Dedap an thi thu DHSPHNlan 6

TRUONG 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
t
3x+'+y+r=o
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
@

oAp 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
-3m-
lml
-3m+
lml
: T;

x2=
z
.
Xs X2
+oo
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
(:+
J6-m:5
+re-tg+
s)
:
n/S:s1+y-6;
1*;
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
:

.
N6uy=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
,+
/6;Tm[5):y-3
€ y2
-2s:y'-6y+3
<+
6y:28c)
y=
T.t,ncnlo4i.
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)
=
i"' Tf
t2lnt.dt
=
f"'-

1rqJ,"'nr.d(t')=
9;'-
f,rr"tl;
.Tf,lt'.arrno
8
.
16
^
16
ne
8, 16
,te
I
,
16
,
16 40e3-16
=-e erT-
I t=dc=-e'+-t,l_
=
-e,+
-ej3 9
9'L
9
27
tr
9 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-
:
-
cotr_,.
o(ac(.
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
-
I
^n
atlz
{cotez
t-
t=-7.
c(
Vay cotg
t
=
V3
suy
ra o
=
600.
CAu5.(l,0di€m)Tac6hQpt

fg ,'yf.*
(v+m

-r)'=
(2m-1)' (1)
'-'ir-
(3x*4y*1=0
(2)
(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
\tFp
s
lzm-11

<+
r2m
+
5r
s
5r2m
-
'r
*
lrTiJjjs(a*
-t?,
*
[::6
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
=
_i:;

[i
=
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
(?
*
f-l1(q
^,trong
d6 :
'
(MA
l. IA
(MA.IA
=
0'
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:{
V
=
3
*t
. Giaihg
nirytaduo.c Ut*;
?i.
(zx
+
y-
3
=
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.
,
1 210
1,.
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
f1
;:;'
=+
P=-x-
<
3-

=1
"tt;;
-t-v+z=
1+1
-''
Dingthticxiyrakhi
x=2,y=z=
l.K6tlu4n : MaxP= I khi x= 2,y
:z=
l.