Free Praxis® Mathematics Practice Questions

Sample Practice Questions

Try these 10 sample questions from the Praxis® Mathematics practice bank. Each includes the answer and explanation. Continue with the interactive practice above to review the full question set.

1. 1Let r be rational and s be irrational. Which conclusion about r + s is justified?

   1. A.It is rational if r is nonzero.
   2. B.It is irrational for every rational r.Correct answer: B
   3. C.It is rational if s is positive.
   4. D.It is irrational only when r is an integer.

   Explanation: The standard states that "the sum of a rational number and an irrational number is irrational". If r + s were rational, subtracting rational r would make s rational, a contradiction. Neither positivity nor integrality changes this result.

   Reference: Common Core State Standards for Mathematics — p. 60, The Real Number System, rational and irrational sums

2. 2Which set of ordered pairs defines a single-valued function with x as its input?

   1. A.{(1, 2), (1, 3), (2, 4)}
   2. B.{(1, 2), (2, 2), (3, 4)}Correct answer: B
   3. C.{(0, 1), (0, −1), (2, 3)}
   4. D.{(−1, 0), (2, 1), (2, 5)}

   Explanation: A function "assigns to each element of the domain exactly one element of the range." B assigns each input one output; different inputs may share output 2. A, C, and D each assign two different outputs to one input.

   Reference: Common Core State Standards for Mathematics — p. 69, Interpreting Functions, definition of a function

3. 3Two straight lines intersect. One angle measures 68°. What is the measure of its vertical angle?

   1. A.22°
   2. B.68°Correct answer: B
   3. C.112°
   4. D.136°

   Explanation: The source states "vertical angles are congruent". Therefore the opposite angle also measures 68°. Twenty-two is the complement, 112 is an adjacent supplementary angle, and 136 doubles the given measure.

   Reference: Common Core State Standards for Mathematics — p. 76, G-CO.9

4. 4A school wants to estimate the proportion of all its students who favor a new schedule. Which sampling plan best supports that population inference?

   1. A.Invite students who visit the principal to volunteer.
   2. B.Survey only members of the student council.
   3. C.Randomly select students from the complete enrollment list.Correct answer: C
   4. D.Survey the first students arriving at the library.

   Explanation: The standard states "random sampling tends to produce representative samples and support valid inferences." Sampling the complete enrollment list gives students across the target population an opportunity for selection. The other plans rely on availability, leadership membership, or location and can systematically favor particular opinions.

   Reference: Common Core State Standards for Mathematics — p. 50, Statistics and Probability, random sampling and representative inference

5. 5A student claims that the product of a rational number and an irrational number must be irrational. Which condition on the rational factor is needed to make the claim true?

   1. A.It must be positive.
   2. B.It must be an integer.
   3. C.It must be nonzero.Correct answer: C
   4. D.It must be greater than 1.

   Explanation: The source specifies "the product of a nonzero rational number and an irrational number is irrational." Zero must be excluded because its product with any irrational number is zero, which is rational. Negative and fractional nonzero factors are allowed.

   Reference: Common Core State Standards for Mathematics — p. 60, The Real Number System, nonzero rational factors and irrational products

6. 6What is the real domain of f(x) = √(x − 2)/(x − 5)?

   1. A.Real x at least two, including five.
   2. B.Real x greater than two, excluding five.
   3. C.Real x at least two, excluding five.Correct answer: C
   4. D.All real x except five.

   Explanation: The source defines the domain as "all inputs for which the expression defining a function has a value". The square root requires x ≥ 2, and the denominator excludes x = 5. Combining these restrictions gives [2,5) ∪ (5,∞). A includes division by zero, B unnecessarily excludes 2, and D permits negative radicands.

   Reference: Common Core State Standards for Mathematics — p. 67, Functions introduction

7. 7Two parallel lines are cut by a transversal. A pair of alternate interior angles has measures (3x + 7)° and (5x − 21)°. What is x?

   1. A.7
   2. B.14Correct answer: B
   3. C.21
   4. D.28

   Explanation: The source states that for parallel lines "alternate interior angles are congruent". Set 3x + 7 = 5x − 21, so 28 = 2x and x = 14. Other values result from incorrect coefficient or constant handling and do not make the angles equal.

   Reference: Common Core State Standards for Mathematics — p. 76, Congruence, alternate interior angles for parallel lines

8. 8Students in a study are randomly assigned to two tutoring methods. Why is random assignment important when comparing their outcomes?

   1. A.It ensures identical final scores in the groups.
   2. B.It permits a fair comparison by reducing systematic group differences.Correct answer: B
   3. C.It makes every participant representative of all students nationally.
   4. D.It eliminates all sampling and measurement variation.

   Explanation: The source states that "randomly assigning individuals to different treatments allows a fair comparison of the effectiveness of those treatments." Random assignment addresses systematic preexisting differences between treatment groups. It does not guarantee identical outcomes, create a nationally representative sample, or remove all variation.

   Reference: Common Core State Standards for Mathematics — p. 79, Statistics and Probability introduction

9. 9Which property justifies replacing (u + v) + w with u + (v + w) for real numbers u, v, and w?

   1. A.The commutative property of addition
   2. B.The associative property of additionCorrect answer: B
   3. C.The additive identity property
   4. D.The distributive property of multiplication

   Explanation: Table 3 gives "(a + b) + c = a + (b + c)" for the associative property of addition. The grouping changes while the order stays fixed. Commutativity changes order, identity involves zero, and distribution involves multiplication over addition.

   Reference: Common Core State Standards for Mathematics — p. 90, Table 3, associative property of addition

10. 10For f(x) = −(x + 1)² + 6 with domain all real numbers, what is the range?

    1. A.Outputs at least positive six.
    2. B.Outputs at most positive six.Correct answer: B
    3. C.Outputs at least negative one.
    4. D.Outputs at most negative one.

    Explanation: ETS describes the range as "the set of resulting outputs". The nonnegative square makes f(x) at most 6, attained at x = −1; arbitrarily large squares give arbitrarily negative outputs. The other intervals reverse the bound or confuse the vertex input with an output.

    Reference: ETS Mathematics Study Companion — p. 15, Functions discussion, domain and range