orirru rrffiro, TRU'oNG EHSP HA hiQI TRI'ONG TtrIPT CTTUYTI\ - DI{SP Mdn tlri : TOAN Thd'i gian tdm bdi : 180 phtit, kh6ng kd thd'i gian phdt di Ciu 1. (2,0 di€nt ) Chohdms6 r=1"0-1x2+1. 42 l. KhAo s6t su' bi0n thi6n vd vC dd thi (C) cfra him s6. 2. Tim di€m M thudc (C) sao cho tOng cdc khoAng c6ch til di€m M d6n hai trr,rc tga dQ ld nho nhAt CAu 2. 1Z,O aieml 1. Gi6i phuong trinh : cosSx + 3cos4x * 3cos2x : 8sosx.cos3:x - I' 2 ,^,,1{4i4-3 2. Giei bdt phuongtrinh : 32-* + 6.3r-* t G) Cdu 3. (1,0 di€m) Tfnh tich phAn t:, (1 + ex)(1+x2) CAu 4. (1,0 ,iiem ) Tfr dien ABCD c6 ABC vd BCD ld c6c tam gi6e d€u c4nh bing a. G6c gi0'a dtrong thang AD vir rn{t phing (ABC) bing 60". Tinh diQn tich mpt cAu ngo4i ti6p t(r dien. CAu 5. (1,0 diem) Tirn c5c gi6 tri crta m AC ne Uat phuong trinh sau c6 nghiOm : l*'-xy*2y2-x(m lxz-zxy-Zx<m-2 Cflu 6. (2,0 di6n) l. Trong mat phing Oxy, cho hai di€m Fr(- 4; 0), Fz(4; 0) vd di6m A(0; 3). a) Lap phuo'ng trinh chinh tic crha elip (E) c1i qua di6m A vd c6 haiti€r-r di6nr Fr, Fz . b) Tim tga dQ diOm M thuOc elip (E) sao cho MF1 = 3MIrz. 2. Trong kh6ng gian Oxyz, clio hai duong thing : ,1,,4= Y-1 :'-r'. cl',,*: Y*t -'-3 t 2 2 -1 2 -2 Chri'ng minh c/r, dz cit nhau t4i di6m A; vi6t phuong trinh dud'ng thing A di qua M(2; 3; l) tao vo-i d1, d2 mQt tam gi6c c6n tqi A. CduT. (1,0 diem) Giai he phuorrg trinh : n L:4 r-: Hiit a4 , . t a t A / a /^na a oAp An - THANG DIEM rnr rntf on r,AN urt/ga - NANI zorr t tru ai6m 'z a$m) thi (C) cit oy tai A(0; l), ndn @i trsc tsa d0 bang l. E6 thi hdm s6 c6 hai diiim cr,rc ti6u 1- f tJ i ' i r; I I ua nhf,n oy ldm trqc diSi xung, nOn ta chi cAn x6t M(ru; y") € (C) vd 0 < xo'< l. @ M il€n hai trqc tqa d0 ld : | *" I * I y" I = xo * yo = x"+ lAxJ -1/2x"2 + 1 : 1l4xl + 1/2x,(2-""1 + I 2 l, v6imgi x.,: 0(XoS 1, dingthfrc xityrakhivdchikhixo:0+Yo= l' Viy, di6m M(0; 1). ) . Giai phucrng trinh n dtem) ptru'ong trinh dd cho tuong duong v6i pt : 1 cos8x* 3cos4x* 3cos2x=2cosx(cos9x+ 3cos3x)- ; (? cos8x* 3cos4x+ 3cos2x=2cosx.cos9x* 6cosx.coslx- ] € cos8x * 6cos3x.cosx: cos10x*cos8x* 6cos3x'cot*- t 1 <+ cosl0*=; €) 10x:+ n+2kn e *:* **\1, k.Z. 2. (1,0 bi€u kien : x € (- @; -21u [l; + oo) . Bdt phuong trinh tuong duong v6'i : 3.31-* + 6.31-" 7 3z-'1;z*-z <+ 9.31-x ;.32-JizTx4<+33-* 73s-'17+x-z <+ 3-x> 3 t3+x-z '13+x-2 >x (+ D6psd:x€ (-@; -21 U(2;+oo). r( x(0 It"'+x-2zo lr x2o Lt*'+ x-2>xz (+ l"-i;' (1,0 ) . Tinh tich ph6n eI dx r0 dx Ta co I = J-t U6*g = J-t U;r1"*+; r0 dx Xdt t=J_rr*-)r"*r), dAt t=-x =+ dt= rl dx +l J0 (1+x2)(ex+1) -dx (1+t2)(e-t+1) rL etdt -t Jo 1r+t2;1et+r) Tn r(hid6 I=[ du=ulfr= ]. * I-f#,dat x:tanu +O*:-!au; v6'ix:0+t=0, x:1* r:: 0.s0 IV a iIiAm) o (0,50 di€m). Gqi M ld trung di6rn cira BC. . AABC, ABCD ld haitam gi6c dAu bing nhau nen AM I BC, DM t BC vd AM: DM + BC r (ADM) . Khi d6 ke DH I (ABg) thi DH € (ADM) suy ra cludng thing AM la hinh clri6u cira AD tr€n mp(ABC) + fiffi = 60" vd AADM ctAu . Ggi I, J lAn luo-t li tAm cria haitam gi6c dAu AABC vd ABCD. D Suy ra tdm O ctra mdt c6u ngo4i ti6p tiri diQn ld giao di6m cria liai trqc cfia hai tam gi6c ABC vi BCD lAn luqt v€ qua I vi J. o (0,50 dia@. B6n kinh cria mflt cAu (O) bing OA = \AI4 O]z . rac6AM =*,ot::ot :T talitMa IM = ;AM :; , MO : cos3o" : I Do rt6 oA =VAlzTTMz -JMz az a2 a2 -+ 3912 . !3na2 V{y, S,n.:4nR'= 9 _ a./T5 6 'o *1,* -4' * ' l r\ : * * * * * *r* -i - 1,0 V I cli6m) l. Q,0 itidm) HQ bp' e {if _,17:}li;:":,^ x'- 4xy + 4y2 +2(x2 -2x+ 1): cQng v6 v6'i v6 c6c bpt ta du'gc : 3m <+ (x-2y)'+2(x- l)'.3m (*) 0,50 TiL(+) slry ra m> 0. Khi d6 tax6tcip(x, y)th6a rnsn {i-?;J * fr": rt,, R6r.Ang {,*' :"! +.2v' -x= 0 s m Nhr-rvfly 1r;l) ld nghi€m crtrahdbpr. ." tx'-Zxy-2x*2:0<m D6ps6:m20. 0,50 (1,0 tli€m). a) Phuong trinh chinh tic cria (E) , 5 .5: Elip (E) c6 c: 4 e a2 - b2 = 16. Di€m A(0: 3) "€ (Lt) + b2 = 9 + az =25. x2 v2 VAv(E):-+L:1. 259 0,50 /2 VI x . cuem) a)ft Uf'' = 3MF2 vd MFl + MFz :2a=+ 4MF2 = 2a e 4MFr' : a' ++ [(a-xo)2 *yot ] : 25 e 4(16 -8xn +xo2 *yo2)=25 (l). Met khdc M(x.; y") e (E) + *.'* =, Q)' .2s eJ3e . . zs sJ3e . Giai he (l), (2) ta dusc tut'( , ;-; ); Mt(; ;- I ). 0,50 Cqi vecto chi phuong crtad1,dz lAn luqt le [': 0:2;2),6.= (;2;-z)' Giao cti6m cisa d.yd2ld A(l ; I ; I ). 0,25 Ve"trr ph6p tryi5n cira m4t phing (P) chrlia db dyld$ :17|.,6I : (- 8; 4; 0) n€n phucrng trinh (P) ld 2x- y - I : 0, rO rdng M € (P). rac6 lfrl: lfrl nen q:6+6.=(2;4;0),4=G d:(0;0;a) ldhai vectochi phu'ong cira hai dudng phdn gi6c cria hai g6c t4o bdi db d2. Hai vectcr u1= ,6. "6 th6 xem ld hai vecto chi phuong cira hai dudng thing cAn tim. 0,50 Dud'ng thing A1 di qua M nh$n filim vecto chi phuong c6 phuong trinh : Eudng thing Az di qua M nh{n { ldm vecto chi phuong c6 phr.rong trinh: (x=2 ln =, lz=Llt. (x=2*t' lt=3+2t' \z:\. Dubng thing A2 giao v6i dv dztqi di€m A(l; l; I ) khdng t4o thdnh tam gi6c, n6n loAi Az, (x=2 Vay duong thing cAn tim ld A , ly = E lr=t+t. 0,25 VII 'I itiim\ (1,0 di6m). Giei he phuongtrinh . Hq <rd cho tucrng duong ,u,, li##Jt* ^16r_, *= ,rl; Tac6(2) a,fzyT+zy + t+ dzvr-v +t':(2y'+2y + l)-(2yt -y+ l) e,fzyt + zy + t -,fztz - y + t : t e,ffr + zy + t=.fzyz - y + l, +t e 2y' + 2y + 1 :2y2 -y +2 +z.rfzyz - y + t ez.,[zyr-y+t=3y-l 1 3y-L>0 o lnlrr' - y * L) = 9yz - 6y + L *l y>: ('=i * \r' - r, it= o o t['r==? D6p s6 : (x, y) : (22,3). y=3+ x=22. 1,00 . hai vecto chi phuong cira hai dudng thing cAn tim. 0,50 Dud'ng thing A1 di qua M nh$n filim vecto chi phuong c6 phuong trinh : Eudng thing Az di qua M nh{n { ldm vecto. hai duong thing : ,1,,4= Y-1 :'-r'. cl',,*: Y*t -'-3 t 2 2 -1 2 -2 Chri'ng minh c/r, dz cit nhau t4i di6m A; vi6t phuong trinh dud'ng thing A di qua. on r,AN urt/ga - NANI zorr t tru ai6m 'z a$m) thi (C) cit oy tai A(0; l), ndn @i trsc tsa d0 bang l. E6 thi hdm s6 c6 hai diiim cr,rc ti6u 1- f tJ i '