Sat hạch lan 3- Toan 11-A_B Le Xoay 2012

i+ rR{tOF{c rHpr r-,t xoly Og fru rffAO SAf CffAf LUqNG _lAN E NAwt HQ9!{4lor2 MON ioAN ll Krrdi;B Dd chinh thric Thdi gian: 180'phtit (khilng kd thdi gian,giao iti) Ciu I. GiAi c6c phuong trinh lugng gi6c 1. "or'[1"orr-!o)=, \3 3 ) ^ Ssina x-sin 4x-Zcos2x-2 'r_A zcosx-JZ Cffu II. . lz*'+2xy-5x+y-2 1. Giai h9 phuong trinh I :-' - 2. Chring minh rdng trong tam gi6c ABC, n6u -l I I - t ^#'#';F theo thr? ts l6P'h"1 ^^22'2 cAp s6 cQng thi a,b,c citng theo thri qu l6p thenh c6p sti cQng. Trong d6 A, B, C lAn luqt ld ba g6c cira tam gi6c vi a, b, c theo li d0 dai c6c c4nh A6i aien v6i ba g6c A, 8, C, CAU III. 1. Tinh t6ng ,S=Cl, -C)^*C\, *(-I)'C;, vdi n h s6 nguyOn duong th6a m6n Ci +Ci-t +Ci-z =79 2. Cho da gi6c l6i c6 10 canh. L6y ngdu nhi6n ba clinh trong st5 c6c tlinh ctra da gi6c. Tinh x6c sudt d€ dugc mQt tam gi6c kh6ng c6 c4nh ndo cga tam gi6c li canh ctra da giic. Cflu IV. Cho 3 s6 ducnrga, b, cthoamdn a<b<c vit abc=1. Chringminhrdng , l, *rr-l , *-2- =r. a'+abc b'+abc c+abc Cffu V. 1. Cho hinh ch6p S.,48 CD, gqi M ld m}t di6m thay dOi tren SC kh6ng tmng v6i S ve C, (P) H ' m{t phing di qua AM vit.song song voi BD. Xdc ilinh tnitit Oign ctra hinh chtip cit bdi mat phang (P). Chrmg minh ring m{t phing (P) lu6n di qua mgt du}ng thing cO Ai,rh. 2. Cho tu diQn ABCD. Ggi M,.l/ theo thf t.u lA trung di6m AB vd CD. TrOn c4nh BC 6y di6m .[sao cho IB : 2IC, Bi6t mAt phing (]IINI) cht.nn@i -{ Tinh ti rO 4. JD Ciu VI. Tr0n h9 tryc tga dQ Oxy cho dudng thing d: 2x - 3y - 5 :0 vithai ili€m AGl;3.), B(1;1). Tim trOn duong thing d di6m.M sao cho lzun-zuEl nfrO nfrAt. _____r*5t__ (Cdn bQ coi thi khilng gidi th{ch gi thhm) F{qvit6nthisinh: SOUaodanh Lop N6i d "(n 8a) .(r 8zr) COs- l -cos x_- l= I € slnl _cOS.r__ l- 0 \3 3) [3 3) 7t 8tt c) *cos.x - -? = kn <+ cos.r = I +3k,k e Z. JJ Do -l<cosx<1 Suy ra cos.x = -1 .1 n€n -1<8+3fr<1e -3<k 3-:3 k=-3 3 <+.x- n+2ln,leZ. 2 (1.0d) 8sina x - sin 4x - ZcosLx - 2 = 0 e Ssina x-sin 4x-2cas2x-2=0 <+ Ssina x-2sin Zxcos2x-4cost x = 0 e Ssina x+8.sin3.xcos.x-4sinxcosx-4cos2 x = 0 e 8sin3 x(sinx+cosx)-4cosr(sinx+cosx) = 0 e (z sin3 x - cos r)(sin x + cos x) = 0 [sinx+cos.tr=0 (1) <)t [2sin'.r-cosx=0 (2) (1)e sinx+cosx = 0 e x = -L + kn,k eZ (2) e 2tan3 x- (1 + tun") = O <> tan r = 1 <+ * = t * kn,k e Z. fiSt hqp vdi diAu kiQn suy ra .r = x4 * k\x,k eZ. 1 (1.0iI) EK x+y>2 [2"' +ty: t.] f2v=rJ* +v-2 y-2 f(zx-t)(x+y-2)=0 AJ +3 f2y=lJ* +y-2+3 l[*=1 ll 2 1L**v -2=0 t_ L2v = rJ* +v -2 +3 :_I:i ::= Vflv hQ PT c6 nghiQm suy ra "")" (r.3 t \2'2 31 22 . fl r5T-" "^lrt + ) I A' tan - 1 , .or{ "nr{ "or{ -+ lap thnnh csc <+ .+.:+=, .+ tan- sm- sm- sm- 2222 ( t-r\ . InT\/ | B sml -l n, "^^^[ t / _" tott a .^:_ A _,_c B o -:ffi = t-:6 <) zsm-srn- = stn- sm sm- sm- ? 222 ^ ,. B A-C A+C A-C B <D Slll-:- = COS- = COS Sin- 2 2 2 2 -'-'2 0.25 B A_C B A_C <> ZSln -:- = COS- <+ Sin B = COS:CO 2 2 2 2 1 <+ sin 3 = a(sin ,4 + sin C) .2' . A+C A-C - sln ., ,, .L 0.25 Tir d6 suy ra sinA+sinC = 2sin,B e a + c = Zb hay a,b,c l{p thinii CSC. 0.25 m 1 (1.0iI) Ci +Ci-t +C:-' =79 e "*ry= 78 e n' +n-156 = , *l:=?r, -Fvv rs r.=!?. 0.5 Khi d6 S = Clo - Clo + Ci^ + Cii |rlLt T = CL + Ci + Ci, + + Cli = C)o + Curo + Clu + + Cff S, = cL + cln + clo + + cil;s, = c)o + ci + Cj,+ + cji; thitac6 S=Sr-Sz. MFt kh6c r = cl+ + clo + + c:i + cll + cii. . . + c',i = zs, - c)1 T =clo+c'^+ +c;i+cif + ,1 cil =2s, Tir d6 suy ra 2St-C:1= 2Sz <+ ^9 = S, - tr=i,rt 0.5 ,, (1.0d) S0 c6ch l6y ra ba dinh ld Clo =120 . 0.25 Khi lAy ra ba dinh la ludn dugc mQt tam gi6c. C6 l lopi tam gi6c kh6c nhau: LoSi 1: c6 2 c4nh ld c4nh cria da gi6c c6 10 tam gi6c. Lo4i2: chi c6 1,cpnh ld cpnh cria da gi6c c6 6*10 = 60 tam gi6c. Lo4i 3: Kh6ng c6 cpnh ndo ld cqnh cria {la gi6c c6 120 -7Q = 50 tam gi6c. 0.5 50 5 VAv x5c suAt li P : = - .7J ' '1,20 't'L 0.2s IV (1.0rD tl2- _ f J - z' ', a'+abc b'+abc c+abc 112cll2 A-!-a-A-J-a- " a, +l' b2 +l- c+l-' a2 +l' b2 +l- ab+l 0.5 .l * .1 az +l b? +l- ab+\ '' az +1 ab+l' b2 +l ab+l- " I - I - I - I .n ^t -' a'+l ab+l b2 +l ab+l- - e @-4 (b-a)(!b-1).< 0 e @-?)'("0.:t) . .o - (ob +r) (a'+ r)(a' * t) = " * (ou +t)(a'+ r)(a' * t) - " Md theo gin thitit abc=l,a<b<c n€n c >-!>ab=l=1 suy ra b6t ^. c cuol cung tuon oung. EEng thtc xAy ra khi a: b hoflc c : 1. Ta c6 diAu minh. dnng thfrc ph6i chimg 0.5 v I (1.0ir) '/l\' li\ f_[ /t\ JT-, (Vc hinh dring* m6i cti6il didD - \rur I ra glao-qrem AU va trD. J la glao dlem AM va SI. Tr€n m{t phdng (sBD) k6 duong thang quu J song song v6i BD cit sB, SD lin Se$ip;e-1" $uy_r3"tlif ig-ergilgsergtgeippjE_l4f, 0.5 Lauray (r/ or qua-drem A c0 drnh va song song v6i BD c6 dinh n6n (p) Iu6n di qua mOt duong thing d c6 dinh di qua a va song song vdi duone tfrine gn - 0.5 2 (1.0d) (V€ hinh d,i4g* m6i ch6m di6mj Uor "b la glao di€m cta MI vd AC thi N, J , E th6ng hdng hay MI, NJ, AC d&rg quy t4i E. a Do IB=2IC n6n suy ra CA=CE. (Do CE = 2MF=AC) 0.s lu U Ke ducrrrg th6ng song song vdi NJ c6t AD tpi K. Suy ra KA: KJ MaNC: ND n€n JK = JD. 1 f,t y6,v JD=!JA 6u, t4 =2 3JD 0.s VI (r.0d) o,{*=t:t.' ^ Gii sri u(t+3t;-t+2t). ly = -l+Zt fu = (-z-zt;+ -zt); uE = (-tt;z-a) + zffi-tffi = (tt - +;zt +z) 0.5 f - -11 lzMA-3MBl =nf -16t+20 Suy ra lzue*zunl nrto "rt6t 8 e-t 13 suyra -(#,*) 0.5 i duyQt Neudi r;l Ngud'i du1 Td truong gudi b"( ' t-4 t 45. ( )=r I /./ L "' u Nguy6n ThiHpnh Hodng Drlc Trudrng . ', a'+abc b& apos;+abc c+abc 112cll2 A-!-a-A-J-a- " a, +l' b2 +l- c+l-' a2 +l' b2 +l- ab+l 0.5 .l * .1 az +l b? +l- ab+ '' az +1 ab+l' b2 +l ab+l-. giic. Cflu IV. Cho 3 s6 ducnrga, b, cthoamdn a< ;b& lt;c vit abc=1. Chringminhrdng , l, *rr-l , *-2- =r. a'+abc b& apos;+abc c+abc Cffu V. 1. Cho hinh ch6p S.,48 CD, gqi M ld. " I - I - I - I .n ^t -' a'+l ab+l b2 +l ab+l- - e @-4 (b- a)( !b- 1).< 0 e @-?)'("0.:t) . .o - (ob +r) (a'+ r)(a' * t) = " * (ou