Thời gian làm bài 180 phút
Câu I (2 đim)!""#$%&
'
' ( ) y x x m x= + + + +
*+,-."/0#(
$
)
! "1%.2#23,-."/"#$%&4"5$6!
! 7$"8995.:/*;<$,=,>?9."@9
y x= +
*A.,-."/(
$
).B5',5=$C"DE5F.(G)HIH
%<*"**.5JC.KLJ*;<(
$
).B5I2#2K99+*2M5"<K!
Câu II (2 đim)!
1. 515"FC">N9.:7"
' '
' '
x y
y x
+ − =
+ − =
2. 515C">N9.:7"
*%
( %5O)( .< )
%5O *%
x
x
x
−
− + =
−
Câu III (2 đim)
! P"."=.P*"2Q.."=.:RO<L%5":<4"5SK<L"7"C"@995M5"BET5,>?9.:R.D$(G)E
4P"U6SK<".:V*"#"!
! :94"995<*""<5,5=$HI*&,/"H,WX#5,BI6<Y!O2#IL0#"<5Z<
,>?9."@92K99+*2M5"<K2#*[92K99+*2M5I!:\O2#IL0]L"<5,5=$2#%<
*"6E(2M5E0#$W.%&*".:>M*2#EY<)!
<) ^*,/".D$2#E4P"$_.*`K9B5.5JC.aX5FI!
E) ^*,/"2/.:P*;<2#%<*".aX5FI*+."=.P*"0M"].!
Câu IV (1 đim).7$$,=E].C">N9.:7"%<K*+9"5F$
'
( ) ( ) x m x m x+ ≥ + + − +
!
bMỗi thí sinh chỉ chọn câu Va hoặc Vb
Câu Va (3 đim)!Chương trình cơ bản
! :9$_.C"@9.c<,WOLH*",5=$('G)2#"<5,>?9."@9*+C">N9.:7"0#
d x y− + =
2#
' d x y+ − =
!d5J.C">N9.:7",>?9."@9,5SK<2#.B2M5"<5,>?9
."@9
#d v d
$W..<$95**D*+,e"0#95<,5=$*;<
#d v d
!
! :94"995<2M5"F.c<,WOLfH*""<5,>?9."@9
x y z− + −
∆ = =
−
'
x y z+ − −
∆ = =
−
2#$_.*`K( )O
gL
gf
ghOiLgjfgk6!d5J.C">N9.:7"$_.
C"@9.5JCOl*2M5$_.*`K( ),-9."?5%9%92M5
∆
2#
∆
!
'! 7$C"`."m**;<%&C"a*
( )
n
i−
Câu Vb. (3 đim).Chương trình nâng cao
! :9$_.C"@9.c<,WOLH*""LCoE0
( )H
*+C">N9.:7"
x y
a b
− =
2#0#,5=$E].4p
."KW*()!c5X
HX
0#**,>?9."@9,5SK<2#%9%92M5**,>?9.5F$*Q*;<()!
"a9$5":q9"7"E7""#".BET5X
HX
2#**,>?9.5F$*Q*;<()*+X5F.P*"4"9,r5!
! :94"995<2M5"F.c<,WOLfH*"',5=$(G'Gi)HI(GG)H(GG)!7$.c<
,W*;<,5=$%<*"
gI
g
,B.95.:/"s"].!
'! 515E].C">N9.:7"
h
Hk j
09 h09 h 09x x x+ ≤ −
tJ.t
(EW*5."54"99515."P*"97."\$)
uuv
DK w W5XK9 5=$
^x6
¡
mE5J."5\Ly6'O
gjOg'6'(Og)
x
≥ ∀ ∈
¡
#$%&,-9E5J.:\
¡
#$%&4"9*+*m*.:/
5M5"B
05$
x
y
→±∞
= ±∞
I19E5J."5\
O
−∞
+∞
Ly gg
L
+∞
−∞
-."/ L
i O
k
'
k
k
">N9.:7""#",W95<,5=$*;<,-."/(
$
)2#,>?9."@9L6Og
O
'
g'O
g$O6
'
x
x x m
=
⇔
+ + =
>?9."@9L6Og*A.,-."/(
$
).B5',5=$C"DE5F.4"52#*"e4"5
( ) ' g x x x m= + + =
*+"<59"5F$C"DE5F.4"*4"52#*"e4"5
n
n h
h
()
m
m
g m
m
∆ > − >
<
⇔ ⇔
≠ ≠
≠
(z)
"5,+"#",W*;<I2#."s<${
'
!
B C
B C
x x
x x m
+ = −
=
()
<*+|y(O)6'O
gjOg$g
F%&9+***.5JC.KLJ.B5I2#0#|y(O
I
)6
' j
B B
x x m+ + +
H|y(O
)6
' j
C C
x x m+ + +
*.5JC.KLJ.B5I2#2K99+*4"52#*"e4"5|y(O
I
)!|y(O
)6i
⇔
(
' j
B B
x x m+ + +
)!(
' j
C C
x x m+ + +
)6i
⇔
(
' j
B B
x x+
)(
' j
C C
x x+
)g($g)(
' j
B B
x x+
g
' j
C C
x x+
)g($g)
6i
⇔
n( ! ) } ( ) 'j '( )(( ) ( ))
B C B C B C B C B C B C B C
x x x x x x x x m x x x x x x+ + + + + + − + +
g$
g$g6
J."~C2M5().<,>~*
n$
•kh$g'j$g'($g)(n•$i')g$
g$g6
⇔
h$
•'$g6C">N9.:7"29"5F$!
dQL4"9*+95.:/*;<$."s<${L\K*`K!
k
k
k
k
5€K45F
'H 'x y≤ ≤ ≤ ≤
<*+
' '
' '
x y
y x
+ − =
+ − =
⇔
(' )
(' )
x y x y
y x y x
− + − =
− + − =
(' ) (' )
(' )
(' )
x y y x
x y
y x
⇒ − + − =
− =
⇔
− =
⇔
'
'
x
y
x
y
=
=
=
=
5=$.:<.<."]L
'
H
'
x x
y y
= =
= =
."s<${!
J.0KQF*+"F*+"<59"5F$
'
H
'
x x
y y
= =
= =
k
k
k
k
4
%5O *%
*%
x
x
≠
≠
!
">N9.:7".>N9,>N92M5
( %5O) *% ( %5O) *%
% %5O *% %5 %5O *%
*%
(%5O *% ) ( %5O)( *% )
%5O %5O *%
%5O *% %5O!*% (%5O )(*% )
%5O
*%
x x
c x x x x
x
x x
x
x x x
x
− − − −
= ⇔ =
− − −
−
⇔ = ⇔ − = + −
+ −
⇔ − + − = ⇔ − + =
=
⇔
= −
g)%5O64"9."s<${4274"5,+*%O6!
g)*%O6i
Hx k k
π π
⇔ = + ∈¢
."s<${!
J.0KQ">N9.:7"*+"c9"5F$
Hx k k
π π
= + ∈¢
k
k
k
k
>?9.:R.D$(G)E4P"U6*+C">N9.:7"(Oi)
g(Li)
6
5€K45F
'x≤ ≤
!•C">N9.:7",>?9.:R.<*+
( )y x= + − −
2M5
y ≥
d#
( )y x= − − −
2M5
y <
!
"=.P*"2Q.."=*`.7$
(
)
(
)
' '
'
( ) ( )
} ( )
V x dx x dx
x dx
π π
π
= + − − − − − −
= − −
∫ ∫
∫
_.O•6%5.2M5
x
π π
− ≤ ≤
!
k
k
O6
⇒
.6
π
−
O6'
⇒
.6
π
XO6*%.X.H
( )x− −
6*%.!
dQLd6
} %c tdt
π
π
π
−
∫
6
h ( % ) hc t dt
π
π
π π
−
+ =
∫
k
k
<) c50#.:K9,5=$*;<!
<*+
HBN AB BN AM BN BM⊥ ⊥ ⇒ ⊥
O
X,+.<$95*I2K9.B5I%KL:<I6
MN
H
.>N9.m6
MN
!
dQL6I666
MN
6
b
D$$_.*`K9B5.5JC0#HE4P"U6
b
I
E)_.6OYHI6LY
<$95*I2K9.B5I\
6I
gI
L
<$95*2K9.B5\
6
g
KL:<E
6
6
gI
gI
6O
g<
gL
x,+O
gL
6E
•<
4"9,r5!
aX5FI*+,L0#IH,>?9*<I!
"=.P*"
<O ( ) ( )
j
V y a x y a b a= ≤ + = −
!
"=.P*".aX5FI0M"].4"56I6
b a−
!
k
k
k
k
d
<*+
'
( ) ( ) x m x m x+ ≥ + + − +
( )( ) ( ) ( )x x x m x x x⇔ + − + ≥ − + + +
5€K45F
x
≥ −
"5,+E].C">N9.:7".>N9,>N92M5
x x
m
x x x x
+ +
≥ +
− + − +
⇔
x x
m
x x x x
+ +
− ≥
− + − +
(z)
_..6
x
x x
+
− +
HE].C">N9.:7"(z)*+XB9.•.
m≥
()
<*+
‚ H ‚ '
( )
x x
t t x
x
x x
x x
− − +
= = ⇔ = − +
+
− +
− +
!
I19E5J."5\
k
O
i
'− +
+∞
.y g i
.
' '
'
+
dQL
ƒ '
G
'
t
+
∈
!I].C">N9.:7",{*"*+9"5F$4"52#*"e4"5()*+9"5F$
ƒ '
G
'
t
+
∈
4"52#*"e4"5
ƒ '
„G …
'
<O ( )
t
M f t m
+
∈
≥
2M5|(.)6.•.
|y(.)6•.
I19E5J."5\
.
' '
'
+
|y(.) g i
|(.)
h
•E19E5J."5\
ƒ '
„G …
'
<O ( )
h
t
M f t khi x
+
∈
= =
dQLL*E..>N9,>N92M5
h
m ≤
k
k
k
d< '
>?9."@9,5SK<('G)*+XB9
<O ' by a b∆ + − − =
>?9."@9
∆
.B2M5"<5,>?9."@9
#d v d
$W..<$95**D*+,e"0#95<*;<
#d v d
4"52#*"e4"5
( H ) ( H )d d∆ = ∆
%( HX ) %( HX )c c⇔ ∆ = ∆
k( ) k( )
a b a b
a b a b
a b a b
− +
⇔ =
+ +
⇔ − = +
a b a b
⇔ − = +
"_*
a b a b
− = − −
' a b⇔ − =
"_*
' a b+ =
"c<6'HE6"_*<6HE6i'!
dQL*+"<5,>?9."@9."s<${L\K*`KE#5.'OgL•62#O•'L6
k
k
k
k
do*.N*"eC">N9*;<
x y z− + −
∆ = =
−
2#
'
x y z+ − −
∆ = =
−
0`0>~.0#
(G G)H ( GG)u v= − = −
r r
!_.C"@9()%9%92M5
∆
2#
∆
"Q*_C2o*.N
k
(G G)H ( GG)u v= − = −
r r
0#$*_C2o*.N*"eC">N9!
<*+
( )
G G 'G jG '(GG)
n
− −
= = − − = −
÷
− −
r
!_.C"@9()*+XB9
()OgLg$6!
_.*`K( )*+.D$(iGGi')2#E4P"U6'!
_.C"@9().5JCOl*2M5( )4"52#*"e4"5
( H( )) ' ' k
m
d I P R m
− + +
= ⇔ = ⇔ = ±
+ +
dQL*+"<5$_.C"@9."s<${L\K*`K0#OgLg
' k
62#OgL•
' k
6!
k
k
k
'
<*+
( ) ( )
( )
h
n
h
h h h
( )
( ) ( )
( )
i i i
i i
i i
− = − −
= − −
= − = −
dQLC"`."m**;<%&C"a*.:\0#
h
k
k
k
k
dE '
*,>?9.5F$*Q*;<()*+XB9
b
y x
a
= ±
!c5(O
GL
)
( )
x y
H
a b
∈ ⇒ − =
()
*,>?9."@9X
GX
*+C">N9.:7"0#
#
b b b b
y x y x v y x y x
a a a a
= + − = − + +
*95<,5=$*;<X
GX
2M5**,>?9.5F$*Q(
G
x y
a b
y x
b a
+ +
)HI(
G
x y
a b
y x
b a
− −
)!
x5F.P*""7"E7""#" 6
I
6!I!%5
AOB
∠
6
! ( ! )OA OB OAOB−
uuur uuur uuur uuur
<*+
!OA OB
uuur uuur
6
(( ) ( ) )(( ) ( ) )
x y x y
a b a b
y x y x
b a b a
+ + + − + −
6
( )
h
x y
a b
a b
+
−
÷
6
h
a b
+
÷
(."o())
( ! )OA OB
uuur uuur
6
(( )( ) ( )( ))
x x y y
a a b b
y y x x
b b a a
+ − + + −
6
( )
h
x y
a b
a b
−
−
÷
6
h
a b
−
÷
(."o())
x,+
ab
S =
4"9,r5!
k
k
k
k
c<,W.:c9.D$.<$95*I0#(GG)
<*+
k
k
( ) ( ) ( )
I
' ( )
'
MG GA MG GB MG GC
MG GA GB GC MG GA GB GC
MG GA GB GC
GA GB GC
+ +
= + + + + +
= + + + + + +
= + + +
≥ + +
uuuur uuur uuuur uuur uuuur uuur
uuuur uuur uuur uuur
dQL
gI
g
≥
GA GB GC+ +
X]KEq9O1L:<4"52#*"e4"5
M G
≡
k
k
'
<*+
( )
h
j
h
Hk j Hk
h
Hk j
h 09
09 h09 h 09 09 h09
09 h09 h 09
h 09
09 09
09 09 j }09 09
x
x x x x x
x x x
x
x x
x x x x
x
− ≥
+ ≤ − ⇔ + ≥
+ ≤ −
− ≥
+ ≥
⇔
+ ≤ − +
>
}
k
}
09
k
09
h
x
x
x
x
x
≤ ≤
⇔
≤ −
>
≤ ≤
⇔
< ≤
QC9"5F$*;<E].C">N9.:7"0# 6
}
k
„G … (G …
h
∪
k
k
k
k