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Question 76

Two positive irrational numbers, $x$ and $y$ , have a rational sum and a rational product.

Jamie is trying to prove that $x^2 + y^2$ is rational.

Here is Jamie's proof:

Step 1: $x^2 + y^2 = (x + y)^2$

Step 2: $x + y$ is rational, so $x + y)^2$ is rational.

Step 3: Therefore, $x^2 + y^2$ is rational.

ai.

Identify Jamie's mistake.

[1]

aii.

Write down a correct version of the proof that $x^2 + y^2$ is rational.

[3]

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

[4]

Markscheme

1.1 Proof Questions

A Level
Maths
1.1 Proof

132 exam-style questions on OCR (MEI) A Level Maths 1.1 Proof, covering 1.1.1 Structure of mathematical proof, 1.1.2 Disproof by counter example, 1.1.3 Proof by contradiction (A-level only), and 1.1 Proof. Each one has a worked solution and a mark scheme showing where the marks go.

1.1 Proof

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1.1 Proof

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Question 76

Two positive irrational numbers, $x$ and $y$ , have a rational sum and a rational product.

Jamie is trying to prove that $x^2 + y^2$ is rational.

Here is Jamie's proof:

Step 1: $x^2 + y^2 = (x + y)^2$

Step 2: $x + y$ is rational, so $x + y)^2$ is rational.

Step 3: Therefore, $x^2 + y^2$ is rational.

ai.

Identify Jamie's mistake.

[1]

aii.

Write down a correct version of the proof that $x^2 + y^2$ is rational.

[3]

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

[4]

Markscheme

1.1 Proof Questions

A Level
Maths
1.1 Proof