100+ Free GRE Math Subject Test Practice Questions

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If f(x) = arctan(x), find f'(x).

−1/(1 + x²)

1/√(1 − x²)

1/(1 + x²)

1/(1 − x²)

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GRE Verbal Reasoning

Practice Questions100 questions

2026 Statistics

Key Facts: GRE Math Subject Test Exam

~66 questions

Exam Length

ETS GRE Subject Test Content & Structure

2 hrs 50 min

Time Limit

ETS GRE Subject Test Overview

200–990

Score Scale

ETS GRE Score Reporting

~50% Calculus

Largest Content Area

ETS GRE Mathematics Content Outline

$150

Registration Fee (U.S.)

ETS GRE Subject Test Fees 2025

3× per year

Test Dates

ETS GRE Subject Test Schedule (Sept, Oct, Apr)

The GRE Mathematics Subject Test contains approximately 66 five-choice questions and lasts 2 hours 50 minutes, with scores reported on a 200–990 scale (ETS, 2025). Calculus accounts for roughly 50% of questions, algebra (including linear, abstract, and number theory) for ~25%, and additional topics (real analysis, topology, complex variables, discrete math, probability) for ~25%. The test is administered by ETS on paper at designated test centers in September, October, and April. Top PhD programs in mathematics typically look for scores in the 80th percentile or above (~800+).

Sample GRE Math Subject Test Practice Questions

Try these sample questions to test your GRE Math Subject Test exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Find the derivative of f(x) = x³ ln(x).

A.3x² ln(x) + x²

B.3x² ln(x) + x

C.x² ln(x) + x²

D.3x ln(x) + x²

Explanation: By the product rule, f'(x) = 3x² · ln(x) + x³ · (1/x) = 3x² ln(x) + x². The product rule states (uv)' = u'v + uv'.

2Evaluate ∫₀¹ x·eˣ dx.

A.1

B.e − 1

C.e + 1

D.2e − 2

Explanation: Integration by parts with u = x, dv = eˣ dx gives uv − ∫v du = x·eˣ − ∫eˣ dx = x·eˣ − eˣ. Evaluating from 0 to 1: (e − e) − (0 − 1) = 0 + 1 = 1.

3Let f(x) = (x² − 4)/(x − 2). What is lim_{x→2} f(x)?

A.0

B.2

C.4

D.undefined

Explanation: Factor the numerator: x² − 4 = (x−2)(x+2). So f(x) = x+2 for x ≠ 2. The limit as x→2 is 2+2 = 4.

4If f(x) = arctan(x), find f'(x).

A.1/(1 + x²)

B.1/√(1 − x²)

C.−1/(1 + x²)

D.1/(1 − x²)

Explanation: The standard derivative formula is d/dx[arctan(x)] = 1/(1 + x²). This follows from implicit differentiation of tan(y) = x.

5Find the area enclosed between y = x² and y = x over [0, 1].

A.1/6

B.1/3

C.1/2

D.1/4

Explanation: On [0,1], x ≥ x², so the area = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.

6Which of the following is the general solution to the ODE dy/dx = 2xy?

A.y = Ce^(x²)

B.y = Ce^(2x)

C.y = C·x²·e^x

D.y = Ce^(x² + C)

Explanation: Separating variables: dy/y = 2x dx. Integrating: ln|y| = x² + K, so y = Ce^(x²) where C = ±e^K is an arbitrary constant.

7For the series Σ_{n=1}^{∞} 1/n², which of the following is true?

A.It diverges by the comparison test

B.It converges to π²/6

C.It converges to 1

D.It converges but its sum cannot be determined

Explanation: The series Σ 1/n² is a p-series with p = 2 > 1, so it converges. By the Basel problem (Euler), the sum equals π²/6 ≈ 1.645.

8Find ∂f/∂y for f(x, y) = x³y² + sin(xy).

A.2x³y + x·cos(xy)

B.3x²y² + y·cos(xy)

C.2xy + x·cos(xy)

D.3x²y² + cos(xy)

Explanation: ∂f/∂y = x³·(2y) + cos(xy)·(x) = 2x³y + x·cos(xy). Treat x as a constant when differentiating with respect to y.

9Determine all critical points of f(x) = x³ − 3x + 1.

A.x = 1 only

B.x = −1 only

C.x = ±1

D.x = 0 only

Explanation: f'(x) = 3x² − 3 = 3(x²−1) = 3(x−1)(x+1). Setting f'(x) = 0 gives x = 1 and x = −1. Both are critical points.

10What is the radius of convergence of the power series Σ_{n=0}^{∞} nˣⁿ / n!?

A.0

B.1

C.e

D.∞

Explanation: By the ratio test: |a_{n+1}/a_n| = |(n+1)x^{n+1}/((n+1)!) · n!/nxⁿ| = |x|/(n+1) → 0 for all finite x. Since the limit is 0 < 1 for every x, the radius of convergence is ∞.

About the GRE Math Subject Test Exam

The GRE Mathematics Subject Test is a computer-delivered exam of approximately 66 multiple-choice questions designed to assess undergraduate mathematics knowledge. It is offered in September, October, and April each year. The test covers calculus (~50%), algebra (~25%), and additional topics such as real analysis, topology, complex variables, and probability (~25%). Most applicants are mathematics, statistics, or physics majors applying to graduate programs.

Questions

66 scored questions

Time Limit

2 hours 50 minutes

Passing Score

Scored 200–990 (no universal passing score; programs set their own cutoffs)

Exam Fee

$150 (U.S.) (Educational Testing Service (ETS))

GRE Math Subject Test Exam Content Outline

~50%

Calculus and Applications

Single-variable calculus (limits, derivatives, integrals, arc length, optimization), multivariable calculus (partial derivatives, gradients, double/triple integrals, Jacobians, Green's/Stokes' theorems), differential equations (separable, linear first-order, second-order linear with constant coefficients, Bernoulli equations), and sequences and series (convergence tests, Taylor/Maclaurin series, power series, radius of convergence).

~25%

Algebra

Linear algebra (vector spaces, subspaces, rank-nullity theorem, eigenvalues and eigenvectors, determinants, diagonalization, characteristic polynomials), abstract algebra (groups, subgroups, Lagrange's theorem, cyclic and symmetric groups, rings, ideals, fields), and number theory (primes, Euclidean algorithm, modular arithmetic, Fermat's little theorem, Chinese Remainder Theorem).

~25%

Additional Topics

Real analysis (completeness, Cauchy sequences, uniform continuity, compact sets, Heine-Borel, MVT, IVT, Bolzano-Weierstrass), complex variables (analytic functions, Cauchy-Riemann equations, contour integrals, Cauchy's theorem, residue theorem), topology (open/closed sets, compactness, connectedness, Hausdorff spaces, fundamental groups), probability and statistics (distributions, expected value, variance, Bayes), and discrete mathematics (combinatorics, graph theory, Cayley's formula).

How to Pass the GRE Math Subject Test Exam

What You Need to Know

Passing score: Scored 200–990 (no universal passing score; programs set their own cutoffs)
Exam length: 66 questions
Time limit: 2 hours 50 minutes
Exam fee: $150 (U.S.)

Keys to Passing

Complete 500+ practice questions
Score 80%+ consistently before scheduling
Focus on highest-weighted sections
Use our AI tutor for tough concepts

GRE Math Subject Test Study Tips from Top Performers

1Master the fundamental theorems: IVT, MVT, Heine-Borel, Cauchy-Riemann, Lagrange's theorem — expect questions that ask you to identify which theorem applies.

2Drill convergence tests for series: ratio test, root test, comparison, integral test, and alternating series test — know which to apply and why.

3Review all standard group types (cyclic, symmetric, dihedral, quotient groups) and their orders, generators, and homomorphisms.

4Practice contour integration using the residue theorem — at least one complex-variable contour integral appears on most administrations.

5Time yourself strictly: 66 questions in 170 minutes = about 2.5 minutes per question — learn to skip and return to hard questions.

Frequently Asked Questions

Who should take the GRE Mathematics Subject Test?

Mathematics, statistics, applied mathematics, and some physics graduate school applicants — especially those applying to competitive PhD programs that explicitly require or recommend it. Many programs made it optional after 2020 but competitive programs still value high scores.

What score do I need on the GRE Math Subject Test?

There is no universal passing score. Top PhD programs (MIT, Harvard, Princeton, Stanford) typically see admitted students scoring in the 90th percentile (~880–990). Most strong programs look for 700+ (roughly 50th–60th percentile). Check each program's stated expectations.

How is the GRE Math Subject Test scored?

Scores are reported on a 200–990 scale in 10-point increments. ETS uses formula scoring: you earn one point per correct answer and lose 1/4 point per wrong answer on the paper-based version. The raw score is converted to the scaled score.

When is the GRE Mathematics Subject Test offered?

ETS offers the Mathematics Subject Test three times per year: September, October, and April. The test is computer-delivered at Prometric test centers. Register several weeks in advance to secure a preferred date and location.

What undergraduate topics should I review?

Prioritize single and multivariable calculus (the largest section), then real analysis, linear algebra, abstract algebra, and differential equations. Spend secondary effort on complex variables, topology, number theory, probability, and discrete mathematics.

How long should I study for the GRE Math Subject Test?

Most serious test-takers spend 100–200 hours reviewing all undergraduate math. Start with the ETS official practice book and work through actual GRE math past exams. Focus on understanding proofs and not just computation, since the test requires conceptual mastery.

100+ Free GRE Math Subject Test Practice Questions

GRE Subject Test: Mathematics