An engineer is studying two irrational vibration frequencies, $f_1$ and $f_2$ (measured in kHz). It is known that the difference $f_1 - f_2$ is a rational number and the product $f_1 f_2$ is also a rational number.

The engineer proposes the following proof to show that $f_1^2 + f_2^2$ is rational:

Step 1: $f_1^2 + f_2^2 = (f_1 - f_2)^2$

Step 2: Since $f_1 - f_2$ is rational, $f_1 - f_2)^2$ is rational.

Step 3: Therefore, $f_1^2 + f_2^2$ is rational.

(i) Identify the error in the engineer's proof.

(ii) Write down a correct proof that $f_1^2 + f_2^2$ is rational.

Prove by contradiction that the difference $x - y$ is irrational, where $x$ is any irrational number and $y$ is any rational number.

The engineer proposes the following proof to show that $f_1^2 + f_2^2$ is rational:

Step 1: $f_1^2 + f_2^2 = (f_1 - f_2)^2$

Step 2: Since $f_1 - f_2$ is rational, $f_1 - f_2)^2$ is rational.

Step 3: Therefore, $f_1^2 + f_2^2$ is rational.

(i) Identify the error in the engineer's proof.

(ii) Write down a correct proof that $f_1^2 + f_2^2$ is rational.

Prove by contradiction that the difference $x - y$ is irrational, where $x$ is any irrational number and $y$ is any rational number.

1.1 Proof by Contradiction

Question 81

Markscheme

1.1 Proof by Contradiction

Question 81

Markscheme

1.1 Proof by Contradiction

Question 81

1.1 Proof by Contradiction Questions

1.1 Proof by Contradiction

Question 81

1.1 Proof by Contradiction Questions