Thi thu DH-Toan D-2012-lan 2- Le Xoay

\ \ rRtroNG rHpT rp xoav on rnr KHAO sAt cnAr L{toNG Kr{or t2 -LAN rr Ndrn hsc 201r'2a12 rndi eion,Yilp:rT?til"rnuk3;i", giao di) Cf,u 1: ChohdmsO y= mxt -3xz +(*'-3m+2)x+4 (1) 1. Kh&o s6t su bi6n thi€n vd vE d6 thi hnm sti dA cho vdi m = 1. 2. X6c dinh m AC AO thi hdm s6 1f) c6 hai diOm cr,rc dai, cuc titiu nim ve hai phia cira tnrc tung. Ciu2l. t. Gi6i phuong trinh: cos2 x = (1+ sinx)(sin3x - 2sin x + l) . lZ*' -9y3 = (x - y)( xy -I) 2. Giai he phucrng trinh: j , ^ u ; "." ' l*'-3'Y* Y" =-l Ciu 3: 1. Giei phuirngtrinh: x13]5-1 = l1l.frIl+Jts-rlf"-rl 2. Tim s6 nguy€n ducrng n nho nhAt sao cho: trong khai triOn (l + x)'theo cOng thtic nhi thric Newton, c6 hai sO hpng Ueri titip mA ry lQ cria hai hQ s0 bing 5i9. Cfiu 4: ll=-Eoog mFt phdng u,qi he trgc top <lQ Oxy, cho hai duong thtng; d1: x * y + I : 0; d2; 2x* y + 1 : 0.vi di€m M (1; 1). X6c dinh to4 d$ di0m A thuQc d1 vd B thuQc d2 sao cho M li trung di6m cua dopn thbng AB. 2. Cho hinh ch6p S.ABC c6 chi c4nh b€n dAu nghi0ng v6i d6y mQt g6c 45'. D6y la tam gi6c ABC c6 AB = AC = a; eEC = ,q,eB = a .*H ld hinh chi6u cira S l€n (ABC). a. CMR: HA: HB = HC = HS? Tinh SH? b. Tinh th6 tich hinh ch6p S.ABC C6u 5: Bitlt x, y,z e R;x23;y> 4;z> 5. Chtmg minh rlng : ytJi-z**t.[y-+nryJl-s._L* I - l_ xyz =tE- 4-tG http://www.haimathlx.violet.vn BAp Ax EE Tlu xnAo sar cnAr iuql{c xlr6r rz MON: rOAr.r- rH6t n, o V5i rn = t. Ta c6 hdm s5: y = xt -3x2 + 4 Cdc gi5'i han: lim y = -crr; lim y = a[ N f,++€ Sg bitln thi€n: r^) V = 3x- -6x v'-0<>1"=o - l 1 L^_L Hdm s6 d0ng bi6n tr€n (-oo; 0) vd (2; +m) Hdrn sO nghich bi6n tr€n (0; 2) Hdm s6 dfi CE tqiy = 0; yio:4 EBq$ dpt CT tpi >1 : Z: v", = O 0,25 Bing bi6n thi€n 0,25 4+* ,/ \ ,1 ,/ \ ,/- -oo \ o/ Eths cft oy tpi (0; 4) cdt ox tai (-1 ;0); (2;0) Nhfln I (1;2) ldm t6m <ftii xrirrg f(x)=x^3-3-x^2i.4t : : :, .,,::t.:i | :/ '."-:- - i*-: -t : l i:r ., ,: : 1 I , I I t \J: : ! : *_ '-:__- a- j:- *, i l \r, : l \:,l gfr i 1: I i ii: r! !l li I ; =., , ll]* i, i, i ili i :i : -i :-:; -: -;-, -i *- : .r | : I _i _,1: : r/ i\.;r ri . - j + j Jj , - i I t! 2r','t I tl *-r : i., i , ,, ,. .i: : 1 i i ,,.: , i: : ol -zl : : i-,, :_ : ti 0,25 !'=3mx2 -6x+{m2 -3m+2) D€ ilths (l) c6 cuc dai, cgc t tr6i dAu. thoi miny€u cAu bdi toa http://www.haimathlx.violet.vn l3*(*t ()< lm*0 *3m+2) <0 *[ v0y m<0 1<m<2 m e (-*;0) U(1;2) cost e(l It lr I = (1 + sin x)(sin3r - 2sinx + t) - sin 2 .r) = (l + sin x)(sin 3.r - 2 sin .lc + l) +sinx=0 -sinx=sin3x-2sin.r+1 lsin x = -l cri lsin 3x = sin x r, I E/ lr=-'72 elx=tr L.=%* + k2r ln,/ /2 (k;leZ) eZx3 -9yt = xt - yt) <) x' = 8/' o x =2y (3) Th6 (3) vdo (2): - y' = -t <+ y = +1 VOv h0 tl6 cho c6 2 y =1> x=2 /=-l)x=-2 Edt #; = u;Ji]l =, e s u;v < 2) Ta duoc: v2 +l + 3u = | + 3u + uv DK: 1 <x{5 v2 +3u-3v-uv=0 (u -v)(3 - v) = 0 fu=v [v=3 Do rti6u ki6n n6n u _ v +n/j- = .f,J <+ 5 _ x = x _f *,, =l-(til6 D6p s6: nghi€m pt ld x = 3 0,25 (1) (2) = (x - y)(4xy -t) 2 - l y r 4xy + x2 -3xy + y2) y)(x' + xy- y') Gi6i he: {':' -"' t2 I.r - Jxy + rh6 (2) vdo (t): 2x3 -9y3 = (r - yX e 2xi -9y3 = (x - 0,25 0,25 http://www.haimathlx.violet.vn 3.2 lal' 1.I Theo cOng thirc nhi thrlc Newton (l + x)' = Cl + C)x + Clx2 + + C!xk 2 h$ s6 ctra 2 s6 hpng li€n ti€p lit: C!;C!.l (0<f <n;keZ) 0,25\ Theo gii thi€t: ck +Xdt: -' c:*' 9 kt(n-k)r. 9 (k+1)!(n-r-1)! e9(k+l) = 5(n-k) elt-zk+l+4k+4 5 0,25 n ld s0 nguyen ducrng nho nh6t = Gk ! O ld s5 nguy€n duong nh6 nh6t =,t=4=n=13 0,25 fk+t S *Xdt: -n' = " c:e ES: z: l3 . Tuong t.u c6 k=8, n=l3. 0r25 4.1 1rI 1d Vi I e d, + A(tr;-t, *1) B e d, + B(t2;2t, +l) 0,25 M(1;*1) + A, (t - t,;t,); MB(t, - l;2t, + 2) 0;25 (r l tL-1,=1,^-l Theo ycbt - AM = MB = 1- 't '2 It, =Zt, +Z ={,, Ir' -0 -t 0,25 > A(2;-3);B(01) 0,25 4.2 2d 2d 4.2a. SA c6 hinh chi6u h HA tr€n (ABC) '=> 96c gig_a SA v,q (ABC) l4 &4i? = 45' => ASAH vu6ng cdn t4i H => HA = HS Tuong t.u : HB = HS; HC = HS VQy HA: HB = HC: HS Theo tr€n -> H ld tdm duOng tr&n ngo4i ti6p AABC;iheo dinh ly Sin: AB/SinC{AH => AH = a(Zsino):> SH: a(2sino) 4.2b. I V*u, = iSH.SMBC l^1 S-ac = =en.AC.sinBAC =*a'.sin(l80" -2a) 22 1 =!a2.sin2r;, 2-" '-"'-* 4,25 0,25 0,25 Ar25 0,25 0,25 http://www.haimathlx.violet.vn JE I vsnc = +#:'z sin2a= 1 o,. ro, o viitrai -G-3 *F *B xyz ?/,\ Jt- X6t hdm s6: f Q) = tf e 2 a;a> 0) ; a ld hing sd. -f'(t) = BT: T a 2a +co f(t) +0_ f(t) J;/ ,/2o *Jt-3,"ly-4 J= | I I * -; * - , * *;- = ui+ V.ft (Ddu ':' xtty rakhi x=6; 58; z:10) n- .Jl .,[ru .Jo -x. - 6' y - I f t: ',1 Z- ) _ V: zl0 http://www.haimathlx.violet.vn . mxt -3xz +(*'-3m+2)x+4 (1) 1. Kh&o s6t su bi6n thi n vd vE d6 thi hnm sti dA cho vdi m = 1. 2. X6c dinh m AC AO thi hdm s6 1f) c6 hai diOm cr,rc dai, cuc titiu nim ve hai. x * y + I : 0; d2; 2x* y + 1 : 0.vi di€m M (1; 1). X6c dinh to4 d$ di0m A thuQc d1 vd B thuQc d2 sao cho M li trung di6m cua dopn thbng AB. 2. Cho hinh ch6p S.ABC c6. -9y3 = (r - yX e 2xi -9y3 = (x - 0,25 0,25 http://www.haimathlx.violet.vn 3.2 lal' 1.I Theo cOng thirc nhi thrlc Newton (l + x)' = Cl + C)x + Clx2 + + C!xk 2 h$