so GrAo Duc vA EAo rao HA Nor ne rm rrnl DAr HQC EoT r NAtvt Zgt tThdi gian ldm bdi: 180 phrfit, khdng tC ttrOi gian giao dA.. DA thi gom ol trang.. Chrmg minh rdng mdi ti6p tuy6n cira C
Trang 1so GrAo Duc vA EAo rao HA Nor ne rm rrnl DAr HQC EoT r NAtvt Zgt t
Thdi gian ldm bdi: 180 phrfit, khdng tC ttrOi gian giao dA
DA thi gom ol trang.
I pHAN cHUNG cHo rAr cA cAc rni sINH 1l,o eiom;
Cffu I (2,0 tli6m) Cho hdm s6 y: xt -3x' +1
1 Kh6o s6t sp biiSn thiOn vd v€ dO thi (C) ctrd nam s6
2 Chrmg minh rdng mdi ti6p tuy6n cira (C) chi ti6p xric vdi (C) tei dung mQt di6m
Cfru II (2,0 tli6m)
1 Giai phuong trinh 9'io" * 4,gcosz x= 13 +,*'z'+l-3'o'2'.
lx+ Y =g
2 Giai he Phuong trinh I r ^-
r -:-[r/xZ + 9 *^ly'+9 =10 (x'yeR)'
cflu IItr (1,0 tli6m) Tinh tich phdn 1 :pt 4*.
ix'
CAu IV (1,0 ili6m) Cho hinh chop S.ABCD c6 d6y ln htnh vudng cpnh c , SAL(eACn) vd SA=a.
Ggi A',8',C' vit D' lan luqt ld trung ,rliOm cira !C,SD,SA vit SB Chimg minh rdng AA',BB',CC' vh
DD' d6ng quy; Tfnh th6 tictr ctra hinh ch6p ^S'.1'B' C' D' theo a, v&i ,S' h tam cfu hinh w0ng ABCD '
CiuV(1,0tli6m)Xicdinh m saocho xa -2x3 +8**l)*'-2mx+m'-4>A, Vxe [-f1]
II PHAN RItNG (3,0 tli6m) Thf sinh chi iluqc chgn mQt trong hai phin (phin A ho{c phin B)
A Phin A (theo chucrng trinh Chuin):
Cfiu VI.a (2,0 ili0m)
1 Trong mat phang tqa d0 Oxy , cho hinh r.u6ng c6 mQt dinh l(- 1;2) vA m$t duong ch6o nim trOn
ttucmg thing c6 phucmg trinh 2r - y -l = 0 Tim tqa d0 c6c dinh cdn lgi cira hinh vu6ng
2 Vi6t phuong trinh m[t cAu (C) cO t6m thu$c dudng thlng (A) c6 phuong trinh
lx-vtz=o J"
lZx+ Y+22-I=0 vdtitip xric voi hai m{t phang (a): 2x +2y - z + 6 =O va (B) ; 2x +2y - z-6 = 0
Cffu VII.a (1,0 rli6m) Cho zr,z, ldhai nghiQm phrlc cria phuong frnh zz -22 +5 = 0 Tinh gi6 tri cua
bi€u thirc P =lr?l*l':l
B Phin B (theo ehuong trinh Ning cao):
Cflu VI.b (2,0 tli6m)
1 Trongm{tphingtqad0 Oxychodudmgtdn(C): *2+y2-4x+2y-5=0.fhlrtA6euerngthing 4
d*! x - my=0 cat ducrng trdn (C) t?i hai di6m A, B pherr-biQt, sao cho dQ dii do4n AB nhtt nh6t
z Trong khdng gian tsa dQ Oxyz cho c6c tli6m l(l;O;t), f(tt;O), CQ;l;*l) vi m{t phdng (a) cO
phucrng trinhx + y + z-1 = 0 Tim to4 d0 diem M sao cho khoing c6ch tir M dln (a) Uang khoang
c6chtu M danm6iei6m A,B,C.
(r- \3
Ciu VII.b (1,0 tli6m) Tim sd phtrc z ,Ai6t Z =42-!
Trang 2
-n6t-so crAo Duc vA o.A,o rAo HA NOI
of rnr rrulDAr Hgc - DgT r nim zort
M6n Tofn - KhAi A
I
(2,0 ili6m)
t rt.O tli6m)
T0P x6c dinh: B'
Sg bi6n thi6n
- Gi6i han: lirn ! = -c; lim Y - +60.
0,25
0,25
- Chi€ubii5nthi€n: !'=3x2 -6x; y'-0ex=0 hotrc x=2.
.y'>0e x<0 hoflc x>2; .y'<0<]0<x<2 HAm sO AOng bi6n tr€n cfc khoang
(-*,0), Q,**) vA nghich bi6n trOn ktroang (O,Z).
- Cuc tri: Hdm s5 dat cgc ct4i tq.i x = Q;yru = 1, d?t cgc tiAu tqi x:2i/cr =-3.
:-iia;s L-i6iitiliit;
4,25
EO thi
y"=6x-6; /"=0(i r=1
ximg cta AO tfri hnm s6
Dd thi hdm sO c6 di6m uOn l(t,-l) vd n6 la tdm d6i
Ar25
Trang 3Ciu Eip 6n Di6m
ai di6m: Mo(to,Yo) vd M1(x1,Y1) Khi d6 phucrng trinh cria ti6p tuy6n li
y=6*8-6xoh-'r-3'+3xfi+I vd ,:b*? -e'r! -z*l *?t:! 0'5
' -'. 0,25
0,25
#ii;fi;d'd #dil dcn ;ils i;-i,-6;ong liiirtr
"t imqt tiiSp tuv6n n€n 3r& - 6xs =zxl - 6x1 ,
-2*3+lxfr +l=1xl +3xl +t .
Giai hQ trOn ta dugc xe = 11, do d6 ta c6 dpcm
II
(2,0 tli6m)
T (1"0 dtffi)
Phuong trinh dd cho tucrng duong vfi
9sin2.r * 4,gl-sn'zx =13+ 93/2-Zsintx -31-2sin2x
<) 9sin2r * gsin'x 31 = 13 + 3-
-:-92sn'r gstn- r
<+ 9sin" * jl - -4 - 13 = o'
9sin2 x
(nrrr r ),
Dflt r =9''" , 1 < r < 9, ta nhfln dugc phuong trinh , *+ -T -13 = 0
e (t *l)(t -3X/ - 9) = 0 <+ r =l;t = 3;t = 9
Ar25
0025 Phrrcmstrinhddchotuonsduonsv6i sin2r=0 ho6c sin2x=t hoFc qr-41 f -=-!!-?
a o
sin2x=0<+ x=kn.
sin2 r= 1 <> cosx = 0 e r = n l2+ kn 11:
0,25
sin2 x =ll2e cos2x= Q 49 x = t I 4+ ktr 12
Vfly phucmg trinh dd cho c6 nghiQm * = k7 (k 4.
4
z d.o tli6m)
-a c'6 y = 8 - x, thO vdo phuong trinh thrl hai cira hQ
0,25
0,25
€x2-8x*+t+@-59 e@=-x2+8x+9
f-*'*8x+920
*
tft' * eh' -t6x +n)=l *' *sx+ef
f-ts x <v
<>{
[x'-8x+16:0
(3x=4.
Ix=4
Suy ra he dA cho c6 mQt nghiQm duy nhdt j ., _ ,
l"v 4,25
o:rs
III
(1,0 tli6m) Ta c6 t =2'pI4*
ix'
2
Trang 4Cflu Edp 6n Ei6m
0,25
I
4 Vqy I =)-*
e
IV
(1,0 tli6m)
LSAC ta c6 AA',CC',,9,S' ld c6c dudng trung tuyOn n6n
G, cintam gi6c vi
,scr - 2cts', (l)
AA' t4r ftqng x6t
tdm
'i\
ic\
'lt{ \
-i".A
0,25 Tucrng tp, trong LSBD ta cfing c6 BB' cdt DD' t4i trqng t6m G, vd
^sG, - 2c25',, (2)
TiI (1) viL Q)suy ra G, = Grhay AA',BB',CC' vd DD' d6ng quy.
Tt gia thii5t ta suy ra A'B'll =Jrcn, 2 ' B'C'll =!ne,C'D'll 2 =Iut
2
D'A'll =!ac Do d6 (A'B'c'D;)tt(aaco) vit A' B'c'D' h hinh vudng eqnh
2 Hcnrnir4 s'A'/l= ]s,a, mir Sl L('encn) nOn ,S'l'I {A'B'C'D').
z
va a
2 0r25
Ydy vs,.u,u,.,r, =
i t' A' ft{A' B'C'r) = i
23
doa
_.-=+
24 24
0,25
v
(1,0 tti6m)
Ta c6
xo -zxt +8*+l)x2 -Zmx+m'*4>-a, vxe[*1,1]<>
(*' - * * *f r-4vx e [*r,r]o Hfi(r' - * * *)' > 4 (3) DAt t = xt -x, tac6 t'=2x-I.
'r
0r25
4,25
Trang 5Din 6n
,{
:fr
Do d6 xe [-t,t]€>/€ rl
;,21 surra (3) <+
tl1fik
+ m)' > 4.
X6t g(r)=(r+ m)'; s'(t)=2Q+m).
11
.-m
44
-m>2em<-2
-1< 44 -m<2eJ<*"L
Q* *)'
4
Trang 6Cflu Din rin Di6m
nhu *, !
4
nOu- 2<m<!
4 nilam < -2.
tTf +*)'
( i)'
0
(**z)'
Suy ra niq(r + *)' 2 4 e
L+'l
lf( *-1.j' = o
llt ,o) l*.2
ll*'a * | 4
l it z)' > + L* <-4'
l.l*.-Z
hoac '4 m>2.
v0v; cdn tim ld m < -4
02s
VI.a
1z,o oi6m; L (1,0 iliSm)
Ggi hinh w6ng cAn tim li ABCD, do n(*t,Z) kh6ng thuQc duong th5ng
2x * y-1 = 0 nOn dudmg thing ld phdi ld <lu,crng ch6o BD Ta c6 C la <fi6m d6i
xrlng cua A qua BD, ggi 1 ld tdm cira hinh vudng BD c6 vdc to chi phuong
uQ,2), do AC I BD n6n v6c to ph6p tuy6n oiua AC W n(t,Z) V{y phuffrg trinh cia
AC liL
(x +t)+ Z(y -Z)= 0 €) x +2y-3 = 0.
4,25
Ta c6 tqa dQ cfia I li nghiQm cua hQ
[x+2Y =3
lz*-Y:1
Suy ra tqa d0 ctra C(3,0)
lx:1
c>i
nei, U,2 = 5 n6n clulng trdn tdm I bfunkrrth IA c6 phucrng trinh
(t - 1)' +(y -1)' = 5. Ar25 Tqa d0 cria B,D ld nghi$m cria hQ
{G-t)'*(v-r)'=5 e, l2x- Y =1
v{y B(0,- t}, c (2.,2) ho4c B(2,,3} c(0,-t).
Jx=0
l.v=-t
[x=2
Lv=1.
0,25
2 fl.o tli€m)
V6c to ph6p tuy6n ctta (a\(B) n iQ,z,-l), F" a (u)tt(p} Ta c6 a(o,o,e) (o),
I-d-el
a/;(d) =4+a((a\,(il)=q.
+|;l)'
0r25
Trang 7Cflu Eip 6n Di6m
a\\al,lp))rz,v[y R=2
Gqi / ld tdm cua (c), khi d6
aQ ;@)) = d (r ;(B)) olLx + 2 v z + 6l
3
lzx+2y-z-61
M4t kh6c I e A, nOn toa d0 cira li nghiQm cua hg
lx-/+z=o
l2x+y+22=1 €
lzx+zy-z:o
I
I ]n=*IJ
l, =!
Le
n-lJ'
3J vfly: M[t cAu (c) c6 phuong o* ('
+)' ( (,-t)' =o
0r25
YILa
(1,0 iti6m)
Phuong trlnh zz -22 + 5 = 0 c6 hai nghiQm phtrc z, =l+2i vit z, =1-2i 03s
YI.b
(2,0 tli6m)
l.(1,0 tli6m)
- Dudrrg th6ng d* di qya di€m o c6 einrr nim trong dtd'nttidn, Ao Ait A;-i"?tn;Ai
: Iraplg4q dUqriDO dei AB nh6 nh6t khi vd chi khi ,qE IOt 0,25
- Trlc le d* di qua O vi nhfln OI(2;-l) ldm VTPT
2 (1.,0 di6m)
Ta c6 MAz = MBz = MCz = d'z(M;(d)\ MA2 = MB2 e | = z (5) 0,25
MAz = d' (u;("))o 3(x -l)'? +3y' #(r -1)2 = (x + y + z -1)2 . 0,25 Thay (5) va (6) vdo phrrong trinh cui5i tren ta nhdn dugc 6z - J e s - I
6
VII.b
(1,0 ili6m) zJt -6i-3J1+i
_ -(Jl -zi * si * sJi) _ - eJi -ti = _2,D _ i
'll