ĐÁP ÁN TOÁN L4-2011(DDT HN)

TRCSTqG TEtrT SAG FUV T'EI *Ap AN - T'E{ANG Bstreg s'sgs sg{€t PAH E{Qa E"AN Hv G7 tfiztzw1l} e{0lN , T'o6nu, umoi a N$i dung cho - ^ 2x-5 -,.) 1- ,.,- I = , 0. Khi m:2:) y= / Y - x-Z " x-2-' (x-2)' TiQrn c$n dimg x = 2, tiQm cin ngang y :2'Ei0m dac Uiat ( J;O) ; O;|) Phucrng hinh: xY e (x+ y-4)m+ .ltoy-4=0 O< '[3-xY=g - my = rnx - 4m + 3 ching Vre 3 - xY = 0 dirng v6i Ym +1' I lr=1 lx+y-4=o ltr=, otr' -4x+3=o€lJ"='' LL'''=t cO dinh lA A (1; 3); B (3; 1 - Phuong trinh dudng thing qua A c6 hg s0 g6c i =1r+1. rcrt \t -3 = 1G -l) e y - :x *;. (d t) 2\-"' r 2 2 - Fhuong trinh clubng thdng qua B c6 h0 sA gOc 1 b y + =){r- 3) <+ , =1* -f,. Orl- Giao diAm ciia (d1) vdi ox ld c (-1 ; 0), cria (d2) vdi ox ld D (1;o). J Khoang c6ch gita (d1), (d2) cfing ohinh ld chiAu cao cta hinh thang Vpy, diQn tich hinh thang Phf,i tim li: 5 =(AC +BD\t=(Ji3.*'#= Xdt hai trudrng hgP: 2a"[j; aaa <2x e {zx-z> o J1x' +x+4 <2x-2o l-r"'+x+4 > o I f-:"t * x+4<(7x-2)'z=4x2 -Bx+4 4n;5 20 {lJ.=-=-' 3 J13 3 :- i"t t4 o1-1{x{; fzr'-0" r o (4 lt'''=, is 4 s 4 *1 s' o\1.,=; (v\ t<1e27<28:duns-) IJC>- lz 2 BiiII/I 1 1 0.5 ong duong v6i: 2a'[4; a xa > 2x a't4r' + x + 4 > 2x -2' Nh$n xdt rAng khi x < 0 thi 2x-2 < 0 n6n b6t phuortg toinh trOn sE tiring khi -3x' + x+ 4>-0 <+ -1 = t 11.Vi x < 0 :> -1 < x < 0' a J 0.5 a l"orzr=21lopi1 ,-'l z o f'orr' = 1, ,nuu mdn (*) vi sin2x : xf * o 'l bieu kign sin2x r 0. (*). Vcyi tfi6u kiQn ndy, phucrng trinh ffiong ouong vo.t l-2sin2xcos'x I 1 - o < = - coszx-l €' cos' 2x - 5 coslx + I = o' -s z-""''" I '- 4 i CAU HI BAi XV r = [t [(1 + cos x; in(l + s inx) - lrr(l + cos;r][dx : {i nrt+ s inx)d;r * [i ror r ln(l + sinx)d;c - [i fn(f + cos x)dx. Chri y ring nrSu .lo ", ' Jo "'- ""- / -:i- Jo lr AaTt: x '2 ^Toro'no trri fi lnlr + cos x)d:r = Ii t, + sin l)(-)dr = f m{t + sin t)dt = f m{l + s inx)dx' Y 4y, I= j'u "out lntr * lin*Vt' D?t t : l+5inx, ta c6: Jcosx ln(l + s inx)dx = J tn{t + s inx). d(l+sinx) : llntdt =lnt.t - l r.L.a, = tlnt -t +c= (1 + sinx)ln(l+ s inx) - (1 + sinx) +c. JJt l+ :> I : (l+sinx)ln(l+sinx)-lsinxl =21n2-1. t' Gqi ffiacdi6md6i ximg cria C qua B vd qua D. S C F LEFC +A le trung ctiam cria EF*(MNP) di qua A. Theo da bdi' ta ph6i tinh ti T/ -A ' SAPMN JU Vrou., 0r5 005 v"".," s,4.si/.sP 22 4 l/,u*u 2 .fac6: /sANp =Dtar)iv'Di ::-a-'3'sANl' -1. !auv' /rnr,r- sl.sB.st 3'3 9 vrro,,,, 9 Vrrr,, =l-Vrnrr, - '- Vr^rr,, 9 Vrnur o Vdy, ti sO hai phdn trOn vd duOi bane j 11) ') =-, = - 323 I (x=l DE th6y: ]- - ' Id nghipnn cfia hQ v6iYm. Ngodi ra n6u (xo; Yo) 1d nghigm ly =m cria hQ thi (ys; x6) ctng ld nghiQrn cria h6. vfly, da hQ c6 nghiQm duy nhSt thi m = ; 2"+2'-2"), N (2b+2 ; 2b ; -b+1), P (2c ; c ; c+l )' Gii sir M thudc (dl) c6 tqa d0 M (a+l Ba di6m M, N, P thing hdng khi m trung diiim MP, tatim dugc M ( -14 ; cing phucrng vot MP. Sir dgng gin thi5t N ld 11 -28 ; 30 ) ; N (-17 ; -1s ; ?) ; P ( -20 ; -10 ; -9 )' '?," X6t trudng hqp: ") x> 1: Bdtphuongtrinh ban dAu <+ fnt'f e f @)=lnx-Jx*f 'O' (") I 1 I -i zJi-x-r Ta c5: -f '(x)=t ^ v z = 7-' x '/Jx 2 ZxJx Theob6tdingthr?ccdsi: x+ i.>zJ;=.f '(x)<0 khix> 1' f(x) nghfch biiin trom [r;+o) + f(x) < f(1) : 0 khi x ) 1 :) Bat d$ng thirc (*) CAU Vla Ggi B, C ld hai dinh cdn lai crla tam gi6c dAu thi B ( -m; n)' C (m; n)' Tam gi6c ABC dAu nQi ti6p elip (E) khi vd chi khi: l*' *!' =l lntz +4n' =16 i16 4 €j l'- ^ +4 l3m'=n'-4n+4 l4nt' : nt'+n'-4n Tri he tr€n tim du-o.c : ,=-3 (n:2lopi vi A= B =C), tt d6 nz=J€ no* 16.,8 o -,"-^rr#-76BJt *) 0 . x < 1: Bdt ddng thftc ban ddu <> rnx ,#.e f (x):lnx-G*;;'0. (**) 2",[i -x-1 Gi6ng tr6n ta c6 f '(x) = < 0 + Hdm s6 nghleh biiln tr6n ( o; 1):' f(x) > f(l): 0:) eat dang th{rc (8+) tl6ng" Cdu vIb 2,0S I @ (n r 0 ) ld hai dinh con l4i cria tarngtilc otsc. Khi d6 tam gibcoBC d6u nQi ti6p (p) e It:. r::;:;rrim tiuo'c ffi : 5, n : 2J1. T* d6 Soec: nJ\. I,00 2 Kho6ng c6ch gifra dr vd dz O** # Gqi q ld goc gita d3 vi m$t phang @) ta t; v/ co Slnq= 3 . 1,S0 C6u vHb I,0s Xdt hem s6 f(x): x'- 3x. ,.t I a co bang Dlen mlen: :> f(x) =3x'- 3:0 € +1. "+ + ,f'{" @,25 Xetba trulng hgp: *)u<-l *v <-1. -Vihdmf(x)ldd6ngbi6nh6n [-oo;-1) n6nf(u) <"f (v)'f(v) 14:] ut-3u' 'l^ v Jv+4. *v>-1. - Vi hdm f(x) c6 mQt cpe hi duy ntr6t tai x : I ndn: (v) > f (l) = -2, (u) < f (-1) -L. :>(u)-(v)<2-(-21:4. *)-1.u(1:>v>-1. U,75 5 Vi hdm f(x) nghich bi*5n h6n [_t;t] nen f(u) . f (-1) :2' Ngodi ra tr6n khoang (-1;+*) hdm sO c6 mQt cuc tri duy nh6t tai x : I ndn f(v) > f(1) : -2' VAy f(u) - f(v)<2-{-21=4. *) u >l=v>1. Vi hdm f(x) d6ng bitfn fr€n [1;**; ndnrf(u) < f(v) + 4. =] u3 - 3u . o3 - 3v + 4. ,Ju,-JSr;: VrX f/"e;