rnu0Nc EHSp HA Nor KHor rHPT csuytN THI rutl o4.t Hgc t Aw lv tlAtvt zooq Mdn thi: To6n "': ::::l1T.?::: ::: ono', cau r (2 di6m): cho hdm sri , = ff t'i ',,'l' Tim t6t cA circ giittri cia m d6 hdm st5 c6 cyc d4i, cuc titiu. Chrlmg minh ring trung di6m cria do4n thing n6i cric dii3m cgc d4i, cgc ti6u cria d6 thi hdm s0 in{c6 O;ntr khi m thay d6i. ,12. Kf hiQu (C) la dd thi cria hdm sd ilng vdi m = 2. Tim cdc di6m M thuQc (C) c6 hodnh dQ l6n hon I sao cho khodng cdch tir M d6n giao di6m cira hai duong tiQm c{n cira (c) nh6 nh6t. Ciu 2 (2 <ti6m). ,/ l. ciai b6r phuong trinh : ri I + 2. GiAi phuong trinh : @or2* ffi=o CAu3(l di'3m)" Tinhdi€ntich hinh phing gi6i h4n boi hai parabol : y = I - x2, !=axz v6i a>0. Cnu 4 (1 di6m). Cho hinh lap phuong ABCD.A'B'C'D' c6 dd ddi canh bing a. Cgi K ia trung di€m cua cqurh * , / tC vd I Id tdm cia hinh vu6ng CC'D'D. Tinh th6 tich cria c6c khoi da diQn cto m4t phing (Aklt) chia ra tr€n hinh l4p phuong. icau 5(1 di6m). chrmgminh ringphuongtrinh 2x3 - 3x - 6\6F=x+ 1+6=0kh6ngc6nghigm iim. ^7 CAu 6 (2 di€m). ' l) Trong m{t phing oxy, cho elip (E) , + .'i = t ,udirim M {J; t). viiir phuomg trinh cdc dudng thdng v tli qua M vd c6t (E) t4i hai tti€m A vd B sao cho M ld trung tli€m cua AB . / 2) Trong k:h6ng gian oxyz, cho cric di€m S(0; 0; 2), A(0; 0; 0), B( r ; z; 0), c(0; 2; 0). Gqi E vd F ran rugt ld hinh chiliu vu6ng g6c cria A l€n SB vd SC. Chung minh ring 5 di6m A, B, C, E, F cirng thuQc mQt m4t cdu. Virlt phuong trinh m4t cAu d6. CAu 7 (l <li€m) Chung minh tling thric : Cloro - Clo.o + Cloro - + (- rlC35iot + - CiBl6 + CrzBlB = z,uu' Dtr kiiin itgt thi thii Ifrn sau vdo cdc ngdy t6,17/s/200g. 8,3x 3x + zx 9(3x- Zx) - 3x , @ http://wwww.violet.vn/haimathlx oAp AN ivtoN roAN IAN rv cAu I. (z,o ei6m). l. (1,0 diem). TSp xdc dinh : R\ { I }. xz-zx+z-m ( x+ ! Tac6y' = 6r- =t y' =Q:[*, _ Zx * 2 _m = 0 (1) Hdm si5 c6 cgc d4i, cgc tiiiu khi vd chi khi pt ( l) c6 hai nghiQm phdn bigt khric i. ( m+1 ual: [4, = I_ (2_m) > o e+ n]> l. Gid sri A(xr, yr), B(xz, yz) Id c6c di6m CD. CT crla tl6 thi vd E(xe, yE) ld trung di6m crla AB. Khid6 xl,X2tdnghigmcrla(l)vdxs= |t*, *xz)=l.Suyradi6m EthuQcduongthing x= lc6tllnh. 2. Voi m=2. Phuongtrinhctia(C)duo.c vi6tthdnh : y: x- I * +. x-1 D6thi(C)c6tiQmc{ntltmg x=l vAti€mcflnxiOn y= x-l.GiaocriahaitiEmcdnldl(l;0). Di6m M e (C) <=+ M( x*; xy- I * fr I Nh4n xdt : IM nhd nh6t khi vd chi khi IM2 nh6 nhdt. ' xM-l Tac6, IM2 =(xv- l)2+(x"- I +# )2 = 2(xru- t)t +o;h +z> 2^12+2, dlubingxiy ra khivdchikhi 2(xy-l)'= *hae'(xr,r-l)'= i =*" =l++(vi xv>l). Vay di€m M cAn tim c6 tsa tlo (l ++ , tTf ', cAu rr. (2,0 di6m) \t2' tz )' 1. (1,0 di6m). Bdt phuong trinh dd cho du-oc Uiiln dOi thdnh : ''(;). - (i).*' ;m-,i: ^r-' 'l\z/ ^l \21 Bt t +1 Dpt 1= 0- r 0, t + l. Khi d6 bdt phuong trinh trd thdnh : ,(r-r) s ; €t+l- 8t >o<+ t2-s>oe[,tt3 ,*i 6-= t e(r_l)-; r(r_1) Lo<r<t lo.(f)- 2. (1,0 diCm). EiAu ki€n sin4x * = + Khi d6 pt tuong duong vdi pt : 2sin4x - \E - 2sin2x + Zt[1 cos2x = 0 <+ 4sinZx.cosZx-2sin2x +2{1cos2x'- VS=O e Zsin2x(Zcos2x- l)+ Vg(Zcos2x- t;:g <=+(2cos2x-r)(2sin2x+V:)=0,=[ ZcosZx- 1=o *=r [ .t:tz" =t!,^ LZsinZx + y'3=9 lsinZx= -,13/2 r [ 2x =:* 2kn . Cos2x - - <=+ I z " lZx= -n* Zkn' .,tr [2x = -I* 2kn rSin2x '"erl 3 z - l2x= +* zkn' ,Ttzft E6ps6: x= -*kTr, *=T*nt, k6t ho. p v6'i diAu kiQn suy ra x: +kn, keZ. k6t trqp v6i di6u ki6n suy ra x : 4 * on, o * r. i ke Z. ' t;'' \:; t.,,, T ).'t,i :itr.::ii, @ 3 [* > log13 ++l z <r I xco. lt 6 http://wwww.violet.vn/haimathlx CAU UI. ( 1,0 di6m).Hoinh dQ giao ctidm cria hai parabol li nghiQm cta phuong trinh : 1 ' Jt*a 7'. ,/Ga Dohai parabol ddu nhdn tryc oy lAm trycd6i xirng vd I -x2 > ax2, v * .(-#, ;ft i' nen s=2IJ'(1 -xz -axz)dx=z.l? - f,' .ul*'lt =2xr-3,,*u1 d= #-#"=# cAuw. ( l,otli6m). Gqi F ld giao di6m cua AK vA CD. Duong thing'Fl cttcc'vd DD' tan tuqt tei M vA N. M[t phing (AKJ) chia hinh lflp phucrng thAnh hai t<trtii aa diQn ln mr6i ctrOp cut tam giric ADN.KCM vn khiii da dien ANMKBB'A'D'C', Br Vi KB: KC n€n CF = AB, do tl6 CF = CD. Trong AFC'D, FI vi C'C lA cic ilulng trung ruy€n n€n M li trgng tdm cria tam gi6c d6. 11 Do d6 CM =; CC' =; a. B Vi I le trung di6m cta CD' n€n D'N : CtU = ] a. 3 . I -xt=axt <=+(1 +a)xz= 1 <+ 2 Vay, DN :; a. Ta c6 :Vr :VeDN.Kcvr:1 n," + B' + y'E-E ;,trong tl6 h : CD : a; a a2 az a2 zas Ii€n V'= -t- ,t 3'3 rz 6) 3l Ggi Vz li th6 tictr cria kh6i da diQn cirn l4i, khi dd : Vz : a3 - Vr : at - cAu v. ( l,o di6m ). 11 Dat f(x)=:x*- ; x-V5xz - x* 1+ l,hAms6xrictlinhvoimgi x e R. 32 Tac6 f(x)= *'-1- # 2 2{Sxz-x+7 lsGit:;;T-+g:!1 vd f'(x) = 2x - - zJsxz*x+t -r*- 2(5x2- x +1) F a2 a2 B : SnoN - -; $,': Srccrur : E zas z9a3 3536 L9 4(5x2- x+r1rF x+r Nhfn thdy f '(x) < 0 vdi mqi x S 0, n6n f (x) nghich biiln trong khoring (- c"; 0). Suy ra f (x) > f(0) = 0 v6i moi x < 0. Vay hdm s6 f(x) d6ng bi6n trong khodng (- *; 0). Do il6 (x) < f(0) = 0 voi mgi x < 0. V4y, phuong trinh di cho kh6ng c6 nghi€m im. :j ^it: http://wwww.violet.vn/haimathlx cAu vI. ( z,o ei6m). l) (1,0 tti€m). Duong thing x = I di qua M cit (E) tai haidi6m o,t, f ) vn B(l; -* I Rd rdng M kh6ng lA trung di€m cria AB. ' Xdt tludng thing (d) di qua M c6 hQ sti g6c k. Ta c6 phuong trinh cua (d) : y : k(x - I ) + I ( I ). Thay (l) viro phuong trinh cfia (E) ra duqc : 4x? +9J11x- l) + Il'z= 36 <=+ (9k2+4)x2 + l8k(l -k)x+9(l -kF-36= 0 (2). -Dulng thing (d) cit (E) tai trai ei6m A, B thda mdn MA = MB khi vd chi khi phuong trinh (2) c6 hai nghiQm Xa, xs sao c' xA + xB -18k(1-k) 4 hoiXy: , -6=lerk= rac6 9(l -k)'-36 =9(l *il'-36<0 c6n 9k: +4>0, Vd'i k:- 4 ;, Dod6,voi k: -!. pt(Z)c6 hainghi€mphAn bi6txa,xsrhoamdn: xy:*ol-t =,. T6m lai, c6 m6t dudng thing di qua M thoa mdn y6u ciu cira bdi torin ld d : 4x+ 9y - l3 = 0. 2) (l,0 di€m). ra irhfn thdy E3.IE:43. ed = R.eC: o $, + AS 1(ABC) vi AC J- BC + BC t_ (SAC) ,a AF J- (SBC) '+ AF 1 BF. Lai c6 AE J_ SB (theo gt), n€n n5m iliiim A, B, C, E, F cirng nim tr€n mQt m4t ciu duong kfnh AB. Gqi I ld trung di6m cria AB thi I (1; l; 0) lA tdm mdt cAu 2 A t; vd b6n kinh R = IA : l- {+' Vdy phuong trinh m{t ciu ld : } 1. s (*-;) +(y-l)'+z==;. CAU Vn ( 1,0 di6m). Theo khai tri6n nhi thric Niu-Tsn, ta c6: (l+r" =cg+icfi +lcf,+ +r'Cff = cfi+,cl_ci-jCfl+cf +icfi_Cf -rcfr,.r =(l -ci + c* -cf + )+r(c* -c; + c; -c; + ). Met khdc (l+ i)" tlugc viiit v6 d4ng luong gidc : (r+ i)": rF lcosf + isinf) : @*"7 + t,l7:sinT . Theo tfnh ch6t cria hai sr5 phrlc bing nhau, r{p dl,lng cho n : 201 0, ta suy ra: cloro - cSoro + c!o.o + (-r)k't.c3[;J * -cZEiI+ cSBlB :.,[ffi"i]llf" :2roos http://wwww.violet.vn/haimathlx http://wwww.violet.vn/haimathlx . minh ring trung di6m cria do4n thing n6i cric dii3m cgc d4i, cgc ti6u cria d6 thi hdm s0 in{c6 O;ntr khi m thay d6i. ,12. Kf hiQu (C) la dd thi cria hdm sd ilng vdi m =. rnu0Nc EHSp HA Nor KHor rHPT csuytN THI rutl o4.t Hgc t Aw lv tlAtvt zooq Mdn thi: To6n "': ::::l1T.?::: ::: ono', cau r (2 di6m):. Clo.o + Cloro - + (- rlC35iot + - CiBl6 + CrzBlB = z,uu' Dtr kiiin itgt thi thii Ifrn sau vdo cdc ngdy t6,17/s/200g. 8,3x 3x + zx 9(3x- Zx) - 3x , @ http://wwww.violet.vn/haimathlx