1
The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution (x, y) to this system?
Đáp án và giải thích
Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the nonlinear equation shown intersect at the point (4,5) . Thus, the solution to the system is (4,5) .
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
2
On the first day of a semester, a film club has 90 members. Each day after the first day of the semester, 10 new members join the film club. If no members leave the film club, how many total members will the film club have 4 days after the first day of the semester?
Đáp án và giải thích
Choice B is correct. It’s given that the film club has 90 members on the first day of a semester, and 10 new members join the film club each day after the first day of the semester. This means that after 4 days, 4 x 10, or 40, new members will have joined the club. Adding 40 members to the original 90 club members yields 130 members. Thus, the film club will have 130 total members 4 days after the first day of the semester.
Choice A is incorrect. This is the number of members that will have joined the film club 4 days after the first day of the semester if 100 new members, not 10, join the film club each day. Choice C is incorrect. This is the number of members the film club will have 4 days after the first day of the semester if 1 new member, not 10, joins the film club each day. Choice D is incorrect. This is the number of members the film club has on the first day of the semester.
3
The graph of the linear function f is shown, where y = f(x). What is the y intercept of the graph of f ?
Đáp án và giải thích
Choice D is correct. The y-intercept of a graph is the point where the graph intersects the y-axis. The graph of function f shown intersects the y-axis at the point (0,8). Therefore, the y-intercept of the graph of f is (0,8).
Choice A is incorrect. This is the point where the x-axis, not the graph of f, intersects the y-axis. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.
4
s + 7r = 27
r = 3
What is the solution (r, s) to the given system of equations?
Đáp án và giải thích
Choice B is correct. The second equation in the given system is r=3. Substituting 3 for r in the first equation in the given system yields s+7(3)=27, or s+21 = 27. Subtracting 21 from both sides of this equation yields s=6.
Therefore, the solution (r, s) to the given system of equations is (3, 6). Choice A is incorrect. This is the solution (s,r) , not (r, s) , to the given system of equations. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.
5
The table shows selected values from function f.
Which of the following is the best description of function f ?
Đáp án và giải thích
Choice B is correct. The given values show that as x increases, f(x) also increases, which means that f is an increasing function. Furthermore, f(x) increases at a constant rate of 1 for each increase of x by 1. A function with a constant rate of change is linear. Thus, the function f can be described as an increasing linear function.
Choice A is incorrect. For a decreasing linear function, as x increases, f(x) decreases rather than increases. Choice C is incorrect. For a decreasing exponential function, for each increase of x by 1, f(x) decreases by a fixed percentage rather than increases at a constant rate. Choice D is incorrect. For an increasing exponential function, for each increase of x by 1, f(x) increases by a fixed percentage rather than at a constant rate.
6
The graph of a system of linear equations is shown.
The solution to the system is (x, y). What is the value of x ?
Đáp án và giải thích
The correct answer is 4. A solution to a system of equations must satisfy each equation in the system. It follows that if (x,y), is a solution to the system, the point (x,y), lies on the graph in the xy-plane of each equation in the system. According to the graph, the point (x,y), that lies on the graph of each equation in the system is (4,1) . Therefore, the solution to the system is (4,1) . It follows that the value of x is 4.
7
23, 27, 29, 32, 35, 36, 52
What is the range of the 7 scores shown?
Đáp án và giải thích
The correct answer is 29. The range of a data set is the difference between its maximum value and its minimum value. For the data set shown, the maximum score is 52 and the minimum score is 23. The difference between those scores is 52-23, or 29. Therefore, the range of the 7 scores shown is 29.
8
In the figure, line m is parallel to line n, and line k intersects both lines. Which of the following statements is true?
Đáp án và giải thích
Choice C is correct. Vertical angles, or angles that are opposite each other when two lines intersect, are congruent. It’s given that line k intersects line n. Based on the figure, the angle with measure x° and the angle with measure 145° are vertical angles. Therefore, the value of x is equal to 145.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
9
The equation x + y = 1,440 represents the number of minutes of daylight (between sunrise and sunset), x, and the number of minutes of non-daylight, y, on a particular day in Oak Park, Illinois. If this day has 670 minutes of daylight, how many minutes of non-daylight does it have?
Đáp án và giải thích
Choice B is correct. It’s given that the equation x+y =1,440 represents the number of minutes of daylight, x, and the number of minutes of non-daylight, y, on a particular day in Oak Park, Illinois. It’s also given that this day has 670 minutes of daylight. Substituting 670 for x in the equation x+y =1,440 yields 670 + y = 1,440. Subtracting 670 from both sides of this equation yields y=770. Therefore, this day has 770 minutes of non-daylight.
Choice A is incorrect. This is the number of minutes of daylight, not non-daylight, on this day. Choice C is incorrect and may result from conceptual or calculation rrors. Choice D is incorrect. This is the total number of minutes of daylight and non-daylight.
10
Scott selected 20 employees at random from all 400 employees at a company. He found that 16 of the employees in this sample are enrolled in exactly three professional development courses this year.
Based on Scott’s findings, which of the following is the best estimate of the number of employees at the company who are enrolled in exactly three professional development courses this year?
Đáp án và giải thích
11
If 4x − 28 = −24, what is the value of x − 7 ?
Đáp án và giải thích
12
For a snowstorm in a certain town, the minimum rate of snowfall recorded was 0.6 inches per hour, and the maximum rate of snowfall recorded was 1.8 inches per hour. Which inequality is true for all values of s, where s represents a rate of snowfall, in inches per hour, recorded for this snowstorm?
Đáp án và giải thích
Choice D is correct. It’s given that for a snowstorm in a certain town, the minimum rate of snowfall recorded was 0.6 inches per hour, the maximum rate of snowfall recorded was 1.8 inches per hour, and s represents a rate of snowfall, in inches per hour, recorded for this snowstorm. It follows that the inequality 0.6 ≤s≤ 1.8 is true for all values of s.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
13
y=4x
\tag{hi} y=x^{x}-12
A solution to the given system of equations is (x, y),
where x > 0. What is the value of x ?
Đáp án và giải thích
14
A store sells two different-sized containers of blueberries. The store’s sales of these blueberries totaled 896.86 dollars last month. The equation 4.51x + 6.07y = 896.86 represents this situation, where x is the number of smaller containers sold and y is the number of larger containers sold.
According to the equation, what is the price, in dollars, of each smaller container?
Đáp án và giải thích
The correct answer is 4.51. It’s given that the equation 44.51x+ 6.07y = 896.86 represents this situation, where x is the number of smaller containers sold, y is the number of larger containers sold, and 896.86 is the store’s total sales, in dollars, of blueberries last month. Therefore, 4.51x represents the store’s sales, in dollars, of smaller containers, and 6.07x represents the store’s sales, in dollars, of larger containers. Since x is the number of smaller containers sold, the price, in dollars, of each smaller container is 4.51.
15
A right circular cylinder has a base diameter of 22 centimeters and a height of 6 centimeters. What is the volume, in cubic centimeters, of the cylinder?
Đáp án và giải thích
16
Đáp án và giải thích
17
At a particular track meet, the ratio of coaches to athletes is 1 to 26. If there are x coaches at the track meet, which of the following expressions represents the number of athletes at the track meet?
Đáp án và giải thích
Choice B is correct. It’s given that at a particular track meet, the ratio of coaches to athletes is 1 to 26. If one number in a ratio is multiplied by a value, the other number must be multiplied by the same value in order to maintain the same ratio.
If there are x coaches at the track meet, multiplying both numbers in the ratio by x yields 1(x) to 26(x) , or x to 26x. Therefore, the expression 26x represents the number of athletes at the track meet.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
18
Kaylani used fabric measuring 5 yards in length to make each suit for a men’s choir. The relationship between the number of suits that Kaylani made, x, and the total length of fabric that she purchased y, in yards, is represented by the equation y − 5x = 6.
What is the best interpretation of 6 in this context?
Đáp án và giải thích
Choice D is correct. It’s given that the equation y-5x=6 represents the relationship between the number of suits that Kaylani made, x, and the total length of fabric she purchased, y, in yards. Adding 5x to both sides of the given equation yields y=5x+6. Since Kaylani made x suits and used 5 yards of fabric to make each suit, the expression 5x represents the total amount of fabric she used to make the suits. Since y represents the total length of fabric Kaylani purchased, in yards, it follows from the equation y=5x+6 that Kaylani purchased 5x yards of fabric to make the suits, plus an additional 6 yards of fabric.
Therefore, the best interpretation of 6 in this context is that Kaylani purchased 6 yards more fabric than she used to make the suits.
Choice A is incorrect. Kaylani made a total of x suits, not 6 suits. Choice B is incorrect. Kaylani purchased a total of y yards of fabric, not a total of 6 yards of fabric. Choice C is incorrect. Kaylani used a total of 5x yards of fabric to make the suits, not a total of 6 yards of fabric.
19
Đáp án và giải thích
20
Đáp án và giải thích
21
The function g is defined by g(x) = (x + 14)(t − x), where t is a constant. In the xy-plane, the graph of y = g(x) passes through the point (24, 0). What is the value of g(0) ?
Đáp án và giải thích
The correct answer is 336. By the zero product property, if (x+14)(t-x)=0, then x+14=0, which gives x=-14, or (t-x)=0, which gives x=t. Therefore, 9(x)=0 when x=-14 and when x=t. Since the graph of y=g(x) passes through the point (24,0) , it follows that g(24)=0, so t=24. Substituting 24 for t in the equation g(x)=(x+14)(t-x) yields g(x)=(x+14)(24-x). The value of g(0) can be calculated by substituting 0 for x in this equation, which yields g(0)=(0+14)(24-0), or g(0)=336.
22
Đáp án và giải thích
23
A right rectangular prism has a height of 9 inches. The length of the prism’s base is x inches, which is 7 inches more than the width of the prism’s base. Which function V gives the volume of the prism, in cubic inches, in terms of the length of the prism’s base?
Đáp án và giải thích
Choice D is correct. The volume of a right rectangular prism can be represented by a function V that gives the volume of the prism, in cubic inches, in terms of the length of the prism’s base. The volume of a right rectangular prism is equal to the area of its base times its height. It’s given that the length of the prism’s base is x inches, which is 7 inches more than the width of the prism’s base. This means that the width of the prism’s base is x-7 inches. It follows that the area of the prism’s base, in square inches, is x(x-7) and the volume, in cubic inches, of the prism is x(x-7)(9). Thus, the function V that gives the volume of this right rectangular prism, in cubic inches, in terms of the length of the prism’s base, x, is V(x)=9x(x-7) .
Choice A is incorrect. This function would give the volume of the prism if the height were 9 inches more than the length of its base and the width of the base were 7 inches more than its length. Choice B is incorrect. This function would give the volume of the prism if the height were 9 inches more than the length of its base. Choice C is incorrect. This function would give the volume of the prism if the width of the base were 7 inches more than its length, rather than the length of the base being 7 inches more than its width.
24
Đáp án và giải thích
25
The result of increasing the quantity x by 400% is 60. What is the value of x ?
Đáp án và giải thích
26
Đáp án và giải thích
27
Đáp án và giải thích
