Pure Mathematics Year 2 (A Level) Unit Test 1: Proof
1 It is suggested that the sequence produces only prime numbers.
a Show that , and produce prime numbers. (2 marks)
b Prove by counter example that the sequence does not always produce a prime
number. (2 marks)
2 Prove by exhaustion that for positive integers from 1 to
6 inclusive. (3 marks)
3 Use proof by contradiction to prove the statement: ‘The product of two odd
numbers is odd.’ (5 marks)
4 Prove by contradiction that if n is odd, n3 + 1 is even. (5 marks)
5 Use proof by contradiction to show that there exist no integers a and b for which
25a + 15b = 1. (4 marks)
6 Use proof by contradiction to show that there is no greatest positive rational
number. (4 marks)
7 Use proof by contradiction to show that, given a rational number a and an
irrational number b, a − b is irrational. (4 marks)
8 Use proof by contradiction to show that there are no positive integer solutions to
the statement (5 marks)
9 a Use proof by contradiction to show that if n2 is an even integer then n is also an
even integer. (4 marks)
b Prove that is irrational. (6 marks)
10 Prove by contradiction that there are infinitely many prime numbers. (6 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
1 It is suggested that the sequence produces only prime numbers.
a Show that , and produce prime numbers. (2 marks)
b Prove by counter example that the sequence does not always produce a prime
number. (2 marks)
2 Prove by exhaustion that for positive integers from 1 to
6 inclusive. (3 marks)
3 Use proof by contradiction to prove the statement: ‘The product of two odd
numbers is odd.’ (5 marks)
4 Prove by contradiction that if n is odd, n3 + 1 is even. (5 marks)
5 Use proof by contradiction to show that there exist no integers a and b for which
25a + 15b = 1. (4 marks)
6 Use proof by contradiction to show that there is no greatest positive rational
number. (4 marks)
7 Use proof by contradiction to show that, given a rational number a and an
irrational number b, a − b is irrational. (4 marks)
8 Use proof by contradiction to show that there are no positive integer solutions to
the statement (5 marks)
9 a Use proof by contradiction to show that if n2 is an even integer then n is also an
even integer. (4 marks)
b Prove that is irrational. (6 marks)
10 Prove by contradiction that there are infinitely many prime numbers. (6 marks)
© Pearson Education Ltd 2017. Copying permitted for purchasing institution only. This material is not copyright free. 1
