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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Proof

Proof by deduction, by exhaustion, disproof by counterexample, odd and even arguments.

12 original questions, 44 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Prove that the sum of any three consecutive integers is a multiple of 3

2 marks
Show answer
(n - 1) + n + (n + 1) = 3n, which is a multiple of 3 because n is an integer.
2

Jamal says

“If x and y are real numbers with x > y, then x2 > y2”

Use a counterexample to show that Jamal is wrong.

2 marks
Show answer
For example x = 1, y = -2: 1 > -2 but 12 = 1 is not greater than (-2)2 = 4, so the statement is false.
3

Prove by exhaustion that every prime number between 30 and 50 is either one more or one less than a multiple of 6

3 marks
Show answer
The primes are 31, 37, 41, 43, 47: 31 = 30 + 1, 37 = 36 + 1, 41 = 42 - 1, 43 = 42 + 1, 47 = 48 - 1, so every one is 6k ± 1.
4

Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8

3 marks
Show answer
(2n + 3)2 - (2n + 1)2 = 8n + 8 = 8(n + 1), a multiple of 8.
5

Prove that x2 − 6x + 11 > 0 for all real values of x

3 marks
Show answer
x2 - 6x + 11 = (x - 3)2 + 2, and (x - 3)2 ≥ 0, so x2 - 6x + 11 ≥ 2 > 0.
6

Prove that n2 + 5n + 3 is odd for every integer n

4 marks
Show answer
n even (n = 2k): 2(2k2 + 5k + 1) + 1. n odd (n = 2k + 1): 2(2k2 + 7k + 4) + 1. Both odd, so it is always odd.
7

For each statement, either prove that it is true or use a counterexample to show that it is false.

  1. The sum of any two consecutive square numbers is odd.
  2. n2 − n + 11 is prime for every positive integer n
5 marks
Show answer
a) True: n2 + (n + 1)2 = 2(n2 + n) + 1, odd b) False: n = 11 gives 121 = 11 × 11, not prime
8

Ava is asked to prove that n3 − n is a multiple of 6 for every positive integer n. She writes

“n = 1 gives 0, n = 2 gives 6, n = 3 gives 24 and n = 4 gives 60. These are all multiples of 6, so the statement is true.”

  1. Explain why Ava has not proved the statement.
  2. Prove that n3 − n is a multiple of 6 for every positive integer n
4 marks
Show answer
a) Only four cases checked; it must hold for all (infinitely many) positive integers b) n3 - n = (n - 1)n(n + 1), a product of three consecutive integers, so divisible by 2 and by 3, so by 6
9

The points A, B and C have coordinates (1, 2), (5, 4) and (3, 8) respectively.

Prove that triangle ABC is a right-angled isosceles triangle.

4 marks
Show answer
Gradient AB = 1/2, gradient BC = -2, product -1 so angle ABC = 90°. AB = BC = √20, so the triangle is right-angled and isosceles.
10
  1. Prove that x2 + y2 ≥ 2xy for all real numbers x and y
  2. Hence prove that, for all positive real numbers a and b, a + b2 ≥ √(ab)
5 marks
Show answer
a) (x - y)2 ≥ 0 so x2 + y2 ≥ 2xy b) x = √a, y = √b in a) gives a + b ≥ 2√(ab), so (a + b)/2 ≥ √(ab)
11

Prove that one more than the product of any four consecutive integers is a perfect square.

4 marks
Show answer
n(n + 1)(n + 2)(n + 3) + 1 = (n2 + 3n)(n2 + 3n + 2) + 1 = (n2 + 3n + 1)2, a perfect square.
12

Prove that the cube of any integer is either a multiple of 9, one more than a multiple of 9 or one less than a multiple of 9

5 marks
Show answer
n = 3k: 9(3k3). n = 3k + 1: 9(3k3 + 3k2 + k) + 1. n = 3k - 1: 9(3k3 - 3k2 + k) - 1. Every integer is one of these forms.

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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