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ACT 74H Math Answer Explanations

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1. The 32-member French Club is meeting to choose a student government representative. The members decide that the representative, who will be chosen at random, CANNOT be any of the 5 officers of the club. What is the probability that Luis, who is a member of the club but NOT an officer, will be chosen?

The number of members who are not officers is 32-5=27.

Luis is one of the 27 members who are not officers. Therefore the probability that he will be chosen is \frac{1}{27}.

Choice C.

1. The 32-member French Club is meeting to choose a student government representative. The members decide that the representative, who will be chosen at random, CANNOT be any of the 5 officers of the club. What is the probability that Luis, who is a member of the club but NOT an officer, will be chosen?

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2. The statement AABC ADEF is true. Which of the following statements must be true?

Since the two triangles are congruent. The following must be true:

$$\angle A\cong\angle D\\
\angle B\cong\angle E \\
\angle C\cong\angle F\\
\overline{AB}\cong\overline{DE}\\
\overline{BC}\cong\overline{EF}\\
\overline{AC}\cong\overline{DF}
$$

Choice K.

2. The statement AABC ADEF is true. Which of the following statements must be true?

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3. In the figure below, C lies on AD, the measure of ZBAC is 65°, the measure of ZBCD is 100°, and the measure of LABC is x°. What is the value of x ?

Choice .

3. In the figure below, C lies on AD, the measure of ZBAC is 65°, the measure of ZBCD is 100°, and the measure of LABC is x°. What is the value of x ?

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4. A group of students was surveyed about what types of music each had listened to during the previous week. Students could choose 1, 2, or 3 types of music (Rap, Rock, or Country), or they could respond

The area where 8 falls into is an overlap of rap and rock.
 
Choice J.

4. A group of students was surveyed about what types of music each had listened to during the previous week. Students could choose 1, 2, or 3 types of music (Rap, Rock, or Country), or they could respond “none of these music types.” Only 10 students responded with “none of these music types,” and 18 students responded that they had listened to all 3 types. In the figure below, the 8 regions show the numbers of responses in each category. One region had exactly 8 responses. W lat type(s) of music did those 8 students respond they had listened to during the previous week?

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5. What is the value of 2|2 — 9| — 3(4 + 2) ?

$$
\begin{align}
2|2-9|-3(4+2) & = 2|-7|-3(6) \\
& = 2\cdot7-3\cdot6 \\
& = 14-18 \\
& = -4
\end{align}
$$

Choice E.

5. What is the value of 2|2 — 9| — 3(4 + 2) ?

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6. Portions of the graphs represented by the functions —2x + y = 2 and x + y = 5 are shown in the standard (x,y) coordinate plane below. Although only a portion of each graph is shown, the domain of each function is all real numbers. If it can be determined, at what point do the graphs intersect?

$$
\left\{
\begin{aligned}
-2x+y &=2 \\
x+y&=5
\end{aligned}
\right.
\ \ \Rightarrow\ \
\left\{
\begin{aligned}
x &=1 \\
y&=4
\end{aligned}
\right.
\ \ \Rightarrow\ \
Point \ of\ intersection\ is\ (1,4)
$$

Choice G.

6. Portions of the graphs represented by the functions —2x + y = 2 and x + y = 5 are shown in the standard (x,y) coordinate plane below. Although only a portion of each graph is shown, the domain of each function is all real numbers. If it can be determined, at what point do the graphs intersect?

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7. On a map, 1/2 inch represents 10 actual miles. Two

 $$\frac{x}{4\frac{1}{2}}=\frac{10}{\frac{1}{2}} \Rightarrow x=90$$
 
Choice E.

7. On a map, 1/2 inch represents 10 actual miles. Two

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8. The cost of a long-distance call to a certain city is $1.05 for the first minute and $0.15 for each additional minute or part thereof. What is the cost of a 15-minute call to this city'?

$$Cost=$1.05+$0.15\times(15-1)=$3.15$$
 
Choice H.

8. The cost of a long-distance call to a certain city is $1.05 for the first minute and $0.15 for each additional minute or part thereof. What is the cost of a 15-minute call to this city’?

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9. Property valued at $56,000 is assessed at 3/4 of its value. If the yearly tax is calculated as $3 per $100 of assessed value, what is the yearly tax on this property?

$$Assessed\ value=$56,000 \times \frac{3}{4}=$42,000\\
\Rightarrow Yearly\ tax=$3 \times\frac{$42,000}{$100}=$1,260$$
 
Choice C.

9. Property valued at $56,000 is assessed at 3/4 of its value. If the yearly tax is calculated as $3 per $100 of assessed value, what is the yearly tax on this property?

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10. Tammy will draft 1 player at random from a list of 20 players for her fantasy football team. Each player in the list plays only l position. The number of players who play a particular position is given in the table below. What is the probability that the player Tammy drafts will be a kicker or a receiver?

The probability is
 
$$\frac{4+8}{20}=\frac{3}{5}$$
 
Choice K.

10. Tammy will draft 1 player at random from a list of 20 players for her fantasy football team. Each player in the list plays only l position. The number of players who play a particular position is given in the table below. What is the probability that the player Tammy drafts will be a kicker or a receiver?

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11. Ben is saving money to buy a TV that costs $495, including tax. Ben opens a savings account with a deposit of $75 and deposits $65 at the end of each month. What is the minimum number of months Ben will need to make deposits until he has enough money in his account to buy the TV ?

Let the number of month be m.
 
$$$75+$65\cdot m=$495\\
\Rightarrow m = 6.46\approx7$$
 
Choice C.

11. Ben is saving money to buy a TV that costs $495, including tax. Ben opens a savings account with a deposit of $75 and deposits $65 at the end of each month. What is the minimum number of months Ben will need to make deposits until he has enough money in his account to buy the TV ?

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12. What is the slope of HM, shown in the standard (x,y) coordinate plane below?

$$Slope\ of\ \overleftrightarrow{HM} = \frac{y_2-y_1}{x_2-x_1} =\frac{-4-2}{2-(-2)} =\frac{-6}{4}=-\frac{3}{2}$$
 
 
Choice F.

12. What is the slope of HM, shown in the standard (x,y) coordinate plane below?

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13. The polynomial 45x2 + 26x — 8 is equivalent to the product of (5x + 4) and which of the following binomials?

The questions asks what the (   ?   ) should be in:
 
$$45x^2+26x-8=(5x+4)(\quad?\quad)$$
 
To solve for the (  ?  ), we need to perform the following:
 
$$
\begin{pmatrix}5&4\\\color{red}{\frac{45}{5}}&\color{red}{\frac{-8}{4}}\\ \end{pmatrix}
\Rightarrow \begin{pmatrix}5&4\\\color{red}{9}&\color{red}{-2}\\ \end{pmatrix}\\
\Rightarrow (\quad?\quad) = (9x-2)$$ 

 
Choice B.

13. The polynomial 45×2 + 26x — 8 is equivalent to the product of (5x + 4) and which of the following binomials?

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14. Given that sin2x = —4 what is cos2x ?

We know that

$$sin^2x\ + cos^2x=1\\
\Rightarrow cos^2x=1-sin^2x=1-\frac{4}{13}=\frac{9}{13}$$

Choice H.

14. Given that sin2x = —4 what is cos2x ?

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15. In the figure below, all of the small squares are equal in area, and the area of rectangle ABCD is 1 square unit. Which of the following expressions represents the area, in square units, of the shaded region?

Let each side of small squares be k:

Area of rectangle region = 6k\cdot8k=1\Rightarrow k^2=\frac{1}{6\cdot8}

Therefore, area of shaded region = 7k\cdot5k = 7\cdot5\cdot k^2=\frac{7\cdot5}{8\cdot6}

Choice E.

15. In the figure below, all of the small squares are equal in area, and the area of rectangle ABCD is 1 square unit. Which of the following expressions represents the area, in square units, of the shaded region?

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16. In ABC shown below, the measure of LA is 48°, and AB AC. What is the measure of LC ?

It is given that \overline{AB} \cong \overline{AC}, thus \angle B =\angle C.

Because \angle A+\angle B +\angle C = 180^\circ;\ \angle A=48^\circ, we have:

$$\angle C = \frac{180^\circ-48^\circ}{2}=66^\circ$$

Choice J.

16. In ABC shown below, the measure of LA is 48°, and AB AC. What is the measure of LC ?

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17. The first 5 terms of a sequence are given in the table below. The sequence is defined by setting al = 9 and a„= an _1+ (n — 1)2 for n ?_ 2. What is the sixth term, a6, of this sequence?

It is given that a_5=39.

All you need to do is plug in 6 for n:

$$\begin{align}
a_6 & =a_5+(6-1)^2\\
&=39+25\\
&=64
\end{align}$$

Choice B.

17. The first 5 terms of a sequence are given in the table below. The sequence is defined by setting al = 9 and a„= an _1+ (n — 1)2 for n ?_ 2. What is the sixth term, a6, of this sequence?

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18. In the standard (x,y) coordinate plane, the line represented by which of the following equations goes through (0,7) and is parallel to the line represented by y = —2x — 4 ?

Because the line is parallel to the line represented by y = -2x – 4, the line has the same slope -2. Thus we can write the line as:

$$y = -2x + b$$

Because the line passes through (0,7):

$$7=-2\cdot0 +b\\
\Rightarrow b=7\\$$

Therefore, the line is:

$$y = -2x + 7$$

Choice G.

18. In the standard (x,y) coordinate plane, the line represented by which of the following equations goes through (0,7) and is parallel to the line represented by y = —2x — 4 ?

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19. A scale drawing of Corinne's bedroom floor is shown below. All given dimensions are in feet, and all intersecting line segments shown are perpendicular. Corinne wants to completely cover the floor with square hardwood tiles. Each tile has a side length of 1 foot, and no tiles will be cut. How many tiles will Corinne need to cover the floor?

The bedroom floor can be seen as being consisting of one rectangle and a square with lengths as below:

Thus the area is 5\times5+6\times9=79. The number of tiles needed is $$\frac{79\ ft^2}{1 \ ft^2}=79$$

Choice D.

19. A scale drawing of Corinne’s bedroom floor is shown below. All given dimensions are in feet, and all intersecting line segments shown are perpendicular. Corinne wants to completely cover the floor with square hardwood tiles. Each tile has a side length of 1 foot, and no tiles will be cut. How many tiles will Corinne need to cover the floor?

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20. A survey about 3 issues affecting Bluff City Park was given to 60 residents. The results of the survey are shown below.

We are told that the survey accurately predict how the 1,200 resident would respond:

$$\require{cancel}\frac {x}{1200}=\frac {12}{60}\\
\Rightarrow x=\frac {12\times\cancel{1200}20}{\cancel{60}}=240
$$

Choice F.

20. A survey about 3 issues affecting Bluff City Park was given to 60 residents. The results of the survey are shown below.

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21. In the standard (x,y) coordinate plane, the graph of the line 3x — 4y = d passes through the point (-5,6). What is the value of d ?

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21. In the standard (x,y) coordinate plane, the graph of the line 3x — 4y = d passes through the point (-5,6). What is the value of d ?

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22. Given u and v such that (a2)1 = a12 and (a')2 = a8 for all positive a, what is au'

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22. Given u and v such that (a2)1 = a12 and (a’)2 = a8 for all positive a, what is au’

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23. In the standard (x,y) coordinate plane, a translation maps a point (x,y) to its image (x — 5, y + 3). To what image does this translation map (-3,-2) ?

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23. In the standard (x,y) coordinate plane, a translation maps a point (x,y) to its image (x — 5, y + 3). To what image does this translation map (-3,-2) ?

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24. Given x = 4a+ b which of the following expressions is equivalent to b ?

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24. Given x = 4a+ b which of the following expressions is equivalent to b ?

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25. Matt purchased a 60-foot-long roll of chain-link fence. He used the entire roll of fence to construct a rectangular pen for his dog. Given that the pen is 12 feet wide, what is its length, in feet?

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25. Matt purchased a 60-foot-long roll of chain-link fence. He used the entire roll of fence to construct a rectangular pen for his dog. Given that the pen is 12 feet wide, what is its length, in feet?

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26. Angle LJKL is shown below with the given lengths in coordinate units. What is the measure of LJKL in radians?

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26. Angle LJKL is shown below with the given lengths in coordinate units. What is the measure of LJKL in radians?

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27. A rectangle is 31/-5- meters wide and 51/-S meters long.

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27. A rectangle is 31/-5- meters wide and 51/-S meters long.

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28. Which of the following operations will produce the largest result when substituted for the blank in the

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28. Which of the following operations will produce the largest result when substituted for the blank in the

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29. In Westville on Tuesday, the high temperature was 30°C and the low temperature was 20°C. What was the difference between the high and low temperatures, in degrees Fahrenheit ?(Note: The relationship between the temperature c, in degrees Celsius, and the temperature f, in degrees Fahrenheit, is given by c = (f — 32).)

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29. In Westville on Tuesday, the high temperature was 30°C and the low temperature was 20°C. What was the difference between the high and low temperatures, in degrees Fahrenheit ?(Note: The relationship between the temperature c, in degrees Celsius, and the temperature f, in degrees Fahrenheit, is given by c = (f — 32).)

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30. A committee will be selected from a group of 12 women and 18 men. The committee will consist of 5 women and 5 men. Which of the following expressions gives the number of different committees that could be selected from these 30 people?

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30. A committee will be selected from a group of 12 women and 18 men. The committee will consist of 5 women and 5 men. Which of the following expressions gives the number of different committees that could be selected from these 30 people?

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31. Which of the following expressions is equal to 3 	9

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31. Which of the following expressions is equal to 3 9

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32. When written in scientific notation, which of the following expressions is equal to the number of blocks used to construct the pyramid?

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32. When written in scientific notation, which of the following expressions is equal to the number of blocks used to construct the pyramid?

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33. The perimeter of the pyramid's base was 920 meters when construction was completed. At that time, which of the following values is closest to the length, in meters, of each diagonal of the base?

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33. The perimeter of the pyramid’s base was 920 meters when construction was completed. At that time, which of the following values is closest to the length, in meters, of each diagonal of the base?

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34. Which of the following expressions is equal to the length, in meters. of DC ?

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34. Which of the following expressions is equal to the length, in meters. of DC ?

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35. In AABC, AB = 6 cm, AC = 12 cm, inLA = 600, and	AC is the longest side. Which of the following statements about the measures of the angles in AABC must be true? (Note: mZX denotes the measure of angle X.)

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35. In AABC, AB = 6 cm, AC = 12 cm, inLA = 600, and AC is the longest side. Which of the following statements about the measures of the angles in AABC must be true? (Note: mZX denotes the measure of angle X.)

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36. Erika is landscaping her front yard. The yard, which is level, has the shape of a rectangle that is 60 feet wide by 80 feet long. To cover the yard with a layer of topsoil having a uniform depth of 4 inches ( 1 foot), Erika needs to use how many cubic feet of topsoil?

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36. Erika is landscaping her front yard. The yard, which is level, has the shape of a rectangle that is 60 feet wide by 80 feet long. To cover the yard with a layer of topsoil having a uniform depth of 4 inches ( 1 foot), Erika needs to use how many cubic feet of topsoil?

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37. Suzanne and Chad are going to bake and deliver cookies to college students during final exam week. They estimate it will cost $4 for the ingredients to make each batch of cookies and $50 to buy the mixer, bowls, and other utensils they will need. They decide to sell the cookies for $5 per batch. Assume they have no other expenses. Which of the following equations represents the profit, P dollars, they will make on b batches of cookies?

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37. Suzanne and Chad are going to bake and deliver cookies to college students during final exam week. They estimate it will cost $4 for the ingredients to make each batch of cookies and $50 to buy the mixer, bowls, and other utensils they will need. They decide to sell the cookies for $5 per batch. Assume they have no other expenses. Which of the following equations represents the profit, P dollars, they will make on b batches of cookies?

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38. The sum of the measures of LA and LB is 90°. The sum of the measures of LA and LC is 180°. The sum of the measures of LB and ZD is 180°. What is the sum of the measures of LC and ZD ?

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38. The sum of the measures of LA and LB is 90°. The sum of the measures of LA and LC is 180°. The sum of the measures of LB and ZD is 180°. What is the sum of the measures of LC and ZD ?

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39. The yard will be enclosed by a fence and the back side of the house. The fence will begin at one back corner of the house and will end at the other. What is the minimum number of feet of fencing needed to enclose the yard?

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39. The yard will be enclosed by a fence and the back side of the house. The fence will begin at one back corner of the house and will end at the other. What is the minimum number of feet of fencing needed to enclose the yard?

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40. The area of the garden is what percent of the area of the yard?

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40. The area of the garden is what percent of the area of the yard?

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41. Beginning next year, Mr. Santana will increase the number of bulbs in the garden each year so that the numbers form a geometric sequence. In 3 years, there will be 162 bulbs in the garden. By what factor will the number of bulbs be multiplied each year?

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41. Beginning next year, Mr. Santana will increase the number of bulbs in the garden each year so that the numbers form a geometric sequence. In 3 years, there will be 162 bulbs in the garden. By what factor will the number of bulbs be multiplied each year?

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42. Suspended from the ceiling is a weight on a large spring that is oscillating up and down. The distance, d inches, between the location of the center of the mass of the weight after t seconds and the weight's equilibrium location at t = 0 is modeled by the function d= 5 sin(4itt). What is the amplitude of the function?

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42. Suspended from the ceiling is a weight on a large spring that is oscillating up and down. The distance, d inches, between the location of the center of the mass of the weight after t seconds and the weight’s equilibrium location at t = 0 is modeled by the function d= 5 sin(4itt). What is the amplitude of the function?

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43. Given that i2 = —1 and that k is a positive integer, what is the value of i (4k + ?

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43. Given that i2 = —1 and that k is a positive integer, what is the value of i (4k + ?

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44. Ling asked 11 people how many text messages each of them sent last week. Each of the 11 responses was in one of the intervals given in the table below. Which interval contains the median of the data?

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44. Ling asked 11 people how many text messages each of them sent last week. Each of the 11 responses was in one of the intervals given in the table below. Which interval contains the median of the data?

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45. For all real numbers a, b, and c, which of the following expressions is equal to la—b—cH

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45. For all real numbers a, b, and c, which of the following expressions is equal to la—b—cH

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46. Given g(x) = , which of the following expressions x- is equal to g(x — 1) for all x in its domain?

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46. Given g(x) = , which of the following expressions x- is equal to g(x — 1) for all x in its domain?

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47. A circle with radius 10 cm is divided into 3 congruent arcs. What is the length, in centimeters, of each of the 3 arcs?

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47. A circle with radius 10 cm is divided into 3 congruent arcs. What is the length, in centimeters, of each of the 3 arcs?

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48. Consider all positive integers that are multiples of 20 and that are less than or equal to 300. What fraction of those integers are multiples of 15 ?

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48. Consider all positive integers that are multiples of 20 and that are less than or equal to 300. What fraction of those integers are multiples of 15 ?

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49. In the figure below, ABCD is a trapezoid with AE perpendicular to AB; AE is 10 units long; and DC is 28 units long. If the area of right triangle LEBA is 60 square units, what is the area, in square units, of trapezoid ABCD ?

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49. In the figure below, ABCD is a trapezoid with AE perpendicular to AB; AE is 10 units long; and DC is 28 units long. If the area of right triangle LEBA is 60 square units, what is the area, in square units, of trapezoid ABCD ?

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50. The fraction is equivalent to 0.285714. What is the digit in the 1,001st decimal place of 0.285714 ? (Note: The digit in the 4th decimal place of 0.285714 is 7.)

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50. The fraction is equivalent to 0.285714. What is the digit in the 1,001st decimal place of 0.285714 ? (Note: The digit in the 4th decimal place of 0.285714 is 7.)

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51. The 3 lines with equations y = 4, x = —3, and y = x, respectively, bound a unique triangular region in the standard (x,y) coordinate plane. Which of the following descriptions is the best classification of this triangle?

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51. The 3 lines with equations y = 4, x = —3, and y = x, respectively, bound a unique triangular region in the standard (x,y) coordinate plane. Which of the following descriptions is the best classification of this triangle?

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52. At what point in the standard (x,y) coordinate plane do

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52. At what point in the standard (x,y) coordinate plane do

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53. The employees at a hotel reservation center assign an 8-digit confirmation number (CN) to each customer making a reservation. The first digit in each CN is 8. The other 7 digits can be any digit 0 through 9, and digits may repeat. How many possible 8-digit CNs are there?

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53. The employees at a hotel reservation center assign an 8-digit confirmation number (CN) to each customer making a reservation. The first digit in each CN is 8. The other 7 digits can be any digit 0 through 9, and digits may repeat. How many possible 8-digit CNs are there?

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54. Which of the following number line graphs represents all values in the domain of the function y = loglo(x2 — 4x + 3) ?

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54. Which of the following number line graphs represents all values in the domain of the function y = loglo(x2 — 4x + 3) ?

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55. What is the determinant of the matrix shown below?

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55. What is the determinant of the matrix shown below?

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56. At Wafer Technologies, identification codes each consist of the following sequence: 1 digit, 4 letters, 1 digit. For any 1 code, the digits (0-9) may be the same, but the letters, each from the English alphabet, must all be different. Which of the following expressions gives the probability that a randomly selected identification code contains the word MATH, spelled correctly?

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56. At Wafer Technologies, identification codes each consist of the following sequence: 1 digit, 4 letters, 1 digit. For any 1 code, the digits (0-9) may be the same, but the letters, each from the English alphabet, must all be different. Which of the following expressions gives the probability that a randomly selected identification code contains the word MATH, spelled correctly?

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57. What is the distance, in coordinate units, between 2 + 6i and —4 + 3i in the complex plane?

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57. What is the distance, in coordinate units, between 2 + 6i and —4 + 3i in the complex plane?

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58. What is the minimum value of

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58. What is the minimum value of

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59. Which of the following data sets has the greatest standard deviation?

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59. Which of the following data sets has the greatest standard deviation?

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60. The circle with equation x2 + (y — 1)2 = 1 is graphed in the standard (x,y) coordinate plane below. Suppose the circle rolls along the positive x-axis for 2 rotations and then stops. Which of the following is an equation of the circle in its new position?

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60. The circle with equation x2 + (y — 1)2 = 1 is graphed in the standard (x,y) coordinate plane below. Suppose the circle rolls along the positive x-axis for 2 rotations and then stops. Which of the following is an equation of the circle in its new position?

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