Goi H Id truc tdm tarn giiic ABC, M r,,ti N tuong u,ng la truns diOrn ag vi CH.. a Chring rninh ring A'MB'N ld hinh vu6ng..[r]
Trang 1so Grio nuct vA DAo r,{0
I}EN'TITE
DE Tt{I HOC SINII (;I0I L()I'9'I'RUN(; IIOC CO S()
t,{AN,{ IIOC 2(}t3 - 20t4
MOn: f},iX TIri'i gian: 150 phrit (kh6ng k0 lrhf t etO)
iri tri cira bieLr
1\
+) -Y -r+r
Ciu I (5 di0rn)
lu) I{Lrt r:t.rn biCu thric: A =
{
\
h) Cho l.r = 5 + viiJ Ti'h g
5- ll:x -1
Chu 2 (4 di6m)
:l
t/r'r/x -18 rtvi x)Jts
thirc:
-0
Giai he phuong trinh :
Ciu 3 (3 di6rn)
Tirn nghi0rn nguy6n duong cua phuo-ng trinh: r, + 1 , - I 3(x - l/) = 0
Cf,u 4 (3 di0rn)
CtLo hinh thang ABID (AB // CD), giao (i€rn hai dubng tlreo la O Dubr-rg rh3 g
qua O song song vdi AB cit aO va BC IAn luqr rai M vd N
a) Chung minh: Y*Y AB CD =r.
b) BiCt Soou = m2i Scoo = n2 Tinh Sorc, theo m vi n (v6i Sn,,u, Sc,rr, Sno.,, lAn luqt la dign tich tam gi6c AOB, di0n tich tam gidc COD, dion tich t['gi6c ABCD) CAu 5 (5 di6m)
Cho tam gi6c nhon ABC n6i ti6p trong dudng tron tdrn O vit IdB = 450 K6 c6c du"crng cao AA' vd BB' Goi H Id truc tdm tarn giiic ABC, M r,,ti N tuong u,ng la truns
diOrn ag vi CH
a) Chring rninh ring A'MB'N ld hinh vu6ng
b) Chring rninh ring A'B', MN, OH d6ng quy.
HET
l-t-Vx-JIs-Jx*Jtu
Trang 2SO GI,qO DIIC VA DAO TAO
TiEN'I]I1E
HTJoNG DAN crrAnn
xV rut cHoN rroc srNn (;ror
Lop e rnuNc Hec co sO ruAm Hec 2013 -zatl
lll0n: TO,.ilV
I
;hi l
:hri I
NEiiruiig-Rrit gon bi€r-r thu'c:
A:{ \ Vx- JtS -Vx+JtS /' 1rlt, /x' -lti rtr.i r ) vTs
Di6rn
(2,5d)
i;iii-chri
la
A2:2(x'-x'+ts)
Taco:
-A: :(Jx-vTs -J* *JT) (- - Ji,r)
o' : (* -vffi *x + J8 -z,F - rs;( rl=;
a' : (2" -2J.' -ls)( 6t- 1 8)
t2
-tr
fl -Jo
Nhung do theo gi6 thitit ra rh6y
tT
-\
o : (J.-Jr8 - J, Jtr)J* - J- - ru .9
+A:-6
0.5 0.5 0.25 0.5
0.25 0.5 1b Cho 2x:5 + 13 Tinh gi6 tri c0a bi6u thf;:
B:x5 - 5x4 * 4x3 - 2; - nx * 2023
(2.sd)
2x = 5* JiJ :)2x- 5 - .,fiJ
= (2* -5)' = 13
1y2-5x+3=0
B_
B:2014
0.5 0.5
0.5
0.5 0.5
Trang 3- :.r')' - I T("r'' - 9.1,' ) - +(t + -i.r.)'
)
,r+J.l'_r -) ' .r *3r,
I 5(.r - :r,)' - I 7(x' * 9,),'
) - u( t * i.,,)' -r+3r,* 3 *=5
Di6u
Ndu
=0 (/)
.r-iv
ki6nxiicdinh:r+3_i,.
f s4r;.r'' = o
x= r\.thi (/)o{ I <=
I
-\
| -J
I .)-.
I /)',
I ) vo nghi6m.
FN6u x *+3y Chia2
(;r +:y)(, - 3).) ra c6
vd phuong triph (l ) cho
:
* I :hePT(
l0
4-1 -
f-_
35
0.5
0.5
0.25
0.25
{,'t,
-3y)'
lrc* -3Y
^l/
-l
I x+3y+
t
- - ?,'
Edtt-n '/ ' x+3y th\
t
A
+Voi /=- n thi
J
=[,-i)[,.+)=o<+/=
-21v
Thay vio (**) Ta co :
-l8r'-l=5.6-r'= -8y 4'' i - -1
36
a
-J-0.25
Trang 4I
I l
I
I
i
I
I
I I I
I
i
I
I
Vo'i,r, = l=., = 1 ( thoa rnirn DKXD)
' -l(; 12
r = j thi t-]l'
= l= -r = 2.1, 1-5ay, vao (*,F)
-[,,, =
_i'
-) =
i
L)'=
y =:* $.tt, = r - , i.,tr
y=*-*J8s=.r-,-iJ"
vay hq phuong rrinh c6 4 nghie," ,
[?,+j, [*,-+),
(, lJss, I aGl,(, - !,6j,1 - a*;)
() l-i
0.25
0.25
0.25
0.2s
0.5
+ vo'i
CO:
1
1'
V0i
VCIi
'l-a
( thoa man DKXD)
( thoa mdn DKXD)
lt
- +
- Vti-5
2t0
1-a;*t
2 r0
3 Tim nghidm nguydn ducrng cria phuong trinh:
*' + y' -13(x -t)=0
(3d)
*'+y' -13(x -/)=o
<+ (x + y)' + (* - y)' -2.t3(x - y) +132 = 1
<+ (x + y)' + (13 -x -t y)' =169 =122 + 52
Vi x,yeZ* =0< x+y<13,0<13-x+y
| **!=lZ [ ,+y=5
=( fl3-x*y=5 v{ [13-x+y=12
[x=10 (x=3
-Jt
-/1 lv=2 IJ _ V<lv=2IJ
69
<13
1.0
0.5 0.5 0.5 0.5
4 cho hinh thang ABCD (AB ll cD), giao di€m hai duong cheo ta o
Duong thing qua o song song v6i AB cit ao vd BC lAn lucrt tai M
vA N.
(3d)
rI
o Vti'i r = -I. > \ - -,1 1tlrolr nrirr I)K\l))
z{+
I'
r 015 l
l
ii
Trang 5,d'l\/ jl1A' _ + - _
.4t] ct)
b) ili0t S.,.,,,, : n'rl; S(.r,, = ltl 'l'inlr
Sn,,,, S,.,.,,, S,\r(.r) liin lrrot Ii cliCn tfch
tam giiic COD di.^n rfch ru.giiic AIICID).
,,\lr(,r) thcrl nr r li n (r'tii
tanr giac AOI] di0rr lfch
I'linh vC
,'\ - ^\
Nr F- ->( -\ r
4a)
4b)
IVIO AM MO MD
laC()-=-.-= CD AD' AB AD
o - - MO MO AM + MD AD
TNONO lu(rngtu,ta + 1
t)t
/r\.,A,.\^., MO+IVO MO+IVO
(r)\,'a(l)su-! q_-)
SU\,ra 'CDAB _+ 2
0.5
0.5
0.5 0.25
0.25
Suou
=oB S.ro,
-oA.
snoo oD'Scoo oc'
oB oA Srou_Sooo
-:-_ oD oc s.nro sro,
= Sloo = m2 n2 1 S ooo = m.n
Tuong tU Suo = m.n V4y Souc, = m, + n2 + 2ntn = (m * n)2
0.25 0.25
0.2s
0.25
5 \-no ram glac nnon AIJU nQr ti€p trong dudng tron t6rn o vd
^
ACB = 45o Ke cric duong cao AA,, BB, Goi H ld t.rc t6rn
tam giitc ABC, M vaN tucrng irng ld trung di6m AB vd CH
a) Chfng minh r6ng A'MB,N ld hinh vu6ng
b) Chrine minh rins A'B' MN OH r{Ano n,,,
(sd)
5a Hinh vE
Trang 6
TacoMA,=MB-: !as il)
2
NA' : NB' : lcu (2)
2 Niat klrac ,i FiB= 45" ncn LCAA' va ,\BA,l/ vu0.g cirr
n€n LCA'H = LAA'B(c.g.c)
= CH = AB = NA'= MA' (3)
= MA'N = AA'C = 900 (4)
Ti.r (l), (2), (3) va (4) suy ra A'MB'N la hinh vu6ns (*)
l -l
i
l
I I
I
I
I
I
I
I
I
I
I
I
I
I
I
I I
I
I
I
I
0.,5
0.5
0,5
0.5 0.s
5h Do O vd A' thuQc dudng trung truc cfia c4nh AC
N0n 0l' L AC
= O,4'll B' H (5) Tucrng tu, ta co OB'll A' H (6)
Tt (5) vn (6) suy ra A'FIB'O ld hinh binh henh (**)
Tir (*),(**) suy ra A'B', OH, MN clt nhau tqi trung di6m
m6i duong
1.0 0.5 0.5 0.5