TRI'ONG TIIPT DAo DUY TTI - IIA NoI of rrn rrnlDAr Hgc r,Arv r (2010 2011) nndx, roAN rn6r a Thdi gian: 180 phtu (kh6ng *6 nm g*n giaa di) CAu I. Cho him s5 y: x3 - 3x (l)- / U Xnaosft sg bi6n thi€n vi v€ eO tni narn sO (t;. '.t u 2/ Tl^m etii phuong trinh x3 -3x = + c6 3 nghiQm phdn biQt. m" +l 3/ V6iO(0; 0) vd A(2;2) h hai ttiiAm nim tr6n AO tni hdm sd (1), rim di€rn lvi nim trOn cung OA crha dd thihAm sO (t) sao cho khodng c6ch tu M cli5n OA ld ld.n nh6t, Cffu II. 1/ Cho b6t phuong trinh: JT3x+2) 7n-rt7 1x+4 l, v al Giai bdt phuong trinh khi m :4. b/ Tim tdt cir chc giittri cria m ee UAt phucrng trinh tr€n nghiQnr ding viii riiqi x >:i. ,,1 2/ Gi6i phucrng trinh: tanx+cosx-cos2x =sinx(l +tan!-tarrx). Cffu III. V ttTrCn m{t phdne tqa d0 Oxy, tim phuong trinh tludrng thingr di qua di6rn M(l; 3) sao cho dudng theng d6 ctrng v6i2 dudrng thang (d1): 3x-{- 4y + 5 : {j vi (d2): 4x + 3y - I : 0 t4o th'nnh mQt tam gitlc cdn c6 dinh ld giao di6m cria (d1) ,,,d (d2). ( 2l Cho hinh ch6p SABC c6 dhy ABC li tam gi6c cdn, c6 AB : AC : ?,a; RC - 2a c6c m{t bon ctrng tao vdi ttax,m0t g6c 600. Hinh chitiu H cia dinh s xu6ng (ABC) nam O trong tam gi6cABC. '/ at cntng minh t6ng H h tem dulng trdn nQi ti6p cria tam gi6c ABC. b/ Tfnh thiS tich cria trl di0n SABC Cffu IV. Cho tflp hW A gdm n phAn tri (n > 4). nii5t ra"g sO tgp con gOm 4 pfran tri cria A bing}} Hn sO tdp con g6m 2 phan tfr cria A. Tim k e {1,2,3, ,n} sao cho si5 tfp con g6m k phAn trl cria A ln lcrn nhdt. Cf,uY, Cho 3 sO kh6ng 6m a,b,c. Chung minh ring: a3 +bt +c'> a'Jbc +b'Ji +t'^[on . _x oAp An vA rHANc orru rrrr ruuD+r rrgc IAN r - ivr0N roAN Caulf NQi dung cho tli6m oi6m al Khio s6t str bi€n thi6n vi ve d6 thj hdm sd y = x3 - 3x 1.00 TXD: R Gioih4n: L im1x3-3x;= to r+ t@ Strbitinthi6n: y'=3*-3. Tac6y'=0<=>x= t I Bnng bi6n thi6n: HAm sii tt6ng biiSn tr0n (-co; -l) vi (l; +o); Nghlch biiSn ffin Gl; t) Hnm si5 d4t cyc d?r t?r (-l;2),cgc ti6u 6(l; -Z) x -dr -1 I 1-@ v' +0 0 + v *oo -2 Ei0m u6n: Y" = 6x - 0 <:> x:0 . Di6m u6n U(0; 0) VE hinh: 2 Ditim u5n: y" = 6x = 0 <:> x = 0 . Ei6m u6nU(0; 0) Ve hinh a2 Tim m ttti phuong fiinh x3 -3x = + c6 3 nghiQm phdn biQt. m" +l 1.00 SO nghigm cua phuong trinh x3 -3,c =#bdng sd giao <ti6m cria 2 dO thf : [y=r'-3t lz^ l'= *u 0.25 DU,a vio dO thi ta th6y dO phuong trinh t€n c6 3 nghigm phdn bigt thi +. ?* q)q=7-rn'-l<m<m2 +l m'+l 0.50 (=) -mr -m- I < 0 < m' -m + I (Lu0n dung voi mgi m) V{y voi mgi m phuong trinh ludn s[ ]nghiQm phan bipt. 0.25 43. 1.00 Gpi M(xo; yo) voi 0 < xo < zlb,mOt diem bat kj' n[m tr0n cung OA ciad6 thi. X6t duong thdng (d) qua M vi song song voi OA. Ta c6: d(M; oA) = d((d); oA); Khoang c6ch nay lcrn nh6t khi (d) cdch xa OA ntrAt <=> (d) chinh b ti6p tuytin song sonl OA tar ti6p di6m M. Phuong tinh duong thdng OA li y = x => Tiiip tuy6n t4i M song song voi OA c6 hQ sd g6c f (x6): I <=> 3xo2- 3 = I =) Xo = # r**(hrfl ,! S 4*a2, a -^ti aa a (.r < -2;x 2 -l) 1.00 THt.Ntiu 4-Jf 4x+4=ol-=t 16<xz -3x+4 Biit phuong tinh lu6n th6a m5n. f _, t*.F <-> x, -ix-r2zo<=>l ^-, :* lx( LZ rH2,Ni5u q-J*z*+q>g4=1 +.*.* 0 Khi d6 hai vii cua BPT di cho ludn duong, binh phuong hai vi6 ta c6: BPT <=> x' -3x+2r-16-sJ* 1r+4 +x2 -3x+4 [*. u-s, a=2 4"{y, -{J )-9 <=> x, -*-!. o .=r l ^ - o, : (*uu mfln dk x6c ttir 16 I 6+J53 l_r>- L+ riit rrq,p f) ta c6,.e.P'L#f y rS:ftYf, r6t hq,p THl, ta c6 tl6p s5 x e,-*,tftrt*,**l IUlb. 1.00 . BPT quy v0 drrr1e J f a* *;+ .fir' 3r * 4 > m A xet nam so f (x) ='[x' 1x +, +'[77)c + 4 (**) => /(x) = 2x-3 2x-3 D6 th6y f '(x)>0 Vx > 3 n€n (x) <l6ng biiSn ffin (3; +o1 EO 1**1 c6 nghiQm Vx 2 3 <-> M! f (x)> m <+ f (3) =2+ ,[2 > m un. 1.00 tanx+cosx-cost x =sinx(l*t*{t*r) DKcosx. cos x/2 *0 '2 '7 sinl.sinr PT <=> tanr+cosx-cos'a =pinr(l *-4-) '' cos cos.r 2 xx cosr.cos-+sm slnI (=) tflox* cosx -cost x = sinx(4) x cos cosr (n) ,O'I (=> tanx*cosr-cos'x = sinx 2 .orI ort 2 <=>cosx-cos2x-0<->cosx= I <->x:k2 r $e Z) Nghr€m thoa m6n diAu kiQn (*) IIu1. 1.00 Phuong trinh duong thdng (d) qua M(l; 3) c6 d4ng A(x * 1) + B(y - 3) = 0 <:)Ax+By-A-38=0 Tac6 i(A;n'1,112;+S,ir(a;3) Hn luqt h cric vdcto ph6p tuy6n cua (d), (dr), (dz). E6 (q t4o voi (dr); (dz) mQt tam gi6c cdn tai dinh ld giao cua (dr); (dz) <=> t(d ; d,) = t(d ; d ) , I 3 A ! 4 B ! - ft A f n ! 11o,* o.:= o : :t u^ ^.=rl n = ' ' s"[t +E y17;g - L3A+48=4A-38 LA= Chgn B : I ta c6 A: I ho{c -1. Vfly phuong ttnh ttuong thlttg (d) c6 d4ng: x + y - 4= 0 ho{c x - y - 2 : 0- Go. i MN, P lan luqt h hinh chi6u cria H l€n c6c c4nh BC, AC, AB Tac6 ISMH = ISNH = ISPH=600 => HM : HN : HP = SH/tan600, H h diAm trong dudmg trdn n6n H ld t6m ducmg trdn nQi ti6p cua tam gi6c ABC. I Tac6 V*u, =;SH.SABc li ducrng cao, dudng trung tuy6n cria tam gi6c Sesc= %ANI.vc= |Jst -+A.za=a2Ji Ta l+i c6 Sesc : pr: P. HN =) HN : ry = =>sH=HN.tan60o =$.a:+ 4 j4 =) vsnsc=:+tJ3=f" <=> (n - 2).(n - 3) = 240 <:> n : 18 X6t ddy sO {Cl,hf e {0;1;2; ;18}. Trlgiathi6t tae6 cl =20c: n(n-r)(rL -2)(n-3) =2gn(n,-l) 242 \ \ Xdt day sO {Ci};lr e {0;1;2; ;18}. Ta c6 T : -* = .\*t=< I <=> /r < 8,5 1=) k= 0;l;2 ;g ciJ' 18-k 'v v'r''"''e Tuongt.uT> | 4=)k=9; l0; ;18 Tt d6 ta nh{n duo. c c,! Ma,r k{ri k = 8 ho{c 9. so srinh t.uc titip nhfn dugc k = p vl. 1.00 at +b3 +ct > a2Jbc +b'.[* +t'JoO j Ta c6 theo BDT cdsi: 2vP : z1a2 J-bc + b' Ji + t' J ony < a2 (b + c) + b2 (a + c) + c, 1a + < ab(a + b) + bc(b + c) + ca(a + c) (l) Lai c6 dE dang chimg minh ttugc a3 +b3 >-ab(a+b) cq(Do a3+ b3 -ab(a+b) = (a+b)(a-b)' > 0 voi mgi q b > 0) Tuong t:ty b3 +c3 >bc(b+c);ct +at > ca(c+a) => 2VT ab(a + b) + bc(b + c) + ca(c + a) (2) Tir (1) vn (2) ta nhAn dugc dpcm. Ddu ":'xiy ra khi a = b =c. r7