A security analyst is verifying a property of an encryption key generation algorithm. The analyst needs to prove the following statement:

'For every non-zero rational number $α\alpha$ and every irrational number $β\beta$ , the result of the calculation $5αβ\frac{5\alpha}{\beta}$ is always an irrational number.'

Identify the correct starting assumption for a proof by contradiction of this statement.

There exists a non-zero rational number $α\alpha$ and an irrational number $β\beta$ such that $5αβ\frac{5\alpha}{\beta}$ is rational.
For all non-zero rational numbers $α\alpha$ and all irrational numbers $β\beta$ , the value $5αβ\frac{5\alpha}{\beta}$ is rational.
There exists a non-zero rational number $α\alpha$ and an irrational number $β\beta$ such that $5αβ\frac{5\alpha}{\beta}$ is irrational.
For all non-zero rational numbers $α\alpha$ and all irrational numbers $β\beta$ , the value $5αβ\frac{5\alpha}{\beta}$ is irrational.

A security analyst is verifying a property of an encryption key generation algorithm. The analyst needs to prove the following statement:

'For every non-zero rational number $α\alpha$ and every irrational number $β\beta$ , the result of the calculation $5αβ\frac{5\alpha}{\beta}$ is always an irrational number.'

Identify the correct starting assumption for a proof by contradiction of this statement.

There exists a non-zero rational number $α\alpha$ and an irrational number $β\beta$ such that $5αβ\frac{5\alpha}{\beta}$ is rational.
For all non-zero rational numbers $α\alpha$ and all irrational numbers $β\beta$ , the value $5αβ\frac{5\alpha}{\beta}$ is rational.
There exists a non-zero rational number $α\alpha$ and an irrational number $β\beta$ such that $5αβ\frac{5\alpha}{\beta}$ is irrational.
For all non-zero rational numbers $α\alpha$ and all irrational numbers $β\beta$ , the value $5αβ\frac{5\alpha}{\beta}$ is irrational.

1.1.4 Proof by contradiction (A-level only)

Question 46

Markscheme

1.1.4 Proof by contradiction (A-level only)

Question 46

Markscheme

1.1.4 Proof by contradiction (A-level only)

Question 46

1.1.4 Proof by contradiction (A-level only) Questions

1.1.4 Proof by contradiction (A-level only)

Question 46

1.1.4 Proof by contradiction (A-level only) Questions