6 đề-đáp án thi thủ ĐHSPHN-2008-2009

EL
TRLIONC
DHSP
Ha
NOI
ru6r
rHpr
cHUyEN
nt
rru
THrt
DAr
Hec
naOn
roAN
IAN
I
trAwt
Hec
2oo8
_
2oo9
(Th6.i
gian
IB0phtit)

t***
Ciu
l.
(Z,O
Aiem).

Chohdms6
y=n*r*,*+
l)x2+1m2*4m*3)x+1
l.
Khdo
s6r
vi
v€
d6 thi
cria
hdm
sti
khi
m =
-
3.
2.
v'i
gi6
tri
ndro cta
m,
hdrm
s5 c6
clrc
d?i, cgc
tiAu?
Ggi
x1,
x2

li
hai
di6m
cgc
tl4i,
cgc
tiiiu
cria
hi'n
s5,
hdy
tirn gid
tri
lon
nhdt
crja
bi6u
thric
A
=
i*r'i-ri"'i
-rl
f
Cflu
2.
(2,0
di€m)
y
l.
Gini

phuong
trinh
:
. cos2x
*
cos5x
-
sin3x
-
cosSx
=
sinl0x.
Z.
Giili
bdt
ptru,rng
trinh
:
Cdu
3.
(1,0
di€m)
.
^74os'@jslog,
(x-)
Tim
hq
cdc
nguy6n
him

cria
him
s5
:
f(x):
'
xa-1
x(x+-s;6xs-sx+1)
'
Cffu
4.
(2,0
diOrn)
cho
hinh
ldng
tr-tr
tam giiic
d6u
ABC.A'B'C'
c6
dO
dai
canh
d6y
bing
a, g6c
gita
cluo*g
thing

AB'
vdrn{t
ptrang@e,C'C)
bing
a.
e'
6vv
l.
Tinh
d0
ddi rtoen
thing
AB'
rheo
a
vd
s.
2'
Tinh
di$n
tich
rn{t.ciu
ngo4i
titip
hinh
ldng
t4r
AIIC.A'8,C,
theo
a vdi

a.
g,Cdu
5.
(1,0
di€rn)
cieihephuungrrinh
[r;?;
!2n
,"ur,
,,!
gCAu
6. (1,0
diem)
Chirng
minh
ring
:
i
'{
i
*+fr+
++-1oos/
1
,-
1

-
ioos
ciaos
i*Ft

-
roo,
til
+-
EJ*o
+
"
+
(Trong
d6
Cl
IA.s6
tO
h-op
chflp
k crja
n
phnn
tri)
<pCflu
7.
(1,0
didm)
\
Trong
m{t
phing
vdi
rr€
r-o:

dg
oxy,
cho
tam
gi6c
ABC
vdi
A(?;
-1),
B(r; -2)
vd
rrg*g
tdrn
G
cfra
tarn giric
niim
tr€n
duon!
tning
d:
x-+
y
-
2
= o.
iray
tim
tga
dQ

diem
c,
bi,it
rang
di-6n
tich
tam
gidrc
bing
j.
i
i\rtt
t
-
2i
I !
|

-
'
, .J'
L]
\i I
r.
2
1\
-l
r200a I'
uzoog/
trdt

a9
f-T-"i:iti:%
/>

6 2008-2009 ĐHSP HỌC ĐẠI THỬ THI ĐỀ
Sưu Hải Minh Nguyễn tầm:
oAp
AN
vA
rueNc
DrEM
l.
(1,25
Aiemt
.
Gidi
hqn:
limr,-*_
y
=
+ co
,
limrr-__
Jr=
-
.,b.
.
.
Su
bii5n

thi€n:
y,
=
?x2
-
4{

,y:'=
io
U
- O.
f,,ia.
x
=
2.
y'>o*
f
;:
vd
y,<o<+o<
x<2.
Do
tl6
hdm
s6
d6ng
biiSn
trong
nrdi
khod.ng

G*;
0)
vit (2;+oo),
nghich
biiin
hong
Vdim=-3,thi
t=:*t
.
Tap
x6c
dinh
:
R
khoang
(0;2).
.
Cgc
tri
:
Hdm
si5
y
dat
cgc
d4i
tai
x:
0 vd
yc,p

=
y(0)
:
;,
Hdm
sti d4t
cgc
ti6u
t4i
x:2
vA.!c.r
=
l(2)=
-
13.
.
Beng
bitin
thi€n
xl
o
z
+co
1
r
2\
?.o
_*,/
\_
!/

Dg
Ai
.
(Hqc
sinh
tu
ue
t,
?t,*r"::
r
!.','
=
Of
O,c6
y"(l):0
vd
y,,
dr5i
d6u
khi
di qua
x
= l,
n€n
di6m
(l;
-; )
le
diiSm

u6n
yd
cfing
ti
tem
d6i
xring
cria
d6
thi.
?6
Ai cit
trsc
tung
r4i
ai6m
qo;
]
).
?o
d6
A
ton
ntrdt
bing
3
khi
m
=
-

+.
Ta
c6 y'=
2*
I::1::"Lgu1:oc
ti6u
nri
vd
chi
*,
ri:0
c6
hai
nghiQm
ph6n
bigt
x1,
x2
hay
a': (m
+
l)2
-2(m2
*
4m
+
3)
>
0
(+

m2
*
6*;;-;
il:ffi
.:;.
$eo
dinh
lf
Viet,
ta
c6
x,
1x2
=
-
(m +
l),
xr.xz
=lm2
+
4m
+
3y.
l"l:l
A=
li(*'
r
4m *
3)
+2(m

+
l)l
=|l*,
*
a**
z1
Tanhinth6y,vdi
m
e(-5;
-l)thi:
-
9S
mf+
gm+
7=(m*4)2_9
<0.
1.
(1,0
di6m)
CAU
II
Phuong
trinh
dusc
viiSt
ve
ftng
cosSx
-
cos2x

+
sin3x
+
sinl0x
_
cos5x
=
0
<+
-
2sin5x.sin3xI
sin3x +
2sin5x.cos5x_cos5x=0
c+
cos5x(2sin5x_
l)
_
sin3x(2s
@
in5x-1)=0

Voi
sinsx=
1
0 [
u*=r+Zkr
z

[s*=n_]*zkn
(+

Vdi
cos5x
= sin3x
(+
cos5x
= cos(
_
fx;
J4f
tfln
nglri0m
crlaphuong
uinh
la
S
={
I
+
3II
t30'
S
t
t __
n
.
kn
_+_
[.
=
-'i*

i,,G'
4'
I
*
T,
*
*
T,-l
* r,,).
2.
(l,o
ci5m1
Bdtphuong
trinh
itugc
virit
vA
dang
J2*
tog3(x
-
il
=
log,
(x
_
)
trl
DFt
t:

log.
(x
-
i),*r
d6 (l)
trd
thanh
,lffist
o
[z
iii
t,
e
[,,
_l]
I
=
o
suyra
togr(x-il=
2ex-*-no
*>?.
Vly
t4p
nghiQm
cria
bdt phyone
trinh
H
S

=
tf,;
+
o).
rac6
f-*$ffi-Iaffi:/ffi
-
fy4-r\'lv t-4 1\ t
i;ffit
*.
++
(cos5x
-
sin3x)(2sin5x _
l)
:
0
I
sin5x
-
1
.Hlz
lcosSx
=
sin3x
;
E
ll
r.
*

.\
.i
!.
I
:r
'
.?
.l
i.
t'
tl:
(:
>:.
:.
t'
l.
t-
+.
;a-
E
CAU
rv
l.
Gqi
M
H
trur_rg
dirim
cria
BC,

thi
AM
1BC,
AM
J-
BB'
n€n
AM
J.
mp@B,C,C),
do
d6
,$fu
=
a.
Tqong
tam
gi6c
rnr6ng
AB,M,
tac6
AB,=
N
- "€
.
sina
2sina
1.
Gqi
I

vi
I'lAn
lugt
li
t6m
hai
ttriy
ABC
vi
A,B,C,
Klti
CO,
rtng
di6m
O
cria
n,
lA
tankh6i
ciu
neoei
-
tifo
Utotr
langtru
.
Ta
c6II'
=
BB'.

BB,2
= AB,2
_
AB2
=
3az
-2
a2q3-+sinz
a1
-;ri"{-a-=ffi
BB.'=*;lffi
a^[j
3
4'i
S l

-;-
.
-__t_
suy
r4
trong
tam
giric
vu6ng
oiA,
c6
c-
E ,-1-
,l

:
-;ri""
lg
-

sinz
a,
IA=

va
oA2=ot2+tAr=
#(g-+sin2a)-
f
Gqi
R ld
brin
kinh
m[t
ciu,
thi
*i=
#;
(z
_
+sinz
a)
+
Khi
d6
dign

tich
mflt
c6u
ngo4i
ti6p
hinh
lang
trr.r
ld
:
?2
3
s
=
+z'
(#
cs
-
+
rin'"1
+
*)
=
ara2ffi
+
5.
Tt
he phuong
trinh
suy

ra
x
)
0,
y,
0.
Cfing
tu
hQ
phuongtrinh
vd
theo
Uit
aang
th&c
C6si,
ta
c6
.
6V3
=
2^/7
+
y
=,/F
+
lF
+
y
>3W

= 3\/74=
6iE
Deng
thric
xdy
ra
khi
vd
chi
khi
G
=y
=2W:+
x
=
\876.
Vdy
nghi€m
cria
h€
phuong
trinh
la'
x:
ffi vity
-
21/i.
Trudc
ti6n
ta

chring
minh
c6ng
tfrri*
t
n+t/
L 1 \
.F=;*,l.ffi
*.ht-Il
(r)
Thft
vfy,
e#+#=HP.ffi
_
k!.(n_k)t.(n+1_k+k+1)
(n+r)l
Hffi=#+
I
C6ng
thric
(l)
dugc
chu-ng
minh.
Ap
dpng
(l)
vdri
k
di

rir
O
Adn
ZOOS;G
m=;ffi(il.a;)
"
=4q(#.*)
uz^ooe
20Lt
L
2OOg
m=#(m.4''t
Do
6$0,
=
crool,n6n
I6y
tong
tone
vd
cria
2009
deng
thric
tr6n
ta dusc
12008
I
2OOg
/

't
i
ar=-cim
=ffi.r.(ffi
+
t+
+ffi)
i'^ 1 r
vd''
*;+*.
'.#t=ffi(eh
+
#
+
+
eh)
a)

Clu
VII
Tir
gin
thi6t
ta
suy ra Segc
=
3Snea
=+
Sesc
:1

uU dO
dei
AB
=
r,E.
Phuong
trinh tluong
thit
g
AB
:
x-y
-
3
:
0.
0ls
Gii
sri
G(xc;
2
-
x6),
khi
tt6
khoang
cich tir
G
ttdn AB la
1

:
l2xq'sl
'
,tz
suyra
segc
=i*
n
+
l2xc-51
:
t
*
[};
I
3
0,25
Ta
c6
tga
tlQ
aiAm
C(+c;
ys)
dugc tinh theo
c6ng thric
f*o
=f
t*^
+

xs
*
xs)
lto=lct^*vB+vc)
Vdi
xc:2
thi
yc
-
0,i khi d6 thay
s6 ta
dugc Xc:3,
Yc:3.
V6i
xc:
3 thl
yc
=
-1,
kfii
d6 thay
sd ta
ttirgc
.xc
:
6:
Yc:
0.
V$y
c6 hai

<tiAm C
th6a
mfln
bii to6n:
C1(3; 3) vn
Cz(6;
0).
0'5
x
E
E
F
It
,::
H
:i
.1i
i::
li
t:
f:
t.
i.:
4
J:
i:
ri
i.
s:
ri.

t:
1.
i;
:l
t-

@
'rRU'a,NG
DHSp
r{A
NOl
Dt THr rrrrl DAr Hgc rAN
rr NAvr
zoog
rliol
T'HPT ciruvtlx
M6n
thi:
To6n
Thoi
gian
ldm bii: 180
ph0t
lr**
clu
I
(2
di€m):
Cho
hdm

sd
r
=-lP
f
rl
1) KhAo
s6t
vi
ve
d6 thi
(C)
crha him
s5 khi m
=
0.
?)
Tim
nr AE A6 tbi
hAm sO
(t)
cit tryc
Ox
t+i
hai di,im
phdn
biQt
c6
hoinh d0
ldn luqt
li x1,

x2
sao cho
r
=
I
xr
-
x2
I
dat
gii
tri nh6 nhAt.
C6u
2
(2
di6nr).
i.
GiAi
phuong
trir*r :
2sin2
(x
-
.5
=
2sin2x
-
tarx
.
2. V6'i gi6

tri
ndo
cia
m,
phuong
trinh
sau c6'nghiQm
duy
nhdt :
2log'
(mx
+
28)
=
-
log5(12
-4x
-
x2).
Cdu 3
(l
di€nr).
Tinh tich
phan
:
Cnu 4
(1
di€m).
-a'

Tan
gi6c
MNP c6
dinh P nim trong
mflt
phang (a),
hai dinh
M
vi
t'f nirn
vB
mQt
phia
cia
(o)
c6
hinh chiiiu
vu6ng
g6c
tren
(s)
Dn luqt
li M' vi
N' sao cho
PM'N' li
tam
gi6c
dAu
canh
a.

CiAsirMM'=
2NN'=
a.
j
-
Tinh diQn
tfch tam
gi6c
PMN, tu d6
suy
ra
gi6
tri eua
g6c
gita
hai mflt
pheng
(c)
vA
(MNP).
-
:
Ciu
5
(l
ditim).
Cho tlp hpp
A
c6
l0

phan
*.
H6i c6
bao
nhi€u cich chia tfp
hqp A thenh
hai tip
-/.
:
cau 6
(2
dirim).
/
:./
1)
Trong
m{t
phing
voi
hQ
tga dQ
Oxy, cho
elip
@)
c6
phuong
rriot, r
{
*.*
=

,.
9
'4
:
MQt
g6c
vu6ng
tOv
quay
xung
quanh
di6m
O c6 c6c
canh
Ot
vi
ov
cit
(E)
lan luqt t4i
M
viN.
chil11113ng
mrnn rang:
6F
" ON,
=
36
.
Trong

kh6ng
gian
v6i
hQ toa d0
Oryz,
cho ducrngthang
O'
T=?=
|
tamit
phturg
CI)
:
x
+
!
+
z- 3
=
0.
Vi6t phuong
trinh
tluong
thang
A nim tong
mit
phang
(P),
vu6ng g6c
vsi

d vi
c6
khoang cach d6n
d mQt khoing
h=
'#
.
Ciu
7 (l
di€m).
C6c
s6 thpc
x,
y
thay d6i sao
cho x*
y
=
2.
Hdy
tim
gi6
tri
lon nh6t
cua
bi€u
thric
:
P
=

1x3
+
4(f
+
4.
-
,.li xdx
l=l
'
J1
x+y;l['


M4t
khdc
lim*-s+
f(x)
=
+ oo
vi
lim*_e-
f(x)
=
_
-
,
Tac6f(x)>0v6i
-4>x>
-6vdf(x)<0voi
x

e(-4;0)
u (0;2)
.
Bing
bi€n
thi€n
:
Nhu
vfy,
tu
bing
biiin thi€n
suy
ra
phuong
trinh
(3)
hay
ciing
Ii
phuong
tdnh (2)
c6 nghiQm
duy
nhAt thuQc ( -
6; 2) \
{0}
khi
vA
chi

khi
:
cAum.
(
1,0
di6m).
|
-^>
t!.
lm
<
_L4
l ;,=l-rT
L-m=-4
L
m=4.
3,13-
2,12
-t
3
rac6
r=
1€$ff=
Jfxzdx
-
J€x\Fldx
e.,6-1
lis-,y'
=
't-'

zfi
=
t'lf
-itf
r.,-
r)ia(*,
-
1)
=+
-*ic.,-,)-lf
cAu
rv.
(
1,0 di6m).
K6o dAi
MN cit
M'N'tai
E,
khi d6
NN'
li duong
trung
binh
trong
AEMM',
mi
M,N'
=
pN'=
a

n€n
EN'
=
4
suy ra
APEM'
li tam giac
n?ng
tei
p
vi
EP
=
rGMryffi7?
=
816
,
d6ng
tiroi
Ep .t-
pM.
Trong
tam.gi6c
vu6ng
c6n
pMM',
c6
pM
=
a.,12

,
nAn FP PM
=
a.E
"^17-
:
^2-17
Ta c6
.966p
=
2Suup +
Suup=
|
fe.ru
=l*^f, .
Viy
S,r.1up
=I^'#
Vi
EP
la
giao
tuy6n
cria
hai
mpt
pheng
(a)
ve
@Ia}Q

vi
EP
1PM,
n€n
g6c
a
giiia
t.rai
m{t phing
nay
bAng
g6c
frFFf
=
450
Cht
)t
;
C6
th6
tinh
g6c
a
bing
c6ch
sri
dsng
c6ng
thrlc
SpM,N,

=
Spyy.cos
g.

cAU v.
(
1,0 didm).
GiA
sri
k li
sii'cdch
chia
r{p
A rh6a
man
y€u
c6u
bAi
to6n.
Ta
nhin th{y
ring,
'''si
mdi
c6ch chia ta dugc
hai
tip
con
kh6c
r6ng cua

A.
Suy
ra
s6
cdc
t6p
con
kh6c
r6ng
cria
bing
2k.
Tri d6 ta
c6
:
2k
=
Cls*Crzo* +Cio
=zta
-Z
s
ft=2e-
I
=511.
Viy,
si5 c6ch
chia theo
y€u
cAu
bii toan

bang
5l L
cAu
u.
(
2,0
di,im).
l)
(1,0
di6m).
Dat
@;ffi1=a
(0
S
oS2Tr)
vA
(d;
}]f)=c+l
Tac6:
Mf*"=oMcosc
"'tYr,r
=
OMsina,
Do
Me(E)ndn'
xft*Yil-,
4
OM?cos2c
OM2sinza
,

;-= i
.
1
coszrt
sin2cr
-m=J-=
4'
1
sin2q
cos2cr
.r
usrg
rU', ra
cung co
&
=
T
*
T
^
7
1_
_
coszd.
sinzc
*
sin2c
,
cos2c
:1

-
1
suYra
6fr?*o=il,
s
r'i+:r=;*;.
1113
_T_
=
_
oMz
0N2
36'
2)
(1,0
ditim).
cia sir aa
dgmg
duo.
c A th6a
mEn
bii to64
tlf
a
se nin
trong
m{t
phang
(e
vu6ng g6c

v6i
d,n€n
m(Q)
nh4n vdc
t?hi phusnC
cria
d
lA
i(-2;3;2)
tAm v6c
to ph6p
tuy6n.
Phuong
trinh
cria
*(O
g
:
-2x+3Y+22+a=0
(1).
GqiA li
giao
di6m
cuad vr5'i
mf(fi,
thi
tga
ttg
giao
diiim

cria
A tn
nghi€in
ctia
he phucrng
ft+3 v-9 z-6
r-:-:_
trinh
:
J
-2
3
2
e+
A(3:
0:
0)
(2).
(x*y*z-3=0
Ke
AB
J.
A, B
e
A.
Ggi
C
ld
giao
di6m cria

m(e) voi
d
vi g
ld
g6c
giGa
d
vA
(P)
thi
g
=
ffie
,tac6
fi(l;
l; t) h
mQt
vdc
to
ph6p
tuy6n
cria
(p),
Khi d6 :
-,
t-2+3+21
li l;
sne
=
JE!t7-

=
{;
+
Ianp
=
J14-
@

Ta
c6
BC li duong
vu6ng
g6c
chung
cira
d vi
A,
d6ng
thoi
dg
dai
troqn
BC
=
h
=
'#
.
Suy
ra:

Ac
=
:
Bc
<=+
AC
-z'iE'
. E
=
g
tane
11
'
.,,/
a
rry'?'
"1*"
AC cfing
tA
khoang
c6ch
hr
A dAn
m(e,
n€n
tir
(r)
vi
(2)
ta c6 :

o.=#=ffi
ea-6=*#,
Do
A
nim trong
mf@),
n€n
A li
giao
ruyiin
cua
hai
mft
pheng (P)
va
(e.
T6m
lai ta
c6
hai
rtuong
theng
A th6a
m6n
bAi torin
lA:
,",.,.f
x+y+z-3=
[
x+y+z-3=0

tv'i
:
[-zx+3y+
2z*6*#=
o
ua (a)'[-zx+3yr
zz*
6_#=
o
cAu
ylr.
(
l,o
di6m).
Tac6
P
=x3y3
*2(x3
+f;
+
4=*tf +Z(x+yXxz-xy
+yt1++
=
x3y3
+
2(x
+
y)[(x +y)2
_
3xy

]
+
+
Theo
giithitit
x
+
y
=
2
n€n
p
=
x3y3
-
l2xy
+2A.
D6t
1
=
xy, do
(x
+
y)2:
4xy
n6n t
<
l.
DAtf(t)
=f -lzt+20,

te(-oo;
U,thif(D=3f
-
12=0et=-2.
Ta c6
f( 2)
=
36,lim,*-*
f(g
=
-
o,
f(r)
=
9
vi
f(t)
>
0 voi
t
<
-'
2
c6n
f
(t)
.
o voi -2
<
t

<
l
Tir
c6c
t6t
qua
tr€n,
suy
ra
maxf(t)
:36
khi t
=
-
2.
Vsi
i
=
-2,tac6
hQ
phuong
trinh
:
(x*y=2
IuL-l
e
x=lt16;y=lTG.
Y
4y,
gtdtri

lsn
nh6t
cfra
p
b6ng
36,
khi
x
=
I
*
16,
y
=
I
_
16
ho4c
x
=
I
_y'3,
y
=
!
*
.fi
.
Dy
kiiin

k)
thi
tht?
tin
sau
sE vdo
cdc
ngdy
2b
-
29
thdng
3 ndm
2009
'5
Dtt
TRUONC
DHSP HA NQI
rcr6r rHPT cnuvtN
CAu I
(2
di€m):
Cho
hdm
sii
y
=
NT THI
THU
DAI HOC I,AN

III NAN,r
ZOOS
Nidn thi,
To,in
Thdi
gian.lAm
bdi:
180
phrit
*
**.
x2- zmx+
m2
x-1
(l)
v
l. X6c dinh
tAt
ce cac
gi6
trf cira
m d6 ham
s5
d4t cgc
ti6u
tai
x
=
2.
'Jz.

Tim c6c
gi6
tri cua m d6 tr6n d6
thi crla hdm
sii
1t;
tdn
tei it nh6t mQt di6m
mA
ti6p
tuy6n cria d6 thi
tei
dii5m d6 vu6ng
g6c
vdi dudng thing
y
=
x.
Ciu
2
(2
di6m).
V
t. Gidi
phuongtrinh:
. aX aX
stn";
-
cos";
L

,+"i"-
:3cosx'
,,1
2.
Giai he
phuong
trinh :
v
CAu 3
(l
di6m).
Tinh tich
ph6n
: t
=
[/3
x2+r+.,/@Txf
r/Cau
a
(i
diem).
Cho tri diQn
SABC c6
g6c
AB'C
=
90", SA
=
fift
=

2a,BC= a.,,/3
v.d SA vu6ng
g6c
vdi
rn4r
phing (ABC).
Gqi M li
tlitim
trdn duong
thing
AB, sao cho
AM
=
2Md.
Tinh khodng
cdch tir tlitim B dffor
mp(SCM).
I
CAu5(l
<li€m).
Cho0
<a<b<c
<i<e
vd a+b+c*d*e=
L
Chung minhbAtdingthric
'
'4U"
+
be

+cd+de)
+cd(b
+
e-a)
S
+
,_
25
CAu
6
(2
di6m).
qo
l)
'l
rong
mflt
phing
v6i hQ tga
dQ Oxy, cho
tam
gi6c
ABC c6
ctinh A(-2;3), duong cho
CH nim
trdn duong
'/
thing
: 2x+y
-7

=0
viduongtrungtuy6n
BM
nimtrdndudngthing
:
2x-y+l
=0.
Hay vi6t phuong
trinh
c6c cqnh vd
tim tga
d6
trgng
tdm
G
cira tam
gi6c
ABC.
t
'/ 2)
Cho hinh
hgp ABCD.A'B'C'D'.
Tr€n ducrng thing
AC
l6y di6m
M
vA trdn
duong thing
C'D t6y diiim N
I

/cM
I
sao cho
MN // BD'. Tinh ti
tU
;'
Jcart
(l
di€m)
Xdc
dinh
tep
hsp cdc di6m trong
m{t
phnng phftc
bi€u
di6n cdc
s6
phric
z.th6a
min diAu ki6n :
lz+il
l-l
=
l.
I z-3i I
f
t*t
*y=4+,tW
ll,*'

-ztg2=te1+h
D4r kiiin
thi th* Idn
tdi vdo
ctic ngdy 18,19/4/2A09

:
DAF
AN
rovr
r,lr
mON
roAN
(Thi
thti'DH
IAn
III
- 2009)
CAU
I.
xz-2x+2m-m2
1.
(t,O
Oiem).
Tgp
xdc
dinh:
R\
{l}.
Ta

c6
y'=-
(x-1f
Gii
sir
him s6
d4t cgc
ti6u
tei x
=
2,
suy
ray'(2)
=
0 hay
4
-
4
+Zm-mz
-
0
<+
m
=
0 ho{c
m=
2.
2,
tad6u
c6

y'
J
#=+
y'
=o
<+x=
o
ho4c
x=2'
Mat
khric
y'
>
0 khi
x
e
(-
.o;
0)
u
(2;
+
co)
vd
y'
<
0 khi
x
e (0;
l

)
u
(l;
2)'
Do
d6 x
=
2 ld
tliOm cuc
ti6u
cria him
s5.
Viy, d6
th6a
mdn
bdi
to6n
thi : m:0
hodc
m =
2.
'
x?-2x+Zm-mz
2.
(l,0di6m)R\{l}.
Tac6Y' = 7;17
T6n
tai
tii5p tuy6n
cua

<16
thi hnm
s6 vu6ng
g6c
v6i
tluong
thing
y
=
X khi
vd chi
khi
phuong
trinh
sau
c6 nghiQm
:
x2-zx+zm'mz
(x-1)2
I
- -l
{=r
x2
-
Zxt1m-m2
=
-(x-1)2,x+
I
€
2x2

-4x*2m-m2
+
1=
0,x+
I
(*)
Do
phuong
trir,h
(*)
c6 it
nhAt
mgt
nghiQm
kh6c
1, nCn
ta
c6
hai
trucrng
hqp
sau
:
a)
Phuong
trinh
(*)
c6 hai
nhiQm
phdn bi-Ot'

hay
L'=4-
2(-m'+2m+
1)>0c+
2m2-4m+2>0
++m+
I'
(L'
=
2(m-
1)2
=
0
b)
phuong
trinh
(*)
c6
nghiQm
kdp
x *1,
diAu
niy
tusng
duong
voi
I
-
-a
a

1
(loa)-
'
T6m
lai' m *
1'
^l
e
-
"
txt
=
xz
=

1
Ghi
chir : Niiu
thi sinh
giii
bdng
c6ch
:
[r,orrioo
trong
d6
f(x)=2*z
-
4x*2m'-
m2 +

1,
thitr.'
0,5
di6m.
CAU
il.
l.
(1,0
c1i6m).
Phuong
trinh
dugc biiSn
d6i
thdnh
:
ci{ -
."ixr
+,i,{.,o{)
=}
z+rin")"or*
*=Gi{
-
.or)tr
+1sinx;
=!r*sinx)(co{
-
'i"iAi{
*
'oJr)'
.

V6irinl
-
"or|=0c+sin(|
-fl=0<=+x
=|*lkrc,kez'
22La'z
t
t'

x
x. x
x
3
, r ^
r
Voi
r
+
jsinx
=-i(t+sinx)(sinj
+
cos)<+sini
+
co5
=-;(ptndyv6nghi€m).
VAy,
nghiQm
cira
phucrng trinh
ld

'
*
=
;
+zkn,k
eZ'
w.

2
.(l,Odiem). DiAuki€n
x * 0,
y>
-
2 .
Phusng
trinh
(2)
<+
lxl =
4
+
2y
.thay
vio
pt(l)
ta dugc
:
4+3Y=
q+JfrT
{+3Y=

't7+'z
-
.(
v>0
7
:
tv,
;;=ey,
o
Y=;
+x=t5'
fx=S ft=
-5
vay
hQ
phuong
rrinh
c6 hai nghiQm
,
l,
=
*
'
t
o
=
1.
cAu
rn.
(

l,o dii!m).
Tac6
,=
[/t
,G
xdx
-,'o
6qnffi
xdx
DAt r=
,@-*z
ql
thi
dt=
#|v6i
x:0thit=l;
v6'i1=y'Jthit=2.
Khi d6 t:
I:#
=
frr+r1*aqt*
1)
:fr
a
* t)+,11,:zbf3
-
.lz).
cAu
ry.
(

1,0
di6m).
Do M
=2W,
n€n
khoang c6chttrA
tliin mdSCM)
beng
2
lAn
khodng
cSch tu B d6n mp(SCM)
.
Tir
gia
thi6t ta
suy ra AM:
+,
BM
=
+
,
3' 3
arlsl
1
CM::. Ke AK
l- CM, thi
CK J- mp(SCM).
3-'^
Do

goc
ffi
=
90'+
Fm
>
90o n€n
tti6m M nim
giea
CvdK.
Ta c6dAKM
-
AcBM
=+
AK:
tt=l"
=
n"13.
cM
^lTl'
Ke AH
J-
SK, thi
AH I
mp(SCM)
vd AH
ld khoing c6ch tu A ddn mp(SCM)
tL1131
43+^/7a
Taco:

;;;=Asz
*1;;=
q^r*&=G
=
AH=
E.
V{y, ktroing crich
tt B
ddn mp(SCM)
bing
+
-
{43
CAUv.
(
l,0rli€m).
Ttgiathii5tsuyra
0<"s
i.e6tdingthricduscbiiSndoithdnh:
-fL/^
a[b(c+e)
+d(c+e)] +cd(b+e
-u)=
*
o
a@+d)(c+e)+cd(b+e-a)S
+
2S
x2+r+.,ftr+xf
Theo

bAt
ding
thtic
C6-Si,
ta
c6
:
) ,
a(b
+
d)(c
+
e)
+
cd(b
+
e
-a)
=
^
(ry=)'*
1.*o*!*._")3=
a(r:a)z
*,'=}|,',
Xethdms6:
f(a)
=+*ry,vdi
0<as
f
.rac6

f(a)=frf
-
5a2-4a+1)>0,vutto;f)'
Suy
ra
(a)
d6ng
bicn
trcn
(0;
Jl
*
it"l
s
(;
)
=f
tan'*)
cAu
vt.
(
2,0
di€m).
-
l)
(1,0
di6m).
Dubng
thdng
chira

c4nh
AB
vu6ng
g6c
vsi
CH
n€n
nhen
vectq
il(2;
1)
lim
vecto
chi
piiuong'
Do
d6
dudng
thing
AB
c6
phuong
trinh
:
x
-2y
+
8:0'
Suy
ra,

tqa
d0
diem
B
ld
nghiQm
criahQ
phuong
irinh
:
|
{"-r,
*
I
=
o
*1i,.=_3-B(2;s).
/
l2x-Y*1=o
(Y=
^Yr
-'
-^
*-'-^
'+;A
'-?
Et
thu6c
BM'
Gqi

C(x;
y)
thuQc
CH,
suy
ra
trung
di€m
c0a
AC
Ie
M
t7"',
'
(
2x+
v-7
=
o"
o
*
[;lf
+c(:;r).
Tqa
d0
cria diSm
Cli
nghigm
cria
hQpt,

\,
*
-
#
*,
Suyra,
BC
:
4x+y-
13 =0
vd
AC
:
2x+5y-11
=0'TrgngtdmG(1;
3)'
2)
(1,0
di6m).
D[t
EA=
d
,EF
=E
'EC
=
d
'
Tac6
EF

:
d +6+
d,
vi
MN
i/
BD'
n€nffi
=Ufr
v[y
Mfr =
kd.
+tci
+
Pt
(1)
Mat
kh6c
Mfr =ft
+7d
*Vfr
=
nVt
+fr
+
^V6
Trong
d6
At =
i

- d,7?
=6,
c6
=
d
-E'
Suy
ra
ffi =
n(c-
-
d)
+E
+
n1d
-87
:
(m
-n)
d."11
-m)
6
+
nt
(2)-
.
(m-n
=
So
sdnh

(l) ve
(2)
ta
c6
hQ
phucrng
trinh
:
|
1
-
m
=
t n=k
t4
CM
1
vQy
tvtc
=;
AC
h"Y
1;
:
J
CAUvll
(
1,0tli6m).
Df;t
z=x+yi

+
z-3i=x+g-f)i
vit
z*i
=
x*(y+l)i
Di€uki€n:x*0vdY*3-
D6
dang
chimg
minh
du-oc
tinh
chAt
l:l:
#
Suv-
l#l=l
elx
+
(v
+1)il2=[x
*
(v-3)il2
e.x2+(v+l)t=x2*(v
-3)2-ev=1'
K6t
lu{n:
Tip
hqp

c6c
di€m
trong
m{t
phdng
phrlc bi6u
di6n
c6c s6
phfc zthbamtutdi|u
kign
:
l#l=
ttno"*^fur=t
-
k fk=713
t-lm=2/3
(
n=1/3
,,@

8!
rnu0Nc
EHSp
HA
Nor
KHor rHPT
csuytN
DE
THI
rutl

o4.t
Hgc
t Aw
lv tlAtvt
zooq
Mdn
thi: To6n
"':
::::l1T.?:::
:::
ono',
cau r
(2
di6m):
cho
hdm
sri
,
=
ff
t'i
',,'l'
Tim
t6t cA
circ
giittri
cia
m d6
hdm
st5 c6 cyc

d4i,
cuc
titiu.
Chrlmg
minh
ring
trung
di6m
cria do4n
thing
n6i cric dii3m
cgc
d4i,
cgc ti6u
cria
d6 thi
hdm
s0
in{c6
O;ntr
khi
m thay
d6i.
,12.
Kf
hiQu
(C)
la dd
thi cria
hdm

sd ilng vdi
m
=
2. Tim
cdc
di6m
M thuQc
(C)
c6
hodnh
dQ
l6n
hon
I sao
cho
khodng
cdch tir
M d6n
giao
di6m
cira hai
duong
tiQm
c{n
cira
(c)
nh6 nh6t.
Ciu
2
(2

<ti6m).
,/
l.
ciai
b6r
phuong
trinh
:
ri
I
+
2.
GiAi
phuong
trinh
:
@or2*
ffi=o
CAu3(l
di'3m)"
Tinhdi€ntich
hinh
phing
gi6i
h4n
boi
hai
parabol
:
y

= I
-
x2,
!=axz
v6i
a>0.
Cnu
4
(1
di6m).
Cho hinh
lap phuong
ABCD.A'B'C'D'
c6 dd
ddi
canh
bing
a.
Cgi
K
ia
trung
di€m
cua
cqurh
* ,
/
tC vd
I Id
tdm

cia
hinh vu6ng
CC'D'D.
Tinh
th6
tich
cria
c6c
khoi
da
diQn
cto m4t phing
(Aklt)
chia
ra
tr€n
hinh
l4p
phuong.
icau
5(1
di6m).
chrmgminh
ringphuongtrinh
2x3
-
3x
-
6\6F=x+
1+6=0kh6ngc6nghigm

iim.
^7
CAu 6
(2
di€m).
' l)
Trong
m{t
phing
oxy,
cho
elip
(E)
,
+
.'i
= t ,udirim
M
{J;
t). viiir phuomg
trinh
cdc
dudng
thdng
v
tli
qua
M vd
c6t
(E)

t4i
hai tti€m
A vd
B
sao
cho M
ld
trung
tli€m
cua
AB
.
/
2)
Trong
k:h6ng
gian
oxyz,
cho
cric
di€m
S(0;
0; 2),
A(0;
0;
0),
B(
r
;
z;

0),
c(0;
2;
0).
Gqi E vd
F
ran
rugt
ld
hinh
chiliu vu6ng
g6c
cria
A
l€n
SB
vd
SC.
Chung
minh
ring
5
di6m
A, B,
C,
E,
F
cirng
thuQc
mQt

m4t
cdu. Virlt phuong
trinh
m4t
cAu
d6.
CAu 7
(l
<li€m)
Chung
minh
tling
thric
:
Cloro
-
Clo.o
+
Cloro
-

+
(-
rlC35iot
+
-
CiBl6
+
CrzBlB
=

z,uu'
Dtr
kiiin
itgt
thi thii
Ifrn
sau
vdo
cdc
ngdy
t6,17/s/200g.
8,3x
3x
+ zx
9(3x-
Zx)
-
3x
,
@

oAp AN ivtoN roAN IAN rv
cAu I.
(z,o
ei6m).
l.
(1,0
diem).
TSp
xdc dinh :

R\
{
I
}.
xz-zx+z-m
(
x+ !
Tac6y'
=
6r-
=t
y'
=Q:[*,
_
Zx *
2
_m
=
0
(1)
Hdm si5 c6 cgc d4i, cgc tiiiu khi vd
chi khi
pt
(
l) c6 hai nghiQm
phdn
bigt khric i.
(
m+1
ual:

[4,
=
I_
(2_m)
> o
e+
n]>
l.
Gid sri A(xr,
yr),
B(xz,
yz)
Id c6c di6m CD. CT crla
tl6 thi vd E(xe,
yE)
ld trung
di6m
crla AB.
Khid6
xl,X2tdnghigmcrla(l)vdxs=
|t*,
*xz)=l.Suyradi6m
EthuQcduongthing x= lc6tllnh.
2.
Voi m=2. Phuongtrinhctia(C)duo.c
vi6tthdnh
:
y:
x-
I

*
+.
x-1
D6thi(C)c6tiQmc{ntltmg
x=l
vAti€mcflnxiOn
y=
x-l.GiaocriahaitiEmcdnldl(l;0).
Di6m M
e (C)
<=+
M( x*;
xy- I
*
fr
I
Nh4n xdt : IM nhd nh6t khi vd chi khi IM2 nh6 nhdt.
'
xM-l
Tac6, IM2
=(xv-
l)2+(x"- I
+#
)2
=
2(xru- t)t
+o;h +z>
2^12+2,
dlubingxiy ra
khivdchikhi 2(xy-l)'=

*hae'(xr,r-l)'=
i
=*"
=l++(vi
xv>l).
Vay di€m M cAn tim c6 tsa
tlo
(l ++
,
tTf
',
cAu rr.
(2,0
di6m)
\t2'
tz
)'
1.
(1,0
di6m). Bdt
phuong
trinh
dd cho
du-oc
Uiiln dOi
thdnh
:
''(;).
-
(i).*'

;m-,i:
^r-'
'l\z/ ^l
\21
Bt
t +1
Dpt 1=
0-
r
0, t + l. Khi d6 bdt
phuong
trinh
trd thdnh :
,(r-r)
s
;
€t+l-
8t
>o<+
t2-s>oe[,tt3
,*i
6-=
t e(r_l)-; r(r_1)
Lo<r<t
lo.(f)-
2.
(1,0
diCm). EiAu ki€n
sin4x
*

=
+
Khi d6
pt
tuong
duong vdi
pt
: 2sin4x
-
\E
-
2sin2x
+
Zt[1 cos2x
=
0
<+
4sinZx.cosZx-2sin2x
+2{1cos2x'-
VS=O
e
Zsin2x(Zcos2x- l)+
Vg(Zcos2x-
t;:g
<=+(2cos2x-r)(2sin2x+V:)=0,=[
ZcosZx- 1=o
*=r
[
.t:tz"
=t!,^

LZsinZx
+
y'3=9
lsinZx=
-,13/2
r
[
2x
=:*
2kn
.
Cos2x
-
-
<=+ I
z
"
lZx=
-n*
Zkn'
.,tr
[2x =
-I*
2kn
rSin2x '"erl
3
z
-
l2x=
+*

zkn'
,Ttzft
E6ps6:
x=
-*kTr,
*=T*nt,
k6t ho.
p
v6'i
diAu kiQn suy ra
x:
+kn,
keZ.
k6t trqp v6i
di6u ki6n suy ra x
:
4
*
on, o
*
r. i
ke Z.
'
t;''
\:;
t.,,,
T
).'t,i
:itr.::ii,
@

3
[*
>
log13
++l z
<r
I
xco.
lt
6

CAU
UI.
(
1,0 di6m).Hoinh
dQ
giao
ctidm cria
hai
parabol li nghiQm cta
phuong
trinh :
1
'
Jt*a
7'.
,/Ga
Dohai
parabol
ddu

nhdn
tryc oy lAm
trycd6i xirng
vd I
-x2
>
ax2,
v *
.(-#,
;ft
i'
nen
s=2IJ'(1
-xz
-axz)dx=z.l?
-
f,'
.ul*'lt
=2xr-3,,*u1
d=
#-#"=#
cAuw.
(
l,otli6m).
Gqi
F ld
giao
di6m
cua
AK vA CD.

Duong thing'Fl
cttcc'vd
DD'
tan tuqt
tei M
vA
N.
M[t
phing
(AKJ)
chia
hinh lflp
phucrng
thAnh
hai
t<trtii aa
diQn
ln mr6i ctrOp
cut tam
giric
ADN.KCM
vn khiii
da dien ANMKBB'A'D'C',
Br
Vi KB:
KC n€n CF
=
AB,
do tl6
CF

=
CD.
Trong
AFC'D,
FI vi C'C
lA
cic
ilulng
trung ruy€n
n€n M
li trgng
tdm cria tam
gi6c
d6.
11
Do d6 CM
=;
CC'
=;
a.
B
Vi
I le trung di6m cta CD'
n€n D'N
:
CtU
=
]
a.
3

.
I
-xt=axt
<=+(1
+a)xz=
1
<+
2
Vay,
DN
:;
a.
Ta c6 :Vr
:VeDN.Kcvr:1
n,"
+
B'
+
y'E-E
;,trong
tl6 h
:
CD
:
a;
a a2 az a2
zas
Ii€n V'=
-t-
,t

3'3
rz 6) 3l
Ggi
Vz li th6 tictr cria kh6i da diQn
cirn l4i, khi
dd
: Vz
:
a3
-
Vr
:
at
-
cAu
v.
(
l,o di6m
).
11
Dat
f(x)=:x*-
;
x-V5xz
-
x*
1+
l,hAms6xrictlinhvoimgi x
e
R.

32
Tac6 f(x)= *'-1-
#
2 2{Sxz-x+7
lsGit:;;T-+g:!1
vd
f'(x)
=
2x
-
-
zJsxz*x+t
-r*-
2(5x2- x +1)
F
a2
a2
B
:
SnoN
-
-;
$,': Srccrur
:
E
zas z9a3
3536
L9
4(5x2- x+r1rF
x+r

Nhfn
thdy
f
'(x) <
0 vdi
mqi x
S
0, n6n f
(x)
nghich biiln trong khoring
(- c";
0).
Suy
ra
f
(x)
>
f(0)
=
0
v6i moi x
<
0.
Vay
hdm
s6
f(x) d6ng bi6n
trong khodng
(-
*;

0).
Do il6
(x)
<
f(0)
=
0
voi mgi x
<
0.
V4y, phuong
trinh
di cho kh6ng
c6 nghi€m
im.
:j
^it:

cAu vI.
(
z,o
ei6m).
l)
(1,0
tti€m).
Duong
thing
x
=
I

di
qua
M cit
(E)
tai haidi6m
o,t,
f
)
vn
B(l;
-*
I
Rd rdng
M
kh6ng lA
trung
di€m
cria AB.
'
Xdt
tludng thing
(d)
di
qua
M
c6 hQ
sti
g6c
k.
Ta

c6
phuong
trinh cua
(d)
:
y
:
k(x
-
I
)
+
I
(
I
).
Thay
(l)
viro
phuong
trinh
cfia
(E)
ra
duqc :
4x?
+9J11x-
l)
+
Il'z=

36
<=+
(9k2+4)x2 +
l8k(l
-k)x+9(l
-kF-36=
0
(2).
-Dulng
thing
(d)
cit
(E)
tai trai
ei6m
A, B thda
mdn
MA
=
MB khi
vd chi khi
phuong
trinh
(2)
c6
hai
nghiQm Xa,
xs
sao
c'

xA
+
xB
-18k(1-k)
4
hoiXy:
,
-6=lerk=
rac6
9(l
-k)'-36
=9(l
*il'-36<0
c6n 9k:
+4>0,
Vd'i k:-
4
;,
Dod6,voi
k:
-!.
pt(Z)c6
hainghi€mphAn
bi6txa,xsrhoamdn:
xy:*ol-t
=,.
T6m
lai, c6
m6t dudng
thing

di
qua
M
thoa mdn
y6u
ciu
cira bdi
torin ld
d : 4x+
9y
-
l3
=
0.
2)
(l,0
di€m).
ra irhfn
thdy
E3.IE:43.
ed
=
R.eC:
o
$,
+
AS 1(ABC)
vi
AC
J- BC

+
BC
t_
(SAC)
,a
AF
J-
(SBC)
'+
AF 1 BF.
Lai c6
AE J_
SB
(theo
gt),
n€n
n5m
iliiim
A, B,
C, E, F
cirng nim
tr€n mQt
m4t
ciu
duong
kfnh
AB.
Gqi I
ld trung
di6m

cria
AB thi
I
(1;
l;
0) lA tdm
mdt cAu
2
A
t;
vd
b6n kinh R
=
IA
:
l-
{+'
Vdy
phuong
trinh
m{t ciu
ld :
}
1.
s
(*-;)
+(y-l)'+z==;.
CAU
Vn
(

1,0
di6m).
Theo
khai tri6n
nhi thric
Niu-Tsn,
ta
c6:
(l+r"
=cg+icfi
+lcf,+ +r'Cff
=
cfi+,cl_ci-jCfl+cf
+icfi_Cf
-rcfr,.r
=(l
-ci +
c* -cf
+
)+r(c*
-c;
+
c;
-c;
+
).
Met khdc (l+
i)" tlugc
viiit v6
d4ng

luong
gidc
:
(r+
i)":
rF
lcosf
+
isinf)
:
@*"7
+
t,l7:sinT
.
Theo
tfnh ch6t
cria
hai
sr5
phrlc
bing
nhau,
r{p
dl,lng
cho
n
:
201 0,
ta
suy ra:

cloro
-
cSoro
+
c!o.o

+
(-r)k't.c3[;J
*

-cZEiI+
cSBlB
:.,[ffi"i]llf"
:2roos
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!


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.

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