ACT Form 74H Math Answer Explanation


2016 December(74H) Math Question 1

Probability Practice

- One Event

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?

Choice C

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 K

Since the two triangles are congruent. The following must be true: $$\angle A\cong\angle D\\$$ $$\angle B\cong\angle E \\$$ $$\angle C\cong\angle F\\$$ $$\overline{AB}\cong\overline{DE}\\$$ $$\overline{BC}\cong\overline{EF}\\$$ $$\overline{AC}\cong\overline{DF} $$

2016 December(74H) Math Question 3

Angles Practice

- Collinear Points & Triangles

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 C

$$\angle BCD+\angle BCA = 180°$$ $$\Rightarrow\angle BCA=180°-100°=80°$$ $$\angle CAB+\angle ABC+\angle BCA=180°$$ $$\Rightarrow 65°+X°+80°=180°$$ $$\Rightarrow X°=35°$$

Choice J

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

Choice E

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

Choice G

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

Choice E

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

Choice H

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

Choice C

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

2016 December(74H) Math Question 10

Probability Practice

- One Event

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?

Choice K

The probability is \(\frac{4+8}{20}=\frac{3}{5}\)

2016 December(74H) Math Question 11

Word Problems Practice

- Cost

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 ?

Choice C

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

Choice F

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

2016 December(74H) Math Question 13

Quadratic Equations Practice

- Factoring

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

Choice B

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 H

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

Choice E

74h

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 J

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: \(\angle C = \frac{180^\circ-48^\circ}{2}=66^\circ\)

2016 December(74H) Math Question 17

Sequences and Patterns Practice

- General Sequences

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?

Choice B

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 G

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: $$y = -2x + 7$$

2016 December(74H) Math Question 19

Perimeter, Area, Volume Practice

- Mixed Shapes

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?

Choice D

74h

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 F

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 A

We are told that the …

Choice G

$$(a^2)^u=a^{12}=(a^2)^6\ \Rightarrow u=6\\$$ $$(a^v)^2=a^{8}=(a^4)^2\ \Rightarrow …

Choice B

The shifting rule is that: …

Choice F

$$x=\frac{4a+b}{3}\\$$ $$\Rightarrow 3x=4a+b\\$$ $$\Rightarrow b=3x-4a$$

Choice B

Let the length of the …

2016 December(74H) Math Question 26

Angles Practice

- Circles

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

Choice J

We need to know \(\pi …

Choice A

Area of rectangle =$$3\sqrt5 \times …

Choice H

11 either divided or multiplied …

Choice D

And, by rearranging the expression, …

2016 December(74H) Math Question 30

Probability Practice

- Combinations

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?

Choice E

The key of the committee …

Choice E

To rewrite the expression, we …

Choice F

We need to know: $$1\ …

Choice D

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In a square, the lengths …

Choice K

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In the right triangle \( …

2016 December(74H) Math Question 35

Angles Practice

- Collinear Points & Triangles

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.)

Choice B

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The length of BC can …

Choice F

The volume of the layer …

2016 December(74H) Math Question 37

Word Problems Practice

- Fixed Cost + Variable Cost

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?

Choice E

The net profit per batch …

Choice J

The workings are as follows: …

Choice C

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The fence is indicated by …

Choice G

$$ Area \ of\ garden …

Choice B

The number of bulbs in …

2016 December(74H) Math Question 42

Trigonometry Practice

- Graphing Trig Functions

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?

Choice J

The amplitude of the function …

Choice B

You can substitute the value …

2016 December(74H) Math Question 44

Statistics Practice

- Mean, Median

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?

Choice J

The median would be the …

Choice D

Because the absolute value equation …

Choice F

All we need to do …

Choice B

The circumference ofthe circle = …

2016 December(74H) Math Question 48

Sequences and Patterns Practice

- General Sequences

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 ?

Choice F

The question is asking for …

2016 December(74H) Math Question 49

Perimeter, Area, Volume Practice

- Mixed Shapes

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 ?

Choice D

$$ \begin{align} Area \ of\Delta …

2016 December(74H) Math Question 50

Sequences and Patterns Practice

- General Sequences

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.)

Choice F

The decimal repeats in chunks …

Choice C

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As can be seen in …

Choice J

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The vertical asymptote \(x=3\) is …

2016 December(74H) Math Question 53

Probability Practice

- Combinations

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?

Choice C

The confirmation number will be …

2016 December(74H) Math Question 54

Logarithms Practice

- Logarithm Expressions

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

Choice K

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The range of values can …

2016 December(74H) Math Question 55

Matrices Practice

- Determinants

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

Choice C

For a matrix A = …

2016 December(74H) Math Question 56

Probability Practice

- More than One Event

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?

Choice G

The identification code will be in the following format: $$\underline{Digit_1}\ \underline{Letter_1}\ \underline{Letter_2}\ \underline{Letter_3}\ \underline{Letter_4}\ \underline{Digit_2}$$ \(\underline{Digit_1}\) and\(\underline{Digit_2}\) can be any digits from 0-9, while the letters can not be repeated from the 26 English alphabet. Thus the total number of possible codes is: $$10\cdot (26)\cdot (26)\cdot(25)\cdot(24)\cdot(23)\cdot 10$$ of which the number of codes with the fixed letters "MATH" $$ \underline{Digit_1}\ \underline{M}\ \underline{A} \ \underline{T} \ \underline{H}\ \underline{Digit_2} $$ is $$10\cdot 10$$ Therefore, the probability of randomly selecting such a code is: $$\frac{10\cdot 10}{10\cdot (26)\cdot (26)\cdot(25)\cdot(24)\cdot(23)\cdot 10}$$

2016 December(74H) Math Question 57

Number Types & Properties Practice

- Complex Number

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

Choice D

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The complex plane is shown below.

With the coordinates A (-4, 3i), C(2, 6i), the distance between A and C is simply $$b=\sqrt{BC^2+AB^2}=\sqrt{3^2+6^2}=\sqrt{45}$$

Choice F

To make f(x) as small …

2016 December(74H) Math Question 59

Statistics Practice

- General Statistics

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

Choice A

You don’t have to compute …

2016 December(74H) Math Question 60

Conics Practice

- Circles

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?

Choice K

The coordinates of the center …