ACT Form 72G Math Answer Explanation

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2014 December(72G) Math Question 1

Word Problems

- Cost

1. Bella sells real estate. She is paid 3% of the sale price of every house she sells. What is Bella paid for selling a house at the sale price of $250,000 ?

Choice D

$$$250,000\times 3\%=$7,500$$

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2014 December(72G) Math Question 2

Quadratic Equations

- Factoring

2. For what value(s) of x is (x + 3)(x — 4) = 0 true?

Choice F

Solving for the equation, we have x = -3 and 4.

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2014 December(72G) Math Question 3

Number Types & Properties

- Order of Values

3. Which of the following inequalities orders the numbers 0.1, 0.04, and 1/6 from least to greatest?

Choice B

$$\frac{1}{6}\approx0.167$$

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2014 December(72G) Math Question 4

Algebraic & Function Expressions

- Solve Algebraic Expressions

4. Given that 2x + 1 = 5, what is the value of (x + 3)^3 ?

Choice H

$$2x+1=5\Rightarrow x=2 \\ $$ $$\Rightarrow (x+3)^3=(2+3)^3=5^3=125$$

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2014 December(72G) Math Question 5

Graphing Properties

- Vectors

5. What vector is the result of adding the vectors (1,-2), (2,3), and (-4,2) ?

Choice B

<1+2-4,-2+3+2>=<-1,3>

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2014 December(72G) Math Question 6

Algebraic & Function Expressions

- Solve Algebraic Expressions

6. What value of x makes (2\3)x + 5 = 11 true?

Choice H

$$\frac{2}{3}x=6\\$$ $$ \Rightarrow 2x=18 \\ $$ $$\Rightarrow x=9$$

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2014 December(72G) Math Question 7

Statistics

- Mean, Median

7. In Summit City, the daily low temperatures, in degrees Fahrenheit (°F), during the first week in December were —7, —11, 15, 23, 45, 23, and 2. To the nearest 1°F, what was the mean daily low temperature for that week?

Choice B

$$\frac{-7-11+15+23+45+23+2}{7}=13°C$$

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2014 December(72G) Math Question 8

Inequalities

- Inequality Expressions

8. Assume that a, b, and c are real numbers. If a > b, and (a + c) > (b + c), then what is true of c ?

Choice K

$$(a+c)>(b+c) \\$$ $$\Rightarrow a>b$$

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2014 December(72G) Math Question 9

Equation of a Line

- Equations that Describe a Line

9. What is the slope of the line given by the equation 2x + 7y = —12 ?

Choice C

$$7y=-2x-12 \\$$ $$ \Rightarrow y=-\frac{2}{7}x-\frac{12}{7}$$

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2014 December(72G) Math Question 10

Trigonometry

- Pythagorean Theorem & sin cos tan

10. One leg of a right triangle is 4 meters long and the other leg is 8 meters long. The hypotenuse of the triangle is how many meters long?

Choice K

$$h=\sqrt{4^2+8^2}=\sqrt{80}$$

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2014 December(72G) Math Question 11

Word Problems

- Distance, Speed, Time

11. Students studying motion observed a cart rolling at a constant rate along a straight line. The table below gives the distance, d feet, the cart was from a reference point at 1-second intervals from t = 0 seconds to t = 5 seconds.

Choice C

Only the equation in C fulfils all sets of d and t in the table.

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2014 December(72G) Math Question 12

Ratios, Proportions, Percentages, Fractions

- Proportions & Ratios

12. The ratio of 30 to 24 is the same as the ratio of 40 to what number?

Choice G

$$\frac{30}{24}=\frac{40}{x} \\ $$ $$\Rightarrow x=32$$

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2014 December(72G) Math Question 13

Probability

- One Event

13. A box contains 3 red, 2 white, and 5 blue beads, each of which is a solid color. Miguel will randomly remove 1 bead from the box, record the color of the bead, and return the bead to the box. Miguel will complete this process 50 times. How many times should Miguel expect to record a bead color that is NOT white?

Choice D

$$50\times \frac{3+5}{3+2+5}=40$$

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2014 December(72G) Math Question 14

Algebraic & Function Expressions

- Write Algebraic Expressions

14. Whenever 3(a + 6b) — c = 0, which of the following expressions is equal to a ?

Choice F

$$3(a+6b)=c \\ $$ $$\Rightarrow a+6b=\frac{c}{3} \\$$ $$ \Rightarrow a=\frac{c}{3}-6b$$

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2014 December(72G) Math Question 15

Exponents, Radicals, & Scientific Notation

- Radical Expressions

15. Given that /2x —9 = 1, x = ?

Choice E

$$\sqrt{2x}=10 \\$$ $$ \Rightarrow 2x=100 \\ $$ $$\Rightarrow x=50$$

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2014 December(72G) Math Question 16

Quadratic Equations

- Factoring

16. A high school ice hockey conference that consists of 10 teams requires that every conference team must play each of the other conference teams at least once in a season. For n teams in the conference, the number of conference games played in a season must then be at least n 2— n . What is the minimum number of conference games that must be played in a season for this conference?

Choice H

$$\frac{10^2-10}{2}=45$$

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2014 December(72G) Math Question 17

Trigonometry

- Pythagorean Theorem & sin cos tan

17. For an angle with measure a in a right triangle, sin a = 113 15 and tan a = 112/15. What is the value of cos a ?

Choice A

$$\tan\alpha=\frac{\sin\alpha}{\cos\alpha} \\ $$ $$\Rightarrow \cos\alpha=\frac{\sin\alpha}{\tan\alpha}=\frac{\frac{112}{113}}{\frac{112}{15}}=\frac{15}{113}$$

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2014 December(72G) Math Question 18

Similar Triangles & Types of Triangles

- Triangle Types

18. In isosceles triangle ABC, AB is congruent to BC and the measure of the vertex angle, B, is 75. What is the measure of A?

Choice H

$$\angle A = \frac{180°-75°}{2}=52.5°$$

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2014 December(72G) Math Question 19

Algebraic & Function Expressions

- Solve Function Expressions

19. Cowling's rule is a method of calculating a medication dosage for a child by using the child's age and the adult dosage. The rule uses the formula d = a( t + 1 / 24 ), where d is the child's dosage in milligrams, a is the adult dosage in milligrams, and t is the child's age in years. Dato is taking a 150-milligram dosage of medication that has an adult dosage of 300 milligrams. According to this rule, which of the following values is closest to Dato's age in years?

Choice B

$$150=300(\frac{t+1}{24}) \\$$ $$ \Rightarrow t\approx 11$$

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2014 December(72G) Math Question 20

Graphing Properties

- Interpret Graphs or Figures

20. Each side of a square is 3 cm long. One vertex of the square is at (4,2) on a square coordinate grid marked in centimeter units. Which of the following points on the grid could be another vertex of the square?

Choice F

72g

As can be seen from the graph, only (7,2) is 3 cm from (4,2).

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2014 December(72G) Math Question 21

Angles

- Circles

21. The Ferris wheel at a state fair has a radius of 60 feet, rotates at a constant speed, and completes 1 rotation in 4 minutes. How many degrees does the Ferris wheel rotate in 30 seconds?

Choice D

$$\frac{30}{60\times4}\times360°=45°$$

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2014 December(72G) Math Question 22

Algebraic & Function Expressions

- Systems of Equations

22. At Gussie's Pizzeria, Odetta spends $8.75 (before tax) on the purchase of 2 appetizers and 3 slices of pizza. The price of each slice of pizza is p dollars. The price of each appetizer is twice the price of a slice of pizza. Which of the following systems of equations, when solved, gives the price, a dollars, of an appetizer and the price, p dollars, of a slice of pizza at Gussie's?

Choice F

"Odetta spends$8.75 on the purchase …

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2014 December(72G) Math Question 23

Perimeter, Area, Volume

- Other Shapes

23. The perimeter of a parallelogram is 76 inches, and the length of 1 side is 10 inches. If it can be determined, what are the lengths, in inches, of the other 3 sides?

Choice C

Since it is a parallelogram, the length of the other side is 10 inches, and that: $$\frac{76-10-10}{2}=28$$

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2014 December(72G) Math Question 24

Word Problems

- Cost

24. John told Raquel that if he spent up to $30 from his savings account, his savings account would have at least 3/4 as much in it as it has now. From John's statement, Raquel can deduce that the least amount of money that John could have in his savings account now is:

Choice K

$$x-30\geq \frac{3}{4}x \\ $$ $$\Rightarrow …

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2014 December(72G) Math Question 25

Probability

- One Event

25. At East High School, the 20 cast members of the spring musical sold 2 types of tickets—student and adult—for the Friday night and Saturday night performances. The price of each student ticket was $2 and the price of each adult ticket was $3. The table below gives the number of tickets sold, by type and by night. The stem-and-leaf plot below shows the number of tickets, regardless of type, sold by each of the 20 cast members. The auditorium where the spring musical will be performed has 10 seats in the 1st (front) row. Each row behind the 1st row has 4 more seats than does the row in front of it. Suppose 1 cast member will be picked at random from the 20 cast members who sold tickets to receive a prize. What is the probability of picking a cast member who sold more than 30 tickets?

Choice A

$$\frac{4}{20}=\frac{1}{5}$$

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2014 December(72G) Math Question 26

Word Problems

- 1 Variable Equations

26. For which night was the total amount collected for the tickets greater, and by how many dollars was it greater?

Choice J

$$100\times2+110\times3-122\times2-90\times3=16$$

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2014 December(72G) Math Question 27

Sequences and Patterns

- Arithmetic Sequences

27. How many seats are in the 8th row of the auditorium?

Choice A

$$10+4\times7=38$$

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2014 December(72G) Math Question 28

Perimeter, Area, Volume

- Triangles, Squares, Rectangles, Circles

28. In rectangle ABDE shown below, C lies on BD and F lies on AE such that rectangles ABCF and CDEF are similar (ABCF — CDEF), the length of AB is 4 cm, and the length of BC is 8 cm. What is the area, in square centimeters, of ABDE ?

Choice H

$$\frac{CD}{DE}=\frac{AB}{BC} \\$$ $$ \Rightarrow \frac{CD}{4}=\frac{4}{8} …

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2014 December(72G) Math Question 29

Ratios, Proportions, Percentages, Fractions

- Proportions & Ratios

29. Points A, B, C, and D lie on the real number line as shown below. The coordinate of B is 0, AC is 15 units long, BD is 20 units long, and AD is 32 units long. What is the coordinate of C ?

Choice A

$$AD=AB+BD=AC-BC+BD=32 \\$$ $$ \Rightarrow 15-BC+20=32 …

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2014 December(72G) Math Question 30

Exponents, Radicals, & Scientific Notation

- Scientific Notation

30. In a certain country, last year's population was estimated to be 3.6 x 10^6 people, and last year's public debt was estimated to be 1.8 x 10^9 dollars. Based on these estimates, what was last year's public debt per person in this country?

Choice H

$$\frac{1.8\times10^9}{3.6\times10^6}=500$$

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2014 December(72G) Math Question 31

Perimeter, Area, Volume

- Triangles, Squares, Rectangles, Circles

31. A landscaper will install a triangular garden near the intersection of 2 streets. In the plan of the garden shown below, the angle nearest the intersection is a right angle. The landscaper has purchased the following materials for the garden: 1 bag of fertilizer at a price of $15.50, 50 plants at a price of $0.70 each, and 3 bags of mulch at a price of $4.00 each. An 8% sales tax was added to the total price of these materials. The landscaper will also install a water sprinkler that will be located an equal distance from the 3 vertices of the garden. The landscaper calculated the area of the garden before purchasing the fertilizer, plants, and mulch. What is the area, in square feet, of the garden?

Choice C

$$\frac{9\times12}{2}=54$$

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2014 December(72G) Math Question 32

Word Problems

- Cost

32. What was the total cost, including sales tax, of the fertilizer, plants, and mulch that the landscaper purchased?

Choice G

$$(15.5+0.7\times50+4\times3)\times1.08=67.5$$

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2014 December(72G) Math Question 33

Graphing Properties

- Interpret Graphs or Figures

33. The plan of the garden will be aligned in the standard (x,y) cdordinate plane so that the right angle is at (0,0) and the other 2 vertices are at (12,0) and (0,-9). What coordinates give the location of the water sprinkler?

Choice E

72g

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2014 December(72G) Math Question 34

Perimeter, Area, Volume

- Triangles, Squares, Rectangles, Circles

34. The landscaper has decided to enclose the garden by installing a decorative fence of uniform height along the garden's perimeter. What is the minimum length, in feet, of fencing required to enclose the garden?

Choice K

$$h=\sqrt{9^2+12^2}=15 \\ $$ $$\Rightarrow 9+12+15=36$$

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2014 December(72G) Math Question 35

Similar Triangles & Types of Triangles

- Similar Triangles

35. Shown below are right triangles ACD and ABE with lengths given in inches. What is the length, in inches, of AB ?

Choice D

$$\frac{AB}{AC}=\frac{BE}{CD} \\ $$ $$\Rightarrow \frac{AB}{20}=\frac{5}{15} …

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2014 December(72G) Math Question 36

Perimeter, Area, Volume

- Triangles, Squares, Rectangles, Circles

36. The circumference of a circle is 50 cm. What is the length, in centimeters, of the diameter of the circle?

Choice G

$$2\pi r=2\pi \frac{d}{2}=50 \\$$ $$ …

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2014 December(72G) Math Question 37

Equation of a Line

- Distance & Midpoint

37. Which of the following values is closest to the distance, in inches, between point A and point B on the ruler shown below? (Note: The ruler shown is NOT actual size.) Back to top↑

Choice C

$$3.75-0.25-\frac{0.25}{2}=3.375$$

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2014 December(72G) Math Question 38

Algebraic & Function Expressions

- Solve Algebraic Expressions

38. If r + 5 = m and r + 6 = n, what is the value of n — m ?

Choice J

n-m=r+6-(r+5)=1

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2014 December(72G) Math Question 39

Angles

- Parallel Lines

39. In the figure below, lines l and m are parallel. Lines a and b intersect l at the same point. What is the value of x?

Choice C

$$(4x+12)+(3x+18)+(5x-18)=12x+12=180 \\$$ $$ x= 14$$

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2014 December(72G) Math Question 40

Perimeter, Area, Volume

- Mixed Shapes

40. The line y = 1/2x + 3 and the line y = 3x — 12 intersect at (6,6), as shown in the standard (x,y) coordinate plane below. The shaded region is bounded by these 2 lines, the x-axis, and the y-axis. What is the area of the shaded region, in square coordinate units?

Choice G

$$Area = 6\times6-\frac{6\times3}{2}-\frac{6\times2}{2}=21$$

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2014 December(72G) Math Question 41

Angles

- Vertex Angles & Bisectors

41. In the figure below, AD, BE, and CF all intersect at point Z. The measures of 2 angles are given. What is the measure of CZE ?

Choice E

$$\angle CZE=180-51=129$$

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2014 December(72G) Math Question 42

Word Problems

- Distance, Speed, Time

42. Emma and Kristin each ran 10 laps (4,000 meters) around a 400-meter track without stopping. Emma ran at a constant speed of 250 meters per minute. Kristin ran at a constant speed of 200 meters per minute. Both Emma and Kristin began running at the same instant. How many laps did Kristin have left to run when Emma had completed her run?

Choice J

$$\frac{4000-200\times\frac{4000}{250}}{400}=2$$

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2014 December(72G) Math Question 43

Equation of a Line

- Equations that Describe a Line

43. Three linear equations are given below. y= x y= x + 1 y = —x + 1 In terms of being parallel, being perpendicular, or intersecting, how are the graphs of these equations related in the standard (x,y) coordinate plane?

Choice D

Lines with the same slope …

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2014 December(72G) Math Question 44

Algebraic & Function Expressions

- Systems of Equations

44. For what value of a would the following system of equations have an infinite number of solutions?

Choice G

$$4a=8\times3 \\$$ $$ \Rightarrow a=6$$

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2014 December(72G) Math Question 45

Ratios, Proportions, Percentages, Fractions

- Proportions & Ratios

45. Given that x = 2 when y = 12 for the proportion x/6 = k/y, what is x when y = 8 ?

Choice C

$$\frac{2}{6}=\frac{k}{12} \\$$ $$ \Rightarrow k=4 …

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2014 December(72G) Math Question 46

Sequences and Patterns

- Logical Expressions

46. Two events are independent if the outcome of one event does not affect the outcome of the other event. One of the following statements does NOT describe independent events. Which one?

Choice K

The fact that 7 is …

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2014 December(72G) Math Question 47

Quadratic Equations

- Factoring

47. Which of the following binomials is a factor of 3x^2 + 11x — 4 ?

Choice E

$$3x^2+11x-4=(x+4)(3x-1)$$

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2014 December(72G) Math Question 48

Trigonometry

- Pythagorean Theorem & sin cos tan

48. The figure below shows a lighthouse keeper looking down at a rowboat on the sea through a navigational instrument. The instrument is 60 feet above sea level and indicates an angle of depression of 53° to the rowboat. Which of the following is closest to the horizontal distance, in feet, between the navigational instrument and the rowboat? (Note: sin 53°=0.80, cos 53°=0.60, tan 53°=1.33)

Choice G

$$\tan53°=\frac{60}{x} \\$$ $$ \Rightarrow x\approx …

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2014 December(72G) Math Question 49

Statistics

- Mean, Median

49. Each of 20 students in a class took a test and received a whole number score. The median of the scores was 87. None of the students received a score of 87, and 30% of the students received scores of 90 or above. How many students received scores of 88 or 89 ?

Choice B

$$10-20\times30\%=4$$

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2014 December(72G) Math Question 50

Perimeter, Area, Volume

- Other Shapes

50. The radius and height of the right circular cylinder shown below are given in centimeters. Which of the following is closest to the volume, in cubic centimeters, of the cylinder?

Choice K

$$\pi r^2 h=\pi 10^2 \times7\approx …

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2014 December(72G) Math Question 51

Logarithms

- Logarithm Expressions

51. What value of x satisfies the equation below?

Choice D

$$\log_{2^4}{x}=-\frac{3}{4} \\$$ $$ \Rightarrow \frac{1}{4}\log_{2}{x}=-\frac{3}{4} …

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2014 December(72G) Math Question 52

Number Types & Properties

- Rational & Irrational Numbers and Integers

52. Let x be a negative odd integer. The expression xy3 is a positive even integer whenever y is any member of which of the following sets?

Choice K

y must be negative even …

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2014 December(72G) Math Question 53

Graphing Properties

- Reflection, Rotation, Symmetry, Translation

53. Point I(3,4) is shown in the standard (x,y) coordinate plane below. Point I is reflected across the line x = 7 and, after the reflection, is labeled J. What are the coordinates of J ?

Choice E

72g

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2014 December(72G) Math Question 54

Trigonometry

- Pythagorean Theorem & sin cos tan

54. The angle measures of CDE are shown below. Point B of ABC lies on CD, and point C lies on AE. What is the value of cos(a + b) ?

Choice G

$$\cos(\alpha+\beta)=\cos(60°)=\frac{1}{2}$$

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2014 December(72G) Math Question 55

Matrices

- Determinants

55. The determinant of any 2 x 2 matrix is ad - bc. The determinant of is equal to 0. What are all possible values of x?

Choice A

$$(x+3)(x-2)-7\times2=0 \\$$ $$ \Rightarrow (x+5)(x-4)=0 …

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2014 December(72G) Math Question 56

Absolute Values

- Absolute Value Expressions

56. Which of the following inequalities is equivalent to (|x|+1)2<=4 ?

Choice K

$$|x|+1\leq 2 \\$$ $$ \Rightarrow …

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2014 December(72G) Math Question 57

Trigonometry

- Graphing Trig Functions

57. Given that 2 sin a = 2 and 2 cos(- b) = 2, which of the following could be a value, in radians, of a + b ?

Choice D

$$\sin \alpha =1 \Rightarrow \alpha …

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2014 December(72G) Math Question 58

Conics

- Ellipse

58. One of the following equations determines the graph in the standard (x,y) coordinate plane below. Which one?

Choice F

The center of the ellipse …

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2014 December(72G) Math Question 59

Perimeter, Area, Volume

- Triangles, Squares, Rectangles, Circles

59. Cubes each having a side length of 0.5 cm are put together to form a rectangular solid with 8 layers. Each layer has 6 cubes. What is the volume, in cubic centimeters, of the rectangular solid?

Choice A

$$0.5^3\times(6\times8)=6$$

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2014 December(72G) Math Question 60

Algebraic & Function Expressions

- Nested Functions

60. Consider the 2 functions f(x) = 2x + 5 and g(x) = 3x + b,

Choice J

$$f(g(x))=g(f(x)) \\$$ $$ \Rightarrow 2(3x+b)+5=3(2x+5)+b …

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