2007 GRE Math-Sub Nov 3, 2007 Test time: 170 minutes

2007 GRE Math-Sub Nov 3, 2007 Test time: 170 minutes

... zqs... 10 66 60... fz zqs vonneumann vonneumann sub... Bless by Luobo June 21, 2008

1. 2. g(x) = e 2x+1, cos 3x 1 lim x 0 x 2 g(g(x)) g(e) lim x 0 x 3. d x 4 e t2 dt dx x 3 4. y = f(x) R y 4 = 3(x 1) f 1 g(x) = x A.f (1) = 3 B.(f 1 ) (4) = 1 3 C.(gf) (1) = 5 D.(g f) (1) = 1 2 E.?? 5. g(x) 3x 5 + 96 = x 6. A. x 3 = 10 x B. x 2 + 4x + 7 = 8x + 15 C. 3x 10 = 7 x D. e x = x E. sec x = e x2 7. cos 97x = x c g(t)dt, c 8. 9 2m 3 9. f(x) y = sin x 0, π 4, π 2 f(π) π 4 10. (cos x + 1 + x 2 cos 3 x sin 3 x)dx π 4 11. R R R R A. B. R C. a 0 a 2 = 0 D. R E. R 1

12. 3 5 6 2 13. A. B. ( 5, 1, 1, 1, 0) C.?? D.?? E. λ ( 5, 1, 1, 1, 0) 14. V P P 2 = P? (I) P (II) P (III) P = I 15. f(x) y + xy = x f(0) = 1 lim + e ax e bx 16. 0 (1 + e bx )(1 + e ax ) dx x + f(x) 17. F = xi + yj + zk z = 1 x 2 y 2 Flux 18. (x 3) 2 + (y 2) 2 = x 2 + (y + 2) 2 19. A 100 3 B 0.02 4 B A A 4 100 20. S R A.?? B.?? C.?? D. {x R open set U, s.t. x U S} E. {x R open set U, s.t. x U S c } 2

21. 5 22. Z + d (I) n Z + {n} (II) Z + (III) Z + 0 x = y d(x, y) = 1 x y 23. 2 1 2, 3 1 3, 6 1 6 24. f(x) = A. 1 1+x B. 1 1 x C. 1 (1+x) 2 + n=1 D. 1 (1 x) 2 E.?? x n n x ( 1, 1) f (x) 25. f(x) ɛ > 0, δ > 0, s.t. f(x) f(1) ɛ when x 1 δ A. lim x f(x) = 0 B.?? 3

C. f D. f E. f 26. (I) C, s.t. log x C x, x R + n (II) C, s.t. k 2 Cn 2, n Z + k=1 (III) C, s.t. sin x x C x 3, x R 27. f, f 28. log x = Cx 4 C 29. 2 30. n n 31. f(x, y, z) = 3xy 2 + z f (0, 0, π) (1, 2, 3) A. 0.2 B. 0.8 C. 1.4 D. 1.6 E.2.0 32. f(x)11 g(x) = f(x 10 ), f (0) = f (0) = 1 g (11) (0) A. 0 B. 1 C. 10! D. 11! E. 4

33. f 2x 2 f(x) = 3x 2 x Q x / Q A. f, B. f x = 0 C. f x = 0 f x = 0 D. f E. f 34. x = 5 cos θ y = 5 sin θ z = θ L(θ) (5, 0, 0) P (θ) = (x(θ), y(θ), z(θ)), D(θ) P (θ). θ 0 > 0 L(θ 0 ) = 26, D(θ 0 ) =? 35. (1 + i) 10 36. 10 20 37. 1 10 10 e 2 38. e 3 dx x log x 39. Z 17 0 (I) 5 (II) 8 (III) 16 40. x, y 3x 5 mod 11, 2y 7 mod 11 x + y? mod 11 41. A. Z B. Z\{0} 5

C. R D. R + E.?? 42. 43. 0 x z = 0 44. 2x y + 3z = 5 45. y = x 2, x (0, 1) y 46. y = 2 x 2, y = x 2 z = y + 3 z = 0 47. A = 1 2 3 4 5 0 2 3 4 5 0 0 3 4 5 0 0 0 4 5 0 0 0 0 5 A. A B. Ax = x C.?? D.?? E. det A = 120 48. f(x + iy) = e x sin y + ig(x, y) g(3, 2) g(1, 2) z 2 49. lim z 0 z 2 =? A.0 B.1 C. D.?? E. 50. f(x) (0, 3) f (x) 1 f(3) = 5 f(0) 6

51. A, B, C A B C C A. A B B. A B C. A B D. A, B E. A, B 52. R S, : (1) xry yrx (2) xry yrz xrz R (I) x, y, s.t. xry (II) x, s.t. xrx A. (I) R (II) B. (II) R (I) C. (I) (II) R D. (I) (II) R E. (I) (II) R 53. f (x) f(0), f(2), f(4) 54. 3 11 1 7

55. x = t 2 + 2t t > 0 d2 y (8, 80) dx2 y = 3t 4 + 4t 3 56. X [0, 3] Y [0, 4] X Y P (x < y) 57. Z 10 S = {0, 2, 4, 6, 8} A. S Z 10 B. S C. S D.?? E. S 58. V W 7 4 V W A.0 B.1 C.2 D.3 E.4 59. f, g R g f A. f B. g C. f D. g E. g f 60. {x n } x n (0, 1 n ) (I) lim n x n = 0 (II) f (0, 1) lim n f(x n) (III) g (0, 1) lim n g(x n) 61. A, B, C A B B A C A B A. B C A B. A B C C.?? D.?? E. C A B A 8

62. R 2 T = {(x, y) x / Q or y / Q} A. T B. T C. T D. T E. T 63. S 5 A.5 B.6 C.8 D.24 E.120 64. T, S T S = {ts t T, s S} A, B sup A > sup B sup(ab) A. sup A sup B B. max{sup A sup B, inf A inf B} C. max{sup A sup B, sup A inf B} D. max{sup A sup B, sup B inf A} E. max{sup A sup B, sup B inf A, inf A inf B} 65. 2, 3, 5 2 5 11 66. A 3 A. n > 0 (A I) n = 0 B. v 1, v 2, v 3 Av i = 0 C. 0 A D. A E. Ax = y y 9