I'R{'GNG PTTE{ EAO NUIT TTI HA NOn Bn rrur B4r FIgc L.&N rn-lt^Ana Hqc 2010 - zfitt, na6m roAn E[ec xa6i a Thdi gianldmbdi: I8Tphfit +/- ( KH1NG XE rnU CUU rn irDE) / efurn: cho hdrn 16 y: (m+l)x+m x+m 1) Khno sit vi ve dd thi hem sO ttri m : 1. Tim tr6n dd thi nhimg di6m c6 t6ng khoing c6ch d€n hai tiQm can nhO nhAt. 2) Chtmg minh rdng v6i moi m l0 aO tni hdm sd lu6n tii5p xric voi mQt duimg thing cO a6n. @I!: Gi6i phucrng trinh: "'/i+ ti"" + Ji _siil = 2cosx . CAU ItrI: 1) Giai cdc phuong trinh sau: , a) (4x - l) fi'? +1 :2x2 +2x+ 1. b) log*+r p - Jt-zx*; =)- 2) Tim m d€ phuqng trinh sau c6 nghiQm duy nhAt. 3m-z= h C6u XV: 1) Trong m4t ph6ng voi h€ tga dQ Oxy cho tanl gi6c ABC c6 A(-2; l), B(3; 5), C(2;1). Tim tqa t10 truc tem H, t6m I ctra dudngfirdn ngo4i ti6p vd trgng tem G cira tam gi6o ABC. Chrmg minh H, I, G tha"g hang . 2) Ctro tam gi6c ABC vu6ng t4i A, AE : a, BC :2a.Haitia Bx vi Cy cring vuOng g6c voi m{t pb5-ng (ABC) -rir nim v0 mOt phia d6i v<ri mit phang d6. Tr6ri Bx, Cy lAn luqt 16y c6c diAm B', C' sao cho BB' : a, CC' : m. a,Yrn gi6 tri nio ctra m thi AB'C' ld tarn giAc vudng. b, Khi tam gi6c AB'C' vuOng t4i B', ke AH vu6ng g6c voi BC. Chrng minh ring B'C'H idr tarn gi6c vu0ag, Tinh gbc, glftahai m4t phAng ( ABC ) vd (AB'C'). e6g-Y' Chr.rng miiih ring khi a l0 thi hQ phuong trinh sau c6 nghiQm duy nhAt: 7 a =y -F- v a' a" - L'T I,*, i,,' ;- GiAm thi coi tfui kFr0ng gidi thich gi thdm EAP AN - Ttr{ANG $TEM TTTT TTTTT EAI IIQC LAN TT nnol.{: Todn, rtr6i a {Jdvl"! N$i dung cho di6m Ei6m 1/l 1.00 .2x+l^l laco:y= / ' x+l .r+l . Tgp xic dlnh : D: R\ {-1} ' Su bi6n thi€n I + Ta c6 : y'- > 0,V.x e D ' (x+t) + FIim si5 lu0n ddng bii5n tr6n D + Hdm sO kh6ng c6 c\rc hi 0.25 + Gicli nun' ri*, (, - t:) = - = Dudrng thing d1 : x = -l ld tiQm cfln Ctimg ',- t)=, tin [' x+t) = Euong thing dz I =2ld tiQm cgn ngang + Bing bi6n thi€n: X co -l *co v' + + +e) .l 2 2 -o 0.25 " DO thi: + Giao di6m voi t4rc ox, (-*,.) + Giao di€m voi truc Oy : (o,t) + DO thi nh$n di0m I(-1,2) ldm tdm AOi xtmg + VE hinh: 8.25 "1. '' . +i -=_'- -i1=it . t. ;t 1:"( .!. .:::{::-'::.'i';., v J 4 3 2 OI -t -2 -J -4 6.x I -4 -3 -2 + Gei vr(";z *) , ( voi a * -t) te di€m biit ry thuQc tlo thihim s5 + KhoAng cich tu M itiSn dr ld : hr : d(M,dr) = la+ tl + Khoing c6ch tu M tltin dz ld : hz : d(M,dz) = # th,*h,=la'{-fr=rffi=t +lt,+h, nh6nhitbing zkfi la*{ = # + Vdy c6 hai diAm tr€n ed tUftnOa mdn dii bei la [a=0 el la= -2 Mr(0;l) va LLt?!). 0.25 v2 1.00 +Tac6 1 y=@:-lEl! =fti*t X- x+m x+m + Goi ! = ax+b li dudng thing phii tim. Khi d6, hQ phuong trinh sau c6 nghiQm voi Vrz re 0. l**t- m' =ax+b ) x+m lm' [1' *o,y =' (l) (2) 8.25 (2)a m' =o(r+*) , (3) x+m + Trtr (l) va (3) theo vi5 ta dugc: 2mz m+l =It-Am x+lti 0.25 2nt? x+n7 1 I x+nl o,{o -r l)+ I - b m(a+1)+1-b a2 Lln ;ir 'lj 'rl (4) + Tir (2) ve ( ) suy ra: nr' [n (a + r)+ r,- aJ' 4m1i', o fu("+ 1)+ I - b7' = 4o*' e (" - 1)'*' -z{u -t\a+ t}n + (a - t)' = s (s) 0.?s Phucrng trinh (5) dtng vcri Ym * 0khi vd chi khi: V{y vdi Vm * A AO tni hdm s6 lu6n ti€p xric voi tluong thing cO dinh: 7 = x+7 ( C,5 th6 giai theo phucrng phrip nghiQm kdp ) 0.25 II 1.00 l+sinx 1fst"t=4cos2x (l) (2) r+Jcostl =Zcosz x 0.75 0.25 <'}2cos2r-cos.l:-l=0 -l I-cos =l i t. (3l I fcos x = -it u lkhongth6am6n(l)) 4.25 €>cosx=l € x=Zkn , keZ + V$y phuong trinh c6 nghiQm li x=2kr , keZ 0.25 III/1 a (+"-r)]FT =zxz +Zx+t tri 1.00 +E{t t=^[r'4>1 + xz - t2 -1, thay vdo (l) ta dugc (+'-r)r =2Q' -l)+2.x+l <+Ztz -(+"-l)+2x-l =0 0-25 + Ta c6 : A, = (+x - t)' - S(Zx - t)= (+; - f)' 0.?5 ft=2r-l I -l I 'lr:-<l l2 (lo4i 4.25 : :-' \".ii r1: i'- :n .: ::al - Vdi l<x<4: e .,1;* =4-x <) r+ I =(a- rY <;>xt -9x+13=0 I s+JE- el'= n I n -Jzs l- = z - o.6t*f =2x-l hx-t>0 otr'a1=(zx-l)' ['=l ; e{ 2 [rr'-4x=o 4 ot=J + V$y phuong trinh c6 nghiQm li ;-, $-'tl:;;;l)= j ,t, + Di€u kiQn : {l: l-i x+3>0 JL-rr+7=3-lx-ll'o *x>-3 e*2<x<4 (*) <x<4 + Khi116: e log,*,(3 -l'-rD= <+ 3-lx-tl =(x+r)i = .{x$ Q) +V6i -2<x<l: e rE*l =x+2 (? r+ 3=(x+Z)t -l e x' +3x+1=0 r-r: | -3+{) I t2 <>l I -r-Js I L2 -r+.'6 K6ttrqpv0i -2<x<l tadugc x=-r-' K€ttrgp vtri 1 <x < 4 tadugc -E -s +' Vay phuong tiinh co nghiQm lir :.v = t , x= e -J2s 2 tl.u2 1.G0 O5 Ur6y n6u x-1 ld nghiQm duy nhAt crla phuong n'inh thi l-x ctng ld nghiQm vi It - xl = l" - tl . iviuf,n phuong trinh c6 ngliiOm duy nh6t ta phii c6: x-l : l-x =+ x: I :+3m -2:7 - rn: l. Ngugc lgi, ni5u m : I thi phuongtrinh trcr thinh: 1: l - o lt-ll : 0 ex=1. zlx -11 Edy ld nghiQm duy nh6t. t. uapso:m: l. ( C6 th6 gidi theo nhi6u c6ch kh6c. Vi du nhu dga vAo db thi. ) 1.00 wn 1.00 * rrong,a* c[uJ) 4.25 + Dulng thing d1 tli qua ,0. (- Z;i), vu6ng g6c voi Cn = (t;+) c6 phucrng hinh : t.(x+2)+ q{y -t)- 0 e x + 4y -2 = A + Eudmg thang <12 tli qua n (:;S) , vu6ng g6c vdi fr = (+;O) cO phucmg trinh li : a.(* -3)+ 0.(y - 5) = 0 <> x -3 =0 H : drn clz = toa dQ di€m H lA nghigm cria hQ phuong trinh fx+4y-2=o lx4 < <3 I I lx-3=O ly:-4 + vqy "[r'-;) 4.25 + Gei M, N tan luE ld trung di6m criaBC vd AC = M[;'3),w(ol) + Dubns trung hsb A, cria cqnh BC di qua "[j t) , c6 vecto ph6p tuy6n cs = (t;+)nen c6 phucng kinh ld: l.(" - i). ^r-3) = 0 e 2x + 8y -2e = 0 -F Ducrng trung tr.uc A, cira qnh AC cli qua N(o;t), c6 vecto ph6p tuyiin eC =(+;o) nOn c6 phucrng hinh li: +.(x - 0)+ 0,C - 1) = 0 q) I = Q I : A, n A, = toa dQ di6m I li nghiQm cria hQ phuong trinh 0.25 .l:E'i:i:-:-A-;.ir: '.i:rr4'+-i ''':lr.::*-l - {i'=.r-' -2e=t-{;:? + v0y '(''?) = IH =3IG ; = Hai vects 7fr,7G cirng phuong ; :+ Ba tliiim I, G, H thdng hing ( Dudmg th6ng di qua ba tti6m I, G, H tluo. c gqi h ttudng thang Euler cta tam gi6c ). 0.25 IYl2a r.00 +Tac6 : AB'2 : AB2+BB'2: 2a2 ; AC2 :BC' -ABz : 3a2 AC'2 : AC2+ a%t= 3az +mz > AB'2 B'c'2= BC'+lai'-ccl'= 4a2 +lo-^l' 0.2s + Vi AC'2 > AB'2 n€n tam girlc AB'C' khdng thiS vu6ng tpi C'. 0.25 + N6u tam gi6c AII'C' vu6ng Qi A, khi tl6 ta c6 : B'C'2= AB" +AC" o 4az +la-ml' =5az +m? ezl-0 e C =C' 0.25 ? NiSu tam gi6c AE}'C' vu6ng t4i B', khi d6 ta c6 : AC'2=AB'2 +B'C'? e3az +m' =Zaz +4az +lo-^lt €m=2a. 0.25 rvt2b 1.00 + Khi m =2a ta c6 : B'C'2 = 5a' 0.2s + Trong tarn gi6c vudng ABC c6 : AB ? =HB.BC =HB - ABz -a BC2 HC:BC-lIB: 3a 2 +Tac6:HB'2:HB2 +BB'2 (;)' (+)' . 5az +a- 4 l-lC'? : HC2 + CC'z : 3 HC'2 =HB'2 +B'C'2 + Tam gi6c HB'C' vu6ng tei B' -rQo)'=+ 0.25 +Tac6 I S_uc : |en.ec =!o.oJ! =ry |nu'o'c'=|'A'JS =+ c_ o MB'c.'- 0.25 + Ggi a ld g6c gita hai m$t phing (ABC) vi (AB'C'). Vi tam gidc AJ3C h hinh chi6u vu0ng g6c cria tam giric AB'C' bdn md.t pheng (ABC) ndn ta c6 : S -rc = So,rr'."cosa = cosa _ s-u. = _E S-u,., V t o :] fi = u.""or-E Vto ^7 a.2s v 1.00 F [' 2 a =y+- v at =Jr+ x . fx*O + Di€u kiQn: i y*0 Vi x vd 4 .r,ng d6u, mdr ** 4 =Zyz > 0 xx H€ phuong trinh diu bii<+ [:O:= Y' + a' 12ry'=x" +a' Tru tung ve hai phusng binh tr6n ta c6: Zxy.(x- y ) = ( y - x).( y + x ) e(x-y).(Lxy+x+y):0 x>0.Tuongtuy>0. 0) 0.75 [r:y ( V(rxy +x*Y) > 0 khi x> 0, Y> 0) Khi d6, thay vio hQ phuong trinh (I) ta dugc: 2x3-x2:a2 +SO nehiem criS hQ phuong trinh (l) chfnh li sO aiem chung cria db thi hnm sd 77x);:zx3 - x2, v6i x > 0 va tlulng thing y o *' [x =O Ta c6 : ./'(t) = 6xz -2x =2xQx-l)= 0 o I - -1 L'=J * xet tta* so 7(T ) =Zxj -'r2 ' v6i x > o JjL(r'' -"')= *- Bang biiSn thi6n : 0 € T 3 r ,f'(r) f(') cit ao thihim sd f(r)=2xt -x' t4iilirng mQt diOm duy nh6t' =a He phuong trinh (l) lu6n c6 nghiQm duy nhitkhi a * 0 ' r . miiih ring khi a l0 thi hQ phuong trinh sau c6 nghiQm duy nhAt: 7 a =y -F- v a' a" - L'T I,*, i,,' ;- GiAm thi coi tfui kFr0ng gidi thich gi thdm EAP AN. Ctimg ',- t)=, tin [' x+t) = Euong thing dz I =2ld tiQm cgn ngang + Bing bi6n thi n: X co -l *co v' + + +e) .l 2 2 -o 0.25 " DO thi: + Giao di6m voi t4rc ox, (-*,.) +. bi6n thi n I + Ta c6 : y'- > 0,V.x e D ' (x+t) + FIim si5 lu0n ddng bii5n tr6n D + Hdm sO kh6ng c6 c c hi 0.25 + Gicli nun' ri*, (, - t:) = - = Dudrng thing