1. www.VNMATH.com
rl
/t . ,^
$&i
I l.]i:r t:
ll;c.(.,)
pn rm KSCL DAr HQC NAM 20:-2LAN ]'HrJ'I
MOn: TOAN; tcnOi g
Tld'i gian ldm bdi; 180 philt, kh6ng k€ thdt gian phdt di
D0 g6m: 02 trang
]..1!.'::
I. PHAN CHUNG crro rAr cA cAc rni srNH (7,0 diam)
r: i.,rjr,? - .r: :
.:l:-,:.,, .:.,' i :.
;:,,." L,au r (2,0 ilim)Cho hdm s6 y - -t
'22-'l- +3*' -! (t).
: 1. Khio s6t sU bi6n thi6n vd v6 dO thi (C) cua hdm s6 (t).
",,,'2" Cho di6m Mthu6c dO thi (C) co xM =m.Tlm c6c giltrf thUc ciratham s6 *
de tiep tuy6n tai M cit d6 thi (C) tai hai di€m phAn bi6r A. B sao cho MA:3MB
(B nirn gifr'a A vn M). - = J Jt-
{'iu II (2,0 ttihnt)
L=
*f +["T ;
lu lc^a
i . Gi6i phuong trinh: 4sin2 x.cos x -Zsinzx -sin" = .or"..
;:. Ciei bAt phuong tr:inh: *'.2" + x.z"nt +IZ <3.2" + 4x2 +gx . L1'.fi)U{l;D-)
Cfru III 1l,b mAnr) lllr' gi6i han , = SYa 'l
Cf,u IV (1,0 cli,dnt)
L
Cho hinh ch6p S.ABCD c6 d6y ABCD ld hinh vuOng cpnh bing a. C4nh bOn SA vqdlg g6c
!'r "y -
'vOJ lnat pltang da vA SiA: . Goi B', D' lAn luqt ld hinh chi€u w6ng goc cua ctiiim l.
"Ji
tr0n canh SB vA SD. Chirng minh rdng cqnh SC vu6ng g6c v6'i m5t phing (AB'D'). Gsi C'
Jr'-L;;;t;;il;. ffi,?;"r;;'ffitilj:16*;,;;:: :,:S
r*f5rsvurvurvL{cL.r.rraL}rudrrtirrD JJ / vulgalulDL. lrnntllellcncllaKnoicnop).u'L.'D'. *17
ra giao di6m .uu ,oat
Cffu Y (100 tli6nt) Cho bas0thucldr6ngAm x)y,zth6aman x+y*z:Z0Iz, Timgiritri u#-
,rirAt criabi6uthrc: p -{F(",*l| .iFb} +,).iF1; .4> ,[b+V
I[. ]tI{41 RIENG (3,0 diam)
'.1'h{ sinh clri dngtc chgn nrpt trong hai. phfrn A ho{c B:
A. 'Ihco chu'o'ng frinll chuin
(lf,rr YIa. (2.0 dirn) Trong mdt phing Oxy cho di6m Me;a)vd'ducrng
tr.on (C) c'o
phirong tniih: x' + y' *
=2x 6y + 6 = 0.
i.l.
Vi6t phuorrg tr"inh duong thdng d di qua ditim M cit ducrng tron (C) rai hai di6rrr p.
e
sao cho M la trung di6m cua PQ. )(1Y
1r -0= D
2. vi6t phuong trinh duong trdn (c') d6i x#g uii ao*g troq(c ) quu di6m M.
Cffu VIIa .(1,0 dinl Chrmg minh ring: ^-J) + (t- {) = +
(zn+1)3" =2'".C),,..1+2.22,-,.Cln*,+3.22,-2.C:,,*r *.., +(2n+t)cliil(r-*),
,1 I
x'
I
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oii.,rrung rn4t phang Oxy, cho tam gi6c'ABC'c6 A(l;3). Ducrng phdn gi6c trong cria goc B
:nim tr6n duong thing d, c6 phuong trinh ld: x - y -l = 0 va'dulng "uo *n6t ph6t tt dinh C
lina* tr6n ducmg th8ng drc6 phucrng trinh lA:x + 3y + 2- 0 . Vi6trphuong trinh duhng thang
.. ::.i.-i,"
.chria cpnh BC cira tetm gi6c ABC. *- ly-q>O : ,:
i: r.
l"r.:.'j:',
:
0"'" 2.Cho elfp (E):
r /
2 2
l* * =t, gqi F,(-.;0) ld,mQt ti€u di6m cira (E). Tim di6m M tr6n (E) sao
..'-'.,. 25 16
,.r ,cho dO ddi FrM co gi6 tri 16n nnat. [t '4
. Cf,u VIIb. (1,0 itidm) Cho mQt da gi6c l6i ArA2...A:2. Hoi c6 bao nhi6u tam gi6c c6 ba dinh
l6y trr 3 trong 32 di6m Ar, Az, ..., A:z sao cho mdi canh cira lutn gi6c khdng c6 cqnh nho lir
frrq
tlat
canh cita da gi6c l6i ArA2. ..An.
': I
L.rE'T]
lrri I -----
'..::,:,.-i, r I i
i{,. .F
:'; l l.
+cfwri rr., ',. - '- .Jl.l
.,
(Cdn bd coi thi khdng gidi thich gi th€m)
. ilr 'l
I.j-rl.-
! I
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PAp AN VA THANG EIEM
,T
I I. (1,0 tli1m
'(2 ili0m ) T0px6cdinh: D=R,
Su bi6n thi6n:
- Chi6u bi6n thi6n:
!'= -2x3 + 6x;
!'=0g;s=0; x=--rE hoAc x=$. 0,25',
Hdm
'6
AOng bi6n trdn c6c kho6n* (-*'-..6) va (o;"-6); nehich bicn
tron c6c khoang (-16'o) va (..6;**).
- CUc tri: Hdrm sO dat cuc dpi t4r x= -..,6 vdL x: 1f, 1 lcn = 4 .
I
Hdm sO dat cgc ti6u tpi x:0 vd lcr
2' 4,25
- Gi6i han: lim y: x-)+o ! =-oo.
-t-+-co
lim
- BAng bi€n thi6n
+ : 0 - 0 ,+ '0.',,_
4,25
r DO thi
0,25
2. (1 iti
Ta c6 diOrn
Phuong trinh ti6p tuyiln ei M: ! = -Zm(^' -3)x+
|*- *U*, *I
AL
.
Xdt phuong trinh hodnh d9 giao dii5m:
-^4
-**
1
2 3xz -l=-z*(,n'
2 -3)x+ 1*^ -3,r'
2' -!.
z',
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<+ (, - *)' (r' -r 2rnx +3rnz -6)=o
.'-tt
f*=*
-- I
L"t +Zmx + 3m2 -6=0 (-)'
EAt /(x)
= x2 1- 2mx +3m2 - 6 -
Ti0p tuy6n t4i M clt 1C; t4i 2 di6rn A, B phdn bi6t kh6c M khi vd clii khi
phucrng trinh (*) c6 2 nghiQm ph6n bipt kh6c m, hay :
Iaco:'
l*n'*u
:3m2 - (t.z)6
Do IWt : 3MB vd B nim gitaA vd M n6n M{ = 3MB , tri d6 suy ra :
""A
-?- -_1 -^fr r
1- - )xB =
-2m 0.:)
rr (1.1), (r .rr,rr,r)^?::r)r*r*
[". : _2m
),i,,*"=r*'-u *l;: =Q
I" " l'6
3x, : -2nt
1", - l* - tJt
Do vAy * = -Ji; m:J7 (tnOu mdn didu kiQn (**)).
YQy m = -Ji ; m = rlz n cilc gi|tri cAn tim.
(2,0 c6:
*.4
Ta 4sin2 x.cos x - Zsin2x - sinx = cosx
Cllem)
{+ cos"(4rin" - 1)- sinx(zsinx + 1)= g
<* (zsinx + l)[cosx(2sin" - 1)- sin*] = o
1.4,.'t1t e (2sinx + t)[zcosx.sinx * (cosx + sinx)] -o
l1
lsinx---
c+l 2
(l)
[z ,o, x. sin x - (cos x +:sin .r) -0 (2)
T
iiiii,f :,1
I *--L+kur
Giai (1): sinx=-t ol 6
(tr e z) . (th6a mdn).
2 I *=A+k2n
,
4,25
L6
Gi6i (2): 2cosx.sinx - (cosx + sinx)= 0
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+2sinr.cos *=t'-'1.
Khi d6, (2) tro thinh t2 - t- I= 0 e t -LS
2
Taco: .or["-z'']-1-S
---("" 4 2J,
)
*=o -u,""or[t-f.) +k2n
el|
I
4 zJz) J,25
) ,_'., ,(k=z).6naamdn)
l*-o*ur".orf!-fl +kzn
4
| 2J2)
Vpy phuong trinh dE cho c6 c6c nghigm ln
*= -1+ k2n, * =!+ k2tr, x=L- ur""orf!:gl
-[zJz
) --'
+ katr,
6 6 4
n (t pr-
-^ *,25
.rr=1+arccosl
4 l+kzn (kez)
lzJz)
2. (1,0 clidm)
Ta c6: *'.2" + x.2"*1 +r2 <3.2*' + 4x2 +Bx
o *'.2"' +2.x.2" +12-3.2*' -4x2 -Bx<o
l
+,25
!i': ,, :
e 2*' (x' + 2x 3) a.(*t + 2x 3) . 0 ]
I
o (r' +2x -t)(r' - o) . o ,).2s
l
'*--l 1
l*'"*2'x'-3<o I[-3<x<1 <+'-3<rc<-Ji.
1 ci l
lzu -4>o
-
[l'l'.D
r.),25 ]
Truong hgp 2:
,^ llx<-3
[*'*Zx-3>o ll'"- - <+I<*<Ji.
_4<o<+{1"t1
1"",-'"
l2*
[u.o
Vfly tap nghiQm'cfra bdt phuong trinh dd cho ld S = (-:;-.,n) , (r;rD ) ,
i],25
ili 1,0 &i6m
(l,r) lleea _ 1
di6m) Ddt r= +997 x n1t
'J yLJ
997
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Khi x-+0 thi t-+1. 0,25 I
Dovfly r=1,T?
997 997 i
W+=lig lee3 +tteez +...+r+1 1994 2
0,25
vay L=r 0,25
2'
I/ 1,0 fri4nt
,(1ro
* sc -1,(da'D')
,di6m) laco:
BC LAB
sa t(encn)* srl , ur] * BC' L(s'll)
a BC LAB, Ita- ^'
--{,. 0,25
Ir pl1 r'
Mi lB'r ^s,
j I t--
', :: lt::::
AB'L (SBC) + AB' I SC . i, .+',1>/
*-*--
t |.l
+ tt
I /'
Chrmg minh tucrng tg, ta co =+ AD'L SC . i,: ,l ',
0,25
Do d6, ,sc r (aa'n')
* Tinh th6 tich cria kh6i chop S.B'C'D'
Trong tam gi6c vu6ng SAB, ta c6
AB = a, SA = ali, SB = ali vd AB'L SB
o t2
Suy ra SB':
ot zall
=
^sB 3
n,li
Tuong ty SD'= '-J
),Jj :
Ke eC' tsc .+ ,4C' . (eA' n') .
M[t khdc, tam gi6c SAC li tam gi6c vu6ng cdn n6n C' li trung di6m cria
SC, do do SC'= rz.
zali zalj
v' u.r,o, _.sB'.sc'.sD'
Ia co = I ='o'
Irr.oro ,SB.,SC.,SD oJS.Zo.oJi
2
9
0,25
'
t'ad. vs BCD =! stIBC"BD =o'-&- (d.v.t.t)
suy ra v, u,,,o, =t + =
+(d.v.t.t). 0,25
V
lu 1,0 iti6m
i (1,0 V6i vi, y 2 o ta lu6n co (* - y)'(x + y) > o
I *. ;.
*.;.
I cjrenu
cjren
I
I
o ("' - r')(*+y)> o Vx,y 2 o.
I
<+ x'+ ),t 2xzy+xy' e:.("'* yt)>1.(*ty+xy')
l----
I
i,
I
I
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n 4.(*' +y')> (r*y)r-
e'.fi7[r*r]
>2(**y).
0,25
Tucrng tg ta c6: ,.fiZp' + >-z(y + z)
"J
,S7pt;;, >2(z+x).
4,25
Do d6, P >-a(x+ y + t) ep > 8048.
D6.u ':' xay rakhi vA chi khi x = ! = , ='Or" .
4,2:5
vfy gi6 tri nho nh6t cua bi6u thric r uirs@thi vd chi khi .
y-L--.
20t2 4,25
e-
-a
J
VIa. l.(1,0 cti6m)
Dulng tron (C) c6 t6m I(1;3), b6n kinh R:2.
:
Md IM Jr<R + M ndm.trong dulng trdn (C). 0,25
Dunng thing.qua M lu6n cdt du*g trdn (c) tai 2 di€m phan biet p, e. Dc
M le trung di€m cria pe th pe t tit . a,25
Duong ina"e pq di il; M ,'ha;lfr:f] u* 16; tr pi;ap i"t&.
;r i -T- ;6*' 5,r5-
phuong trinh:
€> Jr+ y-6=0. 'Jr25
VAy phucrng trinh duong thlng,cAn tim ld x + y -6 = 0 ,
2. (1,0 itihni
Eucrrrg trdn (C) c6 tAm I(1;3) vd b6n kinh R: 2.
Gqitdm cua dudng trdn (C') le I' vd bdn kinh cua (C') la R', ta co
fi,lr-
Md M(2;4)
:+i <+{' . €I'(3;5).
1../,' = ZYr - Y, lY,,= 5 - ^r25'
Vpyphuongtrinhcila.dudngtrdn(C,):(,_3),+(y_5),=4
t)"25
:*.-:"-l
VIIa. 1,0 di4m I
T a c6 : (2 + *)"*r
= Cln*r22n+1 * C),*rZ2' x * Cl,*rTzn-t *2 * :
. . * Cl:il *r,.t . 0,25
LAy dao hdm 2 u6 tu duoc:
(Zn+f )(Z + *)'" = C),*,2'".+2.C1,u22"-t *+... + (Zn +t)Ci;ilx,, tii,25
"-****l I
Clrgn x =7ta duoc:
,) -'){
(zn + l)3" = C)nnt22" + Z.Cl,*,22n-t * 3.C|,u22n-2 * ... + (zn + t) ciiil. i
e (2n+r)32" -22",c),*t +2.22'1.C|,*r+3.22n-2.C,*r+...+ (zn+r)c::i:.(r. ru).
0.25 i
VIb. 1. (1.0 cli€m
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l'.-l;';:
' ;. ')r
iF.;Y+jr.''1'j
iiir').'tL.t': i:.1
(2,,,A Gc.ri A'
ld di6m ddi xring vdi A qua dr.
di6m) Phuong trinh duong thing d di qua A vh vu6ng g6c vdi dlco dqng
i."i., t. '.
;.,t :
iii.'-i
.
i.:;, ,.
x+y+c=0.
li;;,r.,::
Md A thudc d ndn ta co e: -4.
: ij.
!r+_;l.i t-"-
.: r I r 4,25
iijls;,liri.
i.:.j,-ii.
, ;','
i-rliii:.li;:. .i::
Do c16, phu<rng trinh cira dudng th[ng d: x+ y-4=0.
,;!*iji:::1.,,i,.
Gqi I ld giao di6m cira d1 vd d.
l, .:' :: r' I(hi d6, to? d0 cria I ld nghiQm cua h9
'-.,t..;;., . I
i*i)l:J. i phuong trinh:
".*,:;,',,t'
f ." _s
i:i!: j,:. :
(r*y-4=0 l'-t
eIl,(5 3)
'i.;1.,L.'.; .:!,,
I "
EZ.,!.rij..
<+{ -:-1.
:f:.i:.i.:lr '; r
lx-v-1=0 I 3
r' 2'2)
Iy=-
t-2
r "'l Do d6, tqa d0 cua A'(4;0). 0,25
Ta co cl2 vu6ng g6c vdi AB n6n phucrng trinh cria dubng th6ng AB c6 dang
3x-y+c'=0,
Do A thuQc ducrng thdng AB n6n ta tinh dugc c' : 0.
Phuong trinh cira dubng thang AC: 3x *.! = 0.
0,25
D" B it gilo die;iCtri AJmg thdng d1 vli ducrng thing AB n6ntos clQ cria
B 1A nghiQm cua hQ phucrng trinh:
_ r_ I
l"--; L <+Bl( 1 3
[3r-v=o €{
{--- r
Y
"t,-l:-"1.2)'
L*-.,,-1=0 - 1..- 3 - 2'
"t'2 I - -- V
Mil A' ln diOm d6i xirng vdi A qua d1 n€n A' thuQc'ducmg thing BC.'
V4y phuong trinh cria duong thlng BC: x -3y -4 = 0. .
0,25
2; (1.A di€m)
Ta c6 c:3; Ggi M(x;y) thuQc (E).
-, (3
4M'= (x + 3)2 *y'-l ;x+5-' | 0,25
5 )
{
^
Suvra F,M
JI =|x+5.
) 0,25
Do MthuQc elip (E) n6n -5 (x ( 5 + 2s 4M <8. 0,25.
Do viy, F,M sg. Diiu ':'
x6y ra khi vd chi khi x: 5.
Vdv d0 tliri cua doan FrM dat ei|tri lcrn nh6t khi vd chi khi M(5;0). 4,25
VIIb. 1,0 cti6m
1,0 diiirn +)S6tamgi6cc63dinhld3trongs632dinhcfiadagi6cl6iddchold 4,25
9jz:129!.0:..---*-.-.*. ia;e"h;,fi:du
;iffi
atim Jdtu* t C?nh;t f .a"n i:i:-
*
siil;6 eia" i6i 0,25
canh.
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