1、 绝密 启用前 2015 4 22 15 16 150 120 10 5 50 . . 1. | 1| 2 A x x 2 | lg( 2) B x y x x R A C B A 3, 1) B. 3, 1 C. 1,1 D. ( 1,1 2. 1 zi i 2 z z A B. C. D. 3. 500 500 0 H “ ” 2 2 0 H 1% 2 K A 7.897 B. 6.635 C. 5.024 D. 3.841 ( ) 4. 2 cos ( ) 2 yx A ( , ) 2 k k k Z B. ( , ) 2 k k k Z C. (2 , 2 ) k k k Z D. (
2、2 , 2 2 ) k k k Z 5. . S 11 12 A. 6 n B. 6 n C. 6 n D. 8 n 6. 1 A. 32 3B. 64C. 32 3 3D. 64 37. 2 0 () P K k 0.05 0.025 0.010 0.005 0.001 0 k3.841 5.024 6.635 7.879 10.83 A 48 B 72 C 78 D 84 8 P ABC , 3 APC BPC APB , M ABC , 9 PA PB PB PC PC PA uur uur uur uuu r uuu r uur ,| PM uuur A. 2 B. 43 3C. 6D
3、. 229 C 22 22 1 xy ab 0 , 0 b a , A O A C Q P, 60 PAQ 4 OQ OP uuu r uuu r C A 23 3B 7 2C 2 13 5D 310. , 0, ) xy 22 42 x y x y ax e e a A. 1 4B. 1 C. 2 D. 1 2 6 5 5 25 . . . 1114 11. 2 sin cos 3 2 1 tan 2sin sin 2 . 12. 2015 2 2015 0 1 2 2015 ( 2) x a a x a x a x , 2 0 2 4 2014 1 ( ) ( a a a a a 2 3
4、5 2015 ) a a a . 13. ) (x f R ( 2) y f x (2,0) , uv ( ) ( 1) 0, ( 1) 0, f u f v f u v 22 uv 14. () y f x 0 , x a b 0 () () b a f x dx fx ba 0 x () fx , ab “ ”. (1) 3 ( ) 3 f x x x 2, 2 (2) 2 ( ) 1 g x x mx 1,1 “ ” m . 15 16 2B . 15 . 15. （选修 4-1：几何证明选讲） AB O , CF O OC AB F O FD AB D CF AB E 3, 3 OA
5、DB DE 16. 4-4 3 cos 4 sin 0 a 1 cos sin x y a . 6 75 . . 17. 12 ABC A B C a b c 2 3 cos cos 3 b c C A a . A gkstk 6 B BC 7 AM ABC . 18. 12 n a n n S 11 3, 3 ( ) n nn a a S n N . 3 n n S n a 1 a . 19. 12 5 1 2 3 4 5 10 1 2 3 4 5 6 5 7 9 8 4 8 9 7 7 ( ) ( ) 1 2 X Y X Y . 20. 12 1 1 1 ABC A B C 1 AA AB
6、C 90 BAC 2, 6 AB AC , D 1 BB 1 1 3 BD BB 11 A C AC E DE ABC 1 ADC ABC 7 cos 7 1 AA 1 ADC ABC l l DE 21. 13 22 22 : 1( 0) xy C a b ab (0, 1) P P 2 . Q | PQ l O: 1 2 2 y x C A B OA OB uur uuu r 4 3 3 2 AOB S 22. 14 R () fx 2 2 2 (1) ( ) 2 (0) 2 x f f x e x f x 2 1 ( ) ( ) (1 ) 24 x g x f x a x a a R .
7、 () fx () gx x y m | | | | x m y m x y m . 2 a 1 x e x 1 x ea ln x 2015 1 2 3 4 5 6 7 8 9 1 0 A C B A A C D A A C D B A C B C D B D D B A 11. 9 5 12. 113. 1 214. (1) 3,0 ;(2) , 44 15. 3316. 217. I 2 3 cos cos 3 b c C A a 2sin 3sin cosC cos 3sin BC A A 2sin cos 3sin( ) 3sin B A A C B 3 sin 0 B 3 cos
8、2 A (0, ) A 6 A 6 II 6 B 6 A ab ACM 2 2 2 7 21 4 cos 32 b b b 2 b 10 ABC 13 2 2 3 22 ABC S 12 18. I 11 n n n a S S 1 23 n nn SS 1 1 1 3 2 3 n n n n S S 4 1 30 a 3 n n S 1 3 a 2 6 II I 1 1 3 ( 3) 2 nn n Sa 1 1 ( 3) 2 3 nn n Sa 2 n 21 11 ( 3) 2 2 3 nn n n n a S S a 8 n a 1nn aa * nN 2 n 1 1 ( 3) 2 2 3
9、 nn a 21 1 ( 3) 2 2 3 nn a 22 1 3 2 12 ( ) 3 0 2 nn a 2 n * nN 1 9 a 10 2 1 1 3 a a a 1 a ( 9,3) (3, ) . 12 19. ( ) 7 2 1 2 s 2 2 14 5 s 5 .6 ( ) X 0,1,2 2 1 1 ( 0) 5 2 5 PX 3 1 2 1 1 ( 1) 5 2 5 2 2 PX 3 1 3 ( 2) 5 2 10 PX X X0 1 2 P 1 51 23 10 1 1 3 11 0 1 2 5 2 10 10 EX . 9 Y 0,1,2 3 1 3 ( 0) 5 5
10、25 P Y 3 4 2 1 14 ( 1) 5 5 5 5 25 P Y 2 4 8 ( 2) 5 5 25 P Y Y Y0 1 2 P 3 2514 258 25 3 14 8 6 0 1 2 25 25 25 5 EY . 12 20. A xyz 1 AA h 11 2,0,0 , 0,6,0 , 2,0, , 0,0, , 0,6, , 0,3, 32 hh B C D A h C h E .2 1 AA ABC 1 0,0,1 n ABC 1 2,3, 0,0,1 0 66 hh DE n 3 DE ABC 4 1 ADC 2 , n x y z 2 21 , , 2,0, 2
11、0 33 , , 0,6, 6 0 hh n AD x y z x z n AC x y z h y hz 5 6 z x y h 2 , , 6 n h h 6 12 12 2 12 67 cos cos , 7 1 2 36 nn nn nn h 7 63 h 1 63 AA 8 11 BCC B 1 CB C D F AF AF 1 ADC ABC 9 1 / BD CC 11 11 = 33 BD BB CC 1 1 3 BF BD FC CC 1 2 BF CB 11 2,0,0 2, 6,0 3, 3,0 22 AF AB BF AB CB 10 63 h 2,3, 2,3, 3
12、6 h DE 15 5 cos , 2 8 3 2 4 AF DE AF DE AF DE 11 l DE 55 22 88 12 21. (1) 1 2 2 2 y x ) , ( y x Q 22 ( 1) PQ x y 22 2(1 ) ( 1) yy 2 ( 1) 4( 1 1) yy 1 y max 2 PQ 5 (2) l n my x ( R m ) l 0 n my x O: 1 2 2 y x 1 1 | | 2 m n 1 2 2 m n A B ( 1 x 1 y ) ( 2 x 2 y ) 0 2 2 2 2 y x n my x 0 2 2 ) 2 ( 2 2 2 n
13、 mny y m 2 2 2 2 1 m mn y y 2 2 2 2 2 1 m n y y 8 ) 2 ( 8 ) 2 )( 2 ( 4 4 2 2 2 2 2 2 n m n m n m ) 2 ( 2 2 2 2 1 m mn y = OB OA = 2 1 2 1 2 1 2 1 ) )( ( y y n my n my y y x x 2 2 2 3 ) ( ) 1 ( 2 2 2 2 2 1 2 1 2 m m n n y y mn y y m 2 1 2 2 m m 2 2 2 ) ( 2 1 sin | | | 2 1 OB OA OB OA AOB OB OA S AOB|
14、 | 2 1 1 2 2 1 y x y x | ) ( ) ( | 2 1 1 2 2 1 y n my y n my | ) ( | 2 1 1 2 y y n 2 2 2 2 ) 2 ( 1 2 2 | | 2 1 m m m n 2 1 2 1 2 2 2 2 m m m 1 2 1 2 1 2 2 2 m m m 2 1 2 2 m m 3 2 4 3 AOB S ) 1 ( 2 4 6 3 2 22. 22 ( ) (1) 2 2 (0) x f x f e x f (1) (1) 2 2 (0) f f f (0) 1 f 2 (1) (0) 2 f fe 2 (1) 2 fe
15、22 ( ) 2 x f x e x x 4 22 ( ) 2 x f x e x x 2 2 2 1 1 1 ( ) ( ) (1 ) (1 ) ( 1) 2 4 4 4 xx x g x f x a x a e x x x a x a e a x . 5 () x g x e a 0 a ( ) 0 gx R 6 0 a ( ) 0 x g x e a ln xa ,ln xa ( ) 0 gx () gx ln , xa ( ) 0 gx () gx 0 a () gx ( , ) 0 a () gx ln , a ,ln a 8 1 ( ) ln , ( ) ln x e p x x
16、q x e a x x 2 1 ( ) 0 e px xx () px 1, ) x ( ) 0 pe 1 xe ( ) 0 px xe ( ) 0 px . 1 1 ( ) x q x e x 1 2 1 ( ) 0 x q x e x ( ) qx 1, ) x (1) 0 q 1, ) x ( ) 0 qx () qx 1, ) x ( ) (1) 2 0 q x q a . 1 xe 1 | ( ) | | ( ) | ( ) ( ) x e p x q x p x q x e a x 1 () x e m x e a x 1 2 ( ) 0 x e m x e x () mx 1,
17、) x ( ) (1) 1 m x m e a ) (x g2 a ( ) 0 mx | ( ) | | ( ) | p x q x e x 1 x e + a ln x . xe 11 | ( ) | | ( ) | ( ) ( ) 2ln 2ln xx e p x q x p x q x x e a x e a x 1 ( ) 2ln x n x x e a 1 2 ( ) x n x e x 1 2 2 ( ) 0 x n x e x ( ) nx xe 1 2 ( ) ( ) 0 e n x n e e e () nx xe 1 ( ) ( ) 2 0 e n x n e a e | ( ) | | ( ) | p x q x e x 1 x e + a ln x . 2, 1 ax e x 1 x e + a ln x . 14
