Podcast
Questions and Answers
What concept is tested in TMUA 2021 related to integers?
What concept is tested in TMUA 2021 related to integers?
- Functions
- Factors of Integers (correct)
- Rational Numbers
- Irrational Numbers
Which TMUA exam year features prime numbers as a topic?
Which TMUA exam year features prime numbers as a topic?
- TMUA 2022 and TMUA 2021 (correct)
- TMUA 2023 only
- TMUA 2021 and TMUA 2023
- TMUA 2022 only
Which of the following was a focus in both TMUA 2021 assessments?
Which of the following was a focus in both TMUA 2021 assessments?
- Counterexamples (correct)
- Sequences and Series
- Probability Theory
- Polar Coordinates
What type of statements are examined in TMUA 2023 regarding mathematical expressions?
What type of statements are examined in TMUA 2023 regarding mathematical expressions?
Which content area was deemed medium difficulty across two different TMUA years?
Which content area was deemed medium difficulty across two different TMUA years?
Flashcards
Counterexample (TMUA 2023)
Counterexample (TMUA 2023)
An example that proves a statement is false.
Counterexample (TMUA 2022)
Counterexample (TMUA 2022)
A specific case that shows a statement is incorrect.
Counterexample (TMUA 2021)
Counterexample (TMUA 2021)
Example showing a statement is NOT always true.
Counterexample (TMUA 2021)
Counterexample (TMUA 2021)
Signup and view all the flashcards
Prime Numbers
Prime Numbers
Signup and view all the flashcards
Factors of Integers
Factors of Integers
Signup and view all the flashcards
Statements with Exponents
Statements with Exponents
Signup and view all the flashcards
Study Notes
Counterexamples: Statements with Exponents
- Claim: For all positive real numbers x and y, √x = x√y
- Counterexample I: x = 1, y = 16. √1 = 1 and 1√16 = 1*4 = 4. This is not a counterexample as it does not satisfy the claim.
- Counterexample II: x = 2, y = 8. √2 ≠ 2√8 = 2*2√2. This is a counterexample.
- Counterexample III: x = 3, y = 4. √3 ≠ 3√4 = 3*2=6. This is not a counterexample.
- Correct Answer: II only
Counterexamples: Prime Numbers
- Claim: If n is prime, then n² + 2 is not prime.
- Counterexample I: n = 2. 2² + 2 = 6, which is not prime. This is not a counterexample, as the claim is satisfied.
- Counterexample II: n = 3. 3² + 2 = 11, which is prime. This is a counterexample.
- Counterexample III: n = 4. 4² + 2 = 18. This is not prime. Not a counterexample.
- Correct Answer: II only
Counterexamples: Factors of Integers
- Claim: If a is a factor of bc, then a is a factor of b or a is a factor of c.
- Counterexample I: a = 5, b = 10, c = 20. 5 is a factor of 10*20, and 5 is a factor of 10, so this is not a counterexample.
- Counterexample II: a = 8, b = 4, c = 4. 8 is a factor of 4*4. 8 is not a factor of 4. This is a counterexample.
- Counterexample III: a = 6, b = 7, c = 12. 6 is a factor of 7*12 = 84, and 6 is a factor of 12, so this is not a counterexample.
- Correct Answer: II only
Counterexamples: Sequence of Positive Integers
- Sequence: u₁ = 15, u₂ = 21, u₃ = 30, u₄ = 37, u₅ = 44, u₆ = 51, u₇ = 59
- Claim: If n is a prime number, then uₙ is a multiple of 3 or uₙ is a multiple of 5.
- Counterexample: n = 5. u₅ = 44 is not a multiple of 3 or 5. This is the smallest counterexample.
- Correct Answer: 5
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
