Đề thi thử môn toán THPT 2016 là tài liệu dành cho các bạn chuẩn bị ôn thi đại học.Đây là đề thi rất hay và các câu hỏi sát với cuộc thi hơn.Bộ đề gồm 5 đề thi toán THPT có đáp án rất hay.Chúc các bạn có tài liệu ôn thi thật tốt .
http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! DeThiThu.Net Truy c p http://dethithu.net th ng xuyờn c p nh t nhi u Thi Th THPT Qu c Gia, ti li u ụn thi THPT Qu c Gia cỏc mụn Toỏn, Lý, Húa, Anh, V n, Sinh , S , a c DeThiThu.Net c p nh t h ng ngy ph c v s t ! Thi Th THPT Qu c Gia - Ti Li u ễn Thi: Like Fanpage http://facebook.com/dethithu.net c p nh t nhi u thi th v ti li u ụn thi h n Tham gia Group: ễn Thi H Toỏn - Anh cựng h c t p, ụn thi: http://facebook.com/groups/onthidhtoananhvan Thi Th K THI THPT QUC GIA MễN TON Web: http://dethithu.net Mail: dethithu.net@gmail.com Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Mc lc s 06 s 07 s 08 s 09 s 10 P N ỏp ỏn s 06 ỏp ỏn s 07 13 ỏp ỏn s 08 16 ỏp ỏn s 09 20 ỏp ỏn s 10 24 Cỏc Nh ng thi th v ỏp ỏn s 1, 2, 3, , cỏc b n vui lũng xem t i: http://dethithu.net thi th Like Fanpage ú website ó c p nh t ng riờng r t tr c nờn khụng g p chung vo thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! thi th s http://dethithu.net THI TH I HC (THPT QG) MễN: TON thi th s Thi gian lm bi: 180 phỳt Cõu 1.(2 im) Cho hm s y = x x + a) Kho sỏt s bin thiờn v v th (C ) ca hm s b) Tỡm m phng trỡnh x x + = m cú ba nghim thc phõn bit Cõu 2.(1 im) http://dethithu.net a) Gii phng trỡnh: sin x + cos( x ) = b) Tớnh mụun ca s phc z bit z = + 2i + i + 1+ i 2 Cõu 3.(0,5 im) Gii phng trỡnh: log x + log x = ( )( ) y x + y y +1 = Cõu 4.(1 im) Gii h phng trỡnh: x 2 x + x y + xy = y ( ) Cõu 5.(1 im) Tớnh tớch phõn: I = + cos x dx http://dethithu.net Cõu 6.(1 im) Cho lng tr ng ABC.ABC cú ỏy ABC l tam giỏc vuụng ti A,AB=2a,AC=a,AA = 3a Tớnh th tớch lng tr v khong cỏch gia hai ng thng AB v BC Cõu 7.(1 im) Trong mt phng vi h ta Oxy cho hỡnh ch nht ABCD cú nh B thuc ng trũn x + y = 10 ,nh C thuc ng thng x + y = Gi M l hỡnh chiu vuụng gúc ca B lờn AC Bit rng cỏc im N ; ,P (1;1) ln lt l trung im ca AM ,CD ng thi B cú honh dng,C cú tung 5 http://dethithu.net õm.Tỡm ta cỏc nh ca hỡnh ch nht Cõu 8.(1 im) Trong khụng gian vi h ta Oxyz cho mt cu (S ) : x + y + z x y + z 15 = 0, mt phng (P): x y + z + 13 = Tỡm tõm v bỏn kớnh ca mt cu (S ) Vit phng trỡnh mt phng (Q) song song vi mt phng (P) ng thi tip xỳc vi (S ) http://dethithu.net Cõu 9.(0,5 im) Gi A l hp cỏc s t nhiờn cú ch s ụi mt khỏc c to t cỏc ch s 0,1,2,3,4,5,6 Chn ngu nhiờn mt s t A Tớnh xỏc sut s c chn cú cỏc ch s khỏc ch s v http://dethithu.net tng cỏc ch s l Cõu 10.(1 im) Cho ba s thc a,b,c tha món: log a + log b + log 32 c = Tỡm giỏ tr ln nht ca biu thc P = 1+ a + 1+ b + 1+ c2 Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! http://dethithu.net THI TH K THI THPT QUC GIA s Mụn : TON Thi gian lm bi 180 phỳt x+2 x1 a) Kho sỏt s bin thiờn v v th (C) ca hm s ó cho b) Tỡm m ng thng d : y = x + m ct th (C) ti hai im phõn bit http://dethithu.net Cõu (1,0 im) cos 2x + (1 + cos2 x) tan x = + sin2 x a) Gii phng trỡnh cos x b) Tỡm hp cỏc im biu din s phc z tha |z i| = |2z 2i| Cõu (2,0 im) Cho hm s y = Cõu (0,5 im) Tỡm giỏ tr ln nht v giỏ tr nh nht ca hm s y = x (x2 x 1) trờn on [0; 2] http://dethithu.net x +x + =0 x x3 y Cõu (1,0 im) Gii h phng trỡnh 3x + =4 y2 Cõu (1,0 im) Tớnh tớch phõn I = x2 ln (x 1) dx http://dethithu.net Cõu (1,0 im) Cho hỡnh chúp S.ABCD cú ỏy ABCD l hỡnh bỡnh hnh, mt phng (SBD) vuụng gúc vi ỏy, cỏc ng thng SA, SD hp vi ỏy mt gúc 300 Bit AD = a 6, BD = 2a v gúc ADB = 450 Tớnh th tớch chúp S.ABCD v khong cỏch t nh C n mt phng http://dethithu.net (SAD) theo a Cõu (1,0 im) Trong mt phng vi h ta Oxy, im M (2; 1) v hai ng thng d1 : x y + = 0, d2 : x + 2y = Vit phng trỡnh ng thng qua M v ct d1 , d2 ln lt ti A, B cho M l trung im AB http://dethithu.net x1 y+3 Cõu (1,0 im) Trong khụng gian vi h ta Oxyz, cho ng thng d : = = z3 v mt phng (P ) : 2x + y 2z + = Vit phng trỡnh ng thng nm (P ), ct v vuụng gúc vi d http://dethithu.net Cõu (0,5 im) Cú 14 tm th c ỏnh s t n 14 Chn ngu nhiờn tm th Tớnh xỏc sut tm th c chn cú tm th mang s l,4 tm th mang s chn ú cú nht mt tm th mang s chia ht cho http://dethithu.net Cõu 10 (1,0 im) Cho cỏc s thc dng a, b, c tha a + b + c = Chng minh rng : 2ab 3bc 2ca + + c + ab a + bc b + ca http://dethithu.net Ht Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! http://dethithu.net THI TH K THI THPT QUC GIA s Mụn : TON Thi gian lm bi 180 phỳt Cõu (2,0 im) Cho hm s y = x3 + 3x2 a) Kho sỏt s bin thiờn v v th (C) ca hm s ó cho b) Tỡm m ng thng d : y = m (x + 1) ct th (C) ti ba im phõn bit M (1; 0) , A, B cho M A = 2M B http://dethithu.net Cõu (1,0 im) a) Gii phng trỡnh sin 2x + cos 2x cos 4x = 3i 3+i + b) Tỡm phn thc, phn o v mụ un ca s phc z = 1i 2i x+8 2x+3 Cõu (0,5 im) Gii phng trỡnh 3.243 x+8 = 32 x+2 Cõu (1,0 im) Gii h phng trỡnh 2x2 y + y = 2x4 + x6 (x + 2) y + = (x + 1)2 http://dethithu.net Cõu (1,0 im) Tớnh th tớch trũn xoay to thnh quay quanh Ox hỡnh phng gii hn bi th hm s y = sin x, trc honh v hai ng thng x = 0, x = Cõu (1,0 im) Cho hỡnh chúp S.ABCD cú ỏy ABCD l hỡnh thoi cnh a; gúc BCD = 600 ; cnh SA vuụng gúc vi mt phng (ABCD) Hai mt phng (SCB) v (SCD) vuụng gúc vi Tớnh th tớch chúp S.ABCD v khong cỏch t C n mt phng (SBD) Cõu (1,0 im) Trong mt phng vi h ta Oxy, cho tam giỏc ABC cú tõm ng trũn ngoi tip tam giỏc ABC l I(2; 1) v tha iu kin AIB = 900 , chõn ng cao k t A n BC l D(1; 1), ng thng AC i qua im M (1; 4) Tỡm ta cỏc nh A, B bit rng nh A cú honh dng http://dethithu.net x=1+t y = t Cõu (1,0 im) Trong khụng gian vi h ta Oxyz, cho hai ng thng d1 : z=2 y1 z x3 = = Chng minh d1 v d2 chộo Vit phng trỡnh mt phng (P ) v d2 : cha d1 v song song vi d2 http://dethithu.net Cõu (0,5 im) Gi A l hp cỏc s cú ch s khỏc c lp t cỏc ch s 1, 2, 3, 4, Chn ngu nhiờn ba s t A, tớnh xỏc sut ba s c chn cú ỳng mt s cú mt ch s http://dethithu.net Cõu 10 (1,0 im) Cho cỏc s thc x, y tha x2 + y + xy = Tỡm giỏ tr ln nht v giỏ tr nh nht ca biu thc P = x3 + y 3x 3y Ht Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! http://dethithu.net THI TH K THI THPT QUC GIA s Mụn : TON Thi gian lm bi 180 phỳt 2x + x1 a) Kho sỏt s bin thiờn v v th (C) ca hm s b) Tỡm m ng thng d i qua A(2; 2) v cú h s gúc m ct th (C) ti hai im phõn bit thuc hai nhỏnh ca (C) http://dethithu.net Cõu (1,0 im) sin + cos a) Cho cot = Tớnh giỏ tr ca biu thc A = 2 sin cos b) Gii phng trỡnh z + 2z + z 18 = trờn hp cỏc s phc C Cõu (2,0 im) Cho hm s y = x+1 Cõu (0,5 im) Gii bt phng trỡnh log 2x1 > Cõu (1,0 im) Gii h phng trỡnh: http://dethithu.net x3 x2 + x = y y y + x3 + 4x2 + = y e x + ln x dx x2 Cõu (1,0 im) Tớnh tớch phõn I = Cõu (1,0 im) Cho hỡnh chúp S.ABCD cú ỏy ABCD l hỡnh vuụng cnh a Mt bờn SAD l tam giỏc u v SB = a Gi E, F ln lt l trung im ca AD v AB Gi H l giao im ca F C v EB Chng minh SEEB, CHSB v tớnh th tớch chúp C.SEB Cõu (1,0 im) Trong mt phng vi h to Oxy, cho tam giỏc ABC cú A(1; 4), tip tuyn ti A ca ng trũn ngoi tip tam giỏc ABC ct BC ti D, ng phõn giỏc ca ADB cú phng trỡnh x y + = 0, im M (4; 1) thuc cnh AC Vit phng trỡnh ng thng AB http://dethithu.net x y+2 Cõu (1,0 im) Trong khụng gian vi h ta Oxyz, cho hai ng thng d1 : = = x1 y6 z z+4 v d2 : = = Tỡm im A trờn d1 , im B trờn d2 cho ng thng AB 2 http://dethithu.net i qua im M (1; 9; 0) Cõu (0,5 im) Mt t cú hc sinh nam v hc sinh n Giỏo viờn chn ngu nhiờn hc sinh lm trc nht Tớnh xỏc sut hc sinh c chn cú c nam v n Cõu 10 (1,0 im) Cho cỏc s thc khụng õm a, b, c tha a + b + c = Tỡm giỏ tr ln nht ca biu thc : ca ab bc + + P = 3a + bc 3b + ca 3c + ab http://dethithu.net Ht Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! http://dethithu.net THI TH K THI THPT QUC GIA s 10 Mụn : TON Thi gian lm bi 180 phỳt Cõu (2,0 im) Cho hm s y = x3 3x + a) Kho sỏt s bin thiờn v v th (C) ca hm s ó cho b) Tỡm im A thuc (C) cho tip tuyn vi (C) ti A ct (C) ti im B khỏc A tha 2013xB + 2014xA = 2012, ú xA , xB ln lt l honh ca A v B http://dethithu.net Cõu (1,0 im) a) Gii phng trỡnh sin 2x (2 cos x 5) + cos 2x + sin x cos x + = b) Tỡm hp cỏc im biu din mt phng phc Oxy ca s phc w = + i z +2 bit rng s phc z tha |z 1| http://dethithu.net Cõu (0,5 im) Tỡm xỏc nh ca hm s y = log2 + log (x2 + 3x) x + x2 2x + = 3y + y + x2 y 3x + 3y + = Cõu (1,0 im) Gii h phng trỡnh Cõu (1,0 im) Tớnh tớch phõn I = xe2x + xex + dx ex + http://dethithu.net Cõu (1,0 im) Cho hỡnh chúp S.ABCD cú SB = SC = SD = AB = BC = CD = DA = a v gúc gia hai mt phng (SAB) v (SAD) bng 900 Tớnh th tớch chúp S.ABCD Cõu (1,0 im) Trong mt phng vi h ta Oxy, cho hỡnh bỡnh hnh ABCD cú gúc ABC nhn, nh A(2; 1) Gi H, K, E ln lt l hỡnh chiu vuụng gúc ca A trờn cỏc ng thng BC, BD, CD Phng trỡnh ng trũn ngoi tip tam giỏc HKE l (C) : x2 + y + x + 4y + = Tỡm ta cỏc nh B, C, D bit H cú honh õm, C cú honh dng v nm trờn ng thng x y = http://dethithu.net x7 = Cõu (1,0 im) Trong khụng gian vi h ta Oxyz, cho hai ng thng : z1 x1 y+2 z5 y2 = v : = = Tỡm ta giao im A ca v Vit phng 2 trỡnh mt phng () cha v http://dethithu.net Cõu (0,5 im) Tỡm h s ca s hng cha x2 khai trin nguyờn dng tha 2Cn + 2n+1 n 6560 22 23 Cn + Cn + + Cn = n+1 n+1 x+ 24x n , bit n l s http://dethithu.net Cõu 10 (1,0 im) Cho cỏc s thc dng x, y tha x + y < Chng minh rng : + + x y 1xy 36 Ht Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website ỏp ỏn Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! thi th s P N THANG IM THI TH I HC (THPTQG) s Website: http://dethithu.net Mụn: TON (ỏp ỏn thang im gm trang) Cõu ỏp ỏn a (1,0 im) (2,0 +) Tp xỏc nh D = R im) +) S bin thiờn: Chiu bin thiờn: y ' = x x, y ' = x = x = Gii hn: lim y = + , lim y = x + im 0,25 0,25 x Bng bin thiờn: + http://dethithu.net 0,25 + Hm s ng bin trờn cỏc khong ( ;0) v (2;+ ) ,nghch bin trờn khong (0;2) Hm s t cc i ti x = 0, y C = ,hm s t cc tiu ti x = 2, y CT = +) th th hm s i qua cỏc im (0;1),(1;-1),(2;-3) th i xng qua im I ( 1; -1) 0,25 b (1,0 im) S nghim ca phng trỡnh x x + = m chớnh l s giao im ca th (C ) vi ng thng y = m T th suy phng trỡnh cú nghim thc phõn bit v ch m a (0,5 im) (1,0 Ta cú : sin x + cos( x ) = sin x cos x cos x = cos x(sin x 2) = im) ( sin x = khụng xy ra) cos x = x = + k , k Z http://dethithu.net 0,5 0,5 0,25 0,25 b (0,5 im) Ta cú z = + 2i + i (1 + 2i) (1 i ) + i 3i + i + = + = + 1+ i (1 + i) (1 i ) 2 Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 0,25 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! =1 i vy z = 12 + (1) = 0,25 3 (0,5 im) (0,5 k: x > ,Pt log x + log x = (log x 1)(log x + ) = im) x = log x = (TM ) http://dethithu.net x = log x = 25 0,25 0,25 (1,0 k: x im) Ta cú x + x y + xy = y ( x y )(x + xy + y ) = ( x y ) ( x + y )2 + y = [ ] x = y ( x + y ) + y = x = y = khụng tha mn 2 ( ] )( x + 1) = , x (0;1] x )( x + 1) = 3( x + 1) = ( x + + x ) (*) Vi x = y t phng trỡnh u ta cú ( [ x+3 x Khi ú x + Ta thy x = l mt nghim ca phng trỡnh (*) Vi < x < thỡ ( ) x + > cũn ( 0,25 ) x + + x < nờn (*) vụ nghim.Võy (*) cú nghim nht l x = t ú suy h cú nghim nht (x; y) = (1;1) 2 (1,0 Ta cú I = + cos x dx = dx + cos xdx im) 0 ( ) Trong ú: 0,25 0,25 dx = x = 0,25 http://dethithu.net 0 2 0 ( ) Xột K = cos xdx = cos x cos xdx = sin x cos xdx t t = sin x suy dt = cos x.dx , x = t = 0, x = (1 2t ) t + t dt = t t + Vy I = + 15 Th tớch lng tr l (1,0 V = AA'.S ABC = AA' AB AC im) = 3a 2a.a = 3a = 0,5 0,25 (1 t ) dt t = ú K = 2 = 15 0,25 0,5 Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Gi M v M ' ln lt l chõn ng cao h t A v A' cỏc tam giỏc ABC , A' B ' C ' ta cú B' C ' ( AA' M ' M ) nờn ( AB'C ') ( AA' M ' M ) Trong mp ( AA' M ' M ) h MH AM ' thỡ MH ( AB ' C ' ) Khi ú d ( AB ', BC ) = d ( BC ,( AB 'C ' )) = d (M ,( AB 'C ")) = MH 1 1 1 http://dethithu.net = + = + + 2 2 2 MH MM ' AM MM ' AB AC 1 1 49 6 => = + + = => MH = a Vy d ( AB ', BC ) = a 2 7 9a 4a 36a MH a m Gi Q l trung im BM thỡ PCQN l hỡnh bỡnh hnh nờn NP // CQ,mt khỏc Q l trc tõm (1,0 tam giỏc BNC nờn CQ BN suy BN NP im) Ta cú NP = ; l mt vộct phỏp tuyn ca ng thng BN nờn phng trỡnh ng 5 thng BN l x + + x = x + y + = Ta 5 5 B l nghim ca h x = x + y + = y = x x = 2 y = y = 13 x + y = 10 x + x + = suy B (1;3) vỡ B cú honh dng 0,25 0,25 0,25 0,25 Gi C (1 2c; c ) ta cú CB = (2c;3 c ) , CP = (2c;1 c ) CB CP nờn CP.CB = 4c ( + c )(1 c ) = 5c + 2c = c = c = C cú tung õm nờn C (3;1) Suy D( 1;3) , A(3;1) Vy A(3;1) , B (1;3) , C (3;1) , D( 1;3) http://dethithu.net 15 +1+ + =3 (1,0 Mt cu cú tõm I ;1;2 v bỏn kớnh R = 4 im) Do Mp (Q ) song song vi mp (P ) nờn phng trỡnh cú dng x y + z + D = 0, D 13 (Q ) tip xỳc vi (S ) nờn d ( I ,( Q ) ) = R => 11 + D =3 D = D = 13 D = , D 13 nờn ta ly D = vy phng trỡnh cn tỡm l 2x y + 2z = 0,25 0,25 0,25 0,25 0,25 0,25 Ký hiu abc l mt s bt k thuc A 0,25 Ta thy a cú cỏch chn (a 0) ,b cú cỏch chn (b a ) tng t c cú cỏch chn (0,5 http://dethithu.net Vy s phn t ca A l 6.6.5 = 180 im) Xột s abc cú cỏc ch s khỏc ch s v tng cỏc ch s l t cỏc ch s ó cho ta chon c c b s {a; b; c}l {a; b; c} = { ;3;4}v {a; b; c} = { ;2;5} T mi b trờn ta to c 3!= s nờn ta cú 0,25 1 10 Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! t sin x + cos x = t |t| ọ sin x cos x = t2 , phng trỡnh (2) tr thnh : t2 = t2 + 2t = t+ ủ t=1 t = (loi) x = k2 ó Vi t = sin x + cos x = sin x + = x = + k2 (loi) 2 Vy phng trỡnh cú nghim x = + k, x = k2 (k Z ) Cõu 2b (0,5 im) Gi z = x + yi ( x, y R ), ta cú : http://dethithu.net |z i | = |2z 2i | | x + yi i | = |2 ( x yi ) 2i | ( x 2)2 + (y 1)2 = 4x2 + (2y + 2)2 3x2 + 3y2 + 4x + 10y = 10 x + y2 + x + y = 3 Vy hp cỏc im biu din s phc z l ng trũn tõm ; , bỏn kớnh R = 3 ầ Cõu (0,5 im) ủ x=1 x = (loi) Ta cú y(0) = 1, y(1) = e, y(2) = e2 , ú max y = y(2) = e2 , y = y(1) = e o hm y = ex ( x2 x 1) + ex (2x 1) = ex ( x2 + x 2); y =0 [0;2] [0;2] Cõu (1,0 im) iu kin x = 0, y = H ó cho tng ng vi : x + =2 x2 y 3x + = y2 x + Tr theo v (1) v (2) ta cú : x2 + ầ x 1 +2 = x x y y y http://dethithu.net x +2 = x y 3x2 + = y2 2x + ồ2 (1) (2) = x x = y x y = x x = = x+ y x y x x = x thay vo (1) c 4x2 + = x = y = y x x Vi = x + thay vo (1) c 4x2 + = (vụ nghim) y x x http://dethithu.net Vy h cú hai nghim ( x; y) = (1; 1) v ( x; y) = (1; 1) Vi Cõu (1,0 im) Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 13 http://dethithu.net - Website t đ Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! dx du = u = ln( x 1) x1 , ta cú : x3 dv = x2 dx v= x3 ln( x 1) I= 3 = Vy I = 125 ln 3 http://dethithu.net 125 ln x3 dx = x1 3 x3 x2 + + x + ln( x 1) 124 ln 35 = 5ầ dx x +x+1+ x1 124 ln 35 http://dethithu.net Cõu (1,0 im) ữ Ta cú S ABCD = 2SADB = DA.DB sin ADB = a 6.2a sin 450 = 2a2 Gi H l hỡnh chiu ca S trờn BD SH ( ABCD ) (vỡ (SBD )( ABCD )) Khi ú H A, HD ln lt l hỡnh chiu ca SA, SD lờn ( ABCD ) ữ ữ Do ú SAH = SDH ln lt l gúc ca SA, SD vi ( ABCD ) ữ ữ Theo gi thit ta cú SAH = SDH = 300 H A = HD AD Suy tam giỏc H AD vuụng cõn ti H H A = HD = = a Trong tam giỏc SH A vuụng ti H cú SH = H A tan 300 = a = a 3 2a Vy th tớch chúp S.ABCD l VS.ABCD = S ABCD SH = 3 S http://dethithu.net I A B H K D C BD d( H, (SAD )) = d( H, (SAD )) HD Gi K l trung im AD HK AD AD (SHK ) Gi I l hỡnh chiu ca H trờn SK H I (SAD ) H I = d( H, (SAD )) a Tam giỏc H AD vuụng ti H nờn HK = AD = 2 1 1 a 15 Trong SHK cú = + = + HI = H I2 HS2 HK2 a 3a 2a Vy d(C, (SAD )) = d( H, (SAD )) = H I = 3 http://dethithu.net Cõu (1,0 im) Ta cú A d1 A(t; t + 2); M l trung im AB nờn B(4 t; t) Khi ú B d2 nờn t 2t = t = A (1; 3) , B (3; 1) AB = (2; 4) ng thng cn tỡm qua M(2; 1) v cú mt vect phỏp tuyn = (2; 1) n Vy ng thng cn tỡm cú phng trỡnh 2( x 2) + (y 1) = 2x + y = Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 14 Ta cú d(C, (SAD )) = d( B, (SAD )) = http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Cõu (1,0 im) Ta giao im A ca d v ( P) l nghim h x y+3 z3 = = 2x + y 2z + = x =0 y = z=4 A(0; 1; 4) http://dethithu.net = (2; 1; 2) Mt phng ( P) cú mt vect phỏp tuyn n( P) ng thng d cú mt vect ch phng dợ = (1; ú 1) u 2; u n ng thng i qua A(0; 1; 4) v nhn , = (5; 0; 5) lm mt vect ch phng ( P) x = 5t Vy cú phng trỡnh y = z = + 5t d Cõu (0,5 im) Phộp th l chn ngu nhiờn tm th 14 tm th nờn ta cú || = C14 = 3432 Gi A l bin c "trong tm th c chn cú tm th mang s l, tm th mang s chn ú cú nht mt tm th mang s chia ht cho 5", ta cú hai trng hp sau : TH1 : Chn th chia ht cho l th ghi s Ta cn chn thờm th ghi s l th ghi s l (tr th ghi s 5) v th ghi s chn th ghi s chn (tr th ghi s 10) Do ú trng hp ny cú C6 ì C6 = 225 cỏch chn TH2 : Chn th chia ht cho l th ghi s 10 Ta cn chn thờm th ghi s l th ghi s l (tr th ghi s 5) v th ghi s chn 3 th ghi s chn (tr th ghi s 10) Do ú trng hp ny cú C6 ì C6 = 400 cỏch chn Do ú s kt qu thun li cho bin c A l | A | = 225 + 400 = 625 625 | A | = Vy xỏc sut ca bin c A l P( A) = || 3432 Cõu 10 (1,0 im) Bt ng thc cn chng minh tng ng vi : http://dethithu.net 2ab 3bc 2ca + + (c + a) (c + b) ( a + b) ( a + c) (b + c) (b + a) 2ab (1 c) + 3bc (1 a) + 2ca (1 b) (1 a ) (1 b ) (1 c ) ab + 4bc + ca 16abc 1 + + 16 a b c p dng bt ng thc 1 + x y 1 + + a b c c : x+y 4 + a b+c http://dethithu.net 16 = 16 a+b+c Ta cú bt ng thc cn chng minh Ht Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 15 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! THI TH K THI THPT QUC GIA Mụn : TON Thi gian lm bi 180 phỳt http:/ / dethithu.net ỏp ỏn s Cõu 1a (1,0 im) http://dethithu.net Tp xỏc nh : D = R S bin thiờn : + Gii hn ti vụ cc : lim y = ; lim y = + x + y x + Bng bin thiờn : y = 3x2 + 6x = x y + y 3x ( x 2); y 0 + 0 =0 ủ x=0 x=2 x O + U Hm s ng bin trờn (0; 2) Hm s nghch bin trờn (; 0) v (2; +) Hm s t cc i ti x = 2; yC = Hm s t cc tiu ti x = 0; yCT = th : http://dethithu.net + Ct Oy ti (0; 4) + Nhn im un U (1; 2) lm tõm i xng Cõu 1b (1,0 im) Phng trỡnh honh giao im ca d v (C ) l : 2 x + 3x = m( x + 1) ( x + 1)( x 4x + + m) = ủ x = x2 4x + + m = t f ( x ) = x4 4x + + m cú = m v f (1) = m + Do ú vi m < v m = thỡ d ct (C ) ti ba im phõn bit : M(1; 0), A( x1 ; m( x1 + 1)), B( x2 ; m( x2 + 1)) đ x1 + x2 = Trong ú theo nh lý vi-ột ta cú (1) x1 x2 = m + ằ Ta cú MA = ( x1 + 1; m ( x1 + 1)) MA = | x1 + 1|ằ + m2 MB = ( x2 + 1; m ( x2 + 1)) MB = | x2 + 1| + m2 ủ x1 = 2x2 + = | x + 1| + m2 Li cú MA = 2MB | x1 + 1| + m x1 = 2x2 x1 = http://dethithu.net Vi x1 = 2x2 + thay vo (1) ta cú x2 = m = x1 = 11 Vi x1 = 2x2 thay vo (1) ta cú x2 = m = 81 Vy m = hoc m = 81 Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 16 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Cõu 2a (0,5 im) http://dethithu.net Ký hiu phng trỡnh ó cho l (1) ta cú : ọ (1) sin 2x + cos 2x cos2 2x sin2 2x = sin 2x sin 2x + = (1 cos 2x )2 sin x cos x sin 2x + = 4sin4 x sin x sin xcos2 x + cos x 2sin3 x = sin x = 2sin3 x sin xcos2 x cos x = sin x = 2tan3 x 3tan2 x tan x = sin x = tan x = http://dethithu.net x = k (k Z ) x = + k Vy phng trỡnh cú nghim x = k, x = + k (k Z ) Cõu 2b (0,5 im) ọ ọ i (1 + i ) + i (i ) 2+ + = i Ta cú z = i2 2i2 2 2+ , phn o ca z l v |z| = Vy phn thc ca z l 2 7+4 ằ + = + 4 Cõu (0,5 im) iu kin x = 8, x = Phng trỡnh ó cho tng ng vi : 3 10x +15 x +8 = 32 2x +16 x +2 41x +68 4x +32 =3 x = 12 41x + 68 = 62 4x + 32 x+2 x= 41 12 x +2 62 41 http://dethithu.net Kt hp iu kin phng trỡnh cú nghim x = 4, x = Cõu (1,0 im) Xột h 2x2 y + y3 = 2x4 + x6 ằ phng trỡnh ( x + 2) y + = ( x + 1)2 ọ (1) 2x2 y x2 + y3 x2 2x yx y x2 ọ ọ + yx ọ3 ọ =0 ầ + y + x2 ủ y=x x=y=0 ồ2 ọ y2 + x y + x = ọ 2x2 + y2 + x2 y + x4 = y = x2 2x2 (1) , ta cú : (2) + x4 = http://dethithu.net Vi x = y = thay vo (2) khụng tha Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 17 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Vi y = x2 thay vo (2) ta cú ( x + 2) x2 + = x2 + 2x + () t x2 + = t, (t 1), phng trỡnh () tr thnh : ( x + 2)t = t + 2x t(t x ) 2(t x ) = (t x )(t 2) = ủ t=2 t=x Vi t = x2 + = x = y = http://dethithu.net Vi t = x x2 + = x (vụ nghim) ọ ọ Vy h cú hai nghim ( x; y) = 3; v ( x; y) = 3; Cõu (1,0 im) Th tớch trũn xoay cn tớnh l : sin xdx = 2 V= 0 (1 cos 2x ) dx = Vy th tớch trũn xoay cn tớnh l V = ầ x sin 2x 2 = http://dethithu.net Cõu (1,0 im) Theo gi thit ABCD l hỡnh thoi v ữ = 600 BCD l tam giỏc u BCD a2 Do ú BD = a; AC = a 3, suy din tớch ABCD l S ABCD = AC.BD = 2 BD AC http://dethithu.net Ta cú BD (SAC ) BD SC BD SA Gi O = AC BD, (SAC ), k OMSC, M SC SC ( MBD ) a Do ú l gúc gia (SCB) v (SCD ) = 900 OM = BD = BMD BMD 2 a a SA AC AC.OM a Ta cú SAC OMC = = SA = = 2 OM2 3a OM MC OC a4 a3 Do ú th tớch chúp S.ABCD l VS.ABCD = SA.S ABCD = S M H C B O A D Ta cú O l trung im AC nờn d (C, (SBD )) = d ( A, (SBD )) AH SO Trong (SAC ), k AH SO, H SO, ta cú AH (SBD ) AH = d ( A, (SBD )) AH BD 1 a Trong tam giỏc SAO vuụng ti A cú = + = + = AH = AH AS2 AO2 3a 3a a a http://dethithu.net Vy khong cỏch t C n (SBD ) l d (C, (SBD )) = AH = Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 18 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Cõu (1,0 im) Ta cú AIB = 900 ữ = 450 hoc ữ = 1350 ACB ACB ữ T ú suy ACD = 450 , ú tam giỏc ACD vuụng cõn ti D nờn DA = DC Li cú I A = IC nờn ID AC hay AC nhn ID = (1; 2) lm mt vect phỏp tuyn Mt khỏc AC i qua M (1; 4) nờn cú phng trỡnh x 2y + = Ta cú A AC (2t 9; t) DA = (2t 8; t + 1) DA = 5t2 30t + 65 | + + 9| Li cú DA = 2d( D, AC ) = = 10 ủ t=1 30t + 65 = 10 T ú suy 5t http://dethithu.net t=5 Vỡ im A cú honh dng nờn A(1; 5) ng thng DB i qua D (1; 1) v nhn DA = (2; 6) lm mt vect phỏp tuyn Do ú DB cú phng trỡnh 2( x + 1) + 6(y + 1) = x + 3y + = Ta cú B DB B(3m 4; m) IB = (3m 2; m 1); I A = (3; 4) Vỡ AIB = 900 nờn I A IB = 3(3m 2) + 4(m 1) = m = B(2; 2) Vy A(1; 5) v B(2; 2) http://dethithu.net Cõu (1,0 im) ng thng d1 i qua M1 (1; 1; 2) v cú mt vect ch phng = (1; 1; 0) u = (1; 2; 1) ng thng d2 i qua M2 (3; 1; 0) v cú mt vect ch phng u2 ợ ú ợ ú Ta cú , = (1; 1; 1) , M1 M2 = (2; 2; 2) , M1 M2 = = u u u u Do ú d1 v d2 l hai ng thng chộo (pcm) Mt phng ( P) cha d1 nờn i qua M1 (1;ợ1; 2) ú Hn na ( P) song song vi d2 nờn nhn , = (1; 1; 1) lm mt vect phỏp tuyn u u Vy ( P) cú phng trỡnh 1( x 1) 1(y + 1) + 1(z 2) = x + y z + = Cõu (0,5 im) http://dethithu.net = 60 S phn t ca A l A5 S phn t ca A khụng cú ch s l A3 = 24 S phn t ca A khụng cú mt ch s l 60 24 = 36 Phộp th l chn ngu nhiờn ba s t A nờn || = C60 = 34220 Gi A l bin c "ba s c chn cú ỳng mt s cú mt ch s 4" Ta cú s kt qu thun li cho A l | A | = C36 C24 = 9936 9936 2484 | A | Vy xỏc sut bin c A l P ( A) = = = 34220 8555 || Cõu 10 (1,0 im) http://dethithu.net (x + y )2 ( x + y)2 x + y Khi ú P = ( x + y)3 3( x + y) xy 3( x + y) = ( x + y)3 3( x + y)( xy + 1) Hay P = ( x + y)3 3( x + y)[( x + y)2 2] = 2( x + y)3 + 6( x + y) Xột hm s f (t) = 2t3 + 6t trờn [2; 2] cú f (t) = 6t2 + 6; f (t) = t = Khi ú f (2) = 4, f (1) = 4, f (1) = 4, f (2) = Do ú max f (t) = f (2) = f (1) = 4; f (t) = f (1) = f (2) = T gi thit ta cú ( x + y)2 = xy [2;2] [2;2] Vy P t giỏ tr ln nht bng ( x; y) = (1; 1), ( x; y) = (2; 1), ( x; y) = (1; 2) V P t giỏ tr nh nht bng ( x; y) = (1; 1), ( x; y) = (2; 1), ( x; y) = (1; 2) Ht Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 19 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! THI TH K THI THPT QUC GIA Mụn : TON Thi gian lm bi 180 phỳt http:/ / dethithu.net ỏp ỏn s Cõu 1a (1,0 im) Tp xỏc nh : D = R \{1} http://dethithu.net S bin thiờn : + Gii hn, tim cn : lim y = lim y = tim cn ngang l y = x + x lim y = ; lim+ y = + tim cn ng l x = x x + Bng bin thiờn : y = < 0, x D ( x 1)2 x y y y + + I O 2 Hm s nghch bin trờn (; 1) v (1; +) Hm s khụng cú cc tr th : ầ + Ct Oy ti (0; 1) v ct Ox ti ; x + Nhn giao im I (1; 2) ca hai tim cn lm tõm i xng Cõu 1b (1,0 im) ng thng d qua A(2; 2) v cú h s gúc m nờn cú phng trỡnh dng y = m( x + 2) + Phng trỡnh honh giao im ca d v (C ) l : http://dethithu.net đ 2x + x=1 = m ( x + 2) + mx2 + mx 2m = x1 t f ( x ) = mx2 + mx 2m cú = 9m2 + 12m ng thng d ct (C ) ti hai im phõn bit v ch : a =0 >0 f (1) = m =0 9m2 + 12m > = m>0 m< Gi s d ct (C ) ti hai im phõn bit cú honh x1 , x2 ( x1 < x2 ) 2m http://dethithu.net Theo nh lý vi-et ta cú x1 + x2 = 1, x1 x2 = m Khi ú d ct (C ) ti hai im thuc hai nhỏnh phõn bit v ch : x1 < < x2 ( x1 1) ( x2 1) < x1 x2 ( x1 + x2 ) + < 2m + + < m > (tha món) m Vy vi m > thỡ d ct (C ) ti hai im thuc hai nhỏnh phõn bit Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 20 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Cõu 2a (0,5 im) + cot = 13 = Ta cú A = cot 2 4+ Cõu 2b (0,5 im) Phng trỡnh ó cho tng ng vi : http://dethithu.net ọ z=2 z = i Vy phng trỡnh ó cho cú nghim phc z = 2, z = i (z 2) z2 + 4z + = Cõu (0,5 im) x+1 > x > Khi ú bt phng trỡnh ó cho tng ng vi : iu kin 2x x < http://dethithu.net x+1 < 0, t [0; +) Suy hm s f (t) luụn ng bin trờn (0; +), ú () f ( x ) = f (u) x = u Vi x = u y = x2 + thay vo (2) c x=0 ọ x4 x3 2x2 = x2 x2 x = x = x = (loi) http://dethithu.net Vi x = y = 1; vi x = y = Vy h cú hai nghim ( x; y) = (0; 1) v ( x; y) = (2; 5) ọ2 x3 + 4x2 + = x2 + Cõu (1,0 im) e Ta cú I = dx + x u = ln x dx = dx x2 t e ln x e dx = ln | x ||1 + x = dx x x3 v = du 2e3 + 10 Like Fanpage c p nh t Vy I = ln x dx = + x2 e ln x dx x2 , ta cú : http://dethithu.net e e x3 I = 1+ ln x e x3 e3 x3 dx = + x e = 2e3 + 10 http://dethithu.net thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 21 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Cõu (1,0 im) a Tam giỏc SAD u cnh A nờn SE AD v SE = a Tam giỏc AEB vuụng ti A cú EB = AB2 + AE2 = Nhn thy SE2 + EB2 = SB2 nờn SEB vuụng ti E SE EB (pcm) SE EB Ta cú SE( ABCD ) SECH (1) SE AB http://dethithu.net Ta cú ABE = BCF ABE = BCF ữ + ữ = ữ + BCF = 900 CH EB (2) Xột BHF cú BFH FBH BFH T (1) v (2) ta cú CH (SEB) CH SB (pcm) S F A B H E D C 1 a = + = + = BH = BH BF2 BC2 a a a ằ a 2a Trong tam giỏc HBC vuụng ti H cú CH = BC2 BH = a2 = 1 a a a2 15 Li cú din tớch tam giỏc SEB l SSEB = SE.EB = = 2 2 1 2a a2 15 a3 Do ú th tớch chúp C.SEB l VC.SEB = CH.SSEB = = 3 12 a3 Vy SE EB, CH SB v VC.SEB = 12 http://dethithu.net Trong tam giỏc FBC vuụng ti B cú Cõu (1,0 im) A E M K B I M C D http://dethithu.net ữ Gi AI l phõn giỏc ca gúc ữ ta cú : AID = ữ + BAI, I AD = CAD + CAI BAC, ABC ABC ữ Li cú BAI = CAI, ữ = CAD nờn AID = I AD DAI cõn ti D DE AI Do ú AI cú phng trỡnh x + y = Gi M l im i xng vi M qua AI ta cú phng trỡnh MM l x y + = Gi K l giao im ca AI v MM ta cú K (0; 5), suy M (4; 9) Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 22 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Khi ú AB i qua A(1; 4) v nhn AM = (3; 5) lm vect ch phng Do ú AB cú phng trỡnh 5( x 1) 3(y 4) = 5x 3y + = http://dethithu.net Cõu (1,0 im) Ta cú A d1 A(3t1 ; + t1 ; + 2t1 ), B d2 B(1 + t2 ; 2t2 ; t2 ) Suy AM = (1 3t1 ; 11 t1 ; 2t1 ) , BM = (t2 ; + 2t2 ; t2 ) Do ú AM, BM = (12 + 6t1 + 3t2 + 3t1 t2 ; 5t2 + 5t1 t2 ; 9t1 + 13t2 7t1 t2 ) đ 12 + 6t1 + 3t2 + 3t1 t2 = t1 = ng thng AB qua M AM, BM = 5t2 + 5t1 t2 = t2 = 9t1 + 13t2 7t1 t2 = Vy A(3; 1; 2) v B(2; 4; 1) http://dethithu.net Cõu (0,5 im) Phộp th l chn ngu nhiờn hc sinh tng s 11 hc sinh Do ú s phn t khụng gian mu l || = C11 = 165 Gi A l bin c "ba hc sinh c chn cú c nam v n" chn hc sinh cú c nam v n ta chn nam n hoc nam n 1 Do ú s kt qu thun li cho bin c A l | A | = C5 ì C6 + C5 ì C6 = 135 | A | 135 = = Vy xỏc sut ca bin c A l P( A) = || 165 11 Cõu 10 (1,0 im) Theo gi thit v bt ng thc AM GM ta cú : bc bc bc =ằ =ằ 3a + bc a ( a + b + c) + bc ( a + b) ( a + c) ca ca ca =ằ =ằ 3b + ca b ( a + b + c) + ca (b + c) (b + a) ab ab ab =ằ =ằ 3c + ab c ( a + b + c) + ab (c + a) (c + b) http://dethithu.net Cng theo v (1), (2) v (3) ta cú : P ầ bc 1 (1) + a+b a+c ầ ca 1 (2) + b+c b+a ầ ab 1 (3) + c+a c+b bc + ba ab + ac a+b+c ca + cb + + = = ( a + b) (c + a) (b + c) 2 Du bng xy a = b = c = Vy P t giỏ tr ln nht l a = b = c = Ht Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 23 http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! THI TH K THI THPT QUC GIA Mụn : TON Thi gian lm bi 180 phỳt http:/ / dethithu.net ỏp ỏn s 10 Cõu 1a (1,0 im) Tp xỏc nh : D = R S bin thiờn : + Gii hn ti vụ cc : lim y = +; lim y = x + y x + Bng bin thiờn : y = 3x2 = 3( x2 1); y = x = x y + y + + U + 1 O 1 Hm s ng bin trờn (; 1) v (1; +) Hm s nghch bin trờn (1; 1) Hm s t cc i ti x = 1; yC = Hm s t cc tiu ti x = 1; yCT = th : + Ct Oy ti (0; 1) + Nhn im un U (0; 1) lm tõm i xng http://dethithu.net Cõu 1b (1,0 im) ọ Ta cú A (C ) A x A ; x3 3x A + ; y = 3x2 y ( x A ) = 3x2 A A ọ Do ú tip tuyn ti A l y = 3x2 ( x x A ) + x3 3x A + A A Phng trỡnh honh giao im ca tip tuyn v (C ) l : ọ 3x2 ( x x A ) + x3 3x A + = x3 3x + A A ọ 3x2 ( x x A ) = x3 x3 3x + 3x A A A ọ ọ 3x2 ( x x A ) = ( x x A ) x2 + xx A + x2 ( x x A ) A A ọ ( x x A ) x2 + xx A + x2 3x2 + = A A ọ ( x x A ) x2 + xx A 2x2 = A ( x x A ) (( x x A ) ( x + x A ) + x A ( x x A )) = ( x x A )2 ( x + 2x A ) = ủ x = xA x = 2x A http://dethithu.net Vỡ tip tuyn ct (C ) ti B khỏc A nờn x B = 2x A Khi ú 2013x B + 2014x A = 2012 4026x A + 2014x A = 2012 x A = Vy im cn tỡm l A(1; 3) Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 24 x http://dethithu.net - Website Thi Th THPT Qu c Gia - Ti Li u ễn Thi.C p nh t h ng ngy.Truy c p t i ngay! Cõu 2a (0,5 im) Phng trỡnh ó cho tng ng vi http://dethithu.net ọ sin x 2cos2 x cos x + + 2cos2 x cos x + = x = + k2 sin x = x = + k2 (k Z ) 2cos x cos x + (2 sin x + 1) = cos x = x = + k2 cos x = ọ Vy phng trỡnh cú nghim + k2, x = + k2, x = + k2 (k Z ) 6 Cõu 2b (0,5 im) http://dethithu.net ọ w2 Ta cú w = + i z + z = 1+i w3i w2 Do ú |z 1| w i 1+i 1+i t w = x + yi ( x, y R ), ta cú w i x + yi i ( x 3)2 + y 16 ọ Gi (C ) l ng trũn tõm I 3; v bỏn kớnh R = Ta cú hp cỏc im biu din s phc w l phn mt phng nm ng trũn (C ), k c (C ) http://dethithu.net Cõu (0,5 im) x+1 > x > Khi ú bt phng trỡnh ó cho tng ng vi : iu kin 2x x < 1 x+1 < (tha món) Vỡ AB = DC D (4; 1) http://dethithu.net Vy B(4; 3), C (2; 1), D (4; 1) Cõu (1,0 im) x y2 z1 x = = 2 y = Ta giao im A l nghim h x y+2 z5 = = z=5 l A (1; 2; 5) Vy giao im ca v ng thng qua im M(7; 2; 1) v cú vect ch phng = (3; 2; 2) u qua im M (1; 2; 5) v cú vect ch phng u = (2; 3; 4) ng thng ù ũ Mt phng () cha , nờn qua M(7; 2; 1) v nhn , u = (2; 16; 13) lm vect u = phỏp tuyn Vy () cú phng trỡnh 2( x 7) 16(y 2) 13(z 1) = 2x 16y 13z + 31 = Cõu (0,5 im) Xột khai trin (1 + x )n = n k =0 k Cn x k http://dethithu.net Ly tớch phõn hai v cn t n ta cú (1 + x ) n +1 n+1 Khi ú ầ n k Cn = k =0 x+ 24 x ồn x k +1 k+1 ầ n k +1 3n +1 6560 3n +1 k Cn = = n=7 n+1 k+1 n+1 n+1 k =0 1 = x + x 2 ồ7 = k C7 x (7k) k =0 ầ 1 x ồk = k C7 k x2 2k k =0 S hng cha x2 tng ng vi s hng cha k tha k = k = Vy h s ca s hng cha x2 l 21 http://dethithu.net Cõu 10 (1,0 im) Vỡ x + y < nờn x y > 0, ú theo bt ng thc Schwarz ta cú bt ng thc cn chng minh Ht Like Fanpage c p nh t thi th nhi u h n qua Facebook : http://facebook.com/dethithu.net 27