# Study Guide

## Field 004: Mathematics

### Sample Selected-Response Questions

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

##### Competency 0001 Number and Quantity

1. Use the information below to answer the question that follows.

Matrix B is composed of the elements from matrix A, as shown above. Which statement describes the relationship between the determinants of the two matrices?

1. det(B) = det(A)

2. det(B) = det(A)

Correct Response: B. This question requires the examinee to demonstrate understanding of the properties of matrices. Rows 2 and 3 of matrix A have been interchanged to form matrix B. A theorem from linear algebra states: "If matrix B results from interchanging two rows or two columns of matrix A, then det(B) =  det(A)." Alternatively, calculate each determinant. Det(A) = a11a22a33 + a12a23a31 + a13a21a32  a11a23a32  a12a21a33  a13a22a31. Det(B) has the same six products as det(A), but with opposite signs, so det(B) =  det(A).

##### Competency 0002 Algebra

2. Which equation could be a step in finding the points of intersection of the graphs of the curves given by y = 2x squared minus 2x + 8 and y = 4x squared minus 6x + 3 when using the method of completing the square?

1. (x minus 2) squared =

2. (x minus 1) squared =

3. (x minus 1) squared =

4. (x minus 2) squared =
Correct Response: C. This question requires the examinee to rewrite expressions in equivalent forms. To find the points of intersection, write 2x squared minus 2x + 8 = 4x squared minus 6x + 3 and use the properties of equality to rewrite this equation as 2x squared minus 4x minus 5 = 0. Then complete the square so that (x squared minus 2x + 1) minus 2 minus 7 = 0 and 2(x minus 1) squared minus 7 = 0. Finally, write (x minus 1) squared = .

##### Competency 0003 Functions

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

The diagram above shows a wind turbine on a vertical shaft. The height of the shaft is 16 m. The length of a wind turbine blade is 12 m. Point P is located at the tip of one of the wind turbine blades. The wind turbine rotates in the counterclockwise direction at a constant rate of 1 revolution per second. At t = 0, point P is located at a height of 16 m above ground. Which function models the height of point P above ground as a function of t?

1. f(t) = 12 cos (2πt)
2. f(t) = 16 cos (πt)
3. f(t) = 12 sin (2πt) + 16
4. f(t) = 16 sin (πt) + 4
Correct Response: C. This question requires the examinee to model periodic phenomena with trigonometric functions. The height can be modeled by either a sine wave or a cosine wave. Since the length of the blade is 12 m, point P will rise to (16 + 12) m and fall to (16  12) m. Thus, the amplitude, A, of the wave is 12 and the axis of the wave on a t-y coordinate plane is y = 16. The period is 1 second and, using period = , B = 2π, which is the coefficient of t in a sine wave or cosine function. At t = 0, point P on the blade is at the height of the axis of the wave, so the sine function is the better choice. Using the general form of a sine wave, f(t) = A sin (Bt) + k and f(t) = 12 sin (2πt) + 16.

##### Competency 0004 Calculus

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

Graph strait line starting at open parenthesis negative two comma negative two close parenthesis staying constant until open parenthesis zero comma negative two close parenthesis and starts increasing steadily passing through open parenthesis one comma zero close parenthesis then continuing to increas up to open parenthesis three comma four close parenthesis then decreasing steadily down to open parenthesis four comma zero close parenthesis

The graph of a function f(x) is shown above. What is the value of ?

1. 1
2. 5
3. 6
4. 11
Correct Response: A. This question requires the examinee to interpret a definite integral as a net area. The expression is the negative of the area below the x-axis. Using geometry, the area is 4 for the square and 1 for the triangle, so = 5. Similarly = 6, the area of the triangle above the x-axis. Adding the two parts results in = 5 + 6 = 1.

##### Competency 0005 Geometry and Measurement

5. A student is assigned the following proof.

The student reasons as follows:

Draw a line between point A and point C. Then, BCA DAC. Also note that BAC DCA. Now, AC AC so the two triangles are congruent by angle-side-angle.

Which statement best justifies the second and third sentences in the student's response?

1. If two lines are cut by a transversal and same-side interior angles are congruent, then the lines are parallel.
2. If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
3. If two parallel lines are cut by a transversal, then same-side interior angles are congruent.
4. If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Correct Response: D. This question requires the examinee to apply theorems about lines and angles. AC is a transversal intersecting BC and AD. BCA and DAC are alternate interior angles. Because these angles are formed by parallel lines, they are congruent. AC is also a transversal intersecting AB and DC. The same reasoning holds for BAC and DCA.

##### Competency 0006 Statistics and Probability

6. A bottling company uses a machine to fill juice bottles. The quantity of juice that goes into each bottle is normally distributed, with a mean of 471.5 mL and a standard deviation of 1.75 mL. Approximately what percentage of the bottles receives less than 468 mL?

1. 1.75%
2. 2.5%
3. 3.5%
4. 5%