THI TOÁN 2010 (TT CHÙA BỘC HN)

TTBDVH THANG LONG . BACH KHoA Dla didnr hgc: Sd4- 6 Chria B6c DT: 357372s3 - 09I658087A A\ DE THT TOAN ilO? 1 Thdi gian ldnz htii: lB0 phtir, kh6ng kd rhdi gian phtir rld pHAN cHUNG cHo rAr cA rni srNH (?0 didm) ef*I(2,0 didm) Cho him sd -y - 2x3 + 3(m - l)r' + 6(m - 2)x -l 1. Khrio sdt su bidn rhi€n vi v6 dd rhi cria him sd khi m = Z. 2. Tim m dd didm cuc dai vh cu. c tidu c,ria him sd li ddi xrlng nhau qua dudng y - x. 8'UI (2,0 ilid@ L Giai phuongtrinh: 3cos.r-3sinx -rgx.sinx+sin xtgzx =0 2. Tim m dd phuong trinh c6 nghi€m duy nhAt: 25' + (ry -t)S' + 2m + 3 = 0 CAu III (1,0 didd n rrnhtichphan /= i;;" -1 eesJV (1,0 didm) Cho hinh ch6p SABCD- c6 SA=a I (ABCD). Driy ABCD li hinh thang vu6ng 6 A vh B. AB=BC= a, _ AD = 2a.Eld trung didm AD, X6c dinh tAm vh tfnh brln kinh mat cdu-ngoai ildp SCED. CAu V (1,0 didm) Chilng minh rang phuong trinh Zxj -3x -6j5x' - - +l+ 6 = 0 kh6ng c6nghiem am. B. Theo chuong trinh ning cao ib Q,A didm) '\ Trong mat pliing vdi ir€ toa dO Oxy, cho tam giiic cdn dinh A. Canh AB c6 phurrng trinh x- y+ 6 = 0- Canh BCc6 phuongtrinh x +2y- 0. Tim phuong tr)nh,Cudngcaoha ttllcrjatam gi6c. 2. Chc ninh i3p phuong ABCDA'B'C'D'canh a, goi M v) N lh trung didm cua BC v) CC,. I li giao didm cria CD'vd DC'. Qua I vE mdt duong thing cit BN vi DM tai p vi e, rinh d6 dai doan pe. CAu VII.b (i,0 did@. Ciiii phu,rng tr)nh (x + l)togl (x + z)+ 4(; + z). log. (x + 2) = 1 6 Het THr rHir-DoT 2 vAO NGAY NcHi CIl/s h chiduoc lAm mOt trong 2 phdn (phdn A horic B) Oxy, cho tam giiic ABC. PhAn giric trong AD iI x+y+Z=O, . Canh AB di qua didm M( t; I ), dicn tich tam gi6c li ! . n* ,o^ )xyz, cho 2 duo-ng thing r- lx = _3u I Dt|Y =3+2u lr=-t t- song song vh ciich ddu 2 ducrng thing niy. rinh / _ a{ir - J9r' * lSox + soo ll = r 8' 'J pHAN RIENG (3,0 didm): Thi sint A. Theo chuong trinh chudn CAu VIa (2,0 didm) i. Trong mat ph&ng voi h€ toa d6 duorg cao BH ld 2x -y+ I = 0 . d6 ci{c didrn A, B, C. 2. Trong khdng gian v6i hC toa dO O ix=l O,lv=-4+Zt , T' [z=3+r Tim phuong trinh crja mat phing r Cau VIi. a(i,0 didd Tim nghi€m nguy€n cila phuong t t- cosl L E0 GrAo DUC vA sAo rAo TTBDVH THANG LONG pd rrm ruu DAr FIec tvrOt{ ToAN r{AM z0t0 - B{ir z Thdi gian ldm bdi : 180 phfit, khbng kd thdi giaizTtl#{dd FHAi\r cHTJNG cmo r{T cA cAc rgf sxNH (7,0 ExdM) CAu tr (2,0 cliim) Cho him s6: g : st - 2(1 .: m)r2 + 2 1. KhAo s6t vh ve dd thi him so vdi m: -l ;: Tt;";tg ao irrila; ro;i aiC* .,ii rq va tam giiic tao boi 3 didm cuc rri nhy c6 dien tich qs bang 32. CAu II (2,0 tliCm) sin3z cos3c - ".34' 1. Giai phuong t.inf,' ffi.* ffi - ianr-2cosz(:f - r) 2. Tim m dd phucrng hinh sau c6 nghiOm ,r'(1/x + t + J{-) : d -n2 * 2r - 2m2 + 3 rII;\Y$ tItVr-:Ll - V-4 cAu III (1'o didm) I ^ -,).t^2 ^ Tinh tich phan t: f ffi* CAu IV (1,0 didm) : ' ' Cho hinh'l[ng tn] xiOn ABC.A'B'C c6 A'4 = A'B = A'C = 3a. D6y ABC li tam giiic cAn 6 A ndi AB = 3a, BC = 2a. Tinh thd tfch cria khdi da diOn A'BCC'B' . - :' CAu V (1,0 tlidrn) ". biai'it?*pitiong ttinr, ( rs _ Br2 = ar _ Ja _ z )' ,-2.", "a-t 1toe,fi *log' ?i,: (c - 3)3 \ '- PHAN RIENG (S,O DIf,nfl: Thi sinh chi duo.c lirrn mOt.irong hai phAn ( Fhdn A hoec Fhdn B) A. Theo chuong trinh chudn Cau VIa iz,O Aidrn) r. Trong mat phing tga dQ Oxy, cho tam gi6c ABC c6 toa d0 A(2, -3), B(3, -2). Tigng tAm G ctra tam gidc ABC thuOc ciuong thing (d): 3x - y - 8 = 0 vh tam gi6c ABC c6 di0n tfch Uang l Tlm toa d6 c. ' 2.'Trong khdng gian vdi h0 toa dQ Oxyz, cho tam gir{c ABC c6 A(1, 2, 5). Trung tuyOn BM c6 phuong trinh: a-j v-6 z-L Trung fuyen CN c6 phr-roiig uinh: * ,,n :a .2 : =. I-4p , -z-: T : I ' 1-: 4: 1 ' phuong trinh tham sd cria Canh BC. CAu VIIa (1,0 didm) 'Dnr sd iiflii ,'tnaa mdn: (z + 2z)3 :8t B. Theo chuong trinh ning cao Cnu VIb (2,0 diim) r. Tiong mat phing tga dQ Oxy, cho tam gi6c ABC. Phuong trinh canh AB ld: x + y + I = 0. Gqi M(2; 1) lI'trung didm AC vd N lh trung didm BC. Tam gi6c NAB c6 diOn tfch bang qrt. Tim toa dO didm N. 2. Tiong khOng gian v6i hQ tga dd Oxyz, cho tam gi6c ABC c6 A(1, 2, 5). Trung tuyOn BM c6 phuong trinh: "-=t : ';u : + Trung tuydn CN c6 phuong trinh: ' ,n :u-:2 -';'. l-qp f Q ' _2 2 I -f- Q 1 _4 - 1 phuoiig trinh tham sd cria canh BC. Cau VIIb (1,0 didm) _z q^ o TimtoadodidmMtr€nddthihhmso:,:ffisaochokhoAngc6cht}A{ddndudrrgthing C:3sl*A+7:Onh6nhat. Ho vd t€n thi sinh So brio danh a* cIA* r*a,ic vA ar.E* i'au: c)€ TF{r 'FHdi BAI F*sC e4*N F*AH ,\Aft4 z*!# - sGT 3 TTEE)1iH THANG LONG Tizdi giun it)nt bdi l8A ohur, khdtt,q ke' rhoi gian phdr de pHAN cFruFrG ci{o rdT cA cAC THf srNH (7,0 srdu} Cau t i2.S dierni CFro h}rn so: ,: $ tt, r. KhAo silt vh ve dd thi hZim sc (ii. z. Cho hai didm,4(-5,1): F{i.3). Tlm cdc rii€m 1,'ir€n do thi (Ci sao cho ram gi6c }rten vu6ng rai M. C*u E-l (2,0 rliem) L" Gitti phuong trinh: 2 cos2r+sin2riScosr-r-sins+I:0 i. Giai he phirong rrinh sau: {gtriT-;r:i, ?'z ,. ^ 1 Lau trx {r,u dtem} t Tfnh tfch phdn /: f,r"o'".sin*, cosf,d.r. CAu rV (1,s clidm) Cho hinh ch6p trt gidc ddu S. ABCD cd do dhi duong cao bang a vi g6c SB : a, a f 4Eo. Tinh ihd tfch cfia hinh ch6p S.ABCD theo a, ct. ' C6:u V (l,Cr didm) Cho ba sd thuc a, b. c th6a mfur n + b + t: r. Chrrng rninh €ng # - * * # tf{r'*o + 3b+c + 3"+o) PHaN RIENG (3,0 DIEM): Thi sinh chi du-o.c lim mdr rrong hai phdn ( phdn .4 ho6c Fhdn B) A. Theo chuong trinh chudn Crtu Wa (2,0 tlidm) 1. Tiong mat.phing foa dd lxy cho tam gi6c ABC cd A(4, -1), ducrng cao BE cd phuong trinh 2:t - 3y i L2 : O, duirng trung tuyen BM c5 phuong "trinh 2r: * Zy :0. Lap phuong trinh ba canh cIa tam giric. . 2: Irong-hg tryc lOa_dQ 9lrt"rhotti4 chgp S. ABC cd canh ben SA wong g6c v6i m4t d6y (ABC) vi A(0, 0, 0), S(0, 0, 6). Gi6 sr? K(1, 2,0) h tam duong trdn ngoai riep tari lUr nef. f-ap pfiuong Einh mat cdu ngoai tidp hinh ch6p S. ABC . CAu VIIa (1,0 iiidm) Tim sd phrlc z thda mdn lzl:S vd lz- Z+Ail:2. B. Theo chuong trinh n6ng cao Cau VIb (2,0 ttidm) l. Tiong mat phing toa d0 Oxy cho elip (E) , * n { : ,. LAp phucrng hinh duong rhang ({ di qua '94 M(7,7) sao cho cilt @) tai hai didm A, B th6a mdn MA = MB. 2. TlonghQ tnlc toadO Oxyzchomdtcdu (s) '12+y2*22:2b, dubrngthing (A) ,? :+ 1",' Try phTg (Q) ,, + y t z j 1 :0. I-+p phuong tri4 mat phing (P) song song vdi Ouong tfrang (-n), -4t piri.ng.(P) vuOng g6cvlur mit ph&ng (8) vA (4 cilt mat cdu (S) theo giao ruydn lb dubng t6n "O Uan kinh bang 4. r Q CAu vIIb (1,0 didm) Tim m dd dd rhi hhm so s: t-ry:!-f tiep xric v6i truc hohnh. :1: _ Ho vh tOn thf sinh T{ar ; Sd b5o danh. T'F.ildNiG TT{PT n i r'r FrEt-qz .nii' L2 .i.Lj -r t LI i -l u 2. Giiri phuorrg trinh: Ciu IEE" Tiirh: lsinrl+ cos2x = q. \lL\-1+s"{;1 BE THX Tr{U B+r r{GC SgT V (20S8 - 20{-le) ' tyi6n: TOAN; Xndl a Thdi gian ldnt bdi: lB0 phtit, I:hong ki tlii gien phui di +t.+ CAulChohdms,5y=*4-4r2+*. . ' i. Kh6o s6t.s1r bi6n thi€n vi v6 d6 thi hdm s6 v6im:3. 2. Gie su dd th! him'so cit tryc hoinh qi b6n didm ph6n bigt. Hay tim m d€ cho hinh phinq gidi hsn mi ao thi ham.s6 vd tryc hoanh c6 diQn ffi;hA; pliia trdn vd phAn phia du6i tryc hoinh bing nhau. CAu II" 1. GiAi hp phuong trinh: nil*re I\i" ciro hinh ch6p S.ABC;'6 UU, fil.t* gi6c ABC v96ng c6n tai dinh A (l=90u), AB: AC - a. h44t bdn qua canh huydn BC u:6ng g6c vdi m{t ddy, Irai mgt bOn con lai d3u hsp vc'i tu4*. doi, cac g6c aiE'aoll iintt thc tf;h c;; hinh clii'p S.ABC. /' Cfiu V" I{4c <iinli m de hq sa-u c6 ngiriQm: / ft )x" - 5x+4 <0 l^ l3*'' -tnxrlx+i6=0 "-i TZi FF-^ l-i-A !.'1^ , ra -i \-Ad"& \,r. rru.uB zurdr1g giarr vdi itO toa dO D,A c6c.rudng goc Oxyz cho hai di€m A'(t;2;1), B(3; -I;2). iho dodrug thing id) va mat phini'(p) .o cac phuo*g hir.h , _ k I I /*\ nJrir sau: (d) . = "- = ; (P l: 2x- J' l- zi 1 = 0. 1 -r 2 i:" Tim tga iio diOm C ddi;ririig.r,6'i ciidin A qua (F). 2" vidi ohuorig trinh dirong tiring (n) ai qua A, cit (d; vd song song ,rdi (p). tim ioa aO"oieni l,zl thu-5c €t;t; .f ta"g f.froaiug ,ach Ir{A + Iv{B dpt gid tri nh6 nhAt. CAu VXn. Tim gi6 tri ion iih6t vd gi6 trinh6 nhat cira him sd: .(t- j/=sin'x+43cosx. n-\, Gidm ihi coi thi khong giai thich gi thAm oAp Aru - TI{ANG uldm f. { CAU D6p [n Diim I (2,0 didm) t. (1,0 tlidd Khho s6t. + T4p xdc dinh: D = R + SLr biai thi€n: - Chi€u bien thidn: ! = Zxt + 3x2 - I l=6x2 +6x=0 khix=0;-l f" =72x+ 6 = 0'vh ddi dau khi * = -+, , = -t: didm u6n Him so ddng bidn (= o";-l) va (O;+m) nghich bidn r€n (- t;O) '_r _1) Cucclai tai x=-l;y=0.CuctitiLrtai x=0t y= l.Didm u.;n 1[- Z, 2) 0,2s lirn.y rc;lim!=+q .t+-4 .\++4 Hhm so khOng c6 ti6m cAn. Dd thi cdt Ox tai fx=-l ir=l lz ;Oytai y*-l 0,25 Bing bien th x Y- /: v n: _o) -r _1 0 1 L 0 - 0 -0+ cD- cl +co + - +00 0,25 v -1 T i W, / \,|- ,r/ / \ / 'l-' 0,25 2. (1,0 didm) / = 6x2 + 6(m- l)x + 6(m -2) = o {"' = -t lx, = -m+2 0.25 {"' IY' - _t L =-3m+6 {x- t" =-m*2 =mr 9m2 +24m-21 0,25 Dd ddi xfrng qua y = x thi (x. lr l."' =lz =lt 0,25 {-t = 't l- m+2 -9m2 +24m-21 =-3nz+6 m=2 0,25 II Q,A diiim) l. (1.0 diam cbt- { O {x (sin x - cos x)(rg'" - jl = o frgx = I <+t lrgx = t^5 l, lX=-+kn I4 lt=*L+k, L3 (t, e z) l, lx=-+kt l4 i lr=+!apo la LJ (t eZ) /=5''>0 *m <+tr <0</) ft €) /r/. < 0 cr : <0 a Znt+3 < 0: r,,._! ={t2 Hoac: A =m2 -l\m-11 =ol*=-l lm =11 III (1,0 0rem) Tinh tich phAn n r= I _n ) ! )- 't cosxdx + l_=l t d 4-sin2 x -t I': him .f(*)= 4_,m ri rg ndn tfch phan ray v6i hai c6n d.i xring ra = L : hlm .f(r)=, tlf t, lh cha-n n€n +_sln-x n ,, = r] ,G,n{ ,i4-sinrx 2l|. 2+sinxl =-lin-l 4l 2-sinxl =fln3. tlfu.t=1rni L Xric dinh tam-va Uan liintr* ro4 dQ cr{c ditim s(ooa); ,(;+,,,,); r(ooo) o,s2 = +.+ + Go - o1, = 7!* j],, -2 ^2 1 OE2=!-+!-+rl a- ) 4 4 -o=7+z; B. OS2=OE2+2,, 3a 2 4;,+"+) R=OE-oJi CAU D6p 5n Diirn v (1,0 orem) Clirng minh phtro:rg trinh X6t him so ./(x) = 1., - -x' J I x- 2 +l xd Vxe R 0,75 ./'(")= "' _1_ 2 10x-l 2 0,2s ./" (") = 2x - j 19 <0 Vx<0 4(5x'-x+l 0,25 v\.f"(").0 khi x<0nen /'(x)J tni x<0,rfcra /'(:r)t,f'(0)=0 ktri x< 0; vi /'(r)tO nen /(x)t u,i x<0, trlc ra /(x)r,f(0)=0 Vx<0. viy phuong trinh kh6ng cd nghi€m x< 0. 0,2s VIa. (2,0 drem) l. Cho tam siiic ABC Goi N li didm doi ,fx-1'=g 'it* !-rz=o xirng c&a M qua AD rhi MI: x - y - 0 IV t(- r;*r)=r I (- :;-:) / N -D C 0,6 Uanh AC lir duong rhdng qua N vh I BH. (x+v+2=0 ol, *i, ., = o 't(s;-t) ' ca B{'*+Y-3=o Bf-l-,2) l2r-y +l=0 -\Z'-,1 VAyAC: x+2y+9=0 nhABlhAM: 2x+y-3=0 0,25 - s [to, - l,vo, C(-2y,, -9; y,,) c,(:;-o) C,(z;-e) AC2 = 5(y,, +7)2 6 8 xor =3 xoz =7 0,25 AD le phin gi:ic ./(.",;,)=.r*)r*2. trong n€n B vA .f(B) f (c,) <0 C phii nam ue f'ai pt iu cua n€n chi co C,(::-6) li rhich hon AD 0,25 VIa. 2. (1,0 die"d 'Iim mat phing C,(ozt) u,(-lzo) Dung mlt phing u,(t - +t) u.,(oi *z) P ll(Dt) (O, ) vA di qua trung dirinn I c&a M M 0,25 Lu."C,l=nn=(zt-a) - a,2s ,( !_r r) '(z z 2) 0,25 P :2(x - :).:[r * :)-'(,- ;J = 2x + 3y - 6z +]= o 0,25 VIIa, , (1,0 didm) +180x+800 r_Bn2 -2s _gn _40'_ 3n+5 3 9 9(:r+S) 9x =24n-4A - 25 3n+S 3n+5=+1; t5; t25 nguyen =+ 3n + 5 l) Lrdc (t) cfia 5, rL.rc I n=-10 n=-2 n=0 x=-31 ^ l I6n x=-) loai vi x< Vay 'J Cho tam ei6c canl. u i 0,2s tslc e .BC neu t("- P'-,-T=o lt-r*o=o d;( t*r,!.t+)rft;t) BH lA duong thing qua B vh f C-A; + 7/ _ l0 =l vIb (2,0 didm) Cho hinh tAp p[ on hi toa d6 nhLr h)nh c6 A(000), B(a00), C( 00a), B'(aOa), C'(aaa), ve. aaO), D(0a0) D'(0aa) ,( g"g\ l) 't l ,(";,) r('";) *(,,;) N(,-;,) BP =xBN * rn =xx' CAu Ddp :in Didm DA = you Q( nr,o - 4,ol ."\,- 2 ) * = (;, o(, - t),- ;. +) / /(:,, - r, ; - +) 7D=( or-t' aY.a\"( 1 v 1) \ 2 2-t)"1' 0,6 +,1),sQaoo) ,(r, l+="(,-+) lr-t - -dY l2 l" I a t___ lt ) ) t" 2^ "=J,!=/, IP=alQe 1 ;' J Gilira a = 0,25 PQ=OF t _14a2 9 Po2 = r,' *4a2 *a' 99 0,25 VIIb. (1,0 diem) 2. (1,0 died Giai phuons trinh DAt u = log,,(x + 2), phuong rinh thlnh (x *3)u'+ 4(:r + 2\u -16 = 0 4,25 6'= (2r + 8)' 4,25 L u, = -Q' ut = : x+i 4,25 ^ I 16l 4-Lt 8t 8l li nghidm duy nhAt log.,(x +2)= -4 log.,(x +2)= 4 - ./ t\ 0,25 sAp Arq - rrrANG ndpr L CAu I (2,0 didmi 1 (1,0 tlidm). Khio sdr va vE dd thi hhm sd g : 14 - 4r2 + 2 + TXD: R + Su bidn thiOn: * y' = 4r3 - 8r : 4r(r2 - 2) :0 khi r:0; z : -rD;, : J2 + H)'nr sd ddng bidn tion (-v?, e ve (r,tr, *m); hdm s6 nghich bidn rren (-*, -rt) vit (g,.t/i) + Cuc dai tai x = 0; y = 2. CLrc tidu tai hai didm *: _@. + Gi6i han, ,IT*U: *oo; ,IT_U: *m + Bing bidn thiOn: '\t/" o,$t + Hlrn sd cd 3 didm crc fi-i y! - 4r{r2 - l+rn):0 c6 3 nglrigrn phAn bi€t. D:ip sd m < l. + Toa d6 : aidm cqc rri. A(0,2); B(- + Tiung didm qfia BC te H(0, 2 - (t - m)2). Oien tich ,9 : )nC.nU _r,t= *.11 _ *y + .9 : 32. Giai ra duoc nz : -B (rh6t ;en) u t (1,0 didm). + Ureu klen: cos r * -I; sffi + Vdi didu ki0n trdn phuong trinh ruong vdi sin r(I - cosz c) - cos r(1 - sin2 r) .3n t**t" * 1+tin;-=tanr+1+cos(f -2c) + Phudng trinh; (sinr -t cosc)(cosr - 1) : Q + sinr;+CoS,x*: n tu O J + cos;r : 1 [a dLr'o-c nghiOm r : n2r, n e Z (thoa rn6n (*)). [...]... D}.t t : (*) a ut + t e [0;1], phuongtrinh trd thirnh: t(t Xdt ttdm f(t; = f'(t) - t)3 25 t(1- t)', t € [0;1] : (t - t)2 (t - zt) + f'(t): 0 o... ilidnl Cho hlm s6 y = x' -3tnx2 * nt -l ' l Qftu vi v6 dd thi cria ddng bidn Vx < 0 Kh6o s6t su bidn thi n i ^2,, Tim m dd him sd tI li him sd khi m = l (2,0 didm) l+4cosxcos3x= i , r GiAi phuong trinh: ^x /sln2 2 Cau y - tl Jzx+t+J3 -2x =\L*-t\2 2',/ kt Giiri phuong tr)nh: III (2,0 dihn) 'frong lihOng gian v6i hQ to4 dQ Oxyz cho hai duong thing s_x-r_y-r ' trt =-Z = 1 '1 (xeR) Z = z-J | r ot= _x+)... D6 thi hAm sii bdc 4 khdng co ti€m cin D+o hirn: y/ : 4x3 -8x: +^(^2 -z) [^ o _t; - (v: -t) *-l') ! 4 Y - +cc ^{, _U Bing biSn thi n: (f ::) [x: o Y':0 . coi thi khong giai thich gi thAm oAp Aru - TI{ANG uldm f. { CAU D6p [n Diim I (2,0 didm) t. (1,0 tlidd Khho s6t. + T4p xdc dinh: D = R + SLr biai thi n: - Chi€u bien thidn:. s6t.s1r bi6n thi n vi v6 d6 thi hdm s6 v6im:3. 2. Gie su dd th! him'so cit tryc hoinh qi b6n didm ph6n bigt. Hay tim m d€ cho hinh phinq gidi hsn mi ao thi ham.s6 vd. BC. Cau VIIb (1,0 didm) _z q^ o TimtoadodidmMtr€nddthihhmso:,:ffisaochokhoAngc6cht}A{ddndudrrgthing C:3sl*A+7:Onh6nhat. Ho vd t€n thi sinh So brio danh a* cIA* r*a,ic vA ar.E* i'au: