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Đề thi thử đại học năm 2014 môn toán trường chuyên Nguyễn Trung Thiên, Hà Tĩnh

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sở giáo dục và đào tạo hà tĩnh
Tr"ờng THPT Nguyễn Trung Thiên
Đề THi thử đại Học LầN I năm 2014
Mụn thi: Toán - KHI A, A1, B
Thi gian lm bi: 180 phỳt
I. Phần chung cho tất cả thí sinh(7,0 điểm)
Câu I (2,0 điểm) !"# "$% &'
3
1
x
y
x

=
+
() *+ ,"- .!/
01 2"3# &4, &5 6789 ,"7:9 ;$ ;< *+ ,"- .!/ (=> "$% &'1
?1 @78, A"BC9D ,EF9" ,78A ,GH89 (=> .!/ 678, I"#39D (4(" ,J D7># *7K% L (=> ? ,7M% (N9 (=> .!/ *89
,78A ,GH89 6O9D
22
1
Câu II (2,0 điểm)
01 P737 A"BC9D ,EF9"
12sin(2)coscos3
4
x
xx

++=+ 1
?1 QR9"S L T
2


tanx
1os
dx
cx+


Câu III (1,0 điểm) P737 "M A"BC9D ,EF9"S
22
2
41
2
1
xyxyy
y
xy
x

++=


+=+

+


Câu IV (1,0 điểm) !"# "F9" (")A U1VW!X () *4H VW!X Y$ "F9" ,">9D1
Z4H Y[9 VW T ?> \ W! T !X T XV T >\ UV ;G]9D D)( ;[7 *4H^ %_, A"`9D.UW!/ ,a# ;[7 *4H %b,
D)(
60
o

1 QR9" ,"K ,R(" I"'7 (")A U1VW!X ,"c# >1
Câu V (1,0 điểm) !"# d &' ,"5( eBC9D f^ H^ g1 QF% D74 ,E- 9"h 9"i, (=> 67KG ,"j( S

222
222
()()()
333
xyz
Pxyz
yzxzxy
=+++++1
II. Phần riêng (3,0 điểm): Thí sinh chỉ đ"ợc làm một trong hai phần (phần A hoặc phần B)
A. Theo ch"ơng trình chuẩn
Câu VI. a. (1,0 điểm) QE#9D %_, A"`9D ;[7 "M ,k> *b lfH ("# ,>% D74( VW! () ,Ek9D ,m% P .?\n0/1
ZBo9D ,EG9D ,E5( (=> (a9" W! () A"BC9D ,EF9"
d:3xy40=1 ZBo9D ,"`9D VW () A"BC9D ,EF9"
1
:10310dxy++= 1 QF% ,k> *b (4( *p9" V^ W^ !1
Câu VII. a. (1,0 điểm) QE#9D %_, A"`9D ;[7 "M ,k> *b lfH ("# ">7 *7K% V.?\q/^ W.r\s/1 @78,
A"BC9D ,EF9" *Bo9D ,Et9 .!/ ,78A fu( ;[7 ,Ev( "#$9" ,a7 *7K% V ;$ I"#39D (4(" ,J ,m% .!/ *89 W 6O9D
w1
Câu VIII. a. (1,0 điểm ) QF% &' "a9D I"]9D ("j> f ,E#9D I">7 ,E7K9
()
3
2

n
Pxx
x


=+


(0)x > 1
W78, EO9D 9 ,"h> %x9S
67898
2
332
nnnnn
CCCCC
+
+++= 1
B. Theo ch"ơng trình nâng cao

Câu VI. b. (1,0 điểm) QE#9D %_, A"`9D ;[7 "M ,k> *b lfH ("# ,>% D74( VW! ;G]9D (m9 ,a7 V.0\?/1
@78, A"BC9D ,EF9" *Bo9D ,Et9 .Q/ 9D#a7 ,78A ,>% D74( VW! 678, *Bo9D ,"`9D d:xy10= ,78A fu( ;[7
.Q/ ,a7 W1
Câu VII. b. (1,0 điểm) QE#9D %_, A"`9D ;[7 "M ,k> *b lfH ("# ">7 *Bo9D ,"`9D
1
:350dxy++=\
2
:310dxy++= ;$ *7K% L.0\n?/1 @78, A"BC9D ,EF9" *Bo9D ,"`9D *7 yG> L (z,
12
,dd Y{9 YB|, ,a7 V ;$ W
&># ("#
22AB =
1
Câu VIII. b. (1,0 điểm) P737 A"BC9D ,EF9"S
3
2

2
2
loglog2
2
x
x
x


+=




1
nnnnnnnnnnnnnnnnnnnnn Hết nnnnnnnnnnnnnnnnnnnn
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§¸p ¸n K.A gåm cã 6 trang.
L"u ý : Mäi c¸ch gi¶i ®óng ®Òu cho ®iÓm tèi ®a.

C©u §¸p ¸n vµ h"íng dÉn chÊm §iÓm
1 (1,0 ®iÓm)
________________________________________________________________________
} QNA f4( *-9"S
{ }
\1DR=−

} U5 6789 ,"7:9S
2
4

'0
(1)
y
x
=>
+
^
1x∀≠−
^ &GH E> "$% &' *+9D 6789 ,E:9 (4( I"#39D

(
)
;1−∞− ;$
(
)
1;−+∞ 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
} P7[7 "a9S
lim1
x
y
→−∞
= \ lim1
x
y
→+∞
= T• Q7M% (N9 9D>9DS HT0

1
lim

x
y

→−
=+∞\ lim
x
y
→+∞
=−∞ T• Q7M% (N9 *j9DS fTn01
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
} W39D 6789 ,"7:9S

f ∞− n0 ∞+

H’

}
H +∞ 0

0
−∞

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
} Z+ ,"- S
P7># ;[7 lfS .d\q/^ D7># ;[7 lHS .q\nd/1
Z+ ,"- 9"N9 L.n0\0/ Y$% ,m% *'7 fj9D1
x
0
-13
1

-3





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C©u
I.

2,0
®iÓm
2 (1,0 ®iÓm)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
P73 &ö
(
)
00
;Mxy ,"Gb( (C),
0
0
0
3
1
x
y
x

=
+
,
0
1x ≠− .
2"7 *) A"BC9D ,EF9" ,78A ,GH89
∆ ,a7 M Y$S

()
()
0
0

2
0
0
3
4
1
1
x
yxx
x
x

=−+
+
+


()
(
)
2
2
000
41630xxyxx⇔−++−−=
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Q"c# *Ò S

(
)
,22dI∆=


()
(
)
()
0
2
2
00
4
0
4163
22
161
xxx
x
−−++−−
⇔=
++


()()
42
00
181160xx⇔+−++=






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0
0
1
3
x
x
=



=−


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

@[7
0
1x = ^ A"BC9D ,EF9" :2yx∆=−\
@[7
0
3x =− ^ A"BC9D ,EF9" :6yx∆=+1





q^w
1 (1,0 ®iÓm)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PQ
1sin2cos22coscos2
x
xxx⇔++=


2
2cos2sincos2coscos20xxxxx⇔+−=

(
)
(
)
22
2coscossincossin0xxxxx⇔+−−=



(
)
(
)
coscossin1cossin0xxxxx⇔+−+=
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
cos0
cossin0
cossin1
x
xx
xx
=


⇔+=


−=


2
tan1
1
cos
4
2
xk
x

x
π
π
π

=+


⇔=−




+=





~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
2
4
2
xk
xk
xk
π
π
π
π

π

=+



⇔=−+


=




k









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C©u
II.
2,0
®iÓm

2 (1,0 ®iÓm)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Q> ()S
22
tansincos
1coscos(1cos)
xxx
Idxdx
xxx
==
++
∫∫

Z_,
2

costx=
2sincosdtxxdx⇒=−

UGH E>S
1
2(1)
dt
I
tt
=−
+


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

11111
ln
212
t
IdtC
ttt
+

=−=+

+



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

28, YGN9S
2
2
11cos
ln
2cos
x
IC
x

+
=+


1




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C©u
III.
1,0
®iÓm

N"N9 fÐ, HTq I"]9D ,"h> %x9 "M A"BC9D ,EF9"1

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HM ,BC9D *BC9D ;[7
2
2
1
4
2
1
x
xy
y
y
xy
x

+
++=





+=+

+


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Z_,
2
1x
u
y
+
= ^ ; T f } H1 HM ,Eë ,"$9"S
4
1
2
uv
v
u
+=



=+



P737 "M ,> ()S G T0
; T d

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@[7
2
1
1
2
11
3
2
3
5
x
x
y
u
y
v
x
xy
y
=



+

=
==




⇒⇔


=
=−



+=

=









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C©u
IV.
1,0
®iÓm

Pk7 N Y$ ,EG9D *7K% VW1
A
B
C
D
N
60
0

Q> ()S
AN // DC
AN = DC = a



9:9 VX!N Y$ "F9" 6F9" "$9"1
UGH E>S N! T VX T >
T• NV T NW T N! T> ">H
ACB∆
;G]9D ,a7 ! &GH E>
ACBC⊥
1
X# ()SAABCD⊥ 9:9

SABC⊥
1
¸A ev9D *-9" Yý 6> *Bo9D ;G]9D D)( ,> &GH E>
SCBC⊥
1
UGH E>S P)( D7÷> .UW!/ ;$ .VW!X/ Y$
SCA∠
T•
60SCA∠=°

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
M_, I"4(S
NBC∆
*ÒG 9:9
60NBC∠=°


3
3
2
ACABa==

.tan603.33SAACaa=°==
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
2
33
4
ABCD
a
S =

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
QR9" *B|( ,"K ,R(" (")A U1VW!X 6O9D
3
33
4
a
1
















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C©u
V.
1,0
®iÓm
Q> () S
333222
2
3
x
yzxyz
P
xyz

++++
=+



¸A ev9D 6i, *`9D ,"j(

22
2,,ababab+≥∀
222
x
yzxyyzzx⇒++≥++1
(§¼ng thøc x¶y ra khi x=y=z)
333
2
3
x
yzxyyzzx
P
xyz

++++
⇒≥+



333
222
333
xyz
P
x
yz

⇒≥+++++




________________________________________________________________________
XÐ, "$% &'
3
2
()
3
t
ft
t
=+ ;[7 , • q \

2
2
2
'()ftt
t
=− \
4
'()02ftt=⇔= 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
W39D 6789 ,"7:9S

,
q
4
2
∞+

H’


n q }
H
+∞ +∞



4
8
32

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@NH
4
48P ≥ 1 Z`9D ,"j( f3H E> I"7
4
2xyz=== ">H
4
48P = 1




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A. Theo ch"¬ng tr×nh chuÈn
C©u
VI. a.
1,0
®iÓm

Pk7 M Y$ ,EG9D *7K% W!^ ;F
Md∈
9:9 M .%\ d%ns/1
M$
2GAGM=−


9:9 V .rn?%\ wnr%/1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
AAB∈
2m⇒=

(
)
2;2M⇒ ^
(
)
2;7A − 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
W! yG> M ;$ ;G]9D D)( ;[7 e 9:9 () A"BC9D ,EF9" f } dH – 8 T q1
BABBC=∩
9:9
(
)
1;3B − 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
M Y$ ,EG9D *7K% W! 9:9
(
)
5;1C 1

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C©u
VII.
a.
1,0
®iÓm

Pk7
00
(;)Ixy Y$ ,m% (=> *Bo9D ,Et9 .!/1
2"7 *)^ e# .!/ ,78A fu( ;[7 lf ,a7 V 9:9 ;[7
(0;1)i =

Y$ ;c(,C *C9 ;- ,E:9 ,Ev( lf^ ,> ()S
IAi⊥


(
)
(
)
00
1.10.00xy⇔−+−=
0
2x⇔=1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Q"c# D73 ,"78,^ ,> ()S
R T LW – w
2
;25IB =
()()
22
0
26425y⇔−+−=

0
43y⇔−=±
0
0
7
1
y
y
=



=


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~




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@[7
0
7y = ,"F (2;7)7IR⇒=1
@[7
0
1y = ,"F (2;1)1IR⇒=1
@NH ,> () ">7 *Bo9D ,Et9 ({9 ,F%S
()()
22
2749xy−+−=\
()()
22
211xy−+−=



q^w
C©u

VIII.
a.
1,0
®iÓm
¸A ev9D (]9D ,"j(
11
1
kkk
nnn
CCC
++
+
+=^ ,> ()S

6789
33
nnnn
CCCC+++
677889
2()
nnnnnn
CCCCCC=+++++

789
111
2
nnn
CCC
+++
=++

89
22nn
CC
++
=+
9
3n
C
+
=
P73 ,"78, ,BC9D *BC9D ;[7

98
32
2
nn
CC
++
=
3
2
9
n +
⇔=
15n⇔=
1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
2"7 *)
()
3

2

n
Pxx
x

=+




()
15
15
3
15
0
2
k
k
k
K
Cx
x

=

=






305
15
6
15
0
2
k
kk
K
Cx

=
=

1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
U' "a9D I"]9D ("j> f ,BC9D j9D ;[7
305
06
6
k
k

=⇔=1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
U' "a9D A"37 ,F% Y$
66

15
.2320320C = 1




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B. Theo ch"¬ng tr×nh n©ng cao
C©u
VI. b.

1,0
®iÓm

Pk7 L Y$ ,m% (=> *Bo9D ,Et9 9D#a7 ,78A
ABC∆
1
@F
ABC∆
;G]9D (m9 ,a7 V 9:9 L Y$ ,EG9D *7K% W! ;$
AIBC⊥
1
Q"c# D73 ,"78,
()BCd⊥
//dAI⇒


W49 IR9" (=> .Q/ Y$S (,)2RdAd==1
()BCd⊥

W!S f } H } ( T q1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(,)2dAdR==
12
2
2
C++
⇔=
1
5
C

C
=−



=−


UGH E>
:10
:50
BCxy
BCxy
+−=


+−=


ZBo9D (># VL (=>
ABC∆
*7 yG>
(
)
1;2A ;$ &#9D &#9D ;[7 ():10dAIxy⇒−+= 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
N8G :10BCxy+−=
10
:
10

xy
IBCAI
xy
+−=

⇒=∩

−+=



L.q\0/1
UGH E>S
()
2
2
():12Txy+−=1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
N8G
:50BCxy+−=
50
:
10
xy
IBCAI
xy
+−=

⇒=∩


−+=



L.?\d/1
UGH E>S
()()
22
():232Txy−+−=1
@NH () ">7 *Bo9D ,Et9S
()
2
2
12xy+−= ;$
()()
22
232xy−+−=1


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C©u
VII.
b.
1,0
®iÓm
@F
1
Ad∈ ^
2
Bd∈ 9:9 Dk7 ,k> *b (;35)Aaa−−\ (;31)Bbb−−1

(
)
;43()ABbaba=−−−

1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
QJ D73 ,"78,
22AB =
&GH E>S


()()
2
2
4322baba−+−−=

1
Z_,
tba=−
^ ,> ()S
()
2
2
2
348
2
5
t
tt
t
=


+−+=⇔

=


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@[7

2t =

2ba⇒−=
(2;2)AB⇒=−

Y$ ;c(,C ("p A"BC9D (=> ∆ ({9 ,F%1
UGH E> A"BC9D ,EF9" *Bo9D ,"`9D (=> ∆ Y$
12
22
xy−+
=

10xy⇔++=1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@[7
2
5
t =
2
5
ba⇒−= 1
QBC9D ,5 ,> () A"BC9D ,EF9" *Bo9D ,"`9D (=>
∆ Y$ 790xy−−=1
@NH () ">7 *Bo9D ,"`9D ({9 ,F% Y$
10xy++= ;$ 790xy−−=1

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C©u
VIII.
b.
1,0
®iÓm

ZIS
0x >
^
1
2
x ≠ 1
PQ
3
2
2

2
log
2
2
2log2
log2
x
x
x




⇔+=



~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


2
2
2
3log1
1
21log2
1log2
x
x
x



⇔+−=

+

2
2
2
3log1
log0
1log
x
x
x

⇔−=
+

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Z_,
2
logtx= ^ ,> ()S
31
0
1
t
t
t


−=
+

31(1)0ttt⇔−−+=
2
210tt⇔−+=
1t⇔=

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@[7
1t =

2
log1
x
⇒=
2x⇒=
1
@NH A"BC9D ,EF9" () 9D"7M% f T ? 1

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