Study Guide

Mathematics
Sample Questions

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Competency 0001
Understand real numbers and mathematical problem solving.

1. If p and q are prime numbers and what is the value of (p + q)?

  1. 5
  2. 7
  3. 8
  4. 9
Enter to expand or collapse answer.Answer expanded
Correct Response: B. The variables can be isolated by multiplying both sides of the equation by 50q3, which yields 200 = p2q3. If p and q are both prime, then p2q3 is the prime factorization of 200. Since 200 = 25 × 8 = 52 × 23, and 5 and 2 are both primes, p must be 5 and q must be 2, so p + q = 5 + 2 = 7.

Competency 0002
Understand complex numbers, vectors, and matrices.

2. A two-dimensional vector v is drawn in standard position on a coordinate plane. Let θ be the angle made by the vector and the positive x-axis. Which of the following vectors represents v as an ordered pair?

  1. (Ι v Ι cos θ, Ι v Ι sin θ)
  2. (v sin θ, v cos θ)
  3. (v cos θ, v sin θ)
  4. (Ι v Ι sin θ, Ι v Ι cos θ)
Enter to expand or collapse answer.Answer expanded
Correct Response: A. The figure below shows vector v, drawn in standard position on a coordinate plane, and represented by the coordinates (x, y).



Then sin θ = , where Ι v Ι represents the length of v, so y = Ι v Ι sin θ. Also, cosθ = , and x = Ι v Ι cos θ, so v can be represented by the ordered pair (Ι v Ι cos θ, Ι v Ι sin θ).

Competency 0003
Understand relations and functions.

3. Which of the following equations represents the inverse of y = ?




Enter to expand or collapse answer.Answer expanded
Correct Response: D. To find the inverse of a function of the form y = f(x), the original equation is rearranged by solving it for x as a function of y: y = ⇒ y(1 + 3x) = 6x – 4 ⇒ y + 3xy = 6x – 4 ⇒ y + 4 = 6x – 3xy ⇒ y + 4 = x(6 – 3y) ⇒ x = . Exchanging the variables x and y results in the inverse function f –1, y = .

Competency 0004
Understand linear, quadratic, and higher-order polynomial functions.

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

Given the table of orders and total costs above, and that there is a solution to the problem, which of the following matrix equations could be used to find d, p, and g, the individual prices for a soft drink, a large pizza, and garlic bread respectively?




Enter to expand or collapse answer.Answer expanded
Correct Response: D. The system of linear equations can be solved using matrices. Each order can be expressed as an equation, with all three equations written with the variables in the same sequence. The first order is represented by the equation 4d + p + g = 19.62, the second order by 6d + 2p + g = 34.95, and the third order by 3d + p = 16.50. The rows of the left-hand matrix contain the coefficients of d, p, and g for each equation: (4 1 1), (6 2 1), and (3 1 0). The middle matrix contains the variables, d, p, g. The right-hand matrix vertically arranges the constants of the equations.

Competency 0005
Understand exponential and logarithmic functions.

5. Which of the following is equivalent to the equation 3 log10 x – 2 log10 y = 17?

  1. 3x – 2y = 1017

  2. x3 – y2 = 1017

  3. = 1017

  4. = 1017
Enter to expand or collapse answer.Answer expanded
Correct Response: C.





Competency 0006
Understand rational, radical, absolute value, and piecewise defined functions.

6. Which of the following represents the domain of the function f(x) = ?




Enter to expand or collapse answer.Answer expanded
Correct Response: B. Unless otherwise specified the domain of a function is the range of values for which the function has a real number value. A rational function must have a nonzero denominator, and solving the equation 3x + 1 = 0 yields x = . Thus this value must be excluded from the domain. The radical expression in the numerator must have a nonnegative argument and solving the inequality 2x + 3 ≥ 0 yields x ≥ . Putting these two results together results in ≤ x < or x > . The "or" represents the union of the two sets defined by the inequalities, or the union of the two intervals.

Competency 0007
Understand measurement principles and procedures.

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

The shape of the letter B is designed as shown, consisting of rectangles and semicircles. Which of the following formulas gives the area, A, of the shaded region as a function of its height, h?




Enter to expand or collapse answer.Answer expanded
Correct Response: A. The total area of the letter B can be viewed as the area of an rectangle plus the area of a circle with radius minus the area of a circle with radius This simplifies to and further to and and .

Competency 0008
Understand Euclidean geometry in two and three dimensions.

8. Use the incomplete proof below to answer the question that follows.

In the proof above, steps 2 and 4 are missing. Which of the following reasons justifies step 5?

  1. AAS
  2. ASA
  3. SAS
  4. SSS
Enter to expand or collapse answer.Answer expanded
Correct Response: C. The side-angle-side (SAS) theorem can be used to show that ΔABC and ΔCDA are congruent if each has two sides and an included angle that are congruent with two sides and an included angle of the other. In the diagram and are given as congruent, and the missing statement 2 is that is congruent to itself by the reflexive property of equality. The included angles ∠BAC and ∠DCA are congruent because they are alternate interior angles constructed by the transversal that crosses the parallel line segments and . Thus ΔABC and ΔCDA meet the requirements for using SAS to prove congruence.

Competency 0009
Understand coordinate and transformational geometry.

9. The vertices of triangle ABC are A(–5, 3), B(2, 2), and C(–1, –5). Which of the following is the length of the median from vertex B to side AC?

  1. 4
Enter to expand or collapse answer.Answer expanded
Correct Response: C. The midpoint of side AC where its median intersects is computed as The distance from B(2, 2) to (–3, –1) is computed as

Competency 0010
Understand trigonometric functions.

10. Which of the following are the solutions to 2 sin2θ = cos θ + 1 for 0 < θ ≤ 2π?




Enter to expand or collapse answer.Answer expanded
Correct Response: B. Since sin2θ = 1 – cos2θ, 2 sin2θ = cos θ + 1
⇒ 2(1 – cos2θ) = cos θ + 1 ⇒ 2 cos2θ + cos θ –1 = 0 ⇒ (2 cos θ – 1)(cos θ +1) = 0
⇒ cos θ = or cos θ = –1. Thus for 0 < θ ≤ 2π, θ = or π.

Competency 0011
Understand differential calculus.

11. If f(x) = 3x4 – 8x2 + 6, what is the value of ?

  1. –4
  2. –1
  3. 1
  4. 4
Enter to expand or collapse answer.Answer expanded
Correct Response: A. The limit expression is equivalent to the derivative f'(1). Since it is much easier to evaluate the derivative of a polynomial, this is preferred over evaluating the limit expression.
f'(x) = 12x3 – 16x, so f'(1) = 12 – 16 = –4.

Competency 0012
Understand integral calculus.

12. A sum of $2000 is invested in a savings account. The amount of money in the account in dollars after t years is given by the equation A = 2000e0.05t. What is the approximate average value of the account over the first two years?

  1. $2103
  2. $2105
  3. $2206
  4. $2210
Enter to expand or collapse answer.Answer expanded
Correct Response: A. The average value of a continuous function f(x) over an interval [a, b] is . Since the independent variable t represents the number of years, the average daily balance over 2 years will be of the integral of the function evaluated from 0 to 2:

Competency 0013
Understand principles and techniques of statistics.

13. Use the histogram below to answer the question that follows.

Which of the following statements describes the set of data represented by the histogram?

  1. The mode is equal to the mean.
  2. The mean is greater than the median.
  3. The median is greater than the range.
  4. The range is equal to the mode.
Enter to expand or collapse answer.Answer expanded
Correct Response: B. The mean can be calculated as [10(1) + 30(2) + 50(3) + 30(4) + 20(5) + 10(6) + 10(7)] ÷ 160 = 3.5625. The median is the 50th percentile, which is 3. The mode is the most frequent value, which is 3. The range is 7 – 1 = 6. Thus "the mean is greater than the median" is the correct response.

Competency 0014
Understand principles and techniques of probability.

14. The heights of adults in a large group are approximately normally distributed with a mean of 65 inches. If 20% of the adult heights are less than 62.5 inches, what is the probability that a randomly chosen adult from this group will be between 62.5 inches and 67.5 inches tall?

  1. 0.3
  2. 0.4
  3. 0.5
  4. 0.6
Enter to expand or collapse answer.Answer expanded
Correct Response: D. A normal distribution is symmetric about the mean. Thus, if 20% of the heights are less than 62.5 inches (2.5 inches from the mean), then 20% of the heights will be greater than 67.5 inches (also 2.5 inches from the mean). Thus, 100% – (20% + 20%) = 60% and the probability is 0.6 that the adult will be between 62.5 and 67.5 inches tall.