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Algebraic Methods

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Question 141

A cybersecurity expert is verifying a protocol where a security key k k\,k is generated as the quotient k=μσ\displaystyle k = \frac{\mu}{\sigma}k=σμ​, where μ \mu\,μ is a non-zero rational coefficient and σ \sigma\,σ is an irrational physical constant. To establish the security of the protocol, the expert wishes to use proof by contradiction to prove the statement:

'The quotient of a non-zero rational number and an irrational number is always irrational.'

Identify the correct starting assumption for the expert's proof.

  • For all non-zero rational μ \mu\,μ and irrational σ\sigmaσ, the quotient μσ\displaystyle \frac{\mu}{\sigma}σμ​ is rational.
  • There exists a non-zero rational μ \mu\,μ and an irrational σ \sigma\,σ such that the quotient μσ\displaystyle \frac{\mu}{\sigma}σμ​ is rational.
  • There exists a non-zero rational μ \mu\,μ and an irrational σ \sigma\,σ such that the quotient μσ\displaystyle \frac{\mu}{\sigma}σμ​ is irrational.
  • For all non-zero rational μ \mu\,μ and irrational σ\sigmaσ, the quotient μσ\displaystyle \frac{\mu}{\sigma}σμ​ is irrational.
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Markscheme

Algebraic Methods Questions

  1. A Level
  2. /Maths
  3. /Algebraic Methods

368 exam-style questions on Edexcel A Level Maths Algebraic Methods, covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division. Each one has a worked solution and a mark scheme showing where the marks go.

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