017 1 獨家發行 MATHEMATICS QR https://goo.gl/6js7dx 106 106 1 01 16 106 C 99 99 106 6 6 11 B C
100~106 100 13 11 7 4 3 6.67 101 13 11 7 4 3 6.57 10 1 10 7 4 3 6.4 103 13 11 8 5 3 6.67 104 1 10 7 4 3 6.53 105 1 10 7 4 3 6.48 106 11 9 6 3 6.9 106 34 10 ABC D 6 106 106 Ch1 Ch 48AC 15 5 Ch3 E 10 Ch1 AD 5 Ch 71 10 Ch3 F 5 30 Ch4 15 10 Ch1 611 10 Ch 9 5 5 Ch3 BG 10 Ch1 413 7.5 Ch 10 5 Ch3 D.5 0 Ch4 3 5
7 MATHEMATICS 106 1 Ch 1 Ch1 106 106 / / 106 3 106 4 1 3 5 6 8 9 1 10 11 1 D A B 13 C E F G 3
106 1 7 4 8 10 11 D G 杕 106 杌 106 9 100 13 10 6 杈 杝 堅持不懈! 4
MATHEMATICS 106 65 35 杕 r 1, r, r 1 r.,. 坽 夌 奅 妵 妺. 坽 n 1, n, ch4 = n 1r 1 + n r = ( n 1 + n n 1 n,. n 1 + n n 1 + n n 1 n 1 + n ) r 1 + ( n n 1 + n ) r, 杌, a, a. b, 81 3. b 坽 1.7 夌 3 奅 5. 妵 9 妺 81. 奅 b 1 b (b ) = b 4 3 (b 4 ) = b 8, ch3 b 8 = 81 3 = 3 1 1 8 b = 3 = 3 3 = 3 3 3 1. 73 = 5.196 5.. 杈 Γ y a x = 1, l. b (t, t ), t = 0. t > 0,. 坽 Γ, l 夌 Γ, l 奅 l, Γ 妵 Γ, l 妺 l, Γ. 妺 ch4 Γ (i) Γ y a x b = 1 (0, 0),. Γ l (ii) (t, t ) x = t, t, Γ 'y = x, y = t, (0, 0),, Γ ' l (P ), Γ (Q ). 5
杝 A, C, B, D, 1 B, D.. 坽 夌 奅 妵 1 妺 1. 妵 ch ch1 杕, 1,, A(1, 0, 0), B(1, 1, 0), C(0, 1, 0), D(0, 1, 1). x = 1 + 0t = 1 杌 AB y = 0 + 1t = t, t R, P(1, t, 0), z = 0 + 0t = 0 x = 0 + 0t = 0 CD y = 1 + 0t = 1, t R (, t ), z = 0 + 1t = t Q(0, 1, t), PQ = ( 1, 1 t, t), 杈 PQ = PQ = (1) + (1 t) + t = t t + = (t 1 ) + 3, t = 1, PQ = 3. 杍 016 x y... 坽, 夌, 奅, 妵, 妺. 妵 ch4 6
MATHEMATICS 杕, 1 9 8 3 1 3 7 10 15 17 19 0 5 4 6 9 9 1 18 1 4 7 7 17 13 14 1 8 9 11 11 7 7 8 7 杌泝 沴,. 杚 x π x 3π cosx cosx 坽 0 夌 1 奅 妵 4 妺. 坽 π x 3π π x 3π, π 3.14, 1.57 x 4.71, cosx > 0 x,, cosx < 0 x, ch1 cosx cosx, 0. 杋.. ( ) ( ), 坽 5 夌 60 奅 68 妵 76 妺 84. 夌 ch 7
, 泝 3 + 沴 + 3, 泝 3 +, 沴 + 3, C 1 3!! 3 1 = 3 = 1 3!! 3 C 1 = 3 = 1 C 1 1 = 1 = 4 = 8 = 8 = 8 = 8 1 + (1 + 4 + 8 + 8 + 8 + 8) = 60. 30 毐 m, n 4 a, b. f (x) = ax m g(x) = bx n 3,. 坽 m, n a, b 夌 m, n a, b 奅 m, n a, b 妵 m, n a, b 妺 m, n. 8 坽奅 杕 m, n m = 0 n = 0, f (x) = a, g(x) = b, 3, ch m = 0 n =, f (x) = a, g(x) = bx, a = bx, 3 m = n = 0, m = 0 n = 4, f (x) = a, g(x) = bx4, a = bx 4, 3 m = 4 n = 0, m = n =, f (x) = ax, g(x) = bx, ax = bx, 3, m = n = 4 m = 4 n =, f (x) = ax, g(x) = bx 4, ax = bx 4 x (bx a) = 0, x = 0, bx a = 0 x = x = 0, a b. a b, a, b, a b, 永不放棄!
MATHEMATICS 杌 m, n m = 1 n = 1 m = 3, 3, n = 3 m = 1 n = 3 ( m = 3 n = 1 ), f (x) = ax1, g(x) = bx 3, ax = bx 3 x (bx a) = 0, x = 0, bx a = 0 x = a x = 0, b. 杈 m, n a b, a, b, a b,, m = 0 n = 1 m = 1 n = m = 1 n = 4 m =, 3. n = 3 氙 Γ, (0, 0) Γ (, 6) Γ.. 坽 Γ 夌 Γ Γ 10 奅 Γ Γ 10 妵 Γ x Γ x 10 妺 Γ Γ 10. 妺 杕, Γ, (0, 0) Γ P(, 6) Γ, Γ C, d(c, P) < d(c, O). 杌 1, d(c, P) = d(c, O), C L, L OP (1, 3) OP, Lx + 3y = 10. 杈 L,. 坽 Γ,. 夌 Γ,. 奅 Γ 10,. 妵 x 10 A x,. ch 氚 L 1 x 1 = y 1 = z 1, L x y + z = 4, L 3 x + y 4z = 5.. 坽 L 1 L 夌 L 1 L 3 奅 L 1 L 妵 L 1 L 3 妺 L L 3. x = t y = t, t z = 4 + 4t 夌奅妵 ch 9
x = 1 + k L 1 y = 1 + k z = k, k, d 1 = (,, 1), L, (1,, ) (1, 1, 4) = (6, 6, 3) = 3(,, 1), d = (,, 1), z = 0, x y = 4 x = x = + m x + y = 5 y = 3, L y = 3 + m, m, z = m x = t L 3 y = t, t, d 3 = (1, 1, 4), z = 4 + 4t 坽 d 1 = (,, 1) # d = (,, 1),. 夌 d 1 d 3 = (,, 1)(1, 1, 4) = + 4 = 0, d 1 d 3. 奅 d 1 # d, L 1 (1, 1, 0) L, L 1 # L,. x = t 妵 d 1 d 3, L 3 y = t L 1, t 1 = t 1 = 4 + 4t, t = 1, 1 z = 4 + 4t (1, 1, 0) L 1 L 3,. 妺 d d 3, x = + m = t y = 3 + m = t z = m = 4 + 4t 4t m = 4 L L 3, L L 3. t + m = t + m = 5. 汸 ABCDE,.,. 坽 AD = 夌 DAB = 45 奅 BD = 6 妵 ABD = 45 妺 BCD. 坽妵 ch1 坽 AD, ADE, AD = + = 夌 ADE, DAE = 45, DAB = 60. 奅妵 BD, ABD, DH AB H, ADH, ADH = 30, AH = 1 AD =, BH = 6. DH = ( ) = 6, BDH, BD = 6 + 6 = 3, ABD = 45. 妺 CDBCBD = 4 3, BCD BCD = 1 3 = 3. 10
MATHEMATICS 汧 50, 453934,. x, y.. 坽 x + y = 39 夌 y 11 奅 39 x + y 妵 11 妺 7. 夌妺, ch x y 34 39 x 16 39 45 39 = 6 5 50 坽 x + y = 34,. 夌 y 6 + 5 = 11,. 奅 = ( ) = 50 x,. 妵 = x, x 34 x 39, x 34. = 50 x 50 34 = 16, 16,. 妺 y 11, y 11, 坽 x + y = x + 11 = 34, x 34 11 = 3, = 50 x 50 3 = 7,. 汫 ABCD. A D A B A C,. 坽 D B D C = DA A B A C 夌 BAC, BDC 奅 BAC, BDC 妵 BAC, BDC 妺 AB < DA AC < DA, BDC. 奅妺 ch1, A D AB, AD AC, 坽 D B DC = ( DA + AB)( DA + AC ) = D A DA + DA AC + AB DA + AB AC = D A DA + AB AC = D A + A B AC = DA + A B AC,. 夌 BAC A B AC = 0, DB DC = DA + A B AC = DA + 0 > 0, BDC,. 奅 BAC A B A C > 0, D B D C = DA + A B A C > 0, BDC,. 11
妵 BAC A B A C < 0, D B D C = DA + A B A C, 0,,. 妺 AB < DA AC < DA AB AC < DA DA = DA, D B D C = DA + A B A C > AB AC + A B A C = AB AC + AB AC cos(bac) 0, 1 cos(bac) 1 D B D C > 0, BDC. 35 A. a n a n = a n-1 + f (n ), n f (x). a 1 = 1, a =, a 3 = 5, a 4 = 1, a 5 =. 5 ch ch1 杕, n =, a = a 1 + f (0) = 1 + f (0), f (0) = 1. n = 3, a 3 = a + f (1) 5 = + f (1), f (1) = 3. n = 4, a 4 = a 3 + f () 1 = 5 + f (), f () = 7. 杌 f (x), f (x) = ax + bx + c, f (0) = 1 = c, f (1) = 3 = a + b + c, f () = 7 = 4a + b + c. a = b = c = 1, f (x) = x + x + 1, 杈 n = 5, a 5 = a 4 + f (3) = 1 + 9 + 3 + 1 = 5. 4 B., ABC P A P = ( 3, 5 6 ) A 1 P = A 1 B + 5 A C. A, P BC M, A M = (, ). ( 40 1, 5 ) 1 ch3,, 1 A P = A 1 B + 5 A 1 C, + 1 1, B, P, C, 5 A M = k A P = k( 1 A B + 1 5 A C ) = k A B + k 5 A C, B, M, C, k + k 5 = 1, k = 10 7. A M = k A P = 10 7 A P = 10 7 ( 4 3, 5 6 ) = ( 40 1, 5 1 ). 1
MATHEMATICS C. a 5x 3 + (a + 4) x + ax + 1 = 0, a =. 7 ch f (x) = 5x 3 + (a + 4) x + ax + 1,, f (x) x 1, x + 1, 5x + 1, 5x 1, a, f (x) = 5x 3 + (a + 4) x + ax + 1, f (x) = 0, f (x) x + 1, 5x + 1, f (x) = 5x 3 + (a + 4) x + ax + 1 = (5x + 1)(x + 1)(x + 1) = 5x 3 + 11x + 7x + 1, a = 7. D. a 1, a,, a 9 k. k =. a 1 x a y + a 3 z = k + 1 a 4 x a 5 y + a 6 z = k 5 a 7 x a 8 y + a 9 z = k + 9, 5 ch1 ch3 a 1, a,, a 9, a, a + d, a + d,, a + 8d, d, ax (a + d) y + (a + d ) z = k + 1, (a + 3d) x (a + 4d ) y + (a + 5d ) z = k 5, (a + 6d ) x (a + 7d ) y + (a + 8d ) z = k + 9 a (a + 3d) (a + 6d) (a + d) (a + 4d) (a + 7d) (a + d ) (a + 5d ) (a + 8d ) k + 1 k 5 k + 9 ( 1) ( 1) a 3d 6d (a + d) 3d 6d (a + d ) 6d 1d k + 1 k 6 8 ( ) a (a + d) (a + d ) k + 1 3d 3d 6d k 6, 0 0 4k + 0, 4k + 0 = 0, k = 5 a 1, a,, a 9, a 1 + a 7 = a 4, a + a 8 = a 5, a 3 + a 9 = a 6, a 1 x a y + a 3 z = k + 1 泝 a 4 x a 5 y + a 6 z = k 5 沴, a 7 x a 8 y + a 9 z = k + 9 沊泝 + 沊 ( a 1 + a 7 )x ( a + a 8 )y + ( a 3 + a 9 ) z = k + 10 a 4 x a 5 y + 4a 6 z = k + 10 沝沝 沴 0 = 4k + 0, k = 5. 13
E. a, b, x a x b b a = 3. log a, log b log x log x 1 3 log a + 3 log b = 1 3 (1 + log 3 log ) + 3. (4 log + log 3), x 47 ch3 杕 1 + log 3 log = log 10 + log 3 log = log 10 9 = log 45, 4 log + log 3 = log 4 + log 3 = log 16 3 = log 48, 1 3 log a + 3 log b = 1 3 log 45 + log 48 log a = log 45, log b = log 48. 3 log 杌, log log x = 1 3 log 45 + log 48, 45 48, 3 (x 45)(48 x) = 1, x = 47. log log F.,,.. 9 64 ch3, 4 = 4 4 4 4 = 56,,,. 泝 1 1 1 1 4! = 4 沴 0 0 沊 0 0 = 4 + 6 + 6 = 36, = = 36 56 = 9 64. 4!!! = 6 4!!! = 6 14
MATHEMATICS G.. 4, 3., 6.. 14.4 ch3 杕,,, OAB, OP ABOP AB, OAB = 1 OA OB = 1 AB OP 1 3 4 = 1 5 OP, OP = 1 1. 5 杌 = 6 = PQ = 6 OP = 6 1 5 = 7 5 = 14.4.. 努力 ~ 努力加油! 加油! 加油! 15
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