Chfng minh rlng vdi moi gLd tri crla k dudng rhing d lu6n cit P t+i hai didrn A, B phan biQt.. Xdc dinh f*atictra k dd do4n AB ng6n nhflr.. Chrrng minh r[ng: MBC lb tam g!6c vuOng.. X6c
Trang 1Truong THPT L€ Xoay
Nam hoc: 2Ol'1, - 2012
DE THI KHAO SAT CHUYEN DE LOP lO I,AN I
Mdn:Tor{n-KhdiA+AB
j' Thdi gian lim bii: 150 phrit (kh6ng kd thdi gian giao dd)
CAu I (2 ttidm): Cho Parabol (P) c6 phuong trinh: '4y = - '-vd didm MQ,-l) Gqi d li duong thing qua M
vi c6 hQ sd g6c li k
a Chfng minh rlng vdi moi gLd tri crla k dudng rhing d lu6n cit (P) t+i hai didrn A, B phan biQt.
b Xdc dinh f*atictra k dd do4n AB ng6n nhflr
Cau II (3.5 tlidm):
1 Giii phuong trinh sau: xt +rl x' +I I =3I.
.t (z
2 Giai he phuong trinh sau: lf^+ l'1+ 3x - 6v = 0
lx'+xy-3
3 Tim m dd phtrong trinh sau c6 3 nghiOm phan bi€t duong:
x3 +2(m-x)(mx-l)=mx'
CAu III (2.5 tlidm): Cho tam gi6c ABC bidt A(-2,3); B(4,3; vi C(1,0); Gqi G lii trgng tam cfra MBC
vd I ld didm trOn canh AB sao cho L{=2IB
a Chrrng minh r[ng: MBC lb tam g!6c vuOng ]
b X6c dinh tga dQ ctra didin D tren dudng thioe BC sao cho dulng thing ID song song vdi dudng
thing AG
Cau IV (2 diim):
r r sln2c cos'a ' 1 Unung intnn rang: I -:- = Sln d.cos a
1+cota I+tana (Vdi gii thi6t'cdc bidu thrlc dI cho ddu c6 nghla)
2 Cho a, b, c ld c6c sdthuc ducrng th6a mdn: a+b+c =J&
Clrring nrinh rang: ab +bc + caZg(a + b + c)
zags - _-
I{o v} fdn thi sinh: Sd bdo danh:
( Cdn bQ coi thi kh6ng gitii thich gi th€m)
Trang 2oAp AN * THANG udvr
I a.
2
lho Parabol (P) c6 phuong trinh: y =+vi ttidm M(0,-t) Goi a li ttuong thing qua M
rh c6 h€ so g6c li k
IMR: vdi moi gi6 tri cfia k tludng thing d ludn cit (P) tai hai diifn A, B phan bi6t
1.0
- Phuong trinh dudng thang d li: y = lu-|.
- Phuong tdnh hodnh d0 giao didm cira d vi (P) li:
2 -+ =/tx- 1 e x2 + 4lu- 4= 0 (l)
4
C6: A'=4k2 +4>0,VftelR.
Do d6 phuong trinh (1) luOn c6 hai nghiQm phAn bi€t x,,xrv6i Vft e IR -+Duong thing d lu6n
c6t (P) tai hai didm A, B phdn biOr v6i V& e IR
0.25
a.2s
0.25 0.25
- Gii srl A(x,y,);B(x,yr) trong d6 x,,x, li hai nghi0rn ph0n biOt ctra ptr (1)
t = .r;l"r-I; /" = fuz-l
- Ap dgng dinh lf Vi- et ta c6: x,* x, = 4k;x,x, = -4
- Ta c6: AB2,= (*, - *r;' + (r, - yr)' = (t + tc,)(x, - ,r),
= (t + k )[(", *,,,)' - 4*,*,]
DAu"-"xiy ra<+k=0
Vay vdi k=O,thi AB*,n = 4.
0.2s
0.25 0.25
0.25
Dk: VxeR
D+t Jr'z+f f = t, dk: r > 0 -+ xz =tz 11
Khi d6 phuong rinh:d6 cho tr6 rhlnh: tz + t - 42 = 0*
[; = U'
,1,,
,
^
[x=-5
Vdy phuong trinh dd cho c6 hai nghi€m lh: x=5; x= -5
0.25
0.5
0.5
0.25
Trang 3Giei h0 phuong trinh sau;
Ta c6: (r) \ / <+ {ff*+y)+ 3x-6y= o(l)
lx(x+ Y)=3(2)
- Ta thdy x = 0 kh6ng ld nghi€m cira hQ.
-XEt x* 0 Khi a6: (z) e x+ y =1 (3)
The (3) vdo (i) ra dugc prr: U- *Z* - 6y = 0 <+ 3y, +3xz - 6ry = 0
,-,8)
o(t- !)'=0<)x=!
Thay x - y vdo (2) ta duoc: 2y' =3 *> y' 3 =t* Y:llt E r.r=*
V4y hq phuorng trinh d6 cho c6 hai nghiQrn li: (x,y) =
[,E,1EJ,[
E
,rlz
I;
IJ
- ,rlzt_
o.25
0.25
0.2s
4.25
m tli phuong trinh sau c6 3 nghi€m phfln biQt d
Ta c6: (t) * x?(x-m)-2(x-m)(mx-t) = 0
T-
_. <t (r- *)(*' -2mx+2)=o <+ l^;':
lx'-2mx+2=A Q)
Vdy phuong trinh (1) c6 ba nghidm phan biQt duong *, m > 0 vd phuong rrinh (2) c6 hai
nghi€m phdn biOt duong kh6c m
l*> o
lo'to
l.
e{sto '>
I
l"'o
l*'-zm2 +2*0
Vay gi6 tri cdn tim cira mld: m, JZ .
0.25
0.25
Cho t1m gi6c ABC bidt ,4(-2,3);
didm trdn canh AB sao cho IA = ZIB
Ta c6: trB = (6,0), AC = (3, -3), gg = 1-3,-3)
Khi d6: AC.BC = -9+9 = 0 *+ dC t gC .
YQy MBC vuOng rai C
XD tga dQ cria diim D t.Cn
Trang 4f *^ _rn**u**, _=2+4+l _,
L, -le*la*lc =3*3*o =2
- Vi t ttrugc cqnhAB mi IA = 2IB n€n ra c6: Vi lZW
.l*, -xn=2(xu-x,)
t 1
ly,-yn=2(yu-y,)
- Goi toa dQ ctra didm D h: D(x,y)
- Vi D tlruqc dqdng thing BC non 3_D ctrng phucmg vot Ee
Mn BD=(x- 4;y-3); Ee =?3;-3) '-3-3 -+ x-i -+*"-y=l(l)
- Theo gt h c6: ,ID cDng phuong vdi AG
Md,,lD= (.r - 2; y - 3), 7G =(3; -l) - + = + -+, +3,y= I I (2)
- Ta c6 tga dQ trgng tdfn G ctia tam gi6c ABC lh:
Tu (1) vn (2) ta;c6 h€:
0.25
0.25
0.25
0.25
0.25
0.25
Chrrng minh rnng: , sin2 4 cos'a
= Slll 4:COS 4
sln- 4 eos-a
laco: vt -t _+
_r, sinl a + cos3a_, _ (sina+cosq)(sint a +eosta-sina.cosa)
= t-(l-sina.cosa)
= sina.cos a=VP (dpcm)
Cho a, b, c Ii €6c sd thr;c duqng th6a m6n; a +b + c = Jobt
Chung minh ringz ob +bc + ca) 9(a+b + c)
Vdi moi sd thuc x, y, zta lu0n c6: x2 + y'.+ z' 2 xy + yz + zx +(x+y+z)' >3(xy+y+a) .
Ap dqng BDT tr0n v6i x = ab; y = bc; z =ea ta duo c:
-+ (ab + b c + ca) > Ji.J
"b".J a + b + c = Jj.@ -.:]+ b + clJi a a 111
rl
Trang 5Ir , 13
Mrt kh6c ta c6: a +b + c = mZ'
lk,jjj,jf_ = (a +b * dW_
-+ Vc+b+p >3J3(2)
Tt (1) ve (2) ta c6: ,ab + bc+ ca > g(a + b + c) (dpcm)
Dau "-r' f6yrakhi vichilthi:,a= g=s-= ! :'' i'
0.25
0.25
.-( Thf qinh lim theo c6eh khdc ddng v6n cho didm rfii da)
tdtr g
Ra ttd virlipr66- 5o
1
L=
Nguydnfhi Hdng NgwydaThlEqnh
Trang 6so cD& ur viNn pHUC
rnU$gc rHpr r,n xoav
Ciu I.
't
l.
3
Cdu II
ri rnr sAr n4crr LAN r NAM Hec z0tt-2012
on mu nnON roAlr L'Op ro- icnot o Thli gian ldm bdi 150 phrit, kh6ng t<e tnOi gian giao d6
-_;
-2
Tim m dd phuung trinh sau v0 nghiQm
x-tit I_ x-3 _.,
x-l x
cho phucrng trinh: zxz +Z(m+r)x+ mz +4nt+3=0 c6 2 nghiQm
",,x Tim
gi6 tri ldrn nh6t, nh6 nh6t cria bi6u rhric A:2( t * rl)_3*,*,.
Gi6i phucrng trinh:
Js,i + Jut = 4x -9 +zr,\f - 5, *2
(t
i ciai he phucmg trinh:
{o,l._rtiiiul:_;o=0
2 cho h6 phucng trinh:
{.:T;::r=|!, rl_o*, Tim m ae ne c6 nghiQm duy nh6t ( x,y) th6a mdn r*t
=z
Cdu III.
i cho rarn gi6c a.BC, ,'€n BC l6y di6m D sao chorp =1ae Gqi E Id di€m th6a mdn qEA+zEE +rfr :o chrmg minh A, E, o driig rrang
-
2 Nhfln dang tam gi6c biCr I ri" c =
G;-F
lo'@+c-a)=bo +co -ao CSu iV" Cho 2 sti thUc a,b th6a mdn a+b>2
Chring rninh rAng: at +b3 < aa +bo
Cdn b.6 coi thi khing.giirti thfch gi th€m.
Trang 7Toa;tu - lfJ,lh'D
4 ltL+b ,Fr<4
@-*; r * qx' i) (r-a) : Jic t J)
t tL /rz.x- + lt 4t-+J = zzt-;,t_
tL(n+z/ = 3 inl
NQj; nt*e=a qd h=*2- lRsn /1t"a7r*1t/"V/V
1>r da- ct* riui n I #, :
L3
J Dt 1n t,t
I
J
K
d
\ - z : taL+4tu +3
2
A : a(trrrn - )x,)(L = J(\n )
V /\ = *(kM)'-
fr-{
<n 2h= flvt"'+VVn+l+
L 4 => - 3,w'"'* A-0 w,
(n\+n+))
- f au-.t{rru -Jl
l*
* tl
a -,/
)ru aH ttl,
l8 ,r,, 4{r=, (rrt{ &=
b{ t4s LJ, = - t t\' - Lorn- 47 'htt LI, - 4J
Y
.,t
u:t-.t/'
-la*
: G't LTUZA = "'6
.l
t:Ua'
{il," th**A,a 6rN{ll=o tLhtor,'-,1
Trang 8xL? L
l-+ V t-,1
l' ':;''' I
= )rt-t {-L-4 -6-t
t.1
t trq/
14 t'-t-(:0
Vo' t=J -t* co, ,
,l/
T;P
f= - -( ho,
ll Tr L + ll r_a
u
i-f +,6 7ct
'!=) L
*6 -Jl x_+4x.>
; 2L= 2_
,/
&di'tr
I
1
fd)
yt'
tr- -;)44 + [y'= o (,{
}
b'-Aq F=u 4 tL-o ) (o,cr)
o _ U,r.,:a';V€, tUi-rf1l C./* ya
Lo
l- L =4J
XL
,,
L-'J
+'_
'4(xa,
tlVrr-, *l[;; =j
H,Gb + -tlffin, =;!
ll 5r s tt+ z- : { *
LL-'<)' J 6-2_*_z"o
) 3*-_:fr._t,
=
qt /*u, -l
) )(1 4qx_+
Eq
P1- uj l\, ,x
-, )
oz{-Etf
or3'-q
zs-
CI)zi-ry2f
Drt ,z -\ t- rl ,ll
Trang 9O = (144 -4[ra ,-f2
{\
I)Y = t'' -l) [v,,+r1 2
-*r'" t l\^
".t tqL rcI;
&t-4
luk=4
€r f l* (O
Lq7r4
/ ?_ + (tur.4
V= zwrS
)v+'l
I
I I I I I I
I
I
lq,r I I I
I
I I
D,;
I i
i
i
I i I
i
I I
iVse I i
I
i
iqi{
j
I t lo7,i'
d 'a?
.y,t+wGC*r"/\?-+1.'
t tq e (ar+oo)
v;,7t ,
e,
€,)
td
.*))
\=/
,e At E/ t) t/*-{*
o
Trang 10'fii
/sT
I
cd
C,-Xli, C-'
4t
o
d
4 4n4
A) Qa-{o
tl
qd'-b-&:k-[')L
t-pa'4"
Ja+ b
+2a-(12
H4
2-ze-b
4{ttnc- , #*
T
)1b,.e>
gj{rarte t'
za+ L
-:-€-+=J
{:,
A)
=- trt f't =b
0iLf
'r4-
vtI-€
v
"7 r^ -.
I [tt t tG .h
I
I
l,,2)
n* I I
6\ +b
+bJtLr-:/
€
{,, +b-L2/ 0 \
I
*) frl t,*,
).c
l)D otrr,*,r= q.L)i4)u Va.
w
*6
,hfu.u,