-fRU'oNG rr-rpr EAo ouv rrj un Tnr rgtl D4.I HQC LAN lv (2710212011) mON roAr'{ Hqc xHOr a Tttdi gian lam biti lB{) phitt; khong k€ thdi gian phdt di PHAN cHUNG cHo rAr ca cAc rni SmH Ciu I: Cho hirm'6 Y: mx-4m-r3 x-m 1) Lrhao s5t vir vE d0 thi hdm s0 khi m= 2' 2),chf11;;; ;il;;; ili; ;1 "-ao t i .h'r.'? 1u6n di q"i lT di6m c0 dinh A va B. Tri hai di6m A vi B hdy lap phucmg fiinh cua hai duong thangc6 hQ s6 g6c bang 1,5' Tinir diQn tich hinh thang gi6i han bdi AB, hai dumrg thdng niry vd trlic Ox' CAu il; i; GiAi b6t phuong trinh: 2a{4*1ra <L x 2) Giai phucrng trinh: sinox+cosox i ^ 1 5 sin 2x 2 Ssin 2x Ciu III: Tinh tich phAn: ft ? ,. ' rl+cosr l= lln(l+srnx) &. 'l 1 + cosx '0 CAU IV: Cho hinh chop S.ABCD, co da,v ABCD lA hinh vu6ng, dulng cao SA' Gqi M lir trung rli6m SC; N, P lan luqt nam tr€n SB va SD sao cho * = + = 1 ta, phang GvO-lP) chia hinh -'^" sB sD 3 chop thdnh hai ph6n. Tinh ti sO AC tich cuahal pUan dO' Cfiu V: Chimg t6 rang vsi mqi 916 ui cua tham s0 m, hQ phuong trinh sau lufin co nghiQm: l*+*Y+Y=2m+1 i 1xY(x+ !)=m2 +m' X6c dinh m ae ne cO ngtriem duY nhdt. pHAN mtNC ( rgi SINH CHi LAIVI MQT TRONG HAI PHAN A HOAC B ) A. Theo chuong trinh chuAn: CAuVIa: _ j ' l) Tinh diQn tich tanr gi6c dAu nQi tii5p elip (E), ;+'r- = i. nhan dii5m A (u;2) la dinh i,ir trpc turig lirm truc ddi xirng. . ?)Trong kh6ng gian voi h0 uuc tga dQ oxyz, tim ba di6m M, N, P lan lugt thu0c c6c rJucrng thing: (d,) + ='=' =+, (dr) + =*=+t tor) i =+=+ sao cho M' 1 2 -)' \"j/ 2 7 -1' 2 1 i N. P thang hAng, dOng thoi N lir trung di6m ciia doan flrang MP' Ciu VII a: .lnx1 Cho x > 0, x *1. Chimg minh rang: ,_l.G' B" Theo chucrng trinh ning cao Ciu VI b: 1) Tinh di€n rich tam gi6c dAu nQi ti0p paraboi (P): 1p :2x, nhan dinh ctra parabol ldm mQt dinh vir tryc hoanh Ox ldm trgc ddi ximg.' 2) Trong khOng gian voi h0 truc tga dQOxyz: a) Tinh khoang crlch gita hai duong thdng: I x =2-I (o,l? =+:-i,^ o', 1r-::t*, l.! - r b) Tinh goc gitadudrng therrg (dr) + =+=+ voi m{tphang (a):x+y-z+Z:0' \r' 4 | -2 Cfru VII b: Gi6su u1v. Chrmgminhring:u'-3u < v3-3v+4' Gi6m thi coi thi khdng gi6i thich gi th6m-: www.laisac.page.tl 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; . -2 Cfru VII b: Gi6su u1v. Chrmgminhring:u'-3u < v3-3v+4' Gi6m thi coi thi khdng gi6i thich gi th6m-: www.laisac.page.tl TRCSTqG TEtrT SAG FUV T'EI *Ap AN -. 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. (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 thi5 t N ld 11 -28 ; 30 ) ; N (-17 ;