Math Subject GRE Questions
1. Find the sum of the series
∞
X
n
4n+1
n=1
2. A rectangle has dimensions a units by B units with a > b. A diagonal divides the rectangle
into two triangles. A square, with sides parallel to those of the rectangle, is inscribed in each triangle. Find the distance between the vertices (of the squares) that lie in the interior of the rectangle.
3. Find the index of the subgroup generated

1
 ↓
3
by the permutation

2 3 4 5
↓ ↓ ↓ ↓ 
1 2 4 5
in the alternating group A5 .
4. In a pure radioactive decay model, the rate of change in the mass, M , satisfies the differential
equation
dM
1
=− M
dt
10
If the initial value of M is M0 , find M in terms of M0 after 20 units of time, t, have elapsed.
5. Let f (x) = x2n − x2n−1 + . . . + x4 − x3 + x2 − ax + b. Which value for the pair (a, b) will insure
that the x-axis will be tangent to the graph of f (x) at x = 1?
6. Let the set S be infinite and let the set T be countably infinite. Let S denote the complement
of S. Is S and T are both subsets of the real numbers, which of the following pairs of sets must be
of the same cardinality?
(A) T and S ∩ T ,
(B) S and S ∪ T ,
(C) T and S ∪ T ,
(D) Both (A) and (B), (E) Both (A) and (C)
7. Find the maximum value of f (x) = 5 sin(7x) + 12 cos(7x).
8. In a computer program, separate loops with distinct indices produce M and N operations respectively. If these reside internally in a loop with an independent index producing P operations,
find the total number of operations represented by the three loops.
9. Given that the Taylor series for g(x) is
∞
X
ax xn and f (x) = [g(x)]3 , find f 00 (0).
n=1
x
, find f (f (f (x))).
10. If f (x) =
1−x
∞
\
k 11. Define Sk =
k = k, k + 1, . . . , for k = 1, 2, . . .. Find
Sk .
j
k=1
1
2
12. Evaluate lim
n→∞
∞
X
k=1
1
.
n 1 + nk
Z
13. Evaluate the definite integral
0
3
a
x 2 + b2
dx.
x 2 + a2
2
14. Let f (x, y) = x − axy + y − x. Find the greatest lower bound for a so that f (x, y) has a
relative minimum point.
15. Given that S and T are subspaces of a vector space, which of the following is also a subspace?
(A) S ∩ T , (B) S ∪ T , (C) 2S, (D) Both (A) and (C), (E) Both (B) and (C)
16. Find the point x + iy at which the complex function f (x + iy) = x2 + y 2 + 2xi is differentiable.
17. Define a function f (x) to be differentiable redundant of order n if the n-th derivative f (n) (x) =
f (x) but f (k) (x) 6= f (x) when k < n. For easy examples, in this context ex is of order 1, e−x is of
order 2, and cos(x) is of order 4. Which of the following functions is differentiable redundant of
order 6?
√ √ 3
−x
x/2
−x
−x/2
(B) e + cos(x),
(C) e sin 23 x ,
(A) e + e
cos 2 x ,
(D) Both (A) and (C),
(E) All of (A), (B) and (C)
18. In a homogeneous system of 5 linear equations in 7 unknowns, the rank of the coefficient matrix
is 4. What is the maximum number of independent solution vectors to the system?
19. Let X, Y, Z represent statements. The compound statement (X ∨ Y ) ∧ (X∧ ∼ Z) is true under
which of the following conditions?
(A) X=T, Y =F, Z=F, (B) X=T, Y =F, Z=T, (C) X=F, Y =T, Z=T,
(D) X=T, Y =T, Z=T, (E) X=F, Y =T, Z=F.
20. From the set of integers {1, 2, . . . , 9}, how many nonempty subsets sum to an even integer?
21. How many topologies are possible on a set of 2 points?
22. In the (ε, δ) definition of the limit, lim f (x) = L, let f (x) = x3 + 3x2 − x + 1 and let c = 2.
x→c
Find the least upper bound on the δ so that f (x) is bounded within ε of L for all sufficiently small
ε > 0.
23. Find the remainder on dividing 320 by 7.
24. In the partial fractions decomposition of
s2 + 1
, what is the numerator of the fraction
(s2 − 2) (s2 + 3)
with denominator s2 + 3?
25. For how many distinct real coefficients, a, will the system of equations y = ax2 + 2 and
x = ay 2 + 2 admit a solution with y = 1?
3
26. A subgroup H in a group G is invariant if gH = Hg for every g ∈ G. If H and K are both
invariant subgroups of G, which of the following is also an invariant subgroup?
(A) H ∩ K, (B) HK, (C) H ∪ K, (D) Both (A) and (B), (E) Both (B) and (C)
27. Let R be a ring such that x2 = x for each x ∈ R. Which of the following must be true?
(A) x = −x for all x ∈ R,
(B) R is commutative,
(C) xy + yx = 0 for all x, y ∈ R
(D) Both (A) and (C),
(E) All of (A), (B), and (C)
√ 28. In the integral domain D = r + s 17 r, s integers , which of the following is irreducible?
√
√
√
√
√
(A) 8 + 2 7, (B) 3 − 17, (C) 9 − 2 17, (D) 7 + 17, (E) 13 + 17
29. For which value of k is xk a solution for the differential equation x2 y 00 − 3xy 0 − 4y = 0?
30. Which of the following is a factor of√x4 + 1?
√
(A) x + 1, (B) x2 + 1, (C) x2 − 2x + 1, (D) x2 + 2x − 1, (E) None of these
31. The surface given by z = x2 − y 2 is cut by the plane given by y = 3x, producing a curve in the
plane. Find the slope of this curve at the point (1, 3, −8).
32. If A is an n × n matrix with diagonal entries all a, and all other entries b, then one eigenvalue
of A is a − b. Find another eigenvalue of A.
2
8d
33. In ancient Egypt, the formula A =
was used for the area of a circle of diameter d.
9
Using the correct multiple relating the volume of a sphere to the area of a circle, what should the
Egyptian formula be for the volume of the sphere of diameter d?
34. Find k so that the matrix

k 1 2
A= 1 2 k 
1 2 3

has eigenvalue λ = 1.
1+i
35. Express z 14 in the form a + bi if z = √ .
2
36. Which number is nearest a solution of e−x/100 − e−x = e−1 ?
1
1
1
(A) 1 − ln(4), (B) 1 − ln(3), (C) 1 − ln(2), (D) −1, (E) 0
2
2
2
1
37. In the Laurent series f (z) =
centered at z = 1, what is the coefficient for (z − 1)−2 ?
z−4
Z x2
dt
38. If
, then find f 0 (2).
3
1
+
t
1
4
∞
X
1
must
n · 3n
n=2
(A) converge to a value greater than 1/4, (B) converge to a value greater than 1/9,
(C) converge to a value less than 1/18,
(D) converge to a value less than 1/12, (E) diverge
39. The series
40. If A is a square matrix of order n ≥ 4 and ai,j = i + j represents the entry in row i and column
j, then the rank of A is always
(A) 1, (B) 2, (C) n − 2, (D) n − 1, (E) None of these
41. In the finite field Z17 , what is the multiplicative inverse of 10?
42. The system of equations x2 − y = a and y − x = b has exactly one value of x in its solution(s).
What is this value of x?
43. log4 (64) is identical to
log10 (64)
(A) log7 (343), (B)
,
log10 (4)
(C) log8 (256),
(D) Both (A) and (B),
(E) Both (A) and (C)
44. Let S = {tan(k) | k = 1, 2 . . . }. Find the set of limit points of S on the real line.
45. Given the sequence an = sin(n), n ≥ 1, find lim inf{an } − lim sup{an }
n→∞
n→∞
46. Find the length of a diagonal of a rectangular pentagon of side length 1.
47. Which of the following is a solution to the differential equation y 00 = 2y 0 + 8y?
(A) e2x − e−4x , (B) xe4x , (C) e2x , (D) e4x − e−2x , (E) xe−2x
48. Let f (x) be defined and strictly increasing on (a, b]. Find the maximum value of g(x) = −f (x)
on (a, b].
49. Find the number of distinguishable permutations of six colored blocks if one is red, two are
yellow, and three are blue.
50. Let f (x) = 4x2 − x + 2 on the closed interval [−3, 3]. Find a value of c ∈ [−3, 3] such that the
slope of f (x) at x = c is equal to the slope of the secant line on [−3, 3].
2
2
2
51. The
p volume V of a region in space bounded above by the surface x + y + z = 4 and below
z = − x2 + y 2 is represented by a triple integral in spherical coordinates as
ZZZ
ρ2 sin(φ) dρ dφ dθ.
V
Find the upper limit of integration for φ.
52. The reliability of component C is R. A system is designed as a series of n subcomponents each
of which is doubly redundant using two components, C. Find the reliability of the system.
5
53. A family of curves is represented by the differential equation ydx − xdy = 0. Which of the
following best describes the family of orthogonal trajectories to this given family?
(A) all parabolas with vertex at (0, 0), (B) all hyperbolas with centers at (0, 0),
(C) all line through (0, 0),
(D) all circles with centers at (0, 0),
(E) all lines parallel to the y-axis
54. Determine which of the following sequences is a solution for the difference equation
xn+2 + 36n = xn+1 + 6xn
(A) 3n + 6n,
(B) 2n + 6n + 1,
(C) 6n + 6n,
(D) 3n + 6 + n,
(E) (−1)n 2n+1 + 6n + 1
55. Let f be a mapping from a topological space X onto itself. Which of the following is true for
continuous f .
(A) Every open set in X is the image of an open set B in X.
(B) f −1 (B) is bounded for each bounded set B in X.
(C) f is one-to-one
(D) Both (A) and (B)
(E) Both (A) and (C)
56. At the point (2, −1, 2) on the surface z = xy 2 , find a direction vector for the greatest rate of
decrease of z.
→
−
→
−
i −2j
→
−
→
−
→
−
→
−
→
−
→
−
→
− →
−
(A) i − 2 j , (B) i − 4 j , (C) √
, (D) − i + 4 j , (E) i + j
17
57. Let G be a polyhedron with 27 vertices and 40 edges. Find the number of faces on G.
58. Fine the coefficient of x2 y 3 in the binomial expansion of (x − 2y)5 .
59. In the evaluation of the integral
dx
, what is the coefficient of ln (|2 + 3x|)?
x(2 + 3x)2
60. A biased coin is tossed repeatedly until the first ‘tail’ occurs. The expected number of tosses
required to produce the first tail is estimated as T . Assuming this is true, find the probability of
at least two tails in 3T tosses.
61. The vertices of a quadrilateral are (0, 0), (1, 4), (3, 2), and (5, 5). Find the first coordinate of
the centroid of the region.
62. Let R be a ring and x 6= 0 be a fixed element in R. Which of the following is a subring of R?
(A) {r |xr = 0},
(B) {x |x−1 exists in R },
(C) {xn |n = 1, 2, . . .},
(D) {nx |n is an integer }, (E) Both (A) and (D)
63. If f (x) = ln(x), find
f (2 + m + n) − f (2 + m) − f (2 + n) + f (2)
lim lim
m→0 n→0
mn
6
64. Let S = {x1 , x2 , . . . , xn , . . .} be a topological space where the open sets are Un = {x1 . . . . , xn },
n = 1, 2, . . .. Let E = {x2 , x4 , . . . , x2k , . . .}. Find the set of cluster points of E.
65. Given that
∞
X
an converges to L, which conclusion is valid for
n=1
(A) It may diverge,
(D) It converges to M > L,
∞
X
a2n ?
n=1
(B) It converges absolutely,
(E) It converges to L2
(C) It converges to M < L,