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. TruOng Chuyen U Quy Don BR - VT DE TID TmJ D~ HQC - CAO DANG LAN 1 (2010 - 2011) Man: Toan - Kh8i: A & B . DE cHiNH THiJ'c Thitigian liun btli: 180 phut. I. Ph in chnng cho tit ca cac thi sinh (7.0 diem): Can I (2.0 diem): Cho ham 56 Y = x 3 - 3x 2 • 1) Khao sat S\l bien thien va ve db thi eua ham 56. 2) Tim m9i gia tri eua tham s6 m d8 trong khoang (1; +cc), phuong trinh x 2 1x - 31 =m cO hai nghi~m phan bi~t. Can II (2.0 diim): 1) Giai phuong trinh: eot 2x - 2 tan 3x = sin 2x(1 +tan x tan 2x) . 2 2 2) Giai bdt phuong trinh: 2(1 J x - 2x) S; x +.J x - 3x + 2 . Can III (1.0 diem): Tinh tich phan I ln (X + ~) dx . I X Can IV (1.0 diim): Cho Wnh ch6p S.ABCD c6 m~t ben (SAB) vuong g6c vai ~t day. Hinh thoi ABCD c6 g6c ABC = 60~ , tam giac, SAB vuong can t1ili S va SD = a. Tinh theo a th~ tich Clla kh6i ch6p S.ABCD va di~n tich cua m~t cau ngo~ tiep t(r di~n SACD. Can V (1.0 diim): Cho cac s6 th\lc x, y thay d6i trong d01ilD [1;2]. Tim tdt ca gia tri cua 56 th\lc z d~ bi~u thuc P = (x + yz)( x - y) + xyz c6 gia tri 16n nMt la M thoa man M? 2 . x2 _xy+ y2 II. Phin rieng (3.0 diem) : Thi sinh chi ilu(1c lam m(jf trong hai phdn (phdn A holjc phdn B). A. Theo chvO'Dg trinh Chnin: . Can VI.a (2.0 diem): 1) Trong m~t pMng vai h~ t9a dQ Oxy, cho tam giac d~u ABC nQi ti8p duang tron (C): x 2 + y2 - 2x + 4Y - 4 = 0 va duang tMng AB ~o vai dubng tha.ng d: x - y +1= 0 g6c 45° . Viet phuong trinh duang tha.ng AB. 2) Trong khong gian v6i h~ t9a dQ Oxyz, cho lang nv tam giac d~u ABCAIB1CI cO dinh AlJ3;l;l); hai dinh B, C thuQc nvc Oz va AAI =1 . Tim t01il dQ cac dinh A, B, C. . Can VII.a (1.0 diem): Cho s6 nguyen duong n thoa man 2 n - 4 (C;-2 -C!_2 -n) = C:~12. Tim h~ s6 Clla s6 h1ilDg chUa x~ trong khai tri€n nhj thuc Newton cU. ( <Jx' + ~ rv(ri x> O. B. Theo chlfO'Dg trinh Nang cao: Can VI.b(2.0 iliim): 1) Trong m~t ph~ng vai h~ t9a dQ Oxy, cho hai di~m A(l ;2) va BO ~-2). Tim t01il dQ di8m C tren duang tha.ng d 1 : x - y -1 a sao cho duang tron ngo1ili ti8p tam giac ABC ti~p xuc vai dubng tha.ng d 2 : x + y - 3 = a . 2) Trong khong gian vai h~ t9a dQ Oxyz, vi8t phuong trinh m~t d.u (S) c6 tam thuQe duang tha.ng d : ~ = ~1 :;;;; z; 1 , (S) di qua giao di8m cua d vai m~t pha.ng Oxy va cat m~t pha.ng Oxy theo giao tuy8n la duang tron e6 ban kinh bkg J5 . • {27(X+2 Y )=4.3 X - Y Can VII.b(1.0 diem): Giro h~ phuong trinh . 31og 2 (1 + x.2- Y ) = 2x-5y IIItT Thi sinh kbong dU'q'c sl1 dy.ng tai Iitu. Can bQ coi thi khong giai thich gi them. B9 va. ten thi sinh: .•• • ••••••••• •••••••.•• •.•• ••• Sa bao danb: . www.MATHVN.com www.mathvn.com I Tru'01lg Chuy@n U Quy Bon BR VT. BAp AN VA HlfONG DAN CHAM nt THI THU B~I HQC - CAO BANG LAN 1 (2010 - 2011). Mon: Toan Kh6i: A & B y Cau N{U dung Thang Ghi eho di~m 3 1. Khao sat va ve etA thi: y =x - 3x 2 * T~p xae dink D = lR 0.25 Gi&i h~: lim y = +00; lim y = -00 . x-++co x-+~ * D~o ham: y' = 3x 2 - 6x 0.25 y' = 0 ~ x = 0; x = 2 * Bimg bi~n thien: x -00 0 2 +00 0.25 y' + 0 - 0 + 0.25 * Db thi: y" = 6x -6, di~m u6n 1(1;-2). Db thi qua 0, A(3 ;0), B( -1 ;-4) Db thi nh~ I lam tam d6i xUn . 2. Tim tham s6 m: x 2 1x-31 = m (1) 0.25 * S6~ghi~m trong khoang (1; +(0) eua (1) hI s6 giao di~m e6 hoanh dl) > 1 eua duemg th~ng d: y =m va (C*) : y = x 2 1x - 31. 0.25 * Ve db thi (C*): y x 2 1x - 31 (suy tir (C)). va db thi d: y = m tren eimg M trve. *K8t qua: 0 < m :::; 2 ho"e m 4. 0.5 Cach khac: C6 th~ l~p bang bi8n thien eua ham s6 y = x 2 1x - 31 tren (1; +(0) . www.MATHVN.com www.mathvn.com II III 1. Giiii pbU'O'llg trinb: cot 2x - 2 tan 3x = sin 2x(1 + tan x tan 2x) (1) * (1) ¢::> cot 2x 2 tan 3x = tan 2x 7r 7r cos7x X =-+k-'kEZ *¢::> =o¢::> 14 7' sin4xcos3x { . 4 3 0 (2) sm xcos X;f; * :E>i~u ki~n (2)<=:> X;f; m 7r ;X;f; 7r + m 7r ,(m E Z) 4 6 3 7r 7r 7r 7r 7r * -+k-;f;-+m-'m-¢::>k;f;7/+3·/EZ. 14 7 6 3' 4 • V~y nghi~m eua (1): x = ~ + k; ;k E Z va k;f; 71 + 3;1 E Z. *Cach khac: (1) <=:> cot2x-tan2x = 2tan3x <=:> cot4x =cot(7r -3x) 2 7r jx = ~ + k 7r { 7r 7r 4x= 3x+k7r 14 7 X= +k- <=:> 2 ¢::> <=:> 14 7 { 4X;f;m7r x;f;m!!'- k;f;7/+3 4 2 2 2. Giiii bit pbU'O'llg tnnb: 2(1 J x - 2x) :s; x +.J x - 3x +2 (1) * T~p X3c dinh eua bAt phuong trinh: D = (-00; 0] u [2; +00 ) . (1) <=:> 2.J x 2 - 2x + x - 2 + .J x 2 - 3x + 2 ;;::: 0 : thoa v6'i mQi x;;::: 2 . * V6'i x:S; 0 : (1) <=:> (J2-x r -2h.J2-x:S; Jl-x'\/2-x ¢::> J2-x:s; 2h +.Jl-x. * ¢::> 4x + 1 :s; 4.Jx 2 - x (2): thoa v6'i mQi x:S; _.! 4 " 1 0 (2) 1 * Vm < x:s;: <=:> x:S; 4 24 1 V~y t~p nghi~m eua (1) 1a S =(-00; - 2 4] u [2; +00). Tinb tich philn: 1= I ln (X + ~) dx 1 X ~ 1 1 1 * u=ln(x+vl+x2)~ u'= Jl ;v':::::-~V= 2 3 . x 2x 1 2 1.J3 1.J3 dx * I= 2 ln (x+.Jl+x ) +- J ~. 2 1+ x 2 2x 1 2 I x 1r 7r 7r.J3 dx 3" cost 1 ~ * d~t x =tant, < t < - J ~ = J-'-2- dt = 2 2 1 x 2 1+x 2 !!. sm t sintl~ 4 1 r;; 1 ~ J2 J3 * I=-ln(l+ ,2)- 1n(2+,,3)+ 2 6 2 3 1 .J3 - 1 1 *Clich khac: _dx== _ J x dx - J tdt = Jt2;l11 1 -1~X2-1 r x 2 .J1 -2+ 1 -fi"j T3 x 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 I www.MATHVN.com www.mathvn.com 0.25 0.25 IV Tinh the tich khoi chop SoAReD va S mit cAu * GOi H la trung diam Clla AB, chUng minhSH J (ABCD). GOix =AB 2SH thi VS,ABCD = x:: . 2 2 * SD 2 = SH 2 + HC 2 +CD 2 nen x =~ 2 a 3 J6 B V~yVS,ABCD 48 * GQi G la tarn dtron (ACD), d\ffig dUOng thAng d (qua G, song song SH) la 1:n,lc cua dUOng tron (ACD). D\ffig N sao cho SHAN la hinh binh hanh, chUng minh dugc (CRN) la m~t trung tnrc do~ SA .D\ffig 0 E d sao cho ON II AG, vi AG II HC nen 0 E (CHN). V~y 0 la tarn m~t cAu ngolJ,i ti~p SACD. v Tim gia tri z: P =(x+ yz)(x- y)+ xyz x2 _xy+ y2 2 x x +(2z l) z x 1 * Chia tir va m~u cho ita co: P = : c;: = , d~t t =Y ~ t E ["2 ;2] 2 P = f(t) =t +(2z -1)t - z . t 2 -t+ 1 * j{t) h\ ham lien tvc tren [1/2 ;2J nen cO Maxf(t) = M . I le[2;2j , J, (2 +(2z-1)t-z , 'A V~y : M ~ 2 <=> bat phuang trinh (an t) : 2 ~ 2 co nghlym t -t+1 tren [1/2 ;2J. 2 A A t -t+2 , h' (1/2 2J * <=> z ~ co ng Hil m tren ;. 2t-1 t 2 -t+2 1 Lap BBT cua get) = E (- ;2]. . 2t-l 2 *T'hd II' () J7 S .,. A, • I' >J7 m ugc .J.Y.lmg t = - . uy ra gla tr! can tIm cua z a: z _ . ~!~ 2 2 2 PHANT CHON VLa A.Theo chU'O'llg trinh chuan: 1. Binh hoc toa dO phAng: * (C) cO tarn(1 ;-2), ban kinh R =3. AB tlJ,o vOi d goc 45° nen tim dugc VTPT la n= (1; 0) ho~c n (0;1). V~y phuang trinh AB co d~ng: x + C =0 ho~c Y + D =0 . * Tarn giac ABC d~u nQi ti~p (C) nen khoang cach d(I,AB) Rsin300 = l 2 * C = 1/2 ho~c -5/2 . Phuang trinh AB: 2x + 1 =0; 2x - 5 = O. * D = 7/2 ho~c 1/2. Phuang trinh AB: 2y + 7 = 0; 2y + 1 O. 0.25 0.25 0.25 0.25 0 25 . 025 . 0.25 0.25 0.25 0.25 www.MATHVN.com www.mathvn.com I 2. mnh hoc toa d6 kh6ng gian: * ~(.J3;l;l)co hinh chi~u tren Oz lit 1(0;0;1), ciing lit trung diem BC. All =2, AAI = 1 nen AI .J3. V~y IB =IC = 1. *Suy ra B(O;O;O), C(0;0;2) ho~c C(O;O;O), B(0;0;2). *Ptrinh m~t phing (ABC) co d{Ulg: ax + by + cz + d O. (a l + b 2 + c 2 > 0) (ABC) chua Oz vit d( AdABC)) 1 cho c =d =0, a = .J3, b = -1 ho~c c =d = 0, a 0, b = 1. V~y pt (ABC): .J3x- y =0 ho~c y = O. *A lit hinh chiSu cua Al tren (ABC) nen A(.J3 ;~;1) ho~c A( .J3;O;l) . 2 2 VILa T6 h9l!: 2 C l C n *2 n - 4 (c n - - ) - - - 2 (1) D' A k'~ l~n: n E tl.J '71+ > 3(*) n - n-2 n n-l • leu ,n_ V6i di~u ki~n (*), (1) ~ 2 n - s (n-4) == 1. *n = 3, n 4: VT < 0; n> 5: VT > 1; n 5 lit nghi~m duy nhAt. 2)15 15 5 k 15 * V +- = LCI~2IS-kX3 - . ( X k=O 5 * S6 h~g chua X SI3 khi k = 10. Yay s6 h~g cAn tim lit: CII~25 x 3 VLb B.Theo chtrO'llg tnnb nang cao: 1. mnh hoc toa d9 pbing: * Ta co A thuQc d 2 nen hili toan ttrong duong vcri vi~c vi~t phuong trlnh dUOng tron (T) di qua B vit ti~p xiic d 2 4J.i A, sau do tim C lit giao di~m cua (T) vit d I. *Tam I cua (T) lit giao diem cua trung tnrc do{Ul AB (y = 0) vit dth d qua A, vuong goc d 2 (d: x - y + 1 =O);Vay 1(-1;0). *Ban kinh cua (T): R IA = .J8 , v~y phuong trinh (T): (x + Il + i = 8. * To~ dQ C : ( J3; J3 -1);(.J3;.J3 -1). 2. mnh hoc toa d6 kh6ng gian: * d n Oxy = A(2;-1;0). GQi J lit hinh chi~u cua tam m~t cAu 1 tren Oxy thi AJ lit ban kinh giao tuy~n = -J5 vit iiJ :::: (d,'iiXy) . * sine d, 0;;) == lcos(d; k)1 = ~ v~y ban kinh m~t cAu lit R = A~ = J6 . v6 cosliU * fA = J6 & lEd nen 1(0;0;-1) ho~c 1(4;-2; 1). *Yay phuong trinh m~t cAu lit: x2+y2+(z+1)2 =6 ho~c (X_4)2 +(y+2i+(z-li =6. VILb {27(X + 2 Y ) = 4.3 x - y (1) Giii he phU'O'llg tnnb: 31og 2 (1 +x.2- Y ) = 2x - 5y (2) , ( 4 )x;Y 4 *The vito (1): - =-~ x- y=3 (4) 27 27 *(3) ~ 2 Y +Y = 1 ~ y = 0 (pp so sinh ho~c dUng hitm ) * V~y nghi~m cua h~: x = 3, y = o. 0.25 0.25 - Co tM tim A trui)-c. Tinh a1P;1c AB = AC 0.25 = 2. r6i suy ra B. C 0.25 0.25 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 www.MATHVN.com www.mathvn.com . 2 C l C n *2 n - 4 (c n - - ) - - - 2 (1) D' A k'~ l~n: n E tl.J '71+ > 3(*) n - n-2 n n-l • leu ,n_ V6i di~u ki~n (*), (1) ~ 2 n - s (n-4) == 1. *n =. * V6'i x:S; 0 : (1) <=:> (J2-x r -2 h.J2-x:S; Jl-x'/2-x ¢::> J2-x:s; 2h +.Jl-x. * ¢::> 4x + 1 :s; 4.Jx 2 - x (2): thoa v6'i mQi x:S; _.! 4 ". sintl~ 4 1 r;; 1 ~ J2 J3 * I=-ln(l+ ,2 )- 1n(2+,,3)+ 2 6 2 3 1 .J3 - 1 1 *Clich khac: _dx== _ J x dx - J tdt = Jt2;l11 1 -1 ~X 2-1 r x 2 .J1 -2 + 1 -fi"j T3 x 0.25 0.25 0.25