# Test Information Guide

## Field 09: Mathematics

Sample Multiple-Choice Questions

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

FormulasCandidates taking the Mathematics test (field 09) 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 0002**

Understand the properties of real and complex numbers and the real and complex number systems.

One. Use the proof below to answer the question that follows. Given x plus z equals y plus z. Prove x equals y Proof Start Layout first row x plus z equals y plus z second row open parenthesis x plus z close parenthesis plus open parenthesis negative z close parenthesis equals open parenthesis y plus z close parenthesis plus open parenthesis negative z close parenthesis third row x plus open brace z plus open parenthesis negative z close parenthesis close brace equals y plus open brace z plus open parenthesis negative z close parenthesis close brace fourth row x plus zero equals y plus zero fifth row x equals y. end layout. Which of the following properties of the real numbers is used to justify one of the steps in the above proof?

- associative
- distributive
- commutative
- transitive

- Answer
**Correct Response: A.**

Correct Response: A.

**Objective 0003**

Understand the principles of number theory.

2. Use the diagram below to answer the question that follows. Two gears. Gear A and Gear B. Gear A is larger than Gear B. Gear A has fourteen teeth labeled one to fourteen. Gear B has twelve teeth labeled one to twelve. The tooth labeled one on Gear B is in the groove on Gear A between tooth fourteen and tooth one. Gear A is moving clockwise and Gear B is moving counter clockwise.

- 2
- 6
- 42
- 84

- Answer
**Correct Response: B.**

Correct Response: B.

### Patterns, Relations, and Algebra

**Objective 0005**

Understand the properties of functions and relations.

3. Which of the following graphs represents a one-to-one function?

graph with two curves, first curve is in the third quadrant starting at the left close to the x access and decreasing slowly at first then very quickly as it gets closer to the y axis second curve is in the first quadrant starts high close to the y axis then decreases quickly at first then decreases much slower as it approaches the x axis

graph with a curved continuous function starting in third quadrant increasing quickly past the x axis into the second quadrant where it slows its increase to a peak then starts to decrease slowly then quicker as it passes the y axis into the first quadrant past the x axis into the fourth quadrant where it slows its decrease to a low peak then starts to increase quickly past the x axis back up into the first quadrant where it continues to increase.

- Answer
**Correct Response: C.**

Correct Response: C.

**Objective 0008**

Understand the properties and applications of quadratic relations and functions.

4. **Use the diagram below to answer the question that follows.**

The four corners of a square *PQRS* have been cut off to form a regular octagon, as shown in the diagram above. If each side of the square is 1 cm long, which of the following equations should be solved to find the length of each side of the octagon, *x*?

*x*^{2}+ 2*x*– 1 = 0*x*^{2}– 4*x*+ 2 = 0- 3
*x*^{2}+ 2*x*– 1 = 0 - 7
*x*^{2}– 8*x*+ 2 = 0

- Answer
**Correct Response: A.**

Correct Response: A.

**Objective 0010**

Understand the properties and applications of exponential and logarithmic functions and relations.

5. If log_{x}10 = 0.9603, which of the following is the best approximation of *x*?

- 7
- 9
- 11
- 13

- Answer
**Correct Response: C.**

Correct Response: C.

### Geometry and Measurement

**Objective 0013**

Prove theorems within the axiomatic structure of Euclidean geometry.

6. **Use the geometric proof below to answer the question that follows.**

For the proof, there is a diagram of a circle with center point, P. Point R is on the circle at 12:00, and point S is on the circle at about 8:30. Radius PR and radius PS are drawn. A line is drawn between points R and P. Point Q is drawn on line RS, and a line is drawn between point Q and center point P.

Given: Circle P;

Line PQ is a median of triangle PRS.
Prove: Triangle RPQ is congruent to triangle SPQ.

Statements and Reasons.

Statement 1, Circle P; line PQ is a median of triangle PRS.
Reason, Given.

Statement 2, Line RQ is congruent to line QS.
Reason, Definition of median.

Statement 3, Line PR and line PS are radii of P.
Reason, Definition of radius.

Statement 4, Line PR is congruent to line PS.
Reason, All the radii of a circle are congruent.

Statement 5, Question mark.
Reason, Question mark.

Statement 6, Triangle RPQ is congruent to triangle SPQ.
Reason, SSS postulate.

Which of the following statements and reasons would be most appropriate in step 5 of this proof?

- Answer
**Correct Response: D.**

Correct Response: D.

**Objective 0015**

Understand the principles and properties of coordinate and transformational geometry and characteristics of non-Euclidean geometries.

7. The center of a town is located at (0, 0) on an *x-y* coordinate system with each grid unit measured in miles. A radio tower is located at the point (15, –20). The radio signal is strong enough to reach homes within a 40-mile radius. Which of the following inequalities represents all ground locations within 40 miles of the radio tower?

- (
*x*– 15)^{2}+ (*y*+ 20)^{2}≤ 40 - (
*x*+ 15)^{2}+ (*y*– 20)^{2}≤ 40 - (
*x*– 15)^{2}+ (*y*+ 20)^{2}≤ 1600 - (
*x*+ 15)^{2}+ (*y*– 20)^{2}≤ 1600

- Answer
**Correct Response: C.**

Correct Response: C.

### Data Analysis, Statistics, and Probability

**Objective 0016**

Understand the principles and concepts of descriptive statistics and their application to the problem-solving process.

8. **Use the diagram below to answer the question that follows.**

Bar graph showing the number of cars on the y axis and the age of cars on the x axis where the standard deviation sigma equals two point eight six and the mean mu equals six point four eight. Now the data of cars and their ages. two cars that are one year old, five cars that are two years old, two cars that are three years old five cars that are four years old, six cars that are five years old, three cars that are six years old, four cars that are seven years old, seven cars that are eight years old, six cars that are nine years old and ten cars that are ten years old.

The graph shows the distribution of the age of 50 cars in a parking lot. Rounded to the nearest whole number, the mean, μ, is 6 years and the standard deviation, σ, is 3 years. How many cars have an age within one standard deviation of the mean?- 17
- 20
- 33
- 41

- Answer
**Correct Response: C.**

Correct Response: C.

### Trigonometry, Calculus, and Discrete Mathematics

**Objective 0019**

Understand the properties of trigonometric functions and identities.

9. **Use the graph below to answer the question that follows.**

Graph showing continuous curve starting in the fourth quadrant increasing through the origin up to y equals one half then decreasing down to the x axis at pi over two back into the fourth quadrant down to y equals negative one half then increasing back up to the x axis at pi and continuing into the first quadrant.

The graph of which of the following equations is shown in the figure above?*y*= cos (2*x*)*y*= 2 cos*y*= sin (2*x*)*y*= 2 sin

- Answer
**Correct Response: C.**

Correct Response: C.

**Objective 0022**

Understand integral calculus.

10. **Use the graph below to answer the question that follows.**

Graph showing velocity in miles per hour on the y axis and times in hours on the x axis. Starting at open parenthesis zero comma one hundred close parenthesis there is a steadily increasing line up to open parenthesis one half comma six hundred close parenthesis where the velocity stays constant until three point five hours where it steadily decreases until open parenthesis four point five comma one hundred close parenthesis

The graph above approximates an airplane's velocity,*v*, in miles per hour, during a four-hour-and-thirty-minute flight. The plane left the ground at a velocity of 100 mph and took 30 minutes to reach its cruising speed of 600 mph. One hour before landing it began to descend and landed at a velocity of 100 mph. Which of the following is the best estimate of the total distance in miles that the airplane flew while in the air?

- 1875
- 2175
- 2250
- 2325

- Answer
**Correct Response: D.**

Correct Response: D.