Đề thi thử toán lần 2 - ĐH Vinh pptx

TRNGIHCVINH KHOSÁTCHTLNGLP12LN2,NM2011
TRNGTHPTCHUYÊN MÔN:TOÁN;Thigianlàmbài:180phút
I.PHNCHUNHCHOTTCTHÍSINH(7đim)
CâuI. (2,0đim)
1. Khosátsbinthiênvàvđth(H)hàms
x 1
y
x 2
− +
=
−
.
2. Tìmtrên(H)cácđimA,BsaochođdàiAB=4vàđngthngABvuônggócviđngthngy=x.
CâuII(2,0 đim)
1. Giiphngtrình
( )
sin 2x cos x 3 cos 2x sin x
0
2sin 2x 3
+ − +
=
−
.
2. Giihphngtrình
4 2 2
2 2
x 4x y 4y 2
x y 2x 6y 23

+ + − =



+ + =


.
CâuIII.(1,0đim).Tính dintíchhìnhphnggiihnbiđthhàms
( )
2
x ln x 2
y
4 x
+
=
−
vàtrchoành.
CâuIV.(1,0đim). ChohìnhchópS.ABCDcóđáylàhìnhchnhtviAB=a,AD=
a 2
,gócgiahaimt
phng(SAC)và(ABCD)bng60
0
.GiHlàtrungđimcaAB.BitmtbênSABlàtamgiáccân tiđnhSvà
thucmtphngvuônggócvimtphngđáy.TínhthtíchkhichópS.ABCDvàbánkínhmtcungoitip
hìnhchópS.AHC
CâuV.(1,0đim)Chocácsthcdngx,y,zthomãn
2 2 2
x y z 2xy 3(x y z) + + + = + + .Tìmgiátrnhnht
cabiuthc
20 20
P x y z
x z y 2

= + + + +
+ +
.
II.PHNRIÊNG(3,0đim)
a.Theochngtrìnhchun
CâuVIa.(2,0đim)
1. TrongmtphngtođOxychotamgiácABCcóphngtrìnhchađngcaovàđngtrungtuynk
tđnhAlnltcóphngtrìnhx –2y –13=0và13x –6y –9=0.TìmtođB,Cbittâmđng
trònngoitiptamgiácABClàI(5;1).
2. TrongkhônggiantođOxyzchođimA(1;0;0),B(2;1;2),C(1;1;3)vàđngthng
x 1 y z 2
:
1 2 2
− −
∆ = =
−
.Vitphngtrìnhmtcucótâmthucđngthng ∆ ,điquađimAvàctmt
phng(ABC)theomtđngtrònsaochođngtròncóbánkínhnhnht
CâuVIIa.(1,0đim) Tìmsphczthomãn
z 3i 1 iz − = −
và
9
z
z
− làsthuno.
b.Theochngtrìnhnângcao
CâuVIb(2,0đim)
1. TrongmtphngtođOxychođngtròn(C):
2 2
x y 4x 2y 15 0 + − + − = .GiIlàtâmđngtròn(C).

ngthng ∆ điquaM(1;3)ct(C) tihaiđimAvàB.Vitphngtrình đngthng ∆ bittam
giácIABcódintíchbng8vàcnhABlàcnhlnnht.
2. TrongkhônggiantođOxyzchođimM(1;1;0)vàđngthng
x 2 y 1 z 1
:
2 1 1
− + −
∆ = =
−
vàmtphng
(P):x +y+z  2=0.TìmtođđimAthucmtphng(P)bitđngthngAMvuônggócvi ∆ và
khongcáchtAđnđngthng ∆ bng
33
2
.
CâuVIIb.(1,0đim)Chocácsphcz
1
,z
2
thomãn
1 2 1 2
z z z z 0 − = = > .Tính
4 4
1 2
2 1
z z
A
z z
   
= +

   
   
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TRNGIHCVINH
TRNGTHPTCHUYÊN
ÁP ÁNKHOSÁTCHTLNGLP12LN2,NM2011
MÔN:TOÁN;Thigianlàmbài:180phút
Câu ápán im
1.(1,0đim)
a.Tpxácđnh: }.2{\R =D
b.Sbinthiên:
*Chiubinthiên:Tacó
2,0
)2(
1
'
2
≠ ∀ >
−
= x
x
y
.
Suyrahàmsđngbintrêncáckhong )2;(−∞ và );2( ∞ + .
*Giihn:
1
2
1
limlim − =
−

+ −
=
+∞ → +∞ →
x
x
y
xx
và
1
2
1
limlim − =
−
+ −
=
−∞ → −∞ →
x
x
y
xx
;
+∞ =
−
+ −
=
− −
→ →
2
1
limlim

22
x
x
y
xx
và
−∞ =
−
+ −
=
+ +
→ →
2
1
limlim
22
x
x
y
xx
.
*Timcn:thcóđngtimcnnganglà 1 − =y ;đngtimcnđnglà
2 =x
.
0,5
*Bngbinthiên:
x ∞ − 2 ∞ +
'y
+ +
y

∞ +
1 − 1 −
∞ −
c.th:

thh
àms
cttrcho
ànhti(1;0),
ct trctung ti
)
2
1
;0( − và nhn giao
đim )1;2( −I cahaitimcnlàmtâm
đixng.
0,5
2.(1,0đim)
Vì đ
ngthng
ABvuônggócvi xy = nênphngtrìnhcaABlà mxy + − = .
HoànhđcaA, Blànghimcaphngtrình
mx
x
x
+ − =
−
+ −
2
1

,hayphngtrình
2,012)3(
2
≠ = + + + − xmxmx (1)
Dophngtrình(1)có mmmmm ∀ > + − = + − + = ∆ ,052)12(4)3(
22
nêncóhainghim
phânbit
21
,xx vàchainghimđukhác2.TheođnhlíViettacó
12;3
2121
+ = + = + mxxmxx
0,5
I.
(2,0
đim)
Theogithitbàitoántacó 16)()(16
2
12
2
12
2
= − + − ⇔ = yyxxAB
.130328)12(4)3(
84)(8)(16)()(
22
21
2
21

2
12
2
12
2
12
− = ∨ = ⇔ = − − ⇔ = + − + ⇔
= − + ⇔ = − ⇔ = − + + − + − ⇔
mmmmmm
xxxxxxmxmxxx
*Vi
3 =m
phngtrình(1)trthành
23076
2
± = ⇔ = + − xxx
.SuyrahaiđimA,
Bcntìmlà
)2;23(),2;23( − − + .
*Vi
1 − =m
tacóhaiđim A,Bcntìmlà
)22;21( − − + và )22;21( + − − .
VycpđimTM: )2;23(),2;23( − − + hoc )22;21( − − + , )22;21( + − − .
0,5
1.(1,0đim)
II.
(2,0 iukin:
π
π


kxx + ≠ ⇔ ≠
62
3
2sin và
.,
3
Z ∈ + ≠ kkx
π
π

x
O
1
1 −
2
y
I
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Khiđópt 32sin2)sin2(cos3cos2sin − = + − + ⇔ xxxxx
0)2cos3)(sin3cos2(
0)2cos3)(3cos2()3cos2(sin
03cos2cos3sin32sin
= − + + ⇔
= − + + + ⇔
= − − + + ⇔
xxx
xxxx
xxxx
0,5







+ =
+ ± =
⇔






=






+
− =
⇔

π
π
π
π

π

2
6
2
6
5
1
3
sin
2
3
cos
kx
kx
x
x
ichiuđiukin,tacónghimcaphngtrìnhlà
Z ∈ + = kkx ,2
6
5

π
π

.
0,5
2.(1,0đim)
H






= + +
= − + +
⇔
236)2(
10)2()2(
2
222
yyx
yx
t .2,2
2
− = + = yvxu Khiđóhtrthành



= − = +
= = +
⇔



= + +
= +
⇔




= + + + −
= +
67,12
3,4
19)(4
10
23)2(6)4)(2(
10
2222
uvvu
uvvu
vuuv
vu
vvu
vu
0,5
đim)
TH1. 67,12 = − = + uvvu ,hvônghim.
TH2.



=
= +
3
4
uv
vu
,tacó




= =
= =
3,1
1,3
vu
vu
*Vi



=
=
1
3
v
u
tacó



=
± =
⇔



=

=
3
1
3
1
2
y
x
y
x
*Vi



=
=
3
1
v
u
tacó



=
− =
3
1
2
y

x
,hvônghim.
Vynghim(x,y)cahlà ).3;1(),3;1( −
Chúý:HScóthgiitheophngphápth
2
x theoytphngtrìnhthhaivàophng
trìnhthnht.
0,5
III.
(1,0
đim)
Tacóphngtrình



− =
=
⇔ =
−
+
1
0
0
4
)2ln(
2
x
x
x
xx

.Suyrahìnhphngcntínhdintíchchính
làhìnhphnggiihnbi cácđng
.0,1,0,
4
)2ln(
2
= − = =
−
+
= xxy
x
xx
y
Dođódintíchcahìnhphnglà
.d
4
)2ln(
d
4
)2ln(
0
1
2
0
1
2
∫ ∫
− −
−
+ −

=
−
+
= x
x
xx
x
x
xx
S
.
t
x
x
x
vxu d
4
d),2ln(
2
−
−
= + = .Khiđó
2
4,
2
d
d xv
x
x
u − =

+
= .
Theocôngthctíchphântngphntacó
.d
2
4
2ln2d
2
4
)2ln(4
0
1
2
0
1
2
1
0
2
∫ ∫
− − −
+
−
− =
+
−
− + − = x
x
x
x

x
x
xxS
0,5
www.VNMATH.com
t
.sin2 tx =
Khiđó
ttx dcos2d =
.Khi ;
6
,1

π

− = − = tx khi .0,0 = = tx
Suyra
.3
3
2)cos(2d)sin1(2d
2sin2
cos4
d
2
4
0
6
6
0
0

6
2
0
1
2
− + = + = − =
+
=
+
−
=
∫ ∫ ∫
−
−
−
−

π
π
π
π

ttttt
t
t
x
x
x
I
Suyra

.
3
322ln2

π

− + − =S
0,5
+)Tgithitsuyra ).(ABCDSH ⊥
V )( ACFACHF ∈ ⊥
ACSF ⊥ ⇒
(đnhlíbađngvuônggóc).
Suyra .60
0
= ∠SFH
K ).( ACEACBE ∈ ⊥ Khiđó
.
32
2
2
1 a
BEHF = =
Tacó = =
0
60tan.HFSH
.
2
2a
Suyra
.

3
.
3
1
3
.
a
SSHV
ABCDABCDS
= =
0,5
IV.
(1,0
đim
+)Gi J,
r
l
nl
tlàtâmvàbánkínhđngtrònngo
itiptamgiác
AHC
.Tacó
.
24
33
2
..
4
.. a
S

ACHCAH
S
ACHCAH
r
ABCAHC
= = =
K đng thng ∆ qua J và
.// SH ∆
Khi đó tâm I ca mt cu ngoi tip hình chóp
AHCS.
làgiaođimcađngtrungtrcđon SHvà ∆ trongmtphng(SHJ).Tacó
.
4
2
2
22
r
SH
JHIJIH + = + =
Suyrabánkínhmtculà
.
32
31
aR =
Chúý:HScóthgiibngphngpháptađ.
0,5
Tgithittacó .)(
2
1
)()(3

222
zyxzyxzyx + + ≥ + + = + +
Suyra 6 ≤ + + zyx .
0,5
V.
(1,0
đim
Khiđó,ápdngBTCôsitacó
2
2
11
4
2
8
2
8
)2(
88
)( −








+
+
+

+








+
+
+
+ + +






+
+
+
+ + =
yzxyy
y
zxzx
zxP
.26
2
28

222
)2)((
8
1212
4
≥
+ + +
+ ≥ −
+ +
+ + ≥
zyxyzx
Duđngthcxyrakhivàchkhi 3,2,1 = = = zyx .
VygiátrnhnhtcaP là26,đtđckhi 3,2,1 = = = zyx .
0,5
1.(1,0đim)
Ta có ).8;3( − −A Gi M là trung đim BC
AHIM// ⇒
.Tasuyrapt .072: = + − yxIM
SuyratađM thamãn
).5;3(
09613
072
M
yx
yx
⇒



= − −

= + −
0,5
VIa.
(2,0
đim)
Ptđngthng .011205)3(2: = − + ⇔ = − + − yxyxBC
⇒ ∈ BCB
).211;( aaB − Khiđó
0,5
B
A
H
M
I
C
B
A
S
D
C
E
F
J
I
K
H
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



=
=
⇔ = + − ⇔ =
2
4
086
2
a
a
aaIBIA
.Tđósuyra )7;2(),3;4( CB hoc ).3;4(),7;2( CB
2.(1,0đim)
Tacó ).3;1;2(),2;1;1( − − − ACAB Suyrapt .01:)( = − − − zyxABC
Gitâmmtcu
⇒ ∆ ∈I
)22;2;1( tttI + − .Khiđóbánkínhđngtrònlà
.2
3
6)1(2
3
842
))(,(
22
22
≥
+ +
=
+ +
= − =
ttt

ABCIdIAr
Duđngthcxyrakhivàchkhi
.1 − =t
0,5
Khiđó .5),0;2;2( = − IAI Suyraptmtcu .5)2()2(
222
= + + + − zyx
0,5
t ).,( R ∈ + = babiaz Tacó |1||3| ziiz − = − tngđngvi
|1||)3(||)(1||)3(| aibibabiaiiba − − = − + ⇔ − − = − +
2)()1()3(
2222
= ⇔ − + − = − + ⇔ babba .
0,5
VIIa.
(1,0
đim)
Khi đó
4
)262(5
4
)2(9
2
2
9
2
9
2
23
2

+
+ + −
=
+
−
− + =
+
− + = −
a
iaaa
a
ia
ia
ia
ia
z
z là s o khi và
chkhi 05
3
= − aa hay
5,0 ± = = aa .
Vycácsphccntìmlà
iziziz 25,25,2 + − = + = = .
0,5
1.(1,0đim)
ngtròn (C)cótâm ),1;2( −I bánkính .52 =R Gi H
làtrungđim AB.t
).520( < < = xxAH Khiđótacó
2
4

1
. 8 20 8
2 (ktm vì )
2
x
IH AB x x
x AH IA
=

= ⇔ − = ⇔

= <

nên
.24 = ⇒ = IHAH
0,5
PtđngthngquaM: )0(0)3()1(
22
≠ + = + + − baybxa
.03 = − + + ⇔ abbyax
Tacó baabaa
ba
ba
IHABId
3
4
00)43(2
|2|
2),(
22

= ∨ = ⇔ = − ⇔ =
+
+
⇔ = = .
*Vi
0 =a
tacópt .03: = + ∆ y
*Vi
.
3
4
ba = Chn
3 =b
tacó
4 =a
.Suyrapt .0534: = + + ∆ yx
Vycóhaiđngthng ∆ thamãnlà 03 = +y và .0534 = + + yx
0,5
2.(1,0đim)
Gi(Q)làm
tphngqua
M
vàvuônggócvi ∆ .Khiđópt .032:)( = − + − zyxQ Tacó
).1;1;1(),1;1;2(
PQ
nn − TgithitsuyraA thucgiaotuyn dca(P)và(Q). Khiđó
)3;1;2(],[ − = =
QPd
nnu và dN ∈)1;0;1( nênptca






− =
=
+ =
tz
ty
tx
d
31
21
: .
Vì
dA ∈
suyra ).31;;21( tttA − +
0,5
VIb.
(2,0
đim)
Gi Hlàgiaođimca ∆ vàmtphng(Q).Suyra
).
2
1
;
2
1
;1( −H
Tacó

7
8
1016214
2
33
),(
2
= ∨ − = ⇔ = − − ⇔ = = ∆ ttttAHAd .
Suyra )4;1;1( − −A hoc
).
7
17
;
7
8
;
7
23
( −A
0,5
VIIb.
(1,0
đim)
t w
z
z
=
2
1
tađc 0||||||

2222
> = = − zwzzwz .Hay 1|||1| = = − ww .
Gis ),( R ∈ + = babiaw .Khiđótacó
0,5
M
H
B
I
A
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1)1(
2222
= + = + − baba hay .
2
3
,
2
1
± = = ba
*Vi .
3
sin
3
cos
2
3
2
1
π π


iiw + = + = Tacó
3
4
sin
3
4
cos
4

π π

iw + = và
.
3
4
sin
3
4
cos
1
4

π π

i
w
− =







Dođó
1
3
4
cos2 − = =

π

A .
*Vi
iw
2
3
2
1
− = ,tngttacngcó 1 − =A .
Chúý: HScóthgiitheocáchbinđitheodngđiscasphc.
0,5
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