• 40 questions

MFT Math Practice Test

Unofficial questions on calculus, algebra and eight further topics, in the shares of ETS's Major Field Test in Mathematics.

What is the double integral of x² + y² over the unit disk x² + y² ≤ 1?

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 25 questions. Share this set copies a link to the same questions in the same order.

Exam mode: 25 questions, 60 minutes.

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Unofficial practice questions, not affiliated with ETS. Bank checked October 12, 2026.

Next step More calculus: calculus practice testGRE Mathematics Subject practice test

All 40 questions

Calculus 12

  • For f(x) = x³ on the interval [0, 3], which number c satisfies the conclusion of the mean value theorem? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "Mean value theorem".
  • For which real x does the power series Σ xⁿ/n (n = 1, 2, 3, …) converge? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "Radius of convergence".
  • For which values of p does the improper integral of 1/xᵖ from 1 to ∞ converge? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "Improper integral".
  • Let f(x) = x² ln x. What is f′(e)? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "Product rule".
  • Let f(x) = x³ − 3x. What does f have at x = −1? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "Derivative test".
  • Let F(x) be the integral of sin(t²) from t = 0 to t = x². What is F′(x)? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "Fundamental theorem of calculus".
Show all 40 questions (34 more)

Calculus, continued

  • Let f(x, y) = x²y + y³. What is the gradient of f at the point (1, 2)? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), B. Multivariable calculus. Wikipedia, "Gradient".
  • Let f(x, y) = x³ − 3x + y². What kind of critical point is (−1, 0)? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), B. Multivariable calculus. Wikipedia, "Second partial derivative test".
  • What is the coefficient of x⁴ in the Maclaurin series of cos 2x? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "Taylor series".
  • What is the double integral of x² + y² over the unit disk x² + y² ≤ 1? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), B. Multivariable calculus. Wikipedia, "Multiple integral".
  • What is the limit of (eˣ − 1 − x) / x² as x approaches 0? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "L'Hôpital's rule".
  • What is the value of the integral of x eˣ from x = 0 to x = 1? ETS, Major Field Test in Mathematics test description: I. Calculus (~30%), A. Single-variable calculus. Wikipedia, "Integration by parts".

Algebra 12

  • A 3 × 3 matrix has eigenvalues 1, 2 and 4. What are its determinant and its trace? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), A. Linear algebra, eigenvalues and eigenvectors. Wikipedia, "Eigenvalues and eigenvectors".
  • A linear map T from ℝ⁷ to ℝ⁴ is onto. What is the dimension of its kernel? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), A. Linear algebra, linear transformations. Wikipedia, "Rank-nullity theorem".
  • For which value of k does the system x + 2y = 1, 3x + ky = 3 have more than one solution? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), A. Linear algebra, systems of linear equations. Wikipedia, "System of linear equations".
  • G is a group of order 20. Which of these numbers cannot be the order of a subgroup of G? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), B. Abstract algebra, elementary theory of groups, rings and fields. Wikipedia, "Lagrange's theorem (group theory)".
  • In the group ℤ₁₂ of integers modulo 12 under addition, what is the order of the element 4? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), B. Abstract algebra, elementary theory of groups, rings and fields. Wikipedia, "Order (group theory)".
  • The linear transformation T rotates every vector of the plane by 90° counterclockwise about the origin. What is T(3, 1)? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), A. Linear algebra, linear transformations. Wikipedia, "Rotation matrix".
  • What are the eigenvalues of the matrix with rows (1, 2) and (3, 2)? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), A. Linear algebra, eigenvalues and eigenvectors. Wikipedia, "Eigenvalues and eigenvectors".
  • What is the dimension of the span of the vectors (1, 2, 3), (4, 5, 6) and (7, 8, 9)? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), A. Linear algebra, vector spaces. Wikipedia, "Linear independence".
  • What is the inverse of the 2 × 2 matrix with rows (3, 2) and (7, 5)? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), A. Linear algebra, matrices. Wikipedia, "Invertible matrix".
  • What is the remainder when 2⁵³ is divided by 11? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), B. Abstract algebra, elementary topics from number theory. Wikipedia, "Fermat's little theorem".
  • Which of these rings of integers modulo n is a field? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), B. Abstract algebra, elementary theory of groups, rings and fields. Wikipedia, "Modular arithmetic".
  • Which of these sets is a subspace of the vector space of all 2 × 2 real matrices? ETS, Major Field Test in Mathematics test description: II. Algebra (~30%), A. Linear algebra, vector spaces. Wikipedia, "Linear subspace".

Additional topics 16

  • A function y satisfies y′ = 2y and y(0) = 3. What is y(1)? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), B. Differential equations. Wikipedia, "Exponential growth".
  • A graph has five vertices, of degrees 3, 3, 2, 2 and 2. How many edges does it have? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), C. Discrete mathematics (including graph theory and combinatorics). Wikipedia, "Handshaking lemma".
  • In how many different orders can the six letters of the word COFFEE be arranged? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), C. Discrete mathematics (including graph theory and combinatorics). Wikipedia, "Permutation".
  • In the real line with its usual topology, what is the closure of the set ℚ of rational numbers? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), F. Point-set topology. Wikipedia, "Dense set".
  • Let aₙ = (−1)ⁿ + 1/n for n = 1, 2, 3, … What are the limit superior and the limit inferior of the sequence? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), H. Real analysis. Wikipedia, "Limit inferior and limit superior".
  • Let f map the integers to the integers by f(n) = 2n. Which description of f is correct? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), D. Foundations (including logic, proofs, sets, functions and relations). Wikipedia, "Injective function".
  • Two fair dice are rolled and the sum is 8. Given that, what is the probability that both dice show 4? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), G. Probability and statistics. Wikipedia, "Conditional probability".
  • What is (1 − i)⁶? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), A. Complex analysis. Wikipedia, "De Moivre's formula".
  • What is the angle between the vectors (1, 0, 1) and (0, 1, 1) in ℝ³? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), E. Geometry. Wikipedia, "Dot product".
  • What is the distance from the origin to the plane x + 2y + 2z = 6? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), E. Geometry. Wikipedia, "Distance from a point to a plane".
  • What is the general solution of y″ + 4y = 0? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), B. Differential equations. Wikipedia, "Linear differential equation".
  • What is the integral of 1/z around the unit circle |z| = 1, taken once counterclockwise? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), A. Complex analysis. Wikipedia, "Residue theorem".
  • What is the negation of the statement "For every ε > 0 there is a δ > 0 such that P(ε, δ) holds"? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), D. Foundations (including logic, proofs, sets, functions and relations). Wikipedia, "Universal quantification".
  • Which of these collections of subsets is not a topology on the set X = {a, b, c}? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), F. Point-set topology. Wikipedia, "Topological space".
  • Which of these functions is uniformly continuous on the open interval (0, 1)? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), H. Real analysis. Wikipedia, "Uniform continuity".
  • X is the number shown by one roll of a fair six-sided die. What is the variance of X? ETS, Major Field Test in Mathematics test description: III. Additional Topics (~40%), G. Probability and statistics. Wikipedia, "Discrete uniform distribution".

How this list was compiled: Our own questions, written to the test description ETS publishes for the Major Field Test in Mathematics (https://www.ets.org/content/dam/ets-org/pdfs/mft/math-test-description.pdf, read 2026-10-12): Calculus about 30%, Algebra about 30%, Additional Topics about 40%. The bank holds 12, 12 and 16 questions; the eight additional topics have two each. The theorem or definition each question rests on was read on 2026-10-12 in the Wikipedia article named under it, and every answer that is a calculation was worked out a second time. Licence: Questions, choices and explanations are original work of this site.

How it works

This math Major Field Test practice has 40 questions of our own in the three content areas of ETS's test, each with a worked answer. It is unofficial and not affiliated with, endorsed by or sponsored by ETS.

Read the full guide

Three areas, ten topics

ETS's test description, read October 12, 2026: calculus about 30%, algebra about 30%, additional topics about 40%. This bank holds 12, 12 and 16 questions. The additional topics are complex analysis, differential equations, discrete mathematics, foundations, geometry, point-set topology, probability and statistics, and real analysis, two questions each.

Routine first

ETS rates about 55% of the real questions routine, 25% nonroutine and 20% applied. Most questions here are routine: one or two definitions and a short calculation.

Half length, real pace

The timed exam here draws 25 of our questions in 60 minutes, half the real test at its own pace.

  • ETS says the mathematics needs no calculator. Every answer here comes out by hand.
  • Algebra here means linear algebra plus groups, rings, fields and number theory, so review eigenvalues and Lagrange's theorem alike.

Questions and answers

Is there a study guide for the Major Field Test for mathematics?

What ETS itself offers is the test description, with the outline of all three areas, and a set of sample questions. This page is built on that outline, with questions of its own.

What is on the MFT math test?

50 questions from the usual undergraduate courses: three semesters of calculus, linear and abstract algebra, and topics such as differential equations, discrete mathematics, probability and real analysis. Some may be grouped in sets based on diagrams and graphs.

How long is the Major Field Test in Mathematics?

Two hours. It may be split into two sessions, the program chooses when and where to give it, and a proctor must be present.

How is the MFT in mathematics scored?

ETS reports a total score for each student. Calculus, algebra, and routine, nonroutine and applied questions are reported as indicators for the group only, when at least five students test.

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