Tlnli di€rr tich tarri gi6c ABC.. Tim toa dQ tAm vir b6n kinh cria drLdng trdn C ld giao crira m{t ptring p vd mat cAl S... oAp AN:- THANG DItMrm trrrl nn r,Ax rntl gar NAna zorr .I z dt
Trang 1T'hdi gian tdm bdi : 180 philt, kh6ng k€ thdi gian phdt di
CAU 1 ( 2,0 di|m )
| 2x+L
Cho hanr s0 V =
-.
- x-2
l'I(hios6tsLr.bi6nthi0nviv€d6th!(C)cirahims6.
2 Tim t6t cd c6c gi6tri cttam d6 cluongtiring y = m(x-2)+ Z cit AO thi (C) t4i hai di6m phAn biQt
A, B sao cho doan AB c6 dO dei nho nhAt.
C0u 2 (2,0 diilm)
t Giii phuo'ng trinh : sin2x.(l + tanx) : 3sinx.(cosx - sinx) + 3
2 Giai bAt phuong trinh: 3sx-, -4>5.3sx-,
Cffu 3 ( 1,0 di6m )
-rinh tich phan I =;€$o*
Ciu 4 ( 1,0 rtiAm )
Cho hinh lAp phuo'ng ABCD.A'B'C?D' c6 clQ ddi canli bing a vi di6m N4 thu6c canh CC' sao cho
1c
CM = T 3 tUat phing (a) di qua,t, M vir song song e o vdi BD chia khOi l4p phu'o'ng thdnh hai khoi
da cli6n Tinh thO tich hai t<tr6i Aa diQn d6
CAu 5 ( 1,0 diem)
Ba s6 duo'ng thay d6i a, b, c till0c doan [a, F] md B - o {2 Chri'ng minh ring :
GETf +/[611a1ffi1 1> a+b+c.
C6u 6 ( 2,0 dient)
l Trong mdt phing toa dg Oxy, cho tam gi6c ABC c6 C( I : 2), hai rh-rong cao xuAt ph6t til.A vd B
iin luqtc6 phuongtrinh ld x * y : 0 vir 2x-y+ I =0 Tlnli di€rr tich tarri gi6c ABC.
2 Trongkhonggiantoa116 oxyz, cho mf,tphing(p)c6phuo-ngtrinh: \-zy +zz+ l=0vdrn[t
cAu (S) c6 phuong trinh : x2 + y2 + z2 -4x + Sy + 6z+ l7 : 0.
Tim toa dQ tAm vir b6n kinh cria drLdng trdn (C) ld giao crira m{t ptring (p) vd mat cAl (S)
C6u7.(l,0diim)
Giai h9 phuong trinh :
f*t+xyz=40y
ty'+xzy=10x TI6t
Dqr kiiin ki thi thfr Dgi hgc tfrn thft 3 sd itwqc fi chftc vdo ngdy X\,Z0/S/2011
Trang 2oAp AN:- THANG DItM
rm trrrl nn r,Ax rntl gar NAna zorr
.I
(z dtim)
l (1.0 iti6ml Hoc sinh tu ei6i
Eucrng thing y = m(x-Z) + 2 ciitao tni (c) tai hai cli€m phdn biQt <+ pt '4 = m$-2) + 2
c6 hai nghiem phdn biQt <+ pt mx2 - 4mx + 4m - 5 = 0 (*) c6 hai nghiQm phdn bi€t kh6c 2 0,25
(m*0
(=) lo'=4m1 -m(4m-s)>o om>o
Gi6 sfr A(xr,y,), B(xz;yz) trong d6 x1, X2 lA hai nghi$m c0a (*)
Khi d6 yr = IIrXr - 2m + 2 vit y2= ttlX2 - Zrn + 2
Ta c6 AB2 = (xz - 4r)2 + (y: - yr)2 = (xz - xr)2(m2 + 1) = [(x2 + x1)2 - 4xrx2](m2 + I )
0,25
= tr6-{Tj)l(m' + tl = 3P=T =40 vdi msi m > 0 Eing thric xdy ra
khi vd chi khi m : l VQy, v6i P: I thi AB ngfn nhAt Uing .86-.
0,25
II
Q dianl
l ( 1,0 iliAm) GiAi phuong trinh .
Di6u kiQn : cosx # 0 Phuong trinh dd cho tuong cluong v6i pt :
ffi i un* + l) I T(l-Pnx) + ft er tan2x (1+ tanx) = 3tanx(l- tanx) + 3(l+ tan2x) 0,50
€ tan2x (l+ tanx) = 3(liianx) * [nXI;= ,t
, ;.
oreu Klgn Dar toanJ.
[x = -r+ kn
e | + glz)(th6amdn
lx=+-+kTr
2 (1,0 ilidm) Giai bet phuong trinh
DiAu ki€n , *+';.'
BAt phuong trinh cl6 cho tuong iluong v6i bpt ' S;= - 4 z s.32-# 0,25
D{tt=3s:l= , t>0 Bpttr€ntrdthinh t'?- $-45>0 =+ t:9(dot>0) 4,25 (+ lffi:9 6 s*-r *+3-27 zz oEaxS s Eripsd : xell;f,1 0,50
III
Q,tlidm)
(1,0 iliAm) Tinh tich phdn
Tac6 l= - f,rr"ffi d 1 = - 1 1ntr1-52 lf [*i d(rnVTTF)
= tnfi - *'"f lr"j#*=#r"z*1rrfra*. 0,50
1
Trang 3=+dt=fta,=(l
Vdi x= I thi,=1,'*=y'3thi t:l * l=1f,at=,li 4raLL =*.
E6ps6 ' tt=f-21n2+4 2J3
L2-0.50
U
Q iIiAm)
o (0,50 diA@.Dr,rng thitit diQn c0a m{t phing di qua A, M vA song song v6i BD
Gqi O=AC o BD, O'=A'C' n B,D' vd I=AM nOO' eual k€tluongthEngsong
song v6i BD c8t BB', vd DD' lan lugr t4i K, N Khi d6 AKMN la tni6t diQn cAn dgng.
DAt Vr = Ve.scrrx-f Va.oo"rN , V2 = Vnaco.e,a'c,o'- Vr
-^-^ ? - oI AO 1
c(0,50i1i6@.Tac6 ff=#: I + DN=BK=Ot=;a"=;.
Hinh ch6p A.BCMK c6 chi,iu cao li AB = 4 I
tl6y ld hinh thang BCMK
1 ^ Ot BC(BK+CM) a3 Suy ra Ve.ecrpr = -AB.Ssgy* =; -T =i B'
Hinh ch6p A.CDNM c6 chidu cao ld AD = a
d6y lA hinh thang CDNM
suyraveco*u=lRo.s"o"n 33t26 =T cD(NP+cM) ={ K
V6y, V, =: , r33Vz =
a-Ir0
V
Q iliAm)
l Q,0 ilidm)
'1 Tirgiethi6tsuy ra la-blSB- s.aZ +(a-b)t< 4=r(a+b)2-4ab< 4 0,50 Suy ra a* b < zr/ffi, tuongtgta cfing c6 : b + c S 2rlIEEE, a* c szlffi
VI
(2 itiim)
(1,0 iliiim) Tinh diQn tich tain gi6c
-Duong tlrdng BC c6 vecto chi phuong il ld vecto ph6p t
il =(l; l) Phuongtrinh cta Bc : [; : I ll tu, ra B(l+t ; 2+t) B thuQcduongcao xu6t
ph6ttirB n€ntgaclOthdamdnphuongtrinh:2(l +t)-2-t+ l=0=rt=- L Vgy B(0; l).
ruong t6 phuong trinh AC
' [; : r*-?' vd A(- 5; 5)
0,50
Tac6: BC=fr
Eudng thing BC viiit ve d4ng tdng qu6t : x - y +
Gei AH Ii duong cao, ta c6 AH = -# = #
-0.
t9 Vay Sasc = IAH.BC= 1.
0,50
Trang 4MAt geu (S) c6 tdm IQ;- 3; -3) vd b6n kinh R = \8.
I(hoangcrich.tir1-di5nmp(P)ldh=w=l<R'=G,n6nmp(P)cat
.l
rn{t cdn (S) theo mQt <tucrng trdn c6 b6n kinh bing r = r/ffi = 2 .
g6c vdi mp(P) vecto chi phuong ctia etulng thing d li vecto ph6p tuyiSn cira mp(p), n€n
il;=(f ;1:2)vd phuongtrinhc0a o,{;:?{!r, lz=-3*Zt Dod6 K(2+t ;-3-2t;-3+zt)
Tga tlgtdm K cfra(C)"th6am6n phuongtrinh: Z+t+6+4t-6 +4t+ I =0 =r a =-l
qTLL
Vfy tdm r( i ; -; '-; ), b6n kinh r = 2
0,50
N6u x = 0thi y = 0 =+ (0 ; 0) ld mQtnghiQm cfra hQ phuongtrinh
Niiu x # 0 DAt y = tX, khi d6 hQ pt trd thAnh :
fx3 + xetz = 4otr fx3lr + t2) = 4otx
tt3x3+x3t=10x - [xr11l*t)=16* (+
,i !*r(t* t2) = 4s1 f *' = # (1)
t *'(t' * t) = 1e H l*HP = to (z)
Tir(l) + t>0,k6thgpv6i(2) :+ t=).rnuv t= j uao(l)ractuqc x=*4 + y=.-2.
Drip si5 : H€ pt c6 3 nghi€m (0; 0), (4 Zi , F 4;-2).
(1,0 Gidi he phuong trinh