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1 2 3 4 2
2004 20044 2005 2006 5 2007 5 20085 20094 2010 4.. 20112116. 3
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1 14 14 15 15 16 17 16 18 18 19 19 20 21 17 20 22 21 23 5
15 1 2 15 6
1.. 2 2 1 y = cc y = x y = x y =. x. n n 1 C = 0 C ( x ) = nx, nn + (sin x) = cos x x x x x 1 1 (cos x) = sin x (e ) = e ( a ) = a lnaa>0, a 1(ln x) = (log x) = loga ea>0, x x a 1. 1[ ux ( ) ± vx ( )] = ux ( ) ± vx ( ). 2[ uxvx ( ) ( )] = ux ( ) vx ( ) ± uxvx ( ) ( ). ux ( ) ux ( ) vx ( ) uxvx ( ) ( ) 3[ ] = ( vx ( ) 0). 2 vx ( ) v( x) 3.. 4.. 1 2 y = cc y = x y 2 = x y 3 = x 1 y = y = x x. f ax+b. 5.. 7
20708 07 4 4416 11 11444 6 90 21 12 12560 4 4520 6 70 22 12 12560 4 4416 6 74 22 10 10550 6 6530 5 70 21 08 8 8540 7 5630 6 80 10 10550 5 4520 6 80 14 14570 6 90 20 22 40 4 21 8
2009 4 4416 14 14456 5 78 23 12 12560 4 4520 5+1 70 21+1 12 12560 4 4416 6 74 22 10 10550 7 7428 5 72 22 10 10550 6 6424 6 76 22 10 10550 5 5525 6 75 21 14 14570 6 90 20 8 10 10 12 8540 10550 10550 12560 7 5 5 4 6530 5420 4520 4416 6 80 21 5+1 80 76 21 22 9
2010 4 4520 14 14456 5 74 23 12 12560 4 4520 5+1 70 21+1 12 12560 4 4416 6 74 22 10 10550 7 7428 5 72 22 10 10550 6 6424 6 76 22 10 10550 5 5525 6 75 21 8 8540 7 7535 6 75 21 8 8540 6 6530 6 80 21 8 10 10 12 8540 10550 10550 12560 7 5 5 4 6530 5420 4520 4416 6 80 21 6 80 76 21 22 10
3 4 10 10 14 5 20 10 5 5 5 5 5 2010 11
4 10 2010 10 5 20 10 5 5 5 12
2010 6 37.5% 49 37.12% 9 37.5% 61 36.31% 5 31.25% 39 29.55% 6 25% 39 23.21% 3 18.75% 30 22.73% 5 20.83% 41 24.4% 2 12.5% 14 10.6% 2 8.3% 17 10.12% 0 0 0 0 1 4.17% 5 2.98% 0 0 0 0 1 4.17% 5 2.98% 16 100% 132 100% 24 100% 168 100% 13
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12010 10{ abcd,,, } a b c d a a b c d b b b b b c c b c b d d b b d a b c d a a a a a b a b c d c a c c a d a d a d d ( a c) = A a Bb Cc D d 15
212010 15 Ω Ω Ω Ω 4 16
209 11 τ 1, τ 2, τ 3, τ 4 Aτ 1 > τ 4 > τ 3 Bτ 3 > τ 1 > τ 2 Cτ 4 > τ 2 > τ 3 Dτ 3 > τ 4 > τ 1 τ 1 = 2 2 τ 2 = π τ 3 = 3 τ 4 = 2 3τ 4 > τ 2 > τ 3 > τ 1 C. 17
3.1998 11 10 H. V h 4.07 8 h 1 h 2 h 3 h 4 A h 2 > h 1 > h 4 B h 1 > h 2 > h 3 C h 3 > h 2 > h 4 D h 2 > h 4 > h 1 18
31. 2010 16 2 cos 2α = 2cos α 1 4 2 cos4α = 8cos α 8cos α + 1 6 4 2 cos 6α = 32cos α 48cos α + 18cos α 1 8 6 4 2 cos8α = 128cos α 256cos α + 160cos α 32cos α + 1 10 8 6 4 2 cos10α = mcos α 1280cos α + 1120cos α + ncos α + pcos α 1. m n+ p = 1 3 2= 2, 8= 2, p = 50 m n+ p= 962. 5 32 = 2, 7 128 = 2, 9 m = 2 = 512 n = 400 22010 12 3 3 2 1 + 2 = 3 1 3 + 2 3 + 3 3 = 6 2 1 3 + 2 3 + 3 3 + 4 3 = 10 2. i 1 i+1 1+2+...+i+1 1 3 + 2 3 + 3 3 + 4 3 + 5 3 + 6 3 = 21 2. 19
42010 14 1 PABC x. P x y y = f ( x) f ( x) y = f ( x) x. PABC x x x. x A B x B PABC x. A x A x x 1 4 π + 1. A B C P 20
07 51. 08 2 2 cos 2x= 2cos x 1 x R (cos2 x) = (2cos x 1) ( sin2 x) 2= 4cos x ( sin x) sin 2x= 2cos x sin x 1 1x n C 0 + C 1 x+ C 2 x 2 + + C n x n x R n 2 1 n[(1 + x) n 1] n n n n n k = 1 kc 2 n 3 k k 1 nx i n k = 1 k ( 1) kc k n 0ii n k 2 k ( 1) k Cn k = 1 0iii n n+ 1 1 k 2 1 Cn = k = 1 k+ 1 n+ 1 21
6109 17 API, 365 API [0,50] (50,100] (100,150] (150, 200] (200,250] (250,300] x 2. 5 7 = 78125 2 7 = 128 3 + 2 + 7 + 3 + 8 = 123 365 = 73 5 1825 365 1825 1825 9125 9125 22
22010 19. 12 6 6 C 8 6 10 C. 64 42 54 C. 2.5 4? x, y z z = 2.5x+ 4y 3x+2y x+y=7 12x+ 8y 64, 3x+ 2y 16, 3x+5y=27 6x+ 6y 42, x + y 7, 6x+ 10y 54, 3x+ 5y 27, 2.5x+4y=0 xy, 0. xy, 0. l:2.5x+ 4y = 0l l O 0 l 0 C z. 3x+ 5y = 27, x= 4, C(4,3) x+ y = 7, y = 3. z = 2.5 4 + 4 3 = 22 4 3 z=22. y A B C x 23
32010 17 AE H( m BC h=4mabe=α ADE= β. α β tanα =1.24tan β =1.20 H d m α β. 125m d α - β. 1471 Christian Roder 24
42010 17 4. P EFGH ABCD EFGH. 5 6. 1 w.w.w.k.s.5.u.c.o.m 2 3 BD PEG. 25
71. 2010 12 y = sin 2x π π y = sin( x+ ) y = sin( x ) 6 3 x x A B x x C D 26
2. 2010 12 t St ( )( S (0) = 0) y = S () t C B AD A... 27
81.0917 MN ABABMN AB. MN.. 28
A51.2 2 AB AB. A5 A BCα βhbc. 29
A MN α 1 β 1 B MN α 2 β 2 AB d. 1AMN. AMAN MN. 2BMN. BMBN MN. 3. d β α 1 α 2 β 2 1 BMBN MN. 30
A MN α 1 β 1 B MN α 2 β 2 AB d. 1AMN. AMAN MN. 2BMN. BMBN MN. 3. d β α 1 α 2 β 2 1 BMBN MN. 31
4. MN. 5. y β α 1 α 2 β 2 1 x MN MN. 32
207 18 4 3 163 4 163 163 xoy P( 2, 1) 3 x + 4y = 0... 33
34
2007 3. 35
111 9 13 9 4 3 3 4 5 A 1 B 67 66 C 47 44 D 37 33 19 1998 9 1998 8. 36
211 15 n. n 4 n = 1 n = 2 n = 3 n = 4 n = 6. n 1, 2,, n. F n n. F 1 = 2 F 2 = 3 F = F ( 3) 2 + F 1 n F 5 = 13 F 6 = 21. n n n n = 6 2 6 = 64 n = 6 64 21 = 43. 2143. 37
311 10. 137 t 30 t Mt () = M0 2 M 0 t = 0 137. t = 30 137 10ln 2 / M (60) = A5 B. 75ln 2 C. 150ln 2 D150 M 0 = ( ) ln 2 t M 0 ln 2 30 M () t 2 30 M 0 = 600 M 30 = = 10ln 2 60 60 30 M (60) = 600 2 = 150 D. 411 15 M M = lg A lg A0 A A 0 1000 0.001 9 5 1 M = lg A lg A0 = lg1000 lg 0.001 = 6 A 2 M = lg A lg A0 lg A = M + lg A0 A= A0 10M M = 9 M = 5 9 = A = A 10 5 A1 A0 10 2 0 = = 10 5 A A 10 A 10 9 1 0 2 0 4. 610 4. 38
411 7V 1 V 2. AV 1 V 2 BV 1 V 2 CV 1 V 2 DV 1 V 2 4 3 3 3 3 a V1 = π( a) = πa V 3 2 2 V V 3 2 = a 1 2 V2 3 = π 1 1.72 C 2 511 4 y 2 = 2 px ( p > 0) n A n = 0 B n = 1 C n = 2 D n 3 AB x 12. y A M y A O A F B 1 B x M N N A O F B 2 B x 39
2007.11 2007 60721 mn., m x > 1 (1 + x) 1 + mx 1 n 1 n 6 (1 ) < n + 3 2 m n 1 m (1 ) < ( ), m= 1, 2,, n n + 3 2 3 n + 4 n + + ( n+ 2) n = ( n+ 3) n n. Bernoulli 1 n lim(1+ ) = e n n Escott. 40
709 15{ a n } an, an a1 = m m an+ 1 = 2 3an + 1, an a 6 = 1 m. 3x+1. A 2-2 A 4 98. 41
4 5 32 2 16 x 1 8 4 3 1 2 2 1 1 0.18. y=1 42
08.1112 1008 10 P F P F P F. 2c1 2c 2 2a 1 2a 2 a 1 + c 1 = a 2 + c 2 a 1 c 1 = a 2 c 2 c c ca 1 2 > ac 1 2 1 < 2. a a 1 2 A. B. C. D. 43
110913. 2008 36000km. 6400km km. 44
810 7 r S n n lim S n = A. C. n 2 6πr B. 8 π 2 3 r D. B r i i 2 4πr 2 2πr 3 3 3 3 r r r r r r r r 2 2 2 2 2 1 =, 2 = 1 =, 3 = 2 = ( ),, 3 3 ( ) 2 2 i 1 ri = ri 1 = = r n n n 2 3 i 1 2 2 3 i 1 2 3 n Sn = πri = π (( ) r) = π r ( ) = 4π r (1 ( ) ) 2 4 4 i= 1 i= 1 i= 1 3 lim Sn lim 4π r (1 ( ) ) 4πr n 4 2 n 2 = = B n r o. 0.69. 45
, 120910. 1 3 6 10 1 4 9 16 1 2 113610 214916. A289 B1024 C1225 D1378 46
, 910 15 a> 0, b> 0 2ab a+ b abc, AB AC = a, CB = b O AB AB C AB D D OD, AD, BD C OD E OD ab, ab, ab, E A B A 5. O C 2 a+ b a + b 6.2 3 a b ab 1 1 + 2 2 a b A 2 2. 0.52. 47
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1 f (t) 200412 y = y t 0 t 24. 0 24 t y t 0 3 6 9 12 15 18 21 24 y 12 15.1 12.1 9.1 11.9 14.9 11.9 8.9 12.1 f (t) y = y = k + Asin( ω t + ϕ) A C π π y = 12 + 3sin t, t [0, 24] B y = 12 + 3sin( t+ π), t [0, 24] 6 6 π π π y = 12 + 3sin t, t [0, 24] D y = 12 + 3sin( t+ ), t [0,24] 12 12 2. 4. 0.84 0.43 0.56 0.59. 53
200519 2 4 4. 0.60.7 0.80.9. ξ ξ. 2 3 15. 1 10. ξ. P=0.72D=0.68 54
200616 3 f( x ) = a ( b+ c ). a = (sin x, cos x) b = (sin x, 3cos x) c = ( cos x,sin x), x R. 1 f( x) 2 y = f( x) d d. P 92 B 13 2 2 y sin x 2sin xcos x 3cos x = + + x R 1 23 y = 2sin2 x( x R) sin x cos x 2 2 f ( x) sin x 2sin xcos x 3cos x = +..P0.65D0.67 55
200717 4 100 [1.381.501.40 [1.30,1.341.32.. 3 P 29 1.4 100 1 2 3 0.760.65. [1.301.34 4 [1.341.38 25 [1.381.42 30 [1.421.46 29 [1.461.50 10 [1.501.54 2 100 56
200816 1 t 5 f() t = ( ) cos (sin ) sin (cos ) 1+ t gx= x f x+ x f x (, 17π x π 12 ]. g( x) Asin( ωx+ ϕ) + B ( A > 0, ω > 0, ϕ [0,2π) ) g( x). P 100 B 3 1 sinα 1 cosα cosα + sin α. 1+ sinα 1+ cosα 0.53. 57
200920 6 y 2 = 2 px ( p > 0) F MN MN l M 1 N 1. FM 1 FN 1 FMM1 FM1N1 FNN 1 S 1 S 2 S 3 S 2 2 4S1S3 =. y 2 = 2 px ( p > 0) A( a, 0) ( a > 0) M N M N l: x = a M 1 N 1. p a = AM 1 AN 1 2 AMM1 AM1N1 ANN1 S 1 S 2 S 3. λ a > 0 S 2. = λs S. λ 2 1 3 58
8.5 7 y 2 = 2 px ( p > 0) F y 1 y 2 yy 2 1 2 = p. 8.6 6 P Q P M MQ. B 2 y 2 = 2 px ( p > 0) F A B A B A B AFB = 90. 2001 y 2 = 2 px ( p > 0) F F AB C 59 BCx AC O
201019 7 am 2, 10 b m 2. 30% b 1.1 5 = 1.6 B 8 P 151 am 2 10%. 10 x 1.1 10 2.6 10 1 19 12 12 0 4.44 0.37 0.68 60
201119 * 8{ a n } n S n a 1 = a ( a 0) a = n 1 rsn ( n N +, r R, r 1). { a n } * k N S k + 1 a m + 1 m k * S S k + 2 m N m 2 a a m+ 2. 3.4 4 B 5 P 151 61
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