TMUA · Paper 2: Mathematical Reasoning

Proof structure

8 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 8 questions

A proof by contradiction that $2$ is irrational begins by assuming the opposite. Which assumption is made?
In the proof that there are infinitely many primes, we suppose that $p_{1}, p_{2}, \dots, p_{n}$ is a complete list of the primes. Which number is then considered to reach a contradiction?
To prove *'if $n^{2}$ is even, then $n$ is even'* by contradiction, which assumption should be made at the start?
In a proof by contradiction, once a contradiction has been reached, what can be concluded?
A proof by contrapositive of *'if $P$ then $Q$ '* consists of proving which statement?
Which of the following is sufficient to show that the statement *'For every real number $x$ , $x^{2} + 1 > 2 x$ ' is false*?
Which of the following is sufficient to show that the statement *'There is a real number $x$ with $x^{3} = x^{2} + 4$ ' is true*?
A student checks that $n^{2} + n + 41$ is prime for $n = 1, 2, \dots, 39$ and concludes that it is prime for all positive integers $n$ . Which comment is correct?

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