SO GD-DT IIA TINH TRU,ONG THPT NGUYEX TRUNG TIIITX DE THr rntlDAr Hec LAN 2 (Kh6i a,n; Nim hgc : 2010-2011 MOn : Tofn Thdi gian : 180 phut c6u 1 (2 tli6m) Cho hdm s6 ! = x3 - 3x', + 2 1) Kh6o s6t sr,r biiSn thi6n vd vE AO tfri ( C ) cfa hdm s6. 2) Tim tr0n dudng thing (d): y : -2 nhffng di€m md tir d6 ke dugc Z tiep tuy{in t6i 1C;, eOng thoi 2 ti6p tuy6n do vu6ng g6c v6i nhau. Ciu 2 ( 2 tli6m ) Giii chc phuong trinh vd hQ phuong trinh sau: + logr*, (*' - 2x + 1) = 6 4) =l 2) 2cos' "r + cos 2x + sinx - 0 C6u3(2tti6m'l 1) Tinh tich phdn +4x+4 . 2) Hoi co bao nhi€u s6 tU nhi6n gO- 7 chii s6, tuo cho chfi s6 dring sau nho h<vn chir sO dring liOn tru6c n6. i -aau!-( z rX!€aq) Cho hinh ch6p tri g d6y ABCD ld hinh binh hdnh. Goi B', D' lAn luo,t ld trung di6m cira SB, SD. 'D') cit SC tai C'. 1) Chring minh SC:3 SC' 2)Tinhttr0 tich cria S.AB'C'D' theo th0 tictr V cria S.ABCD. Cffu5 (2tli6m) 1) Trong mat phdng tea dQ oxy cho : diemiu r(2;2), s, f?;1), H, (1,?l tu 3 chan \)/).'5,5,, 5,5, dunng cao cfia tam gi6c ABC lAn luqt ha tri A, B, C. Tim tqa dQ c6c dinh A, B, C 2) Cho hai s6 thgc avd b th6a a-b + 1 : 0. Chring minh r6ng: ffi + ^ta'+b'-4a-4b+8 ::::::::::::::::::::::::::-: ii€r :::::::: 4 I I x4+2 ? x ) 2 dx trongxuanht@yahoo.com sen to www.laisac.page.tl nUoNc oAN cnAvr roAN rnr rHrl on r,Ax rntl2 - KHor A-B l.Gi:ii hG pt 2x+1) I l 6 (1) (2) log ,*r(l - x)' : 6 + 2) + log ,*r(l - x ,, (x- 2)'+ , (Y y+ lop _"( <> ) 2)- +4 -1 )Br-, 6€ :* og y+ ,r(* y+ +k :)= o ov+ -). 2), c)' -x -L a + (1 v- 5)- ,r! ,(1 ) ) v+2 I X ;dr v 4 Lg ( )n. .r0t o bl- r9t .u OT ki, T llc 3ul (1) +2 let (r l+ -1, Di +, pt -) .? v : !=xt -3x'+2 +, TQp xd, Tim dugc y'; Tim c6c di6m tdi han; Tim y" 0,25 +, Khoang don diQir, gioi h?n, bang biOn thiOn 0.25 +, Tim c6c di6m d4c biQt: CUc dai (0;2); Cyc ti€u (272):; Di m u6n (1;0) Oiem cdt tryc ox: (1-".6;0); (l;0); 1f *",6;0); Di6m c6 tsa dQ nguy€n (3;2); (-17 2) 0,25 +, Ve dO thi dgp, tron, c6 tinh d6i xring, di qua c6c di€m <l{c biet 0,25 +, Gid sri c6c Oiem cAn tim ld X(m;-2); Dudng ttrang (A ) di qua X v6i hp s6 g6c k co pt ld: y : k(x-m)-2; khi d6 (l ) tiCp xirc ( C ) niSu vd chi n6u hQ : ll*2 -6x-k ) )3-r2 [x'-3x'+2 k(x-m)-2 (1),. co nghiQm ( 6n x ) (2) (/ 0,25 ThC k tir (l) vdo (2) ta c6 pt: x' -3x' +2= (3r' - 6x)(x - m) - 2 <) x' -3x' +4=3x(x-2)(x-*) <> HoA. x:2,hoirc 2x'-(3m-l)x+2=0 (3) 0.25 rO rdng kh6ng co ti0p tuyOn nao cdu d0 ra thi pt (3) phai co 2 (3*i - 6*,) .(3*', - 6xr) - -1 Vdi x : 2 thl ta duqc ryQt tiOp tuy0n co pt: y : -2, vu6ng goc vdi ti6p tuyOn ndy. NOn mu6n thoa yOu nghi0m pb x,, x2 thoa: y' ( x, ) .V'(*r) - - 1. Tuc : <+ gxixl - 18r, xr(x, * xr) + 36x,x, - -1 0,25 Do x, , xrld nghiQm cria (3), theo Viets ta c6:. xt.xz =l;x, + x, = 55 NOn dugc 9 -9(3m-1) + 36: -1 Hay m=a; Gi6 tri m ndy thoa 27' nghiQm. k6t 1u0n eicm cdn tim tex1fi;-21 2\op. (-xv -2x + v+ 2) + los^ (x' -2x +1\ = 3m-1 2 pt (3) co 0.25 i iDat t: log,-" (y + 2) ta co Pt sau: -2 e t- 1 'e logr-r(Y'+Z)-I e Y: -x- 1 t+- t 0,25 2 D[t t-x+- x ) ta co dt - (1 - i>a* ; dOi cQn t x A9 :J+- 2 -) J : 44 +, Thay gr;tri y vao pt (2) vd giai thi dugc x: 0 ; x - -2 +, Ket hqp diou kiEn ta c6 nghiQm |d : x: -2, y : 1 1 | 2cos'x + cos2x + sinx = 0 +, Thay 2cos' x = cosx.2cost t = cos'x(co sLx +1) thi pt vii5t lai ld: Cosx. Cos2x+ cosx* cos2x+ sinx:0 ecos2x(cos" i11+titt"*cosx:0 e (cosx*sinx) ftro, x -sin x)(cos x + 1) + 1l-0 <> (cosx+sinx) ft.o, " - sin x)(cos x + 1) + 1.1:0 ; i;;-;;;ii- sin'x + cosx(l- sinx) + (1- sinx)l:0 (cosx*sinxx 1 -sinxx 1 *sinx*cosx+ 1 ) - 0 ept co 2hqnghiQm x- -+ + kn;x -! *2kn 42 (ke z) ( Ttry vho lflp lufln tI6 cho tti6m thirnh ph6n chfnh x6c tl6n 0,25 ) 1,00 l I 0 )5 \t )- -' Cflu 4 (2di6m) I,Xtrc dinh tdm O ctra hbh ABCD +, Xd giao O' cfia B'D' vdi SO +, Xd giao C' cira AO' v6i SC ' +,KeCPllSO(P€AS) +, Xd giao K cira AC' vdi CP +,Chung minh K lA trung diOm CP +,Chung minh S ld trung diOm AP +, Suy ra C' ld trgng tdm HaySC-3SC'. 2. Chia th6 tich cria S.AB'C'D' thinh2 kh6i chop tam gt6c: S.AB'C' vd S.AC'D', tdc6: 0,25 0,25 0,25 0,25 0,5 0 0,25 0,25 0,25 0,25 v(s.AB'c') ^sBo.,sc' Tuong tU thi: v(s.AC'D') v(s.ACD) 6 11 v(s.ABC) SB.,SC 23 6 Suy ra : v(S.AB'.'o'):* ir6.nc) +v(s.ACDll : f rfs .ABCD) 1, Tam gi6c ABC C6 AH1, BHz, CH3 ld c6c dudng cao. Nguoc l4i tam giac H'FzHr se c6 AHr, BFtrz , CH3 ld c6c dudng ghdn gi6c. ( C6c em dga vdo tinh ch6t ndy thi ldm dugc nhfing bdi to6n nhu th€. ) +, Ri€ng bdi to6n ndy cho d{c biQt hcrn: < H ,H,H 3cAn tpi Hr.*, Dinh A nim tr€n trung tn;c(d) cria H2H3. Dinh B vd C nim tr€n ducrng ttring (d') di qua H1 vd vu6ng g6c v6i (d). +, Vi6t pt ducrng phdn gi6c cria goc do I{zHr ho. p v6i HzH: ( c6 2 pt ) +, Giao ctra dt co pt thf nh6,tvdi (d) cho ta di6m A( 0; 0 ) j giao vdi (d') cho ta OiCm B( 1; 3 ),giao cria dt co pt thft 2 v6i (d') cho ta C( 2, Ta c6 : ^lo' +b' -za+6b+10 +^la' +b' -4a-4b+8:"/(a-1)'+ (b+3)' *,l@-2)' +(b-2)' Trong mpt phEng tqa dO Oxy"ta lBp tli€m A(1;-3), eiem BQ;2) vd duong thang (d) c6 pt: x - y +1 : 0 Khi d6 di6m M(a,b) ndm trOn (d) thi cap (a;b) thoa a-b+1:0 Ta co lfo- 1)' + (b +3)' + J @ -2)' + (b -2)' :AM + BM L6y B' d6i xung vdi B qua (d) suy ra : B'(1;3) 0.25 0.25 0,25 0,25 Ta c6 ngay AM + BM > AB' : 6 . Suy ra tliiju phf,i chfne minh QJ . v+ 2) + los^ (x' -2 x +1 = 3m-1 2 pt (3) co 0 .25 i iDat t: log ,-& quot; (y + 2) ta co Pt sau: -2 e t- 1 'e logr-r(Y'+Z)-I e Y: -x- 1 t +- t 0 ,25 2 D[t t-x +- x ) ta. cria S.AB'C'D' thinh2 kh6i chop tam gt6c: S.AB'C' vd S.AC'D', tdc6: 0 ,25 0 ,25 0 ,25 0 ,25 0,5 0 0 ,25 0 ,25 0 ,25 0 ,25 v(s.AB'c') ^sBo.,sc' Tuong. C( 2, Ta c6 : ^lo' +b' -za+6b+10 +^la' +b' - 4a- 4b+8:"/ (a- 1)'+ (b+3)' *,l@ -2 ) ' +(b -2 ) ' Trong mpt phEng tqa dO Oxy"ta lBp tli€m A( 1 ;-3 ),