Tnrong Cbuyen Le Quy Don BR VT DE THI TmJ D~I HQC - CAO DANG LAN 3 (2010 - 2011) Mon: Toan - Kb8i: B DE CHiNH THUC ThiJi gian lam bai: 180 philt. I. Ph§n chung eho tit eli cae thf sinh (7.0 ([iim): Cau 1(2.0 ([iim): Cho ham s6 y:::; ~ (*). x l O 1) Khao sat S\I' bi~n thien va ve d6 thi (C) eua ham s6 (*). 2) Vi~t phuang trinh tiep tuy~n d eua (C) bi~t d e~t hai tIVe to~ dQ t~i cae di8m A va B thoa man tr9 n g tam tam giae OAB thuQe dUOng thing: x 4y = 0, (0 la g6e to~ dQ). Cau II (2.0 ([iim): q· h 'nh 2sinxeos2x+sin2x 1 1) tal P uang trl : . = +eosx. -2smx 2) OiM b~t phuang trinh: x log~ x + 6 < 3x log2 X + log.fi x . Cau III (1.0 ([iim): Tinh tieh ph§n 1= j£x+i dx. o 8x+3 Cau IV'(1.0 ([iim): Cho hinh chop S.ABCD co ABCD la hinh vuong e~nh a va m~t ben (SAB) vuong goe v6i m~t day. Tam giae SAB vuong t~i S, goe gifra SB va m~t phfutg (ABCD) bing 30° .Tinh theo a th8 tich cua kh6i chop S.ABCD va khoang each gifra hai dUOng thfutg AB, Sc. Cau v (1.0 ([iim): T ' +f t ' " t' , th J- h h~ h t 'nh {2X 2Y +9X = my2 'd' b h'~ 1m La ca gta r! eua am so m sao COy P uang n co ung a ng lym. 2 xy2 +2y = x II. Ph§n rieng (3.0 ([iim) : Thf sinh chi aU(lc lam m6t trong hai phdn (phdn A hoi;ic phdn B). A. Theo ehU'ong trinh Chuin: Cau VI.a (2.0 ([iim): 1) Trong m~t phing v6i h~ t9a dQ Oxy, eho tam giae d€u ABC n9l ti~p dUOng trim (C): x 2 + y2 -4y-4 0 va c~nh AB co trung di€m M thuQc dUOng th~g d: 2x y-l = O. Vi~t phuang trinh dUOng thing AB va tim to~ dQ di8m C. 2) Trong khong gian v6i h~ t9a dQ Oxyz, cho eac di€m A(3;-I;I), B(-1;0;-2), C(4;1; 1) va D(3;2;- 6). Vi~t phuang trinh m~t cAu tiep xuc v6i hai dUOng thing AC va BD l§n luqt ~i A va B. Cau VII.a (1.0 ([iim): Trong eac s6 phuc z thoa man I; 2-3il 5, tim s6 phue z sao cholz+4-5il d~t gia tri nh6 nh~t. B. Theo chU'ong trinh Nang cao: Cau VI.b(2.0 ([iim): 1) Trong m~t phfutg v6i h¢ t9a dQ Oxy, cho tam giac ABC co dUOng thfutg AB di quadiSm 0; phan giac trong goc A thuQe duemg th~g d l : x y + 1 = 0 va dUOng eao ke tir B thuQC dUOng th~g d 2 : 4x+ 3y 48:::; O. Tim to~ dQ cae di€m A va B. 2) Trong khong gian v6i h~ t9a dQ Oxyz, cho hinh thoi ABCD co tam I thuQe duemg thing d: x-I = y - 3 z - 2 va hai dinh A(3; -1; 1), B( -1; 1;3). Vi~t phuang trinh duang thfutg CD. 1 1 1 ~ , , (.Jj -if ' , Cau VII.b(1.0 ([iem): Tim cae gia tri nguyen duang eua n de so phue z = 1 ~ i.Jj la so thuan ao. IIItT Thi sinh khong dUQ'c SIT d\lng tili li~u. Can be} coi tbi kbong giiii thich gi them. HQ va ten thi sinh: sA bao danb: ••.• ••.•• www.MATHVN.com www.mathvn.com II [ X = 7l' + k27l' ~ * <=> 7l' 7l' {\ ;(k E Z) x = + k7l" X = + k7l' 6 ' 3 gjl1-;(::- ~\1il Tij. [ '''/s x = + * K& hQ'p di6u ki~n, ta co nghi~m: x = 76 0.25 > , TJ."trirng Chuyen Le Quy Bon BR - VT. BAP AN vA HITONG nAN CHAM BE THI THU B~I HQC - CAO BANG LAN 3 (2010 - 2011). Mon: Toan - Khtii: B Ciu I N(U dung 1 Kh • 't' _;l-A thO X . ao sa va ve uO I: Y = x-I Thang • di~m Ghi eM * n.p xac dinh: D = IR. \ {I} 0.25 Y I = -1 2 ; ham s6 nghich bi8n trong timg khoang ( 00; 1), (1; +00) (x-I) . * Ti~m c~: lim Y = 1 ~ TCN : y = 1; lim y = ±oo ~ TCD: x = 1. 0.25 x-*±«> x-*J± *Bang bi8n thien: 0.25 * I'll> thi' 1 +00 y' 1 +00 y I~ ~I -00 tJ 1 JJ ) lJ 2. Viet phU'01l2 trinh tiep tuyen: * T(a;~) lati8p di€m, pmd4tiT: y= 1 2 (x-a)+~. a-I (a-I) a-I 2 * Giao di€m d va Ox: A(a 2 ;O) , giao di€m d va Oy: B(O; a 2 J. (a-I) TrQng tam cua tam giac OAB : G (~; a 2 2 J ' di6u ki~n a *0; 1 3 3(a-I) * G thuQc dt: x-4y 0 <=> a =O;a = -I;a =3 (lo~i a 0). 1 1 1 9 * Pm: y = x+- hoac y = x+- 44' 4 4 0.25 0.25 0.25 0.25 0.25 www.MATHVN.com www.mathvn.com 2. Giai bit phU'O'ng trinh: x log; x +6 < 3x log2 X +log,fl x (1) J ' I * (1) <=> (log2 X - 3)(x log2 X - 2) < 0 r>s 2 (1) 0.25 * ¢:::> {::':<7<O / 2 (I/) / log2 x -;>0 / 0.25 * /(x)=log2 x - 2 d6ngbiantren (O;+«» vaf(2)=Onen /(x)<0<=>0<x<2;/(x»0 x * V~y nghi~m cua (1) : S = (2;8). 0.25 III h A T ' h ' h p an: I = 4J.J2X+ Idx lD tic o 8x+3 * D~t t = .J2x+ 1 => dx = tdt; d6i c~: x = 0 t = 1; x = 4 t = 3. 0.25 3 2 */ f~t 0.25 14t -1 * _3 n .!.+_1___ 1 l l (1 1 ln12t-11J3 *1== .!.,lm~. 0.25 - tl4 8(2t-1) 8(2t+l)[t= "4t+16 2t+1 1 21; 7 0.25 IV Tinh the rich kh6i S.ABCD va khoang each: s * GQi H la hlnh chi~u cua S tren AB, chUng minhSH -L (ABCD) 0.25 va SBA 30·. *Tinh SA = a 'SB =aJ3 'SH = a.fj. VA V a 3 .fj 0.25 2' 2 ' 4 ~y 12 S_ABCD *AB, SC cheo nhau va AB II (SCD)::J SC nen d(AB;SC) == d(H;SCD 0.25 D\ffig va CM duqc d(H;SCD) = HK ( HM II AD va HK -L SM). *T' hHK- ~ 0",~. -HI<- - Q~. HN 0.25 III - a - {,' ,\, I 1.!l- ,r::- 19 . l\ \( ::: f ~ • Lt, ) ~ \ ~ Til , v} C 1. v Tim gia tn tham s8 d~ he pt co 3 nghiem: {2X 2Y +9X ml "''9' \6 , Lt x 2 xy2 +2y * N~u x == 0 suy fa y = 0 va nguqc ll;li. (x;y) = (0;0) 1a m¢t nghi~m cua h~, veri mQi m 0.25 2 X2 9 x - 2 2 *N~u xy'f::O: H~ ¢:::> -y+ -l-m;d~t u=_x ,v=-y =>X=}jU2v,Y=~UV2 l y y x -+2-=1 0.25 x 2 X , 1 veri m6i (u;v) veri uv 'f:: 0 cho ~¢t ngh~m (x;y) thoa xy 'f:: 0 2U+ 9 =m , Bai toan <=> tim m d~ h~ pt 2v co dUng hai nghi~m (u;v) thoa UV'f:: 0 { v+-=l , , u \ Hay phuong trinh 2u + ~ = m co dUng hai nghi~m u khac 0;2. u-2 , 9 u * L~p BBT ciIa /(u) = 2u+_ -;u E R \{0;2}. 0.25 u-2 . * K~t qua: mE (-00;0) D (0; 1) u (25; +«» 0.25 '\ www.MATHVN.com www.mathvn.com C 0.25 'h1 ::; }ir -::) M ( 11 '.1 / 'J I ~ PHAN TV' CHON VLa A.Then chllO'Dg tranh chuan: 1. moh hoc toa 46 phing: *(C) co tam I(0;2), ban kinh R = 2.J2 . ABC d~u nQi tiap (C) nen 1M = .J2 ie- 1l r: ~r2 .;- '(:::: ) i)y _ *M thuQc d nen M(m;2m - 1), 1M = .J2 ~ m = l;m = 2. . 1: l~ - l\'\1' ~n\-3) o£.AN ~, t~ 5 I ~ -:. _ ?- * AB qua M va co VTPT ta 1M',IC = -21M nen i:) N) -\ 4~1- \~-r ~- ,.::::) '2-~ m = 1 ::::> AB : x - y 0 va C(-2 ;4). tv\ (1\.) 1\ ). i'"'N:::. ( 1\. r 11)1- -\ 'il1'V\ tk tJ 7 -14 AiYl v ~ ,-II) L. I * m=5::::>AB:7x-Y-8=O;C(-5-;~·t· 1(£ :::- (-g.} 2.) 2. moh hoc toa 46 khong giao: -'~m. __ ._ ~ ~0tJ-~-m ___ - * mp(p) qua A va vuong goc AC : x + 2Y - 2z+ 1=° 1 ~ ~ (~'.)' 1. 'J _ ~ ) * mp(Q) qua B va vuong goc BD : 2x + y - 2z - 2 0 * M~t cAu dn tim co tam I thu9c (P) va (Q) nen I( 3 + 2t ;2t ;2+3t) *IA = IB ¢:> t = -1. V~y 1(1 ;- 2 ;-1) va R = 3; phuong trinh mcAu : (x-l)2+(y+2)2+(z+1)2 =9 VIla { So phUc: Cach 1: *z x + iy, v&i x, Y E IR X = 2 + 5 cos t ~ 2-3i=5¢:>(x-2i+(y+3)2 25¢:> . ;tE[O;27l') I I Y -3+5smt * Iz+4-5il = )(X+4)2 +(y-5i =~125+20(3cost-4sint) * 3cost -4sint ;::: -5, 'v't E [O;27l') nen Iz +4 - 5il ;::: 5, 'v'z * Dkg thuc xay ra khi cost =-3/5;sint = 4/5. V~y z = -1 + i. Cach 2: . * Tgp h(YJJ cac iii€m M(x ;y) biJu diln z trong mp Oxy la (C): (x - 2)2 + (y + 3)2 = 25. * Iz +4 ,.5i! == )(X+4)2 + (y _5)2 == AM, vaiA(-4;5) nd:m ngocli (C). AMnhO nh&t khi M la giao iiiim (gdn A han) cila (C) va lA, vai 1(2;-3) la tam cua (C). * lA: 4x + 3y + 1 = 0, giao iii€m cila (C) va lA fa B(-1; 1) va C(5 ;-7). *lB<lC. V0-'Z -1 +ilast5phii:cthoayeucauMitoan. VLb B.Theo chllO'ng trinh nang cao: 1. moh hoc toa 40 phing: *D d6i xUng 0 qua d j : D(-I;I). *D thu9c AC (tc phdn giae) va AC vuong g6c d 2 nen AC: 3x 4y + 7 = O. *A ta giao cua AC va d 1 : A(3 ;4) *B ta giao eua OA vii d 2 : JB(6 ;8) 2. moh hoc toa 40 khoog giao: \ *I thu9c d nen I( 1 +t;3+t;2+t) * ABCD la hinh thoi nen IA.IB == 0 ¢:> t = -1 .V~y 1(0 ;2 ; 1) *C, D d6i xung v&i A,B qua I nen C(-3;5; 1) va D(l ;3;-1) x-I y-3 z+1 *Phuong trinh CD: = = 4 -2 -2 VII.b sa ph rc: '-;;3 . 2( -7l' -7l') 1 ''-;;3 2( 7l' .' 7l') *"j -z == cos-+zsm-; +lVj = cos-+zsm- 6 6 3 3 * (J3 - iY = 211 (cos -n7l' + isin -JVr) 6 6 (J3-it 2 11 - 1 [ (-m'C 7l') ., (-n7l' 7l')] • * V~y z==' = cos - +zsm l+iJ3 6 3 6 3 ., -n7l' 7l'. ~T' * z thuan ao ¢:> cos( ) = O,n EN¢:> n = 6k- 5, k E H 6 3 I 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 ·0.25 0.25 www.MATHVN.com www.mathvn.com