Thi thử ĐH chuyên Nguyễn Huệ-lần 1-2011

rci
rHr rnrl
DAr Heqr,AN rHrI
NnAl
xAnn
Hec 2010-_
zltt
DE
THI
MON:
TOAN
xlr6t.L,n
Thoi
gian
ldm bii:
180
phirt,
kh6ng kC
thoi
gian giao
d6
Cffu I:
(2,0
itiAm) Cho him
sd
!
=
x3
-3(nt+1)x2
+9x-m,vti
m ldtharnsd

thgc.
l.
Kh6o
s6t sy bi6n thi6n
vh vE OO
ttri cria hdm
sti
ea cho ring v6i
m
=1.
2. XAc
dinh
m
ee fram
sO ea cho
d4t cuc
tri t4i
xr,x2 sao
cho
la
-*rl:2.
Cflu
II:
(2,0
iti6m)
l.
Gi6i
phucrng
trinh:
1

+ 3 cos
x + cos 2x
-
2cos 3x
=
4sin
x.sin 2x
2.
Giai
he
phuong
trinh:
lxt
+2x+y'+y:3-xy
j
^
"
(x,y€R)
l*y+*+Zy-1
"'
cf,uIII:
(1,0ili0m)
r\m
J
ff-;d.
sinx.sinl
x+: I
\
4)
Cf,u

IV:
(1,0
ili6m) Cho
l6ng tru tam
gi6c
ABC.ATBTCT c6
tdt ch
cdc canh
bang
a,
g6c
tao b&i
canh b€n vi
mdt
phing
d6y
bang 300.
Hinh
chidu
H cfra didm
A
tr€n
mat
phftng
(ArBrCr)
rhuoc
doong thing
B,C,. Tfnh thd tfch
khdi ldng tru
ABC.ATBTC, vi

tinh
khoang cdchgifa
hai dudng
thing AA,
vi
B,C, theo a.
Cffu
V:
(1,0
ifiAm)X6t
chc
sd
thgc
ducrng a,b, c
thoa mdn
di€u kiQn
a+b+c=1.
Tim
gie
d
nh6 nh6t cria
:
":m
?
. h
Cflu VI
(2,0
iiidm)
'
I

1. Trong mat
phing
vdi hQ
tga
ttQ
Oxy cho hai dudng
trdn
:
.
(C1):
x2
+
f
,:13
vd
(C2):
(x:
6)'+
t'
,:
ZS
cit nhau tqi
A(2;3).
Vi6t
phuong
trinh ducrng thdng
ili
qua
A vi 16n
lugt

cdt
(Cr),
(Cz)
theo hai
ddy cung
phAn
biQt
c6 elQ dii
blng
nhau.
2. Trong
kh6ng
gian
v6i h0 tqa d6
Oxyz
cho
tam
gi6c
vu6ng
c0n ABC
c6
BA
:
BC. Bi6t
A(5
;
3
;
- 1),
C

(2
;
3
;
-
0vd
B ld ditim nim
tr6n m{t
phing
c6
phuong
trinh
:
x+
y
-
z
-6
:0.
Tim tga c10 tli€m B.
Cflu VII
(1,0
iti6m)
Gi6i
phucmg
trinh
:
(z
-
tog,

x)logn,,
-;ft;
=
t
utlt
TRUONG THPT
CHUYfi,N
NGUYEN HUE
Thi
sinh kh6ng duqc
s* d4ng
tdi li€u.
Cdn bQ coi thi
kh6ng
gidi
th{ch
gi
thent
TRIIONG THPT
cnuvnN
NCUYNN
HUE
IIrtoNG
uAx
cHAvr
rnr rnrt
DAr
Hec
r,AN
rntl

unAr
.
NAnnFAC
2a!o
-
2ott
EE THI
MON:
TOAN
KIIOI
A, B
CAU
N(
)I
DT]NG
D
I-1
(tei6m)
Ydi
m=
1 ta cd
y
=
xt
-6x'
+9x-1.
*
Tflp
xdc
dinh:

D
=
R
x
Su bidn
thiOn
.
Chidu-bidn
thi€n:
!'=3x2
-l2x+9:3(x2 -4x+3)
[x>3
Tac6
_y'>0<+l
,t,
_y'<0<+1<
x<3.
Lx.
Do d6:
+
Him
sd ddng
bidn tr€n
m6i khoAng
(-*,1)
vd
(3,
+
oo)
+

Hdm sd nghich
bidn trOn
khoing(1, 3).
0,25
0,25
.
crlc tri:
Him sd dat
cuc dai
tai x=7
vd
!co=y(1):3;
dat cuc
tidu tai
x=3
vd
!ct.
=
/(3)
:
-l
.
o
Gi6i
nan:
"l]\/
-
-co;
1im
y

-
+a.
o
Biing bidn thiOn:
0,25
0,25
*
Dd thi:
}lA rhi .3t frrrn frrnd toi
fiid*
rsrr6
rllr ulvltl
(0,
-l).
t-2
(ldi6m)
Him
sd dat cuc dai,
cuc tidu
tai xt,
x2 <+ phuong
trinh
y,=
0
c6 hai
nghi€m
pb
li x,, x,
<+
R x2

-21m+l)x+3=0
c5hainghiOmphdnbi6t
ld
x1, x2.
<+ A'-
(m
+1)2-
3
>
0
ol*>
-t
+
'6
lm<-t-Ji
(l)
0,25
0,25
0,25
0,25
+)
Theo dinh
lf
Viet
ta c6 x, +
x,
:2(m
+1);
xrxr:
3.

Khi d6
1",
-"rl
-
2
e
(x, + xr)'
-
4*r*,
-
4
e
a(m
+l)'
-lz
:
4
^
f m:-3
e
(m
+l)'
:1o
|
(2)
Lm=l
Til
(1)
vn
(2)

suy
ra
gir{
tri
ctn m- - 3
;
m
=
I
II-1
(1
di6m)
PT
e
1+3cosr+cos
2x-2cos(2x+x)
=
4sinx.sin2r
<+
1 +
3cosx+ cos
2x
-2(cosx.cos2x -sinx.sin
2x)
=
4sin
x.sin2x
4,2:
0,2:
<>

I + 3 cosx
+
cos
2x
-2(cos.r.cos2x+
sin
x.sin 2x)
=
0
<+
1 +3cosx+
cos
2x
-2cosr
=
0 <+
I
+ cos.r+
cos 2x
=
0
e
2costx+cosx=0
[cosx
=
o
ol
l
I
cosx

=
-;
0,2
0,2
[":
*
l":
7r
-+Klr
2
*?o
*y2,
J
tt-z
(l
iliem)
[*'
*2x+
y2
t
y
:3-
xy
e[*t
* xy +
y2
+2x+y
=
3
(1)

lxy
+ x+2y
:I
l*y
* *
*2y
=l
(2)
CQng
(1)
vn
(2)
theo vti dugc
(*+
y)'
+3(x+
y)-4
=0
0,2
0,2
0,2
0,2
Suy ra
fx+y=1
l**r=u
V6i
x+
y:l
thay vno
(2)

dugc
-y2
*2y
=g
Iip {y".:(py)-:
(l;o}
(x;y)
=
(:l; 2)
V6i x+
!
=
-4
thay vdo
(2)
duqc
-y2 -3y-5
:0
Phuong
trinh vd nghiQm
Hq c6 2
nghiQm
(x;y):
(l;0); (x;y): (-l;2)
ilI
(1
di6m)
0,2
0,2
-Jil"otI*t

-ra(cotx)
r
cotx*l
0,2:
0,2:
Ji
(-*t
r+
h
lcot
x + rl)+C
IV
(1
tti6m)
Do
AH L(A,B.C,)
nOn
g6c
AAIH
bang 300.
4H
li
g6c gifra
AA, vi
(A,B,C,),
theo gii
thidt
thi
g5c
{\//

/
"
'i/
/
=/y",
0,2
X6t
tam
gi6c vuOng AHA' c6
AA'
=
a,
g6c
4H
=30"
=+
AH
:
I
a a2
Jt
o"li
AtBtc
3 z
4
24
I
2
V
ur"nrurr,

:
!
dn
's
X6t
tam
gi6c vuOng
AHAr
c6
AA,
=
a,
g6c
AAtH=300
=+
AtH
:+
Do tam
gi6c
AlBlCr
lI
tam
gi6c
ddu canh
a, H
thudc B,C,
vlL
AlH
=
{ ^r^A,H

vuong g6c vdi
B,C,.
MAtkhfc
AH
LB,C,
nOn BtCt L(AA.H)
0,25
Ta
c6 AAT.HK
=
ATH.AH
+ HK
=
AtH
'!H
=
oJi
Mt4
2
Tucrng
tg c6: I-bc>
1-ca)
,
(r
*").,(r
*oXr*a)
(r
*r)u(r+"[r+a)
2
1r

*a;n(r
*"1r *";
b2
r'ra,
[r.:)(r.;)[r.:)
-[r.#)'
=
o' Do
d6 min
p
:
8 d4t clusc khi a
:
b
:
c
_l
J
V
(1
ili6m)
Ggi
giao
di6m tht hai cria duong
thing cdn
tim
vdi
(Cr)
vi
(Cz)

ldn luqt li M
vd
N
Ggi M(x;
V)e
(C,)
+ x2 +
y2
:13
(l)
Vi A h trung di6m
ctra MN n€n
N(4
-
x; 6
-
y).
Do
N e
(Cr)
=
(2+
x)2 +(6-
y)2
=25
(2)
ouc"e itrins
;a; ti-
aiq; A ;t M
a6

ph;o"s tiintr
: i
-
at
+
i
:
0
Tri
(l
)
vd
(2)
ta c6 hQ
{*'
*
v'
^=13
'
f(z*
x)z +(6-!)2
:25
ciei
hetadusc
(x:2; y:
3)
(lo4i
vi trungA) vdC
:#
t

r:: ).
Vay*f
{
t
!l
VI- 1
(l
tli6m)
\
\
YT-2
(1
di6m)
AC
:
3J2
suy ra BA:
BC
:
3
l{*-s>'
+(y
-3)'
+(z
+t)'
=9
1@-z)'
+(y-3)'
+(z+4)'
=9

I
L"*,
-z-6:0
0,2:
0,2
0,2
0,2
lf*-s>'+(y-3)'
+(z+t\'
:9
l{*-s)'
+(4-2x)2
+(2-x)2
:s
<+jx+z-l:O
o|t=l-x
L"*y
-
z
-6:0
ll
=7
-2x
Tim dusc:
8(2;3;-1)
hoac
B(3;1;-2)
vII.
(1rti6m)
Dk:x)0,x+3,*+!

9
0,2:
0,2:
Q
-toerx)logr*,
-
*fu
-
|
4a
(z-
tog,.)G;-
jr*
=,
2
-los"
x 4
aJ
"'
-
-1
D{t:
t:log3x
ptthinh,? 4
=1,
h+1't *-z
ft
=-l
2+t t-t
*lt'

-3t-4-geLi
:o'
0,2
So siinh diAu
kiQn duoc
2 nghiQm
x
=81
I
J
0,2
?