高校生の就職への数学II

Size: px
Start display at page:

Download "高校生の就職への数学II"

Transcription

1 II O Tped b L A TEX ε

2 . II oboetene/plan/ 7 9 i

3

4 iii

5 ( ( ( ( ( ( iv

6 ... A B A B ( + 3. ( ( ( ( ( ( ( + ( (3 ( ( 3 ( (4 ( ( ( (5 (p 3 + p 4p 4 (p ( (6 ( ( + ( (7 ( (4 3 + ( (8 (8a + a 3 a 4 a + 3 (3 a a (JFE (9 ( ( (JFE

7 A B Q R Q A = BQ + R B A R R 0 B B B = B ( = B ( B 0 B = + 3. A B ( A 3 + ( B 3 + 5

8 3.. C 0 A B = A C B C. ( 5a b 5 3b 5a 0a 3 b = b 3 3 a 5a b 3 = 3b a A C B C = A B ( + 3 = ( + 3 ( ( + ( = ( 54a 0 b 6 c 8 6a 4 b cd ( ( 3 + ( ( ( ( ( (5 ( ( + ( + ( + ( 3 ( (6 a + b a b b + a b a (

9 4.3 ( A B C D = AC BD A B C D = A B D C = AD = ( + ( ( = ( + BC ( ( ( 3( + 5 = + 5 ( 3( 4 ( + 5 = ( ( 3( + 5 ( 3( 4 ( + 5 = 4.4 ( 3 4a b 3 8a3 b 9 3 ( ( ( ( (3 5a a 6 (NOK 9 (4 ( ab ( ( a b 3 ( (5 3 4 ( (6 + ( (7 + 4 (

10 5.5 ( ( ( ( (3 a a + 30 a 6a + 9 a 3a a 5a ( ( ( ( ( ( ( ( ( ( ( ( (

11 6. ( ( ( = ( = = ( = (3 ( + = ( = 3 ( ( 3 ( + ( = ( 3( + ( ( + ( (3 ( ( ( + ( = ( + 6 (3 5 + ( ( + ( = = ( ( + ( ( ( 4 ( ( + ( = 4 ( ( +.6 ( a a b + b b a ( ( ( (3 + ( ( ( + ( ( (5 a a + a + a a + a (

12 7.7 ( 4 ( ( ( (3 + + (NEC (4 + 8 ( 4 (5 + + ( ( ( (7 + ( (8 + b a b ( (9 + (

13 8.8 ( a + b ab b + c bc a + c ac ( ( 3 + ( ( ( (4 a + (a + (a + + (a + (a + (a + 3 ( (5 a + (a + (a + (a + (a + 3 ( ( ( ( (JR ( (

14 9..3 a + b + c = a + b + c a = a, b = b, c = c a + b + c = 0 a = b = c = 0.3 a( + + b( + c( + ( = + 3 a b c (a + b + c + (a b c = + 3 a + b + c = a b = 0 c = 3 a = b = c = 3.9 ( A( ( B( + C = A, B, C ( ( = a + b + a, b ( (3 A, B, C ( 3 9 ( ( = A + B + + C

15 a + b + c = 0 a b = bc + c c = a b a b (bc + c = a b b( a b ( a b = a b + ab + b (a + ab + b = 0 a b = bc + c.0 a + b + c = 0 ( (a + b(b + c(c + a + abc = 0. a + b + c = 0 ( a ( b + c + b ( c + a + c ( a + b + 3 = 0

16 .5 a b = c d a + 3c b + 3d = a 3c b 3d a b = c d = k a = bk c = dk a + 3c b + 3d a 3c b 3d a + 3c b + 3d = bk + 3dk b + 3d = bk 3dk b 3d = a 3c b 3d = k(b + 3d b + 3d = k = k(b 3d b 3d = k. a b = c d ( ab + cd ab cd = a + c a c.3 a = b (JFE ( a = + a ab + b ( pa + r qa + s = pb + r qb + s

17 .. a a 0 a = 0 a b a + b 0 a = 0 b = 0.6 a + b 4(a b a + b 4(a b = a 4a b + 4b + 4 = (a + (b + 0 a + b 4(a b a = 0 b + = 0 a = b =.4 ( a + b (a + b ( ( a ab + b 0 ( (3 + + z + z + z (

18 3 a + b a b a b a > 0 b > 0 ab a b a > 0 b > 0 a + b ab a = b a + b ab.4 a > 0 a + 9 a 6 a > 0 9 a > 0 a + 9 a a 9 a = 9 = 6 a + 9 a 6.5 ( a > 0 a + a ( ( ( a > 0, b > 0 (a + b a + 4 b ( ( a (3 a > 0, b > 0, c > 0, d > 0 b + c ( b d a + d 4 c (

19 4... i i a b a + bi 3 a b c d a + bi = c + di a = c b = d a + bi = 0 a = 0 b = 0. a b (3 + ia ( ib = + 4i (3a b + (a + bi = + 4i ( 3a b a + b 3a b = a + b = 4 a = b =. ( 3 + ( 3i = 7 i ( ( ( + 5i + (7 i = 5 i ( (3 ( i + ( + i = 4 (

20 5. ( (3 + i + (4 3i ( ( + i(3 i (3 (3 + i (4 ( + 3i( 3i ( (3 + i + (4 3i = ( ( 3i = 7 i ( ( + i(3 i = 3 i + 6i 4i (3 (3 + i = 9 + i + 4i = {3 4 ( } + ( + 6i = 7 + 4i = {9 + 4 ( } + i = 5 + i (4 ( + 3i( 3i = (3i = 4 9i = 4 9 ( = 3. ( (4 + 3i( + i ( ( ( + 3i(4 i ( (3 (4 3i(3 + 5i (NEC (4 (5 3i(4 + i ( (5 (4 i(4 + i ( (6 (3 i(6 + 4i ( (7 ( i 3 ( (8 5i 3 5i 5 ( (9 i i + i 3 i 4 + i 5 ( (0 ( + i i 3 ( ( ( 7 3 i( i ( ( ( + 3 i(3 7 i ( (3 ( + i 3 ( (4 ( + 3 i 3 (

21 6 a + bi a bi i 3i 4 + 3i (4 + 3i( + 3i = 3i ( 3i( + 3i = 5 + 5i 0 = + 3 i = 4 + i + 3i + 9i i a + bi ( 4 3i.4 ( + 3i 4i ( + i i (3 + i + i (4 i + i (5 i 3i (6 3i 3 + i (7 + i i (8 (9 ( i( + 3i + i (3 i(4 + 3i 3 + i (0 i + i + i + + i + i ( ( ( ( ( ( ( ( ( (NEC

22 7 a > 0 a = a i = i.4 a ± a = ± a i ( 4 ( 4 9 (3 = 5 ( ± 4 = ± 4 i = ± 6 i ( 4 9 = i 3i = 6i = 6 (3 = ± 5 = ± 5 i (4 ( = 4 = ± 4 = ± i (4 ( = 4.5 ( 4 5 ( ( 8 ( (3 ( + ( (JFE (4 ( (.6 (t = 0 (

23 8.. a + b + c = 0 I b 4ac.5 a + b + c = 0 = b ± b 4ac a ( = 0 ( = 0 ( = 4 ± = 4 ± i 6 = 4 ± 8 6 = ± i 3 ( = ( 6 ± ( = 6 ± i 4 = 3 ± i = 6 ± ( = 0 ( ( = 0 ( (3 3 = 3 ( + ( 4

24 a +b+c = 0 = b ± b 4ac a b 4ac b 4ac a + b + c = 0 D a + b + c = 0 D D > 0 D = 0 ( 9 D 0.6 D < 0 ( + 3 = 0 D = 4 ( 3 = 5 > 0 ( = 0 D = ( = 0 ( = 0 D = ( = 3 < 0.8 ( = 0 ( ( 4 5 = 0 ( ( = 0 ( (4 + + = 0 ( ( = 0 ( (6 4 m + 9m = 0 (m (

25 0. ( + a a + 3 = 0 a ( ( + k + 8k + 9 = 0 k ( (3 (a = 0 a ( ( k + = 0 k ( ( + a a + 3 = 0 D = a 4 ( a + 3 = a + 4a = (a + 6(a D > 0 (a + 6(a > 0 a < 6 < a ( + k + 8k + 9 = 0 D = (k 4 (8k + 9 = 4(k 8k 9 = 4(k + (k 9 D = 0 (k + (k 9 = 0 k = 9 (3 (a = 0 D = { (a + } 4 4 = a + 4a = (a + 6(a D < 0 (a + 6(a < 0 6 < a < ( k + = 0 D = 6 4 (k + = 8k + 3 D 0 8k k 4

26 .9 ( 4 + (k + = 0 k ( ( (k 4 + k = 0 k ( (3 + a + a 3 = 0 a ( (4 k k k + = 0 k ( ( k = 0 k ( (6 + 4 m 5m = 0 m + m 6 = 0 m (

27 ..3 a + b + c = 0 α β α + β = b a, αβ = c a = 0 α β α + β = 3, 6 αβ = 3 =.0 ( + 3 = 0 (NTT ( = 0 ( ( = 0 ( = 0 α β ( α + β ( α 3 + β 3 (3 (α β (4 α β + β α α + β = 3 αβ = 4 ( α + β = (α + β αβ = ( 3 4 = ( α 3 + β 3 = (α + β 3 3αβ(α + β = ( ( 3 = 9 (3 (α β = (α + β 4αβ = ( = 7 (4 α β + β α = α αβ + β αβ = α + β αβ = 4

28 3. ( = 0 α, β (NTT (i α + β (ii αβ (iii α + β ( = 0 α, β ( (i α + β (ii αβ (iii α + β (3 + = 0 α, β ( (i α + β (ii αβ (iii α + β (iv (α β (4 + = 0 α, β α + β ( ( = 0 α, β (i α β + αβ (ii α + β ( ( = 0 α, β ( (i (α + β 4 (ii (α β 4 (7 + 6 = 0 α, β α + α β (NTT ( (8 3+5 = 0 α, β β + α ( β β α + α ( β

29 m = 0 m ( 3 : 4 ( 3 ( 3α 4α 3α + 4α = 7 3α 4α = m 7α = 7 α = m α = m = ( = 3α = 3( = 3 4α = 4( = 4 ( m = 3 4 ( α α + 3 α + (α + 3 = 7 α(α + 3 = m α + 3 = 7 α(α + 3 = m α = 5 m = 5( = 0 α + 3 α + 3 = = ( m = 0 5. ( k = 0 k ( ( m + m + = 0 m (JFE (3 + a + 4 a = 0 : 3 a (

30 5.4 + p + q = p q p = 6 q = = p = q.3 ( + p + q = 0 5 p, q ( ( + a b = a b ( α β α β (α + β + αβ = 0 p q p + q = i 3 i (3 + i + (3 i = 6 (3 + i(3 i = 9 4i = = (

31 = 0 α β α + β = 0 α + β = 3 αβ = 5 (α + + (β + = (α + β + 4 = = 7 (α + (β + = αβ + (α + β + 4 = = 5 α + β = 0.5 ( 5 4 = 0 α, β α, β ( ( = 0 α, β ( (i α β (ii α β ( = 0 α, β α + β, α + β (

32 7... P ( k P (k ( P ( a + b P b a.9 ( P ( = P ( P ( = = ( P ( = P ( P ( = ( ( + ( + 4 = 6.6 ( ( ( ( ( (

33 8.6 3 m m (NTT P ( = 3 m + 4 P ( = 0 ( 3 m ( + 4 = 0 m =.7 ( 3 a + + a (NTT ( 3 + p p ( (3 4 p 3 + p 4 + p ( (4 3 p p p (

34 a + + b + a b P ( = 3 + a + + b + = ( ( + P ( ( ( + P ( = 0 P ( = 0 P ( = a + + b = 0 a + b = P ( = 0 ( 3 + a ( + ( + b = 0 4a + b = 0 a = 4 b = 6.8 ( a + b a b ( ( 3 + p + q 5 p q (NHK (3 3 + a + b a b (

35 30.8 P ( P ( ( ( + P ( ( ( + a + b Q( P ( = ( ( + Q( + a + b P ( = a + b P ( = a + b 7 P ( = P ( = 5 a + b = 7 a + b = 5 a = 4 b = ( ( ( 5 ( ( f( f( ( ( + (

36 3 P ( k P (k = P ( = P ( = ( ( 5 ( 6 = P ( = ( + ( = ( + ( + ( ( (JFE ( 3 3 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

37 3... ( = 0 ( = 0 (3 ( + ( + 3 = 0 ( = 0 ( + ( + 4 = 0 + = = 0 = ± 3 i ( = 0 ( ( + 4 = 0 = 0 = ± ±i + 4 = 0 (3 + = X X X 3 = 0 (X + (X 3 = 0 ( + + ( + 3 = 0 ( + ( ( + 3 = 0 = ( 3. ( 3 = 0 ( ( = 0 ( ( = 0 ( ( = 0 ( (5 ( 3 ( 3 8 = 0 ( (6 ( 3 8( 3 0 = 0 ( (7 ( + ( + ( 5( 6 = 44 (

38 = 0 P ( = ( P ( = ( ( + 3 ( = 0 P ( + P ( = ( + ( + P ( = 0 + = 0 + = 0 = ±. ( = 0 ( ( = 0 ( ( = 0 ( ( = 0 ( ( = 0 ( ( = 0 (TDK ( = 0 (NEC ( = 0 (

39 a + b = 0 3 ( a b ( ( a + b = 0 ( ( 3 + a ( 3 + b = 0 a + b + 5 = 0 3a + b + 9 = 0 a = b = 6 ( ( = 0 ( ( + 3( + = a + 5 = 0 ( ( a ( ( a + b = 0, a, b (

40 A(a B(b AB AB = b a m n A P B A(a B(b AB m : n P Q m < n n na + mb P Q m A B m + n m > n na + mb m Q n m n A B Q a + b 3. ( O A(5 ( A(4 B( 3 ( OA = 5 = 5 ( AB = 3 4 = 7 = 7 3. ( O A( 3 ( A( B(7 (3 A(6 B( 4

41 36 3. A( B(6 AB ( 3 : P 3 : 4 Q (3 M ( P = = 4 ( Q (3 M + 6 = 7 = 4 = 4 3. A( B(6 AB ( 3 : 5 P ( : Q (3 3 : 7 R (4 M

42 A(, B(, AB AB = ( + ( O A(, OA OA = A(, 3 B(5, AB AB = (5 + { ( 3} = = 5 = 5 O A( 4, OA OA = ( 4 + ( = 0 = ( A(7, 5 B(4, ( ( A( 3, B(3, 6 ( (3 A(, 0 B(0, 6 ( (4 A(8, 7 B( 4, ( A(5, B(, 3 C(0, 3 ABC 3 (

43 ABC ( A(6, 5 B(5, 0 C(, 4 ( ( A(, B(, C(5, 3 ( 3. A(3, B(, 6 P P P (0, PA = PB PA = PB (3 0 + ( = ( 0 + (6 0 = 30 P (0, 3 = A(, B(6, 3 C (

44 39 A(, B(, AB m : n P Q ( n + m P m + n, n ( + m n + m Q, n + m m + n m n m n ( + AB, A(8, 7 B(, 8 AB : P ( 8 + +, 7 + ( 8 (6, + 3 : Q ( ( 8, ( 0, ( A(3, B(8, 5 AB 3 : ( ( A( 4, 0 B(0, 3 AC B C ( (3 A( 7, 0 B(3, 5 AB 3 : C AB ( (4 A(, 3 B(5, 0 ( (i AB (ii AB : (iii AB : 4

45 ( (, m = m( 3.5 (, 5 3 ( 5 = 3( = ( (, 5 3 ( ( (0, ( (3 (0, 3 ( (4 (, ( (5 3 (, 5 ( (6 (5, 60 (

46 4 ( (, (, = ( = = 3.6 (, 4 (, 7 ( 4 = 7 4 ( = ( (, 3, (5, 0 ( ( (, 5, (4, (NTT (3 (, 3, (5, 7 ( (4 (4, 5, (, 7 ( (5 (0, (, ( 3.0 ( ( ( (3 (, 6 ( 3, 5 (3, 4 O O 4 O 5

47 = m + k = m + k m = m m m = 3. ( ( = 3 5 ( 3 + = 5 (3 3 5 = 8 (4 3 = 5 (5 6 = + 7 (6 = 3 3. ( (, 3 = + 3 ( ( A( 4, B(4, 5 P(3, 4 ( (3 (3, = 4 ( (4 (, = 0 ( ( = 0 (4, 6 ( (6 (, 5 ( 3, 3, (, 5 ( 3.3 m 7 = 0, (m = 0 m m (

48 43 3. A(, 3 B(4, AB AB M ( + 4, 3 + ( (3, AB 3 4 = AB m m = m = O A(, 3 M B(4, = ( 3 = ( A B (0,, (4, AB ( ( A(, 0 B(4, 3 (NTT

49 = 0 l l A(, B B (s, t l AB t s AB l t s = B(s, t A(, s + t 4 = 0 ( s + AB, t + l s + t + s t + 7 = 0 + = 0 s = t = 3 B (, 3 l = 0 l l A(, 5 B

50 (3k + (4k + 5k 4 = 0 k k k k ( k + ( + 4 = 0 k = 0, + 4 = 0 = = (, 3.6 ( k + ( k = 0 k ( ( = m + (3m + m ( (, a + b + c = 0 d d = a + b + c a + b 3.7 (, = 0 d d = = 0 5 = 0 5 = 3.7 ( 4 5 = 0 (NTT ( (5, 3, (, (NTT

51 (a, b r ( a + ( b = r r + = r 3.8 ( (, 3 ( ( (, 4 ( (3 (, 5 ( (4 (3, 0 (NHK (5 (3, 4 (7, 7 ( (6 (3, 4 (

52 3.5 A(3, 4 B(5, C r C AB A(3, 4 ( 3 + 5, 4 + ( (4, C r = CA = (3 4 + (4 = O 0 47 ( 4 + ( = ( 0 B(5, ( 4 + ( = ( A( 3, 4 B(5, AB ( ( (3,, (, 5 ( (3 (, 3, (4, 5 (

53 = 0 ( ( = ( + + ( = 5 (, ( = 0 (JFE ( = 0 ( (3 + 6 = 0 ( ( = 0 ( ( = 0 ( ( = 0 ( ( = 0 ( (8 + = + ( = 0 (4, (

54 A(, 7 B(, C(6, l + m + n = 0 A ( + 7 l + 7m + n = 0 B + ( + l m + n = 0 C 6 + 6l + n = 0 l + 7m + n + 50 = 0 l m + n + 8 = 0 6l + n + 36 = 0 l = 4 m = 6 n = = ( (0, 0, (3,, (, ( ( (3, 5, (3,, (4, 0 (

55 = 0 = { + = 0 + =0 = 0 + ( = 0 3 = 0 =, 3 0 = = = 3 = 3 = (, 3 (3, O ( + = 5 + = 0 ( ( ( + = 5, = 5 (3 + = ( (4 + = + = ( (5 ( + ( = 8 + = 3 (

56 5 ( a +b+c = 0 D = b 4ac D D > 0 D = 0 D < 0 a + b + c = 0 ( = 5 = m+5 m + = 5 = m + 5 (m + + 0m + 0 = 0 D = (0m 4 (m + 0 = 0(m 4 D = 0 m 4 = 0 m = ± 3.4 ( + = 4 60 (NEC ( = 0 ( ( = 0 (

57 5 ( O r l O l d d r d < r d = r d > r l l r r d d O O l d r O = k = 0 k r k = 0 d d = k = k 5 d = r k 5 = 3 k = ±5 3.5 ( = a + = a ( ( + = = c c (

58 53 + = r P(a, b a + b = r 3.0 A(, + = P(a, b P a + b = P a + b = A(, a + b = a O 5b 4b = 0 4 b = 0, 5 + = A(, b = 0 a =, b = 4 5 a = 3 5 (, = = ( 3 5, = = ( (3, + = 5 ( ( (3, + = (

59 P (, P P 3. A(, 0 B(4, 0 : P P (, P AP : BP = : BP = AP 4BP = AP AP = (+ +, BP = ( 4 + 4{( 4 + } = ( = 0 ( 6 + = 4 A O B 4 P(, P ( 6 + = 4 P(, (6, O A(5, 0 3 : P

60 3.3. = m + k l > m + k 55 l > m + k l < m + k l m + k m + k l + < r + = r + > r + = r + r + r + = r r r O O + < r r < m + k + > r 3. { + > > 0 + = 6 = O r

61 ( ( A(4, 4 O B(6, 0 ( ( O (3 ( (i (ii 5 O 4 4 ( 4, 3 O (4, 3 5 (4 ( (i (ii O

62 3.9 ( < 4 < > 57 ( ( + 4 (JFE (3 < + < 4 ( (4 { + ( (5 { 3 3 ( (6 { + 4 (

63 ( + (3 > 0 ( + (3 > 0 { + > 0 3 > 0 { + < 0 3 < 0 O 3.30 ( ( ( < 0 ( ( ( + 4( + < 0 ( (3 ( + 4( > 0 ( 3.3 (, ( ( 0 + ( 0 + O. ( + 0

64 , 0, + 3 5, A A 4 ( ( 8 (0, 0 3, (, 0, 3 5 (, 3 A + = k O A k (0, 0 k = 0 (, k = 3 ( 8 3, 0 ( 0, A 0 < k < 3 + = = 3 = 0 = k = 8 3 k = , 0, 3 + 9, + 8 +

65 θ θ OX θ OP θ P 60 O X O 45 X P O P 40 X 4. ( 5 ( 70 (3 660 OP OX α OP α n = = ( 690 = ( O P X

66 6 ( π π π = 3 4 π 4.3 ( 5 ( 60 (3 8 5 π (4 5 π r θ( l S l = rθ, S = r θ S = lr π π = π 4 8 π 4 = 8π 4.4 l S π ( 8 ( π

67 6 4.. θ r P(, P(, r sin θ = r ( θ r cos θ = r O ( r r tan θ = ( r 4.5 π 3 r = = = 3 ( sin π = 3 3 = 3 ( cos π = 3 ( tan π = 3 = 3 3 O 3 π ( tan 0 ( ( sin( 0 ( (3 cos 960 (NTT (4 cos( 70 ( (5 sin π 6 cos 5 6 π ( (6 sin 45 cos( sin 50 cos 405 (

68 63 = sin θ π α π π O α π 3 π π π 5 π 3π θ = cos θ π α π π O α π π π 3 π 5 π 3π θ = tan θ π θ = π θ = 3 π α π π O α π π 3 π π 5 π θ

69 64 ( 4.6 = cos θ = cos θ + π 3 O π π θ ( = cos θ + π = cos θ θ π 3 3 O π π θ 4.6 ( ( = sin π ( ( ( = cos π 6 (

70 4.7 = cos θ = cos θ θ π 65 π π O π π 3 3π π 5 7 π θ 4.7 = sin θ ( = 3 sin θ O π π θ ( = sin 3θ O π θ

71 tan θ = sin θ cos θ 3 + tan θ = cos θ sin θ + cos θ = 4. θ 4 sin θ = 4 5 cos θ tan θ cos θ = sin θ = ( 4 = θ 4 cos θ > 0 9 cos θ = 5 = 3 5 tan θ = sin θ ( cos θ = = ( θ 4 cos θ = 3 sin θ (NTT ( θ 3 cos θ = 3 4 sin θ tan θ ( (3 θ sin θ = 3 5 cos θ tan θ ( (4 θ 3 sin θ = 3 5 cos θ tan θ (

72 ( cos θ = (0 θ < π ( tan θ = 3 ( θ 0 θ < π θ = 3 π, 4 3 π 4 π 3 O π 3 ( θ = 3 0 θ < π θ = 3 π θ = 3 π + nπ (n O 3 θ 3 cos θ = θ = 3 π + nπ θ = 4 3 π + nπ (n 4.9 ( sin = (0 90 ( ( 3 3 tan = 0 (0 π ( (3 tan( 0 = 3 (0 360 ( (4 cos = (0 90 (

73 θ < π sin θ + 5 cos θ + = 0 ( cos θ + 5 cos θ + = 0 cos θ 5 cos θ 3 = 0 ( cos θ + (cos θ 3 = 0 cos θ cos θ + = 0 cos θ 3<0 0 θ < π cos θ = 4. θ = 3 π, 4 3 π 4.0 ( sin sin = 0 (0 360 ( ( sin θ + 3 sin θ = 0 (0 θ < π ( (3 cos θ + sin θ = (0 θ < 360 ( (4 cos θ + sin θ + = 0 (0 θ < 360 ( (5 cos + sin = (0 80 ( (6 sin + cos = 0 (0 < 360 ( (7 sin θ + 5 cos θ 4 = 0 (0 θ < 360 (

74 θ < π sin θ > 0 θ < π sin θ = θ θ = π 6, 5 6 π O π 6 < θ < 5 6 π 5 π 6 π π cos > ( 4. 0 π (NEC ( sin sin = 0 ( sin sin < = 4 cos θ + sin θ 3 (

75 sin(α + β = sin α cos β + cos α sin β sin(α β = sin α cos β cos α sin β 3 cos(α + β = cos α cos β sin α sin β 4 cos(α β = cos α cos β + sin α sin β 4.8 cos 5 cos 5 = cos(45 30 = cos 45 cos 30 + sin 45 sin 30 = = = ( sin 75 ( ( cos 75 ( (3 sin 5 ( (4 sin 65 ( (5 sin 5 + cos 45 ( (6 cos 75 sin 5 (

76 7 5 tan(α + β = 6 tan(α β = tan α + tan β tan α tan β tan α tan β + tan α tan β 4.9 tan 5 tan 5 = tan(60 45 = tan 60 tan 45 + tan 60 tan 45 3 = + 3 = ( 3 ( 3 + ( 3 = ( = 4 3 = tan 75 ( < α < π π < β < π cos α = 3 sin β = 3 cos(α+β 0 < α < π π < β < π sin α > 0 cos β < 0 sin α = ( 5 cos α = = 3 3 cos β = ( sin β = = 3 3 cos(α + β = cos α cos β sin α sin β = 3 ( = sin α =, sin β = 3 cos(α β 0 < α < < β < 80 (

77 7 4.7 ( ( sin( + + sin( = sin cos ( ( sin 75 cos ( sin(45 + α + sin(45 α = cos α ( ( sin A sin B = sin(a + B sin(a B ( 4.9 ABC B = 45 C = 60 BC = 0 AB ( p + q = 0 tan α, tan β tan(α + β p, q (

78 sin α = sin α cos α cos α = cos α sin α cos α = sin α cos α = cos α cos α= sin α sin α= cos α 4.6 (sin α + cos α = + sin α = sin α + sin α cos α + cos α = (sin α + cos α + sin α cos α = + sin α cos α + sin α cos α = + sin α (sin α + cos α = + sin α sin α 4. + cos α = tan α ( 4. ( cos θ = 4 5 sin θ θ ( ( sin θ + cos θ = 6 sin θ (

79 74 sin α = cos α, cos α = + cos α 4.0 sin α = 0 < α < π cos α = 3 sin α cos α ( = α 3 3 cos = ( + = 3 3 sin α > 0 cos α > 0 sin α = 3 cos α = π < α < 3 π cos α = 3 5 sin α cos α tan α = 4. tan α = 3 tan α = tan α tan α, α tan = cos α + cos α tan α tan α = 3 3 = ( tan α = tan α ( π < α < π cos α = 4 5 tan α

80 θ < π cos θ + sin θ = ( sin θ + sin θ = sin θ(sin θ = 0 sin θ = 0 0 θ < π sin θ = 0 θ = 0, π sin θ = 0 sin θ = θ = π π θ = 0, π, < π 3 sin cos = ( 4.6 = sin + cos (0 < π ( a sin θ + b cos θ a sin θ + b cos θ = a + b sin(θ + α cos α = a a + b, sin α = b a + b 4. r sin(θ + α π < α < π ( 3 sin θ + cos θ ( sin θ cos θ ( ( 3 sin θ + cos θ = sin θ + π 6 O π 6 P( 3, 3 ( sin θ cos θ = ( sin θ π 4 O π 4 P(,

81 < π sin + 3 cos = ( sin + π = 3 ( sin + π = 3 0 < π π 3 + π 3 < 7 3 π + π 3 = 3 4 π + π 3 = 9 4 π = 5 π, 3 π 4.7 ( 3 sin + cos = 3 (0 < 360 ( ( 3 sin θ cos θ = (0 < θ < 80 ( (3 3 cos θ sin θ = (0 θ < π ( 4.8 = cos sin (0 < π (

82 a 0 a`n a 0 n a 0 = a`n = a n a` = a = 5 = 5 0 = 0 = ( ( ( 3 0 ( ( 3 ( 3 ( a 0 b 0 m n a m a n = a m+n am a n = am`n 3 (a m n = a mn 4 (ab n = a n b n = 3 6+( 4 = 3 = 9 ( 3 = ( ( 3 = 3 = 8 5. ( ( ( 3 4 ( ( 5 3 (JFE

83 78 a > 0 b > 0 m n n a n b = n n a a ab n = n b b 3 ( n a m = n a m 4 m n a = mn a = = = 5 ( 5 3 = 5 3 = = ( ( ( ( ( 3 ( 7 a > 0 m n r a n = n a, a m n = ( n a m = n a m, a`r = a r = = 3 5 = = 8 3 = ( 3 8 = = a.5 ( 5.5 ( 7 3 ( (3 5.5 (NTT ( ( (4 ( 5 4 ( , (

84 79 ( a > 0 b > 0 r s a r a s = a r+s ar a s = ar`s 3 (a r s = a rs 4 (ab r = a r b r 5. ( ( (8 6 4 ( ( = = 9 = (3 = 3 ( = 3 = 3 ( (8 6 4 = = 8 3 = ( 3 3 = 3 ( 3 = = = 4 ( = = = = 5.7 ( (64 6 ( ( ( ( ( (4 (5 { ( } ( ( ( ( (6 ( 3 3 ( n a n = a (n = 3 ( 3 = 5 = 5 ( 5 = (

85 = a a > a 0 < a < a O O (0, (, a a > 0 < a < 5.9 ( ( = ( = O O = a 3 a > r < s a r < a s 0 < a < r < s a r > a s

86 5.6 3 ( 3 5 ( ( ( 9 ( ( 3 = 3 = = 3 5 = < 3 5 < 3 < 5 9 < 3 ( ( = 3 ( > 3 ( > 3 ( ( 3 ( ( ( 8 5. ( ( 3 ( < 8 ( 5 = 5 ( ( 3 ( < > 3 =( 5. ( 3 < ( 7 ( 3 8

87 8 5.3 ( 9 = 7 ( 8 = + ( 3 = 3 3 = 3 = 3 ( 3 = + 3 = + = 5. ( = 6 ( = 3 ( (KDDI (3 3 + = 7 (4 5 = 5 ( = 65 ( ( ( (6 ( 5 = 5 ( ( = 3 ( (8 3 = 9 (NTT (9 9 = 7 ( (0 36 = (

88 = 0 4 = ( + = 4 ( 4 3 = 0 ( + 4( 8 = > 0 8 = 0 = 8 = 3 = X X 4X 3 = (X + 4(X ( 4 + = 0 ( ( = 9 ( ( = 3 3 ( ( = 0 ( ( = 0 ( ( = 5 +4 (

89 { = 3 + = 7 3 = X 3 = Y 3 + = 3 3 = XY { X + Y = XY = 7 Y = X 3 3 > 0 3 > 0 X > 0 X > 0 0 < X < 4 3 X( X = 7 X X + 7 = 0 (X 3(X 9 = 0 4 X = 3, 9 3 { { { X = 3 Y = 9 X = 9 = Y = 3 { = = = 5.4 { + = 40 ( + = 56 ( ( { = = 3 9 (

90 a > 0 a M > 0 M = a p log a M = p log a M a M M = a 0 a = a log a = 0 log a a = 5.7 ( log 36 = ( log = 5 ( log 36 = 36 = ( log = 5 = 5 > 0 = 6 = ( log 6 = 3 ( ( log 8 = ( log 5 5 ( log 6 log a M = p ( log 5 5 = log = 3 ( log M = a p log a a p = p ( 4 6 = log = 4 ( 5.6 ( log 8 (NTT ( log 8 ( (3 log 0 (

91 ( log ( log ( log 8 = log = log 3 3 = 3 ( log = log 0 00 = log 0 0 = log 0 0 = 5.7 ( log ( ( log 0 0. ( (3 log ( (4 log ( (5 log 3 8 ( (6 log ( (7 log cos 60 ( 5.8 ( log a = a log a a = 3 (a b 3 = a 3 3a b + 3ab b 3 4 D A + C B = AD AB + AC AB 5 A m A n = A mn

92 87 M > 0 N > 0 k log a MN = log a M + log a N log a M N = log a M log a N 3 log a M k = k log a M 5.0 ( log log 6 4 ( log 7 log 56 (3 log 0 5 log log 0 6 ( log log 6 4 = log 6 (9 4 = log 6 36 = log 6 6 = 7 ( log 7 log 56 = log 56 = log 8 = log 3 = 3 (3 log 0 5 log log 0 6 = log 0 5 log log 0 6 = log = log 0 0 = ( log log 3 8 ( ( log log 0 ( (3 log 4 log 3 (NTT (4 log 6 log 3 4 ( 3 (5 log 8 log 4 + log 6 ( (6 log 3 (4 7 + log 3 (4 + 7 ( (7 log 0 + log 0 5 log0 0.6 (NTT 4 (8 log log 0 5 ( (9 log 0 5 log 0 ( (0 log log log 0 5 (

93 88 0 log 0 5 log 5 5. log 5 + log 4 = log(5 4 = log 0 = 5.0 ( log log 5 log 8 3 ( ( log 45 3 log 3 + log 3 5 ( (3 log 5 3 log log 3 + log 8 ( a b c a c log a b = log c b log c a 5. log 3 8 = log 8 log 3 = log 3 log = ( log 9 7 ( ( log 5 5 (NTT (3 log 8 ( (4 log 3 4 ( (5 log ( (6 log ( 4 (7 log 3 9 (

94 log 3 log 4 7 = log 3 log 3 7 log 3 4 = log 3 3 log 3 = 3 5. ( log 3 log 8 9 ( ( log 3 log 3 5 log 5 7 log 7 8 ( (3 (log 3 + log 4 9(log log 9 (NTT 5.3 log 3 = a, log 3 = b log a, b ( 5.4 log A =, log B =, log C = z,, z ( ( log A B ( log AB C (3 log C A B 5.4 ( 3 log 3 5 ( 5 3 log 5 log a M = p M = a p M = a log a M ( 3 log 3 5 = 5 ( 5 3 log 5 = 5 log 5 8 = log (

95 = a = = log a a > 0 < a < = a O a = a O a = = log a = = log a (, 0 (a, a > 0 < a < 5.6 ( = log ( = log 3 O O = log a a > 0 < p < q log a p < log a q 3 0 < a < 0 < p < q log a p > log a q

96 9 5.6 ( log 3 ( log ( 3 > ( log log 3 8 ( log ( 3 > log 0 < 3 < =log =log ( 3 3 < < ( log 3 < ( log ( = ( 6 = 0 ( + ( 3 = 0 + > 0 3 = 0 = 3 = log ( = 8 3 ( ( = 0 (

97 9 5.8 log + log ( 6 = 4 > 0 6 > 0 > 6 log ( 6 = 4 ( 6 = 4 ( + ( 8 = 0 + > 0 = ( log 0 ( 3 = (NTT ( log( + + log 5 = ( (3 log 6 + log 6 ( 5 = ( (4 log + log( 3 = ( (5 log + log( 5 = ( (6 log( + + log( = ( (7 log( + + log( + 5 = ( (8 log( + + log( = 0 ( (9 log 0 3 log log 0 4 log 0 3 = ( (0 log( + log( 9 = ( ( log 0 (4 3 + log 0 (3 4 = ( ( log( + 6 = log( 4 ( (3 log( 5 = log( log( 7 + log ( (4 log(4 + + log( + = ( log 3 + log 5 log 3 (KDDI (5 log( log( = 0 ( (6 log ( 4 log 4 ( = 0 (

98 { 3 = log + log = = 3 > 0 3 > 0 > 3 log = log 0 = 0 (3 = 0 ( (3 + 5 = > 0 = = 5 ( = = { = 3 ( log + log = ( ( { + = 9 log + log = (KDDI (3 { log( + log(7 8 = log( log( + = (

99 log = ( log ( log ( log = log 0 ( = log log = = ( log = log 0 ( = log log 0 0 = = log 0 = log 0 3 = ( log 0 6 ( ( log 0 0 ( (3 log 60 ( (4 log 6000 ( (5 log log 0 ( (6 log 0 8 ( (7 log 6 ( (8 log.5 ( (9 log 0. ( (0 log 0.5 ( ( log.08 ( ( log 864 (NHK (3 log 5 (

100 5.3 log 8.5 = ( ( log 85 ( log 850 (3 log log = ( ( log ( log log 0 = log 0 3 = log = 30 log 0 6 = 30(log 0 + log 0 3 = 30( = < log < 4 log < log < log < 6 30 < ( 3 0 log 3 = (KDDI ( 3 8 log 3 = ( (3 log = 0.300, log 3 = (

101 % 0 log 0 = log 0 3 = n 0. n 0 log 0. n log n log 0 0 n( log 0 + log 0 3 log 0 + log 0 3 = = n log 0 + log 0 3 = = ( log =.0453, log 0 =.300 ( ( 0.08 log = 0.300, log 3 = ( ( log 7 = log = (

102 = f( a = f( b f(b B f(b f(a = b a f(b f(a f(a = a = b A f( b a AB O a b 6. f( = + = = 3 f(3 f( 3 ( = ( { ( + ( } 3 ( = 5 5 = 3 6. = = = ( 6. lim ( + 3 lim ( + 3 = + 3 = 4 6. ( lim 3 3(5 ( ( lim ( (

103 lim lim + 6 = lim ( ( ( + 3( = lim + 3 = + 3 = ( lim ( lim (3 lim (4 lim 0 + (5 lim + (6 lim 3 (7 lim (8 lim ( lim ( lim h 0 (a + h 3 a 3 h ( (NEC ( ( ( ( ( ( ( (

104 99 6. a b + a + b lim 3 + = 5 lim ( + a + b = lim + a + b = 5 0 b = a 4 + a + b lim 3 + = lim { + a + b 3 + ( a a = lim + a + = a + 4 } = lim ( ( + a + ( ( a + 4 = 5 a = b = 6 ( a = b = a, b ( lim a + + b = 3 ( ( lim + a + b = 5 3 (

105 00 f( = a f 0 (a = lim h!0 f(a + h f(a h f(a + h = f( f(a A A = f( A(a, f(a O f( = a a a + h f (a 6.4 f( = = 3 f (3 = lim h 0 f(3 + h f(3 h 6h + h = lim h 0 h = lim h 0 (6 + h = 6 = lim h 0 h(6 + h h = lim h 0 (3 + h 3 h = (3, 9 f (3 = f( = f (a (

106 0 6.. f 0 ( 6.5 f 0 ( = lim h!0 f( + h f( h f( = 5 ( f ( ( f 5( + h 5 ( = lim h 0 h = lim h 0 (0 + 5h = 0 ( f ( f ( = lim h 0 0h + 5h h ( f ( = 0 = 0 f ( = 0 ( = 0 = f( 0 d d (5 d d (5 (5 = 0 d d (5 = 0 d d (3 f( = f ( = lim h 0 ( + h 3 3 h = lim h 0 6 h + 6h + h 3 h d d (3 = 6 d d (53 = lim h 0 ( h + 3h + h 3 3 h = lim h 0 (6 + 6h + h = 6 (

107 0 n n ( n 0 = n n` c (c 0 = 0 n =,, 3 ( = ( = ( 3 = 3 k = kf( 0 = kf 0 ( = f( + g( 0 = f 0 ( + g 0 ( 3 = f( g( 0 = f 0 ( g 0 ( 6.7 = = ( 3 4( + 5( + (3 = = ( = ( ( = ( (3 f( = ( (4 = ( 6.9 ( d d ( ( = 3 d d ( (

108 = ( + 3 ( + 3 = ( = = = ( = (5 + 4( + ( ( = ( 3(3 ( (3 = ( 3 ( (4 = ( + ( 5 (NEC (5 f( = ( + ( ( (6 = ( ( ( (7 = ( ( + + ( (8 = ( + ( (9 = ( + 5(3 + ( + 4 ( 6. = (3 + d d (NEC 6. = ( f( = ( 3

109 = f( A(a, f(a f(a = f 0 (a( a 6. = 3 = l f( = 3 f ( = 3 f( = 3 = 3 f ( = 3 = l ( 3 = ( = ( = 3 (3, 0 ( 3 O l ( = (, 4 ( (3 = 3 = ( 6.4 = 6 + (JFE ( A(5, 4 ( A (3

110 = C(, = = 3 (a, a 3a + 4 a 3 (a 3a + 4 = (a 3( a C(, (a 3a + 4 = (a 3( a a 4a + 3 = 0 (a (a 3 = 0 a =, 3 a = = ( a = 3 4 = 3( 3 ( = + 3 = 3 5 C O (, 3 = (

111 = (0, 3 ( 6.7 = + 3 (, 0 ( (0 (NEC (, 0 = + 3 A(a, a + 3 A f( = + 3 f ( = f(a = a + 3 f (a = ( A ( (a + 3 = (3 ( a = (4 (, 0 a a = (5 a = (6 = (7 = (8 (9 (0

112 f 0 ( f( f( f ( > 0 f ( < 0 f( f( = a = f( f( = a f(a = b f( = b f(b O a b 6.4 = = = 3( ( 3 = 0 =, = = 3 O 3

113 ( f( = ( ( = ( (3 = ( (4 = ( (5 = 5 (3 3 9 ( 6.9 = 3 3 ( 6.0 = (JFE ( ( (3 6. f( = 3 + a + b (, 4 3 ( ( a, b ( f( (3 = f(

114 f( = 3 + a + b + c = = 3 a b c f ( = 3 + a + b f ( = 0 =, 3 ( + 3 = a 3, ( 3 = b 3 a = 3 b = 9 f( = c f(3 = c = c = 5 f( = ( 3 3( 9( + 5 = 0 ( a = 3 b = 9 c = = 3 + a + b + c = 0 = ( ( a, b ( 6.3 = 3 + a + b + c = 3 = 3 ( 6.4 = a 3 + b + c + d = 44 = 4 64 a, b, c, d (

115 0 3 = a 3 +b +c+d = 0 3a +b+c = 0 D D/4 = b 3ac D/4 > 0 0 D/4 = 0 0 a D/4 < 0 a D/4 = b 3ac > 0 D/4 = b 3ac 0 3 = a 3 + b + c + d (a 0 = 3a + b + c D/4 = b 3ac D/4 D/4 = b 3ac > 0 D/4 = b 3ac = 0 D/4 = b 3ac < 0 = 0 α, β α a > 0 O α β O α O a < 0 O α β O α O = a 5 a = a = 0 D D/4 = 6 3a D > 0 a < 6.5 = 3 + a a (

116 = 3 ( 3 5 = 3 = 3( + ( = 0 =, = 5 65 = 6 6 O ( = ( 3 ( = ( 4

117 6.8 8cm 8cm cm (8 cm cm cm cm 3 > 0 8 > 0 0 < < 9 = (8 = 4( = ( + 7 = ( 3( = ( 3 cm 6.7 ( 30cm ( ( 5cm 8cm ( (3 5cm 40cm cm (

118 = a a = 3 6 = 3 = 3( 4 = a = 3 6 O 4 a = a a < 3, 0 < a = a 3 a = 0, 3 3 < a < ( = a 3 a ( 3 3 = a a

119 f( = ( f ( = f ( = 3( f( 0 0 f( 0 f( = f( 0 ( = ( (

120 f( F ( = f( f( d = F ( + C C n n = 0,, d = + C n d = n + n+ + C d = + C d = C F ( = f(, G ( = g( kf( d = kf ( + C k {f( + g(} d = F ( + G( + C 3 {f( g(} d = F ( G( + C 6.9 ( ( (3 5 d = 5 + C = 5 + C (6 + 3 d = C = C ( + ( d = ( + d = C

121 ( d ( ( d ( (3 ( + d ( (4 ( d ( (5 ( 3 d ( (6 ( d ( (7 ( + d ( (8 ( d (

122 7 6.3 ( f( = 3 + ( ( f( = ( ( + ( 6. f ( = ( + (3 f( = 3 f( f( f ( = ( + (3 f( = ( + (3 d = (3 + d = C f( = ( 3 + ( ( + C = C + C + = 3 C = f( = f( =, f ( = ( 3( + f( (NEC

123 F ( = f( 6.0 ( ( 4 3 b a f( d = [ ] b F ( = F (b F (a a [ ] 3 4 d = = = [ ] 3 (4 3 d = 3 = ( {( 3( } = ( 3 3 d ( ( (3 (4 (5 ( (3 d ( ( + d ( ( + 3( d ( ( + d ( ( + 4( d ( (7 3 ( + ( + d ( (8 ( d (NEC

124 9 3 b a b a b a k f( d = k b a {f( + g(}d = {f( g(}d = f( d b a b a f( d + f( d k b a b a g( d g( d 6. p q ( ( ( (p + q d ( (p + q d = p ( + 3 d 3 d + q 0 3 ( + 3 d [ 3 d = p 3 = p q 3 = p + 4q 0 ( 3 d = = ] 3 ( 3 d [ + q {( + 3 ( 3 } d [ ] d = 6 0 ] 3 = = k ( (k + d ( 3 ( + ( + 4 d 3 ( + d

125 0 a a f( d = 0 a b b f( d = f( d a b a c a b c 3 f( d = f( d + f( d 3 a b c (3 + 4 d = = (3 + 4 d 3 0 (3 + 4 d (3 + 4 d (3 + 4 d [ ] (3 + 4 d = = ( = 3 (3 + 4 d 6.35 ( 3 3 ( d ( 4 ( + 3 d 4 ( + 3 d

126 a f(t dt f( a d f(t dt f(t dt d a a 6. f(t dt = a f( a f( = = a 0 0 = a a a =, f( = a =, 6.36 ( f( = 0 (3t t dt ( g( a a g(t dt = +

127 6.3.3 ( a b f( 0 = f( = a = b S = f( S = b a f( d O a S b 6.3 = + = + = = S [ ] S = ( 3 + d = 3 + ( { } 3 ( 3 S = ( = 6 O ( (NEC = O 3 ( = + 4 = 4 ( (3 = + 3 = = 3 ( (4 = 3 + ( (5 = 4 (

128 3 ( a b f( 0 = f( = a = b S S = b a { f(}d O a = f( S S b = f( 6.3 = = 0 =, O S S = 3 { ( 4 + 3} d S 3 ] 3 = [ = ( ( = ( = ( ( = ( (3 = a (a > 0 36 a (NEC

129 4 6.4 = = = 0 = 0,, 0 0 O 0 S S = 0 ( d + [ ] 4 [ = ( 4 = = 4 { ( } d ( ] ( ( = ( ( = ( ( ( 3 (

130 5 (3 a b f( g( = f( = g( = a = b S = f( S S = b a {f( g(}d O a = g( b 6.5 = 3 = 3 3 = 3 = 0, 4 S S = = = {( 3 ( 3} d ( + 4 d [ ] 4 0 = O ( = = + ( ( = + 5 = ( (3 = ( = + (

131 6 6.4 ( 4 = = ( { 0 ( + 0 ( (3 = l + m = n (i l, m, n (ii ( (6, ( 3, 3 O (4 = 4 + 3, = 6 + (

132 7 6.6 = (, 5 = = = 5 = ( = + 3 ( ( + 3 = ( ( + 3 = 3, 3 ( ( S S = = 3 3 {( ( + 3} d ( d [ 4 = ] ( 4 = { ( ( } ( 3 + 3( 3 = ( = 3 3 = 0 ( ( = = (NEC (3 = (, (

133 8 α β ( α( β d = (β α3 6 a + b + c = 0 α β β (a + b + c d = a ( α( β d = α = β α β α ( α( β d = a (β α3 6 { (α + β + αβ} d β β d (α + β d + αβ d α α = 3 (β3 α 3 (α + β(β α + αβ(β α = 6 (β α{(β + βα + α 3(β + α + 6αβ} = 6 (β α( β + βα α = (β α3 6 α β a + b + c = 0 { ( a + b + c = a b + c } = a{ (α + β + αβ} a a = a( α( β S = 3 = { ( 4 + 3} d = ( (3 3 = S = 4 0 = 3 {( 3 ( 3} d = ( (4 0 3 = ( ( 3 d 4 0 ( 4 d

134 (. ( 3 0 ( + 3 ( ( (5 p + 3p + 0 (6 6 + (7 3 5 (8 a 3a + 0 ( ( A = ( B = ( b4 c ( (3 + (4 3a 4 d ( a.4 ( ( 3 ( (4 4 (5 3 3b 3 8a a 4 b.5 ( ( (3 a 6 ( + ( + (4 a 3 ( ( (7 +.6 ( ( (3.7 ( (7 + 4 (8 + ( + ( ( + (.8 ( ( c ( + ( + 4 (6 (7 ( + (9 3 (4 (5 + ( (3 0 (4 + 4 a (8 (9 a b + (3 0 (4 (8 3 (5 (5 (3 + ( + ( a + (a + (a + 3 ( + ( + ( ( 3 (6 (7 ( ( + ( + 3( 4 (5 (5 (6 + ( ( a + 3 (6 (6 +.9 ( A = B = 5 C = ( a = b = (3 A = 3 B = C =.0 a + b + c = 0 a + b = c b + c = a c + a = b (a + b(b + c(c + a + abc = ( c( a( b + abc = 0 9

135 . a + b + c = 0 a + b = c, b + c = a, c + a = b ( + b c + + c a ( a b + c = a b + a c + b c + b a + c a + c b + 3 = b + c a + c + a b + a + b c = a a + b b + c c + 3 = + 3 = a b = c d = k a = bk c = dk ab + cd ab cd ( a + b + 3 = bk b + dk d bk b dk d = k(b + d k(b d = b + d b d a + c a c = (bk + (dk (bk (dk = k (b + d k (b d = b + d b d ab + cd ab cd = a + c a c.3 a = b = k = ak, = bk ( a = (ak a = k + a ab + b = (ak ak bk + (bk a ab + b a = + a ab + b ( pa + r qa + s pb + r qb + s pa + r qa + s = k (a ab + b a ab + b = k = pa + r ak qa + s ak = pb + r bk qb + s bk = pb + r qb + s 30 a(p + rk = a(q + sk = p + rk q + sk b(p + rk = b(q + sk = p + rk q + sk

136 .4 ( a + b (a + b = a a + + b b + = (a + (b 0 a + b (a + b a = 0 b = 0 a = b = ( a ab + b = a ab + b b ( = a b b 0 a ab + b 0 a b = 0 b = 0 a = 0 b = 0 (3 + + z ( + z + z.5 = {( + + ( z + z + (z z + } = {( + ( z + (z } z + z + z = 0 z = 0 z = 0 = = z ( a > 0 a > 0 a + a a + a a a = 3

137 ( ( (a + b a + = + a b b + b a + = a b + b a + a b > 0 b a > 0 (3 a b + b a a + b b a + = 4 ( (a + b a + 4 b ( a b + c ( b d a + d = + ad c bc + bc ad + = ad bc + bc ad + ad bc > 0 bc ad > 0 ad bc + bc ad + ( a b + c d ( b a + d c ad bc bc ad + = 4 4 (. ( = 5 = 4 ( = = 3 (3 = =. ( + 7i ( 4 + 8i (3 7 + i (4 3 7i (5 7 (6 6 (7 i (8 5 (9 i ( i ( 0 ( i (3 + i ( i.4 ( 3 4 i ( i (3 3 i ( i (5 5 3 i (6 i (7 i 3 (8 (9 5 ( i.5 ( 0 ( 6 (3 4 + i ( i.6 t = 3 ± 3i.7 ( = ± 9i 5 ( = 5 ± 7 i 6 (3 = ± 35 i 3 3

138 .8 ( ( (3 (4 (5 (6.9 ( k < 3 5 < k ( k < 8 < k (3 a = 6 (4 0 < k < (6 < m < 4.0 α β (5 k 5 ( α + β = αβ = 3 ( α + β = 3 αβ = 5 (3 α + β = αβ =. ( (i 4 (ii (iii 4 ( (i 5 (ii (iii 9 (3 (i (ii (iii (iv 4 (4 (5 (i 3 (ii 3 (6 (i 8 (ii 6 (7 63 (8 4. ( k = 5 ( m = (3 a = 0 = 4 6 a = ( p = 7 q = 0 ( a = b = = 0.5 ( = 0 ((i = 0 (ii = 0 ( = 0.6 ( 0 ( 7 ( ( a = ( p = 6 (3 p = (4 p =.8 ( a = 5, b = 0 ( p = 5 q = (3 a = 5 b = 9.9 ( + ( ( ( (+ ( (+ ( (3 ( ( (4 ( ( (5 ( ( 7 (6 ( + ( 3 (7 ( ( + ( 3 (8 ( + ( (9 ( ( + 3( + 4 (0 ( + ( 3( ( ( + ( (3 ( ( + ( ( + ( 3. ( = ± 3 i ( = ± ±i (3 = ± ±i (4 = ± ±3 (5 = ± 4 (6 = ± 5 (7 = ± 5 ± 5. ( = ( ( = ± 3 (3 = ( (4 = 3 (5 = 3 (6 = 5 (7 = ± i (8 = ( a = 3 ( = ± i.4 a = 8 b = 8 = ± 3 i 33

139 ( 3. ( 3 ( 5 ( ( ( 4 (3 8 (4 3.3 ( 5 ( 0 (3 0 ( AB = 37 BC = 37 CA = ( BC = CA ( CA 3.6 C(5, 0 ( ( 6, (8, 5 (4(i 3 5 (ii (3, (iii ( 3, 4 ( C(4, 6 (3 C(, 3 AB = ( = 3 + ( = (3 = + 3 (4 = + 5 (5 = 3 4 (6 = ( = + 5 ( = (3 = (4 = (5 = ( = 5 0 ( = 3 + (3 = (//(6, (4//(5, ( (, ( (6 3. ( = 7 ( = (3 = (4 = (5 = 3 (6 = m = 3 m =, 3.4 ( = ( = B(4, ( (, ( ( 3, 3.7 ( 5 7 ( ( ( +( + = 9 ( ( +( + = 6 (3 ( +( = 5 (4 ( 3 + = 9 (5 ( 3 + ( + 4 = 5 (6 ( 3 + ( 4 = ( ( +( 3 = 7 ( ( +(+ = 3 (3 ( 3 +(+ = ( (4, 3 ( ( 3, 4 (3 (3, 0 (4 (, 6 (5 (, 3 (6 (, ( (7 (, 0 (8, 3. ( + ( + = 4 3. ( = 0 ( = 0 34

140 3.3 ( (3, 4, (5, 0 ( 4 5 ( (4 ( 5, 4 (5 (, ( = 3 + 4, = 3 4 ( = (3 =, = ( a = ±5 ( 5 < c < 5 ( ( + = 5 = 5 ( = (0, 3 4 = 5 5, (9, 0 6 > 0 < 3.8 ( < ( + < + + < 4 (3(i (4(i (ii (ii 3 > < ( ( (3 4 3 O 3 4 O 4 O (4 (5 ( O 3 O O

141 3.30 ( ( (3 4 O O O = = 3 5 = 0 = 0 0 ( 4. ( ( (3 P P 5 O X ( π ( π (3 88 ( ( l = π S = 4π ( l = 4π S = 0π ( ( (3 3 (4 0 (5 3 4 ( π 4.6 ( = sin O X O 660 (6 0 = sin P X π π O π π 3 π π 36

142 ( ( = cos π π = cos 6 6 π 3 O π 6 3 π 7 6 π 5 3 π 4.7 ( = 3 sin θ = sin θ θ 3 π O π π θ ( = sin 3θ = sin θ θ 3 3 π O π θ 37

143 4.8 ( sin θ = 5 3 ( sin θ = 7 4, tan θ = 7 3 (3 cos θ = 4 5, tan θ = 3 4 (4 cos θ = 4 5, tan θ = ( = 30 ( = π 6, 7 6 π (3 = 70, 50 (4 = ( = 0, 30, 50, 80, 360 ( θ = π 6, 5 6 π (3 θ = 80 (4 θ = 70 (5 = 90 (6 = 0, 0, 40 (7 θ = 60, < 3 π, 4 3 π < π 4. ( = π, 7 6 π, 6 π ( 0 < π, π < < 7 6 π, 6 π < π , ( ( ( ( (5 ( (6 3 4 sin( + = sin cos + cos sin sin( = sin cos cos sin sin( + + sin( = sin cos ( = 75, = 5 sin 90 + sin 60 = sin 75 cos 5 sin 75 cos 5 = (sin 90 + sin 60 =

144 4.8 ( sin(45 + α = sin 45 cos α + cos 45 sin α sin(45 α = sin 45 cos α cos 45 sin α sin(45 + α + sin(45 α = sin 45 cos α = cos α ( sin(a + B sin(a B =(sin A cos B + cos A sin B(sin A cos B cos A sin B =(sin A cos B (cos A sin B = sin A( sin B ( sin A sin B = sin A sin B p 4.0 q 4. sin α cos α = + ( cos α = sin α cos α cos α = sin α cos α = tan α 4. ( 4 5 ( sin α + cos α = tan α 4.3 sin α = 5 cos α = ( tan α = 4 ( tan α 3 = = π 6, 5 6 π 4.6 = π 6, 5 6 π 3 = 3 π ( = 30, 90 ( θ = 0 (3 θ = π 6, 3 π

145 ( 5. ( ( ( 3 ( ( ( a ( 9 ( > (4 5 ( ( 7 ( (3 4 (4 4 (5 5 ( ( ( O O 5.0 ( < 3 4 < ( ( > > ( ( < 3 ( 4 5. ( = 4 ( = 9 (3 = 4 (4 = (5 = 7 (6 = 3 (7 = (8 = (9 = 3 (0 = ( = ( = 0, 3 (3 = (4 =, (5 = (6 = ( (, = (3, 5, (5, 3 ( =, = ( = 6 ( = ( 3 ( 3 ( ( 3 ( (3 3 (4 (5 4 (6 ( ( 5 ( (3 3 (4 3 (5 6 (6 (7 (8 log 0 4 (9 (0 5.0 ( ( 0 (3 40

146 5. ( 3 ( 3 (3 3 (4 6 (5 3 4 (6 ( ( 3 ( 3 ( a + ab + ab 5.4 ( ( z + ( z 5.6 ( ( O O ( 0 < < 3 ( ( = log 3 ( =, log 5.9 ( = 03 ( = 9 (3 = 9 (4 = 5 (5 = 0 (6 = 4 (7 = 0 (8 = (9 = 0 (0 = 3 ( = ( = 35 (3 = 8 (4 = (5 =, 3 (6 = ( ( (, = (5, ( (, = (5,, 4, (3 (, = (7, ( (.300 (3.778 ( (5.079 ( ( ( ( ( ( (.9363 ( (.994 (.994 ( ( ( ( 5 ( 9 ( ( 7 ( 0 (3 34 ( ( 6 ( 4

147 6.3 ( 6 ( 3 (3 5 (4 (5 3 (6 5 (7 ( ( 08 ( 3a 6.5 ( a =, b = ( a = 3, b = 6.6 f (a = a ( = 4 3 ( = (3 f ( = 3 +8 (4 = ( ( ( = ( = 3 (3 = 8 (4 = (5 f ( = (6 = 3 (7 = (8 = (9 = f ( = ( = 4 ( = 9 4 (3 = 6.4 ( = 4 4 ( = = 6 9, = ( = 7 3, = ( a ( a (3 a (4 a a + 3 (5(6 3 (7( (9(0 (3, (, ( = 3 9 = 3 ( = 0 = (3 = 3 = 3 (4 = = 6 (5 = = O 3 4

148 6.0 ( (0, ( = 5 = 3 (3 5 3 O 6. ( a = 3, b = 6 ( = 0 6 = (3 6 O 6. ( a = 3, b = 0 ( = a =, b = 3, c = 4, d = a < 3, 3 < a 6.6 ( = 0, 3 = 8 ( = 9 = ( 5cm 000cm 3 ( 5 cm (3 5cm ( 0 < a < 4 ( a <, < a 43

149 6.9 ( f( = ( f ( = 3 3 = 3( + ( f ( f( f( 0 f( = 0 0 f( 0 ( = ( f( = ( f ( = 3 + = 3( f ( f( f( 0 f( = f( 0 ( = ( + C ( 4 + C (3 + + C ( C ( C ( C ( C ( C 6.3 ( C ( C 6.3 f( = ( 0 ( 5 (3 4 ( ( 3k + 3 ( ( 0 ( 5 (5 39 (6 0 3 (7 54 (8 8 44

150 6.36 ( f ( = 3 ( g( = a = 6.37 ( 8 ( (3 3 (4 3 ( ( 4 3 ( 9 (3 a = ( 8 ( 6.40 ( 9 ( 5 ( ( 8 ( (3 (i l =, m = 6, n = (ii 8 ( ( 4 ( = 4 = 0 (

151 (

152 (

153 II 8 ( TEL ( FAX (

1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1

1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1 ABCD ABD AC BD E E BD : () AB = AD =, AB AD = () AE = AB + () A F AD AE = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD AB + AD AB + 7 9 AD AB + AD AB + 9 7 4 9 AD () AB sin π = AB = ABD AD

More information

熊本県数学問題正解

熊本県数学問題正解 00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (

More information

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π 4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan

More information

A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6

A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6 1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67

More information

さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a

さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a ... A a a a 3 a n {a n } a a n n 3 n n n 0 a n = n n n O 3 4 5 6 n {a n } n a n α {a n } α {a n } α α {a n } a n n a n α a n = α n n 0 n = 0 3 4. ()..0.00 + (0.) n () 0. 0.0 0.00 ( 0.) n 0 0 c c c c c

More information

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

More information

76 3 B m n AB P m n AP : PB = m : n A P B P AB m : n m < n n AB Q Q m A B AQ : QB = m : n (m n) m > n m n Q AB m : n A B Q P AB Q AB 3. 3 A(1) B(3) C(

76 3 B m n AB P m n AP : PB = m : n A P B P AB m : n m < n n AB Q Q m A B AQ : QB = m : n (m n) m > n m n Q AB m : n A B Q P AB Q AB 3. 3 A(1) B(3) C( 3 3.1 3.1.1 1 1 A P a 1 a P a P P(a) a P(a) a P(a) a a 0 a = a a < 0 a = a a < b a > b A a b a B b B b a b A a 3.1 A() B(5) AB = 5 = 3 A(3) B(1) AB = 3 1 = A(a) B(b) AB AB = b a 3.1 (1) A(6) B(1) () A(

More information

(1) (2) (1) (2) 2 3 {a n } a 2 + a 4 + a a n S n S n = n = S n

(1) (2) (1) (2) 2 3 {a n } a 2 + a 4 + a a n S n S n = n = S n . 99 () 0 0 0 () 0 00 0 350 300 () 5 0 () 3 {a n } a + a 4 + a 6 + + a 40 30 53 47 77 95 30 83 4 n S n S n = n = S n 303 9 k d 9 45 k =, d = 99 a d n a n d n a n = a + (n )d a n a n S n S n = n(a + a n

More information

1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 :

1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 : 9 ( ) 9 5 I II III A B (0 ) 5 I II III A B (0 ), 6 8 I II A B (0 ), 6, 7 I II A B (00 ) OAB A B OA = OA OB = OB A B : P OP AB Q OA = a OB = b () OP a b () OP OQ () a = 5 b = OP AB OAB PAB a f(x) = (log

More information

B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13:

B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13: B. 41 II: ;; 4 B [] S 1 S S 1 S.1 O S 1 S 1.13 P 3 P 5 7 P.1:.13: 4 4.14 C d A B x l l d C B 1 l.14: AB A 1 B 0 AB 0 O OP = x P l AP BP AB AP BP 1 (.4)(.5) x l x sin = p l + x x l (.4)(.5) m d A x P O

More information

OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P

OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P 4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e

More information

17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,

17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1, 17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ

More information

70 : 20 : A B (20 ) (30 ) 50 1

70 : 20 : A B (20 ) (30 ) 50 1 70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................

More information

入試の軌跡

入試の軌跡 4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf

More information

a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552

a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552 3 3.0 a n a n ( ) () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 55 3. (n ) a n n a n a n 3 4 = 8 8 3 ( 3) 4 = 8 3 8 ( ) ( ) 3 = 8 8 ( ) 3 n n 4 n n

More information

( )

( ) 18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................

More information

O E ( ) A a A A(a) O ( ) (1) O O () 467

O E ( ) A a A A(a) O ( ) (1) O O () 467 1 1.0 16 1 ( 1 1 ) 1 466 1.1 1.1.1 4 O E ( ) A a A A(a) O ( ) (1) O O () 467 ( ) A(a) O A 0 a x ( ) A(3), B( ), C 1, D( 5) DB C A x 5 4 3 1 0 1 3 4 5 16 A(1), B( 3) A(a) B(b) d ( ) A(a) B(b) d AB d = d(a,

More information

1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +

1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 + ( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n

More information

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 = 5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =

More information

高等学校学習指導要領解説 数学編

高等学校学習指導要領解説 数学編 5 10 15 20 25 30 35 5 1 1 10 1 1 2 4 16 15 18 18 18 19 19 20 19 19 20 1 20 2 22 25 3 23 4 24 5 26 28 28 30 28 28 1 28 2 30 3 31 35 4 33 5 34 36 36 36 40 36 1 36 2 39 3 41 4 42 45 45 45 46 5 1 46 2 48 3

More information

1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a +

1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a + 6 ( ) 6 5 ( ) 4 I II III A B C ( ) ( ), 5 7 I II III A B C ( ) () x π y π sin x sin y =, cos x + cos y = () b c + b + c = + b + = b c c () 4 5 6 n ( ) ( ) ( ) n ( ) n m n + m = 555 n OAB P k m n k PO +

More information

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

More information

18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B (

18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B ( 8 ) ) [ ] [ ) 8 5 5 II III A B ),,, 5, 6 II III A B ) ),,, 7, 8 II III A B ) [ ]),,, 5, 7 II III A B ) [ ] ) ) 7, 8, 9 II A B 9 ) ) 5, 7, 9 II B 9 ) A, ) B 6, ) l ) P, ) l A C ) ) C l l ) π < θ < π sin

More information

ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2

More information

29

29 9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n

More information

(1) 1 y = 2 = = b (2) 2 y = 2 = 2 = 2 + h B h h h< h 2 h

(1) 1 y = 2 = = b (2) 2 y = 2 = 2 = 2 + h B h h h< h 2 h 6 6.1 6.1.1 O y A y y = f() y = f() b f(b) B y f(b) f() = b f(b) f() f() = = b A f() b AB O b 6.1 2 y = 2 = 1 = 1 + h (1 + h) 2 1 2 (1 + h) 1 2h + h2 = h h(2 + h) = h = 2 + h y (1 + h) 2 1 2 O y = 2 1

More information

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63> 電気電子数学入門 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/073471 このサンプルページの内容は, 初版 1 刷発行当時のものです. i 14 (tool) [ ] IT ( ) PC (EXCEL) HP() 1 1 4 15 3 010 9 ii 1... 1 1.1 1 1.

More information

t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ

t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ 4 5 ( 5 3 9 4 0 5 ( 4 6 7 7 ( 0 8 3 9 ( 8 t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ S θ > 0 θ < 0 ( P S(, 0 θ > 0 ( 60 θ

More information

koji07-02.dvi

koji07-02.dvi 007 I II III 1,, 3, 4, 5, 6, 7 5 4 1 ε-n 1 ε-n ε-n ε-n. {a } =1 a ε N N a a N= a a

More information

名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト

名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト 名古屋工業大の数学 年 ~5 年 大学入試数学動画解説サイト http://mathroom.jugem.jp/ 68 i 4 3 III III 3 5 3 ii 5 6 45 99 5 4 3. () r \= S n = r + r + 3r 3 + + nr n () x > f n (x) = e x + e x + 3e 3x + + ne nx f(x) = lim f n(x) lim

More information

0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (,

0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (, [ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =

More information

i

i i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,

More information

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s [ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =, [ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b

More information

IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a

IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a 1 40 (1959 1999 ) (IMO) 41 (2000 ) WEB 1 1959 1 IMO 1 n, 21n + 4 13n + 3 2 (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b =

More information

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

More information

untitled

untitled 0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.

More information

.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc +

.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc + .1 n.1 1 A T ra A A a b c d A 2 a b a b c d c d a 2 + bc ab + bd ac + cd bc + d 2 a 2 + bc ba + d ca + d bc + d 2 A a + d b c T ra A T ra A 2 A 2 A A 2 A 2 A n A A n cos 2π sin 2π n n A k sin 2π cos 2π

More information

(, Goo Ishikawa, Go-o Ishikawa) ( ) 1

(, Goo Ishikawa, Go-o Ishikawa) ( ) 1 (, Goo Ishikawa, Go-o Ishikawa) ( ) 1 ( ) ( ) ( ) G7( ) ( ) ( ) () ( ) BD = 1 DC CE EA AF FB 0 0 BD DC CE EA AF FB =1 ( ) 2 (geometry) ( ) ( ) 3 (?) (Topology) ( ) DNA ( ) 4 ( ) ( ) 5 ( ) H. 1 : 1+ 5 2

More information

さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n

さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n 1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1

More information

2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l

2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l ABCDEF a = AB, b = a b (1) AC (3) CD (2) AD (4) CE AF B C a A D b F E (1) AC = AB + BC = AB + AO = AB + ( AB + AF) = a + ( a + b) = 2 a + b (2) AD = 2 AO = 2( AB + AF) = 2( a + b) (3) CD = AF = b (4) CE

More information

春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,

春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16, 春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16, 32, n a n {a n } {a n } 2. a n = 10n + 1 {a n } lim an

More information

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy

More information

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P 6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P

More information

mobius1

mobius1 H + : ω = ( a, b, c, d, ad bc > 0) 3.. ( c 0 )... ( 5z + 2 : ω = L (*) z + 4 5z + 2 z = z =, 2. (*) z + 4 5z+ 2 6( z+) ω + = + = z+ 4 z+ 4 5z+ 2 3( z 2) ω 2 = 2= z+ 4 z+ 4 ω + z + = 2 ω 2 z 2 x + T ( x)

More information

( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +

( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x + (.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d

More information

(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n

(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n 3 () 3,,C = a, C = a, C = b, C = θ(0 < θ < π) cos θ = a + (a) b (a) = 5a b 4a b = 5a 4a cos θ b = a 5 4 cos θ a ( b > 0) C C l = a + a + a 5 4 cos θ = a(3 + 5 4 cos θ) C a l = 3 + 5 4 cos θ < cos θ < 4

More information

function2.pdf

function2.pdf 2... 1 2009, http://c-faculty.chuo-u.ac.jp/ nishioka/ 2 11 38 : 5) i) [], : 84 85 86 87 88 89 1000 ) 13 22 33 56 92 147 140 120 100 80 60 40 20 1 2 3 4 5 7.1 7 7.1 1. *1 e = 2.7182 ) fx) e x, x R : 7.1)

More information

2009 I 2 II III 14, 15, α β α β l 0 l l l l γ (1) γ = αβ (2) α β n n cos 2k n n π sin 2k n π k=1 k=1 3. a 0, a 1,..., a n α a

2009 I 2 II III 14, 15, α β α β l 0 l l l l γ (1) γ = αβ (2) α β n n cos 2k n n π sin 2k n π k=1 k=1 3. a 0, a 1,..., a n α a 009 I II III 4, 5, 6 4 30. 0 α β α β l 0 l l l l γ ) γ αβ ) α β. n n cos k n n π sin k n π k k 3. a 0, a,..., a n α a 0 + a x + a x + + a n x n 0 ᾱ 4. [a, b] f y fx) y x 5. ) Arcsin 4) Arccos ) ) Arcsin

More information

iii 1 1 1 1................................ 1 2.......................... 3 3.............................. 5 4................................ 7 5................................ 9 6............................

More information

さくらの個別指導 ( さくら教育研究所 ) A 2 2 Q ABC 2 1 BC AB, AC AB, BC AC 1 B BC AB = QR PQ = 1 2 AC AB = PR 3 PQ = 2 BC AC = QR PR = 1

さくらの個別指導 ( さくら教育研究所 ) A 2 2 Q ABC 2 1 BC AB, AC AB, BC AC 1 B BC AB = QR PQ = 1 2 AC AB = PR 3 PQ = 2 BC AC = QR PR = 1 ... 0 60 Q,, = QR PQ = = PR PQ = = QR PR = P 0 0 R 5 6 θ r xy r y y r, x r, y x θ x θ θ (sine) (cosine) (tangent) sin θ, cos θ, tan θ. θ sin θ = = 5 cos θ = = 4 5 tan θ = = 4 θ 5 4 sin θ = y r cos θ =

More information

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

More information

limit&derivative

limit&derivative - - 7 )................................................................................ 5.................................. 7.. e ).......................... 9 )..........................................

More information

さくらの個別指導 ( さくら教育研究所 ) a a n n A m n 1 a m a n = a m+n 2 (a m ) n = a mn 3 (ab) n = a n b n a n n = = 3 2, = 3 2+

さくらの個別指導 ( さくら教育研究所 ) a a n n A m n 1 a m a n = a m+n 2 (a m ) n = a mn 3 (ab) n = a n b n a n n = = 3 2, = 3 2+ 5 5. 5.. a a n n A m n a m a n = a m+n (a m ) n = a mn 3 (ab) n = a n b n a n n 0 3 3 0 = 3 +0 = 3, 3 3 = 3 +( ) = 3 0 3 0 3 3 0 = 3 3 =, 3 = 30 3 = 3 0 a 0 a`n a 0 n a 0 = a`n = a n a` = a 83 84 5 5.

More information

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,, 01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,

More information

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ 1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c

More information

1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8..............................

More information

untitled

untitled 1 17 () BAC9ABC6ACB3 1 tan 6 = 3, cos 6 = AB=1 BC=2, AC= 3 2 A BC D 2 BDBD=BA 1 2 ABD BADBDA ABC6 BAD = (18 6 ) / 2 = 6 θ = 18 BAD = 12 () AD AD=BADCAD9 ABD ACD A 1 1 1 1 dsinαsinα = d 3 sin β 3 sin β

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11

More information

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,, 6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,

More information

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g( 06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,

More information

1 I

1 I 1 I 3 1 1.1 R x, y R x + y R x y R x, y, z, a, b R (1.1) (x + y) + z = x + (y + z) (1.2) x + y = y + x (1.3) 0 R : 0 + x = x x R (1.4) x R, 1 ( x) R : x + ( x) = 0 (1.5) (x y) z = x (y z) (1.6) x y =

More information

zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 {

zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 { 04 zz + iz z) + 5 = 0 + i z + i = z i z z z 970 0 y zz + i z z) + 5 = 0 z i) z + i) = 9 5 = 4 z i = i) zz i z z) + = a {zz + i z z) + 4} a ) zz + a + ) z z) + 4a = 0 4a a = 5 a = x i) i) : c Darumafactory

More information

i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................

More information

2000年度『数学展望 I』講義録

2000年度『数学展望 I』講義録 2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53

More information

sekibun.dvi

sekibun.dvi a d = a + a+ (a ), e d = e, sin d = cos, (af() + bg())d = a d = log, cosd = sin, f()d + b g()d d 3 d d d d d d d ( + 3 + )d ( + )d ( 3 )d (e )d ( sin 3 cos)d g ()f (g())d = f(g()) e d e d ( )e d cos d

More information

5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 = 4. () = 8 () = 4

5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 = 4. () = 8 () = 4 ... A F F l F l F(p, 0) = p p > 0 l p 0 P(, ) H P(, ) P l PH F PF = PH PF = PH p O p ( p) + = { ( p)} = 4p l = 4p (p 0) F(p, 0) = p O 3 5 5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 =

More information

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x . P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +

More information

f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y

f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y 017 8 10 f : R R f(x) = x n + x n 1 + 1, f(x) = sin 1, log x x n m :f : R n R m z = f(x, y) R R R R, R R R n R m R n R m R n R m f : R R f (x) = lim h 0 f(x + h) f(x) h f : R n R m m n M Jacobi( ) m n

More information

I II

I II I II I I 8 I I 5 I 5 9 I 6 6 I 7 7 I 8 87 I 9 96 I 7 I 8 I 9 I 7 I 95 I 5 I 6 II 7 6 II 8 II 9 59 II 67 II 76 II II 9 II 8 II 5 8 II 6 58 II 7 6 II 8 8 I.., < b, b, c, k, m. k + m + c + c b + k + m log

More information

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2 θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =

More information

4 5.............................................. 5............................................ 6.............................................. 7......................................... 8.3.................................................4.........................................4..............................................4................................................4.3...............................................

More information

漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト

漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,

More information

13 0 1 1 4 11 4 12 5 13 6 2 10 21 10 22 14 3 20 31 20 32 25 33 28 4 31 41 32 42 34 43 38 5 41 51 41 52 43 53 54 6 57 61 57 62 60 70 0 Gauss a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x

More information

koji07-01.dvi

koji07-01.dvi 2007 I II III 1, 2, 3, 4, 5, 6, 7 5 10 19 (!) 1938 70 21? 1 1 2 1 2 2 1! 4, 5 1? 50 1 2 1 1 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 3 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k,l m, n k,l m, n kn > ml...?

More information

(1) θ a = 5(cm) θ c = 4(cm) b = 3(cm) (2) ABC A A BC AD 10cm BC B D C 99 (1) A B 10m O AOB 37 sin 37 = cos 37 = tan 37

(1) θ a = 5(cm) θ c = 4(cm) b = 3(cm) (2) ABC A A BC AD 10cm BC B D C 99 (1) A B 10m O AOB 37 sin 37 = cos 37 = tan 37 4. 98 () θ a = 5(cm) θ c = 4(cm) b = (cm) () D 0cm 0 60 D 99 () 0m O O 7 sin 7 = 0.60 cos 7 = 0.799 tan 7 = 0.754 () xkm km R km 00 () θ cos θ = sin θ = () θ sin θ = 4 tan θ = () 0 < x < 90 tan x = 4 sin

More information

Chap10.dvi

Chap10.dvi =0. f = 2 +3 { 2 +3 0 2 f = 1 =0 { sin 0 3 f = 1 =0 2 sin 1 0 4 f = 0 =0 { 1 0 5 f = 0 =0 f 3 2 lim = lim 0 0 0 =0 =0. f 0 = 0. 2 =0. 3 4 f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0.

More information

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 = #A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/

More information

122 6 A 0 (p 0 q 0 ). ( p 0 = p cos ; q sin + p 0 (6.1) q 0 = p sin + q cos + q 0,, 2 Ox, O 1 x 1., q ;q ( p 0 = p cos + q sin + p 0 (6.2) q 0 = p sin

122 6 A 0 (p 0 q 0 ). ( p 0 = p cos ; q sin + p 0 (6.1) q 0 = p sin + q cos + q 0,, 2 Ox, O 1 x 1., q ;q ( p 0 = p cos + q sin + p 0 (6.2) q 0 = p sin 121 6,.,,,,,,. 2, 1. 6.1,.., M, A(2 R).,. 49.. Oxy ( ' ' ), f Oxy, O 1 x 1 y 1 ( ' ' ). A (p q), A 0 (p q). y q A q q 0 y 1 q A O 1 p x 1 O p p 0 p x 6.1: ( ), 6.1, 122 6 A 0 (p 0 q 0 ). ( p 0 = p cos

More information

直交座標系の回転

直交座標系の回転 b T.Koama x l x, Lx i ij j j xi i i i, x L T L L, L ± x L T xax axx, ( a a ) i, j ij i j ij ji λ λ + λ + + λ i i i x L T T T x ( L) L T xax T ( T L T ) A( L) T ( LAL T ) T ( L AL) λ ii L AL Λ λi i axx

More information

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35

More information

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y 5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x

More information

(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0

(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 (1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP

More information

2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta

2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta 009 IA 5 I, 3, 4, 5, 6, 7 6 3. () Arcsin ( (4) Arccos ) 3 () Arcsin( ) (3) Arccos (5) Arctan (6) Arctan ( 3 ) 3. n () tan x (nπ π/, nπ + π/) f n (x) f n (x) fn (x) Arctan x () sin x [nπ π/, nπ +π/] g n

More information

4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X

4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X 4 4. 4.. 5 5 0 A P P P X X X X +45 45 0 45 60 70 X 60 X 0 P P 4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P 0 0 + 60 = 90, 0 + 60 = 750 0 + 60 ( ) = 0 90 750 0 90 0

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

L1-a.dvi

L1-a.dvi 27 Q C [ ] cosθ sinθ. A θ < 2π sinθ cosθ A. A ϕ A, A cosϕ cosθ sinθ cosθ sinθ A sinθ cosθ sinθ +cosθ A, cosθ sinθ+sinθ+cosθ 2 + 2 cosθ A 2 A,A cosθ sinθ 2 +sinθ +cosθ 2 2 cos 2 θ+sin 2 θ+ 2 sin 2 θ +cos

More information

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B 1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n

More information

2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a

More information

arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = =

arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = = arctan arctan arctan arctan 2 2000 π = 3 + 8 = 3.25 ( ) 2 8 650 π = 4 = 3.6049 9 550 π = 3 3 30 π = 3.622 264 π = 3.459 3 + 0 7 = 3.4085 < π < 3 + 7 = 3.4286 380 π = 3 + 77 250 = 3.46 5 3.45926 < π < 3.45927

More information

(2000 )

(2000 ) (000) < > = = = (BC 67» BC 1) 3.14 10 (= ) 18 ( 00 ) ( ¼"½ '"½ &) ¼ 18 ¼ 0 ¼ =3:141596535897933846 ¼ 1 5cm ` ¼ = ` 5 = ` 10 () ` =10¼ (cm) (1) 3cm () r () () (1) r () r 1 4 (3) r, 60 ± 1 < > µ AB ` µ ±

More information

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) 2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x

More information

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

More information

K E N Z U 01 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.................................... 4 1..1..................................... 4 1...................................... 5................................

More information

f (x) x y f(x+dx) f(x) Df 関数 接線 x Dx x 1 x x y f f x (1) x x 0 f (x + x) f (x) f (2) f (x + x) f (x) + f = f (x) + f x (3) x f

f (x) x y f(x+dx) f(x) Df 関数 接線 x Dx x 1 x x y f f x (1) x x 0 f (x + x) f (x) f (2) f (x + x) f (x) + f = f (x) + f x (3) x f 208 3 28. f fd f Df 関数 接線 D f f 0 f f f 2 f f f f f 3 f lim f f df 0 d 4 f df d 3 f d f df d 5 d c 208 2 f f t t f df d 6 d t dt 7 f df df d d df dt lim f 0 t df d d dt d t 8 dt 9.2 f,, f 0 f 0 lim 0 lim

More information

III

III III http://www.manabino-academ.com . = k...................................... = k p + q................................. = a + b c + d.................................. 4.4..........................................

More information