TRUONG DAI HQC KHOA HQC TU NHIEN TRUONG THPT CHUYEN KHTN nE rlrr tnU DAr Hoc NAvt zoro Mon: TOAN (Ddt s) Thdi gian ldm bd,i: 180 phi,t, khOng *i4 tnal gi.an phd,t di (c^). CAu I. Cho hbm s5 y: 2x-2m*7 r-rn-I 1) Kh6o s6t srt bi6n thi€n vb, ve dd thi ciia hd,m s6 da cho vdi m: l. Z; Cto A(1,2). Tim c6,c gi6 tri crtamsao cho tdn tai mOt tlrrdng thing qua A c6t dd thi (C^) tai hai di6m phan biqt M,N mb" c5,c ti6p tuy6n ta M,I/ cria dd thi song song vdi nhau. l Cdu II. 1) Giei phrrong trinh tanz t:* 9 cor2 * n\31t! : r+. 2) Giei phuong trinh log! 3z * log, 2r :lo1z3r * Iog, SrLogr2r. Clu III. 1) Tim gi5, tri ldn nh6t cria hb,m s6 {r + {r4 a: {i+{T-; sinrdr 2) Tinh nguyOn hA,m t:l sin2z-3cos2lc-7 CAU IV. i) 1. Cho ISng tru ABC.A'B'C',bi6t A'ABC Ib, ch6p tam gi6c ddu c6 canh d5,y a vL khoring c5.ch girla canh b€n vd canh dey d6i diqn bbng ,k. Tinh th6 tich l5ng trg. 2. Trong hq tqa dQ Oraz cho //(1;3;2). Vict phrtong trinh m5.t ph8,ng (P) di qua H c6t Ox,Og,Oz tai A,B,C sao cho 1{ I}, trUc tdm tam gi6c ABC. 3. Tlong he tqa dQ Oxy cho drrdng trdn (C) t (r - 1)' + (g + 1)2 : 25. Vi6t phrrong trinh dr-tdng th&ng qua M(7;3) cdt (C) tpi A, B sao cho M A: 3M B. CAu V. Ciio da giSc ddu 12 dinh. H6i c6 bao nhi€u tam gi6c ttr c6 dinh ld 3 dinh cua rla gi6r: <14, r:tro. - 11fi1_ $A rHAl'H TCf i ilT:091229i145fi oAp Aw DoT 3 (rom tdt) L Cdu I. t)-Vdi rn : I ta c6 y : T+ * V' : ffi a 0 vb. lim"-aaoog : 2,lim"-2+ g : *m,lim,rr- U : -oo. TCD r : 2,TCN g : 2. HLm s6 nghich bi6n tr€n cac khoing (-"";2),(2;*oo) vb. kh6ng q6 cuc tri. Dd thi nhdn I(2,2) lhm tdn d6i xrlng. L -@ 2 *cc f'(r) f(") ,\ -oo o*- -2 2) Tac6 y' :6jry. Gi6,stl M(q,ur),N(rz,ar) e C^(qfr2).Ti6p tuy€n t4i M vit N song song <+ 6;*,;y : i;*ry # rt * rn - ! : - (*, - rn - l){* ri + rz : 2m +2 (I). MAt khac do A,M,N thing hbng + #: fi* me h-2:W - 2:;$1 n€n ':athu dugc (o1 - 1)("t -rn*L):(s2-L)(cz-rn- 1) vbchri i,rr-m-L: -(rr- nr,- 1) sriy raol - 1: -(rr- 1) =+21 +.rz:2. Cingvdi (1) suy ra m:0. CAu II. 1) Chf !, 2cot2r: cotz - tanc "e #* : tano * cotc. Phrrong tr)nh tuong duong tan2 r * 9 cot2 c * (cot c - tan r) * 2 (tan r * cot r) : 16 <+ (tano * 3cotz)2 f (tanr*3cotr) - 20:0 I tanz * 3cotc :4e tanr : lhocLanr:3 o I tunz f 3cotc : -5 <+ tanx : (-s * r/n) 12. 2) Dk z > 0. Ptrinh ++ (log, 3u - log, 2n) + (logr3r log3 2r - logl3r) : g <+ (log2 3o - 1) (log2r - log2 3o) : fl . I Sr :2 + r:213 o L tor, 2x :lo1zSa : t i 2r :3t,3r : 2t + st+L - 2t+L=+ t : :1 + r : Ll6. CduIII. i) Dk0 ( r ( 1. Khid6theob5,tdingthircBunhiacopxki (Srr+ W4)2 <z(r/i + \/7i4) "i, (,/i + ^E-;)2 < 2 @ * 2 - a) : 4 + {, + .{r-S 2, d6u bbng xAy ra <+ r : 1. MErkh6cdo0 < {i,<E- < 1=+ Ji+ {T-" > u +1 -x:!+ T,#= ( 1, d6u b5,ng x8 v r& e r: t ho[c r :0. Vd.y maxy - 2 khi r = 1. 2) Ta c6 sin2:c-3cos2,x - 1 :2 (sinzcosu - 2cos2r +sin2 r) =Z(sinc - cosr) (sinr * 2cosz). Bidu di6n sin tr : A (sinc - cos r)+B (sin r * 2 cosa) + A+ B : l, -A+28 : 0 .+ A * 213, B : 1/3. Do cl6 61 = J;;*-!t-* + Jm#i* : t#xJ + J#ft, : ,fttnltan(! -;)l+ l/5t" ltan (f + 5.)l+ c ( trong d6 sin p:2/t/i,co,p: l/JE). Cdu IV. 1) Gqi M li. trung d\6n BC, MK L AA', {, t (ABC). C6 M K.AA' : A' H.AM E K.\f A'2 + * : A' H.+ * A'H: ffi,sell: + 1vlss.1's'"': ffiw(dvtt). 2) II Id tr{c t6,m, OH L (ABC) n€n m{,t ph&ng (P) qua Il(1;3;2) c6 vecto phSp tuy6n Id OII : (1;3; 2) c6 phrrong trinh: z * 3g * 2z - 14 0. 3) Gqi .IJ li brung diiim .4,B. (C) c6 tam I(1;-1),R: 5. C6 MA.MB : MIz - R2 e 3MB2 : 27 + MB :3,MA: 9 =+ MH :6 + IH :4. L6p A qua M(7;3) c6fi.: (A;B),A2+82l 0 cach l mOt dopn bling4: A(r-7)+B(9-3) :0 d(I, L) : 4 e l3A + 2Bl : 2t/pl@ # A -0 hotc A : -12,8 : 5. Phuong trinh A : y * 3 : 0 holc -12r *59 * 69 : 0. Cdu V. X6t 1 dinh .4 cria da giac ddu, ta s€ tim s5 c5c tam giSc bir c6 g6c ti tai dinh ,4. DSnh s6 c6c dinh ti6p theo cria da gi6r: ddu theo chi6u kim d6ng hd li 2,3,4, ,12. Chf f Id. vdi hai dinh o, b mh. la - bl : e thi ob ld drrdng ch6o chinh vb A.4ab vuOng tq.i A. Khi d6 tam gi6c Aab (2<o<b <12) lb, tir tai Aeb- a > 6 + 8 < o*6 < b < 12. suy rac6 c! cach chgn c5c dinh a,b (a t 6 vd, b nhQn 2 trong 5 gi6 tri 8,9, 10, 11, 12). Vdy s6 tam giac cdn tim IiL 12CZ : I20. . Kh6o s6t srt bi6n thi n vb, ve dd thi ciia hd,m s6 da cho vdi m: l. Z; Cto A(1,2). Tim c6,c gi6 tri crtamsao cho tdn tai mOt tlrrdng thing qua A c6t dd thi (C^) tai hai. A c6t dd thi (C^) tai hai di6m phan biqt M,N mb" c5,c ti6p tuy6n ta M,I/ cria dd thi song song vdi nhau. l Cdu II. 1) Giei phrrong trinh tanz t:* 9 cor2 * n31t! : r+. 2). 2. HLm s6 nghich bi6n tr€n cac khoing (-"";2),(2;*oo) vb. kh6ng q6 cuc tri. Dd thi nhdn I(2,2) lhm tdn d6i xrlng. L -@ 2 *cc f'(r) f(") , -oo o*- -2 2) Tac6 y'