A Level Edexcel Maths 2 hours 39 questions
1.1 Proof
1.1.1 Language of Proof / 1.1.2 Proof by Deduction / 1.1.3 Proof by Exhaustion /
1.1.4 Disproof by Counter Example
Easy (9 questions) /17
Medium (10 questions) /42 Scan here to return to the course
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Hard (10 questions) /42
Very Hard (10 questions) /41
Total Marks /142
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Easy Questions
1 In a mathematical argument, how are three consecutive integers usually denoted
algebraically?
(1 mark)
2 (i) In a mathematical argument, how is an even number usually denoted?
(ii) Similarly, how is an odd number usually denoted?
(2 marks)
3 Prove that the sum of two odd numbers is even.
(2 marks)
4 Explain why (x − 3) 2 ≥ 0 for all real values of x .
(1 mark)
5 Prove that the product of two even numbers is a multiple of 4.
(2 marks)
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6 Use a counter-example to show that (x 2 ) ≠ x .
(2 marks)
7 Prove by exhausting all possible factors that 11 is a prime number.
(2 marks)
8 Prove that k 2 − 6k + 9 > 0 for all real values of k ≠ 3 .
(2 marks)
p
9 Show that 0.6 can be written in the from , where p and q are integers.
q
What does this tell you about the type of number 0.6 is?
(3 marks)
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Medium Questions
1 Prove that the sum of any three consecutive integers is a multiple of 3.
(3 marks)
2 Prove that x 2 + 2 ≥ 2 for all values of x .
(2 marks)
3 Prove that the square of an even number is a multiple of 4.
(3 marks)
4 The set of numbers S is defined as all positive integers less than 5.
Prove by exhaustion that the cube of all values in S are less than 100.
(3 marks)
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5 Use a counter-example to prove that the difference between any two square numbers is
not always odd.
(2 marks)
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6 (a) Express 18 as a product of its prime factors.
(2 marks)
(b) Write down all prime numbers between 1 and 13.
(1 mark)
(c) By dividing 13 by each of the prime numbers found in part (b), prove that 13 is a prime
number.
(3 marks)
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7 (a) Factorise n 2 + 3n + 2 .
(1 mark)
(b) Hence show that n 3 + 3n 2 + 2n = n (n + 1) (n + 2) .
(1 mark)
(c) Given that n is even, write down whether (n + 1) and (n + 2) are odd or even.
(2 marks)
(d) Hence deduce whether n 3 + 3n 2 + 2n is odd or even. Justify your answer.
(2 marks)
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8 (a) By writing it as a fraction in its lowest terms, show that 0.35 is a rational number.
(2 marks)
m p
(b) Two rational numbers, a and b are such that a = and b = where m, n , p , q are
n q
integers with no common factors and n , q ≠ 0 .
Find an expression for ab .
(3 marks)
(c) Deduce whether or not the product ab is rational or irrational.
(2 marks)
9 Prove that a triangle with side lengths of 8 cm, 6 cm and 10 cm must contain a right-
angle. You may use the diagram below to help.
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(4 marks)
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10 (a) A standard chess board has 64, 1x1- sized squares.
It also has 1, 8x8 - sized square.
How many 2x2 - sized squares are there on a standard chess board?
(1 mark)
(b) Write down the number of 3x3 - sized and 4x4 - sized squares there are on a standard
chess board.
(2 marks)
(c) Hence show that there are 204 squares in total on a standard chess board.
(3 marks)
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Hard Questions
1 Prove that the sum of any three consecutive even numbers is a multiple of 6.
(4 marks)
2 Prove that f (x ) ≥ 4 for all values of x , where f (x ) = (3 − x ) 2 + 4 .
(3 marks)
3 Prove that the square of an odd number is always odd.
(3 marks)
4 The set of numbers S is defined as all positive integers greater than 5 and less than 10.
Prove by exhaustion that the square of all values in S differ from a multiple of 5 by 1.
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(4 marks)
5 Use a counter-example to prove that not all integers of the form 2n − 1 , where n is an
integer, are prime.
(2 marks)
6 By considering all possible prime factors of 17, prove it is a prime number.
(3 marks)
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7 (a) Fully factorise n 3 + 6n 2 + 8n .
(2 marks)
(b) Prove that, if n is odd, n 3 + 6n 2 + 8n is odd and that if n is even, n 3 + 6n 2 + 8n is even.
(3 marks)
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m p
8 (a) Two rational numbers, a and b are such that a = and b = , where m, n , p , q are
n q
integers with no common factors and n ,q ≠ 0 .
a
Find expressions for ab and .
b
(4 marks)
a
(b) Deduce whether or not ab and are rational or irrational.
b
(4 marks)
9 Prove that the exterior angle in any triangle is equal to the sum of the two opposite
interior angles. You may use the diagram below to help
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(4 marks)
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10 (a) A standard chess board has 64, 1x1 - sized squares.
It also has 1, 8x8 - sized square.
How many 2x2 - sized and 3x3 - sized squares are there on a standard chess board?
(2 marks)
(b) Hence show that there are 204 squares in total on a standard chess board.
(4 marks)
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Very Hard Questions
1 Prove that the sum of any three consecutive even numbers is always a multiple of 2, but
not always a multiple of 4.
(3 marks)
9x 2 +12x +4
2 Prove that f (x ) ≥ 0 for all values of x , where f (x ) = .
5
(3 marks)
3 Prove that the (positive) difference between an integer and its cube is the product of
three consecutive integers.
(3 marks)
4 The elements, x , of a set of numbers, S, are defined x ∈ℕ, x < 6 .
Prove that every element of S can be written in the form 3n − 2 m where n , m ∈ℕ .
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(4 marks)
5 Give an example to show when the following statement is both true and false.
The square of a positive integer is always greater than doubling it.
(2 marks)
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6 (a) Prove that 23 is a prime number.
(4 marks)
(b) Briefly explain why only prime factors need to be tested for, in order to prove a number
is prime.
(2 marks)
7 Prove that, if n is negative, n 4 − n 3 is positive.
(4 marks)
8 Prove that the sum of two rational numbers is rational.
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(6 marks)
9 Prove the angle at the circumference in a semi-circle is a right angle.
You may use the diagram below to help.
(4 marks)
10 A standard chess board has 64 1x1 - sized squares.
It also has 1 8x8 - sized square.
Prove that there are 204 squares on a standard chess board.
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(6 marks)
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