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

Proof by contradiction

Proving that root 2 is irrational, that there are infinitely many primes, and other results by assuming the opposite.

10 original questions, 45 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

Prove by contradiction that, for any integer n, if n2 is even then n is even.

3 marks
Show answer
Assuming n2 is even and n is odd gives n2 = 2(2k2 + 2k) + 1, which is odd: a contradiction, so n is even.
2

Prove by contradiction that if a is a rational number and b is an irrational number, then a + b is irrational.

3 marks
Show answer
Assuming a + b = p/q leads to b = (pn - mq)/(qn), a rational number: a contradiction, so a + b is irrational.
3

Prove by contradiction that for all positive real numbers x,

x + 9x ≥ 6

3 marks
Show answer
Assuming x + 9/x 0 gives (x - 3)2 < 0, which is impossible, so x + 9/x ≥ 6.
4

You may assume that, for any integer n, if n2 is even then n is even.

Prove by contradiction that √2 is irrational.

5 marks
Show answer
Assuming √2 = a/b in lowest terms forces both a and b to be even, which contradicts the fraction being in lowest terms, so √2 is irrational.
5

Prove by contradiction that there are no integers x and y such that

x2 − y2 = 10

4 marks
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Assuming a solution exists, x - y and x + y have the same parity and multiply to 10; they cannot both be odd, and if both are even the product is a multiple of 4, which 10 is not: a contradiction.
6

Prove by contradiction that log2 5 is irrational.

4 marks
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Assuming log₂ 5 = p/q gives 2p = 5q with p, q positive integers; the left side is even and the right side is odd: a contradiction.
7

A student is asked to prove by contradiction that

sin x + cos x > 1   for all values of x with 0° < x < 90°

The student begins:

Assume that there is an angle x with 0° < x < 90° such that sin x + cos x ≤ 1
Both sides are positive, so (sin x + cos x)2 ≤ 1

Complete the proof.

3 marks
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Expanding gives 1 + 2 sin x cos x ≤ 1, so sin x cos x ≤ 0; but sin x > 0 and cos x > 0 for an acute angle: a contradiction, so sin x + cos x > 1.
8

Prove by contradiction that there are infinitely many prime numbers.

5 marks
Show answer
Assuming a finite list p₁, ..., pₙ of all primes, N = p₁ p₂ ... pₙ + 1 leaves remainder 1 on division by every prime, so it has a prime factor not in the list (or is prime itself): a contradiction.
9
  1. Prove that, for any integer n, if n2 is a multiple of 3 then n is a multiple of 3
  2. Hence prove by contradiction that √3 is irrational.
7 marks
Show answer
a) shown: if n is not a multiple of 3, n2 leaves remainder 1 b) shown: √3 = a/b in lowest terms forces 3 to divide both a and b
10
  1. Prove that, for any integer p, if p3 is even then p is even.
  2. Hence prove by contradiction that 3√2 is irrational.
  3. Deduce, using a further proof by contradiction, that 5 − 3√2 is irrational.
8 marks
Show answer
a) shown: an odd number cubed is odd b) shown: ∛2 = a/b in lowest terms forces a and b both even c) shown: if 5 - ∛2 = r were rational, ∛2 = 5 - r would be rational

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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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