Date | November Example questions | Marks available | 6 | Reference code | EXN.2.AHL.TZ0.8 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Prove | Question number | 8 | Adapted from | N/A |
Question
Prove by contradiction that log2 5 is an irrational number.
Markscheme
* 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∈ℕ where q≥1 such that log2 5=pq M1A1
Note: Award M1 for attempting to write the negation of the statement as an assumption. Award A1 for a correctly stated assumption.
log2 5=pq⇒5=2pq A1
5q=2p A1
EITHER
5 is a factor of 5q but not a factor of 2p R1
OR
2 is a factor of 2p but not a factor of 5q R1
OR
5q is odd and 2p is even R1
THEN
no p, q∈ℕ (where q≥1) satisfy the equation 5q=2p and this is a contradiction R1
so log2 5 is an irrational number AG
[6 marks]