TRUONG EHSP HA NOI KH6I TTIPT CIIUYTN DE THr rHrI DAr HgC L.A.N THI' VI NAM 2009 Mdn thi :TOAN Thoi gian ldm bdi : 180 phrit CAu 1. (2 ili6m): Cho hdm s6 y = 2*'* 9mx2 * l2m2x + I (l). l. Kh6o s6t vd v€ tl6 thi ( C) cria hdm sO (t) t<tri m: t. 2. Tim m O6 nam si5 c6 cgc d4r, cgc ti6u tt6ng thoi xle = xcr. Ciu 2. (2 di6m). r l. Gi6i phuong trinh : sinsx - costx = l.or'z* - ] "orz*. 22 2. ciei hc ohuons trinh : {1Cz:Ev + 15 + '[FTM- CAu 3. (l dicm). Tinh tich phen : I = lru fi ln2xdx. Cfiu 4. (l di6m). Cho hinh ch6p tri gi6c ttdu S.ABCD c6 qnh tl6y bing a vd g6c TSE = a. Gqi O ld giao di6m hai dudng ch6o cta tl6y ABCD. Hdy x6c tlinh g6c a d€ m4t ciu tdm O tli qua ndm iliiim S, A, B, C, D. CAu 5. (l diem). Xdc tlinh m ee frg sau c6 nghiQm : (x' + y' + 2(m - 1)y - 4mx * m2 + Zm = 0 t 3x+'+y+r=o 1 CAu 6. (2 di6m). l) TrongmAtphingvdihgtgactQOxy,choduongtron(C)c6 phuongtrinh: x2 +y'-h-6y+6=0vd v di6m fuf(-f ; l). Gqi A vd B ld cric tiiip di6m cria cric ti6p tuy6n k6 tir M d6n (C). Tim toa d6 tli6m H ld hinh chiiiu vu6ng g6c c0a dii3m M l€n duong thing AB. 2) Trongkh6nggianvdihQtqadQOxyz,chomdtphing(P): x+2y-z+5=0 vd rludngthing . x*1 y+1 z-3 G: -Z-= 1 : 1 HEy viiit phuong trinh m{t'phinC (Q) chria tluong thing d qo v6i m4t phing (P) m$t g6c nh6 nh6t. CAu 7. (l di6m). C6c s6thgc duongthay tl6i x,y, zthbaman : fiJ +,[y I + ^[z- t: t. Tim gi6 tri lon nh6t cta bi€u thirc : P @ http://wwww.violet.vn/haimathlx oAp AN u6x roAN r,AN vr cau r -(2.rli,5m). l. ( 1,0 ai6m; . Hsc sinh tu gidi. 2. (l,0tlitim).Tac6: y':6x2+ lSmx+12m2; y':0<+ x2+3mx+Zm2=0 (1) Him s6 c6 cgc dgi, cgc ti6u khi vi chi khi pt (l ) c6 hai nghiQm phdn biQt'hay A : m2 > 0 <+ m:t 0. Khi d6 pt(l) c6 hai nghiQm : x1 -3m- lml -3m+ lml : T; x2= z . Xs X2 +oo D6u cfra y' : *) Ni5um>0 thi (*)<+ 4m2:-m (vdnghiQm). *) Niium<0thi (*)e m2=-2me m*-2. t^ Dapsoi m: -2. CAu2:(2,0cli6m). 1. (l,0di6m). PT <+ (sinax* cosax)(sinax- cosax) :)"or'z*- ]"orz*. <+ - cos2x (F:sinz2x): J.or2x.( cos2x- 1) e - cos2x(2- sin22x) =cos2x(cos2x - l) <+ - cos2x(l+ cos22x): cos2x(cos2x -l) e cos2x(cos22x + cos2x):0 h e I cos2x: o_ *= lz"J ] + rn : [* = I*Y ke z. lcos2x = -1 [2x =tr * 2kn [x = ] + t<n 2. ( 1,0 tli€m) . X€t phuong trinh : 3logae(49xz) - logr(y3) = 3 DiBu kiQn tt€ pt c6 nghia ld : x * 0, y > 0 (l), Phuong trinh tr6n duo. c virit thdnh : |rlgr(lx)z - 3log7 y - 3 (+ logT(7lxl)- logTy-1 <+ logT$=t o lxl= y. Thay lxl = y vdo phuong trinh .,fiz=y + 15 + JF +7Fi - 15 = .,/4*f rgy + 18 ta ducr. c pt : fit- By + 1s +,[FTz, - rs: JZ]r - 1By + 18 (:+ J6-m:5 +re-tg+ s) : n/S:s1+y-6; 1*; DiAukiQnd6phuongtrinh(*) c6nghla ld: y <- 5, y:3, yt 5. Tir<ti6ukiQn (l), suy ra di,SukiQn criay ld: y = 3, y > 5 (2). D6 gidi pt (*), ta xdt c6c trudng hgp sau : http://wwww.violet.vn/haimathlx . N6uy=3, 16rirngy=3ldmQtnghiQmctrapt(*). Doct6 lxl=y= 3ld nghiQmcriahQptddcho. . NiSu y > 5, khi tt6 (*) tuong duong voi phuong trinh : Ji=E +10+s:Jq=Z c:t v-s+v+s+2fi)r-sXJ'+5) = 4y-6 ,+ /6;Tm[5):y-3 € y2 -2s:y'-6y+3 <+ 6y:28c) y= T.t,ncnlo4i. T6m l4i nghiQm cta h€ phuong trinh ld x: *3, y = 3. Ciu 3: (1,0 tti6m) Efltt=fi + x: t2 + dx:2tdt. VoiX:lthi t:l;x:e2thit:e,n€ntac6: t: z Ii*Lnlrdt: sf, tzrnzto,= : f rn,t.d(t3) : I cr,t li : f t3d0nzt) = i"' Tf t2lnt.dt = f"'- 1rqJ,"'nr.d(t')= 9;'- f,rr"tl; .Tf,lt'.arrno 8 . 16 ^ 16 ne 8, 16 ,te I , 16 , 16 40e3-16 =-e erT- I t=dc=-e'+-t,l_ = -e,+ -ej3 9 9'L 9 27 tr 9 27 27 27 CAu 4: (1,0 di6m). Gqi H ld trung di6m cta AB, do tam gi6c ASB c6n n6n ta c6 : o( ASH:- ;. Khid6 SH = AH cotg- : - cotr_,. o(ac(. Dudng cao cia hinh ch6p ld : s6:y'j112 _[112 : f-r1.;.G =+ so = lJ.o,r, |- r . M4t cdu tdm O di qua ndm di6m S, .O!, ", D ktri vd chi khi cotgzl-1. aJT SO: 2 -t - I ^n atlz {cotez t- t=-7. c( Vay cotg t = V3 suy ra o = 600. CAu5.(l,0di€m)Tac6hQpt fg ,'yf.* (v+m -r)'= (2m-1)' (1) '-'ir- (3x*4y*1=0 (2) (1) le phuong trinh dubng trdn (C) c6 t6m ld I (2m; l- m) vd b6n kinh p: l2m - 11, (2) ld phuong trinh ttudng thing d : 3x + 4y + I = 0. Oe ne cO nghiQm thi khoing c6ch tir tem I diin d nhd hon hoac bdng R: l2m - 1l, hay 13.2m+4(1-m)+11 a hav '2 \tFp s lzm-11 http://wwww.violet.vn/haimathlx <+ r2m + 5r s 5r2m - 'r * lrTiJjjs(a* -t?, * [::6 5 ' VAy, vdi m > ; hoac m < 0 thi hQ phuong trinh c6 nghiQm. Cflu 6. (2,0 tli6m). l) (1,0 di6m). Dudng trdn (C) c6 tdm I(l; 3) vd ban kinh R = 2; MI =2rl-5> 2 = R, n€n M nim ngodi duong trdn. C6ch 1. Gqi H (x; y).Ta c6 ifr(* -t; y - 3), iMt- +; - 2) vd,nhan th6y hai vecto ifr vi iM cirng chiAu, n6nin =t. iM (t > o) € {i _ 1 = _i:; [i = tr _ii . Theo hQ thttc lugng trong AAMH vu6ng, ta c6 : iH'ifr = IH.IM = IA2 = R2 I - 1. 13, e -4(x -l)-2(v -3)=4 e -4(-4t) -2(-2t):4 e t:;' Vav: H(i; T)' crlch 2. Gi6 sri tli€m A(x"; y") ld ti6p di6m , thi f5 (? * f-l1(q ^,trong d6 : ' (MA l. IA (MA.IA = 0' MI1x"+ 3; yo - l;, Id(x" - l; y" - 3). Do d6 ta c6 : f \!,+v3_Zxo_6yo+6:0 .*["rty3 _2x, tr,. U:O_ Zxo+yo_3=0. l(x. + 3)(xo - 1) + ( yo - 1)( yo - 3) = 0 t x2' + yZ t Zxo - 4yo = 0 Suy ra <tuong thing AB c6 phuong trinh 2x + y - 3 = 0. Dudng thing MI c6 phuong trinh : []== ilit Do MI vr.r6ng g6c vdi AB, n€n tsa dQ cta di€m H ld ( *: t+Zt nghiQm criahQphuongtrinh:{ V = 3 *t . Giaihg nirytaduo.c Ut*; ?i. (zx + y- 3 = o 2) (1,0 di6m).Xdt m{t phing (Rj thay O6i ai qua dudng thing d, cit mp(P) theo giao tuy6n A. Khi d6 A chria iti6mA=dn(P).L6ydi6mt<cOAlnntr€nd(K+A).Gqi Hldhinhchi6ucriaKtr€nmp(P), Ilehinh chitiu cria H tr6n A thi HI vd ru cinllhuOng g6c vdi A n6n FFI ld g6c gifa (P) va (R). Ta c6 tanKiit : H *U KH khdng ati Wri (R) thay eoi va gt < HA n€n ffiH nno nh6t <+ tanKIH nh6 nh6teHllonnh6t<+ItrilngAhayAIdtaiA,tircldAnimtr6n(P),diquaAvdvu6ngg6cvoid. , 1 210 1,. Ditim A(-;t - ;r ; l. rni d6, A c6 vdctochi phuonguj =;tuj npl = Cl; l; l). t- co , i[uai ua] = (0; -1; 1) n€n (Q) c6 vecto ph6p tuyr5n ld f = iQ; -l; 1). Viy mp(O c6 phuong trinh : y - z * 4 = 0. CAuT:(l,0cti6m) DidukiQn :x> l, y> l, zZ1. Tiritangthfcgiathiiittasuyra.fiT<l=+ x <2.vAytu.o f1 ;:;' =+ P=-x- < 3- =1 "tt;; -t-v+z= 1+1 -'' Dingthticxiyrakhi x=2,y=z= l.K6tlu4n : MaxP= I khi x= 2,y :z= l. http://wwww.violet.vn/haimathlx http://wwww.violet.vn/haimathlx . +y'-h-6y +6= 0vd v di6m fuf(-f ; l). Gqi A vd B ld cric tiiip di6m cria cric ti6p tuy6n k6 tir M d6n (C). Tim toa d6 tli6m H ld hinh chiiiu vu6ng g6c c0a dii3m M l€n duong thing. ln2xdx. Cfiu 4. (l di6m). Cho hinh ch6p tri gi6c ttdu S.ABCD c6 qnh tl6y bing a vd g6c TSE = a. Gqi O ld giao di6m hai dudng ch6o cta tl6y ABCD. Hdy x6c tlinh g6c a d€ m4t ciu tdm. m4t phing (P) m$t g6c nh6 nh6t. CAu 7. (l di6m). C6c s6thgc duongthay tl6i x,y, zthbaman : fiJ +,[y I + ^[z- t: t. Tim gi6 tri lon nh6t cta bi€u thirc : P @ http://wwww.violet.vn/haimathlx