L Anh Tun n thi i hcTR NG THPT TH NG LONG LM H

E ON THI ai hoc MON TOANGIAO VIEN :LE ANH TUANNAM HOC 2010 - 2011Trang 1n thi i hcL Anh Tun s 1I. PHN CHUNG CHO TT C TH SINH (7,0 im)Cu I (2 im)Cho hm s 3 23 2 y x x + (C)1) Kho st s bin thin v v th (C).2) Tm trn ng thng (d): y = 2 cc im m t c th k c ba tip tuyn n th (C).Cu II (2 im) 1) Gii phng trnh:22 3 1 3 2 2 5 3 16 x x x x x + + + + + + . 2) Gii phng trnh:32 2 cos 2 sin 2 cos 4sin 04 4x x x x _ _+ + + , ,.Cu III (1 im) Tnh tch phn:24 4 6 60(sin cos )(sin cos ) I x x x x dx + +.Cu IV (2 im) Cho hnh chp S.ABC, y ABC l tam gic vung ti B c AB = a, BC = a3, SA vung gc vi mt phng (ABC), SA = 2a. Gi M, N ln lt l hnh chiu vung gc ca im A trn cc cnh SB v SC. Tnh th tch ca A.BCNM.Cu V (1 im) Cho a, b, c, dl cc s dng. Chng minh rng: 4 4 4 4 4 4 4 4 4 4 4 41 1 1 1 1abcd a b c abcd b c d abcd c d a abcd d a b abcd+ + + + + + + + + + + + + + +II. PHN RING (3,0 im) A. Theo chng trnh chun.Cu VI.a (2 im) 1) Trong mt phng vi h to Oxy, gi A, B l cc giao im ca ng thng (d): 2x y 5 = 0v ng trn (C): 2 220 50 0 x y x + + . Hy vit phng trnh ng trn (C) i qua ba im A, B, C(1; 1). 2) Trong khng gian vi h trc ta Oxyz, cho im A(4; 5; 6). Vit phng trnh mt phng (P) qua A, ct cc trc ta ln lt ti I, J, K m A l trc tm ca tam gic IJK.Cu VII.a (1 im) Chng minh rng nu ( )na bi c di + +th 2 2 2 2( )na b c d + + .B. Theo chng trnh nng caoCu VI.b (2 im) 1) Trong mt phng vi h to Oxy, cho tam gic ABC c din tch bng 32, A(2; 3), B(3; 2), trng tmca ABC nm trn ng thng (d): 3x y 8 = 0. Vit phng trnh ng trn i qua 3 im A, B, C.2) Trong khng gian vi h trc ta Oxyz, cho bn im A(4;5;6); B(0;0;1); C(0;2;0); D(3;0;0). Chng minh cc ng thng AB v CD cho nhau. Vit phng trnh ng thng (D) vung gc vi mt phng Oxy v ct cc ng thng AB, CD.Cu VII.b (1 im) Gii h phng trnh: 2 24 4 424 4 4log ( ) log (2 ) 1 log ( 3 )log ( 1) log (4 2 2 4) log 1x y x x yxxy y y xy + + + _'+ + + ,Trang 2L Anh Tun n thi i hc s 2I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I. (2): Cho hm s 3 23 9 7 y x mx x + c th (Cm).1. Kho st s bin thin v v th hm s khi0 m .2. Tm m (Cm) ct trc Ox ti 3 im phn bit c honh lp thnh cp s cng. Cu II. (2):1.Gii phng trnh: 2 2 2 2sin 3 cos 4 sin 5 cos 6 x x x x 2.Gii bt phng trnh:12 2 102 1x xx +Cu III. (1) Tnh gii hn sau: 2 317 5lim1xx xAx+ Cu IV (1): Cho hnh chp S.ABCD c y ABCD l hnh ch nht; SA (ABCD); AB = SA = 1; 2 AD . Gi M, N ln lt l trung im ca AD v SC; I l giao im ca BM v AC. Tnh th tch khi t din ANIB.Cu V (1): Bit ( ; ) xy l nghim ca bt phng trnh:2 25 5 5 15 8 0 x y x y + + . Hy tm gi tr ln nht ca biu thc 3 F x y +.II. PHN T CHN (3)A. Theo chng trnh chun:Cu VI.a (2)1. Trong mt phng vi h to Oxy, cho elip (E): 2 2125 16x y+ . A, B l cc im trn (E) sao cho: 1 2F 8 A BF + , vi 1 2; F F l cc tiu im. Tnh 2 1AF BF +.2. Trong khng gian vi h to Oxyz, cho mt phng ( ) : 2 5 0 x y z v im (2;3; 1) A . Tm to im B i xng vi A qua mt phng ( ) .Cu VIIa. (1): Gii phng trnh: ( ) ( ) ( )2 3 31 1 14 4 43log 2 3 log 4 log 62x x x + + +B. Theo chng trnh nng cao:Cu VI.b (2)1. Trong mt phng vi h to Oxy, vit phng trnh ng trn i qua (2; 1) A v tip xc vi cc trc to .2. Trong khng gian vi h to Oxyz, cho ng thngd : 1 1 22 1 3x y z + v mt phng: P1 0 x y z . Vit phng trnh ng thng i qua (1;1; 2) A , song songvi mt phng ( ) P v vung gc vi ng thngd .Cu VII.b (1) Cho hm s: 2 2 3( 1) 4 mx m x m myx m+ + + ++ c th ( )mC.Tm m mt im cc tr ca ( )mCthuc gc phn t th I, mt im cc tr ca ( )mCthuc gc phn t th III ca h to Oxy.Trang 3n thi i hcL Anh Tun s 3 I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I: (2 im)Cho hm s 3 23 1 y x x + c th (C).1. Kho st s bin thin v v th (C).2. Tm hai im A, B thuc th (C) sao cho tip tuyn ca (C) ti A v B song song vi nhau v di on AB = 4 2.Cu II: (2 im) 1. Gii phng trnh: 84 821 1log ( 3) log ( 1) 3log (4 )2 4x x x + + .2. Tm nghim trn khong 0;2 _ , ca phng trnh: 2 234sin 3sin 2 1 2cos2 2 4xx x _ _ _ + , , ,Cu III: (1 im) Cho hm s f(x) lin tc trn R v 4( ) ( ) cos f x f x x + vi mi xR. Tnh:( )22I f x dx .Cu IV: (1 im) Cho hnh chp S.ABCD c y ABCD l mt hnh vung tm O. Cc mt bn (SAB) v (SAD) vung gc vi y (ABCD). Cho AB = a, SA = a2. Gi H, K ln lt l hnh chiu ca A trn SB, SD .Tnh th tch khi chp O.AHK.Cu V: (1 im) Cho bn s dng a, b, c, d tho mn a + b + c + d = 4 .Chng minh rng:2 2 2 221 1 1 1a b c db c cd da a b+ + + + + + +II. PHN RING (3 im) A. Theo chng trnh chun.Cu VI.a: (2 im) 1) Trong mt phng vi h to Oxy, cho tam gic ABC c din tch bng 32, A(2;3), B(3;2). Tm to im C, bit im C nm trn ng thng (d): 3x y 4 = 0.2) Trong khng gian vi h ta Oxyz, cho ba im A(2;4;1),B(1;1;3) v mt phng (P): x 3y + 2z 5 = 0. Vit phng trnh mt phng (Q) i qua hai im A, B v vung gc vi mt phng (P).Cu VII.a: (1 im) Tm cc s thc b, c phng trnh 20 z bz c + + nhn s phc 1 z i +lm mt nghim.B. Theo chng trnh nng caoCu VI.b: (2 im)1) Trong mt phng vi h to Oxy, cho tam gic ABC c trng tm G(2, 0) v phng trnh cc cnh AB, AC theo th t l:4x + y + 14 = 0; 2 5 2 0 x y + . Tm ta cc nh A, B, C.2) Trong khng gian vi h to Oxyz, cho cc im A(2,0,0); B(0,4,0); C(2,4,6) v ng thng (d) {6 3 2 06 3 2 24 0x y zx y z + + + . Vit phng trnh ng thng // (d) v ct cc ng thng AB, OC.Cu VII.b: (1 im) Gii phng trnh sau trong tp s phc:4 3 26 8 16 0 z z z z + .Trang 4L Anh Tun n thi i hc s 4I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I (2.0 im).Cho hm s 4 25 4, y x x +c th (C).1. Kho st s bin thin v v th (C).2. Tm m phng trnh 4 225 4 log x x m + c 6 nghim.Cu II (2.0 im).1. Gii phng trnh: 1 1sin 2 sin 2cot 22sin sin 2x x xx x+ (1)2. Tm m phng trnh sau c nghim x 0; 1 3 1 + ]:( )22 2 1 (2 ) 0 m x x x x + + + (2)Cu III (1.0 im).Tnh 402 11 2 1xI dxx++ +Cu IV (1.0 im). Cho lng tr ng ABC.A1B1C1 c AB = a, AC = 2a, AA1 2 5 a v 120oBAC . Gi M l trung im ca cnh CC1. Chng minh MB MA1 v tnh khong cch d t im A ti mt phng (A1BM).Cu V(1.0 im). Cho x, y, z l cc s dng. Chng minh: 3 2 4 3 5 x y z xy yz zx + + + +II. PHN RING (3.0 im)A. Theo chng trnh Chun.Cu VI.a. (2.0 im). Trong khng gian vi h ta Oxyz, cho cc im ( 1; 3; 0), (1; 3; 0), (0; 0; ) B C M a vi a > 0. Trn trc Oz ly im N sao cho mt phng (NBC) vung gc vi mt phng (MBC). 1. Cho 3 a . Tm gc gia mt phng (NBC) v mt phng (OBC).2.Tm a th tch ca khi chp BCMN nh nhtCu VII.a. (1.0 im).Gii h phng trnh: 2 12 12 2 3 1 ( , )2 2 3 1yxx x xxyy y y + + +'+ + +B. Theo chng trnh Nng cao.Cu VI.b. (2.0 im). Trong khng gian Oxyz cho hai im A (1; 3; 2), B (3; 7; 18) v mt phng (P): 2x y + z + 1 = 01. Vit phng trnh mt phng cha AB v vung gc vi mp (P).2. Tm ta im M (P) sao cho MA + MB nh nht.Cu VII. b. (1.0 im). Gii bt phng trnh: 24 2(log 8 log ) log 2 0xx x + Trang 5n thi i hcL Anh Tun s 5I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I(2 im)Cho hm s 2 11xyx+ c th(C).1. Kho st s bin thin v v th hm s .2. Vi im M bt k thuc th (C) tip tuyn ti M ct 2 tim cn ti Av B. Gi I l giao im hai tim cn . Tm v tr ca M chu vi tam gic IAB t gi tr nh nht. Cu II (2 im) 1. Giiphng trnh: 3sin 2 2sin2sin 2 .cosx xx x (1)2. Gii h phng trnh : 4 2 22 24 6 9 02 22 0x x y yxy x y + + '+ + (2)Cu III(1 im) Tnh tch phn sau:22sin 30.sin .cos . dxxI e x x Cu IV (1 im) Cho hnh chp t gic u S.ABCD c cnh bn bng a, mt bn hp vi y gc . Tm th tch ca khi chp t gi tr ln nht.Cu V (1 im) Cho x, y, zl cc s dng. Tm gi tr nh nht ca biu thc:3 3 3 3 3 33 3 32 2 24( ) 4( ) 4( ) 2x y zP x y x z z xy z x _ + + + + + + + + ,II. PHN RING (3 im) A. Theo chng trnh chunCu VI.a (2 im)1. Trong mt phng vi h to Oxy, cho hnh ch nht ABCD c tm I(12; 0) . ng thngcha cnh AB c phng trnh x 2y + 2 =0, AB = 2AD. Tm to cc nh A, B, C, D,bit nh A c honh m .2. Trong khng gian vi h to Oxyz, cho 2 ng thng 1( ) d v 2( ) dc phng trnh: 1 21 1 - 2 - 4 1 3( ); ;( ) :2 3 1 6 9 3x y z x y zd d + . Lp phng trnh mt phng (P) cha (d1) v 2( ) d .Cu VII.a (1 im) Tm m phng trnh sau c 2 nghim phn bit :

2 210x 8 4 (2 1). 1 x m x x + + + +(3) B. Theo chng trnh nng cao Cu VI.b (2 im) 1. Trong mt phng vi h to Oxy, cho hnh vung ABCD bitM(2;1); N(4; 2); P(2;0); Q(1;2) ln lt thuc cnh AB, BC, CD, AD. Hy lp phng trnh cc cnh ca hnh vung.2. Trong khng gian vi h to Oxyz, cho 2 ng thng () v ( ) c phng trnh:3 2 2'( ) : 1 2 ;( ) : 2'42 4 'x t x ty t y tz z t + + + ' ' + Vit phng trnh ng vung gc chung ca() v ( ).Cu VII.b (1 im)Gii v bin lun phng trnh: 2 2 3 21 .( 2 2) 3 4 2 mx mx mx x x x + + + + (4)Trang 6L Anh Tun n thi i hc s 6 I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu 1 (2 im): Cho hm s33(1) y x x

1) Kho st s bin thin v v th (C) ca hm s (1).2) Chng minh rng khi m thay i, ng thng (d):y = m(x +1) + 2 lun ct th (C) ti mt im M c nh v xc nh cc gi tr ca m (d) ct (C) ti 3 im phn bit M, N, P sao cho tip tuyn vi th (C) ti N v P vung gc vi nhau.Cu 2 (2 im):1) Gii ptrnh: 2 1 1 15.3 7.3 1 6.3 9 0x x x x + + +(1)2) Tm tt c cc gi tr ca tham s m h phng trnh sau c 2 nghim phn bit:233 322( 2x 5)log ( 1) log ( 1) log 4 ( )log ( 2x 5) log 2 5 ( )xx x ax m b ++ >' + (2)Cu 3 (1 im):Gii h phng trnh:3 23 23 29z 27( 1) ( )9x 27( 1) ( )9 27( 1) ( )x z ay x bz y y c ' (3)Cu 4 (1 im): Cho hnh chp S.ABCD c y l hnh ch nht, AB =2a, BC= a, cc cnh bn ca hnh chp bng nhau v bng 2 a. Gi M, N tng ng l trung im ca cc cnh AB, CD; K l im trn cnh AD sao cho 3aAK . Hy tnh khong cch gia hai ng thng MN v SK theo a.Cu 5 (1 im) Cho cc s a, b, c > 0 tho mn:a + b + c =1. Tm gi tr nh nht ca biu thc: a b cTa b c 1 1 1 + + .II. PHN RING (3 im) A. Theo chng trnh chun Cu 6a (2 im)1) Trong mt phng vi h to Oxy, cho im A(0; 2) v ng thng d: x 2y + 2 = 0. Tm trn d hai im B, C sao chotam gic ABC vung ti B v AB = 2BC.2) Trong khng gian vi h trc Oxyz, cho mt cu (S) c phng trnh: x2 + y2 + z2 2x + 4y + 2z 3 = 0 v mt phng (P): 2x y + 2z 14 = 0. Vit phng trnh mt phng (Q) cha trc Ox v ct mt cu (S) theo mt ng trn c bn knh bng 3.Cu 7a (1 im) Tm cc s thc a, b, c c: 3 22(1 ) 4(1 ) 8 z i z i z i + + + = 2( )( ) z ai z bz c + + T gii phng trnh: z i z i z i3 22(1 ) 4(1 ) 8 0 + + + trn tp s phc. Tm mun ca cc nghim .B. Theo chng trnh nng caoCu 6b (2 im)1) Trong mt phng vi h to Oxy, cho ng trn (C): x2 + y2 6x + 5 = 0. Tmim M thuc trc tung sao cho qua M k c hai tip tuyn ca (C) m gc gia hai tip tuyn bng 600.2) Trong khng gian vi h to Oxyz, cho hai ng thng: (d1) : {x t y t z 2; ; 4 ; (d2) :{3 ; ; 0 x t y t z Chng minh (d1) v (d2) cho nhau. Vit phng trnh mt cu (S) c ng knh l on vung gc chung ca (d1) v (d2).Cu 7b (1 im) Cho s thc b ln2. Tnh J = xln10b3xe dxe 2 v tm b ln2lim J.Trang 7n thi i hcL Anh Tun s 7I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I (2 im): Cho hm s3 22 ( 3) 4 + + + + y x mx m xc th l (Cm).1) Kho st s bin thin v v th (C1) ca hm s trn khi m = 1.2) Cho (d) l ng thng c phng trnh y = x + 4 v im K(1; 3). Tm cc gi tr ca tham s m sao cho (d) ct (Cm) ti ba im phn bit A(0; 4), B, C sao cho tam gic KBC c din tch bng 8 2. Cu II (2 im): 1) Gii phng trnh: cos 2 5 2(2 cos )(sin cos ) x x x x + (1)2) Gii h phng trnh: 3 3 32 28 27 184 6xy yxy x y + '+ (2)Cu III (1 im): Tnh tch phn:I =2261sin sin2x x dx +Cu IV (1 im): Cho hnh chp S.ABC c gc gia hai mt phng (SBC) v (ACB) bng 600, ABC v SBC l cc tam gic u cnh a. Tnh khong cch t B n mp(SAC).Cu V (1 im) Tm cc gi tr ca tham s thc m sao cho phng trnh sau c nghim thc:2 21 1 1 19 ( 2)3 2 1 0x xm m+ + + + + (3)II. PHN RING (3 im) A. Theo chng trnh chun:Cu VIa (2 im): 1) Trong mt phng vi h ta Oxy, cho ng trn (C) c phng trnh 2 2( 1) ( 2) 9 x y + + v ng thng d: x + y + m = 0. Tm m trn ng thng d c duy nht mt im A m t k c hai tip tuyn AB, AC ti ng trn (C)(B, C l hai tip im) sao cho tam gic ABC vung.2) Trong khng gian vi h ta Oxyz, cho im A(10; 2; 1) v ng thng d c phng trnh: 1 12 1 3x y z . Lp phng trnh mt phng (P) i qua A, song song vi d v khong cch t d ti (P) l ln nht.Cu VIIa (1 im): Cho ba s thc dng a, b, c tha mnabc = 1. Chng minh rng: 3 3 34 4 43(1 )(1 ) (1 )(1 ) (1 )(1 )a b cb c c a a b+ + + + + + + +(4)B. Theo chng trnh nng cao:Cu VIb (2 im): 1) Trong mt phng vi h to Oxy, cho im A(2;3), B(3;2), tam gic ABC c din tch bng 32;trng tm G ca ABC nm trn ng thng (d): 3x y 8 = 0. Tm bn knh ng trn ni tip ABC.2) Trong khng gian vi h to Oxyz, cho ng thng (d) l giao tuyn ca 2 mt phng (P): 2x 2y z + 1 = 0, (Q): x + 2y 2z 4 = 0 v mt cu (S): x2 + y2 + z2 + 4x 6y + m = 0. Tm m (S) ct (d) ti 2 im M, N sao cho di MN = 8.Cu VIIb (1 im): Gii h phng trnh : 2 22 22 2log ( ) 1 log ( )3 81 + + +'x xy yx y xy (x, y R)Trang 8L Anh Tun n thi i hc s 8I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I: (2 im)Cho hm s 4 2 2( ) 2( 2) 5 5 + + + f x x m x m m (Cm)1) Kho st s bin thin v v th (C) ca hm s vi m = 12) Tm m (Cm) c cc im cc i, cc tiu to thnh 1 tam gic vung cn. Cu II: (2 im) 1) Gii bt p trnh sau trn tp s thc: 1 12 3 5 2 x x x+ (1)2) Tm cc nghim thc ca phng trnh sau tho mn 131 log 0 + x :sin . tan 2 3(sin 3 tan 2 ) 3 3 x x x x + (2)Cu III: (1 im) Tnh tch phn sau: ( )1012 ln 11xI x x dxx _ + + ,Cu IV: (1 im) Cho hnh chp S.ABCD c y ABCD l hnh thoi vi 0120 A, BD = a >0. Cnh bn SA vung gc vi y. Gc gia mt phng (SBC) v y bng 600. Mt mt phng () i qua BD v vung gc vi cnh SC. Tnh t s th tch gia hai phn ca hnh chp do mt phng () to ra khi ct hnh chp. Cu V: (1 im) Cho ba s thc dng a, b, c tho mn+ + abc a c b . Hy tm gi tr ln nht ca biu thc:2 2 22 2 31 1 1Pa b c ++ + +(3)II. PHN RING (3 im )A. Theo chng trnh chunCu VI.a: (2 im)1) Trong mt phng vi h to Oxy, cho tam gic ABC cn, cnh y BC c p trnh d1: 1 0 + + x y. Phng trnh ng cao v t B l d2: 2 2 0 x y . im M(2; 1) thuc ng cao v t C. Vit phng trnh cc cnh bn ca tam gic ABC. 2) Trong khng gian vi h to Oxyz, vit phng trnh ng thng (d) i qua M(1;1;1), ct ng thng ( )12 1:3 1 2+ x y zdv vung gc vi ng thng ( )2: 2 2 ; 5 ; 2 + + d x t y t z t( t R ).Cu VII.a: (1 im) Gii pt: 1 2 3 23 7 ... (2 1) 3 2 6480n n n nn n n nC C C C + + + + B. Theo chng trnh nng caoCu VI.b: (2 im)1) Trong mt phng vi h to Oxy, cho Elip (E): 2 25 5 + x y , Parabol 2( ) : 10 P x y . Hy vit phng trnh ng trn c tm thuc ng thng ( ) : 3 6 0 x y + , ng thi tip xc vi trc honh Ox v ct tuyn chung ca Elip (E) vi Parabol (P).2) Trong khng gian vi h to Oxyz, vit phng trnh ng thng (d) vung gc vi mt phng (P): 1 0 x y z + + ng thi ct c hai ng thng ( )11 1:2 1 1x y zd + v 2( ) : 1 ; 1; + d x t y z t , vi t R .Cu VII.b: (1 im) Gii h phng trnh sau trn tp s thc: 242 2 11 6log ( )2 2 ( )x xx y ay y b+ +' +. Trang 9n thi i hcL Anh Tun s 9I. PHN CHUNG CHO TT C TH SINH (7,0 im) CuI (2 im) Cho hm s y = x3 + (1 2m)x2 + (2 m)x + m + 2 (m l tham s) (1)1) Kho st s bin thin v v th ca hm s (1) khi m = 2.2) Tm cc gi tr ca m th hm s (1) c im cc i, im cc tiu, ng thi honh ca im cc tiu nh hn 1.CuII (2 im)1) Gii phng trnh: 3 32 3 2cos3 cos sin3 sin8x x x x+ (1)2) Gii h phng trnh: 221 ( ) 4( 1)( 2)x yy x yx y x y + + + '+ + (x, y )(2)Cu III (1 im) Tnh tch phn: 622 1 4 1dxIx x+ + +Cu IV (1 im) Cho hnh hp ng ABCD.ABCD c cc cnh AB=AD = a, AA = 32a v gc BAD = 600 . Gi M v N ln lt l trung im ca cc cnh AD v AB. Chng minh rng AC vung gc vi mt phng (BDMN). Tnh th tch khi chp A.BDMN.CuV (1 im) Cho x,y l cc s thc tha mn iu kin x2+xy+y2 3 .Chng minh rng:

2 24 3 3 3 4 3 3 x xy y +II. PHN RING (3 im)A. Theo chng trnh chunCu VI.a (2 im) 1) Trong mt phng vi h ta Oxy, cho tam gic ABC c nh A thuc thng d: x 4y 2 = 0, cnh BC song song vi d, ptrnh ng cao BH: x + y + 3 = 0 v trung im ca cnh AC l M(1; 1). Tm ta cc nh A, B, C.2) Trong khng gian Oxyz, cho mt phng (): 3x + 2y z + 4 = 0 vhai im A(4;0;0) , B(0;4;0) .Gi I l trung im ca on thng AB. Xc nh ta im K sao cho KI vung gc vi mt phng (), ng thi K cch u gc ta O v ().Cu VII.a (1 im) Gii h phng trnh: {2 2ln(1 ) ln(1 ) ( )12 20 0 ( )x y x y ax xy y b+ + + B. Theo chng trnh nng cao Cu VI.b (2 im) 1) Trong mt phng vi h ta Oxy cho ABC D c cnh AC i qua im M(0; 1). Bit AB = 2AM, phng trnh ng phn gic trong AD: x y = 0, phng trnh ng cao CH: 2x + y + 3 = 0. Tm ta cc nh ca ABC D.2) Trong khng gian vi h to Oxyz, cho mt phng (P): 4x 3y + 11z = 0 v hai ng thng d1: 1x = 32y = 13z +, 41x = 1y = 32z . Chng minh rng d1 v d2 cho nhau. Vit phng trnh ng thng nm trn (P), ng thi ct c d1 v d2.Cu VII.b (1 im) Gii phng trnh:14 2 2 2 1 2 1 2 0x x x xy ( )sin( )++ + + .Trang 10L Anh Tun n thi i hc s 10I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I (2 im). Cho hm s 2 12xyx++ c th l (C). 1) Kho st s bin thin v v th ca hm s.2) Chng minh ng thng d: y = x + m lun lun ct th (C) ti hai im phn bit A, B. Tm m on AB c di nh nht.Cu II (2 im)1) Gii phng trnh: 9sinx + 6cosx 3sin2x + cos2x = 82) Gii bt phng trnh:2 2 22 2 4log log 3 5(log 3) x x x > Cu III (1 im). Tm nguyn hm 3 5sin .cosdxIx xCu IV (1 im). Cho lng tr tam gic ABC.A1B1C1 c tt c cc cnh bng a, gc to bi cnh bn v mt phng y bng 300. Hnh chiu H ca im A trn mt phng (A1B1C1) thuc ng thng B1C1. Tnh khong cch gia hai ng thng AA1 v B1C1 theo a.Cu V (1 im). Cho ba s thc khng m a, b, c tha mn: a2009 + b2009 + c2009 = 3. Tm gi tr ln nht ca biu thc:P = a4 + b4 + c4.II. PHN RING (3 im)A. Theo chng trnh chunCu VIa (2 im).1) Trong mt phng vi h to Oxy, cho 2 ng thng (d1): 7 17 0 x y + , (d2): 5 0 x y + . Vit phng trnh ng thng (d) qua im M(0;1) to vi (d1), (d2) mt tam gic cn ti giao im ca (d1), (d2). 2) Trong khng gian vi h to Oxyz, cho hnh hp ch nht ABCD.ABCD c AO, B(3;0;0), D(0;2;0), A(0;0;1). Vit phng trnh mt cu tm C tip xc vi AB.Cu VIIa (1 im). C bao nhiu s t nhin c 4 ch s khc nhau v khc 0 m trong mi s lun lun c mt hai ch s chn v hai ch s l.2.Theo chng trnh nng cao (3 im)Cu VIb (2 im)1) Trong mt phng vi h ta Oxy, cho im M(1; 0). Lp phng trnh ng thng (d) i qua M v ct hai ng thng (d1): x + y + 1 = 0, (d2): x 2y + 2 = 0 ln lt ti A, B sao cho MB = 3MA.2) Trong khng gian vi h to Oxyz, cho im M(0;1;1)v 2 ng thng (d1), (d2) vi: (d1):1 23 2 1x y z + ; (d2) l giao tuyn ca 2 mt phng (P):1 0 x + v (Q): 2 0 x y z + + . Vit phng trnh ng thng (d) qua M vung gc (d1) v ct (d2).Cu VIIb (1 im) Tm h s ca 8xtrong khai trin Newtn ca biu thc :2 3 8(1 ) P x x + .Trang 11n thi i hcL Anh Tun s 11I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I: (2 im) Cho hm s 11xyx+(C).1) Kho st s bin thin v v th (C) ca hm s.2) Tm trn trc tung tt c cc im t k c duy nht mt tip tuyn ti (C).Cu II: (2 im)1) Gii phng trnh: 2 2 2 2 2log ( 1) ( 5) log( 1) 5 0 x x x x + + + 2) Tm nghim ca phng trnh: 2 3cos cos sin 2 x x x + + tho mn :1 3 x < Cu III: (1 im)Tnh tch phn:120I ln( 1) x x x dx + + Cu IV: (1 im) Cho hnh lng tr ng ABC.ABC c ABC l tam gic vung ti B v AB = a, BC = b, AA = c (2 2 2c a b + ). Tnh din tch thit din ca hnh lng tr b ct bi mt phng (P) i qua A v vung gc vi CA .Cu V: (1 im) Cho cc s thc , , (0;1) xyz v 1 xy yz zx + + . Tm gi tr nh nht ca biu thc:2 2 21 1 1x y zPx y z + + II. PHN RING (3 im): A. Theo chng trnh chun:Cu VI.a: (2 im) 1) Trong khng gian vi h to Oxyz, cho ng thng (d) c phng trnh: {x t ;1 2 y t +;2 z t + ( t R ) v mt phng (P): 2 2 3 0 x y z .Vit phng trnh tham s ca ng thng nm trn (P), ct v vung gc vi (d). 2) Trong mt phng vi h to Oxy, cho elip (E): 2 219 4x y+ . Vit phng trnh ng thng d i qua I(1;1) ct (E) ti 2 im A v B sao cho I l trung im ca AB.Cu VII.a: (1 im) Gii h phng trnh sau trn tp s phc: {2 281z w zwz w + B. Theo chng trnh nng cao:Cu VI.b: (2 im)1) Trong khng gian vi h to Oxyz, cho 4 im A(2;4;1), B(1;4;1), C(2;4;3), D(2;2;1). Tm ta im M MA2 + MB2 + MC2 + MD2 t gi tr nh nht.2) Trong mt phng vi h ta Oxy, choABC cn c y l BC. nh A c ta l cc s dng, hai im B v C nm trn trc Ox, phng trnh cnh : 3 7( 1) AB y x . Bit chu vi ca ABC bng 18, tm ta cc nh A, B, C.Cu VII.b: (1 im) Gii h phng trnh: 2 12 12 2 3 1 ( , )2 2 3 1yxx x xxy Ry y y + + +'+ + +Trang 12L Anh Tun n thi i hc s 12I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I: (2 im) Cho hm s 3 23 2 y x mx m +(Cm).1) Kho st s bin thin v v th hm s khi m = 1 .2) Tm m (Cm) v trc honh c ng 2 im chung phn bit.Cu II: (2 im)1) Gii phng trnh: (sin 2 sin 4) cos 202sin 3x x xx + +2) Gii phng trnh: 3 18 1 22 1x x++ Cu III: (1 im)Tnh tch phn:230sinI(sin cos )xdxx x+Cu IV: (1 im) Cho khi chp S.ABC c SA (ABC), ABC vung cn nh C v SC = a. Tnh gc gia 2 mt phng (SCB) v (ABC) th tch khi chp ln nht.Cu V: (1 im) Tm m phng trnh sau y c ng 2 nghim thc phn bit:2 2 (2 )(2 ) x x x x m + + II. PHN RING (3 im): A. Theo chng trnh chun:Cu VI.a:(2 im) 1) Trong mt phng vi h to Oxy, cho im M(3;1). Vit phng trnh ng thng d i qua M ct cc tia Ox, Oy ti A v B sao cho (OA+3OB) nh nht. 2) Trong khng gian vi h to Oxyz, cho hai im A(1;2;3) v B(3;4;1). Tm to im M thuc mt phng (P): 1 0 x y z + MAB l tam gic u.Cu VII.a: (1 im) Tm h s ca 20xtrong khai trin Newton ca biu thc 532nxx _+ ,, bit rng: 0 1 21 1 1 1... ( 1)2 3 1 13n nn n n nC C C Cn + + + +B. Theo chng trnh nng cao:Cu VI.b:(2 im) 1) Trong mt phng vi h to Oxy, cho 4 im A(1;0), B(2;4), C(1;4), D(3;5). Tm to im M thuc ng thng ( ) : 3 5 0 x y sao cho hai tam gic MAB, MCD c din tch bng nhau. 2) Trong khng gian vi h to Oxyz, cho ng thng 1( ) c phng trnh { 2 ; ; 4 x t y t z ; 2( ) l giao tuyn ca 2 mt phng ( ) : 3 0 x y + v ( ) : 4 4 3 12 0 x y z + + . Chng t hai ng thng 1 2, cho nhau v vit phng trnh mt cu nhn on vung gc chung ca 1 2, lm ng knh.Cu VII.b: (1 im) Cho hm s 2 2(2 1) 42( )x m x m myx m+ + + + ++. Chng minh rng vi mi m, hm s lun c cc tr v khong cch gia hai im cc tr khng ph thuc m. s 13Trang 13n thi i hcL Anh TunI. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I: (2 im) Cho hm s ( )3 12 x 4x mym m+ + +c th l (Cm) (m l tham s)1) Kho st s bin thin v v th (C) ca hm s khi m = 0.2) Xc nh m sao cho ng thng (d): y = x + m ct th (C) ti hai im A, B sao cho di on AB l ngn nht. Cu II: (2 im) 1) Gii phng trnh:sin cos 4sin 2 1 x x x + .2) Tm m h phng trnh: ( )2 22 224xy x ymx y xy + '+ c ba nghim phn bit.Cu III: (1 im) Tnh cc tch phn 13 201 x I x xd ;J = 11x( ln x)e xxxedx e++Cu IV: (1im) Cho hnh lp phng ABCD.A'B'C'D' cnh bng a v im M trn cnh AB sao cho AM = x, (0 < x < a). Mt phng (MA'C') ct BC ti N. Tnh x theo a th tch khi a din MBNC'A'B' bng 13th tch khi lp phng ABCD.A'B'C'D'.Cu V: (1 im)Cho x, y l hai s dng thay itho iu kin 4(x + y) 5 = 0. Tm gi tr nh nht ca biu thc S = 4 14 x y+.II. PHN RING (3 im) A. Theo chng trnh Chun :Cu VI.a (2 im)1) Trong mt phng vi h ta Oxy, cho hai ng thng 1: 3 4 5 0 x y + + ; 2: 4 3 5 0 x y n n. Vit phng trnh ng trn c tm nm trn ng thng d: x 6y 10 = 0 v tip xc vi 1, 2.2) Trong khng gian vi h ta Oxyz, cho hnh chp A.OBC, trong A(1; 2; 4), B thuc trc Ox v c honh dng, C thuc Oy v c tung dng. Mt phng (ABC) vung gc vi mt phng (OBC), tan 2 OBC . Vit phng trnh tham s ca ng thng BC.Cu VII.a (1 im)Gii phng trnh: 22(2 ) 7 4 0 z i z i + + + trn tp s phc.B. Theo chng trnh Nng cao :Cu VI.b (2 im)1) Trong mt phng vi h ta Oxy, cho cc im M1(155; 48), M2(159; 50), M3(163; 54), M4(167; 58), M5(171; 60). Lp phng trnh ng thng d i qua im M(163; 50) sao cho ng thng gn cc im cho nht. 2) Trong khng gian vi h to Oxyz, cho ba im A(2;0;0), C(0;4;0), S(0; 0; 4).Tm ta im B trong mp(Oxy) sao cho t gic OABC l hnh ch nht. Vit phng trnh mt cu i qua bn im O, B, C, S.Cu VII.b (1 im) Chng minh rng : 4 28a 8a 1 1 + , vi mi a thuc on [1; 1]. s 14I. PHN CHUNG CHO TT C TH SINH (7,0 im) Trang 14L Anh Tun n thi i hcCu I. (2 im) Cho hm s 2 11xyx+ (C) 1) Kho st s bin thin v v th (C) .2) Tm cc im M thuc th (C) sao cho tng cc khong cch t M n hai tim cn ca (C) l nh nht.Cu II. (2 im) 1) Tm m h phng trnh c nghim: 11 3x yx x y y m+ '+ .2) Gii phng trnh:cos23x.cos2x cos2x = 0.Cu III. (1 im) Tnh tch phn: 220( sin ) cos I x x xdx +.Cu IV. (1 im) Trn cnh AD ca hnh vung ABCD c di l a, ly im M sao cho AM = x (0 m a). Trn na ng thng Ax vung gc vi mt phng (ABCD) ti im A, ly im S sao cho SA = y (y > 0). Tnh th tch khi chp S.ABCM theo a, y v x. Tm gi tr ln nht ca th tch khi chp S.ABCM, bit rng x2 + y2 = a2.Cu V. (1 im) Cho x, y, z l cc s dng tho mn: 1 1 11x y z+ + . Chng minh rng: 1 1 112 2 2 z y z x y z x y z+ + + + + + + +.II. PHN RING (3 im)A. Theo chng trnh chun Cu VI.a. (2 im)1) Oxy, cho im C(2; 0) v elip (E): 2 214 1x y+ . Tm to cc im A, B thuc (E), bit rng hai im A, B i xng vi nhau qua trc honh v tam gic ABC l tam gic u.2) Trong khng gian vi h to Oxyz, cho mt cu(S): x2 + y2 + z2 2x + 2y + 4z 3 = 0 v hai ng thng 1 21 1: , :2 1 1 1 1 1x y z x y z . Vit phng trnh tip din ca mt cu (S), bit tip din song song vi hai ng thng 1 v 1.Cu VII.a. (1 im) Gii h phng trnh: 2. 5. 905. 2. 80x xy yx xy yA CA C + ' B. Theo chng trnh nng cao Cu VI.b. (2 im)1) Trong mt phng Oxy, cho parabol (P): y2 = 8x. Gi s ng thng d i qua tiu im ca (P) v ct (P) ti hai im phn bit A, B c honh tng ng l x1, x2. Chng minh: AB = x1 + x2 + 4.2) Trong khng gian Oxyz, cho hai im A(1;5;0), B(3;3;6) v ng thngc phng trnh tham s{1 2 ; 1 ; 2 x t y t z t + . Mt im M thay i trn ng thng , xc nh v tr ca im M chu vi tam gic MAB t gi tr nh nht.Cu VII.b. Tnh o hm f (x) ca hm s ( )31( ) ln3f xx v gii bt phng trnh sau:206sin2'( )2tdtf xx>+ s 15Trang 15n thi i hcL Anh TunI. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I (2 im):Cho hm s:33 y x x 1) Kho st s bin thin v v th (C) ca hm s.2) Tm trn ng thng y = x cc im k c ng 2 tip tuyn ti th (C).Cu II (2 im): 1)Gii phng trnh.: 3sin 2 2sin2sin 2 .cosx xx x2) Tm m phng trnh sau c nghim: ( 1) 4( 1)1xxx x mx + Cu III (1 im): Tnh tch phnI=22sin 30.sin .cos . dx.xe x x Cu IV (1 im): Cho hnh nn nh S, ng trn y c tm O v ng knh l AB = 2R. Gi M l im thuc ng trn y v 2 ASB , 2 ASM . Tnh th tch khi t din SAOM theo R, v .Cu V (1 im): Cho: 2 2 21 a b c + + . Chng minh: 2(1 ) 0 abc a b c ab ac bc + + + + + + + II. PHN RING (3 im)A. Theo chng trnh chunCu VI.a (2 im)1) Trong mt phng vi h ta Oxy, cho ng trn (C): (x 1)2 + (y + 1)2 = 25 v im M(7; 3). Lp phng trnh ng thng (d) i qua M ct (C) ti hai im A, B phn bit sao cho MA = 3MB.2) Trong khng gian vi h to Oxyz, cho cc im A(1;0;0); B(0;2;0); C(0;0;2). Gi H l hnh chiu vung gc ca O trn mt phng (ABC), tm ta im H.Cu VIIa (1 im)Gii phng trnh: 22 2log ( 7) log 12 4 0 x x x x + + B. Theo chng trnh nng caoCu VI.b (2 im)1) Trong mt phng vi h to Oxy, cho hnh bnh hnh ABCD c din tch bng 4. Bit A(1;0), B(0;2) v giao im I ca hai ng cho nm trn ng thng y = x. Tm ta cc nh C v D.2) Trong khng gian vi h ta Oxyz, choABC vi ta nh C(3; 2; 3) v phng trnh ng cao AH, phng trnh ng phn gic trong BD ln lt l: 12 3 3:1 1 2x y zd , 21 4 3:1 2 1x y zd . Lp phng trnh ng thng cha cnh BC caABC v tnh din tch caABC .Cu VII.b (1 im) Gii phng trnh:2008 2007 1xx + . s 16Trang 16L Anh Tun n thi i hcI. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I: (2 im) Cho hm s2 41xyx +.1) Kho st s bin thin v v th (C) ca hm s.2) Tm trn (C) hai im i xng nhau qua ng thng MN bit M(3;0) v N(1; 1) Cu II: (2 im)1) Gii phng trnh: 4cos4x cos2x 1 3cos 4 cos2 4xx += 722) Gii phng trnh:3x.2x = 3x + 2x + 1 Cu III: (1 im) Tnh tch phn:K =201 sin.1 cosxxe dxx+ _ + ,Cu IV: (1 im) Cho hnh chp tam gic u S.ABC c di cnh bn bng 1. Cc mt bn hp vi mt phng y mt gc . Tnh th tch hnh cu ni tip hnh chp S.ABC.Cu V: (1 im) Gi a, b, c l ba cnh ca mt tam gic c chu vi bng 2. Chng minh rng:2 2 2522 227a b c abc + + + 0)Cu II:(2 im)1) Gii bt phng trnh:2 23 2 2 3 1 1 x x x x x + + 2) Gii phng trnh :3 3 2cos cos3 sin sin34x x x x + Cu III: (1 im):Tnh tch phn:I=2307sin 5cos(sin cos )x xdxx x+Cu IV: (1 im): Cho hnh chp t gic u S.ABCD c di cnh y bng a, cc mt bn to vi mt y gc 60o. Mt phng (P) chaAB v i qua trng tm ca tam gic SAC ct SC, SD ln lt ti M, N. Tnh thtch khi chp S.ABMN theo a.Cu V: (1 im) Cho 4 s thc a, b, c, d tho mn: 2 21 a b + ; c d = 3. Chng minh:9 6 24F ac bd cd+ + II.PHN RING (3.0 im )A. Theo chng trnh Chun Cu VI.a: (2 im)1) Trong Oxy, cho tam gic ABC vi A(3; 7), B(9; 5), C(5; 9), M(2; 7). Vit phng trnh ng thng i qua M v tip xc vi ng trn ngoi tip ABC. 2) Trong khng gian vih to Oxyz, chohai ng thng: 1:1 1 2x y zd v21 2:1x td y tz t ' +Xt v tr tng ica d1 v d2. Vit phng trnh ng thng qua O, ct d2 v vung gc vi d1Cu VII.a: (1 im) Mt hp ng 5 vin bi , 6 vin bi trng v 7 vin bi vng. Ngui ta chn ra 4 vin bi t hp . Hi c bao nhiu cch chn trong s bi ly ra khng c c ba mu?B. Theo chng trnh Nng cao :Cu VI.b: (2 im)1) Trong mt phngOxy, cho tam gic ABC c nh A(1; 3) v hai ng trung tuyn ca n c phng trnh l: x 2y + 1 = 0 v y 1 = 0. Hy vit phng trnh cc cnh ca ABC. 2) Trong khng gian vih to Oxyz, cho hai im A(0; 0;3), B(2; 0;1) v mt phng (P) c phng trnh: 3 8 7 1 0 x y z + + . Vit phng trnh chnh tc ng thng d nm trn mt phng (P) v d vung gc vi AB ti giao im ca ng thng AB vi (P). Cu VII.b: (1 im) Tm h s x3 trong khai trin22nxx _+ , bit n tho mn: 1 3 2 1 232 2 2... 2nn n nC C C+ + + Trang 35n thi i hcL Anh Tun s 35I. PHN CHUNG CHO TT C TH SINH (7,0 im) Cu I (2 im) Cho hm s y =22 3xx++ (1). 1) Kho st s bin thin v v th ca hm s (1).2) Vit phng trnh tip tuyn ca th hm s (1), bit tip tuyn ct trc honh, trc tung ln lt ti hai im phn bit A, B sao cho OAB cn ti gc ta O.Cu II (2 im) 1) Gii phng trnh: cot 3 tan 2cot 2 3 x x x + + + . 2) Gii phng trnh: 2 22( 1) 3 1 2 2 5 2 8 5 x x x x x x + + + + .Cu III (1 im) Tnh tch phn : 40cos sin3 sin 2x xI dxx.Cu IV (1 im) Cho hnh lp phng ABCD.A B C Dcnh a. Gi M, N ln lt l trung im cc cnh CD, A D . im P thuc cnh DD sao cho PD= 2PD. Chng t (MNP) vung gc vi (A AM) v tnh th tch ca khi t din A AMP.Cu V (1 im) Cho a, b, c l 3 cnh ca tam gic c chu vi bng 3. Tm gi tr nh nht ca biu thc: 3 3 3( ) ( ) ( )3 3 3a b c b c a c a bPc a b+ + + + + .II. PHN RING (3 im)A. Theo chng trnh chun Cu VI.a. (2 im) 1) Trong mt phng vi h ta Oxy, cho ng trn (C): (x 1)2 + (y + 1)2 = 25 v im M(7; 3). Lp phng trnh ng thng (d) i qua M ct (C) ti A, B phn bit sao cho MA = 3MB. 2) Trong khng gian vi h ta Oxyz, cho mt phng (P) : x 2y + 2z 1 = 0 v hai ng thng 1 : 1 91 1 6x y z + + ;2 : 1 3 12 1 2x y z + . Xc nh ta im M thuc ng thng 1 sao cho khong cch t M n ng thng 2 v khong cch t M n mt phng (P) bng nhau.Cu VII.a (1 im) Gi z1 v z2 l 2 nghim phc ca phng trnh:22 10 0 z z + + . Tnh gi tr ca biu thc:2 21 2A z z + .B. Theo chng trnh nng caoCu VI.b(2 im) 1) Trong mt phng vi h ta Oxy, cho tam gic ABC c A(3; 3), B(2; 1), C(11; 2). Vit phng trnh ng thng i qua A v chia ABC thnh hai phn c t s din tch bng 2. 2) Trong khng gian vi h ta Oxyz cho, ng thng 1 2:1 2 1x y zd v mt phng (P): x + 3y + 2z + 2 = 0. Lp phng trnh ng thng di qua im M(2; 2; 4), song song vi mt phng (P) v ct ng thng d.Cu VII.b (1 im) Gii phng trnh: ( )32 7log 1 log x x + .Trang 36L Anh Tun n thi i hc s 36I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 4 2 22( 1) 1 y x m m x m + + (1)1) Kho st s bin thin v v th (C) ca hm s khi m = 1.2) Tm m th ca hm s (1) c khong cch gia hai im cc tiu ngn nht.Cu II (2 im): 1) Gii phng trnh:22cos 3 4cos 4 15sin 2 214x x x _ ,2) Gii h phng trnh:3 2 2 36x 9x 4 02x y y yx y x y + ' + + Cu III (1 im): Tnh tch phn: I = ln 6 2xln 46 5x xedxe e+ Cu IV (1 im): Cho khi chp S.ABCD c y ABCD l hnh ch nht, vi AB = 2AD = 2a, snh SA vung gc vimt phng (ABCD), cnh SC to vi mt y (ABCD) mt gc 045 . Gi G l trng tm ca tam gic SAB, mtphng (GCD) ct SA, SB ln lt ti P v Q. Tnh th tch khi chp S.PQCD theo a.Cu V (1 im): Cho x v y l hai s dng tho mn 2 x y + . Tm gi tr nh nht ca biu thc: P = 3 2 2 32 23 32x 2x y x yy x y+ ++ + +II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong mt Oxy, cho hnh thoi ABCD c cnh bng 5 n v, bit to nh A(1; 5), hai nh B, D nm trn ng thng (d): 2 4 0 x y + . Tm to cc nh B, C, D.2) Trong khng gian vi h to Oxyz, cho mt phng (P): 2x 1 0 y z + v hai ng thng (d1): 1 2 32 1 3x y z + , (d2): 1 1 22 3 2x y z + . Vit phng trnh ng thng () song song vi mt phng(P), vung gc vi ng thng (d1) v ct ng thng (d2) ti im E c honh bng 3.Cu VII.a (1 im): Trn tp s phc cho phng trnh 20 z az i + + . Tm a phng trnh trn c tng cc bnh phng ca hai nghim bng4i .2. Theo chng trnh nng caoCu VI.b (2 im): 1) TrongOxy, cho ng trn (C): 2 26x 2 5 0 x y y + + v ng thng (d): 3x 3 0 y + . Lp phng trnh tip tuyn vi ng trn (C), bit tip tuyn khng i qua gc to v hp vi ng thng (d) mt gc 045 .2) Oxyz, cho hai ng thng (d1): 3 11 1 2x y z + , (d2): 2 21 2 1x y z + . Mt ng thng () i qua im A(1; 2; 3), ct ng thng (d1) ti im B v ct ng thng (d2) ti im C. Chng minh rng im B l trung im ca on thng AC.Cu VII.b (1 im): Tm gi tr m hm s 2 2 2( 1)1x m x m myx+ + ng bin trn cc khong ca tp xc nh v tim cn xin ca th i qua im M(1; 5).Trang 37n thi i hcL Anh Tun s 37I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 3 21 83x3 3y x x + (1)1) Kho st s bin thin v v th (C) ca hm s.2) Lp phng trnh ng thng d song song vi trc honh v ct th (C) ti hai im phn bit A, B sao cho tam gic OAB cn ti O (O l gc to ).Cu II (2 im): 1) Gii phng trnh:21(1 4sin ) sin3x2x 2) Gii phng trnh:2 2 23x 1 tan 16x x x + + +Cu III (1 im): Tnh tch phn: I = 25 2 22( ) 4 x x xdx+ Cu IV (1 im): Cho hnh chp u S.ABCD c cnh y bng a, cnh bn hp vi y gc 060 .Gi M l im i xng vi C qua D, N l trung im ca SC. Mt phng (BMN) chia khi chp thnh hai phn. Tnh t s th tch ca hai phn .Cu V (1 im): Cho x, y, zl cc s dng tho mn 2 2 21 x y z + + . Chng minh:P = 2 2 2 2 2 23 32 xx y zy z z x y+ + + + +II. PHN T CHN (3 im)1. Theo chng trnh chun Cu VI.a (2 im):1) Trong Oxy, cho ng trn (C): 2 2( 1) ( 2) 9 x y + + v ng thng d: 0 x y m + + . Tm m trn ng thng dc duy nht mt im A m t k c hai tip tuyn AB, AC ti ng trn (C) sao cho tam gic ABC vung (B, C l hai tip im).2) Trong khng gian vi h to Oxyz, vit phng trnh mt phng (P) qua O, vung gc vi mt phng (Q): 0 x y z + + v cch im M(1; 2; 1) mt khong bng 2.Cu VII.a (1 im): Tm h s ca 8xtrong khai trin nh thc Niutn ca( )22nx +, bit: 3 2 18 49n n nA C C + (n N, n > 3).2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong Oxy, cho ng thng d: 1 0 x y v hai ng trn c phng trnh: (C1): 2 2( 3) ( 4) 8 x y + + , (C2): 2 2( 5) ( 4) 32 x y + + Vit phng trnh ng trn (C) c tm I thuc d v tip xc ngoi vi (C1) v (C2).2) Trong khng gian vi h to Oxyz, cho im A(3; 1; 1), ng thng : 21 2 2x y z v mt phng (P): 5 0 x y z + . Vit phng trnh tham s ca ng thng d i qua A, nm trong (P) v hp vi ng thng mt gc 045 .Cu VII.b (1 im): Gii h phng trnh: 2 2 22lg lg lg ( )lg ( ) lg x.lg 0x y xyx y y +' + Trang 38L Anh Tun n thi i hc s 38I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s4 2x 1 y x m m + (Cm)1) Kho st s bin thin v v th (C) ca hm s khi m = 2.2) Chng minh rng khi m thay i th (Cm) lun lun i qua hai im c nh A, B. Tm m cc tip tuyn ti A v B vung gc vi nhau.Cu II (2 im): 1) Gii h phng trnh:23 2 25x 93x 2x 6x 18x yxy y + + '+ + + 2) Gii phng trnh:21sin sin 2 1 cos cos2x x x x + + +Cu III (1 im): Tnh tch phn: I = 82311xdxx+Cu IV (1 im): Cho hnh lp phng ABCD.A B C Dcnh a. Gi K l trung im ca cnh BC v I l tm ca mt bn CC D D. Tnh th tch ca cc hnh a din do mt phng (AKI) chia hnh lp phng.Cu V (1 im): Cho x, yl hai s thc tho mn 2 22 x xy y + . Tm gi tr nh nht v gi tr ln nht ca biu thc:M = 2 22x 3 x y y + .II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong mt phng vi h to Oxy, cho tam gic ABC c im M(1; 1) l trung im ca cnh BC, hai cnh AB, AC ln lt nm trn hai ng thng d1: 2 0 x y + v d2: 2x 6 3 0 y + + . Tm to cc nh A, B, C.2) Trong khng gian vi h to Oxyz, cho mt cu (S): 2 2 22x 2 4z 2 0 x y z y + + + v ng thng d: 3 32 2 1x y z . Lp phng trnh mt phng (P) song song vi d v trc Ox, ng thi tip xc vi mt cu (S).Cu VII.a (1 im): Gii phng trnh sau trn tp s phc:2 4 2( 9)( 2z 4) 0 z z + + 2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong mt phng vi h to Oxy, cho tam gic ABC c A(2; 3), B(3; 2), din tch tam gic bng 1,5 v trng tm I nm trn ng thng d: 3x 8 0 y . Tm to im C.2) Trong khng gian vi h to Oxyz, cho hai ng thng d1: 1 12 1 2x y z + v d2: 2 11 1 2x y z . Lp phng trnh ng thng d ct d1 v d2 v vung gc vi mt phng (P): 2x 5z 3 0 y + + + .Cu VII.b (1 im): Cho hm s 2x 1x 1x m mym+ + + (m l tham s). Tm m hm s lun ng bin trn tng khong xc nh ca n.Trang 39n thi i hcL Anh Tun s 39I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 2x 11yx+.1) Kho st s bin thin v v th (C) ca hm s.2) Gi M l giao im ca hai ng tim cn ca (C). Tm trn th (C) im I c honh dng sao cho tip tuyn ti I vi th (C) ct hai ng tim cn ti A v B tho mn: 2 240 MA MB + .Cu II (2 im): 1) Gii bt phng trnh:3 12 2x 1 x x + +2) Gii phng trnh:3sin 3tan2cos 2tan sinx xxx x+ Cu III (1 im): Tnh tch phn: I = 2 2217x 12xdxx +Cu IV (1 im): Cho ng trn (C) ng knh AB = 2R. Trn na ng thng Ax vung gc vi mt phng cha (C) ly im S sao cho SA = h. Gi M l im chnh gia cung AB. Mt phng (P) i qua A v vung gc vi SB, ct SB, SM ln lt ti H v K.. Tnh th tch ca khi chp S.AHK theo R v h.Cu V (1 im): Cho a, b, cl nhng s dng tho mn: 2 2 23 a b c + + . Chng minh bt ng thc:2 2 21 1 1 4 4 47 7 7 a b b c c a a b c+ + + ++ + + + + + II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong mt phng vi h to Oxy, cho tam gic ABC c nh 4 7;5 5A _ , v phng trnh hai ng phn gic trong BB : 2 1 0 x y v CC : 3 1 0 x y + . Chng minh tam gic ABC vung.2) Trong khng gian vi h to Oxyz, cho hai ng thng 18 6 10( ) :2 1 1x y zd+ v 2( ) : 24 2x td y tz t ' +. Vit phng trnh ng thng (d) song song vi trc Ox v ct (d1) ti A, ct (d2) ti B. Tnh AB.Cu VII.a (1 im):Tm phn thc v phn o ca s phc 3(2 2 )(3 2 )(5 4 ) (2 3 ) z i i i i + + .2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong mt phng vi h to Oxy, cho tam gic ABC vung cn ti A, bit cc nh A, B, C ln lt nm trn cc ng thng d: 5 0 x y + , d1: 1 0 x + , d2: 2 0 y + . Tm to cc nh A, B, C, bit BC = 5 2.2) Trong khng gian Oxyz, cho im M(2; 1; 0) v ng thng : 1 12 1 1x y z + . Lp phng trnh ca ng thng d i qua im M, ct v vung gc vi .Cu VII.b (1 im): Gii h phng trnh:{2 25 39x 4 5log (3x 2 ) log (3x 2 ) 1yy y + .Trang 40L Anh Tun n thi i hc s 40I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 3 22 x ( 3) 4 y x m m x + + + +(Cm).1) Kho st s bin thin v v th (C) ca hm s khi m = 1.2) Cho im I(1; 3). Tm m ng thng d: 4 y x + ct (Cm) ti 3 im phn bit A(0; 4), B, C sao cho IBC c din tch bng8 2 .Cu II (2 im): 1) Gii h phng trnh:2 01 4 1 2x y xyx y ' + .2) Gii phng trnh:1 2(cos sin )tan cot 2 cot 1x xx x x+ Cu III (1 im): Tnh gii hn:A = 20cos sin tanlimsinxx x xx xCu IV (1 im): Cho hnh lp phng ABCD.A B C Dcnh bng a. Gi M, N ln lt l trung im ca AB v C D . Tnh th tch khi chp B .A MCN v cosin ca gc to bi hai mt phng (A MCN) v (ABCD).Cu V (1 im): Cho x, y, zl nhng s dng tho mn: 2 2 2x y z xyz + + . Chng minh bt ng thc:2 2 212 zx y zx yz y x z xy+ + + + +II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong mt phng Oxy, cho hai ng trn (C1): 2 213 x y + v (C2): 2 2( 6) 25 x y + . Gi A l mt giao im ca (C1) v (C2) vi yA > 0. Vit phng trnh ng thng d i qua A v ct (C1), (C2) theo hai dy cung c di bng nhau.2) Gii phng trnh:( ) ( )325 1 5 1 2 0x x x+ + + Cu VII.a (1 im): Chng minh rng vi n N*, ta c: 2 4 22 2 22 4 ... 2 42n nn n nnC C nC + + + . 2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong mt phng vi h to Oxy, cho hnh ch nht ABCD c din tch bng 12, tm 9 3;2 2I _ , v trung im M ca cnh AD l giao im ca ng thng d: 3 0 x y vi trc Ox. Xc nh to ca cc im A, B, C, D bit yA > 0.2) Gii bt phng trnh:23 1 13 3log 5x 6 log 2 log 3 x x x + + > + Cu VII.b (1 im): Tm a th hm s 2x x ayx a + ++ (C) c tim cn xin tip xc vi th ca hm s (C ): 3 26x 8x 3 y x + .Trang 41n thi i hcL Anh Tun s 41I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 3 23 1 y x x mx + + +c th (Cm) (m l tham s).1) Kho st s bin thin v v th (C) ca hm s khi m = 3.2) Xc nh m (Cm) ct ng thng d: y = 1 ti 3 im phn bit C(0; 1), D, E sao chocc tip tuyn ca (Cm) ti D v E vung gc vi nhau.Cu II (2 im): 1) Gii phng trnh:2cos3 3sin cos 0 x x x + + 2) Gii h phng trnh:3 3 32 28 27 7 (1)4 6 (2)xy yxy x y + '+ Cu III (1 im): Tnh tch phn: I = 2261sin sin .2x x dx +Cu IV (1 im): Tnh th tch ca khi chp S.ABC, bit y ABC l mt tam gic u cnh a, mt bn (SAB) vung gc vi y, hai mt bn cn li cng to vi y gc .Cu V (1 im): Cho x, y, z l cc s dng tho mn: 1 1 12010x y z+ + . Tm gi tr ln nht ca biu thc: P = 1 1 12 2 2 x y z x y z x y z+ ++ + + + + +II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im): 1) Trong mt phng vi h to Oxy, cho phng trnh hai cnh ca mt tam gic l 5 2 6 0 x y + v4 7 21 0 x y + . Vit phng trnh cnh th ba ca tam gic , bit rng trc tm ca n trng vi gc ta O.2) Trong khng gian vi h to Oxyz, tm trn trc Ox im A cch u ng thng (d) : 1 21 2 2x y z + v mt phng (P): 2 2 0 x y z . Cu VII.a (1 im): Cho tp hp X ={ }0,1, 2, 3, 4, 5, 6, 7. T X c th lp c bao nhiu s t nhin gm 5 ch s khc nhau i mt, sao cho mt trong ba ch s u tin phi bng 1.2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong mt phng vi h to Oxy, cho ng trn (C): x2 + y2 6x + 5 = 0. Tm im M thuc trc tung sao cho qua M k c hai tip tuyn ca (C) m gc gia hai tip tuyn bng 600.2) Trong khng gian vi h to Oxyz, cho hai ng thng: (d1): 24x ty tz' v (d2) : 30x ty tz '. Chng minh (d1) v (d2) cho nhau. Vit phng trnh mt cu (S) c ng knh l on vung gc chung ca (d1) v (d2).Cu VII.b (1 im): Gii phng trnh sau trn tp hp s phc: 4 3 26 8 16 0 z z z z + .Trang 42L Anh Tun n thi i hc s 42I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 2x 41yx +.1) Kho st s bin thin v v th (C) ca hm s.2) Tm trn th (C), hai im i xng nhau qua ng thng MN, bit M(3; 0), N(1; 1).Cu II (2 im):1) Gii phng trnh:41 3 74cos cos 2 cos 4 cos2 4 2xx x x + 2) Gii phng trnh: 3 .2x 3 2x 1x x + +Cu III (1 im): Tnh tch phn: I = 201 sin1 cosxxedxx+ _ + ,Cu IV (1 im): Tnh th tch khi chp S.ABC, bit SA = a, SB = b, SC = c, 0AS 60 B , 0S 90 BC , 0S 120 CA .Cu V (1 im): Cho cc s dng x, y, z tho mn: xyz = 8. Tm gi tr nh nht ca biu thc: P = 2 2 22 2 2log 1 log 1 log 1 x y z + + + + +II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong mt phng vi h to Oxy, cho 2 ng thng d1: 1 0 x y + + v d2: 2x 1 0 y . Lp phng trnh ng thng d i qua M(1; 1) v ct d1, d2 tng ng ti A, B sao cho 2 0 MA MB + uuur uuur r.2) Trong khng gian vi h to Oxyz, cho mt phng (P): 2 2z 1 0 x y + + v hai im A(1; 7; 1), B(4; 2; 0). Lp phng trnh ng thng d l hnh chiu vung gc ca ng thng AB ln mt phng (P).Cu VII.a (1 im): K hiu x1, x2 l cc nghim phc ca phng trnh 22x 2x 1 0 + . Tnh gi tr cc biu thc 211x v 221x.2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong mt phng vi h to Oxy, cho ng trn (C): 2 22x 2 3 0 x y y + v im M(0; 2). Vit phng trnh ng thng d qua M v ct (C) ti hai im A, B sao cho AB c di ngn nht.2) Trong khng gian vi h to Oxyz, cho 3 im A(1; 0; 0), B(0; 2; 0), C(0; 0; 3). Tm to trc tm ca tam gic ABC.Cu VII.b (1 im): Tm cc gi tr x, bit trong khai trin Newton ( )5 lg(10 3 ) ( 2) lg32 2xnx +s hng th 6 bng 21 v 1 3 22n n nC C C + .Trang 43n thi i hcL Anh Tun s 43I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 2x 11yx.1) Kho st s bin thin v v th (C) ca hs.2) Gi I l giao im hai tim cn ca (C). Tm im M thuc (C) sao cho tip tuyn ca (C) ti M vung gc vi ng thng MI.Cu II (2 im): 1) Gii phng trnh:3cos cos cos sin 2 02 6 3 2 2 6x xx x _ _ _ _ + + + , , , ,2) Gii phng trnh:4 2 21 1 2 x x x x + + + Cu III (1 im): Gi (H) l hnh phng gii hn bi cc ng: (C): 2( 1) 1 x y + , (d): 4 y x +. Tnh th tch khi trn xoay to thnh do hnh (H) quay quanh trc Oy. Cu IV (1 im): Cho hnh chp S.ABCD c y ABCD l hnh thoi, cnh a, 060 ABC , chiu cao SO ca hnh chp bng 32a, trong O l giao im ca hai ng cho AC v BD. Gi M l trung im ca AD, mt phng (P) cha BM v song song vi SA, ct SC ti K. Tnh th tch khi chp K.BCDM.Cu V (1 im): Cho cc s dng x, y, z tho mn: 2 2 21 x y z + + . Chng minh:2 2 2 2 2 23 32x y zy z z x x y+ + + + +II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong mt phng vi h to Oxy, cho ng trn (C) c tm O, bn knh R = 5 v im M(2; 6). Vit phng trnh ng thng d qua M, ct (C) ti 2 im A, B sao cho OAB c din tch ln nht.2) Trong khng gian vi h to Oxyz, cho mt phng (P): x 3 0 y z + + + v im A(0; 1; 2). Tm to im Ai xng vi A qua mt phng (P).Cu VII.a (1 im): T cc s 1, 2, 3, 4, 5, 6 thit lp tt c cc s t nhin c 6 ch s khc nhau. Hi trong cc s c bao nhiu s m hai ch s 1 v 6 khng ng cnh nhau.2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong mt phng vi h to Oxy, cho tam gic ABC c nh C(4; 3). Bit phng trnh ng phn gic trong (AD): 2 5 0 x y + , ng trung tuyn (AM): 4x 13 10 0 y + . Tm to nh B.2) Trong khng gian vi h to Oxyz, cho hai ng thng: (d1): 23 810 4x ty tz t + +' v (d2): 3 22 2 1x y z + . Vit phng trnh ng thng (d) song song vi trc Oz v ct c hai ng thng (d1), (d2).Cu VII.b (1 im): Tm a h phng trnh sau c nghim:242 23 4 51 log ( ) log ( 1)xxa x x '+ +Trang 44L Anh Tun n thi i hc s 44I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 2(2 1)1m x myx .1) Kho st s bin thin v v th (C) ca hm s khi m = 1.2) Tm m th ca hm s tip xc vi ng thng y x .Cu II (2 im):1) Gii phng trnh:22 3 cos 2 sin 2 4cos 3 x x x + 2) Gii h phng trnh:2 222x1yx yx yx y x y+ + +'+ Cu III (1 im):Tnh tch phn: I = 230sin(sin cos )xdxx x+Cu IV (1 im): Cho hnh lng tr tam gic ABC.A B C c y l tam gic u cnh bng a, A M (ABC), A M = 32a (M l trung im cnh BC). Tnh th tch khi a din ABA B C.Cu V (1 im): Cho cc s thc x, y. Tm gi tr nh nht ca biu thc:P = 2 2 2 24 4 4 4 4 x y y x y y x + + + + + + + II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong Oxy, cho elip (E): 2 21100 25x y+ . Tm cc im M (E) sao cho 01 2120 FMF(F1, F2 l hai tiu im ca (E)).2) Trong khng gian vi h to Oxyz, cho 3 im A(3; 1; 1), B(7; 3; 9), C(2; 2; 2) v mt phng (P) c phng trnh: 3 0 x y z + + . Tm trn (P) im M sao cho 2 3 MA MB MC + +uuur uuur uuuur nh nht.Cu VII.a (1 im): Gi a1, a2, , a11 l cc h s trong khai trin sau: 10 11 10 91 2 11( 1) ( 2) ... x x x a x a x a + + + + + + . Tm h s a5.2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong mt phng Oxy, cho ng trn (C): 2 2( 3) ( 4) 35 x y + v im A(5; 5). Tm trn (C) hai im B, C sao cho tam gic ABC vung cn ti A.2) Trong khng gian vi h to Oxyz, cho im M(2; 1; 2) v ng thng d: 1 31 1 1x y z . Tm trn d hai im A, B sao cho tam gic ABM u.Cu VII.b (1 im): Gii h phng trnh: 20103 32 22log 2yx yxx yx yxy _ ,'+ + s 45I. PHN CHUNG (7 im)Trang 45n thi i hcL Anh TunCu I (2 im): Cho hm s 22x 3xy++(1).1) Kho st s bin thin v v th (C) ca hm s (1).2) Vit phng trnh tip tuyn ca th (C), bit tip tuyn ct trc honh, trc tung ln lt ti hai im phn bit A, B v tam gic OAB cn ti O.Cu II (2 im): 1) Gii phng trnh:(1 2sin ) cos3(1 2sin )(1 sin )x xx x+ 2) Gii h phng trnh:32 3 2 3 6 5 8 0 x x + Cu III (1 im):Tnh tch phn: I = 23 20(cos 1) cos . x x dxCu IV (1 im): Cho hnh chp S.ABCD c y ABCD l hnh thang vung ti A v D; AB = AD = 2a, CD = a; gc gia hai mt phng (SBC) v (ABCD) bng 060 . Gi I l trung im ca AD. Hai mt phng (SBI) v (SCI) cng vung gc vi mt phng (ABCD). Tnh th tch khi chp S.ABCD theo a.Cu V (1 im): Cho cc s thc dng x, y, z tho mn: ( ) 3 xx y z yz + + . Chng minh:3 3 3( ) ( ) 3( )( )( ) 5( ) x y x z x y x z y z y z + + + + + + + +II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong Oxy, cho hnh ch nht ABCD c giao im hai ng cho AC v BD l im I(6; 2). im M(1; 5) thuc ng thng AB v trung im E ca cnh CD thuc ng thng : 5 0 x y + . Vit phng trnh ng thng AB.2) Trong khng gian Oxyz, cho mt phng (P): 2x 2 4 0 y z v mt cu (S) c phng trnh: 2 2 22x 4 6z 11 0 x y z y + + . Chng minh rng mt phng (P) ct mt cu (S) theo mt ng trn. Xc nh tm v tnh bn knh ca ng trn .Cu VII.a (1 im): Gi 1 2, z z l cc nghim phc ca phng trnh: 22z 10 0 z + + . Tnh gi tr ca biu thc: A = 2 21 2z z + .2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong Oxy, cho ng trn (C): 2 24x 4 6 0 x y y + + + + v ng thng c phng trnh: 2 3 0 x my m + + . Gi I l tm ng trn (C). Tm m ct (C) ti hai im phn bit A v B sao cho din tch tam gic IAB ln nht.2) Trong khng gian vi h to Oxyz, cho mt phng (P): 2 2z 1 0 x y + v hai ng thng 1, 2 c phng trnh 1: 1 91 1 6x y z + + , 2: 1 3 12 1 2x y z + . Xc nh to im M thuc ng thng 1 sao cho khong cch t M n ng thng 2 bng khong cch t M n mt phng (P).Cu VII.b (1 im): Gii h phng trnh: 2 22 22 2log ( ) 1 log ( )3 81x xy yx y xy + + +' s 46Trang 46L Anh Tun n thi i hcI. PHN CHUNG (7 im)Cu I (2 im): Cho hm s3 212 3 .3y x x x + .1) Kho st s bin thin v v th (C) ca hm s.2) Vit phtrnh tip tuyn ca th (C), bit tip tuyn ny i qua gc ta O.Cu II (2 im):1) Gii phng trnh:2 sin 2 3sin cos 24x x x _+ + + ,.2) Gii h phng trnh:2 23 32 12 2y xx y y x ' Cu III (1 im):Tm cc gi tr ca tham s m phng trnh:22 2 2 x x x m + + c 2 nghim phn bit.Cu IV (1 im): Cho hnh chp t gic u S.ABCD c tt c cc cnh u bng a. Tnh theo a th tch khi chp S.ABCD v tnh bn knh mt cu tip xc vi tt c cc mt ca hnh chp .Cu V (1 im): Vi mi s thc x, y tha iu kin( )2 22 1 x y xy + + . Tm gi tr ln nht v gi tr nhnht ca biu thc: 4 42 1x yPxy++.II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Gii phng trnh:2.27 18 4.12 3.8x x x x+ + .2) Tm nguyn hm ca hm s( )2tan1 cosxxxf+ .Cu VII.a (1 im): Trong khng gian vi h ta Oxyz, cho im( )1; 2;3 I . Vit phng trnh mt cu tm I v tip xc vi trc Oy.2. Theo chng trnh nng caoCu VI.b (2 im): 1) Gii bt phng trnh: 34 log243xx+>.2) Tm m hm s 21 mxyxc 2 im cc tr A, B v on AB ngn nht.Cu VII.b (1 im): Trong mt phng ta Oxy, cho ng trn ( )2 2: 2 0 C x y x + + . Vit phng trnh tip tuyn ca ( )C , bit gc gia tip tuyn ny v trc tung bng 30o. s 47I. PHN CHUNG (7 im)Trang 47n thi i hcL Anh TunCu I (2 im): Cho hm s4 2 2 42 2 y x mx m m + +(1), vi m l tham s.1) Kho st s bin thin v v th (C) ca hm s khi m = 1.2) Chng minh th hm s (1) lun ct trc Ox ti t nht hai im phn bit, vi mi0 m < .Cu II (2 im):1) Gii phng trnh:2sin 2 4sin 16x x _+ + ,2) Tm cc gi tr ca tham s m sao cho h phng trnh 21y x my xy '+ c nghim duy nht.Cu III (1 im):Tm nguyn hm ca hm s ( )( )241x)2 1(xxf+ .Cu IV (1 im): Cho khi t dinABCD. Trn cc cnh BC, BD, AC ln lt ly cc im M, N, P sao cho4 BC BM ,2 BD BN v3 AC AP . Mt phng (MNP) chia khi t din ABCD lm hai phn. Tnh t s th tch gia hai phn .Cu V (1 im): Vi mi s thc dng ; ; xyztha iu kin 1 x y z + + . Tm gi tr nh nht ca biu thc:1 1 12 P x y zx y z _ + + + + + ,.II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Gii phng trnh: 4 2log log2 8x xx .2) Vit phng trnh cc ng thng ct th hm s 12xyx ti hai im phn bit sao cho honh v tung ca mi im u l cc s nguyn.Cu VII.a (1 im): Trong mt phng vi h ta Oxy, cho ng thng ( ): 2 4 0 d x y . Lp phng trnh ng trn tip xc vi cc trc ta v c tm trn ng thng (d).2. Theo chng trnh nng caoCu VI.b (2 im):1) Gii bt phng trnh:( )2 4 82 1 log log log 0 x x x + + 0, n > 0. Tnh khong cch t A n mt phng (SMN). T suy ra mt phng (SMN) tip xc vi mt mt cu c nh.Cu VII.b (1 im): Gii bt phng trnh: ( )1224 2.2 3 .log 3 44xx x xx+ > s 53I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 2 11xyx.1) Kho st s bin thin v v th (C) ca hm s.Trang 53n thi i hcL Anh Tun2) Lp phng trnh tip tuyn ca th (C) sao cho tip tuyn ny ct cc trc Ox ,Oy ln lt ti cc im A v B tha mn OA = 4OB.Cu II (2 im): 1) Gii phng trnh:sin cos2tan 2 cos 2 0sin cosx xx xx x++ + 2) Gii h phng trnh:3 2 2 32 2(1 ) (2 ) 30 0(1 ) 11 0xy y xy y xyxy x y y y + + + + '+ + + + Cu III (1 im):Tnh tch phn: I = 1011xdxx++Cu IV (1 im): Cho lng tr ng ABC.A B Cc y ABC l tam gic vung vi AB = BC = a, cnh bn AA= a 2 . M l im trn AAsao cho 1AA'3AMuuuur uuuur. Tnh th tch ca khi t din MA BC .Cu V (1 im): Cho cc s thc dng a, b, c thay i lun tha mn1 a b c + + . Chng minh rng:2 2 22.a b b c c ab c c a a b+ + ++ + + + +II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong mt phng vi h to Oxy, cho im E(1; 0) v ng trn (C): 2 28 4 16 0 x y x y + . Vit phng trnh ng thng i qua im E ct (C) theo dy cung MN c di ngn nht.2) Trong khng gian vi h to Oxyz, cho 2 im A(0; 0; 4), B(2; 0; 0) v mt phng (P): 2x 5 0 y z + + . Lp phng trnh mt cu (S) i qua O, A, B v c khong cch t tm I ca mt cu n mt phng (P) bng 56.Cu VII.a (1 im): C bao nhiu s t nhin gm 7 ch s, bit rng ch s 2 c mt ng hai ln, ch s 3 c mt ng ba ln v cc ch s cn li c mt khng qu mt ln?2. Theo chng trnh nng caoCu VI.b (2 im):1) Trong mt phng vi h to Oxy, cho tam gic ABC cn ti A, bit phng trnh ng thng AB, BC ln lt l: 2 5 0 x y + v 3 7 0 x y + . Vit phng trnh ng thng AC, bit rng AC i qua im (1; 3) F .2) Trong khng gian vi h to Oxyz, cho hai im A(1; 5; 0), B(3; 3; 6) v ng thng : 1 12 1 2x y z + . Tm to im M trn sao cho MAB c din tch nh nht.Cu VII.b (1 im): Tm tt c cc gi tr ca tham s a phng trnh sau c nghim duy nht: 5 5log (25 log )xa x s 54I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 4 2 22 1 y x mx + + (1).1) Kho st s bin thin v v th (C) ca hm s khi m = 1.Trang 54L Anh Tun n thi i hc2) Chng minh rng ng thng 1 y x + lun ct th hm s (1) ti hai im phn bit vi mi gi tr ca m.Cu II (2 im):1) Gii phng trnh:2 22sin 2sin tan4x x x _ ,2) Gii h phng trnh:( )2 2 23 3 32log 4 3 log ( 2)log ( 2) 4 x x x + + Cu III (1 im):Tnh tch phn: I = 320sincos 3 sinxdxx x+Cu IV (1 im): Cho tam gic vung cn ABC c cnh huyn AB = 2a. Trn ng thng d i qua A v vung gc mt phng (ABC) ly im S sao cho mp(SBC) to vi mp(ABC) mt gc bng 600. Tnh din tch mt cu ngoi tip t din SABC.Cu V (1 im): Tm gi tr nh nht ca hm s: 4 3 224 8 8 5( )2 2x x x xf xx x + + + II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong Oxy, choelp (E) c tiu im th nht l( ) 3; 0 v i qua im 4 331;5M _ ,. Hy xc nh ta cc nh ca (E).2) Trong khng gian vi h to Oxyz, cho im A(0; 1; 3) v ng thng d: 12 23x ty tz +'. Hy tm trn ng thng d cc im B v C sao cho tam gic ABC u.Cu VII.a (1 im): Chng minh: 2 1 2 2 2 3 2 2 21 2 3 ... ( ).2n nn n n nC C C n C n n+ + + + + , trong n l s t nhin, n 1 v knCl s t hp chp k ca n.2. Theo chng trnh nng caoCu VI.b (2 im): 1) Trong mt phng vi h to Oxy, cho tam gic ABC c A(2; 7) v ng thng AB ct trc Oy ti E sao cho E 2 A EB uuur uuur. Bit rng tam gic AEC cn ti A v c trng tm l 132;3G _ ,. Vit phng trnh cnh BC.2) Trong khng gian vi h to Oxyz, cho ng thng d: 1 13 1 1y x z + v mt phng (P): 2x 2z 2 0 y + + . Lp phng trnh mt cu (S) c tm nm trn ng thng d c bn knh nh nht tip xc vi (P) v i qua im A(1; 1; 1).Cu VII.b (1 im): Gii h phng trnh: 3 32 24 161 5(1 )x y y xy x + +'+ +. s 55I. PHN CHUNG (7 im)Cu I (2 im): Cho hm s 3 23 2 y x x + .1) Kho st s bhin v v th (C) ca hs.2) Bin lun theo m s nghim ca phng trnh : 22 21mx xx .Trang 55n thi i hcL Anh TunCu II (2 im): 1) Gii phng trnh:52 2 cos sin 112x x _ ,2) Gii h phng trnh:2 82 2 2 2log 3log ( 2)1 3x y x yx y x y+ + '+ + Cu III (1 im):Tnh tch phn:424sin1xI dxx x+ +Cu IV (1 im): Cho hnh chp S.ABCD c y ABCD l hnh ch nhtvi AB = a , AD = 2a . Cnh SA vung gc vi mt phng y, cnh bn SB to vi mt phng y mt gc 060 . Trn cnh SAly im M sao choAM =33a, mt phng (BCM) ct cnh SD ti N. Tnh th tch khi chp S.BCNM.Cu V (1 im): Cho x , y , z l ba s thc tha mn :55 5 1x y z + + .Chng minh rng : 25 25 255 5 5 5 5 5x y zx y z y z x z x y + + ++ ++ + +5 5 54x y z+ +II. PHN T CHN (3 im)1. Theo chng trnh chunCu VI.a (2 im):1) Trong Oxy, cho tam gic ABC vi A(1; 2),ng cao : 1 0 CH x y + , phn gic trong : 2 5 0 BN x y + + . Tm to cc nh B, C v tnh din tch tam gic ABC.2) TrongOxyz, cho hai ng thng : 12 1:4 6 8x y zd + , 27 2:6 9 12x y zd a) Chng minh rng d1 v d2 song song . Vit phng trnh mt phng (P)quad1 v d2 . b) Cho im A(1; 1; 2), B(3; 4; 2). Tm im I trn t d1 sao cho IA + IB t GTNN.Cu VII.a (1 im): Gii phng trnh sau trn tp s phc: 24 31 02zz z z + + + 2. Theo chng trnh nng caoCu VI.b (2 im): 1) Trong mt phng vi h to Oxy, cho hnh ch nht ABCD c din tch bng 12, tm Il giao im ca ng thng 1: 3 0 d x y v 2: 6 0 d x y + . Trung im ca mt cnh l giao im ca d1 vi trc Ox. Tm to cc nh ca hnh ch nht.2) Trong Oxyz, cho hai ng thng: 12 1:1 1 2x y zd v 22 2: 3x td yz t 'a) CMRd1 v d2 cho nhau v vit ptrnh ng vung gc chung ca d1 v d2. b) Vit phng trnh mt cu c ng knh l on vung gc chung ca d1 v d2. Cu VII.b (1 im): Tnh tng: 0 4 8 2004 20082009 2009 2009 2009 2009... S C C C C C + + + + +Trang 56