BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 4

Question 1

What is the order of the group of integers modulo 12 under addition, ℤ₁₂?

Accepted Answer

12

Answer

The group ℤ₁₂ consists of the 12 residues {0,1,2,...,11} under addition modulo 12, so its order is 12.

Question 2

Which of the following correctly describes the Big-O notation f(n) = O(g(n))?

Accepted Answer

There exist positive constants C and N such that |f(n)| ≤ C·g(n) for all n > N

Answer

Big-O describes an asymptotic upper bound: f grows no faster than a constant multiple of g beyond some threshold.

Question 3

In complex analysis, a function is holomorphic if:

Accepted Answer

It is complex-differentiable at every point in its domain

Answer

Holomorphic functions are complex-differentiable on an open set, which (remarkably) implies they are infinitely differentiable and analytic there.

Question 4

Which of the following is an example of a non-Euclidean geometry?

Accepted Answer

Hyperbolic geometry, where the parallel postulate fails

Answer

Hyperbolic geometry replaces Euclid's parallel postulate with the statement that infinitely many parallel lines can pass through a point outside a given line.

Question 5

The Pigeonhole Principle states that if n+1 objects are placed into n containers, then:

Accepted Answer

At least one container holds more than one object

Answer

The Pigeonhole Principle is a basic combinatorial result: distributing more objects than containers forces repetition.

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 4