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Praxis Math 5165

Question 7

Praxis Math 5165. In a group of 100 high school students, 30% are on the high honor roll and 70% are on at least one athletic team. A student is to be selected at random from the group. If the two events “selected student is on the high honor roll” and “selected student is on at least one athletic team” are independent, what is the probability that the student selected will be on both the high honor roll and at least one athletic team?

  • 0.10
  • 0.21
  • 0.37
  • 1.00

Question 8

If x > 8, which of the following expressions is equivalent to the reciprocal of the expression (x^2 + 5x – 14) / (x^2 – x – 56)?

  • (x + 8) / (x + 2)
  • (x + 8) / (x – 2)
  • (x – 8) / (x + 2)
  • (x – 8) / (x – 2)

Question 9

When simplifying the expression sqrt(x^2 – y^2), Tonya is certain that the answer should be x – y. She claims that you can simply take the square root of each term inside the radical and then subtract the roots.

Which of the following questions illustrates Tonya’s misconception and would best help her realize the claim is false?

  • Are the values of the expressions sqrt(30 – 20) and 30 – 20 equal?
  • Are the values of the expressions sqrt(30 – 20) and sqrt(30) – sqrt(20) equal?
  • Are the values of the expressions sqrt(25 – 16) and 25 – 16 equal?
  • Are the values of the expressions sqrt(25 – 16) and sqrt(25) – sqrt(16) equal? Praxis Math 5165

Question 10

Function f is defined by f(x) = x^3 – 5x^2 + 3x – 2 for all real numbers x. What is the value of f'(10), the derivative of f at x = 10?

  • 203
  • 528
  • 1,000
  • 2,030

Question 11

A rectangular piece of cardboard 20 inches by 20 inches was used to make an open box by cutting squares with side lengths of x inches out of the corners and folding along the dotted lines shown in the following figure.

Praxis Math 5165 What is the volume, in cubic inches, of the box in terms of x?

  • x(20 – 2x)
  • x(20 – 2x)^2
  • x^2(20 – 2x)
  • x^2(20 – 2x)^2

Question 12

A student was asked to determine the domain of the function f defined by f(x) = (x^2 – 4) / (x + 2) for all real numbers x for which f(x) is a real number. The student drew a linear graph in the xy-plane.

The student then stated, “Since x – 2 is equivalent to (x^2 – 4) / (x + 2), the graph of f is a straight line. The domain of f is all real numbers because the graph of f is a straight line.”

Which of the following statements describe a flaw in the student’s work or response? Select ALL that apply.

  • The student stated that x – 2 is equivalent to (x^2 – 4) / (x + 2), but the two expressions are not equivalent when x = -2.
  • The student stated that the domain of the function f is all real numbers, but the function f is not defined when x = -2.
  • The student drew the graph of f as a straight line, but the graph of f is not defined when x = -2.

Question 13

The following table shows the cost, c, of a first-class postage stamp for seven values of t, where t represents the number of years after 1990.

Cost of a First-Class Postage Stamp, 1990 to 2013 | Time t (years) | 0 | 3 | 6 | 10 | 15 | 20 | 23 | | :— | :— | :— | :— | :— | :— | :— | :— | | Cost c (cents) | 25 | 29 | 32 | 33 | 37 | 44 | 46 |

What is the average rate of change, in cents per year, of the cost with respect to time from 1990 to 2013?

  • 1.167
  • 1.095
  • 0.913
  • 0.876

Question 27

Praxis Math 5165. A probability experiment consists of two steps. Step 1 is to choose a ball from bowl 1, which contains 7 red balls and 3 white balls. Step 2 is to choose a ball from bowl 2, which contains 10 red balls and 15 white balls. What is the probability that a white ball will be chosen from bowl 1 and a red ball will be chosen from bowl 2?

  • 0.04
  • 0.12
  • 0.42
  • 0.70

Question 28

A student’s work to graph the function f(x) = (x^2 – 4) / (x + 2) is shown. The student simplified the function to f(x) = x – 2 and then graphed the line y = x – 2. Which of the following statements describe a flaw in the student’s work? Select all that apply.

  • The expressions (x^2 – 4) / (x + 2) and x – 2 are not equivalent at x = -2.
  • The domain of the function f is not all real numbers.
  • The graph of the function f is not a solid line.

Question 32

A clothing store charges a flat fee of 6 to ship an order of up to 5 coats. For an order of more than 5 coats, the store charges the flat fee plus 3 for each coat in the order beyond the first 5. Which of the following functions represents the shipping charge, in dollars, for an order of k coats, where k > 5?

  • f(k) = 6 + 3k
  • f(k) = 6 + 3(k – 5)
  • f(k) = 5 + 3(k – 6)
  • f(k) = 3 + 6(k – 5)

Question 33

The table shows the probability distribution for the daily demand for rental cars at a local agency. What is the mean daily demand for rental cars?

  • 10.0
  • 10.7
  • 11.4
  • 12.0

Question 34

A student was asked to find the product of (5x + 4) and (-2x – 4). The student’s work is shown. In which step did the student make a mistake? Step 1: 5x(-2x) + 5x(-4) + 4(-2x) + 4(-4) Step 2: -10x^2 – 20x – 8x – 16 Step 3: -10x^2 – 28x – 16

  • Answer: The student performed the calculation correctly (no mistake in steps shown).

Question 35

In the figure shown, segment AB is parallel to segment CD and segment AB is congruent to segment CD. To prove that triangle ABC is congruent to triangle CDA, a student wrote a proof. Which of the following is the best justification for statement 4?

  • Side-Side-Side congruence postulate
  • Side-Angle-Side congruence postulate
  • Angle-Side-Angle congruence postulate
  • Angle-Angle-Side congruence theorem

Question 37

The graph of the quadratic function f(x) = a(x – 1)^2 + 6 is shown in the xy-plane. What is the value of a?

  • -2
  • -2/3
  • 2/3
  • 2

Question 38

A student is creating a tree diagram to show the possible outcomes of spinning each of the two spinners shown one time. Which of the following is a true statement about the tree diagram?

  • The tree diagram will have 4 paths.
  • The tree diagram will have 6 paths.
  • Since the outcomes are not equally likely, the tree diagram should include probability labels for each branch of the diagram.

Question 39

The line shown in the xy-plane is the graph of y = f(x). For what value of x is f(x) = f^{-1}(x)?

  • -8
  • -6
  • -4
  • 10

Question 40

Based on the table, approximately what percent of the students who completed the review sheet failed the test?

  • 12%
  • 16%
  • 19%
  • 46%

Question 41

A regular octagon ABCDEFGH is inscribed in a circle with center O. The octagon is transformed by first reflecting it across diameter AE, and then rotating the reflected octagon 90 degrees clockwise. Which of the following points corresponds to the image of C as a result of the two transformations?

  • A
  • E
  • F
  • H

Question 42

Which of the following limits can be evaluated with direct substitution but without the use of additional algebraic manipulation or the use of trigonometric identities? Select all that apply.

  • lim x -> 0 (4x – x^2)
  • lim x -> 1 (x^2 – 1) / (x – 1)
  • lim x -> -2 (x^2 + 7x + 10) / (x + 2)
  • lim x -> pi/2 tan(x)cos(x)
  • lim x -> 10 sqrt(x – 8)
  • lim x -> 9 (x – 9) / (sqrt(x) – 3)

Question 43

The complex numbers z1 and z2 are represented by the two points in the xy-plane. Which of the following is equal to z1 + z2, where i = sqrt(-1)?

  • 4 – i
  • 2 – i
  • -1 + 4i
  • -1 + 2i

Question 44

Mary deposited 400 into a new investment account that pays interest at a fixed annual rate, compounded continuously. The amount of money in the account after 10 years was 1,000. Which of the following best approximates the annual interest rate on the account?

  • 8.5%
  • 8.7%
  • 9.2%
  • 10.5%

Question 45

The graphs of the functions f and g are shown in the figure. What is the value of the left-sided limit lim x -> 1^- (f(x) + g(x))?

  • 6
  • 7
  • 8
  • 9

Question 46

A student completed the square in the equation -2x^2 + 16x = 30. Which of the following exhibit the same misconception that the student’s work does?

  • 3x^2 – 18x = 48
  • 2x^2 + 36x = 38
  • -4x^2 + 16x = -20 -> -4(x^2 + 4x) = -20

Question 47

The figure represents a plan for a bridge across a small lake. Approximately how long, in miles, will the bridge be?

  • 0.8
  • 1.0
  • 1.2
  • 1.4

Question 48

The daily profits of a small business are normally distributed with a mean of 1,500 and a standard deviation of 200. What is the approximate percent of the distribution of daily profits that are either less than 1,100 or greater than 1,700?

  • 13%
  • 18%
  • 27%
  • 32%

Question 49

A rectangular quilt consists of 10 rows of square patches, with 12 patches per row. Each square patch has a side length of 9 inches. There is a uniform 1-inch-wide strip of material between squares and a uniform 2-inch-wide border around the outer edge of the quilt. What are the outside dimensions of the rectangular quilt?

  • 90 inches x 108 inches
  • 101 inches x 122 inches
  • 103 inches x 123 inches
  • 104 inches x 125 inches

Probability Question

In a group of 100 high school students, 30% are on the high honor roll and 70% are on at least one athletic team. A student is to be selected at random from the group. If the two events “selected student is on the high honor roll” and “selected student is on at least one athletic team” are independent, what is the probability that the student selected will be on both the high honor roll and at least one athletic team?

  • 0.10
  • 0.21
  • 0.37
  • 1.00

Question 49

A rectangular quilt consists of 10 rows of square patches, with 12 patches per row. Each square patch has a side length of 9 inches. There is a uniform 1-inch-wide strip of material between squares and a uniform 2-inch-wide border around the outer edge of the quilt. What are the outside dimensions of the rectangular quilt?

  • 90 inches x 108 inches
  • 101 inches x 122 inches
  • 103 inches x 123 inches
  • 104 inches x 125 inches

Question 50

Ms. Day’s students tried to solve several systems of linear equations but found that systems such as the following had no solutions: y = 4x + 3 2y = 8x + 5 In the systems with no solutions that were studied, the students noticed that the ratios of the coefficient of x to the coefficient of y for the two equations are equal, as in the example 4/1 = 8/2. Ms. Day would like to ask a follow-up question that would help the students realize that if a system has no solution, then the graphs of the two equations in the xy-plane are parallel lines. Which of the following is the best question that Ms. Day can ask the students to accomplish her goal?

  • What do the equal ratios tell you about the relationship between the slopes of the two lines representing the two linear equations in the system?
  • What are all possible values of the ratios for all systems of equations with no solutions?
  • What number do you need to multiply the first equation by so that the left-hand side of the resulting equation will be equal to the left-hand side of the second equation?
  • Are the ratios of the coefficient of y to the coefficient of x for the two equations also equal?

Question 51

The complex number 1 + sqrt(3)i is a root of the quadratic polynomial p(x), which has real-number coefficients. What is the sum of the roots of p(x)?

  • 0
  • 2
  • i
  • 2sqrt(3)

Question 52

At the beginning of a lesson on parallel lines, a student proposed the following definition: “Two lines are parallel if they never intersect.” The teacher wants to select an example of a pair of lines that would help guide the student to modify the proposed definition. Which of the following examples would best accomplish the teacher’s goal?

  • The two lines in a picture representing the two rails of a railroad track
  • A line on the ceiling running north to south and a line on the floor running east to west
  • Two lines drawn on the board, intersected with a transversal and having congruent alternate exterior angles
  • The line at the intersection between a wall and the ceiling and the line at the intersection between the same wall and the floor

Question 53

When designing a curve on a highway, engineers use the following equation, which models the proper banking angle theta for a car making a turn of radius r feet at a speed of v feet per second: tan(theta) = v^2 / 32r If an engineer is designing a curve with a radius of 1,000 feet, and the greatest possible speed on the curve will be 60 miles per hour, approximately at what angle should the curve be banked? (Note: 1 mile = 5,280 feet)

  • 10.0 degrees
  • 12.0 degrees
  • 13.6 degrees
  • 15.0 degrees

Question 54

The shaded region in the preceding xy-plane is the solution set of which of the following systems of inequalities?

  • y <= -x – 1 and y <= x/2 – 2
  • y <= -x – 1 and y >= x/2 – 2
  • y >= -x – 1 and y <= x/2 – 2
  • y >= -x – 1 and y >= x/2 – 2

Question 55

Marie solved for the value of x in a triangle with a 30 degree angle and a hypotenuse of 9. Marie claimed that x = 18. Which of the following statements from her classmates are correct counterarguments to Marie’s claim? Select ALL that apply.

  • “We know that the larger the angle inside a triangle, the longer the side that is across from it. Since the box marks a 90 degree angle, the side across from it must be larger than the side across from the 30 degree angle. So x must be less than 9.”
  • “We know that in a right triangle, the hypotenuse, the side across from the right angle, is always longest. If x is 18, the hypotenuse, which is 9, is not the longest side. This is a contradiction.”
  • “We can use the equation sin(30) = x/9 to solve for x. Since sin(30) = 1/2, x = 4.5, not 18.”

Question 56

A study of a deer population in a national park found that the number of deer, P(x), can be modeled by a sinusoidal curve over a 20-year period. The study began in 1975. In 1980 the number reached a minimum of 400. In 1990, it reached a maximum of 900. Which of the following functions correctly models the number of deer x years after 1975?

  • P(x) = 250sin(pi/10 * x + pi) + 650
  • P(x) = 650sin(pi/10 * x + pi) + 250
  • P(x) = 400sin(pi/10 * x + pi) + 900
  • P(x) = 250sin(pi * x + pi/10) + 650

Question 57

The endpoints of line segment L are (-2, 5) and (6, -3). Which of the following is an equation of the perpendicular bisector of L?

  • y = x – 1
  • y = x + 8
  • y = -2x + 5
  • y = -x + 3

Question 58

The graph of the function f(x) = sqrt(|x|) is shown. Which of the following is true of the function f at x = 0?

  • It is neither continuous nor differentiable.
  • It is continuous but not differentiable.
  • It is differentiable but not continuous.
  • It is both continuous and differentiable.

Question 59

A teacher wants to select a function f that does not have an inverse but that has an inverse when the domain is restricted. Which of the following functions best accomplishes this goal?

  • f(x) = sqrt(x – 4) + 1
  • f(x) = e^(x+2) – 3
  • f(x) = ln(x + 1) – 5
  • f(x) = (x – 3)^2 + 2

Question 60

Two transformations are performed on f(x) = x^2. First, it is translated 2 units in the positive y-direction. Then, it is translated 3 units in the positive x-direction to obtain g(x). Which is the equation for g(x)?

  • g(x) = x^2 + 6x + 11
  • g(x) = x^2 – 6x + 11
  • g(x) = x^2 + 4x + 7
  • g(x) = x^2 + 3x + 2

Question 61

Which of the following is equivalent to 3^x * 4^(x+2) for all values of x?

  • 8 * 7^(2x)
  • 8 * 12^(2x)
  • 16 * 7^x
  • 16 * 12^x

Question 62

In triangle ABC, triangle ABE has an area of 20. The length of AE is 8. In triangle BEC, BE = BC, and EC = 12. If angle BDE is a right angle, what is the area of triangle BDE?

  • 12 square inches
  • 15 square inches
  • 18 square inches
  • 20 square inches

Question 63

If cos(theta) > 0 and tan(theta) < 0, where 0 < theta < 2pi, which of the following must be true?

  • 0 < theta < pi/2
  • pi/2 < theta < pi
  • pi < theta < 3pi/2
  • 3pi/2 < theta < 2pi

Question 64

The graph of f passes through (1, 2). Which of the following is equal to f'(1)?

  • lim h->0 [f(h + 2) – f(2)] / h
  • lim h->0 [f(h – 2) – f(2)] / h
  • lim h->0 [f(h – 1) – f(1)] / h
  • lim h->0 [f(h + 1) – f(1)] / h

Question 65

Ms. Duncan wants to select a set where sums and products of irrational numbers include both rational and irrational values. Which set best fits this goal?

  • {sqrt(12), 1 + sqrt(2), sqrt(3)}
  • {8 – sqrt(2), 1 + sqrt(2), 1 + sqrt(3)}
  • {1 – sqrt(3), sqrt(3), sqrt(12)}
  • {-sqrt(3), sqrt(3), sqrt(12)}

Question 66

Acceleration is given by a(t) = -32. Which of the following could be the position function s(t)?

  • s(t) = -32t + 10
  • s(t) = -32t^2 + 10t – 3
  • s(t) = -16t + 10
  • s(t) = -16t^2 + 10t – 3
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Isaura K

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