Giaib6tphucrng trinh: t rfuJ ='-r.. Trong m{t phdng vdi he tryc top ctQ Oxy.. LQp phuong trinh dudng cao BH cta tam gi6c ABC.. Cho hinh hQp clung ABCD.A'B'C'D' c6 ddy ld hinh thoi cqnh
Trang 1rRuoNc Nim hqc rHpr 2010-201I LE xoAy nt 1'ffr XSAO SAr M$W: 1OAN_ CUAT fH6t I.{tgNC g, b XHO1 tz
Thdi gian: 180 phtit khdng kE thdi gian giaa di
Cfru 1:
1 Khi c6 he n:0, sO 1,.g6c nh6 vitit phuolg trinh nhdt tii5p tuy6n vdi dd thi hdm sO UiCt tii5p tuy0n ct6
2 Tim m d€ ao tni hdm sd cdt duong thing y: 1 tai 3 eli6m phan biQt c6 hodnh d0 l{p thinh
"6p sO nhdn.
Cdr 2l
1 Giai phuong trinh: 6 tun 2* * 1 = -Z- .
cos2x cot2 x- l'
/\2
2 Giaib6tphucrng trinh:
t rfuJ ='-r.
Cdu 3:
Cf,u 4:
1 Trong m{t phdng vdi he tryc top ctQ Oxy Cho tam giSc ABC cdn tpi A, c6 phucrng trinh 2 c4nh AB, BC lan luqt ld 4x + 3y - 8 : 0; 2x- y - 4 : 0 LQp phuong trinh dudng cao BH cta tam gi6c ABC.
2 Cho hinh hQp clung ABCD.A'B'C'D' c6 ddy ld hinh thoi cqnh a; g6ca2o=60o Ducrng ch6o lcm cria d6y bing duong ch6o nho cira hinh
hop.
a Tinh AA' 1;heo a.
b Gqi I ld trung didm cria AA' Tinh khoing c6ch tt I tdi (BDD').
Cffu 5:
Cho 2 s6 duong a,b: a+ b :4 Tim gi6 trinh6 nhAt cria bii5u thrlc:
M =(r*o*!).+1r+r+j),
l**L =,
1 Gi6i h€: ' ] lu*+= J' z
l' ' J;-'
2 Tim c6c s5 hang khOng dm cria ddy {x,,} co sO hang t6ng qu6t:
_ 30 AI.o
n, -T- , r, (v6i n:1;2;3 )
rn+2
Trang 2Z- a;-r ::-;iti :' i:Ft
: r: ,l;,i r.iJ
or rru xnAo sAr cuAr rtIgNG rcH6r rz
rvr6N: roAN- ru6r B;t
Vdi m = 0 Hdm s6 trd thinh: y = x3 -3*z ,r2, -1
hs tpi
) -at
(xo;/o)c6 hg s6 g6c k = y, (xJ.
cAn tim c6 hO s5 k= -l vdi ti6p di€m M (1: -l
Pt hodnh rtQ giao rliiim:
x' - (m+3)x2 + (2 + 3m)x -2m = 0 e(x-l)(x-2)(.r -m)=0
0,25
0,25 0,25 0,25
y? l, ?,1-l0p thdnh c6p s6 nhdn, ta c6 c6c truong hgp:
Cdp sd nhdn ld 1,2, m =) ln = 4.
C6p s5 nhdn ld l, m,2 => m = *Ji
Cdp sO nhdn ld 2, I, m =) m: Yz
"0 /cos2.r
* 0
4,25
0,25
0,25
Phuong trinh: c+ .'6 t*?t + I
cos2x cos2x
2sin2 x
=%
cosz x-sin2 x
<+ r6sin2r = -cos2x
etanZr=-*
'13 1r kn
C),I= +- 122 Thod m6n dlk
7t knt D/S: X = +- 122
DK:x>-4.
)' > r-8 <+ xz {2::!r+ + it' ).r-8
( '2+tlx+4 : )' r r-8 <+ "'!3::x
Trang 3€4-ala+x+4+x>x-g
9"{4'rra4
<> x<12
K6tlgp{iAu ki€n ta c6: - 4 < x < 12
DK:x>0;y>0
Dat: Ji = o:Ji = bv6i(a, b > 0) ta dugc:
at +! u, =z o)
6z aL-=2
He o) {"'o *t =2b (l)
lb'a +l = Za (2)
L6y (l) tru (2) theo v6 ta du_o c :
ab(a - b) +2(a * 6) = 0
fa=b
<>l
lab =-2
Vdi a = b, thi5 vdo (l) ta rtugc: at -Za+ t = O
lo ='
x=y=l
1-t +.,6;, 3 -16
'42
V0y h€ c6 nghi€rn: 1,rry,11:f ,T,
_ _ 30 Al.o -n, -7n+lg
=
" P" P - nt
>0c> n2 +7n-1gs0<+-g sn<2=lr=t
Vfly dey d6 c6 2 s6 hanemdG6ll
n=l c6 x1:10
n=2 c6 xr:g
AB c6 vec to ph6p tuyiSn
BC c6 vec to phrip tuyiln
nnu(4;l Gin sri AC c6 vec to phii
fi*(2';-t)
;b\fu2 +b2 *o
Trang 4Do MBC c6'n+ AEC = Bel<+ cos(frrr, fiu") =
"or1i*,41
o,[7* =lza-nl
e3a2 -4ab=0
la=0
0,25
+ khi a : 0 , AC c6 I vecto pfrap tuyfi co toq d6 (0;l).
BH.L AC n9n BH qua B vd c6 mQt vecto ph6p tuyiin c6 to4 dQ(l;0) pt BH: x:2
+ldri a = 4b/3, AC c6 l vecto ph6p tuyiin.b tq oo(q;:).0oai vi ec'trtrng an;.
D6p s6: Pt BH: x:2
a.Gid thi6t suy ra AI^BD ddu
+ BD = q ) B'D,= a
Ap dgng dinh lf cos cho MBC ta co
AC = all
Do duong chdo lfn cria d6y bing
dudrng chdo nho cria hinh hQp nen
BD'- AC = alj
- AA'- BB'= ^lBD,z -pt pr2 = ali
0r25 0,25
0,25 0,25
b Ggi M ld giao cta A'C' vd B'D,
A' M L (BDD') + d(A,,(BDD,)) = A, M
Md AA' lt( Rlll)t\ r /l(I (/lf\r)t\\ - )lilt /Dnnr\\ tt t t lt ^t t^ OJj
Md AA'//(BDD') + d(I,(BDD')) = d(A, ,(BDD')) = A,M = A,C,/Z = o!^t .
2
ddng thric C6si cho ba s6 duilgta co
.(;)'.[;)' "T('*o*!
ng ual
t\'
+-l a)
Ap du
[t* "
0,25
0,25
0,25
0,25
[' *' * ;)' .[1)' (1)',' T(' u.*)
2+(a+b\+
z3 a2
+ M +L>2.!- 24
ll4
M{t kh6c dE chung mrnn - *: aba+b2
* M >343
4
D6u bing xiy ra khi vd chi khi a=b=2
vgy M ttat gi6 tri l6n nh6t Ld343/4 khi a=b=2