$&i I l.]i:r t: r: i.,rjr,? - .r: : .:l:-,:.,, .:.,' i :. ;:,,." L,au ,1 I x' I rl /t . ,^ ll;c.(.,) pn rm KSCL DAr HQC NAM 20:-2LAN ]'HrJ'I MOn: TOAN; tcnOi g ] 1!.':: I. PHAN CHUNG : 1. Khio s6t sU bi6n thi6n vd v6 dO thi (C) cua hdm s6 (t). ",,,'2" Cho di6m Mthu6c dO thi (C) co xM =m.Tlm c6c giltrf thUc ciratham s6 * de tiep tuy6n tai M cit d6 thi (C) tai hai di€m phAn bi6r A. B sao cho MA:3MB (B nirn gifr'a A vn M). - = J Jt- {'iu II (2,0 ttihnt) \ L= i . Gi6i phuong trinh: 4sin2 x.cos x -Zsinzx -sin" = .or" \ ;:. Ciei bAt phuong tr:inh: *'.2" + x.z"nt +IZ <3.2" + 4x2 +gx . Cfru III 1l,b mAnr) lllr' gi6i han , = SYa Cf,u IV (1,0 cli,dnt) Cho hinh ch6p S.ABCD c6 d6y ABCD ld hinh vuOng cpnh bing a. C4nh bOn SA vqdlg g6c ' !' r - "y vA SiA: "Ji . Goi B', D' lAn luqt ld hinh chi€u w6ng goc cua ctiiim l. vOJ lnat pltang da tr0n canh SB vA SD. Chirng minh rdng cqnh SC vu6ng g6c v6'i m5t phing (AB'D'). Gsi C' ra giao di6m .uu ,oat Jr'-L;;;t;;il;. ffi,?;"r;;'ffitilj:16*;,;;:: :,:S r*f5rsvurvurvL{cL.r.rraL}rudrrti\rrD JJ / vulgalulDL. lrnntllellcncllaKnoicnop).u'L.'D'. *17 Cffu Y (100 tli6nt) Cho bas0thucldr6ngAm x)y,zth6aman x+y*z:Z0Iz, Timgiritri u#- ,rirAt criabi6uthrc: p -{F(",*l| .iFb} +,).iF1; .4> ,[b+V I[. ]tI{41\ RIENG (3,0 diam) '.1'h{ sinh clri dngtc chgn nrpt trong hai. phfrn A ho{c B: A. 'Ihco chu'o'ng frinll chuin (lf,rr YIa. (2.0 di\rn) Trong mdt phing Oxy cho di6m Me;a)vd'ducrng tr.on (C) c'o phirong tniih: x' + y' =2x * 6y + 6 = 0. i.l. Vi6t phuorrg tr"inh duong thdng d di qua ditim M cit ducrng tron (C) rai hai di6rrr p. e 1r sao cho M la trung di6m cua PQ. )(1Y -0= D 2. vi6t phuong trinh duong trdn (c') d6i x#g uii ao*g troq(c ) quu di6m M. Cffu VIIa .(1,0 di\nl Chrmg minh ring: \^-J) + (t- {) = + (zn+1)3" =2'".C),, 1+2.22,-,.Cln*,+3.22,-2.C:,,*r * , +(2n+t)cliil(r-*), *f +["T ; lu lc^a L1'.fi)U{l;D-) \'l L Tld'i gian ldm bdi; 180 philt, kh6ng k€ thdt gian phdt di D0 g6m: 02 trang crro rAr cA cAc rni srNH (7,0 diam) r (2,0 ili\m)Cho hdm s6 y - -t +3*' -! (t). '22\-'l- Trung Il2 www.VNMATH.com oii.,rrung rn4t phang Oxy, cho tam gi6c'ABC'c6 A(l;3). Ducrng phdn gi6c trong cria goc B :nim tr6n duong thing d, c6 phuong trinh ld: x - y -l = 0 va'dulng "uo *n6t ph6t tt dinh C lina* tr6n ducmg th8ng drc6 phucrng trinh lA:x + 3y + 2- 0 . Vi6trphuong trinh duhng thang ::.i i," i: r. l"r.:.'j:', 2 2 : 0"'" 2.Cho elfp (E): l* * =t, gqi F,( ;0) '-'.,. r \ / 25 16 ,.r ,cho dO ddi FrM co gi6 tri 16n nnat. . Cf,u VIIb. (1,0 itidm) Cho mQt da gi6c l6y trr 3 trong 32 di6m Ar, Az, , A:z tlat canh cita da gi6c l6i ArA2. An. ': I ' ::,:, i, r I i i{,. .F +cfwri rr., ',. - '- .Jl.l :'; l l. ., . ilr 'l bao nhi6u tam gi6c c6 ba dinh lutn gi6c khdng c6 cqnh nho lir .chria cpnh BC cira tetm gi6c ABC. *- ly-q>O : ,: ld,mQt ti€u di6m cira (E). Tim di6m M tr6n (E) sao [t '4 l6i ArA2 A:2. Hoi c6 sao cho mdi canh cira frrq L.rE'T] lrri I (Cdn bd coi thi khdng gidi thich gi th€m) Trang2/2 I.j ! -rl I www.VNMATH.com I. (1,0 tli1m 2. (1 iti PAp AN VA THANG EIEM ,T I '(2 ili0m ) T0px6cdinh: D=R, Su bi6n thi6n: - Chi6u bi6n thi6n: !'= -2x3 + 6x; !'=0g;s=0; x= rE hoAc x=$. Hdm '6 AOng bi6n trdn c6c kho6n* (-*' 6) va (o;"-6); nehich bicn tron c6c khoang (-16'o) va ( 6;**). - CUc tri: Hdrm sO dat cuc dpi t4r x= ,6 vdL x: 1f, 1 lcn = 4 . Hdm sO dat cgc ti6u tpi x:0 vd lcr - Gi6i han: lim y: lim ! =-oo. -t-+-co x-)+o - BAng bi€n thi6n r DO thi Ta c6 diOrn 0,25', 4,25 4,25 0,25 I 2' + : 0 - 0 ,+ '0.',,_ Phuong trinh ti6p tuyiln ei M: ! = -Zm(^' -3)x+ |*- *U*, *I . AL Xdt phuong trinh hodnh d9 giao dii5m: -^4 1 -** 3xz -l=-z*(,n' -3)x+ 1*^ -3,r' -!. 2 2 2' z', Trang Il7 www.VNMATH.com <+ (, - *)' (r' f*=* .'-tt I L"t +Zmx + -6)=o (-)' EAt /(x) = x2 1- 2mx +3m2 - 6 - Ti0p tuy6n t4i M clt 1C; t4i 2 di6rn A, B phdn bi6t kh6c M khi vd clii khi phucrng trinh (*) c6 2 nghiQm ph6n bipt kh6c m, hay : Iaco: ' l*n'*u :3m2 - 6 (t.z) Do IWt : 3MB vd B nim gitaA vd M n6n M{ = 3MB , tri d6 suy ra : 1- -?- -_1 -^fr r\ ""A - )xB = -2m 0.:) rr (1.1), (r .rr,rr,r)^?::r)r*r* [". : _2m ),i,,*"=r*'-u *l;: =Q I" " l'6 1", - 3x, : -2nt l* - tJt Do vAy * = -Ji; m:J7 (tnOu mdn didu kiQn (**)). YQy m = -Ji ; m = rlz n cilc gi|tri cAn tim. Ta c6: 4sin2 x.cos x - Zsin2x - sinx = cosx {+ cos"(4rin" - 1)- sinx(zsinx + 1)= g <* (zsinx + l)[cosx(2sin" - 1)- sin*] = o e (2sinx + t)[zcosx.sinx * (cosx + sinx)] - o l1 lsinx (l) c+l 2 [z ,o, x. sin x - (cos x +:sin .r) - 0 (2) Giai (1): (tr e z) . (th6a mdn). , Gi6i (2): 2cosx.sinx - (cosx + sinx)= 0 -r 2rnx +3rnz 3m2 -6=0 (2,0 *.4 \ Cllem) 1.4,.'t1t iiiii,f :,1 T I * L+kur sinx=-t ol 6 2 I *=A+k2n L6 Trang2lT 4,25 www.VNMATH.com !i': ,, : +2sinr.cos *=t'-'1. Khi d6, (2) tro thinh t2 - t- I = 0 e t -LS 2 Taco: .or["-z'']-1-S ("" 4 ) 2J, I *=o -u,""or[t-f.) +k2n | 4 \zJz) el ) ,_'., ,(k=z).6naamdn) l*-o*ur".orf!-fl +kzn | 4 \2J2) Vpy phuong trinh dE cho c6 c6c nghigm ln *= -1+ k2n, * =!+ k2tr, x=L- ur""orf!:gl + katr, 6 6 4 -[zJz ) ' n (t r-\ .rr=1+arccosl -^ p l+kzn (kez) 4 lzJz) \ J,25 *,25 2. (1,0 clidm) Ta c6: *'.2" + x.2"*1 +r2 <3.2*' + 4x2 +Bx o *'.2"' +2.x.2" +12-3.2*' -4x2 -Bx<o e 2*' (x' + 2x 3) a.(*t + 2x 3) . 0 o (r' +2x -t)(r' - o) . o l*'"*2'x'-3<o [-3<x<1 1 ci I - <+'-3<rc<-Ji. lzu -4>o [l'l'.D Truong hgp 2: , ^ llx<-3 [*'*Zx-3>o ll'"- - 1"",-'" <+{1"t1 <+I<*<Ji. l2* _4<o [u.o Vfly tap nghiQm'cfra bdt phuong trinh dd cho ld S = (-:; ,n) , (r;rD ) , i],25 l +,25 ] I l ,).2s 1 '* l l r.),25 ] ili (l,r) di6m) 1,0 &i6m Ddt r= lleea _ 1 997 +997 x n1t 'J yLJ Trung3/7 www.VNMATH.com I\/ ,(1ro ,di6m) Khi x-+0 thi t-+1. Dovfly r=1,T? W+=lig 997 997 i 1994 2lee3 +tteez + +r+1 vay L=r 2' 0,25 I 0,25 0,25 1,0 fri4nt * sc -1,(da'D') laco: BC LAB sa t(encn)* srl a BC LAB, Mi lB'r ^s, , ur] * BC' L(s'll) + AB'L (SBC) + AB' I SC . Chrmg minh tucrng tg, ta co =+ AD'L SC . Do d6, ,sc r (aa'n') I {,. ta- \^' Ir pl1 r'\ j I t ', \:: lt:::: t |.l , *-* i .+',1>/ I tt /\' i,: ,l ', 0,25 0,25 * Tinh th6 tich cria kh6i chop S.B'C'D' Trong tam gi6c vu6ng SAB, ta c6 AB = a, SA = ali, SB = ali vd AB'L SB o t2 zall Suy ra SB': ot = ^sB 3 ,Jj : ) n,li Tuong ty SD'= '-J Ke eC' tsc .+ ,4C' . (eA' n') . M[t khdc, tam gi6c SAC li tam gi6c vu6ng cdn n6n C' li trung di6m cria SC, do do SC'= rz. zali zalj Ia co v' u.r,o, _.sB'.sc'.sD' = I ='o' ' Irr.oro ,SB.,SC.,SD oJS.Zo.oJi 2 9 0,25 t'ad. vs BCD =! stIBC"BD =o'-&- (d.v.t.t) suy ra v, u,,,o, =t + = +(d.v.t.t). 0,25 V (1,0 *. ;. cjrenu 1,0 iti6m V6i vi, y 2 o ta lu6n co (* - y)'(x + y) > o o ("' - r')(*+y)> o Vx,y 2 o. <+ x'+ ),t 2xzy+xy' e:.("'* yt)>1.(*ty+xy') lu i (1,0 I *.;. I cjren I I I I l i, I I :. I Trang4lT www.VNMATH.com n 4.(*' +y')> (r*y)r- e'.fi7[r*r] >2(**y). Tucrng tg ta c6: ,.fiZp' + "J >-z(y + z) ,S7pt;;, >2(z+x). 0,25 4,25 Do d6, P >-a(x+ y + t) ep > 8048. D6.u ':' xay rakhi vA chi khi x = ! = , ='Or" . vfy gi6 tri nho nh6t cua bi6u thric r uirs@thi vd chi khi 20t2 e- y-L -a J 4,2:5 . 4,25 VIa. l. (1,0 cti6m) Dulng tron (C) c6 t6m I(1;3), b6n kinh R:2. Md IM : Jr<R + M ndm.trong dulng trdn (C). Dunng thing.qua M lu6n cdt du*g trdn (c) tai 2 di€m phan biet p, e. Dc M le trung di€m cria pe th\ pe t tit . Duong ina"e pq di il; M ,'ha;lfr:f] ; r i u* 16; tr pi;ap i"t&. -T- ;6*' phuong trinh: €> Jr+ y-6=0. VAy phucrng trinh duong thlng,cAn tim ld x + y -6 = 0 , 0,25 a,25 5,r5- 'Jr25 2. (1,0 itihni Eucrrrg trdn (C) c6 tAm I(1;3) vd b6n kinh R: 2. Gqitdm cua dudng trdn (C') le I' vd bdn kinh cua (C') la R', ta co Md M(2;4) :+i <+{' . €I'(3;5). 1 /,' = ZYr - Y, lY,,= 5 - Vpyphuongtrinhcila.dudngtrdn(C,):(,_3),+(y_5),=4 fi,lr- ^r25' t)"25 VIIa. 1,0 di4m :* :"-l I T a c6 : (2 + *)"*r = Cln*r22n+1 * C),*rZ2' x * Cl,*rTzn-t *2 * : . . * Cl:il *r,.t . LAy dao hdm 2 u6 tu duoc: (Zn+f )(Z + *)'" = C),*,2'".+2.C1,u22"-t *+ + (Zn +t)Ci;ilx,, Clrgn x =7ta duoc: (zn + l)3" = C)nnt22" + Z.Cl,*,22n-t * 3.C|,u22n-2 * + (zn + t) ciiil. e (2n+r)32" -22",c),*t +2.22'1.C|,*r+3.22n-2.C\,*r+ + (zn+r)c::i:.(r. ru). 0,25 tii,25 I "-****l ,) -'){ i 0.25 i VIb. 1. (1.0 cli€m Trang 517 www.VNMATH.com t*t:.J1lli:i.f ,,. Si.i:?.i+r: '.'.:,i l' l;';: ' ;. r iF.;Y+jr.''1 iiir').'tL.t': i:.1 i."i., t. '. ;.,t : . iii.'-i i.:;, ,. li;;,r.,:: : !r+_;l.i t-"- .: r I r iijls;,liri. i.:.j,-ii. , ;',' i-rliii:.li;:. .i:: ,;!*iji:::1.,,i,. l, .:' :: r' ' ,t ;;., . I i*i)l:J. i ".*,:;,',,t' i:i!: j,:. : 'i.;1.,L.'.; .:!,, EZ.,!.rij :f:.i:.i.:lr '; r r "'l ,t ') 'j (2,,,A di6m) ij. Gc.ri A' ld di6m ddi xring vdi A qua dr. Phuong trinh duong thing d di qua A vh vu6ng g6c vdi dlco dqng x+y+c=0. Md A thudc d ndn ta co e: -4. Do c16, phu<rng trinh cira dudng th[ng d: x+ y-4=0. \ 4,25 Gqi I ld giao di6m cira d1 vd d. I(hi d6, to? d0 cria I ld nghiQm cua h9 phuong trinh: f ." _s (r*y-4=0 l'-t ,(5 3) I " <+{ eIl -:-1. lx-v-1=0 I 3 \2'2) \ r' Iy=- t-2 Do d6, tqa d0 cua A'(4;0). 0,25 Ta co cl2 vu6ng g6c vdi AB n6n phucrng trinh cria dubng th6ng AB c6 dang 3x-y+c'=0, Do A thuQc ducrng thdng AB n6n ta tinh dugc c' : 0. Phuong trinh cira dubng thang AC: 3x *.! = 0. 0,25 D" B it gilo die;iCtri AJmg thdng d1 vli ducrng thing AB n6ntos clQ cria B 1A nghiQm cua hQ phucrng trinh: r_ _ I [3r-v=o l" ; ( 1 3\ { r Y €{ L <+Bl -l:-"1. L* ,,-1=0 - 1 3 - "t, 2' 2)' I V - "t'2 Mil A' ln diOm d6i xirng vdi A qua d1 n€n A' thuQc'ducmg thing BC.' V4y phuong trinh cria duong thlng BC: x -3y -4 = 0. . 0,25 2; (1.A di€m) Ta c6 c:3; Ggi 4M'= (x + 3)2 ^ { Suvra F,M =|x+5. JI ) M(x;y) thuQc (E). -, (3 -\' *y'-l ;x+5 | \5 ) 0,25 0,25 Do MthuQc elip (E) n6n -5 (x ( 5 + 2s 4M <8. Do viy, F,M sg. Diiu ':' x6y ra khi vd chi khi x: 5. Vdv d0 tliri cua doan FrM dat ei|tri lcrn nh6t khi vd chi khi M(5;0). 0,25. 4,25 VIIb. 1,0 diiirn 1,0 cti6m +)S6tamgi6cc63dinhld3trongs632dinhcfiadagi6cl6iddchold 9jz:129!.0: * *. ;iffi atim Jdtu* siil;6 t C?nh;t f .a"n ia;e"h;,fi:du eia" i6i i:i:- * canh. 4,25 0,25 Trang6lT www.VNMATH.com Trang 717 www.VNMATH.com . yLJ Trung3/7 www.VNMATH.com I/ ,(1ro ,di6m) Khi x-+0 thi t- +1. Dovfly r =1, T? W+=lig 997 997 i 19 94 2lee3 +tteez + +r +1 vay L=r 2' 0,25 I 0,25 0,25 1, 0 fri4nt * sc -1, (da'D') laco: BC LAB sa t(encn)*. bd coi thi khdng gidi thich gi th€m) Trang2/2 I.j ! -rl I www.VNMATH.com I. (1, 0 tli1m 2. (1 iti PAp AN VA THANG EIEM ,T I '(2 ili0m ) T0px6cdinh: D=R, Su bi6n thi6 n: -. :3m2 - 6 (t.z) Do IWt : 3MB vd B nim gitaA vd M n6n M{ = 3MB , tri d6 suy ra : 1- -?- - _1 -^fr r ""A - )xB = -2m 0.:) rr (1. 1), (r .rr,rr,r)^?::r)r*r* [".