1. KhAo s6t sU'bi6n thi6n vi ve d6 thi ( C ) cira hdm s6. 2. M ld di€m b6tty o'.tren (C) TicpjuytSn cira ( C) tai M cdt c6c dudng ti€m c4n cria (C) tai A vi B; gqi I ld giao di6m 2 duong tiQrn c{n. Tim toa dQ M sao cho dudng trdn ngoq.i tirip tam !ij. iae c6 di€n tichbing 2n CAu II( 2 di€m): io'crAo Dvc & DAO TAO HATPHONG TIIU,ONG THPT THAI PHTEN i. PHAN CFILN{G CHO TAT CA THi SINH , '^ 7 Lau l(z olem) , 2x-3 Lno ham so y = ;-;.( C) I-L cos2x + 5 sin 1) Giei phuong trinh : DE THI THII EAI HQC LAN II IYAM 2011 M6n : TOAN _ KIIoI A Thdt gian : 180 phrit ( kh6ng nC tndt gtan phdt di ) =- / 2) Gieibdt phuo'ng trinh : -E < x+Zt P - Jn.u)" I x=2+t az:ll=-3+3t felR. I z=t J7T \ iT- I )l cAu rrr ( 1 di€m) : Tinh , 's*(rr, -+)* d ( J+-*') c6u IV ( I di6m) : Trong kh6ng gian cho ldng tru dtrr.g ABC.AB:C, eo AB =.a,AC =Za,AA, =2ali vit ti =120" . Gqi M li.trungditim cira canh CC, . Hdy chfrng minh MB L MAl vi tinh khoing c6ch tir A toi 1.ndt phing (A.BM ). Cdu V ( I di€m): Tim ntde hC phuo,ng trinh sau c6 nghiQm v6,ix > ,, {111 =- fVx'+3*^ly'+5=m rr. PHAN RrENG( 3 diem) A. Theo chuong trinh chuAn CduVLa: ( 2 di€m) l) Trong mat phing v6i h0 trsc tea dQ Oxy cho tam gi6c ABC c6 dinh A rhugc (d):x - 4y -2=0; dudng thing BC songsongv6'i(d);phuongtrinhdudngcaoBH:x*y*3=0.vdtungdi6mcanhabtaM(f;1).ti-mtqadgcac dinh A, B, C. 2) Trongkhdnggianv6'ihQtrqctqad0Oxyzchom{tphing(P) :x -2.y+Zz+2=1vdhaidi€mA(4;1;3)vh B(2;-3;-1).Hdy tim di€m M thuQc (P) sao cho MAz + MB.2 c6 gi6 ni nh6 nh6t . C6.uVil.a( ldi6rn) .t Trongc6c s0 phuczthoamdnphucrngtrinh: liz -Zl=1, -2 - il haytimstiphri'czc6acgumen bingTr B. Theo chuong trinh ning cao : 4 Cdu VI.b: ( 2di€m): l) Trong mdtphingOxyclio di6m C( 2;0)vit Elip(E) c6phuongtrinh: t* r= =1. Timcricdi6m A,B thuQc 4' @) biet ring hai di6rn A,B dOi xn'ng nhau qua tmc irodnh vir tam gi6c ABC ddg. 2) Trong kh6ng gian vdi h€ tga dQ Oxyz, cho hai dudng thdng: r .x-4 y-I z+5 At | ;- = -; = :- Va : J-i-2 - Viet phuong trinh mflt cdu c6 b6n kfnh nhd nhAt ti6p xirc voi chhai duong thing d, vd d2. Cdu VII.b: (1 diCm) Tim gi6 tri nt d€ hdm s6 y = x2 +(m2 -r)x-m2 +m x-L zr) x+- I 3) tan cira d6 th! di qua di€m M(l; 5). d6ng bi6n trdn cdc khoing cria tflp x6c dinh vd ti€m cdn xi€n www.MATHVN.com www.mathvn.com NOi du oAp Ax vA BIEU DIEM THI THITE C LAN II - KIIOI A -2011 l)Khio sit vi vc a6 tni hirm s5 -2x-3 x-2 1. TXD : R\{z} 2. Su biSn thi6n : + Gini hpn - TiQm cgn +D6thi: Gitii phu'o'ng trinh : lim y =a6p ; hn-l_ ! =-q= d6 thi c6 ti6rn cf.1 dirng litx:2 x+2- x-+2- li+ y =2 = dd thi c6 tiQm cf.n ngang lity = 2 + y' = J-< 0 Vx ;e 2 = hdrn sii nghich bi6n tr€n (- -; 2);( 2; +oo) ' (*-2)' v E? thigiao vdi trpc Oy tai di6m A(0;312); D6 thi giao v6i fi'uc Ox t4i di6m B (3/2 ;0) D6 thi nhdn Di6m I Q:2) lA giao cria 2 tiQm c6n lALrn tAm d6i xilng. 2) co: M[",?;), xs * 2,y'(xo) = 6+ Phuong frinh ti6p ruyt5n A voi ( C) t4i M: t:y =-:! "(r-ro;+?&:1 l*o -2)- xo - z To4 dq giao didm A, B cta (A) vd hai tidm cfln n(rje4)t Bea -z;z) \ x\-z ) .Mdt khdc l(?;2) vit AIAB vu6ng t4i I n€n duorg h'dn ngo4i ti6p AIAB duoTrg tr-on cd b6n kinh R:AB/2. Md theo gt, diQn tich ducng trdn bdng 2x > R= Ji o AB =2Ji | -, (z*-z r'?l , [x^=1 f < " - tr' -l?; -, ).] = t -,* - t)' - G+ =, * Ll =', a M(t: t) vd M (3: 3) cos2x+ssin(x+ =-z(1). Ekxd: 0,25 0,25 0,2s ,un[,- 3n 2 z\ ( r\ -l.tanln+-l 6) l. 3/ f sin(x - a J cos(x - a lsin(x+,r Lcos( r + z l6)*0 l6)*0 /3)*0 /3)*0 l'* *] l'* r ktt -+ - 62 r kr + - 2t ( zo.1^ ^ ) e costx +5srnIrc* 2 )=2eZcos' 4,25 0,25 4,25 0,25 r ^ ,, ., ; 2r I cosx=J ltoat) l r=-;+hZ|T <+l r <+l ' I cosr = I 2n L 2 L"=- : +kt/, KOt hgp Ekxd phuo'ng trinh c6 ngiriQm : (g lx> e{ 2 t- l^ [x + u Cgc tri : kh6ng c6 + Bing bi€n thi6n (tt\(n\ ra c6 : t"[r-AJ ,*["*JJ=-l n6n (t) x-5cosx-3=0 2r 3 l2x+9> 0 1 ,- f 3-V9 +2x +0 www.MATHVN.com www.mathvn.com ^ / ? 2x'13 + JS +zx)- t -t2 r \2 (3-Je -2x) 13+Je+zx)- . 2x2 bpte l = <x+21 \3-J9+2x) 2x219 +6^19 a2* +9 +2x\ -T <x+21 e18 +2x + 6.,1; . 2. <2x + 42 Ktit h-op vdi di6u ki6n x6c dinh x +21 a Jg+z* .4 er<i 2 ta c6 nghipm cta b6t phuong trinh ld : lg j l <.r< - l1 la2 [,*o t- ( I - lxl,2' Jl 0\ I I, = [xe?'dx 0 lx*y=3 lt=3-x ? - z) fJ;r' + 3 * rly' *5 = n.' lJx' +: + D[t /(x) =.,6t;+.ft-rt' * s = -+)0.='l*r',d*-'l-L* '14-x') i irl+_*? Dat u = x,dy =e2t clx - tr=f Ir L='[+- oo, ,=t[4-] +at=P, x= 0=>r =z; x=1=>r=v5 o tf4-x' ^14-*/ 2 + r,= le-f)at=f 4Ji . e2+l 16 +I=-:''*' 3Jj 43 lv{A,t = A,C,2 +C,Mt =7zo)t *("Ji)' =9a2;BC2 = AB2 + AC2 -zAB.AC3osl20" =7a2 i BM2 = N +a,f =7d *(rJ t)' =tfr;48 =A4' +zE =(uJs)' +d =zti Suy ra A,B' = MAr2 + MBz + MB L MAl. Hinlr ch6p MBAAT vd GABA, c6 chung d6y ld tam gi6c BAA, vd du6ng cao birrg nhau n€n th6 tich bing nhau v-t/ -r/ -l - i !a.za.sin:20"=ltJE v = yMB,t,t, = yc'ae,t, = jA4'S^rr. =:2atl 5' 2 3 ^ a'JE =d(A,(A,BM))=#=ffi:m=+ l(r 1 . - A. I \- ,f(:*1\s=,, tt, , X X-3 :.:,=:-: .J x' +3 ./1: - x;'? + s .f '(x)=6a;"nf4afi =(3-DJx'1 +3 o{"-'=t l2x" +I\x-27 =0 Phuong trinh thft hai c6 A,=81+54=135=9.15, vd hai nghiQm: ,r=2Y. hainghi€m ndy ddu bi loai vi nh6 hon 2. vfy, dg.o hiim cria hdm s6 kh6ng rhe d6i O5 t<icm tra ring cd d6u tr6n 12;.o) , rigodi www.MATHVN.com www.mathvn.com VIa r) VIb 1) ra f '(3) > 0 n6n .f '(*) > A,Yx2Z . Do d6, gi6 tri nh6 nirdt ciia f (x) khix>2 phuong trinh d6 cho c6 nghi€m (vdi liL f (2)=^11 +Je .Ctng d6 th6y tyif Q)= co , Tir. d6 suy ra: h€ x>2) khi vd chi khi *>"G *J7 . -4a-l -a+I 2 .( 2 2\ 1 =___ €d= __= ,[_: , _ j ) 817 __0 _+b DoBC//(il.'+=+ A( 4a+2i a) suy ra eUe+a-1;-a+1). Do AM LBH M tdnunghek,q,c ,a" c(2,!\ l3'? / \' - / :x+ y +3 =0 n1n: Be (d):x+y+3=0 n€n B(b;-b-3) e b=-4e B(-4;l) e,Kr1f 4y-lf =17 +1o*y <> ir - -2y -1* z=(-Zy -1) + yi -(q 1'7 \ =BC I Y-n :1-+b l. \3 '3 ) Gqi I ld trung ditSm AB suy ra I( 3 ;-1 ;1). Ap dgng hg thric trung ruy6n ftong tam gi6c MAB n6n ta c6 : MAz +IvB2 =2742 *AB2 2 + MA2 + W2 nhdnhd e tufr nnl nndi e l,fr L (p) ndn M ld hinh chi6u ctia i tr€n (p) Gqi (d) ld duong thing vu6ng g6c v6'i (p) va di qua I suy ra phuong trinh cria (d) Id : M ld. giao di6m crla (d) va @) n6n ta c6 M(2;1;-1) Ddt z =x+ yi(x,y e R) li(x + yi)- :l =l(x + yi) - z- rl o l-y - 3 + xil=le - 2) + (y - \rl x-3 y+1 z -1 -=_=_ 1aa l z:] =r^4=4 (i) Acgumencio,M)gfLellcz:'-,,"' *" 4 Jz t l f- =r-4=-l {3) l^lt-zy -1)' + y' a 'Jz Ctng Q)vd (Z)cho ta y - -1, thi tai th1amdncd 1t1vd 1zy Ydy z =1-i Goi A(x;y)=B(x;-y). A,B phdn bi|tn€n y* 0. Ae( Do C (2;0) n€n ACAB cdn rai C. =ACAB ftieCA=AB €(z-x)t + t, =4x2 (2) lx=-2 >y=0(loqi) Gi;ih€(DvA e)ta itdc: i to ' , iru L^ - 13 -'Y - 6'76 n,,.n*^e r\ ,\(to tz) lto -tz) ta co z dtem A,b can rm ta l,'; n I "" lO; n ) 0,25 0,25 0,25 0,25 0,25 0,25 0,25 0,25 0,25 0,25 0,25 4,25 www.MATHVN.com www.mathvn.com (^ I Y: l-Lf I a, : I I = -3 + 3r = (d, di qua M r(2; -3:0) c6 WCp ur(I; 3; t). I z =t ra rttlu : ,41.M/14, =60 +0+ dr;drch/o nhau.Goi MN litfoqn vutng g6cchung cila d,;d, rh1 Mdt ct cobenkinh nh| nh{t trcl rhe void,d, titmit ciu ahig tciin uw M e d,+ M(4+3r;I-r;-5 -Zt) ; N e dr+ N(Z+ t,;-3 +3t,;r) I tutY .^ =O l-tqr -2t'=lZ lr = -I \m.r,=o o \ ,r* rtt'=9o {r'=t +M(r;2;-3);N(3;0;1) M{t ctu &to-ng ki;h MN cJ ftn r (2;1; -t1 c J u ah nin n = J e P hndng rriih mdt c d7t t d : (x - 2)' + (y- 1)t + (z + I)2 = 6 TiQm cAn xi€n (A): y = x + m2 . t,m y =1 ;>0,Vx+1:> (x -r)' a Y4y m:_Z 0,25 TU M(1; 5) e (A):+ m = 12. *2 -2* +1-m {t -1)2 <+A'<0 em<O >OVx *Ie x2 -2x+1-m>0Y x +1 www.MATHVN.com www.mathvn.com . / ? 2x'13 + JS +zx)- t -t2 r 2 (3-Je -2x) 13+Je+zx)- . 2x2 bpte l = <x +21 3-J9+2x) 2x219 +6^19 a2* +9 +2x -T <x +21 e18 +2x + 6.,1; . 2. <2x + 42 Ktit h-op. "" lO; n ) 0 ,25 0 ,25 0 ,25 0 ,25 0 ,25 0 ,25 0 ,25 0 ,25 0 ,25 0 ,25 0 ,25 4 ,25 www.MATHVN.com www.mathvn.com (^ I Y: l-Lf I a, : I I = -3 + 3r = (d, di qua M r (2; -3:0) c6 WCp ur(I;. - 62 r kr + - 2t ( zo.1^ ^ ) e costx +5srnIrc* 2 )=2eZcos' 4 ,25 0 ,25 4 ,25 0 ,25 r ^ ,, ., ; 2r I cosx=J ltoat) l r=-;+hZ|T <+l r <+l ' I cosr = I 2n L 2 L"=-