1. www.VNMATH.com
KV THI KSCL TTII DAI Hq}C xAIvI IOTIL,AN rr*tI l
XA
nn rnr u0n : ToAr, KI{or A
Thdi gian ldm bdi; 180 phtit, kh6ng ka tha gian phat di
nA tni g6rn: 02 trang
*.||',' . ffi-r*6 gh-".%
fr€' .
'|.
psAw cguNc cno TAT ce Tui swn ( 7,0 arcm
1
Cdu I. 12,0 dihml:
1. Khio s6t sp bi6n thi6n vA v€ dO rhi (C) ctra hdm s6 y _2*+rl .
x-1
2. X6c dfnh m d6 duong thEng: y = x -2mc6t (C) tai hai di6m phAn bier M, N sao cho MN
:4112.
Cffu II. (2,0 iti6m):
1. Giai phucrng trinh: cos2x*5: Z(Z-cosx)(sinx-cosx) :
2. Giai phuong trinh: x'- x * 1006..[+ 8048x = 1006
ciu trrl. (1,0 di€m): Tinh gi6i han : I : ,' **o - J 20L?I - -204i
x+l X-l
c6u IV. (1,0 ctihm): cho hinh ch6p S,ABCD cir d6y ABCD ld hinh chfr nh6t v6i AB : a
AD : 2a . Canh SA vu6ng g6c v6i mat ph&n g diry, c?nh b€n SB tao vcyi mat phang clay m6t
,
g6c 600. TrOn canh SA I6y di6m M sao cho AM :a.6
=:i, ,-X+ ^r.i*
mat phang ( tsCM) c6t cpnh SD tpi
N . Tinh th€ tfch kh6i chop S.BCNM
cf,uv. (1,0iti€m): chox,y, zldbasOthucthoam6n: x'-xy+y'=1.,Tim giitrildnnh6t
vd gi6 tri nho nhAt ctra bi6u thilc: p : xo + yo
-4
. x. +y'_3
PHA|{ RIENG ( 3,0 tti,m)
Th{ sinh cht itwgc ldm mpt trong hui phan ( phhn A hoqc B)
A.Theo chwons trinh Chudn
C6u VIa. (2,0 ctiA@
1. Trong mdt phing vdi h€ toa d6 oxy, cho ducyng tron (c): x2 +yz -2x+6y
-15 0. ri6t :
phuong trinh ducmg thing A vu6ng goc v6i ducrng thing : 4x-3y+2:0
vd c6t duong rr-on
(C) tpi A, B sao cho AB 6. :
2. "Irong m4t phdng v6i hd truc toa dO Oxy cho hinh chfr nh4t ABCD c6 di6n
tich bing 12, vd,
I
c6 tdm ld giao di6m ctra dudng thing d, :x-y-3=0 vdi cl, :x +y-6:0.
Trung di6m cua
mQt cpnh li giao diiirn cria d1 vdi truc Ox. Tim toa c16 c6c dinh ctra hinh
chfr nhat.
cdu vlra . (1,0 din) : cho n ld so tq nhi6n l6n hcyn 2, tinht6ng :
S=Cl, +2.Ci+3.C] +a.Cl +...+n.Cil
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I
B,Theo chwons trinh Ndns cao
Cfiu VIb. (2,0 dihm):
1. Trong m{t pheng vdi hQ to4 d$ Oxy, cho hinh binh hdnh ABCD c6 di€n tich bing +. Bi6t
A(1;0), B(0;2) vd giao didm I cua hui c,ro'g chdo nim tr6n ducrng thil; y f ;. ffi;;;
c6c dinh C vd D.
2. Trong m4t phing vdi hQ top d0 Oxy cho parabol (p): y : x2 -2x vitelip (E): :r.
** r,
Chirng minh ring (P) giao @) tai b6n di6m phdn biQt cung nim tr6n m6t ducrng tron.
Vi6t phuong trinh ducrng trdn di qua 4 di6m d6. l
cdu vlrb. (1,0 rtihm): cho n ld s6 ru nhi6n l6n hon 2, tittht6ng :
s = cl, * 2.C:+ 3.c] * 4.ci+... + (-l)"-'.n.ci
'
-________-H6t_-____-_-_
-,t ,
:
(Thf sinh kh6ng dwqc sir &,mg tdi liQu. Cdn b6 coi thi kh6ng gidi thich gi th6m..) i
kang2 /2

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oAp Ax - BrEU ornvr
CAU DAP AN DIEM
I 1.(1,0 tli6m)
(2"r9 +) Tap x6c tlinh : n {t}
ili6m) +) Su bi6n thi6n
- Chi6u bi6n thi6n ' y'= < 0, Vx e (-"o;1) u (1;+*)
*h 0r25
Him s6 nghich bi6n tr€n c6c kho6ng Coo;l) vd (1;+oo)
-Hdm sii kh6ng c6 cpc tri
-Gi6i han vh tiQm cpn :
lim v:
lim y:2 :
' lim v = -oo vd lim v:
*co
x-++.o" x-+-"oJ x-+l-' x--+l*'
OO tfri hdm s5 c6 tiQm cin ngang:y:2 vi ti6m cfln dimg : x :,1 0,25
BAng bi6n thi€n
0,25
+)E6 thi :
clrtruc Ox t4i di6m (-l ; O )
1
pO ttri hdm sd clt trpc Oy tai di€m ( 0 ; -1
),
2
Giao di6m cria hai tiQm cQn ld tdm dOi xring cua d6 thi.
0r25
2.0,0 tli6m)
Dudng thdng Y = x -2mcat (C) tpi hai di6m phdn biet M, N e phuong trinh :
2x+l
x-l
<>Phuong trinh r,' -(3 + 2m)x +2m-1 = 0 (*) .o 2 nghiOm phan biQt kh6c 1 0,25
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,_-,[o= (3 + 2^)' - 4(2 m-1)>0
- ft'-(:* 2m)t+2m -1+0
(+{l4*' + 4m+ l3 > 0 thoA m6n VmER.
[*3*0
cqi ip; d6;iil iiidrn M i[i rraii,lil: i,")lrvt.;H:]*) . Khi ciit;;;*
li "tN (*).
hai nghiOm cria phuong trinh
0r25
[,n=-1
Theo gi6 thirit : MN - 4"lt e4m2+4m-3-oel 2
It
lm=-
L2
i , )
VOy: l2i cfic gia tri cdn tim. 4,25
n 1. (1.0 di6m
(4,0
iIi6m)
*
[:::x-:1tr I rt t"u".,"-,'". : ;i; ;;;* ;';-,r' = o ;-----
;.lz"i"ft.olj*'tu(":fr;:;i4
vdyphuong trinh dd cho c6 nghi€m : x:
f+k2n;x =n
+k2n (kez)
2. (t,0 tti6m
Ei6u kiQn x6c dinh cira phucrng trinh , >< t- 1
8048
Phuong trinh :
D4t 1+ JG 8048x =2y, y > 1 tu d6 ta duoc
"/ 2
:
4y' - 4y +1 <) yr - y - 2012x
1+ 8048x =
Mdt kh6c, thay I + Ji+ 8048x = 2y vao (*) ta duoc : x, x _2012y
-
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tVdytacohQ : {'t. ;n- itt-iii
(;t: z-vL/'!
= t0
- .---- .. .. .' . '
0,25
rrtr liictroiti" a"q;lGryjG;FioTli= o---- ".."-'-'.'..
€x=y (dox+y+2011>0)
Thay vdo (2) ta duoc :
y' -2013y=Oo[t =:o'3 vr1
lo?inghiQmy: 0 ui J
r tr'?5
LY=u
voi'p2oitt"4",; :iiili#"r*,t - :--'
2
Vdy phuong trinh dd cho c6 nshiCm : x =201?
v*_1
IN
(1,0
0r25
tli€m)
0,25
{},25
_:_--___,_--__-
0,25
II/
G
(1,"0
tti6m)
l
" *ff,_ .7"
i
A
k' a : ,'.':B,
/
vl"-AD7/BC;fi ( BcM)il At-."y ;t.p( BcMt;ti.mpasADi ih;"
tuy6n MN // AD , MN//BC, SA r (ABCD;= Se r eC
sd;--
, fecIAB
Iaco: { I
LBC I SA =BCI(sar)=BCl_BM.
--f----
--.-----------------:-
rri giric BCMNithl"riiri*s il6;s ;6 BMUa.'t*e.;; :-&t'ithid[;hi6; I o,,u
ru6ng g6c ctra SB l6n (ABCD) do vdy g6c hcro bdi SB v6i
: AB tan,ffi :AB tan60o : u.l!
(ABCD) ld ffi
Ia c5 SA
I
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MN-sM
Ivil DIVI IVIN sA-AM
.T=:-(=)-= .MN JA - AJVI
a6'* z
I 7 4a
4.n
AD SA 2a SA u.,6 j
BM:",ffi[AM'-+
{3
(^' '4a
secxrr,r - BC tMN BM :
l'+l+ : ry*
,
[ )r' tJi
Hp SH IBM ,vi BC I(SAB) =+ BC r SH.
-Y-?y_.$_H__+._LP_-CN0 s_ye_l_+& .9"1{_ 1.?.4s.ge__'_eg.
= :-=: ^^n :a:: n sb_Qp _s_,F_c_NM
tunffiF AM "-5
.^=;=; =+=++Affi 300 >
= m= 300, sB = 2a, SH l
's'vll-z" =153 =u
AB 3
Goi V 1A th€ tich ch6p S.BCNM ta c6 1sH.s".r, :ry
V
(1ro
Ta c6 : x' -xy+y' -l g7 1+3xy=(x+y)' >0=xy=-]
J
rli6m) :
MiI |.h.fP.:g j x' - xy + y' = (* - y)' * xy ) xy 0,25
1
+y' -3 =xy -2
Cfrngtir : x' -xy+ y' =r= x' *y' =xy+l={*'
-
[xo *yo -4- -("y)' +2xy_3
(xY)'+2xY-3
suyra: P= 0,25
xy, -2
Xdt hdm so : f(t) --( t*2
:2:-3 r,6i t . [-1,r-l
L 3')
.
f'(t)- 3 -(t_ z'
"'' -:z'f'(t;= oelft-z- VJ
t;
(-z)' )
t;
It-Z+ VJ
Ta c6 bing bi6n thi6n :
-f
I 1€'t'
(r)=2, |{iq r(t) - ..5) = zJi -z
|vtalf(t) =f j
= f (2
L-t"l L-;r
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/
- Tir d6 ta c6 :
*k
MaxP =2 uri,[*=Y=1
Ix=Y=-1
r r----E
--
|
*_Vt-lJi -"JJl-r,,_"17_:JI+J"r6_i
l^-T't=T
|
I_t
i
I
- ,,_!t
I*-Jz-rS*J.,6-i 'r-T t"1-t-JFL
MinP-zJj-z ru'il 2 I
I
-Jz-rS -J.-6-r .._ -#-tT+JG{ l^
|
l^ " =
2 't:T ^_
1o3s
/--Fr--=--i
| * _ -Vz-rJr +VJ:-i .._ -,lt -tJi -JS-r j
l^--,Y=f i
Wa 1. (1,0 di
(ro E,-:*
:'t"" :-3-y__l_1,6__lrurr .,, uu raril r( r;_J ); oan Klnn t(:).
tIi6m) /----:-- -or-?5-
b glu,s*-q-i o* 4P thi AH=3 vd IH eAB suy ra IF{ = ./nt - ALII _ 4 0,25
=9g"lg
vr ouCIng rhang A vu6ng g6c v6i dudng th6ng : 4x_3y+2:o
cira A c6 dang : 3x*{y+g:g "e"piiucrngiiinh
, lc-el [c--11
4=IH= d(I; A)= t- -t <>1"- rr
- { l^-tot' 0,25
'-j-r-:--------- ---. ---- -- : ----------l-u-.-
V0y A : 3x*4y*)e:0 hodc l: :xi+v_i-i:0--*
0,25
,. rr, rtttttl,
Ta c6: a, nO
f. _g
f*-y-3=0
vov'(Z'ij
i;;:o€)1 :
' tv--
t"2
Do vai trd A,B,G,D la nhu nhau n€n gi6 sri M rd trung di€m
M = d, nox suy ra M( 3; o).
Q --'- -' canh AD I
ra c6 : AB:2rM=rtre?T =3Jj
rheo gia trri6t , Soncn:AB.AD =IZ=AD:+S" = 12 :t^6
AB 3Jr-', "
Vi I vd M cung thu6c duong thing dr 3 dr I AD
Ducrng thdngAD diqua M ( 3; 0) vd vu6ng g6c v6i cl1 n6n AD nhfln i(t;t)
dm v6c to phdp tuytin. Phucrng trinh cua AD: x * y _ 3
=0
.l
m
r rang 5/'/

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Lai c6: MA:MD:$-O VA di€m A, D ng thuQc AD n6n:
2
Ix *y-3:0
Toa d6 A, D ld nghiQm cua h€ :
l t;
(x-t)2 +rz
L!
[y:-x+3 e],[y:-x+3 l]:'-*
ll":'
llu=t
c>l .j €)]lx-3:1
=r - -t)'*(3-x)2 -2* <) ir.'
[(*-l)'*y2 l(" ll,. ,: , ll": o
lL
llv=-t
L"
L
Y-ey-.A( ?;.0..P( $ :_!)
fqong tu I cfing lir trung di€m cira BD n6n ta c6 B( 5; a) 0,25
V0y toa d0 c6c dinh cua hinh cht nhdt ld: (Z; t ), (S; +), (
?ry_lt," !{y_itr_gs?!: {11911: ci p"C1,*,'.;.;t::1.;:-._" ,-_
Chonx : n.(1 +1)'-' =Cl + 2Ci +...+.nCi
_._ _
1
=
Vdy: S=n.2n-r
VIb 1. (1,0 di6m
Q$ J5 ' Ph.'o''g trinh AB: z
=
tIi6m) 3:-"-, S-:-tI-?)=AB
I e (d) : y : x si(ilit. I itiffis A;ffi ;il AC ;tBit ;a;;---
C(zt - 1 ; 2t), D (2t; Zt - Z)
Hodnh dd giao di cua (E) vd (P) li nghi6m i phucrnf t.inh .
"t
X1*,*' -2x)2 =1c)9xa.36x3 +37x2-9:0 (x)
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X6ti(;i=da_3o"5_h7;i:q;-ir"iiic"ir,.t'o"n"o'-r(riifi0i.0
(0X(1) < 0,,(1X2) .0, < 0 suy ra (*) c6 4 nghiQm phdn bi€t,
D"ge "(2X(3)
q41-e) Bi-l_4ieql_p_b4s_b_iet _-_,_...__ ._ 0,25
-(Pl i ------
I Y=x2
-2x
"
Top d6 giao di€m cua (E) ve (P) tho6 mdn h€
*,
i| -=-*Y2:1
,
Ie
[8*t-t6x:8v . ^
<> {-; _ ";- --' > 9x'+9yz^ -l6x-8y-9:0
lx'+9y' :9
0,25
<+ x2 +y'-**-*y-t = O
' g g' /
(**)
(**) te phur:ng trinh <tuong hon c6 t6m r: b6n kinh R :g
[3,f ),
Do d6 4 giao di€m cua (E) vd (P) cung nim tr6n ttudrng trdn c6 phuong trinh :
xz+y2-f"-fv-r:o 4,25
v'ub X6t khai tri6n: (1+ x)" = Cl + xCl + x'Cf, +... + x"Cl Ar25
(1ro
Dao him hai vti ta dugc: n.(1+ = Cl + 2xcl + ... + nxo-rCi 0'?5
ili6m) ")n-t
Chgnx : : 1 + n.(l-1)"-t - Cl - ZCI+...+ (-l)".nCi 0,25
Vdv : S:0 0.25
Chri i : - Gi6m kh6o tru6c khi chAm thdng nh6t <hp An.
- Thi sinh ldm theo c6ch khdc md dring vdn cho cti6m t5i cla.
- Cdu V thi sinh c6 thti lam theo cdch x6t :
% - %.r.* - SM'SN -i- q'{
"" .u.o -
%.*o sA.sD sD
- C6u VIIa vd Cdu VIIb thi sinh c6 th€ cti chimg minh,cOng thirc :
kcl = ncl-i
Sau d6 thay lAn lucrt k :1,2,3, . . ..,n
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