# Study Guide

## Field 004: Mathematics

### Sample Selected-Response Questions

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

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**Competency 0001**

Number and Quantity

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

Two matrices, A and B Matrix A start three by three matrix first row first column A sub eleven second column A sub twelve third column A sub thirteen second row first column A sub twenty one second column A sub twenty two third column A sub twenty three third row first column A sub thirty one second column A sub thirty two third column A sub thirty three end matrix Matrix B start three by three matrix first row first column A sub eleven second column A sub twelve third column A sub thirteen row two first column A sub thirty one second column A sub thirty two third column A sub thirty three third row first column A sub twenty one second column A sub twenty two third column A sub twenty three end matrix.

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?

- det(
*B*) = det(*A*) - det(
*B*) = det(*A*)

- Enter to expand or collapse answer. Answer expanded
**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*) =*a*_{11}*a*_{22}*a*_{33}+*a*_{12}*a*_{23}*a*_{31}+*a*_{13}*a*_{21}*a*_{32}*a*_{11}*a*_{23}*a*_{32}*a*_{12}*a*_{21}*a*_{33}*a*_{13}*a*_{22}*a*_{31}. Det(*B*) has the same six products as det(*A*), but with opposite signs, so det(*B*) = det(*A*).

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**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* = 2*x* squared minus 2*x* + 8 and *y* = 4*x* squared minus 6*x* + 3

- (
*x*minus 2) squared = - (
*x*minus 1) squared = - (
*x*minus 1) squared = - (
*x*minus 2) squared =

- Enter to expand or collapse answer. Answer expanded
**Correct Response: C.**This question requires the examinee to rewrite expressions in equivalent forms. To find the points of intersection, write 2*x*squared minus 2*x*+ 8 = 4*x*squared minus 6*x*+ 3 and use the properties of equality to rewrite this equation as2 Then complete the square so that (*x*squared minus 4*x*minus 5 = 0.*x*squared minus 2*x*+ 1) minus 2 minus 7 = 0 and 2(*x*minus 1) squared minus 7 = 0. Finally, write( *x*minus 1) squared = .

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**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*?

*f*(*t*) = 12 cos (2π*t*)*f*(*t*) = 16 cos (π*t*)*f*(*t*) = 12 sin (2π*t*) + 16*f*(*t*) = 16 sin (π*t*) + 4

- Enter to expand or collapse answer. Answer expanded
**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 (2π*t*) + 16.

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**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
- 5
- 6
- 11

- Enter to expand or collapse answer. Answer expanded
**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.

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**Competency 0005**

Geometry and Measurement

5. A student is assigned the following proof.

The student reasons as follows:

Draw a line between point

Aand pointC. Then,BCADAC. Also note thatBACDCA. Now,ACACso the two triangles are congruent by angle-side-angle.

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

- If two lines are cut by a transversal and same-side interior angles are congruent, then the lines are parallel.
- If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
- If two parallel lines are cut by a transversal, then same-side interior angles are congruent.
- If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

- Enter to expand or collapse answer. Answer expanded
**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*.

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**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?

- 1.75%
- 2.5%
- 3.5%
- 5%

- Enter to expand or collapse answer. Answer expanded
**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 × 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.