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a and b are two positive irrational numbers - AQA - A-Level Maths Pure - Question 7 - 2020 - Paper 2
Question 7
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 \( \frac{1}{a} + \fra... show full transcript
Worked Solution & Example Answer:a and b are two positive irrational numbers - AQA - A-Level Maths Pure - Question 7 - 2020 - Paper 2
Step 1
7(a)(i) Identify the error lies in step 1 without contradiction.
Answer
In Caroline's proof, the mistake in Step 1 is the incorrect expression used: she wrote ( \frac{1}{a} + \frac{2}{a + b} ) instead of the correct form ( \frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab} ). This misrepresentation leads to errors in the subsequent steps.
Step 2
7(a)(ii) Complete the proof.
Answer
To prove ( \frac{1}{a} + \frac{1}{b} ) is rational, we start with the correct expression:
Since both a and b are positive irrational numbers, their sum ( a + b ) is rational, and the product ( ab ) is also non-zero and rational. Therefore, ( \frac{1}{a} + \frac{1}{b} ) is rational because it is the ratio of two rational numbers.
Step 3
7(b) Prove by contradiction that the difference of any rational number and any irrational number is irrational.
Answer
To prove this statement by contradiction, assume the opposite. Let ( r ) be a rational number and ( x ) an irrational number such that their difference is rational.
Assume: [ r - x = d ] where ( d ) is rational.
This implies: [ x = r - d ] Since ( r ) and ( d ) are both rational, their difference, ( r - d ), must also be rational, which contradicts our assumption that ( x ) is irrational.
Thus, we conclude that the difference between any rational number and any irrational number must be irrational.