1. j
TRUONG PTTH CHUYEN NGOAI NCTI . Ep rm THUDAT Hgc
MOn: Toan - tctrOi D vd Al
Thoi gian: 180 phrit (kh6ng kC th<ri gian ph6t d6)
Cffu I (2,0 di6m)
Chohdms5y=x3+3x2-4.
1. Kh6o sdt sg bi6n thidn vd v6 OO ttri (C) cria hdm s6.
2. Tim m d6 phuong trinh (x +2)' =ffi cd dring 4 nghigm thpc ph6n bi6r.
C6u II (2,0 tti6m)
1. Giei phuung trinh: I * (
. (X+-3nl.
= 4COSI
cos x
coslx *
3n 4)
z)
2. Gieiphuong trinh: g+g.36_gJ?-. +3G>5(x€R).
Cf,u III (1,0 rli6rn)
t
Tinh tich ph6n: I = j
jsin2x+3sinx.cosx+i dx.
CAu IV
1l,O ai6iry
Cho hinh ch6p S.ABC c6 cl6y ABC li tam gi6c vu6ng tai A, g6c ABC bing 30"; canh
bdn SC"t?o vdi d6y (ABC) mQt g6c bing 60o vd SC: 4a..G ld trong tdm tam gi6c
ABC. Hai
mflt phrng (SAG] vd (SBG) r""g vu6ng g6c vdi m[t phing (ABC] Thhthf J.n .tt. rrmr,
ch6p S.ABC theo a.
C6u V (1,0 tli6m)
cho ba sd x, y, zld cdc so ttrilc ducxlg vi x + y * z =6 . chtrng minh rang:
#1 *Y" *T't+
y" +2 z'+Z x'+2
C0u VI (2,0 di6m)
1. Trong mit phdng toa d6 O"y, cho hinh chfi nh4t ABCD c5 diQn tich bing 6, duang
ch6o
AC c6 phuong trinh x +2y -9 = 0, dudng thang AB di qua di6m E(5; 5), A"#"g'th$"g"en Ai
qua di6m F(5; 1). Vitit phuong trinh c6c c4nh BC vd CD cria hinh chf nhAt, Ui6t Oi6L A
c6
tung dO ,2 hun 1 ua di€m B c6 hodnh dq i(III hun 3.
lcyn
2. Trong khdng gian v6i hQ tQa d6 oxyz, cho hai duong thang (dr): x*i y+2 z*2
) r -))
(d,,): x-2=Y-3-=,
z-4
_l vdmdtphdng (P): x-y+ z-6-0. Tim rli6m E thu6c (d2) sao cho
I
dudng thang (A) qua E vd song song v6i (d1) cat mdt rhlng (p) tpi F sao cho EF : 9.
ef,u WI (2,0 di6m)
c$i - ---' x-v
rre phucrng trinh:
'*' {:.-' : (x' Y e R)
t,or,--,,3*logvj(*y-x-y+t)=,
Her _
Tht sinh khdng daqc str dqrng tdi li6u. cdn bp coi thi kh6ng gitti thlch gi th6m

2. +.
TRUONG PT CHUYEN NGOAI NGU
oAp AN * THANc orEHl
oE rHr THU'DAr HQC
MOn: To6n - fn6i P
(d6p 6n - thang di6m g6m 06 trang)
Ciu Dfp rin Di6m
I l. (1,0 didrn) Khao sat.
(2,0 di6ni) y=x3+3x2-4
TXD: R D:
Su bi6n thi0n:
o)5
+ Gi6i han vd cuc vh gi6i han taiv6 cuc cira hdm s6:
limy--oo ; limy=+oo
x+-6 X-++O
BAng bi6n thiOnr y'= 3x + ox; Y'= 0<>x - 0 hodc
a
0
0 0,25
0
+ Hdm sd ddng bi6n tr0n mdi khodng (- oo; -2) vd (0; + m)
+ Hdm s6 nghich bi6n tr6n khoing (a;0)
+ Hdm sQ dat cgc d4i tai di6m x: -2; gi|tri cuc dai cira hdm sO ta y1-Z; : O 0.25
+ Hdm sO dpt cuc ti6u tai di€m x: 0; gi6tri cuc ti6u crja hdm rO ta v(O) - 4
-
. D6 thi:
+ Di6m u6n: y" : 6x * 6
y":0taidi0mx:-1 vlr
ddi ddu qua di€m x :
..
y" -i. ,A
-l 0,25
-+ ditim U(-1 ; -2) ld di6m u6n
-.i
+ Ciao di6m cua dd thi vdi truc
hodnh ld di€m (-2;,0) vd (1; 0)
+ Giao di6m cfia d6 thi vdi truc
tung la (0;-a)
+ E6 thi nhan di6m u6n ld tAm
-4,
dor xung
T
c pr: (x+z)'=l:gr- (l) ;Dk: {x+r ' I
lx-ll ltn>0 I
I
e lx -11(x +z)' :logrm (2)
I
e Dit: y : lx - 1l(x + 2)' co dd thi (C') I 0,25
y = log, m la dudng thang (d) cing phuong Ox, cit OV I

3. tai M(0;log, m)
. 56 nghi€m ctra phucmg trinh (2) liL sd giao diom c6 hodnh d6 kh6c I cua (d)
vd (C')
C6ch vE (C') suy tir (C):
4, x>1 o)5
*Co: y=lx-11(x +2)' =
fr'*3x2_
t-,u' +3x2 - 4), x<1
+ vc (c'):
" cifi nguy6n phAn d6 thi (C) ring v6i x > 1, duoc phAn d6 thi (C,)
. Ldy d6i xring phAn dO thi (C) irng vdi x < I qua truc Ox, dugc phAn d6
thi (c2). i:
,- (C): (C') u (Cz)
i /:
:
?
D6 thi: ;{l
,ti
I
.!1,
I
.'l
,l
',!
!t
i1 ..
'. I
i i :
! :!
';,1
t'o-
**".-.Fr+:r€* .?s -:F*:€
f --.-,...;:>
;
I
t
i
4"25
',
.a
:
?
''t' t
l,ll
'il I
I
i
. Cdn cir tr6n d6 thi, d6 thoa mdn y6u cAu dd bdi, khi vd chi khi:
0<logrm<4 1<m<16 (t/m) <) 0,25
il L (1,0 di€m) Gi6i phucrng trinh.
Pt<+ I * .l =-4sin[.*1)
(2,0 di€rn)
(1)
cosx slnx 4)
fcin v =r O
0,25
Dk: I <> sin2x * 0
i.cosx + 0
ln) +
Sln XJ COS X ( n
Pt (1) <> =-4srnl x+- |
sln
s1llx. cos X
:.c)OS 4)
/ r
I
3 sin X ft COI X
in>K+ COS =-4sinl x+-i lsinx. COS X
4)
J' sin l" !l)= -rrin[* * r)rin 2x
S x+-
4t) 4) 0,25
.( x+-n)l=0
r *. srnl
4)
(2)
<= sinf **11(O* 2sin2x)-o c>
4) ,E
SIN ZX (3)(tmdk)
2

4. o Giei phuong trinh (2):
(2)e <> *=-I+kru (tmdk)
"++=kn
44
phuo'ng trinh (3):
t-r_
l2*=-L+k2n I x=-1+k1
|
I
4 ^l
t-/l
8
l(rtF
l2*=2!+k2n lr=11+k1
l+Ls
0,25
Vdy nghidm cia phuong trinh ld:
2. (1,0 di Giai bat phuong trinh...
Edt: 36 =t, vi't4-x)o -+ 3Jt; >1 + t>l
J;+Bt-t' +t>5
16+8t-t' >5-t
[Js-t<o 0,25
I ln*Br-t2 >0
j1s-,=o
Lln*8t-t2
o Gi6i he (r):
frts
(l)<>{ " ei [t>s <> 5<t<9 (1)
[t'-8t-9<0 [.-1<t<9
o Gi6i hO (II):
(ll)e{ [t<s e{ [t<s lt<s
e{"- c> 1< t<5 (2)
l2t'-181+16<0 lt'-9t+8<0 [t <t<B
o Tir(1)vd (2),ta.U' [f <t<g <> 1<t<9
Ll <t<5
v6i 1<t<g <> 1< 3E <g
<>l<36<3'
<> o <J4 -* <2 0,25
l+-x>o
I
-1
[4-x<4
<+ 0 <x<4

5. m Tinh tich phAn..
( 1 ,0 di6m)
0,25
L
I
2 t
'.|-o*= Ilcot2x+3cotx+2 -J-a^
1
I
1
/t+lcotx+1+cot2x SlN- X
SlN" X 0,25
4 7
Ddt: cot x=t,c6 d(cotx) = dt 1,
-.1-o*
SlN. X
= o, -' r ClX= -dt
SlN. X
DOi cAn:
TE 1T
X
4 2
t
0,25
o C6: I = i--_l--(-at) = '1'--J-6,
It'+3t+2 it'+3t+2
=Jt,,*, - ' *rnl,*zll
*2)d,= J*dt-if-dt=tnlt.rll
o)5
= ln2- ln 1 - (tr,: - ln2) = 2ln2 - 1n3 = In!
IV Tinh the tfch
/t O rt;A'.''i cA. lan /-'1 J| /AT)/^1
vr. rltf'rJ/ flJ-,,
(sBG) r (ABC)
(SAG) n (SBG): SG
sG I
(ABC): c
SGIGC 0,25
vi sc r (ABC) : G

6. .|
n Ggi M ld trung diem AB 1
CM=iCC=3u 0,25
=
c Xdt tam gi6c vu6ng ABC c6 CBA = 30o :2AC
BC
AB = Ga; - ACt =
=
=
r ^,EAc
Xdt tam gi6c vu6ng AMC;
AC2 + AM2 = cM2 <> AC2 .[+)' =ru,
%,rec =;SG.t*rI| .., t. lgJja,
ZJ3a.!21!"-= 36aj
(dvrt)
V
+
( 1,0 di6m) Ap dung t ddng thric C6-si cho Uu ,O a""ng, tu
.o:
*x2+2- 3,s.-
2,-..,-/_t^ + x, + d >1W
^!{r.*'
+ 4) =1(*'
2 x._)*Vqx
+ Tuong tV, y' +2>)rW;
z2 +2.)r+n
s"y,^'
#* ;+* :*,:W . !W* .L
t ddng th,i
tE" q.^y =!{t"+ e tr"'f =!tr* + 2xy)
xy + xy)
+ rucrns t;'
{iyV =!t , +2yz) ; {z** ,!Or+2zx)
xY' yz2 zxt I-
" vi*7i- * +2< nLz(x + v+ z)+2("v- vz+ zx)l
e Md (" -y)' *(y- r)' *G_x), > 0 e> + y' + z, > xy +yz+ zx
^'
.t (i+y+ r)'>3("y +yz+zx)
<> xy + yz+ r*.{ - p
3
xY-
urr2 yz'
.,-2
1, zx'2
- 7i+fir*?i<-(t2+24)=4 0,25
. D6u bing xhy rakhi vd chi khi x : y : z:

7. VI 1. (1,0 di6m) Vi0t phuong trinh..
(2,0 di6m)
AC: x +2y*9=0 :>A(-2a+9;a),ut1
'2 B
. 4t --fra- 4;5 - a) ; AF: (2a- 4; | - a)
Vi AE.AF =0
0,25
e (2a - 4)' + (s - a)(r - a) = o
[a=3 (thoamAn)
c>5a'_ 22a+21=oel 't ? - D C
I a:- < (loai)
L s2
e Vdi a:3 -+ A(3; 3)
. AB di qua A(3; 3), E(5; 5) + phuong trinh AB: x - y:0
r AD di quaA(3; 3),F(5; 1)+phuongtrinhAD: x+y-6:0
o ViBe AB->toad0B(b; b)(b>3)
D e AD + tga d0 D(d; 6-d)
0,25
z 2 *:l
d
-> rrungdi6m cuaBD h rrq+,b
)
o Vi re ACn6nra.6' b*d*2(b-d*g)-9=0<>d*3b_.6
2 (z )
+ D(gu - 6;-3b +12)
AB2 = 2(b_3)2 ; N)2 :2(3b-9)'
Sor.o:$ 43 AB.AD=$ 43 AB2'AD2 =36
0,25
e z(b _t)' .z(ga -9)':36 c> (o -:)o -1
Vi b> 3 -+ b-3 > 0 + b -3 - i <>b: 4
o b: 4 -+ B@; $ ; D(6; 0)
I phucrng trinh CB: x +y -B = 0 0,25
phucrng trinh CD: x -y - 6 0 :
2. (1,0 diOrn) Tirn c6c di0m..
o
lx=2-t
Ptts cua (d2): I y:3 + t
I
0,25
lz:4 + t
' vi E e (d2) -) toa dqE(2 -t;3 +t; 4 + t)
f*=r-t+2n
" Vi (A) di quaE va (A) song song v6i (d1) ptts (A)' j y =3+t+m
=
lz=4+t-2n
. Vi F e (A) -) toa d0 F(2 -t+ 2m;3 +t+ m; 4 + t-2m)
r Vi F e (P) n6nta co'.2-t+2m-(:+t+m) +(++t-2m)-6=0
<> -t-m-3 :0 0,25
<> m :-t-3

8. > F(- 3t- 4;0; 3t + 10)
= PP (2t+6;3 +t;-2t-6)
r ViEF:ge(zt+6)'+(:+t )- + l-2t - 6)' = 8l <> 9(t +3)' = 81
,, t ,)
<> (t +3)' =q<>['
*3=3 [t=0 <)t
It *3=-3 [t=-6
t:0 -+E{2;3;D; F,(-4; 0; 10) (tmdk)
t: -6 -+ E2(B; 4; a) ; F2(14;0; -8) (tmdk)
WI
(1,0 di6m) o GiAi hd phucrng trinh:
[*-r]r
I
le''-A'':X"*y
0,25
llo*,-_,, (ty- x - y + 1) =
3 + logu,
[.-' -eY-r =(x -l)-(V- f)
- Ito*,._,3 + log* [(" - 1)(v - 1)] =
[x-t>0
rt [x>t
o Dk: jx-l+t ex+2
L(*-lXv-l)r0 |.v't
' He(II)el -(x-1)=e'-r-(v
'
[e--r
_ '"r
3 +2logr[(x -
Llot(" r)
o Xdthdms6 f(t)=e'-t v6it>0
*Co f'(t):e'-1
+Vit>0-+et>1->et-1>0
-_> l'(t) > o, vt > o
-+ hdm sO f ddng bi6n tr6n (0; +m)
r Md phuong trinh (1) c6 dpng: f(x-1) =f(v-1) (v6ix-1>0; y-1>0)
x-1 = y-1 (vi f ddng bi6n)
X:
ThO x :
y viro phuong trinh (2), ta co
logr*_,t 3 +2logr(x -l)' : J s> 1og,*_,, 3 + logr(x-1) =5 (3)
. Dit: 1og,(x-1): t,t+0 vi (x- 1) * 1
0,25