w
TRU'0NG
DHSP
HA
Nol
KH6I
THPT
CHUVEN
sE THI
THIIFJA.I
IIQC
LAN
V I-,IAM
2009
Mdn
fhi:
To6n
Thdi
gian
ldm bdi:
180
ph0t
Cflu [.
(2
dii3m):
Cho
hdm
sii
y:
x3
+

3x2
+
rnx
+
I
(l)
(m
Id tham si5)
'
'
L Khio
s6t vi
vE d6 thi
hdm sii
1t ;
thi
m
=
0.
2.
TimmA6euO'ngthingy=
lcatd6tlri
hdmsO(t)tai
3
di6m
phdnbiQt: I(0;,1),A
vdB. Vd'i
gi6tri
nio
cria m,

c6c tiiip
tuy6n crla
d6
thi hdm
sO
(t
)
tai
c6c di6m A vd B
vu6ng
g6c
v6i
nhau.
CAu II.
(2
di6m).
l. Giiiphuongtrinh:
8cosx.cos($-x).cos($
+x)+
I
:0'
((x-a)(x+1)-y(y+s)
JA
2.
Gieihephuongtrinh:
i
loro_r,0
+D=T
c6u
rrr.

(l.di6m). Tinh
tich
phdn
: t
=
i"vz
-G#r),
,
Cdu
IV.
(l
di€m).
Cho
hinh ch6p
tir
gidc
d6u S,ABCD c6 d0 ddi c4nh
driy bing a,
cAnh b€n tao vdi
m4t
driy
mQt
g6c
bing
30". Tinh
diQn tich
mgt cAu ngoai titlp
hinh
ch6p.
Cffu V.

(l
ditim). C6c
s6 thgc
duong thay dili x,
y,
z th6a md.n
: x2
+
y2
+
z2 = 3.
Tim
gi6tri
nh6
nhAt cira bi6u
thti'c , P
=
-i-
*
+
*
+
xy+1
yz+L
zx+\
CAu
VI.
(2
di€m).
L

l)
Trong
mflt
phing v6{ he tqa d9
O*y,
cho
hai di6m A(5;
-
2), B(- 3;
4) vd
dud'ng thing d c6
phuong
\
/
trinh:
x
-2y
+
I = 0. Tim
tga'dQ
di6m C tr€n
dudng
thing d sao
cho tam
gi6c
ABC
vu6ng
tai C.
2) Trong kh6ng
gian

v6i hQ tqa
dp Oxyz
cho
hai
duong thing
,
(
*=1+:
x+4
y-8 z-B
d,:{ v=1-t vA d,:-
(
z=-Z+Zt
'
2 1
-1
a) Chung minh
ring
hai
du'dng
thing
d1 vd
d2 chdo
nhau.
b) Gpi MN ld duo'ng
vu6ng
g6c
chung
crla
d1 vd d2

(M e
d1
,
N
e
d2). Hdy viiit
phuong
trinh
mat
cAu duong kinh MN.
CAu VII.
(l
di€m).
Tinh
t6ng
'
s
=
'4=
*=+ *;+-
+

+
*+
+
-:
.
2!.20071
41.20051
6!.2003!

2006!.3! 20o8t.1!
-

oAp
AN
mox
roAN
r,AN v
cAu
r.
1z,o
a;em;.
l.
(1,0
didm;.
Voi
rn
=
0, ta c6
y
:
x3
+
3x2
+
l.
'
T?R
x6c
dinh

: D
=
R'
.
Subi6nthi6n:
y'=3x:+6x,
y'=0c+x=0
hoflc
x:-2.
Tac6y(0):1,y(-2)=5.
Bdng
bi6n
thi€n
:
yco:
y(-2): 5,
ycr
:
Y(0)
:
l.
.
Dd
thi
:
(
hqc
sinh
qu
vE

hinh
).
2.
(1,0
di6m).
Eudng
thing
y
=
I
cit d6
tlri hdm sd tai 3 didm
phdn
biet khi vd
chi
khi
pt
sau c6 3
nghiQrn
ph6n
biqt :
xj
+
3x2
+
mx
+
l
=
I

<.+
x'
+
3xt
+
mx
:
0
<+
x(x2
+
3x
+
m)
= 0
e+
[
-r
*
r-i=*o= 0
(2)
(=
pt
(2)
c6
2
nghiQm
phin
biet
ktritc 0, hoy

[a
=
l-*oT
) 0
=
{|; I
(*)
Gii sri A(x1;
l), B(xe;
l). Khi
d6 h9 sil
g6c
cua ti6p tuy6n tai A li
:
ke=y'(xe)
=3x2l-+
6x4
+
6:3(xl
+
3xa+61
-3xr-
2m=
-3xa-2rn.
Tuong f.u, ta cr)ng
c5 : ke
:
-
3xs
-

2ln .
C6c titip ruytin
t4i A
vd
B
vu6ng
g6c
vdi nhau
khi vi chi
khi :
ke.ka:-
1
e (3x6+
2m)(?B+2m):
-
l
e
gxaxs*6m(xs+xs)+4m2+
l=0
(3).
Theo h€ thirc Viet, ta c6 :
x1
+
xe
:
3, xaxs
:
m, thay vdo (3)
ta duoc
:

f,n=:tG
4m2
-9m+
i
=
0
<=
|
,_h
.U hai
giri
n'i niy cria m thoa mdn dk
(*)
Lm
=
-;-
cAu
tt.
(2,0
tii6m)
I
(r'0
di€m)'
*"*'-::
-",il::-;:,Tl4cosx
(
2cos?x
-
| I
+

r
:
o
<=
'
3 2'
2(4cos3x-3cosx)=-
I
c"a
cos3x:
-
1r*"=++
**,
Urz.
293
2.
(1,0di6m).Ei6uki€n
x-2>
0.x-2* l,y+ 2>0,y+0.
Ta
c6 :
(x
-
4)(x+ l)
=y(y+5)
e (x
+y+
lXx-y-4)=
0.=
[.::].:-n.

IX=-y-l
.
Vdi x=-y- I
e
x-2 y-3>0=+
y<-3.
lt2

Theo di6u kign
thi
y
>
-
2,
n€n truong
hq
p
ndy
bi
lo4i.
.
V<ii x
=
y+4;rhayvAoptrhirhai
tacluoc
:
logr.*r(y+D=*aV2
=
[;
:

|
,
*.'.o diiu
kiQn
chi
co x
= 6 th6a
mdn.
Vay : NghiQm
cta
phuong
trinh
li
x=
6,y
=2.
:c6 [: f
t%-
d*-=
ltoVZ
*td*
J1
x(xro+1)2,
.r1
x10(x10+1)z'
D4t
t=xro,
vdi
x= I
thi t=

l; x:'W
thi
t=2,vd
dt= l0xedx.
Khid6 ,_
r
i2
dt
1,2f1+t-t)dt t
12 dr t.2 dt
ro,r
t(t+l)z:
iJr
i(*LF
=
GJr
(r-r)
-
-Jr
Gf
=
L*r:(i-#dt
+;fu1?:*''1,*l
l?.*(i
-
)=frrrn+-rn:)-fr
cAu
ry.
(
r,o di6nr).

Coj iI {C tim cua
hinh vu6ng ABCD,
thi
SH
1
(ABCD)
vi
IiE
=
30". Do AC: uVZ, ,uy ra
SA
=
t'
"u
rn
=
*
Tdm O
qia
cAu ngo4i
ti6p hinh
ch6p
sC
fa'giuo
Aie,r,
giE"u
m4t
phing
trung truc cia doan thing
SA vi dudng

thdng
SH. Gqi M.ld trung di6m cria doqn
thing
SA, ta c6
4 di6m
A,
M, H, O cirng nirl tr€n dudng
tr'6n dudng
kinh
AO,
n€n
A
SA.SM
=
sH.so
<+
R: so
=
s1.s,V
=
4
. huu uen
sH 3
' '
'-
kfrrh
cia mqt ciu ngo4i tiiip hinh ch6p
li
R
=

+
.
3
Vay,
Sr.
=
4nR2
=
cAu v.
(
t,o
di6m
).
Tru6ch6ttachirng
minh bdtdingtlrfic
'
1*1*1-
-1*-,
vdia, b,cduong.
a
b
c-a+b+c'
+b+sx:.;.fr=,.(l
.
f;).(:.r;).
(l
*
f,):r*
z+z+z=s.
I119

Suvra:.
-
+
-+
->
:-,ddubingxdyrakhia=b=c.
(ttpcrn)
'
a b c-a+b+c
Ap dung
Udt eing thric tr6n vd bdt
ding
thuc
a2
+
bz
>
2ab.
7LL93
T^
^:
-
D-

-
y-+y!+L
y"!r.
+t
x.+2.
+1-

x2+y2+22+3
2,
222
Ddu
bing xdy ra khi vi chi khi x:Y
=z: l.
3
Vey,
P*:;,khix=y=z=1.
-y-2=t,-[t;;
@
(^i\
/.\\t
/
"'',
t'.,H
-
/ / \
j-
L \ r\
/

\
:
\-
: '
\
|
\


cAu
vL
(
2,0
tlitirn).
I)
(1,0didrn).Cdchl GidsfLdi€mCed,khid6C(2t-l:),Te
(2t-6;t+2)vAEC1zr+2;t-4).
G6cIeB:90'eTe
.Ee
:0
e
(2t-6)(2t+2)+(r +2){t-4)=0
+.+
t2-2.t-4:0<+t=l
+1/6
Vayc6haidi6mCrCndthoamdrryduciubiitorln: C1(1+
ZrlS;l+r/5)uaC:( I
-26;
f
-r/5).
Cdch 2. Gid su
di6m
C
€
d, khi d6 C(2t
-
i
;
t).

G6c TeB
:90'
suy
ra
di6m C
nirn tr€n dudng trdn dudng
kinh AB.
Ta c6 trung di6m
I cria do4n thing
AB c6
tga tI$
(l;
l)
vA
AB
:
10.
N6n : IC
=
5
c=+ (2t-42
+
(
t- l)2=25.
Gidi
ptndy ra s€
dugc
k6t
qun
nhu cdch l.

2)
(1,0
di6m).
a)
(0,5
di6m).
Duongthingd1diquaA(I;
I;
-2)6c6uectochiphuongE=tf
;-
I;2).
Euong thing d2
tti
qua
B(-4;
8; 8) vA c6 r,ecto
chi
phusng
@'
=
1l;
l;
-
l).
Ta c6
ffi,41=
(-l;
5;\, TE
=
(-

5; 7; l0). Suy
ratL;f,.,6).
af,
=
1-l)(-s)
+
s.7
+
3.10:70
+
0.
Vfly hai
dulng thing
d1
vd
d2 chdo nhau.
b)
(0,5
di6m).
DitlmM
e
dl
+
M(l+
t; l-
t; -2+ 2Q. Di€m N
e
d2+
N(- 4
+

2t';
8
+
t'; 8-t').
Suy ra MF
:
(-5
-
t+ 2t':7
+
t
+
t';
10
-
2t- t').
Tac6Mfi
tuleMrt.q.=0<+6t+t'= 8; Mfr r7;,eMfr.q=0<+t+
6t'= 13.
Giaihe {6rt!=8_
e
il=1
_ f
M(Z;o;o)
.rt+6t,=13
tt,=2
1
lru(O;fO;O)
Suy
ra

mit
cAu dudng kinh
MN c6 tArn
I(l; 5;
3), bdn kinh R:
tr.
Phuong trinh
cria n6 Ii :
(x-
l)t
+
(y
-
5)2
+
(z-
l)2
=
35
cAu vu.
(
l,o
di6m).
rac6
2ooe!.s
:
ffi-;##.##+
.
+ffi
.ffi

:
C\.oog+
Cloos+
CSoog
+

+
Ci333
*
Crt333
.
&
Bing
crich
khai tli6n
1l
+
x
;200e
ue 6hqn X
:
-1,
ta
dnoc
ding
thirc :
Cloos+
C3.oog+C!.o0"+

*C3333*

C3333:Cloor+C3oor+Clooe+

*C3331*
C3333,
NgoAi
ra, chgn x
:
I ta cluo.'c tling
thri'c :
cloos+
cztoos+
C3.oog
+

*
C:339
*
ci333
=
2200e
Ttrd6
suy ra : C|oo,
+
C\.oor+ Cloor+ C!oo,
+

*
C3333
+
Ci333

-
22008
-200s
^
L
-I
v av
?r,
-
2009!