IIB 数学演習問題

1

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. Ԡ New 2 ͭͷ์෺ઢ C1 : y = x² C2 : y = x² − 6x + 15 ͷڞ௨઀ઢͷํఔࣜΛٻΊΑɻ
2. Ԡ New f(x) = ax² + bx + c g(x) = x³ + dx ͱ͢Δɻ ࣍ͷ 2 ͭͷ৚݅ (a)ɼ(b) Λͱ΋ʹຬͨ͢Α͏ʹఆ਺ a, b, c, d ͷ஋ΛఆΊΑɻ (a) ۂઢ y = f(x) ͸఺ (1, −3) Λ௨Δɻ (b) ۂઢ y = f(x) ͱۂઢ y = g(x) ͸఺ (2, 6) ʹ͓͍ͯڞ௨ͷ઀ઢΛ΋ͭɻ

ʲղ౴ʳ

1. y = 2x − 1
2. (a, b, c, d) = (2, 3, −8, −1)

2

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. New ؔ਺ f(x) = x³ + ax² + 3x + 4 ͕ۃ஋Λ΋ͭΑ͏ʹɼఆ਺ a ͷ஋ͷൣғΛఆΊΑɻ
2. New ؔ਺ y = x³ − 3ax² + 3x + 1 ͕୯ௐʹ૿Ճ͢ΔΑ͏ʹɼఆ਺ a ͷ஋ͷൣғΛٻΊΑɻ
3. Ԡ New ؔ਺ f(x)=2x³ − 3(a + 2)x² + 12ax ͕͋Δɻ (1) f(x) ͕ۃ஋Λ΋ͨͳ͍Α͏ʹɼఆ਺ a ͷ஋ͷൣғΛఆΊΑɻ (2) f(x) ͕ۃ஋Λ΋ͭͱ͖ɼۃେ஋Λ a Ͱදͤɻ

ʲղ౴ʳ

1. a < −3 , 3 < a
2. −1 ≤ a ≤ 1
3. (1) a = 2 (2) a < 2 ͷͱ͖ −a³ + 6a² 2 < a ͷͱ͖ 12a − 8

3

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. New ࣍ͷؔ਺ͷάϥϑΛ͔͚ɻ (1) y = x⁴ − 2x² (2) y = x⁴ − 6x² − 8x
2. New ؔ਺ y = −½ x⁴ − 2x + 3 ͷ −2 ≤ x ≤ 3 ʹ͓͚Δ࠷େ஋ͱ࠷খ஋ΛٻΊΑɻ

ʲղ౴ʳ

1. چ BW ɹؔ਺ͷ஋ͷมԽʢ̎ʣ࿅श໰୊ (1)(2) ͷղ౴Λૠೖ͍ͯͩ͘͠͞ɻ 9 ஋େ࠷ 2
2. (x = −1) ɹ࠷খ஋ − 87 2 (x = 3)

4

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. Ԡ New ࣮ ਺x, y ͕ x² + y² = 4 Λຬͨ͢ͱ͖ɼxy² − 2x² ͷ࠷େ஋ͱ࠷খ஋Λߟ͑Δɻ (1) x ͷऔΓ͏Δ஋ͷൣғΛٻΊΑɻ (2) xy² − 2x² ͷ࠷େ஋ͱ࠷খ஋ΛٻΊΑɻ
2. Ԡ New 0 ≤ x ≤ π ͷͱ͖ɼؔ਺ y = 4 sin³θ + cos²θ − 2 sinθ + 1 ͷ࠷େ஋ͱ࠷খ஋ΛٻΊΑɻ

ʲղ౴ʳ

1. (1) −2 ≤ x ≤ 2 (2) ࠷େ஋ 40/27 ɼ࠷খ஋ −8
2. ࠷େ஋ 3ɼ࠷খ஋ 5/4

5

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. Ԡ New a > 0 ͱ͢Δɻؔ਺ f(x) = x³ − 3x + 1 (0 ≤ x ≤ a) ʹ͍ͭͯɼ࣍ͷ໰͍ʹ౴͑Αɻ (1) ࠷খ஋ΛٻΊΑɻ (2) ࠷େ஋ΛٻΊΑɻ
2. Ԡ New a > 0 ͱ͢Δɻؔ਺ f(x) = x³ − 3a²x (0 ≤ x ≤ 1) ʹ͍ͭͯɼ࣍ͷ໰͍ʹ౴͑Αɻ (1) ࠷খ஋ΛٻΊΑɻ (2) ࠷େ஋ΛٻΊΑɻ

ʲղ౴ʳ

1. (1) 0 < a < 1 ͷͱ͖ a³ − 3a + 1ɼ1 ≤ a ͷͱ͖ −1 (2) 0 < a < 3 ͷͱ͖ 1ɼ3 ≤ a ͷͱ͖ a³ − 3a + 1
2. (1) 0 < a < 1 ͷͱ͖ −2a³ɼ1 ≤ a ͷͱ͖ 1 − 3a² (2) 0 < a < 1/√3 ͷͱ͖ 1 − 3a²ɼ 1/√3 ≤ a ͷͱ͖ 0

6

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. Ԡ New ۂઢ y = x³ + 3x² ʹ͍ͭͯɼۂઢʹ఺ (1, p) ͔ΒҟͳΔ 3 ຊͷ઀ઢ͕Ҿ͚ΔΑ͏ͳఆ਺ p ͷ஋ͷൣғΛٻΊΑɻ
2. Ԡ New 3 ࣍ํఔࣜ x³ − 3a²x + 4a = 0 ʹ͍ͭͯɼҟͳΔ 3 ݸͷ࣮਺ղΛ΋ͭ a ͷ஋ͷൣғΛٻΊΑɻ

ʲղ౴ʳ

1. −4 < p < 4
2. a < −√2 , √2 < a

7

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. New x ≥ 0 ͷ͢΂ͯͷ x ʹରͯ͠ɼෆ౳ࣜ 2x³ − 12x² + 18x + k ≥ 0 ͕੒ΓཱͭΑ͏ͳఆ਺ k ͷ஋ͷൣғΛٻΊΑɻ
2. Ԡ New x ≥ 0 ͷͱ͖ɼෆ౳ࣜ x³ + 32 − px² ͕੒ΓཱͭΑ͏ͳఆ਺ p ͷ஋ͷൣғΛٻΊΑɻ

ʲղ౴ʳ

1. k ≥ 0
2. p ≥ 6

8

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. Ԡ New ౳ࣜ f(x) = x + 3 + ∫₋₁ˣ (x − t)f(t) dt Λຬͨؔ͢਺ f(x) ΛٻΊΑɻ
2. x ∈ [1, 2] ∫₁ˣ f(t) dt = x² − kx + 9 Λຬͨؔ͢਺ f(x) ͱఆ਺ k ͷ஋ΛٻΊΑɻ
3. New ࣍ͷ৚݅Λຬͨ͢ 2 ࣍ؔ ਺f(x) ΛٻΊΑɻ f(1) = 1 , f(−1) = −1 , ∫₋₁¹ f(t) dt = 4

ʲղ౴ʳ

1. f(x) = 3x + 1
2. f(x) = x² − 10x + 9 , k = 10
3. f(x) = −3x² + x + 3

9

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. Ԡ New ۂઢ y = −x² + 2x ͱ x ࣠Ͱғ·Εͨ෦෼ͷ໘ੵΛɼ௚ઢ y = (−a + 2)x ͕ 2 ౳෼͢ΔΑ͏ʹɼఆ਺ a ͷ஋ΛఆΊΑɻ
2. Ԡ New ఺ (−1, 2) Λ௨Δ܏͖ m ͷ௚ઢͱɼ์෺ઢ y = x² Ͱғ·Εͨਤܗͷ໘ੵΛ S ͱ͢Δɻ m ͕͢΂ͯͷ࣮஋਺ΛͱͬͯมԽ͢Δͱ͖ɼS ͷ࠷খ஋ͱͦͷͱ͖ͷ m ͷ஋ΛٻΊΑɻ

ʲղ౴ʳ

1. a = √3/4 4 ஋খ࠷ 2
2. (m = −2)

10

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. ์෺ઢ C : y = x² − 5x + 8 ʹ఺ (3, 1) ͔Β 2 ຊͷ઀ઢΛҾ͘ͱ͖ (1) ͜ͷ 2 ຊͷ઀ઢͷํఔࣜΛٻΊΑɻ (2) ์෺ઢͱ 2 ຊͷ઀ઢͰғ·Εͨ໘ੵΛٻΊΑɻ
2. Ԡ New 2 ͭͷ์෺ઢ C1 : y = x² − 5x + 7 , C2 : y = x² + 3x − 1 ͷ྆ํʹ઀͢Δ௚ઢΛ  ͱ͢Δɻ (1) ௚ઢ  ͷํఔࣜΛٻΊΑɻ (2) C1.C2,  Ͱғ·Εͨਤܗͷ໘ੵΛٻΊΑɻ

ʲղ౴ʳ

1. (1) y = −x + 4 , y = 3x − 8 (2) 2/3
2. (1) y = x − 2 (2) 16/3

11

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. New ࣍ͷۂઢͱ௚ઢͰғ·Εͨ໘ੵΛٻΊΑɻ (1) y = x²(x − 2) ͱ x ࣠ (2) y = x(x − 1)(x − 2) ͱ x ࣠
2. Ԡ New ۂઢ C : y = x³ − x + 1 ʹ͍ͭͯɼ࣍ͷ໰͍ʹ౴͑Αɻ (1) ఺ (0, −1) Λ௨Δ઀ઢ  ͷํఔࣜΛٻΊΑɻ (2) ۂઢ C ͱ௚ઢ  Ͱғ·Εͨਤܗͷ໘ੵΛٻΊΑɻ

ʲղ౴ʳ

1. (1) 4/3 (2) 1/2
2. (1) y = 2x − 1 (2) 27/4

12

New ͕෇͍͍ͯΔ໰୊͸ɼ͜ͷतۀͰ৽ग़ͷࣄ߲Ͱ͢ɻ෮शͷࡍॏʹࢹ͍ͯͩ͘͠͞ɻ

1. Ԡ New ۂઢ C : y = |x² − 2x| ͱ௚ઢ  : y = x Ͱғ·Εͨ 2 ͭͷ෦෼ͷ໘ੵͷ࿨ΛٻΊΑɻ
2. ൃల໰୊ New ؔ਺ f(t) Λ f(t) = ∫₋₁¹ |x² − t²| dx (t ≥ 0) Ͱఆٛ͢Δɻ (1) f(t) ΛٻΊΑɻ (2) t ≥ 0 ʹ͓͚Δ f(t) ͷ࠷খ஋ΛٻΊΑɻ

ʲղ౴ʳ

1. 13/6
2. (1) 0 ≤ t ≤ 1 ͷͱ͖ f(t) = 8/3 t³ − 2t² + 2/3 ɼ1 < t ͷͱ͖ f(t)=2t² − 2/3 (2) 1/2