Dtt TRUONC DHSP HA NQI rcr6r rHPT cnuvtN CAu I (2 di€m): Cho hdm sii y = NT THI THU DAI HOC I,AN III NAN,r ZOOS Nidn thi, To,in Thdi gian.lAm bdi: 180 phrit * **. x2- zmx+ m2 x-1 (l) v l. X6c dinh tAt ce cac gi6 trf cira m d6 ham s5 d4t cgc ti6u tai x = 2. 'Jz. Tim c6c gi6 tri cua m d6 tr6n d6 thi crla hdm sii 1t; tdn tei it nh6t mQt di6m mA ti6p tuy6n cria d6 thi tei dii5m d6 vu6ng g6c vdi dudng thing y = x. Ciu 2 (2 di6m). V t. Gidi phuongtrinh: . aX aX stn"; - cos"; L ,+"i"- :3cosx' ,,1 2. Giai he phuong trinh : v CAu 3 (l di6m). Tinh tich ph6n : t = [/3 x2+r+.,/@Txf r/Cau a (i diem). Cho tri diQn SABC c6 g6c AB'C = 90", SA = fift = 2a,BC= a.,,/3 v.d SA vu6ng g6c vdi rn4r phing (ABC). Gqi M li tlitim trdn duong thing AB, sao cho AM = 2Md. Tinh khodng cdch tir tlitim B dffor mp(SCM). I CAu5(l <li€m). Cho0 <a<b<c <i<e vd a+b+c*d*e= L Chung minhbAtdingthric ' '4U" + be +cd+de) +cd(b + e-a) S + ,_ 25 CAu 6 (2 di6m). qo l) 'l rong mflt phing v6i hQ tga dQ Oxy, cho tam gi6c ABC c6 ctinh A(-2;3), duong cho CH nim trdn duong '/ thing : 2x+y -7 =0 viduongtrungtuy6n BM nimtrdndudngthing : 2x-y+l =0. Hay vi6t phuong trinh c6c cqnh vd tim tga d6 trgng tdm G cira tam gi6c ABC. t '/ 2) Cho hinh hgp ABCD.A'B'C'D'. Tr€n ducrng thing AC l6y di6m M vA trdn duong thing C'D t6y diiim N I /cM I sao cho MN // BD'. Tinh ti tU ;' Jcart (l di€m) Xdc dinh tep hsp cdc di6m trong m{t phnng phftc bi€u di6n cdc s6 phric z.th6a min diAu ki6n : lz+il l-l = l. I z-3i I f t*t *y=4+,tW ll,*' -ztg2=te1+h D4r kiiin thi th* Idn tdi vdo ctic ngdy 18,19/4/2A09 http://wwww.violet.vn/haimathlx : DAF AN rovr r,lr mON roAN (Thi thti'DH IAn III - 2009) CAU I. xz-2x+2m-m2 1. (t,O Oiem). Tgp xdc dinh: R\ {l}. Ta c6 y'=- (x-1f Gii sir him s6 d4t cgc ti6u tei x = 2, suy ray'(2) = 0 hay 4 - 4 +Zm-mz - 0 <+ m = 0 ho{c m= 2. 2, tad6u c6 y' J #=+ y' =o <+x= o ho4c x=2' Mat khric y' > 0 khi x e (- .o; 0) u (2; + co) vd y' < 0 khi x e (0; l ) u (l; 2)' Do d6 x = 2 ld tliOm cuc ti6u cria him s5. Viy, d6 th6a mdn bdi to6n thi : m:0 hodc m = 2. ' x?-2x+Zm-mz 2. (l,0di6m)R\{l}. Tac6Y' = 7;17 T6n tai tii5p tuy6n cua <16 thi hnm s6 vu6ng g6c v6i tluong thing y = X khi vd chi khi phuong trinh sau c6 nghiQm : x2-zx+zm'mz (x-1)2 I - -l {=r x2 - Zxt1m-m2 = -(x-1)2,x+ I € 2x2 -4x*2m-m2 + 1= 0,x+ I (*) Do phuong trir,h (*) c6 it nhAt mgt nghiQm kh6c 1, nCn ta c6 hai trucrng hqp sau : a) Phuong trinh (*) c6 hai nhiQm phdn bi-Ot' hay L'=4- 2(-m'+2m+ 1)>0c+ 2m2-4m+2>0 ++m+ I' (L' = 2(m- 1)2 = 0 b) phuong trinh (*) c6 nghiQm kdp x *1, diAu niy tusng duong voi I - -a a 1 (loa)- ' T6m lai' m * 1' ^l e - " txt = xz = 1 Ghi chir : Niiu thi sinh giii bdng c6ch : [r,orrioo trong d6 f(x)=2*z - 4x*2m'- m2 + 1, thitr.' 0,5 di6m. CAU il. l. (1,0 c1i6m). Phuong trinh dugc biiSn d6i thdnh : ci{ - ."ixr +,i,{.,o{) =} z+rin")"or* *=Gi{ - .or)tr +1sinx; =!r*sinx)(co{ - 'i"iAi{ * 'oJr)' . V6irinl - "or|=0c+sin(| -fl=0<=+x =|*lkrc,kez' 22La'z t t' x x. x x 3 , r ^ r Voi r + jsinx =-i(t+sinx)(sinj + cos)<+sini + co5 =-;(ptndyv6nghi€m). VAy, nghiQm cira phucrng trinh ld ' * = ; +zkn,k eZ' w. http://wwww.violet.vn/haimathlx 2 .(l,Odiem). DiAuki€n x * 0, y> - 2 . Phusng trinh (2) <+ lxl = 4 + 2y .thay vio pt(l) ta dugc : 4+3Y= q+JfrT {+3Y= 't7+'z - .( v>0 7 : tv, ;;=ey, o Y=; +x=t5' fx=S ft= -5 vay hQ phuong rrinh c6 hai nghiQm , l, = * ' t o = 1. cAu rn. ( l,o dii!m). Tac6 ,= [/t ,G xdx -,'o 6qnffi xdx DAt r= ,@-*z ql thi dt= #|v6i x:0thit=l; v6'i1=y'Jthit=2. Khi d6 t: I:# = frr+r1*aqt* 1) :fr a * t)+,11,:zbf3 - .lz). cAu ry. ( 1,0 di6m). Do M =2W, n€n khoang c6chttrA tliin mdSCM) beng 2 lAn khodng cSch tu B d6n mp(SCM) . Tir gia thi6t ta suy ra AM: +, BM = + , 3' 3 arlsl 1 CM::. Ke AK l- CM, thi CK J- mp(SCM). 3-'^ Do goc ffi = 90'+ Fm > 90o n€n tti6m M nim giea CvdK. Ta c6dAKM - AcBM =+ AK: tt=l" = n"13. cM ^lTl' Ke AH J- SK, thi AH I mp(SCM) vd AH ld khoing c6ch tu A ddn mp(SCM) tL1131 43+^/7a Taco: ;;;=Asz *1;;= q^r*&=G = AH= E. V{y, ktroing crich tt B ddn mp(SCM) bing + - {43 CAUv. ( l,0rli€m). Ttgiathii5tsuyra 0<"s i.e6tdingthricduscbiiSndoithdnh: -fL/^ a[b(c+e) +d(c+e)] +cd(b+e -u)= * o a@+d)(c+e)+cd(b+e-a)S + 2S x2+r+.,ftr+xf http://wwww.violet.vn/haimathlx Theo bAt ding thtic C6-Si, ta c6 : ) , a(b + d)(c + e) + cd(b + e -a) = ^ (ry=)'* 1.*o*!*._")3= a(r:a)z *,'=}|,', Xethdms6: f(a) =+*ry,vdi 0<as f .rac6 f(a)=frf - 5a2-4a+1)>0,vutto;f)' Suy ra (a) d6ng bicn trcn (0; Jl * it"l s (; ) =f tan'*) cAu vt. ( 2,0 di€m). - l) (1,0 di6m). Dubng thdng chira c4nh AB vu6ng g6c vsi CH n€n nhen vectq il(2; 1) lim vecto chi piiuong' Do d6 dudng thing AB c6 phuong trinh : x -2y + 8:0' Suy ra, tqa d0 diem B ld nghiQm criahQ phuong irinh : | {"-r, * I = o *1i,.=_3-B(2;s). / l2x-Y*1=o (Y= ^Yr -' -^ *-'-^ '+;A '-? Et thu6c BM' Gqi C(x; y) thuQc CH, suy ra trung di€m c0a AC Ie M t7"', ' ( 2x+ v-7 = o" o * [;lf +c(:;r). Tqa d0 cria diSm Cli nghigm cria hQpt, \, * - # *, Suyra, BC : 4x+y- 13 =0 vd AC : 2x+5y-11 =0'TrgngtdmG(1; 3)' 2) (1,0 di6m). D[t EA= d ,EF =E 'EC = d ' Tac6 EF : d +6+ d, vi MN i/ BD' n€nffi =Ufr v[y Mfr = kd. +tci + Pt (1) Mat kh6c Mfr =ft +7d *Vfr = nVt +fr + ^V6 Trong d6 At = i - d,7? =6, c6 = d -E' Suy ra ffi = n(c- - d) +E + n1d -87 : (m -n) d."11 -m) 6 + nt (2)- . (m-n = So sdnh (l) ve (2) ta c6 hQ phucrng trinh : | 1 - m = t n=k t4 CM 1 vQy tvtc =; AC h"Y 1; : J CAUvll ( 1,0tli6m). Df;t z=x+yi + z-3i=x+g-f)i vit z*i = x*(y+l)i Di€uki€n:x*0vdY*3- D6 dang chimg minh du-oc tinh chAt l:l: # Suv- l#l=l elx + (v +1)il2=[x * (v-3)il2 e.x2+(v+l)t=x2*(v -3)2-ev=1' K6t lu{n: Tip hqp c6c di€m trong m{t phdng phrlc bi6u di6n c6c s6 phfc zthbamtutdi|u kign : l#l= ttno"*^fur=t - k fk=713 t-lm=2/3 ( n=1/3 ,,@ http://wwww.violet.vn/haimathlx . 2. 'Jz. Tim c6c gi6 tri cua m d6 tr6n d6 thi crla hdm sii 1t; tdn tei it nh6t mQt di6m mA ti6p tuy6n cria d6 thi tei dii5m d6 vu6ng g6c vdi dudng thing y = x. Ciu 2 (2 di6m). V t mdn bdi to6n thi : m:0 hodc m = 2. ' x?-2x+Zm-mz 2. (l,0di6m)R{l}. Tac6Y' = 7;17 T6n tai tii5p tuy6n cua <16 thi hnm s6 vu6ng g6c v6i tluong thing y = X khi. dii!m). Tac6 ,= [/t ,G xdx -,'o 6qnffi xdx DAt r= ,@-*z ql thi dt= #|v6i x:0thit=l; v6'i1=y'Jthit=2. Khi d6 t: I:# = frr+r1*aqt* 1) :fr a * t)+,11,:zbf3 - .lz). cAu