topic
TMUA proof and counterexample questions
All 24 proof and counterexample questions from the TMUA past papers, 2016 to 2023, newest first, each with a worked solution.
2023
- P2 Q3 · Consider the claim: For all positive real numbers x and y, √(x^y) = x^√(y) Which of the following…
- P2 Q4 · A student attempts to answer the following question. What is the largest number of consecutive odd…
- P2 Q19 · In this question, f(x) is a non-constant polynomial, and g(x) = x,f'(x). f(x) = 0 for exactly M…
2022
- P2 Q3 · Consider the following statement about the positive integer n: if n is prime, then n² + 2 is not…
- P2 Q7 · Consider the following claim: The difference between two consecutive positive cube numbers is…
- P2 Q17 · A student answered the following question. a and b are non-zero real numbers. Prove that the… hard
2021
- P2 Q5 · On which line is the first error in the following argument?
- P2 Q10 · The first seven terms of a sequence of positive integers are: u₁ = 15, u₂ = 21, u₃ = 30, u₄ = 37,…
- P2 Q11 · A student attempts to solve the following problem, where a and b are non-zero real numbers: Show…
2020
- P2 Q3 · A student makes the following claim: For all integers n, the expression 4((9n+1)/(2) - (3n-1)/(2))…
- P2 Q4 · Consider the following statement: Every positive integer N that is greater than 6 can be written…
- P2 Q8 · A student is asked to prove whether the following statement (*) is true or false: (*) For all real…
2019
- P2 Q4 · Consider the following conjecture: If N is a positive integer that consists of the digit 1…
- P2 Q6 · A student attempts to solve the equation cos x + sin x tan x = 2sin x - 1 in the range 0 ≤ x ≤ 2π.…
- P2 Q7 · For which one of the following statements can the fact that 12² + 16² = 20² be used to produce a…
2018
- P2 Q3 · Consider the following statement: A car journey consists of two parts. In the first part, the…
- P2 Q6 · Which one of the following functions provides a counterexample to the statement: if f'(x) > 0 for…
- P2 Q8 · The diagram shows an example of a mountain profile : a zig-zag path made of strokes above a dashed…
- P2 Q9 · Consider the following attempt to solve the equation 4x√(2x-1) = 10x - 5: 4x√(2x-1) = 10x - 5…
- P2 Q13 · The following is an attempted proof of the conjecture: if tanθ > 0, then sinθ + cosθ > 1. Suppose…
2017
- P2 Q5 · Consider the following three statements: 1 10p² + 1 and 10p² - 1 are both prime when p…
- P2 Q9 · Consider the following attempt to prove this true theorem: Theorem: a³ + b³ = c³ has no solutions…