THI TOÁN L5-2011(ĐDT HN)

TRUONG TT{PT EAO DUY TU 3coso x+4sinz r f nt rur rrilt DAr Hec r,AN TEflJ v ( z01s-z0tl) rvr'ox: ioAx HQc rH6r a / Thdi gian: 120 phrfit; kh6ng t<6 tnUi gian ph{t d6. PHAN CIITJNG DANH CHO TAT CACAC TIfr SN*TI Ciu I: : Cho hdm s6 y = 2x1 + mxz -llx-13. l) KhAo s6t s1r bi6n thi€n vi ve d6 thi hnm s6 khi m : 3. . 2) Vdi gi6 tri ndo cta m thi db thi hem s6 c6 tfidm cuc d?i vn didm cgc ti6u sao cho c6c tli€m ndy crich il€u tryc tung. CAU II: 1) ciai hQ phuong t inr,, {t' + v2 -3x+4v =l ' F E Llrt -Zy, -lx-gl = 3' 2) ciei phuong 6in6. fii1p1a]i-ffi = 1. Cflu IIIr Gqi @) li miAn kin t€n mft pheng tqa d0 ttugc gi6i han bdi c6c du&ng y : fi , y = 2- x, y = 0. Tfnh diQn tfch cria (D) vn th6 tich kh6i tron xoay duqc teo thanh khi ta quay ptiquanrr tryc 6ry. I Ciu VI: - - Cho hinh ch6p SlqC dqh S, el6y li tam gi6c c6n ABC c6 AB = AC : 3a" BC = 2a- Bi6t rltgc{c f4! ben (SAB), (SBC),,(SCA) d+ hsp vdi mflt pherrg ttiy (ABC) mQr g6c eOo. (e du]ng cao SH crL linl .h:P' H:q d phtur mflt phang duw gifi h+n b& r+. te,rrt cria tam gidc. Chrmg;tilffil;rr" tluong bon nQi ti6p tam gi6c ABC vi sA r BC. Tinh thd tfch htnh ch6p. Ciu V: Tim tham s6 m tld phuong uinh: .6+ ,tg =r[;l+gx+m c6 nghiQm. PHAN nltuc ( TIil ssTH CIfr tAM MQT IRoNG HAI PHAjY A HoAc B ) A. Theo Chuomg trinh co bfin: CAU VI a: - Trong kh6ng gian vdi hQ tga d0 DA c6c vudng g6c Oxyz, cho di6m | (1,2,-2) vi mSt ph6ne p): 2x +2y+z*J=g- . l) Lfp phuorg tinh mft cAu (S) tAm I saocho giao cta (S) vi m{t phene p) hclu}ngbbn c6 chu vi bdng 8 z. 2) Chung minh rang m{t cAu (S) ndi trong phln I ti6p xric vdi tludng rhdng L:Zx-l= y+3= z. r .^. . . 3) LA? PhY*g trinh m{t cAu chua tluong thang A n€u trong phAn 2 vd tidp xfc vdi mdt cAu (S) tim dusc d phAn l. Ciu VII a: Tim gi6 tri lon nh6t va gi6 tri nhd nhfu cria bi6u thtc: y - 3sinax+2cos2x E" Theo chuerng trinh nfing cao: CAu VI b: cho hQ huc tga tlQ oxyz vi cho hinh l?p phucrng ABCDA'B'c'D, c6 dinh A tmng v6i g6c tqa d0' dinh B c6 tga d0 (1, 0, 0), dinh D c6 tga doio, l, 0)-vd dinh A, c6 tga do (0, o, t;-cec-a1!fil, ii thay d6i tr€n tloSn thing AB', BD ruong tmg sao cho AM = BN = a (0 < ^)'Jtl.' I ) vi6t dtYgs !1!auane theng MN. Tim a d6 hai i.p"r "*r MN ddng thdi m6ng g6c vfi hai iluong thdng AB' vi BD. 2)xfuc.cintt.u-a€.coan theng MN.c6 dO dai bd nhAt vd tinh dO dei ue nh6t do. 3) Chtmg minh rrng lhi a thay'd6t ml :* tludng thdng lrff.l i"o" ,ong ,ong vfi mQt m{t pheng c6 Cintr. Hay vi6iph"*g ui-nf, .ua mgt phlng tl6. Ciu VII b: Tim s6 phrrc z c6 m6 ttun nhd nhAt th6a man: )", J5 \/ t {(;t-r;;tp+r : Vr,{ PhA TRtiffiG TI{PT neo nuv rtl sAp Arq - TI{ANG nrsns rmr fiil'DAr Hec r,Ax v Gzrc3nsfi) nnON : Tofn, *0,i, O .Bhiry NQi dung cho tti6m Piem Bni I 2.00 I m:3 :7 y =2x3 +3x2 -l2x-13. T{p x6e dinh: R. Su bitin thi€n: ,. r. ,,^ 3 12 13. ? 12 13. ]i- y= .lim x'12+:-5-t)=-*. lim y= iim xr(2 +1-]-i)=**. x->{ x->€ X X- X r_>+@ .r->+o X X- X. f r *) y':6x2 +6x-I2;y'=Q q-y 6x2 +6x-12:0 =tl" -' lx: -Z Bdng bi6n thi6n: BAng bidn thi6n X -@ a .L 1 +oo y' I 0 0 + v o T 7 L- -20 '-'-F +q Ham s6 d6ng bitin trdn c6c khoang ( o; -Z) ; (f ; +*) . Hdm sO nghich bi6n trdn ( -2;l). Hdm s6 dat cuc d4i t4i xco: -2)ycs:7. Hdm s6 dgt cgc ti6u t4i xcr: 1; ycr: -20. *) y":12x+6; y":0 (* x : -l =t y =-!3 . 22 Do thidi qua (-1;0); ,*P,or,,a*;0);(0;-r3) vanhan r c),-|>n^ di6m u6n. 1,00 2 X6t phucmg trinh y' : 0 f,) 6x2 +Zmx *12 = 0. (*). pe nam sO dat cgc tlpi vd cuc ti6u thi phucrng trinh (*) c6 hai nghiOm phan bi6t. Ta c6: L' = m2 +72> 0 Vm. :> Vm tl6 thi lu6n c6 tli€m cr,rc dai A (xr;yr) vd diOm cyc ti6u B (xz; yz). Hai iliem ndy sc c6ch tl6u tryc tung e lr,l= lrrl .=t l'-'= *' (Loai) LXt = -xz ,rt' <-> xr 4 xz: 0 A -+= 0 =) rn = 0. Diip.si5: m:0. t I CAU II 7 I ('' E,oluuIr*fr, Giai hC ta dusc bdn nghiQm cria hQ', [= ) [ 2 lr' * y' -3x + 4y =1 l(r= -t*)+ (l' + 4Y) = | trr' - 2y' -9x-By = r'=tl:1r' -3r)- 2(t' + 4y)=3 D{ru= x' - 3x; v : t2 +4y.Tac6 he: {1*v^= I ^.=r{u='^ l3u -2v =3 [v = 0 I xt -3x =7 , -'\rt +4Y =d -4) ) I 2 ul1-ffia/-"otv =1. <+ 2-(sinx +cosx)+Z{(t -sinx)(l -cos:r) = I el-(sinx+cosx)+z@=0. E{t sinx + iosx : t. Di6u kiQn lll < -lZ. =>sin x cos x =" :t 2 Phucrng trinh clAu bdi c5 dqng: a JtV-ll =t-1. Di€u kiQn: r > 1. Phucrng hinh tr€n trdthdnh: Ji (t - t1= / - 1 <=) / = I <=> s inx + cos x = I <=> .,1-z"o{ *- #l = t \. 4/ .:, *-o =+L+k2zr. 44 fx=k2n .l :> I 7r (k eZ). I x=1+k2n' lz 1,00 C6u lll I,00 - ['[ =t /t lJ*=-z (Loai) ll, f ",.1l ,_2,1 7 l'*[,.'- , ),=J*t=6.( X6t phuctng hinh s: f' J"a" * [,' tz * x) crx .,lX=/ X= ) ) Don vf diQn tich *)Tinh th6 tich Voy. Goi Vr le thC tich v6t th6 sinh rakhi quay hinh thang OABC quanh Oy, Vz ld thO tich vflt thd sinh ra khi quay phdn mflt phing gi6i han boi c5c duong x:0; y:0 ; y: l;y = .6 quanh Oy:> th0 tich cAn tim: V : Vr - Vz : ,1,{z - il'ay - o lo:,0 at = o( o, - 4+. +ill-" + l',=tr, -1, t' 2 3)l sl" 3 s rich.) 32n = - . (tJOn Vl fien 15 CAu IV 1,00 1) Gqi H lA hinh chi6u rudng g6c cira S xu6ng d6y (ABC), E le hinh chi6u r,u6ng goc ciia H xu6ng BC. Theo dinh li ba ducrng vu6ng g6c ta c6: SEIBC :> G6c SEH: G6c gita (SBC) vd d6y (ABC):600:> EH: SH .cot 600:> Khoang c6ch tu H d6n BC ld SH.cot 600. ,k / l\\ / i\ \\ / ll \\ / rl \\ / rl \ \ / it \ \ ^(q :=lt I \ - - -7" \ H-i-' :V/ \ \./' v/ Tucrng tg, H cflng c6ch AC vd Ats m6t khoang SH. Cot 600. H c6ch dAu 3 cqnh tam giiic ABC :> H ld €m duong trdn nQi tit5p ( hoac bdng ti6p ctra tam gi6c ABC ). Vi gi6 thi6t c6 H nim trong tam giitc:> H ld tAm duong trdn nQi ti€p tam gi6c. 2) Do tam gi5c ABC cAn t4i A:> FI, A, E thing hang :> f ac'ttr { ^_ IBC r st I j) V: :,SFI.,S^," 'a J r ; . s*,, : .,lp(p -za)(n -za)(r -za) (F{6 r6ng) P= 2a+3a+3a , r, ^ ^.t17 s 2atn[i :4Ct.=) 5 =l4A.lA.u.A : lA-",1 l:) In:- - 2p4a =oJt 2 => ,sr1 : HE.ta uo' =of J1 =+ :>v :Ir"'O*='or* .(Dcm vi diQn tich.) Cdu V 1,00 Ji * Jg:; = J-; ag*a 111 Di6u kipn: 0 < x < 9. (r ) e (J; . Je -;)'? = (-'' + ex + m) <:> o * z.{i (o - *) = x (e-') + m. (2). oAt ,[1sJ =t =]t2 =9x-xt = f (x). 81 8l l ql :> 0< l6)< 4=rr'=T =)I€Lr,,l (2) (} g+2t t2 +m <+ 9 _-m:t2 -2t. Phucmgtrinhddchoc6 l- ql m6t nghiCm r e I O;a l. Xet hdm g(t) : tt -2t. Co l(x) :2t -2 . L zl nghi€m <+ (2) c6 ft nrr6t (2) co nghidm l- ql relo:al L '21 -9 _<m<T0. 4 -@029+m 2 o 7'+oo 2 tsiiVtra 2,A0 ll tr-t?-?r(l I c6ch (P): 2x + 2y + z + 5 :0 mQt khoang c6ch h : l''' -=-'' = -1-'l: 3. Duong trdn chu " J22 +22 +72 vi 8 n c6 b6n kinh r : 4. Gqi R li brin kinh mdt cAu c6n tim, ta c6: k : h2 + 12 -* 9+!6 -*> R: 5. M{t cAu (S) c6 tem I ( 1;2; -2);R:5 c6 phucrngtrinh: (x-1)t * (y-Z)t + (z+2)2:25. 0.5 2 phuong trinh: 2x -2: y Chuy€n (l) vd tham s0: phuong trinh mdt cAu. +(r -3-z)' *(r +z)' :25. [' . I -')' Theo gii thi6t, (A)c6 (I lx=l*- l2 I iv:-3+t. Thavvao t" lz=t I 0.5 <+ (31-4)2: 0:> l=1 Vi uey (s 5.4) d6ti6ntli6mA Ia: - '. \3' 3'3 ) (a) va mdt cAu (S) cd mQt ditim chung duy nh6t v6i tga J M{t phdng cdn tim ld mdt phdng vudng g6c vdi IA tAi A, cfing ld rn{t phdng tli qua A fl -f 1) vd c6 v6c to phdp il n(1t ii{)r* i(z;-ri;r0). phucmg trinh cin \3' 3'3) [3', l: tim ld: ,( . -1)-r'[r.i] +to(,-1'l = 0 <=>2x -ty +r0z-35 :0 f. 3i \.' 3) \ 3i I,00 Cdu VIIa 1,00 u - 3cosl x + 4sin2 x - : (t, sint xr + +sin' -x . EAt t = sin2x. Diiju kien: 0 < / < 1. Khi ' 3sino x+2cos2 x 3sina x+z(l-sint x) uu. 3t2 -2t +3 I y =#=l+ _ . X6t hdm sO f(t): 3t2 -2t+ 2; (0 <t <1.) - 3t' -2t +2 3t' -2t +2 f(t): 6t_ 2:o o r: ]. 3 Bing bidn thi0n t I @0-1+co J f(x)' -0+ f(x) / 1' ' ./'l,t =t l. f (t\ <3<=> 1+l r I *- ' :1+ t .=t g >, >!. 3 "" s f(t) 3 s - 3 VAy, MinV: { f.f.ri sinzx: 1 => cos x: 0:> r=!+kr(k e z). Maxy: I r.rri sin2x: l=tror2r =!=roro,oe(0;a) =>x=xlo+kr.(kez) ' 5 3 3 ', \',' 2 1,00 CAu vIb 2$A [,00 5 Ta c6 : ru e(xoz) =) iu = 0. AM = a =) xr,t = Zbt =+ =r r(+,r,+) lnxv tAB' ln r an,=o l(t-'t)-$=o J, \^ rBD -'\m un=0'='1_t,_ "J4.+=o' a= 3 - rucrne ,u, tt(t-"Ji oJi ) -f - n.oJi - "Ji\ ( 2 ;, ;u)='*'lt-ar2;;t-;) *1 ea' (t;o;t) ; ra (-t; t; o). 2 YOi a =€ ,n, MN le doan thing vu6ng g6c chung cria AB' vd BD n6n c6 d0 dai b6 3 nhdt. Khi ffi =f1 I -1'l =>lMNl= 6. [3'3' 3) I I 3 0,5 3 .MN NB u MB'ND.IZ-A (Dinh li Talet d6o ). Do CD ll AB; A'B' //AB:> (A'B'CD) ll AB. Do tl6 n6 song song MN. Vdy, khi a rhay dOi, il,ffrt lu6n llm{t phing c6 Ainfr (A,B'CD). Ta c6: l' O(o;t;-l); nC(f;O;O; ld cdp vdc tcr chi phuong:> ;(0;l;t). vay, phucmg trinh m{t phing (A'B'CD): y + z -1 : 0. AB, MN, DB' nim tr6n ba mat phAng song song vdi nhau 0.5 Cdu VII b I,00 Gi6 sri z: x* iy. Tir gi6 thi61 -> l(x + t) + (y - 5)tl = lx +3 -(r + 1)il <=> (x + l)' + (t, -s)? = (x +:)' + (s, +t)'. € x + 3y :4. Ta c6: 16 = (x+3y)' <10(*' * y') =10lzl' => $.lrl.=> Minlrl- o - J0 -t"t'-'"""1"t - Jm Ddu ":" xAyra (} l= I=r r=3.r. md x+3y =4 =)10x=4=> x =?=r r=9. fni - I 3 ' ' -/ 5'"r 5"^"' tl6: .,2 , 2 (z\' .(o\' 40 r , 4 ,,, 2 6. x'+y' =[;J *[;J =-=r lrl=ffi VAy,s6phricthoam5n ld: 7=1y:i. 1,00 . s6 y = 2x1 + mxz -llx-13. l) KhAo s6t s1r bi6n thi n vi ve d6 thi hnm s6 khi m : 3. . 2) Vdi gi6 tri ndo cta m thi db thi hem s6 c6 tfidm cuc d?i vn didm cgc ti6u sao. y':6x2 +6x-I2;y'=Q q-y 6x2 +6x-12:0 =tl" -' lx: -Z Bdng bi6n thi6 n: BAng bidn thi6 n X -@ a .L 1 +oo y' I 0 0 + v o T 7 L- -20 '-'-F +q Ham s6. (*). pe nam sO dat cgc tlpi vd cuc ti6u thi phucrng trinh (*) c6 hai nghiOm phan bi6t. Ta c6: L' = m2 +72> 0 Vm. :> Vm tl6 thi lu6n c6 tli€m cr,rc dai A (xr;yr) vd