Math Trivia Questions for College Students — page 2

A group G in which (ab)^2 = a^2b^2 for all a, b in G is necessarily

• Acyclic
• Babelian
• Cof order two
• Dfinite

If f is a continuously differentiable real-valued function defined on the open interval (-1, 4) such that f(3) = 5 and f'(x) ≥ -1 for all x, what is the greatest possible value of f(0)?

• A8
• B5
• C3
• D4

Which of the following is true?

• ABoth (a) and (b).
• BNeither (a) nor (b).
• CEvery compact space is complete
• DEvery complete space is compact

The curves y = (sin x + cos x)^4 and y = (sin x - cos x)^4 intersect at x = 0. If a is the smallest positive value of x at which these curves next intersect, determine the area between these curves from x = 0 to x = a.

• Aπ
• B4
• C3π/2
• D2

If f is a linear transformation from the plane to the real numbers and if f(1, 1) = 1 and f(-1, 0) = 2, then f(3, 5) =

• A-5
• B8
• C9
• D0

A total of x feet of fencing is to form three sides of a level rectangular yard. What is the maximum possible area of the yard, in terms of x ?

• Ax^2
• Bx^2/9
• Cx^2/8
• Dx^2/4

Consider the system of linear equations w + 3x + 2y + 2z = 0 w + 4x + y = 0 3w + 5x + 10 y + 14z = 0 2w + 5x + 5y + 6z = 0 with solutions of the form (w, x, y, z), where w x, y, and z are real. Which of the following statements is FALSE?

• AThe system has infinitely many solutions.
• BThe sum of any two solutions is a solution.
• CEvery solution is a scalar multiple of (-5, 1, 1, 0).
• DThe system is consistent.

A discrete graph is complete if there is an edge connecting any pair of vertices. How many edges does a complete graph with 10 vertices have?

• A20
• B25
• C10
• D45

A tree is a connected graph with no cycles. How many nonisomorphic trees with 5 vertices exist?

• A4
• B1
• C3
• D2

X is a finite set with the following property: The number of subsets of X that contain exactly 3 elements is 14 more than the number of subsets of X that contain exactly 2 elements. How many subsets of X contain exactly 4 elements?

• A28
• B25
• C35
• D24

The maximum number of acute angles in a convex 10-gon in the Euclidean plane is

• A4
• B3
• C2
• D1

Let f(x) be a function whose graph passes through the origin. If f(2n) = n^2 + f[2(n - 1)] for every integer n, what is the value of f(8)?

• A32
• B30
• C36
• D24

If A is a subset of the real line R and A contains each rational number, which of the following must be true?

• AIf A is open, then A = R.
• BIf A is uncountable, then A = R.
• CIf A is uncountable, then A is open.
• DIf A is closed, then A = R.

Let V be the set of all real polynomials p(x). Let transformations T, S be defined on V by T:p(x) -> xp(x) and S:p(x) -> p'(x) = d/dx p(x), and interpret (ST)(p(x)) as S(T(p(x))). Which of the following is true?

• AST - TS is the identity map of V onto itself.
• BST = 0
• CST = TS
• DST = T

If a real number x is chosen at random in the interval [0, 3] and a real number y is chosen at random in the interval [0, 4], what is the probability that x < y?

• A2/3
• B5/8
• C7/12
• D1/2

Let V be the real vector space of all real 2 x 3 matrices, and let W be the real vector space of all real 4 x 1 column vectors. If T is a linear transformation from V onto W, what is the dimension of the subspace kernel of T?

• A5
• B2
• C4
• D3

For which p is the norm ||x||_p expressable as an inner product?

• A1
• Binfinity
• Cp >= 1
• D2

For which of the following rings is it possible for the product of two nonzero elements to be zero?

• AThe ring of continuous real-valued functions on [0, 1]
• BThe ring of complex numbers
• CThe ring of integers modulo 11
• DThe ring {a + b * sqrt(2) : a and b are rational numbers}

If the finite group G contains a subgroup of order seven but no element (other than the identity) is its own inverse, then the order of G could be

• A35
• B37
• C27
• D28

Let S, T, and U be nonempty sets, and let f: S -> T and g: T -> U be functions such that the function g ∘ f : S -> U is one-to-one (injective). Which of the following must be true?

• Ag is one-to-one.
• Bf is one-to-one.
• Cf is onto.
• Dg is onto.

What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?

• Asqrt(2)
• Bpi
• C1/2
• D1

A fair coin is to be tossed 100 times, with each toss resulting in a head or a tail. If H is the total number of heads and T is the total number of tails, which of the following events has the greatest probability?

• AH >= 48 and T >= 48
• B51 <= H <= 55
• CH = 50
• DT >= 60

In a game two players take turns tossing a fair coin; the winner is the first one to toss a head. The probability that the player who makes the first toss wins the game is

• A2/3
• B1/3
• C1/2
• D1/4

Which of the following is NOT a group?

• AThe nonzero real numbers under multiplication
• BThe complex numbers under addition
• CThe integers under addition
• DThe nonzero integers under multiplication

Let y = f(x) be a solution of the differential equation x dy + (y - xe^x) dx = 0 such that y = 0 when x = 1. What is the value of f(2)?

• A2e
• B1/e
• Ce^2/2
• D1/(2e)

If f: R -> R is a bounded function that is Lebesgue integrable, then which of the following must be true?

• AThere does not exist a countably infinite E of R such that f is nondifferentiable at every x \in E.
• BNone of the above.
• Cf is Riemann integrable.
• DThere does not exist an uncountable subset E of R such that f is nondifferentiable at every x \in E.

If a polynomial f(x) over the real numbers has the complex numbers 2 + i and 1 - i as roots, then f(x) could be

• Ax^3 + 5x^2 + 4x + 1
• Bx^4 + 7x^2 + 10
• Cx^3 - x^2 + 4x + 1
• Dx^4 - 6x^3 + 15x^2 - 18x + 10

What is the volume of the solid in xyz-space bounded by the surfaces y = x^2, y = 2 - x^2, z = 0, and z = y + 3?

• A32/3
• B8/3
• C104/105
• D16/3

For which integers n such that 3 <= n <= 11 is there only one group of order n (up to isomorphism)?

• AFor no such integer n
• BFor 3, 5, 7, 9, and 11 only
• CFor 3, 5, 7, and 11 only
• DFor 4, 6, 8, and 10 only