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

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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