In the figure shown below, \(\overline{YZ}\) and \(\overline{MB}\) intersect at O and \(\overline{XO}\) is perpendicular to \(\overline{YZ}\). What is the value of 3p + 4s − 2t?

F. 15◦
G. 35◦
H. 55◦
J. 135◦
K. 150◦

Solution

Since t is the supplement of the 35 degree angle, t = 180 − 35, or 145°.

Because s is the supplement to t, s = 180 − 145, or 35°.

Since p is the complement of s, p = 90 − 35, or 55°.

Therefore, 3p + 4s − 2t = 3(55) + 4(35) − 2(145) = 165 + 140 − 290 = 15°.

On the cube in the figure shown below, each of the following points is the same distance from R as it is from S EXCEPT:

A. M
B. O
C. T
D. Q
E. N

Solution

The distance between R and N is the length of a diagonal of the cube, whereas the distance from S to N is the length of a side of the cube.

The length of the diagonal is longer than the length of the side.

Therefore, the distance from N to R is not the same as the distance as N to S.

Let x = 2y + 3z − 5. What happens to the value of x if the value of y decreases by 1 and the value of z increases by 2?

F. It decreases by 2
G. It is unchanged
H. It increases by 1
J. It increases by 2
K. It increases by 4

Solution

You are given that x = 2y + 3z − 5.

If the value of y decreases by 1, then the new value of y is y − 1; if the value of z increases by 2, then the new value of z is z + 2.

Substitute these values into the original equation to see the effects of the changes on the value of x:

x = 2(y − 1) + 3(z + 2) − 5

x = 2y − 2 + 3z + 6 − 5 = 2y + 3z − 1

Therefore, x now equals 2y + 3z − 1, which is 4 more than 2y + 3z − 5.

In the figure shown below, s⊥r and x > 90. Which of the following must be true?

A. s // t
B. r⊥t
C. y = 90
D. y > 90
E. y < 90

Solution

You are given that x>90, which means that y must be less than 90, because it is a complimentary angle to x.

The sum of complimentary angles is 180. None of the other answer choices must necessarily be true.

In the xy-coordinate system, if (r,s) and (r + 2, s + t) are two points on the line defined by the equation y = 4x + 5, then t = ?

F. 4
G. 5
H. 8
J. 9
K. 11

Solution

To solve this problem, you need to recognize that the slope of the line is equal to 4.

In the standard equation for a line, y = mx +b, m is equivalent to the slope.

The slope is equal to the change in y-values over the change in x-values; set up the following equation to solve for t:

slope = (s + t) − s/(r + 2) − r

4 = t/2

8 = t

A bag contains only quarters, dimes, and nickels. The probability of randomly selecting a quarter is 1/6. The probability of randomly selecting a nickel is 1/4. Which of the following could be the total number of coins in the bag?

A. 15
B. 24
C. 30
D. 32
E. 40

Solution

Since you cannot have a partial coin, the total number of coins in the bag must be divisible by both 6 and 4 (1/6 are quarters and 1/4 are nickels).

The only answer choice that is divisible by both 6 and 4 is 24.

If m, n, and p are positive integers such that m + n is even and the value of (m + n)² + n + p is odd, which of the following must be true?

F. m is odd
G. n is even
H. p is odd
J. If n is even, p is odd
K. If p is odd, n is odd

Solution

You are given that m + n is even and the value of (m + n)² is also even.

However, because (m + n)² + n + p is odd, the sum n + p must be odd.

A sum of two positive integers is odd only when one is even and one is odd.

Therefore, it must be true that if n is even, p is odd.

If a + b = 25 and a > 4, then which of the following must be true?

A. a = 22
B. b < 21
C. b > 4
D. b = 0
E. a < 25

Solution

The correct answer will be the statement that is always true.

Because a is greater than 4, and 25 − 4 = 21, b must always be less than 21.

Let the function g be defined by g(x) = 3(x²−2). When g(x) = 69, what is a possible value of 2x − 3?

F. −7
G. −5
H. 2
J. 5
K. 7

Solution

To solve this problem, first find the values of x for which g(x) = 69 by solving the equation 69 = 3(x² − 2), as follows:

69 = 3(x² − 2)

23 = x² − 2

25 = x²

x = 5 or x = −5

The possible values of 2x − 3 are therefore 7 or −13, of which 7 is the only available answer choice.

Let S be the set of all integers that can be written as 2n² − 6n, where n is a nonzero integer. Which of the following integers is in S?

A. 6
B. 30
C. 46
D. 64
E. 80

Solution

To solve this problem, start with n = 1 until you reach an answer choice: when you reach n = 8, you will get 2(8)² −6(8) = 80, answer choice E.

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