Thi thử chuyên KHTN-lần 2-2011

,I'R,UONC DAI HOC KHOA HQC TU NHIEN TRUdNG THPT CHUYEN KHTN pE tllr tuu n4r rloc t+Ant rc':,o M6n: TOAN (Dor 2) Thdi. gian ld,m bdi,: 180 phrit, hhAng kd thdi gi,an p,ttdt di Cdu I. Cho hbm s6 y - s3 +3(rn+ 1) 12 +3rn(m+2)s+rnl *3rn2. 1).Kh&o s5,f sU bi€n thi€n vb, v6 dd thi cira hbrn s6 d5, cho vdi nz:0. 2) Chrfi:g rninli rbng r-di inqi rn hd.m s6 lu6n c6 2 cqc tri, ddng thdi kho6,ng c6.ch giria 2 didm nA,y kh6ng phq thuQc vh,o rn. Cdu II. i) GiAi phrrong trinh {1,r tarrc)cos5z = silr * cos3j + 2cos4c -2cosZr. 2) Giei pirrtong tr)nl.r log, (z + 31"e6 ") : iogu ". CAu ril. 1) 'fim t5.t, cA c6c giir i,ri crle i,ham s6 rn de plrrtong trinh sinr * cosc + JTTdiii: n, c6 nghiQm. 2) Tinh t6ng rto fiz r-4 arj n2OlO c' _ v2010 , u2010 t \'2QIO , v2010 ,' , v2010 " - -llT- - J.r-t - TF T 1V{ T -i- tgifiIoG Cdu trV. 1) vi6t phr;ong tr)nh tiudng trbn iii qua hai didrn ,4 (4;5),8 (5;-2) vd, ti6p x-ic dudng rh6.irg (d) :Y = -4. 2) cho cd,u (,9) t6,m o, c6 AB = 2R ) 0 lb ciudng kinh c6 dinh. Di6m r di dQng tren do6i: o-8, ,-;+ ^L*-* /D\ ^,,n L,i ,,^ 4a^ f\D -{+ /o\ rL^^ -:^^ ! : r: i-J\ r-r- t^\ A:1 iiiaL piletrlE (il qQer i va vUong gOC UD CO,r 1u7 urrsu Erd/ uuyvl id, riiidfig'LIOfi (U.i. Uia S',i'nOi:r (/f) c6 dinh A, d6y lb. dqdng trbn (C) vdi tluc d6i x-rlng,4I. X6,c dinh dO dai 01 theo & dd ihd tich n6n (l/) ldn nhdt. 3) cho h)nh ch6p Lit gittc S.ABCD c6,9,4 vuong g6c m5t philn1 (ABCD), tld,y ABCD lb. hinli chrl nh$t dO dei AB : fro,BC = a. Gqi 14 lb. trung clidm cioan CD, bict rang gOc girra hai m5t ph&ng (ABC D) vd (SBtul) Ib. cp 60". a) Chinrg minh rEng mat phfing (SBM) vu6ng g6c mEr phing (SAC). b) Tini rhd rtch rri dign SABM theo a. CAu V. YOi r,t1,z ld nhrlng s6 thgc drrong th6a m6,n s *y * z.: xyz, chtngnrinh rb.ng 2 -,- 1 .9 \nT? \trW !ET;': 4' _HET_ DAP AN (tom tdt) CduI. 1) Tac6U':3r2*6x,y':0<+r=0iro6,ca:-2vir, Iimr*._*U:-F, limr*+*g:*cc Hdm s6 dong bien tr€n cd.c khoAng (-*, -2) , (0, +oo) va. ngirich bicn-tren khoangi(-{0) st.y ra tCD : -Z,yCo : 4:xcr : A,aCr :0 -oo -2 0 +oc f'(x) 0+0- f(n) 4+* ,/\ ,/\,/ ,/ ,/ '/"/' ,/ \,, -oc 0 2) Ta c5 a' : 3n2 - 6(ra+i)r * 6tn.(n't*2J,g' : 0 +> r : _2- riz hod.c t : -rrL.Hb.5r s6 dorig bi€n Lr'€n c5.c khoAr,g (-ec,-2 _ m),(,-m.,*co); nghich bi€n tron kho6.ng (-2-nt,-nz) va rcp : -2 - m,Ucp : Airc'r : -"ti1'1 .Irc:' : 0. K-iri d6 kho6.ng c6.ch girla hai di6m ciJc tri Id /- ,/ (-2 - rn * n)z + (4 - l)' :2,/8. CAu ltr. 1)Chri;i cos5c: ct;s5r*cos3c-cos3c-cocr+cosc : cosr (2cos4r - 2cos Zr + i)phr:o1g trinh decho <+ (sinz * cose) ffi* - sinr*cosc**Y-1 t oy (#T - 1) (sinr + cosr - 1) :0, 2)Dkc>Lr.Del,k-rgoc=gsuyralog2(6v+3v) =A+6v*3u:Zo#(3)r+(B)"=1.DeLlr6y ve tr5.i lb. h).m ti6ng biSn n€n phrrong frinh conghi€m rJuy lha,t g: -t t,: dO ,: t. cAuIrI. 1) De.tt:sinrD+cosr,tel-it,'/z)+L-sin2x:z-t2.suyraiaca.ntjm;;;r.id plrrrong trinh t + \/r=7 : m e6 lghiQm t e i-,/t, ^f;\. Xet f ft) : t +,/T4,t € l- Jj, {4 tac6 f' (t)'= 1 - #V,t e ?rt,Ji);f'(i) : O +* f : I Suy ra f ddng bi6r: tr€n kho&ng (-f,t) vd nghich bi6n tr€n (t,Jl). Me i (f) li€n ruc rr€n ?rt,,tq tt d6 maxf (t) : f(1) : 2,min f (t) : f ?rt) : -,fi. v1 v *^/2 S rn 52. 2) Ddr n : 2010. Ta c6 fi1CTk : ftrc!\\r + n/2 n/2 g- \-' ! f'2le - t \ tn2k+r u - ./-, ?IETTIZFUn : ;+1 L i'rt-i+'t' : b=O L-n - , € 1 nlk*!- z tlr,,1,,n*l @ kn'FE+'ivn+r : G+l)t L\r f i/ lt 1in*tl i.+l 1 c2oll r - (r - il J :lffi: 2fur7a#. Cdu IV. 1) Di6u ki€n 1A : IB e E -TA*6 : 0. Didu ki€n IA: R <+ (g+ 4)" = (r-4), +(y-5)2 <+ 18y -g= (n_ 4)2. Giei he c6 189 -9: (Ta - 10)1++ 49a2 - 15Bs * 109: 0 f' u:r,t:1,/?=b; I, g : 109149,r = 6T fZ,R : 305/49. 2) DAi OI) = a: c6 r.2 = R2 - x2. Co V : nr2(r + R)13 : r/3,(R2 - :r2)(r +,?). X6t /(o) : -t3 - Rrz * R"n +R3. C6 f,(x): -!x2 -Znr'+ R";f,(z):0 +) s: _R hoir.c r: R/3. I/r,o* tai n.: R/3. 3) a) Do ACLBM nEn BML(SAC) tfc ld. (SBM)r(.9,4C). - _ b) AC c6t BM rai 1. C_6- AI :2|JAC =2r/Ja/3. Vpy.9A - ,6U : zo. V(SAM B) : (I/3).2a.a'l'/z = ,/2as /2. Cdu V. DAt o: i,b: i,s: ltac6 ab+bc*ca:1, cling thrlc cAn chrlng minh trd lhA.nh p : ffi + # + ffi < t. cnri i ld i + a2 : ab* bc + ca + a2 : (o * 6)(a * c) vb hai .{*' rt".+,,rr?/,hd+,*ts,asu-vrap=_ 29 _+_ ?L__ I Z. ( t r \, vu\'16 v'.r voDur i6. - ffi - ;i.m - ffi s a(;iT +#i + o (**" + #) +" (aG+o + *) -,e. D&ng thric x&y ra, khi b - c : *ahay e - z = 7r : \m. v TRIJONG DAI HQC KHOA TI9C TU NHIEN TRUdNG THPT CHUY€N KHTN Ciu I. Cho hbm s6 Y : 2r-2m*I fr-rn-L DE rHr THO DAr HQc NAtu zoro Mon: TOAN (Dqt 3) Thdi gian tdm bd.i: 180 phrit, khdng tcd tnat gian ph6't' di (c^). 1) Kh6,o s6,t sg bi6n thicn vb vE dd thi cria hdm f d5' cho vdi rn: l' ;i Ct; , (1,2) . Tim cd,c gi6, tri ctn m sao cho tdn tai m.t dudns thlng qua A cdt dd rhi (C-) tpi frui aia*"phanbigt M,N md, c6c ti6p tuy6n t4'i M,N cria dd thi song song vdi nhau. Cdu II. 1) Giei Phrrong trinh tan2r*9 ca*r+Y!} :u. 2) Giei phrrong trinh log! 3r * 1og, 2a : lo$23r * log, 31 1og3 2r' Cdu III. 1) Tim gi6 tri l6n nh6t cria hb.m s6 {i + {r=i ' {r+{\-r 2) Tinh nguy€n hh,m sinrdx t:I sin2s-3cos2r-I CAU IV. 1) 1. cho l5,ng trq ABC.tB'c" bi6t A',ABC lb. ch6p tam gi6c ddu-c6 canh d6'v a vd khoing c6,ch girla canh b€n vb, canh ddry d6i diQn bx.ng k. Tinh th6 tfch l6'ng trg' 2. Trong hQ tqa aO Oia, cho I/(L;3;2). Vi6t phuong trinh mat phang (P) di qua H c6t d*,Oat,Oz t?,i A, B,C sao cho I{ l}r' tr*c t6m tam gi6'c ABC ' 3. Ttong hQ tqa dA br, 4ro drrdng trdn (C) : (r - 1)' + (v + I)' :25' Vi6t phrrong trinh drldng th&ng qua M (7 ;3) c6't (C) tai A, B sao cha M A : 3M B ' CAu V. Cho da giSc ddu 12 dinh. H6i c6 bao nhieu tam gi5.c tir c6 dinh lb 3 dinh cria rla gi6t: tih. t*ro. a _HET- ffi 07^, ffr nt^ t rN? 17 DA THANH 't't)rft \6r DT:0912291)450 oAp AN o0r a (tom tit) L Cdu I. l)-Vdi rn : I t1 e6 v : +:* * y' : ffi a 0 vh. lim"-aaoV : 2,lirn,-2+ y : *oo,.lim"*2- U: oo. TCD c:2,TCN y:2. Hdm s6 nghich bi6n tr€n c6c khoing ( ;2) , (2; *m) vb. kh6ng q6 cr.rc tri. Dd thi nh&n I (2,2) llul tdm <t6i xrlng. t -@ 2 +oo f'(") r@) ,\ -oo +oo_\\_ *2 2) Ta c6 s! = 6$ry. Gid, srt M (rr,a) ,N (rz,vr) e C*(q t' q). Ti6p tuy6n tq,i M fa N song song <+ 6#ry: e;#:ry 4* rr - rn-l: -(rr- rn- 1) (* rr + rz:Znt+2 (1). Mat khec do A,M,N thing hlng =+ #: #llire"ut-2:# - 2: ;$1 n6n iathu drrgc (c1- 1)("t -nl- 1): (22 -L)(rr-m- 1) vh.chri ! q*rn-l: -(rr- rn- 1) siiy ra 11 -!: -(r"- 1) =+cl *tz - 2. Ctng vdi (1) suy Iarn:0. Cnu II. 1) Chfi ! 2cot2r: cottr - tanrr "d #,; : tans *cotc Phuoug trinh tuong duong tan2 r *9cot2 s * (coto - tanc) * Z(lana* coto) : 16 <+ (tanr * 3cotz)z * (tanc * 3cotc) - 20 :0 I tanc * 3cotr :4€ tanfr: lhoetanr:3 o I t*" f 3cots : -5 (+ tanr : (-s + '/n) 12. 2) Dk c > 0. Ptrinh <+ (logr 3r - logr 2c) + (logr Sclog, 2t - log|3c) : g ++ (logr 3o - f ) (log2r - log2 3o) : I I gr :2 + r:2/3 o L tor, 2r :lo1zlL : t * 2u :3t,32 : 2t + }t+t - 2t+r"+ t : :1 + r : l/6. Ctu III. l) Dk 0 S r < l. Khid6 theo b6t ding thfc Bunhiacopxki ({i + i/r4)'? Sz(,/i + '/7.:l) "i, (Ji + t/T412 < 2(r * 2 - r) : 4 + {i.1 fr tS 2, d6u bhng.xiy ra <+ o : t. MAt kh6cdo 0 S {i,$=Z < 1=+ ,/a+.itr4 2 r*1-x:t* Wk ( 1, d6u bling xiy ra r+ o : t ho6.c o :0. Vfly maxy - 2 khi r: 1. 2)Tac6sin2c-3cos2r-I: {sinccosr -2cos2o+sin2c) :2(sinr-cosc)(sino-1-2coso). Bidu di6n sin c : A (sinr - cos i) +,a (sin o * 2 cosr) => A+ B - L, - A+28 : 0 =+ A = 2/8, B = 1/8. Do <t6 6r : ,f m** r/m;#* - J*ffi + ,f #t-% : ,/i,rnlr* (; - g)l + {st"lt* (; + €)l + c ( trong d6 sine :2/'/i,cas,p:11{il . Cdu IV. 1) Gqi .tV li trung didm BC, MK L AAt, A'H L (ABC). C6 MK.AA' : A' H.AM E K.\[* + + - A'H."{ + A'H: ffi,s.qnc: # *vnpg-n'e'c' : iJ#TF(dwt)' 2) fI Ie tn/c td.m, OH L (ABC) n€n m6t phlng (P) qua Il(1;3;2) c6 vecto ph6p tuySn llr OH : (l;3;2) c6 phrrong trinh: c * 3V * 2z - 14: 0. 3) Gqi .iT lb trung diiim .4B. (C) c6 tdrn I(1; -1),.R - 5. C6 MA.MB : MIz - rtz <+ SMBZ = 27 + MB = 3,MA: 9 =+ MH : 6+ IH - 4. Ldp A qta M(7;3) c6 d : (A; B), A2 +82 { 0 c6ch.I mQt doq,n bing 4: A(r-T)+B(U-3) : 0 d(I,L):4 +r l3A+2Bl=2'/PJ@ fi z4:0 ho{c A: -t2,8:b. Phuong trinh A : y - 3 : 0 hopc -t2r *5y * 69 = 0. Cdu V. X6t I dinh .4 cfra da giac ddu, ta sd t)m s6 c5c tam gi6c tir c6 g6c ti tai dinh .4. D6nh s6 cac dinh tiep theo cria da gi6.c ddu theo chiiu kim d6ng hd lil 2,3,4, ,12. Chri f ld v6i hai dinh a, b mlu la - 6l : 0 thi ab lb drrdng ch6o chinh vh A.AaD vu6ng tai .4. Khi d6 tam giSn Aab (2 S " < b < 12) lh. tir tai A e b - a ) 6.+ 8 < a + 6 < b < 12. Suy ra c6 C! cSch chgn cdc dinh a,.b (a* 6 vir. b nhQ.n 2 trong 5 gi5. ir! 8,9,10, i1, i2). V6.y s6 tam giac cdn tim lb"I2C3: 120. . Cho hbm s6 y - s3 +3(rn+ 1) 12 +3rn(m+2)s+rnl *3rn2. 1).Kh&o s5,f sU bi€n thi n vb, v6 dd thi cira hbrn s6 d5, cho vdi nz:0. 2) Chrfi:g rninli rbng r-di inqi rn hd.m s6. 180 phrit, khdng tcd tnat gian ph6't' di (c^). 1) Kh6,o s6,t sg bi6n thicn vb vE dd thi cria hdm f d5' cho vdi rn: l' ;i Ct; , (1,2) . Tim cd,c gi6,. rhi (C-) tpi frui aia*"phanbigt M,N md, c6c ti6p tuy6n t4'i M,N cria dd thi song song vdi nhau. Cdu II. 1) Giei Phrrong trinh tan2r*9 ca*r+Y!} :u. 2) Giei phrrong