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Mathematics Test - 2 - SAT MCQ


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30 Questions MCQ Test - Mathematics Test - 2

Mathematics Test - 2 for SAT 2024 is part of SAT preparation. The Mathematics Test - 2 questions and answers have been prepared according to the SAT exam syllabus.The Mathematics Test - 2 MCQs are made for SAT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Mathematics Test - 2 below.
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Mathematics Test - 2 - Question 1

If m3 =where n > 0, what is t he value of m in terms of n ?

Detailed Solution for Mathematics Test - 2 - Question 1

For this question, we need to know two Laws of Exponentials from Chapter 9: Law #8 and Law #9. First, we use Law #9 to translate the radicals into exponents.
Given equation:
Apply Law of Exponentials #9:
Apply Law of Exponentials #9 again: 
Apply Law of Exponentials #8: m3 = n1/4
Raise to the 1/3 power: 
Apply Law of Exponentials #8 again: m = n1/12

Mathematics Test - 2 - Question 2

How many solutions to the equation 4 cos x = 1 lie between x = 0 and x = 3π

Detailed Solution for Mathematics Test - 2 - Question 2

In order to solve this without a calculator, we need to know how to analyze this problem in terms of the unit circle. First, let’s solve for cos x: 4 cos x = 1
Divide by 4: cos x = 1/4
the cosine of any angle corresponds to the x-coordinate of the corresponding point for that angle on the unit circle:

Notice that there are exactly two points on the unit circle that have an x-coordinate of 1/4. Now let’s think about the angle. We are told that x goes from 0 to 3π. Remember that a full trip around the circle is 2π radians; therefore, a journey from x = 0 to x = 3π is 1.5 trips around the circle counterclockwise starting from the positive x-axis. If you trace with your finger 1.5 times around the circle starting from the point (1, 0), you’ll hit our “points of interest” exactly three times.

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*Answer can only contain numeric values
Mathematics Test - 2 - Question 3

What is the smallest positive integer value of x such that is less than 1?


Detailed Solution for Mathematics Test - 2 - Question 3

Given inequality: 
Multiply by 2x: 12 + 1 < 2x
Simplify: 13 < 2x
Divide by 2: 6.5 < x
The smallest integer that is greater than 6.5 is 7.

Mathematics Test - 2 - Question 4


In the figure above, rectangle ABCD is inscribed in the circle with center O. What is the area of the circle?

Detailed Solution for Mathematics Test - 2 - Question 4


Since ABCD is a rectangle, we can find the length of its diagonal using the Pythagorean Theorem: 102 + 242 = d2. Even better, we can notice that the two legs are in a 5:12 ratio, and therefore triangle BCD is a 5-12-13 triangle. In either case, we find that DB = 26. Since DB is also a diameter of the circle, the radius of the circle is 26/2 = 13, and therefore, the area of the circle is πr2 = π(13)2 = 169π.

Mathematics Test - 2 - Question 5

In the figure above, AB = BC. If has a slope of m andhas a slope of n, what is the value of mn ?

Detailed Solution for Mathematics Test - 2 - Question 5

If AB = BC, then triangle ABC is isosceles and therefore the two base angles are congruent and the triangle has a vertical axis of symmetry at the line x = 3. This implies that the slopes of linesandare opposites. We can calculate the slope of BC from its endpoints:

Therefore, the slope of  is 3, and so mn = (3)(-3) = -9.

Mathematics Test - 2 - Question 6

A total of 300 tickets were sold for a performance of a school play. The ticket prices were $5 for each adult and $3 for each child, and the total revenue from tickets was $1,400. Solving which of the following systems of equations would yield the number of adult tickets sold, a, and the number of children's tickets sold, c?

Detailed Solution for Mathematics Test - 2 - Question 6

Let a = # of adult tickets sold, and c = # of child tickets sold. If 300 tickets were sold altogether: c + a = 300
The revenue for a adult tickets sold at $5 each is $5a, and the revenue for c child tickets sold at $3 each is $3c. Since the total revenue is $1,400: 5a + 3c = 1,400

Mathematics Test - 2 - Question 7

Which of the following equations represents a parabola in the xy-plane with a vertex that lies on the x-axis?

Detailed Solution for Mathematics Test - 2 - Question 7

The general equation of a parabola in the xy-plane is y = a(x - h)2 + k, in which (h, k) is the vertex. Now let's express each choice in precisely this form.
(A) y = (x - 3)2 + 2 y = 1(x - 3)2 + 2 a = 1, h = 3, k = 2
(B) y = 2(x - 3)2 y = 2(x - 3)2 + 0 a = 2, h = 3, k = 0
(C) y = 2x2 - 3 y = 2(x - 0)2 - 3 a = 2, h = 0, k = -3
(D) y = 3x2 + 2 y = 3(x - 0)2 + 2 a = 3, h = 0, k = 2
If this vertex is on the x-axis, then k = 0. The only equation in which k = 0 is (B).

Mathematics Test - 2 - Question 8

What number is the same percent of 225 as 9 is of 25?

Detailed Solution for Mathematics Test - 2 - Question 8

Set up a proportion: 9/25 = x/225
Cross multiply: 2,025 = 25x
Divide by 25: 81 = x

Mathematics Test - 2 - Question 9


The figure above shows a polygon with five sides. What is the average (arithmetic mean) of the measures, in degrees, of the five angles shown?

Detailed Solution for Mathematics Test - 2 - Question 9

The sum of the measures if the interior angles of any polygon is (n - 2)180°, where n is the number of sides in the polygon. Since this is a 5-sided polygon, the sum of its interior angles is (5 - 2)(180°) = 3(180°) = 540°. Therefore the average of these measures is 540°/5 = 108°.

Mathematics Test - 2 - Question 10

The number of states that joined the United States between 1776 and 1849 is twice the number of states that joined between 1850 and 1900. If 30 states joined the United States between 1776 and 1849 and x states joined between 1850 and 1900, which of the following equations is true?

Detailed Solution for Mathematics Test - 2 - Question 10

Choice B is correct. To fit the scenario described, 30 must be twice as large as x. This can be written as 2x = 30.
Choices A, C, and D are incorrect. These equations do not correctly relate the numbers and variables described in the stem. For example, the expression in choice C states that 30 is half as large as x, not twice as large as x.

Mathematics Test - 2 - Question 11

kx − 3y = 4
4x − 5y = 7
In the system of equations above, k is a constant and x and y are variables. For what value of k will the system of equations have no solution?

Detailed Solution for Mathematics Test - 2 - Question 11

Choice A is correct. If a system of two linear equations has no solution, then the lines represented by the equations in the coordinate plane are parallel. The equation kx − 3y = 4 can be rewritten as y =where k/3 is the slope of the line, and the equation 4x − 5y = 7 can be rewritten as y = where 4/5 is the slope of the line. If two lines are parallel, then the slopes of the line are equal. Therefore, 4/5 = k/3, or k = 12/5. (Since the y-intercepts of the lines represented by the equations are and the lines are parallel, not identical.)
Choices B, C, and D are incorrect and may be the result of a computational error when rewriting the equations or solving the equation representing the equality of the slopes for k.

Mathematics Test - 2 - Question 12

What are the solutions to 3x2 + 12x + 6 = 0 ?

Detailed Solution for Mathematics Test - 2 - Question 12

Choice A is correct. Dividing each side of the given equation by 3 gives the equivalent equation x2 + 4x + 2 = 0. Then using the quadratic formula, with α = 1, b = 4, and c = 2, gives the solutions x = −2 ± √2.
Choices B, C, and D are incorrect and may be the result of errors when applying the quadratic formula.

*Answer can only contain numeric values
Mathematics Test - 2 - Question 13

At a lunch stand, each hamburger has 50 more calories than each order of fries. If 2 hamburgers and 3 orders of fries have a total of 1700 calories, how many calories does a hamburger have?


Detailed Solution for Mathematics Test - 2 - Question 13

The correct answer is 370. A system of equations can be used where h represents the number of calories in a hamburger and f represents the number of calories in an order of fries. The equation 2h + 3f = 1700 represents the fact that 2 hamburgers and 3 orders of fries contain a total of 1700 calories, and the equation h = f + 50 represents the fact that one hamburger contains 50 more calories than an order of fries. Substituting f + 50 for h in 2h + 3f = 1700 gives 2(f + 50) + 3f = 1700.
This equation can be solved as follows:
2f + 100 + 3f = 1700
5f + 100 = 1700
5f = 1600
f = 320
The number of calories in an order of fries is 320, so the number of calories in a hamburger is 50 more than 320, or 370.

Mathematics Test - 2 - Question 14

The table above shows some values of the linear function f. Which of the following defines f ?

Detailed Solution for Mathematics Test - 2 - Question 14

Choice C is correct. The graph of y = f(n) in the coordinate plane is a line that passes through each of the points given in the table. From the table, one can see that an increase of 1 unit in n results in an increase of 3 units in f(n); for example, f(2) − f(1) = 1 − (−2) = 3. Therefore, the graph of y = f(n) in the coordinate plane is a line with slope 3. Only choice C is a line with slope 3. The y-intercept of the line is the value of f(0). Since an increase of 1 unit in n results in an increase of 3 units in f(n), it follows that f(1) − f(0) = 3. Since f(1) = −2, it follows that f(0) = f(1) − 3 = −5. Therefore, the y-intercept of the graph of f(n) is −5, and the slope-intercept equation for f(n) is f(n) = 3n − 5.
Choices A, B, and D are incorrect because each equation has the incorrect slope of the line (the y-intercept in each equation is also incorrect).

Mathematics Test - 2 - Question 15

Graphene, which is used in the manufacture of integrated circuits, is so thin that a sheet weighing one ounce can cover up to 7 football fields. If a football field has an area of approximately acres, about how many acres could 48 ounces of graphene cover?

Detailed Solution for Mathematics Test - 2 - Question 15

Choice C is correct. It is given that 1 ounce of graphene covers 7 football fields. Therefore, 48 ounces can cover 7 × 48 = 336 football fields. If each football field has an area of acres, than 336 football fields have a total area of 336 × = 448 acres. Therefore, of the choices given, 450 acres is closest to the number of acres 48 ounces of graphene could cover.
Choice A is incorrect and may be the result of dividing, instead of multiplying, the number of football fields by. Choice B is incorrect and may be the result of finding the number of football fields, not the number of acres, that can be covered by 48 ounces of graphene. Choice D is incorrect and may be the result of setting up the expressionand then finding only the numerator of the fraction.

Mathematics Test - 2 - Question 16

Mr. Kohl has a beaker containing n milliliters of solution to distribute to the students in his chemistry class. If he gives each student 3 milliliters of solution, he will have 5 milliliters left over. In order to give each student 4 milliliters of solution, he will need an additional 21 milliliters. How many students are in the class?

Detailed Solution for Mathematics Test - 2 - Question 16

Choice D is correct. Let c be the number of students in Mr. Kohl’s class. The conditions described in the question can be represented by the equations n = 3c + 5 and n + 21 = 4c. Substituting 3c + 5 for n in the second equation gives 3c + 5 + 21 = 4c, which can be solved to find c = 26.
Choices A, B, and C are incorrect because the values given for the number of students in the class cannot fulfill both conditions given in the question. For example, if there were 16 students in the class, then the first condition would imply that there are 3(16) + 5 = 53 milliliters of solution in the beaker, but the second condition would imply that there are 4(16) − 21 = 43 milliliters of solution in the beaker. This contradiction shows that there cannot be 16 students in the class.

Mathematics Test - 2 - Question 17


In the equation above, k is a constant. If x = 9, what is the value of k?

Detailed Solution for Mathematics Test - 2 - Question 17

If x = 9 in the equation  - x = 0, this equation becomes  - 9 = 0, which can be rewritten as  = 9 . Squaring each side of  = 9 gives  = 81, or k = 79. Substituting k = 79 into the equation  - 9 = 0 confirms this is the correct value for k.
Choices A, B, and C are incorrect because substituting any of these values for k in the equation k + 2 - 9 = 0 gives a false statement. For example, if k = 7, the equation becomes  - 9 = 3 - 9 = 0, which is false.

Mathematics Test - 2 - Question 18

If a2 + b2 = z and ab = y, which of the following is equivalent to 4z + 8y?

Detailed Solution for Mathematics Test - 2 - Question 18

Substituting a2 + b2 for z and ab for y into the expression 4z + 8y gives 4(a2 + b2) + 8ab. Multiplying a2 + b2 by 4 gives 4a2 + 4b2 + 8ab, or equivalently 4(a2 + 2ab + b2). Since (a2 + 2ab + b2) = (a + b)2, it follows that 4z + 8y is equivalent to (2a + 2b)2.
Choices A, C, and D are incorrect and likely result from errors made when substituting or factoring. 

Mathematics Test - 2 - Question 19

Alan drives an average of 100 miles each week. His car can travel an average of 25 miles per gallon of gasoline. Alan would like to reduce his weekly expenditure on gasoline by $5. Assuming gasoline costs $4 per gallon, which equation can Alan use to determine how many fewer average miles, m, he should drive each week?

Detailed Solution for Mathematics Test - 2 - Question 19

Since gasoline costs $4 per gallon, and since Alan’s car travels an average of 25 miles per gallon, the expression 4/25 gives the cost, in dollars per mile, to drive the car. Multiplying 4/25 by m gives the cost for Alan to drive m miles in his car. Alan wants to reduce his weekly spending by $5, so setting 4/25 m equal to 5 gives the number of miles, m, by which he must reduce his driving.
Choices A, B, and C are incorrect. Choices A and B transpose the numerator and the denominator in the fraction. The fraction 4/25 would result in the unit miles per dollar, but the question requires a unit of dollars per mile. Choices A and C set the expression equal to 95 instead of 5, a mistake that may result from a misconception that Alan wants to reduce his driving by 5 miles each week; instead, the question says he wants to reduce his weekly expenditure by $5.

*Answer can only contain numeric values
Mathematics Test - 2 - Question 20

Intersecting lines r, s, and t are shown below.

What is the value of x?


Detailed Solution for Mathematics Test - 2 - Question 20

The intersecting lines form a triangle, and the angle with measure of x° is an exterior angle of this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. One of these angles has measure of 23° and the other, which is supplementary to the angle with measure 106°, has measure of 180° - 106° = 74°. Therefore, the value of x is 23 + 74 = 97.

Mathematics Test - 2 - Question 21

The weight of an object on Venus is approximately 9/10 of its weight on Earth. The weight of an object on Jupiter is approximately 23/10 of its weight on Earth. If an object weighs 100 pounds on Earth, approximately how many more pounds does it weigh on Jupiter than it weighs on Venus?

Detailed Solution for Mathematics Test - 2 - Question 21

The weight of an object on Venus is approximately 9/10 of its weight on Earth. If an object weighs 100 pounds on Earth, then the object’s weight on Venus is given by The same object’s weight on Jupiter is approximately 23/10 of its weight on Earth; therefore, the object weighs  pounds on Jupiter. The difference between the object’s weight on Jupiter and the object’s weight on Venus is 230 − 90 = 140 pounds. Therefore, an object that weighs 100 pounds on Earth weighs 140 more pounds on Jupiter than it weighs on Venus.
Choice A is incorrect because it is the weight, in pounds, of the object on Venus. Choice B is incorrect because it is the weight, in pounds, of an object on Earth if it weighs 100 pounds on Venus. Choice D is incorrect because it is the weight, in pounds, of the object on Jupiter.

Mathematics Test - 2 - Question 22

The density d of an object is found by dividing the mass m of the object by its volume V. Which of the following equations gives the mass m in terms of d and V?

Detailed Solution for Mathematics Test - 2 - Question 22

The density d of an object can be found by dividing the mass m of the object by its volume V. Symbolically this is expressed by the equation d = m/V. Solving this equation for m yields m = dV.
Choices B, C, and D are incorrect and are likely the result of errors made when translating the definition of density into an algebraic equation and errors made when solving this equation for m. If the equations given in choices B, C, and D are each solved for density d, none of the resulting equations are equivalent to d = m/V.

Mathematics Test - 2 - Question 23

A polling agency recently surveyed 1,000 adults who were selected at random from a large city and asked each of the adults, “Are you satisfied with the quality of air in the city?” Of those surveyed, 78 percent responded that they were satisfied with the quality of air in the city. Based on the results of the survey, which of the following statements must be true?
I. Of all adults in the city, 78 percent are satisfied with the quality of air in the city.
II. If another 1,000 adults selected at random from the city were surveyed, 78 percent of them would report they are satisfied with the quality of air in the city.
III. If 1,000 adults selected at random from a different city were surveyed, 78 percent of them would report they are satisfied with the quality of air in the city.

Detailed Solution for Mathematics Test - 2 - Question 23

Statement I need not be true. The fact that 78% of the 1,000 adults who were surveyed responded that they were satisfied with the air quality in the city does not mean that the exact same percentage of all adults in the city will be satisfied with the air quality in the city. Statement II need not be true because random samples, even when they are of the same size, are not necessarily identical with regard to percentages of people in them who have a certain opinion. Statement III need not be true for the same reason that statement II need not be true: results from different samples can vary. The variation may be even bigger for this sample since it would be selected from a different city. Therefore, none of the statements must be true. 
Choices B, C, and D are incorrect because none of the statements must be true. 

Mathematics Test - 2 - Question 24


The figure on the left above shows a wheel with a mark on its rim. The wheel is rolling on the ground at a constant rate along a level straight path from a starting point to an ending point. The graph of y = d(t) on the right could represent which of the following as a function of time from when the wheel began to roll?

Detailed Solution for Mathematics Test - 2 - Question 24

The graph on the right shows the change in distance from the ground of the mark on the rim over time. The y-intercept of the graph corresponds to the mark’s position at the start of the motion (t = 0); at this moment, the mark is at its highest point from the ground. As the wheel rolls, the mark approaches the ground, its distance from the ground decreasing until it reaches 0—the point where it touches the ground. After that, the mark moves up and away from the ground, its distance from the ground increasing until it reaches its maximum height from the ground. This is the moment when the wheel has completed a full rotation. The remaining part of the graph shows the distance of the mark from the ground during the second rotation of the wheel. Therefore, of the given choices, only choice D is in agreement with the given information.
Choice A is incorrect because the speed at which the wheel is rolling does not change over time, meaning the graph representing the speed would be a horizontal line. Choice B is incorrect because the distance of the wheel from its starting point to its ending point increases continuously; the graph shows a quantity that changes periodically over time, alternately decreasing and increasing. Choice C is incorrect because the distance of the mark from the center of the wheel is constant and equals the radius of the wheel. The graph representing this distance would be a horizontal line, not the curved line of the graph shown.

Mathematics Test - 2 - Question 25

The expression where x > 1 and y > 1,is equivalent to which of the following?

Detailed Solution for Mathematics Test - 2 - Question 25

Choice D is correct. For x > 1 and y > 1, x1/3 and y1/2 are equivalent toand √y, respectively. Also, x−2 and y-1 are equivalent to 1/x2 and 1/y , respectively. Using these equivalences, the given expression can be rewritten as 
Choices A, B, and C are incorrect because these choices are not equivalent to the given expression for x > 1 and y > 1.
For example, for x = 2 and y = 2, the value of the given expression is  the values of the choices, however, are and 1, respectively.

*Answer can only contain numeric values
Mathematics Test - 2 - Question 26

If 2x + 8 = 16 , what is the value of x + 4 ?


Detailed Solution for Mathematics Test - 2 - Question 26

The correct answer is 8. The expression 2x + 8 contains a factor of x + 4. It follows that the original equation can be rewritten as 2(x + 4) = 16. Dividing both sides of the equation by 2 gives x + 4 = 8.

Mathematics Test - 2 - Question 27

In the equation (αx + 3)2 = 36, a is a constant. If x = − 3 is one solution to the equation, what is a possible value of α?

Detailed Solution for Mathematics Test - 2 - Question 27

Choice C is correct. Since x = −3 is a solution to the equation, substituting −3 for x gives (−3α + 3)2 = 36. Taking the square root of each side of this equation gives the two equations −3α + 3 = 6 and −3α + 3 = −6. Solving each of these for a yields α = −1 and α = 3. Therefore, −1 is a possible value of α.
Choice A is incorrect and may be the result of ignoring the squared expression and solving −3α + 3 = 36 for α. Choice B is incorrect and may be the result of dividing 36 by 2 instead of taking the square root of 36 when solving for a. Choice D is incorrect and may be the result of taking the sum of the value of x, −3, and the constant, 3.

Mathematics Test - 2 - Question 28

A software company is selling a new game in a standard edition and a collector’s edition. The box for the standard edition has a volume of 20 cubic inches, and the box for the collector’s edition has a volume of 30 cubic inches. The company receives an order for 75 copies of the game, and the total volume of the order to be shipped is 1,870 cubic inches.Which of the following systems of equations can be used to determine the number of standard edition games, s, and collector’s edition games, c, that were ordered?

Detailed Solution for Mathematics Test - 2 - Question 28

Choice A is correct. The total number of copies of the game the company will ship is 75, so one equation in the system is s + c = 75, which can be written as 75 − s = c. Because each standard edition of the game has a volume of 20 cubic inches and s represents the number of standard edition games, the expression 20s represents the volume of the shipment that comes from standard edition copies of the game. Similarly, the expression 30c represents the volume of the shipment that comes from collector’s edition copies of the games. Because these volumes combined are 1,870 cubic inches, the equation 20s + 30c = 1,870 represents this situation. Therefore, the correct answer is choice A.
Choice B is incorrect. This equation gives the volume of each standard edition game as 30 cubic inches and the volume of each collector's edition game as 20 cubic inches. Choice C is incorrect. This is the result of finding the average volume of the two types of games, using that average volume (25) for both types of games, and assuming that there are 75 more standard editions of the game than there are collector’s editions of the game. Choice D is incorrect. This is the result of assuming that the volume of each standard edition game is 30 cubic inches, that the volume of each collector's edition game is 20 cubic inches, and that there are 75 more standard editions than there are collector’s editions.

Mathematics Test - 2 - Question 29

If α − b = 12 and b/2 = 10 , what is the valueof α + b ?

Detailed Solution for Mathematics Test - 2 - Question 29

Choice D is correct. If b/2 = 10, then multiplying each side of this equation by 2 gives b = 20. Substituting 20 for b in the equation α − b = 12 gives α − 20 = 12. Adding 20 to each side of this equation gives α = 32. Since α = 32 and b = 20, it follows that the value of α + b is 32 + 20, or 52.
Choice A is incorrect. If the value of α + b were less than the value of α − b, it would follow that b is negative. But if b/2 = 10, then b must be positive. This contradiction shows that the value of α + b cannot be 2. Choice B is incorrect. If the value of α + b were equal to the value of α – b, then it would follow that b = 0. However, b cannot equal zero because it is given that b/2 = 10. Choice C is incorrect. This is the value of α, but the question asks for the value of α + b.

Mathematics Test - 2 - Question 30


The scatterplot above shows the numbers of grams of both total protein and total fat for eight sandwiches on a restaurant menu. The line of best fit for the data is also shown. According to the line of best fit, which of the following is closest to the predicted increase in total fat, in grams, for every increase of 1 gram in total protein? 

Detailed Solution for Mathematics Test - 2 - Question 30

Option C is correct. The predicted increase in total fat, in grams, for every increase of 1 gram in total protein is represented by the slope of the line of best fit. Any two points on the line can be used to calculate the slope of the line as the change in total fat over the change in total protein. For instance, it can be estimated that the points (20, 34) and (30, 48) are on the line of best fit, and the slope of the line that passes through them is  or 1.4. Of the choices given, 1.5 is the closest to the slope of the line of best fit.
Choices A, B, and D are incorrect and may be the result of incorrectly finding ordered pairs that lie on the line of best fit or of incorrectly calculating the slope.

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