Proof by contradictionEdexcel A-Level Maths: Mind map

Exam questions on Proof by contradiction

1. A student begins a proof that $\sqrt{2}$ is irrational by assuming that $\sqrt{2}=\frac{a}{b}$, where $a$ and $b$ are integers and $b\neq0$.
   Given that $a^2=2b^2$ and that $a$ is even, complete the proof that $\sqrt{2}$ is irrational.2 marks
2. Euclid's proof that there are infinitely many primes begins by assuming that there is a finite number of primes, $p_1,p_2,\ldots,p_n$, and then considers $N=p_1p_2\cdots p_n+1$.
   Complete the proof that there are infinitely many primes.2 marks
3. Assume, for a contradiction, that $\log_2 3=\frac{p}{q}$, where $p$ and $q$ are positive integers.
   Show that $3^q=2^p$.3 marks