【GR1768】 The Solution for GRE Mathematics Practice Test [61-66]

The Solution for GRE Mathematics Practice Test [61-66]

 

추천글 : 【GRE】 Solution for GRE Mathematics Practice Test 

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61. A tank initially contains a salt solution of 3 grams of salt dissolved in 100 liters of water. A salt solution containing 0.02 grams of salt per liter of water is sprayed into the tank at a rate of 4 liters per minute. The sprayed solution is continually mixed with the salt solution in the tank, and the mixture flows out of the tank at a rate of 4 liters per minute. If the mixing is instantaneous, how many grams of salt are in the tank after 100 minutes have elapsed?

 

 

⑴ answer: E

 

 

 

62. Let S be the subset of ℝ² consisting of all points (x, y) in the unit square [0, 1] × [0, 1] for which x or y, or both, are irrational. With respect to the standard topology on ℝ², S is

 

 

⑴ answer: C

 

 

63. For any nonempty sets A and B of real numbers, let A · B be the set defined by

 

A · B = { xy : x ∈ A and y ∈ B }.

 

If A and B are nonempty bounded sets of real numbers and if sup (A) > sup (B), then sup (A · B) = 

 

 

⑴ answer: E

⑵ Not only positive numbers, but also negative numbers should be considered.

 

 

64. What is the value of the flux of the vector field F, defined on ℝ³ by F(x, y, z) = xi + yj + zk, through the surface z = √ (1 - x² - y²) oriented with upward-pointing normal vector field? (Note: i, j, and k are the standard basis vectors in ℝ³.)

 

 

⑴ answer: E

⑵ Gauss' divergence theorem

 

 

 

65. Let g be a differentiable function of two real variables, and let f be the function of a complex variable z defined by

 

f(z) = e^x sin y + i g(x, y),

 

where x and y are the real and imaginary parts of z, respectively. If f is an analytic function on the complex plane, then g(3, 2) - g(1, 2) =

 

 

⑴ answer: E

⑵ If a complex function is analytic on a region R, it is infinitely differentiable in R. (reference)

⑶ complex analytic function : Cauchy-Riemann equation. For f(z) = u(x, y) + iv(x, y),

 

 

⑷ Thus, g(x, y) = -e^x cos y + C, making g(3, 2) - g(1, 2) = (e - e³) cos 2.

 

 

66. Let ℤ₁₇ be the ring of integers modulo 17, and let ℤ₁₇^× be the group of units of ℤ₁₇ under multiplication. Which of the following are generators of ℤ₁₇^×?

 

 

⑴ answer: B

⑵ Ⅱ. 8¹ ≡ 8, 8² ≡ -4, 8³ ≡ 2, 8⁴ ≡ -1, ··· → {1, 2, 4, 8, -1, -2, -4, -8}

⑶ Ⅲ. 16¹ ≡ -1, 16² ≡ 1, ··· → {1, -1}

 

입력: 2023.01.22 00:23