Đề-đáp án thi thử lần 3-ĐHSPHN

Dtt
TRUONC
DHSP HA NQI
rcr6r rHPT cnuvtN
CAu I
(2
di€m):
Cho
hdm
sii
y
=
NT THI
THU
DAI HOC I,AN
III NAN,r
ZOOS
Nidn thi,
To,in
Thdi
gian.lAm
bdi:
180
phrit
*
**.
x2- zmx+
m2
x-1
(l)
v

l. X6c dinh
tAt
ce cac
gi6
trf cira
m d6 ham
s5
d4t cgc
ti6u
tai
x
=
2.
'Jz.
Tim c6c
gi6
tri cua m d6 tr6n d6
thi crla hdm
sii
1t;
tdn
tei it nh6t mQt di6m
mA
ti6p
tuy6n cria d6 thi
tei
dii5m d6 vu6ng
g6c
vdi dudng thing
y

=
x.
Ciu
2
(2
di6m).
V
t. Gidi
phuongtrinh:
. aX aX
stn";
-
cos";
L
,+"i"-
:3cosx'
,,1
2.
Giai he
phuong
trinh :
v
CAu 3
(l
di6m).
Tinh tich
ph6n
: t
=
[/3

x2+r+.,/@Txf
r/Cau
a
(i
diem).
Cho tri diQn
SABC c6
g6c
AB'C
=
90", SA
=
fift
=
2a,BC= a.,,/3
v.d SA vu6ng
g6c
vdi
rn4r
phing (ABC).
Gqi M li
tlitim
trdn duong
thing
AB, sao cho
AM
=
2Md.
Tinh khodng
cdch tir tlitim B dffor

mp(SCM).
I
CAu5(l
<li€m).
Cho0
<a<b<c
<i<e
vd a+b+c*d*e=
L
Chung minhbAtdingthric
'
'4U"
+
be
+cd+de)
+cd(b
+
e-a)
S
+
,_
25
CAu
6
(2
di6m).
qo
l)
'l
rong

mflt
phing
v6i hQ tga
dQ Oxy, cho
tam
gi6c
ABC c6
ctinh A(-2;3), duong cho
CH nim
trdn duong
'/
thing
: 2x+y
-7
=0
viduongtrungtuy6n
BM
nimtrdndudngthing
:
2x-y+l
=0.
Hay vi6t phuong
trinh
c6c cqnh vd
tim tga
d6
trgng
tdm
G
cira tam

gi6c
ABC.
t
'/ 2)
Cho hinh
hgp ABCD.A'B'C'D'.
Tr€n ducrng thing
AC
l6y di6m
M
vA trdn
duong thing
C'D t6y diiim N
I
/cM
I
sao cho
MN // BD'. Tinh ti
tU
;'
Jcart
(l
di€m)
Xdc
dinh
tep
hsp cdc di6m trong
m{t
phnng phftc
bi€u

di6n cdc
s6
phric
z.th6a
min diAu ki6n :
lz+il
l-l
=
l.
I z-3i I
f
t*t
*y=4+,tW
ll,*'
-ztg2=te1+h
D4r kiiin
thi th* Idn
tdi vdo
ctic ngdy 18,19/4/2A09



:
DAF
AN
rovr
r,lr
mON
roAN
(Thi

thti'DH
IAn
III
- 2009)
CAU
I.
xz-2x+2m-m2
1.
(t,O
Oiem).
Tgp
xdc
dinh:
R\
{l}.
Ta
c6
y'=-
(x-1f
Gii
sir
him s6
d4t cgc
ti6u
tei x
=
2,
suy
ray'(2)
=

0 hay
4
-
4
+Zm-mz
-
0
<+
m
=
0 ho{c
m=
2.
2,
tad6u
c6
y'
J
#=+
y'
=o
<+x=
o
ho4c
x=2'
Mat
khric
y'
>
0 khi

x
e
(-
.o;
0)
u
(2;
+
co)
vd
y'
<
0 khi
x
e (0;
l
)
u
(l;
2)'
Do
d6 x
=
2 ld
tliOm cuc
ti6u
cria him
s5.
Viy, d6
th6a

mdn
bdi
to6n
thi : m:0
hodc
m =
2.
'
x?-2x+Zm-mz
2.
(l,0di6m)R\{l}.
Tac6Y' = 7;17
T6n
tai
tii5p tuy6n
cua
<16
thi hnm
s6 vu6ng
g6c
v6i
tluong
thing
y
=
X khi
vd chi
khi
phuong
trinh

sau
c6 nghiQm
:
x2-zx+zm'mz
(x-1)2
I
- -l
{=r
x2
-
Zxt1m-m2
=
-(x-1)2,x+
I
€
2x2
-4x*2m-m2
+
1=
0,x+
I
(*)
Do
phuong
trir,h
(*)
c6 it
nhAt
mgt
nghiQm

kh6c
1, nCn
ta
c6
hai
trucrng
hqp
sau
:
a)
Phuong
trinh
(*)
c6 hai
nhiQm
phdn bi-Ot'
hay
L'=4-
2(-m'+2m+
1)>0c+
2m2-4m+2>0
++m+
I'
(L'
=
2(m-
1)2
=
0
b)

phuong
trinh
(*)
c6
nghiQm
kdp
x *1,
diAu
niy
tusng
duong
voi
I
-
-a
a
1
(loa)-
'
T6m
lai' m *
1'
^l
e
-
"
txt
=
xz
=

1
Ghi
chir : Niiu
thi sinh
giii
bdng
c6ch
:
[r,orrioo
trong
d6
f(x)=2*z
-
4x*2m'-
m2 +
1,
thitr.'
0,5
di6m.
CAU
il.
l.
(1,0
c1i6m).
Phuong
trinh
dugc biiSn
d6i
thdnh

:
ci{ -
."ixr
+,i,{.,o{)
=}
z+rin")"or*
*=Gi{
-
.or)tr
+1sinx;
=!r*sinx)(co{
-
'i"iAi{
*
'oJr)'
.
V6irinl
-
"or|=0c+sin(|
-fl=0<=+x
=|*lkrc,kez'
22La'z
t
t'

x
x. x
x
3
, r ^

r
Voi
r
+
jsinx
=-i(t+sinx)(sinj
+
cos)<+sini
+
co5
=-;(ptndyv6nghi€m).
VAy,
nghiQm
cira
phucrng trinh
ld
'
*
=
;
+zkn,k
eZ'
w.

2
.(l,Odiem). DiAuki€n
x * 0,
y>
-
2 .

Phusng
trinh
(2)
<+
lxl =
4
+
2y
.thay
vio
pt(l)
ta dugc
:
4+3Y=
q+JfrT
{+3Y=
't7+'z
-
.(
v>0
7
:
tv,
;;=ey,
o
Y=;
+x=t5'
fx=S ft=
-5
vay

hQ
phuong
rrinh
c6 hai nghiQm
,
l,
=
*
'
t
o
=
1.
cAu
rn.
(
l,o dii!m).
Tac6
,=
[/t
,G
xdx
-,'o
6qnffi
xdx
DAt r=
,@-*z
ql
thi
dt=

#|v6i
x:0thit=l;
v6'i1=y'Jthit=2.
Khi d6 t:
I:#
=
frr+r1*aqt*
1)
:fr
a
* t)+,11,:zbf3
-
.lz).
cAu
ry.
(
1,0
di6m).
Do M
=2W,
n€n
khoang c6chttrA
tliin mdSCM)
beng
2
lAn
khodng
cSch tu B d6n mp(SCM)
.
Tir

gia
thi6t ta
suy ra AM:
+,
BM
=
+
,
3' 3
arlsl
1
CM::. Ke AK
l- CM, thi
CK J- mp(SCM).
3-'^
Do
goc
ffi
=
90'+
Fm
>
90o n€n
tti6m M nim
giea
CvdK.
Ta c6dAKM
-
AcBM
=+

AK:
tt=l"
=
n"13.
cM
^lTl'
Ke AH
J-
SK, thi
AH I
mp(SCM)
vd AH
ld khoing c6ch tu A ddn mp(SCM)
tL1131
43+^/7a
Taco:
;;;=Asz
*1;;=
q^r*&=G
=
AH=
E.
V{y, ktroing crich
tt B
ddn mp(SCM)
bing
+
-
{43
CAUv.

(
l,0rli€m).
Ttgiathii5tsuyra
0<"s
i.e6tdingthricduscbiiSndoithdnh:
-fL/^
a[b(c+e)
+d(c+e)] +cd(b+e
-u)=
*
o
a@+d)(c+e)+cd(b+e-a)S
+
2S
x2+r+.,ftr+xf

Theo
bAt
ding
thtic
C6-Si,
ta
c6
:
) ,
a(b
+
d)(c
+
e)

+
cd(b
+
e
-a)
=
^
(ry=)'*
1.*o*!*._")3=
a(r:a)z
*,'=}|,',
Xethdms6:
f(a)
=+*ry,vdi
0<as
f
.rac6
f(a)=frf
-
5a2-4a+1)>0,vutto;f)'
Suy
ra
(a)
d6ng
bicn
trcn
(0;
Jl
*
it"l

s
(;
)
=f
tan'*)
cAu
vt.
(
2,0
di€m).
-
l)
(1,0
di6m).
Dubng
thdng
chira
c4nh
AB
vu6ng
g6c
vsi
CH
n€n
nhen
vectq
il(2;
1)
lim
vecto

chi
piiuong'
Do
d6
dudng
thing
AB
c6
phuong
trinh
:
x
-2y
+
8:0'
Suy
ra,
tqa
d0
diem
B
ld
nghiQm
criahQ
phuong
irinh
:
|
{"-r,
*

I
=
o
*1i,.=_3-B(2;s).
/
l2x-Y*1=o
(Y=
^Yr
-'
-^
*-'-^
'+;A
'-?
Et
thu6c
BM'
Gqi
C(x;
y)
thuQc
CH,
suy
ra
trung
di€m
c0a
AC
Ie
M
t7"',

'
(
2x+
v-7
=
o"
o
*
[;lf
+c(:;r).
Tqa
d0
cria diSm
Cli
nghigm
cria
hQpt,
\,
*
-
#
*,
Suyra,
BC
:
4x+y-
13 =0
vd
AC
:

2x+5y-11
=0'TrgngtdmG(1;
3)'
2)
(1,0
di6m).
D[t
EA=
d
,EF
=E
'EC
=
d
'
Tac6
EF
:
d +6+
d,
vi
MN
i/
BD'
n€nffi
=Ufr
v[y
Mfr =
kd.
+tci

+
Pt
(1)
Mat
kh6c
Mfr =ft
+7d
*Vfr
=
nVt
+fr
+
^V6
Trong
d6
At =
i
- d,7?
=6,
c6
=
d
-E'
Suy
ra
ffi =
n(c-
-
d)
+E

+
n1d
-87
:
(m
-n)
d."11
-m)
6
+
nt
(2)-
.
(m-n
=
So
sdnh
(l) ve
(2)
ta
c6
hQ
phucrng
trinh
:
|
1
-
m
=

t n=k
t4
CM
1
vQy
tvtc
=;
AC
h"Y
1;
:
J
CAUvll
(
1,0tli6m).
Df;t
z=x+yi
+
z-3i=x+g-f)i
vit
z*i
=
x*(y+l)i
Di€uki€n:x*0vdY*3-
D6
dang
chimg
minh
du-oc
tinh

chAt
l:l:
#
Suv-
l#l=l
elx
+
(v
+1)il2=[x
*
(v-3)il2
e.x2+(v+l)t=x2*(v
-3)2-ev=1'
K6t
lu{n:
Tip
hqp
c6c
di€m
trong
m{t
phdng
phrlc bi6u
di6n
c6c s6
phfc zthbamtutdi|u
kign
:
l#l=
ttno"*^fur=t

-
k fk=713
t-lm=2/3
(
n=1/3
,,@