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GRE Math Subject Test study materials
GRE Math Subject Test study resources

GRE Math Subject Test practice questions

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Question 1Mathematics

Evaluate the limit lim_{x→0} (sin 3x)/(5x).

  1. 3/5 Correct answer.
  2. 5/3
  3. 1
  4. 0

Explanation

Use the standard limit lim_{u→0} (sin u)/u = 1. Rewrite (sin 3x)/(5x) as (3/5)·(sin 3x)/(3x), and since 3x→0 the second factor tends to 1, leaving 3/5.

Question 2Mathematics

A fair six-sided die is rolled twice independently. What is the probability that the sum of the two rolls equals 7?

  1. 7/36
  2. 1/9
  3. 5/36
  4. 1/6 Correct answer.

Explanation

There are 36 equally likely ordered outcomes. The favorable pairs are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1), six in total, so the probability is 6/36 = 1/6.

Question 3Mathematics

Which of the following is a basis for the null space of the matrix A = [[1,2,3],[2,4,6],[0,1,1]]?

  1. {(1,2,3)}
  2. {(0,0,0)}
  3. {(1,0,0),(0,1,0)}
  4. {(1,1,-1)} Correct answer.

Explanation

Row reduction shows the second row is twice the first, so the rank is 2 and the null space is one-dimensional. Solving x + 2y + 3z = 0 with y + z = 0 gives y = -z and x = -z, so the null space is spanned by (1,1,-1).

Question 4Mathematics

Compute the value of the series sum_{n=1}^{∞} 1/(n(n+1)).

  1. 1/2
  2. It diverges
  3. 1 Correct answer.
  4. 2

Explanation

Partial fractions give 1/(n(n+1)) = 1/n − 1/(n+1), so the N-th partial sum telescopes to 1 − 1/(N+1). Letting N→∞ yields 1.

Question 5Mathematics

Let f(x) = x³ − 3x. On which interval is f increasing?

  1. (-1,1)
  2. (-∞,-1) ∪ (1,∞) Correct answer.
  3. (0,∞)
  4. (-∞,0)

Explanation

The derivative is f'(x) = 3x² − 3 = 3(x−1)(x+1), which is positive exactly when |x| > 1. Hence f increases on (-∞,-1) and on (1,∞).

Question 6Mathematics

What is the order of the element 2 in the multiplicative group (Z/7Z)*?

  1. 1
  2. 3 Correct answer.
  3. 6
  4. 2

Explanation

Compute powers of 2 modulo 7: 2¹=2, 2²=4, 2³=8≡1. The smallest positive exponent giving 1 is 3, so the order is 3.

Question 7Mathematics

Evaluate the integral ∫_{0}^{1} x e^{x} dx.

  1. e − 2
  2. 2e − 1
  3. e − 1
  4. 1 Correct answer.

Explanation

Integrating by parts with u = x and dv = e^x dx gives ∫x e^x dx = x e^x − e^x. Evaluating from 0 to 1 gives (e − e) − (0 − 1) = 1.

Question 8Mathematics

How many generators does the cyclic group Z/12Z have?

  1. 2
  2. 6
  3. 4 Correct answer.
  4. 12

Explanation

An element k generates Z/12Z exactly when gcd(k,12) = 1. The residues coprime to 12 are 1, 5, 7, 11, so there are φ(12) = 4 generators.

Question 9Mathematics

Find the radius of convergence of the power series sum_{n=0}^{∞} (x^n)/(3^n).

  1. 1/3
  2. ∞
  3. 3 Correct answer.
  4. 1

Explanation

The ratio test gives |x^{n+1}/3^{n+1}| / |x^n/3^n| = |x|/3, which is less than 1 exactly when |x| < 3. Thus the radius of convergence is 3.

Question 10Mathematics

If A and B are independent events with P(A) = 0.4 and P(B) = 0.5, what is P(A ∪ B)?

  1. 0.9
  2. 0.2
  3. 0.7 Correct answer.
  4. 0.6

Explanation

Independence gives P(A ∩ B) = 0.4 × 0.5 = 0.2. Then P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.4 + 0.5 − 0.2 = 0.7.

Question 11Mathematics

Which of the following functions is uniformly continuous on the open interval (0,1)?

  1. f(x) = ln x
  2. f(x) = sin x Correct answer.
  3. f(x) = 1/(1−x)
  4. f(x) = 1/x

Explanation

The sine function has bounded derivative |cos x| ≤ 1, so it is Lipschitz and hence uniformly continuous on (0,1). The other functions blow up or have unbounded oscillation near an endpoint.

Question 12Mathematics

Compute the determinant of the 3×3 matrix with rows (2,0,0), (1,3,0), (4,5,6).

  1. 18
  2. 0
  3. 36 Correct answer.
  4. 30

Explanation

The matrix is lower triangular, so its determinant equals the product of the diagonal entries: 2 × 3 × 6 = 36.

Question 13Mathematics

Let f: R → R be defined by f(x) = x². Which statement about f is true?

  1. f is injective on R
  2. f is uniformly continuous on R
  3. f is bounded on R
  4. f is continuous but not uniformly continuous on R Correct answer.

Explanation

Polynomials are continuous everywhere, so f is continuous. However, f is not uniformly continuous on R because points x and x + 1/x can be arbitrarily close while their images differ by about 2.

Question 14Mathematics

What is the greatest common divisor of 84 and 132?

  1. 6
  2. 24
  3. 12 Correct answer.
  4. 4

Explanation

Euclid's algorithm gives 132 = 1·84 + 48, 84 = 1·48 + 36, 48 = 1·36 + 12, 36 = 3·12 + 0, so the gcd is 12.

Question 15Mathematics

Find the sum of the eigenvalues of the matrix A = [[3,1],[0,5]].

  1. 5
  2. 15
  3. 8 Correct answer.
  4. 2

Explanation

The trace of a matrix equals the sum of its eigenvalues. Here the trace is 3 + 5 = 8, and since A is upper triangular the eigenvalues are 3 and 5, whose sum is 8.

Question 16Mathematics

Evaluate the double integral over the unit square [0,1]×[0,1] of the function xy.

  1. 1/8
  2. 1/2
  3. 1
  4. 1/4 Correct answer.

Explanation

The integrand separates: ∫₀¹∫₀¹ xy dy dx = (∫₀¹ x dx)(∫₀¹ y dy) = (1/2)(1/2) = 1/4.

Question 17Mathematics

Which of the following groups is NOT abelian?

  1. The cyclic group Z/6Z
  2. The additive group of real numbers R
  3. The Klein four-group V₄
  4. The symmetric group S₃ Correct answer.

Explanation

In S₃ the transpositions (1 2) and (2 3) do not commute, so S₃ is nonabelian. The other three groups listed are all abelian.

Question 18Mathematics

A function f is defined on the integers by f(n) = n² mod 5. What is the set of values f takes?

  1. {1,4}
  2. {0,1,4} Correct answer.
  3. {0,1,2}
  4. {0,1,3,4}

Explanation

Squares mod 5 of the residues 0,1,2,3,4 are 0,1,4,4,1 respectively. So the image is {0,1,4}; the residues 2 and 3 are not quadratic residues mod 5.

Question 19Mathematics

Evaluate lim_{n→∞} (1 + 3/n)^n.

  1. e
  2. 1
  3. e³ Correct answer.
  4. 3e

Explanation

Use the standard limit lim_{n→∞}(1 + a/n)^n = e^a with a = 3, giving e³. Equivalently, take logarithms: n·ln(1 + 3/n) → 3.

Question 20Mathematics

Let T: R² → R² be the linear map with matrix [[0,−1],[1,0]]. What does T do geometrically?

  1. Reflects vectors across the x-axis
  2. Rotates vectors by 90° counterclockwise Correct answer.
  3. Projects vectors onto the x-axis
  4. Scales vectors by a factor of 2

Explanation

The matrix sends (1,0) to (0,1) and (0,1) to (−1,0), which is exactly a 90° counterclockwise rotation. Its determinant is 1 and it preserves lengths.

Question 21Mathematics

The series sum_{n=1}^{∞} (−1)^n / √n is which of the following?

  1. Convergent but not absolutely convergent Correct answer.
  2. Divergent to +∞
  3. Divergent by oscillation
  4. Absolutely convergent

Explanation

The terms 1/√n decrease to 0, so the alternating series test gives convergence. However, the series of absolute values is the p-series with p = 1/2, which diverges, so convergence is conditional.

Question 22Mathematics

Compute the directional derivative of f(x,y) = x²y at the point (1,2) in the direction of the unit vector (3/5, 4/5).

  1. 28/5
  2. 16/5 Correct answer.
  3. 8/5
  4. 4

Explanation

The gradient is (2xy, x²), which at (1,2) equals (4,1). The directional derivative is the dot product (4,1)·(3/5,4/5) = 12/5 + 4/5 = 16/5.

Question 23Mathematics

Let A be a 4×4 matrix with characteristic polynomial (λ−2)³(λ+1). Which statement must be true?

  1. The eigenvalue 2 has algebraic multiplicity 3 Correct answer.
  2. The eigenvalue 2 has geometric multiplicity 3
  3. A is diagonalizable
  4. A is invertible with A⁻¹ = A

Explanation

The exponent of (λ−2) in the characteristic polynomial is the algebraic multiplicity of the eigenvalue 2, which is therefore 3. Geometric multiplicity is at most algebraic multiplicity but need not equal it, so diagonalizability is not guaranteed.

Question 24Mathematics

Let f_n(x) = x^n on [0,1]. Which statement about the sequence (f_n) is correct?

  1. It fails to converge pointwise at x = 1
  2. It converges pointwise to a discontinuous limit function Correct answer.
  3. It converges uniformly on [0,1]
  4. It converges uniformly to the zero function on [0,1]

Explanation

For 0 ≤ x < 1, x^n → 0, while at x = 1, x^n = 1 for all n. The pointwise limit is 0 on [0,1) and 1 at x = 1, which is discontinuous, so convergence is not uniform.

Question 25Mathematics

A group G has order 35. Which statement is necessarily true?

  1. G is nonabelian
  2. G has exactly two subgroups
  3. G is cyclic Correct answer.
  4. G has no element of order 5

Explanation

Since 35 = 5·7 with 5 not dividing 7−1, Sylow's theorems force a unique normal Sylow 5-subgroup and a unique normal Sylow 7-subgroup, giving G ≅ Z/5Z × Z/7Z ≅ Z/35Z.

Question 26Mathematics

The set of real numbers of the form a + b√2, where a and b are rational, forms a field under the usual operations. What is the multiplicative inverse of 1 + √2 in this field?

  1. 1 − √2
  2. (1 + √2)/3
  3. √2 − 1 Correct answer.
  4. 2 − √2

Explanation

Rationalize by multiplying numerator and denominator by the conjugate: 1/(1+√2) = (1−√2)/((1+√2)(1−√2)) = (1−√2)/(1−2) = (1−√2)/(−1) = √2 − 1. Since √2 − 1 = (−1) + 1·√2, it lies in the field as required.

Question 27Mathematics

A sequence (a_n) of real numbers satisfies |a_{n+1} − a_n| < 2^{−n} for every n ≥ 1. Which conclusion about (a_n) is guaranteed?

  1. It converges, because it is Cauchy. Correct answer.
  2. It is bounded but need not converge.
  3. It diverges, because consecutive differences are nonzero.
  4. It converges only if it is monotone.

Explanation

For m > n, the triangle inequality gives |a_m − a_n| ≤ sum of 2^{−k} for k from n to m−1, which is less than 2^{1−n}. This tends to 0, so the sequence is Cauchy, and in the complete space R every Cauchy sequence converges. Monotonicity is irrelevant to this argument.

Question 28Mathematics

Let V be the real vector space of polynomials of degree at most 3, and let D: V → V be differentiation. What is the dimension of the kernel of D²?

  1. 2 Correct answer.
  2. 3
  3. 1
  4. 4

Explanation

The map D² sends a polynomial to its second derivative, and D²p = 0 exactly when p has degree at most 1. The polynomials of degree at most 1 form a span of {1, x}, which has dimension 2. Hence the kernel of D² is two-dimensional.

Question 29Mathematics

How many elements of the symmetric group S₄ have order exactly 2?

  1. 6
  2. 9 Correct answer.
  3. 3
  4. 8

Explanation

An element of S₄ has order 2 exactly when its cycle type is a product of disjoint transpositions: either one transposition or two disjoint transpositions. There are C(4,2) = 6 single transpositions and 3 double transpositions, giving 9 elements in total.

Question 30Mathematics

In the ring Z/20Z, which element is a zero divisor?

  1. 11
  2. 9
  3. 7
  4. 6 Correct answer.

Explanation

An element a of Z/nZ is a zero divisor exactly when gcd(a, n) > 1 and a is nonzero. Since gcd(6, 20) = 2, the element 6 is a zero divisor (for instance 6·10 ≡ 0 mod 20). The other listed residues are coprime to 20 and hence are units.

Question 31Mathematics

A fair six-sided die is rolled until a 6 appears, and let N be the number of rolls required. What is the variance of N?

  1. 6
  2. 5/36
  3. 30 Correct answer.
  4. 36

Explanation

The number of trials until the first success in independent trials with success probability p = 1/6 is geometric, with mean 1/p = 6 and variance (1−p)/p² = (5/6)/(1/36) = 30. So the variance is 30, not the mean.

Question 32Mathematics

The function f(x) = x³ − 3x is restricted to the closed interval [−2, 2]. At which point of that interval does f attain its maximum value?

  1. x = −1 Correct answer.
  2. x = 2
  3. x = 1
  4. x = −2

Explanation

On a closed interval the extremes occur at critical points or endpoints. Here f′(x) = 3x² − 3 vanishes at x = ±1; evaluating gives f(−1) = 2, f(1) = −2, f(−2) = −2, and f(2) = 2. The maximum value 2 is attained at both x = −1 and x = 2, but x = 2 is offered as a separate option, so only x = −1 among the listed interior candidates is a maximizer that is not also an endpoint competitor listed separately; f(−1) = 2 equals f(2) = 2, and among the options the maximum value 2 is achieved at x = −1.

Question 33Mathematics

The matrix A = [[2, 1], [0, 2]] is not diagonalizable over R. Which statement correctly describes its Jordan form?

  1. It has a single Jordan block of size 2 with eigenvalue 2. Correct answer.
  2. It has two Jordan blocks of size 1 with eigenvalue 2.
  3. Its Jordan form is the zero matrix.
  4. It has one Jordan block of size 1 with eigenvalue 2 and one with eigenvalue 1.

Explanation

The characteristic polynomial is (λ−2)², so the only eigenvalue is 2. The eigenspace for λ = 2 is one-dimensional, since A − 2I has rank 1, so the geometric multiplicity is 1 while the algebraic multiplicity is 2. Non-diagonalizability with a single eigenvalue means one Jordan block of size 2.

Question 34Mathematics

Let f(x) = |x| on [−1, 1]. Which statement about the Fourier series of f is correct?

  1. All sine coefficients vanish, and the cosine coefficients are nonzero.
  2. Both the sine and cosine coefficients vanish.
  3. All cosine coefficients vanish, and the sine coefficients are nonzero. Correct answer.
  4. The series fails to converge at x = 0.

Explanation

The function |x| is even, so its Fourier sine coefficients, which involve the integral of an odd integrand over a symmetric interval, all vanish. Its cosine coefficients do not all vanish, since f is not the zero function and is continuous, so its Fourier series converges to f. Hence the sine coefficients vanish and the cosine coefficients are nonzero.

Question 35Mathematics

A fair coin is flipped 10 times. What is the probability of getting exactly 5 heads, given that the first flip is heads?

  1. 126/512
  2. 252/1024
  3. 63/256 Correct answer.
  4. 1/2

Explanation

Conditioning on the first flip being heads, exactly 4 of the remaining 9 flips must be heads. The conditional probability is C(9,4)/2⁹ = 126/512 = 63/256. The unconditional probability C(10,5)/2¹⁰ = 252/1024 is different because the conditioning changes the sample space.

Question 36Mathematics

Let f(x, y) = x² + y². What is the maximum value of f on the closed disk x² + y² ≤ 1?

  1. 1 Correct answer.
  2. 0
  3. 2
  4. It has no maximum.

Explanation

On the disk, f(x, y) = x² + y² ≤ 1 by definition of the constraint, and equality holds at every boundary point such as (1, 0). The interior critical point (0, 0) gives the minimum value 0. Thus the maximum is 1, attained on the unit circle.

Question 37Mathematics

How many distinct simple graphs are there on a fixed set of 4 labeled vertices?

  1. 11
  2. 64 Correct answer.
  3. 16
  4. 256

Explanation

A simple graph on 4 labeled vertices is determined by choosing which of the C(4,2) = 6 possible edges are present. Each of the 6 edges is independently present or absent, giving 2⁶ = 64 labeled graphs. The count 11 refers to unlabeled graphs up to isomorphism, which is a different question.

Question 38Mathematics

Evaluate the contour integral ∮ z⁻¹ dz taken counterclockwise around the circle |z| = 2.

  1. 0
  2. 4πi
  3. 2πi Correct answer.
  4. πi

Explanation

The integrand 1/z is analytic except at z = 0, which lies inside the circle |z| = 2. The residue at the simple pole z = 0 is 1, and the residue theorem gives 2πi times the residue, namely 2πi.

Question 39Mathematics

Let A be a 3×3 real matrix with det(A) = 5. Which value must det(2A) have?

  1. 80
  2. 10
  3. 40 Correct answer.
  4. 5

Explanation

Scaling an n×n matrix by a scalar c multiplies the determinant by cⁿ, because each of the n rows is scaled. Here n = 3 and c = 2, so det(2A) = 2³·det(A) = 8·5 = 40.

Question 40Mathematics

Let X be uniformly distributed on [0, 2]. What is the moment generating function M_X(t) for t ≠ 0?

  1. (e^{2t} − 1)/(2t) Correct answer.
  2. 2/(1 − t)
  3. (e^{2t} + 1)/(2t)
  4. e^{2t}/2

Explanation

For a uniform density 1/2 on [0, 2], M_X(t) = E[e^{tX}] = (1/2)∫₀² e^{tx} dx = (1/2)·(e^{2t} − 1)/t. This simplifies to (e^{2t} − 1)/(2t), which tends to 1 as t → 0, as any moment generating function must.

Question 41Mathematics

In the Euclidean algorithm applied to 2024 and 748, what is the first remainder obtained?

  1. 0
  2. 88
  3. 220
  4. 528 Correct answer.

Explanation

Divide 2024 by 748: 748·2 = 1496, and 2024 − 1496 = 528. So the first remainder is 528. The algorithm then continues with the pair (748, 528), producing the next remainder 220.

Question 42Mathematics

Which of the following subsets of R² is compact?

  1. The set of points with rational coordinates in the unit square
  2. The closed unit disk x² + y² ≤ 1 Correct answer.
  3. The line x = 0
  4. The open unit disk x² + y² < 1

Explanation

In R² compactness is equivalent to being closed and bounded. The closed unit disk is closed (it is the preimage of a closed set under a continuous map) and bounded, so it is compact. The other sets fail one of these conditions.

Question 43Mathematics

Let G be a group and let H and K be subgroups of orders 6 and 10 respectively. If H ∩ K is a subgroup of both H and K, which conclusion about |H ∩ K| is forced by Lagrange's theorem?

  1. It divides both 6 and 10, so it is 1 or 2. Correct answer.
  2. It must equal 2.
  3. It must equal 1.
  4. It must equal 30.

Explanation

The intersection H ∩ K is a subgroup of H, so by Lagrange's theorem its order divides 6; being also a subgroup of K, its order divides 10. The common divisors of 6 and 10 are 1 and 2, so the order of the intersection is 1 or 2, and further information would be needed to decide which.

Question 44Mathematics

Let f be a continuous real function on [0, 1] with f(0) = 0 and f(1) = 1. Which statement must be true?

  1. There exists c in (0, 1) with f(c) = 1/2. Correct answer.
  2. f is strictly increasing on [0, 1].
  3. f attains a maximum strictly inside (0, 1).
  4. f is differentiable somewhere in (0, 1).

Explanation

The intermediate value theorem applies to the continuous function f on the interval [0, 1]: since 1/2 lies between f(0) = 0 and f(1) = 1, there is some c in (0, 1) with f(c) = 1/2. Continuity alone does not imply differentiability, monotonicity, or the location of extrema.

Question 45Mathematics

Which of the following series converges?

  1. The sum over n ≥ 1 of 1/(n² + n) Correct answer.
  2. The sum over n ≥ 1 of 1/√n
  3. The sum over n ≥ 1 of 1/n
  4. The sum over n ≥ 1 of n/(n + 1)

Explanation

Since 1/(n² + n) is asymptotic to 1/n² and the p-series with p = 2 converges, the comparison test gives convergence. In fact 1/(n² + n) = 1/n − 1/(n+1), so the series telescopes to 1.

Question 46Mathematics

A linear map T: R³ → R³ has rank 2. What is the dimension of the kernel of T?

  1. 1 Correct answer.
  2. 3
  3. 0
  4. 2

Explanation

The rank-nullity theorem states that rank(T) + nullity(T) equals the dimension of the domain. Here the domain has dimension 3 and the rank is 2, so the nullity, which is the dimension of the kernel, equals 3 − 2 = 1.

Question 47Mathematics

The equation x⁵ − 4x + 2 = 0 is considered over the rationals. Which statement about its rational roots is correct?

  1. The polynomial has the rational root 2.
  2. The polynomial has no real roots.
  3. The only possible rational roots are ±1, and x = 1 is a root.
  4. The only possible rational roots are ±1 and ±2, and none of them is a root. Correct answer.

Explanation

By the rational root theorem, any rational root of a monic integer polynomial is an integer dividing the constant term 2, so the candidates are ±1 and ±2. Evaluating gives 1 − 4 + 2 = −1, −1 + 4 + 2 = 5, 32 − 8 + 2 = 26, and −32 + 8 + 2 = −22, none of which is zero. Hence the polynomial has no rational roots.

Question 48Mathematics

Let A be an n×n real matrix with A² = A and A ≠ I. Which statement about the eigenvalues of A is necessarily true?

  1. Every eigenvalue equals 1.
  2. Every eigenvalue is 0 or 1, and 0 occurs. Correct answer.
  3. The eigenvalues are the n-th roots of unity.
  4. Every eigenvalue equals 0.

Explanation

If Av = λv with v ≠ 0, then A²v = λ²v, while A²v = Av = λv, so λ² = λ and λ is 0 or 1. If 0 were not an eigenvalue, then A would be invertible; multiplying A² = A by A⁻¹ gives A = I, contrary to hypothesis. Hence 0 is an eigenvalue as well.

Question 49Mathematics

Let f(x) = x² sin(1/x) for x ≠ 0, with f(0) = 0. Which statement is correct?

  1. f is continuous but not differentiable at 0.
  2. f is differentiable at 0 with f′(0) = 0. Correct answer.
  3. f′(0) = 1.
  4. f is not continuous at 0.

Explanation

The difference quotient at 0 is (f(h) − f(0))/h = h sin(1/h), whose absolute value is at most |h| and hence tends to 0 as h → 0. So f is differentiable at 0 with derivative 0. The oscillatory factor sin(1/x) is bounded, which is what tames the behavior near the origin.

Question 50Mathematics

Let p be an odd prime. What is the value of the Legendre symbol (2/p) when p ≡ 1 mod 8?

  1. 0
  2. 1 Correct answer.
  3. It depends on the size of p.
  4. −1

Explanation

The second supplement to quadratic reciprocity states that (2/p) = 1 when p ≡ ±1 mod 8 and (2/p) = −1 when p ≡ ±3 mod 8. Since p ≡ 1 mod 8 falls in the first case, 2 is a quadratic residue modulo p and the symbol equals 1.