BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 3 β Questions and Answers
Question 1: What is the fundamental theorem of calculus (Part 1) about?
- If F(x) = β«βΛ£ f(t) dt and f is continuous, then F'(x) = f(x) (Correct answer)
- Every continuous function has an antiderivative that is also continuous
- The definite integral equals the limit of Riemann sums
- The derivative of a polynomial is one degree lower
Correct answer: If F(x) = β«βΛ£ f(t) dt and f is continuous, then F'(x) = f(x)
Part 1 of the FTC states that differentiation and integration are inverse operations when f is continuous.
Question 2: In graph theory, a tree with n vertices has exactly how many edges?
- n β 1 (Correct answer)
- n
- n + 1
- 2n β 2
Correct answer: n β 1
A tree is a connected acyclic graph; removing any edge disconnects it, and adding any edge creates a cycle, forcing exactly nβ1 edges.
Question 3: Which of the following is NOT a valid axiom in Zermelo-Fraenkel (ZF) set theory?
- The Axiom of Choice (Correct answer)
- The Axiom of Extensionality
- The Axiom of Regularity
- The Axiom of Power Set
Correct answer: The Axiom of Choice
The Axiom of Choice is independent of ZF; the system including it is called ZFC, not ZF.
Question 4: The Euler characteristic Ο of a convex polyhedron satisfies:
- Ο = V β E + F = 2 (Correct answer)
- Ο = V + E β F = 2
- Ο = V + E + F = 2
- Ο = V β E β F = 0
Correct answer: Ο = V β E + F = 2
Euler's formula for convex polyhedra states vertices minus edges plus faces equals 2.
Question 5: What does it mean for a set S in a metric space to be dense?
- Every point in the space is a limit point of S or belongs to S (Correct answer)
- S contains all its limit points
- S is closed and bounded
- S has the same cardinality as the ambient space
Correct answer: Every point in the space is a limit point of S or belongs to S
S is dense if its closure equals the whole space, meaning every open ball around any point of the space meets S.
Question 6: Which theorem guarantees that every bounded sequence of real numbers has a convergent subsequence?
- Bolzano-Weierstrass theorem (Correct answer)
- Heine-Borel theorem
- Monotone convergence theorem
- ArzelΓ -Ascoli theorem
Correct answer: Bolzano-Weierstrass theorem
The Bolzano-Weierstrass theorem states that every bounded sequence in ββΏ has a convergent subsequence.
Question 7: In linear algebra, the determinant of a product of square matrices satisfies:
- det(AB) = det(A) Β· det(B) (Correct answer)
- det(AB) = det(A) + det(B)
- det(AB) = det(A) β det(B)
- det(AB) = det(Aα΅) Β· det(B)
Correct answer: det(AB) = det(A) Β· det(B)
The determinant is multiplicative: det(AB) = det(A)det(B) for any two square matrices of the same size.
BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 3
What is the fundamental theorem of calculus (Part 1) about?