PSAT Club

Math – Multiple Choice

General Hints

Look at the answer choices before you begin to work on each question.

Read each question carefully, even if it looks like a question you don't think you can answer. Don't let the form of the question keep you from trying to answer it.

If your answer isn't among the choices, try writing it in a different form. You may have the same answer in a different mathematical format.

Directions

Directions:Solve each problem. Then decide which is the best of the choices given and fill in the corresponding oval on the answer sheet.

Practice Question 1

If y = (x + 3)2, then (-2x - 6)2 must equal which of the following?

(A) -4y2(B) -2y2(C) -4y(D) 2y(E) 4y

Answer 1

If y = (x + 3)2, then (-2x - 6)2 must equal which of the following?

(A) -4y2 (B) -2y2 (C) -4y (D) 2y (E) 4y    CORRECT ANSWER Explanation:

The expression (-2x - 6)2 can be rewritten as [-2(x + 3)]2, which equals 4(x + 3]2. Since y = (x + 3)2, it follows that (-2x - 6)2 = 4(x + 3)2 = 4y. The correct answer is choice (E).

Practice Question 2

In the figure above, AD is a diameter of the circle with center O and AO = 5. What is the length of arc BCD ?

(A) (B) (C) (D) (E)

Answer 2 In the figure above, AD is a diameter of the circle with center O and AO

= 5. What is the length of arc BCD ? (A) (B) (C) (D)    CORRECT ANSWER (E)

Explanation: To solve this problem, it is helpful to draw segment OB in the figure.

Since OB and OD are both radii of the circle, they both equal 5. Therefore, the angles opposite these congruent sides of BOD are congruent and OBD = 36°. The third angle of the triangle, BOD, equals 180°- 36°- 36° = 108°. Arc BCD is a fraction of the circumference of the circle and more specifically equals , which equals The correct answer is choice (D).

Practice Question 3

If 0 < a  < b  < c  < d  < e  in the equation above, then the greatest increase in S  would result from adding 1 to the value of which variable?

(A) a(B) b(C) c(D) d(E) e

Answer 3 If 0 < a  < b  < c  < d  < e  in the equation above, then the

greatest increase in S  would result from adding 1 to the value of which variable?

(A) a    CORRECT ANSWER (B) b (C) c (D) d (E) e Explanation: When the denominator of a fraction is increased, the value of

the fraction decreases. Therefore, adding 1 go b, d, or e will decrease the sum S. Increasing one of the numerators, either a or c, will increase S. Adding 1 to a changes     to    , thereby increasing S by    . Adding 1 to c changes to , thereby increasing S by . Since b<d, then . Therefore, adding 1 to a will result in the greatest increase in S. The correct answer is (A).

Practice Question 4

If m and p are positive integers and (m + p) x m is even, which of the following must be true?

(A) If m  is odd, then p  is odd.(B) If m  is odd, then p  is even.(C) If m  is even, then p  is even.(D) If m  is even, then p  is odd.(E) m  must be even.

Answer 4

If m and p are positive integers and (m + p) x m is even, which of the following must be true?

(A) If m  is odd, then p  is odd.    CORRECT ANSWER (B) If m  is odd, then p  is even. (C) If m  is even, then p  is even. (D) If m  is even, then p  is odd. (E) m  must be even. Explanation: If m is even, then the expression (m + p) x m will always be

even and it cannot be determined whether p is even or odd. This eliminates choices (C) and (D). If m is odd, then (m + p) x m will be even only when m + p is even and m + p will be even only when p is odd. The correct answer is (A) since the truth of statement (A) also eliminates choices (B) and (E).

Practice Question 5

If xy  = 2 and xy 2 = 8, what is the value of x ?

(A) (B) 2(C) 4(D) 8(E) 16

Answer 5

If xy  = 2 and xy 2 = 8, what is the value of x ? (A)    CORRECT ANSWER (B) 2 (C) 4 (D) 8 (E) 16

Explanation: Substituting xy  = 2 into the equation xy2 = 8, you will

obtain (xy)y = 2y = 8, thus y = 4. To find x, substitute y = 4 into one of the two original equations to obtain x = . The answer to this problem is (A).

Practice Question 6

In the incomplete table above, the sum of the three integers in each row, column, and diagonal is the same. If the numerical values in four of the blocks are as shown, what is the value of w ?

(A) -6(B) -5(C)  2(D)  5(E)  8

Answer 6

In the incomplete table above, the sum of the three integers in each row, column, and diagonal is the same. If the numerical values in four of the blocks are as shown, what is the value of w ?

(A) -6 (B) -5 (C)  2    CORRECT ANSWER (D)  5 (E)  8 Explanation: Since the sum of the integers in each row, column, and diagonal is the

same, it follows that w  - 2 + a  = 3 + a  - 3. Thus w  - 2 = 0 so that w  = 2. The answer to this problem is (C).

Practice Question 7

If n  is an odd integer, which of the following must be an odd integer?

(A) n - 1(B) n + 1(C) 2n(D) 3n + 1(E) 4n + 1

Answer 7 If n  is an odd integer, which of the following must be an odd

integer? (A) n - 1 (B) n + 1 (C) 2n (D) 3n + 1 (E) 4n + 1    CORRECT ANSWER Explanation: If n is an odd integer, both one more and one less than n  will be

even integers, eliminating choices (A) and (B). Any even multiple of n  will be an even integer, eliminating choice (C). However, 4n  is even, making 4n +1 an odd integer. The answer to this problem is (E). Note that 3n  + 1 is even if n  is odd and it is odd if n  is even. Since the question asks, "Which of the following MUST be an odd integer," (D) cannot be the correct answer.