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

Question 98

Markscheme

1.1 Proof

Question 98

Markscheme

1.1 Proof

Question 98

1.1 Proof Questions

1.1 Proof

Question 98

1.1 Proof Questions