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
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1 A6, 1) B1, 1) 65 C A, 1) B, ) C = 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, ) C0, k) 66 1 p, p) 1 1 A B AB A 67 1) B ) P, ) PA = PB PB = PC P ABC
2 1 64 AB = 6 1) + 1 1) = = 169 = 1 65 C 0, k) CA = CB 0 ) + k 1) = 0 + ) + k ) 4 + k k + 1) = 9 + k 4k + 4) k = 4 C0, 4) A B AB AB 1 5 AB 1 ), AB = ) + = C 0, 4) = 0 P 1 p, p) PA = PB 1 p 1) + p 1) = 1 p ) + p 0) 4p + p p + 1) = 4p + 8p + 4) + p p = 10 P 8 5, ) 10 P AB AB 1 AB ), 1 AB = ) + 1 = 7 P + 1 = 0 = = 0
3 1 67 1) G , ) G 8, ) ), ) ) + ) = 1) + ) = 4) = = { = { 4 + = = = 11 = 7 14, = P 7 14, ) AB BC P ABC
4 , ) 5, 1) 69 A 1, ) B : P 1, 1) B 70 A, 1) B, 4) AB : A 1, 1 ) B, ) AB m : n C ) n1 + m n, 1 + m m + n m + n AB m : n m \= n) C ) n1 + m n, 1 + m m n m n m > 0 n > 0 m n A 1, 1 ) 68 AB 1 : 1 C B, ) m A 1, 1) n C B, ) 69 B 70
5 1 68, ) 5, 1), ) + = = + 1 = 5 69 B, ) AB : P 1, 1) 1) + = 1 { + + = 5 ) = 5 = 1 + = 4 = B B4, ) AP 5 : B ) 1) ) + 5 1), 5 5 4, ) A P 5 B 70 A, 1) B, 4) AB : ) ) + ) 1 + 4, 5, 10)
6 , 7), 1) = 7 1, ), ), ) 7, ) = 5 = = 74 l 1 : a + 4) + a 1) + 4a = 0 l : = 0 a = l 1 l a = 75 = 1, + = 1, k = k 1 k,, k = 0 k k
7 1 71 A 1, 1) B, ) P, ) B, ) 1 \= P APQ ABC A 1, 1) PQ C Q AQ = BC AC 1 = 1 = ) = 1) 1 1 = = 1 1 ) 1 ) = 1 ) 1 ) I) = m 1 + n 1 = m + n : m 1 = m : m 1m = 1 II) a 1 + b 1 + c 1 = 0 a + b + c = 0 : a 1b b 1a = 0 : a 1a + b 1b = 0 75 i) i ) ii) 1 ii) 76 a + b + c + d + e + f = 0 p + q + r)p + q + r ) = 0 1
8 1 71 1, 7), 1) 1 7) = 5 1) 7) = 5{ 1)} = 5 7, ) 1) = 1) 1 = = 4 1, ), ) = + 1) + = + 1 = = 7, ) = 5 = ) = 9 1 = 1 ) = a + 4 a 1 = 4 4a + 4) = a 1) a = 19 5 l 1 l a + 4 ) ) = 1 a 1 4 a + 4) = 4a 1) a = 8 15 II) a + 4) 4 a 1) = 0 a = 19 5 a + 4) + a 1) 4 = 0 a = 8 15
9 1 75 = 1 = = 1 = + 6 k = k 1 = k + 1 k 1 i ) 1 // k = 1 ii) // k = iii) = + 6 = 1, ) = k + 1 k k = i ) ii) iii) k =, 1, 76 + k 4) ) = 0 = k 4) ± D 4 1 D = k 4) ) = k 80) + 89 k) 1 D 1 D D 1 D D = 0 D k 80 \= 0 { k \= ±4 5 D 4 = 169 k) = 0 k = 9 k = 9
10 = 0 A 4, 0) B 78 = = = = = 79 1) = 1 = ) = 1 = = 77 l A B { AB l A l AB l 78 = f) = f) = f ) = f ) = = f) B 79 = f) = k k = f)
11 1 ) p 4 q 77 B p, q) AB, + = 0 p 4 q + = 0 p q = = 0 AB 1 q p + 4 = 1 p + q = 4 1 p = q = B, ) 78 = 1 1 = = + 1 = ) = + 1 = = = 1 +, ), ) =, ), ) = f), ) 79 1) = 1 1, ) X, Y ) { { X = = X Y = = Y X, Y ) 1 Y = X) X) 1 = X + X 1 = + 1 ) 1, ) = X, Y ) { X = = X + Y = = 4 Y 1 4 Y = X X 1 Y = X + X + 5 = = 1) 1, ) 1) 1, ) 1, ) = + 1) = + 1 ) 1, ) 1, 6) = 1) + 6 = + + 5, ) X, Y ), )
12 a + b + c = 0 0, 0 ) a 0 + b 0 + c a + b 81 1) 1, ) = 0 ) A, ) + +a = 0 1 a 80 0, 0 ) a + b + c = 0 d d = a 0 + b 0 + c a + b 0, 0) d a + b + c = 0 81
13 1 80 a + b + c = 0 l P 0, 0 ) P l H 1, 1 ) PH = 1 0 ) ) X = 1 0 Y = 1 0 PH l b 1 0 ) a 1 0 ) = 0 bx ay = 0 1 H l a 1 + b 1 + c = 0 ax + 0 ) + by + 0 ) + c = 0 ax + by = a 0 + b 0 + c) 1 X Y a0 + b0 + c) X = a a + b, Y = b PH = X + Y = = a0 + b0 + c a + b 81 1) = b 0) a 0) = 0 a0 + b0 + c) a + b H 1, 1) { a) + b) } a 0 + b 0 + c) a + b ) P 0, 0 ) l ) 1 + ) + a 1 + ) = 1 a + = a + =, a =, 4 +
14 ) = + 1 = + 10 = 0 ) = = = 0 8 A, 1) B4, 7) Ct + 1, 10 t) ABC 10 t = t = 8 1) = 1 ) ) ) 8 I) 0, 0) A 1, 1 ) B, ) AB = II) A 1, 1 ) B, ) C, ) ABC = 1 1 ) 1 ) II) AB = 1, 1 ), AC = 1, 1 ) 1 ) 1 ) A I) A 1, 1) B, ) B, ) C, ) A 1, 1 )
15 1 8 1) = + 10 = {5 1)} 4 = = 0 1 ) = = A0, 4) B, 0) C ) 4, 1 D 1, 0) 4 A ABC = ABD BCD = 1 { 1)} 4 1 { 1)} 1 = 8 = 6 ) E 0, 4 5 ABC = ACE + ABE = 1 AE = ) ) 4 + = AE 8 ABC )10 t 1) 7 1)t + 1 ) = t + 4 = 10 7t 1 = ±10 t = 7, 7 C E 1 D 1 B ABC 1 A B C A 0, 0) B, 6) C t 1, 9 t) ABC = A B C ABC t) 6t 1) = 10
16 k k + ) + k + 1) k 5 = 0 k 85 5k + ) + k + 1) + k 1) = 0 P P, 5) = 86 1 = = 0 A B 1 ) 87 + = 1 + = 5 + = 6 84 ka 1 + b 1 + c 1) + a + b + c ) = 0 k { a1 + b 1 + c 1 = 0 a + b + c = 0 85 k P 84 P P, 5), ) =, 5) k a 1 + b 1 + c 1 = 0 a + b + c = 0 ka 1 + b 1 + c 1 ) + a + b + c ) = 0 k
17 1 84 k k + 1) = 0 k { + 1 = = 0, 1) 85, 5) 5k + ) + k + 1) 5 + k 1) = 0 k = = 0 = k 1) = 0 A B1 ) k 6 1) = 0 k = ) = = 0 87 k + 1) + + 5) = = 1 + = 5 1 : k + ) + k + 1) k 5 = = 6 k + ) k + 1) = 0 k = 5 + 1) ) = = = 6 k + ) + k + 1) = 0 k = ) 1 + 5) = 0 7 = 0
18 = 0, ) 89 1) 0, 0), 0), 4) ), 0) 0, ), 5) 90 1) 1, ) ) a 4, 0) 0, ) ) 1 = ) 7, 1), 6) ) + = 4 = 1, 1)
19 88 Ca, b) P, ) r CP = r a) + b) = r 89 a) + b) = r + + l + m + n = 0 ) ) ) ) a) + b) = k k > 0 a, b) k k = 0 1 a, b) k < 0 ) 90 1) r> 0) r ) a) + b) = r a ) = 1, ) r r 91 1) 1, 1 ), ) 1 ) ) + 1 ) ) = 0, ) 1, 1), ) ) = 0 ) 4 p, p) p
20 = 0 ) 4 + 1) 1 = 0 ) + 1) = 5, 1) l + m + n = 0 1) 0, 0) n = 0, 0) 4 + l = 0 l =, 4) m = 0 m = = 0 1) + ) = 5 1, ), 5 ), 0) 0, ), 5) 9 + l + n = m + n = l + 5m + n = = 0 ) 5, 5 l = 5, m = 5, n = = 5 ) + 5 ) = ) 1, ) 4 4 r > 0) r, r) r) + + r) = r 1, ) 1 r) + + r) = r r 6r + 5 = 0 r 1)r 5) = 0 r = 1, 5
21 ) a, b) r> 0) a) + b) = r 4, 0) 0, ) { 4 a) + b = r 1 a + b) = r 1 4 a) + b = a + b) 4b = 8a 1 b = a 1 r = 4 a) + a ) = 5a 0a + 5 a) + a + ) = 5a 0a + 5 ) 1 1 p, 1) p > 0) = 0 p ) = 1 p 1 = ± p > 0 p = ) + 1) = = 91 1) 7, 1),, 6) 7) ) + 1) + 6) = = 0 5, 5 ) ) ) = + 5 ) = 65 4 ) 4 p, p) = 5) + p p) + p) = p + ) 1 1) p) + 1 p) = p + ) 4p 14p + 6 = 0 p = 1, ) + 1 ) = 5 4, + ) + ) = 5 p + = p +
22 . 9 A0, ) B4, 0) Cc, 0) AC = BC c = ), ABC 9 l 1 : + = 0 l : + 14 = 0 l : 7 10 = ) l : = A4, 0) B ) B, ) AB ), , 0) A1, ) B0, 4) C4, ) A B B C
23 9 AC = BC C AB ACB 9 4 α, β) l 1 l l α, β) a + b + c = 0 aα + bβ + c a + b 94 1) AB l AB l l : = = tan 60 ) AB AB = 10 B = A ) l AB 95
24 9 AC = BC c + = c 4) c = 7 8 AC = BC ACB AB AB 4 CM A 4 CM : = 4 7 ) 8 = N C M 4 B ABC N N : NA = B : AB = 4 : 5 N = 4 9 A = 4 NB N B = 1 NB : = CM NB ABC = = ), α, β) α β ) = α + β 14 7α β 10 = ) α, β) l 1 l, l β > α +, β < α + 14, β < 7α 10 1 α β + ) = = α + β 14) 7α β β l l 1 { α β + = α + β 14 l = 5α + β 14) = 7α β 10 α = 4, β = ) = α 4) + 8) =
25 94 1) l 60 AB = 10 B = 4 B 4 cos 10, 4 sin 10 ) B, ) Ba, b) AB ) a + 4, b l b = a + 4 b = a + 4) 1 AB l b a 4 = 1 b = a B, ) ) A = l P, ) AB, ) P = 4 95 AB B = 1 B = BC BC α, β) B C α + β = α + β 4) = α 4) + β ) 0 = 8β + 16 = 8α 4β + 0 B B B l A P A A l C β =, α = = ) + = 5
26 = = + k k 97 C : = 0, ) m + ) m 6 = 0 C m 98 = = = 0 4 = 0
27 96 97 d r d r d r I) II) I) AB = AH = A H B H A
28 96 ) ) + = 0, 0) + k = 0 k 1 + 1) k = + k + = D k) = + k + k = 0 D 4 = k k ) = 4 k 0 k = 0 ) + + ) = 5 C, ) 5 m + ) m 6 = 0 C C m + ) m 6 = 0 m + ) m ) 6 m + ) + m) = 5 5m = 5m + 6m + 9) 5m = 5m + 6m + 9) m m = 0 m )m + 1) = 0 m = 1
29 98 + = 4 0 0) A B H +1 = 0 H H = ) = 1 H A ) AH = 1 = 7 AB AB = AH = 14 B = 0 ) + + ) = 9 P, ) = 0 = = 1 +k + k = P 1 = 5 + ) k = k + = 5 k + = ±5 k = 1, = 0, = 0 P 1
30 = a + b = 0 8, ) a = b = 101 1, 1) 1 5, 1) = = 10 + = 6) + = a) + b) = r 0, 0) 0 a) a) + 0 b) b) = r
31 = 0 4) + + 1) = 5 P4, 1) 5 A8, ) AP 1) = = 4 8) + = a = 4, b = 8 P A8, ) 4) + + 1) = 5 8, ) 8 4) 4) + + 1) + 1) = 5 = , 1) 1 5, 1) = m 5) + 1 m 5m + 1 = 0 1, 1) 1 m 1) 5m + 1 = 1 m + 1 m + 1 4m + ) = m m 16m + = 0 m = 8 ± = 8 ± = 8 ± = 8 ± ± ±
32 10 + = 6) + = 4 A6 0) 4 0) 1 = = m + n n m + 1 = 6m + n 1, m + 1 = 4 1 n = 6m + n n = ±6m + n) n = 6m, m n = 6m 1 6m m + 1 = 9m = m + 1 m = ± 4, n = ± n = m 1 m m + 1 = m = m + 1 m = ± 4 ±
33 .5 10 P Q + 16 = 0, + + = 0 PQ 104 8) + 15) = 5 10 PQ P Q 104 d r 1 r i ) ii) iii) r 1 r d r 1 r d r 1 r d d < r 1 r d = r 1 r r 1 r < d < r 1 + r iv) v) r 1 r r 1 r d d d = r 1 + r d > r 1 + r
34 10 + = 16 4 ) + 1) = 1, 1) = A B C PQ AC = = 7 PQ AB = 4 1 = 1 C 1 A B C 0 : 8) + 15) = 5 A8, 15) 5 C 0 B B = A AB = = 1 + = 1 C 0 C C = A + AC = = + = B A C
35 , 7) 1, 7) 8, 0) C 1 C 1 + = 4 C = 106 a C 1 : + 1 = 0 C : 6a + 8a + 4 = 0 C a C 1 C a 107 C + = 4 ) = 0 C, ), 108 C 1 : = 0 C : = 0 C 1 C P Q PQ
36 f, ) = 0 g, ) = 0 mf, ) + ng, ) = 0 m n ) m n 105 C 1 f, ) = 0 C g, ) = 0 m = 1 n = 1 ) 1 ) 106 C : a) + 4a) = 5a 4 5a 4 > 0 C 5a 4 = 0 C a, 4a) 5a 4 < 0 C C C 1 C f, ) = 0 g, ) = 0 f, ) = 0 g, ) = 0 { f, ) = 0 1 g, ) = 0 { f, ) = 0 1 f, ) g, ) = C + 4) + k ) = PQ ) ) = 0 C 1 C PQ P Q
37 105 C 1 : + + p + q + r = 0 7, 7) 1, 7) 8, 0) 7p + 7q + r + 98 = 0 1 p + 7q + r + 50 = 0 8p + r + 64 = 0 1 6p + 48 = 0 p = 8 r = 0 p { = 8, q = 6, r = 0 C1 : = 0 4 C : + 4 = = = ) 1 7) 8 0) 106 C : 6a + 8a + 4 = 0 a) + 4a) = 5a 4 5a 4 > 0 a < 5, 5 < a C 1 C C 1 C + 1) 6a + 8a + 4) = 0 6a + 8a 5 = 0 C 1 C C 1 ) < ) 5 6a) + 8a) < a < 1 a > 1 a < 1, 1 < a
38 = 0 C k ) = k + k + 5k 4 = 0 + k) + + k) = 10k 5k k 5k + 4 = k + 1)k 1) = 0 k = 1, 1 k, k) ), 1), C 1 C P Q C 1 C ) ) = 0 + = 0 C : = 0 ) + + 5) = 5 C, 5) 5 C A A PQ H A + = 0 AH AH = + 5) = PH = 5 5 ) = 5 H PQ PQ PQ = PH = 5 { C1 : = 0 1 C : = 0 1 = = 0 ) 7) = 0 =, 7 C 1 C, 0) 7, 5) PQ 7) + {0 5)} = 5 C A P Q H C 1
39 , 6) 5, ) = = 1 + = 1 1 P 111 A 1, 0) B, ) 1 : P P P P, )
40 109 P, ) A1, 6) B5, ) PA = PB 1) + 6) = 5) + ) = , 6) 5, ), 9 ) = 4 = 4 ) + 9 = P, ) + = = = + + ) 1 + ) + = = ± + + ) P = 1, = 111 P, ) AP : PB = 1 : PB = AP PB = 4AP ) + ) = 4{ + 1) + } = + ) + + 1) = 8, 1)
41 . 11 = Pa, a ) Qb, b ) b = a + PQ 11 0, 0) 1 C A1, 0) C P A P G G 114 m = 0 1, + m m = 0 P m P 1, ) ) + = ) 11 b = a + Qb, b ) Pa, a ) a PQ a 11 Aa 1, a ) Bb 1, b ) Cc 1, c ) ABC G ) a1 + b 1 + c 1 a, + b + c G X, Y ) P G 114 P m m P 1 m
42 11 PQ R, ) = a + b, = a + b b = a + a + a + ) = = a = a + a + ) = a + a + 1 a = 1 = 1) + 1) + = + 1 = + 1 = P R Q = 11 P, ) AP G X, Y ) X = + 1 { = X 1 Y = = Y 1 P, ) AP \= 0 P, ) G + = 1 1 X 1) + Y ) = 1 Y \= 0 X 1 ) + Y = 1 Y \= ) + = 1 0, 0) 9 ), 0 A 1
43 114 1 m i ) = 0 1 = 0 ii) \= 0 1 m = 0 m = + = 0 + = 0 1) + 1 ) = 5 4 \= 0 0, 0) 0, 1) i ) ii) 1) + 1 ) = , 1)
44 . 115 t { = t + 1 = t C : = l : = m 1) A B 1) m ) m AB 117 = a = + P Q P, Q R1, 0) G a G = 1 P +, )
45 115 t 116 = m 1) α β, ) = α + β, = α + β, ) = m 1) = m 1) m f,, m) = 0 g,, m) = 0 { f,, m) = 0 ) g,, m) = 0 m ), ) ) m { X = + Y = + = 1 X Y t Xt + Y = 0 X Y
46 115 { = t = t t = 1 = = ) C : = l : = m 1) = m 1) m + m = 0 1 D D = m 4m = mm 4) > 0 m m < 0, 4 < m ) A B α β 1 α + β = m AB = α + β = m m = = m 1) = 1) 1) < 0, > 4 < 0, < = 1) < 0, < 117 = a = + a + ) + = 0 1 D D = a + ) 8 = a + 4a 4 > 0 a <, a > + 1 α β α < β) α + β = a + P Q = a Pα, aα) Qβ, aβ) PQR G, )
47 = α + β + 1 = a + 4 aα + aβ aα + β) aa + ) = = = a, 4 a = 5 ) + ) = = 1) 1) <, > + < 1 G, > 1 + = 1) 1) < 1, > PX, Y ) { X = + Y =, t Xt + Y = 0 D, D = X 4Y 0 Y 1 4 X + = 1 + ) = 1 X Y = 1 Y = 1 X 1 P = = = = 1 1
48 { ) + 1) ) > 0 ) + 119, ) > f) < f)) = f) a) + b) > r < r ) a) + b) = r 10 AB 0 { A 0 B 0 { A 0 B ) 4 > 0 < < )
49 ) + 1 1, 0) = π = π ) + 1) 0 { ) + 1) { ) + 1) ) > 0 < < + ) ) < 0 { > < { < > = = < > 0 > 0 )
50 .5 1 t, t) t 1, 1 t) l t 1) l t ) t l t 1 t l t : = t t 1) P l t 1 P ) t t 1 l t, ) 1 l t : f,, t) = 0 t 1 1 f,, t) = 0 t, 1 ) 1 t t 1 t l t 1) 1) t
51 1 1) l t t 1 t) = t t 1) t) + t = t 1) t + t ) 1) t t + 1)t + + = 0 t 0 + 1) + ) l t 1 l t t 1 = t t + = 0 ft) = t t + 1) ft) = 0 1 D D 4 = = 0 = = ) ft) = 0 t 1 1 t = i ) 1 1 D 0 ii) 1 < < 1 f 1) 0 f1) i ) ii) 1) l t = t 1 l t
52 ) D 1 + ) D, ) , ) { A B 100g A 0g 5g g B 10g 10g g A B 40g 0g
53 14 D + k + = k, ) D + = k D 1 = + k D k 15 = m = m + 7 ) m 7, 0 ) + 7 1) D m = k + ) = k + 4 k + 4 > 0 k + 4 0, ) 17 A B g g
54 14, + 8 0, + 4 0, 1 + 4, = k k = + k 1 k, ) 0) : 5 : { 0) 15 1) = = < 0) D ) + 7 = m = m + 7 ) \= 7 ) m 7 ), 0 m 4 k k =, ) + 8 = 0 8 = + k + 4 = 0 = = D 7 1 = 1 + D = 1 + m + 7 ) = m + 7m 6) = 0 1 D 4 = m 7m 6) = m 1)m 6) = 0 m = 1, 6 1 = m D 1 7 m m 7 1 m = 1 m 0, 0) m m =
55 16 D D + 4 = k + ) = k + 4 0, ) D + ) > 0 A0, ) k + 4 D k A + 1 = ) = 1 k k + 4 = 1 k = 1 4 = 7 A 1 1 = 1 Pt, t 1) 1 t ) AP = t + t ) = t 4 5t + 9 = t 5 ) t 4) AP t = = AP 4 k + 4 = k = 9 4 = 5 5, 7 17 A g B g k g k = = k 4 1 D D 4 k k D ) k 4 400, 400 k = = = = =
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(
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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
高校生の就職への数学II
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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
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70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................
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
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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
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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
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koji07-01.dvi
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０４年度LS民法Ⅰ教材改訂版.PDF
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