Thi thử chuyên KHTN-lần 2-2011

,I'R,UONC
DAI
HOC KHOA
HQC
TU NHIEN
TRUdNG THPT CHUYEN
KHTN
pE
tllr tuu n4r
rloc
t+Ant
rc':,o
M6n:
TOAN (Dor
2)
Thdi.
gian
ld,m
bdi,:
180
phrit,
hhAng
kd thdi
gi,an
p,ttdt
di
Cdu I.
Cho
hbm s6
y
-

s3
+3(rn+
1) 12
+3rn(m+2)s+rnl *3rn2.
1).Kh&o s5,f sU bi€n thi€n vb,
v6
dd thi cira
hbrn s6 d5, cho
vdi
nz:0.
2)
Chrfi:g
rninli rbng r-di
inqi
rn
hd.m s6
lu6n c6 2 cqc tri,
ddng thdi kho6,ng
c6.ch
giria
2
didm nA,y
kh6ng
phq
thuQc
vh,o
rn.
Cdu II.
i) GiAi
phrrong

trinh
{1,r
tarrc)cos5z
=
silr
*
cos3j
+
2cos4c
-2cosZr.
2)
Giei
pirrtong
tr)nl.r
log,
(z
+ 31"e6
")
:
iogu
".
CAu
ril.
1)
'fim
t5.t, cA c6c
giir
i,ri
crle i,ham s6 rn
de

plrrtong
trinh
sinr
*
cosc
+
JTTdiii:
n,
c6 nghiQm.
2)
Tinh t6ng
rto fiz r-4 arj n2OlO
c' _
v2010
,
u2010
t
\'2QIO
,
v2010
,' ,
v2010
"
-
-llT-
-
J.r-t
-
TF
T

1V{
T
-i-
tgifiIoG
Cdu trV.
1) vi6t
phr;ong
tr)nh tiudng
trbn iii
qua
hai didrn ,4
(4;5),8
(5;-2)
vd, ti6p x-ic
dudng rh6.irg
(d)
:Y
=
-4.
2)
cho
cd,u
(,9)
t6,m o, c6 AB
=
2R
) 0
lb
ciudng kinh
c6 dinh. Di6m

r
di dQng
tren do6i:
o-8,
,-;+
^L*-*
/D\
^,,n
L,i
,,^
4a^
f\D
-{+
/o\ rL^^
-:^^
! :
r:
i-J\ r-r- t^\ A:1
iiiaL
piletrlE
(il
qQer
i va vUong
gOC
UD CO,r
1u7
urrsu
Erd/
uuyvl id, riiidfig'LIOfi
(U.i.

Uia S',i'nOi:r
(/f)
c6 dinh A,
d6y
lb. dqdng trbn
(C)
vdi
tluc d6i x-rlng,4I.
X6,c
dinh
dO dai
01
theo
& dd ihd
tich n6n
(l/)
ldn nhdt.
3) cho
h)nh ch6p Lit
gittc
S.ABCD c6,9,4 vuong g6c
m5t
philn1
(ABCD),
tld,y
ABCD
lb. hinli
chrl
nh$t
dO dei AB

:
fro,BC
=
a.
Gqi
14 lb.
trung
clidm
cioan
CD,
bict rang
gOc girra
hai m5t
ph&ng (ABC
D)
vd (SBtul)
Ib.
cp
60".
a) Chinrg minh rEng
mat
phfing (SBM)
vu6ng
g6c
mEr
phing
(SAC).
b) Tini rhd rtch rri dign
SABM theo a.
CAu

V. YOi r,t1,z
ld nhrlng s6 thgc
drrong th6a
m6,n
s
*y
* z.:
xyz, chtngnrinh
rb.ng
2
-,-
1
.9
\nT?
\trW
!ET;':
4'
_HET_
DAP
AN
(tom
tdt)
CduI.
1) Tac6U':3r2*6x,y':0<+r=0iro6,ca:-2vir,
Iimr*._*U:-F,
limr*+*g:*cc
Hdm
s6
dong
bien

tr€n
cd.c khoAng
(-*,
-2)
,
(0,
+oo)
va. ngirich
bicn-tren
khoangi(-{0)
st.y ra
tCD
:
-Z,yCo
:
4:xcr
:
A,aCr
:0
-oo
-2
0
+oc
f'(x)
0+0-
f(n)
4+*
,/\
,/\,/
,/

,/
'/"/'
,/
\,,
-oc
0
2)
Ta c5
a'
:
3n2
-
6(ra+i)r
*
6tn.(n't*2J,g'
:
0
+>
r
:
_2-
riz hod.c
t
:
-rrL.Hb.5r
s6
dorig bi€n Lr'€n
c5.c khoAr,g (-ec,-2
_
m),(,-m.,*co);

nghich
bi€n tron kho6.ng
(-2-nt,-nz)
va
rcp
:
-2
-
m,Ucp
:
Airc'r
:
-"ti1'1
.Irc:'
:
0.
K-iri
d6
kho6.ng
c6.ch
girla
hai
di6m
ciJc tri
Id
/-
,/
(-2
-
rn

*
n)z
+
(4
-
l)'
:2,/8.
CAu
ltr. 1)Chri;i
cos5c:
ct;s5r*cos3c-cos3c-cocr+cosc :
cosr
(2cos4r
-
2cos
Zr
+
i)phr:o1g
trinh decho <+
(sinz
*
cose)
ffi*
-
sinr*cosc**Y-1
t
oy
(#T
-
1)

(sinr
+
cosr
-
1)
:0,
2)Dkc>Lr.Del,k-rgoc=gsuyralog2(6v+3v)
=A+6v*3u:Zo#(3)r+(B)"=1.DeLlr6y
ve
tr5.i
lb. h).m
ti6ng biSn
n€n
phrrong
frinh
conghi€m
rJuy
lha,t g:
-t
t,: dO
,:
t.
cAuIrI. 1)
De.tt:sinrD+cosr,tel-it,'/z)+L-sin2x:z-t2.suyraiaca.ntjm;;;r.id
plrrrong
trinh t
+
\/r=7
:
m e6

lghiQm
t
e
i-,/t,
^f;\.
Xet
f
ft)
:
t
+,/T4,t
€
l-
Jj, {4
tac6
f'
(t)'=
1
-
#V,t
e
?rt,Ji);f'(i)
:
O
+* f
:
I
Suy ra
f
ddng bi6r:

tr€n
kho&ng
(-f,t) vd
nghich
bi6n
tr€n
(t,Jl).
Me
i
(f)
li€n
ruc
rr€n
?rt,,tq
tt d6
maxf
(t)
:
f(1)
:
2,min
f
(t)
:
f
?rt)
:
-,fi.
v1 v
*^/2

S
rn
52.
2)
Ddr n
:
2010.
Ta
c6
fi1CTk
:
ftrc!\\r
+
n/2
n/2
g-
\-'
!
f'2le
-
t
\ tn2k+r
u
-
./-,
?IETTIZFUn
:
;+1
L
i'rt-i+'t'

:
b=O L-n
-
, €
1
nlk*!-
z
tlr,,1,,n*l
@
kn'FE+'ivn+r
:
G+l)t
L\r
f
i/
lt 1in*tl i.+l
1 c2oll r
-
(r
-
il
J
:lffi:
2fur7a#.
Cdu IV.
1) Di6u
ki€n 1A
:
IB
e E

-TA*6
:
0.
Didu ki€n
IA: R
<+
(g+
4)"
=
(r-4),
+(y-5)2 <+
18y
-g=
(n_
4)2.
Giei
he c6
189
-9:
(Ta
-
10)1++
49a2
-
15Bs
*
109:
0
f'
u:r,t:1,/?=b;

I,
g
:
109149,r
=
6T
fZ,R
:
305/49.
2)
DAi
OI)
=
a:
c6
r.2
=
R2
-
x2.
Co V
:
nr2(r
+
R)13
:
r/3,(R2
-
:r2)(r
+,?).

X6t
/(o)
:
-t3
-
Rrz
*
R"n
+R3. C6
f,(x):
-!x2
-Znr'+
R";f,(z):0
+)
s:
_R
hoir.c
r:
R/3.
I/r,o*
tai n.:
R/3.
3)
a) Do
ACLBM
nEn BML(SAC)
tfc
ld.
(SBM)r(.9,4C).
-

_
b)
AC c6t BM
rai 1.
C_6- AI
:2|JAC
=2r/Ja/3.
Vpy.9A
-
,6U
:
zo.
V(SAM
B)
:
(I/3).2a.a'l'/z
=
,/2as
/2.
Cdu
V.
DAt o:
i,b:
i,s:
ltac6
ab+bc*ca:1,
cling
thrlc
cAn
chrlng

minh
trd
lhA.nh
p
:
ffi
+
#
+
ffi
<
t.
cnri
i
ld
i
+
a2
:
ab*
bc
+
ca
+
a2
:
(o
*
6)(a
*

c) vb
hai
.{*' rt".+,,rr?/,hd+,*ts,asu-vrap=_ 29 _+_
?L__
I
Z.
(
t
r
\,
vu\'16 v'.r voDur
i6.
-
ffi
-
;i.m
-
ffi
s
a(;iT
+#i
+
o
(**"
+
#)
+"
(aG+o
+
*)

-,e.
D&ng
thric
x&y
ra,
khi
b
-
c
:
*ahay
e
-
z
=
7r
:
\m.
v
TRIJONG
DAI
HQC
KHOA
TI9C
TU NHIEN
TRUdNG
THPT
CHUY€N
KHTN
Ciu

I.
Cho
hbm
s6
Y
:
2r-2m*I
fr-rn-L
DE
rHr
THO
DAr
HQc
NAtu
zoro
Mon:
TOAN
(Dqt
3)
Thdi
gian tdm
bd.i:
180
phrit,
khdng
tcd tnat
gian
ph6't' di
(c^).
1)

Kh6,o
s6,t
sg
bi6n
thicn
vb
vE
dd
thi
cria
hdm
f
d5'
cho
vdi rn:
l'
;i Ct;
,
(1,2)
.
Tim
cd,c
gi6,
tri ctn
m
sao
cho
tdn
tai
m.t

dudns
thlng
qua A cdt
dd
rhi
(C-)
tpi
frui
aia*"phanbigt
M,N
md,
c6c
ti6p
tuy6n
t4'i
M,N
cria
dd thi
song
song
vdi
nhau.
Cdu
II.
1) Giei
Phrrong
trinh
tan2r*9
ca*r+Y!}
:u.

2) Giei
phrrong
trinh
log!
3r
*
1og,
2a
:
lo$23r *
log,
31
1og3
2r'
Cdu
III.
1)
Tim
gi6
tri
l6n
nh6t
cria
hb.m
s6
{i
+
{r=i
'
{r+{\-r

2) Tinh
nguy€n
hh,m
sinrdx
t:I
sin2s-3cos2r-I
CAU
IV.
1) 1.
cho
l5,ng
trq
ABC.tB'c"
bi6t
A',ABC
lb. ch6p
tam
gi6c
ddu-c6
canh
d6'v
a
vd khoing
c6,ch
girla
canh
b€n
vb, canh
ddry
d6i

diQn
bx.ng
k.
Tinh
th6
tfch
l6'ng
trg'
2.
Trong
hQ tqa
aO
Oia,
cho
I/(L;3;2).
Vi6t
phuong trinh
mat
phang
(P)
di
qua
H
c6t d*,Oat,Oz
t?,i
A,
B,C
sao
cho
I{ l}r' tr*c

t6m
tam
gi6'c
ABC
'
3.
Ttong
hQ tqa
dA
br,
4ro
drrdng
trdn
(C)
:
(r
-
1)'
+
(v
+
I)'
:25'
Vi6t
phrrong
trinh
drldng
th&ng
qua M
(7

;3)
c6't
(C)
tai
A, B sao
cha
M A
:
3M
B
'
CAu
V. Cho
da
giSc
ddu
12
dinh.
H6i c6 bao
nhieu tam
gi5.c
tir c6
dinh
lb 3
dinh
cria
rla
gi6t:
tih.
t*ro.

a
_HET-
ffi
07^,
ffr nt^ t rN?
17
DA
THANH
't't)rft
\6r
DT:0912291)450
oAp
AN o0r
a
(tom
tit)
L
Cdu
I. l)-Vdi
rn
:
I t1
e6
v
:
+:*
*
y'
:
ffi

a
0
vh.
lim"-aaoV
:
2,lirn,-2+ y
:
*oo,.lim"*2-
U:
oo.
TCD
c:2,TCN
y:2.
Hdm
s6 nghich bi6n
tr€n
c6c
khoing ( ;2)
,
(2;
*m)
vb.
kh6ng
q6
cr.rc tri. Dd
thi
nh&n
I
(2,2)
llul tdm <t6i

xrlng.
t
-@
2 +oo
f'(")
r@)
,\
-oo
+oo_\\_
*2
2) Ta c6
s! =
6$ry.
Gid,
srt
M
(rr,a)
,N
(rz,vr)
e
C*(q
t'
q).
Ti6p
tuy6n tq,i
M
fa
N
song
song <+

6#ry: e;#:ry
4* rr
-
rn-l:
-(rr-
rn-
1)
(*
rr
+
rz:Znt+2
(1).
Mat khec do A,M,N
thing hlng
=+
#:
#llire"ut-2:#
-
2:
;$1
n6n
iathu drrgc
(c1-
1)("t
-nl-
1):
(22
-L)(rr-m-
1) vh.chri
!

q*rn-l:
-(rr-
rn-
1)
siiy ra 11
-!: -(r"-
1)
=+cl
*tz
-
2.
Ctng
vdi
(1)
suy Iarn:0.
Cnu
II. 1)
Chfi
!
2cot2r:
cottr
-
tanrr
"d
#,;
:
tans
*cotc Phuoug
trinh tuong
duong

tan2 r
*9cot2
s
*
(coto
-
tanc)
*
Z(lana*
coto)
:
16
<+
(tanr
*
3cotz)z
*
(tanc
*
3cotc)
-
20
:0
I
tanc
*
3cotr
:4€
tanfr:
lhoetanr:3

o
I
t*"
f
3cots
:
-5
(+
tanr
:
(-s +
'/n)
12.
2)
Dk
c
>
0.
Ptrinh
<+
(logr
3r
-
logr
2c)
+
(logr
Sclog,
2t
-

log|3c)
:
g
++
(logr
3o
-
f
)
(log2r
-
log2 3o)
:
I
I
gr
:2
+
r:2/3
o
L
tor,
2r
:lo1zlL :
t
*
2u
:3t,32 :
2t
+ }t+t

-
2t+r"+
t
:
:1
+
r
:
l/6.
Ctu III.
l) Dk
0
S
r
<
l. Khid6 theo
b6t ding thfc Bunhiacopxki
({i
+ i/r4)'?
Sz(,/i
+
'/7.:l)
"i,
(Ji
+
t/T412
<
2(r
*
2

-
r)
:
4
+
{i.1
fr
tS
2,
d6u bhng.xiy
ra
<+ o
:
t.
MAt kh6cdo
0
S
{i,$=Z
<
1=+
,/a+.itr4
2
r*1-x:t*
Wk
(
1,
d6u
bling
xiy ra
r+

o
:
t ho6.c o
:0.
Vfly
maxy
-
2 khi
r: 1.
2)Tac6sin2c-3cos2r-I:
{sinccosr
-2cos2o+sin2c)
:2(sinr-cosc)(sino-1-2coso).
Bidu
di6n sin
c
:
A
(sinr
-
cos i)
+,a
(sin
o
*
2
cosr)
=>
A+ B
-

L,
-
A+28
:
0
=+
A
=
2/8,
B
=
1/8.
Do <t6 6r
:
,f
m**
r/m;#*
-
J*ffi
+
,f
#t-%
:
,/i,rnlr* (;
-
g)l
+
{st"lt*
(;
+

€)l
+
c ( trong
d6
sine
:2/'/i,cas,p:11{il
.
Cdu IV. 1)
Gqi
.tV li
trung
didm
BC,
MK
L AAt,
A'H L
(ABC).
C6
MK.AA'
:
A'
H.AM E
K.\[*
+
+
-
A'H."{
+ A'H:
ffi,s.qnc:
#

*vnpg-n'e'c'
:
iJ#TF(dwt)'
2) fI
Ie tn/c td.m,
OH
L
(ABC)
n€n
m6t
phlng
(P) qua
Il(1;3;2)
c6
vecto
ph6p
tuySn
llr
OH
:
(l;3;2)
c6
phrrong
trinh: c
*
3V
*
2z
-
14:

0.
3) Gqi .iT
lb
trung
diiim
.4B.
(C)
c6 tdrn I(1;
-1),.R
-
5.
C6
MA.MB
:
MIz
-
rtz <+
SMBZ
=
27
+ MB
=
3,MA: 9
=+
MH
:
6+ IH
-
4.
Ldp A

qta
M(7;3) c6
d
:
(A;
B), A2
+82
{
0
c6ch.I
mQt doq,n
bing 4:
A(r-T)+B(U-3)
:
0
d(I,L):4
+r
l3A+2Bl=2'/PJ@
fi
z4:0
ho{c
A:
-t2,8:b.
Phuong
trinh A :
y
-
3
:
0

hopc
-t2r
*5y
*
69
=
0.
Cdu V.
X6t I
dinh .4 cfra
da
giac
ddu, ta sd
t)m
s6
c5c tam
gi6c
tir c6
g6c
ti tai dinh .4.
D6nh
s6 cac
dinh tiep theo cria
da
gi6.c
ddu theo chiiu
kim
d6ng hd
lil
2,3,4, ,12.

Chri
f
ld v6i
hai
dinh a,
b mlu
la
-
6l
:
0
thi
ab
lb drrdng
ch6o
chinh
vh A.AaD
vu6ng
tai .4. Khi d6
tam giSn Aab
(2
S
"
<
b
<
12) lh. tir
tai A e
b
-

a ) 6.+ 8
<
a
+
6
<
b
<
12.
Suy ra c6
C!
cSch chgn
cdc
dinh
a,.b
(a*
6
vir.
b
nhQ.n 2 trong
5
gi5. ir! 8,9,10,
i1,
i2).
V6.y
s6
tam
giac
cdn
tim lb"I2C3:

120.