Toan pt.de057.2012
- 1. S畛GD&TH畉ID働NG TR働畛NGTHPTNINHGIANG 畛THI TH畛畉IH畛CL畉N1NM2012 M担nthi: TON,Kh畛iAvB Th畛igianlmbi:180ph炭t,kh担ngk畛th畛igiangiao畛 PH畉NCHUNGCHOT畉TC畉THSINH(7,0i畛m) C但uI(2,0i畛m) Chohms畛 - + = + 2 x m y x c坦畛th畛l(Cm) 1) Kh畉os叩ts畛餌bi畉nthi棚nvv畉渋畛th畛c畛ahms畛khi 1m = . 2) T狸mc叩cgi叩tr畛c畛am畛動畛ngth畉ngd: 2 2 1 0x y+ - = c畉t(Cm)t畉ihaii畛mAvBsaochotam gi叩cOABc坦di畛nt鱈chb畉ng 1(Olg畛ct畛a畛). C但uII(2,0i畛m) 1) Gi畉iph動董ngtr狸nh 2 2 sin sin 3 tan 2 (sin sin3 ) cos cos3 x x x x x x x + = + 2) Gi畉iph動董ngtr狸nh 2 2 2 1 1 3x x x x x+ + + - + = . C但uIII(1,0i畛m)T鱈nht鱈chph但n 1 3 2 0 ( 1) 2x x x dx- -嘆 C但uIV(1,0i畛m)Choh狸nhlngtr畛ヂ畛ng . ' ' 'ABC A B C c坦叩yABCltamgi叩cc但nt畉iC,c畉nh叩yAB b畉ng2avg坦cABCb畉ng300 .T鱈nhth畛t鱈chc畛akh畛ilngtr畛ヂ . ' ' 'ABC A B C bi畉tkho畉ngc叩chgi畛ahai 動畛ngth畉ngABv 'CB b畉ng 2 a C但uV(1,0i畛m)Chox,ylc叩cs畛th畛cthay畛ith畛am達ni畛uki畛n 2 2 1x y xy+ + = .T狸mgi叩tr畛l畛n nh畉tvgi叩tr畛nh畛nh畉tc畛abi畛uth畛c: 2 2 S x y xy= - PH畉NRING(3,0i畛m)Th鱈sinhch畛動畛clmm畛ttronghaiph畉n(Ph畉nAho畉cB) A. Theoch動董ngtr狸nhchu畉n C但u VI.a(2,0i畛m) 1) Trongm畉tph畉ngOxy,chotamgi叩cABCc坦動畛ngcaoAH,trungtuy畉nCMvph但ngi叩ctrong BD.Bi畉t 17 ( 4余1), ( 余12) 5 H M- vBDc坦ph動董ngtr狸nh 5 0x y+ - = .T狸mt畛a畛畛nhAc畛atam gi叩cABC. 2) Trong kh担ng gian Oxyz, cho 動畛ng th畉ng 1 1 : 2 3 1 x y z+ + D = = - v hai i畛m (1余2余 1),A - (3余 1余 5)B - - . Vi畉t ph動董ng tr狸nh動畛ng th畉ng d i qua i畛m A v c畉t 動畛ng th畉ng D sao cho kho畉ngc叩cht畛B畉n動畛ngth畉ngdll畛nnh畉t,nh畛nh畉t. C但uVII.a(1,0 i畛m)T鱈nhm担unc畛as畛ph畛c z ,bi畉t 3 12z i z+ = vzc坦ph畉nth畛cd動董ng. B. Theoch動董ngtr狸nhn但ngcao C但uVI.b(2,0i畛m) 1) Trong m畉t ph畉ng Oxy, cho 動畛ng tr嘆n (C): 2 2 ( 2) ( 3) 4x y- + + = v 動畛ng th畉ng d: 3 4 7 0x y m- + - = .T狸mm畛tr棚ndc坦duynh畉tm畛ti畛mMmt畛坦k畉斬動畛chaiti畉ptuy畉n MA,MBt畛i(C)(A,Blc叩cti畉pi畛m)saochog坦cAMBb畉ng1200 . 2) Trong kh担ng gian Oxyz, cho i畛m A(1余4余2) v hai 動畛ng th畉ng l畉n l動畛t c坦 ph動董ng tr狸nh 1 2 1 1 1 : , ': 1 1 2 2 1 1 x y z x y z- + - + - D = = D = = - .Vi畉tph動董ngtr狸nh 動畛ngth畉ngdiquai畛mA, c畉t動畛ngth畉ng Dvc叩ch動畛ngth畉ng 'D m畛tkho畉ngl畛nnh畉t. C但uVII.b(1,0i畛m) T狸mm畛ph動董ngtr狸nhsauc坦2nghi畛mtr叩id畉u ( 3)25 (2 1)5 1 0x x m m m+ + - + + = H畉tβ Thi th畛 畉i h畛c www.toanpt.net
- 2. PNVBI畛UI畛MCH畉M C但u N畛idung i畛m Kh畉os叩ts畛餌bi畉nthi棚nvv畉渋畛th畛(C)c畛ahms畛 - + = + 1 2 x y x 1,00 TX: 臓 . 2 3 ' 0, 2 ( 2) y x x - = < " 孫 - + 0,25 Hms畛ngh畛chbi畉ntr棚n( 余 2) v ( 2余 )-促 - - +促 lim 1 x y 速賊促 = - TCN: 1y = - 2 2 lim , lim x x y y- + 速- 速- = -促 = +促 TC: 2x = - 0,25 L畉pBBT 0,25 1 畛th畛 4 2 足2 足4 足5 足1 O 1足2 0,25 2 2 1 0x y+ - = c畉t(Cm)t畉ihaii畛mAvB... 1,00 1 2 2 1 0 2 x y y x+ - = = - .Pthonh畛giaoi畛mc畛adv(Cm)l - + = - - + - = 孫 - + 21 2 2 0 (1), 2 2 2 x m x x x m x x 0,25 Dc畉t(Cm)t畉i2i畛mA,B (1)c坦2nghi畛mpbkh叩c足2 2 ' 1 4(2 2) 0 9 2 8( 2) ( 2) 2 2 0 m m m D = - - >狸 - 孫 <鱈 - - - + - 孫樽 0,25 G畛i 1 2,x x l2nghi畛mc畛a(1).Khi坦 1 1 2 2 1 1 余 , 余 2 2 A x x A x x 脱 旦 脱 旦 - -巽 歎 巽 歎 竪 淡 竪 淡 2 2 2 2 1 1 2 1 2 1 2( ) ( ) 2 ( ) 4 2(9 8 )AB x x x x x x x x m辿 湛= - + - = + - = -谷 短 0,25 I 2 1 1 1 1 7 . ( , ) 2(9 8 ). 9 8 1 2 2 4 82 2 OABS AB d O d m m m= = - = - = = - (tm) V畉y 7 8 m = - 0,25 Gi畉iph動董ng tr狸nh 2 2 2 1 1 3x x x x x+ + + - + = 1,00II 1 2 2 1 0x x+ + > v 2 1 0,x x x- + > " 臓 TX: 臓 TH1. 0x 贈 .Ptlu担nTM 0,25
- 3. TH2.x>0.PT 2 2 1 1 1 1 2 1 3 x x x x + + + - + = .畉t 1 , 0t t x = > Ta動畛c 2 2 2 2 2 1 3 2 3 1t t t t t t t t+ + + - + = + + = - - + 2 2 2 2 2 9 1 6 1 3 1 4t t t t t t t t t + + = + - + - - + - + = - 0,25 2 2 2 1 4 0 4 7 9(1 ) 16 8 8 7 0 8 t t t tt t t t t t =辿- 続 贈狸 狸 棚 鱈 鱈 棚 = -- + = - + - - =樽 樽 谷 0,25 畛ichi畉uv畛it>0ta動畛c 1 1t x= = Th畛l畉ith畉yx=1th畛am達npt.V畉yptc坦nghi畛mduynh畉tx=1 0,25 Gi畉iph動董ngtr狸nh 2 2 sin sin 3 tan 2 (sin sin3 ) cos cos3 x x x x x x x + = + 1,00 K:cos 0,cos3 0x x孫 孫 Pt tan sin tan3 sin3 tan 2 (sin sin3 )x x x x x x x + = + 0,25 sin( ) sin( ) (tan tan 2 )sin (tan3 tan 2 )sin3 0 sin sin3 0 cos cos2 cos3 cos2 x x x x x x x x x x x x x x - - + - = + = 0,25 sin 0 sin3 sin 0 cos3 cos x x x x x =辿 棚 棚 - = 谷 0,25 2 sin 0 sin 0 ( ) sin 2 0 cos 0 ( ) x x TM x x L = =辿 辿 棚 棚= =谷 谷 0,25 T鱈nht鱈chph但n 1 3 2 0 ( 1) 2x x x dx- -嘆 1,00 1 1 3 2 2 2 0 0 ( 1) 2 ( 2 1) 2 ( 1)I x x x dx x x x x x dx= - - = - + - -嘆 嘆 . 畉t 2 2 2 2 2 (1 ) . (0) 0, (1) 1t x x t x x tdt x dx t t= - = - = - = = 0,25 1 2 0 (1 ) ( )I t t t dt= - -嘆 0,25 1 1 5 3 4 2 0 0 ( ) 5 3 t t t t dt 脱 旦 = - = -巽 歎 竪 淡 嘆 0,25 III 1 1 2 5 3 15 - = - 0,25 T鱈nhth畛t鱈chkh畛ilngtr畛ヂ . ' ' 'ABC A B C 1,00 G畛iM,Nl畉nl動畛tltrungi畛mc畛aABv A'B'.Tamgi叩cCABc但nt畉iC suy ra AB ^ CM. M畉t kh叩c AB ^ CC' ( ') ' ' ( ')AB CMNC A B CMNC ^ ^ . K畉斬 ( ). ( ') ' ' ( ' ')MH CN H CN MH CMNC MH A B MH CA B^ ^ ^ 0,25 mp( ' ')CA B ch畛a 'CB vsongsongv畛iABn棚n ( , ') ( ,( ' ')) ( ,( ' ')) 2 a d AB CB d AB CA B d M CA B MH= = = = 0,25 IV Tamgi叩cvu担ng 0 .tan 30 3 a BMC CM BM = = Tamgi叩cvu担ng 2 2 2 2 2 2 1 1 1 4 3 1 CMN MN a MH MC MN a a MN = + = + = 0,25
- 4. T畛坦 3 . ' ' ' 1 . .2 . . 2 3 3 ABC A B C ABC a a V S MN a a= = = N M A' B' C A B C' H 0,25 T狸mgi叩tr畛l畛nnh畉tvgi叩tr畛nh畛nh畉tc畛abi畛uth畛c: 2 2 S x y xy= - 1,00 2 2 2 2 2 ( ) ( ) ( 2 ) ( ) (1 3 )S xy x y S xy x y xy xy xy= - = + - = - 0,25 畉tt xy= 2 2 2 1 1 1 3 ( ) 0 3 x y xy xy x y t+ + = - = - 続 贈 2 2 2 1 ( ) 1 0 1x y xy x y xy t+ + = + = + 続 続 - . 0,25 2 2 1 ( ) (1 3 ), 1余 3 S f t t t t 辿 湛 = = - -棚 炭谷 短 . 2 0 '( ) 2 9 0 2 9 t f t t t t =辿 棚= - = 棚 = 谷 21 2 4 ( 1) 4, (0) 0, 4 2 2 3 9 243 f f f f S S 脱 旦 脱 旦 - = = = = 贈 - 贈 贈巽 歎 巽 歎 竪 淡 竪 淡 0,25 V 2 1, 1 max 2 2 1, 1 min 2 S x y S S x y S = = - = = = - = = - = - 0,25 T狸mt畛a畛畛nhAc畛atamgi叩cABC 1,00 t DquaHv ^BDc坦pt 5 0x y- + = . (0余5)BD I ID = . 0,25 Gi畉Ｂ s畛 'AB HD = . Tam gi叩c 'BHH c坦 BI l ph但n gi叩c v c滴ng l 動畛ngcaon棚n 'BHH c但n Iltrungi畛mc畛a ' '(4余9)HH H . 0,25 ABiquaHvc坦vtcp 3 ' 余3 5 u H M 脱 旦 = = -巽 歎 竪 淡 r uuuuuur n棚nc坦ptl5 29 0x y+ - = . 0,251 T畛a畛Blnghi畛mc畛ah畛 5 29 (6余 1) 5 x y B x y + =狸 -鱈 + =樽 .Mltrungi畛mc畛a AB 4 余25 5 A 脱 旦 巽 歎 竪 淡 0,25 Vi畉tph動董ngtr狸nh 動畛ngth畉ngdiquai畛mAvc畉t動畛ngth畉ng Dsao chokho畉ngc叩cht畛B畉n動畛ngth畉ngdll畛nnh畉t,nh畛nh畉t. 1,00 G畛idltiquaAvc畉t D t畉iM ( 1 2 余3 余 1 )M t t t - + - - ( 2 2 余3 2余 ), (2余 3余 4)AM t t t AB= - + - - = - - uuuur uuur 0,25 VI.a 2 G畛i H l h狸nh chi畉u c畛a B tr棚n d. Khi 坦 ( , )d B d BH BA= 贈 . V畉y ( , )d B d l畛n nh畉t b畉ng BA H A 尊 . i畛u ny x畉y ra 0,25
- 5. . 0AM AB AM AB ^ = uuuur uuur 2( 2 2 ) 3(3 2) 4 0 2t t t t - + - - + = = (3余6余 3)M - .Ptdl 1 2 1 1 2 1 x y z- - + = = - M畉tph畉ng(P)ch畛adv Dc坦ptl: G畛iKlh狸nhchi畉uc畛aBtr棚n(P) BH BK 続 .V畉y ( , )d B d nh畛nh畉t b畉ngBK H K 尊 .L炭c坦dl動畛ngth畉ngiquaAvK 0,25 T狸m 動畛cKvvi畉tptd 0,25 T鱈nhm担unc畛as畛ph畛c z ,bi畉t 3 12z i z+ = 1,00 Gi畉Ｂs畛 , ,z x yi x y= + ﾂ . 3 3 12 ( ) 12z i z x yi i x yi+ = + + = - 0,25 3 2 3 2 2 3 2 3 3 (1) 3 (3 12) 3 12 (2) x xy x x xy x y y i x yi x y y y 狸 - =誰 - + - + = - 鱈 - + = -誰樽 0,25 Do 2 2 0 (1) 3 1x x y> = + .Th畉紳vo(2)ta動畛c 2 3 3 3(3 1) 12 2 3 0y y y y y y+ - + = - + + = (3) 0,25 VII.a Gi畉ipt(3)ta動畛c 2 1 4y x= - = .Dox>0n棚nx=2 V畉y 2 5z i z= - = 0,25 T狸mm 畛tr棚ndc坦duynh畉tm畛ti畛mMmt畛坦k畉斬動畛chaiti畉ptuy畉n MA,MBt畛i(C)(A,Blc叩cti畉pi畛m)saochog坦cAMBb畉ng1200 1,00 0,25 0,25 0,25 1 0,25 Vi畉tph動董ngtr狸nh 動畛ngth畉ngdiquai畛mA,c畉t動畛ngth畉ng Dv c叩ch動畛ngth畉ng 'D m畛tkho畉ngl畛nnh畉t. 1,00 0,25 0,25 0,25 VI.b 2 0,25 ( 3)25 (2 1)5 1 0x x m m m+ + - + + = c坦2nghi畛mtr叩id畉u 1,00 0,25 0,25 0,25 VII.b 0,25