Math Trivia 3

15 Points Per Correct Answer - No time limit  

1. Arithmetic: What is the remainder when 25 is divided by 8? 1 3 2 4

2. Basic Algebra: Which of the following is the correct way to factor x^2 - 100? (x - 10)^2 (x - 10)(x + 10) (x + 10)^2 x(x - 10)

3. Geometry: A triangle has one side of length 8 and another side of length 10. Which of the following COULD NOT be the length of the third side? 20 3 16 6

4. Analytic Geometry: Which of the following conics have asymptotes? The ellipse The hyperbola The circle The parabola

5. Calculus: Suppose F(x) and G(x) are antiderivatives of a continuous function f(x). What can you say about their difference F(x) - G(x) ? It's equal to the derivative of f(x). It's always 0. It's a constant. It's equal to f(x).

6. Analytic Geometry With Vectors: Let a and b be two non-zero vectors in R^3 and let c = a x b denote their cross product. Which of the following is equal to the dot product of a and c? a c b 0

7. Basic Set Theory: If A and B are sets, let A - B denote the set of all elements of A that are not in B. Which of the following is TRUE? (A - B) is a subset of A All of these (A - B) and B are disjoint The union of (A - B) and (A intersect B) is A

8. Logic: Which of the following statements is logically equivalent to p AND q? [Note: in the answers, NOT means "negation"] NOT( (NOT p) AND (NOT q) ) NOT( (NOT p) OR (NOT q) ) NOT( p OR q ) NOT( p AND q)

9. Linear Algebra: Let A be a 2 by 2 matrix with real entries that has 3 as one of its eigenvalues, and let v be an eigenvector belonging to the eigenvalue 3. Which of the following must equal Av? (1/3)v the zero vector 9v 3v

10. Combinatorics: Let (n choose k) denote the number of k element subsets of an n element set (also called a binomial coefficient). Which of the following is equal to (9 choose 5) + (9 choose 6) ? (10 choose 7) (10 choose 5) (10 choose 6) (9 choose 7)

11. Elementary Number Theory: Let x and y be integers and let S denote the sum of their squares. Which of the following could S NOT be congruent to mod 4? In other words, if we divided the sum of the squares of two integers by 4, which of the following could not be a remainder? 0 3 1 2

12. Real Analysis: Let a_n denote the sequence 2 + (-1)^n. What is the lim inf (as n goes to infinity) of a_n? 1 The lim inf does not exist 3 2

13. Group Theory: Which of the following groups is abelian? The symmetric group on 4 letters, S4 The cyclic group of order 6, Z6 The dihedral group of order 8, D8 The quaternion group, Q

14. Ring Theory: Let R be a ring (non-commutative) and let x and y be elements of R which satisfy yx = 3xy. Which of the following is equal to (x + y)^2? x^2 + y^2 x^2 + 4xy + y^2 x^2 + 2xy + y^2 x^2 + 3xy + y^2

15. Basic Topology: Let X and Y be topological spaces, let U be an open set of X, V an open set of Y, and let f be a continuous function that maps X into Y. Which of the following must be true? The inverse image of V under f is an open set of X f(U) is a closed set of Y f(U) is an open set of Y The inverse image of V under f is a closed set of X