.fRUONG D}.iSP I-IA NOI TR.UOI\G TT{PT' CHUVEF{ . }}FISP pn :r'm rFIU'F]Ar ri-oc LAI{ Ir ruAnr zolt lvidii ilii : TOAI{ 7'hd'i gian ldm hdi : I B0 phtit, khong kA thdi gian phfrt di CAu 1. ( 2,0 diAnt ) Cho hdnr sd .y: ?+ I. I(hAo s6t su bi6n rhi6n vA v6 c16 thi (C) cfra hArn s6. 2' Timtdt citcircgi6tri cirantd6drrongtirang y:m(x_ 2)+2citci6rhi (C)tai hai di6rnphAnbi€t A, B sao cho doan AB c6 dQ dii'nho nhAt. Cdrr 2. (2,0 diem) l. Giei phLlo-ng trinh : sin2x.(l + tanx) : 3sinx.(cosx _ sinx) + 3 2. Gi6i bat phuong rrinh : 3*= - 4> 5.3# Cflu J. (, 1,0 cliem ) - .V3ln\fizTJ Tinh tich phAn I : J, " il-J '6* Cf,u 4. ( 1,0 diem ) Cho hinh lAp plruo'ng ABCD.A'B'C'D' cci do cldi canh bing a vd cii6rn M t5Lr6c ca'6 CC, sao cho cM : ? *Ut phang @) diqLra A, M vd song song v6'i BD chia kn6i tap phu.ong thdnh hai l<h6i da cti6n. Tinh th€ tich hai I<h6i da diQn d6. Cdtr 5. ( t,0 dietn ) tla s6 clu'ong thay d6i a, b, c thu6c doan [', B] md F - o {2. crrfi.ng mi'h rdrig : Gilf +/66a1+rtra+f > a*b*c. Cflu 6. ( 2,0 didm ) l' Trong rndt phlng toa d0 oxy, cho tam gi6,eABC c6 c( I : 2), hai dildng cao xu6r ph6t tLr A vd B lAn lu'otc6 phLLongtrinh IA x + y : 0 vd 2x-y + I = 0. Tinh cli6n tic6 ta'r gi6cABC. 2. Trong l<h6ng gian toad6 Oxyz, cho mdtphing (p) c6 phLrong trinh: x*2y 1,22+ l= 0 vd rndt cAu (S) c6 phLrong trinh : x2 + yz + z2 - 4x + gy + 6z+ 17 :0. Tinr toa dd tam vd b6n kinh crla du'o'ng tron (c) la giao cira niir plidng (p) vd rnat cAir (S). Cf,u 7. ( t,0 ttient ) Giei he phLLong trinh : nat t*t + xyz :40y Lyt+x'y=10x DqE kiare ki tki thft fr$i hpc rftn tratu s sE dwgc t6 chs?c vda rcgdy rg,z#/s/zL!i sap Ani - THANc DIEM rnr rrrtl oH r.An rnU nar NAivr zor r CAU oAp AN olBtvt I 7z AiAml L (1.0 man. Hoc sinh tu sidi. i. (t,o aiiiml . Tim c6c gi|tri m Euong thang y = m(x-2) + 2 cttd0 thi (C) tai hai di'5m phdn bi6t <+ c6 hai nghiQm phdn bi€t <+ pt nrx' - 4mx + 4m - 5 = 0 (*) c6 hai 2x+1 Pt;=m(x-2)+z nghiQm phdn biQt khbc2 0,25 (m*0 c+ Jo' : 4m? -m(4m-s) > o o m >o I \+m-Bm*4m-5+o 0,2s Gi6 srlr A(x,,y,), B(xr;yr) trong d6 xr, Xz ld hai nghiQm c0a (E). Khi d6 yr = lnXr -Zm*Z vit y2= lrlx2 - 2m+2 Tac6 AB2 = (*, - 4,)t + (y, -y,)2 = (xz - xr)2(m2 + l) : [(x2 + x1)2 - 4x1x2](m2 + l) 0,2 5 4(4m-5) . 20(m2 + 1) 2o,2m : Il6 -=;= X*t + l) = :-:il"-:1/ Z"#:40 vdi mgi m > 0. Eing thri'c xdy ra khi vd chi khi m : L Vdy, v6'i rh: I thi AB ngdn nhat Uing V40 . 0,25 II (2 cti6m) l. ( 1,0 clilnt\ . Gi6i phLro-ng trinh . Ei6u kiQn : cosx f 0. Phuong trinh dd cho tuong duong v6i pt : sinzx ^ sinx . 3 ) ,, l , # (tun* + 1):3:lla(l- tanx) + c+ tan2x (l+ tanx):3tanx(l-tanx) + 3(l+ tan x,t cos'x' ' cosx cos'x 0,50 (+ tan2x (l+ tanx) = 3(l+ tanx) e ). dreu Klgn Dal toanJ. [tanx = -1 [* = Itan2x=3 H l*: (kez) ( th6a rndn -i+tn *I+kn -3 0,50 2. (1,0 cli1nt). Giai bat phuong trinh . DiAuki€n:xl BAt phu'o'ng trinh dE cho tuong duo'ng v6'i bpt x+3 . x+3 3sx-, - 4,5.3"-t*u ') ; 0,25 AfJ D{t1=3sx-2, t>0.'Bpttr6ntrdthanh t2 -4t 45>0 + t:9(dot> 0) 0,25 x+3 ATJ 'isx-z> ge - 5x-z 27 >2 e -<x< - 5 -9 Ddp sd : a1 x€(-:-l '5'9' 0,50 III (1,tti€nl (1,0 diAm). Tinh tich phAn rrc6 r- - I,fitnVTJ?d1= -11nur1 1p lf -llr"1otrn.,rTT7; = rn,/z - *^o * lr/t*=a* :#rnz *f "lo*. 0,50 L Tas€tinh.l =f*clx, ddt x:tanr =r dt:-+ dr:(l .of,)d, , x= r/3 thi IV (1 ili1nt) o (0,50 ilid@. Dung thirit diQn cira mat phing di qua A, M va song song v6.i BD. Gqi o=AC o BD, o' =A'c' n B'D' va I =AM n oo'. eual k6tluo.ngth6ngsong song v6'i BD cit BB' vri DD' lan luo.t t4i K, N. Khi d6 AKMN la thi€t di€n can dung. DAt Vr = Ve.scvr* Ve,.oouru , Vz= Veaco.a,s.c.o,- Vt. t (0,50 tlie@.racO ff =#= 1 =+ DN=BK=Ot =;a* =; . Hinh ch6p A.BCMK c6 chi€u cao ld AB = a, I ddy ld hinh thang BCMI(. 1 Suy ra vo.scrr, ::aB o - AB Bc(BK+cM) a3 3 '.recvrc:T z :T B' Hinh ch6p A.CDNM c6 chi€Lr cao ld AD = a, tf6y ld hinh thang CDNM. 1 . - AD CD(ND+CM) a3 Suy ra Ve.coN" =:AD 3 '.Jcolrlr:T , =; r( a3 .3 t^3 V?y, Vr =-:- , Vz: ai -a' -2a" B '333 t. (1,0 TtLgi6thi6tsuyra lg llSF-o <2 +(a-b)t< 4=+(a+b)2 _4abs 4 suy ra a* b < 2lr+a, tlro'ngtu'tacfingc6 : b+c <2fi1fi , a+ c szrlr+ ac Dod6: a*b*c<V1 +ab+/1 +bc+/1 +a;(dpcnr) (1,0 ttiAm). Tinh di6n tich tarn gihc VI (2 tli€n) DLld'ng thdng BC c6 vecto chi phLro-ng il la vecto phap cira dLLdtrg thing x +y : 0, n€n d=(l; l). Phuo'ngrrinh cira*a' [i:1il tu, ra B(l+t;2 +t). B rhu6cdr-rongcaoxu6t phSttrrB n€ntead6th6amdrr phuongtrinh:2(l +t)-z-r+ I -0 +t:- l. vay B(0; I). ruo'ng ts, phuong trinh AC , [; = i:i' vd A(- 5; 5) Tac6: BC=J2 Duo'ng thing BC vitit va d4ng t6ng qurit : x - y + l-s-s+r I 9 Goi AH ld dudng cao, ta c6 AH - :-: - I -j - - 'lZ ,/Z (1,0 tti€m). 'lim toa d0 tdrn vd bAn kinh . Mat cAu (S) c6 tdrn IQ; - 3; -3) vd bdn kinh R : \i5 . I(ho6ngc6chtiL1t1iinmp(P)lith=w=l<R=V5,n€nmp(P)cdt rn{t cAu (S) theo mQt tfud'ng trdn c6 b6n kinh bing r : ,lP P = 2. 0,50 Tdm K cria dudng trdn (C) ld giao di€m cria mp(P) vdi dub'ng thingd tli qua di€m 1vd vudng g6c v6{ mp(P). Vecto chi phuong.c0a duong thhng d Li vecto phrip tuytin cria mp(p), n6n ui:rt;1 :2)vi phuongtrinh cria o,[;='-{ir, Dod6 K(2+t ;-3 -2t;-3 +zt) \z: -3 { 2t. Tqad0tem K cira(C).th6amdn phuongtrinh: 2+t+6+4t-6+4t + I :0 + t =-+., \ 7 11 Vay tAm I((il ;- ). bdnkinh r:2 'J', 3' 3" 0,50 WI ,Q iliem) (1,0 cli€nt). Giei he phLro'ng trinh N6u x=0thi y:0 =+ (0;0) ld mQtnghiQm crhahQ phuongtrinh. Ntiu x I 0. D?t y = tX, khi d6 hQ pt tro thAnh : fx3 + x3t2 = 4otx fx3(t + tt) : 4otx tt3x3+*tt=10* : ["t1rt+tj=1gx (+ (x.(t +t') :4ot ( "t = # (1) I *'(r' * t) = 1s € lt## : to (z) Tt(l) + t>0,k6tho.pv6'i(2) - t::.Thay t= ) ,eo(l)raduo-c x=+4 + y=r2. Drip sti : HQ pt c6 3 nghiOm (0; 0), (4; Zj , e a; - 2). I,00 . "lo*. 0,50 L Tas€tinh.l =f*clx, ddt x:tanr =r dt:-+ dr:(l .of,)d, , x= r/3 thi IV (1 ili1nt) o (0,50 ilid@. Dung thirit diQn cira mat phing di qua A, M va song song v6.i BD. Gqi. sinh tu sidi. i. (t,o aiiiml . Tim c6c gi|tri m Euong thang y = m(x-2) + 2 cttd0 thi (C) tai hai di'5m phdn bi6t <+ c6 hai nghiQm phdn bi€t <+ pt nrx'. vdi mgi m > 0. Eing thri'c xdy ra khi vd chi khi m : L Vdy, v6'i rh: I thi AB ngdn nhat Uing V40 . 0,25 II (2 cti6m) l. ( 1,0 clilnt . Gi6i phLro-ng trinh . Ei6u