Tim to4 ctQ c6c iti6m B,C sao cho MBCvudng tai l.
Trang 1TRUd}vG TIIPT T"f XOEY
xAu Hec 2oro - 2o1r nt rnr rcrAo sAr crruytN prt r,AN rr
lvtoN roAN LOP IO-BAN KHTN
Thli gian: 150 pMt (kh6trg t6 ttroi gian giao d6) Ciu I1Z Oi6m;
t Giai b6t phucng o,no, €F < I (x e.R)
2x+5
a,pthda mdn h€
.ghfng minh ring sinl = Zsin(A+2c)
*.i [r'"1, +2otor)=1,o.(-*,*)
oreu Kten'
l"or1r* z',n) =|, o.("r+)
[z,laa2t.gy +3x=s,tto-x-+y ++ , -\
lx"+xY'=x'Y+6Yj
CAu II (2.0 tli6m)
Cho phuong trinh: x2 +Zmx-ilm(m+t)-Sl=0 (l) Tim lzd€ phuong trinh (llc6 2 nghiQm thgc
phdn bi6t 4,x, thoi mdn 4 +2x, =3
CAu III 1}.0 aieml
1 Trong m[t phing tqa dQ Orycho MBCvhi ,l(-t;t), hai tli6m B,C thir tg nim tr€n dudng thing
L,:x-y+4=0, Lr:3x-y-8=0, trung di€m Mctn c4nh ABnim tr€n cluong thang
Lr:2x+3y-4= 0 Tim to4 ctQ c6c iti6m B,C sao cho MBCvudng tai l t
2 Trong mft phing tea dQ Orycho tluong tron (C):(x-l)'+(7-15)'=3vd dii:n M(-l;3) Lgp phuong trinh dudng thing A qua M vit ti}p xric vdi (C).
Cfiu IV (2.0 di6m)
l Cho MBCthatmin mo=g
Cflu V (1.0 tli€m)
Cho 3 s6 thgc duong x,y,zthbamdn di€u ki€n r+y * z =3 Tim gi6 tr!nh6 nh6t cria bi6u thric:
D
- - 'i- ! -L- r
i x+Zya'y+Zza'z+2xa
**:-Het-'-H9 vi t6n thi sinh: , S0 b6o danh:
Cdn bQ coi thi khdng giii thich gi th0m
Trang 2- oAp An vA rHANcnErr
Lrru f:
+ nS hn tlwacacht;hdc cl,ingvdn cho.dftmtiit tla + Ddp dn gon 04 trang
I
i;
I
I
I
ciei b& phuong trinh' &;fi1l < I (x e n)
1.0
I s l-,6
l +.rS'-+
ol 2 n 2
l:) - ;
LZ
| , -zt+JrsT
I
{e-e x'-3x+l<4xz +20x+25 e3x2 +23x+24>0*l '
_ _',_6.,,i31
r
rS-o - -B+JigT - -3-& t t,/
tiu}ra
oXdt -rkti*#<0<t 3.x<-jtnou'an t)($',**)
xz -3x+132x+5
0.25
t.2s
0.25
0.25
2
(r,ye R)
't
Giii h$.phuangtrinh: [z,t+ -ztltt+ 3x = 5
lt' *t' =x'y+6yt
1.0
_.! {++Zx+8y>0 Ei€u ki€n: ' {p0-x-4y>0 Tac6 (z)ox'-8.yt +ryz -Tyt =*y-4yt
<a (r - 2y) (x2 +Zry + +y' * y') = y(x -}y\(x +2y)
<+ (r - zy) (x2 + xlt +3"'
) = o c+ {.r* 2y)l(".i")'
Khi do (r) o 2Jffi *31 = 5.ffiJ+4
a 2(J4 + 6x- +) + I (x - z) + s{z -.,,66:'5i) = o
*!orl=oc) x=2y
4' J
2+
0.25
0.25
025
I
Trang 3v?y h$ c6 nghiQm duy nh6t (4y)=(Z;t) 0.25
u
Cho phtrong trinh: x? +2nx*3m(m+t)-Sl=0(1) Tim n di! phumg trinh
(t) c6 2 ngtri$m thpc phnn biQt r,,.r, &d min x, +2x, = ) 2.4
PT (l) cd 2 nghi$m Jhgc phnn bi-€t r,, r; <+ A' = m' -l-ln(n + t) - Sl] > O
a 4m2 +3m+59 > 0<+ Ym e R
Khi d6 theo viet , [''*"'=4m (2)
lt'" = -3m2 -Jm-59(3)
Theo gii thiiit x, +2x, =3 (4)
(r rin (z),(a) = tl, ==roir;t *ay veo (3)
+ (-+m - 3)(l + zm) = -3m' - 3m - 59
e m' +3m -10= 0 e
l; = _r(th6a man) vdv
l,l, = _s
0.25 0.25 0.25 0.25 0.25 0.25
0.25
0.2s
ilI
I Trong rnlt ph*ng tqa d0 Oxyeho MBCvli A(-t;t), hai di6m B,C tht t.u nim
trOn dudng thing A, :x-y+4=0, Ar;3x-y-8=0, trung diEm Mcta cgnh
I
ABnimtr0n du&ng thing A, :2x+3y-4= 0 Tim top ttQ c6c diilm B,C sao cho MBCvu}ngt?i A.
8f
Tir giA thiiit + B(b;n+a) e L,, C (c;3c -8) e A,
rheo crtrung diiim + M =(+r+)
z.b-l*3.0*5-4=oeb M{tkh6cMeL,r>
t l.tE =(o:z\
Dod6{* \' /
llA =(c+l;3c-9)
Lpi c6 M8Cvu6ngt1i A
d'dt
e eE.,{C = 0 e 0.(c+ t) + 2.{3c -9) = 0 <> c =
K6t luin: ^a(-t;r), C(3;t)
0.5
0.5
0.5
0.5
2 2 Trong m{t phdng tqa d0 Orycho tludmg trOn (C):(r-l)'?+(r-15)'=3vi
di6m M(-l;3) LAp phuong trinh dudngthdng Aqua M virtiilpxric vdi (C).
Trang 4Ta co iC) :(* - l)' +(1 - I 5)i :A"o rent 1 = itr; tS), R = J3
Drrong theng Aqua M(- t;3f
+ A:ặr+ l)+D(y-3)== Q 4=y coc+hy+ a-ib =0(a, +b, > Ol
A tidp xlic vfi (c) = d\r,
^: )= R c+ lq'I
n 4i
111?- l'1 = n,1
JaT +b' e> 3(a2 + b2) = (2a +llb)' c> az + 4gb^a +l"i!b2 = 0 c) a =, -b[b t b-j $s
+ Chon b=l= a=-24t Jqii5 .
Vay co rtuong thing can rim : a
tlho A4Stlthoi nrdn ffio = c " Clirmg minh rnng sin ,{ = ls:ip (,a +?.C'l
'ra c6 :;in 4 = 2 sin {r + zr} | <+ sin r,= 2 sin (tr - B+ ci .= : sin (r - c}
M*tkft6c fro=ce4nf =4e,2 l;92b2 +2c2 -4, =4cz c>o, =-'2(b, -"ti(1.) 'Iheo DL sin c6
(*) e2(sint .B- sin'C) = *in' ;t <+ {l - cos2l})-(t* *,rs 2r-') ,= sin2 ;
e -2r;in(C+D)sin(C:*8)= sin2.4 e Zsin(8,-C) - r,in r do
si;r(B rC)= rinl > 0:+Dpcrn.
| 'Ifnh s,n(rz +zlt)
L
-z(il a r'i,
1, ọ (-
iO, -")
Zonrt:I,0.(":+)
l !
I
srn td -r
Ioosl'f +
12 Cho c6c siS a, pthbamen h.6 rfiiu kiQn
I a c6 sin(a +ZAWI\= I *r sind,= | =.r,rt s =l -]- ="
ll
zJt ( 3n \
= cosd = -, do 4 €
l- 2-;_" l rili5 Mpt khric: cos(ig+'2011n')=ị= cosp- -ị +sin; pr=t
it* =l?
*, sin 8=
v:!y
5n' I
t!€i tr'.-"I
- i\ 7')
sin (a | 2 P\= sin a c6s 2p + cos cv sin Z f, = : { -:'l * jig
3t.8.i J
fỊ,,
I -it;:,J'n
24
_:-_t
Trang 5"/
Tac6 sin(a+ mtflz)=|osiRa =J*"o*rr =t-i=;
M{t kh6c: cos(p+ 20 llr} = !(+ cosp = 1=sintd=l- I =15
'I
r :+ sin p = -+= l''n t'=
lcos2l =
f-l)=.8
[4/ 8
do pe(",+) B-r=!-t=-T
'88 v0v
sin (a + z: p)= sin a cos 2B + cos a sin, O =
: (-I)" + * =
*7 *2^m
0.25
0.25
0.2s
0.25
v
cho 3 s6 thyc duong x,y,zthlamxn diAu ki€n r+y+ z = j.Tim gi6 tri nh6 nhAt
cria bi6u thrlc: P =
-+
- +-x+2y' y+Zza '
z+Zxa
1.0
Theo BDTC6sic6 : x' _x(x+Zyo)-zry' _ _ 2ry, 2ryn
,, -
#T =, -1d;,ry.y,,
" -f,w *,y + y,) =, -|(2, * f)
rucrng t$ suv r^,
Tri d6 + P> (x+ y + z)-|t,* r, +2yz + z2 +Zzx + xr)
= (x + y + z) -|b *, * r)' = 3 - Z= I Diu ding thric xiy ra khi x = ! = z = |
Ki5t lufln: minP= Ikhi 16= y - s -l
0.25
0.2s
0.25
0.25
-H
Td trudng chuydn m6n thim dlnh
.{f
Nguy6n Thi HSnh
Wnh tudng, ngay 08 tfuing 05 ndm 201I Gi6o vi€n ra it€ vi Iflp ildp 6n
Bni Hii Quang