TRLIONC DHSP Ha NOI ru6r rHpr cHUyEN nt rru THrt DAr Hec naOn roAN IAN I trAwt Hec 2oo8 _ 2oo9 Th6.i gian IB0phtit ... Tinh d0 đi rtoen thing AB' rheo a vd s... oAp AN vA rueNc DrEMl...
Trang 1TRLIONC DHSP Ha NOI
ru6r rHpr cHUyEN
nt rru THrt DAr Hec naOn roAN IAN I
trAwt Hec 2oo8 _ 2oo9
(Th6.i gian IB0phtit)
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Ciu l (Z,O Aiem) Chohdms6 y=n*r*,*+ l)x2+1m2*4m*3)x+1
l Khdo s6r vi v€ d6 thi cria hdm sti khi m = - 3.
2 v'i gi6 tri ndro cta m, hdrm s5 c6 clrc d?i, cgc tiAủ Ggi x1, x2 li hai di6m cgc tl4i, cgc tiiiu cria hín s5, hdy tirn gid tri lon nhdt crja bi6u thric A = i*r'i-ri"'i -rl f
Cflu 2 (2,0 di€m)
y l Gini phuong trinh :
cos2x * cos5x - sin3x - cosSx = sinl0x.
Z Giili bdt ptru,rng trinh :
Cdu 3 (1,0 di€m) .
^74os'@jslog, (x-) Tim hq cdc nguy6n him cria him s5 :
x(x+-s;6xs-sx+1) ' Cffu 4 (2,0 diOrn)
cho hinh ldng tr-tr tam giiic d6u ABC.ÁB'C' c6 dO dai canh d6y bing a, g6c gita cluo*g thing AB' vdrn{t ptrang@e,C'C) bing ạ é 6vv
l Tinh d0 đi rtoen thing AB' rheo a vd s
2' Tinh di$n tich rn{t.ciu ngo4i titip hinh ldng t4r AIIC.Á8,C, theo a vdi ạ
g,Cdu 5 (1,0 di€rn)
cieihephuungrrinh
[r;?; !2n ,"ur, ,,!
gCAu 6 (1,0 diem) Chirng minh ring : i '{ i
*+fr+ ++-1oos/ 1 ,- 1 -
EJ*o + " + (Trong d6 Cl IẠs6 tO h-op chflp k crja n phnn tri)
<pCflu 7 (1,0 didm)
\
Trong m{t phing vdi rr€ r-o: dg oxy, cho tam gi6c ABC vdi Ẳ; -1), B(r; -2) vd rrg*g
tdrn G cfra tarn giric niim tr€n duon! tning d: x-+ y - 2 = ọ iray tim tga dQ diem c, bi,it
rang di-6n tich tam gidrc bing j.
i
i\rtt.-t - -2i
! | - - '
, J' L]
\i r I -.2
1\
-l
r200a Í uzoog/
trdt
a9
Trang 2oAp AN vA rueNc DrEM
l (1,25 Aiemt
Gidi hqn: limr,-*_ y = + co
, limrr- Jr= - .,b .
Su bii5n thi€n: y, = ?x2 - 4{ - ,y:'= io U - O f,,ia x = 2.
Do tl6 hdm s6 d6ng biiSn trong nrdi khod.ng G*; 0) vit (2;+oo), nghich biiin hong
Vdim=-3,thi t=:*t Tap x6c dinh : R
khoang (0;2).
Cgc tri : Hdm si5 y dat cgc d4i tai x:
0 vd yc,p = y(0) :
;,
Hdm sti d4t cgc ti6u t4i x:2 vA.!c.r = l(2)= - 13.
Beng bitin thi€n
1
Dg Ai (Hqc sinh tu ue t,
?t,*r":: r !.',' = Of O,c6 y"(l):0 vd y,, dr5i d6u khi di qua x = l, n€n di6m
(l; -; ) le diiSm u6n yd cfing ti tem d6i xring cria d6 thi
?6 Ai cit trsc tung r4i ai6m qo; ] )
?o d6 A ton ntrdt bing 3 khi m = - +.
Ta c6 y'= 2*
I::1::"Lgu1:oc ti6u nri vd chi *, ri:0 c6 hai nghiQm ph6n bigt x1, x2 hay a': (m + l)2 -2(m2 * 4m + 3) > 0 (+ m2 * 6*;;-; il:ffi :;.
$eo dinh lf Viet, ta c6 x, 1x2 = - (m + l), xr.xz =lm2 + 4m + 3y.
l"l:l A= li(*' r 4m * 3) +2(m + l)l =|l*, * a** z1
Tanhinth6y,vdi m e(-5; -l)thi: - 9S mf+ gm+ 7=(m*4)2_9 <0.
1 (1,0 di6m)
CAU
II Phuong trinh dusc viiSt ve ftng
cosSx - cos2x + sin3x + sinl0x _ cos5x = 0
<+ - 2sin5x.sin3xI sin3x + 2sin5x.cos5x_cos5x=0
c+ cos5x(2sin5x_ l) _ sin3x(2s
@ in5x-1)=0
Trang 3Voi sinsx= 1 z 0 [ u*=r+Zkr
Vdi cos5x = sin3x (+ cos5x = cos( _ fx;
J4f tfln nglri0m crlaphuong uinh la S ={ t30' I + 3II
S t
[ = -'i* i,,G' 4'
I * T, * * T,-l * r,,).
2 (l,o ci5m1
Bdtphuong trinh itugc virit vA dang
J2* tog3(x - il = log, (x _ ) trl DFt t: log (x - i),*r d6 (l) trd thanh
,lffist o [z iii t, e [,, _l] I
= o
Vly t4p nghiQm cria bdt phyone trinh H S = tf,; + o).
rac6 f-*$ffi-Iaffi:/ffi
- fy4-r\'lv t-4 1\
t i;ffit *.
++ (cos5x - sin3x)(2sin5x _ l) : 0
I sin5x - 1
.Hlz lcosSx = sin3x
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CAU
rv l Gqi M H trur_rg dirim cria BC, thi
AM 1BC, AM J- BB' n€n AM J mp@B,C,C),
do d6 ,$fu = a Tqong tam gi6c rnr6ng AB,M,
tac6 AB,= N - "€
1 Gqi I vi I'lAn lugt li t6m hai ttriy ABC vi A,B,C,
Klti CO, rtng di6m O cria n, lA tankh6i ciu neoei
-tifo Utotr langtru .
Ta c6II' = BB'.
BB,2 = AB,2 _ AB2 = 3az -2 -;ri"{-a-=ffia2q3-+sinz a1
BB.'=*;lffi
a^[j
4'i
S l
-;- - t_
suy r4 trong tam giric vu6ng oiA, c6 c- ,l : -;ri"" E ,-1- lg - sinz a, IA=
Trang 4va oA2=ot2+tAr=
#(g-+sin2a)- f Gqi R ld brin kinh m[t ciu, thi *i= #; (z _ +sinz a) +
Khi d6 dign tich mflt c6u ngo4i ti6p hinh lang trr.r ld :
?2 3
s = +z' (# cs - + rin'"1 + *) = ara2ffi + 5.
Tt he phuong trinh suy ra x ) 0, y, 0.
Cfing tu hQ phuongtrinh vd theo Uit aang
th&c C6si, ta c6 .
6V3 = 2^/7 + y =,/F + lF + y >3W = 3\/74= 6iE
Deng thric xdy ra khi vd chi khi G =y =2W:+ x = \876.
Vdy nghi€m cria h€ phuong trinh la' x: ffi vity - 21/i
Trudc ti6n ta chring minh c6ng tfrri*
.F=;*,l.ffi *.ht-Il (r)
Thft vfy,
e#+#=HP.ffi
_ k!.(n_k)t.(n+1_k+k+1)
(n+r)l
Hffi=#+I C6ng thric (l) dugc chu-ng minh
Ap dpng (l) vdri k di rir O Adn ZOOS;G
m=;ffi(il.a;)
" =4q(#.*)
uz^ooe 20Lt
L 2OOg
m=#(m.4''t
Do 6$0, = crool,n6n I6y tong tone vd cria 2009 deng thric tr6n ta dusc
12008 I 2OOg / 't i
vd'' *;+* '.#t=ffi(eh + # + + eh)
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Trang 5VII
Tir gin thi6t ta suy ra Segc = 3Snea =+ Sesc :1 uU dO dei AB = r,E
Phuong trinh tluong thit g AB : x-y - 3 : 0.
0ls
Gii sri G(xc; 2 - x6), khi tt6 khoang cich tir G ttdn AB la 1 : l2xq'sl
' ,tz
suyra segc =i* n + l2xc-51 : t *
[}; I 3
0,25
Ta c6 tga tlQ aiAm C(+c; ys) dugc tinh theo c6ng thric
f*o =f t*^ + xs * xs) lto=lct^*vB+vc)
Vdi xc:2 thi yc - 0,i khi d6 thay s6 ta dugc Xc:3, Yc:3.
V6i xc: 3 thl yc = -1, kfii d6 thay sd ta ttirgc xc : 6: Yc: 0.
V$y c6 hai <tiAm C th6a mfln bii to6n: C1(3; 3) vn Cz(6; 0)
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