【GR1768】 The Solution for GRE Mathematics Practice Test [01-20]

The Solution for GRE Mathematics Practice Test [01-20]

 

추천글 : 【GRE】 Solution for GRE Mathematics Practice Test 

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1. 

 

⑴ answer: E

 

 

 

2. What is the area of an equilateral triangle whose inscribed circle has radius 2?

 

 

⑴ answer: C

 

 

 

3. 

 

⑴ answer: D

 

 

 

4. Let V and W be 4-dimensional subspaces of a 7-dimensional vector space X. Which of the following CANNOT be the dimension of the subspace V ∩ W?

 

 

⑴ answer: A

⑵ V = span [{e₁, e₂, e₃, e₄}], W = span [{e₁, e₂, e₃, e₄}] → dim(V ∩ W) = 4

⑶ V = span [{e₁, e₂, e₃, e₄}], W = span [{e₁, e₂, e₃, e₇}] → dim(V ∩ W) = 3 

⑷ V = span [{e₁, e₂, e₃, e₄}], W = span [{e₁, e₂, e₆, e₇}] → dim(V ∩ W) = 2

⑸ V = span [{e₁, e₂, e₃, e₄}], W = span [{e₁, e₅, e₆, e₇}] → dim(V ∩ W) = 1

 

 

5. Sofia and Tess will each randomly choose one of the 10 integers from 1 to 10. What is the probability that neither integer chosen will be the square of the other?

 

 

⑴ answer: E

⑵ the number of cases that either integer chosen will be the square of the other : (1, 1), (2, 4), (3, 9), (4, 2), (9, 3)

⑶ the probability : 1 - (5 / 100) = 0.95  

 

 

6. Which of the following shows the numbers 2^1/2, 3^1/3, and 6^1/6 in increasing order?

 

 

⑴ answer: C

⑵ 6 < 2³ = 8 ⇔ 6^1/6 < 2^1/2

⑶ 2³ = 8 < 3² = 9 ⇔ 2^1/2 < 3^1/3 

 

 

7. The figure below shows the graph of the derivative f ' of a function f, where f is continuous on the interval [0, 4] and differentiable on the interval (0, 4). Which of the following gives the correct ordering of the values f (0), f (2), and f (4)?

 

 

⑴ answer: C

⑵ f (0) = f (2) - ∫ f ' dx (x = 0 to 2) < f (2) + ∫ f ' dx (x = 2 to 4) = f (4)

⑶ f (2) + ∫ f ' dx (x = 2 to 4) = f (4) < f (2)

 

 

8. Which of the following is NOT a group?

 

 

⑴ answer: B

⑵ group theory 

⑶ In case of (B), no inverse element always exists, so it is not a group.

 

 

9. Let g be a continuous real-valued function defined on ℝ with the following properties.

g '(0) = 0

g ''(-1) > 0

g ''(x) < 0 if 0 < x < 2.

Which of the following could be part of the graph of g?

 

 

⑴ answer: A

⑵ A : g '(0) = 0, g ''(-1) > 0 (convex), g ''(x) < 0 if 0 < x < 2

⑶ B : g '(0) = 0, g ''(-1) > 0 (convex), a point of inflection exists at 0 < x < 2

⑷ C : g '(0) > 0, g ''(-1) > 0 (convex), g ''(x) < 0 if 0 < x < 2

⑸ D : g '(0) = 0, g ''(-1) > 0 (convex), a point of inflection exists at 0 < x < 2

⑹ E : g '(0) = 0, g ''(-1) < 0 (concave), g ''(x) < 0 if 0 < x < 2 

 

 

10. In the xy-plane, the set of points whose coordinates satisfy the equation below is

 

 

⑴ answer: A

 

 

 

11. The region bounded by the curves y = x and y = x² in the first quadrant of the xy-plane is rotated about the y-axis. The volume of the resulting solid of revolution is

 

 

⑴ answer: B

 

 

 

12. For which integers n such that 3 ≤ n ≤ 11 is there only one group of order n (up to isomorphism)?

 

 

⑴ answer: B

⑵ There is only one group of order n if gcd(n, φ(n))=1, φ being Eular's totient function.

○ Euler's totient function

 

 

○ φ(3) = 3 × (1 - 1/3) = 2

○ φ(4) = 4 × (1 - 1/2) = 2

○ φ(5) = 5 × (1 - 1/5) = 4

○ φ(6) = 6 × (1 - 1/2) × (1 - 1/3) = 2

○ φ(7) = 7 × (1 - 1/7) = 6

○ φ(8) = 8 × (1 - 1/2) = 4

○ φ(9) = 9 × (1 - 1/3) = 6

○ φ(10) = 10 × (1 - 1/2) × (1 - 1/5) = 4

○ φ(11) = 11 × (1 - 1/11) = 10 

 

 

13. If f is a continuously differentiable real-valued function defined on the open interval (-1, 4) such that f (3) = 5 and f '(x) ≥ -1 for all x, what is the greatest possible value of f (0)?

 

 

⑴ answer: D

 

 

 

14. Suppose g is a continuous real-valued function such that

 

 

for each x ∈ ℝ, where c is a constant. What is the value of c?

 

 

⑴ answer: B

 

 

 

15. Let S, T, and U be nonempty sets, and let f : S → T and g : T → U be functions such that the function g ∘ f : S → U is one-to-one (injective). Which of the following must be true?

 

 

⑴ answer: A

⑵ if f is not one-to-one, g(f(x)) = g(f(y)) even if x ≠ y.

 

 

16. Suppose A, B, and C are statements such that C is true if exactly one of A and B is true. If C is false, which of the following statements must be true?

 

 

⑴ answer: B

⑵ XOR truth table

 

 

 

17. Which of the following equations has the greatest number of real solutions?

 

 

⑴ answer: B

⑵ A : one solution. The graph shows that there is one solution.

⑶ B : two solution. The root formula.

⑷ C : one solution

⑸ D : no solution. For x < 0, e^x > 0 > x. For x > 0, (e^x)' = e^x > e⁰ = 1.

⑹ E : one solution at x = 0

 

출처 : 이미지 클릭

 

 

18. Let f be the function defined by

 

 

for all x such that -1 < x < 1. Then f '(x) =

 

 

⑴ answer: A

 

 

 

19. If z is a complex variable and z̄ denotes the complex conjugate of z, what is

 

 

⑴ answer: E

 

 

 

20. Let g be the function defined by g(x) = e^2x+1 for all real x. Then

 

 

⑴ answer: E

⑵ the definition of derivation 

 

 

입력: 2023.01.14 16:34