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.. Tinh diQn tich mpt cAu ngo4i ti6p tr dien... Gqi M ld trung di6rn cira BC.. Ggi I, J
Trang 1orirru rrffiro, TRU'oNG EHSP HA hiQI
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= t Y-1 2 :'-r' 2 cl',,*: -1 Y*t 2 -'-3
-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
Trang 2oAp An - THANG DIEM rnr rntf on r,AN urt/ga - NANI zorr
'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
(1,0 ) Tinh tich ph6n
Ta co I = J-t U6*g = J-t U;r1"*+;
Xdt t=J_rr*-)r"*r), dAt t=-x =+ dt=
rl dx +lJ0
(1+x2)(ex+1)
-dx (1+t2)(e-t+1)
rL etdt -t Jo 1r+t2;1et+r)
Trang 3r(hid6 I=[ du=ulfr= ].
0.s0
IV
aiIiAm)
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
!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)
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
Trang 4VI
-.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
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