[分享] Sample PSAT/NMSQT Questions---The Math Sect

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Correct Answer: A
Explanation:

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question 10 of 13

If a and b are integers greater than 100 such that a + b = 300, which of the following could be the exact ratio of a to b ?


(A) 9 to 1
(B) 5 to 2
(C) 5 to 3
(D) 4 to 1
(E) 3 to 2
(from the October 12, 1999 test)

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Correct Answer: E
Explanation: To solve this question, you need to look at the answer choices. For any of the answer choices to be the ratio of a to b, some multiple of the sum of the two numbers must evenly divide 300. For example, if the ratio of a to b equaled 9 to 1, then a would equal 9x and b would equal x for some number x. Furthermore, 9x + x would have to equal 300. This is possible since 10x = 300 yields an integer solution, namely x = 30. However, if x = 30, then a would equal 270 and b would equal 30. Although the sum of these numbers equals 300, they do not satisfy the other condition in the problem. That is, both of these numbers are not greater than 100. Therefore, choice (A) can be eliminated.

Answer choices (B) and (C) can be eliminated since neither the sum of the two numbers in (B) nor the sum of the two numbers in (C) evenly divided 300. (5x + 2x = 300 does not yield an integer solution, nor does 5x + 3x = 300.)

Although answer choices (D) and (E) are possible ratios of a to b (both 4x + x = 300 and 3x + 2x = 300 yield integer solutions), (D) results in a = 240 and b = 60 and can be eliminated since 60 is not greater than 100.

Only choice (E) gives a correct ratio of a to b that satisfies all of the conditions in the problem. For (E), a = 180 and b = 120, and both integers are greater than 100.

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question 11 of 13


In the figure above, a square is inscribed in a circle with diameter d. What is the sum of the areas of the shaded regions, in terms of d ?

(A)  
(B)  
(C)  
(D)  
(E)  
(from the October 12, 1999 test)

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如下

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(B)

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(A)

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(C)

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(D)

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(E)

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Explanation: The sum of the areas of the shaded regions is the area of the circle minus the area of the square. The area of the circle is pi*r^2 where r=d/2 Therefore, this area is pi*(d/4)^2. To find the area of the square, you need the length of one of the sides. Since the diagonal of the square is d, each side equals . Therefore, the area of the square is . The sum of the areas of the shaded regions is  which equals . The correct answer is (A).

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question 12 of 13


Two seniors, Abby and Ben, and two juniors, Cathy and Dave, are to be assigned to the 3 lockers shown above according to the following rules.

All 3 lockers are to be assigned.
Abby and Ben cannot share a locker with each other.
A senior cannot share a locker with a junior.
The locker assignments of all four students can be determined from the assignments of which of the following pairs?

I. Abby and Ben
II. Ben and Cathy
III. Cathy and Dave


(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
(from the October 16, 1999 test)

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Correct Answer: D
Explanation: Since the students are to be assigned to the lockers shown, each "assignment" is the pairing of a student with a specific locker (locker #46, #47, or #48), not the pairing of a student with another student. The conditions of the problem allow you to deduce which students will share a locker, but they are not enough to allow you to deduce the specific locker assignments. For example, knowing that Cathy and Dave will share a locker does not tell you to which locker they will be assigned.

First consider what knowing the assignments of Abby and Ben will tell you about the locker assignments of the remaining two students. Since Abby and Ben are seniors and they cannot share a locker with each other or with any juniors, you know that Cathy and Dave must share the third locker. Since you know the specific locker assignments of all four students, (I) is correct.

If you know the assignments of Ben and Cathy, you know that Abby is in the third locker and Dave must share Cathy’s locker. Therefore, (II) is correct.

If you know the assignments of Cathy and Dave (they must share the same locker), you only know to whom one of the lockers is assigned. You will not know specifically to which lockers Abby and Ben are assigned -- you will only know that they do not share a locker. The correct answer is choice (D).

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question 13 of 13

For positive integers a and b, let a ^ b be defined as ab +1. If x and y are positive integers and x ^ y = 16, which of the following could be a value of y?
I. 1
II. 2
III. 3


(A) I only
(B) II only
(C) I and III only
(D) II and III only
(E) I, II, and III
(from the October 16, 1999 test)

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Correct Answer: C
Explanation: For this question, you are given that x ^ y = 16 where x^  y is defined as xy +1. You are asked which of three values are possible for y when xy +1 = 16.

The value of y could be 1 if x = 4, since 41+1 = 42 = 16. So I is correct. The value of y could be 3 if x = 2, since 23+1 = 24 = 16. So III is correct. Since there is no integer that can be raised to the (2 + 1) or 3rd power to obtain 16, II is not correct. The correct answer is (C).