Luyenthipro.vn TRNG THPT KHOI CHU CHNH THC THI KHO ST CHT LNG LN I Nm hc 2015 2016 MễN: TON LP 12 Thi gian lm bi: 150 phỳt, khụng k thi gian giao ( thi gm 01 trang) Cõu 1( 2,0 im) Cho hm s y x3 3x2 (C) a) Kho sỏt s bin thiờn v v th ca hm s (C) b) Tỡm m ng thng i qua im cc tr ca th (C) to vi ng thng : x my mt gúc bit cos Cõu 2(1,0 im ) Tỡm cỏc ng tim cn ca th hm s y 2x x 2015 Cõu 3( 1,0 im) Xỏc nh h s ca s hng cha x khai trin x5 x Cõu 4(1,0 im) Gii phng trỡnh sin2 x sin x cos x 2cos2 x a Cõu 5(1,0 im) Cho hỡnh chúp S.ABCD, ỏy ABCD l hỡnh thoi cnh a, SA , SB a , BAD 600 v mt phng (SAB) vuụng gúc vi ỏy Gi H, K ln lt l trung im ca AB, BC Tớnh th tớch t din KSDC v tớnh cosin ca gúc gia ng thng SH v DK Cõu 6(2,0 im) Trong mt phng vi h ta Oxy, cho hỡnh ch nht ABCD cú DC BC , tõm I( - ; ) Gi M l trung im ca cnh CD, H( - 2; ) l giao im ca hai ng thng AC v BM a) Vit phng trỡnh ng thng IH b) Tỡm ta cỏc im A v B Cõu 7( 1,0 im) Gii phng trỡnh 2x 2x x x2 4x2 4x 2x trờn s thc x y z Cõu 8( 1,0 im) Cho ba s thc x, y, z thay i tha 2 x y z Tỡm giỏ tr ln nht ca biu thc P x3 y3 z3 - Ht Thớ sinh khụng c s dng ti liu Cỏn b coi thi khụng gii thớch gỡ thờm H v tờn thớ sinh: ; S bỏo danh: TRNG THPT KHOI CHU HNG DN CHM KSCL LN I MễN: TON LP 12 (Hng dn gm 04 trang) Chỳ ý: Hc sinh lm cỏch khỏc m ỳng thỡ cho im ti a phn ú im ton bi khụng lm trũn CU P N TX: D S bin thiờn: y 3x2 6x 3x x IM 0.25 x y x Hm s ng bin trờn cỏc khong ; v 2; Hm s nghch bin trờn khong 0;2 Hm s t cc tiu ti x = yCT , cc i ti x = yCẹ 0.25 Gii hn lim y , lim y x Bng bin thiờn x x - y 0 + 1a) (1,0 ) + - + + 0.25 y -4 - th y f(x)=x^3-3*x^2 0.25 x -4 -2 -2 -4 -6 ng thng i qua C, CT l : 2x y VTPT n1 2;1 ng thng ó cho : x my cú VTPT n2 1; m 1b) (1,0 ) Yờu cu bi toỏn cos ; cos n1; n2 25 m2 4m 5.16 m2 11m2 20m m m 0.25 0.25 0.25 (1,0 ) m m 11 2x 2x ( hoc lim ) nờn x 2015 l Vỡ lim x2015 x 2015 x2015 x 2015 tim cn ng ca th hm s 2x nờn y = l tim cn ngang ca th hm s Vỡ lim x x 2015 Xột s hng th k + khai trin Tk C x k (1,0 ) k x 0.5 0.5 k 0.25 Tk1 C9k 59k.x7k18 Vỡ s hng cha x3 nờn 7k 18 k Vy h s ca s hng cha x3 khai trin l C93.56 1.312.500 0.25 0.25 0.25 PT sin2 x cos2 x sin x cos x cos2 x 0.25 sin x cos x sin x 2cos x (1,0 ) 0.25 sin x cos x sin x 2cos x 0.25 tan x x k k tan x x arctan2 k k 0.25 0.25 S 0.25 B C K H M (1,0 ) A T gi thit ta cú AB = a, SA D a a , SB nờn ASB vuụng ti S 2 AB SAH u Gi M l trung im ca AH thỡ SM AB Do SAB ABCD SM ABCD SH 0.25 1 Vy VKSDC VS.KCD SM SKCD SM SBAD 3 a a.a a3 (vtt) 2.2 32 0.25 Gi Q l im thuc AD cho AD = AQ HQ KD nờn SH , DK SH , QH Gi I l trung im HQ MI AD nờn MI HQ M SM ABCD SI HQ SH ,QH SHI 0.25 Trong tam giỏc vuụng SHI cú: 6a (1,0 ) 1 a HQ DK HI 4 cosSHI a a a SH 2 IH 1; 0.25 0.5 Nờn ng thng IH cú phng trỡnh x y A 0.5 B I H D C M T gi thit ta suy H l trng tõm ca BCD IA 3HI A(2;5) 6b (1,0 ) 2 BC BC BM BC2 MC , HC AC 3 3 2 HB HC BC nờn BM AC BM i qua H( -2; ), nhn IH 1; lm VTPT cú phng trỡnh Ta cú HB x y ta B cú dng B( t; - t - ) 0.25 0.25 0.25 Li cú IA IB nờn 18 t t t 4t t Do ú t 2 B 2;1 2 B 2;1 2 0.25 K: x Phng trỡnh 2 (1,0 ) 2x 2x 2 2x 12 2x 12 (*) 2x 2x 2 0.25 Xột hm s f t t t trờn 0; cú f t 2t t 0; nờn hm s f(t) ng bin trờn 0; 2x 12 Do ú pt (*) tr thnh f 2x 2x f f ủo ng bieỏ n 0.25 2x 2x 2x 2x 2x 2x 2x 2x 2x 2x ( **) 2x a t thỡ phng trỡnh (**) tr thnh x b a b a2 b2 4a2b2 (1) a b a2 b2 a2 b2 a2 b2 0.25 T (1) a b 16 4a2b2 a b a2b2 a2 b2 2ab 16 8a2b2 a4b4 (***) t ab = t t thỡ pt (***) tr thnh 16 8t 16 8t t t t t 2t t x t loaùi 2x 2x Vy t = t loaù i x 2x 2x t loaùi 0.25 Chỳ ý: HS cú th gii theo cỏch khỏc nh sau t a x x Phng trỡnh ó cho tr thnh a a a 2a a 8a 8a Cú x y z z x y P x3 y3 x y 3xyz T x2 y2 z2 x y 2xy z2 2z2 2xy xy z2 0.25 Vy P 3z z2 4 x y z2 z2 z 2 3 4 t P f z 3z3 3z vi z ; K 3 z K Cú f z 9z , f z z K Do x2 y2 z2 (1,0 ) 4 Ta cú: f ,f 3 Do vy max P z 4 2 , f ,f 3 3 ;x y 0.25 0.25 0.25 ... cos ; cos n1; n2 25 m2 4m 5 .16 m2 11 m2 20m m m 0.25 0.25 0.25 (1, 0 ) m m 11 2x 2x ( hoc lim ) nờn x 2 015 l Vỡ lim x2 015 x 2 015 x2 015 x 2 015 tim cn ng ca... bin thi n x x - y 0 + 1a) (1, 0 ) + - + + 0.25 y -4 - th y f(x)=x^3-3*x^2 0.25 x -4 -2 -2 -4 -6 ng thng i qua C, CT l : 2x y VTPT n1 2 ;1 ng thng ó cho : x my cú VTPT n2 1; m 1b) (1, 0... lim x x 2 015 Xột s hng th k + khai trin Tk C x k (1, 0 ) k x 0.5 0.5 k 0.25 Tk1 C9k 59k.x7k18 Vỡ s hng cha x3 nờn 7k 18 k Vy h s ca s hng cha x3 khai trin l C93.56 1. 312 .500 0.25