# 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?

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

**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*^{2} – 2*x* + 8 and *y* = 4*x*^{2} – 6*x* + 3

- (
*x*– 2)^{2}= - (
*x*– 1)^{2}= - (
*x*– 1)^{2}= - (
*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 2*x*^{2} 2*x*+ 8 = 4*x*^{2} 6*x*+ 3 and use the properties of equality to rewrite this equation as2 Then complete the square so that (*x*^{2} 4*x* 5 = 0.*x*^{2} 2*x*+ 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*?

*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

- 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 (2π*t*) + 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
- 5
- 6
- 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

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.

- 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?

- 1.75%
- 2.5%
- 3.5%
- 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.