Question 1

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: Factor the equation to . Set both equal to and you get and .

Question 2

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: The factorial sign (!) just tells us to multiply that number by every integer that leads up to it. So, can also be written as: To make this easier for ourselves, we can cancel out the numbers that appear on both the top and bottom:

Question 3

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: A geometric sequence is one in which the next term is found by mutlplying the previous term by a particular constant. Thus, we look for an implicit definition which involves multiplication of the previous term. The only possibility is:

Question 4

1. (correct answer)
2. 
3. 
4. 

Explanation:

How do you find the area of a hexagon?

1. In a regular hexagon, split the figure into triangles.
2. Find the area of one triangle.
3. Multiply this value by six.

Regular hexagons:

In this figure, the center point, , is equidistant from all of the vertices. As a result, the six dotted lines within the hexagon are the same length. Likewise, all of the triangles within the hexagon are congruent by the side-side-side rule: each of the triangle's share two sides inside the hexagon as well as a base side that makes up the perimeter of the hexagon. In a similar fashion, each of the triangles have the same angles. There are in a circle and the hexagon in our image has separated it into six equal parts; therefore, we can write the following:

Now, let's look at each of the triangles in the hexagon. We know that each triangle has two two sides that are equal; therefore, each of the base angles of each triangle must be the same. We know that a triangle has and we can solve for the two base angles of each triangle using this information.

Each angle in the triangle equals . We now know that all the triangles are congruent and equilateral: each triangle has three equal side lengths and three equal angles. Now, we can use this vital information to solve for the hexagon's area. If we find the area of one of the triangles, then we can multiply it by six in order to calculate the area of the entire figure. Let's start by analyzing . If we draw, an altitude through the triangle, then we find that we create two triangles.

Let's solve for the length of this triangle. Remember that in triangles, triangles possess side lengths in the following ratio:

Now, we can analyze using the a substitute variable for side length, .

We know the measure of both the base and height of and we can solve for its area.

We have solved for the area of a regular hexagon with side length, . If we know the side length of a regular hexagon, then we can solve for the area.

If we are not given a regular hexagon, then we an solve for the area of the hexagon by using the side length(i.e. ) and apothem (i.e. ), which is the length of a line drawn from the center of the polygon to the right angle of any side. This is denoted by the variable in the following figure:

Alternative method:

If we are given the variables and , then we can solve for the area of the hexagon through the following formula:

In this equation, is the area, is the perimeter, and is the apothem. We must calculate the perimeter using the side length and the equation , where is the side length.

Solution:

In a hexagon the number of sides is and in this example the side length is .

The perimeter is .

The area is .

Question 5

1. (correct answer)
2. 
3. 
4. 

Explanation: To find the perimeter of a square you must multiply the side length by . To do this we plug the side length, , into the equation Then multiply the side length by , The answer is .

Question 6

1. (correct answer)
2. 
3. 
4. The limit does not exist.
5. 

Explanation: This is a graph of . We know that is undefined; therefore, there is no value for . But as we take a look at the graph, we can see that as approaches 0 from the left, approaches negative infinity. This can be illustrated by thinking of small negative numbers. NOTE: Pay attention to one-sided limit specifications, as it is easy to pick the wrong answer choice if you're not careful. is actually infinity, not negative infinity.

Question 7

1. Median (correct answer)
2. Mean
3. Mode
4. Range
5. Interquartile range

Explanation: A box and whisker plot separates the data into quartiles so that each quartile has an equal number of data points. The box indicates the interquartile range, that is, the top line of the box is the third quartile and the bottom line of the box is the second quartile. The line separating the second and third quartiles indicates the median. The lines outside of the box indicate the outer-quartiles (first and fourth).

Question 8

1. (correct answer)
2. 
3. 
4. 
5. No triangle can exist with these characteristics.

Explanation: Since we are given , , and , and want to find , we apply the Law of Sines, which states, in part, . Substitute and solve for : Take the inverse sine of 0.6355: There are two angles between and that have any given positive sine other than 1 - we get the other by subtracting the previous result from : This, however, is impossible, since this would result in the sum of the triangle measures being greater than . This leaves as the only possible answer.

Question 9

1. (correct answer)
2. 
3. 
4. 

Explanation: The perimeter of a polygon is found by calculating the sum of all of the side lengths. In this instance, the polygon is regular, so the perimeter can be found by mulitplying the length of one side by the total number of sides:

Question 10

1. Vertical line test (correct answer)
2. Horizontal line test
3. Calculating zeroes
4. Calculating domain and range

Explanation: The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one (or ) value for each value of . The vertical line test determines how many (or ) values are present for each value of . If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function. The horizontal line test can be used to determine if a function is one-to-one, that is, if only one value exists for each (or ) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function. Example of a function: Example of an equation that is not a function:

Question 11

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: The range is the simplest measurement of the difference between values in a data set. To find the range, simply subtract the lowest value from the greatest value, ignoring the others.

Question 12

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: Change the left side to and the right side to so that both sides have the same base. Apply log and then set the exponential expressions equal to each other (). Thus, .

Question 13

1. (correct answer)
2. 
3. 

Explanation: An arithmetic sequence is one in which the common difference is added to one term to get the next term. Thus, if the first term is 3, we add 5 to get the second term, and continue in this manner. Thus, the first four terms are:

Question 14

1. (correct answer)
2. 
3. 
4. 

Explanation: To find the perimeter of a rectangle you must find the sum of all of the side lengths. To do this, we plug the base and height into the equation . With a base of and a height of our equation becomes . Multiply and add the numbers together: The answer is .

Question 15

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: .

Question 16

1. 0 (correct answer)
2. -1
3. -1/2
4. 1/2
5. 1

Explanation: The domain of a function includes all of the values of x for which f(x) is real and defined. In other words, if we put a value of x into the function, and we get a result that isn't real or is undefined, then that value won't be in the domain. If we let x = 0, then we will be forced to evaluate , which is equal to 1/0. The value of 1/0 is not defined, because we can never have zero in a denominator. Thus , because f(0) isn't defined, 0 cannot be in the domain of f(x). The answer is 0.

Question 17

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: Add up the 's and divide in half, which results in 5. Do the same to the 's and you get 6. Put the and in an ordered pair so that your answer is (5, 6).

Question 18

1. 243 (correct answer)
2. 2.41
3. 9
4. 3
5. 27

Explanation: An exponent written as a fraction can be rewritten using roots. can be reqritten as . The bottom number on the fraction becomes the root, and the top becomes the exponent you raise the number to. is the same as . This will give us the answer of 243.

Question 19

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: Multiplying top and bottom by the complex conjugate eliminates i from the denominator

Question 20

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: . However, cannot be simplified any further because the terms have different exponents. (Like terms are terms that have the same variables with the same exponents. Only like terms can be combined together.)

Question 21

1. 12 centimeters
2. 15 centimeters (correct answer)
3. 18 centimeters
4. 24 centimeters
5. 9 centimeters

Explanation: The length and width of the rectangle are in a ratio of 3:4, so the sides can be written as 3_x_ and 4_x_. We also know the area, so we write an equation and solve for x: (3_x_)(4_x_) = 12_x_2 = 108. x2 = 9 x = 3 Now we can recalculate the length and the width: length = 3x = 3(3) = 9 centimeters width = 4x = 4(3) = 12 centimeters Using the Pythagorean Theorem we can find the diagonal, c: length2 + width2 = c2 92 + 122 = _c_2 81 + 144 = c2 225 = c2 c = 15 centimeters

Question 22

1. (correct answer)
2. 
3. 
4. 

Explanation: When multiplying polynomials you add the powers of each like-termed polynomial together to find the answer. In this example the powers are and which add to . Therefore our answer is

Question 23

1. (correct answer)
2. 
3. There is no horizontal asymptote.
4. 
5. 

Explanation: Look at the exponents of the variables. Both our numerator and denominator are . Therefore the horizontal asymptote is calculated by dividing the coefficient of the numerator by the coefficient of the denomenator.

Question 24

1. (correct answer)
2. 
3. 
4. 

Explanation: To find the area of a circle you must plug the radius into in the following equation. In this case, the radius is , so we plug into .

Question 25

1. (correct answer)
2. 
3. 
4. 
5. 

Explanation: In order to find the surface area of a right cylinder you must find the area of both bases (the circles on either end) and add them to the lateral surface area. The area of the two circles is easy to find with but remember to multiply by 2 for both bases . Next find the lateral area. The lateral area if unrounded would be a rectangle with height of 10 m and length equal to the circumference of the base circles. Thus the lateral area is Now add the lateral area to the area of the two bases: