\ Tru'(l'ng Chuyen U Quy Don llR - VT DE THI nIli' D';'I HOC - CAO DANG LAN 3 (2010 - 2011) Mon: Toan - KhBi: D . DE CHiNH THUC "-~ TIlt'; ginll/lull biii: 180 pltut. ) I. Phin chung eho tAt ca de thi sinh (7.0 tlMm): f{;!W!j Cau 1(2.0 tI;im): Cho ham s6 y = x-I (*) x+l 1) Khao sat S\I biSn thien va ve d6 thi (C) cua ham s6 (*). 2) Cho hai di~m A(4;3) va B(-I;-2). Tim t9a de) di~m Mtren (C) sao cho tam giac MAB co tr9ng Him ding thuQc (C). Cau II (2.0 tliim): · h . !, sin 4 x+sin2x cos 4 x '2· 1) G tal P uang tnnJ,.l: = COS- x. tan 2x-l Xl +i - xy(x + y) == 4 2) Giai h~ phuang trinh: . { l+xy=~x+ y-l In2 3x Cau III (1.0 tliim): Tinh tich phful J-2-e-dx. o e X -9 Cau IV (1.0 tliim): Cho hinh chOp S.ABC co day la tam giac ABC vuong t~i C, BC a va BAC 30°. Th~ :1 tich kh6i ch6p S.ABC bAng ~ va SA = SB = SC . Tinh chitu cao eua hinh eh6p S.ABC va eosin cua goc 2 gifra hai dUOng thdng AC, SB. Cau V (1.0 tliim): 2 2 2 ;. d h' 9 'nh::' a b c + 1 Ch ocacsot ' h \IC uang a, b ,c t oa a+ b +c;;:: . Cl mngml . rang: + + ;;:: + 1 +1 10 . b c a II. Ph~n rieng (3.0 tliim) : Thi sinh chi aU(fc lam m<J1 trang hai phdn (phdn A ha(ic phdn B). A. Theo chuong trinh Chu§n: Cau VI.a (2.0 tliim): 1) T rong m~t phing v6i M tQa de) Oxy, cho tam giac ABC co C thue)c dUOng thing d: x - y - 3 :::: 0 ; A( 1 ;2), B(2;7) va chiSu cao AH cua tam giac ABC bAng 1. Tim to~ de) C. 2) Trong khong gian v6i M tQa dQ Oxyz, cho cac di~m A(0;2;-l), B(3;0;2) va C(2;-1 ;0), ViSt phtrang trinh m~t du (S) di qua A, B, eva (S) ti€p xuc v6i m~t phdng (P): x + y + 2z + 2 = 0 . Cau VII. a (1.0 tlMm): Trong m~t phing phuc cho s6 phuc z thoa man Iz - il == 2, tim t~p hgp cac di6m bi~u di~n cua 36 ph~c w =·iz + i-I. B. Theo chU'ong trinh Nang cao: Cau VI.b(2.0 tlMm): 2 1 1) Trong m~t phing v6i M tQa dQ Oxy, cho elip (E): ~5 + ~ =1 co hai tieu di6m la FI va F2. Di~m M tren (E) tho a man ~ == 60° , Tinh di~n tfch tam giac MFIF2. 2) Trong khOng gian v6i M tQa de) Oxyz, vi~t phuang trinh m~t ph~ng (P) di qua di€m A(3;1;2), (P) "d' h! d x-2 y-3 z+l d A h"kh' 'h'- d '(P)b~ 3 song song ven uang tang : = == ong t en oang cac glUa va ang. 2 1 2 Cau VII.b(l.O tliim): Tim mQi gia trj Clla tham s6 m d~ d6 thj ham s6 y::::: (x + 2)(x 2 - mx +m 2 3) ti6p XUC v6i tr\lc hoanh. IIIt1r Thi s~n~ khO,n~ dU'9'c sir dl}ng tili li~u. Can bi} coi thi khOn r g~ai thich gi them. 11(,) va ten thl Sinh: , So bao danh: www.MATHVN.com www.mathvn.com I 'fnrang Chuyen Le Quy Don BR VT. DAp AN VA HtrONG DAN CHAM DE THI THU D~I HQC - CAO DANG LAN 3 (2010 - 2011). Mon: Tolin - KhAi: D CAu , NQidung Thang Ghi chii (fi~m 1 Kh • . , - d" b' X -1 , ao sat va ve 0 t I: y = 0.25 x+I * T~p X3.C djnh: D = IR \ {-I} 0.25 y' = 2 2; ham sil d6ng bien trong timg khoang (-«>; -1), (-1; +00) . (x+l) * Ti~m c~: lim y = 1 ~ TCN : y = 1; lim y:::: +00 ~ TCD: x = 1. 0.25 x-+±«> .<-+-1" *Bang bien thien: x -00 1 +00 y' + + Y 1/ +00 I ~ -00 u' :; " , u L " ! .1.1 <4 4.1 1 ·IJ , V' " (C)Y'~ ,+1 0.25 I l 2. Tim di~m M: *M thuQc (C) nen M(m; m-l), ABM III tam giac <::::> AM,AB khang cung phuong 0.25 m+l <::::>m:;t:O;±I(*) -Niu khdng tlq.t hoq.c kiim 0.25 * TrQngtam G(m+3; 2m J,GthUQC(C) <=> 2m =~. tra tlk cho 3 3(m+1) 3(m+l) m+6 16i tla 0.5tl 0.25 * m=Ohoacm=9. 0.25 * Ki6m tra di~u ki~n (*), ket lu~n M(9;4/5). f-= j " ' 1 + IT 1 G · • . b t' h sin 4 x + sin 2x - cos 4 X 2 2 (1 ) • lal p U'O'Ug nn : = cos x , tan2x-l x:;t: Jr +k Jr 0.25 * :I: ~ k·. 8 2 uleu l¥n: Jr Jr { x;c -+ k-,(k E Z) 4 2 0.25 * V 6i dk tren, (1) <=> cos 2x =cos 2 2x . 0.25 * cos2x = 0 (101;ii) 0.25 I * cos2x = 1 <::::> x =mJr,(m E Z) (thoa man dk) II X2 + y2 -xy(x+ y)=4 2. Giai he phU'O'Ug trinh: ~ { I+xy= x+y-l 0.25 S2 2P - SP = 4 (1) * D~t S=x+y,P=xy;S2-4P~0,S~1,h~trathimh r;:;-:; { 1+P = " S -1 (2) 0.25 * (1) <::::> (S + 2)(S -2-P) = 0 <::::> S:::: P+ 2 (lo'ili S = - 2, do dk). 0.25 * The vao (2): 1 + P :::: 11 + P <=> P :::: 0 V P = -1. Suy ra S = 2 ho~c S = 1 0.25 * N h' A (0'2)' (2'0)' [I+.J5 .1 J5).[l J5 .1+.J5) g l~m , , " 2' 2 ' 2' 2 . www.MATHVN.com www.mathvn.com r:I~II; r ~ : :: r ·- IV v VI. a Tioh ehiSu cao hloh chOp S.ABC va khoang each: * GQi H lit chan dUOng cao eua hinh chop, SA == SB == SC nen H lit tam day, v~y H III trung di~m AB. a 2 Jj a 3 , * AC = aJj => SABC = -2 - ; v = 2 nen SH = afj * D\l'Ilg hinh chii' nh~t ABCD, 5Uy ra g6c (SB;AC) = g6c (BS;BD) = rp, vai SB == SD == 2a; BD = afj *N' fj en cosrp=4' s ., . , , , , , ,'0 : """;:: ,,, : /~ \ ~~ ;! , ,-' \.: A ~ (j_"~"*~"~ __ _ H\ C6th€dtinhgiavitrai: VT'2a+b+c+ 9 ; Saua6xetham f(x) x+~;x~9 a+b+c x PHAN TU CHQN A.Theo chaO'll!! tnoh chuio: 1. mnh hoc toa dO pbing: * Phuong trinh dth BC c6 d{lIlg: a(x - 2) + bey -7) = O;a 2 + b 2 > 0 l la+Sbl *Chieu cao AH = 1 nen 1 = d(A;BC) = .J <:=> b = Ov 5a + 12b = 0 a 2 +b 2 * V~y BC: x - 2:; 0 ho~c BC: 12x 5y + 11 = 0 * Suy ra C(2 ;-1) ho~c C(-2617;- 4717). c 2. mnh hoc toa dO khong gian: * M~t cAu (S) qua A, B, C nen co tam I thuQc hai m~t trung tnrc (M), (N) cua AB, AC : (M) : 3x - 2y + 3z - 4 = 0 ; (N) : 2x - 3y + z = O. * Suy ra I c6 to~ dQ d{mg tham 56: ](1 + 7t; 1+3/; 1 51) B * (S)ti€p xucvai (P) nen d(I;(P» = IA =R<=> R;:: J6 = J(l +7t)2 +(31-1)2 +(2-5t)2 I=Ovt = 12 vAR= J6 83 * V~y phuang trinh (S) : (x - V + (y _1)2 +(z _1)2 == 6 hay (x_ 167 )2+(y 119)2+(z_23 i =6 83 83 83 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 In2 3x e Tioh tfeh phan: 1= S 2 X dx o e -9 2 12 * 1 == eX => dl == eX dx. I = S-2- dt II -9 * 1= 1+ 3 3} t I 2(/-3) 2(/+3) ~ * ] =(t+~lnl/-31)2 2 t+3 1 3 2 * ] =1+-ln- 2 5 0.25 0.25 0.25 0.25 www.MATHVN.com www.mathvn.com VILa Sa phuc: * z x + iy (x;y E lR), ta co x 2 + (y -Ii == 4 .(*) * w = iz + i -1 == -(y + 1) + (x + l)i co di~m bi~u dien M : XM = - Y - * Thay y -XM 1 va x = YM - 1 vao (*) ta duqc (XM + 2)2 + (YM _1)2 * V~y ~p hQP M la dubng tron tam 1(-2 ; 1), ban kinh R =2. C6 thJ d;Jt w = x + iY ; BiJu diln z thea x; y. Tfr Iz - il = 2 suy ra kq nhu tren. 0.25 1, YM = x +1 == 4 0.25 0.25 0.25 VLb VII.b B.Theo chU'O"llg trinh nang cao: 1. mnh hoc toa do phing: *FJ(-4;0), F 2 (4;0) va FJF2 = 8 * Gia sir M(x;y) ta e6 MFJ 5 + 4/5.x; MF2 = 5 - 4/S.x. Theo dli Cosin: 8 2 = (S + 4/S.X)2 + (5 - 4/S.xi - 2(5 + 4/S.x)(S - 4/S.x).cos60 o 3J3 *Gilii tim duqc x 2 13 .25/16 . Suy ra Iy I= 4 * V~y S(MF 1 F 2 ) = 3J3 . 2. mnh hoc toa do khong gian: 2 * Pt (P): a( x - 3) + b( y - 1 ) + e( z - 2) = 0, vcri a 2 + b 2 + c > 0 (P) song song d nen : 2a + b + 2e = O. l-a+2b- 3c l *Khoang cach (d;(P» = d(B;(P» = I 2 2 ' vcri B(2;3;-I) thuQc d. 2 va +b +c * d(d;(P» = 3 <:::;> 2c 2 - ac -lOa 2 =0 *Chon a = l;e = -2 , b = 2 ; ho~c a 2; c = 5; b = -14 V~y phuong trinh m~t ph!ng (P): x + 2y - 2z - 1 = 0 ho~e 2x - 14y + 5z - Tim tham sa m:. * Di~u ki~n ti6p xue eua Ox va db thi ham s6 : (X + 2)(X2 mx+m2 -3)==0 "A co nghl~mx. { X2 -mx+m2 -3+(x+2)(2x-m)=0 * H~ pt tren <:::;> {:;-~+m'-3 0(1) 2 X2 - mx + m - 3 = 0 (II) { (x+2)(2x m)=O * (I) c6 nghi~m khi m = , * (II) co nghi~m khi m = -1; m == ±2 . 2 = O. 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 . In2 3x e Tioh tfeh phan: 1= S 2 X dx o e -9 2 12 * 1 == eX => dl == eX dx. I = S- 2- dt II -9 * 1= 1+ 3 3} t I 2(/ -3 ) 2(/ +3) ~ * ] =(t+~lnl/ -3 1 )2 2 t +3 1 3. Tru'(l'ng Chuyen U Quy Don llR - VT DE THI nIli' D& apos;;'I HOC - CAO DANG LAN 3 (2010 - 2011) Mon: Toan - KhBi: D . DE CHiNH THUC " ;-~ TIlt'; ginll/lull. bao danh: www.MATHVN.com www.mathvn.com I 'fnrang Chuyen Le Quy Don BR VT. DAp AN VA HtrONG DAN CHAM DE THI THU D~ I HQC - CAO DANG LAN 3 (2010 - 2011) . Mon: Tolin - KhAi: