Pat1.52

F F (PAT 1)1. F p, q, r ˈ F F F . F p ⇒ (p ⇒ (q ∨ r)) F p ⇒ (q ∨ r) . F p ∧ (q ⇒ r) F (q ⇒ p)∨∼ (p ⇒ ∼ r) F F 1. . . 2. . . 3. . . 4. . .2. F F = {{1, 2}, {1, 3}, {2, 3}} F F 1. ∀x∀y[x ∩ y ≠ ∅] 2. ∀x∀y[x ∪ y = ] 3. ∀x∃y[y ≠ x ∧ y ⊂ x] 4. ∃x∀y[y ≠ x ∧ y ⊂ x]3. F A = {∅, 1 , {1}} F F 1. ∅ ⊂A 2. {∅} ⊂ A / 3. {1, {1}} ⊂ A 4. {{1}, {1, {1}}} ⊂ A /4. F A = {x x ˈ F x ≤ 100} B = {x x ∈ A 3 x } P(B) F F F 1. 2 16 2. 2 17 3. 2 18 4. 2 195. F S = {x x 3 = 1} F F F S 1. {x x 3 = 1} 2. {x x 2 = 1} 3. {x x 3 = − 1} 4. {x x 4 = x}6. FS ˈ 2x 3 − 7x 2 + 7x − 2 = 0 S F F F 1. 2.1 2. 2.2 3. 3.3 4. 3.5 1

7. F A = {x x − 1 ≤ 3 − x} a ˈ F A F a F F F 1. (0, 0.5] 2. (0.5, 1] 3. (1, 1.5] 4. (1.5, 2]  x2 , x ≥ 08. F f(x) = 3x − 1 g −1 (x) =  2  −x , x < 0 F f −1 (g(2) + g(−8)) F F F 1− 2 1+ 2 1. 3 2. 3 1− 2 1+ 2 3. −3 4. −39. F A = [−2, − 1] ∪ [1, 2] r = {(x, y) ∈ A × A x − y = − 1} F a, b > 0 a ∈ D r , b ∈ Rr F a+b F F F 1. 2.5 2. 3 3. 3.5 4. 410. F f(x) = x2 − 1 x ∈ (−∞, − 1] ∪ [0, 1] g(x) = 2 x x ∈ (−∞, 0] F F 1. Rg ⊂ Df 2. Rf ⊂ Dg 3. f ˈ ˆ F 1−1 4. g F ˈ ˆ F 1−111. F cos θ − sin θ = 5 3 F F sin 2θ F F F 1. 4 13 2. 9 13 3. 4 9 4. 13 912. F ABC ˈ A F 60 , BC = 6 AC = 1 F cos(2B) F F F 1. 1 4 2. 1 2 3. 3 2 4. 3 4 2

13. F −1 ≤ x ≤ 1 ˈ π arccos x − arcsin x = 2552 F F sin( π ) 2552 F F F 1. 2x 2. 1 − 2x 2 3. 2x 2 − 1 4. −2x14. F A = {a F y = ax F y 2 = 1 + x 2} B = {b F y = x+b y2 = 1 − x2 } {d d = c 2 , c ∈ B − A} F F F F 1. (0, 1) 2. (0, 2) 3. (1, 2) 4. (0, 4)15. F F F y 2 − 4y + 4x = 0 F F (a, b) F a+b F F F F 1. 4 2. 5 3. 6 4. 716. F F F (2, 1) F F x = 1 F F 1 3 F F F F 1. (0, 1) 2. (0, 2) 3. (1, 0) 4. (3, 0)17. F F (±3, 0) F (2, 21 2 ) F F F 1. (−4, 0) 2. (0, 5 2 2 ) 3. (6, 0) 4. (0, − 3 2 )18. F 4 x − y = 128 3 2x + y = 81 F F y F F F 1. −2 2. −1 3. 1 4. 2 3

19. log 3 x = 1 + log x 9 F F F 1. [0, 4) 2. [4, 8) 3. [8, 12) 4. [12, 16)20.  4 x+ 9 x = 1 F F  25   25  . F a ˈ F a>1 . F F F F F 1. . . 2. . . 3. . . 4. . .  1 2 −1   21. F A = 2 x 2  x y ˈ   2 1 y  F C 11 (A) = 13 C 21 (A) = 9 F det(A) F F F F 1. −33 2. −30 3. 30 4. 33  −2 2 3   22. F AT =  1 −1 0  2 3 A −1 F    0 1 4  F F 1. −2 3 2. −2 3. 2 3 4. 223. F x, y, z F 2x − 2y − z = − 5 x − 3y + z = − 6 −x + y − z = 4 F F 1. x2 + y2 + z2 = 6 2. x+y+z = 2 3. xyz = 6 4. xy z = −2 4

24. F ABCD ˈ F M ˈ F AD AM = 1 AD 5 N ˈ F AC AN = 1 AC 6 F MN = aAB + bAD F a+b F F F 1. 15 2 2. 1 5 3. 1 3 4. 125. F u v ˈ F F F F u + 2v F 2u + v F u⋅v F F F 1. −4 5 2. 0 3. 1 5 4. 3 526. FS ˈ z2 + z + 1 = 0 z ˈ F F F F S 1. {−cos 120 − i sin 60 , cos 60 + i sin 60 } 2. {cos 120 + i sin 60 , − cos 60 + i sin 60 } 3. {−cos 120 − i sin 120 , − cos 60 + i sin 60 } 4. {cos 120 + i sin 120 , − cos 60 − i sin 60 }27. F z1 z2 ˈ F z1 + z2 2 = 5 z1 − z2 2 = 1 F z1 2 + z2 2 F F F 1. 1 2. 2 3. 3 4. 428. F C ˈ F F x y F F C = 3x + 5y x, y ˈ 3x + 4y ≥ 5, x + 3y ≥ 3, x ≥ 0 y≥0 F F C F F F F F F 1. 21 5 2. 29 5 3. 25 4 4. 27 4 ∞ n29. F lim n2b + 1 = 1 F Σ  ab   2 + b2  F F F n → ∞ 2n 2 a − 1 n=1 a 1. 1 3 2. 2 3 3. 1 4. F F F 5

30. F an ˈ F an + 2 an = 2 n F F F F F 10 2552 Σ a n = 31 Σ an n=1 n=1 1. 2 1275 − 1 2. 2 1276 − 1 3. 2 2551 − 1 4. 2 2552 − 1 ∞31. F a 1 , a 2 , a 3 , ... ˈ Σ an = 4 F F ˈ F n=1 a2 F F F 1. 4 2. 2 3. 1 4. F F F a2 F F F F32. F A ʽ F F F F y = 1 − x2 X B F F F X 2 y = x4 x = −c x = c F c F A = B F F F 1. 2 2. 2 3. 2 2 4. 433. F f(x) = x 4 − 3x 2 + 7 f ˈ ˆ F F F 1. (−3, − 2) ∪ (2, 3) 2. (−3, − 2) ∪ (1, 2) 3. (−1, 0) ∪ (2, 3) 4. (−1, 0) ∪ (1, 2)   f(1 + h) − f(1)34. F f (x) = 1  1 + 1  F F lim F F F 2 x x3  h → 0 f(4 + h) − f(4) 1. 1 2. 16 5 3. 7 5 4. 1 535. F A = {1, 2, 3, 4} B = {a, b, c} S = {f f : A → B ˈ ˆ F } F F F 1. 12 2. 24 3. 36 4. 39 6

36. ˂ 4 F F F F 8 F F ˂ F F F F F F 1. 96 2. 192 3. 288 4. 38437. F F F F 4 F F3 F F 2 F 1 F F 4 F ˈ FF F F F 1. 4 35 2. 35 3 3. 2 5 4. 1 438. 7 F F F F 3 F3 F F 2 F F F F F F ˈ F F F F F 1. 4 21 2. 5 21 3. 8 21 4. 10 2139. Fn ˈ F n F {1, 2, ..., 2n} F F ˈ F F F 1 20 F F ˈ F F 1 F F F 1. 1 20 2. 3 20 3. 9 20 4. 11 2040. F 99 F F ˈ x 1 , x 2 , ..., x 99 F F F F F F F 1. 2. 49 99 49 99 Σ xi = Σ xi Σ (x 50 − x i ) = Σ (x 50 − x i ) i=1 i = 51 i=1 i = 51 3. 4. 49 99 49 99 Σ x 50 − x i = Σ x 50 − x i Σ (x 50 − x i ) 2 = Σ (x 50 − x i ) 2 i=1 i = 51 i=1 i = 51 7

41. F 80 ˈ ( ʾ) 3.5 4 4.5 5 5.5 6 ( ) a 15 10 20 b 5 F F F 4.5 ʾ F F F F F F 1. 165 2. 7 16 3. 9 16 4. 11 1642. F 40 ˈ ( ) 9 - 11 15 12 - 14 5 15 - 17 5 18 - 20 10 21 - 23 5 F x F F F F F 1. x = 17.444 F F 2. x = 14.875 F F 3. x = 17.444 F 4. x = 14.875 F43. F F F a, b, c, d F F FF F a b c d F (z) -3 -0.45 0.45 1 F F 1. −a + 2b + 2c − 3d = 0 2. −a + b + c − 3d = 0 3. a − 2b + 3c + 2d = 0 4. a−b+c−d = 0 8

44. F .6 F F F F 140.6 F 3.01% F F F F F 159.4 F 46.99% F F F F 155 F F 160 F F F F F F F F F 0 z ˈ z 1.00 1.12 1.88 2.00 F F F 0.3413 0.3686 0.4699 0.4772 1. 12.86 % 2. 13.14 % 3. 15.87 % 4. 13.59 %45. F ˆ F F F (X) ʽ F (Y) 100 F F F F F F F ˆ F ˈ Y = a + bX 100 100 100 100 Σ xi = Σ y i = 1000, Σ x i y i = 2000, Σ x 2 = 4000 i i=1 i=1 i=1 i=1 F F F 15 F ʽ F( ) F F F 1. 16 2. 16.67 3. 17 4. 17.6746. 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, ... F 5060 F F F F 1. 1 2. 10 3. 100 4. 100047. Fn ˈ r ˈ n2 F 11 F F ˈ F r F F 1. 1 2. 3 3. 5 4. 7 9

48. F P(x) Q(x) ˈ 2551 F P(n) = Q(n) n = 1, 2, ..., 2551 P(2552) = Q(2552) + 1 F P(0) − Q(0) F F F 1. 0 2. 1 3. −1 4. F F F F49. 6 , , , , F F F F F F 4 F F F F 5 F F F F 1. 2. 3. 4.50. FF F F F 1. F 2 2. F 3 3. F 4. F ******************** 10

ˆ F F F PAT 1 / . . 52F 1 2 . p → (p → (q ∨ r)) ≡ ∼ p ∨ (∼ p ∨ (q ∨ r)) ≡ ∼ p ∨∼ p ∨ q ∨ r ≡ ∼ p ∨ (q ∨ r) ≡ p → (q ∨ r) . p ∧ (q → r) ≡ p ∧ (∼ q ∨ r) ≡ (p∧∼ q) ∨ (p ∧ r) ≡ (p∧∼ q)∨∼ (∼ p∨∼ r) ≡ (∼ q ∧ p)∨∼ (p → ∼ r) ∗ * ∼q∧p ≡ ∼q∨p ≡ q → p /F 2 1 1 F F 2 F {1, 2} ∪ {1, 3} = {1, 2, 3} ≠ 3 F x = {1, 2} F F y F y ≠ x ∧ y ⊂x ˈ ( F {1}, {2}, { }) 4 F x F y ≠ x ∧ y ⊂ x, y ∈ ˈF 3 2 1. ∅ ˈ 2. ∅ ∈A ∴ {∅} ⊂ A 3. 1, {1} ∈ A ∴ {1, {1}} ⊂ A (F F F ) 4. {1} ∈ A F {1, {1}} ∉ A {{1}, {1, {1}}} ⊂ A / 11

ˆ F F FF 4 1 A = {2, 4, 6, 8, ..., 100} B = {6, 12, 18, ..., 96} ∴ n(B) = 16 ∴ P(B) = 2 n(B) = 2 16F 5 2 x3 = 1 F x = 1 → x = 1, − 1 F 1 x3 = 1 →x = 1 F 2 x2 = 1 → x = 1, − 1 F 3 x3 = − 1 → x = − 1 F 4 x4 = x → x 4 − x = 0 → x(x 3 − 1) = 0 → x = 0, 1 F F 2 FF 6 4 2x 3 − 7x 2 + 7x − 2 = 0 F x3 − 7x2 + 7x − 1 = 0 2 2 −  − 7  = 7 = 3.5  2 2F 7 4 x−1 ≤ 3−x F −(3 − x) ≤ x − 1 x−1 ≤ 3−x −3 + x ≤ x − 1 2x ≤ 4 −2 ≤ 0 x≤2 REAL ∩ (−∞, 2] F (−∞, 2] F 2 12

ˆ F F FF 8 1 f(x) = 3x − 1 → f −1 (x) = x + 1 3  x2 , x ≥ 0 g(2) 2 =  2 → 2 = x 2, x ≥ 0  −x , x < 0 →x = 2, − 2 ∴ g(2) = 2  x2 , x ≥ 0 g(−8) −8 =  2 → − 8 = − x2, x < 0  −x , x < 0 → x 2 = 8, x < 0 → x = 2 2 , − 2 2 ∴ g(−8) = − 2 2 f −1 (g(2) + g(−8)) = f −1 ( 2 + (−2 2 )) − 2 +1 = f −1 (− 2 ) = 3F 9 2 x∈A ∴ −2 ≤ x ≤ − 1 1≤x≤2 (1) r ; x−y = −1→ y = x+1 (1) ; −2 + 1 ≤ x + 1 ≤ −1+1 1+1 ≤ x+1 ≤ 2+1 −1 ≤ y ≤ 0 2≤y≤3 (2) F y ∈ A ∴ (2) ∩ A F y = − 1, 2 y y = x + 1 F x = − 2, 1 ∴ D r = {−2, 1}, R r = {−1, 2} a, b > 0 ∴ a + b = 1+2 = 3 2 F y = x+1 x, y ∈ [−2, − 1] ∪ [1, 2] y F r = {(1, 2), (−2, − 1)} a = 1, b = 2 2 ∴ a+b = 3 1 x -2 -1 1 2 -1 -2 13

ˆ F F FF 10 1 f(x) = x 2 − 1, D f = (−∞, − 1] ∪ [0, 1] y f F ˈ ˆ F 1−1 ( 3 ) F x 2 (−1, 0) (1, 0) x -1 1 -1 g(x) = 2 x , D g = (−∞, 0] y g ˈ ˆ F 1−1 y = 2x ( 4 ) 1 x 2 F R f = [−1, ∞) R g = (0, 1] 2 F 1 Rg ⊂ DfF 11 3 5 cos θ − sin θ = → cos 2 θ − 2 sin θ cos θ + sin 2 θ = 5 3 9 2 sin θ cos θ = 1 − 5 9 ∴ sin 2θ = 4 9F 12 4 F sin e F sin 60 6 C = 1 sin B → sin B = 1 2 2  2 1 6 ∴ cos 2B = 1 − 2 sin 2 B = 1 − 2  1 2 2  3  = 4 60 A B 14

ˆ F F FF 13 2 π 2552 = arccos x − arcsin x sin π 2552 = sin(arccos x − arcsin x) ... sin π 2552 = sin( π − arcsin x − arcsin x) 2 arcsin x + arccos x = π 2 sin π 2552 = sin( π − 2 arcsin x) = cos(2 arcsin x) 2 sin π 2552 = 1 − 2 sin 2 (arcsin x) = 1 − 2x 2F 14 3 A = {a / F y = ax F y 2 = 1 + x 2} F y = ax (1) y2 = 1 + x2 (2) F y (1) (2) F (ax) 2 = 1 + x 2 1 a 2 x 2 − x 2 = 1 → x 2 (a 2 − 1) = 1 → x = ± a2 − 1 F y = ax F y2 = 1 + x2 F x F F F a 2 − 1 ≤ 0 → (a − 1)(a + 1) ≤ 0 -1 1 ∴ A F F [−1, 1] F = 1 B = {b / F y = x+b y2 = 1 − x2 } y = x+b → x−y+b = 0 F 2 CP < 0−0+b p < 1 12 + 12 C(0,0) b < 2 y2= 1 - x 2 x2 + y 2 = 1 − 2 <b< 2 ∴ B F F (− 2 , 2) C ∈ B − A = (− 2 , − 1) ∪ (1, 2) d = c 2 = (1, 2) 15

ˆ F F F F 15 3 PARA y 2 − 4y + 4x = 0 y 2 − 4y + 4 = − 4x + 4 (y − 2) 2 = − 4 (x − 1) F (1, 2) C c=1 (a,b) = (2,b) 4c → c = 1 m AB = m AC B(1,2)A(0,0) 2−0 1−0 = b−0 2−0 b = 4 DI : x=2 ∴ a+b = 2+4 = 6 F 16 1 l ml = 1 CP 3 P(1,b) CP 1 r = CP = ( 3 + 1 − 1) 2 + (1 − 2) 2 = 2 m CP = − 3 C(2,1) b−1 = − 3 (x − 2) 2 + (y − 1) 2 = 2 2 1−2 F b = 3 +1 (0, 1) x = 1 F ˈ (0, 1) F x = 1 F F 17 1 P(2, 21 ) 2 = PF 1 + PF 2 F1 (-3,0) F2 (3,0) 2a = 21 (2 + 3) 2 + ( 2 − 0) 2 + (2 − 3) 2 + ( 2 − 0) 2 21 2a = 8 → a = 4 F 1 F 2 = 2c = 6 → c = 3 a2 = b2 + c2 → 42 = b2 + 32 → b2 = 7 y2 F x2 16 + 7 = 1 (−4, 0) F ˈ (−4, 0) F 16

ˆ F F FF 18 2 4 x − y = 128 → 2 2x − 2y = 2 7 F 2x − 2y = 7 (1) 3 2x+y = 81 → 3 2x + y = 3 4 F 2x + y = 4 (2) (1) (2) F y = −1F 19 3 log 3 x = 1 + log x 9 F log 3 x = 1 + 2 log x 3 2 log 3 x = 1 + log 3 x (log 3 x) 2 = log 3 x + 2 (log 3 x) 2 − log 3 x − 2 = 0 (log 3 x − 2)(log 3 x + 1) = 0 F log 3 x = 2 log 3 x = − 1 x = 32 x = 3 −1 = 9 = 1 3 F x = 9, 1 3 = 9 + 1 = 28 = 9.33 3 3F 20 3  4 x+ 9 x  25   25  = 1 F 2 2x = 1 2x + 3 2x 5 2x 5 2 2x + 3 2x = 5 2x F F x = 1 2 F ˈ F a ˈ F a = 1 2 ..... F ( )  4 x+ 9 x = 1  25   25  17

ˆ F F F F  4 x ˈ ˆ F  9 x ˈ ˆ F F  25   25   4 x+ 9 x = 1 F ..... F ( )  25   25 F 21 4 C 11 (A) = 13 C 21 (A) = 9 F M 11 (A) = 13 F −M 21 (A) = 9 −2 2 −1 x 2 1 y = 13 − 1 y = 9 xy xy − 2 = 13 (1) −(2y + 1) = 9 y = −5 F y (1) F x(−5) − 2 = 13 x = −3 −6 − 2 + 20 = + 12  1 2 −1  1 2 −1 1 2   A =  2 −3 2  , det A = 2 −3 2 2 − 3 = + 21 + 12 = + 33    2 1 −5  2 1 −5 2 1 +15 + 8 − 2 = + 21F 22 3  −2 2 3   −2 1 0      AT =  1 −1 0  → A =  2 −1 1  F det A F det A = 3      0 1 4   3 0 4  a −1 ij = 1 C (A) det A ji a −1 23 = 1 C (A) 3 32 = 1 [−M 32 (A)] 3 0 −2 0 a −1 23 = −1 3 2 1 = − 1 (−2 + 0) = 2 3 3 −2 18

ˆ F F F F 23 1 2x − 2y − z = −5 (1) x − 3y + z = −6 (2) −x + y − z = 4 (3)(2) + (3) F −2y = −2 → y = 1(1) + (2) F 3x − 5y = − 11 (4) F y (4) F 3x − 5(1) = − 11 → x = −2 F x y F (3) F −(−2) + 1 − z = 4 → z = −1 1 x 2 + y 2 + z 2 = (−2) 2 + 1 2 + (−1) 2 = 6 2 x + y + z = − 2 + 1 + (−1) = − 2 3 xyz = (−2)(1)(−1) = 2 (−2)(1) 4 xy z = −1 = 2 F 24 1 MN = AN − AM D C MN = 1 6 AC − 1 AD 5 4 = 1 (AB + AD) − 1 AD M 5 6 5 1 = 1 1 AB − 30 AD N 6 1A B ∴ − a + b = 1 − 30 = 5301 = 30 = 15 6 1 4 2 F 25 1 (u + 2v) ⋅ (2u + v) = 0 2 u 2 + 2 v 2 + 5u ⋅ v = 0 2 + 2 + 5u ⋅ v = 0 u⋅v = −45 F 26 4 z2 + z + 1 = 0 −1 ± 1 − 4 −1 ± 3 i 3 3 z = = = −1+ i, − 1 − i 2 2 2 2 2 2 z = cos 120 + i sin 120 , − cos 60 − i sin 60 19

ˆ F F F F 27 3 F vector u + v 2 = u 2 + 2u ⋅ v + v 2 (1) u − v 2 = u 2 − 2u ⋅ v + v 2 (2)(1) + (2), u + v 2 + u − v 2 = 2 u 2 + v 2    z ˈ vector F z1 + z2 2 + z1 − z2 2 = 2 z1 2 + z2  2   5 + 1 = 2 z1 2 + z2  2   ∴ z1 2 + z2 2 = 3 F 28 2 F C = 3x + 5y F 3x + 4y ≥ 5 x + 3y ≥ 3 x ≥ 0 y ≥ 0 F intersect F y (0, 5 ) 4 (0, 1) ( 3,5 ) 5 4 5 x ( 3 , 0) (3, 0) 3x + 4y = 5 x + 3y = 3 F F C(0, 5 ) 4 = 3(0) + 5( 5 ) 4 = 25 4 C( 3 , 4 ) 5 5 = 3( 3 ) + 5( 4 ) 5 5 = 29 5 F C C(3, 0) = 3(3) + 5(0) = 9 20

ˆ F F FF 29 2 lim bn 2+ 1 = 1 2 b → =1 → b = 2a n → ∞ 2an − 1 2a ∞ n ∞ n ∞ ∴ Σ  ab  Σ  2a 2  n = = Σ 2  n=1 a 2 + b2  n = 1 a 2 + 4a 2   n=1 5 2 2 2 2 3 = + 5 5 +  5  + ...... 2 = 5 = 2 1− 2 3 5F 30 2 an + 2 an =2 F a 1 , a 3 , a 5 , .... a 2 , a 4 , a 6 , .... ˈ r=2 10 Σ a n = 31 → (a 1 + a 3 + ..... + a 9 ) + (a 2 + a 4 + ... + a 10 ) = 31 n=1 a 1 (2 5 − 1) a 2 (2 5 − 1) 2−1 + 2−1 = 31 → a 1 + a 2 = 1 = 2552 Σ an (a 1 + a 3 + ..... + a 2551 ) + (a 2 + a 4 + .... + a 2552 ) n=1 a 1 (2 1276 − 1) a 2 (2 1276 − 1) = 2−1 + 2−1 = 2 1276 − 1(a 1 + a 2 ) = 2 1276 − 1F 31 3 ∞ Σ an = 4 → a 1 + a 2 + a 3 + .......... = 4 n=1 a1 1−r = 4 → a 1 = 4(1 − r) → a 1 r = 4r(1 − r) a2 = 4r − 4r 2 ∴ = 4(−4)(0) − 4 = 1 2 a2 F 4(−4)F 32 2 A y A = 2 (2)(1) = 4 3 3 F 1 -1 1 x y = 1 - x2 21

ˆ F F F B y 2 B = ∫ c  x 2  dx = x3 c y = x4 −c  4  12 −c = c3 −  −c  = 3 c3 12  12  6 x -C C FF F A=B F c3 6 = 4 3 c3 =8 ∴ c=2F 33 3 f ˈ ˆ F f (x) > 0 f (x) = 4x 3 − 6x > 0 2x(2x 2 − 3) > 0 2x( 2 x − 3 )( 2 x + 3 ) > 0 3 0 3 2 2 4 F F F 3 (−1, 0) ∪ (2, 3) ˈ F F F f (x) > 0F 34 2 f(x+h) − f(x) lim = f (x) h→0 h f(1 + h) − f(1) lim f(1 + h) − f(1) h→ 0 h lim = h → 0 f(4 + h) − f(4) lim f(4 + h) − f(4) h→ 0 h 11 1 + f (1) 21 1 = f (4) = 11 1 = 2 = 16 5 5 + 22 8 8 22

ˆ F F FF 35 3 A = {1, 2, 3, 4} B = {a, b, c} F ˆ F A B 4! = 1!1!2!2! × 3! = 36 F F F 0→∆ 0→∆ 0→∆ 0F 36 1 F ˂ F F F F 3!(2!) 4 = 96 F FF 37 1 n(S) = F 4 10 =  10  4 = 210 n(E) = F 4 F = 4 3 2 1 1 1 1 1 = 24 F P(E) = 24 210 = 4 35 23

ˆ F F FF 38 1 n(S) = 7 F F = 7 4 2 3 2 2 = 210 n(E) = 7 F F F F F 3 F F F F * = 5 4 2 1 2 2 F * = 5 2 3 2 = 40 F P(E) = 40 = 4 210 21F 39 3 n n P( F F ) =  2n  1 = 20 n F  2n  = 20 n F n = 3 F {1, 2, 3, 4, 5, 6} F 3 3 1 2 P( F F 1 )= 6 3 = 9 20F 40 3 F FF F F F 3 F F F F F F (x 50 ) 24

ˆ F F FF 41 4 µ= Σ x N (3.5)a + (4)(15) + (4.5)(10) + (5)(20) + (5.5)b + (6)(5) 4.5 = 80 F 7a + 11b = 250 ................. (1) a + b + 50 = 80 a + b = 30 ................. (2) (1) (2) F a = 20 b = 10 F (MD) = Σ f N− µ x FF F µ = 4.5 MD = (20)(1) + (15)(.5) + (10)(10)80(20)(.5) + (10)(1) + (5)(1.5) + = 11 16F 42 4 x .( ) (f) d fd 9 - 11 15 0 0 12 - 14 5 1 5 15 - 17 5 2 10 18 - 20 10 3 30 21 - 23 5 4 20 40 65 i Σ fd µ = a+ N µ = 10 + (3)(65) 40 = 14.875 25

ˆ F F F .( ) (f) 9 - 11 15 F 12 - 14 5 F 15 - 17 5 18 - 20 10 21 - 23 5 F F F F x = 14.875 F FF 43 1 x−µ z= σ a−µ FF −3 = σ → a = µ − 3σ b−µ −0.45 = σ → b = µ − 0.45σ c−µ 0.45 = σ → c = µ + 0.45σ d−µ 1 = σ → d = µ+σ a, b, c d F F F 1 ˈF 44 4 F F 3.01% .4699 46.99% 140.6 159.4 z = -1.88 z = 1.88 140.6 − µ F −1.88 = σ .................. (1) 159.4 − µ 1.88 = σ .................. (2) 26

ˆ F F F (1) (2) F µ = 150 σ=5 Z1 = 155 − 150 = 1 5 Z2 = 160 − 150 5 = 2 .4772 - .3413 = .1359 ˈ 13.59% z1 = 1 z2 = 2F 45 2 y = a + bx Σy = bΣx+Σa → 1000 = 1000b + 100a ............. (1) Σ xy = b Σ x 2 + a Σ x → 2000 = 4000b + 1000a ............. (2) (1) (2) F a = − 10 b = 4 3 3 F y = − 10 + 4 x 3 3 F x = 15 F y = − 10 + 4 (15) = 50 = 16.67 3 3 3F 46 2 F a 1 = 1, a 3 = 2, a 6 = 3, a 10 = 4, a 15 = 5 F a n(n + 1) 2 =n F n = 100 F a 5050 = 100 ∴ a 5051 = 1, a 5052 = 2, ............, a 5060 = 10 27

ˆ F F FF 47 4 n n2 n2 F 11 1 1 1 2 4 4 3 9 9 4 16 5 5 25 3 . . . . . . . . . . . . . . . . . . F n2 F 11 ˈ 7 F FF 48 3 F(x) = P(x) − Q(x) P(x) = Q(x) x = 1, 2, 3, ..., 2551 FF P(x) − Q(x) = 0 x = 1, 2, 3, ..., 2551 F F(x) = 0 x = 1, 2, 3, ..., 2551 ∴ F(x) = k(x − 1)(x − 2)(x − 3) ..... (x − 2551) (1) F P(2552) = Q(2552) + 1 → P(2552) − Q(2552) = 1 → F(2552) = 1 ∴ (1) x = 2552 F(2552) = k(2552 − 1)(2552 − 2)(2552 − 3) ..... (2552 − 2551) 1 1 = k(2551)(2550)(2549).....(1) → k = 2551 ! ∴ P(0) − Q(0) = F(0) = 1 2551 ! (0 − 1)(0 − 2)(0 − 3) ..... (0 − 2551) = 1 2551 ! ⋅ (−2551!) = − 1 28

ˆ F F FF 49 3 (1) F 4 F F F F , , F ˈ .... F 3F 50 3 ************************ 29

Pat1.52

by june41 on Sep 20, 2012

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Pat1.52 Document Transcript