EXN.2.AHL.TZ0.8

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

assume there exist $p, q\in \mathrm{\mathbb{N}}$ where $q\ge 1$ such that ${\log }_2 5=\frac{p}{q}$         M1A1

 

Note: Award M1 for attempting to write the negation of the statement as an assumption. Award A1 for a correctly stated assumption.

 

${\log }_2 5=\frac{p}{q}\Rightarrow 5=2^{\frac{p}{q}}$        A1

$5^q=2^p$         A1

 

EITHER

$5$ is a factor of $5^q$ but not a factor of $2^p$        R1

 

OR

$2$ is a factor of $2^p$ but not a factor of $5^q$        R1

 

OR

$5^q$ is odd and $2^p$ is even        R1

 

THEN

no $p, q\in \mathrm{\mathbb{N}}$ (where $q\ge 1$) satisfy the equation $5^q=2^p$ and this is a contradiction        R1

so ${\log }_2 5$ is an irrational number        AG

 

[6 marks]

[N/A]