Thi thu DHSP lan 2-2011

aa TRUONC DFISP HA NQI TRI'OI{G TFIPT CHUYEN - EFTSP rJE THr rHU DAr Fr-oc LAN u ruann zolr tr{l- 'l^: - rzl i xr ' lvluil Lill , lt ftl! T'hdi gian tdm bdi : 180 philt, kh6ng k€ thdi gian phdt di CAU 1. ( 2,0 di|m ) | 2x+L Cho hanr s0 V = - x-2 l'I(hios6tsLr.bi6nthi0nviv€d6th!(C)cirahims6. 2. Tim t6t cd c6c gi6tri cttam d6 cluongtiring y = m(x-2)+ Z cit AO thi (C) t4i hai di6m phAn biQt A, B sao cho doan AB c6 dO dei nho nhAt. C0u 2. (2,0 diilm) t. Giii phuo'ng trinh : sin2x.(l + tanx) : 3sinx.(cosx - sinx) + 3 x+3 gx-7 2. Giai bAt phuong trinh: 3sx-, -4>5.3sx-, Cffu 3. ( 1,0 di6m ) - rinh tich phan I =;€$o* Ciu 4. ( 1,0 rtiAm ) Cho hinh lAp phuo'ng ABCD.A'B'C?D' c6 clQ ddi canli bing a vi di6m N4 thu6c canh CC' sao cho 1c CM = T. tUat phing (a) di qua,t, M vir song song vdi BD chia khOi l4p phu'o'ng thdnh hai khoi 3 e o da cli6n. Tinh thO tich hai t<tr6i Aa diQn d6. CAu 5. ( 1,0 diem) Ba s6 duo'ng thay d6i a, b, c till0c doan [a, F] md B - o {2. Chri'ng minh ring : GETf +/[611a1ffi1 1> a+b+c. C6u 6. ( 2,0 dient) l. Trong mdt phing toa dg Oxy, cho tam gi6c ABC c6 C( I : 2), hai rh-rong cao xuAt ph6t til.A vd B iin luqtc6 phuongtrinh ld x * y : 0 vir 2x-y+ I =0. Tlnli di€rr tich tarri gi6c ABC. 2. Trongkhonggiantoa116 oxyz, cho mf,tphing(p)c6phuo-ngtrinh: \-zy +zz+ l=0vdrn[t cAu (S) c6 phuong trinh : x2 + y2 + z2 -4x + Sy + 6z+ l7 : 0. Tim toa dQ tAm vir b6n kinh cria drLdng trdn (C) ld giao crira m{t ptring (p) vd mat cAl (S). C6u7.(l,0diim) Giai h9 phuong trinh : f*t+xyz=40y ty'+xzy=10x TI6t Dqr kiiin ki thi thfr Dgi hgc tfrn thft 3 sd itwqc fi chftc vdo ngdy X\,Z0/S/2011 http://www.violet.vn/haimathlx Sưu tầm: Nguyễn Minh Hải oAp AN:- THANG DItM rm trrrl nn r,Ax rntl gar NAna zorr Cffu oAp Ai.l ortwt .I (z dtim) l. (1.0 iti6ml. Hoc sinh tu ei6i. Eucrng thing y = m(x-Z) + 2 ciitao tni (c) tai hai cli€m phdn biQt <+ pt '4 = m$-2) + 2 c6 hai nghiem phdn biQt <+ pt mx2 - 4mx + 4m - 5 = 0 (*) c6 hai nghiQm phdn bi€t kh6c 2 0,25 (m*0 (=) lo'=4m1 -m(4m-s)>o om>o In*-8m*4m-5+Q r 0,25 Gi6 sfr A(xr,y,), B(xz;yz) trong d6 x1, X2 lA hai nghi$m c0a (*). Khi d6 yr = IIrXr - 2m + 2 vit y2= ttlX2 - Zrn + 2 Ta c6 AB2 = (xz - 4r)2 + (y: - yr)2 = (xz - xr)2(m2 + 1) = [(x2 + x1)2 - 4xrx2](m2 + I ) 0,25 = tr6-{Tj)l(m' + tl = 3P=T =40 vdi msi m > 0. Eing thric xdy ra khi vd chi khi m : l. VQy, v6i P: I thi AB ngfn nhAt Uing .86 0,25 II Q dianl l. ( 1,0 iliAm) . GiAi phuong trinh . Di6u kiQn : cosx # 0. Phuong trinh dd cho tuong cluong v6i pt : sinzx sinx 3 ffi i un* + l) I T(l-Pnx) + ft er tan2x (1+ tanx) = 3tanx(l- tanx) + 3(l+ tan2x) 0,50 € tan2x (l+ tanx) = 3(liianx) * [nXI;= ,t ,. ;. oreu Klgn Dar toanJ. [x = -r+ kn e | + glz)(th6amdn lx=+-+kTr L -3 0,s0 2. (1,0 ilidm). Giai bet phuong trinh DiAu ki€n , *+';.' BAt phuong trinh cl6 cho tuong iluong v6i bpt ' S;= - 4 z s.32-# 0,25 D{tt=3s:l= , t>0. Bpttr€ntrdthinh t'?- $-45>0 =+ t:9(dot>0) 4,25 (+ lffi:9 6 Eripsd : xell;f,1 *+3-27 s*-r zz oEaxS s 0,50 III Q,tlidm) (1,0 iliAm). Tinh tich phdn . Tac6 l= - f,rr"ffi d 1 = - 1 1ntr1-52 lf . [*i d(rnVTTF) = tnfi - *'"f . lr"j#*=#r"z*1rrfra*. 0,50 1 http://www.violet.vn/haimathlx Sưu tầm: Nguyễn Minh Hải Tas€tinh ,=Jrvt;;hox, d{t x=tant =+dt=fta,=(l +tan2t)dt n tlT Vdi x= I thi,=1,'*=y'3thi t:l * l=1f,at=,li =*. 4raLL E6ps6 tt=f-21n2+4. ' 2J3 L2- 0.50 U Q iIiAm) o (0,50 diA@.Dr,rng thitit diQn c0a m{t phing di qua A, M vA song song v6i BD. Gqi O=AC o BD, O'=A'C' n B,D' vd I=AM nOO'. eual k€tluongthEngsong song v6i BD c8t BB', vd DD' lan lugr t4i K, N. Khi d6 AKMN la tni6t diQn cAn dgng. DAt Vr = Ve.scrrx-f Va.oo"rN , V2 = Vnaco.e,a'c,o'- Vr. -^-^ ? - oI AO 1 c(0,50i1i6@.Tac6 ff=#: I + DN=BK=Ot=;a"=;. Hinh ch6p A.BCMK c6 chi,iu cao li AB = 4 I tl6y ld hinh thang BCMK. 1 ^ Ot BC(BK+CM) a3 Suy ra Ve.ecrpr = -AB.Ssgy* =; -T =i . B' Hinh ch6p A.CDNM c6 chidu cao ld AD = a d6y lA hinh thang CDNM. suyraveco*u=lRo.s"o"n =T cD(NP+cM) ={ K 33t26 a3 , a3 _za3 B V6y, V, =: , Vz = a- r33 Ir0 V Q iliAm) l. Q,0 ilidm) '1 Tirgiethi6tsuy ra la-blSB- s.aZ +(a-b)t< 4=r(a+b)2-4ab< 4 0,50 Suy ra a* b < zr/ffi, tuongtgta cfing c6 : b + c S 2rlIEEE, a* c szlffi Docl6: a+b+c S VrT?E +Vr+:Ec+rma (dpcm) 0,r0 VI (2 itiim) (1,0 iliiim). Tinh diQn tich tain gi6c - Duong tlrdng BC c6 vecto chi phuong il ld vecto ph6p t il =(l; l). Phuongtrinh cta Bc : [; : I ll tu, ra B(l+t ; 2+t). B thuQcduongcao xu6t ph6ttirB n€ntgaclOthdamdnphuongtrinh:2(l +t)-2-t+ l=0=rt=- L Vgy B(0; l). ruong t6 phuong trinh AC ' [; : r*-?' vd A(- 5; 5) 0,50 Tac6: BC=fr Eudng thing BC viiit ve d4ng tdng qu6t : x - y + Gei AH Ii duong cao, ta c6 AH = -# = # -0. t9 Vay Sasc = IAH.BC= 1. 0,50 http://www.violet.vn/haimathlx Sưu tầm: Nguyễn Minh Hải \ \ ). Tim tqa d0 tdm vd b6n kinh MAt geu (S) c6 tdm IQ;- 3; -3) vd b6n kinh R = \8. I(hoangcrich.tir1-di5nmp(P)ldh=w=l<R'=G,n6nmp(P)cat .l rn{t cdn (S) theo mQt <tucrng trdn c6 b6n kinh bing r = r/ffi = 2. . Tdm K cria clulng trd d cli qua tfi€m 1vd vu6ng g6c vdi mp(P). vecto chi phuong ctia etulng thing d li vecto ph6p tuyiSn cira mp(p), n€n il;=(f ;1:2)vd phuongtrinhc0a o,{;:?{!r, Dod6 K(2+t ;-3-2t;-3+zt) lz=-3*Zt. Tga tlgtdm K cfra(C)"th6am6n phuongtrinh: Z+t+6+4t-6 +4t+ I =0 =r a =-l qTLL Vfy tdm r( i ; -; '-; ), b6n kinh r = 2 0,50 N6u x = 0thi y = 0 =+ (0 ; 0) ld mQtnghiQm cfra hQ phuongtrinh. Niiu x # 0. DAt y = tX, khi d6 hQ pt trd thAnh : fx3 + xetz = 4otr fx3lr + t2) = 4otx tt3x3+x3t=10x - [xr11l*t)=16* (+ ,i !*r(t* t2) = 4s1 f *' = # (1) t *'(t' * t) = 1e H l*HP = to (z) Tir(l) + t>0,k6thgpv6i(2) :+ t=).rnuv t= j uao(l)ractuqc x=*4 + y= 2. Drip si5 : H€ pt c6 3 nghi€m (0; 0), (4 Zi , F 4;-2). (1,0 Gidi he phuong trinh http://www.violet.vn/haimathlx Sưu tầm: Nguyễn Minh Hải . +tan2t)dt n tlT Vdi x= I thi, =1,'*=y' 3thi t:l * l=1f,at=,li =*. 4raLL E6ps6 tt=f-21n2+4. ' 2J3 L2- 0.50 U Q iIiAm) o (0,50 diA@.Dr,rng thitit diQn c0a m{t phing. s0 V = - x-2 l'I(hios6tsLr.bi6nthi0nviv€d6th!(C)cirahims6. 2. Tim t6t cd c6c gi6tri cttam d6 cluongtiring y = m(x-2)+ Z cit AO thi (C) t4i hai di6m phAn biQt A, B sao. ) Cho hinh lAp phuo'ng ABCD.A'B'C?D' c6 clQ ddi canli bing a vi di6m N4 thu6 c canh CC' sao cho 1c CM = T. tUat phing (a) di qua,t, M vir song song vdi