TRUONG DHSP HA NQITRUcD{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.. Tim toa dd di6m M tr€n
Trang 1TRUONG 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
Trang 2CAU 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.
Zcos4x ^ sin4x cosz4x - sinz4x :
-
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
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
b-ut / \ b_
uo /
Trang 3CAU III.
Tac6l =-ir"o(*) : -r* li :fi*o*= I i ti(* - *)0,.
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) <+ .=:+ 1-xy (vi x,y>0n€n 1-xy +0).
Dat x=tanq, y=tanB vo-ia,p e ro;|)khi d6 z: tan(a+ ilvdbi€uthrictr6.thdnh:
' P: -: " + r- - -=:* r+tan2a r+tanz1 l+an2(a+p) -""" : Zcos2a+3cos2f-2cos21a+81*
: 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
[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'
Trang 4Cdu Vl.
1) Duongtrdn(C) c6tArn I (3;- i)vdb6n kinh R:5.
3t-6.
Xet M (U# 22 ) e a Dr-rongthEngquaM se ldtieptuydn cna(C)taiT(x;y) khi vdchikhi tWT tf =O
<+ (x - t) (x - 3) + (y - + ) 0 + 1) - 0 <=+ x, - 3x - rx + 3r + y, + y - 3t-6v - t'-u
= 0
<+ x2+ y'-6x+2y+3x-y-tx+3t -ff, T-0<+ l5+3x-y-rx +zt-ffv -tlru =o
16+3t 53r+336
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
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
gifa hai dudng thdng B'C vd AC' bing h: ;;ftr
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