Giai phucrng trinh: cos2x*5: ZZ-cosxsinx-cosx 2.. Tinh th€ tfch kh6i chop S.BCNM cf,uv.. Trung di6m cua mQt cpnh li giao diiirn cria d1 vdi truc Ox.. Tim toa c16 c6c dinh ctra hinh chfr
Trang 1KV THI KSCL TTII XA DAI Hq}C xAIvI IOTIL,AN rr*tI l
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 fr€' gh-".%
psAw cguNc cno TAT ce Tui swn ( 7,0 arcm 1
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
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
g6c 600 TrOn canh SA I6y di6m M sao cho AM :a.6 =:i, ,-X+ mat phang ^r.i* ( 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
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,
c6 tdm I 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 di\n) : cho n ld so tq nhi6n l6n hcyn 2, tinht6ng :
S=Cl, +2.Ci+3.C] +a.Cl + +n.Cil
Trang 1 12
Trang 2B,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):
Chirng minh ring (P) giao @) tai b6n di6m phdn biQt cung nim tr6n m6t ducrng tron
l
Vi6t phuong trinh ducrng trdn di qua 4 di6m d6
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
'
-,t
(Thf sinh kh6ng dwqc sir &,mg tdi liQu Cdn b6 coi thi kh6ng gidi thich gi th6m ) i
kang2 /2
Trang 3oAp Ax - BrEU ornvr
I
(2"r9
ili6m)
1.(1,0 tli6m)
+) Tap x6c tlinh : n \ {t}
+) Su bi6n thi6n
- Chi6u bi6n thi6n ' y'=
*h < 0, Vx e (-"o;1) u (1;+*) 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 :
1
pO ttri hdm sd clt trpc Oy tai di€m ( 0 ; -1 ), clrtruc Ox t4i di6m (-l ; O )
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
Trang 1/7
Trang 4(4,0
iIi6m)
1 (1.0 di6m
*
[:::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
,_-,[o= (3 + 2^)' - 4(2
- ft'-(:* 2m)t+2m
l4*' + 4m+ l3 > 0 (+{
[*3*0
m-1)>0 -1+0
thoA m6n VmER.
cqi ip; d6;iil iiidrn M
"tN i[i rraii,lil: i,")lrvt.;H:]*) Khi ciit;;;*
li hai nghiOm cria phuong trinh (*).
Theo gi6 thirit : MN - 4"lt
l2i cfic gia tri cdn tim
Ei6u kiQn x6c dinh cira phucrng trinh , >< t - 1
8048
Phuong trinh :
D4t 1+ JG 8048x =2y, "/ y > 21 tu d6 ta duoc :
1+ 8048x = 4y' - 4y +1 <) yr - y - 2012x
Mdt kh6c, thay I + Ji+ 8048x = 2y vao (*) ta duoc : x, - x _2012y
[,n=-1
It
lm=-L2
0r25
4,25
Trang2/7
Trang 5t (;t: ;
= t0 itt-iii - .' '
VdytacohQ : {'t n- z-vL/'!
rrtr liictroiti" a"q;lGryjG;FioTli= o " "-'-'.'
€x=y (dox+y+2011>0)
Thay vdo (2) ta duoc :
y' -2013y=Oo[t =:o'3
lo?inghiQmy: 0 ui vr1
voi'p2oitt"4",; :iiili#"r*,t - : '
Vdy phuong trinh dd cho c6 nshiCm : x =201?
0,25
tr'?5
IN
(1,0
tli€m)
v*_1
0r25
0,25
{},25
_:_ _,_ -0,25
II/
(1,"0
tti6m)
G
k'\
A a : ,'.':B,
vl"-AD7/BC;fi ( BcM)il At-."y ;t.p( BcMt;ti.mpasADi ih;"
sd; tuy6n MN // AD , MN//BC, SA r (ABCD;= Se r eC
, fecIAB
Iaco: {
=BCI(sar)=BCl_BM.
LBC I SA
. -:-rri giric BCMNithl"riiri*s il6;s ;6 BMUa.'t*e.;; :-&t'ithid[;hi6;
ru6ng g6c ctra SB l6n (ABCD) do vdy g6c hcro bdi SB v6i (ABCD) ld ffi
Ia c5 SA : AB tan,ffi :AB tan60o : u.l!
l
i
/
I
I o,,u
f I
Trang3/7
Trang 6MN-sM MN sA-AM a\6'* z 4a
.T=:-(=)-=
BM:",ffi[AM'-+
{3
(^' '4a\
Hp SH IBM ,vi BC I(SAB) =+ BC r SH.
-Y-?y_.$_H +._LP_-CN\0
.^=;=; AM "-5 :-=: ^^n :a:: n l
tunffiF =+=++Affi AB 3 = 300 > m= 300, sB = 2a, 's'vll-z"SH =153 =u
Goi V 1A th€ tich ch6p S.BCNM ta c6 1sH.s".r, :ry
Ta c6 : x' -xy+y' -l g7 1+3xy=(x+y)' >0=xy=-]
J
MiI |.h.fP.:g j 1 : x' - xy + y' = (* - y)' * xy ) xy
Cfrngtir : x' -xy+ y' =r= x' *y' =xy+l={*' +y' -3 =xy -2
-
[xo *yo -4- -("y)' +2xy_3
suyra: P= (xY)'+2xY-3
Xdt hdm so : f(t) ( :2:-3 r,6i t
t*2
f'(t)- "'' (-z)' \-:z'f'(t;=)
Ta c6 bing bi6n thi6n :
0,25
0,25
t;
VJ
t;
VJ
[-1,r-l
ft-z-oel
It-Z+
|vtalf(t) =f (r)=2,
L-t"l
-f 1€'t'
5) = zJi -z
|{iq r(t) = f (2
-L-;r j
I
V
(1ro
rli6m)
Trang 4/7
Trang 7-/ Tir d6 ta c6 : *k
MaxP =2 uri,[*=Y=1
Ix=Y=-1
r r E
| *_Vt-lJi -"JJl-r,,_"17_:JI+J"r6_i i
I_t
I
| " = -Jz-rS -J.-6-r _ -#-tT+JG{ l^ ^_
/ Fr = i
| * _ -Vz-rJr +VJ:-i _ -,lt -tJi -JS-r j
Wa
(ro E\,-:*1 (1,0 di
\:'t""
tIi6m) :-3-y l_1,6 lrurr \ \.,, uu raril r( r;_J ); oan Klnn t(:).
=9g"lg b glu,s*-q-i o* 4P thi AH=3 vd IH eAB suy ra IF{ = ./nt / : - ALII _ 4
-or-?5-0,25
0,25
0,25
vr ouCIng rhang A vu6ng g6c v6i dudng th6ng : 4x_3y+2:o
"e"piiucrngiiinh
cira A c6 dang : 3x*{y+g:g
, lc-el [c 11 4=IH= d(I; A)= t- - -t <>1"- rr
{ l^-to '-j-r-: - - : -l-u-.- t' V0y A : 3x*4y*)e:0 hodc l: :xi+v_i-i:0 *
, \rr\, rtttttl,
Ta c6: a, nO
Do vai trd A,B,G,D la nhu nhau n€n gi6 sri M rd trung di€m canh AD
M = d, nox suy ra M( 3; o) Q '- -'
ra c6 : AB:2rM=rtre?T =3Jj
rheo gia trri6t , Soncn:AB.AD =IZ=AD:+S" AB = 3Jr-', 12 :t^6 "
Vi I vd M cung thu6c duong thing dr 3 dr I AD
dm v6c to phdp tuytin Phucrng trinh cua AD: x * y _ 3 = 0
f _g f*-y-3=0
i;;:o€)1 ' tv : vov'(Z'ij
t"2
I
.l
m
r rang 5/'/
Trang 8ng thuQc AD n6n:
di€m A, D
*y-3:0
[y:-x+3 [y:-x+3 l]:'-*
[(*-l)'*y2 =r -
l(" -t)'*(3-x)2 -2*
ll, ,: ,
lL
Lai c6: MA:MD:$-O
2
Toa d6 A, D ld nghiQm cua h€ :
Y-ey-.A( ?;.0 P( $ :_!)
fqong tu I cfing lir trung di€m cira BD n6n ta c6 B( 5; a)
V0y toa d0 c6c dinh cua hinh cht nhdt ld: (Z; t ), (S; +), (
Hodnh dd giao di cua (E) vd (P) li nghi6m
"t i phucrnf t.inh . -2x)2 =1c)9xa.36x3 +37x2-9:0 (x)
VA Ix
l L!(x-t)2 +rz t;
ll":'
llu=t ir.'
ll": o
llv=-t
L L"
0,25
X1*,*'
?ry_lt," !{y_itr_gs?!: {11911: ci p"C1,*,'.;.;t::1.;:-._" _._ _ ,-_
Chonx : 1
Vdy: S=n.2n-r
1 (1,0 di6m
VIb
Q$
tIi6m) 3:-"-, S-:-tI-?)=AB = J5 ' Ph.'o''g trinh AB: z
I e (d) : y : x si(ilit I itiffis A;ffi ;il AC ;tBit
;a;; -C(zt - 1 ; 2t), D (2t; Zt - Z)
Trang 6/7
Trang 9(0X(1) < 0,,(1X2) 0,
"(2X(3) < 0 suy ra (*) c6 4 nghiQm phdn bi€t,
D"ge
-(Pl q41-e) Bi-l_4ieql_p_b4s_b_iet _-_,_ _
i
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
<+ x2 +y'-**-*y-t ' g g' = O (**) \ /
(**) te phur:ng trinh <tuong hon c6 t6m r:
[3,f ), b6n kinh R : g
Do d6 4 giao di€m cua (E) vd (P) cung nim tr6n ttudrng trdn c6 phuong trinh :
xz+y2-f"-fv-r:o
0,25
0,25
4,25
v'ub
(1ro
ili6m)
X6t khai tri6n: (1+ x)" = Cl + xCl + x'Cf, + + x"Cl Ar25
Dao him hai vti ta dugc: n.(1+
")n-t = Cl + 2xcl + + nxo-rCi 0'?5
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'{
- 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
Trang 717