Test Information Guide

Field 63: Mathematics
Sample Multiple-Choice Questions

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The following reference material will be available to you during the test:

Formulas PDF opens in new window.

Candidates taking the Mathematics test (field 63) will be provided with an on-screen scientific calculator with functions that include the following: addition, subtraction, multiplication, division, square root, percent, sine, cosine, tangent, exponents, and logarithms. You may NOT bring your own calculator to the test.

Number Sense and Operations

Objective 0001
Apply knowledge of the properties and structure of the real number system.

1. A student claims that performing an operation of the type the value a over the value b minus the value c over the value d requires determining the smallest integer that is divisible by b and d. For the student's claim to be true, which of the following statements must also be true?

  1. a, b, c, and d are positive integers.
  2. b and d are relatively prime.
  3. a and c are relatively prime.
  4. b and d are both integers.
Answer
Correct Response: D.

Correct Response: D.


Relations, Functions, and Algebra

Objective 0004
Apply the principles and properties of relations and functions.

2. Use the graph below to answer the question that follows.

Given the graph of f of x shown, what is the value of 2 times the function of negative 1 minus the function of 3 all over the function of 0?

  1. undefined
  2. 0
  3. 1
  4. 3
Answer
Correct Response: C.

Correct Response: C.


Objective 0005
Apply the principles and properties of linear, absolute value, and quadratic relations and functions.

3. Over the course of a summer, a child sets up a lemonade stand beside a busy bicycle path. When the child charges $0.25 per cup, 75 cyclists purchase lemonade each day, and when the child charges $0.50 per cup, 60 cyclists purchase lemonade each day. Assuming that lemonade sales can be modeled by a linear function, how much money should the child expect to collect at the end of a day on which the price of lemonade is $0.70 per cup?

  1. $33.60
  2. $39.00
  3. $42.00
  4. $50.40
Answer
Correct Response: A.

Correct Response: A.


Objective 0006
Apply the principles and properties of exponential and logarithmic relations and functions.

4. Use the table below to answer the question that follows.

x y
0.1 1.5
square root of 10 0
1000 negative 2.5

An equation written in the form y = m times log of x + n could represent the relation shown in the table, where m and n are constants. What is the value of n?

  1. 2.5
  2. 0.5
  3. negative 0.5
  4. negative 5.5
Answer
Correct Response: B.

Correct Response: B.


Objective 0008
Apply the principles and properties of trigonometric functions and identities.

5. Which of the following expressions is equivalent to 1 over cosecant theta minus sine theta?

  1. tangent squared theta
  2. tangent theta times secant theta
  3. cotangent squared theta
  4. cotangent theta times cosine theta
Answer
Correct Response: B.

Correct Response: B.


Geometry and Measurement

Objective 0009
Apply the principles, concepts, and procedures related to units and measurement.

6. Two right triangular prisms are similar. Each edge of the smaller prism measures one-third of the length of the corresponding edge in the larger prism. If the total surface area of the larger prism is 216 square inches, what is the total surface area of the smaller prism?

  1. 24 square inches
  2. 36 square inches
  3. 60 square inches
  4. 72 square inches
Answer
Correct Response: A.

Correct Response: A.


Objective 0010
Apply the axiomatic structure of Euclidean geometry.

7. Use the diagram below to answer the question that follows.

Two watchtowers located at points A and B have visual contact with a fire located at point C. The watchtowers are 50 kilometers apart and have views of the fire at the angles shown in the diagram. Which of the following equations can be solved to determine d, the distance between the fire and the watchtower located at point A?

  1. d sine 30 degrees = 50 sine 110 degrees
  2. 50 sine 40 degrees = d sine 30 degrees
  3. d sine 110 degrees = 50 sine 40 degrees
  4. 50 sine 30 degrees = d sine 110 degrees
Answer
Correct Response: B.

Correct Response: B.


Objective 0012
Apply the principles and properties of coordinate and transformational geometry and the characteristics of non-Euclidean geometries.

8. If the graph of the value y squared over 9 minus the value x squared over 16 = 1 undergoes a 90 degrees clockwise rotation about the origin, which of the following points is a focus of the transformed graph?

  1. ( negative 5, 0)
  2. ( negative 3, 0)
  3. (0, 4)
  4. (0, 5)
Answer
Correct Response: A.

Correct Response: A.


Probability, Statistics, Calculus, and Discrete Mathematics

Objective 0013
Apply the principles, properties, and techniques of probability.

9. The owners of an orange grove project a profit for the next growing season of $240,000. However, it is predicted that there is a 12% chance that the grove will experience a damaging insect infestation. In that case, the projected profit is $60,000. Profit-protection insurance of $240,000 is available for $30,000. In order to decide whether to buy the insurance, the owners calculate the expected profit with and without insurance. Without insurance, the expected profit is $218,400. What is the expected profit if the owners do buy insurance?

  1. $217,200
  2. $236,400
  3. $243,600
  4. $270,000
Answer
Correct Response: A.

Correct Response: A.


Objective 0015
Apply principles and techniques of limits, continuity, and differential calculus.

10. Which of the following statements about the function y = 6 x over the quantity x squared plus 1 is true?

  1. No maximum or minimum value of the function exists.
  2. The maximum value of the function is 3.
  3. The maximum value of the function occurs at x = 1 and x = negative 1.
  4. The minimum value of the function is 0.
Answer
Correct Response: B.

Correct Response: B.