Proof (OCR AS Maths: Pure)

Exam Questions

2 hours39 questions
1
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1 mark

In a mathematical argument, how are three consecutive integers usually denoted algebraically?

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2
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2 marks
(i)
In a mathematical argument, how is an even number usually denoted?

(ii)
Similarly, how is an odd number usually denoted?

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3
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2 marks

Prove that the sum of two odd numbers is even.

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4
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1 mark

Explain why open parentheses x minus 3 close parentheses squared space greater or equal than 0 for all real values of x.

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5
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2 marks

Prove that the product of two even numbers is a multiple of 4.

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6
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2 marks

Use a counter-example to show that square root of open parentheses x squared close parentheses end root space not equal to x.

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7
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2 marks

Prove by exhausting all possible factors that 11 is a prime number.

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8
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2 marks

Prove that k squared minus 6 k plus 9 greater than 0 for all real values of k not equal to 3.

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9
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3 marks

Show that 0.6 can be written in the from p over q, where p and qare integers.

What does this tell you about the type of number 0.6 is?

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1
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3 marks

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

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2
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2 marks

Prove that x squared plus 2 greater or equal than 2 for all values of x.

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3
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3 marks

Prove that the square of an even number is a multiple of 4.

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4
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3 marks

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.

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5
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2 marks

Use a counter-example to prove that the difference between any two square numbers is not always odd.

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6a
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2 marks

Express 18 as a product of its prime factors.

6b
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1 mark

Write down all prime numbers between 1 and 13.

6c
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3 marks

By dividing 13 by each of the prime numbers found in part (b), prove that 13 is a prime number.

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7a
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1 mark

Factorise n squared plus 3 n plus 2.

7b
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1 mark

Hence show that n cubed plus 3 n squared plus 2 n equals n left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis.

7c
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2 marks

Given that n is even, write down whether left parenthesis n plus 1 right parenthesis space space a n d space left parenthesis n plus 2 right parenthesis are odd or even.

7d
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2 marks

Hence deduce whether n cubed plus 3 n squared plus 2 n is odd or even. Justify your answer.

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8a
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2 marks

By writing it as a fraction in its lowest terms, show that 0.35 is a rational number.

8b
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3 marks

Two rational numbers, a and b are such that begin mathsize 14px style a equals m over n end style and b equals p over q  where m comma space n comma space p comma space q are integers with no common factors and n comma space q not equal to 0.

Find an expression for a b.

8c
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2 marks

Deduce whether or not the product a b is rational or irrational.

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9
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4 marks

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.

1-1-maths-q9medium-1

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10a
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1 mark

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?

10b
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2 marks

Write down the number of 3x3 - sized and 4x4 - sized squares there are on a standard chess board.

10c
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3 marks

Hence show that there are 204 squares in total on a standard chess board.

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1
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4 marks

Prove that the sum of any three consecutive even numbers is a multiple of 6.

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2
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3 marks

Prove that straight f left parenthesis x right parenthesis greater or equal than 4 for all values of x, where straight f left parenthesis x right parenthesis equals left parenthesis 3 minus x right parenthesis squared plus 4.

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3
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3 marks

Prove that the square of an odd number is always odd.

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4
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4 marks

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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5
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2 marks

Use a counter-example to prove that not all integers of the form 2 to the power of n minus 1, where n is an integer, are prime.

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6
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3 marks

By considering all possible prime factors of 17, prove it is a prime number.

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7a
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2 marks

Fully factorise n cubed plus 6 n squared plus 8 n.

7b
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3 marks

Prove that, if n is odd, n cubed plus 6 n squared plus 8 n is odd and that if n is even, n cubed plus 6 n squared plus 8 n is even.

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8a
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4 marks

Two rational numbers, a and b are such that a equals m over n  and b equals p over q  , where m comma space n comma space p comma space q are integers with no common factors and n comma q not equal to 0.

Find expressions for a b and begin mathsize 14px style a over b end style.

8b
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4 marks

Deduce whether or not a b and begin mathsize 14px style a over b end style are rational or irrational.

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9
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4 marks

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

1-1-maths-q9hard

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10a
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2 marks

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?

10b
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4 marks

Hence show that there are 204 squares in total on a standard chess board.

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1
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3 marks

Prove that the sum of any three consecutive even numbers is always a multiple of 2, but not always a multiple of 4.

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2
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3 marks

Prove that straight f left parenthesis x right parenthesis greater or equal than 0 for all values of x, where begin mathsize 14px style straight f left parenthesis x right parenthesis equals fraction numerator 9 x squared plus 12 x plus 4 over denominator 5 end fraction end style.

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3
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3 marks

Prove that the (positive) difference between an integer and its cube is the product of three consecutive integers.

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4
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4 marks

The elements, x, of a set of numbers, S, are defined x element of straight natural numbers comma space x less than 6.

Prove that every element of S can be written in the form 3 n minus 2 m where n comma space m space element of straight natural numbers.

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5
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2 marks

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.

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6a
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4 marks

Prove that 23 is a prime number.

6b
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2 marks

Briefly explain why only prime factors need to be tested for, in order to prove a number is prime.

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7
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4 marks

Prove that, if nis negative, n to the power of 4 minus n cubed is positive.

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8
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6 marks

Prove that the sum of two rational numbers is rational.

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9
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4 marks

Prove the angle at the circumference in a semi-circle is a right angle.

You may use the diagram below to help.

1-1-edexcel-alevel-maths-pure-q9vhard

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10
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6 marks

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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