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Tim so h,;mg khong chua x trong kh.trien.. So hanq thu k+1 cua kh.trien..[r]
(1)men: loan (thi thu Ian 13/6/2015) -Tnvcmg ChUYE!n TNH DAp AN vA BI~U DI~M Ciiu i Y Bai giai Diem 0.25 y 0.25 - b Viet ohuono trinh tiep tuven cua th] (C) t<;lidiem c6 hoanh x thoa man y"=O o y'=6XL-6x " O Y = ~x=2 o o Ma: y'( '-"-'- - _ - - - - _ - - - - 2)= -~y( 2' 2)= 2 -"'-'._."_ -"-"-"-'"-"-"-"'-"-"-"-' Oketqua(t) y=-2(X- )+2 i la (t): y=y'( - '- - _ _ - ,-,,_ i i i) )(x- )+y( _".- _."- ._ _. '-"-"-"-"-"- - "-"-"-"-' "-"- 1 ~(t):y=-2x+4 (1-2i)Z=X+2y+(y-2X)i _ => (1-2i)z+3(1+i) { 3(1 + I)Z = 3(x + y) + 3(x - y)1 _ -"-"'-"-"-' _ _-.- - '-"-'._".- ._ _ _ _ _ _ - '-"-"- _ I i f T + b, Giai bat phU'O'n9 trinh 0.25 _ _ _ _ - _ ._ _ _ _ _ 0_25 =2+7i_ Tinh O.5d Ii± z =2+7i <=> 4x+5y-2+(x-2y-l)i=O c 0_25 {x 4X + 5y = = ~ => z=3-21 { X - 2y = Y =-2 · - - - _- - - - - -_ - - - - - - - - - - - _ - _ - - - - - - - - - - - _ - - - - - _ - - - - - - _ - - - • z=3-2i => i+ =3+3i => [i+ 1=312 <:=:> z '-'"-"-"-"-"-' 0.25 _ _."_ _ _ _ _ _.'-'._ _ - Cho s6 phlic Z thoa (1-2i)z+3(1+i)z GOi z=x+yi voi X,YE!R va i2=_1 • _ _ _ _ _ _ - _ _ - '-'"-"-" '-"-"-"-" '-"-"-" tiep tuyen voi (C) t<;lidiem co h.do x= _ _ _ _ - - _ - _ - _.- _ - - - - - - - _ Ph.trinh .- .- '-"-"-"-"- Ciiu ~ a, 0.25 -"-"-"-"-"-'"-"-"-"-"-"- ._ 1diem ~ y"=12x-6 _ z 6.I09~ x + _ 5.109,.12 x - s voi x la s6 thvc - _ _ - _ 0.25 0.5 . . .- -. - . -.- .- - -. - - - - . .- - - - - - - - - • Dkxd: x>O • 6.109~ x + 5.109J2 x - s ~ 0.25 3109~ x + 51092 X - s ~ • -2 S 1092 x S - ~ «r: 0.25 - s x s ~2 n cso Tinh: n f sin 2x o(sinx+2)2 1diem dx -n f - 1= 1= sin2x o(sinx+2)2 dx = f2sinxcosx (sinx+2)2 dx 0.25 COS xdx = dt • £)~t: t=sinx+2 ' ' '-_. -_.-_ _ - ~ J sin x (sinx+2)2 . -._ -_ = t - dt _._ . t2 - -_ _- - - - - -_. _ - ._-._ _ -, .-_.,-_._ -_. - ._. _._- I (2) • • x=O => t=Z: x= ~ =>t=3 _ _ _ _ _ - _ ~ _ -.~ - _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - _ _ _ _ - _.' _ - - _ - - _ _ _ • = 2ft t~ dt = 230il!_t 2.~)dt t '-"-"-"-"-"-"-"-"-"-"-"-"-"'- .- 1 + 2(!)\3) t _.'- _ - _.- - - - - - - - - -. - -. -. .- - .- -. - -.'- - '-"-' "-"-"-"- 025 '-"-' "-'._" 0.25 Cau : a, Giai: sinx+sin2x=sin3x • sinx+sin2x=sin3x <=>sin3x-sinx=sin2x x = k.n [ 21t X = m.21t <=> x = n'T • <=> 21t X = <=>2cos2xsinx=2sinxcosx <=> sinx [ cos2x = <=> = cosx voi k,n,m 113cac so nguyen x=1t+k.21t n.3 0.5d 0.25 0.25 - Cho x=O Tim so h,;mg khong chua x kh.trien 'b I _ _ - _ _ _ _ _ - = 2[(lnX)I~ .1=2In2"+4(3"-2")=2In2"-3" 0.25 '-"-"-" cua: (x -1J ' biet: A n2 = Cnn-2 + Cnn-1 0.5d + 4n + • £)kxd: n E N va n~2 •A2 = n n! cn-2 + cn-1 + 4n + <=>A = c-: + 4n + <=>-n+1 (n-2)! n n = n n E N 1\ n ~ {n E N 1\ n ~ {n <=> <=> <=> { 2n(n -1) = n(n + 1) + Sn + 12 n2 -11n -12 = n • Khi n=12 ta dU'Q'C: ( x - (n+1)! 2!(n-1)1 + 4n + <=> E N 1\ n ~ E {-1; 12} <=>n = 12 JX2 )12 k So hanq thu (k+1) cua kh.trien k • T k+1khOng co chtza x ~ • W.y so cau E 113:Tk,1 ~-D = C~2(-2rx-2.x12-k = C~2(-2rx { 24 - 3k = <=>k = S <=>k = S hi;lng k6ng co crura x la: T 9= 28 C~2 .g (?)_co_! .Y.t.pt ~a.~.:=J1;:.1~~),-I/~ dl~~) 0.e~ ~ ~~.1_vt~pJa._~.~J~~.~ ~! _ _ _ _ _ _ _ _ n_ I b, p £)th O ang d co {1vtc :n=(1;-1;2) M(1,-1,2)Ed => Ph trlrln h d: IX=1+t ri t E ," m y -1-t VO'l 2t lZ = + Viet phtrinh m.cau co tam narn tren true Ox va tiep xuc voi mp(P) tai M ME (P) nen Tam cua rn.cau (8) can tim 113giao diem cua d va true Ox ~~~_d~9~_~9.~I Q.~_?y'~.~<?:i_t.=:=.:1._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Ban kinh cua (S) 113R=OM => R2=OM2=6 V$y: (8): x2+l+z2=6 cse 0.25 N,k ~ 12 Cho mat phanq (P): x-y+2z-6=0 va diem M(1 ;-1 ;2) Viet ohvono trlnh dlJ'anq di qua M va vuonq g6c voi mp(P) : : a, 0.25 Cho hinh lang tru ABC.A'B'C' co day 113£lABC vuong can tal C, AB=2a Hinh chieu vuong g6c cua dinh A' tren mat ph~ng day 113H trung di§m cua earth AB, g6c giQ>aA'C va mp(ABC) la 45° Tfnh th§ tlch cua khoi lang tru ABC.A'B'C' va khoang each giQ>adU'ang th~nq BB' va A'C A' / / C' : _ /:A S' 'H , _ - JL , / / c ' _~_ J _ _ _ _ _ _ 9.'2!.!J~.!~~qi.~~.g.~.~I~_~!].?~.q: ~~_~~jl I::!.~~~.~'-' _ _ _ _:._ _ 9.~.~.~~.~9' diem 0.5d 0.25 0.25 0.5d 0.25 0.25 (3) -~ABc~'~6'~;-~-~~-t~iC-;&i"H'i~'tr~~g"di~~'~G;'AB',-AB~2~-;-C'A~C'B~~-J2";~'CH-~;"-"-" 0.25 • H=Hc(ASC)(A') =:> CH=Hc(ASC)(A'C) va 00<,.i\'CH <900 =:> ,.i\'CH =450 _·_H.I::!C!-i~!?~Jy.? ~~~~g'y~.<?.']g_C?_~.t.§l.i t1.? ~.'.Ij:=:~; _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ , ·VASC.A·S·C·= 5MBC.A H = ",a 0.25 ."._B.!?:(IM'_? ~~'.!(~~.f:.~r;L~.~iB.!?:;g.AJ=:=.9.(!?f?.:;(~~~q)2=.q(~~(~'~)l=.~~_(t!;lP:~'_r:;J) • IHIIBC =:> BC.lIH va BC.lA'H =:>BC.l(A'HI) =:>(A'HI).l(AA'C) theo q.tuyen A'I Ma HKc(A'HI) va HK LA'I, nen HK.l(AA'C) =:> HK=d(H;(AA'C)) , " • 6.AHI vuong tal H co • Do do: d(BB';A'C)= _ aJ2 1 HI =:>-=-+ =-+-=-=:>HK HK2 Hf HA'2 a2 a2 a2 =- 0.25 a.J3 0.25 2a.J3 cao (Chu It: H.sinh co th~ qiai each dunq t.chat d.cao cua tLi' dien vuong H.AA'C tc;li H) Cho hlnh thang ABCO vuonq tai A(1;1) va B Tren AB lay diem M cho BM=2AM, N(1;4) la hlnh chi~u cua M tren dU'ang thllng co Tim tea cac dinh B,C,O bi~t CM.lOM, BE d vai d: x+v-z=O A(1;1) di~m " D , , ,.:r., : '" N(1~'I) c B d:x"y.2=O '-O'Cmr:'-BN~AN':-C~c'tc;'gi~c-Bc-N-M ';;~-MNDA~n¢iti~~'tr~ng-d-U;t;;n'gtrb-n~"-'-"-"-'-'-" -.- - - -.-.-.- - - CNB = CMB , rna CMB = M6A do: MC.lOM va OA.lMB; MDA = ANM Do d6: 90 = CNM = CNB + BNM = 8NM + MNA = BNA =:> BN.lAN "-" "-"-"-"-"-"-' " '-"-"-" ' ' "-"-"-"-"-"-"-"-"-"-"-' "-" '-' "-''-"-"-" "-"-"-"-"-"-"-"-"-"'-"nen: 0.25 o o BEd~B(b;2-b); f~~=(b-1;-2-b)=:>NB.lAN~b=_2=:>B(_2;4) lAN = (0;3) _ _. . _ _ _ _ _ _ _ _ _ _ _ _"_"_"_"_'._ GQi: M(X;Y)) o - - - - - - - - - - - C=COnBC 0.25 _ _ _ _ _ _ _ _ _.,_ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _.'_"-0.- _ _ _ _ _ _ rna AM = 2AB ~ {x -1 = -1 AB = (-3;3) lAM=(X-1;y-1) =:>MN = (1;2) - - _ - _-.- -_.- -_ _ _ _ _ _" - - - - c::> y-1=1 {x = =:>M(0;2) y=2 Do d6 CO x+2y-9=0 _ - - - -.- - - - -, - - - - - - - - - _.,- - - _._- 0.25 - - - - - - - _.- va O=COnAO Ma BA = (3; -3) =:> Be x-y+6=0 TOc;ldo C la {x + 2y = ~ x - Y = -6 AD x-y=O TOc;ldo 18 {x + 2y = ~ x - Y= {x = -1 =:>C(-1 ;5); Y= 0.25 {x = =:> 0(3;3) Y=3 V$y: B(-2;4), C(-1 ;5),0(3;3) csu '>'lXY2(~ Giai: (3x I I • 8kxd: + 1) = 3JY2.:9 + 3y .' VO'l -1)Jx y + xy - - 4x X,y E lR di~m + 3x y - 7x = x2y+xy?5 • 3JY2.:9 ma xy2 (~ + 3y = 3(~ + y) > 3(1Y I +y)? 'ily E lR.va ~ + 1) = 3JY2.:9 + 1>0 voi 'ilXE lR 0.25 + 3y =:>x>O Voi x>O va y(x2+x)?5 =:>y>O Do d6 dieu kien cua he la: x>O va y>O Xet h.s6: f(t)= tJt2;"1 + t voi t>O =:>f '(t) = : :~;~~o·::y~~~·~'( J x,~ ~~;)·03J~:: 9·:3 y=~;;x:~ 1:'):7[J(~F,:,r······ 'oN + + ~ + 1> ,'ilt>O =:>f tang tren (0;+00) '-,If + 0.25 I I v« x>O va ~ >0 nen f(x) = f(~J ~ x = ~ vai x>O _ _ _ _ ~ _ _ _ _ _ _ _ _ _ t._ _ _ _ _ _ _ _ _._y_ _ _ _ _ Y._ _ _ _ _._._ ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ' _~! '._9_()_cJ~.:_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ : (4) • 0.25 -J X>O + 1) = 3~y2 +9 + 3y xy2(~ 4x3 + xy - - 3x3y + - - 4x + 9x y= <:::> X ~ (3x -1)-v3x 1 (3x _1)~x2y l <:::> Y=- Y - 7x = <:::> {X > /\ 3x - ;:: : X=- { (3x _1)~x2y + xy - - 4X3 + 3x3y - 7x = X;::3 -/\ X (3x -1)·J3x <:::> = - - 7x = 4X3 9X2 + 7x - «-'-'-'-:-' :-f~~'~-~-~-~-~-«-'-' ~F~t~~'~I'-' . . .-. -.: l l : l -1).J3x - (3x 9X2 + 7x - (3x -1)( y = ~ /\ X ;:: ~ X <:::> = 4X3 l .J3x - (x-.J3x-2)[4x(x+.J3x-2)+3x-1)=0 ! ==> 4x(x+ J3x - (Vl: x z ~ y 2) + x-1 > ==> 4x(x+ J3x - X j xE{1;2) Cho a, b, C Cau dU'O'ng <:::> 0.25 x2-3x+2=0 vt: 2) + x-1 = nqhiern) V$y: T={(x;y)/(1; so thvc 103cac T = -1 )2] y = ~ /\ X ;:: ~ X J3x-2=x X ;:: ~3<:::> {X = v jX = 23 yY -2 = ~ /\ <:::> l (.J3x - x) = 4X[X2 - y = ~ /\ X ;:: ~ X 3<:::> <:::> -«-. . .-.-. - 3),(2; ~ )} Tim gia tr] Ian nhat cua bieu thvc: diem _ 2 +b +c +4 (a + b)J(a + 2c)(b + 2c) Ja Vol: a,b,c>O ta duoc: • a2+b2+c2+4;:: (a + b + c + 2)2 f( • ( a+ b) -va+ )(b 2) (a + b)(a + b + 4c) (3a + 3b)(a + b + 4c) (2 2b 2)2 c + c $ = $a+ + c 0.25 = _ 2(a + b + C)2 Nen: T = _ Ja2 +b2 +c2 +4 < (a+b)J(a+2c)(b+2c) _ - a+b+c+2 27 2(a+b+c)2 Xet ham s6 f(t)= _8_ - 2~ vai t E (0;+0-:) t +2 2t - - _.- - - - - -" - - - - - _.- - -.- - - - - - - - - - - - - - _ 27 {I>O f '( t) = - (t + )2 + t3 ==> f' (t) = <=> (t _ 6)( + 21 t + 18) = <=> t = - - - - - 0.25 - -.- - - _.- ._ _ - - - - _ - ,-, e ! t '-"-'1-"- .i'(tl f(t) .- _ -«_ _ _ - UJ _ _ _ _ «_ •• _ .~ •• ._ - - - _ •• _ •• _ •• _ • _ •• _ - ._ - _ • _ ~ ==> DU'a vao bbt ta duoc: vt, t>O Mo3' _ " • _ •• a b c>O , , _ •• _ •• _ 0.25 • «_ _ _ _ _«_.~_ _ _ _ _«_ _ _ _ _0 _ _ II +x ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - •• _ •• _ ==> •• ==> a+b+c>O _ •• _ •• _ •• _ •• _ •• _ •• _ •• f(t)$- T< _ 27 <~ - a + b + c + 2(a + b + C)2 - - , - ,"_""_ •• _ •• _ •• _ •• _ a,b, c > /\ a va.b.c= O, T$- l •• _ •• _ •• _ •• - •• =b=c a + 2c = b + 2c va T= - <:::> 3a + 3b = a + b + 4c _ •• = _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• _ •• - -0_" • _ •• _ •• _ • _ •• _ •• _ •• _ •• <:::>a=b=c=2 0.25 a+b+c=6 V$y: Gia trj Ion nhat cua T 103 %, khi: a=b=c=2 GHI CHU: - Thi sinh lam dung b~ng each khac voi dap an, GK v~n cho du di~m phan twO'ng LPng - Di~m 56 khong lam tron, Het -: _ •• (5)