Proof by contradictionEdexcel International A Level Maths: Mind map

Exam questions on Proof by contradiction

1. A student begins a proof by contradiction that $\sqrt2$ is irrational.
   Given that $p^2=2q^2$ and so $p$ is even, complete the proof.2 marks
2. Euclid's proof that there are infinitely many primes begins by assuming that there are only finitely many primes, $p_1,p_2,\ldots,p_n$. Let $N=p_1p_2\cdots p_n+1$.
   A student takes the first six primes, $2,3,5,7,11,13$, so $N=30031$. Show that $N$ is not prime, and state what this shows about Euclid's argument.2 marks
3. In a proof by contradiction you assume that the statement is false and show that this leads to an impossible result.
   Prove by contradiction that there are no integers $m$ and $n$ such that $14m+21n=1$.3 marks