Disproof by counter-exampleAQA A-Level Maths: Mind map

Exam questions on Disproof by counter-example

1. A student claims: 'For every positive integer $n$, the number $n^2+n+41$ is prime.'
   The student first tested $n=1,2,3,4,5$ and found a prime each time. Explain why this did not prove the claim, and what is needed to disprove it.2 marks
2. Consider two statements. Statement P: if $x$ and $y$ are irrational numbers then $x+y$ is irrational. Statement Q: if $x$ and $y$ are irrational numbers then $xy$ is irrational.
   Give a counter-example to statement P in which $y\ne-x$, and show that it is a counter-example.2 marks
3. Consider the statement: for every real number $x$, $x^2>x$.
   Show that the statement is false by finding a counter-example and explaining clearly why it works.3 marks