The length of arc XY of a circle is equal to ¹⁄₆ of the circumference of the circle. The length of the arc is 7π inches. What is the radius, in inches, of the circle?

F. 42
G. 21
H. 14
J. 7
K. 3

Solution

To solve this problem, you must remember that the formula for the circumference of a circle is C = 2πr, where r is the radius.

You are given that the length of arc XY is 7π, and that this is 1/6 of the total circumference.

Therefore, the circumference must be 6 × 7π = 42π.

Substitute this value for C in the equation and solve for r, as follows:

42π = 2πr

21π = πr

21 = r

On the number line shown below, t, u, v, w, x, y, and z are coordinates of the indicated points. Which of the following is closest in value to |w − u|?

A. t
B. v
C. x
D. y
E. z

Solution

The absolute value of any number is positive.

Eliminate answer choice A because it is negative.

Since w is close to 1.75 on the number line shown, and u is close to 0.6 on the number line shown, the difference of w and u is close to 1.15 on the number line shown. This corresponds to v.

The greatest integer of a set of consecutive even integers is 12. If the sum of these integers is 40, how many integers are in this set?

F. 5
G. 6
H. 12
J. 20
K. 40

Solution

One quick way to solve this problem is to start with 12 and add each preceding even integer until you get to a sum of 40, and then count the terms:

12 + 10 + 8 + 6 + 4 = 40.

There are 5 terms in the set.

The average of a and b is 6 and the average of a, b, and c is 11. What is the value of c?

A. 21
B. 17
C. 13
D. 8
E. 5

Solution

To solve this problem, first recognize that $\frac{a + b}{2}$ = 6, which means that a + b = 12.

Also, $\frac{a + b + c}{3}$ = 11, which means that a + b + c = 33.

Therefore, 12 + c = 33, so c must equal 33 − 12, or 21.

In the figure below, all line segments are either horizontal or vertical and the dimensions are given in feet. What is the perimeter in feet, of the figure?

F. 16
G. 14
H. 13
J. 12
K. 10

Solution

To solve this problem, you will need to find the two missing dimensions.

Sketch the figure as shown below to help visualize the missing dimensions:

You can see that a must be 1 and b must be 3.

Therefore, the perimeter is 4+3+2+1+1+3 = 14.

What is the matrix product $\begin{bmatrix} 2x\\ 3x\\ 5x \end{bmatrix}$ [1, 0, −1]?

A. $\begin{bmatrix} 2x &0 &−2x \\ 3x &0 &−3x \\ 5x &0 &−5x \end{bmatrix}$
B. $\begin{bmatrix} 2x &0 &−2x \\ 0 &0 &−3x \\ 10x &0 &−10x \end{bmatrix}$
C. [2x 3x 5x]
D. [9x 0 −9x]
E. [0]

Solution

To find the matrix product, simply multiply the terms from a row in the first matrix by the corresponding term from a column in the second matrix, as shown next:

2x × 1 = 2x 2x × 0 = 0 2x × −1 = –2x

3x × 1 = 3x 3x × 0 = 0 3x × −1 = –3x

5x × 1 = 5x 2x × 0 = 0 5x × −1 = –5x

In a certain budget, 30% of the money goes toward housing costs, and, of that portion, 20% goes toward rent. If the amount of money that goes toward rent is $630, what is the total amount of the budget?

F. $1,680
G. $2,100
H. $4,095
J. $7,560
K. $10,500

Solution

To solve this problem, first set the total amount of the budget equal to b. Housing costs are equal to 30 percent of b, or .3b. Of this portion, 20 percent, or .2, goes toward rent.

Therefore, rent, r, is equal to .2(.3b), or .06b. You are given that rent, r, equals $630, so now you can set up an equation and solve for b, as follows:

.06b = 630

b = 630/.06

b = 10, 500

In ABC, AB ≅ AC and the measure of ∠B is 34°. What is the measure of ∠A ?

A. 34°
B. 56°
C. 68°
D. 73°
E. 112°

Solution

A good way to solve this problem is to sketch triangle ABC, as shown below:

You are given that AB is congruent to AC, and that the measure of angle B is 34°. This means that the measure of angle C is also 34°, and the measure of angle A is 180° − 34° − 34° = 112°.

In the figure below, LMNO is a trapezoid, P lies on LO, and angle measures are as marked. What is the measure of angle MON?

F. 15°
G. 25°
H. 30°
J. 35°
K. 45°

Solution

To solve this problem, recognize that MN is parallel to LO and that these lines are cut by the transversal MO.

Therefore, angles OMN and MOL are alternate interior angles and they have the same measure, 30°.

Now, because a straight line has 180°, the measure of angle MON must be 180° − 30° − 105° = 45°.

What is the distance in the standard (x, y) coordinate plane between the points (5,5) and (1,0)?

A. $\sqrt{26}$
B. $\sqrt{41}$
C. 4
D. 6
E. 16

Solution

To solve this problem you could either use the distance formula, or sketch a quick figure like the one shown below:

You can see that the distance between the points is the hypotenuse of the right triangle with side lengths of 5 and 4.

Use the Pythagorean theorem to solve for the hypotenuse, c:

c² = 5² + 4² = 41

c = $\sqrt{41}$

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