• 66 questions

GRE Math Subject Test Practice Exam

Unofficial college mathematics questions in the three content areas ETS lists for the GRE Mathematics Test, each answer worked out.

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How many real solutions does the equation e^x = x + 2 have?

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Unofficial practice questions, not affiliated with ETS. Bank checked October 10, 2026.

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All 66 questions with answers

Calculus 33

  • C is the circle x² + y² = 4, traversed once counterclockwise. What is the line integral over C of (x² − y) dx + (3x + y²) dy? Answer: 16π. Source: ETS, GRE Mathematics Test content: Calculus (50%), Green's theorem. Wikipedia, "Green's theorem". The answer is worked out from what the question prints.
  • For f(x) = x³ on [0, 3], the mean value theorem gives a point c in (0, 3) where f′(c) equals the average rate of change of f. What is c? Answer: √3. Source: ETS, GRE Mathematics Test content: Calculus (50%), the mean value theorem. Wikipedia, "Mean value theorem". The answer is worked out from what the question prints.
  • For which real numbers p does the series with terms 1/(n (ln n)^p), n ≥ 2, converge? Answer: Only for p > 1. Source: ETS, GRE Mathematics Test content: Calculus (50%), the integral test. Wikipedia, "Integral test for convergence".
  • The function f(x, y) = x³ − 3xy + y³ has a critical point at (1, 1). What kind of point is it? Answer: A local minimum. Source: ETS, GRE Mathematics Test content: Calculus (50%), the second-derivative test in two variables. The answer is worked out from what the question prints.
Show all 66 questions (62 more)

Calculus, continued

  • How long is the arc of the curve y = (2/3)x^(3/2) between x = 3 and x = 8? Answer: 38/3. Source: ETS, GRE Mathematics Test content: Calculus (50%), arc length. The answer is worked out from what the question prints.
  • How many real solutions does the equation e^x = x + 2 have? Answer: Two. Source: ETS, GRE Mathematics Test content: Calculus (50%), curve sketching. The answer is worked out from what the question prints.
  • If f(x) = x^x for x > 0, what is f′(e)? Answer: 2e^e. Source: ETS, GRE Mathematics Test content: Calculus (50%), differentiation. The answer is worked out from what the question prints.
  • If y′ + y = e^(−x) and y(0) = 0, what is y(1)? Answer: 1/e. Source: ETS, GRE Mathematics Test content: Calculus (50%), first-order linear differential equations. The answer is worked out from what the question prints.
  • If y′ = 2xy and y(0) = 3, what is y(1)? Answer: 3e. Source: ETS, GRE Mathematics Test content: Calculus (50%), separable differential equations. The answer is worked out from what the question prints.
  • If y″ + 4y = 0, y(0) = 0 and y′(0) = 2, what is y(π/4)? Answer: 1. Source: ETS, GRE Mathematics Test content: Calculus (50%), second-order linear differential equations. The answer is worked out from what the question prints.
  • Let f(x) = x³ + x. What is the derivative of the inverse function f⁻¹ at the point 2? Answer: 1/4. Source: ETS, GRE Mathematics Test content: Calculus (50%), derivatives of inverse functions. The answer is worked out from what the question prints.
  • Let F(x) = ∫ from 0 to x² of √(1 + t³) dt. What is F′(2)? Answer: 4√65. Source: ETS, GRE Mathematics Test content: Calculus (50%), the fundamental theorem of calculus. The answer is worked out from what the question prints.
  • Let f(x, y) = x²y + sin(xy). What is ∂²f/∂x∂y at the point (1, 0)? Answer: 3. Source: ETS, GRE Mathematics Test content: Calculus (50%), partial derivatives. The answer is worked out from what the question prints.
  • The region under y = e^x for 0 ≤ x ≤ 1 is revolved about the x-axis. What is the volume of the solid? Answer: π(e² − 1)/2. Source: ETS, GRE Mathematics Test content: Calculus (50%), volumes of revolution. The answer is worked out from what the question prints.
  • What is the area of the region enclosed by the curves y = x² and y = 2x + 3? Answer: 32/3. Source: ETS, GRE Mathematics Test content: Calculus (50%), area between curves. The answer is worked out from what the question prints.
  • What is the coefficient of x³ in the Maclaurin series of e^x sin x? Answer: 1/3. Source: ETS, GRE Mathematics Test content: Calculus (50%), Taylor series. The answer is worked out from what the question prints.
  • What is the directional derivative of f(x, y) = x² + 3xy at (1, 2) in the direction of the vector ⟨3, 4⟩? Answer: 36/5. Source: ETS, GRE Mathematics Test content: Calculus (50%), the gradient. The answer is worked out from what the question prints.
  • What is the distance from the point (1, 2, 3) to the plane 2x − y + 2z = 15? Answer: 3. Source: ETS, GRE Mathematics Test content: Calculus (50%), coordinate geometry in three dimensions. The answer is worked out from what the question prints.
  • What is the flux of the vector field F(x, y, z) = ⟨x, y, z⟩ outward through the sphere x² + y² + z² = 4? Answer: 32π. Source: ETS, GRE Mathematics Test content: Calculus (50%), the divergence theorem. Wikipedia, "Divergence theorem". The answer is worked out from what the question prints.
  • What is the integral of e^(−x² − y²) over the whole xy-plane? Answer: π. Source: ETS, GRE Mathematics Test content: Calculus (50%), polar coordinates. The answer is worked out from what the question prints.
  • What is the limit as n → ∞ of (1/n) times the sum of (k/n)² for k = 1 to n? Answer: 1/3. Source: ETS, GRE Mathematics Test content: Calculus (50%), Riemann sums. The answer is worked out from what the question prints.
  • What is the limit of (1 + 2/x)^(3x) as x → ∞? Answer: e⁶. Source: ETS, GRE Mathematics Test content: Calculus (50%), limits and the number e. The answer is worked out from what the question prints.
  • What is the limit of (1 − cos 3x)/x² as x → 0? Answer: 9/2. Source: ETS, GRE Mathematics Test content: Calculus (50%), limits. The answer is worked out from what the question prints.
  • What is the maximum value of f(x) = x e^(−x²) on the real line? Answer: 1/√(2e). Source: ETS, GRE Mathematics Test content: Calculus (50%), maxima and minima. The answer is worked out from what the question prints.
  • What is the maximum value of x + 2y on the circle x² + y² = 5? Answer: 5. Source: ETS, GRE Mathematics Test content: Calculus (50%), Lagrange multipliers. The answer is worked out from what the question prints.
  • What is the radius of convergence of the power series whose nth term is (n!/nⁿ) xⁿ? Answer: e. Source: ETS, GRE Mathematics Test content: Calculus (50%), the ratio test. The answer is worked out from what the question prints.
  • What is the slope of the tangent line to the curve x³ + y³ = 6xy at the point (3, 3)? Answer: −1. Source: ETS, GRE Mathematics Test content: Calculus (50%), implicit differentiation. The answer is worked out from what the question prints.
  • What is the sum of the infinite series whose nth term is n/3ⁿ, for n = 1, 2, 3, …? Answer: 3/4. Source: ETS, GRE Mathematics Test content: Calculus (50%), power series. The answer is worked out from what the question prints.
  • What is the value of the iterated integral ∫ from 0 to 1 ∫ from x to 1 of e^(y²) dy dx? Answer: (e − 1)/2. Source: ETS, GRE Mathematics Test content: Calculus (50%), double integrals. The answer is worked out from what the question prints.
  • What is ∫ from 0 to 1 of x² e^x dx? Answer: e − 2. Source: ETS, GRE Mathematics Test content: Calculus (50%), integration by parts. The answer is worked out from what the question prints.
  • What is ∫ from 0 to π/2 of sin³x dx? Answer: 2/3. Source: ETS, GRE Mathematics Test content: Calculus (50%), trigonometric integrals. The answer is worked out from what the question prints.
  • What is ∫ from 0 to ∞ of x e^(−2x) dx? Answer: 1/4. Source: ETS, GRE Mathematics Test content: Calculus (50%), improper integrals. The answer is worked out from what the question prints.
  • Which is the general solution of y″ − 3y′ + 2y = 0? Answer: y = C₁e^x + C₂e^(2x). Source: ETS, GRE Mathematics Test content: Calculus (50%), second-order linear differential equations. The answer is worked out from what the question prints.

Algebra 17

  • A is a 3 × 3 real matrix with det A = 2. What is det(3A⁻¹)? Answer: 27/2. Source: ETS, GRE Mathematics Test content: Algebra (25%), linear algebra, matrix algebra. The answer is worked out from what the question prints.
  • A is the matrix with rows (1, 1) and (0, 1). What is the upper-right entry of A¹⁰? Answer: 10. Source: ETS, GRE Mathematics Test content: Algebra (25%), linear algebra, matrix algebra. The answer is worked out from what the question prints.
  • A real matrix with 4 rows and 6 columns has rank 3. What is the dimension of its null space? Answer: 3. Source: ETS, GRE Mathematics Test content: Algebra (25%), linear algebra, vector spaces. Wikipedia, "Rank–nullity theorem". The answer is worked out from what the question prints.
  • How many elements of the cyclic group Z₁₂ generate the whole group? Answer: 4. Source: ETS, GRE Mathematics Test content: Algebra (25%), abstract algebra, group theory. The answer is worked out from what the question prints.
  • If log₂ x + log₂ (x − 2) = 3, what is x? Answer: 4. Source: ETS, GRE Mathematics Test content: Algebra (25%), elementary algebra, logarithms. The answer is worked out from what the question prints.
  • In the symmetric group S₅, what is the order of the permutation (1 2 3)(4 5)? Answer: 6. Source: ETS, GRE Mathematics Test content: Algebra (25%), abstract algebra, group theory. The answer is worked out from what the question prints.
  • The roots of x³ − 6x² + 11x − 6 are r, s and t. What is r² + s² + t²? Answer: 14. Source: ETS, GRE Mathematics Test content: Algebra (25%), elementary algebra, roots of polynomials. The answer is worked out from what the question prints.
  • What are the eigenvalues of the matrix with rows (4, 1) and (2, 3)? Answer: 2 and 5. Source: ETS, GRE Mathematics Test content: Algebra (25%), linear algebra, eigenvalues and eigenvectors. The answer is worked out from what the question prints.
  • What is (1 + i)⁸? Answer: 16. Source: ETS, GRE Mathematics Test content: Algebra (25%), elementary algebra, complex numbers. The answer is worked out from what the question prints.
  • What is the characteristic polynomial of the matrix with rows (1, 2) and (3, 4)? Answer: λ² − 5λ − 2. Source: ETS, GRE Mathematics Test content: Algebra (25%), linear algebra, characteristic polynomials. The answer is worked out from what the question prints.
  • What is the degree of the field Q(√2, √3) over Q? Answer: 4. Source: ETS, GRE Mathematics Test content: Algebra (25%), abstract algebra, field theory. The answer is worked out from what the question prints.
  • What is the determinant of the 3 × 3 matrix with rows (2, 1, 0), (1, 3, 1) and (0, 1, 2)? Answer: 8. Source: ETS, GRE Mathematics Test content: Algebra (25%), linear algebra, matrix algebra. The answer is worked out from what the question prints.
  • What is the rank of the matrix with rows (1, 2, 3), (2, 4, 6) and (1, 0, 1)? Answer: 2. Source: ETS, GRE Mathematics Test content: Algebra (25%), linear algebra, systems of linear equations. The answer is worked out from what the question prints.
  • What is the remainder when 3¹⁰⁰ is divided by 7? Answer: 4. Source: ETS, GRE Mathematics Test content: Algebra (25%), number theory, congruences. Wikipedia, "Fermat's little theorem". The answer is worked out from what the question prints.
  • Which integer x with 0 < x < 26 satisfies 7x ≡ 1 (mod 26)? Answer: 15. Source: ETS, GRE Mathematics Test content: Algebra (25%), number theory, congruences. The answer is worked out from what the question prints.
  • Which of these groups is not cyclic? Answer: Z₂ × Z₄. Source: ETS, GRE Mathematics Test content: Algebra (25%), abstract algebra, group theory. Wikipedia, "Cyclic group".
  • Which of these sets is a subspace of R³? Answer: All (x, y, z) with x + y + z = 0. Source: ETS, GRE Mathematics Test content: Algebra (25%), linear algebra, vector spaces. Wikipedia, "Linear subspace".

Additional topics 16

  • A connected planar graph is drawn with 8 vertices and 12 edges. Counting the unbounded region, how many faces does the drawing have? Answer: 6. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), discrete mathematics, graph theory. Wikipedia, "Planar graph". The answer is worked out from what the question prints.
  • A fair coin is tossed 400 times. What is the standard deviation of the number of heads? Answer: 10. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), probability and statistics. Wikipedia, "Binomial distribution". The answer is worked out from what the question prints.
  • An algorithm runs an outer loop for i = 1 to 10 and, inside it, an inner loop for j = 1 to i. How many times does the body of the inner loop run? Answer: 55. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), discrete mathematics, algorithms. The answer is worked out from what the question prints.
  • C is the circle |z| = 1, traversed once counterclockwise. What is the integral over C of e^(2z)/z² dz? Answer: 4πi. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), complex variables. Wikipedia, "Cauchy's integral formula". The answer is worked out from what the question prints.
  • f is a continuous real-valued function on the closed interval [0, 1]. Which statement must be true? Answer: f takes a largest value at some point of [0, 1]. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), introductory real analysis, continuity. Wikipedia, "Extreme value theorem".
  • The functions f_n(x) = xⁿ on [0, 1] converge pointwise to a function f. Which statement is true? Answer: f is not continuous, so the convergence is not uniform. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), introductory real analysis, sequences of functions. Wikipedia, "Uniform limit theorem".
  • How many ordered triples (x, y, z) of nonnegative integers satisfy x + y + z = 10? Answer: 66. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), discrete mathematics, combinatorics. The answer is worked out from what the question prints.
  • In the real line with its usual topology, how many subsets are both open and closed? Answer: Two. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), general topology. Wikipedia, "Connected space".
  • Let S be the set of numbers 1 − 1/n for n = 1, 2, 3, …. Which statement is true? Answer: The least upper bound of S is 1, and S has no largest element. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), introductory real analysis, sequences. Wikipedia, "Infimum and supremum".
  • Newton's method is applied to f(x) = x² − 3 starting from x₀ = 2. What is x₂? Answer: 97/56. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), numerical analysis. Wikipedia, "Newton's method". The answer is worked out from what the question prints.
  • What is the area of the triangle with vertices (1, 1), (4, 2) and (2, 5)? Answer: 11/2. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), geometry. The answer is worked out from what the question prints.
  • What is the limit of √(n² + 3n) − n as n → ∞? Answer: 3/2. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), introductory real analysis, sequences. The answer is worked out from what the question prints.
  • Which of these sets is uncountable? Answer: The power set of the positive integers. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), discrete mathematics, set theory. Wikipedia, "Cantor's theorem".
  • Which of these subsets of real numbers is compact? Answer: [0, 1] ∪ {2}. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), introductory real analysis, elementary topology of the real line. Wikipedia, "Heine–Borel theorem".
  • Which statement is the negation of "For every real x there is a real y with y² = x"? Answer: There is a real x such that y² ≠ x for every real y. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), discrete mathematics, logic. Wikipedia, "Universal quantification".
  • X and Y are independent and uniformly distributed on [0, 1]. What is the probability that X + Y is less than 1/2? Answer: 1/8. Source: ETS, GRE Mathematics Test content: Additional Topics (25%), probability and statistics. The answer is worked out from what the question prints.

How this list was compiled: Our own questions, written to the content description ETS publishes for the GRE Mathematics Test (https://www.ets.org/gre/test-takers/subject-tests/about/content-structure.html, read 2026-10-10) and to the topic list on page 3 of its GRE Mathematics Test Practice Book (https://www.ets.org/pdfs/gre/practice-book-math.pdf): Calculus 50%, Algebra 25% (elementary algebra, linear algebra, abstract algebra and number theory), Additional Topics 25% (introductory real analysis, discrete mathematics, and general topology, geometry, complex variables, probability and statistics, numerical analysis). The bank holds 33, 17 and 16 questions. Every answer that is a number or a formula was worked out a second time on 2026-10-10; each named theorem was read on that day in the Wikipedia article named under its question. Licence: Questions, choices and explanations are original work of this site.

How it works

This free GRE Math Subject Test practice exam has 66 questions of our own, spread over calculus, algebra and additional topics in ETS's proportions. It is unofficial and is not affiliated with, endorsed by or sponsored by ETS.

Read the full guide

Three content areas

ETS's content description, read October 10, 2026: calculus 50%, algebra 25% (elementary, linear and abstract algebra, number theory), additional topics 25% (real analysis, discrete mathematics, topology, geometry, complex variables, probability and statistics, numerical analysis). This bank holds 33, 17 and 16 questions.

Full length

Exam mode gives all 66 questions on the real test's clock, as a full mathematics GRE practice test.

Reading the notation

"∫ from 0 to 1" is a definite integral, f′ a derivative, e^(2x) a power and ⟨3, 4⟩ a vector. Z₁₂ is the integers modulo 12. A matrix is written out by its rows.

  • Nothing is taken off for a wrong answer, and ETS's practice book says a guess is better than a blank.
  • ETS warns that questions needing only good precalculus can be among the hardest on the test.

Questions and answers

How many questions are on the GRE Mathematics Test?

About 66 multiple-choice questions on courses that undergraduates commonly take. ETS gives the test on computer, the total testing time is 2 hours 50 minutes, and no section is timed separately.

Is this GRE math subject test practice exam official?

No. All 66 questions were written for this page from ETS's published content areas, and each answer is worked out from what the question prints. None is taken or reworded from a real test or from the test in ETS's practice book.

Where can I find more GRE math subject test practice questions?

ETS publishes a free GRE Mathematics Test Practice Book. It holds one full-length practice test of ETS's own questions, a worksheet for scoring it and a score conversion table.

How is the GRE math subject test scored?

ETS counts the questions you answer correctly and converts that number to a total scaled score from 200 to 990, in ten-point steps. Nothing is subtracted for a wrong answer.

Is the GRE math subject test the same as GRE math?

No. The General Test's Quantitative Reasoning measure stays at high school mathematics. The Mathematics Test is a separate Subject Test on college mathematics. For the first, use the GRE quantitative practice test.

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