• 66 questions

Praxis 5161 Practice Test

Unofficial secondary mathematics practice questions, spread like the ETS outline for Praxis Mathematics (5165).

What is the greatest common factor of 84 and 126?

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

Print covers the next 66 questions. Share this set copies a link to the same questions in the same order.

Exam mode: 66 questions, 180 minutes.

More options

Unofficial practice questions, not affiliated with ETS (Educational Testing Service). Bank checked October 6, 2026.

Next step Praxis Algebra I (5162) practice testCLEP College Math practice test

All 66 questions

Number and quantity 7

  • A cheetah runs 30 meters per second. How many kilometers per hour is that? ETS, Praxis Mathematics (5165) Study Companion: I.A Number and Quantity (10% of the test), reasoning quantitatively and using units
  • An operation on the real numbers is defined by a ★ b = 2a + b. Which statement about ★ is true? ETS, Praxis Mathematics (5165) Study Companion: I.A Number and Quantity (10% of the test), the structure of the real number system
  • Simplify (2 + 3i)(4 − i). ETS, Praxis Mathematics (5165) Study Companion: I.A Number and Quantity (10% of the test), the structure of the complex number system
  • Simplify (4 + 2i)/(1 + i). ETS, Praxis Mathematics (5165) Study Companion: I.A Number and Quantity (10% of the test), the structure of the complex number system
  • Simplify (27x⁶)^(2/3), where x > 0. ETS, Praxis Mathematics (5165) Study Companion: I.A Number and Quantity (10% of the test), properties of radicals and rational exponents
  • What is the greatest common factor of 84 and 126? ETS, Praxis Mathematics (5165) Study Companion: I.A Number and Quantity (10% of the test), the structure of the real number system
  • Which product is a rational number? ETS, Praxis Mathematics (5165) Study Companion: I.A Number and Quantity (10% of the test), the structure of the real number system
Show all 66 questions (59 more)

Algebra 13

  • A line passes through (−2, 7) and is perpendicular to y = (1/2)x + 3. What is its equation? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), linear equations in various forms
  • A student says x² + 25 cannot be factored. Over the complex numbers, which factorization is correct? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), writing algebraic expressions in equivalent forms
  • A student solves x/(x − 2) = 2/(x − 2) by multiplying both sides by x − 2 and reports x = 2. What should the teacher point out? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), justifying the reasoning used to solve equations, with extraneous solutions
  • A student squares both sides of √(x + 6) = x and gets x = −2 or x = 3. Which values solve the original equation? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), justifying the reasoning used to solve equations, with extraneous solutions
  • A theater sells adult tickets for $12 and child tickets for $7. It sells 150 tickets for $1,400. How many adult tickets were sold? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), creating equations and inequalities that describe relationships
  • Add 2/(x − 1) + 3/(x + 2), where the denominators are not zero. ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), rewriting rational expressions
  • For which value of k does x² + kx + 9 = 0 have exactly one real solution? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), techniques for solving equations and inequalities
  • How many points do the line y = 2x − 5 and the parabola y = x² − 4x + 4 share? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), techniques for solving systems of equations and inequalities
  • Solve x² − 6x + 13 = 0 over the complex numbers. ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), techniques for solving equations and inequalities
  • What is the average rate of change of f(x) = 2^x from x = 1 to x = 4? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), rate of change of nonlinear functions
  • What is the quotient when 2x³ − 3x² − 11x + 6 is divided by x − 3? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), writing algebraic expressions in equivalent forms
  • What is the remainder when x³ − 4x² + 2x + 7 is divided by x + 2? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), zeros of polynomial functions and factors of polynomials
  • Which is the vertex form of f(x) = −2x² + 12x − 5? ETS, Praxis Mathematics (5165) Study Companion: I.B Algebra (20% of the test), writing algebraic expressions in equivalent forms

Functions 13

  • A geometric sequence has first term 5 and common ratio 3. What is its 6th term? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), modeling relationships with functions and sequences
  • A sequence has a(1) = 12, and each later term is 5 less than the term before. What is a(20)? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), modeling relationships with functions and sequences
  • Convert 150° to radians. ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), points on the unit circle and values of trigonometric functions
  • f(x) = (x − 1)² is not one-to-one on all real numbers. On which of these domains does f have an inverse function? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), obtaining new functions from existing functions
  • The height in meters of a Ferris wheel car after t seconds is h(t) = 15 sin(πt/20) + 18. How long does one full turn take? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), modeling periodic phenomena with trigonometric functions
  • If f(x) = 2x − 1 and g(x) = x² + 3, what is f(g(2))? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), obtaining new functions from existing functions
  • Solve 2cos(x) − 1 = 0 for 0 ≤ x < 2π. ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), solving trigonometric, logarithmic and exponential equations
  • Solve 4e^(2t) = 28 for t. ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), solving trigonometric, logarithmic and exponential equations
  • Solve log₃(x − 2) = 2. ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), solving trigonometric, logarithmic and exponential equations
  • What is the domain of f(x) = √(x + 2)/(x − 5)? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), functions and function notation
  • What is the exact value of sin(7π/6)? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), points on the unit circle and values of trigonometric functions
  • What is the reference angle for an angle of 235°? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), points on the unit circle and values of trigonometric functions
  • What is the value of log₂(40) − log₂(5)? ETS, Praxis Mathematics (5165) Study Companion: II.A Functions (20% of the test), using logarithms to solve problems

Calculus 7

  • A particle has position s(t) = t³ − 6t² + 9t meters after t seconds. When is its velocity zero? ETS, Praxis Mathematics (5165) Study Companion: II.B Calculus (10% of the test), differentiation and integration techniques
  • A student says the derivative of x² sin(x) is 2x cos(x). What is the correct derivative? ETS, Praxis Mathematics (5165) Study Companion: II.B Calculus (10% of the test), differentiation and integration techniques
  • At which value of x does f(x) = x³ − 3x have a local minimum? ETS, Praxis Mathematics (5165) Study Companion: II.B Calculus (10% of the test), analyzing the behavior of a function
  • What is the limit of (3x² − x)/(2x² + 5) as x grows without bound? ETS, Praxis Mathematics (5165) Study Companion: II.B Calculus (10% of the test), limits of functions
  • What is the limit of (x² − 9)/(x − 3) as x approaches 3? ETS, Praxis Mathematics (5165) Study Companion: II.B Calculus (10% of the test), limits of functions
  • What is the limit of f(x) = |x − 2|/(x − 2) as x approaches 2? ETS, Praxis Mathematics (5165) Study Companion: II.B Calculus (10% of the test), limits of functions
  • What is the value of the definite integral of 3x² + 1 from x = 0 to x = 2? ETS, Praxis Mathematics (5165) Study Companion: II.B Calculus (10% of the test), differentiation and integration techniques

Geometry 13

  • A 6-foot post casts a 4-foot shadow while a tree casts a 22-foot shadow. How tall is the tree? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), congruence and similarity
  • A circle has radius 9. How long is the arc cut off by a central angle of 80°? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), theorems about circles
  • A student claims that tripling every side of a triangle triples its area. A triangle of area 12 is dilated by scale factor 3. What is the area of the image? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), transformations in the plane
  • An inscribed angle of a circle intercepts an arc of 110°. What is the measure of the angle? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), theorems about circles
  • The hypotenuse of a 30°-60°-90° triangle is 12. How long is the longer leg? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), properties of triangles, quadrilaterals and other polygons
  • The point (3, −5) is reflected over the line y = x. What is its image? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), transformations in the plane
  • The radius of a cylinder is multiplied by 2 and its height by 1/2. By what factor is its volume multiplied? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), problems involving solids
  • Two parallel lines are cut by a transversal. Alternate interior angles measure 3x + 10 degrees and 5x − 30 degrees. What is x? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), properties of lines and angles
  • Two sides of a triangle are 5 and 8, and the angle between them is 60°. How long is the third side? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), trigonometry applied to triangles
  • Two triangles have two pairs of congruent sides, and the angles between those sides are congruent. Which criterion proves the triangles congruent? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), congruence and similarity
  • What are the center and radius of the circle x² + y² − 6x + 4y − 12 = 0? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), theorems about circles
  • What is the measure of each interior angle of a regular polygon with 10 sides? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), properties of triangles, quadrilaterals and other polygons
  • Which three lengths can be the sides of a triangle? ETS, Praxis Mathematics (5165) Study Companion: III. Geometry (20% of the test), properties of triangles, quadrilaterals and other polygons

Statistics and probability 13

  • A game pays $10 with probability 0.2, $2 with probability 0.5, and nothing otherwise. What is the expected payout? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), decisions using probabilities and expected values
  • For the data 2, 9, 10, 10, 11, 12, which statement is true? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), summarizing and interpreting data on a single variable
  • If P(A) = 0.5 and P(A and B) = 0.2, what is P(B | A)? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), independence and probabilities of compound events
  • In a random sample of 400 voters, 228 favor a measure. What is the estimate of the proportion of all voters who favor it? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), inferences from samples, experiments and observational studies
  • In how many ways can a committee of 3 be chosen from 10 students? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), probabilities involving finite sample spaces and counting
  • In how many ways can a president, a vice president and a treasurer be chosen from 7 club members? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), probabilities involving finite sample spaces and counting
  • Of 200 students, 80 play a sport, 50 play an instrument and 20 do both. Are "plays a sport" and "plays an instrument" independent events? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), summarizing and interpreting data on two variables
  • Researchers randomly assign 60 volunteers to one of two study methods and then compare the two groups' test scores. What kind of study is this? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), inferences from samples, experiments and observational studies
  • The residual plot for a linear fit shows a clear U-shaped pattern. What does that suggest? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), creating and interpreting linear regression models
  • Test scores are normally distributed with mean 500 and standard deviation 100. About what percent of scores fall between 400 and 600? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), normal distributions
  • What is the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), summarizing and interpreting data on a single variable
  • Which correlation coefficient shows the strongest linear relationship? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), creating and interpreting linear regression models
  • With P(A) = 0.45, P(B) = 0.30 and P(A and B) = 0.10, a student adds the first two and reports P(A or B) = 0.75. What is the correct value? ETS, Praxis Mathematics (5165) Study Companion: IV. Statistics and Probability (20% of the test), independence and probabilities of compound events

How this list was compiled: Our own questions, written to the content topics of the ETS Study Companion for Praxis Mathematics (5165) (https://www.ets.org/pdfs/praxis/5165.pdf, read 2026-10-06; test page https://praxis.ets.org/test/5165.html): Number and Quantity 7 questions (10%), Algebra 13 (20%), Functions 13 (20%), Calculus 7 (10%), Geometry 13 (20%), Statistics and Probability 13 (20%). The bank holds the same 7, 13, 13, 7, 13 and 13. Every computed answer is worked from what the question itself prints, and each was worked out a second time on 2026-10-06. Licence: Questions, choices and explanations are original work of this site.

How it works

This free Praxis 5161 practice test follows the Praxis Mathematics (5165) outline, the secondary mathematics test ETS lists today: 66 questions of our own, each worked out. It is unofficial and not affiliated with, endorsed by or sponsored by ETS.

Read the full guide

What the test covers

ETS lists number and quantity (10%), algebra (20%), functions (20%), calculus (10%), geometry (20%) and statistics and probability (20%). This bank holds 7, 13, 13, 7, 13 and 13 questions, the numbers in that table. About a quarter of the real questions set the mathematics in a teaching task, and some here do too.

Timing

The real test has 66 questions in 180 minutes. Exam mode uses the same 66 and 180.

  • ETS provides an on-screen graphing calculator, and a list of notations, definitions and formulas sits on the Help screen of the real test.
  • The real test also has multiple-answer and numeric-entry questions. Here each question has four choices and one right answer.

Questions and answers

Is Praxis 5161 the same test as 5165?

They carry different codes. On October 6, 2026 the ETS site had no test page and no Study Companion for 5161; the Mathematics test it lists is 5165, and these questions follow that outline. Ask your state agency which code it accepts.

Is this Praxis II math 5161 free practice test the real exam?

No. These are our own questions on the content topics ETS publishes for Mathematics (5165). Nothing is taken or reworded from a real test or from ETS sample questions, and your score here says nothing about your score there.

Is there a Praxis II math 5161 practice test PDF?

Yes. After a round, Print opens a question sheet that can hold all 66 questions, with an answer key if you want one. Choose Save as PDF in the print window to keep it as a file.

What score do you need to pass Praxis Mathematics?

ETS does not set passing scores. Each state or licensing agency sets its own qualifying score, and they vary. Exam mode gives the number right and no pass verdict.

More tests

Show 7 more

Browse Civics and Law