TRUONG DHSP HA NQI TRUcD{c rHpr cHuytN - EHSp DE THI utl DAr Hec r,AN vr xAvr zoro M6n thi : TOAN Thdi gian ldm bdi ; t B0 philr, kh6ng ke thdt gian phdr di CAu I. 1z diafi: Cho hAm s6 y = x4 -2a2x2 +b v6.i a, b ld tham s5 (1) . 1. Khdo s6tsubi6n thi€n vdve dd thl crja hdm sO 1ty mi u= Evd b =4. !2 2. Tim cdc gi6, tri crla a * 0 vd b di5 c6c di6m cuc dai, cuc ti€u cria dO thi hdm s6 1t; tao thdnh tam gi6c d€u. CAu II. (2 dia@. 1) Giii phuong trinh : cot2x - 2tan4x - tan2x = -4^13 . 2) ciei he phucrng trinh : [(x - +)G + ]) = v(v + 5) ,,,uw'6Lrrrrr. I logt"_rl(y +Z) : T C0u III. (t die@, rinh tich ph6n r : f fr, a- CAu IV. (1 die@. Cho hinh ch6p tam gi5c dOLr S.ABC c6 c4nh d6y bing a vd canh b€n tqo v6.i mdr driy m6t g6c 60". M6t mdt cAu tam o ti6p xric v6'i m4t ddy (ABC) tai A va ti6p xrlc v6'i duong thing BS tai H. H6y x6c dinh vi tri tuong coi gina H v6'i hai di6m B, S vd tinh di€n tich mdt cAu t6m o. C6u V. (l di€m). Cho cdc sO duong x, y, zth6a rndn xlz* y + y - z: 0. Tim gi6 tri l6n nh6t cfia bi6u thilc : P= .2 - 3 xz+l y2+1 z2+L' Cdu VI. (2 dia@. 1) Trong mdt phdng oxy, cho ducrng tron (c) : x' + y2 - 6x + 2y - 15= 0. Tim toa dd di6m M tr€n dudng thhngd:3x-22y-6=0,saochotirdi6mMkdduo'ct6'i(a)haititiptuy6nMA,MB(A,Bldcricti6p di€m) md dudng thang AB di qua di6m C(0 ; l). 2) Trongkhdnggianoxyz,chohinhldngtrudrmgABC.A'B'c'v6iA(a;0; 0),8(-a;0; 0),c(0;r;0) vd B'(-a; 0; b), trong d6 a vd b rd hai si5 duong thay d6i nhung ru6n th6a mdn a + u: 6lz.Tim a, b dti khoAng c6ch gifi.a hai duong thing B,C vd AC, l6n nh6t. CAu VII. (1 die@. Cho hai s6 phric Z1 = cosf- isin fi vit z2=_ I +iVT. Hdy x5c dinh d4ng dai s6 crja s6 phric z= (2,.22)ts. H6r CAU I. 1. Hoc sinh tu'gidi. . 2. Tac6: y' = 4x3 -4a2x Bing bi6n thi6n oAp AIv ToM TAT + y' :0 <=+ 4x(x2 - ar) : 0 * [r]=:f trt vd lim"_* y: + oo Dat A(0;b), B(-lal ;b-a4), c(lal ,b_uo). t(hi d6di€mAthu6crrucrungcdnhai ditim ximg nhau qua truc tung, n6n tarn giiic ABC cAn tai A. E€ AABC d6u cAn vd drl ld AB : BC. tuong duong v6'i : a' + at - 4a2 e a6:3 <= a: + V3 . Vdy v6i a : *V3 vd b ld s6 thuc tuy y. Cdu lI : 1. Di6u ki€n : sinSx # 0. cos2x sin2x P I (+ - Ztan4x = _4r8. sin2x cos2x Zcos4x ^ sin4x cosz4x - sinz4x : - = tY5 c=-: _l^,n sin4x -cos4x rvr " sin4x.cos4x Lvr c+ cos8x = -€ sin8x e+ cot 8x: -v5 ( do singx I 0 ). ' €+ 8x=-I*kn',= "= -l l<n 6 +;* u (kez)' f2<x+3 2. Di€ukidn:l y>-z I y*o Tac6 phuorrgrrintr (x-4)(x+ 1):y(y+ 5) ? *, _ 3x _y(y + 5) _ 4:0 (*) Coi(*) ldphuo.ngtrinh An xc6 A =(2y+5)2. n6npt(*) c6 nghi€m le lx = y+4 [x=_y_1 V6i x : y + 4,thay vdo pt thfr liai cria h€, ta dugc : log(v+zt(y +2)=t#*,#:t €+ y2 -y-z:o So s6nh v6'i diOu ki6n chi c6 x = 6 th6a rndn. v6,x=-y-i, dox>2n€n -y-r > 2ey < -3, rar.ngth6amdndi.uki6n. Vdy nghi6rn crja h€ phu'ong trinh ld (x, y) = (6, 2). B, C d6i ^.). ureu nay *' []:;' - ll:Z \ y'ot /+* \ / \ / \ b-ut / \ b_ uo / CAU III. Tac6l .:fi*o*= I .i ti(* - *)0,. =-ir"o(*) : -r* li =1+lrnlrl li = I *1rn1 . 3 4 lx+1| l0 3 4 3 Cdu IV. Gqi G ld trgng tdm crla AABC. I(d OI( I SG, I( € SG. Ifti d6 SEE = 60" n€n SG = GB.tan60o: "G /3 = u. 3 ss :\reEz;ZBz : tr +t : r^,15 . \33 Do BH = BA = a va u. +ndn H nim gifr.a S vd B. 3 Tac6 oH2 +HS2 = oS2 : oI(2 +KS2. OA : oH = GK = R ( R - b6n kinh mdt c6u tAm O ) n6n n, + 1&J3 -a)' = (+)' + (a-R), <+ Do d6 di€n tich m4t cAu la : S,,. = 4ftp12 = CAU V. A (+-J5 )a D : i r- l\ 2t3 (rs -eVr )naz 3 Tac6 xyz*y *y=z<=+x+ y=z(1- xy) <+ .=:+. (vi x,y>0n€n 1-xy +0). 1-xy Dat x=tanq, y=tanB vo-ia,p e ro;|)khi d6 z: tan(a+ ilvdbi€uthrictr6.thdnh: ' P: -: " + r- - -=:* : Zcos2a+3cos2f-2cos21a+81 r+tan2a r+tanz1 l+an2(a+p) -""" * : cos2a+ I + 3cos2/t- [cos(2a + Zb+ 1] : ZsinB.sin(2a + h +3(1- sinrB ). Eat t:sinB thi p< -3t, +2t+3:-3(t-1lr*T=T Edng thri'c xAy ra khi vd chi khi { sinB=+.*l sinB-i l [sin(2c + F) = r - ' lz* =]- 0.+ zkz I sinB - + = fcos'F = f, =!r'=tun'F = * cos?u-sinp Icos2a =1 l*, = tunro=i J2 7 "12) ' 10 uooo f,r*= T rar,chinghan (x, y,z)=(Q '2' Cdu Vl. 1) Duongtrdn(C) c6tArn I (3;- i)vdb6n kinh R:5. . 3t-6. Xet M (U# ) e a. Dr-rongthEngquaM se ldtieptuydn cna(C)taiT(x;y) khi vdchikhi tWT. tf =O 22 <+ (x - t) (x - 3) + (y - + ) 0 + 1) - 0 <=+ x, - 3x - rx + 3r + y, + y - 3t-6v - t'-u = 0 22 ' ) " 22 t 22 <+ x2+ y'-6x+2y+3x-y-tx+3t -ff, T-0<+ l5+3x-y-rx +zt-ffv -tlru =o 16+3t 53r+336 €(3- t)x- n y+ n =0 (*). Nhu'vay c6c ti6p di6nr A, B cria ti6p tuyt5n kd tu M d6n duong tron (C) c6 toa d6 th6a mdrr phuong trinh (*) . Do d6 (*) chfnlr ld phuong trinh dudng thdng AB. Eu'd'ng thing ndy di qua di€m C (0; 1) khi vd chi 1 6+3t 63t+336 76 , 1 A khi - : * :: 0 er t- - Suy radi6m M (-*:-1). 22223'3-/' 2) Tu' giA thi€t suy ra: CCi : BBi + C'(0; 1; b) , - B' C : (a: l; - b), AC' = (- a; l; b). KhoAng c6ch gifi'a hai dud'ng thing B'C vd AC' B bing khoAng c6ch tir di€rn A d6n rndt phdng (o) chila B'C vd song song v6'i AC'. Vecto'ph6p tuyiin cria mp(a) ld i : tE, i, ed t : Qr, - ool, l;u _|,l, l:, 1, l) = rzu, 0; 2a) Suy ra phuong trinh cira mp(u) ld : bx * az= A. Do d6 khoing c6clr tu A d6n mp(c,) cfing ld l<hoAng cdch ., : I - ab gifa hai dudng thdng B'C vd AC' bing h: ;;ftr i r 1t.,: ab ab v'?'5 1 Ap dung bat dang tnuc Co-sl ta co : ffi S Um= ,/, S A. DSng thric xdy ra khi vd chi khi u= 6 = jtfT. VAy khoang c6ch gifi'a hai dud'ng thing B'c vd AC' t6'n nhAt bing 3 khi a : b 3{2. Cf;u VII. rac6: 21 = cos(-;)+i sin(-*) yd z2=r(-)* rf) =ztcosf,*rrinf,). Suyra Zr.zz=z1cos3*;rinf )+ (21.22)tE=ztt(ror')n *,rinf ):2't.i . B' a*b O"E -___: - 1 ) 1^l') /,""" t.o ', "' t. t . TRUONG DHSP HA NQI TRUcD{c rHpr cHuytN - EHSp DE THI utl DAr Hec r,AN vr xAvr zoro M6n thi : TOAN Thdi gian ldm bdi ; t B0 philr, kh6ng ke thdt gian. . 1. Khdo s6tsubi6n thi n vdve dd thl crja hdm sO 1ty mi u= Evd b =4. !2 2. Tim cdc gi6, tri crla a * 0 vd b di5 c6c di6m cuc dai, cuc ti€u cria dO thi hdm s6 1t; tao. Eu'd'ng thing ndy di qua di€m C (0; 1) khi vd chi 1 6+3t 63t+336 76 , 1 A khi - : * :: 0 er t- - Suy radi6m M (-*:-1). 22223'3-/' 2) Tu' giA thi t suy