6 đề-đáp án thi thủ ĐHSPHN-2008-2009

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 6 2008-2009 ĐHSP HỌC ĐẠI THỬ THI ĐỀ 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 @ 'rRU'a,NG DHSp r{A NOl Dt THr rrrrl DAr Hgc rAN rr NAvr zoog rliol T'HPT ciruvtlx M6n thi: To6n Thoi gian ldm bii: 180 ph0t lr** clu I (2 di€m): Cho hdm sd r =-lP f rl 1) KhAo s6t vi ve d6 thi (C) crha him s5 khi m = 0. ?) Tim nr AE A6 tbi hAm sO (t) cit tryc Ox t+i hai di,im phdn biQt c6 hoinh d0 ldn luqt li x1, x2 sao cho r = I xr - x2 I dat gii tri nh6 nhAt. C6u 2 (2 di6nr). i. GiAi phuong trir*r : 2sin2 (x - .5 = 2sin2x - tarx . 2. V6'i gi6 tri ndo cia m, phuong trinh sau c6'nghiQm duy nhdt : 2log' (mx + 28) = - log5(12 -4x - x2). Cdu 3 (l di€nr). Tinh tich phan : Cnu 4 (1 di€m). -a' Tan gi6c MNP c6 dinh P nim trong mflt phang (a), hai dinh M vi t'f nirn vB mQt phia cia (o) c6 hinh chiiiu vu6ng g6c tren (s) Dn luqt li M' vi N' sao cho PM'N' li tam gi6c dAu canh a. CiAsirMM'= 2NN'= a. j - Tinh diQn tfch tam gi6c PMN, tu d6 suy ra gi6 tri eua g6c gita hai mflt pheng (c) vA (MNP). - : Ciu 5 (l ditim). Cho tlp hpp A c6 l0 phan *. H6i c6 bao nhi€u cich chia tfp hqp A thenh hai tip -/. : cau 6 (2 dirim). / :./ 1) Trong m{t phing voi hQ tga dQ Oxy, cho elip @) c6 phuong rriot, r { *.* = ,. 9 '4 : MQt g6c vu6ng tOv quay xung quanh di6m O c6 c6c canh Ot vi ov cit (E) lan luqt t4i M viN. chil11113ng mrnn rang: 6F " ON, = 36 . Trong kh6ng gian v6i hQ toa d0 Oryz, cho ducrngthang O' T=?= | tamit phturg CI) : x + ! + z- 3 = 0. Vi6t phuong trinh tluong thang A nim tong mit phang (P), vu6ng g6c vsi d vi c6 khoang cach d6n d mQt khoing h= '# . Ciu 7 (l di€m). C6c s6 thpc x, y thay d6i sao cho x* y = 2. Hdy tim gi6 tri lon nh6t cua bi€u thric : P = 1x3 + 4(f + 4. - ,.li xdx l=l ' J1 x+y;l[' http://wwww.violet.vn/haimathlx M4t khdc lim*-s+ f(x) = + oo vi lim*_e- f(x) = _ - , Tac6f(x)>0v6i -4>x> -6vdf(x)<0voi x e(-4;0) u (0;2) . Bing bi€n thi€n : Nhu vfy, tu bing biiin thi€n suy ra phuong trinh (3) hay ciing Ii phuong tdnh (2) c6 nghiQm duy nhAt thuQc ( - 6; 2) \ {0} khi vA chi khi : cAum. ( 1,0 di6m). | -^> t!. lm < _L4 l ;,=l-rT L-m=-4 L m=4. 3,13- 2,12 -t 3 rac6 r= 1€$ff= Jfxzdx - J€x\Fldx e.,6-1 lis-,y' = 't-' zfi = t'lf -itf r.,- r)ia(*, - 1) =+ -*ic.,-,)-lf cAu rv. ( 1,0 di6m). K6o dAi MN cit M'N'tai E, khi d6 NN' li duong trung binh trong AEMM', mi M,N' = pN'= a n€n EN' = 4 suy ra APEM' li tam giac n?ng tei p vi EP = rGMryffi7? = 816 , d6ng tiroi Ep .t- pM. Trong tam.gi6c vu6ng c6n pMM', c6 pM = a.,12 , nAn FP PM = a.E "^17- : ^2-17 Ta c6 .966p = 2Suup + Suup= | fe.ru =l*^f, . Viy S,r.1up =I^'# Vi EP la giao tuy6n cria hai mpt pheng (a) ve @Ia}Q vi EP 1PM, n€n g6c a giiia t.rai m{t phing nay bAng g6c frFFf = 450 Cht )t ; C6 th6 tinh g6c a bing c6ch sri dsng c6ng thrlc SpM,N, = Spyy.cos g. http://wwww.violet.vn/haimathlx cAU v. ( 1,0 didm). GiA sri k li sii'cdch chia r{p A rh6a man y€u c6u bAi to6n. Ta nhin th{y ring, '''si mdi c6ch chia ta dugc hai tip con kh6c r6ng cua A. Suy ra s6 cdc t6p con kh6c r6ng cria bing 2k. Tri d6 ta c6 : 2k = Cls*Crzo* +Cio =zta -Z s ft=2e- I =511. Viy, si5 c6ch chia theo y€u cAu bii toan bang 5l L cAu u. ( 2,0 di,im). l) (1,0 di6m). Dat @;ffi1=a (0 S oS2Tr) vA (d; }]f)=c+l Tac6: Mf*"=oMcosc "'tYr,r = OMsina, Do Me(E)ndn' xft*Yil-, 4 OM?cos2c OM2sinza , ;-= i . 1 coszrt sin2cr -m=J-= 4' 1 sin2q cos2cr .r usrg rU', ra cung co & = T * T ^ 7 1_ _ coszd. sinzc * sin2c , cos2c :1 - 1 suYra 6fr?*o=il, s r'i+:r=;*;. 1113 _T_ = _ oMz 0N2 36' 2) (1,0 ditim). cia sir aa dgmg duo. c A th6a mEn bii to64 tlf a se nin trong m{t phang (e vu6ng g6c v6i d,n€n m(Q) nh4n vdc t?hi phusnC cria d lA i(-2;3;2) tAm v6c to ph6p tuy6n. Phuong trinh cria *(O g : -2x+3Y+22+a=0 (1). GqiA li giao di6m cuad vr5'i mf(fi, thi tga ttg giao diiim cria A tn nghi€in ctia he phucrng ft+3 v-9 z-6 r-:-:_ trinh : J -2 3 2 e+ A(3: 0: 0) (2). (x*y*z-3=0 Ke AB J. A, B e A. Ggi C ld giao di6m cria m(e) voi d vi g ld g6c giGa d vA (P) thi g = ffie ,tac6 fi(l; l; t) h mQt vdc to ph6p tuy6n cria (p), Khi d6 : -, t-2+3+21 li l; sne = JE!t7- = {; + Ianp = J14- @ http://wwww.violet.vn/haimathlx Ta c6 BC li duong vu6ng g6c chung cira d vi A, d6ng thoi dg dai troqn BC = h = '# . Suy ra: Ac = : Bc <=+ AC -z'iE' . E = g tane 11 ' .,,/ a rry'?' "1*" AC cfing tA khoang c6ch hr A dAn m(e, n€n tir (r) vi (2) ta c6 : o.=#=ffi ea-6=*#, Do A nim trong mf@), n€n A li giao ruyiin cua hai mft pheng (P) va (e. T6m lai ta c6 hai rtuong theng A th6a m6n bAi torin lA: ,",.,.f x+y+z-3= [ x+y+z-3=0 tv'i : [-zx+3y+ 2z*6*#= o ua (a)'[-zx+3yr zz* 6_#= o cAu ylr. ( l,o di6m). Tac6 P =x3y3 *2(x3 +f; + 4=*tf +Z(x+yXxz-xy +yt1++ = x3y3 + 2(x + y)[(x +y)2 _ 3xy ] + + Theo giithitit x + y = 2 n€n p = x3y3 - l2xy +2A. D6t 1 = xy, do (x + y)2: 4xy n6n t < l. DAtf(t) =f -lzt+20, te(-oo; U,thif(D=3f - 12=0et=-2. Ta c6 f( 2) = 36,lim,*-* f(g = - o, f(r) = 9 vi f(t) > 0 voi t < -' 2 c6n f (t) . o voi -2 < t < l Tir c6c t6t qua tr€n, suy ra maxf(t) :36 khi t = - 2. Vsi i = -2,tac6 hQ phuong trinh : (x*y=2 IuL-l e x=lt16;y=lTG. Y 4y, gtdtri lsn nh6t cfra p b6ng 36, khi x = I * 16, y = I _ 16 ho4c x = I _y'3, y = ! * .fi . Dy kiiin k) thi tht? tin sau sE vdo cdc ngdy 2b - 29 thdng 3 ndm 2009 '5 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 [...]... ti6p tuy6n k6 tir M d6n (C) Tim toa d6 tli6m H ld hinh chiiiu vu6ng g6c c0a dii3m M l€n duong thing AB 2) Trongkh6nggianvdihQtqadQOxyz,chomdtphing(P): x+2y-z+5=0 vd rludngthing -Z-= y+1 z-3 G: x*1 1 : 1 HEy viiit phuong trinh m{t'phinC (Q) chria tluong thing d CAu 7 (l di6m) C6c s6thgc duongthay tl6i x,y, zthbaman : Tim gi6 tri lon nh6t cta bi€u thirc : P @ qo v6i m4t phing (P) m$t g6c nh6 nh6t fiJ +,[y... Dudng thing MI c6 phuong trinh 2) (1,0 di6m).Xdt m{t phing (Rj thay O6i ai qua dudng thing d, cit mp(P) theo giao tuy6n A Khi d6 A chria iti6mA=dn(P).L6ydi6mt . 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. -6= lerk= rac6 9(l -k)'- 36 =9(l *il'- 36& lt;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. Khio s6t vi vE d6 thi hdm sii 1t ; thi m = 0. 2. TimmA6euO'ngthingy= lcatd6tlri hdmsO(t)tai 3 di6m phdnbiQt: I(0;,1),A vdB. Vd'i gi6tri nio cria m, c6c tiiip tuy6n crla