Proof by contradiction

Expand and simplify (a b)²

Peter thinks that the sum of any rational number and its reciprocal is always greater than 2

Peter checks two examples:

Use a counter example to show that Peter is incorrect.

Given that a and b are distinct positive numbers, use proof by contradiction to prove that

TURN OVER FOR SECTION B

Section B

all questions in the spaces provided.

Assume that a and b are integers such that

Prove that a is even.

Hence, prove that 2b 1 is even and explain why this is a contradiction.

Explain what can be deduced about the solutions of the equation

Millie is attempting to use proof by contradiction to show that the result of multiplying an irrational number by a non-zero rational number is always an irrational number.

Select the assumption she should make to start her proof.

Tick () one box.

Every irrational multiplied by a non-zero rational is irrational.

Every irrational multiplied by a non-zero rational is rational.

There exists a non-zero rational and an irrational whose product is irrational.

There exists a non-zero rational and an irrational whose product is rational.

a and b are two positive irrational numbers.

The sum of a and b is rational.

The product of a and b is rational.

Caroline is trying to prove is rational.

Here is her proof:

Step 2 2 is rational and a b is non-zero and rational.

Identify Caroline’s mistake.

Write down a correct version of the proof.

Prove by contradiction that the difference of any rational number and any irrational number is irrational.

Prove that the sum of a rational number and an irrational number is always irrational.

The three sides of a right-angled triangle have lengths a, b and c, where a, b, c

State an example where a, b and c are all even.

Prove that it is not possible for all of a, b and c to be odd.

A curve has equation

2 2 4

Prove that the curve does not intersect the coordinate axes.

Prove that the curve has no stationary points.

In the case when x 0, find the equation of the tangent to the curve when y 1

Section B

all questions in the spaces provided.

Prove by contradiction that ³ 2 is an irrational number.

Section B

all questions in the spaces provided.