6 đề-đáp án thi thử ĐH-Hà nội -2011

TRU,ONG EHSP HA NOI TRTIONG THPT CUUVNTV - EHSP DE THI rnu DAI HQC t AN I Narr ZOrt Mdn thi : TOAN Thdi gian tdm bdi : IB0 phtit, kh6ng ke thdi gian phdt di Ciu 1. ( 2,0 dilm ) Cho hAm s6 .y: Irt - 1**' + (*2 -3)x, trong do m littham s6. 32 I . I(hAo s6t sg bi€n thiOn vi v€ d6 thi cta hdm s6 khi z = l. diri cdrc canh g6c vudng cira mQt tam giirc vudng c6 dQ ddi c4nh huy€n birg F . "{2 Cdu 2. (2,0 diem) 1. Giii phuong trinh : 2. Gidi phuo-ng trinh : CAu 3. ( 2,0 di1m ) 1. Tim nguyOn hdm cria hdm sO f1x; : tanx.tan(x + ];.tan1x - l;. 3" 3', 2. Tim c6c gi6 tri cta m d6 phuong trinh sau c6 nghiQm duy nh6t : g-lz-xl _ 43-|2-xl= p. Ciu 4. ( 1,0 didm, , Cho, hinh ch6p tu gi6c ttdu S ABCD tt6 UU ddi canh ddy bing a, cgnh b6n bing t' ttnn goc tao boi '2 mdt b€n v6'i mdt d6y vi the tictr t<hOi cau ngopi ti6p hinh ch6p d6. Cfru 5. ( 2,0 diem ) .3 Trong mdt phdng Oxy : L Chohaidi€m A(2;1), B(-l;-3)vdhai cluongthing d1: x+y+3 =0; cl2: x-5y- 16:0. Tim tga dQ c6c cti6m C, D lin luE thuQc clr vd dzsao cho t6'gi6c ABCD ld hinh binh hdnh. 2. Vi6tphuongtrinh tluo'ngthingtiOpxircv6'i duongtron (C), xt+ y?-zr] x+4y+4=0vdtao v6'i trr,rc tung mdt g6c bing 60". CAu 6. ( 1,0 diem ) Xdtcdctam thirc bgc hai f(x):ax2 + bx+ c,trotlgd6 a< bvnf(x)> 0 vd.i moi x € R. Hay tiin gi6 tri nho nh6t cira biriu thric M = TT Het 2. Tim t6t cd cdc gi6 tr! oba m ae ham s6 c6 cuc d4i t4i x6p, c!.rc ti6u t4i xcr ddng thd'i xss, .rcr lh dO z+^[s-x, +12 ,[s-*,1 sinal3x*|l * sinal3x -?=: x2 -l Dy kiAn ki ttti thrb Egi hgc l6n 2 sE ihrqc t6 chrbc vito ngdy 26,27/2/2011 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 TRI'ONG TIIPT DAo DUY TTI - IIA NoI of rrn rrnlDAr Hgc r,Arv r (2010 2011) nndx, roAN rn6r a Thdi gian: 180 phtu (kh6ng *6 nm g*n giaa di) CAu I. Cho him s5 y: x3 - 3x (l)- / U Xnaosft sg bi6n thi€n vi v€ eO tni narn sO (t;. '.t u 2/ Tl^m etii phuong trinh x3 -3x = + c6 3 nghiQm phdn biQt. m" +l 3/ V6iO(0; 0) vd A(2;2) h hai ttiiAm nim tr6n AO tni hdm sd (1), rim di€rn lvi nim trOn cung OA crha dd thihAm sO (t) sao cho khodng c6ch tu M cli5n OA ld ld.n nh6t, Cffu II. 1/ Cho b6t phuong trinh: JT3x+2) 7n-rt7 1x+4 l, v al Giai bdt phuong trinh khi m :4. b/ Tim tdt cir chc giittri cria m ee UAt phucrng trinh tr€n nghiQnr ding viii riiqi x >:i. ,,1 2/ Gi6i phucrng trinh: tanx+cosx-cos2x =sinx(l +tan!-tarrx). Cffu III. V ttTrCn m{t phdne tqa d0 Oxy, tim phuong trinh tludrng thingr di qua di6rn M(l; 3) sao cho dudng theng d6 ctrng v6i2 dudrng thang (d1): 3x-{- 4y + 5 : {j vi (d2): 4x + 3y - I : 0 t4o th'nnh mQt tam gitlc cdn c6 dinh ld giao di6m cria (d1) ,,,d (d2). ( 2l Cho hinh ch6p SABC c6 dhy ABC li tam gi6c cdn, c6 AB : AC : ?,a; RC - 2a c6c m{t bon ctrng tao vdi ttax,m0t g6c 600. Hinh chitiu H cia dinh s xu6ng (ABC) nam O trong tam gi6cABC. '/ at cntng minh t6ng H h tem dulng trdn nQi ti6p cria tam gi6c ABC. b/ Tfnh thiS tich cria trl di0n SABC Cffu IV. Cho tflp hW A gdm n phAn tri (n > 4). nii5t ra"g sO tgp con gOm 4 pfran tri cria A bing}} Hn sO tdp con g6m 2 phan tfr cria A. Tim k e {1,2,3, ,n} sao cho si5 tfp con g6m k phAn trl cria A ln lcrn nhdt. Cf,uY, Cho 3 sO kh6ng 6m a,b,c. Chung minh ring: a3 +bt +c'> a'Jbc +b'Ji +t'^[on . I'R{'GNG PTTE{ EAO NUIT TTI HA NOn Bn rrur B4r FIgc L.&N rn-lt^Ana Hqc 2010 - zfitt, na6m roAn E[ec xa6i a Thdi gianldmbdi: I8Tphfit +/- ( KH1NG XE rnU CUU rn irDE) / efurn: cho hdrn 16 y: (m+l)x+m x+m 1) Khno sit vi ve dd thi hem sO ttri m : 1. Tim tr6n dd thi nhimg di6m c6 t6ng khoing c6ch d€n hai tiQm can nhO nhAt. 2) Chtmg minh rdng v6i moi m l0 aO tni hdm sd lu6n tii5p xric voi mQt duimg thing cO a6n. @I!: Gi6i phucrng trinh: "'/i+ ti"" + Ji _siil = 2cosx . CAU ItrI: 1) Giai cdc phuong trinh sau: , a) (4x - l) fi'? +1 :2x2 +2x+ 1. b) log*+r p - Jt-zx*; =)- 2) Tim m d€ phuqng trinh sau c6 nghiQm duy nhAt. 3m-z= h C6u XV: 1) Trong m4t ph6ng voi h€ tga dQ Oxy cho tanl gi6c ABC c6 A(-2; l), B(3; 5), C(2;1). Tim tqa t10 truc tem H, t6m I ctra dudngfirdn ngo4i ti6p vd trgng tem G cira tam gi6o ABC. Chrmg minh H, I, G tha"g hang . 2) Ctro tam gi6c ABC vu6ng t4i A, AE : a, BC :2a.Haitia Bx vi Cy cring vuOng g6c voi m{t pb5-ng (ABC) -rir nim v0 mOt phia d6i v<ri mit phang d6. Tr6ri Bx, Cy lAn luqt 16y c6c diAm B', C' sao cho BB' : a, CC' : m. a,Yrn gi6 tri nio ctra m thi AB'C' ld tarn giAc vudng. b, Khi tam gi6c AB'C' vuOng t4i B', ke AH vu6ng g6c voi BC. Chrng minh ring B'C'H idr tarn gi6c vu0ag, Tinh gbc, glftahai m4t phAng ( ABC ) vd (AB'C'). e6g-Y' Chr.rng miiih ring khi a l0 thi hQ phuong trinh sau c6 nghiQm duy nhAt: 7 a =y -F- v a' a" - L'T I,*, i,,' ;- GiAm thi coi tfui kFr0ng gidi thich gi thdm ,/ T'R{IffiG TE#T s}ao puv T{l, pm s's{x rmalo4r noc x,AN rm, xAla F{EC 2010 - 20Il nndlol : Todn, m6i a Thdi gian ldm bdi : 180 phrit ( kh6ng te tn}i gian phdt d0 ) tsAi t. (2 di}m) Cho hdm s0 ! = x3 +3x2 -mx+2 c6 AO mi (C*). 1" Kh6o sdt vd ve dd thi hdm s6 vdi m:0. ' 2. Timm'd}hdm s6 c6 cgc d4i, cgc ti6u sao cho khoing-crlch tu trung di€m cfa do4n thdng n,5i hai di6m cuc tri cria (C*) d6n tiiSp tuytin cria (C-) tai cii€m c6 hoinh elQ bing 1 le lorr nhdt. Eei It. ( 3 diCm ) Giai c6c phucrng trinh, hQ phucrng trinh sau : - fry-3x- 2y+6:A L i, ., l*" +y'-2x-4y+3=0. 2. 4sin2x-cotx= Jj - 3. 2losr(3x+5)+logo(lx+l)8 = 4logr(l2x+8). Bii III. ( I diem ) Tim sd hang khdng chfta x trong khai triiSn nhi thric Newton cria r' Bni Iv. ( 3 di€m ) 1 . Trong m{t phing vdi hQ,fa dQ O*y , cho du}ng thing d : 3x - 4y +2011 = 0, vd duong trdn (C) : ,' + y' -6x -2y+1= 0 . ViCt phucrng trinh dudrng thing A song song v6i d vd cft (C) theo mQt dAy cung c5 dQ dni bing 4. 2. ChotudiQnABCD c6AB :a,AC- Za,AD= 3a vit A)C =C)n:fAn=600 . Tinh th6 tfch tri di6n ABCD. 3. Trong kh6ng gian v6i hQ tga dQ Oxyz,cho m{t phing (P) : r+ 2y- "-4 :0 vd duong thing a ' * :4 = + Tim ditim M tr€n tnp oz sao cho cdc khodng 2t-1 c6ch tir M d6n m{t phAng (P) vd tludmg thdng d bing nhau. tsni V" ( i diem ) Cho c6c sd thuc duon g a,b"c th6a m6n a+b +c =3 . Chring minh ring : G.trh-b+c a6.111-rao +r.l,h-o+b.3. HA NOI P(x)= (r*r-+l' \ r-l. Gi6m thi coi thi kh6ng gi6i thfch gi th6m rci rHr rnrl DAr Heqr,AN rHrI NnAl xAnn Hec 2010-_ zltt DE THI MON: TOAN xlr6t.L,n Thoi gian ldm bii: 180 phirt, kh6ng kC thoi gian giao d6 Cffu I: (2,0 itiAm) Cho him sd ! = x3 -3(nt+1)x2 +9x-m,vti m ldtharnsd thgc. l. Kh6o s6t sy bi6n thi6n vh vE OO ttri cria hdm sti ea cho ring v6i m =1. 2. XAc dinh m ee fram sO ea cho d4t cuc tri t4i xr,x2 sao cho la -*rl:2. Cflu II: (2,0 iti6m) l. Gi6i phucrng trinh: 1 + 3 cos x + cos 2x - 2cos 3x = 4sin x.sin 2x 2. Giai he phuong trinh: lxt +2x+y'+y:3-xy j ^ " (x,y€R) l*y+*+Zy-1 "' cf,uIII: (1,0ili0m) r\m J ff-;d. sinx.sinl x+: I \ 4) Cf,u IV: (1,0 ili6m) Cho l6ng tru tam gi6c ABC.ATBTCT c6 tdt ch cdc canh bang a, g6c tao b&i canh b€n vi mdt phing d6y bang 300. Hinh chidu H cfra didm A tr€n mat phftng (ArBrCr) rhuoc doong thing B,C,. Tfnh thd tfch khdi ldng tru ABC.ATBTC, vi tinh khoang cdchgifa hai dudng thing AA, vi B,C, theo a. Cffu V: (1,0 ifiAm)X6t chc sd thgc ducrng a,b, c thoa mdn di€u kiQn a+b+c=1. Tim gie d nh6 nh6t cria : ":m ? . h Cflu VI (2,0 iiidm) ' I 1. Trong mat phing vdi hQ tga ttQ Oxy cho hai dudng trdn : . (C1): x2 + f ,:13 vd (C2): (x: 6)'+ t' ,: ZS cit nhau tqi A(2;3). Vi6t phuong trinh ducrng thdng ili qua A vi 16n lugt cdt (Cr), (Cz) theo hai ddy cung phAn biQt c6 elQ dii blng nhau. 2. Trong kh6ng gian v6i h0 tqa d6 Oxyz cho tam gi6c vu6ng c0n ABC c6 BA : BC. Bi6t A(5 ; 3 ; - 1), C (2 ; 3 ; - 0vd B ld ditim nim tr6n m{t phing c6 phuong trinh : x+ y - z -6 :0. Tim tga c10 tli€m B. Cflu VII (1,0 iti6m) Gi6i phucmg trinh : (z - tog, x)logn,, -;ft; = t utlt TRUONG THPT CHUYfi,N NGUYEN HUE Thi sinh kh6ng duqc s* d4ng tdi li€u. Cdn bQ coi thi kh6ng gidi th{ch gi thent . tr6n dd thi nhimg di6m c6 t6ng khoing c6ch d€n hai tiQm can nhO nhAt. 2) Chtmg minh rdng v6i moi m l0 aO tni hdm sd lu6n tii5p xric voi mQt duimg thing cO a6n. @I!: Gi6i. 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. cluongthing d1: x+y+3 =0; cl2: x-5y- 16: 0. Tim tga dQ c6c cti6m C, D lin luE thuQc clr vd dzsao cho t6'gi6c ABCD ld hinh binh hdnh. 2. Vi6tphuongtrinh tluo'ngthingtiOpxircv6'i