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510.76 B450D JYEN VAN NHO - LE BAY - NGUYEN VAN THO MOIi BO DE THI ^Tii LUAN TOAN HOC Danh cho thi sinh Icfp 12 on tap va thi Dai hoc, Cao dang Bien soan theo ngi dung va cau true de thi cua Bg Glao due - Dao tao 0 Ha N O I DVL.009154 NHA XUAT BAN DAI HQC QUOC GIA HA NOI N G U Y E N V A N N H O - LE B A Y - N G U Y I N V A N T H O BO Df THI T U LUA TO^n HOC ^ Danh c h o thi sinh I6p 12 on tgp v a thi Dqi h o c - C a o d a n g ^ Bien soqn theo noi dung v a c d u true d4 thi c u a Bp GD&DT NHA XUAT BAN DAI HOC QUOC GIA HA NOI Jltue lue Lifinoiddii ^ PHAN 1: T H I T U Y E N SINH D ^ I HQC - C A O DANG 3 6 C. De thi minh hoa 4 B. Mot so dieu can luTu y 3 A. Cau true de thi de thi tuyen sinh Dai hoc, Cao dang 2009 r 8 Dap an - thang diem 6 De thi tuyen sinh Dai hoc, Cao ding, khoi A 15 Dap an - thang diem 14 De thi tuyen sinh Dai hoc, Cao d^ng, khoi B, D 273 E. Mirdi de thi i\i luyen tap 21 D. Ba mi/di de thi CO IcJi giai : PHAN 2: T H I T O T N G H I E P T R U N G H Q C PHO THONG 289 291 B. De thi minh hoa 289 A. Cau true de thi Tot nghiep THPT 2009 314 D. De thi tham khao 297 C. De thi mau TNTHPT 292 Dap an - thang diem 318 De thi tot nghiep THPT phan ban 2008, Ian 2 316 De thi tot nghiep THPT phan ban 2008, Ian 1 315 De thi tot nghiep THPT phan ban 2007 314 De thi tot nghiep THPT phan ban 2006 Pfi^nl: THI TUY^py SINH a/^l HQQ, SAG BANG K. CMi TRUC DE THI TUYEN SINH DAI HOC, CAO DANG NAM 2 0 0 9 I. PHAN CHUNG CHO TAT CA THf SINH (7,0 di^m) Cdu I A^^i dung kien thvlc Diem — Khdo sat, ve do thi ciia ham so. — Cdc bdi loan lien quart den tint; dun^ ciia dao ham va do thi ciia ham so: Chieu bie'n thien cua ham so. Cifc tri. Gia tri Idn nha't va nho nhat cua ham so'. Tie'p tuyen, tiem can 2,0 (d\Jng va ngang) cua do thi ham so'. Tim tren do thi nhi^ng diem CO tinh chat cho trtfdc; ti/dng giao giCfa hai do thi (mot trong hai do thi la du"5ng thing);... III II IV V PhU(fn^ trinh, bat phu ke. - Bat ddn}> thvtc. Cuc tri ciia bleu thi'fc dgi 2. Theo chUOitg trinh Nang Cdu 1,0 so. cao: NQi dung kien thi'tc , Diem * PhUtfng phdp toa do trong mat phdni* vd trong khong gian: - Xac djnh toa dp c u a d i e m , vecttf. - Dxiiing trdn, ha diTdng c o n i c , mat c a u . Vl.b 2,0 - V i e t phiTPng trinh mat p h i n g , dtfcJng t h i n g . - T i n h g o c ; tinh k h o a n g c d c h lit d i e m d e n difdng thang, mat p h i n g ; k h o a n g e a c h - giffa hai di/dng l h a n g . V i tri tiTdng d o i ciia dUtJng thang, m a t phang va mat c a u . - Sophi'/c. , •, ^ , . , , , / , - Do thi ham phan thUc huu ti dang: + bx + c v~ va mot ii'x + b' Vll.b so yen to lien quan. - Su tiep xi'tc ciia hai dudng 1.0 cong. ke. - To h(/p, xdc sud't, thong Idgarit. - He phU(fng trinh mii vd - Bat dang thUc. Cuc tri ciia hieu thi'fc dai B. MOT S 6 D I E U C A N so. LUU Y: DiTa vao cau true cua de thi va npi dung giijra hai bp sach theo chi/Png trinh Chuan va chi/c^ng trinh Nang cao , chung ta can lifii y mot so van de nhU"sau : I. P H A N C H U N G CHO T A T CA THf SINH (7,0diem) Cdu I: Trong phan nay chiing ta chi khao sat va ve do thj , cung nhif chi xet cac bai toan l i e n quan d e n do thj cua ba loai ham so'. 4 , • Hc^m bac 3 : y-cuc'+ hx^ + cjc + J , (« ^ O) . • Ham bac 4 (dang trung phiTcfng): y - ax^ + hx^ + c , (a O) . • Ham phan thuTc dang : y = ifilA ^ ^ o, at/ - be ^ O) cx^-d * Khi khao sat tinh chat cua ham so , tinh loi, 16m va viec tim diem uo'n ciaa do thi CO the bo qua khong can xet (neu can thi chi can tim diem uon cua ham bac 3 de suy ra tarn doi xiJng cua do thi , con ham bac 4 thi nen bo qua hoan toan phan nay). * Cac bai loan ve sir tiep xuc cua hai diTfJng cong cung se khong diTdc de cap tdi trong phan chung nay . * Cac bai toan ve tiem can cung chi de cap den liem can dtfng va tiem can ngang Cdu III: * Viec uTng dung tich phan de tinh the tich cua khoi tron xoay chi c6 cac khoi khi cho hinh phing quay quanh true Ox. II. PHAN RIENG (3,0diem) ... u 1. Theo chittfitg trhih Chiidn: Cdu Via: * Trong phan nay doi vdi phiTdng phap tpa do trong mat phing ta chi can on lai cac bai toan c6 lien quan den di/dng thang , du'dng tron va elip . cac bai toan ve hypebol va parabol khong du'dc de cap tdi trong chi/dng trinh chuan , cac bai toan ve tiep tuyen cua elip'cung du'dc bo qua . * Doi vdi phi/dng phap toa dp trong khong gian phan khoang each tif diem den dirdng thang va khoang each giffa hai diTdng thang cheo nhau cung diTdc bo qua. Cdu Vila: * Phan so phffc ehi c6 ctic bai toan c6 lien quan den cac phep toan ve so phffc va viec giai cac phUdng trinh bac hai co he so Ihffc , khong de cap den can bac hai cua so phuTc , cung nhu" viec giai phffdng trinh eo he so phffc va cac bai toan CO lien quan den dang lifdng giae eiia so phffc . 2. Theo chUcfng trinh Ndng cao: Cdu VIb: Cac bai toan c6 lien quan den tie'p tuyen cua cac du'dng conic cung khong dc cap tdi trong cau true de thi mdi nay . Nhff vay doi vdi cac dffdng conic chi can on lai cac dang toan ve viet phifdng trinh ehinh tac , tim cac diem nlim trcn conic thda tinh cha't nao do va cac bai toan ve mot so tinh chat dac trffng ci.i tifng du'dng conic ^ 5 C. DE THI MINH HOA DUcfi day la hai de thi minh hoa va dap an - thang dientchi tiet cua Bg Gido due va Dao tao, cdc ban cdn xem ki de Met dUctc nhUng yeu cdu cdn dat ditdc khi lam bdi. TiT do rtit ra dUcfc each trinh bay Wi gidi mgt de thi cho ngdn gon nhitng ddy du vd chinh xdc. THI T U V ^ N SINH 961 HQC, CflO OANG - K H 6 | fl (Thcfi gian Idm bdi: 180 phut) I. PHAN CHUNG CHO TAT CA THf SINH (7,0 diem) Cau I (2,0 die'm) Cho ham so y = -x^ + + 4 , trong do m la^tham so thiTc. 1. Khao sat sif bien thien va ve do thi cua ham so da cho, vdi m = 0. ,2. Tim tat ca cac gia tri cua tham so m de ham so da cho dong bien tren khoang (0;+«)). Cfiu II (2,0 di^'m) 1. Giai phU'dng trinh: V3(2cos^ A-+ cosj:-2J + (3-2cosx)sin-;r = 0 2. Giai phiTdng trinh: logj {x + 2)-3 + log^{x-5)^ + log, 8 = 0 2 CfiuIII(l,0di6'm) Tinh dien tich hinh phang gidi han bdi do thi ham so y = yje^ + 1 , true hoanh hai duTcJng thing x = In3 , x = In 8. CauIV(l,Odie'm) Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a, SA^SB = a, mat phang (SAB) vuong goc vdi mat phing (ABCD). Tinh ban kinh mat cau ngoai tie'p hinh chop S.ABCD. CSu V (1,0 di^m) ' Xet cac so'thufc difdng x,y,z thoa man dieu kien: x + y + z - l . Tim gia tri nho nhaft cua bieu thiJc: yz zx xy II. PHAN RIENG (3,0 diem) Thi sink chi diMc lam mot trong hai phdn (phdn 1 hoqc phdn 2) 1. Theo chUcfng trinh Chudn: au VLa (2,0 diem) -^ i > 1. Trong mat phang toa do Oxy, cho di/dng tron ( C ) : + - bx + 5 = 0. . Tim diem M thuoc true tung sao cho qua M ke difdc hai tiep tuyen cua (C) ma goc giSa hai tiep tuyen do bang 60". 2. Trong khong gian vdi he toa do Oxyz, cho diem M ( 2 ; 1 ; 0) va ';c = l + 2r {d):ly = -l + t . ,\ ^ . diTdng thang: " • i i Viet phufcJng trinh tham so' cua dtfdng thang di qua diem M , cat va vuong goc vdi dufdng thang fc?). Cau VILa (1,0 diem) Tim he so cua j trong khai trien thanh da thtfc ciia bieu thuTc: P = lx^ +x-\f 2. Theo chitctng trinh ndngcao: ,. CSu VI.b (2,0 diem) ' • 1. Trong mat phang toa do Oxy, cho di^dng tron [C): x^ + y^ - 6x + 5 = 0. Tim diem M thuoc true tung sao cho qua M ke di/dc hai tiep tuyen cua (C) ma goc gii?a hai tiep tdyen do bang 60". 2. Trong khong gian vdi he toa dp Oxyz, cho d i l m M ( 2 ; 1; O) va di/dng thang ^ ' 2 1 - I Viet phufdng trinh chinh tic cua dtfcfng thang di qua diem M , cat va vuong goc vcti dU'dng th^ng (d). Cfiu VII.b (1,0 A\im) Tim he so cua trong khai trien thanh da thuTc ciia bieu thtfc: P = X^ + A- \ 7 Cdu I (2,0 diem) \. 0,25 OAP A N - T H R N G DI^M Dap an \Diem diem) V d i m = 0 , ta c6 ham so' y = -.v^ - 3JC" + 4 T a p xac djnh: D = x. Sif bien thien: • C h i e u bien thien: y ' = - 3 . V " - 6 . V . Ta c6: y' = 0 « 'x = -2 x =0 r.v<-2 <=> v'<0<=> • [.r>0 0,50 <=> y ' > 0 < : = > - 2 < A < ( ) . D o do: + H a m so nghjch bien tren m o i khoang ^-co; -2) va ((); +cc) + H a m so dong bien tren khoang ( - 2 ; ( ) ) . CiTc t r i : H a m so dat ciTc tieu tai .v = 2 va ycT = >'(~2) = 0 ; dat cifc dai tai A" = 0 va J C D = y(^0 = ^ • Gic'Jihan: iim y = +co ; 0,25 lim y =- o o . B a n g bien thien: 0 y' -2 X y 0 + 00 0 + 0,25 \ Xo " . X- CO Dolhi: + D o thi c^t true tung tai d i e m /\{0 ; 4 ) , citt true hoiinh tai diem 0,25 ; ()) va tiep xuc v d i true hoanh tai C ( - 2 ; 0 ) . 2. (0,75diem) Ham so da cho nghich bien tren khoang (0 ; +oo) o v' = -?>x^ « 3A-^ - 6A: + m < 0 , VJC> 0 +6x>m , VJC>0 (*) 0,25 Ta CO bang bien thien cua ham so v = 3.v"+6.r tren (0;+a)): s I 0 Tir do, ta di/ffc: (*) o w < 0. II {2,0 diem) 1.(1,0 diem) Phi/rtng trinh da cho tifcfng difdng vdi phi/dng trinh: (2 sin jr - \/3) (V3 sin A-+ cos A-j = 0 <=> sm X = \/3 sin jr + cos A: = 0 x-[-\Y^ + n7t, « e Z . A- = - — + ^;r ,k&'L. 6 2. (1,0 diem) Dieu kien: x > - 2 A-^5 0,50 0,50 (i V(3i dieu kien do, ta c6: Phifcfng trinh da cho tUdng dUtrng vcti phu'cJng trinh: log2[(A- + 2 ) | A - 5 | ] - l o g 2 8 C>(A- f 2)|.v-5|-8 ' A - - 3 A - - 1 8 ) ( . r - 3 v - 2 ) = () 0,50 (1,0 Ill diem) L^-3A:-18 = 0 [A:^-3X + 2 = 0 31^17 <:=>j: = - 3 ; j t = 6 ; j : = — 2 0,50 D o ' i c h i e ' u v d i d i e u k i e n (*), ta dufdc t a t ca c a c n g h i e m c u a phtfctng t r i n h d a c h o l a : x = 6 v a x = ^ • K i h i e u 5 la d i e n tich can tinh. V i V e ' + l > 0 V ; c G [ l n 3 ; l n 8 ] n e n S = J yje' In B&t yje'' +I=t,tac6 +l.d.x 0,25 3 dx^^^ / ' - I 0,25 K h i x = l n 3 t h i / = 2 , k h i .v = l n 8 t h i t = 3 i t - i i t - i j U 0,50 2 r - l 2 ' = 2 +ln/-l 2 TV (1,0 diem) D o SA = S B = +l ' - l n / + l ^=2+l n - . 2 2 a) n e n la tarn g i a c d e u . G o i G va. I tiTcfng tfng l a t a r n c i i a tarn g i a c d e u SAB v a t a m cua hinh v u o n g ABCD. 0,50 G o i O la tarn c u a m a t c a u ngoai tiep hinh chop ta CO OGl{SAB) va S.ABCD, OIl{ABCD). 1 Suy ra: M=p.V-\t°— t r o n g d o H l a t r u n g d i e m c u a AB. + T a r n g i a c OCA v u o n g t a i G. /„ '\''"' b • / 0,25 c 10 K i hieu R la ban kinh cua mat cau ngoai tiep hinh chop S.ABCD, ta c6: 0,25 R = OA = yJOG V (1,0 diem) + GA = . — + V 4 T a c o : P = — + — + ~ + —+ y z z X Nhan tha'y: x^ -xy>xy = 9 — + —. X y , . 6 (*) Vx,y&R 0,50 Do do: J:'^ + j " ^ > jry(jc + y ) , Vjc,y > 0 x^ v2 Hay — + — >x+v,\/x,v>0 y X • ; v' z^ TiTdng ti/, ta c6: — + — > V + z , V y , ^ > 0 z y ' —+—>z + . . '• ••• ' xyx,z>o z X Cong theo tufng ve'ba bat d i n g thuTc viifa nhan dUdc d tren, 0,50 ket hdp v d i (*), ta diTdc: P > 2 ( j c + y + z) = 2 , \/x,y,z>0 Hdn nffa, ta lai CO F = 2 khi va x + y + Z = 1 = V= e =" 3 V i vay, m i n P = 2 . VI.a I. (1,0 diem) (2,0 diem) V i e t l a i phuTdng trinh cua (C) dU'di dang: (A - 3)^ + y^ = 4 0,25 Tir do, (C) CO tarn / (3 ; 0) va ban kinh /? - 2 . Suy ra triic tung khong c6 diem chung v d i difdng tron (C). Vi vay qua mot d i e m bat k i tren true tung luon ke dtfdc hai 0,25 tiep tuye'n den (C). X e t d i e m M[0;m) tuy y thuoc true tung. Qua M ke cac tiep tuyen MA va MB cua (C) (A,B la cac tiep diem). Ta c6: Goc giffa hai du'dng thang MA va MB bang 60" o /\A/B = 6 0 " [AMB = 120" (1) ^ (2) 0,25 . 11 L V i yV//la phan giac cua AMB ncn: AMI = 30" A// = sin „ c^MI 30" ^ = 2R o Vw" + 9 = 4<:=>w = ±\/7 (2) ^ ^ A.W - 60" <=> M / - Dc thay, khong c6 ^'^ „ o A// sin 60" 3 /? 0,25 thoa (*). Vay, C O tal ca hai diem can tim la: (o ; va (o ; . 2. (7,0 tfi^'w) Goi H la hinh chicu vuong goc cua M Ircn J, ta c6 Af// la 0,25 dU'cJng ihtlng di qua M, cat vii vuong goc vc'li d. Vi H ed ncn toa do cua Hc6 diing: (l + 2r; - 1 + / ; - / ) . Suyra =(2r-l;-2 +f VI MH 1 d \ (/ C O mot vecl« chi phi/dng la M" = (2 ; 1 ; - 1 ) , r ncn 2 . ( 2 r ~ l ) + l . ( - 2 + 0,50 /) + (-!).(-/) = 0. Tilf do, la dir«c / - - . V i the, MH =1 - :-~ 3 l3 ;-~ 3 3J Suy ra phu'ctng trinh tham so cua du'dng thang MH la: 1 - 4/ V = 2+/ A- = 0,25 :.^-2t VILa (1,0 diem) Theo cong thuTc nhj thiJc Niu-ldn, ta c6: P.c;:(.v-l)%ctr(.v-l)%... + ctr'(.v-ir.... 0,25 .Ctv'"(.v-1).QV^ Suy ra. khi khai trien P thanh da thu'c, A " chi xua't hien khi khaitrien C^'(A-lf va 0,25 Q'.v-(A^l)^ He so'cua .v^ trong khai trien cua Cll[x-l)^' la: C".C^. 0,25 He so ciia .v"^ trong khai trien cua C^.K^ {x ~ l)^ la: - Q ' . C " . 12 V i vay, he so cua Vl.b trong khai t r i c n P thanh da thi?c la: 0,25 1. (1,0 diem). X c m phan 1. Cau V l . a . {1,0 diem) 2. {1,0 diem) G o i H la hinh vuong goc ciia M i r c n J, la c6 A/// la difrJng thang di qua M . cSl va vuong gc)c v d i d. 0,25 jc = 1 + 2/ D CO phUdng Irinh tham so la: Vi H y = - l+/ ncn tpa dp ciJa H c6 dang: (1 + 2/ ; - 1 + ? ; - / ) . Suy ra MH = {2tV I MH Ld \ ^t \ 0,50 \a d c6 mot veclO chi phu'dng la u = (2 ;1 ; - l ) , ncn 2 . ( 2 / - l ) + l . ( - 2 + /) + ( - l ) . ( - r ) = ( ) . 2 . • (\ Tii do, la diWc / = - . V i the, MH = V3 4 2^ 3 3 Suy ra, phu'dng,lrinh chinh lac ciia duTJng ihflng MH la: .y - 2 _ >• - 1 _ 0,25 1 Vll.b {1,0 diem) Theo cong Ihifc nhj ihiYc Niu-Utn, la co: P =-- (.V - 1 ) % Ql.v^ ( . - 1)-% - . . - ^ ^ (.V - if'' + ••. 0,25 + QU'"(X-I) + QV^ Suy ra, k h i khai trien P lhanh da ihifc. . v ' c h i xua't hien k h i khai tricn C " ( A - 1 ) " va Q ' X " (A - l ) " \ 0,25 He so' ciia .v' trong khai i r i c n ciia C^,' (A - l / ' la: ~ C " . Q ^ . He so'ciia A' trong khai t r i c n ciia Q'.A" (A - 1)"^ la: + Q ' . c ] . V i vay, he so ciia A ' trong khai t r i c n P lhanh da thuTc la: -C^QSQ^.C^^ + IO. 0,25 0,25 13 Bi THI TUVdN SINH OR! HQC . CflO D^N^ - KH6| B.D (Th&i gian lam bai: 180 phiit) I. PHAN CHUNG CHO TAT cA THf SINH (7 di^m) Cau I ( 2,0 di^m ) Cho ham so' y = ^"^^^ . x-2 1. Khao sat sir bien thien va ve do thi (C) cua ham so da cho . 2. Tim tat ca cac gia tri cua tham so m de dtfdng thing y = 2x + m dt (C) tai hai diem phan biet ma hai tiep tuyen cua (C) tai hai diem do song song vdi nhau Cfiu II (2,0 die'm) 1. Giai phufdng trinh : (l + 2cos3jr)sinx + sin2jc = 2'sin^ 2x + — 4; 2. Giai phiTdng tnnh : log, - 2| + logj |JC + 5| + log, 8 = 0 '^ 2 J- • •. Cfiu III (1,0 die'm ) Tinh dien tich hinh phSng gidi han bdi do thi ham so y = x.\n^(x^ + \] j X +1 , true tung, true hoanh va d i T d n g thing x = yle - \ Cfiu IV (1,0 di^m ) Cholangtru /\BC./4'fi'C'c6 day/\5Cla tam gidc deu canh a , AA'= 2a va dirdng thing AA' tao vdi mat phing (ABC) mot g6c bang 60".Tinh the tich khoi tu" dien y4C/i'B'theo a . CSu V (1,0 di^m ) * Tim ta't ca cac gia tri cua tham so a de bat p h i T d n g trinh j ; . ; x^ + 'ix^ -\ ' - z + 5 = 0 1. Tinh khoang each giffa diTdng thang d \k mat phang (P). 2. K i hieu / la h i n h chieu vuong goc cua d tren (P). | • ? J V i e t phi/dng trinh chinh t^c cua di/dng t h i n g / . ' }• I | Cau V I I . b ( 1 , 0 d i e m ) -| Cho so p h u t z = l + V 3 / . H a y v i e t dang lifOng giac cua so phtfc 2^. DAP Cdu I AN-THANG Bap 1.(1,25 f DI^M an Diem diem) [2,0diem) Tapxacdinh : D = R \ { 2 } . ' Sir b i e n t h i e n : n • C h i e u bien thien : y ' = < 0 ,\/xeD . (x-lf Suy ra, ham so nghich bien tren m o i khoang 0,50 {-°o;2) va (2 ; + 0 0 ) • Ctfc trK H a m so khong c6 ctfc tri 15 • Gidi han : lim y= lim y = 2]; Urn >' = +co vh. lim >' = - co. Suy ra do thj hkm so c6 mpt ti$m can dtfng 0,25 la dirdng thing ;c = 2 , mpt ti$m can ngang Ik dtfdng thing >' = 2 . B5ng bien thicn : X -00 2 - y" - 0,25 2 y -00 2 • Do t h i : + Do thi c i t true tung tai diem , cat true hoanh tai 0,25 diem B tr + Do thj nhan diem / ( 2 ; 2 ) (Ik giao Ciia hai difcJng ti^m can) Ikm tarn do'i xtfng . 2. (0,75 diem) Di/dng thing y -2x + m c i t CC) tai hai diem ph§n biet mk tiep tuyen tai 66 song song vdi nhau 2x + 3 <=> — -2x + m c6 hai nghipm phan biet , jc^ thoa man 0,50 dieukien v ' ( ; C | ) = 3''(jC2) < : : > 2 J : ' + ( m - 6 ) j : - 2 m - 3 = 0 c6 hai nghiem phSn bi^t , khac 2 vk thoa man dieu kipn A:, + =4 A = ( m - 6 ) +8(2m + 3 ) > 0 2.2^+(m-6).2-2m-3^0 6-m — o m = -2 0,25 . - 4 II 1. (1,0 diem) (2,0 diem) PhiTdng trinh da cho ti/dng difdng vdi phiTdng trinh : sin X + sm4x - sin 2x + sin 2 = 1 - cos 4x + 2) <=> sin jc + sin 4.x: = 1 + sin 4 J: o sin x 1 c:> x = —TT + k2;r ,keZ 2 2. (1,0 diem) Dieu kien : x:^2 0,50 0,25 0,25 Vdi dieu kiSn 66, PhiTdng trinh da cho tiTdng diTdng vdi 0,50 phifdng trinh: log2 [\x + 2\\x- 5|] = log2 8 ' y £, r - x + 2 x-5 =8 A:^-3;C-18-0 x^-3x <=>X = 0,50 + 2-^0 -3;JC = 6;A: = ^^^Hl , th6a man di^u kiSn (*) III Ki hieu S la dien tich can tinh . (1,0 x.ln'^(x^+i\ 0; diem) Vi x'+l '->oyxe ^jc.ln^fjc^+l) nen 5 = | '-.dx xUl 0 Dat ln{^x^+l^ = t .taco dt = 2x.dx 771 Khi x = 0 thi r = 0,khi x = 4e^. thi r = l Vivay : S - - } / ^ A = -/^ 1 THtJ VIENTINKBINHTHUAN 0,25 0,50 0,25 17 IV (1,0 diem) K i hieu A va V tiTcfng lirng la chieu cao va the ^ch cua kho'i lang tru da cho , ta c6 : V—h.S AAIIC 1 3 V J Goi H la hinh chieu vuong goc cua A' tren mp(ABC), A'H^h ta c6 va A \ 4 ^ = 60" Suyra: h = A'A.sin60"=aj3 Do do, A V = h^^ifQ = av3. V = Dieu kien : x > 1 . V d i dieu kien do bat phi/dng trinh da cho ti/dng difcfng vdi bat phurdng trinh : diem) A;^+3JC^-1 Ta nhan thay , ham so: f{x) / ( 1 ) = 3 V A : > 1 . V i the ton tai x > Ithoa man (*) , hay baft phiTdng trinh da cho c6 nghiem klii va chi khi a > 3. VI.a 1. (1,0 diem) (2,0 diem) Ta CO : + A ( l ; 7 ; 3 ) e v a M = (2 ; 1 ; 4) la mpt vec M chi phifdng cua d . ' + n = (3 ; - 2 ; l ) la mot v6c td ph^p tuyen cua (P). 18 Ma M . n = 2 . 3 - 1 . 2 - 4 . 1 = 0 v a / l « ( P ) 0,25 (do 3 . 1 - 2 . 7 - 1 . 3 + 5 ^ 0 ) nen d//{P). Do do khoang each h giffa d va (P) chinh b^ng khoang each 0,25 tiif/ldenfPj. ^ V i v a y , h- 3.1-2.7-3 + 5 , — 79 + 4 + 1 9714 14 . 0,25 2. (1,0 diem) Ta CO dirdng thang d di qua d i e m v 4 ( l ; 7 ; 3 ) va c6 vee td chi phiTdng M = (2 ; 1 ; 4 ) G o i d' la dtftJng th^ng di qua A va vuong goc v d i (P). Do /i" = (3 ; - 2 ; 1) la mot vee td phap tuyen cua (P) nen n" 0,25 la mot vee td chi phi/dng cua d'. Suy ra , phi/cJng trinh cua d' Ik: x - l ^ y - 7 ^ z - 3 3 -2 -1 G o i /4' la giao dieV1 cua ^ ' v d i fPj , ta x-\ la nghiem ciia he : 3 CO A ' e / . Toa do A' z-2> - 2 - 1 3jc-2y-z + 5 = 0 -.-v 41 40 33 G i a i he tren ta difdc . x = — , v = — , z = — . 14 ' 7 14 Hdn niya , v i dll{P ) nen {d)ll{l). 0,50 V i vay ^ = (2 ; I ; 4 ) la mot vee td chi phu'd ng cua / . Suy ra phu'dng trinh tham so cua di/dng thang / 1 ^ : 41 ^ 14 40 v=— +/ • 7 33 ^ Z = — + 4/ 14 VILa (1,0 0,25 T a c o : Jc(3 + 5/) + y ( l - 2 / f = A:(3 + 5/) + > ' ( - 1 1 + 2/) ^ = ( 3 j t - l l y ) + (5;c + 2>')./ 0,50 diem) 19