TRIJONG EAI HQC VINH TRIIONG THPT CHUYTN of xrrAo sAr cnArr,ugr\c t 6p tzr,An 3, NAna zorr m0n: TOAN; Thli gian lim bhi: IBA phrtt r. rHAN cHUNc cHo rAr cA rHi slr.{H 1z,o a$q 1 Ciu I. 1Z,O ei6m; Cho him s6 y = 1*o -(3m+l)xz +2(m+l), m ldtham s5. "4 l. Kh6o s6t sg bitin thi6n vd vC dO thi hdm sb c16 cho khi z = 0. 2. Tinr llr A6 A6 fti ham sii da cho co 3 tli6m cgc ti l$p thanh mQt tam gf6c e6 trgng t6m ld g6c toa d0. CAu II. (2,0 tti6m) I . Giai phuorg trinh 2Iogo(l a ,l2y a1= logz (5 - r) + log , (3 - x). 2 2. Gieiohuongtrinh lsinZx-cos2x)tanr* sin3x =sinr+cosx. Ciu III. (1,0 di6m) Tinh thc tich khdi trdn xoay tu":iah khi quay hinh phang gidi han boi d6 thi hem 1(:.1 ')L- s6 y = E,trqc hoanh vi dudmg thdng x = 1 xung quanh truc hoanh. A* e- * -l- t-4r ttl Ciu IV. (1,0 di6m) Cho hinh Hng tru dtmg ABC.A' B'C' c6 AC = a, BC =2a, ZACB = 1200 vd <tudmg ttrang A'C t4ovoim{tphdng (ABB'A') g6c 300. GgiMldtrungdi6m BB'. TinhthCtichk*r6i teng trU dd cho vd khoang c6ch gita hai ttuong theng AM, CC' theo a. CAu V. (1,0 iti6m) Tim a dO hq phucrrg trinh sau c6 nghiQm II. PHAN nrtNc e,o iti6m) a. Theo chuong trinh Chu6n CATVIa (a0 di6n$ l*' ^[y a -2xy -2x =1 ft' -rr-: x! = a+2 Thi sinh chi tlugc ldm mQt trong hai phdn (phin a, hoic b) l. Trong mat phdng tga ilQ Oxy, chodudrng thang d:2x+y+3=0 vd elip (E) ,t*t-=l. Virit '41 phuong trinh dudrng theng A vu6ng g6c voi d vit cht (D t+ihai di6m A, B saocho diQn tich tam gifuc OAB bing 1. 2. trong kh6ng gi* tqu dg oxyz, cho m{t pheng Q):2x-y+22+9:0 vd hai di6m Ae;-l;2), B(1;- 5; 0). Tim tqa d0 cta diOm MthuQc (P) sao cho ffi.uE d4t gid tri nh6 nhAt. Cf,u VIIa. (1,0 tli6m) Vitit ng6u nhi€n mQt s6 tw nhi6n ch8n gdm.4 ght s6 eoi mQt kh6c nhau l€n bang. Tinh x6c su6t e6 si5 vira vii5t th6a mdn trong s6 eo m5i crrt so dAu lon hcrn chit s6 e,mg tru6c n6. b. Theo chrorrg trinh Ning cao Cf,u VIb. (z,o ai6n; 1. Trong mat ph6ng tga d0 Oxy, cho parabol (P): y' = 4x c6 ti€u diOm F. Gqi M h <ti6m th6a man didu kiQn Ffr = -3fu; d ld ttuong th5"g U6t ti tli qua M, d cgt(P) tai hai di6m phdn biQt A vit-y. Chtmg minh reng tam gi6c OABliLtam gi6c vu6ng. 2. Trong kh6ng gian tga dO Oxyz,cho dudrng thang d,**=l ='.4 =l vitc6c <tii5m Ae;2;7), -2t2 B(l; 5; 2), C(3;2; 4). Tim tqa d0 di6m MthuQc d sao ebo MA2 - MBz - MCz dat gi6 tri lcrn nh6t. Ciu VIIb (1,0.tli6m) Hai ban An vd Binh thi it6u voi nhau mQt t'{n b6ng ban. Hq quy u6c choi v<yi nhau nrlAu.nh6t 5 s6c, ai theng tru6p 3. s6c li ngyd th*g cuQc vn tren rliu k6t trtir. rinfr:<a. r"aida trat d6u t6t ttnic sau s6c thf tu, bitit rang xac su6t An thing trong m8i s6c ld 0,4 vd s6c ndo ctng c6 nguoi th5"g. .L d- /n- n6t Ghi chrt: L Bfg s€ trd bdi vdo cdc ngdy 21, 22/05/2AI I. DA nhSn iluqc bdi thi, thi sinh phdi ngp lqi phidu du thi cho BTC. trongxuanht@yahoo.com sent to www.laisac.page.tl -['ltu'*i\i{i S,{i l-it}C !'i}ili 'trR{i'#.}iil T'l-{P I' il}"ILryEN DAP EN+ BE K}IAO M$F{: It $AT CIL-{T L.u-'#io{G L{3P 12 LAN 3, i\.4,.h,{ ?{}11 TO,LN; Thcri gian ihm bii: J8 i) pltfit I, (?,0 rr^ \ {ttcm) Khi m - 0 harn sd trcr thanh y =+t 4 a. T?p xic dinh : D = ffi ;1l le hirm sO b. Su bi0n thi€n: * ChiAu bi6n thi6n: Ta c6 y'- xt * 2 [x=0 y,=0 e I \r7 Lt = +Ji' Suy ra hdrn sd ddng bi0n tr0n c6c o '* r' +2 chiu. x. I t-x > J, ['. -Ji y'>0<+l ,_ ;y'<0el r l-J1<x<o/'/ Lo.x<Ji khoing ( D; O) vir d1;+ oo); him sO nghich bi6n trOn cAc 0'5 khoing (-*; - Ji) uit (O; Jz). *Cuctr!:Hdmstidatcgctl4it4i x=0 vdi y.u=2;hdmsi5datcqctitiutqi x= Ji ve r=-Ji vbi yr, =1. c" Dd thi: ss-tfri 'itdiii s# nir$n trgc tung limn trui: d6i xr?ng. Hilm s& dd sho c6 3 di6m cgc tri € 3 nghiQm phfin bipt {=} x3 - 2(3 ln + l)x - 0 c6 3 nghiQm Khi d6 3 di0rn cuc tri cria .I'= 0 c6 phAn bipt d6 thi 1 QM .a J le AQ;2m+2), B(- (1) -9nr2 - 4m+ 1) va gi6c ABC a !a+2!n - 0. Hay 2m+2+2(-9m'-4m+1)-0 egmz + I(€t hqp vo'i (1) suy ra gifttri cua m|d m -+. 3 Diou kiqn: 1= x < 3. 2 Khi do phuong trinh dd cho e logr(1 + e logz(1 + JZx -_I) - log, (5 - x) - logz (3 - x) Jzx-r)-log r= 3-x 5 -x -#e^lTx-l - ? f m Zl3 3m-2- 0 e I L* -Il3 fidn tin 1. (1,0 diint 2" ffn# di6m) L. (1,0 ili€m / I I I -"{; 6nt + 2; 2x -I e1+ :G:.^,-r * i'-,',+ rj ''' :' "- I I I I I I I I i I I I I i I I I I I | - -' i r ta {x *3}'{?x * 1} - { c* {x * t}{?x: -* I tx + 13} * * E=* X.6t hqp di0u kign ta co nghigm cua phuong trinh la x: l, 1t-Jr? 'r,- 4 1 o , Jo * ., r i. t Vt / l'r' - {-tt rw 4) t'u A .T tlrJ 2. $,8 iti€nt) DiAu kiEn: cos.r * 0 +> x *L* kn,.k eZ. ')" V6i di€u kiQn d6 phuong trinh tuong cluong v6i sin2.xsinx - cos2;rsinx + sin 3x = cosx(sinx + cosx) e sin 2xsinx - cos2-rsinx + sin 2xcosx + cosixsinx = cosr(sinx + cosx) e cosx(2sinx - l)(sinx * cosx) = Q e (2sinx - l)(sinx + cosx) = 0, vi cosx ;t 0 . 0'5 015 * 2sinx-l = 0 <+ sinx=!o * =L + k2nv x =5T + kyn 266 * sinx+cosx - 0 e tan.x = -l e x = -! + kn. 4 Vfy nghiem cta phuong trinh li , =[+ k2n,. =+ + k2r;, = -L + kn, k eZ. Chri f: HS c6 tnc viet nghiQm cria PT: r : (- ry +* kn ;x : -t+ kn, k eZ. r[[" t1,0 iIi6m) r- m , vxg' Ta cd + € x - 0" Suy ra hinh phing da cho la st+1 ixe' Y- ,!:0,x-0 ve x:t" r et +l r Do d6 &$ tfch kiroi triiii xilay le V - n'{-g ^ dx. i {-n. -i- 1}' hinh thang cong dugc gi6i han bdi c6c du'crng (1) 015 015 Edt u: x. dv = "' = d*. Khi d6 du = dx.u : l ' (e'+l\t e'+1 Theo c6ng thirc tich phAn tung phAn ta c6 'l-g ^d*: , ' l' n';d* -l *'{r )* 'fG,+tt'* - "' *11' - J"' *t - ".1- J[' -'\1 f' c. =-!*rl' -,n1", *rJ' : :-_tn "l l. . e+l lo -lo e+I 2 Thay vio (l) ta iluqc th6 tich kh6i trdn xoay li v =' ,( - " tn ' * 1.). '"[e+1 "' z )' IV. (1ro di€rn +) Ke CH L AB. Vi AA'-L(ABC) n6n AA,LCH * CH LTABB,A,) * /,CA' H = (A'C, {ABB, A,)): 300. +) S* dUng dinh li cosin ve cdng thirc diOn tich cho LABC ta c6 AB - aJ| , CH -ZS nuc - a'}o'sinl}}o - tr AB ffi-:ol,' +C4=1CH =z"E + AA'= -a + ) Th€ tich ldng trg li V - AA'.5 ABC - a az Jj _ a3Jl os 214 - ACz t 7 : 7 0'5 \u-< A' +) MAt phlng (ABB' A') chua AMvi song song = d(AM,CC') = d(C,(ABB' A')) = CH - c 5 '11 - CC' ,ln 7 0'5 V. (1ro di0m { t _ " xz? * I D+t, = Jtn I . r 2 0 rrQ trcr rrranir l;, t :utr; == o + Z Rd ran g z. S khOng thoa rnfln hg. VEi e > 0, d?t x - tz hQ tro thantr f tt U' *zt1 -1 fr'(r';3r)= a+Z (1) (2) Suy ra BBT DUa viro BBT suy ra he c6 nghiqm hay 0n chin g6m 4 chf s0 duo-c viet ra th6a mdn m6i chfr sd 16n hon chfi' s dung tru6c n6. Khi d6 Q = {abcd '. a + 0, d e {0, 2, 4, 6,8\\; :- 3 S,5- fa>4 I l1 la< L2 lo*2>6 el 3 I a+2<- L2 f'(t) \. (1,0 di6m. (?,0 iIi6m) VIa. l*)AJ-d=ptAcodpn +) Tqa dQ A, ,B le nghiQrn d cit (r1 tai hai diom A, B +) Ggi A(2yt-m; !t), B(Zyz . mi !) trong d6 h' lz ldnghiQm ctia (1) + !,:!,:+'!:!: .# . : ;j.: , * ABz = 5(yz - yr)' =5[(vr + vr)' - 4vrvr)- 8P ] AB ='li E7 +) Euong cao oH =d(qL)=#- roou=loH"e'n-I.'YA =1 € m2 =4 € m=t2 (th6a man (*)). suy ra phuong t inh A.t-2y* 2 = 0 hoa" t g x-2y + m=0. lx -2y + , ,A I cua hQ 1 x'' ) l-+ Y- t4 J € h9 c6 2 nghiQm m-q [,r_ Zy-m (+{ -1 ls"ut -4my+m2-4:o (1) phen biet€ 3 2- 4m' > 0 e -zJz <m <zJt. (*) 2. (7,0 ttidm. 015 +) Gpi / le tt""g di6m AB.Khi d6 I(2;- 3; 1) vi fr,*78 = 0- +) Ta c6 fr4.M8 = (Mi +fr>fffi +787 = tui + u>fat - u) = MI' - IA2 . +ffi.uE datgi6trinhonhAt e MInh6nhAtldo u' =lf khongd6i)' .? 4-44.bit'I qt'i9r-v'l-ole eeg gll+-{lt-"1-(1)'- - - - lx=2+2t I +) Chqn G =6 =(Z;-t;2) + phuong trinh tU:ll =-3-t. Thay vio phucrng trinh (P) suy ra lz =l+2t t=-2* M(-2;-1;-3). i *rr 11 \ i ri ltltxl,] n.{ ={r;l;*.r.t':0<{r <h {{- 1dt, De tinh ifil ta xrit cac tnrcvng hqp sau +) d =0. Trudng hqp niy c6 , j s6. : +) d e{2,4,6,8}. Truong hqp niy "6 (4 - efi.+ sA. suyralol=4 ++(4-4>72221___ _ __._ _.___-__ _ pe tintr lQn I ,u x€t c6c trubng hqp sau +) d - 4. Truoug hqp niy cd I s6. +) d = 6. Truung hgp nay cd Ci s6. +) d :8. Truo'ng hqp nay c6 C; sS. Suyra lCInl=t *Cl.+Cl -46- Do do P(A)=l# : :? r= 0,02, -/ lCIl 2296 ' ie d ln s* cl-i1n ) . \ l. (1,0 tti6m. Ii}' | .l (P) : yz - 4x c6 p :2 + ti6u di€m r(l; 0) olai"l l.rl.t€u d r ox*pt d:x=e.Tthe {f |' ^rYs [x-4 =ffi.og: 16- 16 - o + aoB: 9oo. +)NCu d LOx* pt d:y-k{x-4)- * M(4; la$; =+{ LB(A; 0), 4) -4) Tqa 6Q A, B la nghiQm cria h$ li;la;G;'a;aist{uii,;iaiffi-pifi;i,i-ditii-(it;ltt;si;Gil;i'd;tilct c+l';-d: _.2 ,2 Gii sri LtLq; yr), B(?; yr) trong d6 /rr v, li nghi$rn cria (2) ) !t!z = -i6. Ta c6 d.oE = 1Wz'rz. + lrtz = ?4)' -16 = 0 = AOB= 900. '4 Suv ra O,4 vu6ne e6c v6i OBhay tam gi6c OAB vu}ngtrong mgi trudmg hqp. Tu "9 !pg!n. {v-b-4k l*-Y: i'^ <+{ 4 Ly'==4x ' , A., 1r Itty'-4y-r6k-o (1) 2. (1,0 iti€nr @thfcsau4s6c;Anbi6nc6Anlingudithdngchung,cuQc;l;1ibi6n ;6 A; itt"ne te"rhi t; B auii5n "6 Binh li ngudi thdng chung cu6c vi.a, tiui6n c5 ninn thing sdc tht i, i :1,2,3,4. Khi cl6 ta c6 H=AwB; A : "Trong3 s6c tliu nn *ring 2 sdc vi s6c thfr tu An thing" = ([Azh w ArBrA, w BtArAt) Ao ; B : "Trong 3 s6c ciiu Binh thing 2 s6c vd s6c thri tu Binh thing" = (B1B2A3 v BrArB, w ArBrBt) B o. ' tt ;# iildt ;l,r * "i; i+ i,, ti,; :i :e' ;al' ; : r,;;,',i ' - "' -' - Theo c6ng thirc tinh x6c su6t ta c6 P(l) = 3.19 ,412 .0,6.0,4:0,1152, P (B) =3'10,6;2'0'4'0'6 = 0'2592' Suy ra P(H) - PtA)+ P(B) = 0,3744- YIIb. (1,0 tIi6m) l*'!,4! Lrr.j:! . -_ I) - log, (5 - x) - logz (3 - x) Jzx-r)-log r= 3- x 5 -x -# e^lTx-l - ? f m Zl3 3m- 2- 0 e I L* -Il3 fidn tin 1. (1,0 diint 2" ffn# di6m) L. (1,0 ili€m / I I I -& quot;{; 6nt. khongd6i)' .? 4-4 4.bit'I qt'i9r-v'l-ole eeg gll +-{ lt-" 1-( 1)&apos ;- - - - lx=2+2t I +) Chqn G =6 =(Z;-t;2) + phuong trinh tU:ll = -3 - t. Thay vio phucrng. I cua hQ 1 x'' ) l-+ Y- t4 J € h9 c6 2 nghiQm m-q [,r_ Zy-m (+{ -1 ls"ut -4 my+m 2-4 :o (1) phen biet€ 3 2- 4m' > 0 e -zJz <m <zJt. (*) 2. (7,0