rnu0Nc
EHSp
HA
Nor
KHor rHPT
csuytN
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