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(1)TRU'ONC DIISP I'iA NQI
TRU'cr.,rc THPTcrquv0ru - flHsi)
Dtr TFII.rrNti DAI F{ec r-AN trtI zuenq zorz M6rr thi : TOAN
Thdi gian titru biri ; lB0 plnit, khong ke thc)'i gicrn phil cli
tri nAo c&a m, dLLo'ng thang )'= - x * n't
CAtr l (2,0 di?ut )
2x-1 t
Cho hAm so y: ;1
L l(hao sdt str bi6n thi€n vd v0 dd th! (C) cfra hdm s6
2 Coi 1ld giao didm hai clLro'ng ti$rn c6n cira (C) Vd'i gid
cit (C) tai hai di6rl pli6n biet A,B vh tanr gi6c IAB d€u. C6u 2 (2,0 dient)
l Ciai phLlong trinh ;h - ( cosx + sinx.tan j ) =
./rr\.fi
srnlx-;/+ cos (;- *.1
2 Tim citc grhtri cira thanr s6 o d6 phLrong trinh sau c6 dilng hai nghi6m phdn bi6t :
a f:-' ;
log3x" -n.l togzxu+a+ l:0. C6u 3 {},0 dietn }
-n 2sin2 rI - x) j'inhticir ohdn t: fo + 4"
' ; t.) coszx
Cdu 1, (1,0 iliitn )
Tf'cliQn ABCDc6c4nh AB:6,canhCD:Svdc6ccanhconlai bing 174.HsytinlrdiOntich
nrdt cAu ngo4i titip tri'diQn ABCD.
C6u 5 (t,0 diAnt )
Clioc6cs6cluro'ng a,b,c,m,n,p thdamdn ctf nr:b+ n:c+ P:k
Clrirng nrinh ring : an-r htrt * cm < k2
Cdu 6 (2,0 diAm)
I Cho cli6m.M(0; 2) vd hyperbol @ , + -+ :1 Lap phuro'rrg trinh duLong thing (r/) cli qua
5-di€n A4 cht (m tai hai cli6m phAn biQt A, B sao cho MA:;Mtr.
2 Trorrgkh6ng gianOxyz,chorn{tcAu (.}, x'+ y2 +22 +6x-2y-22- l4:0.
- ', !
Vi6t phu'o'ng trinh m{t phdng (P) chfi'a truc Oz vdr cht nrdt cdu theo mdt dLrirng tron co bdn l<inh r : 4.
Cf,u (t,0 diim )
Ciai he b6t phLlo'ng trinh
COSX
-''-13' 't '
4il(i[{,$d5r:r,.,'&
@/a-:"r
( tog{z - xz) < 0
1
lx6 + 4(t - *213 ;' 1
r
tf
H6t
(2)rlAp Ax - TFIAI{G B}BM nur tnU BH LAN tll - nAtvq zorz
N6ua+ l=0 <+ u=- r+ttrltrothdnh tt -2t=0* [l= ! troail
N6u a+- I,khi d6r=0kh6ngldnghigmcia(l).DCpt(l)cirdtngniQtnghiQmcluongthi
*)Trrdng hfp L Pt(l) c62 nghi$mtr6i diu <+ a+ I <0 e a< - l'
t) )q
il )i l (1,0 aliini Hoc sinh ttr
2 (1,0 dilm) 7int m
I
Q ttidni
t-tJ,O aiim), Giai Phuottgtrinh
Didu ki0n : cosx 10, cc,s j #
r sinx.sinl l)hrxrnc trinh dd cho e ; ( cosx +
# )
=
cos.x LU5;
1 ^ ?X
<+ -;- - ( co:i)i + Islll ;./ -aoszx
* l- r^.L-(l-2sin2I+ 2sin?l) =Vs.,un* <+ tan2x - J3.tanx =
I tanx=0
I tanx = VJ
lx=kn
lx=i+t<,t (kez).
So siinh vdi diiu kiQn nghiQm cira phr'rongtrinh ld : x 2h, * =I+ lm'keZ'
-/ft\-[
sinlx-;J+ srn (; + xJ
-;;;-zsi n x.cosl
c osx
0,50
0.50
II
(2 ttiim)
S phuo'ng trinh sau co hai nghiQm phdn
2x-1- ( x+1
bi0t x1.x2 : l_f:-x*tn <+ [xr+ (f-nr)x*m-1=0
lzi trLrng di0nr ctia
'lIJ'
r(hi c1(r ;;, = - -r, + nt ( i = t' 2; vit H(T, #l.m : t #' l-l' 7E (*' - x1 i xr - xz)
(lA=tB (lA2=lBz
\/.-r8diu *
trn =E-AB (=
\tu, =1rc, (**) Tacd 141 :lN e (x1 -x2)[x1 +x2-(m-l)] :0'
l)o.r, + \2: tl: - I r,On ding tl.rirc ndy clirng vdi moi rrr thoa (t)
Tac6 (+*) * gY:t.t*r-xr)2 e (nt -3)2:3;(xr+ xz)2-4xrx:l:3[(nr - l)'?-4(n - l)l a t,r' - 6n+ 3 = Q e nt:3 trG Citc gi|tri ndy cLran ddu thoamln (*)'
Dip sii : m =3 !,,18
Z- W iiiml Tim gia tt"! eia tham sd a "'"
Grtign : log3x8:0 <+ lxl> l
pT<+ log3 xr+Za,flQlp +a+ l=0.DAtt=Jtg,y" Z0.Ptdacho,trothdnh t2+2at+a+l=0 (l) Nhanx6t:V6i m6i t>0, pt v4f,-g.F=t <+log3x2=t2e x2=3t' (* xr.z=+JAAthoaminxr lxz.
Suyra ptcldchoc6d0nghai nghifmphdnbi€tkhi vdchi pt( l)c6dfingmdtnghiEmkhongdm'
(3)1 2/3/201 2
orrr itrsttoTtzpt(,1.,,,'ri***n*-;-*[f:==.i,=9out'=0o{n,_i, tr=s€,r=r:'(
' t -'fi
l)apso:it.-l.a='t'.
III (t itidn)
L (1,0 di2ntl 'finh tich phdn
"I r- cosl|- zx; "I r- si.zx ,l (cosx - sirrx)2
l-aco - l l-u r0 cos2x ,''- tlx= J0 le '',"'""ilr=l-6-rlr
cos2x - ' J0 (cosx - sinx)(cosx + sinx) '' 0,5u
r:cosx-sinx r:d(cosx+sinx)r , t rsinxllf :',.,G*t
t=Jo'.*-*.i,*ut=Jouffi= lnlcosx ,lo z
0.50
IV (t tliim)
(1,0 diAm) Tinh diQn tich mdt cdu
Tlreo giti thidt DA : DB = CA = CB = ,[74, tarn gi6c ACB cdn n€rr tdm I cua dLrorg tldn ngoqi ti6p AACB thuQc cludng cao CE Ta c6 ACAB = ADAB
d6 EC = ED + ACED cdn + ctuirng cao EF cria ACED lA du6ng vudng g6c
chuug ctia AB vd CD d6ng thdi ld trung t4rc crha AB, CD Vdy tdm O hinh cALr ngo4i titip t['diQn ABCD nim tr0n EF
0,5u
Tac6 EF"= ED,_ DF" mir ED' = 74 _() = 65 =r E.F,=65 _ l6:49 =+ EF =
Mat l<hic tlF: OE + OF: .l pt - 9 +
^/ n2 - te
7
Ciiii phurrn-e trinh ./Rt : 9 +,'l R' =G : 7 dugc R : 5 Do db clien tich m4t cAu la S = 4nR2 = l00n (dvdt)
0,50
V
(t itidnl t
(1,0 diim) Chting minh riing
l'a c6 : k' =(a+ mxb+ n)(c+ p)= abc mnp+ abp+ can + anp + bcnr + brnp r cmn
M4tkhdc k(an + bp + crn): an(c+ p)+ bp(a+ m)+cm(b+ n): abp+ can + anp + bcm + bmp+ 0n1n
Vril l<r= abc4 mnp+ k(an+ bp+cm)> k(an+bplS;r1)
<+ l<2 > an + bp + cnr (clpcm)
t,0{)
VI Q rlidm)
l (1,0 diAm) Vidt phuons trinh dud'ng thdns
Nh{nxdt:DuongthangdiquaM(0:2)songsongv6'itr,ucCrykh6ngcii(fl.-Khi d6 (d) : y : /.x + Toa dd giao di6rn cria (d) voi (f0 ld nghi€m c(ra h6 phrcrng rrinh :
ti =H.;4 + 14k2- l)x2+ l6kx+20:o (l).
De((iln(ff) =\A,.Bl<e(t)c6hainghiemphinbiet ,' (4lcz - 1+ 0 ( t' + +! " t A,> o *+ iro _ ,u,.rt, o*
.l
k++--''o (2) tkt<l:2
0,50
l(hi d6 ( I) c6 hai nghiQm phdn biQt x1 x2 lir hodnh dd cria zl 5.- s
Titdi0uliicrn ly'l=rMB *X' =-x2,khi cldtac6:
{1",*x,=-#u^f", ah _ 36 t2
J g"-t- 20 I, s^z:4krfl o1 1xz-n[z-l-t- Lz-+Gkr-r)r: -tc+l(=+L(thoanrinl2)) Vdy c6 hai dudng thing thoa mdn bdi to6n : (dy) : y = x + 2, (dz) y = -x + 2.
(ra
lx *x.
vaiB,thoamin J ::'^'
I tr.t, = 4kr_1
0,sa
2
r,
(4)12/3/2012
TII|aiil,,llii! l'lttrotz trinlt ntd! pltattg "
vlr|6r', (5);r5 t6'-r-r l,i G 3l lr l) vd bin l<inh R : 5'
(iQi //(.r : b: o) lir hinlr chi0Lr cria / ldri mat plrfng (P) Mat phang (/') chila truc o n€rr o6 vcct0 phiip tr-ry0rr :.- ,,r,ra ii=(-b;n;Q) voi a:r br+0.
rt lfr,OHl.trong d6 i< (O: O: l) vd oH (a: b; c)' SL''
-Suy phucrng trinh m[t phdng (P) co ci4ng : - bx + a)' '= 0' VI
(2 rtiim)
ffin c6 b6n kinh r: 4 * tH: \f R2 - 12 : i' l3b + al_
= <+ 9b2 + 6ab + .f = 91,t2 + 9a2 Nlrtr vdy khoang c6cli tir i den (P) bang J € ffi,
la=0
c+ 8a2-6ab=oel-=11 Lu- q"
V{r} c6 hai m4t plrdng(P) lin luotcir phuongtrinh ld: x=0 ' 4x-3y:0' (1,0 tliznt) G,"L!19
sl.
l.
+ 411
e [0;
':l<+lxl
voi lsls
dnh g(t) = tl ).
z +l=-;3
0<+2-x'
411 - xrlt
f1x) tro th
tt_ -2 e l,_
[r-:l<
xt'+
I rhi
-0
Ta cir log' (2 ,, - x
Xdt ht\ur s6 tlx) =
Dat t=xt,0<t<
o'(tt=0 et?:4(
VII
Q,0 didnt) ra co g(f) =f *tol = a, g(r) : I'
Su1,rarzing(l) =!+ ntinf(xf =f 'Suyrabdtphuongtrinh x T6nr l4i : 1'4p nghiQm cua hQ b6t phucrng trinlr ld S : [- l; l]'
6 +
4( I - *'l'> i nghiQm dirng vx€ [-l; l]'