epataHHe KpHTepHJI BaJlJlLI
BepHO IIOCTpOeHa MaTeMaTHqeCKaH MO,n;eJIb, perneHHe CBe,n;eHO K HCCJIe,n;OBaHHIO1
3TOM Mo,n;eJIH, IIPH 3TOM perneHHe MomeT 6bITb He 3aBeprneHO
PerneHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, IIepeQHCJIeHHbIX BbIrne 0
M aKCUMaAbJ-lblU 6aAA 3
a
[!DHaH,lJ;HTe Bce 3Ha�eHH.H a, IIPH Ram,lJ;OM 1'13 ROTOpbIX JIHHHH y=alx- 2 1+l a l-2 1'1
Y=2:OrpaHHqHBalOT MHoroyrOJIbHHK, IIJIO�a,n;b KOToporo He 60JIee 0,5.
PemeHHe.a
1) eCJIH JIHHHH Y= alx- 2 1+l a l-2 1'1 Y
=2: orpaHH�HBaIOT qmrypy, TO ee IIJIOru;a,n;b
a
H
Y==2:;6y�eT paBHa IIJIO�a,n;H cpHrYPbI, OrpaHHqeHHOH JIHHHHMH y==alxl+lal-2
2) IIPH a == 0 JIHHHH IIapaJIJIeJIbHbI H MHoroyrOJIbHHK He 06pa3YIOT.
� a > O .
JIHHHH y==alxl+a-2 H
a
y==
2:OrpaHHqHBalOT cpHrypy
a
(TpeyrOJIbHHK) IIPH ->a-2, oTKy�a a<4 (CM. pHC.).2
HaH�eM a6c�HccbI TOqeK IIepeCeqeHHH JIHHHH:
a 2 1
alxl+a-2==-, oTKy�a Ixl==---.2 a 2
IIo3ToMY IIJIO�a�b IIOJIyqeHHOrO TpeyrOJIbHHKa paBHa
( �_
� }( ; -a+2)= (2-!JPeIIIHM HepaBeHCTBO
(2-� J< 0,5; a
2-lOa + 16 < O.
a
IIOJIyqHM 2 < a < 8.
e yqeTOM YCJIOBHH 0 < a < 4 IIOJIyqHM 2 < a < 4.
ID a < O .
a
JIHHHH Y= a I xl- a - 2 1'1 Y
=
2:orpaHH�HBaIOT clmrypy
a 4(TpeyrOJIbHHK) IIPH
- < -a-2, oTKy�a a < - - (CM. pHC.).2 3
HaH�eM a6c�HccbI TOqeK IIepeCeqeHHH JIHHHH:
a 2 3
alxl-a-2==-, oTKy�a Ixl==-+-.2 a 2
IIo3TOMY IIJIO�a�b IIOJIyqeHHOrO TpeyrOJIbHHKa paBHa
( 3aJ( !+ 3
).
( _a_2_a
) =
2+2
a 2 2-a
y� � I I I I I I__
a>O y=alxl+lal-2\ /
--
, IJ
\ / a
\ Iy=--
2_
\ / --
\ 0 J x
\ /
\ /'If
a-2I I
- 'riy� � I , I
a< -a-2J �
/ \
J \
/ \ --
I 0 \ x
/ \
/ \ a
I \ Y=2:I \ I I
-
y=alxl+lal-2 _
I , I I I I
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230 Er3. TVinOBblE 3K3AMEHAL.J.VlOHHbIE BAPVlAHTbl
( 2 +3a
JPemHM HepaBeHCTBO
2< 0,5; 9a
2+ 26a + 16 < o.
-a
8IIoJlY"tIHM - 2 < a < - -
.
9
4 4C yqeTOM YCJlOBHH a < - -
nOJlY"tIHM- 2 < a < - -
.
3 3
OTBeT: [- 2; -
: ) u [2; 4).
CO,ZJ;epataHHe KpHTepHH
060CHOBaHHO IIOJI�eH BepHbIH OTBeT
C IIOMOm;blO BepHoro paccym,n;eHHH IIOJIyqeHO MHomeCTBO 3HaqeHHH a, OTJIHqalO-
m;eeCH OT HCKOMoro TOJIbKO HCKJIlOqeHHeM TOqeK a = -2 H/HJIH a = 2
B pemeHHH BepHO HaH,n;eHbI Bce rpaHHqHble TOqKH MHomeCTBa 3HaqeHHH a
( a = - 2, a = -
: 'a = 2, a = 4 ), HO HeBepHO orrpe�eJIeHbI rrpOMemYTKH 3HaqeHHH a,
HJIH
IIOJIyqeH HeBepHbIH OTBeT H3-3a BblqHCJIHTeJIbHOH omH6KH, HO IIPH 3TOM BepHO
BbIIIOJIHeHbI Bce marH pemeHHH
BepHo paCCMOTpeH XOTH 6bI O,n;HH H3 CJI�aeB pemeHHH, H IIOJI�eH XOTH 6w O,n;HH
H3 rrpoMemyrKoB [- 2; -
: ) HJIH [2; 4 ), B03MomHO, C BKJIIOqeHHeM/HCKJIIOqeHHeM
rpaHHqHbIX TOqeK,
BaJIJILI
4
3
2
1
HJIH
3a,n;aqa BepHO CBe,n;eHa K HCCJIe,n;OBaHHIO B3aHMHoro paCIIOJIOmeHHH ,n;BYX JIyqeH
C 06m;eH BepmHHoH H IIpHMOH (aHaJIHTHqeCKH HJIH rpa<pHqecKH)
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, IIepeqHCJIeHHbIX Bblme 0
M aKCUMaJtbllblU 6aJtJt 4
� M3,n;aTeJlbCTBO Ha BbIcTaBKY npHBe3J10 HeCKOJlbKO KHHr ,n;JlH npo,n;amH (Kam,n;ylO KHHry
npHBe3J1H B e,n;HHcTBeHHoM 3K3eMnJlHpe). QeHa Kam,n;oii KHHrH-
HaTyparrbHoe "tIHCJlO
py6J1eii. ECJlH �eHa KHHrH MeHbIIIe 100 py6J1eii, Ha Hee npHKJleHBalOT 6HPKY «BbIro,n;HO».
O,n;HaKo ,n;o OTKpbITHH BbICTaBKH �eHY Kam,n;oii KHHrH YBeJlH"tIHJlH Ha 10 py6J1eii,H3-3a "tIero KOJlH"tIeCTBO KHHr C 6HpKaMH «BbIro,n;HO» YMeHbIIIHJlOCb.
a) MorJla JlH YMeHbIIIHTbCH cpe,n;HHH �eHa KHHr C 6HpKOii «BbIro,n;HO» nocJle OTKpbITHH
BbICTaBKH no cpaBHeHHIO co cpe,n;Heii �eHoii KHHr C 6HpKOii «BbIro,n;HO» ,n;o OTKpbITHH
BbIcTaBKH?
6) MorJla JlH YMeHbIIIHTbCH cpe,n;HHH �eHa KHHr 6e3 6HpKH «BbIro,n;HO» nocJle OTKpbITHH
BbICTaBKH no cpaBHeHHIO co cpe,n;Heii �eHoii KHHr 6e3 6HpKH «BbIro,n;HO» ,n;O OTKpbITHH
BbICTaBKH?
B) M3BecTHo, "tITO nepBOHa"tlarrbHO cpe,n;HHH �eHa Bcex KHHr COCTaBJlHJla 110 py6J1eii,
cpe,n;HHH �eHa KHHr C 6HpKOii «BMro,n;HO� COCTaBJlHJla 81 py6J1b, a cpe,n;HHH �eHa
KHHr 6e3 6HpKH- 226 py6J1eii. IIocJle YBeJlH"tIeHHH �eHbI cpe,n;HHH �eHa KHHr
C 6HpKOii «BMro,n;HO� COCTaBHJla 90 py6J1eii, a cpe,n;HHH �eHa KHHr 6e3 6HpKH
210 py6J1eii. IIpH KaKOM HaHMeHbIIIeM KOJlH"tIeCTBe KHHr TaKoe B03MomHo?
@ 2021. 000 «ll3,ZJ;aTeJIbCTBO «Ha:U;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY ,l];rrO «MD;HMO.)KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX :u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIap;aTeJIeii He ,ZJ;OIIYCKaeTCH
PEWEHVI� VI KPVlTEPVIVI OU.EHVlBAHVI� 3A.L1,AHVl1I113-19 tan
PemeHHe.
a) IIpe,n;nOJlOmHM, "tITO npo,n;aBaJIHCb Bcero TPH KHHrH, KOTopbIe nepBOHa"tlaJIb-
HO CTOHJlH 120, 94 H 20 py6J1eii. Cpe,n;HHH �eHa «BbIro,n;HbIX» KHHr COCTaBJlHeT
94+20u
= 57 py6J1eH. IIocJle YBeJlH"tIeHHH �eHbI KHHrH CTaJlH CTOHTb 130, 104 H 30 py6-2
Jleii. Tenepb cpe,n;HHH �eHa «BbIro,n;HbIX» KHHr COCTaBJlHeT 30 py6J1eii.
6) B npHMepe H3 npe,n;bI,n;ym;ero nYHKTa nepBOHa"tlaJIbHO cpe,n;HHH �eHa «HeBbIro,n;-
120 6u
130+104117 6
u
HbIX» KHHr COCTaBJlHeT py JleH, a nocJle YBeJlH"tIeHHH= py JleH.
2
B) IIYCTb Bcero npHBe3J1H n KHHr. IIepBOHa"tlaJIbHO «BbIro,n;HbIX» 6bIJlO x KHHr, no-
cJle YBeJlH"tIeHHH �eHbI BbIro,n;HbIX CTaJIO Y KHHr. Cpe,n;HHH �eHa Bcex KHHr nocJle YBe-
JlH"tIeHHH COCTaBJlHeT 120 py6J1eii. IIoJlyqaeM ,n;Ba ypaBHeHHH:
110n = 226(n-
x) + 81x H 120n = 210(n-
y) + 90y,
oTKy,n;a 116n = 145x, TO eCTb 4n = 5x, H 90n = 120y, TO eCTb 3n = 4y. II03TOMY "tIHCJlO n
KpaTHO 4 H 5, TO eCTb KpaTHO 20. TaKHM o6paaoM, n > 20.
IIoKameM, "tITO B03MomeH cJlY"tlaii n = 20.
IIYCTb nepBOHa"tlaJlbHO 6bIJlO 15 KHHr no 80 py6J1eii, o,n;Ha KHHra-
no 96 py6J1eii
H "tIeTbIpe KHHrH no 226 py6J1eii. Tor,n;a cpe,n;HHH �eHa Bcex KHHr 110 py6J1eii, cpe,n;HHH
n;eHa KHHr C 6HpKOii «BbIro,n;HO» 81 py6J1b, a cpe,n;HHH �eHa KHHr 6e3 6HpKH- 226 py6-
Jleii. IIocJle YBeJlH"tIeHHH �eHbI cpe,n;HHH �eHa KHHr C 6HpKOii «BbIro,n;HO» COCTaBHJla
90 py6J1eii, a cpe,n;HHH �eHa KHHr 6e3 6HpKH- 210 py6J1eii. Bce YCJlOBHH BbInOJl-
HeHbI.
OTBeT: a) ,n;a; 6) ,n;a; B) 20.
CO,ZJ;epataHHe KpHTepHH BaJIJILI
BepHO rrOJIyqeHbI Bce rrepeqHCJIeHHbIe (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTbI 4
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 O,lJ;HH H3 CJIe,lJ;YIOID;HX pe3YJIbTaTOB:- 060cHoBaHHoe pemeHHe rrYHKTa a;
- 060cHoBaHHoe pemeHHe rrYHKTa 6; 1
-
HCKOMa-H On;eHKa B rrYHKTe 8;
-
rrpHMep B rrYHKTe 8, 06eCrreqHBalOID;HH TOqHOCTb rrpe,lJ;bI,lJ;YID;eH On;eHKH
PemeHHe He COOTBeTcTByeT HH O,lJ;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aKCU.MaJtbllblU 6aJtJt 4
@ 2021. 000 «ll3,ZJ;aTeJIbCTBO «Ha:U;HOHaJIbHOe 06pa30BaHHe». @ 2021. QOY ,l];rrO «M�HMO»
KOIIHpoBaHHe, paCilpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMep'tleCKHX :u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIap;aTeJIeii He ,lJ;OIIYCKaeTCH
232 Er3. TVinOBblE 3K3AMEHAL.J.VlOHHbIE BAPVlAHTbl
BapuaHT 21
OD a) PemHTe ypaBHeHHe COSX+2COS( 2x- ; )=.J3sin2X-l.6) Han,D;HTe Bce KOpHH 3Toro ypaBHeHHH, rrpHHa,D;Jlemarn;He OTpe3KY [- 51t; -
7
2
1t
].PemeHHe.
a) IIpeo6paayeM ypaBHeHHe:
cosx+cos2x+.J3 sin2x ==.J3 sin2x-l;
cosx+2cos2
x-I == -1;
cos x . ( 1 + 2 cos x ) == O.
1 n 2n3Ha"tlHT, cosx==O HJlH cosX==--, oTKy,n;a x==-+nk, x== + -+2nk, r,n;e kEZ.
223
6) C IIOMOID;blO "tIHCJlOBOH oKpymHocTH oT6epeM KOpHH,
rrpHHa,D;Jlemarn;He OTpe3KY [- 51t; -
7
2
1t
].9n 7n 14n
IIoJly"tlHM "tIHCJla -
-,- - H --
2 2 3 -5Jt
3aMe1lallUe. OT6op KopHeH MomeT 6bITb 06oCHoBaH H JlI06bIM
,n;pyrHM cIIoco6oM: C IIOMOID;blO rpa<l>HKa, perneHHeM JlHHeH-
HbIX HepaBeHCTB H T. II.
14Jt--
3
9Jt--
2
n 2nOTBeT: a) -+nk, + -+2nk, r,n;e kEZ; 6)
2 3
9n--
2'
7n--
2'
14n--
3·
CO.D;epataHHe KpHTepHH BaJIJILI
060CHOBaHHO rrOJIyqeHbI BepHbIe OTBeTbI B 060HX rrYHKTax 2
06ocHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe a,
HJIH
rrOJIyqeHbI HeBepHbIe OTBeTbI H3-3a BbIqHCJIHTeJIbHOH omH6KH, HO rrpH 3TOM 1
HMeeTCH BepHaH rrOCJIe�OBaTeJIbHOCTb Bcex marOB pemeHHH 060HX rrYHKTOB:
rrYHKTa a H rrYHKTa 6
PemeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aKCU.MaJlbHblU 6aJlJl 2
ODTO"'lKH A, B H C JlemaT Ha oKpymHocTH OCHOBaHHH KOHyca C BepmHHon S, rrpH"'leM
A H C ,n;HaMeTpaJlbHO IIpOTHBOIIOJlOmHbI. TO"tlKa M -
cepe,n;HHa BC.
a) �oKamHTe, "tITO IIpHMaH SM o6paayeT C IIJlOCKOCTblO ABC TaKoH me yroJl, KaK
H IIpHMaH AB C IIJlOCKOCTblO SBC.
6) HaH,n;HTe yroJl Mem,n;y IIpHMOH SA H IIJlOCKOCTblO SBC, eCJlH AB == 6, BC == 8
H SC == 5.J2.
@ 2021. 000 (<li3.n;aTeJIbCTBO «Ha:U;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY ,l];rrO «MIJ;HMO»
KOIIHpoBaHHe, pacIIpocTpaHeHHe H HCIIOJIb30BaHHe B KOMMep'tleCKHX :u;eJIHX 6e3 IIHCbMeHHoro pa3perneHHH IIpaBOo6JIa.n;aTeJIeH He .n;OIIYCKaeTCH
PEWEHVI� VI KPVlTEPVIVI OU.EHVlBAHVI� 3A.L1,AHVl1I113-19 �.a.:��
PemeHHe.
a) IIpoeK�HH TO"tlKH S Ha IIJlOCKOCTb OCHOBaHHH KOHyca-
�eHTp 0 ero OCHOBaHHH. TaK KaK OM 1- BC H SM 1- BC ,
yroJl HaKJlOHa SM K ABC -
STO yroJl SMO. STOT me yroJl
HBJlHeTCH yrJloM Mem�y IIpHMOH OM H IIJlOCKOCTblO SBC.
YroJl Mem�y IIpHMOH AB H SBC TaKoH me, TaR KaK
IIpHMbIe OM H AB IIaparrJleJlbHbI.
h h6) CHHYCOM HCKOMoro yr Jla HBJlHeTCH "tIHCJlO - ==
,
SA 5.J2
r�e h -
paCCTOHHHe OT TO"tlKH A �o IIJlOCKOCTH SBC. Pac-
hCTOHHHe OT TO"tlKH 0 �o IIJlOCKOCTH SBC paBHO
- (TaK KaK
2
o -
cepe�HHa CA). STO paCCTOHHHe-
BbICOTa OT IIpHMO-
SO-OMyrOJlbHOrO TpeyroJlbHHKa SOM, KOTopaH paBHa
SM
ABOM == - == 3-, SM == -J SC
2- MC
2== .J34
2
h SO-OM 15
2 SM
-
.J34·
s
A B
H SO == -J SM2
- M02
== 5 , IIOSTOMY
h 3CJle.n;oBaTeJlbHO, Clmyc HCKOMoro yrJla paBeH
5.J2-
m·
OTBeT: 6) arcsin � .
17
CO,ZJ;epataHHe KpHTepHH BaJIJIbI
HMeeTCH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a, H 060CHOBaHHO rrOJIY-2
"tIeH BepHbIH OTBeT B rrYHKTe 6
HMeeTcH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a,
HJIH1
060cHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6, B03MomHO, C HCrrOJIb30BaHHeM
YTBepm,n;eHHH rrYHKTa a, rrpH 3TOM rrYHKT a He BblrrOJIHeH
PerneHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepe"tIHCJIeHHbIX Bblrne 0
M aKCU.MaJtbHblU 6aJtJt 2
�PernHTe HepaBeHcTBo 41og� ( sin
3x ) + 81og 2 (sin x ) > 1 .
PemeHHe.
IIpeo6paayeM BbIpameHHe 9log� (sin x )+8Iog 2 (sin x )-1 > 0 IIPH YCJlOBHH sin x > O.
1
IIycTb log2(sinx)==t, Tor�a 9t2
+8t-1 > 0, oTKy�a t < -1 HJlH t > -.
9
CJle�OBa TeJlbHO, 0 < sin x < 0,5 HJlH sin x > �. BTopoe HepaBeHCTBO HeB03MomHO,
a H3 HepaBeHCTBa 0 < sin x < 0,5 IIoJly"tlaeM:
@ 2021. 000 «ll3.n;aTeJIbCTBO «Ha:U;HOHaJIbHOe 06pa30BaHHe». @ 2021. QOY ,1];IIO «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX :u;eJIHX 6e3 IIHCbMeHHoro pa3perneHHH IIpaBOo6JIa.,n;aTeJIeH He .n;OIIYCKaeTCH
234 Er3. TVinOBblE 3K3AMEHAL.J.VlOHHbIE BAPVlAHTbl
1t 51t21tk < x < -+ 21tk HJlH -+ 21tk < x < 1t+ 21tk, r,n;e k E Z.
6 6
OTBeT: (21tk; � + 21tk ], [5
6
1t
+ 21tk; 1t + 21tk ), r,ll;e k E Z .
Co.n;epataHHe KpHTepHH BaJIJIbI
060CHOBaHHO IIOJI�eH BepHbIH oTBeT 2
06ocHoBaHHo IIOJIyqeH OTBeT, OTJIHqalOm;HHCH OT BepHoro HCKJIlOqeHHeM MHome-
1t 51tCTBa
- + 21tk H/HJIH MHomeCTBa -+21tk,6 6
1HJIH
IIOJIyqeH HeBepHbIH OTBeT H3-3a BbIqHCJIHTeJIbHOH ornH6KH, HO IIPH 3TOM HMeeTCH
BepHaH IIOCJIe�OBaTeJIbHOCTb Bcex rnarOB perneHHH
PerneHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, IIepeqHCJIeHHbIX BbIrne 0
M aICCU.MaJtbHblU 6aJtJt 2
[illHa rHnoTeHyae AB H KaTeTax BC H AC npHMoyroJlbHoro TpeyroJlbHHKa ABC OTMe"tleHbI
TO"tlKH M, N H K COOTBeTCTBeHHO, rrpH"tIeM npHMaH NK napaJlJleJlbHa npHMoH AB
1H BM = BN = - KN. TO"tlKa P -
cepe,n;HHa OTpe3Ka KN.2
-
a) �OKajKHTe, "tITO "tIeTbIpexyroJlbHHK BCP M -
paBHo6e,n;peHHaH Tpanen;HH.
6) HaH,n;HTe nJlOm;a,n;b TpeyroJlbHHKa ABC, eCJlH BM = 1 H L.BCM = 15° .
PemeHHe.
a) IIoCKOJlbKY npHMaH NP naparrJleJlbHa npHMoH
1BM H NP=-KN=BM=BN, "tIeTbIpexyroJlbHHK
2
BNPM -
pOM6. 3Ha"tlHT, npHMaH PM naparrJleJlb-
Ha rrpHMoH BC, a TaK KaK CP -
Me,n;H8;Ha npHMO-
yroJlbHoro TpeyroJlbHHKa KCN, npoBe,n;eHHaH
H3 BepIIIHHbI npHMoro yr Jla, TO
1CP=-KN=BN=PM=BM.
2
CJle,n;oBaTeJlbHO, "tIeTbIpexyroJlbHHK BCP M
16) IIoCKOJlbKY CP = - KN = BN = PM, TpeyroJlbHHK CP M paBHo6e,n;peHHbIH, n03TOMY
2
L.PCM = L.CMP = L.BCM .
A
K
B N c
paBHo6e,n;peHHaH Tpanen;HH.
3Ha"tlHT, CM -
6HcceKTpHca yrJla BCP. Tor,n;a
LABC = L.M BC = L.BCP = 2L.BCM = 2.15° = 30°,
LKNC = LABC = 30°, CN = KN cos 30° = 2..J3
=.J3 ,
2
BC =CN +BN =.J3 +1, AC = BCtg30° = .r�1 .
@ 2021. 000 (dI3,ZJ;aTeJIbCTBO «Ha:U;HOHaJIbHOe 06pa30BaHHe». @ 2021. QOY ,l];rrO «M�HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMep'tleCKHX :u;eJIHX 6e3 IIHCbMeHHoro pa3perneHHH IIpaBOo6JIa.,n;aTeJIeH He ,ZJ;onYCKaeTCH
PEWEHVI� VI KPVlTEPVIVI OU.EHVlBAHVI� 3A.L1,AHVll/113-19 235
CJIe)J;OBaTeJIbHO,
8 ABC =.!AC .BC =.!(.J3 +1)..J3 +1
=
2.J3 +3.
2 2 .J3 3
2.J3 + 3OTBeT: .
3
Co.n;epataHHe KpHTepHH
HMeeTCH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a, H 060CHOBaHHO rrOJIY-
qeH BepHbIH OTBeT B rrYHKTe 6
060cHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6,
HJIH
HMeeTCH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a, H rrpH 060cHoBaHHoM
pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a apHcpMeTHqeCKOH omH6KH
HMeeTcH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a,
HJIH
rrpH 060cHoBaHHoM pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a
apHcpMeTHqeCKOH omH6KH, 1
HJIH
060cHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6 C HCrrOJIb30BaHHeM YTBepm,n;eHHH
rrYHKTa a, rrpH 3TOM rrYHKT a He BbIrrOJIHeH
BaJIJILI
3
2
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aICCU.MaJtbHblU 6aJtJt 3
[ill IIJIaHHPyeTcH BbI)J;aTb JIbrOTHbIH Kpe)J;HT Ha �eJIoe "tIHCJIO MHJIJIHOHOB py6JIeH Ha "tIeTbIpe
rO)J;a. B cepe)J;HHe Kam)J;oro rO)J;a )J;eHcTBHH Kpe)J;HTa )J;OJIr aaeMID;HKa BoapacTaeT Ha 20 0/0
ITO cpaBHeHHIO C Ha"tlarrOM rO)J;a. B KOH�e 1-ro H 2-ro rO)J;OB aaeMID;HK BbIITJIa"tlHBaeT
TOJIbKO ITpO�eHTbI ITO Kpe)J;HTY, OCTaBJIHH )J;OJIr HeHaMeHHO paBHbIM ITepBOHa"tlarrbHOMY.
B KOH�e 3-ro H 4-ro rO)J;OB aaeMID;HK BbIITJIa"tlHBaeT O)J;HHaKOBbIe CYMMbI, ITOrarnaH BeCb
)J;OJIr ITOJIHOCTblO. HaH)J;HTe HaHMeHbrnHH paaMep Kpe)J;HTa, ITPH KOTOpOM o6ID;aH CYMMa
BbIITJIaT aaeMID;HKa ITpeBbICHT 8 MJIH Py6JIeH.
PemeHue.
060aHa"tlHM paaMep Kpe)J;HTa "tIepea 8. B KOH�e 1-ro H 2-ro rO)J;OB aaeMID;HK BbIITJIa-
"tIHBaeT ITO 0,28. Bcero 0,48 aa )J;Ba rO)J;a.
PaccMoTpHM ITOrarneHHe Kpe)J;HTa aa CJIe)J;YIOID;He )J;Ba rO)J;a. B cepe)J;HHe 3-ro rO)J;a
)J;OJIr BoapacTeT )J;O 1,28. 060aHa"tlHM "tIepea x paaMep BbIITJIa"tlHBaeMOH CYMMbI B KOH�e
3-ro H 4-ro rO)J;OB. IIocJIe BbIITJIaTbI B KOH�e 3-ro rO)J;a )J;OJIr paBeH 1,28-
x, a B cepe-
)J;HHe 4-ro rO)J;a OH paBeH 1,2(1,28-
x). B KOH�e 4-ro rO)J;a BeCb )J;OJIr )J;OJImeH 6bITb
ITOrarneH, TO eCTb ITOCJIe)J;HHH BbIITJIaTa paBHa 1,2(1,28-
x) H ITO YCJIOBHIO paBHa x.
3Ha"tlHT,
x =
1448 =
3681,2(1,28
-
x) = x, 2,2x = 1,448,220 55'
H o6ID;HH paaMep BbIITJIaT paBeH o 48 +72
8 =
948 . no YCJIOBIUO,
55 55
948 > 8 948 > 440 .
55'
@ 2021. 000 «M3,ZJ;aTeJIbCTBO «Ha:U;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY ,l];rrO «MD;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMep'tleCKHX :u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6)1a.,ZJ;aTeJIei He ,ZJ;OIIYCKaeTCH
236 Er3. TVinOBblE 3K3AMEHAL.J.VlOHHbIE BAPVlAHTbl
IIpH S == 5 3TO HepaBeHCTBO BepHO, a npH S == 4 OHO HeBepHO, KaK H npH MeHbrnHX S.
OTBeT: 5 MJlH py6J1eH.
CO�ep3taHHe KpHTepHH BaJIJILI
060CHOBaHHO rroRyqeH BepHbIM OTBeT 3
BepHo rrOCTpoeHa MaTeMaTHqeCKaH Mo,n;eRb, pemeHHe CBe,n;eHO K HCCRe,n;OBaHHIOu
STOH Mo,n;eRH, H rrOR�eH pe3YRbTaT:2
-
HeBepHbIM OTBeT H3-3a BbIqHCRHTeRbHoM omH6KH;-
BepHbIM OTBeT, HO pemeHHe He,n;OCTaTOQHO 060cHoBaHo
BepHo rrOCTpoeHa MaTeMaTHqeCKaH Mo,n;eRb, pemeHHe CBe,n;eHO K HCCRe,n;OBaHHIO1
STOM Mo,n;eRH, rrpH STOM pemeHHe MomeT 6bITb He 3aBepmeHO
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCReHHbIX BbIme 0
M aICCU.MaJtb1-lblU 6aJtJt 3
� HaH,lI;HTe Bce 3Ha-qeHH5I a, IIPH Kam,ll;OM H3 KOTOpbIX CHCTeMa ypaBHeHHH
{( ay -ax + 2)(y
-
x + 3a) == 0,
Ixyl ==a
HMeeT' pOBHO rneCTb perneHHH.
PemeHHe.
1I3 BToporo ypaBHeHHH CHCTeMbI nOJly"tlaeM: a > o.
ECJlH a == 0, TO nOJly"tlaeM CHCTeMY
{y-x=o,
OTKY,lI;a x=y=O.xy
== 0,
3Ha"tlHT, a > O. IIOJly"tlHM CHCTeMY
{( a(y - x) + 2)(y
-
x + 3a) == 0,
I xy I == a,
2
oTKy,n;a y-x==-- HJlH y-x==-3a, xy== + a.
a
a
IIpH 3TOM YCJlOBHH rHnep60Jla y== -
rrepeceKa-x
eT Kam,n;ylO H3 npHMbIX B ,n;BYX pa3J1H"tIHbIX TO"tlKax.
:3TO ,n;aeT "tIeTbIpe pa3J1H"tIHbIX perneHHH ,n;aHHoH CHCTeMbI.
Ha Koop,n;HHaTHoH nJlOCKOCTH MbI nOJly"tla-
eM ,n;Be napaJIJleJlbHbIe npHMbIe H ,n;Be rHnep-
60JlbI. ECJlH ,n;�e npHMbIe COBna,n;aIOT, TO rneCTb pa3-
2JlH"tIHbIX perneHHH HeB03MomHO. IIo3ToMY - * 3a,
a
OTKY,lI;a a"* � .
@ 2021. 000 «M3,n;aTeJIbCTBO «Han;HOHaJIbHOe 06pa30BaHHe). @ 2021. qOY ,l];rro «MD;HMO)KOIIHpoBaHHe, pacIIpocTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX n;eJIHX 6e3 IIHCbMeHHoro pa3perneHHH IIpaBOo6JIa,n;aTeJIeii He ,n;OIIYCKaeTCH
PEWEHVls:I VI KPVlTEPVIVI OL.J.EHVlBAHVls:I 3AAAHVl1I113-19 237
a
E�e )J;Ba perneHHH nOJly"tlalOTcH npH nepeCe"tleHHH npHMbIX rHnep6oJlOH y==--
X
B )J;BYX pa3J1H"tIHbIX TO"tlKax. �JlH 3Toro HymHo, "tITo6bI rHnep60Jla )J;Bam)J;bI nepeCeKaJla
TY npHMYIO, "tITO )J;aJIbrne OT Ha"tlaJla Koop)J;HHaT, H He HMeJla 06�HX TO"tleK C npHMoH,
"tITO 6J1Hme K Ha"tlarry Koop)J;HHaT. �JlH 3Toro HymHo, "tITo6bI H3 )J;BYX KBa,n;paTHbIx ypaB-
HeHHH
x2 -!x+a == 0 H x
2-3ax+a == 0
a
O)J;HO HMeJlO pOBHO )J;Ba pa3J1H"tIHbIX KOpHH, a )J;pyroe He HMeJlO KopHeH. �HCKpHMHHaH-
TbI 3THX ypaBHeHHH )J;OJlmHbI 6bITb npOTHBOnOJlOmHbIX 3HaKOB. IIoJly"tlaeM:
( :2- 4a ) ( 9a
2- 4a ) < 0 ·
Y"tIHTbIBaH, "tITO a > 0, npHXO)J;HM K HepaBeHCTBY
(a3
-1 )(9a-4) > 0,
4OTKY)J;a a <- HJlH a> 1.
9
4OTBeT: 0 < a < -; a > 1.
9
Co.n;epataHHe KpHTepHH BaJIJILI
060CHOBaHHO IIOJIyqeH BepHbIH OTBeT 4
C IIOMOm;blO BepHoro paccym,n;eHHH IIOJIyqeHO MHomeCTBO 3HaqeHHH a,
4 3OTJIHqalOm;eeCH OT HCKOMoro TOJIbKO BKJIlOqeHHeM TOqeK a = 1 H/HJIH a =-
9
B pemeHHH BepHO HaH,n;eHbI Bce rpaHHqHbIe TOqKH MHomeCTBa 3HaqeHHH a
( a = 1, a = : ' a = � ), HO HeBepHO orrpe�eJIeHbI rrpOMemYTKH 3HaqeHHH a,
HJIH
IIOJIyqeH HeBepHbIH OTBeT H3-3a BbIqHCJIHTeJIbHOH omH6KH, HO IIPH 3TOM BepHO
BbIIIOJIHeHbI Bce marH pemeHHH
2
BepHO paCCMOTpeH XOTH 6bI O,n;HH H3 CJIyqaeB pemeHHH, H IIOJIyqeHbI ,n;Ba
rrpOMemYTKa (- <X.J; : ) H (1; + <X.J ), B03MomHO, C BKJIIOqeHHeM rpaHHqHbIX TOqeK,
1
HJIH
3a,n;aqa BepHo CBe,n;eHa K HCCJIe,n;OBaHHIO B3aHMHoro paCIIOJIOmeHHH rHIIep60JI
H IIpHMbIX (aHaJIHTHqeCKH HJIH rpacpHqeCKH)
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, IIepeqHCJIeHHbIX BbIme 0
M aICCU.MaJtbHblU 6aJtJt 4
@ 2021. 000 (dla.n;aTeJIbCTBO «HaU;HOHaJIbHOe o6paaOBaHHe». @ 2021. QOY ,1];IIO «MD;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMep'tleCKHX u;eJISlX 6ea IIHCbMeHHoro pa3pemeHHSI IIpaBOo6JIa.n;aTeJIeii He .n;OIIYCKaeTCSI
238 Er3. TVinOBblE 3K3AMEHAL.J.VlOHHbIE BAPVlAHTbl
�M3BecTHo, "tITO B KOIIIeJlbKe JlemaJlo n MOHeT, Kam,ZJ;aH H3 KOTOpbIX MOrJla HMeTb
,ZJ;OCTOHHCTBO 2, 5 HJlH 10 py6J1eii. AHH c,ZJ;eJlaJIa Bce CBOH nOKynKH, paCnJlaTHBIIIHCb
3a Kam,ZJ;YIO nOKynKY OT,ZJ;eJlbHO 6e3 C,ZJ;a"tlH TOJlbKO 3THMH MOHeTaMH, nOTpaTHB npH 3TOM
Bce MOHeTbI H3 KOIIIeJlbKa.
a) MOrJlH JlH Bce ee nOKynKH COCTOHTb H3 6J10I(HOTa 3a 56 py6J1eii H pyrIKH 3a 29 py6J1eii,
eCJlH n == 14?
6) MOrJlH JlH Bce ee nOKynKH COCTOHTb H3 "tIaIIIKH "tIaH 3a 10 py6J1eii, CbIpKa 3a 15 py6J1eiiH nHpomKa 3a 20 py6J1eii, eCJlH n == 19?
B) KaKoe HaHMeHbIIIee KOJlH"tIeCTBO nHTHpy6J1eBbIx MOHeT MOrJlO 6bITb B KOIIIeJlbKe,
eCJlH AHH KynHJla TOJlbKO aJIb60M 3a 85 py6J1eii H n == 24?
-
PemeHHe.
a) �a. HanpHMep, 3anJlaTHB 3a 6J10KHOT 5 ,ZJ;ecHTHpy6J1eBbIx MOHeT H 3 ,ZJ;Byxpy6-JleBbIe MOHeTbI (8 MOHeT), a 3a py"tlKY 1 ,ZJ;ecHTHpy6J1eBYIO MOHeTY, 3 nHTHpy6J1eBbIeH 2 ,ZJ;Byxpy6J1eBbIe (6 MOHeT).
6) IIpe,ZJ;noJlomHM, "tITO AHH c,ZJ;eJlaJIa nOKynKH Tpe6yeMbIM o6pa30M. 3a "tIaIIIKY "tIaH
OHa 3anJlaTHJla JlH60 1 ,ZJ;ecHTHpy6J1eBYIO MOHeTY, JlH60 2 nHTHpy6J1eBbIe, JlH60 5 ,ZJ;BYX-
py6J1eBbIx. 3a CbIpOK AHH ,ZJ;OJlmHa 6bIJla 3anJlaTHTb XOTH 6bI O,ZJ;HY nHTHpy6J1eBYIO MO-
HeTY, a Ha6paTb OCTaBIIIHeCH 10 py6J1eii MomHO O,ZJ;HHM H3 Tpex YKa3aHHbIX BbIIIIe cno-
C060B. 3Ha"tlHT, 3a CbIpOK OHa 3anJlaTHJla JlH60 2, JlH60 3, JlH60 6 MOHeT. CJle,ZJ;oBaTeJlb-
HO, 3a "tIaii H CbIpOK OHa 3anJlaTHJla JlH60 11 MOHeT, JlH60 He 60Jlee 8 MOHeT.
B nepBOM cJly"tlae OHa 3anJlaTHJla 3a nHpomOK 8 MOHeT. OHH He MOrJlH 6bITb Bce
,ZJ;Byxpy6J1eBbIe. 3Ha"tlHT, cpe,ZJ;H HHX JlH60 6bIJla ,ZJ;eCHTHpy6J1eBaH MOHeTa, JlH60 no Kpaii-Heii Mepe ,ZJ;Be nHTHpy6J1eBbIe MOHeTbI. OCTaBIIIHecH 10 py6J1eii HeJlb3H Ha6paTb 6 HJlH
7 MOHeTaMH. IIpHIIIJlH K npOTHBOpe"tlHIO.
Bo BTOpOM cJly"tlae OHa 3anJlaTHJla 3a nHpomOK He MeHee 11 MOHeT. :3TO TaKme
HeB03MomHO, nOCKOJlbKY Tor,ZJ;a nOJlyrIHJlOCb 6bI He MeHee 22 py6J1eii.
IIoJly"tleaHbIe npOTHBOpe"tlHH nOKa3bIBaIOT, "tITO AHH He MOrJla C,ZJ;eJlaTb YKa3aHHbIe
nOKynKH Tpe6yeMbIM 06pa30M.
B) IIYCTb AHH KynHJla aJIb60M 3a 85 py6J1eii, nOTpaTHB 24 MOHeTbI: k ,ZJ;Byxpy6J1e-
BbIX, l nHTHpy6J1eBbIx H m ,ZJ;ecHTHpy6J1eBbIx. Tor,ZJ;a 2k + 5l + 10m == 85 H k + l + m == 24.
OTclO,ZJ;a nOJly"tlaeM k == 24- l- m, 48 - 2l- 2m + 5l + 10m == 85 H 8m == 37 - 3l. 3Ha"tlHT,
37 - 3l ,ZJ;eJlHTCH Ha 8. CJle,ZJ;oBaTeJlbHO, "tIHCJlO l He"tleTHO. IIpH l paBHOM 1, 3 H 5 BbI-
pameHHe 37 - 3l paBHO 34, 28 H 22 COOTBeTCTBeHHO H He ,ZJ;eJlHTCH Ha 8.
IIpHMep k == 15, l == 7 H m == 2 nOKa3bIBaeT, "tITO AHH MOrJla 3anJlaTHTb pOBHO 7 nH-
THpy6J1eBbIx MOHeT.
OTBeT: a) ,ZJ;a; 6) HeT; B) 7.
CO,ZJ;epataHHe KpHTepHH BaJIJILI
BepHo rrOJI�eHbI Bce rrepeqHCJIeHHbIe (CM. KpHTepHH Ha 1 6a.rrJI) pe3YJIbTaTbI 4
BepHo rrOJI�eHbI TPH H3 rrepeqHCJIeHHbIX (CM. KpHTepHH Ha 1 6a.rrJI) pe3YJIbTaTOB 3
BepHo rrOJIyqeHbI ,lJ;Ba H3 rrepeqHCJIeHHbIX (CM. KpHTepHH Ha 1 6a.rrJI) pe3YJIbTaTOB 2
BepHo rrOJI�eH O,lJ;HH H3 CJIe,lJ;YIOm;HX pe3YJIbTaTOB:- 06ocHoBaHHoe pemeHHe rrYHKTa a;
- 06ocHoBaHHoe pemeHHe rrYHKTa 0; 1
-
HCKOMaH On;eHKa B rrYHKTe 8;-
rrpHMep B rrYHKTe 8, 06eCrreqHBalOm;HH TOqHOCTb rrpe,lJ;bI,lJ;ym;eH On;eHKH
PemeHHe He COOTBeTcTByeT HH O,lJ;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme 0
M aICCU.MaJtbHblU oaJtJt 4
@ 2021. 000 «H3,ZJ;aTeJIbCTBO «Ha:U;HOHaJIbHOe o6pa30BaHHe». @ 2021. qOY ,rr;rro «MD;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX :u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIa.,ZJ;aTeJIeH He ,ZJ;OIIYCKaeTCH
PEWEHLt1� Lt1 KPLt1TEPLt1Lt1 OU.EHLt1BAHLt1� 3AAAHLt1� 13-19 239
BapuaHT 27
lliJ a) PemHTe ypaBHeHHe (2 cos2
x + 3sin x-
3) .log2 ( .J2 cos x ) = 0 .
6) Haii,n;HTe Bce KOpHH 3Toro ypaBHeHHH, IIpHHa,n;JIemaID;He OTpeaKY [-5n;- 3n ] .
PemeHHe.
a) IIpeo6paayeM ypaBHeHHe:
(2 -2sin2
x + 3sin x -3) .log2 ( .J2 cosx) = 0;
(2sin2
x-3sinx+l ).log2 (.J2 cosx) = O.
3HaQHT, log2 ( .J2 cos x ) = 0 HJIH 2 sin2
x- 3 sin x + 1 = 0 IIPH YCJIOBHH COS X > O.
1t
PernaH IIepBoe ypaBHeHHe, IIOJIyqaeM .J2 cos x = 1, oTKy,n;a x = + - + 21tk, r,n;e k E Z .
4
.1
PernaH BTopoe ypaBHeHHe, IIOJIyqaeM sinx = 1 HJIH Slnx =-. YtIHTbIBaH YCJIOBHe2
1t
COS x > 0, IIOJIyqaeM: x =-+21tk, r,n;e k E Z.6
6) Haii,n;eM KOpHH, y,n;OBJIeTBOpHIOID;He HepaBeHCTBY -51t < x < -31t. IIoJIyqaeM:
1t 1t
-51t < -- + 21tk < -31t, -51t < - + 21tk < -31t HJIH
4 4
1t
-51t < - + 21tk < -31t .
6
1t 171tx=---41t=-- .
4 4Ma IIepBoro HepaBeHCTBa Haxo,n;HM k = -2 H IIPH 3TOM
1t 151tMa BToporo HepaBeHCTBa: k = -2, a x =--41t = --.
4 4
1t 231tMa T p eTbero He p aBeHCTBa: k = -2 a x = - - 41t = --
.,
6 6
3aMe1lallUe. OT6op KopHeii MomeT 6bITb 06ocHoBaH H JII06bIM ,n;pyrHM cIIoco6oM: C IIO-
MOID;blO rpaQ.:>HKa, C IIOMOID;blO e,n;HHHQHOii oKpymHocTH H T. II.
1t 1t
OTBeT: a) -+21tk, + -+21tk, k E Z; 6)6 4
231t 151t--
6'
-4'
171t--
4
CO�ep3taHHe KpHTepHH BaJIJILI
060CHOBaHHO rrOJIyqeHbI BepHble OTBeTbI B 060HX rrYHKTax 2
060cHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe a,
HJIH
rrOJI�eHbI HeBepHble OTBeTbI H3-3a BblqHCJIHTeJIbHOH omH6KH, HO rrpH 3TOM 1
HMeeTCH BepHaH rrOCJIe�OBaTeJIbHOCTb Bcex marOB pemeHHH 060HX rrYHKTOB:
rrYHKTa a H rrYHKTa 0
PemeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX Bblme 0
M aICCUMaJtbHblU oaJtJt 2
@ 2021. 000 «Ha.n;aTeJIbCTBO «Ha:U;HOHaJIbHOe o6paaOBaHHe». @ 2021. qOY ,l];rro «MD;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIbaOBaHHe B KOMMepqeCKHX :u;eJIHX 6ea IIHCbMeHHoro paaperneHHH IIpaBOo6JI8.,lJ;aTeJIeH He .n;OIIYCKaeTCH
240 Era. TlIInOBblE aK3AMEHALJ.lIIOHHbIE BAPlIIAHTbl
[K]B �HJIHH,n;pe o6pa3yrom;aH rreprreH,n;HKYJIHpHa rrJIOCKOCTH OCHOBaHHH. Ha oKpymHocTH
o,n;Horo H3 oCHoBaHHH �HJIHH,n;pa BbI6paHbI TO"tlKH A, B H C, a Ha oKpymHocTH ,n;pyroro
OCHOBaHHH-
TO"tlKa Cl ' rrpH"tIeM CC
l
-
o6pa3yrom;aH �HJIHH,n;pa, a AC -
,n;HaMeTp oc-
HOBaHHH. M3BecTHo, "tITO LACB == 45°, AB == 3.J2 , CCl
== 6.
a) ,D:oKamHTe, "tITO yrOJI Mem,n;y rrpHMbIMH ACl
H BC paBeH 60°.
6) HaH,n;HTe paCCTOHHHe OT TO"tlKH B ,n;o rrpHMoH ACl
.
PemeHHe.
a) IIycTb BBl-
o6pa3yrom;aH �HJIHH,n;pa. Tor,n;a BBlClC-
rrpHMoyrOJIbHHK, rr03TOMY
yrOJI Mem,n;y rrpHMbIMH ACl
H BC paBeH yrJIY AClB l. C
1
YrOJI ABC orrHpaeTCH Ha ,n;HaMeTp OCHOBaHHH �HJIHH,n;pa, rr03TO-
MY OH rrpHMOH. 3Ha"tlHT, rrpHMaH BlC l , rraparrJIeJIbHaH rrpHMOH BC, BlrreprreH,n;HKYJIHpHa rrpHMbIM AB H BBl. TaKHM o6pa30M, rrpHMaH
BlC l rreprreH,n;HKYJIHpHa rrJIOCKOCTH ABBl , a 3Ha"tlHT, yroJI ABlC l
rrpHMoH.
B rrpHMoyrOJIbHOM TpeyrOJIbH HKe ABlC l
& 4 == BC == AB == 3.J2, A& == � AB2
+ BBf == � AB2
+ cc;f == 3.J6 .
3Ha"tlHT, LA4& == 60°.
6) IIpHMaH AB rreprreH,n;HKYJIHpHa rrpHMbIM BC H BBl. TaKHM o6pa30M, rrpHMaH AB
rreprreH,n;HKYJIHpHa rrJIOCKOCTH BBlC. CJIe,n;oBaTeJIbHO, TpeyrOJIbHHK ABCl rrpHMoyrOJIbHbIH,
rro3ToMY HCKOMoe paccToHHHe paBHO ero BbICOTe h, rrpoBe,n;eHHoH K rHrroTeHY3e. IIoJIyqaeM:
I 2 2 I7S . I 2 2 rn .
AB . B4 I7S
B4 ==" BC + cq == 3" 6, A4 ==" AB + Bq == 6" 2, h == == 1,5" 6 ·
A4
OTBeT: 1,5.J6.
CO,ll;epmaHHe KpHTepHJI BaJIJILI
HMeeTCH BepHoe .n;OKa3aTeJIbCTBO YTBepm.n;eHHH nYHKTa a, H 060CHOBaHHO nOJIY-2
"tIeH BepHbIH OTBeT B nYHKTe 6
HMeeTcH BepHoe .n;OKa3aTeJIbCTBO YTBepm.n;eHHH nYHKTa a,
HJIH1
06ocHoBaHHo nOJIY"tIeH BepHbIH OTBeT B nYHKTe 6, B03MomHO, C HCnOJIb30BaHHeM
YTBepm.n;eHHH nYHKTa a, npH 3TOM nYHKT a He BbInOJIHeH
PemeHHe He COOTBeTcTByeT HH O.n;HOMY H3 KpHTepHeB, nepe"tlHCJIeHHbIX BbIme 0
M aKCUMaJlb1tblU 6aJlJl 2
OUPeIIIHTe HepaBeHcTBo log5( : +2 )-log5(X+2) < lOg5( XX:
1
).PemeHHe.
JIeBaH "tIaCTb HepaBeHCTBa HMeeT CMbICJI rrpH -2 < x < -1,5 HJIH x> 0, rr03TOMY rrpH
3TOM YCJIOBHH HepaBeHCTBO rrpHHHMaeT BH,n;:
2x+3 x+1< .
x(x+2)-
x2 ,
2X2 +3x x2
+3x+2< .
x2
(x+2)-
x2
(x+2),
@ 2021. 000 «H3.n;aTeJIbCTBO «HaU;HOHaJIbHOe o6pa30BaHHe». @ 2021. qOY )];rro «MIJ;HMO»KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH IIpaBOo6JIa,n;aTeJIeH He .n;OIIYCKaeTCH
PEWEHlII� III KPlIITEPlIIlII OU.EHlIIBAHlII� 3AAAHlIIVl13-19 241
00
x2
-2 < 0;
(X-v'2)(X+v'2) < O.
OTKy,n;a- v'2 < x < v'2. Y"tIHTbIBaH OrpaHH"tIeHHe -2 < x < -1,5 HJIH x > 0, rrOJIy"tlaeM:
O<x < v'2.
OTBeT: (0; v'2] .
CO,ll;epmaHHe KpHTepHJI BaJIJILI
060CHOBaHHO IIOJIy"tleH BepHbIH OTBeT 2
060cHoBaHHo IIOJIY"tleH OTBeT, OTJIH"tIalOID;HHCH OT BepHoro HCKJIlO"tIeHHeM TO"tlKH .J2,
HJIH1
IIOJIyqeH HeBepHbIH OTBeT H3-3a BbI"tIHCJIHTeJIbHOH omH6KH, HO IIPH 3TOM HMeeTCH
BepHaH IIOCJIe)J;OBaTeJIbHOCTb Bcex marOB pemeHHH
PemeHHe He COOTBeTcTByeT HH O)J;HOMY H3 KpHTepHeB, IIepe"tlHCJIeHHbIX BbIme 0
M aKCUMaJlb1tblU 6aJlJl 2
OKpymHocTb C �eHTpOM 01 KacaeTCH OCHOBaHHH BC H AD, a TaKme 60KOBOH CTOpOHbI AB
Tparre�HH ABCD. OKpymHocTb C �eHTpOM 02 KacaeTCH CTOpOH BC, CD H AD. MaBecTHo,
"tITO AB == 9, BC == 8, CD == 4, AD == 15.
a) ,D:oKamHTe, "tITO rrpHMaH 0102 rrapaJIJIeJIbHa OCHOBaHHHM Tparre�HH ABCD.
6) HaH,n;HTe 0102.
PemeHHe.
a) TO"tlKa 01 paBHoy,n;aJIeHa OT rrpHMbIX AD H BC. B C
3Ha"tlHT, TO"tlKa 01 JIemHT Ha cpe,n;HeH JIHHHH Tparre-
�HH ABCD. AHaJIOrH"tIHO, TO"tlKa 02 JIemHT Ha cpe,n;-
HeH JIHHHH Tparre�HH ABCD. CJIe,n;oBaTeJIbHO, rrpH-
MaH °1°
2 co,n;epmHT cpe,n;HIOIO JIHHHIO Tparre�HH
ABCD, a aHa"tlHT, rrpHMaH 0102 rraparrJIeJIbHa OCHO-
BaHHHM Tparre�HH ABCD.
6) IIycTb K -
cepe,n;HHa CTOpOHbI AB, a L -
ce-
pe,n;HHa CTOpOHbI CD. TO"tlKa 01 paBHoy,n;arreHa A D
OT rrpHMbIX AB, BC HAD, rr03TOMY JIy"tlH AOI H BOIHBJIHIOTCH 6HcceKTpHcaMH yrJIOB DAB H ABC COOTBeTCTBeHHO. 3Ha"tlHT,
LBAO:t + LABO:t = LBAD;LABC
= 900
,
TO eCTb LAOIB == 900
CJIe,n;oBaTeJIbHO, KOI-
Me,n;HaHa, rrpoBe,n;eHHaH K rHrroTeHyae
AB rrpHMoyrOJIbHOrO TpeyrOJIbHHKa AOIB.AHaJIOrH"tIHO, TpeyrOJIbHHK C0
2D rrpHMoyrOJIbHbIH, a L02
-
Me,n;HaHa, rrpoBe,n;eHHaHK ero rHrroTeHyae CD. TO"tlKH K, 01' 02 H L JIemaT Ha cpe,n;HeH JIHHHH Tparre�HH ABCD.
3Ha"tlHT,
AD+BC AB CD0 1 0 2
== KL-K0 1 -L02 ==
'
---- == 5.222
OTBeT: 6) 5.
@ 2021. 000 «ll3.n;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY )];rrO «MIJ;HMO»KonHpoBaHHe, pacnpOCTpaHeHHe H HCnOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 nHCbMeHHoro pa3pemeHHH npaBoo6JIa.n;aTeJIeH He .n;onYCKaeTCH
242 Er3. TlIInOBblE 3K3AMEHAL..J.lIIOHHbIE BAPlIIAHTbl
CO,ll;epmaHHe KpHTepHJI BaJIJILI
HMeeTCH BepHoe �OKa3aTeJIbCTBO YTBepm�eHHH rrYHKTa a, H 060CHOBaHHO rrOJIY-
qeH BepHbIH OTBeT B rrYHKTe 6
06OCHOBaHHO rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6,
HJIH
HMeeTCH BepHoe �OKa3aTeJIbCTBO YTBepm�eHHH rrYHKTa a, H rrpH 06ocHoBaHHoM
pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a apH<pMeTHqeCKOH omH6KH
HMeeTcH BepHoe �OKa3aTeJIbCTBO YTBepm�eHHH rrYHKTa a,
HJIH
npH 06ocHoBaHHoM pemeHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a apH<pMe-
TH"IIeCKOH omH6KH,
HJIH
06ocHoBaHHo rrOJIyqeH BepHbI� OTBeT B rrYHKTe 6 C HCrrOJIb30BaHHeM YTBepm�eHHH
nYHKTa a, IIPH 3TOM rrYHKT a He BbIrrOJIHeH
PemeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIme
3
2
1
M aKCUMaJlbHblU 6aJlJl
o
3
[ill15 MapTa llJIaHHpyeTcH B3HTb Kpe,n;HT B 6aHKe Ha 26 MeCH�eB. YCJIOBHH ero B03BpaTa
TaKOBbI:
1-ro "({HCJIa Kam,n;oro MeCH�a ,n;OJIr B03paCTaeT Ha 3%
110 cpaBHeHHIO C KOH�OM
IIpe,n;bI,n;Yl.QerO MeCH�a;
co 2-ro 110 14-e "({HCJIO Kam,n;oro MeCH�a He06xo,n;HMO BbIIIJIaTHTb "({aCTb ,n;OJIra;
15-ro "({HCJIa Kam,n;oro MeCH�a C 1-ro 110 25-ii ,n;OJIr ,n;OJImeH 6bITb Ha 40 TbICHq
Py6JIeii MeHbme ,n;OJIra aa 15-e "({HCJIO IIpe,n;bI,n;Yl.QerO MeCH�a;
K 15-MY "({HCJIY 26-ro MeCH�a Kpe,n;HT ,n;OJImeH 6bITb IIOJIHOCTblO IIorameH.
KaKYIO CYMMY llJIaHHpyeTcH B3HTb B Kpe,n;HT, eCJIH 06I.QaH CYMMa BbIIIJIaT 1I0CJIe 1I0JIHOrO
ero lIorameHHH COCTaBHT 1924 TbICH"({H Py6JIeii?
PemeHHe.
IIYCTb CYMMa Kpe,n;HTa paBHa A TbICH"({ Py6JIeii. IIo YCJIOBHIO ,n;OJIr lIepe,n; 6aHKoM
(B TbIC. Py6JIeii) 110 COCTOHHHIO Ha 15-e "({HCJIO Kam,n;oro MeCH�a ,n;OJImeH YMeHbmaTbCH
,n;o HYJIH CJIe,n;YIOI.QHM 06pa30M:
A; A -
40; A -
80; ...; A -
960; A -
1000; O.
IIepBoro "({HCJIa Kam,n;oro MeCH�a ,n;OJIr B03paCTaeT Ha 3 0/0, 3H-a"({HT, IIOCJIe,n;OBaTeJIb-
HOCTb pa3MepOB ,n;OJIra (B TbIC. Py6JIeii) 110 COCTOHHHIO Ha 1-e "({HCJIO TaKOBa:
1,03A; 1,03(A-
40); 1,03(A-
960); 1,03(A-
1000).
CJIe,n;OBaTeJIbHO, BbIIIJIaTbI (B TbIC. Py6JIeii) ,n;OJImHbI 6bITb CJIe,n;YIOI.QHMH:
0,03A + 40; 0,03(A-
40) + 40; 0,03(A-
960) + 40; 1,03(A-
1000).
Bcero CJIe,n;yeT BbIIIJIaTHTb
25.0,03.2A-960
+1000+1,03(A-IOOO)=1,78A-390 (TblC. py6JIeH),2
oTKy,n;a 1,78A- 390 = 1924; 1, 78A = 2314; A = 1300.
3HaQHT, CYMMa, KOTOpyro IIJIaHHpyeTcH B3HTb B Kpe,n;HT, paBHa 1 MJIH 300 TbIC. py6JIeii.
OTBeT: 1 MHJIJIHOH 300 TbICHQ py6JIeii.
@ 2021. 000 «Ha.n;aTeJIbCTBO «HaU;HOHaJIbHOe 06paaOBaHHe». @ 2021. qOY ,[(rro «MIJ;HMO»KonHpoBaHHe, pacrrpOCTpaHeHHe H HCnOJIbaOBaHHe B KOMMep-qeCKHX u;eJIHX 6ea nHCbMeHHoro paapemeHHH npaBoo6JIa.n;aTeJIeH He .n;onYCKaeTCH
PEWEHV1� V1 KPV1TEPV1V1 OU.EHV1BAHV1� 3AAAHV1� 13-19 243
CO,IJ;ep3taHHe KpHTepHJI BaJIJILI
06ocHoBaHHo nORy�eH BepHbIM OTBeT 3
BepHo nOCTpoeHa MaTeMaTH�eCKaH Mo�eRb, perneHHe CBe�eHO K HCCRe�OBaHHIO
3TOM Mo�eRH H nORyqeH pe3YRbTaT:2
-
HeBepHbIM OTBeT H3-3a BbI�HcRHTeRbHoM ornH6KH;-
BepHbIM OTBeT, HO perneHHe He�OCTaTO�HO 06ocHoBaHo
BepHo nOCTpoeHa MaTeMaTH�eCKaH Mo�eRb, perneHHe CBe�eHO K HCCRe�OBaHHIO1
3TOM Mo�eRH, npH 3TOM perneHHe MomeT 6bITb He 3aBeprneHO
PerneHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, nepe�HCReHHbIX BbIrne 0
MaKCUMaJlbHblU 6aJlJl 3
[ill HaH,D;HTe Bce 3HaQeHHSI rrapaMeTpa a, rrpH Ram,D;OM H3 ROTOpbIX CHCTeMa ypaBHeHHH
{x
4+ y2 = a2 ,
x2
+ y= I a + 11
HMeeT pOBHO qeTbIpe pemeHHH.
PemeHHe.
Mcxo,n;HaH CHCTeMa paBHOCHJIbHa CHCTeMe ypaBHeHHH:
{x
4
+(la+ll-x2 t = a2
,
{2 X
4-
2Ia+llx2
2+2a+l = 0,
y= la+ll -x
2; y -la+ll -x .
TaKHM o6paaoM, Hcxo,n;HaH CHCTeMa ypaBHeHHH HMeeT pOBHO qeTbIpe paaJIHqHbIX pe-
meHHH Tor,n;a H TOJIbKO Tor,n;a, Kor,n;a 6HKBa,n;paTHoe ypaBHeHHe
2x4
-2Ia+llx2
+2a+l = 0
HMeeT pOBHO qeTbIpe paaJIHqHbIX KOpHH. 8TO BbIIIOJIHHeTCH, Kor,n;a KBa,zJ;paTHoe ypaBHe-
HHe
2t2
-2Ia+llt+2a+l = 0
HMeeT pOBHO ,n;Ba IIOJIOmHTeJIbHbIX KOpHH.
QTo6bI IIOJIyqeHHOe KBa,n;paTHoe ypaBHeHHe HMeJIO ,n;Ba KOpHH, ero ,n;HCKpHMHHaHT
,n;OJImeH 6bITb IIOJIOmHTeJIbHbIM:
4 ( a + 1)2 - 8 ( 2a + 1 ) > 0; a2
- 2a -1 > 0 ,
oTKy,n;a a < 1-.J2 HJIH a> 1+.J2 .
QTo6bI KOpHH IIOJIyqeHHOrO KBa,n;paTHoro ypaBHeHHH 6bIJIH o,n;Horo aHaKa, cBo6o,n;-
HbIH qJIeH 3Toro ypaBHeHHH ,n;OJImeH 6bITb IIOJIOmHTeJIbHbIM:
2a + 1 > 0 ,
oTKy,n;a a>- 0,5.
QTo6bI KOpHH KBa,n;paTHoro ypaBHeHHH 6bIJIH IIOJIOmHTeJIbHbIMH, K03<P<PH�HeHT
IIPH t ,n;OJImeH 6bITb OTpH�aTeJIbHbIM, TO eCTb a *- -1 .
TaKHM o6paaoM, Hcxo,n;HaH CHCTeMa ypaBHeHHH HMeeT pOBHO qeTbIpe pemeHHH IIPH
- 0,5 < a < 1- .J2 HJIH a > 1 +.J2 .
OTBeT: (-0,5; 1-.J2); (1 + .J2; + 00 ) .
@ 2021. 000 «H3,ZJ;aTeJIbCTBO «Ha�HoHa.rrbHoe 06pa30BaHHe». @ 2021. qOY )];nO «MIJ;HMO»KonHpoBaHHe, pacnpOCTpaHeHHe H HCnOJIb30BaHHe B KOMMepqeCKHX �eJIHX 6e3 nHCbMeHHoro pa3pemeHHH npaBoo6JIa,z:J;aTeJIeH He ,ZJ;onYCKaeTCH
244 Er3. TlIInOBblE 3K3AMEHAL..J.lIIOHHbIE BAPlIIAHTbl
CO,ll;epmaHHe KpHTepHJI
060CHOBaHHO nOJIyqeH BepHbIH OTBeT
C nOMOIIJ;blO BepHoro paccym�eHHH nOJIyqeHo MHomeCTBO 3Ha"tleHHH a, OTJIH"tIa-
lOIIJ;eeCH OT HCKOMoro TOJIbKO BKJIlO"tIeHHeM TO"tleK a = -0,5, a == 1-.J2, H/HJIH
a==l+.J2
BaJIJILI
4
3
B pemeHHH BepHO HaH�eHbI Bce rpaHH"tIHbIe TO"tlKH MHomeCTBa 3Ha"tleHHH a
( a == 1-.J2, a == -0,5, a == 1 + .J2), HO HeBepHO onpe�eJIeHbI npOMemYTKH 3Ha"tle-
HHH a,
MJIM
nOJIy"tleH HeBepHbIH OTBeT H3-3a BbI"tIHCJIHTeJIbHOH omil6KH, HO npH 3TOM BepHO
BbInOJIHeHbI Bce marH pemeHHH
BepHo paCCMOTpeH XOTH 6bI O�HH H3 CJIy-qaeB pemeHHH H nOJIy-qeHbI �Ba npOMemYT-
Ka (- 00; 1-.J2) H (1 + .J2; + 00 ), B03MomHO, C BKJIlO"tIeHHeM rpaHH"tIHbIX TO"tleK,
MJIM
3a)J;a"tla BepHO CBe�eHa K HCCJIe�OBaHHIO B3aHMHoro pacnOJIOmeHHH oKpymHocTH
H npHMbIX (aHaJIHTH"tIeCKH HJIH rpa(}>H"tIeCKH)
PemeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, nepe"tlHCJIeHHbIX BbIme
M aKCUMaJlb1tblU 6aJlJl
2
1
o
4
[illHa ,n;OCKe HalIHcaHO 12 pa3JIHqHbIX HaTyparrbHbIX qHCeJI. Cpe,n;Hee apHcpMeTHqeCKOe
ceMH HaHMeHbillHX H3 HHX paBHO 8, a cpe,n;Hee apHcpMeTHqeCKOe ceMH HaH60JIbillHX
paBHO 16.
a) MomeT JIH HaH60JIbillee H3 3THX ,n;BeHa,n;�aTH qHCeJI paBHHTbCH 18?
6) MomeT JIH cpe,n;Hee apHcpMeTHqeCKOe Bcex ,n;BeHa,n;�aTH qHCeJI paBHHTbCH II?
B) HaH,n;HTe HaHMeHbillee 3HaqeHHe cpe,n;Hero apHcpMeTHqeCKOrO Bcex ,n;BeHa,n;�aTH qHCeJI.
PemeHHe.
a) ECJIH HaH60JIbillee qHCJIO paBHO 18, TO CYMMa ceMH HaH60JIbillHX qHCeJI
He 60JIbille 18 + 17 + 16 + 15 + 14 + 13 + 12 == 105, a HX cpe,n;Hee apHcpMeTHqeCKOe MeHb-
ille HJIH paBHO 15.
6) IIYCTb CYMMa IIHTH HaHMeHbillHX qHCeJI paBHa A, CYMMa illeCToro H ce,n;bMoro
IIO BeJIHqHHe qHCeJI paBHa B, a CYMMa IIHTH HaH60JIbillHX qHCeJI paBHa C. IIpe,n;lIo-
JIOmHM, qTO cpe,n;Hee apHcpMeTHqeCKOe Bcex ,n;eCHTH qHCeJI paBHO 11. Tor,n;a IIOJIyqaeM:
A + B == 56, B + C == 112, A + B + C == 132,
oTKy,n;a B == 36, C == 76. 3TO HeB03MomHO, IIOCKOJIbKY ,n;OJImHO BbIIIOJIHHTbCH HepaBeH-
CTBO C> 2,5B.
B) IIo YCJIOBHIO A + B == 56, B + C == 112. Cpe,n;Hee apHcpMeTHqeCKOe 6y,n;eT HaHMeHb-
illHM, Kor,n;a 6y,n;eT HaHMeHbilleH CYMMa Bcex qHCeJI:
A + B + C == (A + B) +(B + C)- B == 168 - B.
3HaqHT, HymHo HaHTH HaH60JIbillee 3HaqeHHe B.
IIYCTb qHCJIa, HalIHcaHHbIe Ha ,n;OCKe, paBHbI al , a
2 , ..., al2 , IIpHqeM
al
< a2
< ... < al2
.
Tor,n;a �+2 < ag, ar+ 2 < ag, �+3 <ag, ar+ 3 < llto, �+4 < llto' ar+ 4 < lltl'
0f3 +5 < lltl, ar +5 < llt2, a6 +6 < llt2, ar +1 < ag, oTKy,n;a
5a6 +5ar +35 < 2ag +2ag +2llto +2lltl +2llt2; 5B+35 < 2C.
@ 2021. 000 «H3,ZJ;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY ,[(rro «MIJ;HMO»KonHpoBaHHe, pacnpOCTpaHeHHe H HCnOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 nHCbMeHHoro pa3pemeHHH npaBoo6JIa,ZJ;aTeJIeH He ,ZJ;onYCKaeTCH
PEWEHlII� III KPlIITEPlIIlII OU.EHlIIBAHlII� 3A)J,AHlIIVl13-19 245
3HaqHT, 7B < 2B+2C-35 =189; B < 27 .
IIoKameM, qTO qHCJIO B MomeT paBHHTbCH 27. HaIIpHMep, eCJIH Ha ,n;OCKe HaIIHca-
HbI qHCJIa 2, 3, 6, 8, 10, 13, 14, 15, 16, 17, 18, 19, TO YCJIOBHH 3a,n;aqH BbIIIOJIHeHbI
H B = 27. TaKHM o6pa30M, HaHMeHbrnee 3HaqeHHe cpe,n;Hero apHcpMeTHqeCKOrO paBHO
A+B+C 168-B=11 75.
12 12'
OTBeT: a) HeT; 6) HeT; B) 11,75.
CO,ll;ep3taHHe KpHTepHJI BaJIJILI
BepHo nOJIy"tleHbI Bce nepe"tlHCJIeHHbIe (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTbI 4
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 O,n;HH H3 CJIe,n;YIOW;HX pe3YJIbTaTOB:- 06ocHoBaHHoe perneHHe nYHKTa a;
- 06ocHoBaHHoe perneHHe nYHKTa 6; 1
-
HCKOMaH o:u;eHKa B nYHKTe 8;..
-
npHMep B nYHKTe 8, 06eCne"tlHBalOW;HH TO"tlHOCTb npe,n;bI,n;yw;eH o:u;eHKH
PerneHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, nepe"tlHCJIeHHbIX BbIrne 0
M aKCUMaJlb1tblU 6aJlJl 4
BapuaHT 31
OU a) PemHTe ypaBHeHHe 4. 16 cOSX - 9. 4 COSX + 2 = O.
6) Haii,n;HTe Bce ROpHH 3Toro ypaBHeHHH, rrpHHa,n;JIemaru;He OTpe3RY [- 21t; -
; ] .
PemeHHe.
a) IIycTb t = 4 cosx
, Tor,n;a ypaBHeHHe 3aIIHrneTCH B BH,n;e 4 t2
- 9 t + 2 = 0, oTKy,n;a
1t = - HJIH t = 2.
4
1 1
IIpH t =
4rrOJIyqaeM: 4 cOSX
=
4; cosx = -1, oTRy,n;a x = -1t + 21tk, k EZ
.
1 1t
IIpH t=2 rrOJIyqaeM: 4 cosx
=2; cosx=
2' oTRy,n;a x= +
"3+21tn
, nEZ.
6) C IIOMOID;blO qHCJIOBOH oKpymHocTH oT6epeM KOpHH,
rrpHHa,n;JIemaru;He OTpe3RY [- 21t; -
; ] .
51t
IIoJIyqHM qHCJIa:
-3;-1t
·
-1t -21t
3aMellaHue. OT6op KopHeH MomeT 6bITb 060CHOBaH H JII06bIM
,n;pyrHM CIIoco6oM:' C IIOMOID;blO rpacpHKa, perneHHeM JIHHeH-
HbIX HepaBeHCTB H T. II.
1t
2
1t
OTBeT: a) -1t+21tk, k EZ; + -+21tn, n EZ; 6)3
51t--.
-1t.
3'
@ 2021. 000 «M3.n;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe». @ 2021. qOY ,IUIO «MIJ;HMO»KonHpoBaHHe,' pacnpOCTpaHeHHe H HCnOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 nHCbMeHHoro pa3pemeHHH npaBoo6JIa,z:J;aTeJIeH He .n;onYCKaeTCH
246 Er3. TlIInOBblE 3K3AMEHALJ.lIIOHHbIE BAPlIIAHTbl
CO,ll;ep3taHHe KpHTepHJI BaJIJILI
060CHOBaHHO rrOJIyqeHbI BepHbIe OTBeTbI B 060HX rrYHKTax 2
060cHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe a,
HJIH
rrOJIyqeHbI HeBepHbIe OTBeTbI H3-3a BbIqHCJIHTeJIbHOH omH6KH, HO rrpH 3TOM 1
HMeeTCH BepHaH rrOCJIe�OBaTeJIbHOCTb Bcex marOB pemeHHH 060HX rrYHKTOB-
rrYHKTa a H rrYHKTa 6
PemeHHe He COOTBeTcTByeT HH O�HOMY H3 KpHTepHeB, rrpHBe�eHHbIx BbIme 0
M aKCUMaJlbHblU 6aJlJl 2
I14l Ha pe6pe SA IIpaBHJIbHOH qeTbIpexyrOJIbHOH IIHpaMH,lJ;bI SABCD C OCHOBaHHeM ABCD
�OTMeqeHa TOqKa M, IIpHqeM SM: MA = 5 : 1. TOqKH P H Q
-
Cepe,lJ;HHbI pe6ep BC
H )UD COOTBeTCTBeHHO.
a) ,D:oKamHTe, qTO CeqeHHe IIHpaMH,lJ;bI IIJIOCKOCTblO MPQ HBJIHeTCH paBH06e,lJ;peHHOH
TpalIe�HeH.
6) HaH,lJ;HTe OTHorneHHe o6�eMoB MHororpaHHHKOB, Ha KOTopbIe IIJIOCKOCTb MPQ
pa36HBaeT IIHpaMH,lJ;Y.
PemeHHe.
a) IIycTb N -
TaKaH TOqKa Ha pe6pe SB, qTO
SN: NB = 5 : 1. TpeyrOJIbHHKH SAB H SMN lIo,lJ;o6HbI
IIO ,lJ;BYM IIpOlIop�HOHarrbHbIM CTopOHaM H yrJIY Mem,lJ;y HHMH.
3HaqHT, LSAB = LSMN, a IIpHMbIe AB H MN lIaparrJIeJIbHbI,
MN *AB. IIpHMaH PQ TaKme lIaparrJIeJIbHa IIpHMOH AB.
3HaqHT, OTpe3KH MN H PQ II aparrJIeJIbHbI H He paBHbI,
H II03TOMY CeqeHHe IIHpaMH,lJ;bI IIJIOCKOCTblO M PQ-
3TO
Tpane�HH MN PQ.
TpeyrOJIbHHKH MAQ H NBP paBHbI, IIOCKOJIbKY
MA=NB, QA=PB, H LMAQ=LNBP, II03TOMY MQ=NP,
a 3HaqHT, TpalIe�HH MNPQ paBH06e,lJ;peHHaH.
6) IIycTb o6�eM II HpaMH,lJ;bI SABCD paBeH V.
IIHTHrpaHHHK AMQBNP COCTOHT H3 qeTbIpexyrOJIbHOH IIHpaMH,lJ;bI MABPQ C OCHOBaHHeM
ABPQ H TpeyrOJIbHOH IIHpaMH,lJ;bI MBNP C OCHOBaHHeM BNP.
PaCCTOHHHe OT TOqKH M ,lJ;O IIJIOCKOCTH BNP OTHOCHTCH K paCCTOHHHIO OT TOqKH A
,lJ;O 3TOH IIJIOCKOCTH KaK 5: 6, a IIJIO�MH TpeyrOJIbHHKOB BNP H SBC OTHOCHTCH KaK
1 : 12. 3HaqHT, OTHorneHHe o6�eMoBIIHpaMH,lJ; MBNP H ASBC paBHO 5: 72, TO eCTb
5Vo6�eM IIHpaMH,lJ;bI M BN P paBeH
-
144
IIJIO�Mb IIpHMoyrOJIbHHKa ABPQ COCTaBJIHeT IIOJIOBHHY IIJIO�MH KBMpaTa ABCD.
PaccToHHHe OT TOqKH M ,lJ;O IIJIOCKOCTH ABCD OTHOCHTCH K paCCTOHHHIO OT TOqKH S
V
,lJ;O 3TOH IIJIOCKOCTH KaK 1: 6, II03TOMY o6�eM IIHpaMH,lJ;bI MABPQ paBeH12
.
TmmM o6pa30M, 06'beM AMQBNP paBeH17V
, TO eCTb OTHomeHHe 06'beMOB144
MHororpaHHHKOB AMQBNP H CDSNPQM paBHO 17: 127.
S
A B
OTBeT: 6) 17: 127.
@ 2021. 000 «H3,lJ;aTeJIbCTBO «HaU;HOHaJIbHOe 06pa30BaHHe). @ 2021. qOY ,[(rro «MIJ;HMO)KOIIHpoBaHHe, paCIIpOCTpaHeHHe H HCIIOJIb30BaHHe B KOMMepqeCKHX u;eJIHX 6e3 IIHCbMeHHoro pa3pemeHHH npaBoo6JIa,lJ;aTeJIeH He ,lJ;OnYCKaeTCH
PEWEHlII� III KPlIITEPlIIlII OU.EHlIIBAHlII� 3AAAHlIIVl13-19 247
CO�ep3taHHe KpHTepHJI BaJIJILI
MMeeTCH BepHOe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a, H 060CHOBaHHO rrOJIY-2
"tIeH BepHbIH OTBeT B rrYHKTe 6
MMeeTcH BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHHH rrYHKTa a,
MJIM1
060CHOBaHHO rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6 C HCrrOJIb30BaHHeM YTBepm�eHHSI
rrYHKTa a, rrpH 3TOM rrYHKT a He BbIrrOJIHeH
PerneHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrpHBe,n;eHHbIX BbIrne 0
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log a x 4 8
I15l PemHTe HepaBeHCTBO
( )> +
2 a·
�log3 ;7
log3 x log3 X -log3 X
PemeHHe.
IIycTb t = log a x, Tor�a HepaBeHCTBO IIpHMeT BH�:
t 4 8 t2
> +.
-
2'
t - 3 t t - 3t t( t - 3)
t2
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(t-2)2> 0.
t( t - 3)-
,
t( t -
3)
4t-12
t( t -
3)
8> 0 .
-
,
t( t -
3)
OTKy�a t < 0; t = 2; t > 3.
IIpH t < 0 IIOJI�HM: logax < 0, oTKy�a 0 < x < 1.
IIpH t = 2 IIOJIy"tlHM: logax = 2, oTKy�a x = 9.
IIpH t > 3 IIOJI�HM: logax> 3, oTKy�a x > 27.
PemeHHe HCXO�Horo HepaBeHCTBa:
O<x<l; x=9; x>27.
OTBeT: (0; 1); 9; (27; + (0).
CO�ep3taHHe KpHTepHJI BaJIJILI
060CHOBaHHO rrOJIyqeH BepHbIH OTBeT 2
060cHoBaHHo rrOJIyqeH OTBeT, OTJIHqalOID;HHCH OT BepHoro HCKJIlOqeHHeM TOqKH 9,
MJIM1
rrOJIyqeH HeBepHbIH OTBeT H3-3a BbIqHCJIHTeJIbHOH ornH6KH, HO rrpH 3TOM HMeeTCSI
BepHaH rrOCJIe,n;OBaTeJIbHOCTb Bcex rnarOB perneHHH
PerneHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCJIeHHhIX BhIme 0
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248 Er3. TlIInOBblE 3K3AMEHALJ.lIIOHHbIE BAPlIIAHTbl
lliJB Tpane�HH ABCD OCHOBaHHe AD B ,n;Ba pa3a MeHbIIIe OCHOBaHHH BC. BHYTPH Tpane�HH
B3HJIH TO"llKY M TaK, "liTO yrJIbI BAM H CDM npHMbIe.
a) ,D:oKamHTe, "liTO BM == CM.
6) HaM,n;HTe yrOJI ABC, eCJIH yrOJI BCD paBeH 640
,a paCCTOHHHe OT TO"llKH M
,n;o npHMoM BC paBHO CTopOHe AD.
PemeHHe.
a) lIycTb npHMbIe AB H CD nepeceKalOTCH B TO"llKe L.
Tor,n;a TpeyrOJIbHHKH BLC H ALD no,n;o6HbI C K03cpcpH�HeHToM
no,n;o6HH 2, nOCKOJIbKY BC == 2AD. 3Ha"llHT, A H D -
cepe,n;HHbI
BL H CL COOTBeTCTBeHHO. TaKHM o6pa30M, AM H DM -
cepe,n;HHHbIe nepneH,n;HKY JIHpbI B TpeyrOJIbHHKe BLC, a M -
�eHTp onHcaHHoM OKOJIO 3Toro TpeyrOJIbHHKa oKpymHocTH,
n03TOMY BM == CM KaK pa,n;HYCbI 3TOM oKpymHoCTH.
6) lIycTb H -
cepe,n;HHa BC, Tor,n;a M H -
cepe,n;HHHbIH
nepneH,n;HKYJIHp K CTopOHe BC. 3Ha"llHT, TpeyrOJIbHHKH BHM
H CHM paBHo6e,n;peHHbIe npHMoyrOJIbHbIe, n03TOMY
B H c
L
LBMC == 90°. lIo CBOMCTBY BnHcaHHoro yrJIa
LBLC= LB�C-
45°, oTRy,n;a LABC = LLBC = 180° - LBLC - LBCL = 71°.
OTBeT: 6) 71 0.
CO�ep3taHHe KpHTepHJI BaJIJILI
MMeeTC.H BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHH.H rrYHKTa a, H 060CHOBaHHO rrOJIY-
qeH BepHbIH OTBeT B rrYHKTe 6
06ocHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6,
MJIM
3
HMeeTC.H BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHH.H rrYHKTa a, H rrpH 06ocHoBaHHoM
perneHHH rrYHKTa 6 rrOJIyqeH HeBepHbIH OTBeT H3-3a apHcpMeTHqeCKOM ornH6KH
MMeeTC.H BepHoe ,n;OKa3aTeJIbCTBO YTBepm,n;eHH.H rrYHKTa a,
MJIM
rrpH 06ocHoBaHHoM perneHHH rrYHKTa 6 rrOJIyqeH HeBepHbIM OTBeT H3-3a
apHcpMeTH"LIeCKOH ornH6KH, 1
MJIM
06ocHoBaHHo rrOJIyqeH BepHbIH OTBeT B rrYHKTe 6 C HCrrOJIb30BaHHeM YTBepm,n;eHH.H
rrYHKTa a, rrpH aTOM rrYHKT a He BbIrrOJIHeH
PerneHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepeqHCJIeHHbIX BbIrne 0
2
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Y CJIOBHH ero B03BpaTa TaKOBbI:
Kam,n;bIii HHBapb ,n;OJIr YBeJIHqHBaeTCH Ha r%
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ro,n;a;
C cpeBparrH IIO HIOHb Kam,n;oro ro,n;a Heo6xo,n;HMO BbIIIJIaTHTb O,n;HHM IIJIaTemOM qaCTb
,n;OJIra.
Haii,n;HTe r, eCJIH H3BeCTHO, qTO Kpe,n;HT 6y,n;eT IIOJIHOCTblO IIorarneH 3a ,n;Ba ro,n;a, IIpHqeM
B IIepBbIii ro,n; 6y,n;eT BbIIIJIaqeHO 160 000 py6JIeii, a BO BTOpoii ro,n;- 240 000 py6JIeii.
PemeHHe.
r
IIYCTb k = 1 + -. IIo YCJIOBHIO ,n;OJIr IIepe,n; 6aHKoM (B py6JIHX) IIO COCTOHHHIO
100
Ha HIOJIb ,ZJ;OJImeH YMeHbmaTbCH CJIe,n;YIO�HM o6pa30M:
300000, 300000k -
160000, 300000k2
- 160000k - 240000 = 0,
oTKy.n;a 15k2
-8k-12=O; k= : ; r=20.
OTBeT: 20.
CO,IJ;ep3taHHe KpHTepHH BaJlJlLI
060CHOBaHHO rroRyqeH BepHbIM OTBeT 3
BepHo rrOCTpoeHa MaTeMaTHqeCKa.H Mo,n;eRb, perneHHe CBe,n;eHO K HCCRe,n;OBaHHIO
3TOM Mo,n;eRH H rroRyqeH pe3YRbTaT:2
-
HeBepHbIM OTBeT H3-3a BbIqHCRHTeRbHoM ornH6KH;-
BepHbIM OTBeT, HO perneHHe He,n;OCTaTOqHO 060cHoBaHo
BepHo rrOCTpoeHa MaTeMaTHqeCKa.H Mo,n;eRb, perneHHe CBe,n;eHO K HCCRe,n;OBaHHIO1
3TOM Mo,n;eRH, rrpH 3TOM perneHHe MomeT 6bITb He 3aBeprneHO
PerneHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrpHBe,n;eHHbIX BbIrne 0
M aKCUMaJlbHblU 6aJlJl 3
� Haii.n;HTe Bee 3HaqeHHH a, rrpH Kam.n;OM H3 KO TOpbIX ypaBHeHHe
..J 5 -7 x oln( 9x2
-
a2
) = ..J 5 -7 x oln( 3x + a)
HMeeT pOBHO O,n;HH KopeHb.
PemeHHe.
Mcxo,n;Hoe ypaBHeHHe paB HOCHJI bHO ypaBHeHHIO
..J 5 -7 x 0 (In( 3x2
- a2
) -In( 9x + a) ) = 0 .
PaccMoTpHM ,ZJ;Ba CJIyqaH.
IIep BbIii cJIyqaii:
..J 5 -7 x = 0 IIPH YCJIOBHH {3x -
a > 0,3x+a>0.
5IIOJIyqaeM x = -. Y CJIOBHe IIpHHHMaeT BH,n;
7
15--a>O
7'
15 15oTKy,n;a --<a<-.
7 715- + a > 0,
7
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250 Er3. TV1nOBbIE 3K3AMEHALJ.V10HHbIE BAPV1AHTbl
5 15 15To eCTb B aTOM CJIyqae x=- npH --<a<-.
777
BTOpoii cJIyqaii:
In( 9x2
-
a2
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IIoJIyqaeM:
In( 9x2
-
a2
) = In( 3x + a) ;
npH YCJIOBHH 5 - 7 x > 0 .
oTKy�a
a+1x=
3'
8 a+1a < -. To eCTb B aTOM CJIyqae x =
7 3
{(3X -
a )(3x + a) = 3x + a,
{3X -
a = 1,3x + a > 0; 3x + a > 0,
15 _7a+7 > 0,a > -
-. Y CJIOBHe npHHHMaeT BH�2 3
5 a+1
KOpHH ypaBHeHHH x = -
H X =
7 3
1 8npH --<a < -.
2 7
8COBna�aIOT npH a =
;:;.
oTKy�a
15 1TaKHM o6paaoM, HCXO�Hoe ypaBHeHHe HMeeT pOBHO O�HH KopeHb npH - - < a < - -
7 2
8 15H - < a<-.
7 7
15 1 8 15OTBeT: - - < a < - -; - < a < -
·
7 2 7 7
CO�ep3taHHe RpHTepHJI
060CHOBaHHO I10JIyqeH BepHbIH OTBeT
BaJIJILI
4
C I10MOW;blO BepHoro paccym,n;eHHH I10JIyqeHo MHomeCTBO 3Ha�eHHH a,
1 8OTJIH�alOw;eeCH OT HCKOMoro TOJIbKO HCKJIIO�eHHeM To�eK a = - -
H/HJIH a = -
2 7
3
B pemeHHH BepHO HaH,n;eHbI Bce rpaHH�HbIe TO�KH MHomeCTBa 3Ha�eHHH
a ( a = -
1
7
5, a = -
� , a = � , a = 175
), HO HeBepHO orrpe.n;eJIeHbI rrpOMemYTKH
3Ha�eHHH a,
HJIH
BepHO I1poH,n;eHbI Bce 3TaIlbI pemeHHH, HO HeBepHO HaH,n;eHbI rpaHH�HbIe TO�KH
MHomeCTBa 3Ha�eHHH a H3-3a BbI�HCJIHTeJIbHOH omH6KH
2
BepHO paCCMOTpeH XOTH 6bI O,n;HH H3 CJIy�aeB pemeHHH H I10JIy�eH O,n;HH
H3 I1pOMemYTKOB ( _
15.
15
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7
HJIH ( _!. 8
] , B03MomHO, C HCKJIIO�eHHeM
2'71
rpaHH�HOH TO�KH BbIIIOJIHeHbI Bce marH pemeHHH
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, I1epe�HCJIeHHbIX BbIme 0
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PEWEHV1s:1 V1 KPV1TEPV1V1 OLJ.EHV1BAHV1s:1 3A)J,AHV111113-19 251
mJHa �OCKe HanHcaHO 35 pa3JIHqHbIX HaTYPaJIbHbIX qHCeJI, Kam�oe H3 KOTOpbIX JIH60
qeTHOe, JIH60 ero �eCHTHqHaH 3anHCb OKaHqHBaeTCH Ha �H<PPY 3. CYMMa HanHcaHHbIX
qHCeJI paBHa 1062._
a) MomeT JIH Ha �OCKe 6bITb pOBHO 27 qeTHbIX qHCeJI?
6) MorYT JIH pOBHO �Ba qHCJIa Ha �OCKe OKaHqHBaTbCH Ha 3?
B) KaKoe HaHMeHbIIIee KOJIHqeCTBO qHCeJI, OKaHqHBalOID;HXCH Ha 3, MomeT 6bITb
Ha �ocKe?
PemeHHe.
a) HanpHMep, CJIe�YIOID;HH Ha60p qHCeJI Y�OBJIeTBOpHeT YCJIOBHIO 3a�aqH:
3, 13, 23, 33, 43, 53, 63, 73, 2, 4, 50, 52, 56.
6) IIycTb Ha �OCKe HanHcaHO pOBHO �Ba qHCJIa, OKaHqHBalOID;HXCH Ha 3. Tor�a
Ha �OCKe 33 qeTHbIX qHCJIa. Hx CYMMa He MeHbIIIe, qeM CYMMa 33 HaHMeHbIIIHX qeTHbIX
qHCeJI:
68.332+4+...+66= =1122>1062.
2
8TO npOTHBOpeqHT TOMY, qTO CYMMa HanHcaHHbIX qHCeJI paBHa 1062.
B) IIycTb Ha �OCKe HanHcaHO n qHCeJI, OKaHqHBalOID;HXCH Ha 3, H 35 -
n qeTHbIX
qHCeJI. Tor�a CYMMa qHCeJI, OKaHqHBalOID;HXCH Ha 3, He MeHbIIIe
3+ 13+... +( 3+ 10(n -1)) =( 6 + 10( n -1)). n
= 5n2
-
2n,2
a CYMMa qeTHbIX qHCeJI He MeHbIIIe
2 + 4 +... + 2(35 - n) = (35 - n)( 36 -n) = n2
-71n + 1260.
TaKHM o6pa30M,1062 > 6n
2-73n+ 1260; 6n
2-73n+ 198 < 0
,
oTKy�a, yqHTbIBaH, qTO n-
�eJIoe, nOJIyqaeM n > 5.
ECJIH Ha �OCKe HanHcaHO nHTb qHCeJI, OKaHqHBalOID;HXCH Ha 3, H 30 qeTHbIX qHCeJI,
TO HX CYMMa HeqeTHa. 3HaqHT, qHCeJI, OKaHqHBalOID;HXCH Ha 3, 60JIbIIIe nHTH.
IIpHBe�eM npHMep IIIeCTH qHCeJI, OKaHqHBalOID;HXCH Ha 3, H 29 qeTHbIX qHCeJI,
CYMMa KOTOpbIX paBHa 1062:
3, 13, 23, 33, 43, 53, 2, 4, 54, 56, 82.
OTBeT: a) �a; 6) HeT; B) 6.
CO�ep3taHHe RpHTepHJI BaJIJILI
BepHo rrOJIy�eHbI Bce rrepe�HCJIeHHbIe (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTbI 4
BepHo rrOJIy�eHbI TPH H3 rrepe�HCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 3
BepHo nOJIy�eHbI ,n;Ba H3 rrepe�HCJIeHHbIX (CM. KpHTepHH Ha 1 6aJIJI) pe3YJIbTaTOB 2
BepHo rrOJIyqeH O,n;HH H3 CJIe,n;YIOW;HX pe3YJIbTaTOB:- 06ocHoBaHHoe pemeHHe rrYHKTa a;
- 06ocHoBaHHoe pemeHHe rrYHKTa 0; 1
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HCKOMaH On;eHKa B rrYHKTe 6;
-
rrpHMep B rrYHKTe 6, o6ecrre�HBalOW;HH TO�HOCTb rrpe,n;bI,n;yw;eH On;eHKH
PemeHHe He COOTBeTcTByeT HH O,n;HOMY H3 KpHTepHeB, rrepe�HCJIeHHbIX BbIme 0
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Ers. <I>HIIH -
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