1. The document provides information about a math exam, including the exam time of 180 minutes and 6 questions ranging from 1 to 2 points each. The questions cover topics such as solving equations, finding roots of equations, integrals, geometry problems, and systems of equations. 2. The responses provide solutions to each question, showing the steps and reasoning for obtaining the answers. Solutions include solving equations, finding integrals, using geometry relationships, and solving a system of inequalities. 3. Diagrams and calculations are shown to visually depict the solutions to the geometry problems involving shapes, angles, and areas.

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$rn c6n cira (C). Vd'i gid
cit (C) tai hai di6rl pli6n biet A,B vh tanr gi6c IAB d€u.
C6u 2. (2,0 dient)
l. Ciai phLlong trinh
;h - ( cosx + sinx.tan j ) =
./rr.fi
srnlx-;/+ cos (;- *.1
2. Tim citc grhtri cira thanr s6 o d6 phLrong trinh sau c6 dilng hai nghi6m phdn bi6t :
.. a f:-'--;
log3x" -n.l togzxu+a+ l:0.
C6u 3. {},0 dietn }
-n 2sin2 rI - x)
j'inhticir ohdn t: fo --+ 4"
' ; t.) coszx
Cdu .1, (1,0 iliitn )
Tf'cliQn ABCDc6c4nh AB:6,canhCD:Svdc6ccanhconlai bing..174.HsytinlrdiOntich
nrdt cAu ngo4i titip tri'diQn ABCD.
C6u 5. (t,0 diAnt )
Clioc6cs6cluro'ng a,b,c,m,n,p thdamdn ctf nr:b+ n:c+ P:k
Clrirng nrinh ring : an-r htrt * cm < k2.
Cdu 6. (2,0 diAm)
I . Cho cli6m.M(0; 2) vd hyperbol @ , + -+ :1. Lap phuro'rrg trinh duLong thing (r/) cli qua
5-
di€n A4 cht (m tai hai cli6m phAn biQt A, B sao cho MA:;Mtr.
2. Trorrgkh6ng gianOxyz,chorn{tcAu (.}, x'+ y2 +22 +6x-2y-22- l4:0.
- ', !
Vi6t phu'o'ng trinh m{t phdng (P) chfi'a truc Oz vdr cht nrdt cdu theo mdt dLrirng tron co bdn
l<inh r : 4.
Cf,u 7. (t,0 diim )
Ciai he b6t phLlo'ng trinh
COSX
't '
@/a-:"r
( tog{z - xz) < 0
1
lx6 + 4(t - *213 ;' 1
r
H6t..
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2. rlAp Ax - TFIAI{G B}BM
nur tnU BH LAN tll - nAtvq zorz
. N6ua+ l=0 <+ u=- r+ttrltrothdnh tt -2t=0* [l= ! troail
. N6u a+- I,khi d6r=0kh6ngldnghigmcia(l).DCpt(l)cirdtngniQtnghiQmcluongthi
*)Trrdng hfp L Pt(l) c62 nghi$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 # 0 .
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 = 0
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 khi 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 min (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 khi pt( l)c6dfingmdtnghiEmkhongdm'

3. 1 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 ,''- tlx= le '',"'""ilr=l-6-rlr- r0 cos2x J0 cos2x
- ' J0 (cosx - sinx)(cosx + sinx)
'' 0,5u
. r:cosx-sinx r:d(cosx+sinx)r , t--... rsinxllf :',.,G*tt=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 do
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 ra 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 1 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 + 2. 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 o1
-t- Lz-+Gkr-r)r:---tc+l(=+L(thoanrinl2))
I, s^z:4krfl 1xz-n[z-l
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 ra 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_
= 3 <+ 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 htur 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]'