Psat Math Help

When a positive integer n is divided by 5, which of the following CANNOT be the remainder?

1. 1

2. 2

3. 3

4. 4

5. 5

Choice (E) is correct. When division is carried out within the set of integers, the result is a quotient and a remainder. Consider the following division results:

16 Ð"* 5 = 3 remainder 1,

17 Ð"* 5 = 3 remainder 2,

18 Ð"* 5 = 3 remainder 3,

19 Ð"* 5 = 3 remainder 4,

20 Ð"* 5 = 4 remainder 0,

21 Ð"* 5 = 4 remainder 1,

and so on.

When you divide by 5, the remainder can be 0, 1, 2, 3, or 4, but not 5, because the remainder has to be less than the divisor.

In the figure above, if ACDE is a square, what is the area of polygon ABCDE?

1. 22

2. 27

3. 31

4. 37

5. 45

Choice (C) is correct. Right triangle ABC has AB = 3 and BC = 4. The area of this triangle is . By the Pythagorean theorem, the length of the hypotenuse, , is 5 since AC2 = 32 + 42 = 9 + 16 = 25. Since AC = 5, the length of each side of the square is 5, and the area of the square is 52, or 25. The total area of ABCDE is thus 6 + 25 = 31.

If 3 is subtracted from the square root of x, the result is 8. What is the value of x Ð'- 1 ?

1. 10

2. 24

3. 60

4. 66

5. 120

Choice (E) is correct. Translating the information about x gives the equation Ð'- 3 = 8. From this, it follows that = 11, and therefore, x = 112 = 121. The value of x Ð'- 1 is 121 Ð'- 1 = 120.

The number 0.123 is between and for some positive integer n. What is the value of n ?

1. 1

2. 12

3. 13

4. 120

5. 130

Choice (B) is correct. The comparison of 0.123 with and is easier if all three numbers are in the same form, so rewriting 0.123 as will help you see the comparison more clearly. Since < < , it follows that 12.3 is between n and n + 1. The only integer n that satisfies this condition is 12, because 12 < 12.3 < 12 + 1.

Which of the following could be the lengths of the sides of a triangle?

1. 2, 3, and 1

2. 3, 6, and 9

3. 3, 10, and 6

4. 4, 7, and 2

5. 7, 8, and 9

Choice (E) is correct. An important theorem about triangles is the triangle inequality theorem; this theorem states that in any triangle, the sum of the lengths of any two of the sides must be greater than the length of the third side. For example, 2, 3, and 1 cannot be the lengths of the sides of a triangle, since 2 + 1 is not greater than 3. You can try to construct a triangle with sides of lengths 2, 3, and 1, and you will see that it collapses into a line segment. The same thing happens if you try to construct a triangle with sides of lengths 3, 6, and 9. The lengths collapse into a line segment. The numbers 3, 10, and 6 cannot be the lengths of the sides of a triangle, according to the triangle inequality, since 3 + 6 < 10. Likewise, 4, 7, and 2 cannot be the lengths of the sides of a triangle since 4 + 2 < 7.

Among the answer choices, the only set of numbers that obeys the triangle inequality is the set 7, 8, and 9.

In the xy-plane, the line with equation 2x + y = 3 is perpendicular to the line with equation y = mx + b, where m and b are constants. What is the value of m?

1. Ð'-2

2.

3. 0

4.

5. 2

Choice (D) is correct. In the xy-plane, if you have two lines that are not parallel to either the x-axis or the y-axis and the two lines are perpendicular, then their slopes must be negative reciprocals of each other. That is, the product of the slopes must be Ð'-1. The equation 2x + y = 3 can be rewritten as y = Ð'-2x