Proof (Edexcel A Level Maths: Pure): Exam Questions

A student is investigating the following statement about natural numbers.

" is a multiple of 4"

Prove, using algebra, that the statement is true for all odd numbers.

Use a counterexample to show that the statement is not always true.

Prove, using algebra, that

is even for all .

Prove, using algebra, that

is odd for all

In this question and are positive integers with

Statement 1: is never a multiple of 5

Show, by means of a counter example, that Statement 1 is not true.

Statement 2: When and are consecutive even integers is a multiple of 8

Prove, using algebra, that Statement 2 is true.

“If and are irrational numbers, where , then is also irrational.”

Disprove this statement by means of a counter example.

Use proof by exhaustion to show that for

Use algebra to prove that the sum of any three consecutive integers is always a multiple of 3.

Show that is positive for all real values of , where .

"The difference between any two square numbers is always odd."

Disprove this statement by means of a counter example.

Write 18 as a product of its prime factors.

Write down all prime numbers, , such that .

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

Factorise .

Hence, factorise .

Given that is an even natural number, determine whether and are odd or even.

Determine whether  is odd or even, where is a natural number.

Explain your answer clearly.

A student claims that it is possible to have a triangle with side lengths 6 cm, 8 cm and 10 cm that is not a right-angled triangle, as shown below.

Prove that the student is not correct.

Use algebra to prove that the sum of any three consecutive even numbers is a multiple of 6.

Use algebra to prove that the square of an odd number is always odd.

Let be a natural number in the range .

Use proof by exhaustion to show that all possible values of differ from a multiple of 5 by 1.

By considering all possible prime factors of 17, prove by exhaustion that 17 is a prime number.

Prove that the exterior angle in any triangle, , is equal to the sum of the two opposite interior angles, and .

"The square of a positive integer is always greater than doubling the positive integer."

Disprove this statement by means of a counter example.

Prove that for all , is not divisible by .

Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3

Prove that for all positive integers ,

is divisible by

"All integers of the form , where is a positive non-square integer less than 10, are prime."

Disprove this statement by means of a counter example.

Factorise .

Prove that

• if is odd, then is odd,

• if is even, then  is even.

Two rational numbers, and are given by and  where and  are non-zero integers with no common factors.

Find expressions for and in terms of and .

Using the results in part (a), determine whether or not  and are rational.

A function is given by

Show that for all real values of .

Use algebra to prove that the positive difference between an integer and its cube is the product of three consecutive integers.

The elements, , of a set of numbers, , are defined by where .

Use proof by exhaustion to show that every element in the set can be written in the form where .

Use algebra to prove that the sum of two rational numbers is rational.

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