Gqi M li tlitim trdn duong thing AB, sao cho AM = 2Md.. Tinh khodng cdch tir tlitim B dffor mpSCM... Do M =2W, n€n khoang c6chttrA tliin mdSCM beng 2 lAn khodng cSch tu B d6n mpSCM.
Trang 1TRUONC DHSP HA NQI
rcr6r rHPT cnuvtN
CAu I (2 di€m): Cho hdm sii y =
NT THI THU DAI HOC I,AN III NAN,r ZOOS
Nidn thi, To,in Thdi gian.lAm bdi: 180 phrit * **.
x2- zmx+ m2
v l X6c dinh tAt ce cac gi6 trf cira m d6 ham s5 d4t cgc ti6u tai x = 2
'Jz Tim c6c gi6 tri cua m d6 tr6n d6 thi crla hdm sii
1t; tdn tei it nh6t mQt di6m mA ti6p tuy6n cria d6 thi tei dii5m d6 vu6ng g6c vdi dudng thing y = x
Ciu 2 (2 di6m)
V t Gidi phuongtrinh:
aX aX
stn"; - cos"; L ,+"i"- :3cosx'
,,1 2 Giai he phuong trinh :
v CAu 3 (l di6m) Tinh tich ph6n : t = [/3
x2+r+.,/@Txf
r/Cau a (i diem) Cho tri diQn SABC c6 g6c AB'C = 90", SA = fift = 2a,BC= a.,,/3 v.d SA vu6ng g6c vdi rn4r
phing (ABC) Gqi M li tlitim trdn duong thing AB, sao cho AM = 2Md
Tinh khodng cdch tir tlitim B dffor mp(SCM)
I
CAu5(l <li€m) Cho0 <a<b<c <i<e vd a+b+c*d*e= L
Chung minhbAtdingthric
'
'4U"
+ be +cd+de) +cd(b + e-a) ,_ S +25
CAu 6 (2 di6m)
qo l) 'l rong mflt phing v6i hQ tga dQ Oxy, cho tam gi6c ABC c6 ctinh A(-2;3), duong cho CH nim trdn duong
'/ thing : 2x+y -7 =0 viduongtrungtuy6n BM nimtrdndudngthing : 2x-y+l =0.
Hay vi6t phuong trinh c6c cqnh vd tim tga d6 trgng tdm G cira tam gi6c ABC.
t
'/ 2) Cho hinh hgp ABCD.A'B'C'D' Tr€n ducrng thing AC l6y di6m M vA trdn duong thing C'D t6y diiim N I
/cM
I sao cho MN // BD' Tinh ti tU ;'
Jcart (l di€m)
Xdc dinh tep hsp cdc di6m trong m{t phnng phftc bi€u di6n cdc s6 phric z.th6a min diAu ki6n :
lz+il l-l = l.
I z-3i I
f t*t *y=4+,tW
ll,*' -ztg2=te1+h
D4r kiiin thi th* Idn tdi vdo ctic ngdy 18,19/4/2A09
Trang 2DAF AN rovr r,lr mON roAN
(Thi thti'DH IAn III - 2009)
CAU I.
xz-2x+2m-m2
1 (t,O Oiem) Tgp xdc dinh: R\ {l} Ta c6 y'=-
(x-1f Gii sir him s6 d4t cgc ti6u tei x = 2, suy ray'(2) = 0 hay
4 - 4 +Zm-mz - 0 <+ m = 0 ho{c m= 2
2, tad6u c6 y' J
#=+ y' =o <+x= o ho4c x=2'
Mat khric y' > 0 khi x e (- o; 0) u (2; + co) vd y' < 0 khi x e (0; l ) u (l; 2)'
Do d6 x = 2 ld tliOm cuc ti6u cria him s5
Viy, d6 th6a mdn bdi to6n thi : m:0 hodc m = 2
2 (l,0di6m)R\{l} Tac6Y' = 7;17-.
T6n tai tii5p tuy6n cua <16 thi hnm s6 vu6ng g6c v6i tluong thing y = X khi vd chi khi phuong trinh sau c6 nghiQm :
x2-zx+zm'mz (x-1)2
I
- -l
{=r x2 - Zxt1m-m2 = -(x-1)2,x+ I € 2x2 -4x*2m-m2 + 1= 0,x+ I (*)
Do phuong trir,h (*) c6 it nhAt mgt nghiQm kh6c 1, nCn ta c6 hai trucrng hqp sau :
a) Phuong trinh (*) c6 hai nhiQm phdn bi-Ot' hay
L'=4- 2(-m'+2m+ 1)>0c+ 2m2-4m+2>0 ++m+ I'
(L' = 2(m- 1)2 = 0 b) phuong trinh (*) c6 nghiQm kdp x *1, diAu niy tusng duong voi I - -a a 1 (loa)-' T6m lai' m * 1' ^l
-e - " txt = xz = 1
Ghi chir : Niiu thi sinh giii bdng c6ch :
[r,orrioo trong d6 f(x)=2*z - 4x*2m'- m2 + 1, thitr.' 0,5 di6m
CAU il.
l (1,0 c1i6m) Phuong trinh dugc biiSn d6i thdnh :
ci{ - "ixr +,i,{.,o{) =} z+rin")"or*
*=Gi{ - or)tr +1sinx; =!r*sinx)(co{ - 'i"iAi{ *
'oJr)'
V6irinl - "or|=0c+sin(| -fl=0<=+x =|*lkrc,kez'
22La'z
r Voi r + jsinx =-i(t+sinx)(sinj + cos)<+sini + co5 =-;(ptndyv6nghi€m).
VAy, nghiQm cira phucrng trinh ld
' * = ; +zkn,k eZ'
w.
Trang 32 (l,Odiem) DiAuki€n x * 0, y> - 2 .
Phusng trinh (2) <+ lxl = 4 + 2y thay vio pt(l) ta dugc :
't7+'z
- ( v>0 7
: tv, ;;=ey, o Y=; +x=t5'
fx=S ft= -5 vay hQ phuong rrinh c6 hai nghiQm ,
l, = * ' t o = 1.
cAu rn ( l,o dii!m).
Tac6 ,= [/t xdx -,'o ,G 6qnffixdx
DAt r= ,@-*z ql thi dt= #|v6i x:0thit=l; v6'i1=y'Jthit=2.
Khi d6 t: I:# = frr+r1*aqt* 1) :fr a * t)+,11,:zbf3 - lz).
cAu ry ( 1,0 di6m).
Do M =2W, n€n khoang c6chttrA tliin mdSCM)
beng 2 lAn khodng cSch tu B d6n mp(SCM) .
Tir gia thi6t ta suy ra AM: +, BM = + ,
arlsl 1
CM:: Ke AK l- CM, thi CK J- mp(SCM).
3-'^
Do goc ffi = 90'+ Fm > 90o n€n tti6m M nim giea
CvdK.
Ta c6dAKM - AcBM =+ AK: tt=l"
= n"13.
cM ^lTl'
Ke AH J- SK, thi AH I mp(SCM) vd AH ld khoing c6ch tu A ddn mp(SCM)
Taco:
;;;=Asz *1;;= q^r*&=G = AH=
E.
V{y, ktroing crich tt B ddn mp(SCM) bing - + {43
CAUv ( l,0rli€m) Ttgiathii5tsuyra 0<"s i.e6tdingthricduscbiiSndoithdnh:
-fL/^
a[b(c+e) +d(c+e)] +cd(b+e -u)= * o a@+d)(c+e)+cd(b+e-a)S 2S+
x2+r+.,ftr+xf
Trang 4) ,
a(b + d)(c + e) + cd(b + e -a) = ^ (ry=)'* 1.*o*!*._")3= a(r:a)z *,'=}|,',
Xethdms6: f(a) =+*ry,vdi 0<as f .rac6 f(a)=frf - 5a2-4a+1)>0,vutto;f)'
Suy ra (a) d6ng bicn trcn (0;
Jl * it"l s (; ) =f tan'*)
cAu vt ( 2,0 di€m)
- l) (1,0 di6m) Dubng thdng chira c4nh AB vu6ng g6c vsi CH n€n nhen vectq il(2; 1) lim vecto chi piiuong'
Do d6 dudng thing AB c6 phuong trinh : x -2y + 8:0'
Suy ra, tqa d0 diem B ld nghiQm criahQ phuong irinh :
| {"-r, * I = o *1i,.=_3-B(2;s).
^Yr -' -^ *-'-^ '+;A '-? Et thu6c BM'
Gqi C(x; y) thuQc CH, suy ra trung di€m c0a AC Ie M t7"', '
( 2x+ v-7 = o"
o * [;lf +c(:;r).
Tqa d0 cria diSm Cli nghigm cria hQpt,
\, * - # *,
Suyra, BC : 4x+y- 13 =0 vd AC : 2x+5y-11 =0'TrgngtdmG(1; 3)'
2) (1,0 di6m) D[t EA= d ,EF =E 'EC = d '
Tac6 EF : d +6+ d, vi MN i/ BD' n€nffi =Ufr
v[y Mfr = kd +tci + Pt (1)
Mat kh6c Mfr =ft +7d *Vfr = nVt +fr +
^V6 Trong d6 At = i - d,7? =6, c6 = d -E'
Suy ra ffi = n(c- - d) +E + n1d -87
: (m -n) d."11 -m) 6 + nt
So sdnh (l) ve (2) ta c6 hQ phucrng trinh : | 1 - m =
t n=k
vQy tvtc =; AC h"Y
J CAUvll ( 1,0tli6m) Df;t z=x+yi + z-3i=x+g-f)i vit z*i = x*(y+l)i
Di€uki€n:x*0vdY*3-D6 dang chimg minh du-oc tinh chAt l:l: #
Suv- l#l=l elx + (v +1)il2=[x * (v-3)il2 e.x2+(v+l)t=x2*(v -3)2-ev=1'
K6t lu{n: Tip hqp c6c di€m trong m{t phdng phrlc bi6u di6n c6c s6 phfc zthbamtutdi|u kign :
l#l= ttno"*^fur=t
-k fk=713
t-lm=2/3
( n=1/3
,,@