TMUA · Paper 2: Mathematical Reasoning

Proof techniques

5 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 5 questions

To prove that $\frac{a + b}{2} \geq ab$ for all positive real numbers $a$ and $b$ , which expression is it most useful to show is non-negative?
A student wants to prove that the sum of two consecutive integers is always odd. Which way of writing the two integers leads to a complete proof?
A proof by contradiction that there is no largest even integer begins: *'Suppose $N$ is the largest even integer.'* Which statement produces the contradiction?
A student proves that the product of two odd integers is odd. Let the integers be $2 m + 1$ and $2 n + 1$ . Which expression correctly shows that their product is odd?
To prove that $a^{2} + b^{2} \geq 2 ab$ for all real numbers $a$ and $b$ , which observation completes a correct proof?

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