Study Guide

Field 004: Mathematics

Expand All | Collapse All

Sample Selected-Response Questions

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

Formulas PDF document

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)

Answer
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 = 2x2 – 2x + 8 and y = 4x2 – 6x + 3 when using the method of completing the square?

  1. (x – 2)2 =

  2. (x – 1)2 =

  3. (x – 1)2 =

  4. (x – 2)2 =
Answer
Correct Response: C. This question requires the examinee to rewrite expressions in equivalent forms. To find the points of intersection, write 2x2 – 2x + 8 = 4x2 – 6x + 3 and use the properties of equality to rewrite this equation as 2x2 – 4x – 5 = 0. Then complete the square so that (x2 – 2x + 1) – 2 – 7 = 0 and 2(x – 1)2 – 7 = 0. Finally, write (x – 1)2 = .

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
Answer
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.

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

  1. 1
  2. 5
  3. 6
  4. 11
Answer
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.
Answer
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%
Answer
Correct Response: B. This question requires the examinee to demonstrate understanding of a normal probability distribution. Using the given mean and standard deviation, 471.5 – 2 Χ 1.75 = 468, so the values in question are more than 2 standard deviations below the mean. In a normal distribution, approximately 95% of the data falls within 2 standard deviations of the mean, so about 5% falls outside that region. Since the normal distribution is symmetric, 2.5% of the data falls more than 2 standard deviations above the mean and 2.5% falls more than 2 standard deviations below the mean.