Hsc sinh tu gidi... Doct6 lxl=y= 3ld nghiQmcriahQptddcho..
Trang 1TRUONG EHSP HA NOI
KH6I TTIPT CIIUYTN
DE THr rHrI DAr HgC L.A.N THI' VI NAM 2009
Mdn thi :TOAN
Thoi gian ldm bdi : 180 phrit
CAu 1 (2 ili6m): Cho hdm s6 y = 2*'* 9mx2 * l2m2x + I (l).
l Kh6o s6t vd v€ tl6 thi ( C) cria hdm sO (t) t<tri m: t.
2 Tim m O6 nam si5 c6 cgc d4r, cgc ti6u tt6ng thoi xle = xcr.
Ciu 2 (2 di6m)
r
l Gi6i phuong trinh : sinsx - costx = l.or'z* - ]
"orz*.
22
2 ciei hc ohuons trinh : {1Cz:Ev + 15 +
'[FTM-CAu 3 (l dicm) Tinh tich phen : I = lru fi ln2xdx.
Cfiu 4 (l di6m)
Cho hinh ch6p tri gi6c ttdu S.ABCD c6 qnh tl6y bing a vd g6c TSE = a Gqi O ld giao di6m hai dudng ch6o cta tl6y ABCD Hdy x6c tlinh g6c a d€ m4t ciu tdm O tli qua ndm iliiim S, A, B, C, D
CAu 5 (l diem) Xdc tlinh m ee frg sau c6 nghiQm :
(x' + y' + 2(m - 1)y - 4mx * m2 + Zm = 0
1
CAu 6 (2 di6m)
l) TrongmAtphingvdihgtgactQOxy,choduongtron(C)c6 phuongtrinh: x2 +y'-h-6y+6=0vd
v di6m fuf(-f ; l) Gqi A vd B ld cric tiiip di6m cria cric ti6p tuy6n k6 tir M d6n (C) Tim toa d6 tli6m H ld hinh chiiiu vu6ng g6c c0a dii3m M l€n duong thing AB.
2) Trongkh6nggianvdihQtqadQOxyz,chomdtphing(P): x+2y-z+5=0 vd rludngthing
x*1 y+1 z-3
G: -Z-= 1 : 1
HEy viiit phuong trinh m{t'phinC (Q) chria tluong thing d qo v6i m4t phing (P) m$t g6c nh6 nh6t CAu 7 (l di6m) C6c s6thgc duongthay tl6i x,y, zthbaman : fiJ +,[y I + ^[z- t: t.
Tim gi6 tri lon nh6t cta bi€u thirc : P
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Trang 2oAp AN u6x roAN r,AN vr
cau r -(2.rli,5m)
l ( 1,0 ai6m; Hsc sinh tu gidi
2 (l,0tlitim).Tac6: y':6x2+ lSmx+12m2; y':0<+ x2+3mx+Zm2=0 (1)
Him s6 c6 cgc dgi, cgc ti6u khi vi chi khi pt (l ) c6 hai nghiQm phdn biQt'hay A : m2 > 0 <+ m:t 0
Khi d6 pt(l) c6 hai nghiQm : x1 : T; -3m- lml x2= z -3m+ lml.
D6u cfra y' :
*) Ni5um>0 thi (*)<+ 4m2:-m (vdnghiQm).
*) Niium<0thi (*)e m2=-2me m*-2.
t^
Dapsoi m: -2.
CAu2:(2,0cli6m).
1 (l,0di6m) PT <+ (sinax* cosax)(sinax- cosax) :)"or'z*-
]"orz*.
<+ - cos2x (F:sinz2x):
J.or2x.( cos2x- 1) e - cos2x(2- sin22x) =cos2x(cos2x - l)
<+ - cos2x(l+ cos22x): cos2x(cos2x -l) e cos2x(cos22x + cos2x):0
h
e I cos2x: o_ *= lz"J ] + rn : [* = I*Y ke z.
lcos2x = -1 [2x =tr * 2kn [x = ] + t<n
2 ( 1,0 tli€m) X€t phuong trinh : 3logae(49xz) - logr(y3) = 3
DiBu kiQn tt€ pt c6 nghia ld : x * 0, y > 0 (l),
Phuong trinh tr6n duo c virit thdnh : |rlgr(lx)z - 3log7 y - 3
(+ logT(7lxl)- logTy-1 <+ logT$=t o lxl= y
Thay lxl = y vdo phuong trinh ,fiz=y + 15 + JF +7Fi - 15 = ,/4*f rgy + 18 ta ducr c pt :
fit- By + 1s +,[FTz, - rs: JZ]r - 1By + 18
DiAukiQnd6phuongtrinh(*) c6nghla ld: y <- 5, y:3, yt 5
Tir<ti6ukiQn (l), suy ra di,SukiQn criay ld: y = 3, y > 5 (2)
D6 gidi pt (*), ta xdt c6c trudng hgp sau
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Trang 3N6uy=3, 16rirngy=3ldmQtnghiQmctrapt(*) Doct6 lxl=y= 3ld nghiQmcriahQptddcho.
NiSu y > 5, khi tt6 (*) tuong duong voi phuong trinh :
Ji=E +10+s:Jq=Z c:t v-s+v+s+2fi)r-sXJ'+5) = 4y-6
T6m l4i nghiQm cta h€ phuong trinh ld x: *3, y = 3.
Ciu 3: (1,0 tti6m)
Efltt=fi + x: t2 + dx:2tdt VoiX:lthi t:l;x:e2thit:e,n€ntac6:
t: z Ii*Lnlrdt: sf, tzrnzto,=
: f rn,t.d(t3) :
I cr,t li : f t3d0nzt)
8 16 ^ 16 ne 8, 16 ,te I , 16 , 16 40e3-16
=-e -erT- 3 9 9'L I t=dc=-e'+-t,l_ 9 27 tr = -e,+ 9 -ej 27 27
27
CAu 4: (1,0 di6m) Gqi H ld trung di6m cta AB, do tam gi6c
ASB c6n n6n ta c6 :
o(
ASH:-; Khid6 SH = AH cotg- o(ac( : - cotr_,
Dudng cao cia hinh ch6p ld :
s6:y'j112 _[112 :
f-r1.;.G
=+ so = lJ.o,r, |- r .
M4t cdu tdm O di qua ndm di6m S, .O!,
",
D ktri vd chi khi
cotgzl-1.
aJT
SO: .-2
-t
{cotez t- t=-7.
c(
Vay cotg t = V3 suy ra o = 600
CAu5.(l,0di€m)Tac6hQpt '-'ir- - fg (3x*4y*1=0 ,'yf.* (v+m -r)'= (2m-1)' (2) (1)
(1) le phuong trinh dubng trdn (C) c6 t6m ld I (2m; l- m) vd b6n kinh p: l2m - 11,
(2) ld phuong trinh ttudng thing d : 3x + 4y + I = 0
Oe ne cO nghiQm thi khoing c6ch tir tem I diin d nhd hon hoac bdng R: l2m - 1l, hay
13.2m+4(1-m)+11
a
hav'2
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Trang 4<+ r2m + 5r s 5r2m -
5
' VAy, vdi m >
; hoac m < 0 thi hQ phuong trinh c6 nghiQm
Cflu 6 (2,0 tli6m)
l) (1,0 di6m)
Dudng trdn (C) c6 tdm I(l; 3) vd ban kinh R = 2; MI =2rl-5> 2 = R, n€n M nim ngodi duong trdn
C6ch 1 Gqi H (x; y).Ta c6 ifr(* -t; y - 3), iMt- +; - 2) vd,nhan th6y hai vecto ifr vi iM cirng chiAu,
n6nin =t iM (t > o) € {i _ 1
=
=
tr _ii
Theo hQ thttc lugng trong AAMH vu6ng, ta c6 : iH'ifr = IH.IM = IA2 = R2
I - 1 13,
e -4(x -l)-2(v -3)=4 e -4(-4t) -2(-2t):4 e t:;' Vav: H(i; T)'
crlch 2 Gi6 sri tli€m A(x"; y") ld ti6p di6m , ' thi f5 (MA l (? * IA f-l1(q (MA.IA = 0' ^,trong d6 :
MI1x"+ 3; yo - l;, Id(x" - l; y" - 3) Do d6 ta c6 :
f \!,+v3_Zxo_6yo+6:0 *["rty3 _2x, tr, U:O_ Zxo+yo_3=0 l(x + 3)(xo - 1) + ( yo - 1)( yo - 3) = 0 -
t x2' + yZ t Zxo - 4yo = 0 Suy ra <tuong thing AB c6 phuong trinh 2x + y - 3 = 0
Dudng thing MI c6 phuong trinh : []== ilit Do MI vr.r6ng g6c vdi AB, n€n tsa dQ cta di€m H ld
( *: t+Zt
nghiQm criahQphuongtrinh:{ (zx + V = y- 3 *t 3 = Giaihg nirytaduo.c Ut*; ?i.
o
2) (1,0 di6m).Xdt m{t phing (Rj thay O6i ai qua dudng thing d, cit mp(P) theo giao tuy6n A Khi d6 A chria iti6mA=dn(P).L6ydi6mt<cOAlnntr€nd(K+A).Gqi Hldhinhchi6ucriaKtr€nmp(P), Ilehinh chitiu cria H tr6n A thi HI vd ru cinllhuOng g6c vdi A n6n FFI ld g6c gifa (P) va (R)
Ta c6 tanKiit :
H *U KH khdng ati Wri (R) thay eoi va gt < HA n€n ffiH nno nh6t <+ tanKIH nh6
nh6teHllonnh6t<+ItrilngAhayAIdtaiA,tircldAnimtr6n(P),diquaAvdvu6ngg6cvoid.
Ditim A(-;t - ;r ; l rni d6, A c6 vdctochi phuonguj =;tuj npl = Cl; l; l).
t-co , i[uai ua] = (0; -1; 1) n€n (Q) c6 vecto ph6p tuyr5n ld f = iQ; -l; 1)
Viy mp(O c6 phuong trinh : y - z * 4 = 0
CAuT:(l,0cti6m) DidukiQn :x> l, y> l, zZ1.
Tiritangthfcgiathiiittasuyra.fiT<l=+ x <2.vAytu.o "tt;; f1 ;:;' =+ -t-v+z= P=-x- < 3- =1
1+1 -''
Dingthticxiyrakhi x=2,y=z= l.K6tlu4n : MaxP= I khi x= 2,y :z= l.
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