A LevelMathsAQAPureExam QuestionsProofProof by Contradiction

Proof by Contradiction (AQA A Level Maths: Pure): Exam Questions

Exam code: 7357

1 hour18 questions
Easy
Medium
Hard
Very Hard
13 marks

Given that x is an integer such that x squared is odd, use proof by contradiction to show, using algebra, that x is odd.

How did you do?

23 marks

x is an integer such that x squared is even.

Prove by contradiction, using algebra, that x is even.

How did you do?

33 marks

Given that x is an integer such that 3 x is odd, use proof by contradiction to show, using algebra, that x is odd.

How did you do?

42 marks

Below is an attempt at a proof by contradiction to show that there is no largest multiple of 7.

Line 1:

Assume there is a number, S, say, that is the largest multiple of 7

Line 2:

S equals 7 k

Line 3:

Consider the number S plus 7

Line 4:

S plus 7 equals 7 k plus 7

Line 5:

therefore S plus 7 equals 7 left parenthesis k plus 1 right parenthesis

Line 6:

So S plus 7 is a multiple of 7

Line 7:

This is a contradiction to the assumption that S is the largest multiple of 7

Line 8:

Therefore, there is no largest multiple of 7

Both line 2 and line 6 are incomplete.

Complete these lines of the proof.

How did you do?

52 marks

"There are an infinite number of positive multiples of 10."

A proof by contradiction starts as follows:

Proof

Assume there are a finite number of positive multiples of 10.

This means there is a greatest multiple of 10, written as 10 k, where k element of straight natural numbers.

Consider the expression 10 k plus 10.

Write the statements needed to complete the proof.

How did you do?

14 marks

Prove by contradiction that there are an infinite number of positive even numbers.

How did you do?

2a1 mark

A student is attempting to answer the following exam question:

“Prove by contradiction that square root of 2 is an irrational number. You may use without proof the fact that if a number n squared is even, then n must also be even.”

The student’s proof is as follows:

Line 1:

Assume square root of 2 is a rational number.  Therefore, it can be written in the form begin mathsize 16px style square root of 2 equals a over b end style, where a space and space b are integers with b not equal to 0, and where a and b have no common factors.

Line 2:

Squaring both sides gives 4 equals a squared over b squared

Line 3:

Multiplying both sides by b squared gives a squared equals 2 b squared

Line 4:

 

Line 5:

This means a can be written as a equals 2 m, for some integer m

Line 6:

Squaring gives a squared equals left parenthesis 2 m right parenthesis squared equals 4 m squared

Line 7:

Substituting a squared equals 4 m squared into a squared equals 2 b squared gives space 4 m squared equals 2 b squared

Line 8:

Dividing both sides by 2 gives 2 m squared equals b squared

Line 9:

This shows that b squared is even, and therefore b must be even

Line 10:

It has been shown that both a and b are even, so they share a common factor of 2.

Line 11:

This is a contradiction of the assumption that a and b have no common factors.

Line 12:

Therefore, square root of 2 is irrational.

There is an error within the first three lines of the proof.

Find the error and write down the correct line of the proof.

How did you do?

2b1 mark

Line 4 of the proof is missing.

Complete this line of the proof.

How did you do?

33 marks

Given that x cubed is odd, use proof by contradiction to show, using algebra, that x is odd.

How did you do?

44 marks

x and y are two real numbers such that x plus y is an irrational number.

Prove by contradiction that x or y or both are irrational numbers.

How did you do?

53 marks

Prove by contradiction that a triangle cannot have more than one obtuse angle.

How did you do?

16 marks

Prove by contradiction that square root of 11 is an irrational number. 

You may use without proof the fact that if n squared is a multiple of 11, then n is a multiple of 11.

How did you do?

23 marks

Given that x to the power of n is odd where n is a positive integer, use proof by contradiction to show, using algebra, that x is odd.

How did you do?

34 marks

Prove by contradiction that there are an infinite number of positive powers of 2.

How did you do?

41 mark

Below is a proof by contradiction that log subscript 2 7 is irrational.

Line 1: 

Assume log subscript 2 7 is a rational number.  Therefore it can be written in the form log subscript 2 7 equals a over b, where a and b are integers with no common factors, and b not equal to 0. Note that a over b greater than 1 (as a over b equals log subscript 2 7 greater than log subscript 2 2 equals 1) so we can assume that a greater than b greater than 0.

Line 2: 

Rearranging log subscript 2 7 equals a over b gives 2 to the power of a over b end exponent equals 7

Line 3: 

Raising both sides to the power b gives open parentheses 2 to the power of a over b end exponent close parentheses to the power of b equals 7 to the power of b

Line 4: 

Line 5: 

This says that a power of 2 must equal a power of 7

Line 6: 

This is not possible, except when a equals b equals 0 which contradicts a greater than b greater than 0

Line 7: 

Therefore log subscript 2 7 is irrational

Lines 4 is missing.

Complete this line of the proof.

How did you do?

54 marks

Use proof by contradiction to show that, given a rational number x and an irrational number y, x plus y is irrational.

How did you do?

15 marks

Prove by contradiction that there are an infinite number of prime numbers.

How did you do?

25 marks

Without solving the equation directly, use algebra to prove by contradiction that the solutions to the equation 

3 x squared plus 10 x minus 8 equals 0

cannot be written in the form x equals a over b where a and b are both odd integers.

How did you do?

34 marks

A composite number, N, has the following properties:

  • It is a positive integer greater than 1

  • It is not a prime number

  • It has at least two prime factors

  • It can be written as a product of its prime factors, p subscript 1, p subscript 2, ..., p subscript k:

N equals p subscript 1 cross times p subscript 2 cross times... cross times p subscript k

where k greater or equal than 2.

Prove by contradiction that any composite number, N, must have at least one prime factor that is less than or equal to square root of N.

How did you do?