Math

Math Practice Test: Practice Test 2

Practice Test 2 for Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

The center of a circle is and its radius is . Which of the following could be the equation of the circle?

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Question 1

The center of a circle is and its radius is . Which of the following could be the equation of the circle?

(correct answer)

Explanation: The general equation of a circle is , where the center of the circle is and the radius is . Thus, we plug the values given into the above equation to get .

Question 2

What is the perimeter of a rectangle with a base of and a height of ?

(correct answer)

Explanation: To find the perimeter of a rectangle, find the sum of all of the side lengths using the equation: Substitute the given values for base and height:

Question 3

What is the y-intercept of the line ?

(correct answer)

Explanation: The y-intercept is the point at which the line intersects the y-axis. It does this at . We plug in for in our equation, , to give us . Anything multiplied by is , so . Our coordinates for the y-intercept are .

Question 4

Which analysis can be performed to determine if an equation is a function?

Vertical line test (correct answer)
Horizontal line test
Calculating zeroes
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 5

What is the diameter of a sphere with a volume of ?

(correct answer)

Explanation: To find the diameter of a sphere we must use the equation for the volume of a sphere to find the radius which is half of the diameter. The equation is First we enter the volume into the equation yielding We then divide each side by to get We then multiply each side by to get We then take the cubic root of each side to solve for the radius The radius is We then multiply the radius by 2 to find the diameter The answer for the diameter is .

Question 6

Stewie has marbles in a bag. How many marbles does Stewie have?

(correct answer)

Explanation: Simplifying this equation we notice that the 3's, 2's, and 1's cancel so Alternative Solution

Question 7

Which of the following does NOT belong to the domain of the function $f(x)=\frac{x^{-1}+\sqrt{1-x}}{4x^2+1}$ ?

0 (correct answer)
-1
-1/2
1/2
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 $0^{-1}$ , 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 8

What is the value of ?

243 (correct answer)
2.41
9
3
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 9

Simplify:

(correct answer)

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 10

The length and width of a rectangle are in the ratio of 3:4. If the rectangle has an area of 108 square centimeters, what is the length of the diagonal?

12 centimeters
15 centimeters (correct answer)
18 centimeters
24 centimeters
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 11

What is the horizontal asymptote of this equation?

(correct answer)
There is no horizontal asymptote.

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 12

Solve for and .

(correct answer)
Cannot be determined.

Explanation: 1st equation: 2nd equation: Subtract the 2nd equation from the 1st equation to eliminate the "2y" from both equations and get an answer for x: Plug the value of into either equation and solve for :

Question 13

Find the height of a triangle if the area of the triangle = 18 and the base = 4.

9 (correct answer)
6
1
4

Explanation: The area of a triangle = (1/2)bh where b is base and h is height. 18 = (1/2)4h which gives us 36 = 4h so h =9.

Question 14

What is the area of a triangle with a base of and a height of ?

(correct answer)

Explanation: The formula for the area of a triangle is . Plug in our given values.

Question 15

What is the domain of the function below:

**
(correct answer)

Explanation: The domain is defined as the set of possible values for the x variable. In order to find the impossible values of x, we should: a) Set the equation under the radical equal to zero and look for probable x values that make the expression inside the radical negative: There is no real value for x that will fit this equation, because any real value square is a positive number i.e. cannot be a negative number. b) Set the denominator of the fractional function equal to zero and look for probable x values: Now we can solve the equation for x: There is no real value for x that will fit this equation. The radical is always positive and denominator is never equal to zero, so the f(x) is defined for all real values of x. That means the set of all real numbers is the domain of the f(x) and the correct answer is . Alternative solution for the second part of the solution: After figuring out that the expression under the radical is always positive (part a), we can solve the radical and therefore denominator for the least possible value (minimum value). Setting the x value equal to zero will give the minimum possible value for the denominator. That means the denominator will always be a positive value greater than 1/2; thus it cannot be equal to zero by setting any real value for x. Therefore the set of all real numbers is the domain of the f(x).

Question 16

Find the equation (in slope-intercept form) of a line perpendicular to .

(correct answer)

Explanation: First, find the slope of the original line, which is . You can do this by isolating for so that the equation is in slope-intercept form. Once you find the slope, just replace the in the original equation withe the negative reciprocal (perpendicular lines have a negative reciprocal slope for each other). Thus, your answer is

Question 17

In a standard deck of cards, with replacement, what is the probability of drawing two Ace of Hearts?

(correct answer)

Explanation: Probability of drawing first ace of hearts: Probability of drawing second ace of hearts: Multiply both probabilities by each other:

Question 18

The area of a particular rectangle is . If the length of the rectangle is twice the width, what is the width of the rectangle?

(correct answer)

Explanation: The forumla for the area of a rectangle is Length X Width. Here, that would be Width X 2Width (length). Solving for width, you get . Dividing by 2 and taking the square root of each side gives us . Thus, the width of the rectangle is units.

Question 19

Solve the following equation for .

(correct answer)

Explanation: The first step in solving this equation is to distribute the 2 through the parentheses. This gives us: Next, we subtract 6 from both sides, in order to get the variable alone on one side of the equation: Finally, we divide both sides by 2 to solve for :

Question 20

Two rectangles are similar. The perimeter of the first rectangle is 36. The perimeter of the second is 12. If the base length of the second rectangle is 4, what is the height of the first rectangle?

8
2
4
6 (correct answer)
10

Explanation: Solve for the height of the second rectangle. Perimeter = 2B + 2H 12 = 2(4) + 2H 12 = 8 + 2H 4 = 2H H = 2 If they are similar, then the base and height are proportionally equal. B1/H1 = B2/H2 4/2 = B2/H2 2 = B2/H2 B2 = 2H2 Use perimeter equation then solve for H: Perimeter = 2B + 2H 36 = 2 B2 + 2 H2 36 = 2 (2H2) + 2 H2 36 = 4H2 + 2 H2 36 = 6H2 H2 = 6

Question 21

Find the area of the following rhombus:

Screen_shot_2014-03-01_at_8.57.04_pm

The perimeter of the rhombus is .

(correct answer)

Explanation: The formula for the perimeter of a rhombus is: Where is the length of the side Plugging in our values, we get: The formula for the area of a rhombus is: Where is the length of one diagonal and is the length of another diagonal By drawing the diagonals, we create a right triangle with the hypotenuse as and the side as . Since we know that is a phythagorean triple, we can infer that the third side is . Plugging in our values, we get:

Question 22

Function

The above graph depicts a function . Does exist, and why or why not?

does not exist because . (correct answer)
does not exist because
does not exist because
exists because
exists because

Explanation: exists if and only if . As can be seen from the diagram, , but . Since , does not exist.

Question 23

What is the area of a rectangle with width and length ?

(correct answer)

Explanation: The area of a rectangle is found by multiplying the length times the width.

Question 24

The fourth term in an arithmetic sequence is -20, and the eighth term is -10. What is the hundredth term in the sequence?

220 (correct answer)
110
55
210
105

Explanation: An arithmetic sequence is one in which there is a common difference between consecutive terms. For example, the sequence {2, 5, 8, 11} is an arithmetic sequence, because each term can be found by adding three to the term before it. Let denote the nth term of the sequence. Then the following formula can be used for arithmetic sequences in general: , where d is the common difference between two consecutive terms. We are given the 4th and 8th terms in the sequence, so we can write the following equations: . We now have a system of two equations with two unknowns: Let us solve this system by subtracting the equation from the equation . The result of this subtraction is . This means that d = 2.5. Using the equation , we can find the first term of the sequence. Ultimately, we are asked to find the hundredth term of the sequence. The answer is 220.