EL TRLIONC DHSP Ha NOI ru6r rHpr cHUyEN nt rru THrt DAr Hec naOn roAN IAN I trAwt Hec 2oo8 _ 2oo9 (Th6.i gian IB0phtit) t*** Ciu l. (Z,O Aiem). Chohdms6 y=n*r*,*+ l)x2+1m2*4m*3)x+1 l. Khdo s6r vi v€ d6 thi cria hdm sti khi m = - 3. 2. v'i gi6 tri ndro cta m, hdrm s5 c6 clrc d?i, cgc tiAu? Ggi x1, x2 li hai di6m cgc tl4i, cgc tiiiu cria hi'n s5, hdy tirn gid tri lon nhdt crja bi6u thric A = i*r'i-ri"'i -rl f Cflu 2. (2,0 di€m) y l. Gini phuong trinh : . cos2x * cos5x - sin3x - cosSx = sinl0x. Z. Giili bdt ptru,rng trinh : Cdu 3. (1,0 di€m) . ^74os'@jslog, (x-) Tim hq cdc nguy6n him cria him s5 : f(x): ' xa-1 x(x+-s;6xs-sx+1) ' Cffu 4. (2,0 diOrn) cho hinh ldng tr-tr tam giiic d6u ABC.A'B'C' c6 dO dai canh d6y bing a, g6c gita cluo*g thing AB' vdrn{t ptrang@e,C'C) bing a. e' 6vv l. Tinh d0 ddi rtoen thing AB' rheo a vd s. 2' Tinh di$n tich rn{t.ciu ngo4i titip hinh ldng t4r AIIC.A'8,C, theo a vdi a. g,Cdu 5. (1,0 di€rn) cieihephuungrrinh [r;?; !2n ,"ur, ,,! gCAu 6. (1,0 diem) Chirng minh ring : i '{ i *+fr+ ++-1oos/ 1 ,- 1 - ioos ciaos i*Ft - roo, til +- EJ*o + " + (Trong d6 Cl IA.s6 tO h-op chflp k crja n phnn tri) <pCflu 7. (1,0 didm) \ Trong m{t phing vdi rr€ r-o: dg oxy, cho tam gi6c ABC vdi A(?; -1), B(r; -2) vd rrg*g tdrn G cfra tarn giric niim tr€n duon! tning d: x-+ y - 2 = o. iray tim tga dQ diem c, bi,it rang di-6n tich tam gidrc bing j. i i\rtt t - 2i I ! | - ' , .J' L] \i I r. 2 1\ -l r200a I' uzoog/ trdt a9 f-T-"i:iti:% http://wwww.violet.vn/haimathlx Sưu Hải Minh Nguyễn tầm: oAp AN vA rueNc DrEM l. (1,25 Aiemt . Gidi hqn: limr,-*_ y = + co , limrr-__ Jr= - .,b. . . Su bii5n thi€n: y, = ?x2 - 4{ ,y:'= io U - O. f,,ia. x = 2. y'>o* f ;: vd y,<o<+o< x<2. Do tl6 hdm s6 d6ng biiSn trong nrdi khod.ng G*; 0) vit (2;+oo), nghich biiin hong Vdim=-3,thi t=:*t . Tap x6c dinh : R khoang (0;2). . Cgc tri : Hdm si5 y dat cgc d4i tai x: 0 vd yc,p = y(0) : ;, Hdm sti d4t cgc ti6u t4i x:2 vA.!c.r = l(2)= - 13. . Beng bitin thi€n xl o z +co 1 r 2\ ?.o _*,/ \_ !/ Dg Ai . (Hqc sinh tu ue t, ?t,*r":: r !.',' = Of O,c6 y"(l):0 vd y,, dr5i d6u khi di qua x = l, n€n di6m (l; -; ) le diiSm u6n yd cfing ti tem d6i xring cria d6 thi. ?6 Ai cit trsc tung r4i ai6m qo; ] ). ?o d6 A ton ntrdt bing 3 khi m = - +. Ta c6 y'= 2* I::1::"Lgu1:oc ti6u nri vd chi *, ri:0 c6 hai nghiQm ph6n bigt x1, x2 hay a': (m + l)2 -2(m2 * 4m + 3) > 0 (+ m2 * 6*;;-; il:ffi .:;. $eo dinh lf Viet, ta c6 x, 1x2 = - (m + l), xr.xz =lm2 + 4m + 3y. l"l:l A= li(*' r 4m * 3) +2(m + l)l =|l*, * a** z1 Tanhinth6y,vdi m e(-5; -l)thi: - 9S mf+ gm+ 7=(m*4)2_9 <0. 1. (1,0 di6m) CAU II Phuong trinh dusc viiSt ve ftng cosSx - cos2x + sin3x + sinl0x _ cos5x = 0 <+ - 2sin5x.sin3xI sin3x + 2sin5x.cos5x_cos5x=0 c+ cos5x(2sin5x_ l) _ sin3x(2s @ in5x-1)=0 http://wwww.violet.vn/haimathlx Voi sinsx= 1 0 [ u*=r+Zkr z [s*=n_]*zkn (+ Vdi cos5x = sin3x (+ cos5x = cos( _ fx; J4f tfln nglri0m crlaphuong uinh la S ={ I + 3II t30' S t t __ n . kn _+_ [. = -'i* i,,G' 4' I * T, * * T,-l * r,,). 2. (l,o ci5m1 Bdtphuong trinh itugc virit vA dang J2* tog3(x - il = log, (x _ ) trl DFt t: log. (x - i),*r d6 (l) trd thanh ,lffist o [z iii t, e [,, _l] I = o suyra togr(x-il= 2ex-*-no *>?. Vly t4p nghiQm cria bdt phyone trinh H S = tf,; + o). rac6 f-*$ffi-Iaffi:/ffi - fy4-r\'lv t-4 1\ t i;ffit *. ++ (cos5x - sin3x)(2sin5x _ l) : 0 I sin5x - 1 .Hlz lcosSx = sin3x ; E ll r. * .\ .i !. I :r ' .? .l i. t' tl: (: >:. :. t' l. t- +. ;a- E CAU rv l. Gqi M H trur_rg dirim cria BC, thi AM 1BC, AM J- BB' n€n AM J. mp@B,C,C), do d6 ,$fu = a. Tqong tam gi6c rnr6ng AB,M, tac6 AB,= N - "€ . sina 2sina 1. Gqi I vi I'lAn lugt li t6m hai ttriy ABC vi A,B,C, Klti CO, rtng di6m O cria n, lA tankh6i ciu neoei - tifo Utotr langtru . Ta c6II' = BB'. BB,2 = AB,2 _ AB2 = 3az -2 a2q3-+sinz a1 -;ri"{-a-=ffi BB.'=*;lffi a^[j 3 4'i S l -;- . -__t_ suy r4 trong tam giric vu6ng oiA, c6 c- E ,-1- ,l : -;ri"" lg - sinz a, IA= http://wwww.violet.vn/haimathlx va oA2=ot2+tAr= #(g-+sin2a)- f Gqi R ld brin kinh m[t ciu, thi *i= #; (z _ +sinz a) + Khi d6 dign tich mflt c6u ngo4i ti6p hinh lang trr.r ld : ?2 3 s = +z' (# cs - + rin'"1 + *) = ara2ffi + 5. Tt he phuong trinh suy ra x ) 0, y, 0. Cfing tu hQ phuongtrinh vd theo Uit aang th&c C6si, ta c6 . 6V3 = 2^/7 + y =,/F + lF + y >3W = 3\/74= 6iE Deng thric xdy ra khi vd chi khi G =y =2W:+ x = \876. Vdy nghi€m cria h€ phuong trinh la' x: ffi vity - 21/i. Trudc ti6n ta chring minh c6ng tfrri* t n+t/ L 1 \ .F=;*,l.ffi *.ht-Il (r) Thft vfy, e#+#=HP.ffi _ k!.(n_k)t.(n+1_k+k+1) (n+r)l Hffi=#+ I C6ng thric (l) dugc chu-ng minh. Ap dpng (l) vdri k di rir O Adn ZOOS;G m=;ffi(il.a;) " =4q(#.*) uz^ooe 20Lt L 2OOg m=#(m.4''t Do 6$0, = crool,n6n I6y tong tone vd cria 2009 deng thric tr6n ta dusc 12008 I 2OOg / 't i ar=-cim =ffi.r.(ffi + t+ +ffi) i'^ 1 r vd'' *;+*. '.#t=ffi(eh + # + + eh) a) http://wwww.violet.vn/haimathlx Clu VII Tir gin thi6t ta suy ra Segc = 3Snea =+ Sesc :1 uU dO dei AB = r,E. Phuong trinh tluong thit g AB : x-y - 3 : 0. 0ls Gii sri G(xc; 2 - x6), khi tt6 khoang cich tir G ttdn AB la 1 : l2xq'sl ' ,tz suyra segc =i* n + l2xc-51 : t * [}; I 3 0,25 Ta c6 tga tlQ aiAm C(+c; ys) dugc tinh theo c6ng thric f*o =f t*^ + xs * xs) lto=lct^*vB+vc) Vdi xc:2 thi yc - 0,i khi d6 thay s6 ta dugc Xc:3, Yc:3. V6i xc: 3 thl yc = -1, kfii d6 thay sd ta ttirgc .xc : 6: Yc: 0. V$y c6 hai <tiAm C th6a mfln bii to6n: C1(3; 3) vn Cz(6; 0). 0'5 x E E F It ,:: H :i .1i i:: li t: f: t. i.: 4 J: i: ri i. s: ri. t: 1. i; :l t- http://wwww.violet.vn/haimathlx http://wwww.violet.vn/haimathlx . c6 dO dai canh d6y bing a, g6c gita cluo*g thing AB' vdrn{t ptrang@e,C'C) bing a. e' 6vv l. Tinh d0 ddi rtoen thing AB' rheo a vd s. 2' Tinh di$n. eh) a) http://wwww.violet.vn/haimathlx Clu VII Tir gin thi6 t ta suy ra Segc = 3Snea =+ Sesc :1 uU dO dei AB = r,E. Phuong trinh tluong thit g AB : x-y - 3 : 0. 0ls Gii sri G(xc;. DrEM l. (1,25 Aiemt . Gidi hqn: limr,-*_ y = + co , limrr-__ Jr= - .,b. . . Su bii5n thi n: y, = ?x2 - 4{ ,y:'= io U - O. f,,ia. x = 2. y'>o* f ;: vd